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This is to certify that the
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Design and Performance Tradeoffs of High-Gain Observers
with Applications
to the Control of Smart Material Actuated Systems

presented by

Jeffrey H. Ahrens

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Design and Performance Tradeoffs of High-Gain
Observers with Applications to the Control of Smart
Material Actuated Systems

By

Jeffrey H. Ahrens

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements

for the degree of
DOCTOR OF PHILOSOPHY
Department of Electrical and Computer Engineering

2006

ABSTRACT

Design and Performance Tradeoffs of High-Gain
Observers with Applications to the Control of Smart
Material Actuated Systems

By

Jeffrey H. Ahrens

The study of high-gain observers has typically involved properties that are asymptoti-
cally recovered as the gain is pushed higher. In any practical implementation of high-gain
observers one will ultimately encounter performance tradeoffs associated with the choice
of gain. These include tradeoffs between fast reconstruction of the system states, bet-
ter rejection of modeling uncertainty, and closed-loop stability versus amplification of
measurement noise, large transient response amplitude, and computational cost in the
discrete-time case. We propose several high-gain observer designs and examine their ef-
fectiveness at dealing with these tradeoffs.

We examine the tradeoff between closed-loop stability and large observer transient
response by considering a time-varying high-gain observer that is of the form of an ex-
tended Kalman filter (EKF). We highlight an important feature of the Riccati equation
with respect to the observer transient and show closed-loop asymptotic stability for a par-
ticular class of nonlinear systems under EKF feedback. We compare the performance of
the time-varying extended Kalman filter against a fixed-gain high-gain observer in terms
of closed-loop stability and transient response.

To balance the tradeoff between state reconstruction speed during the observer tran-

sient with amplification of measurement noise at steady—state we propose a high-gain

observer that switches between two gain values. This scheme is able to quickly recover
the system states during large estimation error and reduce the effect of measurement
noise in a neighborhood of the origin of the estimation error. We argue boundedness of
the trajectories of the closed-loop system.

Since closed-loop stability for sampled-data systems using high-gain observers follows
for sufficiently small sampling periods, there is a tradeoff between elevated sampling rates
and closed-loop performance. We consider a multirate sampled—data output feedback
control design in order to relax the tradeoff between computational cost and closed-loop
stability. This scheme employs control update rates that are fixed by a state feedback
design with a sufficiently fast measurement sampling rate. We prove practical stabilization
for the closed-loop system under multirate output feedback. We also argue stability with
respect to a set in the presence of bounded disturbances.

For smart material actuated systems, the existence of significant hysteresis nonlinear-
ity inherent in smart materials along with difficulties in measuring system states points
to output feedback control designs employing hysteresis compensation. We apply our
multirate output feedback scheme to a shape memory alloy actuated rotary joint by com-
bining the observer with a hysteresis inversion controller. The rotary joint is modeled as
a hysteresis operator of Preisach type combined with a dynamic system. Experimental
results of the proposed scheme are reported.

This dissertation attempts to address certain criticisms of high-gain observers and thus

may be of interest to both control theoreticians and practicing engineers.

To my family

iv

ACKNOWLEDGMENTS

It’s been said that a wise teacher teaches by example. I’m indebted to my advisor
Dr. Hassan Khalil for being a patient and wise teacher. Many thanks are due to my co-
advisor Dr. Xiaobo Tan for all the help, discussions, and use of the laboratory. I would
like to thank Dr. Percy Pierre and Dr. Barbara O’Kelly for supporting my study at
MSU. My colleagues at MSU especially: Sridhar, Leonid, Shahin, Eduardo, Uche, Zheng,
Yang, Memon, and Shahid. To my father, Don, for teaching me the value of education
and personal responsibility. To my mother, Martha, who’s outlook on life has instilled
in me confidence, optimism, and perseverance. And to my wife, Amy, for the boundless

love, constant support, and infinite patience that have made all this possible.

TABLE OF CONTENTS

LIST OF TABLES viii
LIST OF FIGURES ix
1 Introduction 1

1.0.1 High-Gain Observer Performance Tradeoffs .............. 5

2 The Extended Kalman Filter as a Time-Varying High-Gain Observer 10

2.1 Introduction ................................... 10
2.2 Closed-Loop Stability Under EKF Feedback ................. 12
2.2.1 Boundedness of the Riccati Equation ................. 17
2.2.2 Closed-Loop System Stability ..................... 23

2.3 Comparison ................................... 29
2.3.1 Control Versus Estimate Saturation in the EKF ........... 30
2.3.2 EKF Versus a Fixed-Gain HGO .................... 34
2.3.3 Initialization of the Riccati Equation ................. 37

2.4 Conclusions ................................... 40
3 Switched Gain High-Gain Observers 50
3.1 Introduction ................................... 50
3.2 Motivation .................................... 52
3.3 Switched-gain observer ............................. 57
3.3.1 Switching Scheme ............................ 62
3.3.2 Switching Analysis ........................... 62

3.4 Gain Switching Example ............................ 73
3.5 Nonlinear Differentiation ............................ 76
3.5.1 Deadzone Switching Example ..................... 80

3.6 Conclusions ................................... 8O

4 Multirate Sampled Data Output Feedback Using High Gain Observers 95
4.1 Introduction ................................... 95
4.2 Multirate Output Feedback Control ...................... 99

vi

4.3 Stabilization of the Origin ........................... 108

4.3.1 Boundedness and Ultimate Boundedness ............... 109

4.3.2 Exponential Stability of the Origin .................. 115

4.4 Stabilization with Respect to a Compact Set ................. 123
4.4.1 Definitions and Problem Formulation ................. 124

4.4.2 Boundedness and Ultimate Boundedness ............... 127

4.4.3 The Tracking Problem, Integral Control, and Disturbances ..... 129

4.5 Example ..................................... 134
4.6 Conclusions ................................... 141

5 Application to Smart Material Actuated Systems 142
5.1 Introduction ................................... 142
5.2 Model ...................................... 144
5.2.1 Class of Systems ............................ 144

5.2.2 Preisach Operator ............................ 145

5.3 Output Feedback Control ........................... 147
5.3.1 Simulation example ........................... 153

5.4 Experimental Results: Control of a Shape Memory Alloy Actuator ..... 159
5.4.1 Actuator Model ............................. 163

5.4.2 Experimental Results on Regulation ................. 173

5.4.3 Experimental Results on Tracking ................... 179

5.4.4 PID versus PID with Hysteresis Inversion .............. 188

5.5 Conclusions ................................... 191

6 Conclusions 192
A Kronecker Matrices 197
B Technical Lemma 199
C Inverse Preisach Operator 202
CI Preisach Operator Inversion Algorithm [63] .................. 202
C2 Inversion Error [63] ............................... 204
BIBLIOGRAPHY 206

vii

2.1

2.2

2.3

2.4

3.1

3.2

3.3

51

LIST OF TABLES

Table showing the values of 5 for which the closed-loop systems under EKF
and HGO feedback are stable and unstable for system (2.73) with a=2 and
a saturation level of 6. Shown are the values for the local results with
1:1(0) = 0.9 (left) and the nonlocal results with 231(0) = 4.9 (right).

Table showing the values of s for which the closed-loop systems under EKF
and HGO feedback are stable and unstable for system (2.73) with a=3 and
a saturation level of 6. Shown are the values for the local results with
121(0) = 0.9 (left) and the nonlocal results with x1(0) = 4.9 (right).

Table showing the values of E for which the closed-loop systems under EKF
and HGO feedback are stable and unstable for system (2.78) with a=2 and
a saturation level of 10. Shown are the values for the local results with
321(0) 2 0.9 (left) and the nonlocal results with 121(0) 2 4.9 (right).

Table showing the values of e for which the closed-loop systems under EKF
and HGO feedback are stable and unstable for system (2.78) with a=3 and
a saturation level of 10. Shown are the values for the local results with
131(0) = 0.9 (left) and the nonlocal results with 331(0) :2 4.9 (right).

Comparison of R.M.S tracking errors 81 2 51:1 — 7‘, 62 = x2 — 1*, el steady-
state, and 62 steady-state for the switched-gain scheme and two HGOs for

lvl 3 0.0016 and $0 # (g3. ............................

Comparison of R.M.S tracking errors 61 = .131 — r, 82 = x2 — 7", e] steady-
state, and 62 steady-state for the switched-gain scheme and two HGOs for

['0] 3 0.016 and $0 7é (f) ..............................

Comparison of R.M.S tracking errors ‘31 = $1 — 'r, 82 = 1:2 —— 7*, 61 steady—
state, and 62 steady-state for the switched-gain scheme and two HGOs for

|v| S 0.0016 and 9350 2 gb. ............................

SMA wire physical parameters, where p, c, h, and the diameter are specified

37

37

38

38

75

76

76

by the manufacturer and the length and resistance R are measured. . . . . 165

viii

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

2.11

2.12

3.1

3.2 Plot of the nonlinear gain function. ......................

LIST OF FIGURES

Simulation results showing the output $1, the estimate :33, the control u,
and the gain h3(t) for EKF feedback under control saturation only.
Simulation results showing the output 331, the estimate :23, the control u,

and the gain h3(t) for EKF feedback under estimate saturation. ......

Simulation results showing the output 2:1, for the EKF (solid) and HGO
(dashed) for 5 = 0.01 (left) and the EKF and HGO for 5 = 0.001 (right).
Simulation results showing the output 51:1 and the control u for EKF feed-
back and the state x3 and the control it for HGO feedback for a. = 3.
Simulation results showing the EKF observer gains h1, h2, and h3 for

a=3. ......................................

Simulation results showing the EKF observer estimates 5:1, :82, and £73 for

a=3. ......................................

Simulation results showing the state 173 and the control u for EKF feedback

and the output 11:1 and the control u for HGO feedback for a = 2 .....

Simulation results showing the EKF observer estimates :21, 5:2, and :23 for

a=2. ......................................

Simulation results showing the EKF observer gains h1, fig, and h3 for

a=2. ......................................

Simulation results showing the output .731, and the control u for EKF feed-
back with P(0) = I . The plots on the left are stable, but the plots on the
right become unstable due to an impulsive—like disturbance at t = 203.
Simulation results illustrating peaking in the EKF observer estimates 5:1,
i2, and i3 and the response of the observer gains h1, kg, and h3.

The response of the EKF observer estimates :51, 53:2, and :83 with Riccati
initialization to suppress peaking. Also shown are the observer gains h1,

h2, and h3. ...................................

Diagram illustrating Dead Zone switching scheme. .............

ix

32

33

34

41

48

49

78

3.3 Top: The velocity reference trajectory (7")(dotted) and .732 under the
switched observer (solid). Bottom: Switching behavior of the gain.

3.4 Position tracking error (61 = x1 — 7') for the switched-gain observer (top),
the observer with E2 = 0.01 (middle), and the observer with 51 = 5 x 10—4
(bottom). ....................................

3.5 Tracking error (62 = 3:2 - 7") for the switched-gain observer (top), the
observer with E2 = 0.01 (middle), and the observer with 51 = 5 x 10‘-4
(bottom). ....................................

3.6 Steady—State tracking error (81 = x1 — r) for the switched-gain observer
plotted against the observer with 81 = 5 x 10“4 (top) and the observer
with 82 = 0.01 plotted against the observer with 51 = 5 x 10"4 (bottom).

3.7 Steady-State tracking error (82 = x2 — 7'") for the switched-gain observer
(top), the observer with 52 = 0.01 (middle), and the observer with 51 =
5 x 10_4 (bottom). ..............................

3.8 Behavior of the control for the three cases. .................

3.9 "hacking error (62 = 172—1") for the observer with 52 (top) and the switched-
gain observer (middle) for an impulsive-like disturbance at 2.5s. The bot-
tom plot shows the switching behavior of the gain. .............

3.10 Switching behavior of the gain when Td = 0. ................

3.11 Steady-State tracking error (81 = $1 — r) for the switched-gain observer
plotted against the observer with 51 = 5 x 10-4 (top) and the observer
with 52 = 0.01 plotted against the observer with 51 = 5 x 10—4 (bottom)
with [til 3 0.016. ................................

3.12 Steady-State tracking error (el = .731 — r) for the switched-gain observer
plotted against the observer with 51 = 5 x 10—4 (top) and the observer
with 52 = 0.01 plotted against the observer with 51 = 5 x 10"4 (bottom)
with Ivl 3 0.0016 and (250 = ¢. ........................

3.13 The signal w(t) (top) and its first (middle) and second (bottom) derivatives.

3.14 i3 versus w(2) for Dead Zone switching (top) Linear HGO with g = 188
(middle) and g = 1570 (bottom). .......................

3.15 Tracking error w(2) — 5:3 for Dead Zone switching (top) Linear HGO with
g = 188 (middle) and g = 1570 (bottom). ..................

3.16 Zoomed in tracking error w(2) — 5:3 for Dead Zone switching (top) Linear

HGO with g = 188 (middle) and g = 1570 (bottom). ............
(2)

A

3.17 Tracking error w — :53 (top) and the zoomed in tracking error w(2) — :53

(bottom) for the low noise estimator of [67]. .................

4.1 Block diagram illustrating the multirate output feedback control scheme.

81

82

83

84

89
90

94

97

4.2

4.3

4.4

4.5

4.6

5.1
5.2
5.3
5.4
5.5

5.6

5.7

5.8

5.9

Simulation results showing the output x1 and the control it for sampled-
data state feedback (top) with T = 0.2, single-rate output feedback (mid-
dle) with T 3 = 5 = 0.2, and multirate output feedback (bottom) with
T3 = 0.2 and Tf = E = 0.04. ......................... 136
Simulation results showing the transient response of the multirate observer
estimates (solid) versus the system states (dashed). ............. 137
Simulation results for an impulsive—like disturbance at t = 10. Shown are
the output 171 and the control 1:. for two cases of single-rate sampled-data
output feedback, (1) 5 = 0.1, T3 = 0.1 (2) a = 0.01, T3 = 0.01, and
multirate sampled-data output feedback, where 5 = 0.01, T5 = 0.1, and
Tf = 0.01. ................................... 138
Simulation results for an impulsive-like disturbance at t = 10. Shown are
the multirate observer estimates (solid) versus the system states (dashed)
for the entire simulation run (top) and zoomed in on the transient response
at t = 10 (bottom). .............................. 139
Simulation results for an impulsive-like disturbance at t = 7.087. Shown
are the output $1 and the control 11. for multirate sampled-data output
feedback, where E = 0.01, T5 = 0.1, and Tf = 0.01 (top), and single-rate

sampled-data output feedback, where 8 = 0.01, T3 = 0.01 (bottom). . . . 140
Model structure of a smart actuator and plant ................. 144
Delayed relay. .................................. 145
Diagram of the multirate control scheme with hysteresis inversion ...... 147

Hysteresis loop based on the Preisach operator used in the simulation ex-
ample. ...................................... 153
The state (top), the control (bottom left), and the inversion error (bottom
right) for sampled-data state feedback with exact inversion and T = 0.005. 156
The state (top), the control (bottom left), and the inversion error (bottom
right) for single—rate sampled-data output feedback with exact inversion
and T3 = 0.005 .................................. 157
The state (top), the control (bottom left), and the inversion error (bottom
right) for multirate sampled—data output feedback with exact inversion,
Ts = 0.005, and Tf = 0.001. .......................... 158
The state (top), the control (bottom left), and the inversion error (bottom
right) for sampled-data state feedback with inversion error and T = 0.005. 160
The state (top), the control (bottom left), and the inversion error (bottom
right) for single—rate sampled—data output feedback with inversion error
and T3 = 0.005 .................................. 161

xi

5.10

5.11
5.12
5.13

5.14
5.15

5.16
5.17

5.18

5.19

5.20

5.21

5.22

5.23

5.24

5.25

The state (top), the control (bottom left), and the inversion error (bottom
right) for multirate sampled-data output feedback with inversion error,
T3 = 0.005, and Tf = 0.001. .......................... 162
Robotic joint actuated by two SMA wires. .................. 163
Electrical Diagram. ............................... 164
Plot showing the measured output, current input, and identified hysteresis
nonlinearity .................................... 168
Identified Preisach weighting masses. ..................... 169
Simulation of the identified Preisach operator versus the measured data

(left) and the error between the two (right). ................. 169
Plot showing the results of an open-loop hysteresis inversion experiment. . 170
Experimentally obtained SMA actuator normalized frequency response
(dashed) and the response of the transfer function (5.41) (solid) ....... 171
Plot of an angle regulation experiment for SR (top) and MR (middle) out-
put feedback controllers with inversion and MR without inversion (bottom).
Shown are the angle 9 (solid) versus the setpoint 7‘ (dashed) (left) and the
current 2' (right). ................................ 175
Plot of an angle regulation experiment for slow SR (top), MR (middle), and
fast SR (bottom) output feedback controllers without inversion. Shown are
the angle 9 (solid) versus the setpoint 7" (dashed) (left) and the regulation
error 0 — 7‘ (right). ............................... 176
Plot of the current 2' for an angle regulation experiment for slow SR (top),
MR (middle), and fast SR (bottom) output feedback controllers without
inversion. .................................... 177
Comparison of MR versus slow SR (top) and MR versus fast SR (bottom)
for a regulation experiment. .......................... 178
Plot of a sinusoidal tracking experiment for slow SR (top), MR (middle),
and fast SR (bottom) output feedback controllers with inversion. Shown

are the rotation angle 6 (solid) versus the reference 7" (dashed) ........ 181
Plot of a sinusoidal tracking experiment for slow SR (top), MR (middle),
and fast SR (bottom) output feedback controllers with hysteresis inversion.
Shown are the tracking error 0 — 7‘ (left) and current 2' (right). ....... 182
Plot of a sinusoidal tracking experiment for slow SR (top), MR (middle),

and fast SR (bottom) output feedback controllers without hysteresis inver-
sion. Shown are the rotation angle 6 (solid) versus the reference 7“ (dashed). 183
Plot of a sinusoidal tracking experiment for slow SR (top), MR (middle),
and fast SR (bottom) output feedback controllers without inversion. Shown

are the tracking error 6 — r (left) and the current 2' (right). ........ 184

xii

5.26

5.27

5.28

5.29

5.30

Comparison of MR versus fast SR (top) and MR versus slow SR (bottom)
for the tacking experiment with reference given by (5.46) and controller

without hysteresis inversion. ..........................

Comparison of MR versus fast SR (top) and MR versus slow SR (bottom)
for the tacking experiment with reference given by (5.46) and controller

with hysteresis inversion .............................

Plot illustrating the effect of increasing the ratio T3 / T f for multirate output
feedback with hysteresis inversion (top) and without hysteresis inversion
(bottom). Both plots show two data sets each; one for T3 = 0.01 and

another for T 5 = 0.005 ..............................

Plot of experimental results comparing PID with hysteresis inversion (left)
with PID (right). Shown are the angle 6 (solid) versus the reference r
(dashed) (top), the tracking error 6 — 1‘ (middle), and current 2' (bottom).
Plot of experimental results comparing PID with hysteresis inversion (left)
with PID (right). Shown are the angle 6 (solid) versus the reference 7‘

(dashed) (top), the tracking error 6 —— 7‘ (middle), and current 2' (bottom). .

185

186

187

. 189

190

C.1 Preisach plane with L = 8 ............................ 202

xiii

CHAPTER 1

Introduction

Most nonlinear control design tools assume state feedback (i.e., measurement of all state
variables) to achieve a desired goal. In many applications full state measurement is either
impractical or not possible. In such cases, it is necessary to use alternate methods to
obtain the system state information. One method is the use of observers to estimate the

system states from the output measurements. For linear time-invariant systems

.7': = A$+Bu

y 2 Ca:
a state observer takes the form
3i: 2 A:i:+ Bu+H(y — 0:23)

This observer, referred to as a Luenberger observer, reproduces the right hand side of the
system dynamics and is driven by the output error through the observer gain H. With the
pair (A, C) detectable, H can be chosen to guarantee global convergence of the estimates

5: to the system states :13. For a general class of nonlinear systems

y = W?)

designing an observer to provide convergence outside a small neighborhood of the origin
of the estimation error has proved to be challenging. To achieve nonlocal convergence,
nonlinear observer designs typically exploit the special structure of a certain class of
nonlinear systems. One such nonlinear observer is the high-gain observer. High-gain

observers are applicable to a class of nonlinear systems that have the form

2 z ¢(:::,.:) (1.1)
:i: = Ax+Bd>(:r,z,u) (1.2)
y = Ca: (13)
C = 90/22:) (1.4)

where u is the control input, a: E Kr and z E Re constitute the state vector, and y and C
are the measured variables. The r x r matrix A, the r x 1 matrix B, and the 1 x 7‘ matrix

C are given by

 

 

 

 

P0 1 -0- P0-
0 0 l 0 0
A: g 3 ,3: 5 (1-5)
0 0 1 0
[0 0 0] -1-
C: 1 0 ...() (1.6)

One source for the model (1.1)-(1.4) is the normal form of input—output linearizable sys-
tems, as discussed in [36]. In addition to the normal form, this class of systems also
arises in mechanical and electromechanical systems where the position is measured, but
its derivatives, the velocity and acceleration, are not measured. In [26], Esfandiari and

Khalil introduced a robust output feedback technique based on the use of high-gain ob—

servers. Output feedback is achieved by first considering a globally bounded partial state

feedback controller given by

u = WAC)

that is designed to meet the performance objectives. Then, the state :1: is replaced by the

estimate :3: that is generated by the high-gain observer

.1: = A1: + Bagel}, c, u) + H(y — 0:2) (1.7)

where {/50 is a known nominal model of (b and the observer gain H is designed as

T
01 O2 07‘
H: _,__’......,_ 1.8
5 '52 57‘ ( )
The ai’s are chosen such that the roots of
sr+alsr_1+-~+ar_ls+ar (1.9)

have negative real parts. The output feedback controller is given by

u : 7(571C)

Some of the key features of the high-gain observer can be seen by rescaling the estimation

error according to

($77 — in)
51—1

52' =

With this rescaling we can arrive at the following equation for the estimation error

e: = 1105+ 68W. 2:, 7(1, 0) - «106:, c. 7(i‘, 0)] (1.10)

where A0 is a Hurwitz matrix. This equation shows that the effect of the modeling
uncertainty (cb— (b0) is reduced with smaller 8. Also, this equation shows that the observer
error evolves in a time scale t/s, which is faster then the time scale of the plant. One
attribute of the high-gain observer is what is known as the peaking phenomenon. Due to

the gain structure (1.8), the observer will exhibit a transient response of the form
1
E€$p(—at/€)

Notice that while this term will rapidly decay, the amplitude will be quite large. Thus, the
transient response approaches an impulsive-like behavior as e is pushed smaller. Peaking
in the estimates can lead to large control magnitude which in turn can lead to instability
if not properly handled. One way to do this is to globally bound the control by saturating
it outside a compact region of interest. This will limit the control magnitude so that the
plant does not experience the effects of peaking. Although globally bounding the control
can prevent peaking from driving the system unstable, its effects will still be present in the
control signal. Peaking can cause the control to swing quickly between its minimum and
maximum values. This type of control behavior can lead to excessive power consumption
or mechanical wear. Therefore, it is still desirable to avoid peaking in favor of a more well
behaved control signal.

As we alluded to above, the closed-loop system under high-gain observer feedback will
contain fast and slow time scales. This two—time scale nature of the closed-loop system
allows it to be studied using singular perturbation analysis. The idea of combining a glob—
ally bounded state feedback control with a sufficiently fast high-gain observer was used
by Atassi and Khalil [7] to prove a separation principle for the class of systems under
consideration. They showed that by choosing s sufficiently small, one can guarantee sta-
bility of the closed-loop system. In addition, they showed that as 5 —> 0 the performance
under output feedback using high-gain observers approaches the performance under state

feedback.

In this thesis we will consider continuous-time as well as sampled-data systems. In
sampled-data systems the actual process is continuous-time while the controller is imple-
mented in discrete-time by digital computers. In [19], Dabroom and Khalil considered
discrete—time implementation of the high-gain observer. Various discretization methods
were studied to determine the most suitable algorithm and best choice of observer pa-
rameters. Furthermore, in [20], Dabroom and Khalil showed that an output feedback
controller based on the discrete-time high-gain observer stabilizes the origin of the closed-
loop system for sufficiently small sampling period T. In addition, it was shown that the
performance under sampled-data output feedback asymptotically approaches the perfor-
mance under continuous-time state feedback as T —> 0. In discrete-time, the sampling
period of the observer is chosen proportional to the gain parameter 5. That is, T = 015 for
some positive constant 0. Thus, for the discrete-time high-gain observer, more accurate

estimation of the system states is achieved by faster sampling of the output.

1.0.1 High-Gain Observer Performance Tradeoffs

The tradeoffs that concern us here are the ones that come with the choice of the gain pa-
rameter 5. Choosing smaller values of 5 yields the following benefits: faster reconstruction
of the system states, better rejection of modeling uncertainty, and recovery of the perfor—
mance under state feedback. This comes at the expense of amplification of measurement
noise, larger peaking amplitude, and more computational cost in the discrete-time case.
On the other hand, with larger 5, one can expect reduced susceptibility to measurement
noise, smaller peaking amplitude, and less computational demands in the discrete-time
case. However, this comes with the price of slower state reconstruction and greater sig-
nificance of the modeling error.

With this in mind, we consider the design and performance tradeoffs of closed-loop
systems under output feedback using high-gain observers with application to the control of

systems with smart material actuators. This will be done in four steps. First, we consider

a time-varying high-gain observer that is of the form of an extended Kalman filter (EKF).
We will show that when applied to a class of nonlinear systems similar to (1.1)—(1.4),
the closed-loop system under EKF feedback, when parameterized as a high-gain observer,
is asymptotically stable. Further, we compare the performance of the extended Kalman
filter with a fixed-gain high-gain observer to evaluate whether the added complexity of the
EKF provides advantages in terms of closed-loop stability and the peaking phenomenon.
Second, we highlight the tradeoff between state reconstruction and modeling uncertainty
versus immunity to measurement noise and propose a switched-gain high-gain observer
design to relax this tradeoff. Third, turning our attention to sampled-data systems, we
consider a multirate sampled-data output feedback control design in order to relax the
tradeoff between computational cost and closed-loop stability. Finally, we apply our
multirate output feedback design to smart material actuated systems. We introduce each
of these ideas here; additional background and discussion is given in the introduction to

each chapter.

Extended Kalman Filters

The gain of the high-gain observer, as shown above, is designed by a pole placement ap—
proach. Atassi and Khalil [8] studied the high-gain observer for pole placement, algebraic
Riccati equation, and Lyapunov equation-based algorithms. They were able to show that
each of these three gain design methods, along with a globally bounded control, satis-
fies the separation result of [7]. Another observer for nonlinear systems is the extended
Kalman filter [27]. The extended Kalman filter has been widely used in the areas of con-
trol and signal processing as a state estimator for nonlinear stochastic systems. The EKF
is based on linearization about the current state estimate and on the covariance of the

input and measurement noise, which are typically treated as stochastic processes. The

filter gain, P(t)CTR—1, is obtained from the solution to the Riccati equation
P _ A A T A _ T *1
—— 1(2:(t))P + PA1(:r(t))+ Q PC R CP (1.11)

where Q and R are the input and output error covariance matrices, respectively. The
matrix A1 (1205)) is obtained from the linearization of the nonlinear system about the cur-
rent estimate. In the noise free case, the EKF can be parameterized to function as a
deterministic observer for nonlinear systems. Furthermore, based on a particular choice
of the covariance matrices, the EKF can be designed as a time-varying high-gain observer.
We prove that under EKF feedback the origin of the closed-loop system is asymptotically
stable and the estimation error converges exponentially. Further, we compare the bene-
fits and limitations of observers with time-varying gain versus observers with fixed gain.
This comparison will involve the value of s that provides closed—loop stability and the

susceptibility of the observer with respect to peaking.

Switched-Gain Observers

One common criticism of high—gain observers is their performance in the presence of
measurement noise. We note that high observer gain tends to differentiate noise thereby
degrading the performance of the closed-loop system. To deal with the tradeoff between
state reconstruction speed and suppression of modeling uncertainty versus amplification
of measurement noise we introduce a high-gain observer design where the gain matrix is
switched between two values. We note that during the observer transient, it may be more
desirable to use high-gain to quickly reconstruct the system states when the estimation
error is large at the expense of increased impact of measurement noise. On the other hand,
at steady-state, when the transient has died down, it is more desirable to use smaller gains
to lessen the effect of noise. This is the basic idea behind the switched-gain observer. We
use high-gain during the transient to quickly recover the state estimates, then once the

estimation error has reached steady-state, we switch to a lower gain to reduce the effect

of measurement noise. We prove that under the switched—gain observer, all trajectories

of the closed-loop system are bounded.

Multirate sampled-data output feedback control

Motivated by applications of control to smart material systems we seek to analyze the
performance of a closed-loop system when multirate sampled-data output feedback is
considered. For sampled-data state feedback the sampling period is dictated by the band-
width of the closed-loop system. With discrete-time observers, a more accurate estimate
of the system states can be obtained with faster sampling of the output. Here we consider
a sampled-data system where the input and output are sampled at different rates. Using
discrete-time high-gain observers, we note that the sampling frequency should be chosen
proportional to the observer poles which are located at 0(1/8). Therefore, the output
sampling period decreases as 5 decreases. We seek to balance the tradeoff between fast
sampling rates, needed to guarantee stability under high-gain observer feedback, and the
computational costs associated with elevated sampling rates. We start with a sampling
rate that is chosen based on state feedback design. We show that stability of the closed-
loop system can be achieved by using a sufficiently fast measurement sampling rate and
a control update rate that is fixed by the same state feedback design. We prove practi-
cal stabilization of the origin of the closed-loop system under multirate output feedback.
Further, in the presence of bounded disturbances in the closed-loop system we prove sta-
bilization with respect to a set containing the origin. We show through simulation that

the multirate scheme may be less susceptible to peaking than the single-rate scheme.

Smart Materials Actuators and Control

Finally, we will apply the multirate output feedback control design to the control of sys-
tems that employ smart materials as actuation devices. Smart materials exhibit significant

hysteresis and we consider controller designs that employ hysteresis inversion algorithms

 

such as the one introduced by Tan and Baras in [63]. In general, these inversion algo-
rithms are computationally demanding and controller designs based on them may place
a constraint on the choice of sampling rate. We work with a model for smart material
actuators that consists of a hysteresis operator in cascade with a linear dynamic system
and use hysteresis inversion for feedforward compensation. In the presence of bounded
hysteresis inversion error, we demonstrate the applicability of multirate output feedback
control. Further, we present experimental results for the control of a shape memory alloy
actuated robotic joint. This is done by applying a controller based on hysteresis inversion
with a high-gain observer in the multirate scheme.

This thesis is divided into three parts. In the first part, we consider the extended
Kalman filter as a time-varying high-gain observer and compare the EKF to a fixed-gain
high-gain observer. This is covered in Chapter 2. In Chapter 3 we turn our attention to
the effect of measurement noise and consider a switched-gain observer design. Chapters
4 and 5 consider multirate output feedback using high-gain observers and applications to
the control of smart material actuated systems, respectively. Finally, Chapter 6 discusses

the conclusions and future work.

CHAPTER 2

The Extended Kalman Filter as a

Time-Varying High-Gain Observer

2.1 Introduction

In this chapter we examine a high-gain observer that has time-varying gain. This ob-
server will be of the form of the well-known extended Kalman filter (EKF). Based on a
parameterization of the EKF we provide stability results for closed-loop systems under
EKF feedback. Phrther, through simulation, we study whether a time-varying high-gain
observer is able to balance the tradeoff between closed-loop stability and large peaking
transients in the observer. First, we provide some background on the extended Kalman
filter.

Since the 1970’s, the extended Kalman filter has seen successful application as a state
estimator for nonlinear stochastic systems. See [27] and [62] for an introduction. In the
noise free case, the EKF can be parameterized to function as an observer for deterministic
nonlinear systems. In the 1990’s, study of the stability and convergence properties was
conducted. An early method for constructing deterministic observers as asymptotic limits
of filters appeared in [9]. Additional work on the convergence properties of extended

Kalman filters used as observers has been conducted in [11], [12], [21], [56], [57], [61].

10

Early convergence results were able to show that the EKF converges exponentially for
general classes of systems, but these results were mostly local. Efforts to expand the
domain of attraction appeared in [11] and [56]. In [56] a modification of the linearized
system matrix was introduced to improve stability. In [11] a study of the influence of
the disturbance covariance matrices Q and R on the convergence was conducted for the
discrete-time case. The results of [11] show that choosing Q and R according to a linear
matrix inequality can enlarge the domain of attraction. In [21] it was recognized that, for
a particular parameterization of the covariance matrices, the EKF is a time-varying high-
gain observer that asymptotically approaches a fixed-gain observer as the gain is pushed
higher. Furthermore, it was shown that the EKF is a global exponential observer for a
class of nonlinear systems transformable to the lower triangular form. This argument was
based on a global Lipschitz property for the system nonlinearities.

To this point, analysis of the closed-loop system under EKF feedback has been lim-
ited. A separation result for a Kalman-like observer for a certain class of MIMO nonlinear
systems was presented in [70]. This result made use of certain assumptions on the bound-
edness of the states of the system under control; these assumptions were consistent with
the proposed application of feedback control of polymerization reactors. Global results
were given under global Lipschitz conditions. Aside from very restrictive assumptions on
the nonlinearities, exponential stability of the estimation error does not guarantee the
behavior of the closed-loop system, even when the system under state feedback is expo-
nentially stable [66]. Hence, it seems appropriate to study the behavior of the closed-loop
system when an extended Kalman filter is used as an observer. Toward that end, we relax
the global Lipschitz condition and consider a class of systems transformable to the special
normal form with linear internal dynamics. Based on a parameterization of the Riccati
equation, the closed-loop system under EKF feedback is placed in the standard singu-
larly perturbed form. We note that by relaxing the global Lipschitz condition, difficulties

may arise as a result of the peaking phenomenon. Peaking in the estimates can lead to

11

instability in the closed-loop system. This phenomenon is typically overcome by globally
bounding the control outside a compact region of interest. In our situation, in addition to
globally bounding the control, the time-varying matrices of the Riccati equation must be
globally bounded in order to have a well defined solution. Previous convergence results
relied on assuming that the solution to the Riccati matrix equation is bounded. In [9]
and [61] observability conditions are given that ensure the boundedness of this solution.
In this chapter we argue boundedness by using perturbation analysis. This is done by
making use of standard results on time-invariant Riccati equations.

We begin in the next section by putting the EKF on a theoretical footing. We argue
that the origin of the closed-loop system under EKF output feedback is asymptotically
stable. In addition, we show that the observer error is semiglobally exponentially stable.
In Section 2 .3, we compare through simulation the use of the extended Kalman filter versus
a fixed-gain high-gain observer. We study the performance of the EKF parameterized
as a time-varying high-gain observer in terms of closed-loop stability and the peaking

phenomenon.

2.2 Closed-Loop Stability Under EKF Feedback

Consider the system

2 = Fz+C2r1 (2.1)
:i: = Arc + qu(z, :r, u.) (2.2)
y = C1: (2.3)

where :r 6 RT and z 6 Re are the states, 11 is the input, and y is the output. The function
95 is assumed to be continuously differentiable and satisfies q5(0, 0, 0) = 0. The E x 6 matrix
F is Hurwitz. The r X 7' matrix A, the 7‘ x 1 matrix B, and the 1 x r matrix C are the

same as (1.5)-(1.6). The internal dynamics (2.1) are driven by the output y = .171. Given

12

this structure, the system (2.1)-(2.2) is said to be in the special normal form [34]. Let

X = [z :13]T and rewrite (2.1)-(2.2) as
X = f (X, it)
The extended Kalman filter for this system is given by
52: fee. u) + PCeTR‘lo — CeX")

P = AeP + PAZ + Q — Pegs—10813

where R, Q, and P(0) are positive definite symmetric matrices and

 

C8 : [01x8 Cl
The matrix Ae takes the form
A A
Ae = 11 12
A21 A22
in which
All—F , A12=[G 0 Olfxr
0
where
r _
0 0
0&5 . .
A0 — a 61% — 01"(z,:r,u)
0 0 'z
_ (1951 (1952 (107‘ _

 

 

13

(2.5)

(2.6)

(2.7)

In the forthcoming equations, we will use All in place of F. We consider the state
feedback controller

u = e/(z,:r) (2.9)
The closed-loop system under state feedback is given by
z' = A112 + C231 (2.10)
:1': = A2: + ng(z,a:,'y(z,:1:)) (2.11)
We state our assumptions.
Assumption 2.1
1. The origin (:1: = 0, z = 0) 0f (2.10)-{2.11) is globally asymptotically stable.

2. The function 7 is locally Lipschitz in its arguments and globally bounded in :13. Fur-

thermore, ”)(0, 0) = 0.
In addition, we assume that the closed-loop system satisfies the following 188 property

Assumption 2.2 The system

2 = All?! + Gl‘l (2.12)
:1: = A1: + Bei(z, :13, 7(z + v, 1)) (2.13)
with 12 viewed as the input, is input-to-state stable {155).

Assumption 2.3 The functions

@-
02

 

(22,113,11) and (2’35“)

14

fori = 1, - - - ,r are globally bounded in 2 and 13.1

Assumption 2.3 ensures that the matrices A21 and A22 of the Riccati equation are
bounded. Globally bounding the control protects the plant from peaking. Bounding
A21 and A22 protects the Riccati equation from peaking, ensuring a well defined solu-
tion. Due to the parameterization for Q (see below), peaking will appear only in the
:1: estimates. However, for convenience we bound A21 and A22 in 2 as well. This sim-
plifies the analysis by allowing the Riccati equation to be studied independently of the

estimation equations. We parameterize Q in the following way

Q1 Q2
Q = T 1 1 1 (2.14)
where Q1 and Q3 are chosen to be positive definite symmetric, D = diag[1, e, - - - ,er—l],

and e > 0. We take R = 1. The above parameterization produces a two-time scale
behavior in the solution to the Riccati equation (2.6). We partition and scale P according

to

P1 10212—1
P: 1 T 1 1 1 (2'15)
D" P2 E1) P30—

where P1(0) and P3(0) are chosen to be positive definite and P2(0) is chosen so that P(0)

is positive definite. Then, the observer can be written as
l: _ r, 2 ’1 T *
2—A112+C.11+PZD C (y—Czr) (2.16)

. 1 _ _
:2; = A2 + Bcf)(2,.1‘:, u) + 21) 11330 1CT(y — (:2) (2.17)

The gain lDTIP D_1CT has the structure of a high-gain observer 7 , 21 . This was
5 3

exploited in [21], using a parameterization similar to the above, to show global exponential

 

1Global boundedness can always be achieved by saturating :i: and 2 outside a compact
region of interest.

15

stability of the extended Kalman filter. For the fast estimation error we use the standard
rescaling

 

for i = 1, . --,r. Thus, a: — e = 1325, where 192 = diag[eT—1,eT—2,---,1]. Define the
estimation error for the internal states by 1) = z —— 2. The closed-loop system under output

feedback can now be written in the standard singularly perturbed form

2 = A112 + 0:31 (2.19)
:1': = Ax + qu(z,:1:, 7(2 — 17, :1: — 025)) (2.20)
7'; = Ann + e’"‘1(G — 1320551 (2.21)
55' = (A — P3CTC)§ + 535(2, :13, n, 1925) (2.22)

P1 = A11P1 + 101A?1 + A12P2T + P214}; + Q1 — chTcpéF (2.23)
5P2 = P2(A + 5.405)T — P20T0P3 + [112133 + 52111132
+eP1A7§ID + EQQD (2.24)
12153 _—. (A + 51103103 + P3(A + 12.40,)T + Q3 — P3CTCP3

+52(P§"A§’"ID + 01421192) (2.25)

where 6 = (b(z,:1:, u) — (15(2, :13, u) and

 

 

0 0
A05
0 0
_e"-1d¢1 er-Zdeg an)“

Note that A12D_1 = A12. Equations (2.19)-(2.21) and (2.23) characterize the “slow”
dynamics and (2.22), (2.24), and (2.25) the “fast” ones. In the next two subsections we

present stability results on the closed-loop system under EKF feedback (2.19)-(2.25).

16

2.2.1 Boundedness of the Riccati Equation

We begin by studying the Riccati equation (2.23)-(2.25) alone. To do so, we will treat
A21 and A22 as bounded time—varying matrices and use perturbation theory to argue

that the solutions of (2.23)-(2.25) are bounded. We have the following result.

Theorem 2.1 Consider the closed-loop system (2.19)-(2.25) under output feedback. Let
Assumptions 2.1-2.3 hold and let M and N be any compact subsets of R€+T+e and RT
respectively. Then, for trajectories (2,131)) x 50 starting in M x N there exists 51‘ such
that, for all 0 < e S 51‘, P(t) is bounded and P3(t) is positive definite for allt Z 0,

uniformly in s.

Proof: First, it can be checked that for all (2,:ir) E R€+T the pairs (Ae(:2,:i:),Ce) and
(Ae(2, 2:), \/Q) in (2.6) satisfy the notions of uniform detectability and uniform controlla-
bility given in [9], respectively. As a consequence, there exists a bounded solution P(t) to
(2.6) and also (2.23)-(2.25) through the rescaling (2.15). However, the bounds obtained
have dependence on e. For the analysis we need to show that the solution P(t) of the
rescaled Riccati equation (2.23)-(2.25) is bounded uniformly in e. We begin by viewing

the following equations as a nominal model (5 = 0 on the right hand side) of (2.23)—(2.25)

P1 : Allpl + 151.4:le + £112ng + P2143112 + Q1 — p2CTCPéF (2.26)
5P2 = 152(A -- P30T0)T + A12P3 (2.27)

5P3 = A133 + P3AT + Q3 — P3CTCP3 (2.28)

By standard results on Riccati equations [40], with Q3 positive definite and (A,C) ob-

servable, (2.28) has a unique limiting solution P3+ = PgT > 0 such that

A+ dif A - chTC

17

is a Hurwitz matrix. Moreover, P3(t) approaches P; exponentially fast [14], i.e.,
||P3(t) — P9?” 3 936—o3t/e (2.29)

for some positive constants g3 and 03. For equation (2.27) it is easy to show that the

limiting solution is given by

192+ = —A12P§’(A — chTch = A12 (2.30)

where the second equality follows from A12 2 CC, CAT = 0, and C (I — CTC) = 0.
Rewrite (2.27) as

-_ _ _ T
5P2 : (P2 — 132+) [A+ — (P3 — P;)cTc] (2.31)
By (2.29) and the fact that A+ is a Hurwitz matrix we have that
Mega) — 192+” 3 gee—”zt/E (2.32)

for some positive constants g2 and 02. Since P2(t) is bounded and A11 is Hurwitz, the

solution to the Lyapunov equation (2.26) is bounded. Indeed we have
1151a) — Pin 3 ale—“It (2.33)

where P1+ is the limiting solution to (2.26) and 91 and 01 are positive constants. Hence,
each P,- is bounded uniformly in e for all t Z 0. We point out that equations (2.23)-(2.25)
are e perturbations of (2.26)-(2.28). We argue for the boundedness of P(t) studying the

error

18

between the full system and the nominal system. Since, P(t) is bounded for all t Z 0 we

have that “P(t)” S N for some positive constant 1V. Consider the system

A.

P1 = AIIPI + p1A{1+(A12 — PZCTC)P2T + P2(A12 — P2CTC)T
—1520T0132T (2.34)
23—1317“ _*T~_~T~
E 2 — 2 2 (t) + (A12 P20 C)P3 P20 CP3 + EQQD
+€A11(P2 + P2) + 5(131 + PIMQD + €(f32 + p2)Ag;: (2.35)

8P3 = A2(t)133 + 153/lg“) — P3CTCIB3 + €A0€(P3 + P3)

+e(P3 + P3)Ag’€ + 52(132 + P2)A’§’"lo + 520112163 + 152)T (2.36)

where

112(1) .—. A+ — (P3 — P;)CTC (2.37)

We use the vec operator to write (2.34)-(2.36) as a system of vector equations. This
operator transforms a matrix to a vector by stacking the columns of the matrix from
left to right starting with the first column on top. Let 7r,- = vec(Pz-), 7?,- = vec(P,-),
”2T 2 vec(P2T), 772T 2 vec(P2T), q2 =2 vec(Q2), and q2T = vec(Qg). Also, let

' l
71'2

WE : 7T2T

 

 

[”3

Consider the following identity from Kronecker matrix algebra ([10], Ch. 7)

vec(ABC) 2 (CT ® A)vec(B) (2.38)

19

Using this identify we can arrive at the following standard singularly perturbed vector

equations

71'1 = 111171] + (.7112 + h13(7r2)) n2 + ll’l47T2T (2.39)
5715‘ = (N1+ N2(t) + N3(7l'5))71'5 + E (N47r5 + N477}: + N57r1+ N57_r1)

+€Nq((12, (1211‘) (2'40)

where the matrices M,- and N,- are listed in Appendix A. By standard results on Kronecker

products, we have that A11 and N1 are Hurwitz matrices and

l|M3(7r2)|| S m1l|7r2|| , ||N3(775)|| S n1|l7re|| (2-41)

for some positive constants m1 and n1. Furthermore, as discussed in Appendix A we

have

||N2(t)|| 3 946-0412.- (2.42)

for some positive constants g4 and 04. Also, the matrices Mg, [114, N4, N5, and
Nq(q2,q2T) are bounded with bounds that are independent of 5. Consider (2.40) with

e = 0 on the right hand side
57'1'5 =(N1+ N2(t) + N3(7T5)) W5 (2.43)

It can be shown that the origin

of (2.39) and (2.43) is locally exponentially stable. Let $1 and 85 be the positive definite
solutions to

SlMl + Mirsl = —1

20

and

SgNl + AfiFSg = —1

Using V = V1(7r1) + V2075) with
V1 2 71391771 and V2 2 «$55775
it can be shown that there exist an 51 such that for all 0 < e S 51

- c

v s -C2llrll2 — fume)? (2.44)
. def . . .
1n the set 91 2 {Hall g CI} for some pos1t1ve constants c1, c2, and C3 independent of
8. Since (2.40) is an e-perturbation of (2.43) we can use V as a Lyapunov function for

(2.39)-(2.40) to arrive at
. C .,
v s —C2||7rll2 — (f — c4) liven? + csllrsll + cenwaumu (2.40)

where C4, c5, and C6 are positive constants independent of 5. It is easy to show that
there exist an 52 such that, for all 0 < e S 52 and all (”775” 2 ec7}, V g 0 where c7 is

independent of 5. Hence, we have that for e sufficiently small the set

de
9,: J 1ku 3 sp}

is positively invariant and Qp C (21, where p is some positive constant independent of 5.
Since P(O) = 0, the solution starts in Qp. Also, since [[Pl] 3 c8||77[| for some C8 > 0, it
follows that

IIPIIsIIPII+IIPII sN+epe8 , v 120

Therefore, P(t) is bounded.

21

To show that P3(t) is positive definite for all t 2 0 we can rewrite (2.25) in the

following way
5P3 = A3(t)P3 + P3A3(t)T + P3CTCP3 + Q3 + er3(P2, P3, t, e) (2.46)

where

43(1) 2 4+ — (P3 — P;)CTC — P3CTC

and

‘113 = {SD/121132 + EPgAng + A0€P3 + PgAg;

We note that A3(t) is bounded by some constant L for all t 2 0. It can be shown that

the corresponding state transition matrix satisfies
. . —2L t—
ll‘1’3(tm em! 2 llllle ( ”/5 (2.47)

Let
Q P(t) = Q3 + P3CTCP3 + 5113(132, P3, t, e) (2.48)

From the boundedness of \II3, there exists 53 such that
0 < n11 S QP(t) S H21

for all 0 < e S 53 where R1 and K2 are positive constants independent of 5. Also, since

P3(0) is chosen to be positive definite we have

P3(0) 2 p1 > 0

22

for some positive constant ,0. Using the relation
xTP3(t):1: = xT<I>3(t, 0;e)P3(0)¢g(t, 0;e):1: + E]; xT<I>3(t,r;e)Qp(r)<I):r{(t,r;e):rdr
along with (2.47) we can arrive at

:rTP3(t):1: 2 [pe_2Lt/€ + $71 (1 — (QM/5)] |[:r|[2 (2.49)

Thus, there exist a positive constant C9 independent of e such that
xTP3<t>cc 2 c914”? (250)

Therefore, P3(t) is positive definite for all t 2 0. Taking 5’] S min{51,52,€3} completes

the proof. <1

2.2.2 Closed-Loop System Stability

We are ready to state our results on the stability of the closed-loop system under EKF

feedback.

Theorem 2.2 Consider the closed—loop system (2.19)-(2.25) under output feedback. Let
Assumptions 2.1-2.3 hold and let M and N be any compact subsets of IR€+T+€ and RT,
respectively. Then, for trajectories (z,:1:,77) x :1: starting in M x N there exists 5; such

that, for all 0 < e g 5;, the following holds:

0 the origin (2: = 77 = 0,11: = 6 = 0) of the closed-loop system is asymptotically stable

and M x N is a subset of its region of attraction.

o The origin of the estimation error equations (221)-(222) is exponentially stable.

23

Proof: In Theorem 2.1 it is shown that there exits 51‘ such that, for all 0 < e _<_ 5’], P(t)

is bounded and P3(t) is positive definite for all t 2 0. In particular
1311 S P3“) S 521 (2-51)
where ,61 and 62 are positive constants independent of 5. It can be seen that

—1
532 P3

satisfies

533 = -(A + EA0€)TS3 - 53(A + 81405) + CTC — S3Q3S3

—eZS3(DA21P2 + pg" 451mg; (2.52)

By the argument above S3 will have a bounded, positive definite, symmetric solution for

all t 2 0. Hence,

231 s 830) s an (2.53)

where 63 and B4 are positive constants independent of e.
Boundedness and Ultimate Boundedness

Following analysis similar to [7] we argue that the trajectories (x, 77,.{) are bounded and
satisfy ||(X(t), 77(t))|| + [[§(t)|| g u for any 71 > 0 after some finite time T*(/1). Denote the

right hand side of (2.19)-(2.21) as

f(X, 7), 02E) (2.54)

4 = Am+e’“—1<G—P20T>s1 (2.56)

x.
H

24

With 5 = 0 we have from Assumptions 2.1 and 2.2 that (2.54)-(2.55) has a globally
asymptotically stable equilibrium at the origin. Thus, there exists a positive definite

radially unbounded function V1(X, 77) and a positive definite function U (x, 77) such that

av1 av1
— , —A < — 2.
0): f(x,n,0)+ 071 1177_ U(X,77) ( 56)

for all X and 77. Let M be any compact subset of R€+T+€. Choose a positive constant
c such that c > 7na:1:(X’n)E MV1(X,77). Then M is in the interior of the set CC =
{V1(X,77) S c} C R€+T+€. Let A 2 DC x {W' 3 p52}. Due to the global boundedness of

and 6 in :13, for all ( , ) E QC and g E RT, we have
f X 7?

l|f(X,n,€)|| S k1 a ||5(Z,I,U,D2€)|l S k2 (2-57)

where k1 and k2 are positive constants independent of 5. Furthermore, for any 0 < 5 < 1,

there is L1, independent of e, such that for all (x, 77, €) E A and every 0 < E g e we have,
||f(x,n,€) - f(x,n,0)l| S L1||€|| (2-58)
Letting W'(£) = {TS3fi and using (2.52) it can be shown that

- 1
W s —EgT [0% + s3o3s3 + 5253(0421192 + pgA§,D)s3 + 54353

34534045 + aTeTs3g + {T5386 (259)

Due to the boundedness of P2 from Theorem 2.1 and the global boundedness of A05 and

A21 in it and 2 we have

llAOell S ’63 , IIDA21P2|| S lf4

25

for some positive constants 173 and k4. Thus, (2.59) simplifies to
W s -§(fi§I|Q3H — mile — 2.273311%]? + 2B4II6IHI€II
Therefore, with 54 chosen such that 254,64k3 + 26,21,33k4 S %6§|IQ3[|, we have
V1 S -U(X,77) + L1L2H€II + ET-leHG - P20T|||€1| S -U(X,77) + 57:5 + 67796 (2.60)

-, k k
w s 7an”? + 2134161131 3 751312 + 24mm (2.61)

for all 0 < e S 54, and all (x,77,§) E QC x {W({) S p52}, where k5, k6, and k7 are given

by
p
k=LL —
5 12g3
~ I)
k=Lk —
6 2 133

and k7 = 6§|[Q3|[ with p = 6417362 / k; and L2 an upper bound on [BVl/ax , 8V1 /877]

in QC. Also, from the boundedness of P2
HG — P2CT|| s is (2.62)

for some positive constant ls. Taking e5k5 +egk6 S v, where v = min(X7n)€aQC U(X, 77),
we have that, for every 0 < e S 55, V1 S 0 for all (x,77,€) E {V1(x,77) = c} x {W(§) S
p52} and W S 0 for all (x, 77, E) E QC x {W(§) = p52}. Therefore, A is positively invariant.

For (x(0),77(0),:i:(0)) E M x N, the initial rescaled error {(0) satisfies “6(0)“ S
k3/5r_1 for some nonnegative constant k3 dependent on M and N. Since (x(0), 77(0)) is

in the interior of QC, we have that

l|(X(t),n(t)) - (X(0)JI(0))|| S kit (2.63)

26

while (x(t), 77(t)) E QC. Therefore, there is a finite time T0 such that (x(t), 77(t)) E QC for

all t E [0, T0]. During this time interval
- ls ,
w s —;,%H€|I2 , for w (o 2 p52

Therefore, it can be shown that

, 134% . ’37
W (5(0) S 82(‘r_1)e:rp (‘15???)

Choose 56 small enough that

 

 

. 4 s, k2
T(e) dif L1-ln [3425 S 1T0
[(37 p5 T 2

(2.64)

Such an 5 exists since T(e) -—> 0 as e -—> 0. Therefore, W(§(t)) S p52 for every 0 <

e S 56- Choosing 8,1 = min{e’[‘,é,e4,55.e6} guarantees that, for every 0 < e S 5’1, the

trajectory (x(t),77(t),{(t)) enters A during the time interval [0,T(e)] and remains there

for all t Z T(e). Thus, the trajectory is bounded for all t 2 T(e). Also, for t E [0,T(e)],

the trajectory is bounded by equations (2.63) and (2.64).

To show ultimate boundedness, we begin by noting that inside A the trajectory 5 is

0(5). Thus, we can find a 57 2 87(71) S 5’1 such that, for every 0 < e S 87, we have

||€(t)|l S 11/2

(2.65)

for all t 2 T(e7). For all (x,77,€) E A we have that V1 S --U(X,77) + 5165 + STAB. Thus,

for (m) 4 When) 3 2655 + we dif 145)}

- 1
V13 —§U(X~,Tl)

27

(2.66)

Since U (x,77) is positive definite and continuous, the set { U ( x, 77) S 12(5)} is compact for
sufficiently small 8. Let c0(e) == maXU(XJ})SI/(E) V1(x,77); c0(e) is nondecreasing and
lim5_,0 c0(e) = 0. Consider the compact set {V(X,77) S c0(e)}. We have {U(X,77) S
V(e)} C {V1(X,77) S c0(e)}. Choose 58 = 58(11) S 5’1 small enough such that, for all
e S 581 the set {U(X,77) S 1/(e)} is compact, the set {V1(X,77) S c0(e)} is in the interior
of QC, and

{V1(x,77) S 60(6)} C {||(x,77)|| S 74/2} (267)

Then, for all (x,77) E QC, but (x,77) E {V1(X,77) S c0(e)}, we have an inequality similar
to (2.66). Therefore, the set {V1(X, 77) S c0(e)} x {W(§) S p52} is positively invariant
and every trajectory in QC x {W(§) S p52} reaches {V1(X,77) S c0(e)} >< {W(€) S p52}
in finite time. Thus,

”(Xftlfllftllll S 11/2 (268)

for all t 2 T for a finite time T = T(,u). Taking 5,2 = 5,201) = min{e7,58} it can be
shown that

||(X,~77)l| + Hill S u

for all t 2 T* where T* = max{T(e7), T}.
Exponential Stability of the Estimation Error

From the ultimate boundedness of (z,.r,2,.i:) we can work locally to argue asymp-
totic stability of the closed-loop system. We have that [|6(z, :c, 77, D25)“ S L3[|77[[ + L4||£||
for all (x,77) E B (0, u) x [”5“ S 11} where u is the ultimate bound from above and L3

and L4 are positive constants. Let

V2 2 nTPLn + {T535 (269)

28

be a Lyapunov function candidate for the estimation error, where the positive definite
matrix PL satisfies PLA11 + Ag PL 2 —1. Using (2.55), the first inequality of (2.61),

and (2.62) it can be shown that

. —e"-1£~HPLI) — B4L3 ||77||
V2 3 — [llnll Hill] _1- ,,
—€’" kllPLll - 54L3 2% — 264L4 “5”

It is easy to show that there exist 59 such that for all 0 < e S 59, the matrix above is
positive definite. Thus, for all (x(0), 77(0),§ (0)) starting in M x N the estimation error,

(77, 6), converges exponentially.
Asymptotic Stability

Let C = [77 £]T. Asymptotic stability of the closed-loop system follows from the

composite Lyapunov function

van, a = Wax. 77) + 06(0)” 2 (2.70)

with 6 > 0. From the first inequality of (2.60) we have
V S —9U(X,TI) + 6ksllCll — kQIICII (2-71)

for k8 = L1L2 + er_1L2lc and a positive constant k9. Taking 6 S leg/2kg yields asymp-

totic stability. Finally, choosing e; = min{e’2, 89} completes the proof. <1

2.3 Comparison

In this section we use numerical examples to further study the EKF and compare the EKF

that has a time-varying gain with a HGO that uses a fixed-gain. This is done by dividing

29

the section into three subsections. First, we examine the choice of control saturation or
estimate saturation to globally bound the control in EKF feedback. We then examine the
possible closed—loop stability advantages of using time-varying gains versus fixed gains.
Finally, we present some results on the choice of initial condition of the Riccati equation

to suppress peaking at the initial time instants.

2.3.1 Control Versus Estimate Saturation in the EKF

The stability results for locally Lipschitz nonlinear systems in Section 2.2 came at the
expense of sacrificing global results for semiglobal ones. An essential factor in this sacrifice
is the effect of peaking on the closed-loop system. In high-gain observers, peaking is caused

by the special structure of the observer gain

(2.72)

as
|_—J

For high-gain observers, peaking can be overcome by globally bounding the control outside
a compact region of interest [26]. This can be done by using a saturation function on the
controller. For the case of the extended Kalman filter, globally bounding the control alone
is not enough. Peaking in the estimates may induce numerical difficulties in the solution
to the Riccati differential equation (RDE) as the following example shows. Consider the
system

1131 = 1132, (132 = 1173, 1123 = mg + u (2.73)

and the feedback linearizing controller
u = —:1:§ — :131 — 3:1‘2 — 3.733 (2.74)

where a will be chosen later on. By saturating the control outside a compact region of

interest the effect of peaking can be overcome and the closed-loop system under (fixed-

30

gain) high—gain observer feedback can recover the response under state feedback as e —7 0.
Let a = 3. Using the extended Kalman filter parameterized as in the previous section

we have that the matrix A + 5A0E in (2.25) is

7- u

0 I 0
A + 8/105 = () 0 1 (2.75)

 

 

~.2
L 0 0 35:63 J

During any occurrence of peaking, the estimate .733 will become 0(1/52). Therefore, from
(2.75) with saturation only on the control, the RDE will contain unbounded terms as
e —-> 0. This system was simulated for e = 0.01 with 1:1(0) = 0.9, 1:2(0) 2 333(0) 2 0,
551(0) 2 132(0) = 1133(0) = 0, P(0) = Identity, Q3 2 diag[3,3,1], and with the control

saturated outside (-20,20). Denote the EKF gain by

Hm zip-11246124 = 112(1)

 

 

Figure 2.1 illustrates the response of the system under control saturation. The peaking
in :83 induces a very large gain (from the solution to the RDE) and this gain in turn
exacerbates the peaking in the estimate. Figure 2.1 shows that the saturation of the
control prevents the system states from deviating too much from their initial values, but
the estimate :23 and the gain h3(t) have become prohibitively large. These difficulties
are overcome by saturating the each estimate outside a compact region of interest. This
will globally bound the control and the time-varying terms in the RDE. This approach
is shown in Figure 2.2 where we have saturated 2:1, 732, and 5:3 outside (—2,2). Figure
2.2 shows that the estimate :63 saturates then quickly converges. Also, we see that the
control remains bounded, the gain h3 converges quickly to its steady-state value, and the

output :131 gracefully approaches the origin.

31

Control Sat.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

a Control Sat.
1 10X10 -
0.9 1
“(O
x" 2 5*
X
0.8' +
0.7 - 0 -
0 0.005 0.01 0 0.005 0.01
f 10 x 1019
0 .
3 —10* so) 5 ’
—20.
0 0.005 0.01 0 0.005 0.01
Time Time

Figure 2.1. Simulation results showing the output 3:1, the estimate :63, the control u, and
the gain h3(t) for EKF feedback under control saturation only.

32

Estimate Sat.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Estimate Sat.
1 ~ 1 2_ E
1
#00
><F 0 5 2 0 1
x
_1 1 [
0 ‘2 J
O 5 1O 15 0 0 5 1
6
2 x 10 f
20’
10 ll 1 5
3 0 .cm 1
‘10 F ‘ 0.5
-201
0 -
0 0.5 1 0 0.5 1
Time Time

Figure 2.2. Simulation results showing the output 331, the estimate $3, the control u, and
the gain h3(t) for EKF feedback under estimate saturation.

33

EKF vs. HGO fore = 0.01 EKF vs. HGO fore = 0.001

 

 

 

 

 

 

 

 

 

 

a 1 -
1- 1- 0.5 ’
X X
0 .
O 10 20 0 10 20
Time Time

Figure 2.3. Simulation results showing the output 2:1, for the EKF (solid) and HGO
(dashed) for s = 0.01 (left) and the EKF and HGO for e = 0.001 (right).

2.3.2 EKF Versus a Fixed-Gain HGO

Like the HGO, the EKF achieves faster and more accurate reconstruction of the state :1:
as s —+ 0. It can be shown from analysis similar to the above that the response for the
EKF approaches the response for the HGO as E ——> 0. Here we illustrate this through a

numerical example. We use a HGO with the following fixed-value for the gain matrix
HT 2 [3/5 3/e2 1/53] (2.76)

Figure 2.3 (left) plots the output y = $1 of the closed-loop system for the EKF (solid)
and the HGO (dashed) for :cl(0) = 0.9 and e = 0.01 with all other parameters as above.
Figure 2.3 (right) shows that the two responses have converged for e = 0.001.
Considering the foregoing observation we note that for relatively “large” values of e,
the time-varying terms in the Riccati equation will have more influence over the closed-
loop response. The question of whether the added complexity of the time-varying gain

gives an advantage over a time-invariant gain is examined next. We consider the system

34

(2.73) for two cases. First, let a = 3. Through simulation of the system under state
feedback control for initial conditions in the set {[in S 5}, we selected a saturation level

of 6 for each estimate. We chose

,-

3/e 3/e2 1/53—
P<0>=§D—1P3<0>D—1= 3/e2 8/53 3/24 (2.77)

1/53 3/e4 3/55

 

 

which corresponds to the steady-state solution of the Riccati equation. The time-varying
terms will cause the solution to deviate from this value. For the HGO we use the same
gain matrix H as above, which is the steady-state value of the EKF gain. This system
was simulated for e = 0.01, 331(0) = 4.9. Figure 2.4 shows the response of the output
:61 and the control signal 11 for the closed-loop system under EKF feedback (left) and
the state 3:3 and the control u for HGO feedback (right). The time-varying gain was
able to stabilize the system where as the figure shows the state :63 going unstable for the
response under high-gain observer feedback. Figure 2.5 shows the response of the EKF
gains for this example. The values of the fixed HGO gains are shown as dotted lines
for reference. Here, the time-varying terms in the Riccati equation caused the gains to
become large during the initial transient before settling close to the values of the HGO
gains. This high-gain aided in stabilizing the closed—loop system. The response of the
EKF estimates are shown in Figure 2.6. The estimates experience both positive and
negative saturation, but quickly settle. We compare these observations with the second
case where we now take a = 2 in (2.73) and (2.74). Simulating for e = 0.1 we see that the
fixed-gain high-gain observer was able to achieve stability and the time-varying observer
went unstable as illustrated in Figure 2.7. This is a result of the sensitivity of the Riccati
equation to the transient response of the estimates. Figure 2.8 illustrates this response.
Here the estimates 1:2 and 5:3 experience prolonged negative saturation. Figure 2.9 shows

the effect this has on the gain response. The gains actually decrease below the values

35

of the HGO gains for roughly 2.5 seconds. The estimates were unable to recover from
saturation quickly enough to prevent the system states from blowing up. Again, these
simulations were performed for “large” gains. We emphasize that both the EKF and the
HGO can stabilize each system (a = 2, 3) by making 5 small enough (e.g. e = 0.001).
Due to linearization, we expect systems under EKF feedback to have an added degree
of stability in a neighborhood of the origin of the estimation error (:1: — 1:). To test this we
reexamined the above simulation for a = 2 and 3 with :rl(0) = 0.9, 3:2(0) = 1:2(0) = 0,
and 161(0) = i2(0) = 333(0) = 0. We ran the simulations for e S 1 to determine the values
of e that made the EKF and HGO closed-loop feedback systems stable and unstable. For
a = 2 we found that the closed-loop system under EKF feedback was stable for e S 1.
On the other hand, for HGO feedback the system was stable for e S 0.1 and unstable for
e = 0.2. For a = 3 the EKF feedback system was stable for e S 0.01 and unstable for
E = 0.02. With the HGO, stability was obtained for e S 0.001 and instability resulted with
e = 0.002. These observations are summarized in Tables 2.1 and 2.2. Each of these tables
show the values of e for which the closed-loop system under EKF and HGO feedback are
stable and unstable for both local (:rl(0)——.i:1 (0) = 0.9) and nonlocal (:rl(0)—.i:1(0) = 4.9)
estimation error. The tables show that, for these examples, the EKF does indeed have
a local stability advantage. However, nonlocally the stability advantage depends upon
the system under consideration (a = 2 or 3). As another example, we repeated these

simulations for the following system
:11 = .722, :72 = x3, .63 = .733 + u (2.78)
and the feedback linearizing controller
u = _1121 — 1‘1 — 3.732 — 3.13 (2.79)
where we used a = 2, 3 and P(O) and Q3 were chosen as before. The results are given in

36

 

 

 

 

 

 

 

Local Nonlocal

Stable Unstable Stable Unstable
EKF e S 1 X EKF e S 0.05 e = 0.1
HGO e S 0.15 e = 0.2 HGO e S 0.12 e = 0.15

 

 

Table 2.1. Table showing the values of e for which the closed-loop systems under EKF and
HGO feedback are stable and unstable for system (2.73) with a=2 and a saturation level
of 6. Shown are the values for the local results with 331(0) 2 0.9 (left) and the nonlocal
results with 2:1(0) = 4.9 (right).

 

 

 

 

 

 

 

Local Nonlocal

Stable Unstable Stable Unstable
EKF e S 0.01 e = 0.02 EKF e S 0.01 e = 0.02
HGO e S 0.001 e = 0.002 HGO e S 0.001 e = 0.002

 

 

Table 2.2. Table showing the values of e for which the closed—loop systems under EKF and
HGO feedback are stable and unstable for system (2.73) with a=3 and a saturation level
of 6. Shown are the values for the local results with 2:1(0) = 0.9 (left) and the nonlocal
results with 21(0) = 4.9 (right).

Tables 2.3 and 2.4, again for local and nonlocal estimation error. Again, we find that 10-
cally, the EKF has an added degree of stability. This added degree of stability for the EKF
in a neighborhood of the origin of the estimation error was observed in other examples.
However, from the nonlocal results in Tables 2.1-2.4, we see that nonlocally advantages
to using a time—varying high—gain observer versus a fixed-gain high-gain observer appear

to be at least system dependent.

2.3.3 Initialization of the Riccati Equation

In the previous subsection, the choice of the initial condition of the Riccati equation P(O)
was made to correspond to the fixed-gain values of the HGO. This was done in order to

compare the effect of the time-varying terms in the EKF. Since the choice of P(0) impacts

37

 

 

 

 

 

 

 

Local Nonlocal

Stable Unstable Stable Unstable
EKF 8 S1 X EKF 5S0.17 5:0.18
HGO e S 0.7 e = 0.75 HGO e S 0.19 e = 0.2

 

 

Table 2.3. Table showing the values of e for which the closed-loop systems under EKF
and HGO feedback are stable and unstable for system (2.78) with a=2 and a saturation
level of 10. Shown are the values for the local results with 131(0) = 0.9 (left) and the
nonlocal results with 2:1(0) = 4.9 (right).

 

 

 

 

 

 

 

Local Nonlocal

Stable Unstable Stable Unstable
EKF e S 1 X EKF e S 0.03 5 = 0.04
HGO e S 0.03 e = 0.04 HGO e S 0.01 e = 0.02

 

 

Table 2.4. Table showing the values of e for which the closed-loop systems under EKF
and HGO feedback are stable and unstable for system (2.78) with a=3 and a saturation
level of 10. Shown are the values for the local results with 11(0) = 0.9 (left) and the
nonlocal results with 11:1(0) = 4.9 (right).

the transient response of the observer, it can also influence the stability of the closed-loop
system. To see this, consider a simulation of (2.73)-(2.74) for a = 2, x1(0) = 4.9, the
saturation level equal to 6, and e = 0.1. Under these conditions, and with P(0) chosen as
in (2.77), the closed-loop system under EKF feedback was unstable as shown in Figure
2.7. This time let P(O) = I, where I is the identity matrix. The plots on the left of
Figure 2.10 show the response of the state 231 and the control u. This figure shows that
the closed-loop system is now stable. Now consider the same simulation, but this time
with an impulsive-like disturbance of duration 0.013 and an amplitude of 300 that is
experienced at the input of .731 at time t = 20. At steady-state the solution to the Riccati
equation P(t) will reach a value close to (2.77). Thus, any disturbance that has the effect
of resetting the initial conditions of the system will induce a response similar to the case

where P(0) was given by (2.77). This can be seen in the plots on the right of Figure

38

2.10. Here we see that, after the disturbance at t = 20, the closed-loop system has once
again become unstable. Thus the initial condition of the Riccati equation can improve
stability of the closed-loop, but only for the initial transient. Afterwards, it is susceptible
to steady-state disturbances.

Finally, we examine the effect of the initial condition of the Riccati equation on peaking
in the EKF. Peaking occurs in high-gain observers, not necessarily because the gain is
large, but because of the structure of the gain matrix. In Nonlinear Systems [36], it is
remarked that peaking is “an intrinsic feature of any high-gain observer with h2 >>
hl >> 1.” We consider once again the system (2.73) and control (2.74) for a = 3,
e = 0.01, :rl(0) = 0.9, :1:2(0) 2 333(0) = 0, 11:1(0) 2 122(0) = 5:3(0) = 0, P(0) = 1,
Q3 = diag[3,3,1], and with the estimates saturated outside (—2, 2). Figure 2.11 shows
that the estimates undergo peaking during the initial transient. Also shown are the gains
h1, h2, and h3. Here, the gains quickly approached values where h3 >> h2 >> h1 >> 1.
We can choose the initial condition of the Riccati equation to eliminate peaking during

the initial transient. Consider the following initial condition for the Riccati equation

1x10611
P(0)= 1 10
1 01

 

 

With this choice we have that h1(0) >> h2(0) and h1(0) >> h3(0). Simulation with this
initial condition was carried out and the result is shown in Figure 2.12. Comparison with
Figure 2.11 shows that the peaking in the estimates has been suppressed. The figure also
shows that the gain hl very quickly decreases toward its steady-state value. Initialization
strategies to overcome peaking have been explored for observers with time-varying gains
in [17], [33], and for sampled data output feedback control in [37]. However, as has been
pointed out in [17] and [37], these designs may suffer from peaking through impulsive-like

disturbances that occur after the initial transient. Therefore, the peaking phenomenon is

39

relevant irrespective of the initial gain choice.

2.4 Conclusions

Considering the tradeoff between closed-loop stability and peaking in a high-gain observer,
we have considered the extended Kalman filter parameterized as a high—gain observer with
time—varying gain. We have examined the closed-loop behavior of nonlinear systems in
the special normal form under extended Kalman filter feedback. We have shown that the
origin of the closed-loop system is asymptotically stable and the origin of the estimation
error is exponentially stable. We have seen that in addition to globally bounding the
control, the time-varying functions in the Riccati equation must be globally bounded for
the Riccati equation to have a well defined solution. We have exploited the two-time scale
nature of the partitioned Riccati equation to argue boundedness and positive definiteness
of the solution. Through simulation we have compared the closed-loop performance of the
time-varying EKF versus the time-invariant HGO. We have seen that the EKF with time—
varying gain may provide closed-loop stability advantages over the fixed-gain observer.
Globally this will depend on the particular system under consideration. However, the EKF
appears to have an advantage locally. Furthermore, the Riccati equation can be initialized
to eliminate peaking during the initial transients. Thus, for a particular system, a time-
varying HGO may be able to achieve a better balance between closed-loop stability and

suppression of peaking than a fixed-gain observer.

40

 

Extended Kalman Filter Fixed-Gain HGO

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 . 1000
500»
4.
><F x” 0_——‘"“
2.
—5001
0 —1000 4+ -
0 10 20 0 0.05 0.1
200» . 200l J 1
3 0L 3 ()l
—200 . —2001
0 10 20 0 0.05 011
Time Time

Figure 2.4. Simulation results showing the output 271 and the control u for EKF feedback
and the state 1:3 and the control u for HGO feedback for a = 3.

41

Extended Kalman Filter

 

'- 4OOL

gain h

 

r

 

 

 

......

 

 

 

 

 

 

O 1 2 3 4
Time

 

 

Figure 2.5. Simulation results showing the EKF observer gains h1, h2, and ’13 for a = 3.

42

Extended Kalman Filter

 

l 1 T

 

 

 

 

 

 

 

 

 

 

 

 

 

Time

Figure 2.6. Simulation results showing the EKF observer estimates 5:1, :22, and 5:3 for
a = 3.

43

Extended Kalman Filter Fixed—Gain HGO

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 a
5000i
X” 0 X? Oi
—5000* 5 L
O 5 0 10 20
100 100 f
50' ‘ 50*
3 0» . 3 off
-50- + —50
-100 —100 ‘
0 5 0 1O 20
Tune Tune

Figure 2.7. Simulation results showing the state 3:3 and the control u for EKF feedback
and the output 1:1 and the control it for HGO feedback for a = 2

44

Extended Kalman Filter

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 _ . fi fi
E 0
X
-5 '- 1 1 l 1 l
0 0.5 1 1.5 2 2.5 3
5 r
“N
2 0
X
-5 . . l l I L /
0 0.5 1 1.5 2 2.5 3
5 _—
“('0
2 O
X
—5 F‘ 1 L l l L
0 0.5 1 1.5 2 2.5 3
Time

Figure 2.8. Simulation results showing the EKF observer estimates 5:1, 5:2, and £3 for
a = 2.

45

Extended Kalman Filter

 

..., ..........................................,...-......-..-..q

 

 

 

 

 

 

 

 

 

 

 

 

1 1.5 2 2.5 3
Time

Figure 2.9. Simulation results showing the EKF observer gains h1, fig, and h3 for a = 2.

46

EKF EKF/Dist.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

14
1
6x 0 V
4.
><'_ xF 2
O
—2
0 5 1O 0 20 4O
20 50
O
D 3 .
'9' —20 :3 O”
E E
8 '40 ‘ 8 —50
-60
—80 * —100 ‘
O 5 1O 0 20 40
Time Time

Figure 2.10. Simulation results showing the output 51:1, and the control u for EKF feedback
with P(O) = I . The plots on the left are stable, but the plots on the right become unstable
due to an impulsive-like disturbance at t : 205.

47

EKF

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0
0 (15 0 (15
4
30 4x10
*9, 20 £9:
“5 10 1 .E 2
'5 0 g,
-10 0
0 (15 0 (15
5
500 15X10
a.” 1:” 10
0 .9
'5 g; 5
-500 0
0 (15 0 (15
Time

Time

Figure 2.11. Simulation results illustrating peaking in the EKF observer estimates 3:1,
5:2, and i3 and the response of the observer gains h1, I12, and h3.

48

EKF

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5
1 10x10
‘5"— ; 5
§ 05* 1.§
0
0 A
0 05 0 005 OJ
4
1 4x10
N
.J“ .c
m 0 .E 2
5 a
—1 0
0 05 O 05
5
1
1 15X 0
26” 3°10»
O\/—\ .E
")5 8) 1
-1 0
O 05 O 05
'fime 'fime

Figure 2.12. The response of the EKF observer estimates £1, £2, and 5:3 with Riccati
initialization to suppress peaking. Also shown are the observer gains hl, hg, and h3.

49

CHAPTER 3

Switched Gain High-Gain Observers

3.1 Introduction

It is well known from observer theory [40] that a tradeoff exists between the speed of
state reconstruction and the immunity to measurement noise. The high-gain observer is
known for having the ability to quickly reconstruct the system states and reject modeling
disturbances [26]. In this chapter we examine the tradeoff between fast reconstruction of
the states and rejection of modeling error versus the immunity to measurement noise for a
high-gain observer. Based on this, we introduce a high-gain observer design where the gain
matrix is switched between two values. The idea is to use high-gain during the transient
to quickly recover the state estimates. Then once the estimation error has reached a
steady-state threshold, we switch to a second gain to reduce the effect of measurement
noise. A similar idea was explored in [46] for linear discrete-time filters. The authors
combined two linear filters by switching gains based on the estimation error reaching
a set containing the origin. A large gain is used outside this set to increase the filter
bandwidth which reduces the estimation settling time. When operating inside the set, a
lower gain is used to reduce bandwidth in order to accommodate the measurement noise.
Switched-gain observers have also been considered in [25], where a high-gain observer was

combined with a sliding term for estimation of nonlinear systems. The idea was to use

50

low-gain along with the sliding term to provide stability and avoid the large transients
associated with high-gain. At a predetermined time, after the transient period, the gain
is switched to a high value to provide better estimation and tracking. In this scheme, the
effect of measurement noise was not considered.

The switched-gain scheme proposed in Section 3.3 uses high-gain during the transient
period followed by a discontinuous switching. The switching event takes place when the
estimation error y — :81 reaches a predetermined interval containing the origin. Due to the
observer transient response the design contains a few special features. First, the observer
eigenvalues are assigned to ensure that y — 5E1 decays monotonically towards the switching
interval and reaches it in finite time. Second, a delay time is incorporated into the scheme
that delays switching till after the observer transient response in order to prevent multiple
gain switchings. Third, to avoid peaking after the switching event takes place, the ratio
of the two gains may need to be restricted. Based on this, we study the closed-loop
system under the switched-gain high-gain observer and argue that the trajectories of the
closed-loop system are bounded.

Stability results for closed-loop systems under high-gain observer feedback in the pres-
ence of measurement disturbances can be found in [1] and [6]. For sampled-data high-gain
observers the effects of measurement noise on the performance was studied in [19]. In [1]
the limiting effect of increasing the gain of a high—gain observer on the trajectories of the
closed-loop system in the presence of measurement noise was studied. A reference system
based upon ideal differentiation of the measured output was introduced to compare the
convergence properties of the high—gain observer. Also, in [6](Ch. 4) stability results for a
class of nonlinear systems under HGO feedback in the presence measurement noise were
provided. It was shown that, for sufficiently smooth disturbances, and given a state feed-
back controller that in the presence of the disturbance provides stability of the closed-loop
system, the closed-loop system under HGO feedback can recover the performance of the

state feedback system for sufficiently large observer gain. In [68], estimation error for

the HGO acting as a differentiator for noisy signals was studied. Also, guidelines for the
choice of observer gain in the presence of measurement noise were provided.

In this chapter we also consider the high-gain observer acting as a differentiator for
signals that may have large bandwidth in the presence of measurement noise. Again, the
idea is to use high-gain when the signal of interest is quickly changing in order to achieve
good estimation. Then, at steady-state a lower gain value is used to reduce the effect of
measurement noise. In [67] Tilli and Montanari introduced a continuous switching scheme
based on using a dead zone nonlinearity. Their estimator was designed to estimate lst
and 2nd derivatives of a measured signal corrupted by noise and exploit the variable
bandwidth of this signal. In Section 3.5 we apply this nonlinear switching scheme to the

high-gain observer and provide a numerical example to show its effectiveness.

3.2 Motivation

In this section, we motivate nonlinear switched-gain observers by discussing the tradeoff
between state reconstruction and attenuation of measurement noise. We do so by studying

the observer’s estimation error. Consider the following nonlinear system

2 = 07.1(x,z,'u) (3.1)
:i: = Ax+B¢(.r,z,u) (3.2)
y = 017+?) (3.3)
C = 901332) (3.4)

where u is the control input, I E 26 Q RT and z E Z g R6 are the states, and y and C are
the measured variables. The measurement noise, v(t), is a bounded measurable function.
The functions 1/1 and (2') are locally Lipschitz in their arguments over the domain of interest
and satisfy 21’)(0,0,0) = (0(0, 0.0) = 0. The r x 1" matrix A, the r x 1 matrix B, and the

1 X 1" matrix C are the same as (1.5)-(1.6).

52

The high-gain observer is designed for the system states (3.2) based on measurement
of y. The tradeoff will concern reconstruction of the states :1: and rejection of noise 1). For
this reason, and for simplicity of presentation, we do not consider noise accompanying the
measurement C. In addition, since it is the estimates of the derivatives of the output that
are primarily affected by this tradeoff, we consider the above system where the relative
degree 7‘ is greater than or equal to two. For r = 1, one may simply use the measured

output. The observer for (3.2) has the form

:i‘z Ai+B¢50(i‘,C,u) +H(y—C:i:) (3.5)
The gain is given by
T 01 (12 or
H — — ' — 3 6
e 52 5T ( )

where e is a small positive parameter and the roots of

sr+alsT—1+-~+a.,._1s+a,~=0 (3.7)

have negative real parts.

Assumption 3.1 The function 9:50 is locally Lipschitz in its arguments over the domain

of interest, globally bounded in :r, and satisfies ¢0(0, 0,0) = 0.

we consider the partial state feedback controller

u = “/(17, C) (3-8)

The corresponding output feedback controller is given by

U : ’l/(jv C) (39)

To examine the observation error (:1: — it) we consider the following rescaling
‘-=ei_1(r-—:i:-) (310)
’72 ‘ 2. z '

for 2' = 1...r, This rescaling differs from that of previous work on high-gain observers
due to the presence of measurement noise. W'ith this rescaling we have D—1(e)n = :1: — :i:
where, D(€) = diag[1,5,---,eT—1]. The closed-loop system under output feedback can

be written in the following form

 

 

2 = ui’(x,z,1(:v-D—1(€)n,<)) (3.11)
:z: = Acc+B¢(rv,z/i(:c-D_1(€)n,<)) (3.12)
57') = A077+€TB6(;r,z,D—1(5)7})+B2v (3.13)
where
60:, z, D—1(€)n) = «10:. 2.75.0) — ¢0(i,C,1(i, 0)
and _ .
“0’1 1 0 _ 1
—02 0 1 0
A0_ ,B2—
Tar—1 0 1
-07. 0 0 ' ‘

 

 

Setting 7} = 0 in (3.11)-(3.12) results in the noise—free, closed-loop system under state

feedback

w($,z,1($aC)) (3-14)

it = A$+Bo($,zn’(l‘,0) (3-15)

N.
II

We consider any state feedback control design that satisfies the following properties.

Assumption 3.2

1. The origin (:1: = 0,2. = O) of (3.14)—(3.15) is asymptotically stable and R is its

region of attraction.

2. The function '7 is locally Lipschitz in its arguments and globally bounded in :16. Fur-

thermore, 7(0, 0) = 0.
We operate under the following assumption on the measurement noise.

Assumption 3.3 The measurement noise u(t) is a measurable function that is bounded

by a known upper bound; that is,

|v(t)| S kw

To illustrate the tradeoff between state reconstruction and measurement noise immunity,
we focus on Equation (3.13). Furthermore, these estimation error dynamics will be of pri-
mary concern in presenting and analyzing the switched gain observer design. With this in
mind, let QC be a compact subset of ’72. Due to the global boundedness of 6(x, z, D_1(e)n)
in 11?, we have

l|6(:v,z.D—1(€)n)ll s a

for all (:13, z) E QC and some k5 > 0 independent of e. For the time being we will operate
under the assumption that (1:, z) E QC and study the estimation error. Later we will study
the behavior of the closed-loop system and show that if [:L‘, z) starts in 00 then (:r, 2:) will
remain in QC. Since A0 is a Hurwitz matrix, we have SAO + A35 2 -—I for some positive
definite symmetric matrix S. Take W = nTSn as a Lyapunov function candidate for the

estimation error. It can be shown that
W<_1 2 2T—1- 53k 3 SB k 316
_ Ellnll + e llnllll ll5+€lln|lll 21mm <. >

Furthermore, we have

W s ——w (3.17
25] SI] )

 

55

for

w 2 mus“ (arllssuké + IISBgllkvu)2 (3.18)

where [[5]] = )‘maa‘(S)- Hence, the set
{Ween s 161151 (915811.; + 15321110102} (3.19)

is positively invariant and 77 reaches this set in finite time T(e), where

lim T(€) = 0
e—>O
Inside this set 7) is bounded by
Hall S €611»); + egkvu (3.20)

Aminks) 7 (S)

rescaling (3.10) we have

where c1 = 4]]SB]| ——]LS——|—— and c2 = 4|]SBQ|| Xfl' Referring back to the
mi 1

62 kw (3.21)

“110 — at)“ s ||D_1(€)Ill|n|l s Eclké + EH

 

as an ultimate bound for the estimation error. Consider the initial state ($(t0), :i:(t0)). It

can be shown that the initial error satisfies |]r)(t0)|| S kg for some .133 2 0 dependent on

M and N. From (3.17) and (3.18) we have

W020» s W<n<to>>exp (ii) (3.22)

(’5

whenever (3.18) is satisfied, where 01 = 1/(2]]S]]). Using (3.22) and the rescaling (3.10),

it can be shown that, before reaching the bound (3.21), the error (.i:(t) — :i‘(t)) will be

56

bounded by

 

||:r(t) — at)“ s ”2 exp (£13) (323)

ET—l 25‘

SH

where 02 2 kg ’\min S . The tradeoff between the speed of the reconstruction of
the state and the rejection of the modeling error versus the immunity to measurement
noise is clearly visible from equations (3.21) and (3.23). In the noise free case, the high-
gain observer is designed to quickly reduce the state estimation error to a small value.
However, the presence of measurement noise places a restriction on how large a gain we
can choose. From equation (3.21) we see that for the estimation error to be 0(5) or
0(1) the measurement noise magnitude must be 0(5r) or 0(8r—1), respectively. This
means that for [u] _<_ levy, a can be reduced to Owl/7“) or 0(u1/(r_1)). We can see that
attenuation of measurement noise requires larger values of 5 (low gain). On the other
hand, for fast estimation of the states and rejection of the modeling error we require

smaller values of 5 (higher gain).

3.3 Switched-gain observer

To relax the tradeoff discussed above, we propose a switched-gain observer. The switched-
gain observer design is based on the output error (y — :21) and a known upper bound on
the measurement noise. The idea is to use a sufficiently large gain when the output error
is large. This will provide fast reconstruction of the state estimates at the expense of
increased measurement noise during the transient response. When the output error has
reduced to a small value, we switch the gain to achieve a better balance between the input
and measurement noise. The switching criteria is based upon the output error reaching a
particular interval. Considering that estimates of the higher order derivatives will exhibit
peaking we will have to exercise some care in determining when to switch. If we switch
before the estimates of the higher order derivatives have recovered from peaking, it could

have the effect of driving the output error outside the switching interval. We define the

57

switching interval as Id = [—d, d] for some design parameter (1 > 0. We will discuss the

choice of (1 later. We use the same observer design as before,
5: = .45; + Bq50(:i:, c, u) + H(y — ca) (3.24)

but with the gain matrix, H, taken as

1 1 1
a a
Hszfz _l -22 37,7: (325)
before switching and
T T 02 a2 a2
H = H _. —1 —-2- —" (3.26)
e 2 5r
2 52 2
after switching, where O < 81 < 52. The constants of, j = 1,2, and i = 1,---,r, are
chosen as in (3.7). The different sets of parameters, (121’s and 0122’s allow for the flexibility

of choosing the observer poles at different locations. In the analysis we will consider
the closed-loop system under output feedback for two cases. For the case when the gain
H = H2 we use the same rescaling as above, 77,; = leg—1&2,- — 22-). This will yield the same
system of equations as (3.11)-(3.13) with 8 replaced by 52. When the gain is given by H1

A

we have, using the rescaling 0,- : 521—1 (.132- — 1:2),

2 = zl’(:v,z,1(-r — D‘lenam (3.27)
a: = A2: + Bags, 2,1(27 — D_1(51)6, o) (3.28)
819 = A06 + ETI‘BMJS, Z, D_1(€1)9) + 8221 (3.29)

We will focus on (327)—(329) for the moment. We would like switching of the gain to
be based on detection of the output error entering the switching interval. As mentioned
in the introduction, we need to include a delay between the time (y — .131) enters the

switching interval Id and the time the gain is switched. A delay timer will be initiated

upon detection of (y — £131) entering Id. However, the transient response of the observer
may cause (y — £1) to overshoot the switching interval. Our switching scheme will reset
the delay timer whenever (y — 2:1) exits the switching interval and restart the timer upon
reentry of (y — 5231) into Id. Thus, overshoot of Id may cause starting, resetting, and
restarting of the delay timer. We can avoid this scenario and hasten the gain switching
by designing the observer poles so that (y — 5:1) does not overshoot Id. To see this, write

the observer polynomial (3.7) as
(st—1 + HIST—2 + - . . + 3,43 + fir)(s + K) = 0 (3.30)

where the first polynomial is Hurwitz with 0(1) roots and It >> 1. With this choice of
polynomial roots, the system will exhibit two—time scale behavior. It will have a fast
component that corresponds to the pole located at -K. and (r — 1) slow components that

correspond to the roots of
sT—1+6ls"—2+-~+e,._ls+a =0 (3.31)

Considering this two-time scale choice of observer design, we can represent the estimation
error in the singularly perturbed form. Toward that end, rewrite A0 and 82 in the
following way:

A0 = A015 + A02 (3.32)

and

32 2 320K + 321 (3.33)

59

where

and

A02 =

 

 

b

Tfir—2
_firwl

_31
_52

_Br—l
0

0

1
0

 

 

1320 =

 

1321=

 

 

_fir—l
0

 

d

To transform the system into the singularly perturbed form, we follow the procedure of

[38]. First, notice that the direct sum of the range and null spaces of A01 span RT. Let

the r x (r — 1) matrix M and the r x 1 matrix N be given by

M:

 

 

 

 

_ fir—l 1

The columns of M and N are the bases for the null-space and range-space of A01, re-

spectively. We define the inverse of a transformation matrix T as

T—1 = [M N]

With these matrices T is given by

where the 1 x r matrix Q is

0:11 0 01

and the (r — 1) x r matrix P satisfies PA01 = 0; that is, the rows of P span the left
null-space of A01. Using the transformation matrix T, we consider the following change

of variables

6
91

2T9

According to [38] (Proposition 6.1) the change of coordinates é = P0 and 61 = QB

transforms the system (3.29) into

815 PA02A-I§ + PAO2N91+ EEPBMJI, 3,51: — i‘) + P3212) (3.34)

5161 = QA02M§ + (HQAOIN + QAOQNWI — (n + (311) (3.35)

where we have used the relation 0240111! = 0. It is easy to show that QAOlN = —1,
QAOQJVI = Q, and AOQN = 0. Therefore, we have

515' P21021115 + 5’1‘P36(:c, z, :1: — i) + PB21U (3.36)

519', = {1 —K,01—(K.+fi1)u (3.37)

Note that 61 = x1 — ail and P24021111 is a Hurwitz matrix. Using (3.36) and (3.37) it
can be shown that, for an appropriate choice of d and it, (y — 2:1) will enter, and remain
in, the switching interval. In fact, from (3.37) it can be shown that (y — (271) will reach

the switching interval during a time period [t0, to + T12 (3%)], where T12 (:1) —> O as

C
ft

61

if} ——> 0. We note that if the gain is switched before the transient response of the estimates
of the higher order derivatives has settled it may cause the output error, (y - 21), to
leave the switching interval. This could result in repeated switching of the gain until
the remaining trajectories recover from peaking. To avoid this scenario, once (y — :21)
enters the switching interval we delay switching by a time period T d that depends upon
the peaking period of the observer (3.29) to ensure that switching takes place after the

trajectories of the estimation error 6 have reached an invariant set.

3.3. 1 Switching Scheme

Based on the above discussion, we use the following gain switching scheme for the observer

(3.24):
1. Choose H 2 H1 and reset the delay timer whenever |y — .11] > d.
2. Once (y — 5:1) enters (or begins in) [—d,d] start the delay timer; keep H 2 H1.
3. After the delay time (of duration Td) and while (y—il) E [—d, (1] switch to H 2 H2.

From the foregoing development, we present the following analysis which illustrates the

choice of the delay time duration Td and switching interval (Id) size d.

3.3.2 Switching Analysis

In order to examine the estimation error before switching, we begin by studying the slow
and fast variables of the singularly perturbed system (3.36)-(3.37). Take W1(§,_61) =
W11(E) + W12(61) = {T516 + %6% as a Lyapunov function candidate for (3.36)-(3.37),
where the positive definite matrix 31 satisfies 51(PA02A'I) + (PAOQM)TSl = —I. Let
1811 = Amaasn. Then

. 1 _ 1
w, : —;|l€l|2+2ll€ll 5’1" 115113311“51811382111111

62

1 K 2 (h? +131)
—6 ——6 +—6 Icy.
+€1|1H€1| 51 1+ 5 I1! L.”
We have that W1 g —2%1”€”2 — 2:370? g 0 for
It 2 1 2 1
——0 —— —6 <0 3.38
4511 481111 +€1|1IIE1I_ ( >

1 _ 1
-—|l€||2+2|l£|| 5’1“ 1181138113+5151P8211w :0 (3.39)

(Na + —B__1)
_._gl+

6 k <0 .40
451—I1lvu (3 )

Inequality (3.38) is satisfied for re 2 4. Inequality (3.39) is satisfied for
“EH 2 8€f||51PB||k5 + 8||31P321||kvu (3-41)

and inequality (3.40) is satisfied for

4(K+[31)
it

[91] 2 (Cult

Therefore, W1 3 0 for all

(23,61) 6 [vi/12.01) = 641811 35131108161“ + 18110321112111?

8(n+B 2 de
1) 13112 J91)

+
K2

(3.42)
def , . . . . . .
We have that the set 21 : {111(£,01) _<_ 91} IS posmvely invarlant. Usmg the
inequality

1
Ammsnuafi + 56% 3 W12, 9,)

we have, for (15, 61) in the set $1,

”9]! < IWIIIIEH + ||N||191| < II ”II ,\ ——(—913)+II’VII\/20 (3-43)

min

63

From (3.42) it can be shown that 6 is bounded by

 

“6|! 2 3,1557," + canny (3.44)
where
C4 = 8C3WSTI151PBII (3.45)
can) = C3 (Est/E 11511082111 + he —+—"1)) (3.46)
and C3 = \/)\7|'|I:’::.H(Sl) + fiHNII. For W1(£,61) 2 91 we have
W1 3 —2———€1HSIHW11——1W12_—:—i13W1 (3.47)

where 03 = min {52, W) By the same argument used above, ||l9(t0)]| _<_ kg for some
ks 2 0, dependent on M and N. Therefore, we have ||£(t0)|| S ||P|]k3, [01(t0)] 3 kg,

and

W1 (5(1): 91(0) S 04 ea319(-031/€1) (3.48)

where 04 = (1/2 + [[31]]]|P||2)k§. In addition, we have that (€(t),61(t)) enters the set

21 during the time interval [t0, to + T1(el)] where
e a
T1(51) = —1m (44) (3.49)

03 91

We have from (3.43) and (3.48) that (9 is bounded for all t 2 t0 and ultimately bounded
by (3.44).
Next we study the output error (y — 1:1) by considering the Lyapunov function W12 =

%6% for (3.37). Notice from equations (3.44) and (3.48) and the relation “fill S ]|P|||]6||,

64

that 5 is bounded uniformly in 81 for all 51 < 1 and all t 2 t0. Thus,

IISH S k1

for some positive constant k1 independent of 51. It is easy to show that W12 3 —2%1—0% g
0 for

2
W 2 2 (k1 + (n + may) (3.50)

_ 2 . 2 dEf ' K. 2
Therefore, for all 01 E {ll/”(61) —- ;2[k1+(r£+,61)kvp] -— 912}, W12 3 -2561 _<_ 0.
We have that the set 212 = {W12(61) S 912} is positively invariant. Inside 212, we
have

. 2 2 H3 +13 (16
ly — x11: W + lvl 3 gkl + (47—13 +1) kw =f W) (3.51)

Later we will choose r: sufficiently large so that 7r1(r<.) < d and thus, (y — i1) will be in
the switching interval Id.

Next we compute the time at which (y — 5:1) enters Id. We have for W12(01) Z 012

”/12 S _51W12 (3.52)
1
Therefore,
W12(61(t)) s I’Vlg(t0)e.rp(—nt/51) (3.53)
Note that |61(t0)| 3 k3. Then
1
W12<61<t>> s EkiexM—nt/el) (3.54)

In addition, we have that W12(61) enters the set 212 during the finite time interval

2
5 5 k
T12 (21.) 2 71m (29:2) (3.55)

[t0, to + T12(§,%)] where

 

65

we note that for ||6(t0)|| 3 kg and |61(t0)| E 212, then |y(t0) — :i:I(t0)| S 7r1(r;.) for all
t 2 to. In other words, if d is chosen so that 7r1(r:.) < (1, then for (y(t0) — i'l(t0)) starting
in the switching interval, it will remain in the switching interval for all t 2 t0. Notice
that for large enough K, T12 (5%) < T1(€1). This indicates that when (y - :21) enters
the switching interval, the estimates of the higher order derivatives, i, may not have had
enough time to recover from peaking to reach the invariant set 21. If on the other hand,
T12 (€751) Z T1(51) then we can choose a delay time duration of 0. Therefore, we choose
the switching delay timer duration (Td) according to the following statement
Td > T1 —T12, if T12 < T1

(3.56)
TdZO’ if T122T1

Thus, the delay timer begins when (y — i1) enters Id and lasts for duration Td. After the
time duration Td, the gain is switched. Let T; be the switching time. At the switching

time, using (3.44), we have that the estimation error satisfies

 

 

. 1
1:, — :ril S 7_—1(C4k5€§ + c5(r;)kv,u) (3.57)
’31
After switching we get
53—1
I'm-(Ts)| s ,3, (641.557, + 05(levfll (3.58)
51
and 1
T'-
. _1 5
”77(Ts)” S C4k5€§ 51 + ,._105(h3)kv# (3-59)
1

Next we derive the ultimate bound on the estimation error 17 after switching. For the
high-gain observer operating under the gain H2 we have a result identical to the previous

section (3.11)-(3.13) with e = 52. Therefore, using ”’2 = nTSgn as a Lyapunov function,

66

we have a positively invariant set 22 given by

def ,. 2de
22 = {W2(n)S16H52|l(6’2IIS2Bllk5+||S2lelkvu) =f 92} (3.60)

where the positive definite matrix 52 satisfies 5'on + A382 = —1. Therefore, any 17

starting in 22 will remain there and 17 will be ultimately bounded by
llnll S 656119; + Cgkvu (3.61)

where c1 and 02 are the same as (3.20) with S = 52. It can be shown that, for all
W2 2 02,

W2(77(t)) S 066:1:p(—05t/52) (3.62)

where 05 = 1/(2HSZH) and 06 = ||7](T3)[|2||52||. The trajectory 7} enters the invariant set

22 during the time interval [T3, T3 + T2(€2)], where T2032) is given by

T2(52) = iii—111(3) (3.63)

From (3.61), we have that the output error is bounded by
. 1" def
ly — 331' 3 Hr)“ + lvl 3 ezclké + (1 + c2)kv,u = 7r2(62,u) (3.64)

We choose the size of the switching interval d such that 7T2(€2, ,u) < d.
Referring back to (3.59) we see that if 77(T3) is in the invariant set 22, then 7)(t) will

remain in 22 for all t 2 T5. 011 the other hand, if 7}(T5) is outside 22 then
”TI” 2 4 (€§||S2B||k§ + “5232!th

and

W2 3 0 (3.65)

67

It follows that on the Lyapunov surface l’VZ = c where

r—l
12._1 9500190”
51

E
—1€1+

 

C = ”52” C4k6€g

1V2 S 0. Hence, the set {ll/2(7)) S c} is positively invariant. Then we have

 

1 Er—l
||n(t)|| S C6 0413559” 61+ E_lc5(h:)kvp (3.66)
51
2 ”52”
for all t 2 T3, where C6 Amini‘SZ). From (3.66)
r—l
. _ 5 dc

|y(t) — 1:1(t)| _<_ C6 C4k6€g 151 + Eg—_lc5(rs)kvp + kvp :f 7r3(h:) (3.67)

1

for all t 2 T3. Let £1 = 7r1(K)|r$—+oo and 13 = 713(K)IK;—->oo. Then
£1 = 31'5le

and, using (3.46),

,r—1
2

77 = c C k CT_1€ +c C
—3 ’6 '4 692 1 3E.,-_1
l

 

(8\(ilSIIIIISIPB21i| + \/—8_) kvlt + kvp

With 51, 52, k1), and k75 we choose d > max{7r2,£1,13}. We can then choose K. large
enough such that

771,7r3 < (1

Finally, ||n(t)|| will enter the set 22 in finite time and be ultimately bounded by (3.61).
Choice of 51 and 52

68

Notice from Equation (3.67) that if eff—1 << 63—11(1)”, then 7r3 will be large.
This may yield a large switching interval size compared to the ultimate bound 7r2 in
Equation (3.64). To relax this conservative switching interval size we can restrict the
relationship between 51, 52, and a. First, we choose 52 such that 55 2 kvu. Then
we take 0 < 51 < 52 < 1 and 51 : kgi'g for some positive constant kg. With these
choices 773 will be 0(52). The choice 52 < 1 is made in order to simplify the analysis
and is reasonable in the presence of relatively small measurement noise magnitude. The
theorem to follow guarantees that there exist 5*, dependent on ,u, such that, for all
(kvvu)1/ T S 52 S 5*, all closed-loop trajectories are bounded. Thus, if (5*)r 2 kvu we
can choose (kva)1/ r S 52 S 6* and 51, 82, and k5 can be chosen to relax the state
reconstruction speed and measurement noise tradeoff. With these observations in mind

we state the following theorem.

Theorem 3.1 Let Assumptions {3.1)-(3.3) hold and consider the plant (3. 1)-(3.4), the
output feedback controller (3.9), and the switched-gain observer (3.24) with the switching
scheme as given in Section 3.3.1. Let M be any compact set in the interior of R and
N be any compact subset of RT. Also, let 52 2 (hyml/r and 51 = [655% for positive

constants kv and k5. Then, there exist 5* > 0 such that or every kv t 1/r S e S 8*,
l 2

I. the solutions (z(t),:r(t),i(t)) of the closed-loop system, starting in M X N, are

bounded for all t 2 t0.
2. the estimation error 77 is ultimately bounded by {3.61).

3. the output error (y -— i1) enters the set Id during a finite time interval [t0, t0 +T12]
and remains there for all t 2 T12 and for design parameters re, d, and Td chosen

sufiiciently large.

Proof: We were able to show items (2) and (3) in the foregoing analysis provided that the

69

state (:6, z) remained the compact set QC C R. We argue here that if (2:, 2) starts in QC
then (x, 2) will remain in QC.

For simplicity let x = [3: z]T and denote the right hand side of (3.1)-(3.2) as

d)($’ Z? u)
Arr: + B¢(:c, z, u)

f(X,U-) =

With this notation, the system equation under output feedback control (3.11)-(3.12) is
given by
X = f(X,7(it,C))

and the closed-loop system under state feedback (3.14)-(3.15) is

X = f(x, 7(27, 0) (3138)

From Assumption 3.2, the origin of (3.68) is Asymptotically stable with region of attrac-
tion R. By Theorem 4.17 of [36], there exists a smooth, positive definite function V(X)

and a continuous positive definite function U (x), both defined for all X E R, such that

V(x)—+oo as XHBR

8V
aflxntaO) S -U(X), Vx E ”R

and for any c > 0, {V(X)} S c} is a compact subset of R. Let M be any compact set in
the interior of R and choose positive constants b and c such that c > b > maxXE M V(X).

Then

3C95={V(x)Sblcflc={V(x)Sc}CR

70

Due to the global boundedness of f and 6 in :23, for all x E QC and :i‘ 6 RT we have

llf(x,i(i‘.C))ll 3 k2. “66.2.2: _ an s k,

for some positive constants kg and 165 independent of 5. Also, there exists an L1, inde-

pendent of e such that

||f(X, 7(1‘3, Cl) - f(x,7(l‘, O)” S L1||x - i’ll

Now, consider the set 22 from (3.60). It was argued that this set is positively invariant
provided that (.r, z) E QC. Since a is C(55), n is C(55) inside 22. Let L2 be a upper
bounded for HOV/ax“ over QC. We have

- . L1L2

v s -U(x) + L1L2ll-r — sun 3 -U(x) + Hull 3 —U(X) + c752

T‘—1
E2

 

for all (x, n) E QC x 22 and some positive constant 07 independent of 52. Take 83 = 29/07,
where 19 = mianBQc U(X). It can be shown that, for every 0 < 52 S 53, we have
V(X) S 0 for all (x,77) E {V(X) = c} x 22. Also, from the derivation of (3.60) we have
that W2 S 0 for all (x, 77) E QC x {W2(n) = 92}. Therefore, the set QC x 22 is positively

invariant.

Since x(0) E M, then x(0) is in the interior of QC. We have that

||X(t) - x(0)ll S kzt (3-69)

as long as x(t) E QC. Therefore, there exists a finite time T*, independent of 51 and 82,
such that x(t) 6 DC for all t E [t0,t0 + T*]. We note from the above analysis that the
estimation error 77 enters the set 22 during the finite time period [t0, t0 +T1(51)+T2(52)].

Since T1(el) —> 0 as 51 —+ 0 and T2(€2) —> 0 as 52 ——> 0, there exists 64 such that for all

71

0<€2§€4

t0 + T1061) + T2632) S T*

Hence, the trajectory (x,17) enters the set QC x 22 during the finite time period [t0, to +
T1031) + T2(82)] and remains there for all t 2 to + T1 (51) + T2(52). Prior to entering
this set, x(t) is bounded by (3.69). Taking 5* = min{53,e4} and choosing 0 < 82 S 5*
guarantees that the closed-loop trajectories are bounded. <1

Several remarks are in order.

Remark 3.1 Modification of Theorem 3.1 to include the tracking problem at least re-
quires the use of Lyapunov stability with respect to a set. For the tracking problem, the
ultimate bound on the estimation error will depend on time-varying reference signals in
addition to the noise and modeling error. If the reference signals are large they could have
the effect of moving the output error outside the switching interval. This would cause
repeated gain switching events every Td seconds. One way to deal with this is to take
the maximum values of the reference signals into consideration when choosing 51 and 82
and Id. These values can be chosen large enough that the reference signals do not cause

repeated switching events. This approach is illustrated in the example of Section 3.4.

Remark 3.2 The parameter K. is chosen sufficiently large so that the output error (y—a‘cl)
does not overshoot the switching interval during the transient period. Due to the switching
scheme, this condition on the placement of the observer poles can be relaxed. Without this
condition, as mentioned before, the output error may overshoot the switching interval, but
the switching delay Td along with the resetting of the switching delay when the output
error leaves the switching interval would prevent repeated switching of the gain during
this transient period. However, in this situation it may take longer for the switching
event to occur than the case where K. is chosen to be large. The flexibility in choosing the

location of the observer’s poles allows for either design.

Remark 3.3 we note that at anytime an impulsive-like disturbance that has the effect

72

of resetting the initial conditions can cause the estimation error to leave the switching
interval. When this occurs, from item 1 of the switching scheme above, the gain switches
to the larger value in order to recover the estimates. This will be illustrated with a

numerical example.

3.4 Gain Switching Example

We consider a field controlled DC motor [36] and design a controller based on feedback
linearization so that the shaft angular velocity tracks the reference trajectory shown in

Figure 3.3. The motor equations are given by

41 = 2:2 (3.70)
4:2 : 66,17) (3.71)
43 = yaw) (3.72)
y = $1+v (3-73)
01 = (1:3 (3.74)

where .271 is the rotor position, 232 is the rotor angular velocity, 13 is the armature current,
and control u is the field current. The functions q5(a:, u) = —0.1:z:2 +0.1:r3u and 112(22, u) =

—2:1:3 - 0.22:2u + 200. The estimates, (it, are saturated outside {—100, 100]. For the

observer, we have (light, u) = —0.11:z:2 + 0.13:3u, and we use the following gains
71 70
H? = [— 7] (3.75)
51 51
2 1
H; = — —2 (3.76)
52 52

 

where 51 = 0.0005 and 52 = 0.01. The gain H1 was chosen, using simulation, to ensure

that the estimation error does not over shoot the switching zone. For the switching

73

threshold we use d = 0.05 and a delay time T d = 0.153. The initial conditions for the
system and observer are 5:1(0) = 77, 331(0) = 132(0) = 5:2(0) = 0. The measurement noise
is generated by Simulink’s “Uniform Random Number” block with magnitude limited
within {-0.0016, 0.0016] and sampling time set at 0.0008 seconds. This error magnitude
is consistent with a 1000 c/ r encoder. Figure 3.3 shows the velocity reference 7" (dotted)
and the trajectory :52 (solid) for the closed-loop system under the switched-gain observer.
The two the plots are indistinguishable. The bottom figure plots the value of 5,- versus
time illustrating the switching behavior. Figure 3.4 plots the tracking error, el = x1 — r,
for the closed-loop system under the switched-gain observer (5 = 5,, top), a fixed gain
observer with 5 = 52 = 0.01 (middle), and a fixed gain observer with 5 = 51 = 0.0005
(bottom). Likewise, Figure 3.5 plots the tracking error, 52 = 1:2 — r, for 5 = 5.,- (top),
5 = 52 (middle), and 5 = 51 (bottom). These two plots show that the switched-gain
observer has faster state reconstruction then the fixed—gain case with 5 = 0.01. Figure
3.6 compares the steady-state error 51 of the fixed-gain observer with 5 = 0.0005 against
the switched-gain observer and the observer under fixed-gain with 5 = 0.01, respectively.
This figure shows that the larger gain yields a better performance in the presence of a
modeling error qb—oo. Figure 3.7 zooms in on the steady-state behavior of 52 showing that
more of the measurement noise is attenuated when the observer switches to the smaller
gain. Table 3.1 compares the root mean square (R.M.S) values of the tracking errors for
each of the three observer designs. The rows 51 and 52 list the R.M.S errors calculated
over the entire simulation (0 to 10 seconds). To capture the steady-state behavior, we
exclude the initial transient and calculate the R.M.S tracking errors from 3 seconds to
10 seconds. These values are listed in the rows 51(SS) and 52(SS). The switched-gain
scheme shows better steady-state behavior in 52. Figure 3.8 shows the behavior of the
control signal u under each of the three cases. This plot shows the switched—gain observer
recovers from peaking more quickly than the case with 5 = 0.01, but with greater noise

immunity at steady-state than the case where 5 = 0.0005. Figure 3.9 shows the behavior

74

 

 

 

Switched-gain 5 = 0.01 5 = 0.0005
51 0.0064 0.0741 0.0062
52 0.0761 0.6295 0.0626
51(SS) 0.0013 0.0013 9.37e-4
52(SS) 0.0026 0.0026 0.0108

 

 

 

 

 

Table 3.1. Comparison of R.M.S tracking errors el = x1 — r, 52 = x2 — 1'", e1 steady-
state, and 52 steady-state for the switched-gain scheme and two HGOs for |v| S 0.0016
and (b0 ¢ 45-

of the error 52 = 5172 — r for the switched-gain observer (top) and a fixed-gain observer
with 5 = 52 (bottom) when an impulsive-like disturbance moves the estimation error
(1:1 — :31) outside Id. A measurement disturbance of magnitude 150 and duration 0.001
seconds is experienced at time 2.5 seconds. The observer switches to the larger gain and
is able to recover more quickly than the case when 5 = 0.01. To see the importance of the
delay Td, the behavior of the switched parameter 5 is depicted in Figure 3.10 when the
delay time Td is set to zero. It can be seen that 5 repeatedly switches between 0.01 and
0.0005. Figure 3.11 considers the stead-state tracking error 51 simulated with larger noise
v 6 [-0.016, 0.016]. In this case the noise and modeling error tradeoff favors the smaller
gain at steady-state. Table 3.2 compares the R.M.S values of the tracking errors el and 52
for each of the three cases. This table shows the advantage of the switched-gain observer
in relaxing the noise and estimation tradeoff. We also simulated for the case when d) = 950
with the noise level bounded once again in v E {—0.0016,0.0016]. The error 51 is shown
in Figure 3.12 for this case. The steady-state tracking error e1 is slightly improved at
steady-state for the switched-gain and the fixed-gain observer with 5 = 0.01. Table 3.3
compares the R.M.S values of the tracking errors 51 and 52 for each of the three cases.

Again, the advantage of the switched-gain scheme is evident.

 

 

 

Switched-gain 5 = 0.01 5 = 0.0005
e1 0.0082 0.0741 0.0214
e2 0.1070 0.6154 0.2046
e1(SS) 0.0015 0.0015 0.0208
e2(SS) 0.0248 0.0248 0.2107

 

 

 

 

 

Table 3.2. Comparison of R.M.S tracking errors 51 = 3:1 — r, 82 = $2 — r, 51 steady-
state, and 52 steady-state for the switched-gain scheme and two HGOS for |v| S 0.016
and 0,60 75 (,b.

 

 

 

Switched-gain 5 = 0.01 5 = 0.0005
81 0.0063 0.0741 0.0037
e2 0.0760 0.6294 0.0391
51(SS) 9.49e—5 9.49e-5 9.75e—5
52(SS) 0.0025 0.0025 0.0137

 

 

 

 

 

Table 3.3. Comparison of R.M.S tracking errors 51 = 2:1 — r, e2 = 2:2 — r, 51 steady-
state, and e2 steady-state for the switched-gain scheme and two HGOs for |v| S 0.0016
and 450 = qb.

3.5 Nonlinear Differentiation

Here we consider the high-gain observer used as a differentiator and employ a continuous
nonlinear function to smoothly transition between two gain values. This nonlinear func-
tion is similar to the one used in [67] to deal with measurement noise. The differentiator

takes the form of a linear high-gain observer
5:45+H(y—c:i)

where the gain matrix H is the same as Equation (3.6). The observer estimates the
derivatives of the measured signal y(t) = w(t)+v(t), where w(t) is the signal of interest that

is corrupted by measurement noise v(t). Let G) = [02 0'2 - -- Lair—1)]? ii : x1: _w('i-1)

3

76

1

and consider the rescaling 7),; = 52'— ii. This yields an estimation error equation similar

to (3.13)

57') = A077 + 5rBw(r) + 82v

Consider a second order differentiator (r = 2) and the following transfer functions

 

 

 

 

2
2 5 .. 5a 3 + a
$1 = 2 2 w — 2 2 1 2 v (3.77)
5 s +5als+02 5 s +5als+a2
~ 528 + 801 , (128
:r u} v (3.78)

2 = 5232 + 50113 + 02 — 5232 + 50113 + 072
The tradeoff in choice of 5 can clearly be seen. Accurate estimation is obtained through
smaller values of 5, while rejection of measurement noise requires larger values of 5. Again,
we will employ an observer scheme that transitions between two gain values in order to
relax this tradeoff. For the discontinuous switching scheme of Section 3.3 we had to
introduce a delay time in order to avoid multiple switchings as shown in Figure 3.10.
This simplified the analysis by allowing only one switching event which can be viewed as
a resetting of initial conditions. Rather than a discontinuous switch between two gains, we
use a dead zone nonlinearity to smoothly transition between the gains. This is illustrated
in the block diagram of Figure 3.1. The dead zone function output is zero for y — 531
within the dead zone [—d, d], thus the gain is g2. For y — 53:1 outside the dead zone, the
gain is g1 plus an offset value. This results in a continuous gain function as shown in
Figure 3.2. Mathematically, the nonlinear gain function is given by
9&3, for ly-i1|<d

fly—$1): 2' z. .- . . . (3-79)
91- d(91—9§)819n(y -:L‘1). for ly- $1l2 d

for each power i = 1 - - - r — 1. With

S’IH
[3‘ r—i

77

 

 

y-i1

 

 

 

 

 

 

 

 

 

Figure 3.1. Diagram illustrating Dead Zone switching scheme.

(0
N
031*
‘<
|
X)
—L

 

Figure 3.2. Plot of the nonlinear gain function.

78

the nonlinear high-gain observer is given by

5=Aa+Ga—2flh—2fl (3%)
where _ _
a190/ - i1)
2 A
A 029 (y - $1)
G@—$fl=

 

 

_ (Ir—19T—1fy — i1) _
Referring back to Equation (3.77), the idea behind the nonlinear observer can be described
as follows. Let |v| S kvu. Notice that when a is zero, |§31| S K1kvu for some positive
constant K1. Thus, by choosing the dead zone width to be (1 _>_ K1 kv/J the nonlinear
observer will use the smaller gain value during steady-state operation, when (I) is zero or
very small, thereby reducing the effect of noise. Now, suppose we are operating within

the dead zone and 5 = 52. Then we have that

[ill S K2520} + Klkvu
for some positive constant K2. If w is quickly changing, (I) is large and for a relatively
large choice of 52, the product 5222) may be large enough to move the estimation error :51
outside the dead zone. Once outside the dead zone the nonlinear observer uses the larger
gain to improve the estimation. During the fast transients, the output error may enter
and leave the dead zone, but the continuous nonlinear function provides smooth switching
between gains. The analysis behind the dead zone switching is complicated by the bias
term. Using Lyapunov analysis, it is easy to show that the estimation error is bounded.
However, starting outside the switching zone, it is difficult to show that the estimation
error will reach the switching zone. This remains an open problem and we illustrate this

scheme through a numerical example.

79

3.5.1 Deadzone Switching Example

We consider a signal w(t) whose profile is shown in Figure 3.13 along with the first and
second derivatives. We performed the simulation for three cases of a measurement noise
that is a uniformly distributed function of time between [~3.14e—4, 3.14e—4]. Here we
used Simulink’s “Quantizer” block to generate the noise. The first case considers the
nonlinear high-gain differentiator, Equation (3.80), for 91 = 1570 and 92 = 188. Then,
for comparison, we simulated two linear high-gain observers; one with g = 188 and the
other with g = 1570. Figure 3.14 shows the estimate :23 plotted against w(2) for the
three cases. The figure shows that the dead zone scheme (top) was more effective in
tracking the signal transients then the g = 188 linear observer and better at rejecting
noise at steady-state than the g = 1570 linear observer. The relaxation of the tradeoff is
emphasized in Figures 3.15 and 3.16. These Figures show the tracking error 02(2) — 5:3
for the three cases with Figure 3.16 zoomed in to capture the steady-state behavior.
Notice the difference in the y axis scales. The improved performance with the dead zone
scheme is clear. Also, we remark that the dead zone low noise estimator of [67] used
first order filters and differentiators. Compared to [67], the high-gain observer scheme
exhibits improved performance owing to the low pass filtering characteristic of the third
order high-gain observer used in this example. This can be seen by comparing the dead

zone HGO in Figures 3.15 and 3.16 with the low noise estimator in Figure 3.17.

3.6 Conclusions

We have derived relationships that exhibit the tradeoff between fast reconstruction of the
states and rejection of modeling error versus the immunity to measurement noise for a
high-gain observer. Based on this we have designed a switched—gain version of the high-
gain observer in an attempt to relax the tradeoff between fast state reconstruction and

rejection of measurement noise. To handle the peaking in the estimates we have included a

80

20

 

 

~--—~rdot

 

 

 

 

Velocuty
o a
x

N

 

 

 

 

 

0.01

 

0.005 ~

 

 

 

 

O 2 4 6 8 10
Time

Figure 3.3. Top: The velocity reference trajectory (r)(dotted) and .732 under the switched
observer (solid). Bottom: Switching behavior of the gain.

81

0.4 r
q;— 02 ’

—O.2

0.4

a;- 0.2 '

-0.2

0.4 ’
(D'- 0.2 7

—0.2

Switching

 

T

 

 

 

 

4 6
Fixed with e = 0.01

10

 

 

 

 

 

4 6
Fixed with e = 0.0005

10

 

 

 

 

 

4 6
Time

10

Figure 3.4. Position tracking error (51 = 2:1 — r) for the switched-gain observer (top), the
observer with 52 = 0.01 (middle), and the observer with 51 = 5 x 10"4 (bottom).

82

Switching

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10
N i-
a)
_5 m 4 1 1
O 2 4 6 8 10
Fixed with e = 0.01
10 . .
5
N
a:
0 L
_5 1 L n 1
0 2 4 6 8 10
Fixed with e = 0.0005
10 .
N
a)
_5 1 1 . 1
0 2 4 6 8 10

Time

Figure 3.5. Tracking error (52 = .732 —r) for the switched-gain observer (top), the observer
with 52 = 0.01 (middle), and the observer with 51 = 5 x 10”4 (bottom).

83

 

x 10-3 Switching vs. Fixed with e = 0.0005

fl

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

5 7 |-\ —— Switched-gain ‘
<1;- 0 . i
_5 1 1 4 -
O 2 4 6 8 10
x 10-3 Fixed with 5 = 0.01 vs. Fixed withe = 0.0005
5 .
(D'- 0 _
_5 .
0 2 4 6 8 10

Figure 3.6. Steady-State tracking error (e1 2 1‘1 — r) for the switched-gain observer
plotted against the observer with 51 = 5 x 10‘4 (top) and the observer with 52 = 0.01
plotted against the observer with 51 = 5 x 10'4 (bottom).

switching delay in our scheme. VVe’ve also considered a nonlinear high-gain differentiator
with smooth switching based on dead zone nonlinearity. Numerical examples illustrate

the effectiveness of each design.

84

Switching

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.05 . r
a,“ 0~ W
—0.05 * ‘ ‘ ‘
0 2 4 6 8 10
Fixed with e = 0.01
0.05 . 4 .
a)" 0 - W
—0.05 ‘ ‘ ‘ ‘
0 2 4 6 8 10
Fixed with e = 0.0005
0.05 6 . .
mm 0
—0.05 ‘ ‘ ‘ ‘
0 2 4 6 8 10
Time

Figure 3.7. Steady-State tracking error (52 = :52 —r) for the switched-gain observer (top),
the observer with 52 = 0.01 (middle), and the observer with 51 = 5 x 10_4 (bottom).

 

Switching

 

 

 

 

2 4 6
Fixed with e = 0.01

10

 

 

 

 

 

0 2 4 6
Fixed with 5 = 0.0005

10

 

 

 

 

Time

Figure 3.8. Behavior of the control for the three cases.

86

10

Fixed with s = 0.01

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10 I I I
—1 0 ‘ ‘ ‘ ‘
O 1 2 3 4 5
Switching
10 . I
am 0 k V
_10 I . . .
O 1 2 3 4 5
Switching
0.01 - ' I
‘” 0.005 I
0 1 1 1 1
O 1 2 3 4 5
Time

Figure 3.9. Tracking error (52 = x2 — r) for the observer with 52 (top) and the switched-
gain observer (middle) for an impulsive—like disturbance at 2.53. The bottom plot shows
the switching behavior of the gain.

87

 

0.01

O)
0.005

 

 

 

 

o 0.01
Time

0.02

Figure 3.10. Switching behavior of the gain when Td = 0.

Switching vs. Fixed with s = 0.0005

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0-2 ' — Switched-gain ‘
d" 0.1- '-’-~e=0.0005
OAL'C'.J\V.‘J?AV J. :7 \1. .\“I"\’.\VL‘;1\.'\/.JT’.\W_~.~
O 2 4 6 8 10
Fixed with e = 0.01 vs. Fixed with e = 0.0005
02 2:001 i
(D'- 0.1- . , -‘-'e=0.0005~
0 "\-\/_/.‘,,,.~'A\ /_$_\ /# L} I ‘/ \./ “ 2,—‘v‘r4;\.,\:lf‘\.\d
O 2 4 6 8 10

Time

Figure 3.11. Steady-State tracking error (el = 1:1 — r) for the switched-gain observer
plotted against the observer with 51 = 5 x 10—4 (top) and the observer with 52 = 0.01
plotted against the observer with 51 = 5 x 10_4 (bottom) with [v] S 0.016.

88

X 10-4 Switching vs. Fixed with e = 0.0005

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 ' ' I I
i — Switched—gain
(if-10" l '-'-~e=0.0005 ‘
O . . -
O 2 4 6 8 10
20 x 1074 Fixed with 5 = 0.01 vs. Fixed with e = 0.0005
1. 8 = 0.01
6’10“ 1 ----- e=0.0005‘
0 i- ‘ _ L, . l . 1 . 1
0 2 4 6 8 10
Time

Figure 3.12. Steady-State tracking error (el = $1 — r) for the switched-gain observer
plotted against the observer with 51 = 5 x 10—4 (top) and the observer with 52 = 0.01
plotted against the observer with 51 = 5 x 10_4 (bottom) with [v] S 0.0016 and do = (b.

89

Signal Profile

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20
O '1
8
-20 p 4
—4o . . . 4
O 0.2 0.4 0.6 0.8 1
200 .
1:8 0
-—ZOO L ‘ ‘
0 0.2 0.4 0 6 O 8 1
2000 f T I T
88 O
-2000 ‘ ‘ ‘ L
O 0.2 0.4 0.6 0.8 1

Time

Figure 3.13. The signal w(t) (top) and its first (middle) and second (bottom) derivatives.

Dead Zone Scheme

 

 

 

 

 

 

 

 

 

 

._._..w(2) 4

__ xhat3 _

 

 

 

 

 

 

 

 

""‘C0(2)

— xhat3 _

 

 

 

     

 

 

0 0.2 0.4 0.6 0.8 1 1.2
Time

Figure 3.14. :23 versus w(2) for Dead Zone switching (top) Linear HGO with g = 188
(middle) and g = 1570 (bottom).

91

Dead Zone Scheme

Hm I

(U

5, 0 h. h
61.:

T T i

 

 

 

 

 

 

0.2 0.4 0.6 0.8
g = 188

 

 

 

 

 

 

 

 

 

O 0.2 0.4 0.6 0.8
g = 1570
500 I f T
00
13'
3%
9L
8
-500 ‘ ‘ ‘ '
O 0.2 0.4 0.6 0.8
Time

Figure 3.15. Tracking error (12(2) — 5:3 for Dead Zone switching (top) Linear HGO with
g = 188 (middle) and g = 1570 (bottom).

92

Dead Zone Scheme

 

 

 

 

 

 

 

 

 

 

 

 

20 I f I
C’)
16
i
Al O '-
s
—20 . . . .
0 0.2 0.4 0.6 0.8
g = 188
20 I . . .
CO
is W
5
A. 0- W M
a
s
—20 1 . . 1
0 0.2 0 4 0.6 0.8

g=1570

 

 

 

 

 

0.2 l 0.4 0.6 e 0.8
Time

Figure 3.16. Zoomed in tracking error 62(2) — 5:3 for Dead Zone switching (top) Linear
HGO with g = 188 (middle) and g = 1570 (bottom).

93

Low Noise Estimator

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.2 0.4 0.6 0.8
Time

Figure 3.17. "Hacking error w(2) — :63 (top) and the zoomed in tracking error (12(2) — .733
(bottom) for the low noise estimator of [67].

94

CHAPTER 4

Multirate Sampled Data Output

Feedback Using High Gain Observers

4.1 Introduction

The study of sampled-data systems has received significant attention due to the fact that
modern control systems are almost always implemented digitally [4]. As discussed in [52],
there are primarily three approaches for the design of digital controllers. First, is the
design based on a continuous-time plant model followed by controller discretization and
implementation through sample and hold methods. This approach is the most widely seen
in the literature for nonlinear systems. See [5], [15], [18], [20], [30], [37], [41], [48], [53],
[55], and [74]. Second, is design based on exact or approximate discrete-time plant models
ignoring the intersampling behavior. This method is generally simpler for linear systems
since an exact discrete-time model can be obtained, whereas for nonlinear systems this
is typically not the case and approximate models must be used. See [44], [50], [51], [52],
[54], and [60]. The third and least talked about approach in the literature is to design the
controller based on a sampled-data model of the plant that takes into account the inter-
sampling behavior. We briefly discuss the state of the art of sampled-data state feedback

stabilization. In [18] it is shown that the sampled-data implementation of a controller

95

designed in continuous-time to globally stabilize the origin achieves semiglobal practical
stabilization where the state can be brought arbitrarily close to the origin by choosing
the sampling period T sufficiently small. In [53] and [74], given a continuous-time state
feedback controller that exponentially stabilizes the origin, the sampled-data implemen-
tation can bring the state a: to the origin as t -—> 00. This is shown both locally and
globally, with the global results requiring the restrictive assumption of global exponential
stability. Also, in [5] it is shown that convergence of the state to the origin can be achieved
under sampled-data state feedback, but again for nonlocal results restrictive assumptions
such as global exponential stability and globally Lipschitz functions are needed. In [54],
multirate sampled-data stabilization in the presence of time delay was studied for the
case when the control rate is made faster than the measurement rate. It is shown that
the closed-loop multirate sampled-data system achieves asymptotic stabilization in the
semiglobal practical sense.

Results on sampled-data output feedback and digital observer design can be found in
[2], [20], [32], [37], [41], [49], and [59]. In [59], multirate design of a sliding-mode observer
is considered, where the observer processing rate is higher than the control update rate.
The observer processing rate is selected high enough to allow for accurate estimation.
The control is operated at a lower rate to save on the computational expense. In [2],
the authors study discrete-time observers designed for sampled-data nonlinear systems
for two cases. In the first case the observer is designed for an approximate discrete-time
plant and in the second one it is designed for a continuous-time plant then discretized.
They study under what conditions, and how closely, the two designs achieve convergence
to the exact discrete-time model. In [37], an observer is designed based on a discrete-
time model and it is shown that the performance of a stabilizing continuous-time state
feedback controller can be recovered by a sampled-data output feedback controller for
sufficiently small sampling period. Two schemes for overcoming peaking during the initial

transient are presented, one based on global boundedness of the control law and a second

96

ya)

 

 

u(t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

_ System
ZOH (TS) Dynamics
SD S
Controller L TS

 

. ‘DHGO
x[ka] y[ka] Tf

 

 

 

Figure 4.1. Block diagram illustrating the multirate output feedback control scheme.

based on maintaining the control at a fixed value during the observer transient period.
The class of systems under consideration was the same as that of [20], where an output
feedback controller is implemented by discretizing a controller designed under continuous-
time state feedback and using a discrete-time high-gain observer to estimate the system
states. It was shown that the output feedback controller stabilizes the origin of the
closed-loop system for sufficiently small sampling period T. In addition, it was shown
that the performance under sampled-data output feedback asymptotically approaches the
performance under continuous-time state feedback as T —> 0. For the discrete—time high-
gain observer, more accurate estimation of the system states is achieved by faster sampling
of the output. Based on this observation , we seek to study the stability of a system under
sampled-data output feedback using high-gain observers, where the control rate is fixed
by the sampled-data state feedback design, while the output sampling rate is faster. This
multirate sampled-data output feedback control scheme is illustrated in Figure 4.1 for the
class of systems under consideration. We apply the control signal at the sample period
T5 through a zero—order-hold where the control is held constant in between sampling

points. The output y is sampled at a faster rate where we use the period Tf < T3. The

97

measurement y is a driving term for the observer and it is the sampling rate of this output
that we adjust to achieve more accurate estimation. The measurement C is used in the
control and we sample it with the same period T 3. Consideration of multirate observers
is motivated by asymptotic properties of high-gain observers and applications that may
require computationally demanding controllers such has hysteresis inversion. The idea is
that faster sampling of the output leads to faster and more accurate state estimation. On
the other hand, slower control sampling allows time to process the control signal. This
may be useful when the sampling rate of the control is chosen based on the performance
of the closed-loop system under sampled-data state feedback. Throughout this chapter
and the next we will refer to the sampled-data output feedback scheme that uses the
same control and measurement sampling rates as the single-rate scheme. For single-rate
output feedback using discrete-time high-gain observers, the sampling rate may have to
be pushed higher than that of the sampled-data state feedback case in order to stabilize
the closed-loop system. For the multirate case, stability can be achieved by making
the output sampling sufficiently fast while maintaining the control sample rate that was
designed under state feedback. This scheme may lead to a more computationally efficient
controller. Different sampling rates for the control and the estimation has the additional
benefit of isolating the control signal from the observer’s initial transient. This can be
done by initializing the control signal (i.e. keeping it fixed) until the observer has settled
([37], [4]). We will see that, due to the flexibility of the multirate scheme, initialization
can be performed in as little as one sample period. This chapter is organized as follows.
In Section 4.2 we develop the closed-loop system under multirate sampled-data output
feedback. Section 4.3 provides the analytical results where we consider the stability of the
closed-loop system under multirate output feedback control given a sampled-data state
feedback controller that asymptotically stabilizes the origin. We show that the closed-loop
system under multirate sampled-data output feedback achieves practical stabilization of

the origin using the control sample rate of the state feedback design (T5) and a sufficiently

98

large output measurement rate. In order to bring the trajectories of the closed-loop system
to the origin under multirate output feedback, we need a stronger condition on the stability
of the state feedback controller. We will show that the multirate scheme exponentially
stabilizes the origin of the closed-loop system given an exponentially stabilizing state
feedback controller. As we have seen, it is often the case that sampled-data state feedback
control of nonlinear systems can only be shown to achieve practical stabilization of the
origin. In addition, disturbances may also prevent asymptotic stabilization of the origin.
In Section 4.4 we study the case where the sampled-data state feedback controller can
only bring the trajectories of the closed-loop system to a set containing the origin. We use
the notion of stability with respect to a compact set to show boundedness and ultimate
boundedness of the closed-loop system under multirate output feedback. Also, with a
view toward practical implementation of the multirate observer, we discuss the tracking
problem and integral control. Finally, in Section 4.5 a numerical example seeks to highlight
some of the stability properties of the multirate output feedback controller and its ability

to handle peaking in the estimates.

4.2 Multirate Output Feedback Control

We consider the following class of systems

2 = 46,211) (4.1)
4 = Az+Bo(x,z,u) (4.2)
y 2 Ca: (4.3)
g = O(:1:,z) (4.4)

where :1: E RT and z E Re are the states, u is the input, y and C are the measured outputs.
The functions 96, 7,9, and O are locally Lipschitz in their arguments over the domain of

interest and satisfy 09(0,0,0) 0, it'(0,0,0) = 0, O(0,0) : 0. The r x r matrix A, the

II

99

r x 1 matrix B, and the 1 x r matrix C are the same as (1.5)-(1.6). The multirate output

feedback controller is implemented using the high-gain observer
5=A5+Bamagm+JHy—C@ (4a

where the function 1,00 is locally Lipschitz in its arguments over the domain of interest,

globally bounded in 51:, and satisfies (00(0, 0,0) = 0. The gain matrix is given by

T 01 02 Or
H =-—-—.~.— 46
E E2 57‘ ( )

where 5 is a small positive parameter and the roots of

sr+alsr_1+---+ar_1s+ar=0 (4.7)

have negative real parts. We consider the following partial state feedback controller

u=uaO M&

where 7 is locally Lipschitz in its arguments over the domain of interest and 7(0, 0) = 0.

Let X = [z 1:]T and rewrite (4.1)-(4.2) as

X=FWW) am

where

Wham
HLW=
A3; + B¢(:r, z, u)

Then, we can write the closed—loop system under continuous—time state feedback as

X=Fu) mm)

100

The solution of (4.9) over the sampling period [kT3, kTs + T5] is given by

t

X0) = X“) + (t - li“Ts)F(X(lI‘)I1416))+/kT lF(X(0)iU(k)) - F(X(k))IU(k))ld0 (4-11)

Since the function F is locally Lipschitz we can use the Gronwall-Bellman inequality to

Show that

|lx(t) — xaiu 3 Lil [ea-”9L1 — 1] IIF(X(k),u(k))|| (4.12)

for all t E [kT3, k'Ts + T3] where L1 is a Lipschitz constant of F with respect to X over

the domain of interest. We have
X08 +1) = X08) + TsF(X(k)I “(’0) + T32¢(X(k). 14k), Ts) (4-13)

where (I) is locally Lipschitz in (X, u). This model and equation (4.12) describe the discrete
time plant dynamics and the intersampling behavior, respectively. The discretized high-

gain observer is implemented by first scaling the observer states according to
q 2 Di: (4.14)
where D = diag[l, 5, - - - ,5r_1]. This yields
Ii: :leq + Hoy + Jaw—14.44)] (4.15)
where Ag 2 5D(A — HC’)D—1 = A - HOC and

a1

a2
H0 = EDH =

 

 

Or

101

This nonlinear observer is discretized using the forward difference method. For the dy-
namics under fast sampling we use the index n to indicate the sampling points that are

equally spaced with period Tf. We obtain

q<n+1> = qu<n>+ny<n>+eT—ITfBIIsoiD—lqaicai.um) (4.16)

55(n) = qu(n) (4.17)

We point out that u and C evolve in the slow sampling time k and are constant for all n

where an E [kT5, kTs + T5). The matrices Af, Bf, and Cf are given by

A —I+fiA B —B-H c —D—1
f‘_ 5 07 f_€ 07 f"—

As in [20] we take a = Tf/5 where a is a finite positive value and must be chosen such
that the matrix A f has all its eigenvalues inside the unit circle. This ratio indicates that
the fast sampling frequency increases as the gain increases. The observer estimates are

downsampled for use in the output feedback controller. Using the relation

T3 T3
h = _ = _
Tf 5a
we employ the following notation to denote the value of the vector of estimates under the

slow sampling period

4.3a) = .40th) (4.18)

where [w] indicates the greatest integer less than or equal to w. This gives the following

output feedback controller

u(ft) = «333(k), C(19)) (419)

In the introduction chapter we have seen that this controller will need to be globally

bounded outside a compact region of interest in order to overcome the peaking phe-

102

nomenon. As discussed in [37], peaking in the initial time instant can be overcome by
setting the control to some arbitrary values and then using the observer estimates after
they have recovered from peaking. We take a similar approach here by using the following

control
u(k) = u0(k), for 0 S k < k0

14k) = 7(ris(k)=C(k))i fO’r k 2 l90

(420)

Because of the flexibility of the multirate scheme, we can choose the output sampling rate
sufficiently fast so that the estimates recover from peaking during the period [0, T5]. In this
case we only have to set the'control to some initial value u(0) = “0 (0). Furthermore, the
states of the plant X(t) will not grow by more than 0(Ts) from its initial condition during
this period. As we will discuss later on, the control still needs to be globally bounded
to prevent any peaking that occurs after the initial time. Still, the control scheme (4.20)
prevents controller saturation during the first time instants. In some situations, choosing
the output sampling rate fast enough that the observer peaking passes during the interval
[0, T3] may require Tf to be prohibitively small. Further, the sampling period T f that
guarantees certain closed-loop performance need not be as small. In the forthcoming
analysis we will consider the choice of Tf that is sufficient to guarantee boundedness of
the closed-loop system. Then, we will remark on the choice of T f that avoids peaking.
In what follows, we will need to describe the plant dynamics in the fast sampling time

n. So, consider the solution to equation (4.9) over the fast sampling period [an, an-i-Tf]

t

x<1> = x<n> + <1 — nITf>F<x(II>. 11(4)) + f T iF<x<a>.u<k>> - F<x<n>tu<k>>ida
” f

where again the control u(k) is constant for all an E [kT3, kTS + T5). Also, consider
1 t—nT L
”1(1) — an)“ s L—l [e( f ’ 1 —- 1] ||F(X(n),u(k))ll (4.21)

which describes the intersampling behavior under the fast sampling time n. Then by the

103

same argument used to arrive at (4.13), we get

X(n + 1) = X(n) + TfF(X(n), u(k)) + Tf2<I>(X(n), u(k),Tf)

(4.22)

where (I) is locally Lipschitz in (X, a). For the analysis we will also need a model that

describes the observer error dynamics. This will require a discrete-time model of (4.2)

under the fast sampling rate that makes use of its special structure and the properties

of the matrices A, B, and C. In addition, deriving the observer-error equation requires

rescaling of the estimation error. The details of these derivations are similar to [20] and

are covered next. First, observe that
DB=5T_1B , 50:5, siDAi=AiD , AT=0
The solution to (4.2) can be written as

_,, t
4(1) = 64“ Tf)1~<n)+ / «249—2136419011z<n>.u(k>>
an

t
A(t—r)
+ /7le e BA(r)dr

where A(r) = (,6(:1:(r), z(r),u(k)) — c0(a:(n), z(n),u(lr)). Using (4.23) we have

At 7:1” ' T—l ti ' 1 At/
' — — z : — Z : Er... E
De B — D i—EO 21.4 B 30 5fi'A DB 5 B

and thus

fan+Tf €A<an+Tf—T)BA(T)(1T
an

= 57"_1D”1 /71Tf+Tf eA<an+Tf_T)/€BA(r)dr
7in I

104

(4.23)

(4.24)

(4.25)

(4.26)

We can show that the right-hand side term is 0(52). This is seen from the Lipschitz

property of (Z)
II4<4<1),z(1),u(k>) — 44401). 200.4(4))“ 3 Lynn) — x(n)ll , L, > 0
together with equation (4.21). We arrive at the equation

T
:1:(-n +1) 2 eATf:r(n) +/() f efqtdth/>(z'(n), z(n), u(k))

+57“+1 D_1R(X(n), u(k), 5) (4.27)
where R is locally Lipschitz in (X(n), u(k)) and uniformly bounded in 5 for 5 sufficiently

small. We now consider the following estimation error rescaling

1
Er—l

 

401) = [D(:v(n) — 4(4)) — LDx(n)l

The matrix L is used to eliminate terms in the estimation error dynamics that appear

with negative powers of 5 as will be seen below. It satisfies
AfL + (540 — Af — BfC) — LeAa = 0 (4.28)

Using the property Ar = 0, the solution to this equation can be found to be
r . .
L = Zn — Af)—"(eAa - A, — BfC)(I — eAa)z_1
i=1

Using (4.16),(4.17) and (4.27) we have

{(n + 1) = Afg(n) + fiM/«LD + DeATf — AfD — BfCD — LDeATf]at(n)

 

T ,
+ (1 - L)D/ f 42414434) — rfsao + 52(1 — L)R (4.29)
0

Er—l

Again from (4.23) we have that
r-—1 Ti r—1 ,-
AT f i 01’ 2' Au
De f=DZ—,A 227,—.AD25 D
i=0

i=0

with this identity, (4.25), and (4.28), equation (4.29) simplifies to

{(71 +1) 2 Af§(n) + 5(I — L)D/(;a eASdngb(:r(n), z(n), u(k.))

—5[aB¢O(rt(n), ((4‘), 1106)) — 5(1 - L)R(X(n)i U(k)a€)l
where .7": is given by
5(n) = [1 — D_1LD]:c(n) — 5r—1D—1§(n)

Clearly the term 57‘"1D—'1 is 0(1). Let X = 5A0 - Af — BfC. The term D—lLD,

which is given by
T . .
D—lLD = Z[D_1(I — Af)_lD]"‘1[D_1(I — Af)—1XD][D_1(I — ieAa)D]’—1
i=1

can be shown to be 0(5). This is verified by making use of (4.23) to show that

XD = [eAa _Af - BfC]D = D[6ATf — I—TfA]

. . AT . . . .
From the series expans10n of e f , 1t can be seen that X D 18 0(52). It IS straightforward

to show the following

—1 —1 _ 1 —1 —1

D (I—Af) D——ED A0 D

—1 —1 _1—1—1 _1—1—1AT

D (I—Af) XD—-—-(;D A0 XD——;D A0 D[e f—I—TfA]
D—1(I — eAa)D = (I -- eATf)

106

Also, D—1A0_ 1D as seen from

D'lA—lDz

 

0 0 0 —e"‘1/a,~
1/5 0 0 —5r—201/ar
0 1/5 0 —5T_3ag/ar
_ 0 0 1/5 —a,.__1/ar _

 

is order of 0(1 / 5). From the foregoing, D-ILD is 0(5). Finally, we obtain the following

relation for the estimation error dynamics.

€(n +1) 2

i‘(n)

4,401) + 4904(4). 4(4). <01). 11(4). a) (4.30)
[I —— 5N2(5)].r(n) + N1(5)§(-n) (4.31)

where A f has all its eigenvalues inside the unit circle, 9 is locally Lipschitz in its arguments

and uniformly bounded in 5, for 5 sufficiently small, 9 and 7 are globally bounded in 5:,

and the matrices N1, N2 are analytic functions of 5. Using (4.13) and the control (4.20)

we can write the closed-loop system under multirate output feedback as

X(k+1) =
{(71 +1) 2

538(k) :

X08) + Tsf(X(f€)i€s(k)iTsa5) (4-32)
Af€(n) + 6904001601), ((16), Wt), 6) (4-33)
[1 — 5NQ(5)]x(k) + N1(5)£3(k) (4.34)

where {3(16) 2 E([hk]). Also, the function f, locally Lipschitz with respect to X and {5,

is given by

f(Xiés,Ts.€) = F(X(k), 11(4)) + Ts‘Nka): u-(l€)ITs) (4-35)

with u(k) taken from (4.20) and (4.34).

107

4.3 Stabilization of the Origin

In this section we study the stability of the closed-loop system under multirate output
feedback control. Based on the existence of a single-rate sampled-data state feedback
controller that globally asymptotically stabilizes the origin of the closed-loop system we
show that the multirate output feedback controller achieves ultimate boundedness of all
closed-loop trajectories. We then study the stability of the closed-loop system when an
exponentially stabilizing state feedback controller is considered. The closed-loop system

under the single(slow)-rate sampled-data state feedback control

WC) = 703(k), C(16)) (436)

is given by

X(lc +1) = X01") + TsF(X(/€)I 945601), (00)) + T522606), 703(k), C(19)), Ts)

Observe from (4.34) that by setting 5 = 0 and 5 = 0 we have 533(k) = 510(k) and the control

(4.20) is identical to (4.36). Therefore, from (4.35) we obtain the following reduced system
X(k+ 1) = X(k) +Tsf(X(k),0.Ts,0) (4-37)

which is the closed-loop system under sampled-data state feedback. We make the following

assumption
Assumption 4.1
1. The origin (X = 0) of (4.37) is globally asymptotically stable.
2. The function '7 is locally Lipschitz in its arguments and globally bound in :13.

Global boundedness can be achieved by saturating the control outside a compact region

of interest.

108

4.3.1 Boundedness and Ultimate Boundedness

we have the following result which shows boundedness and ultimate boundedness of the

closed-loop system.

Theorem 4.1 Consider equations (4.32)-{4.34) and the control {4.20). Let Assumption
4.1 hold and let M and N be any compact subsets of R€+T and RT respectively. Then,

for trajectories (z, (r) x :5 starting in M x N the following holds

0 There exists 51‘ such that, for all 0 < 5 S 5’], X(t) is bounded for allt Z 0 and {(n)

is bounded for all n 2 0.

0 Given any a 2 0, there exists 5; > 0, t* > 0 and n* > 0, such that for every

0<5S5§, we have

II€(n)II+le(t)II :11 v 121*. and 4211* (438)

Proof: First, consider the closed—loop system under sampled-data state feedback (4.37).
From Assumption 4.1 and the discrete-time converse Lyapunov theorem of [35] there is a

smooth Lyapunov function V(X) that satisfies

01014’“) S V(X) S 02(IIXII) (4-39)

V(XUJ) + Tsf(X(k)10,Ts, 0)) - V(X(k)) S -03(l|X|l) (4-40)

where 01 and 02 are class [Coo functions and a3 is a continuous positive definite function.
We note that the Lyapunov function V(X) will, in general, depend on the sampling period
Ts, but since we are working with a fixed T3 we can proceed without difliculty. Let 91
denote the compact set {V(X) S c1}. We choose c1 > maxxeM V(X) so that M is in
the interior of 91. We take W(£) = {TPé as the Lyapunov function for the observer

error dynamics where P is positive definite and satisfies A$PA f — P = —I. Also, from

109

([58], Ch. 23, Th. 23.7) we have that HP” > 1 . Consider the following sets
02 = {V(X) S 62}. A = 92 X {ff/(f) S 6352}

where C3 > 0 and (22 is a compact subset of fig—H" for any c2 > 0. We take 52 > Cl so
that M C 511 C 522 and we show that for X(O) starting in M, X(k) remains in {22. Due

to the boundedness of 7 and g in :f: we have for all (X, E) E (21 x RT

[[f(X,€S,T5,E)[[ S K111 “9(ng1 (,14)” S K2

where K1 and K2 are positive constants independent of 5. From equations (4.20), (4.34),

(4.36), and the Lipschitz property of f with respect to :r

“RX-3013.0) — f(x.5s,Ts,e)|| g 5K3

 

 

X06)” + K4||€s(k)l|

for all (X, g) E A and for some positive constants K 3 and K 4 independent of 5. , From the

foregoing, it can be shown that in the set A
V(XUC +1)) = V(X(k) + Tsf(X,€.s-,Ts,€)) S V(X(k)) - 03(||X(k)||) + €TsK5 (441)

for some positive constant K 5. We can choose 5 small enough that X(k) E {V(X(k)) S c1}

cannot leave (22 and thus 02 is positively invariant. This can be seen from
V(X(k +1)) 3 61+ 5T3K5 < 52
de

for V(X(k)) S 51,5 < 51 :f (52 — c1)/T5K5. Likewise, for Cl S V(X(k)) S 52

V(X(k +1)) 3 c2 — K6 + 5T3K5 < 52

110

for 5 < 52 de=f KG/(K5T3) where

K = min a (llxll)
6 ClSV(X(k))SC2 3

For the observer error dynamics (4.33), we can arrive at

1

VV(§(n +1)) S (1— W

) w<e<n>> + swan)” + €sz (4.42)

for some positive constants K7 and K8. For {(n) E {Vi/(£00) S C352}, we obtain

. 1 2 2 / C3 2
W +1 < 1 — — + K —— + K 4.43
(£(n )) _ ( HP“) C38 5 7 )‘minUDl E 8 ( )

It can be seen that for C3 large enough, W’(€(n + 1)) S C352

. Therefore, A is positively
invariant.

With the initial conditions (x(0),i‘(0)) E M x N we have that “5(0)” S l/ET—l,
where 1 depends on M and N. Since M is in the interior of S11, x(0) is in 91. We have
that

”X(k) - X(0)H S TsKlk (4.44)

as long as x(k) E (21. Hence, x(k) remains in 91 for k S Kg/TS for some positive

constant K9. With x(k) E 01 and 5 ¢ {W(€) S C382}, we can rewrite (4.42) as

W(€(n))

2 ..
——+5 K (4.40)
)‘minUDl 8

1 1
W'(£(n+1)) S (1 — —)1V({(n))— —1'V(£(n)) +5K
2||P|I 2|IPH 7
Then for VV(§) 2 C352, we can choose C3 large enough that

W(€(n +1)) S /\W(€(n))

111

for 0 < /\ = (1 —1/(2||P||)) < 1. Hence,

 

 

 

[2
/. n
w (404) s A IlPII€2,._2 (446)
And 6 enters {ll/(é) S C352} for
12
n 2
A IIPll€2.,._2 s a C3
To Show boundedness of trajectories, let
”PHI?
1
77(5) — n (C35 T) (4 47)
_ ln(1/)\) '

and note that E E {W(€) S C352} for all n 2 71(5). From (4.44), x(k) E (22 for all
k S Kg/TS. We can select 53 such that for all 0 < 5 S 53, Tffi(5) < Tsk S K9. This
can be seen by using equation (4.47) to obtain

P 12
501177. (H ”2 ) < K917‘L(1/)\)
C35 7‘

 

where the left-hand side tends to zero as 5 —+ 0. Therefore, x(k) enters {22 during the time
interval [0, 1%“ and is bounded by (4.44) prior to entering this set. The trajectory £(n)
enters the set {W(€) S C352} during the time period [0, ffi(5)]] (where the notation [ft]
denotes the largest integer greater than or equal to 7'1) and remains there for all n 2 73(5).
Furthermore, it is bounded by (4.46) prior to entering this set. From (4.12) and (4.20) it
follows that all closed-loop trajectories are bounded by choosing 51‘ = min(51,52, 53).
Ultimate boundedness follows by an argument similar to [20]. Indeed since {(71) E
{1V (5 (71)) S C382}, then given any n > 0 we can find 54 dependent on )2 such that

||£(n)|| S %;r for all n 2 11(5) and all 0 < 5 S 54. Now, consider the compact set

112

{03(l|x(k)||) S 2€TsK5l and let

u) (5) = max V( )
1 a3(|le|)S2eTsK5

and note that w1(5) —> 0 as 5 —> 0. We have that the set {V(X) S w1(5)} is compact and
{a3(llxl|) S 25T3K5} C {V(X) S w1(5)}. Rewrite equation (4.41) as

V(x(k +1» 3 V(x(k~)) — $401me0 - (gagulmm — eTsKs)

Therefore, with V(X(0)) _>_ w1(5), we have that whenever V(x(k)) 2 w1(5)

k
V(X(k +1» s we» — Z %a3(l|x(’i)ll) s V(X(0)) — 5T5K5(k +1)
220

Thus there exists a finite time k* such that x(k*) enters the set {V(x) S w1(5)}. Now,
consider the time k 2 k*. For x(k) E {V(X) S w1(5)} and a3(llx(k)|[) > 25T3K5 we
obtain

V(X(k +1» 3 41(5) —- %O3(llx(k)ll) < was)

Therefore, x(k + 1) remains in the set x(k) E {V(X) S w1(5)}. On the other hand, for
x(k) E {V(X) S w1(5)} and a3(llx(k)||) S 25T3K5 we have

V(X(k +1» S w1(€) + €T3K5 dif (122(5)

Hence, x(k+1) may leave {V(X) S w1(5)} but remains in a set defined by {V(x) S w2(5)}
where w2(5) —+ 0 as 5 ~—* 0. Now, for x(k) in the set {w1(5) < V(X) S w2(5)}, we have

a3(||)((k)l|) > 25T3K5. Therefore

vmk +1» 3 mac» — $034.an < 42(5)

113

Thus,
V(X(k+1))Sw2(€) , were

In other words, for x(k) belonging to the set {V(X) S w1(5)} at time k = k*, then x(k)

must be in the set V(X) S w (5) for all k 2 k* + 1. From equation (4.39)
2

”an“ s ail-1042(5))

for all k 2 k* + 1. Using this, equation (4.12), and the control equation (4.20) together
with (4.34) then we can find an 55 dependent on M such that Hx(t)|| S %u for all 0 <
5 S 55 and all t 2 t* for a finite time t*. From the foregoing, we obtain (4.38) where

E; = min(54,55). <1

Remark 4.1 Using the control scheme (4.20) we can choose 5 sufficiently small such
that the estimation error 5 reduces to an 0(5) during the time interval [0,T3], thereby
isolating the controller from the observer transient. To do so we must select the fast

sampling period, Tf : 5a, such that
Tf [71(5)] < T3 (4.48)

From equation (4.47) there exists an 5* such that (4.48) is satisfied for all 5 S 5*.
Therefore, the trajectory g (n) recovers from peaking during the interval [0, [7305)“ and
remains bounded and 0(5) thereafter. Furthermore, During [0, T3], the state of the plant

x(t) cannot grow by more than

“X(t) - X(0)|| _<_ K10Ts||F(X(0),uo)||

for some positive constant K10.

114

Here we comment on the recovery of the performance of the multirate sampled—data
closed-loop system under output feedback to the performance under state feedback; both

continuous and sampled-data.

Remark 4.2 In [20] it was shown that the single-rate sampled-data output feedback con-
troller recovers the performance of the continuous-time state feedback controller as the
sampling frequency and the observer gain become sufficiently large. For the multirate

output feedback case, with a fixed T3

||Xr(t) - W)“ S 5(Ts)

for some continuous function 6(T3) and where Xr is the solution of the Closed-loop system
under continuous-time state feedback. When comparing the multirate case to the sampled-
data state feedback case, a similar result holds since the two controllers will be different

over the first sampling period and thus the trajectories will differ by an order 0(Ts).

4.3.2 Exponential Stability of the Origin

In the next theorem, given an exponentially stabilizing sampled-data state feedback con-
troller, we show that the trajectories of the closed-loop system under the multirate scheme
converge to the origin exponentially fast. We study the closed-loop system in the slow
sample time k and show that the discrete—time trajectories converge exponentially. To
do so, we will need a description of equation (4.30) in the slow sampling time k. For

simplicity, we will consider the ratio of the sampling rates

.5

h _
Tf

to be a positive integer. Also, we will work locally, so consider a ball B (0, 91) of radius
91 > 0 around the origin (x,£) 2 (0,0). The results of the previous theorem guarantee

that for 5 small enough, the trajectories will enter this set in finite time. Furthermore,

115

we will assume that the functions F (x, u) and 7(23, C) are continuously differentiable in
a neighborhood of the origin. We begin by studying (4.30) over one slow time period
[k, k + 1]. Consider the discrete-time state equation and the estimation error dynamics in

the fast sampling time n

x(n +1) = x(n) + TfF(x(n), u(k)) + T}<1>(X(n), u(k), Tf) (4.49)

5(71 +1) = Affl") + 890007501)» ((18), “(161,51 (4-50)

The accumulation of (4.50) over the interval hk S n S h(k + 1) is performed as follows,

where for convenience we represent (Us) and u(k) by the index k:

5w.» +1) = Af§(hk) + 59 (X(hk),§(hk), rm)
will + 2) = A2g(hk) + 5A fQ(X(hk), gait), k, 5)
+5Q(x(hk) + TfF(X(hk), k) + T%<I>(X(hk), k, Tf), A f§(hk) + 59, k, 5)

g(hk+h) = A’}g(hk) +5w(x(hk).§(hk),k,5)

Note that h is 0(1 / 5) and that the function w will contain a summation of h terms of
order 0(5) each. Thus, it is not clear that 5w will remain 0(5). To show this, we will
use a change of variables. First, we note that (4.49)-(4.50) has a two—time scale behavior.

For equation (4.50), we have the quasi-steady-state relation

{(71) = Aféin) + EGO/(70,501),C(k),U(k),~€)

We seek a solution to the above equation in the form

€01) = MM"), ((16), U(k),€) (4-51)

116

where «p is a continuously differentiable function of its arguments and 99(0, 0, 0,0) = 0.
This equation describes an (r + l)-dimensional manifold in the (2r + €)-dimensional state
space of (x,§) and is called the slow manifold of (4.49) and (4.50). Evaluating (4.51) at
n + 1 and substituting the difference equations (4.49) and (4.50) results in the following

manifold condition

0 = Af<p(x(n),C(k),U(k)a€)+59(X(N),<P(X(n),C(k),u(k),€)),C,u,€)

—Lp (Mn) + TfF(X(n), u(k)) + T%<I>(X(n), u(k),Tf), C(k), u(k),5) (4.52)
Setting 5 = 0 we have
(I — Af)99(X(n)v ca), ua). 0) = 0 (453)

Because |)\(Af)| < 1, (I —- Af) is nonsingular. Therefore, cp(x,(,u,0) = 0. Using the
implicit function theorem ([72]) we have that cp(x, (, u,5) satisfies (4.52) for 5 sufficiently
small. Furthermore, due to (4.53) this function is 0(5). Consider now the change of

variables

77(71) = €(n) ~ <P(X(n)a C(19). u(k): 5) (4-54)

which along with (4.52) results in

77(‘n + 1) = Afn(n-) + 619(x(n).n(n) + 990d”), C(k), “(19),8), C(k‘), u(16),»3)

We rewrite the right hand side as
77(n +1) = £1an) + 65 (X(n),n(n), C(k'), M1078) (4-55)

where

~

9 (x(n), 0, C(k‘), u(k), 5) = 0 (4.56)

117

Therefore, 77 = 0 is an equilibrium point of the transformed system. The function G is
continuously differentiable with convex domain B(O, 91). From ([36], ex. 3.23, pg. 108)

we have the following relation

~ ,. _ 10¢
9(X(N),77(n),<(k)aMALE) — 0 530.077.911.647 a
dif B<x<n>.n<n>.<<k>,u<k>,e>n (457)

Due to the fact that X and g are bounded and belong to the set B(O, 91), we can treat

the equation

7](n + 1) = [A f + 5B(X(n),7y(n), C(74), u(k),5)] 77(7)) (4.58)

as a time-varying linear system. Let
4(7)) = Af + 530/01), WI), ((16), Wt), 5) (4.59)

Since

|A(Af)] 3 A1 < 1 (4.60)

for some positive constant A1, It follows from ([58], Th. 24.7) that, for 5 sufficiently small,

the state transition matrix, (PAUL, no), of (4.58) satisfies
71—410

“(1)/ammo)” S “1A2 (4-61)

where 0 < A2 < 1 and n1 is a positive constant. In addition, it can be shown that there

exist 51 such that the following properties are satisfied uniformly in 5 for all 0 < 5 S 51

|/\(/l)| S C1 , ”All S C2 , IWN) - Afll S 035 (4-62)

118

where c1, c2, and C3 are positive constants. Now let

n—n
T(n,n0) = ¢A(n,n0) — Af 0

(4.63)

It can be shown that (Appendix A) there exist positive constants 52, 7:2, and 0 < A3 < 1

such that for all 0 < 5 S 52
n—nO

Human s 54243

From (4.58) we have
n—nO

n<n> ——- [4,. + Tuna] n<no>
The accumulation over the period [hk S n S h(k + 1)] is given by

77(hk + h) —_- [4? + rum + h, 7a)] 7)(hk)

Since h is 0(1/5) and l’\(Af)l < 1 we have that

Using this and the fact that T is 0(5), we rewrite (4.65) as

7)(hk + h) = 5G(-r)(hk), x(hk), ((16), u(k), 5)

(4.64)

(4.65)

where the function G is continuously differentiable. We now have an equation that de-

scribes the evolution of the estimation error in the slow sample time 16. Consider the

control (4.20) and equation (4.31)

Uik) = )‘(i(hk)aC(/’€))
C(hk‘) = [I—ElV2]fL‘(hk)+N1€(hk)

119

(4.66)

(4.67)

and substitute these expressions into (4.54)

TIWC) = €(hk) - 99(X(hk), ((10,701 - €N2l$(hk) + N1€(hk), C(16)),5)

Dropping the h notation, we redefine the right-hand side of the above equation as

where (5 is continuously differentiable. By noting that ch = 0 for 5 = 0, the implicit

function theorem shows us that there is an open set V containing (77,5, 5) = 0 such that

{(14) = 1907(k) X0018) (4.68)

where the function 29 is continuously differentiable in V. From the ultimate boundedness
of Theorem 4.1, we can choose 5 small enough to guarantee that we are in V. Now, using
(4.66), (4.67), and (4.68), we write the closed-loop system in the slow sample time in

terms of X and 77 as

X(k +1) = X06) + TsflXUC): 19(77(19): X(k),€),Ts, 5) (4-59)
7107 + 1) = €G(n(k), X(k), ((1018) (4-70)
We have the following result.

Theorem 4.2 Assume that the origin (X = 0) of (4.37) is exponentially stable and the
functions F (x, u) and 7(93, C) are continuously differentiable in a neighborhood of the ori-
gin. Then there ezrists 5; > 0 such that for all 0 < 5 S 53): the origin of (4.69)—(4.’70} is

exponentially stable. That is
‘k <C)\[C 4; <CA‘C Vk>k' 0
[[7l( )1] _ 1 4 7 ”M )[I _ 2 5 , _ >

120

for some positive integer k*, positive constants C1 and c2, [A4] < 1 and [A5] < 1. Fur-

thermore, x(t) decays to zero exponentially fast
Hut)” 3 036—)‘Gt , w 2 o

for some positive constants C3 and A6-

Proof: Again, the results of Theorem 4.1 allow us to work locally. Setting 77 = 0 and

5 = 0 in (4.69) results in the reduced system
X(k +1) =~ X06) + Tsf(x(k), 0, Ts, 0)

which is the closed-loop system under sampled-data state feedback. Linearizing about

X = 0 we obtain

X(k +1) = A11X(k‘)

where ]A(A11)| < 1 by hypothesis. Next, we linearize the full system (4.69)-(4.70) about

(x,77) = (0,0) to obtain

X(k +1) (A11+€A1z)x(k)+A13n(k)+ng1(x,n,Ts,€) (471)

7/(k + 1) 54214416) + 84227206) + 592(01X15) (472)

where g1 and g2 are continuous functions. We have that for any '71 > 0 and 72 > 0 there

exit 92 > 0 such that

ll91ll2 < 71ll(x,n)ll2 , ||92||2 < 72110070112

for all ”(X1 r})|]2 < 92. Since 4,5 is continuous, we can choose 91 so that (x,§) E B(0, 91)
implies [[(X,77)||2 < 92 through the change of variables 7) = 6 — (,5. We need to weaken

the variable 7) in the linear part of equation (4.71). This is accomplished by choosing a

121

matrix .M that satisfies
(A11 + 5A12)M + A13 — 5MAQIM — 5MAQZ 2
Now, consider the composite Lyapunov function
V(XUC), n(k)) = [x — A/‘IanPan - Mal + GnTn

where the matrix P11 is positive definite and satisfies Afi P11A11 — P11 2 —I. It can be
shown that there is 6 sufficiently large and 53 sufficiently small such that the following

relation holds uniformly in 5 for all 0 < 5 S 53.

51l|(x,n)ll2 s v s 62l|(x,n)ll2

where 61, (52 are positive constants. We have

Av s —yTHy
where
k 7r 7T
32: ||x( )II ,n: 1 2
“7706)” 7r2 7r3

7r1 = 1 7* 551(9) — 3271 - 337% - 82/347172 - 5,3572 - 5(186 + 857(9))73

772 = —/\mag;(M) — 588(6)

7T3 = (9 + 47,167,010) - E59(9) - 73271 — 337% - 6213471 7'2 - 85572 - 5(86 + 667(0))73
for some positive constants )8,- that, in general, depend on T3. Also, as shown, 6,; may
depend on 6. Therefore, there exist ’71, dependent on T5 but independent of 5, 54 depen—

dent on 9, and 0 sufficiently large such that for all 0 < 5 S 54 the matrix H is positive

122

definite. Therefore, from the ultimate boundedness of X and 5
,, ,*
132(k)“ s 0114—!» l v k .2 4*

for some positive integer k* and positive constants Cl and A < 1. Using (4.12), (4.20)
and the fact that X(t) is bounded for all t Z 0, it can be shown that X(t) is exponentially

convergent for all 0 < 5 S 5; = min{51,52,53,54}. <1

4.4 Stabilization with Respect to a Compact Set

In the previous section we dealt with the case where a sampled-data state feedback con-
troller was designed to make the origin of the closed-loop system an asymptotically sta-
ble equilibrium point. Here we consider the case when the sampled-data state feedback
controller, rather than stabilize the origin, brings the trajectory X(k) to some forward
invariant compact set. This situation my arise in the presence of a bounded disturbance
where the sampled-data state feedback controller can only achieve stabilization of the
trajectories to a set. Also, as mentioned in the introduction, in many cases sampled-data
state feedback controllers can only be shown to practically stabilize the origin. The results
provided in this section are motivated by applications to the control of smart material
systems, where one may consider hysteresis inversion compensation. Such inversion com-
pensation may be inexact and lead to bounded disturbances in the closed-loop system.
We will also discuss extending the results of this section to the tracking problem and
integral control, both of which can lead to controllers that are unable to asymptotically

stabilize the origin of the Closed-loop system.

123

4.4.1 Definitions and Problem Formulation

Consider the following general nonlinear discrete-time system
$W+¢)=f®W%flH) M73

where k E Z+, x(k) E R”, d(k) E D, with D a compact subset of Rd, and where
f : R” x D —) 1R" is continuous. The function f is locally Lipschitz in x, uniformly in d

if for each compact set Q of IR" there is a constant L such that

||f(1rad) - f(y,d)ll S Llll‘ - 31”

for all x,y E Q and d E D.

Let MD be the set of all functions from Z+ to D and let x(k, k0,x0,d) denote the
solution of (4.73) with initial state x(k0) = x0 and d E MD' The set A is forward
invariant if for each x0 E A, x(k, k0. x0, (1) E A for all k 2 k0. Also, let A be a nonempty

compact subset of R". The distance of a trajectory 19 E 1R" to the set A is given by
WI = inf “19 -77||
A 7)EA

We work with the notion of stability with respect to a set given in [35].

Definition 4.1 The system (4.73) is Uniformly Globally Asymptotically Stable (UGAS)

with respect to a compact invariant set A if the following two properties hold:

1. Uniform Stability: There exist a [C00 function (5 such that for any 5 > 0
|x(k,k0,x0,d)]A S 5 We 2 k0, ‘v’ko E Z+, Vd E MD (4.74)

whenever [‘TOIA S 6(5).
2. Uniform Global Attraction: For any r, 5 > 0, there exist some T E Z+ such that for

124

every d E MD and 160 E Z+

|x(k, k0,x0, dllA < 5 V16 2 k0 + T (4.75)

whenever [xOIA S 6(5).

To facilitate the forthcoming analysis, we derive the system equations once more in order

to illustrate the dependence on the disturbance. So, consider again the system from above

2. = z/J(x,z,d,u) (4.76)
:i: = Ax+B¢(x,z,d,u) (4.77)
y = 0x (4.78)
C = 903.2) (4.79)

where the right-hand side now depends on an unknown but bounded disturbance d. This
disturbance take values in a known compact set D. For the sampled data analysis, we

make the following assumption.

Assumption 4.2 The disturbance d is constant over the period [kT5, kTS + T3)

Remark 4.3 In general, this assumption is restrictive, but as we will see in the next
chapter it is valid when performing Closed-loop, sampled-data, analysis of certain classes
of systems with hysteresis nonlinearity at the input. In Subsection 4.4.3 we will discuss the
extension of the results described here to the case of disturbances that vary continuously

in t.
With the right-hand side written as

., d
F(x.d,u)= Wm, ,u)

Ax + ng(x, z,d, u)

125

we can describe the discrete-time plant dynamics, as in (4.13), sampled at a rate of 1/T3
by
X(k +1) = X(k) + TsF(x(k)~ (1(1), 1104)) + T521100). WC). 11(1), Ts) (4-80)

We consider the partial state feedback controller (4.8) and the discrete-time equivalent

u(k) = 713306), ((13)) (4-81)

This controller together with ( 4.80) satisfies the following assumptions.
Assumption 4.3

1. The function ”y is locally Lipschitz in x and C and globally bound in x.

2. The Closed-loop system (4.80)-(4.81) is uniformly globally asymptotically stable with

respect to the compact invariant set A.
The system (4.76)-(4.79) satisfies:

Assumption 4.4 The functions (25, w, and 9 are locally Lipschitz in x, z, and u uniformly

in d.

As a consequence of this assumption, and by arguments used to arrive at (4.13), the
function (I) is locally Lipschitz in X and u uniformly in d. We use the same observer (4.16)-

(4.17) and change of variables (4.31) as above to arrive at the observer error dynamics

an + 1) = Afan) + 69(x(n).€(n), damn, uam (4.82)

54(7).) : [I — 5NQ(5)]x(n) + N1(5)§(n) (4.83)
Then, the closed—loop system under multirate control is given by

X(k+1) = X(k) +Tsf(x(k),€s(k),d(k),TM) (4-84)
€(n+1) = Af€(n)+89(x(n):€(n),d(k),C(17).11(1):?) (4-85)

126

473(k) :2 [I - 5N2(5)]x(k) + N1(5)§3(k) (4.86)
where the function .7: is locally Lipschitz in X and {5 uniformly in d and is given by
f(x,€s. d,Ts.€) = F(X(k).61(1€),1106))+ TS‘I’iXUC): dUC). u(16),Ts)

with u(k) taken from (4.20).

From assumption 4.3 we have that the discrete-time plant (4.80) under control (4.81) is
uniformly globally asymptotically stable with respect to the compact invariant set A. The
discrete-time converse Lyapunov theorem (Theorem 1 of [35]) guarantees the existence of

a smooth Lyapunov function V(X) that satisfies

041(lXIA) S V(X) S a2(IXIA) (4.87)

AV(X) = V(XUC +1)) - V(X(k)) S —a3(leA) (4-88)

for all X E R814" and d E D where (11 and 072 are class [Coo functions and a3 is a

continuous positive definite function.

4.4.2 Boundedness and Ultimate Boundedness

We have the following result

Theorem 4.3 Consider {4.84)-(4.86) and the control (4.20). Let Assumptions 4.3-4.4
hold and let M and N be any compact subsets of RS” and RT, respectively. Then, for

trajectories (2:, x) x :2: starting in M x N the following holds

0 There exists 5: such that, for all 0 < 5 S 5:, X(t) is bounded for allt Z 0 and £(n)

is bounded for all n 2 0.

*

5 > O, k* > 0 and n* > 0, such that for every

0 Given any )1 Z 0, there exists 5

127

O<5S5§, we have

Han)“ + Ix<k>lA s 11 v k 2 k*. and n 2 n* (489)

Proof: Let ()1 denote the compact set {V(X) S C1} for any c1 > 0. We take W({) = €TP§
as the Lyapunov function for the observer error dynamics where P is positive definite and
satisfies Affl’A f — P = —I and “PH > 1. Let 62 > c1, C3 > 0 and consider the following
sets

Q2 = {V(X) S C2}, A = 92 X {W(€) S 6352}

Due to the boundedness of 7 and Q in x we have for all (X, g) E Q X RT and for all d E D

llf(x.€s,d.Ts,€)ll S K1. l|g(x,5,d,(,u)|| 3 K2

where K1 and K2 are positive constants independent of 5. From equations (4.20), (4.81),

(4.86), and since J: is Lipschitz with respect to x uniformly in d
llf(X' 0? d7 T37 0) — f(X7€-97 Cir/1:375)“ S €K3]]X(k)ll + K4ll€S(k)”

for all (X,€) E Q x RT, all d E D, and for some positive constants K3 and K4 independent

of 5. From the foregoing, it can be shown that in the set A
V(x(k H» = V(X(k) + Tsflx, 53, d, Ts, 6)) S V(x(k)) - a3(l><l,4) + 5T3K5 (4-90)

and
1

W(€(n +1» 3 (1— 1151—1

) W(€(n)) + 5K6||€II + 521(7 (4.91)

for some positive constants K 5, K6, and K7. Repeating arguments as in Theorem 4.1
boundedness of the Closed-loop trajectories follows for both cases of controller design

discussed in the previous section.

128

Ultimate boundedness follows by showing that equation (4.90) satisfies

V(X(k +1» 3 vane» — $416461 A)
for a3(]X(k)|_A) _>_ 25T3K4. Let

111(5) = 25T3K4 + max V(X(k))
a3(|><(1€)|A)S2€T3K4
As in Theorem 4.1, using (4.87) we can show that there is a k* > 0 such that for all
k _>_ 14*

lx(k)|,4 s a1‘1(w(€))

where 52(5) —> 0 as 5 ——+ 0. We have that ||§(n)|] enters the set {W(§(n)) S C352} in finite

time. Using this fact, we can find an 5; such that (4.89) is satisfied for all 5 S 5;. <1

Remark 4.4 We remark on the intersampling behavior of the Closed-loop system. The-
orem 4.3 shows that X(t) is bounded for all t Z 0 and that X(k) and {(n) are bounded
and ultimately bounded. Using equation (4.12), it can be seen that the continuous-time
trajectories of the closed-loop system cannot grow by more than an order of T5 in between

samples. That is

“X(t) - X(k)ll S 0(Ts)

4.4.3 The Tracking Problem, Integral Control, and Distur-

bances

In anticipation of the simulation and experimental results of the next chapter, we consider
the tracking problem and a controller that contains integral action. For the tracking
problem, the closed-loop system will be driven by time-varying reference signals and
stabilization of the origin may not be possible. Therefore, we will consider stabilization

to a set containing the origin. To generalize the discussion of Subsection 4.4.1 we will

129

allow the system to depend on disturbances that vary continuously in t. We derive the
closed-loop system under multirate sampled-data output feedback and discuss extending
the results of the previous Subsection to the case considered here.

Consider the system (4.76)-(4.79). Ideally, we’d like the output y, in the absence
of disturbances, to asymptotically track a reference signal y R(t). We consider reference

signals that satisfy the following

(7")

1. y R(t) and its derivatives up to and including y R (t) are Lipschitz continuous func-

tions of t and belong to the compact set ’R.

2. The signals are available online.

Let YR(t) =2 [yR(t) y'R(t) yg_1)(t)]T and set 5 = x — YR. We can rewrite the

system dynamics in the error coordinates

z = u2(e+YR(t),z,d(t),u) (4.92)

e = A5 + Baa + YR(t), 2, 4(1), u) — yg)(t) (4.93)

The right-hand side depends on an unknown but bounded disturbance d(t). This dis-
turbance take values in a known compact set D C Rd. We further assume that d(t) is

Lipschitz continuous in t. That is

”01(11) - d(12)“ S Ldltl - t2|

for some positive constant Ld' To simplify the notation, let X8 = [z e]T and
DIR = [YR yg)]T. Also, define the right-hand side of (4.92)—(4.93) by the function
F(Xe(t),d(t),yR(t),u). The solution to (4.92)-(4.93) over the period [kT5, kTs + T3] is

given by

t
x49) ——— x66) +111 Fear),doiygvtuandr
. S

130

By adding and subtracting terms to the right-hand side and using the Lipschitz property

of the function F we can arrive at

|lXe(t)-Xe(k)ll S (t-kTs)llF(Xe(k),d(k),yR(k),U(k))ll

t t
+L/ X r)—X kT dT-l-L/ dr—dkT dr
1 kTSH e( e( s)” 2 kTSH ( ) ( s)“

t
+L/ — kT d
3 kTsllyR(T) yRi s)” T

where L1, L2, and L3 are Lipschitz constants of F with respect Xe, d, and 32R, respec-

tively. From the Lipschitz property of 31R and d this simplifies to

||Xe(t) - Xe(k)ll S (t - kTs)||F(Xe(k),d(k),3’12“), u(19))“ + L4(1- kTs)2

t
+erqbuiu>—xaranwv

for some positive constant L4. Applying Gronwall-Bellman to the above equation results

in the following inequality

.Lll. [6(t-kTs)L1 -1] ||F(Xe(k),d(k),yR(k)au(k))ll

2L
+_4 [6(t—kTs)L1 .. 1 _ T3] (4.94)
L1

||Xe(t) - Xe(k)||

|/\

Using the notation d[k] = {d(t) : t E [kT3,kT3 + T3]} and yR[k] 2 {yR(t) : t E

[kT3, kTs + T5]} we can write the discrete-time dynamics in the following way

Xeik +1) = Xeik) + TSF(X€(k)a (WC), 323(1), 1419)) + T92‘1’(Xe(k), 61116123?ka U(k),Ts)
(4.95)
We consider the sampled-data state feedback controller with integral action using the

Forward Euler method

aa+1>= av+iuuv—4Ra» use

131

11(13) = 701(k), WC), ((k),yR(k)) (497)

To study the system using such a controller, we can augment the state equation (4.95)

with ( 4.96)
004 +1) 2 0(1) + T3 4(1) - 43(1)
Xe(k +1) Xer) F(Xe(k), (WC), 303(k), u(19))
+7132 0 (4.98)

(P(XeUC), dlkl, 32121111, “(16), Ts)

We operate under the assumption that the controller (4.97) achieves asymptotic stabi-

lization of a compact set A containing the origin (0, X5) = 0 of (4.98).
Assumption 4.5

I. The function *7 is locally Lipschitz in o, x and C and globally bound in o and x.

2. The Closed-loop system (4.97 )-{4 .98 ) is uniformly globally asymptotically stable with

respect to the compact invariant set A.

Assumption 4.6 The functions a), w, and 9 are locally Lipschitz in x, z, and u uniformly

in d and 32R.

As in the previous subsection, we need a converse Lyapunov result for the analysis. The
proof of the converse Lyapunov Theorem given by Jiang and Wang [35] for UGAS non-
linear systems relies on the fact that the disturbance is discrete and takes on values in a
compact set. Their work parallels the work of Lin, Sontag, and Wang [42] where a con-
verse Lyapunov Theorem was presented for continuous-time systems with disturbances.
It is expected that based on the results of these groups a converse Lyapunov Theorem can
be proven for discrete-time systems affected by continuous-time disturbances that also
takes values in a compact set. We don’t pursue this extension here, however 'we proceed

by making the following assumption

132

Assumption 4.7 Let Xa = [o Xe]T. There exists a smooth Lyapunov function V(Xa)

that satisfies

alilXalA) S V(Xa) S 012(1Xal/1) (4-99)
AV(Xa) = V(Xaik +1)) - V(Xa(k)) S —Ot3(|XalA) (4-100)
for all Xa E R€+T+1, all d E D, and all 31R E ’R. where 0‘1 and 02 are Class [Coo

functions and a3 is a continuous positive definite function.

Based on the foregoing, with the closed-loop system under multirate sampled-data output

feedback using the observer (4.16)-(4.17) and the controller

11(1) = 7(0(k)157(k)a 4(1). 344(8)) (4101)

we have the following result, the proof of which is similar to that of Theorem 4.3.

Theorem 4.4 Consider the state and control, equations (4.98) and (4.101), respectively,
along with the multirate observer (4.16)-(4.17). Let Assumptions 4.5-4.7 hold and let
M and N be any compact subsets of REM" and RT respectively. Then, for trajectories

(z,x) x 2: starting in M x N the following holds

0 There exists 5; such that, for all 0 < 5 S 52;, Xe(t) is bounded for allt _>_ 0, 0(k) is

bounded for all k 2 O, and {(n) is bounded for all n 2 O.

0 Given any n 2 0, there exists 5; > 0, 19* > 0 and 72* > 0, such that for every

0 < 5 S 5;, we have

|]£(n)][ +1Xe(k)lA S u V k 2 k*, and n 2 71* (4.102)

133

4.5 Example

Consider the system

x1 = x2

x2 = x3

x3 = x3 + u
y = 331

together with the state feedback linearizing control law implemented in discrete-time
11(1) = -430.) — 343(k) — 342(k) — 421(k)

We determine through simulation that a sampling period of T = 0.2 is sufficient to
stabilize the origin of the closed-loop system. For initial conditions in the set {1332] S 1}
the control saturation level {[ul S 10} was chosen based on simulation under sampled-data
state feedback. Using the observer (4.16)-(4.17), the closed-loop system under sampled-
data output feedback is simulated for x1(0) = 0.9 and 232(0) = 333(0) 2 531(0) = x2 (0) =
533(0) = 0. With these initial conditions, the control u(0) = u0(0) = 0. Figure 4.2
shows the response of the output and control for the single-rate output feedback case
with T3 = 0.2 and 5 = 0.2 and for the multirate output feedback case where T 5 = 0.2,
Tf = 0.04 and 5 = 0.04. For comparison, the response under sampled-data state feedback
with sampling period T = 0.2 is also given. The plots show that the multirate case,
with the more accurate estimation, is able to stabilize the system. Also, it can be seen
that the control under the multirate scheme avoids peaking. Figure 4.3 shows that the
estimates recover from peaking and converge to the states during the period [0,02],
thereby isolating the control from the observer transient. Clearly, the single-rate case was

unable stabilize the system for this choice of sampling period, but we reiterate the point

134

made in the introduction that for sufficiently small sampling period the single-rate case
will stabilize the closed-loop system. This is shown in the next simulation example where
we examine the transient performance of the single-rate and multirate output feedback
control schemes in the presence of disturbances and initial estimation error. The single-
rate scheme is simulated for T3 = 0.1, 5 = 0.1 and again for T3 = 0.01, 5 = 0.01. For the
multirate scheme we use T5 = 0.1, T f = 0.01, and 5 = 0.01. Figure 4.4 shows that both
responses of the single-rate scheme undergo peaking from the initial estimation error and
consequently, the control response saturates. Consistent with high-gain observer theory
[20], the single-rate scheme with the faster sampling period recovers more quickly. The
multirate scheme however avoids controller saturation. Three fast samples or 0.03 seconds
were enough for the multirate estimates to recover from peaking. In all three cases, the
control was held at a constant value u = 0 over the first input sampling period T3. Figure
4.4 also shows similar behavior in the three cases when an impulsive-like disturbance of
duration 0.0013 and an amplitude of 300 is experienced at the input of 231 at time t = 10.
For the multirate scheme the observer was able to recover from the disturbance before
the next control step was initiated. This can be seen from Figure 4.5 where the estimates
are plotted against the states. The top set of figures show the peaking in each estimate,
and the bottom set focuses on the behavior around t = 105. Here, the estimates can
be seen to experience a transient between slow samples taken at {10,101}. However, if
the disturbance occurs toward the end of the current control step, the next control step
may be computed before the estimates recover from peaking. This situation is illustrated
in Figure 4.6 where the impulsive-like disturbance is felt at t = 7.087. The single-rate
case is also plotted where T 3 = 0.01. The single—rate case experiences negative followed
by positive saturation during successive samples. The multirate case on the other hand
experiences negative saturation during one sample and then settles. These results show
that the multirate scheme compared to the single-rate scheme is less sensitive to the

peaking phenomenon, but still requires a globally bounded control.

135

Sampled—Data State Feedback

 

 

1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x" 0 k 3 0 F
—1 ‘ —1 ‘
0 10 20 O 10 20
Single-Rate Output Feedback
0 1
x'. -50*
—100 ‘
O 2 4
Multirate Output Feedback
1 k 1
><F 0 3 0
—1 —1 ‘
O 1 0 20 O 1 O 20

Time Time

Figure 4.2. Simulation results showing the output x1 and the control u for sampled-data
state feedback (top) with T = 0.2, single-rate output feedback (middle) with T s = 5 = 0.2,
and multirate output feedback (bottom) with T3 = 0.2 and Tf = 5 = 0.04.

136

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2.5~ . 60» . 400_ q
2 . 40- -
200* -
g 1.5- ~ 2 20 1 g
1- N (‘0
x X X 0- _
at) ed at)
XP 1: - .1 XN 0— ._ x00
-200- 4
0.51 ~ —20- -
o 1 —4o- . ‘4°°” "
-O.5 ‘ - ‘ - O ‘
0 0.1 0.2 0 0.1 0.2 0 0.1 0.2
Time Time Time

Figure 4.3. Simulation results showing the transient response of the multirate observer
estimates (solid) versus the system states (dashed).

137

Single-Rate

 

 

10

TS=O.1 ‘ 3 0 [PA [A

 
 
 
 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x'. 0.5
o N —10
0 1O 20 0 10 20
Single—Rate
1 - 1o
TS=O.O1
XI. 0.5 3 0 r
0 -1O
0 10 20 0 1O 20
Multirate Output Feedback
1 - 1
x“ 0.5 1 + 3 0 F F
O —1 r
0 1O 20 0 1O 20
Time Time

Figure 4.4. Simulation results for an impulsive-like disturbance at t = 10. Shown are the
output x1 and the control u for two cases of single-rate sampled-data output feedback, (1)
5 = 0.1, T 3 = 0.1 (2) 5 = 0.01, T5 = 0.01, and multirate sampled-data output feedback,
where 5 = 0.01, T; = 0.1, and Tf = 0.01.

138

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

,_. 0.51 - ,_. 05 ,_. 0.5
s, k 3... 2,,
x 0 x 0 x 0
05 05 Wm 06 WV“
xF :«<N x0
—05 1 -0.5 —05
—1 1 —1 —1
0 10 20 0 10 20 0 10 20
Time Time Time
1 5 5 5
«0-0 1 i l 4—0 co-I
<6 c6 as
.C .C .C
v- N (O
X 05, , x 0 __ x 0 __
o5 o5 ca
><'_ I ><N x”
0 |
—. . -5 - —5 -
9.9 1010.1 9.9 1010.1 9.9 1010.1
Time Time Time

Figure 4.5. Simulation results for an impulsive-like disturbance at t = 10. Shown are
the multirate observer estimates (solid) versus the system states (dashed) for the entire
simulation run (top) and zoomed in on the transient response at t = 10 (bottom).

139

Multi rate

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 10 r
5 1 1
0 5* l
><'_ 3 0 V‘ k?
_5 .
o 1
- _10 .
O 10 20 0 1O 20
Single—Rate
1 - 1O
TS=0.O1 5 .
0.5
xf- 3 0 F‘
_5 .
0 .
- _10 .
O 10 20 O 10 20
Time Time

Figure 4.6. Simulation results for an impulsive-like disturbance at t = 7.087. Shown
are the output 331 and the control u for multirate sampled-data output feedback, where
5 = 0.01, T s = 0.1, and Tf = 0.01 (top), and single-rate sampled-data output feedback,

where 5 = 0.01, T3 = 0.01 (bottom).

140

 

4.6 Conclusions

We have studied multirate sampled-data output feedback control of a class of nonlinear
systems using high-gain observers. Based upon a given asymptotically stabilizing state
feedback controller and input sampling rate, we were able to show that the scheme practi-
cally stabilizes the origin of the closed-loop system. We also examined stabilization with
respect to a compact positively invariant set. We have also seen that initialization of the
controller can overcome the peaking phenomenon as in [37]. Initialization represents the
state of the art of digital control. In [4], Astrom and Wittenmark mention that “it is
important to set the controller state appropriately when the controller is switched on.”
This is done to prevent large switching transients. They go on stating that “for an al-
gorithm with an explicit observer, the controller state may be initialized by keeping the
control signal fixed for the time required for the observer to settle.” As we have seen, both
analytically and through numerical example, the multirate scheme can be designed such
that controller initialization need only be preformed for one slow sample. In addition, the
multirate scheme appears to be less sensitive than the single-rate scheme to impulsive-like

disturbances that may occur at an unknown time.

141

‘—

CHAPTER 5

Application to Smart Material

Actuated Systems

5.1 Introduction

In this Chapter we are motivated by applications to smart material actuated systems
that may employ computationally demanding controllers such as hysteresis inversion al-
gorithms [63]. Control difficulties are listed among the main drawbacks to the use of
smart material actuators [24]. This is due to the nonlinear behavior of the material and
difficulties in measuring the state variables. Smart materials such as piezoelectrics, shape
memory alloys, magnetorheological and electrorheological fluids, and magnetostrictives
represent an area of research that has recently received significant attention due to their
broad application potential. Computer technology has motivated the recent interest in
this area, as they allow for efficient control of these systems. Smart materials exhibit
significant nonlinear behavior, not the least of which is the hysteresis that is inherent in
these materials. Control techniques for systems with hysteresis have recently received re-
newed attention due to the possible application of smart materials in actuator designs [47].
Hysteretic nonlinearities can severely limit a system’s operation resulting in undesirable

inaccuracies, oscillations, and even instability of the closed-loop system [65]. Moreover,

142

tight control is needed when smart material actuators are used in applications such as
micropositioning. For these applications it is necessary to compensate for the hysteresis.

We consider a class of nonlinear systems actuated by a smart material actuator. The
model we use is shown in Figure 5.1, where the actuator portion appears in the dotted
box. This actuator is comprised of a hysteresis operator, denoted by I‘, in cascade with
linear dynamics. Smart material actuator models of this form are discussed in [16] and
[69]. The fundamental approach [65] to dealing with hysteretic nonlinearities, and the
one we use here, is to use an inversion control scheme as shown in Figure 5.3. Here the
hysteresis is preceded by its right inverse, denoted by F—l, so that its effect is canceled.
There are a number of methods available to characterize a hysteretic nonlinearity that
result from physics-based and phenomenological models. Some of these models can be
found in [13], [45], and [71]. These models come with varying levels of complexity and
accuracy. Modeling the hysteretic behavior is complicated by the fact that the output
depends not only on the instantaneous input, but also on the history of its operation.
For a good control design, phenomenological models that characterize the hysteresis must
be sufficiently accurate, applicable for controller design, and efficient enough for real-time
application. In this work we will use the Preisach operator. The reason for this is that the
Preisach operator provides a very general description of the hysteresis. And from [63] a
computationally efficient discrete inversion algorithm is readily available. The analysis in
this case will be complicated by the fact that, in general, the error resulting from inexact
inversion or hysteresis modeling error will be nonvanishing. Hence, the closed-loop state
feedback system may not have an asymptotically stable equilibrium point, but rather will
have an ultimately bounded solution. Thus, we seek to apply the results of the previous
chapter by studying multirate output feedback in the presence of hysteresis inversion
error. We consider a bounded hysteresis inversion error e, and a closed, invariant set A
that contains the origin and whose size is determined by 5,. We start with the assumption

that there exists a sampled-data state feedback controller that achieves stabilization with

143

respect to A in the presence of hysteresis inversion error. With such a controller, we show
that the closed-loop trajectories under multirate output feedback will come arbitrarily

close to the set A.

 

 

 

 

System y .
r 6(5) Dynamics

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 5.1. Model structure of a smart actuator and plant.

This chapter is organized as follows. In Section 5.2 we present the class of nonlinear
systems under consideration. We also discuss the Preisach operator as a model for the
smart material hysteresis. In Section 5.3 we derive the Closed-loop sampled-data system by
including the smart material actuator model and show that this model fits the development
of Chapter 4. We also give a simulation example to illustrate the ideas. In Section 5.4
we present experimental results based on multirate output feedback control of a shape

memory alloy actuated rotary joint. Finally, in Section 5.5 we draw our conclusions.

5.2 Model

5.2.1 Class of Systems

For the system dynamics in Figure 5.1 we consider again the following class of nonlinear

systems

(N2-
|

— 71’1(x,z,u) (5-1)

144

x = Ax+Bq§(x,z,u) (5.2)
y = Cx (5.3)

C = 9(x, z) (5.4)

where x E RT and z E Re are the states, u is the input, y and C are the measured outputs.
The functions ()3, w, and O are locally Lipschitz in their arguments over the domain of
interest and satisfy (b(0,0,0) = 0, w(0,0,0) = 0, O(0,0) = 0. The r x r matrix A, the

r x 1 matrix B, and the 1 x r matrix C are of the same structure as (1.5)-(1.6).

5.2.2 Preisach Operator

The hysteresis is modeled using a Preisach operator, which we review here. Detailed
discussion of hysteresis modeling including the Preisach operator can be found in the
monographs [45] and [71]. The Preisach operator is comprised of delayed rely elements
’j'( b, a) called hysterons. The switching thresholds of these elements are denoted by (b, a)

as shown in Fig. 5.2. The output of the hysteron is described by w(t) = I(b.a)1v’§1’

 

 

1)
<3

 

 

 

 

Figure 5.2. Delayed relay.

W E [0,T], where v is a continuous function on [0,T] and g E {—1,1} is an initial

145

configuration. The Preisach operator can be described as a weighted superposition of all
d.

hysterons. Define the Preisach plane as 730 if {(b, a) E R2 : b S a}. Each pair (b, a)

is identified with the hysteron i)“, a)‘ Let go be an initial configuration of all hysterons,

50 : Do -> {—1,1}. The output of the Preisach operator is given by

114.4(1) = 30444122801164419) (5.5)

where V is called the Preisach measure. If 1/ is nonsingular (see [45]), then (5.5) can be

written as

I‘[v,§0](t) = [PO ,u(b,a)j'(b’a)[v,<0](t)dbda (5.6)

where the weighting u is called the Preisach density. It is assumed that u 2 0 and
u(b, a) = 0 if b < b0 or a > a0, where a0 = —b0 = 30 for some positive constant 50-
Consider the finite triangular area ’P dif { (b, a) E ’Polb Z b0,a S a0}. At any time t,
we can divide ’P into two regions, 73+ and ’P_, where 19+ (’P_, resp.) consists of points
(b, a) such that a“, a) at time t is +1 (-—1, resp.). The boundary between ’P+ and ’P- is
called the memory curve, which characterizes the memory of the Preisach operator. The
set of all memory curves is denoted by A and A0 is called the initial memory curve. We

make use of the following properties of the Preisach operator.
Theorem 5.1 [71]. Let u be continuous on [0, T] and A0 E A.

1. (Rate Independence) If 19 : [0,T] —> [0,T] is an increasing continuous function such

that 19(0) = 0 and 19(T) = T, then F[v(i9(t)), A0] = F[v, A0](t9(t)), VA E [0,T].

2. (Piecewise Monotonicity) If v is either nondecreasing or nonincreasing on some

interval in [0,T], then so is F[v, A0].

The hysteresis nonlinearity can be identified by discretizing the input range into L uni-
form intervals, which generates a discretization grid on the Preisach plane. The Preisach

operator can then be approximated by assuming that inside each cell of the grid, the

146

Preisach density function is constant. This piecewise constant approximation to an un-
known density function can be found by identifying the weighting masses for each cell,
using‘a constrained least squares algorithm, and then distributing each mass uniformly

over the corresponding cell [63], [64].

5.3 Output Feedback Control

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

V(I) 17d (t) “(1) ya)
{(1)
””31 {[an
law—1 SD

 

 

 

 

 

 

 

 

 

[kT] ControllerOL Ts
“d s
‘ DHGO Wk—

)?[ka] Tf

 

 

 

Figure 5.3. Diagram of the multirate control scheme with hysteresis inversion.

The idea behind the control scheme is shown in Figure 5.3. Hysteresis inversion is
used to compensate for the hysteresis nonlinearity in the smart material actuator. This
inversion is subject to modeling error and the sampled-data controller is designed to
stabilize the system dynamics in the presence of this error. Here we sample the output C
with sampling period T5. We apply the control signal at the same rate through a zero—
order-hold (ZOH) where the control is held constant in between sampling points. The
output y is sampled at a faster rate where we use the period Tf < T 5. As discussed

above, we consider a discretization of level L of the Preisach operator. Also, let the

147

Preisach density function ,u be nonnegative and constant within each cell. Given an
initial memory curve A0 and a desired value ud, the inversion problem is to find a value v
such that ud = I‘[v, A0]. This is done by applying the algorithm that is given in [63] and
reproduced in Appendix C. This is an iterative algorithm and it yields the exact solution
in a finite number of iterations. The inversion error results from the error in identifying
the weighting masses and in the level discretization, thus it can be quantified in these

terms. To see this let

u : F—1[ud, A0] (5.7)

114 = 1111116) (5.8)

as shown in Figure 5.3. The inversion error is defined by

d
82' if 21d — ud (5.9)

Given a bounded sequence “d it can be shown (Appendix C) that the inversion error
satisfies a bound of the form

k
lie-illoo s 0361+ 31 (5.10)

for some positive constant 161, where 5,- is the error in identification of the weighting masses
and GS is the saturation of the hysteresis. Saturation is common in smart materials. For
example, magnetostrictives exhibit magnetization saturation and shape memory alloys
exhibit strain saturation. Equation (5.10) shows that the inversion error decreases, with

decreasing identification error and with ever-finer discretization grids (increasing the level

L). Now, let

(Ta = Alaa-l-Blfid (5.11)

ya 2 010a (5.12)

148

fld = F[U,A0] (5.13)

be a realization of the actuator dynamics G(s), where oa E R“. Using (5.7), (5.8), and

(5.9), we can augment the system (5.1)—(5.4) with the actuator dynamics. We have

oa = Aloa + Blud + 8162' (5.14)
2': = 1,1)(x, 2, 010a) (5.15)
x = Ax + B¢(x, 2,0100) (5.16)
y = Crc (5.17)
C = O(x, z) (5.18)

It is this equivalent system that the output feedback controller is tasked with stabilizing
based on the measurements y and C. This is done by first designing a partial state feedback
controller based on the state x and the measurement C. When designing the controller ud
it may be desirable to take into consideration any information about the additive inversion
error 5,, such as a known upper bound. One may apply robust nonlinear control design
tools such as those discussed in Chapter 14 of [36]. The next step in the design is to
implement the same controller in sampled-data with the state x replaced by the estimate
:2 that is calculated from the measurement y using a discrete-time high-gain observer.

The partial state feedback controller is given by

u.) = 112:, o (5.19)

where '7 is locally Lipschitz in its arguments over the domain of interest and 7(0, 0) = 0.

We use the high-gain observer

:i = Ari: + 1330(8, g, ud) + 11(1) — Cx) (5.20)

149

where the function (250 is locally Lipschitz in its arguments over the domain of interest,
globally bounded in x, and satisfies (150(0, 0,0) = 0.

Now, let X = [0a :5 x]T and rewrite (5.14)-(5.16) as
X = P(x, “d + 92') (521)

where _ .

Alaa + Bl (ud + 82')
P(Xa “(1 ‘1' 52') = ib(x, z,010a)

 

 

Ax + B¢(x, z, 0100,)

The control v is applied to the system through a zero-order-hold, thus it is held constant
in between sampling points. Due to the rate independence of the Preisach operator,
11d will also be constant in between sampling points. We denote the signal at the kth
sampling point by ud(k) = ud(k) + 5,-(k). The solution of (5.21) over the sampling period
[kT3, kTs + T3] is given by

X(t) = X(k)+(t-kTs)F(X(k),ud(k)1762119))

t
+[14:r811‘1()<(0)a1141(1)+ e1119)) — F1X1klvud1kl + 6102”“

As was done in Section 4.2 of Chapter 4 we can arrive at the discrete—time system

><(k +1) = X(k‘) + TsF(X(k)a 1101(1)?) + e2:09)) + T92‘1’(X(k)1 14109) + e1(1), Ts) (5-22)

where (I) is locally Lipschitz in (X, ud + 52-). We use the discrete-time high-gain observer

(4.16)-(4.17) that is sampled with the fast sampling period Tf

q(n +1) 2 qu(n) + ny(n) + 5T_1TfB¢O(D_—1q(n), C(k), ud(k)) (5.23)

x(n) : qu(n) (5.24)

150

We point out that u and C evolve in the slow sampling time k and are constant for all n

where an E [kTS, kTs + T 3). The output feedback controller is given by

u617(k) = V(Ii's(k), C(19)) (525)

where (133(k) = x([hk]) and h = Ts/Tf. Following the procedure of Section 4.2 , Chapter

4 we can arrive at the estimation error equation

an + 1) = Afro) + seamen), e111). ca). 11.111), e)

where g is locally Lipschitz in its arguments, uniformly bounded in 5, for 5 sufficiently
small, and globally bounded in x. We arrive at the closed-loop system under multirate

output feedback control

)<(k +1) = X08) + Tsf(x(k),€s(k)a61(k),Ts,€) (526)
€(n +1) = Af€(n) + 690001601), 84(k),C(k)1ud(k),€) (5-27)
53(1) 2 [1 — 5N2(5)]x(k) + N1 (5)5301) (5.28)

where {5(16) 2 C([hk]) and

f(x,€s,e,-,Ts,€) = F(X(k),ud(k) + 82113))

is locally Lipschitz with respect to X and C3. The matrices N1 and N2 are defined as
in Equation (4.31). Furthermore, we have that the inversion error 827(‘1 takes values in a
known compact set D C R. Let MD be the set of all functions from Z+ to D and Let
62' E MD' In the presence of the inversion error, the equations (5.26)-(5.28) fit the closed-

loop system equations (484)-(486) that were developed in Section 4.4.1. Therefore, with

151

a single-rate sampled—data state feedback controller that renders a closed set containing
the origin uniformly globally asymptotically stable we can apply Theorem 4.3 of Section
4.4.

Let A be a compact, invariant subset of Rr+€+a that contains the origin. The size
of A depends on the size of the hysteresis inversion error 52-. Let MD be the set of all
functions from Z+ to D. Using (5.29) it can be shown that the closed-loop system under

the single (slow)-rate sampled-data state feedback control

114(k) = 7(1‘(1‘»‘),C(k)) (5-30)
is given by
X(k +1) = X01") + TsflXUC), 0,e,-(k),T3,0) (531)
we have the following result.

Theorem 5.2 Consider equations (5.26)-{5.28) and the control {5.25). Let 5,- E MD

and suppose that
1. The origin (X = O) of (5.31) is UGAS with respect to the compact set A;
2. The function 7 is locally Lipschitz in its arguments and globally bounded in x.

Let M and N be any compact subsets of R€+r+a and RT respectively. Then, for trajec-

tories (0a, z,x) x :13 starting in M x N the following holds

0 There exists 5’] such. that, for all 0 < 5 S 51‘, X(t) is bounded for allt 2 O and C(n)

is bounded for all n 2 O.

0 Given any I/ 2 0, there exists 5; > 0, 16* > 0 and n* > 0, such that for every

0 < 5 S 5;, we have

[1501)“ +1Xik)lA S V V k 2 19*, and n 2 71*

152

5.3. 1 Simulation example

In this section we illustrate the above analytical results by considering the following

nonlinear system with hysteresis at the input

2 = —23+:L‘1

1:, = 3:2 , 42:13, x3=3x§+z+F[v,A0]
y = 2:1
C = Z

This system fits the class of systems under consideration and we use a controller based
on state feedback design in cascade with a hysteresis inversion operator. The hysteresis
nonlinearity is implemented using a Preisach operator with 10 levels. The weighting
masses are uniformly distributed in each of the 55 cells. Figure 5.4 shows the major loop

of the hysteresis. This hysteresis operator saturates for input values outside [—71, 71]

Hysteresis Loop

 

5O

Operator Output
0

 

 

-50 ‘
—100 O 1 00

Input

 

Figure 5.4. Hysteresis loop based on the Preisach operator used in the simulation example.

153

and has a maximum and minimum output value of i48.812. First, we consider the case
where the hysteresis operator is exactly known and thus there is no inversion error. To

compensate for the hysteresis, we use an inversion operator given by
v = I‘_1[ud, A0] (5.32)

The state feedback controller is designed assuming that the hysteresis operator has been
completely canceled and we use a sliding mode controller to stabilize the origin of the

nonlinear system. This controller is given by

 

2 6
where
sup-.1) = (2| + [1‘2] +2lx3l +3Ix1|3 +1 (5.34)

It can be shown that, with the hysteresis completely canceled, this controller globally
asymptotically stabilizes the origin of the closed—loop system. First, we consider a sim-
ulation based on sampled-data state feedback. We sample each output, y and C, with
sampling period T = 0.005 seconds and simulate the continuous-time plant controlled
by the discrete-time controller. The control signal is held constant between sampling
points by using a zero-order-hold that also uses a sampling period of T = 0.005 seconds.
Figure 5.5 shows that the sampled-data state feedback controller stabilizes the origin of
closed-loop system. The plot of the hysteresis inversion error (bottom right) shows ex-
act inversion has been achieved. To simulate the output feedback control we use the

discrete—time high-gain observer (5.23)-(5.24) with

(150 2 31:13 + C + ud

154

For the observer parameters we choose 01 = 012 = 3, a3 = 1. Also, we take a = 1 so
that T f = 5. Using this observer the single-rate output feedback case is simulated, where
we choose the control sampling period T3 = 0.005 and measurement sampling period
T f = 0.005. Figure 5.6 shows that the system is unstable at this sampling-rate. Using
the multirate output feedback scheme we sample C with period T3 = 0.005 seconds and
y with period T f = 0.001 seconds. The control update rate is the same as the state
feedback design T3 = 0.005. Under multirate output feedback, the closed-loop system is
stable as shown in Figure 5.7. Now consider the case with inexact hysteresis inversion,

where we use the approximate inversion operator
5 = 11711113, )0) (5.35)

Furthermore, we assume that there is at most 20% error in identifying the weighting
masses. Based on this assumption we have the following known bound for the inversion
error

[62'] S 9.7624

In the presence of this error, the control (5.33)-(5.34) can no longer guarantee stability
of the closed—loop system. In order to deal with the hysteresis inversion error, we must
modify the control design. We do so by adding 10 to the function 0 (x, z) to dominate the

inversion error. This function is now given by
snag) = [z] + 1.521 +2153| +315113 +11 (5.36)

and the controller will be robust with respect to the hysteresis inversion error. We simulate
this system where the Preisach weighting masses are randomly perturbed by as much as

20%. The actual hysteresis inversion error satisfies

[8,] S 6.61

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4O 01
20 ,_ 0.051
9
= o “1 o
>
E
-20 1 —0.05’
—40 ‘ -O.1 . -
0 10 20 0 5 10 15
Time Time

Figure 5.5. The state (top), the control (bottom left), and the inversion error (bottom
right) for sampled-data state feedback with exact inversion and T = 0.005.

156

14 20

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1x10 - 1x10
0 0
l
><N —1' I N -1' 1
06 1 “to
xv- "'2b X l x "2’
—3~ 1 ' . -31 _x3
-—--—-X2 ' ""‘Z
—4 - —4 ‘—
O 5 10 0 5 10
4o . 2
20’ 1 h
9 0
3 0 1 ”J
E —2
-2o 1
—40 * -4 -
0 5 10 0 5 10
'fime TMm

Figure 5.6. The state (top), the control (bottom left), and the inversion error (bottom
right) for single-rate sampled—data output feedback with exact inversion and T5 = 0.005.

157

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4O - 0.1
20’ 1 h 0.05)
9
a o “1 o
>
E
-20 —0.05
-40 ‘ —O.1 ‘ *
0 10 20 0 5 10 15
Time Time

Figure 5.7. The state (top), the control (bottom left), and the inversion error (bottom

right) for multirate sampled-data output feedback with exact inversion, T3 = 0.005, and
Tf = 0.001.

Simulating the sampled-data state feedback control, with sampling period T = 0.005
seconds, we see from Figure 5.8 that the new controller is able to stabilize the origin of
the closed-loop system in the presence of the inversion error. This error is visible in the
bottom right plot. Now consider the single-rate output feedback case with T s = 0.005.
With the redesigned controller the closed-loop system is stable, but the control signal
exhibits large oscillatory behavior as shown in Figure 5.9. In addition, a small oscillation
can be seen in the transient response of the state 2. In Figure 5.10 we can see that for the
multirate output feedback case with T3 = 0.005 and Tf = 0.001 the response is stable

and the control signal behaves well.

5.4 Experimental Results: Control of a Shape Mem-
ory Alloy Actuator

In this section, we apply multirate output feedback control to a shape memory alloy
(SMA) actuated robotic joint. SMAs are metallic materials that exhibit coupling between
thermal and mechanical energy domains. The shape memory effect (SME) results from a
transition between two structural phases that is hysteretic in nature. For more detailed
information on the SME consult [23]. The name shape memory results from the materials
ability to “remember” an initial shape. For example, SMA wire can be stretched and upon
heating the wire, it will contract back to its initial shape. Thus, this thermal/ mechanical
coupling has motivated the use of SMA as an actuator. Control of shape memory alloy
actuated systems can be found in [3], [22], [24], and [31]. Also, control and modeling of
hysteresis in SMA has been considered in [28], [29], [43], and [73]. Figure 5.11 shows a
diagram of the rotary joint. This two-wire configuration is referred to as a differential-type
actuator. The rotating joint consists of a 0.5 inch diameter shaft and two Nitinol wires
10.28 inches in length and 0.008 inches in diameter. These wires are stretched by 2%

of their length. Alternate heating and cooling of the two wires provides clockwise (CW)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4O 5
20 a m
2
= o “J o
>'
E
—20
-40 ‘ —5 ‘
10 20 0 10 20
Time Time

Figure 5.8. The state (top), the control (bottom left), and the inversion error (bottom
right) for sampled-data state feedback with inversion error and T = 0.005.

160

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20 h 2
9
3 0 ”J 0 1

>'
E

-20 —2

—40 - —4 ‘

0 10 20 0 1 0 20
Time Time

Figure 5.9. The state (top), the control (bottom left), and the inversion error (bottom
right) for single-rate sampled-data output feedback with inversion error and T3 = 0.005.

161

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

4O . 3.5 -
20 h 3
9
a o “J, 2.5
>
E
-20 2
—40 ‘ 1.5 -
0 1O 20 0 10 20
Time Time

Figure 5.10. The state (top), the control (bottom left), and the inversion error (bottom
right) for multirate sampled-data output feedback with inversion error, T3 = 0.005, and

Tf = 0.001.

162

and counterclockwise (CCW) rotation. Bipolar current is supplied to the actuator where
positive current gives CW rotation and negative current gives CCW rotation. With this
configuration, the actuator can achieve up to 60 degrees rotation in each direction. Figure
5.12 shows a simplified diagram of the electrical setup. We use a PC with a Pentium IV
processor running dSpace ControlDesk, a real-time control and data acquisition software
package. The controller is programmed using Simulink and ControlDesk compiles and
downloads the real-time application to a DSP board for monitoring and control. The
bipolar input current is generated by a voltage controlled current (VCC) amplifier and
a pair of diodes routes the current through the appropriate SMA wire; positive current
through one wire and negative through the other. The joint rotation angle is obtained
through an 8192 counts/rev incremental encoder. The DSP board reads the encoder

measurement after A/ D conversion.

 

 

 

Joint

Figure 5.11. Robotic joint actuated by two SMA wires.

5.4. 1 Actuator Model

We will use a model of the form shown in Figure 5.1 by identifying a Preisach operator
and linear dynamics. For shape memory alloy, there are heat dynamics that relate the

input current to wire temperature. The relationship between the wire temperature and

163

 

 

 

 

  
   

 

dSpace

D/A ControlDesk

 

 

 

 

 

 

  

 

 

AID 8192 c/r
Encoder

 

 

 

 

 

 

 

 

 

 

——|<F

 

\—1>.L
/

VCC AMP

 

SMA / Rotary
Shaft

Figure 5.12. Electrical Diagram.

the resulting force of contraction is hysteretic. Thus, the temperature dynamics typically

precede the hysteresis in a model of the material. We will approximate the temperature

dynamics by using a static gain. We expect this approximation to be valid for relatively

low frequency signals. We begin by deriving the temperature equation. The relationship

between the temperature in the wire and the supplied current can be described by the

following lumped heat transfer equation

dT

pCVFt— = —hA(T — Tamb) + 1242

where

p 2 wire density [kg 771—31

c : wire specific heat [J kg_1 00—11
V 2 wire volume [m3]

T 2 wire temperature [0C]

t = time [s]

164

(5.37)

 

Diameter: 0.008 in.
Length: 10.28 in.
R = 8.37 It

p = 6504.78 kg m—3
c = 837 .1 leg—10C"1
h = 68.97 W m“2 00—1

 

Table 5.1. SMA wire physical parameters, where p, c, h, and the diameter are specified
by the manufacturer and the length and resistance R are measured.

h 2 heat transfer coefficient [1V m—2 00 _1]

A 2 wire surface area [m2]

T

amb = ambient temperature [0C]

R 2 wire resistance [9]

i 2 current [A]

Let T = T — Tamb' We have the following transfer function that describes the heat

dynamics of each SMA wire

 

= *2 GHT(S) (5'38)

5" I) def
2:2 s+a

where, by applying the values given in Table 5.1

a = —0.2404 , b = 181.5618

At (1c, the relationship between current and temperature is given by

165

Next we capture the hysteresis in the actuator by identifying a Preisach operator that
maps the static relationship between the temperature and the measured rotation angle.
To do so, we first map the current to the rotation angle and then use (5.39) to obtain the
temperature values. We limit the input current to values within the range [—0.6A, 0.6A]
and divide this range into 10 equally spaced intervals. The supplied current is shown in
Figure 5.13 (top right). The current is held constant for a period of 45 seconds to allow the
angle to settle to a steady-state value. The measured rotation angle is shown in Figure 5.13
(top left). The measured angle data fluctuates slightly after reaching steady-state for each
45 second period. To obtain a single angular value, the measurement is averaged over each
45 second period. Identification of the Preisach weights is conducted using a constrained
least squares algorithm as discussed in [63] and [64]. We discretized the Preisach plane
into 9 levels and used the input-output data to identify the 45 Preisach weighting masses.
The identified hysteresis nonlinearity is illustrated in Figure 5.13 (bottom left). This

operator maps the input current to the measured angle. Now consider
~ . . b ,2
T = szgn(i)—i (5.40)
a

Using this relationship we can construct a Preisach operator that maps the temperature
T to the measured angle. Since the current is bipolar, the sign function tells us which
wire is being heated. Positive temperature values correspond to the heat of one wire
and negative temperatures to the heat of the other. Based on the values for a and
b, the temperature range is [—262.078°C,262.078°C]. The temperature/angle map is
shown in 5.13 (bottom right) and the identified Preisach weights are shown in Figure
5.14. A simulation of the operator is shown in Figure 5.15. The simulation data and the
experimentally obtained data are indistinguishable, thus the Preisach operator captures
the behavior of the identified hysteresis.

Next we used the identified parameters to construct a hysteresis inversion operator

that maps rotation angle to temperature. In order to test the hysteresis inversion model

166

we conducted an open—loop inversion experiment. We supplied the following sequence of

desired rotation angles to the inversion algorithm
{—40, —30, —20 — 10, 0, 10, 20, 30, 40, 30, 20, 10, 0, —10, —20, —30, —40}

Using the inversion algorithm and (5.40) a set of input current values were generated and
these are shown in Figure 5.16 (top left). Each current value was held constant for a
period of 45 seconds. The measured angle is shown in the top right plot. We averaged
the measured angles over each 45 second period and obtained the values shown in Figure
5.16 (bottom left). These values are plotted against the desired angles for comparison.
Figure 5.16 (bottom right) illustrates how close the hysteresis inversion comes to achieving
linearity by plotting the averaged measured angles against the desired angles. A significant
amount of the nonlinearity has been canceled.

To get a sense of the dynamic behavior of the SMA actuator, we conducted identi-
fication experiments for the linear dynamics. This was done by supplying the actuator
with the sinusoidal current signal i = 0.55in(wt), where the frequency took the following
values

w = {0.02, 0.06, 0.2, 0.3, 0.4, 0.6, 0.8, 1.0, 1.2, 1.4, 1.6, 1.8, 2.0, 2.2}

The resulting frequency response, normalized to obtain unity gain at dc, is plotted in
Figure 5.17. We were able to obtain a reasonably good fit to the identified frequency

response using the following second order system

0.5
TF . = 5.41
f 1‘ .92 + 2.25 + 0.5 ( )

 

as shown in Figure 5.17.

167

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

,1. 0.6 1
8 l
3 0.4l
g, l s: 0.2] 1
< 1 § 0
8 s l
S 0 —0.2‘
g -0.4‘
_60 . . ’0.6 . - ‘
0 1000 2000 3000 0 1000 2000 3000
Time (see) Time (sea)
A A 50
d) d)
s s. 30’
2 2
g) g1 10»
< <
8 8 -1o»
8 8
8 8 -30’
2 2
M . . _50 - - - . -
-0.5 0 0.5 —250-100 0 100 250
Current (A) Temp. (Deg. C)

Figure 5.13. Plot showing the measured output, current input, and identified hysteresis
nonlinearity.

168

 

 

 

 

 

 

 

 

 

 

 

1

50 0.75

S 30 0.5
G)

g: 10 5 0.25

2 -1o , “j 0

cc” —0.25

< —30 _0 5

_50 i—O—-Sim. data '
—*— exper. data ‘0-75
—1
0 50
Data No. Data No.

Figure 5.15. Simulation of the identified Preisach operator versus the measured data (left)
and the error between the two (right).

169

 

 

 

 

 

 

 

 

 

 

 

 

 

60

 

 

 

 

 

 

 

   

 

 

 

 

 

 

05’ x), 8’ 401
A it 1‘1 3
g at ‘ .92 20.
.. 1 2’
5 O] 1 1‘ < 01
s 1 8
o 1 1K 5 —20' 1
1*- (I)
7 3‘11 ‘3 -4o»
—O.5’ *3; ‘ E
0 200 400 600 800 ' 0 200 400 600 800
Time (see) Time (sea)
a 50} 40 —*——Data
0) .
3 A - - - Linear
2 d) 20.
C '0
<2: 61 ‘5,’ 0A
8 E)
‘3 —50» g -20»
‘5 ——111—Meas.
g — e — Desired _40’
—100 A A A 3 A A A A A
0 200 400 600 800 —40 —20 0 20 40
Time (590-) Desired Angle (deg.)

Figure 5.16. Plot showing the results of an open-loop hysteresis inversion experiment.

170

Bode Diagam

 

Magnitude (dB)

1

—25

 

*r 1 1

 

 

Fn'rF

 

 

- 5* -|D data

 

 

 

 

_30 . ...rlrir
10“2 10‘1

Frequency (rad/sec)

Figure 5.17. Experimentally obtained SMA actuator normalized frequency response
(dashed) and the response of the transfer function (5.41) (solid).

171

To close the loop, we use PID as our sampled-data controller in cascade with the

hysteresis inversion operator as in Figure 5.3. This controller takes the following form

0(k+ 1)

ud(k) 2: —kia(k) — kp(i1(k) — r(k)) — kd(:i:2(k) — rd(k)) (5.43)

0(k) + T3(i:1(k) - r(k)) (5.42)

where T3 is the slow sample period, ki: kp, and kd are positive constants, :81 is an estimate
of the joint rotation angle 6 and £32 an estimate for the angular velocity w. Also, 1' and
rd are the angle and angular velocity reference trajectories, respectively. The current

supplied to the actuator is given by

i = s'ign(vz-) M (5.44)

v- = kgf"1[ud(k), A0] (5.45)

where the gain kg will be chosen proportional to %, which is the inverse of the dc gain of
(5.38). This term provides an additional degree of freedom in the control design. The sign
function provides bipolar current for CW and CCW rotation. To test the effectiveness
of the hysteresis inversion, we will present results for PID control with and without the
inversion. In both cases, the control current 2' is saturated outside [—O.7A,O.7A]. To
implement the output feedback controller we use a linear discrete—time high-gain observer,
equations (5.23)-(5.24) with ¢0 = O. This observer estimates the rotation angle 6 and
angular velocity w of the rotary joint and these estimates are used in the controller (5.42)-
(5.43). It is discretized using the forward difference method and the observer parameters
are taken to be (11 = 2 and a2 = 1. In every experiment we choose the relation between
Tf and e to be Tf = 0.35. In the forthcoming experiments we will compare the multirate
(MR) output feedback control scheme with two single-rate (SR) schemes. One SR scheme
will be used where the sampling period is chosen to be the same as the MR scheme’s

control sampling period. We will refer to this scheme as slow single-rate. Similarly, fast

172

single-rate will refer to the scheme that uses a sampling period the same as the MR

scheme’s fast measurement sampling period Tf.

5.4.2 Experimental Results on Regulation

We begin with a regulation experiment, where the controller attempts to rotate the joint
to a desired angle 6d and maintain it there. Thus, 7‘ = 9d and rd = 0 in (5.43). Figure
5.18 shows the results of an experiment with the target angle equal to -15 degrees. The
plot compares the response of a single-rate controller (top) with a sampling period of 0.053
against the response of the multirate controller (middle) where the measurement period
was Tf : 0.0053 and the control sampling period was T3 = 0.053. Hysteresis inversion
was used in the top two plots and the controller parameters were taken to be ki = 0.1,
hp = 30, kd = 4, and kg = 0.028. As can be clearly seen, the single-rate scheme was
unable to stabilize the system under this large sampling period. The response exhibits
an oscillation that can lead to early fatigue of the SMA, which results in a loss of the
shape memory effect. On the other hand, the multirate controller, with the more accurate
state estimation, was able to achieve stabilization. The bottom plot in Figure 5.18 shows
the result of a MR regulation experiment without the hysteresis inversion. Here the PID
gains were taken as kz- = 0, kp = 5, and kd = 0.1. This controller yields good tracking
performance with a much more well behaved control signal. In general, in the case of
regulation to a fixed angle, the controller without hysteresis inversion outperformed the
controller with the inversion. This is may be attributable to the sensitivity of the Preisach
operator to the sensor noise. In all our experiments, computing power limited us to a
maximum of 9 levels. Figure 5.19 compares the response of slow SR with T3 = 0.001, MR
with T3 = 0.001 and Tf = 0.0001, and fast SR T3 = 0.0001 for a PID controller without
inversion. Again, we used the gains ki = 0, My 2 5, and kd 2: 0.1. The steady—state
error shows that each scheme achieves good regulation, where the target angle is now -20

degrees. Figure 5.20 plots the current signal for each scheme. The noise was considerably

173

amplified for the MR and the fast SR.

A series of experiments were conducted for decreasing values of the sampling periods
T3 and Tf. Comparisons between the MR scheme and the fast and slow SR schemes were
made. For the MR scheme, we fixed the slow sampling period TS and ran experiments
for several values of the fast sampling period Tf < T3. We then chose the fast sampling
period from this set that yielded the smallest root mean square (R.M.S) regulation error
(6 -— r). We repeated these experiments for several values of slow sampling period T3.
The resulting R.M.S errors of the MR scheme were compared to the R.M.S errors of the
corresponding slow and fast SR schemes. The results of these experiments are shown
in Figure 5.21. The plot on the top compares MR with slow SR under PID without
inversion. The y axis represents the R.M.S error and the :1: axis represents the sampling
period of the SR scheme which is the same as control sampling period of the MR scheme.
The measurement sampling period of the MR observer is printed next to each data point.
At T3 = 0.01 the SR scheme was not stable, but the MR scheme with Tf = 0.005 was
stable. For T3 _<_ 0.005 the SR scheme is stable and the MR scheme does slightly better
in each case. The plot on the bottom compares MR with fast SR where, the :1: axis is the
sampling period of the SR scheme which is the same as measurement sampling period
of the MR observer. The control sampling period is printed next to each data point for
the MR scheme. The R.M.S errors are close for each case as both schemes achieve good
regulation. Note that the resolution of the encoder is 0.0439 degrees. In general, the
R.M.S error tended to decrease slightly as the sampling period decreased. We point out
that sensor noise tends to increase with faster output sampling. Thus, we expect faster
output sampling to improve tracking performance only up to a point, after which the
noise will begin to deteriorate the performance. This appears to be the case in Figure
5.21 where the performance for all three observers was close for a sampling period less

than 0.005 seconds.

174

SR with lnvserion

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o ' ' 1 r
8 -10 ’ ‘ g 0 f
a WWW/WW s i v
—20’ . "'
- - _1 - - -
0 2 4 6 8 O 2 4 6 8
MR with Inversion
o f ' ' . 1 ' ' f ‘
g, TS=O.05, Tf=0.005 3
8 _10 l E 0 if; l
V < 33’}
a: __ If
-20 A . A —1 A A A
0 2 4 6 8 0 2 4 6 8
MR without Inversion
0 T ' - ‘ 1 I
T9} TS=O.O5, Tf=0.005 3
E _10r ‘ E 0
o _ - T:
—20 _ A A -1 A A A
O 2 4 6 8 O 2 4 6 8
Time (sec) Time (sec.)

 

Figure 5.18. Plot of an angle regulation experiment for SR (top) and MR (middle) output
feedback controllers with inversion and MR without inversion (bottom). Shown are the
angle 6’ (solid) versus the setpoint 7‘ (dashed) (left) and the current 2' (right).

175 ‘

SR without lnvserion

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0 s -
o: 5 m
g -10 3 o A
T
GD _20 CD —O1‘ r..__'___1
0 2 4 6 8 0 2 4 6 8
MR without Inversion ‘ -.
A O TS=1e—3, a 0.1 '
d: _. _ a)
i
(D —20 <D -o.1
O 2 4 6 8 0 2 4 6 8
SR without Inversion
O - - -
,, T =1e—4 g, 0-1
c» S a)
g ‘10 3 o
I
CD —20 ‘9 —o.1
0 2 4 6 8 O 2 4 6 8
Time (sec.) Time (sec.)

 

Figure 5.19. Plot of an angle regulation experiment for slow SR (top), MR (middle), and
fast SR (bottom) output feedback controllers without inversion. Shown are the angle 6
(solid) versus the setpoint 7‘ (dashed) (left) and the regulation error 6 -— 7‘ (right).

176

 

Slow SR without lnvserion

 

 

 

 

 

 

     

 

 

 

 

 

 

1 r I
7:?
Q. A
E A
S,
_1 l 1 1
0 2 4 6 8
MR without Inversion
1 , . ‘1
70‘ H l,
.1. llll
._ A I" a II I 9 II I.
_1 . . .
O 2 4 6 8
Fast SR without Inversion
1 . .
6?
3' o»
:5
_1 1 . .
O 2 4 6 8

Time (sec.)

 

Figure 5.20. Plot of the current i for an angle regulation experiment for slow SR (top),
MR (middle), and fast SR (bottom) output feedback controllers without inversion.

177

 

MR vs. Slow SR Without Hyst. lnvs.

 

‘7 Vi-r *fi

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

h 0.6’ —ar- SR 2* j
2 ~—o— MR ,’
LTJ 0.4- / -
“I 0.2 ,’
2. a: H’-—-—~—-—~o-—--------A—-o --—-—o
0: 0- j\ ,fk-P -1e—4 T= -1e-3 T=5e—3‘
-0.2 A A A A A AAA_2
10'310
Slow Sampling Period Ts
MR vs. Fast SR Without Hyst. lnvs.
_ 0.07t ,ili- _______
g are ----- 3:: ’0. ‘‘‘‘‘ T T“ :
UJ . .z’ \ ‘”
03006 .x" T=Ie-3 T=001fl
2' 0’ ’ s s '
$0.05" T=Se—4 -*-SR-
3 -O-AMR
0.04 . .1 . . . . . . . . l - A L A . a A.
10'4 10‘3 10‘2

Fast Sampling Period Tf

 

Figure 5.21. Comparison of MR versus slow SR (top) and MR versus fast SR (bottom)
for a regulation experiment.

178

5.4.3 Experimental Results on Tracking

We now turn our attention to the tracking problem. We would like the shaft angle to

track a reference trajectory 7‘. To do this we use the control (5.42)-(5.45), where r and

rd : 7" are implemented in discrete—time. First, consider

1 7r 1 0.4 37r
=1_A(.t__) _. ____ .
r 5 [23m 0 4 2 + 23272 (fit 2 )J (5 46)

We use k,- = 2, kp = 8, kd = 0.1, and kg = 0.0028 as the controller gains. Figure
5.22 shows the response of the slow SR with T3 = 0.01, MR with T3 = 0.01 and Tf =
0.001, and fast SR with T 3 = 0.001 under hysteresis inversion control. The plot shows
good tracking for each scheme. Figure 5.23 shows that the error and control signals are
comparable. Now, consider the same experiment this time without hysteresis inversion.
The response is shown in Figure 5.24 and the tracking error and control signals appear
in Figure 5.25. In this case, the MR and fast SR have similar performance, but the slow
SR scheme exhibits an undesirable oscillation in the response.

As with the regulation case, a series of experiments were conducted for the tracking
case for decreasing values of the sampling periods T3 and Tf. Comparisons between the
MR scheme and the fast and slow SR schemes were made for the tracking problem with
reference trajectory (5.46) and the R.M.S tracking error results are plotted in Figures
5.26 and 5.27. Figure 5.26 compares the MR with fast SR (top) and slow SR (bottom)

for the controller without hysteresis inversion. It can be seen that the MR scheme is
comparable to the fast SR scheme. The slow single-rate scheme is unstable at T3 = 0.01
and in general the MR outperforms the slow SR. Figure 5.27 compares the MR with fast
SR (top) and slow SR (bottom) for the controller with hysteresis inversion. Here, the
IVIR scheme appears to do a little better than the fast SR. This is most likely due to the
sensitivity of the inversion scheme to measurement noise. The slow SR scheme appears

to do better than the MR for T3 = 0.01,0.005,0.001, again due to measurement noise.

179

 

 

 

 

From both plots with the hysteresis inversion control, there is a range of sampling periods
where the error increases with decreasing sampling period. This appears to be the result
of sensitivity of the inversion to sensor noise. Thus, there is a range of sampling periods
where faster output sampling does not improve performance.
To examine the effect of increasing the ratio Ts/Tf on the tracking performance for
the multirate scheme, we plot the R.M.S tracking error for a fixed T5 and decreasing
Tf. Figure 5.28 shows the results of such experiments for the controller with hysteresis
inversion (top) and without hysteresis inversion (bottom). The data plotted corresponds
to the tracking experiments with reference (5.46). Both the top and bottom plots show
two data sets each; one for T3 = 0.01 and another for T3 = 0.005. In the top plot
we see that with hysteresis inversion, the error tends to increase in both data sets as
the measurement sampling period is decreased. Conversely, in the bottom plot, for the
case without hysteresis inversion, the R.M.S error tends to decrease in both data sets as
the sampling period is decreased. This seems to support the notion that the inversion
algorithm is sensitive to measurement noise. Again, there is a practical limit on how
small the sampling period can be decreased before the effect of measurement noise begins
to degrade the performance. This was studied in [19] for discrete-time single-rate high-
gain observers. Likewise, there will be a practical limit on how large the ratio TS/Tf
can be chosen before the effect of noise degrades the performance. We note that in any
discrete-time control system, one must be aware of the effect of aliasing when choosing
the measurement sampling rate. For the multirate scheme this is true when choosing the

control sampling rate in addition to the measurement rate since there is downsampling

involved in the scheme.

180

 

 

 

Slow SR with lnvserion

 

I I T

 

 

 

 

 

A 20 - ' .
a W
E 0 ‘ l
a; —20- TS=0.O1 .
O 20 40 60 80 100
MR with lnvserion
20 . ' ' ' ' A

0, r (deg.)

0 W
-20 ’ Ts=0'01’ Tf=1e—3 ~

0 20 40 60 80 100
Fast SR with lnvserion

 

 

 

 

 

 

 

A 20- r ' ' ' ~
g3 ,
E 0 ‘
as —20- TS=1e-3 .
O 20 40 60 80 100
Time (sec.)

 

Figure 5.22. Plot of a sinusoidal tracking experiment for slow SR (top), MR (middle),
and fast SR (bottom) output feedback controllers with inversion. Shown are the rotation
angle 6 (solid) versus the reference 7‘ (dashed).

181

Slow SR with lnvserion

WW

0 so 100 0 50 100
MR with lnvserion

1 F Will
—1 A —1 A

O 50 100 O 50 100
Fast SR with lnvserion

 

 

 

O-r (deg.)
C

I
i (Amps)
.15 o —L

 

 

 

 

 

 

 

9-r (deg )
O

i (Amps)
O

 

 

 

 

 

 

 

 

 

G—r (deg )
_IL 0 _L
i (Amps)
EA

 

- _1 .
0 50 100 0 5 100
Time (sec.) Time (sec.)

 

Figure 5.23. Plot of a sinusoidal tracking experiment for slow SR (top), MR (middle),
and fast SR (bottom) output feedback controllers with hysteresis inversion. Shown are
the tracking error 6 —- 7‘ (left) and current 2' (right).

182

Slow SR without lnvserion

 

 

6, r (deg.)
o

 

 

 

 

 

 

 

 

 

 

 

 

-20 TS=O.O1 .
O 5 1O 15 20 25 30 35 _
MR without lnvserion e.

A 20 - ' r «
a
3 0 .
a5 -20- Ts=0.01, Tf=1e—3 A

0 20 40 60 80 100

Fast SR without lnvserion

A 20 . ' ' T ' 4
8’
a 0 ~
a5 —20- TS=1e-3 .

O 20 40 60 80 100

Time (sec.)

 

Figure 5.24. Plot of a sinusoidal tracking experiment for slow SR (top), MR (middle),
and fast SR (bottom) output feedback controllers without hysteresis inversion- Shown
are the rotation angle 6 (solid) versus the reference 7' (dashed).

183

Slow SR without lnvserion

 

 

1

O—r (deg.)
C
i (Amps)

 

 

 

 

 

 

0 1; 2o 30
MR without lnvserion

 

 

 

 

 

 

 

 

 

 

 

0.2 - 1
3 a
'8 o E" o
‘5 3
GI, ._
—0.2~ A -1 A
0 50 100 0 50 100
Fast SR without lnvserion
0.2 ~ 1
’5 a
3 o ‘51 o
1’ s
Q', ._
-O.2 A — A
0 50 100 O 50 100
Time (sec.) Time (sec.)

Figure 5.25. Plot of a sinusoidal tracking experiment for slow SR (top), MR (middle), and
fast SR (bottom) output feedback controllers without inversion. Shown are the tracking
error 6 — 7‘ (left) and the current 2' (right).

184

 

 

 

 

 

Figure 5.26. Comparison of MR versus fast SR (top) and MR versus slow SR (bottom) for
the tacking experiment with reference given by (5.46) and controller without hysteresis

inversion.

R.M.S Error

R.M.S Error

MR vs. Fast SR Without Hyst. lnvs.

v

 

 

 

 

 

 

0.3— :2: S"; Ts=o.o{
0.2 )3 .
0 1 T =0 001 Ts=0.005 .’ ”#7 1*, 4
. - ______......-~ -0‘ rub—5", .s "
o—wo —-
o- Ts=0 01 Ts=0 01
—0.1 AA A #
1o“t 10'3
Fast Sampling Period 'I'f
MR vs. Slow SR Without Hyst. lnvs.
0.3- - * -SR ,7“
-O- - MR /
0'2 - / Tf=16-4
0.1 - , , - , A ’*‘ l
07:: —————— *:.:*T_._ _ _ -G —-—O
0 _ T “56-5 {K f T =59-5
Tf=5e-5
-0.1 -. A A A AA
10’4 10‘3 10‘2

 

 

 

 

 

 

 

 

 

 

Slow Sampling Period TS

185

 

in.

ii

Figure 5.27. Comparison of MR versus fast SR (top) and MR versus slow SR (bottom)
for the tacking experiment with reference given by (5.46) and controller with hysteresis

inversion.

MR vs. Fast SR With Hyst. lnvs.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ts=5e—4 a - At -SR
8 0‘ ’ I ’ ,, I ’ ‘ ~ * \ ‘—O-‘MR
utJ 0.3 it"‘A‘ ‘ \\\
‘0. ‘ fl‘ \ \ , g :34
E 0.25. l 0” Ts=1e—2
T =5e-2
0.2 e -
10’4 10’3 10‘2
Fast Sampling Period ‘I'f
MR vs. Slow SR With Hyst. lnvs.
0.4 -....-, --..,., .
_, -8R T 5 .4
h _ _ = e
.9: 0'35 0 MR h T—1e-3 ”A
“J i. ,. ’ -. F Tf=5e—3/
0). 03.) ,e”’e ‘*\ \ \ x ’ -1
2. . P4 ,-" Tf=1e—4\‘\\‘O\. ¢ /’ Tf=59-3
I 97T=5e—5 ”Rf: ~ ~ ago
0.25" I '4!
10'4 10'3 10‘2 10‘1

Slow Sampling Period Ts

186

 

 

 

 

MR With Hyst. lnvs.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.45 WT - -
1.. G,\ —-§-Ts=0.01
g 0.4. +T=o.oos‘
UJ \ s
U). 0'35. [\‘\
2' *—___ ‘\
a: 0.3- ——_\5“‘=-~.g-“
F“‘-¥
0.25 -. - -
10‘4 10-3 104
Fast Sampling Period 1”
MR Wlthout Hyst. lnvs.
0.15"‘ ' A... . - A?
é /
“J o.1~ ,’
(D. ,’
0.05. a N w- Ts=o.oosA
10'4 10-3 10""

Fast Sampling Period 1}

Figure 5.28. Plot illustrating the effect of increasing the ratio TS/Tf for multirate output

feedback with hysteresis inversion (top) and without hysteresis inversion (bottom). Both
plots show two data sets each; one for T 3 = 0.01 and another for T3 = 0.005.

 

187

5.4.4 PID versus PID with Hysteresis Inversion

Finally, we compare the PID with hysteresis inversion controller to PID without inversion.

First, consider a tracking problem with
r = 203in(0.1t)

In both cases we use a single-rate controller with sampling period of 0.013. We used
gains of kz- = 0, lop = 5, and kd = 0.1 for the PID case and kz- = 2, 19;; = 8, kd =
0.1, and kg = 0.0028 for the case of PID with hysteresis inversion. Figure 5.29 shows
that the controller with hysteresis inversion achieves stable tracking. For the PID case
without hysteresis inversion, the response is oscillatory and the control signal continuously
saturates. Although stabilization for the PID case can be achieved with faster sampling,
the hysteresis inversion provided added stability at this low sampling rate.

Next, we select a sampling period of 0.0013 and use a single-rate controller to track

the following angular reference

7“ = I53in(0.4t)

We tuned the PID gains to give a similar tracking performance for the two controllers.
The gains chosen were kz- = 4, kp = 6, and kd = 2 for the PID case and k,- = 2, kp = 5,
kd = 0.1, and kg = 0.014 for the case of PID with hysteresis inversion. The plots on the
left of Figure 5.30 show the results of the case with hysteresis inversion and the plots on
the right show those of the case with PID only. From the error plot we can see that both
controllers were able to reduce the steady-state tracking error close the resolution of the
encoder. Again the resolution is 0.0439 degrees. The plot of the current signal shows that
the PID control saturated throughout the experiment. Under the PID with hysteresis
inversion scheme, the control was noisy, but did not saturate. The R.M.S tracking error
for the PID scheme was 0.00914 and the error for the PID with inversion was 0.0319. The

R.M.S of the current was 0.05097 for the PID case and 0.03739 for the PID with inversion.

188

Thus, both controllers achieved good tracking results, but the inversion scheme used less

control effort.

 

 

 

 

 

SR with Hyst. lnv. SR without Hyst. lnv.

2O 20
3 3
m a:

E 0 B 0
co“ a5

—20 —20

0 10 20 0 10 20

 

e—r (deg.)

 

 

 

 

   

 

i (Amps)
i (Amps)

 

 

 

 

 

 

0 10 20 10
Time (sec.) Time (sec.)

20

Figure 5.29. Plot of experimental results comparing PID with hysteresis inversion (left)
with PID (right). Shown are the angle 6 (solid) versus the reference r (dashed) (top), the
tracking error 6 — 7" (middle), and current 2’ (bottom).

189

 

PID Control with Hyst. lnvs. PID Control

 

 

 

 

 

 

 

 

20 20
3 3
a) 0)
E o E 0
<15 <15
—20 —20
100 200 300 400 100 200 300 400
0.1
0.) Q)
a 2 o
(I; a': —o.05
—o

 

.1
100 200 300 400

 

i (Amps)

 

—1 —1
100 200 300 400 100 200 300 400
Time (sec.) Time (sec.)

Figure 5.30. Plot of experimental results comparing PID with hysteresis inversion (left)
with PID (right). Shown are the angle 6 (solid) versus the reference r (dashed) (top), the
tracking error 6 — r (middle), and current 2' (bottom).

190

 

 

5 .5 Conclusions

In this chapter we considered applications of multirate output feedback control to smart
material actuated systems. We have argued that for smart materials modeled as in Figure
5.1 the closed—loop fits the class of systems developed in Chapter 4 and thus we can
apply the stabilization results of that chapter. Furthermore, we presented experimental
results of a shape memory alloy actuated rotary joint under multirate sampled-data output
feedback control. We compared multirate and single-rate output feedback and studied the

effectiveness of the multirate scheme in an experimental application. ”

 

 

191

 

 

CHAPTER 6

Conclusions

This dissertation has been concerned with the design and performance tradeoffs that are
associated with the choice of gain in a high-gain observer. We have considered tradeoffs
between fast reconstruction of the system states, better rejection of modeling uncertainty,
and closed-loop stability versus amplification of measurement noise, large peaking am-
plitude, and computational cost in the discrete—time case. We have proposed several
high-gain observer designs and examined their effectiveness at relaxing these tradeoffs.
We have examined the extended Kalman filter as a time-varying high-gain observer.
Closed-loop stability results for nonlinear systems under EKF feedback have been limited
and we have been able to show that for the special normal form class of systems the origin
of the closed-loop system is asymptotically stable and the origin of the estimation error
is exponentially stable. Many previous theoretical results on extended Kalman filters
assumed the boundedness of the solution to the Riccati equation. From the two-time
scale nature of our problem, we were able to argue boundedness and positive definiteness
of the solution to the Riccati equation. We have shown that an important feature in the
use of the extended Kalman filters is the need for globally bounded time-varying terms
in the Riccati equation in order to guarantee stability. As the EKF has seen wide use

in a variety of applications, we expect that closed-loop stability results will be useful to

practicing engineers.

192

 

 

 

 

Through simulation we have compared the closed—loop performance of the time-varying
EKF versus the time-invariant HGO. We have seen that the EKF with time-varying gain
may provide closed-loop stability advantages over the fixed-gain observer. Furthermore,
the Riccati equation can be initialized to eliminate peaking during the initial transients.
Thus, for a particular system, a time-varying HGO may be able to achieve a better balance
between closed-loop stability and suppression of peaking than a fixed-gain observer. This
comes at the expense of added complexity of the extended Kalman filter when compared
to a fixed-gain high-gain observer.

In response to a common criticism of high-gain observers we have considered a gain
switching scheme to balance the tradeoff between state reconstruction speed and amplifi-
cation of measurement noise. Furthermore, we have proved that when the switched-gain
observer scheme is employed in output feedback, the closed-loop system remains bounded.
We have derived relationships that exhibit the tradeoff between fast reconstruction of the
states and rejection of modeling error versus the immunity to measurement noise for a
high-gain observer. Based on this we have designed a switched-gain version of the high-
gain observer in an attempt to relax the tradeoff between fast state reconstruction and
rejection of measurement noise. To handle the peaking in the estimates we have included a
switching delay in our scheme. We’ve also considered a nonlinear high-gain differentiator
with smooth switching based on dead zone nonlinearity. Numerical examples illustrate
the effectiveness of each design.

We have seen that using high-gain observers in sampled-data feedback control may
require higher sampling rates than the sampling rate that was chosen under a state feed—
back design. Thus, fast sampling guarantees closed-loop stability under the discrete-time
high-gain observer. On the other hand, operating at elevated processing rates is more
computationally demanding and may not be cost effective. This has motivated the study
of multirate sampled-data output feedback control using high-gain observers. This scheme

uses a control update rate fixed by the state feedback design and a faster measurement

193

 

 

 

sampling rate. With our multirate scheme we were able to show that it practically sta-
bilizes the origin of the closed-loop system. Further, with a sampled-data state feedback
design that exponentially stabilizes the origin we have shown that the closed-loop system
under multirate output feedback exponentially stabilizes the origin of the closed-loop sys-
tem. We also examined stabilization with respect to a compact positively invariant set.
This provided a theoretical framework for the application to smart material actuators
where hysteresis inversion compensation is used.

We have designed the control to allow for initialization strategies to handle peaking
in the initial transient. We have seen that the multirate scheme can be designed such
that controller initialization need only be preformed for one slow sample. In addition,
simulation examples show that the multirate scheme appears to be less sensitive than the
single-rate scheme to impulsive-like disturbances that may occur at an unknown time.

Using a shape memory alloy actuated rotary joint we have experimentally tested our
multirate output feedback control scheme. Using a controller based on hysteresis inversion
we have shown that our multirate scheme indeed enjoys closed-loop stability at lower
control update rates than the single-rate scheme. Further, at higher sampling rates we
have seen that the multirate scheme performs well in comparison to both “fast” and
“slow” single rate schemes in the presence of the encoder measurement noise.

The work described in this thesis provides a number of avenues for future research. The
switching schemes introduced here lead to alternate variable gain designs. For example,
an interesting design would explore the use of a nonlinear gain function that is smooth
and results in a continuum of gain values rather than switch between two gain values.
This would allow one to alter the noise and state reconstruction tradeoff depending on
the size of the output estimation error.

For time-varying gains and the extended Kalman filter, expanding the results beyond
the special normal form is complicated by a difficulty in rescaling the partitioned Riccati

equation such that it will be in the standard singularly perturbed form. In addition, con-

194

 

 

sidering a general nonlinear function for the internal dynamics also presents a challenge.
Clearly in that case, a first order approximation of the nonlinear function will only be
valid locally. However, it may be of interest to formulate a stability result that, while it
may be local in the direction of the estimation error of the internal dynamics, would be
regional in the direction of the estimation error of the external dynamics.

For applications to smart materials, an interesting experimental direction may be to
compute the hysteresis inversion and controller operations on an embedded controller
while a high-gain observation can be performed on a separate processor. This additional
computing power will allow more accurate models of the hysteresis and fast sampling
rates for the high-gain observer. Furthermore, processing and communication delays may
present interesting theoretical challenges. Embedded processing of hysteresis inversion is
already an area of active research in the smart microsystems laboratory.

Finally, throughout this thesis we have considered single-input single-output systems
(SISO). The extension to multi—input multi-output systems is fairly straightforward for

systems that have a well-defined vector relative degree. But, for simplicity of presentation

we have only shown the SISO case.

195

 

 

 

 

APPENDICES

 

 

196

 

APPENDIX A

Kronecker Matrices

Using the identity (2.38), the matrices in equations (2.39)—(2.40) are given by

M, = I a All + A118.) 1, Mg = (A12 —— PQCTC) <3) 1, M3 = P20TC a I
M4 = I @0112 — PQCTC)

 

 

 

 

 

 

 

 

 

 

 

 

and _ _ _
N11 0 0 -N12(t) 0 N140) 1
N1 — 0 N21 0 W20) = 0 -N22(t) N240)
_ o 0 N31 1 0 0 N326)
—N13(7r3) o 0 1 MM
N3(7T3) = 0 —N23(773) 0 =Nq " N2802T
_ o 0 N33(7r3) ( 0 _
_ , _ .
N15 + N16 0 0 N17
N4 = 0 N25 + N26 0 aNs = N27
5N36 5N35 N34 _ L o _
where

N11 = A+ a 1, N12 = (P3 — P;)CTC e 1, N13 : P3CTC a I,

197

N14 = I 69 (A12 — PzCTC), N15 2 I a A11, N16 = .405 a 1,
N17 = 13.421 a 1, N18 = D a 1,

N21 = 1 an A+, N22 = I a (P3 — P3+)0TC, N23 = 1 a P3CTC
N24 = (412 — P20TC) ® 1, N25 = A11 ® 1, N26 = I @405

N27 = I (8) 02421, 1V28 = I ® D,

 

N31: 1 a A+ + A+ a 1, N32 = 1a (133 — P;)CTC + (P3 — P3+)CTC® 1
N33 = 1(8) p3CTC, N34 = I ® A05 + A05 ® I, N35 2 DA21® I, (A.1) )

N36 2 I <8} DA21

To arrive at (2.42), consider the submatrices of N2. Using (2.29) and (2.32) and the

equalities A12 = CC and C = CCTC it can be shown that these matrices satisfy
”N12“ S kie—Alt/E, ”NM” 5 kze—AZt/E, ||N22|| S k3e_’\3t/E

||N24|| S k4e—A4t/E, ||N32|| g k5e_)‘5t/5

for some positive constants k,- and A, where, i = 1---5. Using this we can arrive at the

inequality (2.42).

 

198

 

APPENDIX B

Technical Lemma

The following result is a discrete-time analog to ([39],Ch. 5, Lemma 2.2)

Lemma 3.1 Consider equations (4.59), (4.63), and let (4.66)-{4.62) hold. Then, there

exist 5*, rig, and 0 < A3 < 1 such that for all 0 < 5 S 5*
||T(n,n0)|l g sag/\g—no ' (8.1)
Proof: From the definition of T(n, n0) (4.63) we have
T(n, n) = 0

T(n + 1,710) = fl(n)T(-n, no) + (6(a) + Af) A31—TIO

The solution to this equation can be written as

n—l . n—I .
T(n,, no) = Z T(n,i+1) (14(1) — Af) A3,_no + Z A?_i (jig) _ Af) Alf—"0
izno izno

199

 

 

 

 

Multiply through by /\_(n_n0), where 0 < A1 < A3 < 1, and let z(n,n0) :

3
A3_(n—nO)T(n,nO), we have
n—l ——(n+1~——n) ~ i—n
z(n,n0) = 2 A3 0 Z(71,i+ 1) (14(2) _Af) Af O
i=710
”—1 (n "0) ' 1 i n
— ' — n—z— ” . _ _ 0
+227; A3 A f (A(z) Af) A f (13.2)
2 0

Treating z(n, ”0) as a functional on [n0, n] with norm

sup Hz(n,T)H
nOSTSn

we have

z=F(z)+B=T(z)

We show that T is a contraction mapping over the set {2 : |le 5 SP}, for some p chosen

later on. we have

n—l _
—(n—n0) n—nO—l €C3(n — no) A1 n no
22720

 

2 "—1 i—n
e c p /\ 0
“Fig ,3 23631;) 3.2.3ka
z=n0

for some positive constants k1 and k2. Choosing

 

 

200

 

 

 

 

 

where 0 < I/ < 1 we have, for all 5 < 5*, ||T(z)|| 3 5p. Moreover,

"-1 i—n
8C3 A1 0
z — z < —— — —
||T(2) T(1)|| _ ,3}: (,3) Ha 21H

2:710

S €C3k2||22 - z1||< z/||22 __ Zill V6 < 5*
Hence, T is a contraction mapping, and the unique solution to (B2) satisfies

llz(n,no)ll : sp

Therefore,

n—n
||T(n.no)|| 3 m3 0

201

I
.A

 

 

APPENDIX C

Inverse Preisach Operator

C.1 Preisach Operator Inversion Algorithm [63]

 

Figure C.1. Preisach plane with L = 8.

202

 

 

The task of the inversion algorithm is to find 1‘), given A0 and And, such that
fid = 1"[7720]

where F is a Preisach operator with a piecewise uniform density function. Let the input
and output of I‘, corresponding to the memory curve A0, be '00 and “d0, respectively.

Algorithm (21d > ado)

Step 1. um) :2 ’00, 11.510) 2: “(10’ Am) 2: A0, m z: 0;

Step 2.
d(m) :2 min{d(()m), dam), dgm)}
yon—Fl) ;: v(7n) + d(m)
uEim+1) :2 F [v(m+1), [(071)]

where Am is the memory curve after {v(k)};n_1 is applied, and d(()m),d(m),dgm) are

determined in the following way:

-Let d(m) > 0 be such that ”(m) + dgm) equals the next discrete input level;

—Let dgm) > 0 be the minimum amount such that applying ”(m) + dgm) would eliminate

the next corner of the memory curve (See Figure C.1);

-Since the Preisach density is constant within each discretization cell, for d <
min {(1971), dgm)}, we have

P [547") + d, AW] — r (Am), AW] = aém)d2 + agm)d

where 0(m) ,agm) Z 0 can be computed from the density function, and the square term is
(m) > 0

due to the contribution from the triangular region inside the diagonal cell. Let do

203

 

 

 

 

 

be the solution to

2 m
lg, >) ..g 55, >

17d _ F [v(m)a A(m)] = agm) (

If dim) = (18"), go to Step 3; otherwise let m z: m + 1 and go to Step 2;
Step 3. i) :2 'v(m+1) and stop.

The algorithm for ad < ”d0 is analogous to the above.

C.2 Inversion Error [63]

Let V be a nonsingular, nonnegative measure with density u. For a discretization scheme
of level L, let up be a piecewise uniform approximation to V obtained as described in
Section 5.2.2. Denote I}; as the Preisach operator that corresponds to the measure )1.

Let
—1
v = Pup ['ud. A0]

where F 17191 is the exact inverse of Pup. Consider

fl'(l : Full}, )‘Ol

and define the inversion error as ez- : rid — ud. We have the following result the proof of

which can be found in [63].

Theorem C.1 Let ,u g [i for some constant )1. Denote the integral ofu over discretiza-
tion celli as , 1 g i 3 NC, where Nc is the number of cells. Denote by 1/2' the identified

Preisach weighting mass for cell i. Assume that the relative error in identification is 61,
l 0| 85.32

V,'—l/-
i.e., —lO—‘— 3 61, 1 S i S NC. Then ”Gynlloo g 61C3+TQ, where C5 is the saturation

V-
’1,
output of FV, and 30 is as defined in Section 5.2.2.

204

 

 

 

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