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11m Woes-p.14

OUTPUT FEEDBACK SAMPLED-DATA
CONTROL OF NONLINEAR SYSTEMS USING
HIGH-GAIN OBSERVERS '

By

AHMED MOHAMED DABROOM

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

2000

ABSTRACT

OUTPUT FEEDBACK SAMPLED-DATA
CONTROL OF NONLINEAR SYSTEMS USING
HIGH-GAIN OBSERVERS

By

AHMED MOHAMED DABROOM

This thesis presents a separation principle for sampled-data control of a class of
nonlinear systems. The basic ingredients of this technique are: a continuous-time
high-gain observer that robustly estimates the derivatives of the output, a global
bounded state feedback control, and a discrete-time implementation of the high-gain
observer. For sufficiently small sampling period and sufficiently high observer gain,
the sampled-data controller recovers the performance of the continuous-time state
feedback controller. The high-gain observer is discretized using different discretiza—
tion methods to achieve the best discretization algOrithm and parameters choice.

Analysis of the discretized high-gain observer as a numerical difl'erentiator pro-
vides answers to a number of important questions such as: what is the best dis-
cretization method? what are the best choices for the observer parameters and
how do they relate to the sampling period? and what is the effect of the order of
the observer on the estimation of the output derivatives. This is done for noisy as
well as noise-free measurements. We also show how other numerical differentiators

I

are special cases of the high-gain observer discretized by the bilinear discretization

method. The closed-loop analysis shows that the 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. Performance
recovery is shown in two steps. First we show boundedness of trajectories which
come arbitrarily close to the desired equilibrium point as time progress. Second, we
Show convergence to the equilibrium point.

Finally, we experimentally test the use of the discretized high-gain observer in
controlling an electromechanical system of (the pendubot). We Show how saturation
is used to overcome peaking and the effect of increasing the observer gain on the
steady-state error. The experimental results confirm our analysis and show that
the discretized high-gain observer outperforms the Euler formula as a means for

calculating velocities from optical encoder.

To the memory of my father, Mohamed,
and to the great women in my life

my mother, Saidah, and my wife, Ferdous.

iv

ACKNOWLEDGMENTS

In the name of Allah, the most merciful and the most beneficent

All praise and thanks are due to Allah, Lord of the Universe, for His merciful and
divine direction throughout my study. I am indebted to Professor Hassan Khalil,
my advisor and committee chairman, for his valuable time, assistance, and encour-
agement. His kind consideration and understanding have been an incentive for the
completion of this dissertation. Dr. Khalil taught and showed me how I can be
professional and well organized in my thoughts and work.

Sincere appreciation and gratitude are extended to the other members of my
guidance committee: Professors Fathi Salam, Sheldon Newhouse, Steven Shaw , and
Robert Schlueter, for their contributions, advice, and constructive comments to the
study. The author would like to thank Jeddah College of Technology in Jeddah,
Saudi Arabia, for supporting this study.

I would like to extend my deepest thank to my sisters: Fatima, Zahra, Maryam,
and Halima, my brother Fuad, my father in law, and my mother in law for their
support and prayers. Last but not least my greatest thanks are for my most valuable
treasure in the world my children Weaam, Mohammed, and Albara for their patience,

love, and for learning sacrifice at an early eage.

TABLE OF CONTENTS

LIST OF TABLES ix
LIST OF FIGURES x
1 INTRODUCTION 1
1.1 The Class of systems and Observer Design ............... 2
1.2 The Peaking Phenomenon and Bounded Control ........... 7
1.3 Separation Principle Results ....................... 8
1.4 Other High-Gain Observer Designs ................... 9
1.5 Sampled-Data Control .......................... 11
1.6 Overview Of The Thesis ......................... l2

2 Discrete-Time Implementation of High-Gain Observers for Numer-

ical Differentiation 19
2.1 Introduction ................................ 19
2.2 High-Gain Observers ........................... 21
2.3 Digital Implementation .......................... 26
2.3.1 Discretization Methods ...................... 27
2.3.2 Comparison of the Five Methods ................ 29
2.4 Bilinear Implementation ......................... 38
2.5 Numerical Differentiators ......................... 46
2.5.1 Comparison with the Bilinear High-Gain Observer ....... 49
2.5.2 Spline Method Initialization of The High-Gain Observer . . . 52
2.6 Further Considerations . . ; ....................... 52
2.6.1 Modified Observers ........................ 55
2.6.2 Additional Examples ....................... 58
2.7 Conclusions ................................ 67

3 Output Feedback Sampled-Data Control of Nonlinear Systems Us-
ing High-Gain Observers 77

vi

3.1 Introduction ................................ 77

3.2 Continuous-time Design ......................... 78
3.2.1 The Class of Nonlinear Systems ................. 78
3.2.2 Partial State Feedback Control ................. 79
3.2.3 High-Gain Observer (HGO) ................... 80

3.3 Sampled-Data Control .......................... 81
3.3.1 Discretization of the HGO .................... 82
3.3.2 Discrete-Time Closed-LOOp Model ................ 84
3.3.3 The Discretetime Singularly Perturbed Form ......... 87

3.4 Performance Recovery .......................... 89
3.4.1 Boundedness and Ultimate Boundedness ............ 91
3.4.2 Exponential Stability of The Origin and 'Ii'ajectory ConvergencelOl

3.5 Example on the Recovery of the Region of Attraction ......... 109

3.6 Conclusion ................................. 111

Application to the Pendubot 112

4.1 Introduction ................................ 112

4.2 What Is The Pendubot? ......................... 113

4.3 Mathematical Model ........................... 115

4.4 The Equilibrium Manifold ........................ 117

4.5 Controlling The Pendubot ........................ 118

4.6 Hardware Description ........................... 122

4.7 Experimental Results .......................... 122
4.7.1 Confirmation of the Open-Loop Study ............. 126
4.7.2 Confirmation of the Closed-LOOp Study ............. 135

4.8 Conclusion ................................. 151

Conclusions and Future Work 155

5.1 Conclusions ................................ 155

5.2 Future Work ................................ 156
5.2.1 Extending the Scope of the Sampled-Data Observer-Based

Control .............................. 156
5.2.2 Sampled-Data Control ...................... 157
Appendix 159

A. 1 Developing The Singularly Perturbed Form For The Forward Differ-

ence Method ................................ 159

vii

A.2 The Transformation Steps To The Singularly perturbed Form For The

Other Discretization methods ...................... 167
A.2.1 Using Backward Difference Method ............... 167
A22 Using Bilinear Transformation .................. 169
BIBLIOGRAPHY 172

viii

LIST OF TABLES

2.1 Guidelines for choosing T/e for different w,/wm and SNR. .....
3.1 Coefficient of the discrete time implementation of the linear HGO.
4.1 The values of a = T/e for the different discretization methods.

1.1
1.2

2.1
2.2

2.3
2.4

2.5

2.6

2.7

2.8

2.9

LIST OF FIGURES

The closed-loop system a) under state feedback h) under output feedback 10
The closed-loop system a) under state feedback b) under output feed-
back c) sampled-data output feedback ................. 13
The output signal and its first and second derivatives .......... 30
The maximum steady state absolute error in estimating g and ii for
all five discretization methods for the full-order observer; T = 0.01 . . 32
The error in y and 3'] for all five methods;T = 0.01 and e = 0.00667 . . 33
Error in 3') for the bilinear and FOH methods for the full, reduced, and
extended observers ............................. 35
The maximum steady state absolute error in estimating 3] and ii for all
five discretization methods for the full-order observer in the presence

of noise; T = 0.01 and a = 0.0316 .................... 36
Calculating g and i) in the presence of noise; T = 0.01, 6 = 0.033, and
a = 0.0316 ................................. 37
Effect of the ratio of g on the full-order high-gain observer with real
poles at s = —1/6 and T=0.01 ...................... 41
The transient response for different choices of poles: real and complex,
T=0.01 ................................... 42
Comparing the maximum absolute steady-state error for the full, re-
duced and extended-order observers with noise; T = 0.01, a = 0.1 . . 43

2.10 Comparing the signal to noise ratio with T/e and root mean square

error r.m.s. for the full-order observer for T = 0.01 .......... 44

2.11 Comparing full-order observer and Spline interpolant methods; T =

0.001, e = 0.0154, 0'2 = 0.001, and W = 70 ................ 51

2.12 The effect of initializing the high-gain observer by spline interpolant

methods; T/e = 10, T = 0.001, W = 2, and N = 2. .......... 53

2.13 The effect of initializing the high-gain observer by spline interpolant

methods in the presence of noise; T/e = 0.065, T = 0.001, a = 0.01
W=40,N=2anda=0.0316 ..................... 54

2.14 The comparison between the regular, the modified, and the extended

HGO where T = 0.01 ........................... 57
2.15 The output signal and its first and second derivatives .......... 59
2.16 The maximum steady state absolute error in estimating i; and ii for

all five discretization methods for the full-order observer; T = 0.01 . . 60
2.17 The error in g and i) for all five methods;T = 0.01 and 6 = 0.00667 . . 61
2.18 Error in i) for the bilinear and FOH methods for the full, reduced, and

extended observers ............................. 62
2.19 The maximum steady state absolute error in estimating y' and i] for all

five discretization methods for the full-order observer in the presence

of noise; T = 0.01 and a = 0.0316 .................... 63
2.20 Calculating g) and 3'] in the presence of noise; T = 0.1, T/e = 0.28, and

a = 0.0707 ................................. 64
2.21 Effect of the ratio of g on the full-order high-gain observer with real

poles at s = —1/e and T=0.1 ...................... 65
2.22 The transient response for different choices of poles: real and complex,

T=0.1 ................................... 66
2.23 The output Signal and its first and second derivatives .......... 68

2.24 The maximum steady state absolute error in estimating 3'] and ii for
all five discretization methods for the full-order observer; T = 0.1 . . 69

2.25 The error in g) and 3'] for all five methods;T = 0.1 and e = 0.0667 . . . 70
2.26 Error in ii for the bilinear and FOH methods for the full, reduced, and
extended Observers ............................. 71

2.27 The maximum steady state absolute error in estimating 3} and ii for all
five discretization methods for the full-order observer in the presence

of noise; T = 0.1 and a = 0.0707 ..................... 72
2.28 Calculating g] and i] in the presence of noise; T = 0.01, T/e = 0.4, and

0‘ = 0.1414 ................................. 73
2.29 Effect of the ratio of g on the full-order high-gain observer with real

poles at s = —1/e and T=0.1 ...................... 74
2.30 The transient response for different choices of poles: real and complex,

T=0.l ................................... 75
3.1 Recovery of region of attraction: e‘ = 0 (solid), 6’ = 0.007 (dotted),

e: = 0.057 (dashed), and e* = 0.082 (dash—dotted) ........... 110
4.1 Front and side view drawing of the pendubot .............. 114

4.2 The Pendubot model parameters are shown on a pendubot schematic. 116
4.3 The pendubot arm at a: controllable position, b: uncontrollable position119

4.4 Pictorial description of the Pendubot’s interface with its controller . .
4.5 The output signals for both links and their derivatives, T = 0.003 and

a = 2.5 ..................................
4.6 Using the bilinear transformation to calculate the steady state track-
ing error e,, = 2:1 + 2:3 - 1r/2, where T = 0.003sec. and a =
1.0, 1.5, 2.0, 2.5 respectively. ......................

4.7 Using the backward difference method to calculate the steady state
tracking error cu, where T = 0.003sec. and a = 1.2, 1.5, 2.0, 2.5
reapectively. ................................

4.8 Using the forward difference method to calculate the steady state
tracking error cu, where T = 0.003sec. and a = 0.8, 0.9, 1.0, 1.1
respectively. ................................

4.9 The effect of different choices of poles real and complex on the tran-
sient response, T = 0.005 and a = 3.0 discretized by the bilinear
transformation. ..............................

4.10 Tracking error e and control signal for the full, reduced and extended
order HGO using the bilinear transformation, T = 0.01 and a = 2.1.

4.11 Effect of increasing the ratio a = T/e on the full order HGO with real
poles at s = —1/6, T = 0.01 and a = 2.1, 3.0, 5.0, ,6.0 respectively.

4.12 The transient of estimating speed of the first link as 0: increases while
T = 0.01 ..................................

4.13 The control during the transient period as (1 increases while T = 0.01

4.14 The transient speed of link one while T decreases ...........

4.15 The control during the transient period as T decreases ........

4.16 Peaking in the transient of estimating speed of the first link as 0
increases while T = 0.01 .........................

4.17 The effect of the peaking on the control during the transient period
as or increases while T = 0.01 ......................

4.18 Peaking in the transient speed of link one while T decreases .....

4.19 The effect of the peaking on the control during the transient period

as T decreases ...............................
4.20 The sampling time T = 0.003 is fixed while a is increased ......
4.21 Fixing a = 3.0 and decreasing the sampling time T ..........

4.22 Tracking error at steady-state using linear and nonlinear HGO. . . . .
4.23 Ti-acking error at the steady-state using the linear and nonlinear in
another run ................................

xii

123

130

131

132

136

138
139
140
141

145
147
148
149

4.24 Comparison of the different discretization methods of the full order
(FOHG) and reduced order (ROHG) with the Euler formula with and

without averaging ............................
4.25 Comparing the performance of the reduced order HGO discretized by
bilinear method for a = 2 and T = 0.04 and a = 6.5 and T = 0.06
with the Euler method when T = 0.04 sec. ..............

xiii

CHAPTER 1

INTRODUCTION

The use of high-gain observers in continuous-time has evolved as an important tech-
nique for the design of output feedback control of nonlinear systems. The basic
ingredients of this technique are

1. a high-gain observer that robustly estimates the derivatives of the output;

2. a global bounded state feedback control, usually obtained by saturating a con-
tinuous state feedback function outside a compact region of interest, that meets
the design objectives. The global boundedness of control protects the state of
the plant from peaking when the high-gain observer estimates are used instead

of the true states.

The technique was first introduced by Esfandiari and Khalil [16] and since then
has been used by many other researchers. It was used in [16] and [36] to achieve
stabilization and semiglobal stabilization of fully-linearizable systems, in [32, 42, 43]
to design robust servomechanisms for nonlinear systems, in [49] to design variable
structure control, and in [31] to control the speed of induction motors. Teel and
Praly [62, 63] and Lin and Saberi [40] used it in a few papers to achieve semiglobal
stabilization. Jankovic [29] used it in adaptive control problem. Isidori [27] used it

to unify his pioneering work on servomechanisms [28] with Khalil’s work [32]. Jiang,

1

Hill and Guo [30] used a reduced-order high-gain observer to achieve semiglobal
stabilization for a nonlinear benchmark problem.

In most of these papers the controller is designed in two steps. First, a globally
bounded state feedback control is designed to meet the design objective. Second, a
high-gain observer, designed to be fast enough, recovers the performance achieved
under state feedback. This recovery is shown using asymptotic analysis of a singularly
perturbed closed-loop system. The combination of these two steps of [design allows
for a separation approach. This separation approach is used in most of the papers
that utilize high-gain observers. It is proved in a generic form in the work of Teel
and Praly [62], where it is shown that global stabilizability by state feedback and
uniform observability imply semiglobal stabilizability by output feedback. A more

comprehensive separation principle is proved by Atassi and Khalil [8].

1.1 The Class of systems and Observer Design

We consider a multivariable nonlinear system represented by

a: = Az+B¢(x,z,u) (1.1)
2 = 1/)(:r,z,u) (1.2)
y = Ca: (1.3)
C = 902,2) (1.4)

where u e u g R“ is the control input, y E y Q R” and C E R’ are measured
outputs, and a: e X C; R" and z 6 Z <_: R‘ constitute the state vector. The r x 1'

matrix A, the r x 1 matrix B, and p x r matrix 0, given by

A = block diag[A1, . . .,A,,]

B = block diag[B1, . . . ,BP]

0 = block diag [01, . . . , Cp]

 

 

 

 

0 1 0 0
0 0 l 0 0

As: ;B.=
0 0 1 0
0 . 0 l

-r.-xr.- - -r.-xl
Ci=[1 0 0]
1X11

where 1 _<_ i S p and r = r1+. . .+r,,, represent p chains of integrators. The functions

4:, II) and 9 satisfy the following assumption.

Assumption 1.1 The functions if and I]! are locally Lipsehitz and 9 is continuous
over the domain of interest. In addition, ¢(0, 0, 0) = 0, ¢(0, 0, 0) = 0 and q(0, 0) = 0.

This assumption guarantees that the origin is an equilibrium point of the open-100p
system. The main source of the system (l.1)-(1.4) is the normal form of a nonlinear
system having a vector relative degree (11,. . . , r,) [16]. It is well known that if the

nonlinear system

it = f(x)+g(x)u

y = h(x)

has a vector relative degree (r1, . . . , r,), then it can transformed into the form

6 = A:+B[fl(:,z)+gl(:.z)u1
2 = f2(£’z’u)
y = 06

In this case gl(°, ) is nonsingular in the domain of interest, y is the only measured
output, and (1.4) is dropped. Another source where (1.4) is relevant arises when
the dynamics are extended by augmenting a series of integrators at the input side
[33, 62, 66]. Khalil in [33] considers a single-input single-output system modeled by

the 11‘“ — order differential equation

y" = M) + so(-)u‘"""’

where [1 is the input, y is the output, fa and 9,, are functions of
y, y“), . . . , y‘”"”, p, .. ., p(”""1). Augmenting (n — p) integrators at the in-
put side, denoting their states by z; = 11““), setting u = [sh-P) as the control input
of the augmented system, and taking x; = y(“1), results in a system of the form
(1.l)-(1.4) with r = n. In this case all the components of z are measured ; hence
O(:r,z) = z in (1.4). Another example of the use of extended dynamics can be
found in [62]. Reference [62] considers a singleinput single-output nonlinear system
where complete uniform observability guarantees that the state x can be expressed
as x = h(y, ...,y("v),p, . . . ,p(”')) where p is the input, 3] is the output, and h()
is a known function. Furthermore, MW“) = a(x,u,...,u("")) where a is known
function. The dynamics are extended by adding 1., =max{n,,, mu} integrators at the
input side. Taking 1:; = y“), for 1 5 i S ny, z.- = p“), for 1 _<_ i S in, and u = ”(l-+1),

the system can be represented as

:i: = A$+Ba(h(z,z),z)
2 = Aoz+Bou

y=Cz

where (A, B, C) and (A0, 3,) represent chains of n, and In integrators, respectively.
In this case ()5 is independent of u and all the components of z are measured; hence
9(x,z) = z in (1.4).

The model (1.1)-(l.4) may also arise in models of mechanical and electrome-
chanical systems where diSplacement variables are measured while their derivatives
(velocities, accelerations, etc.) are not measured. Examples of such models can be
found in [31, 24, 41, 25, 64, 30]. A model of induction motor [31] can be represented
in the form (1.1)-(1.4) with :1: = [6, 3, (I? where 6 = 9—9"; is the rotor position error,
and z constitutes the rotor flux and stator current. The measured variables y and
C are the rotor position error and stator current, respectively. Examples of models
that can be put in the normal form are the models given in [24] and [41] for the
inverted pendulum-on-a-cart system. These models, taking the cart displacement as
the measured output, have relative degree two but are non-minimum phase. In [25]
and [64], the models given of the ball and beam system fit in the form of (1.1)-(1.4).
These systems can not be represented in the normal form because, taking the ball’s
position as one of the measured outputs, the relative degree is not well defined. A
last example of systems fitting the model (l.1)-(1.4) is the model of the benchmark
rotational/ translational actuator given in [30] where the system has well defined rel-
ative degree with respect to the cart’s position but only locally. The design of the
globally stabilizing state feedback controller of [30] does not transform the system
into the normal form.

The bounded state feedback control u takes the forml

u = 7($,C) (1-5)

 

1 Reference [8] deals with the more general case of dynamic state feedback
controllers.

and is implemented by
u = 7(5r, C) (1-6)

where the state estimate :3 is generated by the high-gain observer (for demonstration
we take p = 1)
ft = Ai+B¢o(§:,(,u)+H(y— 05:) (1.7)

where d), is a known nominal model of 4). The estimation error e = a: — 5: satisfies

the differential equation

.2 = (A — HO) e+ BA(2:, e)

where A(z, e) is due to uncertainty in the state equation and the imperfect cancel-
lation of the nonlinearities due to the use of states estimates. The observer gain H
is designed to render (A - H C') a Hurwitz matrix, as in linear observer theory. In
the presence of A, H needs to stabilize (A - H C) while rejecting the effect of the
perturbation A. Rejection of A is approximately achieved if the transfer fimction
(SI — A+ H C)"lB approaches zero asymptotically as a design parameter 6 tends to
zero. This can be achieved via the singular perturbation approach described in [16].
For the case where (A, B, 0) represents a chain of integrators, the observer gain is

given by
011 02 0r T
e 62 .. . . . . e"

where e is a positive constant to be specified and the positive constants a.- are chosen

such that the roots of

s'+als"l+...+a,_ls+a,=0

are in the Open left-half plane.

For the purpose of analysis, the observer dynamics are replaced by the equivalent

dynamics of the scaled estimation error

Hence, 5: = a: — D(e)n where
D(e) = diag [er-1, . . . , IL"
The closed-loop system is represented by

i- : Ax + B¢(a:, 2, 7(2: - 0(6)”, 0)
2 = 1,0(3, 2,7(1: - 17(5)": 0)

57.7 = A077 + 536(3) 2: D(5)77)

where

5(3t 2, D(€)n) = 441', 2, 7(53'. 0) - ¢e(5=. C, 70%. O)

and

115-4, = D“(e) [A — HC] D(e)

is an r x r Hurwitz matrix.

1.2 The Peaking Phenomenon and Bounded Con-

trol

The high-gain observer design places the observer poles far in the left-half plane. In
particular, they are located at 0(1/6) values that approach infinity as e —+ O. This is

a typical feature of all high-gain observer designs which attempt to reduce the error
transfer function to zero asymptotically. This feature can cause an impulsive like
behaviour known as the peaking phenomena. Designing the state feedback control
u to be globally bounded eliminates the undesirable features of peaking as shown in
[16]. During a short period of time, the excessively large overshot due to peaking
decays to an 0(6) value. This short period of time tends to zero as e —> 0. During
the same interval the control saturates and the state of the plant 2: remains close to
its initial values. After that period, the controller starts to function in a normal way
as if the initial conditions for the fast error equation were of order 0(1). The only
place where peaking could be observed is in the estimates a“: (observer states), which
are internal variables of the controller. The physical variables of the plant a: and u

will be free of peaking.

1.3 Separation Principle Results

It is shown in [8] that output feedback controller (1.6) recovers the performance of
the state feedback controller (1.5) for sufficiently small 6. The performance recovery
manifests itself in three points. First, the origin (a: = 0, z = 0, 1% = 0) of the closed-
lOOp system under output feedback is asymptotically stable. Second, the output
feedback controller recovers the region of attraction of the state feedback controller
in the sense that if ‘R is the region of attraction under state feedback, then for any
compact set S in the interior of ’R and any compact set Q g R', the set S x Q
is included in the region of attraction under output feedback control. Third, the
trajectory of (z, z) under output feedback approaches the trajectory under state
feedback as e -) 0.

Performance recovery is shown in three steps. First, boundedness of trajectories

starting in the specified compact set is established by regulating the parameter c

such that state estimation is fast enough. The global boundedness of the control
function is of paramount importance at this stage. Then, these trajectories are
shown to be arbitrarily close to the origin after a finite time interval; thus a prOperty
of ultimate boundedness is established. Finally, local asymptotic stability of the
origin is argued in three cases: the case where perfect knowledge of the system’s
nonlinearity is available (41,, = 42), the case where the origin under state feedback
control is exponentially stable, and the case where the origin under state feedback
control is asymptotically but not exponentially stable combined with an imperfect
knowledge of the system’s nonlinearity. In the last case, certain conditions were
imposed on the growth of the modeling error due to the imperfect knowledge of the
system’s nonlinearity.

In [6] Atassi and Khalil proved the separation principle for a more general case
where the state feedback control renders a certain compact set positively, invari-
ant and asymptotically attractive. This more general result allows the separation
principle to be applied to a number of control tasks beyond the stabilization of an
equilibrium point. Examples include finite time convergence to a set [14], ultimate

boundedness [63], servomechanisms [32, 42, 43, 27, 35], and adaptive control [33, 4, 5].

1.4 Other High-Gain Observer Designs

Building a controller to stabilize any system needs good measurements of the states
or their estimates. If an observer is used in the implementation, no guaranteed
robustness prOperties hold. Doyle and Stein 1979 [13] showed that the robustness
guaranteed by state feedback can be recovered by the use of a robust observer as in

the following example.

Example 1.1 Consider an nth order linear system that is observable, controllable

and has all the zeros in the left half plane (minimum phase). The closed-loop system

9

 

 

 

 

 

 

 

 

 

 

~ B (SI-A)" C ———>

 

 

 

 

 

 

 

 

 

 

 

(a)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I
I
=0 u ll
rfi X"B (sI-A)'Jr C y=

I _ I

I -u ' ------------------- I
I I
I F B I
I , I
I I
I I
I I
: ,, + + :
: (SI-A) ‘ + ‘ H ‘ I
I - I
I I
l I
I I
I C I
I I
”TTTITTCM----Wm--M_; — y. - notion --- Tc? ,7 -----------------

(I!)

Figure 1.1. The closed-loop system a) under state feedback b) under output feedback

in Figure (l.1a) uses full-state feedback, while the one in Figure (l.1b) uses output
feedback (states estimates). Examining robustness of the two closed-100p systems
at the breaking point x, the 100p transfer functions from the control signal u' to
the control signal u are generally different in the two implementations. They are

identical if the observer dynamics satisfy

H [I + C (s! —- A)" H]‘1 = B [C (31 — A)"1 13]-1 (1.8)

10

where A, B and C are plant matrices and H is the observer gain. Satisfaction of
(1.8) offers a way to adjust observers so that full-state loop transfer characteristics
are recovered at point x. In particular, if the observer gain is parameterized as a

function of a scalar variable q, H (q), such that

H
—7:Z)-—+BW as q—->oo

for any nonsingular matrix W, then (1.8) will be satisfied asymptotically as q —> oo.

Atassi and Khalil [7] reviewed various techniques for the design of high-gain ob-
servers and classified them into three groups. First, pole-placement algorithms which
lead to either a two-time scale structure as in [16] or a multiple time-scale structure
as in [54]. Second, Riccati equation-based algorithms which lead to either an H;
Riccati equation as in [13] and [53, Section 4.4.1] or to an H00 Riccati equation as
in [50] and [53, Section 4.4.2]. Third, Lyapunov equation-based algorithm as in [23].
They showed that separation result similar to those of [8] can be obtained for any
one of the other high-gain observer designs provided the state feedback control is
globally bounded.

1.5 Sampled-Data Control

Due to the advancement in computer technology, the controller is typically imple-
mented using digital computers. Moreover, in most of today’s applications, measure-
ments are available only at sampling points. In recent years there has been some
progress in the analysis and design of sampled-data control of nonlinear systems.
The closed-100p system consists of a continuous—time plant, a digital controller, a
digital-to-analog converter (D/A), and an analog-to-digital (A/ D) converter. In
sampled-data systems, the plant output y(t) is sampled as y(kT) where T and k

11

are the sampling period and sample number, respectively. The output y(kT) is used
to calculate the state estimate :i:(kT) via a high-gain observer. The observer is im-
plemented in discrete-time and u(k) is calculated using a microcontroller or a DSP
board. Then, u(kT) is fed to the continuous-time plant through a zero-order hold
device.

The sampled-data control design is carried out in three steps; see Figure 1.2.
First the control is designed assuming that all states are available for measurements.
Second, a high-gain observer is designed to estimate the states using the measured
outputs. Finally, the observer and the controller are implemented in discrete-time.

Most of the literature on sampled-data control of nonlinear systems assumes state
feedback. The work of Monaco and Normand-Cyrot [45] enables a first parallel
between the continuous-time and discrete-time state space representations. Some
pr0perties of the state equations are not preserved under sampling as studied in [57]
for certain accessibility pr0perties. Also it is shown in [12] that certain properties, like
observer error linearizability [37], are not inherited from the underlying continuous-

time system.

1.6 Overview Of The Thesis

Up to this point, all studies of the closed-loop nonlinear system using high—gain
observers have been limited to continuous-time analysis, despite the fact that such
a system is a combination of a continuous-time system (plant) and a discrete-time
system (high-gain observer and controller). The main goal of this thesis is to analyze
the performance of the closed-100p system when the discrete-time implementation
of the observer and controller is taken into consideration. This will complicate the
analysis compared with the continuous-time case.

We are going to investigate the sampled-data output feedback controller in three

12

 

 

x(l) ”l" rlll-xlll
r—b Controller > Plant >

 

 

 

 

 

 

 

 

stale x(l)

 

 

 

e: Closed-loop for state feedback eontlnuous-tlme control system

”(0 YIIl'xtll)

 

 

X‘ll)
—u

 

 

 

 

 

 

 

 

 

 

 

Observer
state estimates x “(o (”60’

 

 

 

 

 

b:Closed-loopforoulpurfeedbeckcomlnuoustlrrteeomroleystem

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ulkll U'III Mat“)
”IIsWCI,“—uzou s Plant >
mm
m ‘ VK‘
state estimtes m1) (DHGO) ’0‘” M T m

 

c:0loeed-|ooplorourpurteedbeelreernpled-dencomrolsystem

Figure 1.2. The closed-100p system a) under state feedback h) under output feedback
c) sampled-data output feedback

13

steps. First, we study the discretization of the high—gain observer by looking at
the problem as a numerical differentiator design problem (Open-100p system) and
investigate many discretization methods and their effect on reducing the approxima-
tion error. Second, we study the closed-100p system when we apply discrete-time
high—gain observer. Third, we implement the discrete-time via simulation and an
experimental setup.

The fact that the high-gain observer, especially its linear version, is an approxi-
mate differentiator can be seen by examining the transfer function from y to 5?, given
by

G(s) = (.91 — A + HC)‘1H

The following two transfer functions give G (s) for second and third-order systems,

respectively, in the limit when the observer gain 1/6 —) oo.

 

. . 1 6801 + (22 1
111%G(s) = 1133 2 2 =
H «5 es +ales+a2 023 3
(116282 + 0268 + (13 1
l
' = ° — 2 =
[1530(3) [51% d(es) cs (12 + (13.9 s
(1382 I 82

 

 

 

 

where d(es) = 6333 + 016282 + (1268 + 03. The linear high-gain observer can be used
to approximate the derivatives of any smooth signal y(t) by representing the signal

as the output of a normal form driven by the nth derivative of the signal, i.e.,

i: = Az+By(”)(t),

y=C:r:

For any bounded y‘”) (t), the singular perturbation argument used in [16] shows that

14

the estimation error will be of order 0(6) after a short transient period.
lrnplementing the linear high-gain observer in discrete time amounts to numerical
differentiation. In Chapter 2 we concentrate on this issue. There has been a lot of
literature on numerical differentiation, both in digital signal processing and in nu-
merical analysis. We compare the discrete-time linear high-gain observer with them.
The study explores the degrees of freedom we have in choosing the discretization
method, the relationship between the observer bandwidth (observer gain) Us and
the sampling frequency f, = l/T, pole locations, and the dimension of the observer.
The study covers noise-free as well as noisy measurements. The noise is modeled
as white Gaussian noise. We investigate five different discretization methods: for-
ward difference, backward difference, zero-order-hold, first-order-hold and bilinear
transformation methods. For the bilinear transformation method, we investigate the
design of the observer parameters and the choice of the observer order (full, reduced
and extended-order observer). We explore the freedom of choosing real poles verses
complex poles distributed in a butterworth pattern and the effect of that choice on
the transient and steady state errors. Phrthermore, using the extended-order ob-
server gives less error than the other observers. Comparison of the bilinear high-gain
observer with several existing numerical differentiators from the literature shows two
interesting findings. First, most of those differentiators are special cases of the bilin-
ear reduced-order high-gain observer for certain choices of the observer parameters.
Exploiting the freedom to choose these parameters, the bilinear high-gain observer
outperforms those numerical differentiators, especially when measurement noise is
taken into consideration. Second, the bilinear high-gain observer is compared to
a causal spline interpolant method. It is shown that the high-gain observer com-
pares favorably even though it uses less computation time and memory requirements
than the spline method. The chapter ends with a novel idea of using the spline
interpolant method to initialize the high-gain observer, which significantly improves

15

the observer’s transient behavior and eliminate the peaking phenomena. Additional
examples are presented to confirm the chapter’s findings.

After we know how to discretize the high-gain observer we proceed to the anal-
ysis of the interconnected hybrid system. There is no study of the behavior of the
closed-100p system in the presence of the digitally implemented high-gain observer.
References [10] and [20] studied the problem where the high-gain observer was im-
plemented digitally in closed-loop control. However, their analysis concentrated only
on studying the error equation without using a complete model of the closed-loop
system. In Chapter 3 we study the coupled closed loop-system (the plant dynamics
and the error dynamics). Our study of the discrete-time system is guided by the
corresponding study of the continuous-time case. The main tool in the continuous-
time case [8, 16] is the representation of the system in a singularly perturbed form
that shows the slow dynamics (the plant under control) and the fast dynamics (the
observer). The main tool in our analysis is to deve10p a discrete-time version by repre-
senting the closed—loop system in a singularly perturbed form. In the continuous-time
case, the singularly perturbed form is derived by scaling the estimation errors. In the
discrete-time case, however, scaling by itself is not sufficient and we have to perform
additional changes of variables to weaken the slow input into the fast equation.

Moreover, we approximate the continuous-time system (the plant under control)
up to the order of 0(T’)). This approximation complicates the analysis but makes it
more accurate. A lot of mathematical manipulation will be included'in the analysis as
it can be seen in Chapter 3 and the Appendices. The closed-loop analysis is performed
at the sampling points and in between samples. The closed-loop analysis shows
that the sampled data controller using the digitally implemented high-gain observer
recovers the performance of the continuous-time controller as the sampling frequency
and the observer gain become sufficiently large. Performance recovery is shown in

two steps. First we show boundedness of trajectories which comes arbitrarily close

16

to the origin as time progress. Second, we assume that the origin of the continuous-
time system is exponentially stable and show that trajectories enter some ball in the
neighborhood of the origin in finite time and stay thereafter.

The theoretical results developed in Chapter 3 need to be complimented by either
simulation or experimental results to study various factors which the theory may not
reveal, learn how to tune various parameters, or confirm the various observations
we have seen. Also, one of the usual concerns with high-gain observers is their
performance in the presence of noise. This can be addressed either by performing
additional analysis to study the effect of noise or by studying that by simulation
and experiments. In Chapter 2 we use simulations to study the effect of noise. In
Chapter 4, we use high-gain observers in the control of the Pendubot, an electro-
mechanical system consisting of two rigid links interconnected by revolute joints.
The first joint is actuated by a DC—motor while the second joint is unactuated as
a simple pendulum. The pendulum motion is controlled by actuation of the first
link. The experimental results confirm the conclusions of Chapter 2. Moreover, the
experiment confirms the closed-100p analysis of Chapter 3. We compare the use of
high-gain observers with the more common engineering practice of using the Euler
formula (difference method) to calculate speed from position measurements; this
actually is the algorithm provided by the pendubot manufacturer. To compare the
performance of the high-gain observer with the simple Euler method, we determine
what we exactly we gain by using the digital high-gain observer. Our experimental
results show that there is a definite advantage for approaching the problem as a digital
high-gain observer design. This advantage can be seen in two ways. First, high-gain
observers can work for a larger range of sampling periods (or sampling frequencies)
compared to the Euler method. Second, for the same frequency, especially for the
large sampling periods, the high-gain observer produces good performance with less

oscillation in the steady state compared with the Euler method.

17

Finally, Chapter 5 contains our conclusions and directions of future research.

18

CHAPTER 2

Discrete-Time Implementation of
High-Gain Observers for

Numerical Differentiation

2.1 Introduction

In the past few years, high-gain observers played an important role in the design of
nonlinear output feedback control of nonlinear systems. They are mainly used to
estimate the derivatives of the output. All studies of high-gain observers, however,
have been limited to continuous-time analysis, despite the fact that such high-gain
observers are almost always implemented digitally. In this chapter we try to overcome
this deficiency by investigating discrete-time implementation of high-gain observers.
We limit our study to linear high-gain observers so that we can make use of the rich
literature on discretization of linear filters. Our study covers reduced-order, full-order
and extended-order high-gain observers. We emphasize the use of (discrete-time)
high-gain observers as numerical differentiators.

The chapter is organized as follows: Section 2.2 gives background on high-gain

observers and their use as approximate differentiators. Section 2.3 investigates digital

19

implementation of high-gain observers. Five discretization methods are considered:
bilinear transformation (BT), zero-order hold (ZOH), first-order hold (FOH), back-
ward difference (BD) and forward difference (FD). The comparison is carried out for
noise free as well as noisy measurements. Several numerical examples were examined,
but only a few representative cases are included in the chapter. The comparative
study shows that the bilinear transformation method outperforms the other methods.
Section 2.4 concentrates on the Bilinear Transformation method and investigates the
choice of different design parameters. Section 2.5 reviews various numerical differ-
entiators from the signal processing and numerical analysis literature, including the
backward difference method (Euler’s formula), the Spline interpolant method and
several digital filters. It is shown that, except for the Spline interpolant, all these
methods are special cases of the bilinear implementation of full-order or reduced-
order high-gain observers. Optimization of the design parameters, as discussed in
Section 2.4, allows the bilinear high-gain observer to outperform all these methods.
With noisy measurements and under a causality constraint, the bilinear high-gain
observer uses less computation time and less memory compared with the spline in-
terpolant method and results in comparable performance. The section ends with
a novel idea to initialize the high-gain observer by the spline interpolant method,
eliminating the peaking phenomena associated with high-gain observers. In Section
2.6 we discuss a modified high-gain observer by showing via an example how it has
less error than the full order high-gain observer. The section includes also two other
examples to show that the concluded results are common. The conclusions are given

in Section 2.7.

20

2.2 High-Gain Observers

To illustrate the role of high-gain observers as numerical differentiators, let us con-
sider the special case of a single-input—single-output nonlinear system which has a
uniform relative degree equal to the dimension of the state vector. Such a system

has no zero dynamics and can be transformed into the normal form [26]

 

 

 

 

:i: = A2: + B [a(:r:)u + b(:r:)] (2.1)
y = 0:1: (2.2)
where _ q
0 l . 0 0
0 0 1 0 0
A: , B:
0 0 1 0
_0 0mm lenxl
C=[1 o 0]
1er

and a(:r:) # 0. To estimate the state x, we use the observer
5}: = A5: + B[a,(e)¢(e, t) + b.,(:e)] + H(y — Ci) (2.3)

where a,(:r:) and b,(:r:) are nominal models of the nonlinear functions a(z) and b(:r:),
respectively, and the n x 1 matrix H is the observer gain. The estimation error

e = :r: — 5: satisfies the equation

é = (A — HC)e + BA(:1:, e, t) (2.4)

21

where

A(') = [a($) - ao(i')l¢(it t) + W) - 110(5)

The observer gain H is taken as

HT = ] 261 (2.5)

“4‘?
'3]?
I————l

where 6 is a small positive parameter and the positive constants a.- are chosen such
that the roots of
s"+ozls"‘l +-~-+a,._18+a,. =0 (2.6)

have negative real parts. This choice of H assigns the eigenvalues of (A — H C) at
1/6 times the roots of (2.6). Using singular perturbation analysis, it is shown in [16]
that the estimation error will decay to 0(6) values after a short transient period of
the form [0, T1 (6)] where lime...) T1 (6) = 0. During this transient period, the estimate
51‘: may exhibit peaking behavior where the transient response takes the impulsive-
like form (a/6')e‘°‘/‘ for some positive constants r and a. It is shown in [16] that
this peaking phenomenon can be overcome by saturating the control or the state
estimates outside a compact region of interest.

The observer equation (2.3) is nonlinear due to the terms curb and b0. Choosing

the nominal functions a, and b, to be zero results in the linear observer
:i‘:=A:i:+H(y—C:i:) (2.7)

In this chapter, we limit our discussion to the linear high-gain observer (2.7). It
is important to notice that this high-gain observer is essentially an approximate
differentiator. This can be seen by examining the transfer function from y to :5,
given by

G'(s) = (31 — A + H0)“’H (2.8)

22

The following two examples give G(s) for n = 2 and n = 3.

Example 2.1 For n = 2, 0(3) is given by

1 6801 + 012

 

G s =
( ) €232+011€8+02 023

It is clear that 02(3), the transfer function from y to :32, approaches 3 as 6 tends to
zero, which shows that, on a frequency band much smaller than ,/a2/ 6, 5:2 approxi-

mates the derivative g.

Example 2.2 For n = 3, 0(3) is given by

016232 + 0263 + a3

1
0(3) = 2(6—8) 63202 'I' 033

2

 

 

038
t.

where d(6s) = 6333 + (116282 + (1268 + as. From the limit

£2130“) = s

 

 

we see that 5:; and 533 approximate g] and 3'), respectively.

The high-gain observer (2.7) can be used to approximate the derivatives of any
smooth signal y(t) by representing the signal as the output of a normal form driven
by the nth derivative of the signal, i.e.,

:i: = A$+By(”)(t),
y = Cr

23

For any bounded y‘")(t), the singular perturbation argument used in [16] shows that
the estimation error will be of order 0(6) after a short transient period.

The high-gain observers (2.3) and (2.7) are full-order ones. It is also possible to
design reduced-order high-gain observer of order (n — l) [54]. Partitioning the state

 

 

vector 1: as 1
$2
2 = y , where z =
z
xn
we can rewrite equation (2.1) as
l} = A123 (2.9)
2.! = A222 + Bg[a(:r:)u + b($)] (2.10)

where the triple (A22, 82, A12) has the same canonical form structure of the triple
(A, B,C), except that its dimension is (n — 1) instead of n. The reduced-order

observer

ii = (A22 - LA12)(U 'I" Ly) (2.11)

= v + Ly (2.12)

N)

results in the estimation error equation
é2 "'—" (A22 — LA12)82 + 32A(') (2.13)

where e: = z - 2. The similarity between equations (2.4) and (2.13) is clear. There-

fore, the observer gain L takes the same form as H, namely,

LT=]‘-gL £3 1+] (2-14)

24

where 6 is a small positive parameter and the positive constants a.- are chosen such
that the roots of
373—1 + align—2 + . ' ' + 072-23 + 013—] = 0 (2.15)

have negative real parts. The transfer function from y to 2 is given by

03(3) L + (31 - A22 + LAlz)‘1(A22 — LA12)L

= 3(31 _ A2, + LA12)'1L (2.16)

Examination of G 3(3) shows that the reduced-order high-gain observer (2.11) is an
approximate differentiator. The following two examples give G 3(3) for n = 2 and

n=3.

Example 2.3 For n = 2, 03(3) is given by

 

It is clear that limHo G 3(3) = 3; hence, 5:; = 2 approximates the derivative 9.
Compared with 02(3) of Example 2.1, we note that 02(3) is strictly proper while
G 3(s) is only proper. This shows that in the presence of wide-band measurement
noise, the full-order observer should perform better than the reduced-order observer

because its frequency response rolls off at high frequency.

Example 2.4 For n = 3, 03(3) is given by

a] es’iazs
6 s +01¢8+02
s

028
6 s +ata+az

011(8) =

25

[1310 011(3) = 32

Therefore, 532 and :23 approximate g) and 3'], respectively.

2.3 Digital Implementation

In this section, we investigate different discretization methods for high-gain observers.
We start by scaling the observer variables to avoid inherent ill-conditioning of the
realizations (2.7), or (2.11)-(2.12) when 6 is very small. For the full-order observer

(2.7), let
(1,: 64:2,, 2': l,...,n (2.17)
Then
at: = (A—HC)q+Hy (2.18)
j; = D’1(6)q (2.19)
where
HT: [ a1 a2 an ] t D(€)=d538 I1: 6, ”wen-l]

The characteristic equation of (A—HC) is (2.6); hence it is Hurwitz. Equation (2.18)
is a standard singularly perturbed system, whose coefficients are of order 0(1 / 6)
irrespective of the order n of the system. Note that in equation (2.7) some right-
hand side coefficients are of order 0(1/6“'1). The solution of (2.18) does not exhibit
peaking for small 6, which makes it easier to discretize the equation. The effect of
peaking is now contained in the output equation (2.19), and it can be overcome be

saturating the estimates 5: outside a compact region of interest.

26

A similar scaling is applied to the reduced-order observer (2.11)—(2.12) by using

p.-=6‘e,-, i=1,...,n-l (2.20)
Then
61') = (A22 —I'LA12)(p+IZy) (2.21)
2 = r-1(e)(p + 13y) (2.22)
where
ET= ] 011 02 01%,], F(6) =diag [6, ...,6"'1]

2.3.1 Discretization Methods

Discrete-time equivalents of the continuous-time system (2.18) or (2.21) can be found
by standard methods which are well documented in the literature, e.g., [21]. We con-
sider here five of these methods: the bilinear transformation (BT), the zero-order hold
(ZOH), the first-order hold (FOH), the backward difference (BD), and the forward
difference (FD) methods. Formulas for discrete-equivalents of state-space models,
for the first three methods, are given in [21, Chapter 4], and they are available in
the control toolbox of Matlab. It is important to remember that all these methods
are approximations; there is no exact solution for all possible inputs because the
continuous-time system responds to the complete time history of y(t), whereas the
discrete-time equivalent has access only to the samples y(kT). In a sense, the vari-
ous discretization methods make different assumptions about what happens to y(t)
in between the sample points. A brief description of each of the five methods is given
below.

Forward Difference

27

The discretetime equivalent is obtained by the forward rule of numerical integration.
The discrete-time transfer function G (2) is obtained from the continuous—time trans-
fer function G (3) via the mapping 3 = (z — l)/T, where T is the sampling period.
This mapping does not preserve stability of the continuous-time model, except when
T is sufficiently small.

Backward Difference

The discretetime equivalent is obtained by the backward rule of numerical integra-
tion. The discrete-time transfer function G (2) is obtained from the continuous-time
transfer function G(s) via the mapping 3 = (l — z“)/T. This mapping does not
preserve stability of the continuous-time model, except when T is sufficiently small.
Bilinear Transformation

The discrete-time equivalent is obtained by the trapezoidal rule of numerical integra-
tion. The discrete-time transfer function G (z) is obtained from the continuous-time

transfer function G (3) via the mapping 3 = 3- T‘—( 1). This mapping preserves stabil-

2+1
ity of the continuous-time model for any T > 0.
Zero Order Hold

The input signal y(t) is approximated by a stair-case function where the input is kept
constant throughout the sampling period. The output of the discrete-time equivalent
would give the precise values of the state estimates at the sample points, had the input
been a stair—case function. The discrete-time transfer function G (2) is the z-transform
of the continuous—time transfer function G (s)G.,h(s), where Go). (3) = (1 - e’T’) / 3.1
The discrete-time poles map the continuous-time poles according to z = e‘T ; hence
stability of the continuous-time model is preserved.

First Order Hold

This method is similar to the previous one, except that the input y(t) is approximated

 

1The z-transform of a function F(s) is interpreted as Z {f (kT)} where f (t)
5"{(F(s)}; see [211

28

between two sampling points by connecting a straight line from sample to sample .
The discrete-time transfer function G (z) is the z-transform of continuous-time trans-
fer function G(s)Gu.(s), where Glh(s) = T[(l — e‘T‘)/Ts]2. The discrete-time poles

map the continuous—time poles according to z = e”.

2.3.2 Comparison of the Five Methods

Let us consider a third-order nonlinear system in the normal form (2.1), with u = 0

and

11(3) = a($)g.(=v)+(1-a(z)gu(z))

 

3’s:
0(3) _ 2l + z’z:
93(3) = —54221 — 36222 — 9:123

9,,(3) = 54171 — 36172 + 9123

This system was used in [17] to test a numerical differentiator. Figure 2.1 shows
the output and its exact first and second derivatives. We design three high-gain
observers for this system. The poles of all three observers are at s = —1/6; we
will justify this choice of poles later on. The first observer is a third-order full-order
observer with a; = 3, a; = 3, and 03 = 1. The second one is a second-order reduced-
order observer with a1 = 2, and 02 = 1. The third one is a fourth—order full-order
observer with a; = 4, (12 = 6, a3 = 4, and a4 = 1. We will refer to this observer
as the extended-order observer. It corresponds to modeling y(t) as the output of a
fourth-order system driven by y(")(t). All three observers are discretized using each
of the five methods described in the previous section. In all cases, T = 0.01. We did
not specify the value of 6 to examine the effect of the ratio T/ 6 on the performance of
the five methods. The ratio T/6 determines the bandwidth of the high gain observer,

29

 

0W signal ( r)

O O I O
I I O O
_005 ............... , ................................. , ................................. , ............... ..
I
e o , u .
. . . e .
0‘- .............. ................ ........: ................ ' ............... ..
. . . e . .
. o n . .
4,15,. ...........................................................................................
C u a a u
. . . . .
_02b ............................................................................................. d
O O O I O
I O I O
. o o .
_025- ............. .............. ............................. -
e . . . .
_03- ............... '................:....
I I I

 

 

-035 5 l

 

0.5

 

 

1st. demartve ol y

 

 

 

 

 

 

 

 

 

Figure 2.1. The output signal and its first and second derivatives.

30

so increasing or decreasing it will affect the error in calculating the derivatives.

2 in estimating 3} and

Figure 2.2 shows the steady-state root mean square error
3'] for all five discretization methods in the case of the third-order observer. From
the figure it is clear that, for the BD and BT methods, the errors in estimating
the derivatives decrease with increasing T/ 6. The first derivative error reaches a
plateau at T/ 6 = 2, while the second derivative decreases all the way as we increase
T/6. The figure shows also that increasing T/ 6 hurts the ZOH and FD methods for
both derivatives. As for the FOH method, the errors decrease with T/6 until about
T/6 = 2; then they increase. At T/e = 2 both the FOH and BT have the same error.
It appears from the figure that to reduce the steady-state error in the BT method we
should increase T/ 6. However, as T/ 6 —> co, the poles approach -1; thus, increasing
T/6 leads to oscillatory transient behavior. We found that a good compromise is to
take T/ 6 = 2.

From the previous figure, its looks like the FD method has less steady-state error
for the higher derivative. To investigate this point further, we look at the error for
both derivatives at T/ 6 = 1.5. Figure 2.3 displays the error in estimating 3) and ii for
the five discretization methods in the case of the third-order observer, with T = 0.01
and T/ 6 = 1.5.

It shows that the transient behavior of the FD method is more oscillatory com-
pared with the other methods, while the BT and FOH methods give less errors with
relatively shorter transient periods. Figure 2.4 compares the BT and FOH methods
for the three observers when T/ 6 = 2 by looking at the error in estimating 3']. It
shows that the extended-order high-gain observer gives less error in calculating the

second derivative. Moreover, the figure shows that the BT method outperforms the

 

 

2The steady-state root mean square (r.m.s) error is taken as Jfi 2,951 e(t1 + iT)2
where N is the number of samples in an interval [t1, t2], which starts beyond the
transient period and lasts long enough to capture the steady-state behavior.

31

first derivative Full-Order observer T=0.01

l I - l I l l I l

 

 

 

 

 

 

 

 

 

 

r.m.s. OFI'OI'

 

 

 

 

 

0.5 1 1.5 2 2.5 3 3.5 4 45 5
Tlepsilon

Figure 2.2. The maximum steady state absolute error in estimating g and ii for all
five discretization methods for the full-order observer; T = 0.01

32

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ZOH FOH Biinear
02 . - 0.2 .
0.1» «0.1
.C
9!
| 0. v LT OL—g——~——
3‘!
—o.1e t—o.1
-02 ‘ 2
o 2 2 4 o 2 4
TIepsilon=1.5
1.5 1.5 . 1.5
1 1 « 1 t
c 0.5 40.5 ‘05 .
(0
.X
9
es t—o.sL ~ Les » I
-1] . -14 . -11 l
-15 . . - 1.5 - —1.5 - -1.5 -
o 2 4 o 2 4 o 2 4 o 2 4 o 2 4

time it sec.

Figure 2.3. The error in g and i) for all five methods;T = 0.01 and 6 = 0.00667

33

FOH method in estimating ii, and the extended-order high-gain observer gives less
error than the other two observers.

Taking both the transient and steady state behavior into consideration, we see
that the bilinear transformation method provides the best performance among the
five discretization methods. In fact, it provides the best performance for all three
observers. This conclusion is typical of numerical testing of several other examples.

The parameter T/ 6 determines the bandwidth of the discrete-time observer. The
bandwidth increases with T/6. It is clear that in the presence of measurement noise
the observer bandwidth will have to be limited. To study the effect of noise, consider
the noisy measurement ym = y + v, where y... is the measured output, y is the exact
output, and v is white Gaussian noise of variance 0”. We studied the effect of the noise
for different noise level and a representative case is given in Figure 2.5. It shows the
steady-state r.m.s. error in estimating g and j), for all five discretization methods with
o = 0.0316. Increasing T/6, the steady-state r.m.s. error decreases to a minimum;
then it increase with T/6. The same behavior has been observed for different examples
and different noise levels. This is reasonable because there is a certain bandwidth
that allows the high-gain observer to achieve the best trade-off between filtering the
noise and estimating the derivative. On the other hand, decreasing T/ 6 increases
the transient period. Thus, the choice of T/ 6 is a trade-off between reducing the
transient period and minimizing the steady-state error. Taking both the transient
and steady-state behaviors into consideration, we observed, as in the noise-free case,
that the bilinear transformation method outperforms other methods for large signal
to noise ratio. But for small signal to noise ratio, all discretization methods have
the same minimum Steady—state error which occurs at almost the same value of T/ 6.
Figure 2.6 shows this result for a low signal to noise ratio. It is clear from the figure
that almost all methods behave equally and, by the 'help of Figure 2.5, we see that

the minimum steady-state error for the various methods occurs at about T/ 6 = 0.3.

34

 

 

Full order HGO Red. order HGO Ext. Older HGO
. . 0.6 : : 0.6 : I

 

0.6

Bilinear

 

 

 

 

 

 

 

 

 

 

 

 

 

FOH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.4. Error in 3'] for the bilinear and FOH methods for the full, reduced, and
extended observers.

35

first derivative Full-Order observer T=0.01

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 r r r r r ,1 r r r
0.4 '
g 0.3 -
O
*2
E, 0.2 '
0.1 "
0 r r r r r r r r 1
GI 02 0.3 04 0.5 06 07 08 09 I
I I r r
0.8 L _
h 0.6 - ~
g ........ 20.,
«I . - 1:0
604L .
L.’ ...... BD
0 2 _ - - — FOH _
- —— er
0 r r r I r r r r r
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 I

T/epsilon

Figure 2.5. The maximum steady state absolute error in estimating y and i] for
all five discretization methods for the full-order observer in the presence of noise;
T = 0.01 and o = 0.0316

36

Forward ZCI-I Badrward F01 Biinear

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.5 0.5 s 0.5 . 0.5 . . 0.5
n 0 I 0 0» 0 . 0*
9i
06
N
X
—0.5 40.5 4.0.5 4—05 40.5 t
-1 A -1 A —1 -1 -1
0 5 0 5 o 5 0 5 o 5

 

x3&x3h

 

 

 

 

 

 

 

 

limehsec.

Figure 2.6. Calculating g and j} in the presence of noise; T = 0.01, 6 = 0.033, and
o = 0.0316

37

2.4 Bilinear Implementation

Based on the comparative study of the previous section, we adOpt the bilinear trans-
formation as our discretization method. Let us study the design of high-gain ob-
servers with bilinear discrete implementation. The design parameters are the sam-
pling period T, the observer parameter 6 and the a,- coefficients which assign the
roots of the characteristic equation (2.6) or (2.15). The questions that should be
addressed are: How should we choose the observer poles? Should they be all real, or
should we have complex poles? How small should 6 be? What is the proper ratio of
T/ 6?

Regarding the choice of poles, we note that the bilinear high-gain observer can be
made an FIR (Finite Impulse Response) filter or an IIR (Infinite Impulse Response)
filter. An FIR filter has all its poles at the origin 2 = 0. This can be achieved by
assigning all roots of (2.6) or (2.15) at s = —1, which assigns the observer poles at
s = —1/6, and then choosing T = 26. The pole mapping 2 = i—t-g/Lg shows that all
poles map into the origin point z = 0. The only design freedom left in this case is
the choice of the sampling period T (or equivalently the choice of 6). If T is given, all
design parameters would have been determined by the FIR requirement. The FIR

filters corresponding to Examples 2.1 to 2.4 are given, respectively, by

- %(1+z"1) -
0(2) = (2.23)

_ so — z-Ar

 

 

d

38

G(z) = 2-l.1—.(2 — z‘l — 22‘2 + 2‘3) (2.24)

 

 

03(2) = —(1 -— 5‘) (2.25)

and _ l
flT—(3 — 42" + z")
03(2) = (2.26)

 

 

_ 737(1 — 22‘1 + 2") J

In the case of an IIR filter, there is more freedom in choosing the observer poles
and the ratio T/ 6. Let us consider first the case where all poles are real. A real
pole s = —a/6 maps into 2: = (l — 1255-) / (l + %) which lies on the real axis in the
z-plane. The ratio T/ 6 determines the exact location of the pole. Notice that the
limiting cases when T is much smaller than 6 or vice versa drive the pole towards

instability, since

lim (1 —i)-
T/6—+0(1 + 73's)—

and
1— —-)___

T/6—+oo 1 +_ T:

Moreover, if T/ 6 > 2a, the pole will be negative, resulting in oscillatory transient
behavior which is clear from Figure 2.7. For good transient behavior, the pole should
be restricted to [z] _<_ %. This corresponds to the constraint (4/3a) _<_ T/6 S (3/a).
If the roots of (2.6) or (2.15) are real and satisfy —az 5 s 5 —01 for some positive
constants a1 and 02, then the ratio T/ 6 should satisfy (4/3a2) S T/6 S (3/a1). The
limiting case when all roots are assigned at -1 and T = 26 yields the FIR filter as

39

seen before.

In the case of complex poles, we investigated distributing the poles on a semi-circle
of radius r = 1/6. For n = 3, the poles are taken as —l/6 and —(1/6)e*ja. For this
case, the results of Figure 2.8 show that with complex poles the transient response
is oscillatory and the transient period is longer than with real poles. As for steady-
state errors, complex poles result in less errors, but such reduction is insignificant.
Therefore, for IIR filters we limit the choice of poles to real ones.

Another degree of freedom in designing high-gain observers is the choice of the
order of the observer. For example, in Section 2.3 we used a second (reduced),
third (full) and forth (extended)-order observers to estimate the first and second
derivatives of a signal y. Let us use the same example to shed some light on the
choice of the observer order. Figure 2.4 shows that the extended-order observer
outperforms both the full and the reduced-order observer in the case of no noise.
This result is expected because the extended-order observer uses more information
to estimate the derivatives.

For the case of measurement noise, both the reduced-order and extended-order
observers have wider bandwidth compared 'with the full-order observer. This is clear
from the transfer functions in Examples 2.3 and 2.4. This allows more high-frequency
noise to pass. Figure 2.9 shows this result with T = 0.01 and o = 0.1. Notice from
the figure that the minimum steady-state error occurs at different values of T/6, but
the observer which gives the least steady-state error is the full-order observer. We
observed also from different examples that changing the noise level will change the
value of T/ 6 at which the steady-state error is minimum. In general T/ 6 < 2, but it
is important for implementation to find the minimizing values of T/ 6. In conclusion,
we see that, without noise, the extended-order high-gain observer with T/ 6 = 2
outperforms the other two observers. With noise, the full-order observer gives the

best performance.

40

 

 

 

 

 

 

 

 

 

 

 

 

 

 

  

 

 

 

 

w
1
5° é
E 0.5 i?
x“ 0 x"
.5 : 2
E05 -' . E45
0 I Z 0
-1 I
45 A 5 45
1 2 a
u
1
£05 %0
x“ 0 x“ o
.E 5
E45 Es
0 0
4
4% -w

 

Figure 2.7. Effect of the ratio of g on the full-order high-gain observer with real
poles at s = -1/6 and T=0.01

41

error In x2 estim.
O

x2 estlm.
6

error in
.4:
0|

Figure 2.8. The transient response for different choices of poles: real and complex,

T=0.01

 

 

 

 

 

X

 

 

oooooooooooooooooooooooooooooooooooooooooo

 

oooooooooooooooooooo

nnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnnn

 

 

 

1.5 2

42

 

 

 

 

 

 

 

 

 

1.5
1 ........................................ ,
:.E-. 05 .........................................
0)
0
x“ o .....................................
.E
E _05 .......................................
0
_1 ........................................
-15 1
1.5 2
mm.-1Ieps.ande=60
i] ................. t .....................
g 0. ...................................... r
x“ ..................
.5
E_o. .................................... a
0
—1.5 A
1.5 2

 

FII'SI derivative T4101

 

    

 

 

 

 

 

 

 

I r r .l r r ,rfi r
-’ x S
08 ........................................................................................
€05 .........................
O
a;
5.0.4 . . ................. ‘
“ 5 . g ; Full-order
02- .............. --- Ext-order _J
: : 3 : ----- Red-order
o l l l 4 l l
0.05 0.1 0.15 02 025 0.3

 

 

 

 

 

 

 

 

 

.'
B
0
i .............. -
E. .
“ Full-order
--- Ext-order
. . _ . ----- - Red-order
0.05 0.1 0.15 02 025 0.3

T/ebsilon

Figure 2.9. Comparing the maximum absolute steady-state error for the full, reduced
and extended-order observers with noise; T = 0.01, o = 0.1

43

1“ deriv. of the input signal

 

T/ops

 

 

 

 

 

0.35
0.3 '
0.25 *
0.2 '
0.15 .
0.1 '
0.05 '

r.m.s. error

 

 

 

150

2'"deriv.0ltbeinpurslmal

 

T/ops

 

 

 

 

150

 

0.8

.°
a:

r.m.s. error
.°
.5

0.2 >

 

 

 

 

 

Figure 2.10. Comparing the signal to noise ratio with T/ 6 and root mean square

error r.m.s. for the full-order observer for T = 0.01

 

 

 

 

 

 

 

 

 

w,/wm SNR = 80 dB SNR = 60 dB SNR = 40 dB SNR = 30 dB
20 2 2 2 1.55
50 2 2 0.88 0.6
100 2 0.98 0.45 0.3

 

 

Table 2.1. Guidelines for choosing T/6 for different w,/wm and SNR.

The foregoing conclusions depend on the choice of T/ 6, so we need to give some
guidelines on the proper choice of T/ 6 in the presence of noise. We studied differ-
ent examples where the output signals had different frequencies and magnitudes, to
determine the minimizing value of T/ 6 and how it changes with the noise level. We
found out that if the ratio of the sampling frequency to the maximum frequency of
the input signal (w,/wm) is fixed, we get the same curves that describe the relation
between the signal to noise ratio and T/ 6. Figure 2.10 shows that these curves change
with w,/wm. The curves are shown for both the first and second derivatives, for
w,/wm = 20, 50, 100, 150, and 200. The rationale behind choosing these numbers is
how fast we can sample the input signal. It is clear from the figure that increasing
w,/wm more than 100 does not result in a large reduction in the error, especially in
the presence of noise. We note from Figure 2.5 that the convexity of the curve where
the minimum occurs is not sharp, so using a tolerance of :l:10% or 20% for choosing
(T/6) will be reasonable. Also, as the signal to noise ratio decreases, the change of
the value of T/ 6 gets smaller. Rom these curves we propose the guidelines shown in

Table 2. l

45

2.5 Numerical Differentiators

Numerical calculation of the derivatives of a signal is an old problem in numerical
analysis and digital signal processing. The backward difference method (Euler’s
formula) gives one of the earliest and simplest numerical differentiators. Although
it is very crude, its use is quite common in engineering applications. One can also
use formulas from other numerical integration techniques. A different approach to
numerical differentiation is to use interpolation techniques, like the spline interpolant
method [1 7]. In the digital signal processing literature, this problem goes back to
the work of Rabiner and Steiglitz [52] who used digital filters for wide-band digital
differentiators. They used filters of order 16 to 256 (in powers of two). Various
other types of digital differentiators have been designed to minimize both the filter
order and the relative error in the frequency range of interest. Kumar and Dutta
Roy [38] proposed digital differentiators from maximally flat nonrecursive low-pass
digital filters for the narrow frequency band of 0 to 0.2 1r radians, using filters of order
ranging from 9 to 21. Al-Alaoui [1] introduced a new approach that makes use of the
idea that the ideal integrator response lies between the response of the rectangular
and trapezoidal rules. Interpolating these two rules, he gave better approximations
of the ideal differentiator.

In this section, we describe the backward difference method, the spline interpolant
method, and several formulas from the digital signal processing literature. Then, we
study their connection with the bilinear high-gain observer of the previous section.

Euler’s formula for finding the first derivative of a signal y is

err) = gilt/(k) — yrs -1)1 (227)

In the spline interpolant method [17], we start with a window of data {yr—w, . . . , yk}.

46

Let the interpolating polynomial be denoted by
gut) = a0 + a1(t — (k - W)T) + . . . + aN(t — (k — W)T)"

where the coefficients {a0, a1, . . . , a 1v} are determined from the least squares solution

of _ _ _ _ r
1 0 0 co yk_w
1 T T N a _
( ) 1 = y" .W“ (2.28)
Ll WT (WT)NJ .aN. _ y), J

 

 

 

 

 

 

N is the order of the interpolating polynomial, and W x T is the length of the
moving window in time. The estimated derivatives of the polynomial y at time 1' are
calculated by A

53m = grits)
For real time applications we have to use causal differentiators. Therefore, we es-
timate the derivatives at the end of the window. To compare with other methods,
let us consider a third order system which gives first and second derivatives, i.e.,

N = 2, and W = 2. Solving for the coefficients {a0, a1, 02} from equation (2.28) and

substituting in the derivative of e, we obtain

If]: = T‘1(0.5y,,_, - Zinc—r + 1-5yrr) (2.29)
it: = T401542 - Zyk-l + 31):) (2.30)

Various types of Digital Differentiators have been designed using the min max crite-
rion, which amounts to minimization of the relative error over the frequency range
of interest. It is shown by Al-Alaoui in [1] that these nonrecursive differentiators are

outperformed by a digital differentiator which is obtained by interpolating two pop-

47

ular digital integration techniques, the rectangular and trapezoidal rules. The new
differentiator is obtained by taking the inverse of the transfer function of the integra-
tor. The effective range of the differentiator is about 0.8 of the Nyquist frequency.

The resulting transfer function of the new digital differentiator is

G(z): 7——:——:sz + 1%)) (2.31)

This filter was generalized by Bihan [9] to a class of first-order differentiators. The
filter parameters are calculated to satisfy various defined criteria. The minimum

phase digital differentiator is

3 (z-l)
T(1+x)z+(l—x)

 

G (2) = (2.32)

where 0 _<_ x 5 1 . Al—Alaoui [2] pr0posed a stable second-order recursive differentiator

3(z2 — 1)
T(3.7321)(z2 + 0.53582 + 0.0718)

 

G(z) = (2.33)

suitable for applications that require fast differentiation methods. It is obtained from
the Simpson integration rule. The accuracy and the range of the amplitude response
of the pr0posed differentiator is the same as that of the Simpson integrator. Thus,
it is comparable to that obtained by higher order algorithms.

In [3], a new class of stable second-order low-pass differentiators was considered.
These second-order differentiators are obtained by interpolating the traditional Simp-
son and trapezoidal integrators, inverting the resultant transfer function to obtain

the pr0posed differentiator

6(z2 — 1)
Tr1(3 '- a) (Z ‘I' T2)2

 

G(z) = (2.34)

where0<a_<_ 1, r1: (3+a+2x/5)/(3-a), and r2=1/r1

48

2.5.1 Comparison with the Bilinear High-Gain Observer

It is interesting that all the numerical differentiators described in this section can be
viewed as special cases of the high-gain observer. We show this fact for each one of
them.

The bilinear discrete equivalent of the transfer function of Example 2.3 is given
by

012(3) = ——2"‘ x ——z :1.

26 + aiT z + %
Taking or = l and 6/T =3 3/8 results in the filter (2.31). Taking a1 = 1, and T/6 = 2
gives the backward difference method, while the choice a, = 1, T/ 6 = 2/x yields the
filter (2.32). The bilinear discrete equivalent of the transfer function of Example 2.1
with al = 2, a2 = 1 and 6/T = l/Jfi yields (2.33). Ehrthermore, with a1 = 2, a; =
l, and 6/T = m for 0 S a S 1, we obtain (2.34). All the foregoing numerical
differentiators are special cases of high-gain observers with certain choices of the
observer parameters. Such choices are not necessarily the best choices for an observer
design, in view of the results of the previous section on how to choose the observer
parameters. This is particularly so in the presence of noise. Let us elaborate on this
point by considering specific numerical differentiators. In the backward difference
method, the sampling period T is the only design parameter. To calculate (n — l)
derivatives, we end up with a filter whose bandwidth is pr0portional to l /T'"'1. HT
is very small, the backward difference filter will have a large bandwidth. The same
observation holds for the digital filters described by Al-Alaoui and Bihan. They are
special cases of bilinear high-gain observers. All of them have T/6 > 2, and for a
small T, 6 must be small too. Consequently these filters have wide bandwidth. With
such filters, it will be necessary to use low pass filters to filter out the high-frequency
noise, which will add delay to the estimates of the derivative. We saw in the previous

section that high-gain observers can be designed to have a smaller bandwidth by

49

taking T/ 6 < 2.

For the special case of the spline interpolant method with N = 2 and W = 2, it
can be seen that the FIR filter of (2.26), corresponding to the high-gain observer of
Example 2.3, coincides with (2.29)-(2.30). This relationship does not hold for higher
values of N and W. Therefore, by Optimizing the choices of N and W, it is conceivable
that the spline interpolant method may outperform the bilinear high-gain observer.
As for the choice of N, it was observed in [17] that the order of the interpolating
polynomial should be selected as low as possible in order to smooth the noise in the
estimates of the derivatives. Hence, for estimating the first and second derivatives,
we should use N = 2. For a fixed T, this leaves us with the window size W as the
only design parameter that can be adjusted to filter out the noise. For T = 0.001,
6 = 0.0154, 02 = 0.001, and W = 70, Figure 2.11 shows that the bilinear high-gain
observer and the spline interpolant method have the same error for both derivatives.
It shows also that the spline interpolant method has a shorter transient period.
Decreasing the window size allows more high-frequency noise to pass. For very small
T, the window size should be increased to capture more information about the input
signal y in order to smooth the calculated derivatives. For higher noise levels, we
need to increase the window size in order to filter out most of the noise. This will
work up to a certain limit, after which the error becomes independent of the window
size. Increasing the window size increases the computation time and the memory
requirements of the Spline interpolant method. For W = 70, the computation time
for the Spline interpolant method is about four times that of the full-order high-gain
observer (the CPU average time for using the full-order high-gain observer is 7.44
sec., and the CPU average time for the Spline interpolant is 29.79 sec.). Increasing
the window size requires more memory in each 100p for calculation, while the full-
order high-gain observer requires only 11 memory locations. Therefore, we can see

that the high-gain observer can achieve results comparable to the Spline interpolant

50

Full—Order him gain observer

Spline method

0.5

 

0.5

 

 

 

 

 

 

 

 

 

 

 

x3 8 xah

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.1 1. Comparing full-order observer and spline interpolant methods; T =
0.001, 6 = 0.0154, 02 = 0.001, and W = 70.

51

method, with less computation time and memory requirements.

2.5.2 Spline Method Initialization of The High-Gain Ob-
server

One advantage of the spline interpolant method over the high-gain observer is its nice
transient behavior. Starting from zero, it usually approaches its estimates in a fast
monotonic motion. The transient behavior of the high-gain observer, on the other
hand, could be oscillatory and accompanied with the peaking phenomena. We may
use saturation to overcome peaking [16], but the estimates will be degraded during
the saturation period. This drawback of the high-gain observer can be eliminated
by using the Spline interpolant method to initialize the the high-gain observer. Even
though in the presence of noise the window Size needs to be large enough to filter
out the noise, the window width is usually much less than the transient period of the
high-gain observer. Figure 2.12 shows the difference between initializing the high-
gain observer by the Spline interpolant method and without it in the noise-free case,
with T/6 = 10, T = 0.001, W = 2, and N = 2.

It is clear from the figure that initializing the high-gain observer from a spline
interpolant methods has reduced the transient period oscillations, especially in the
estimates of the second derivative. Figure 2.13 shows a Similar comparison in the
presence of noise, with T/6 = 0.065, T = 0.001, W = 40, N = 2, and a = 0.0316.
Notice that the Size of the window has been increased in the presence of noise to give

a good start. Both Figures 2.12 and 2.13 are for the extended-order observer.

2.6 Further Considerations

In this section we study a modified HGO and compare it with the regular HGO. We

also give two more examples to Show that the conclusions drawn from the simulation

52

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

Not initialized by spline lrtirialized by spline
I : : : : I r : :
0.5
g 0
x
on
9" —0.5
-i
-15 —1.5 A A A A
0 0 02 0.4 0.6 0.8 1
T=0.001, eps=0.0001, W=2 Nae
6 ......................... 7 .. . . . . .I .........
4 ................................... I .........
.c 3
‘2 ;
«a s
t') .
" :
_2 i : L : _2 1 1 ; i
0 0.2 0.4 0.6 0.8 1 0 02 0.4 0.6 08 1
time in sec time in see

Figure 2.12. The effect of initializing the high-gain observer by Spline interpolant
methods; T/6 = 10, T = 0.001, W = 2, and N = 2.

53

 

 

 

 

N01 initialized by spline Initialized by spline

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

6 ...........................................

4 .......................................... .
L .c
9 ‘2
no or
0 ('1
x X

_2 1 : i _2 1 1 i
0 0.5 1 1.5 2 0 05 1 1.5 2
time in see time in sec

Figure 2.13. The effect of initializing the high-gain observer by spline interpolant
methods in the presence of noise; T/6 = 0.065, T = 0.001, 0 = 0.01 W = 40 ,N = 2
and o = 0.0316

54

of the third-order example of Section 2.3.2 are common in other examples. Notice
that the main example is the most difficult one because the output signal is fast,
which requires a smaller sampling period. The simulation results will be repetitions

of Figures 2.1-2.8.

2.6.1 Modified Observers

Modified observers were discussed in the literature for both HGO and SMO (sliding
mode observer). In [48], [18] the modification was done only for second-order systems
to estimate 2:2, while in [19] it was done for the SMO for higher-order systems to
estimate it derivatives.

For a second order system the modified HGO is given by

~ A

1111:2131

$2 = ig‘l'kl (151—571)

In this work we are going to extend the modified HGO observer to dimension n as it
was done for the SMO in [19].
Changing the observer output from f: to :5 results in different output equations

~ A

$1=$1

 

SD; = fig-I'm-ll (171—5131), 25157!

c“

which can be written in the vector form

where Cm is n x n and Dm is n x l. The transfer function for the modified linear

55

HGO is
Gm(s) = (31 — A + H Cm)‘1H + D".

Example 2.5 For n = 3, Gm(s) is given by

016282 + 0263 + a3

1
Gm(8) = dlg). 016283 + (12632 + 03.9

026.93 + 0332

 

 

d

where d(es) = 6333 + (116232 + (1268 + (13. Comparing the error in estimating the

derivative for both i: in Example 2.2 and i in this example we get the following:

52—91 1 -€333 _ (116282
Cregular = W, = 3(7) = 0(5)
:2 —s 3 3 2 2
331,—“ —es —ales -O!2€3
and
52-8 —63 3
emodified = W = —1 s = 0(62)
95,—," 01(3) —e3s3 - (116282
a v

This preliminary investigation shows that the error for the linear modified HGO
is of order 0(62) where as for the regular observer it is of order 0(6). Studying the
error equations any“, = :c — :i' and emodified = a: - it in simulation we can see easily
how the modified is better than regular. Including in this comparison the extended
order HGO we can see that the extended order HGO outperforms both the regular
and modified observers. Figure 2.14 shows the error in the highest derivative. The
extended observer has the least error, followed by the modified observer, followed by
the regular observer.

The modified high-gain observer has a larger bandwidth than the regular high-
gain observer. This can be seen from the transfer functions in example 2.5. Therefore,

the modified observer passes more noise than the regular observer, which makes the

56

0.1

 

 

FUI 0w! HGO MOO. HGO EXl. W HGO
I Y 0.1 : z 0.1 v -

 

005. ........ 005. ......... 005.. .........

   

error In x2

-0.05 ........ ,......., ........ '015’" ......... ‘0.05': .........

 

 

 

 

 

 

 

 

 

.61 3 3 —o.1 ; 5 -o.1
o o

 

 

0.5

 

error In x3

   

 

 

 

 

 

 

 

 

 

-o.5 3 *2 —o.5 3 3 -o.5 1 i
o 2 4 6 o 2 4 6 o 2 4 6
timeinsec. timeinsee Iimeinsec.

 

Figure 2.14. The comparison between the regular, the modified, and the extended
HGO where T = 0.01

57

modified observer less desirable in the presence of noise.

2.6.2 Additional Examples

We will consider two more examples to support the conclusions of this chapter.

Example 2.6 Consider third-order nonlinear system in the normal form as follow

:31 = 1‘2
i2 = $3
333 = —x2 cos(:rl) — 4mm: — 2mm? + u
y = 1‘1
where the control u = —w k sin(wt). The parameters are p = 2, k = 0.5, and

w = 0.25. We would like to estimate the states 22 and $3 from the measured output
2:1. We use the discretetime high-gain observer. From the output signal shown
in Figure 2.15 we see that the signal period is about 25 sec; this system is slower than
the system of Section 2.3.2. In particular, comparison of Figures 2.1 and 2.15 shows
that the output of the main example has frequency almost 10 times higher than the
current output. Using three different high-gain observers, described in Section 2.2,
we compare the different discretization methods. Figures 2.16 to 2.22 show the same
trends which were observed from Figures 2.2 to 2.8, respectively. It is important
to notice that for slower systems, in the presence of noise, it is better to use. larger

sampling periods.

Example 2.7 Consider the third-order nonlinear system

522 =$3

58

 

 

 

 

 

I‘defivativeolyuz) foam/sown;

 

 

 

 

 

 

 

 

 

 

 

0.2 r . - s
0.1 .
0 t 4
-0.1 -
A A L A _02 A A A 4
0 10 20 30 40 50 0 10 20 30 40 50
time in see time in sec.

Figure 2.15. The output signal and its first and second derivatives.

59

 

x104 Frst derivative Full—Order observer T=0.01

 

 

 

 

 

 

 

 

 

 

 

 

 

7 I I I 3 I I I I I
.l ° 20”
5W ~FD *
"30
L l
L5 | "FOHi
9 _' —BT
54 l
"I . q
E3 \25 ....................................................... _
2., -r
1_ """""""" -
0‘ l --l‘--;‘ I I I I
05 l 15 2 2.5 3 35 4 45
0.1
0.08'
E066-
0
'l
€0.04“
0.02'
0 I I I I I I I I

 

 

 

Figure 2.16. The maximum steady state absolute error in estimating it and ii for all
five discretization methods for the full-order observer; T = 0.01

60

ZOH
0.2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

x “fir. Dil. x 1WD". x 104F0H x 10mm
2 2 . . 2 - 2 - .
0.11 1 1 1 1
2“
X
I 0* 0 0 OW—v 0 ~‘* -
x“
-0.1’ *-1r r-1 -1L *-1’
—02-64-44--.46-—2+-
0 20 40 0 20 40 0 20 40 0 20 40 0 20 40
T=°-°' x10“1 x 10“ x104 x101," ‘5
1.5 5 . - 5 - . 5 . . 5 . .
1.
2"”
X
' 0’ 0 0] 0 0
x" ]
-0.5
-1 m .
-15 - —-5 - A —5 a - -5 . -5
0 20 40 0 20 40 0 20 40 0 20 40
Iimelnmc.

 

A

 

02040

Figure 2.17. The error in g and 37 for all five methods;T = 0.01 and c = 0.00667

61

Fi:

 

 

Bilinear Tr.

 

 

 

 

 

 

 

(I)
ll.
_4 .......... g .................
_6 i
0 20 40

 

 

 

 

 

 

 

 

 

 

 

.........................
a e

 

 

 

_4 .......... f, ................
'6 "L
0 20 40

 

 

 

 

 

 

 

 

 

Figure 2.18. Error in ii for the bilinear and FOH methods for the full, reduced, and

extended observers.

62

First derivative Full-Order observer T=0.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.2 I .r
‘l
1.1- .I .
.l
1- " r
E
00.9- -
0: -
L e ' FD .
0.7 --so
06'- --Fm-r
—-3T
05 l l l L L l l l
01 0.2 0 3 0.4 0.5 06 0 7 0.8 09 1

Figure 2.19. The maximum steady state absolute error in estimating g and 3') for
all five discretization methods for the full-order observer in the presence of noise;
T = 0.01 and a = 0.0316

63

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T=0.1 (#00707 We s0.28

 

 

 

 

 

 

 

x3 8: :01a

 

—0.1

-0.2

 

 

Figure 2.20. Calculating i1 and i] in the presence of noise; T = 0.1, T/e = 0.28, and
a=QMW

M

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.21. Effect of the ratio of g on the full-order high-gain observer with real
poles at s = —1/6 and T=0.1

65

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

20
time in sec.

 

Figure 2.22. The transient response for different choices of poles: real and complex,
T=0.1

66

T3 = —IL'1—3.’II3 —z3

We use discrete-time high-gain observers to estimate .122 and $3 from the measured
output :51. The exact states of the system are shown as in Figure 2.23. Using
three different high-gain observers, described in section 2.2, we compare the different
discretization methods. Figure 2.24 to 2.30 show the same trends which were observed

from Figures 2.2 to 2.8, respectively.

2.7 Conclusions

We studied discrete-time implementation of high-gain observers and their use as
numerical differentiators. The study covered noisefree as well as noisy measurements.
The noise was modeled as white Gaussian noise. After investigating five different
discretization methods, we concluded that the bilinear transformation method gives
the best performance. This conclusion was also based on numerical testing of several
examples. After adopting the bilinear transformation method, we investigated the
design of the observer parameters and the choice of the observer order (full, reduced
and extended-order observer). We found out that it is preferable to assign the poles as
real ones. In noise-free measurements, we found it advantageous to take T/e = 2 for
two reasons: first, it gives an FIR filter when all observer poles are located at —1/e;
second, it produces the shortest transient period. Furthermore, using the extended-
order observer gives less error than the other observers. In noisy measurements, we
found out that the ratio T/e should be reduced, depending on the noise level. This
reduction of T/e allows the high-gain observer to play a dual role. On one hand,
it acts as a numerical differentiator, and on the other hand, it acts as a low pass

filter that filters out high-frequency noise. In this case the observer which gives the

67

 

 

 

 

 

*1.5 l L l l
0

Firstderivativeolflxz)
1 . . .

 

 

 

 

0 ' 5 10 15 20
timein sec.

-1

 

Seoondderivativeotfixa)

 

0.4

 

 

 

 

—o.4 ‘ - -‘
0 5 10 15 20
time in see

Figure 2.23. The output signal and its first and second derivatives.

68

FlfSl derivative Full—Order observer T=0.1

l I l I F

 

02

     

 

0.15 L

 

 

 

0.1 -

r.m.s. error

0.05 ]

 

 

 

 

p
.5
I

c
7.;
'

 

 

 

 

0.5 1 15 2 25 3 3.5 4 4.5 5
TIE

Figure 2.24. The maximum steady state absolute error in estimating g and i] for all
five discretization methods for the full-order observer; T = 0.1

69

ZOH

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

For. Di1. BacDit. FOH Bilinear
0.5 0.5 0.5 0.5 s 05
N
.C
X
I o o o 0L~———- o ___
x“
—o.5 A .5 A .5 a .5 A .5 _
10 o 10 o 10 o 10 o 10
T=0.1 Tle- 15
0.5 0.5 0.5 ~ 0.5 ~ 0.5
0
.C
X
I o o o o o
X“
-o.5ll A —o.5 A .5 A 5 A 5
o 10 o 10 o 10 o 10
time in sec.

 

 

10

Figure 2.25. The error in 1} and ii for all five methods;T = 0.1 and c = 0.0667

70

 

 

 

Blllnoar Tr.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

.61 A or 3 or A
0 10 20 0 10 20 0 10 20
Ext.0rdeIHGO
0.1 :
.......... .., om
I
o ....................... o .....................
IL
I _005 ........... ............
' -0.1 1
10 20 0 10 20
timehsec. timeiosee

 

Figure 2.26. Error in 3’) for the bilinear and FOH methods for the full, reduced, and
extended observers.

71

r.m.s. error

r.m.s. error
.° .° .°
4; as on

o
to

first derivative Full—Order observer T=0.1

 

 

. I I.

 

 

 

 

 

 

 

 

sgsag

 

 

 

 

1
1 1.5
We

Figure 2.27. The maximum steady state absolute error in estimating g and ii for
all five discretization methods for the full-order observer in the presence of noise;

T = 0.1 and a = 0.0707

72

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.28. Calculating i1 and j] in the presence of noise; T = 0.01, T/e = 0.4, and
0‘ = 0.1414

73

0.01

 

 

 

 

 

 

 

 

0.01
0.005 0.“)5
1:“
X
I
x" 0 0
I
0" : : f
'0.” .......... .......... ,........\ ........... 41W
-o.or 3 .601
0 5 1 0 15 20

0.01

 

 

0.005

92 III-X2 - Xhz
O

-0.005

 

 

-0.01

 

 

 

Figure 2.29. Effect of the ratio of 1;- on the full-order high-gain observer with real
poles at s = -l/e and T=0.1

74

 

 

 

 

 

 

 

 

 

rnag.=1le, 0: 0
0.01 . ,
0.“)5 ..........................................
N
.E
x
I
x" o ......................................
I
N
0
_o.m5 .......................................... 4
-0.01 A i 1
0 5 10 15 20
0.01

 

 

 

 

 

 

Figure 2.30. The transient response for different choices of poles: real and complex,
T=0. 1

75

least error is the full-order observer because it has smaller bandwidth compared with
other observers.

Comparison of the bilinear high-gain observer with several numerical differentia-
tors from the literature showed two interesting findings. First, most of those differ-
entiators are special cases of the bilinear reduced-order high-gain observer for certain
choices of the observer parameters. We found out also that by exploiting the freedom
to choose these parameters, the bilinear high-gain observer outperforms those numer-
ical differentiators, especially when measurement noise is taken into consideration.
Second, the bilinear high-gain observer gives results comparable to a causal spline
interpolant method, especially in the presence of noise. The high-gain observer uses
less computation time and memory requirements than the spline method. Finally,
we proposed a novel idea of using the spline interpolant method to initialize the

high-gain observer, which significantly improves the observer’s transient behavior.

76

CHAPTER 3

Output Feedback Sampled-Data
Control of Nonlinear Systems

Using High-Gain Observers

3.1 Introduction

In Chapter 2, we studied different discretization methods of the high-gain observer
as a numerical differentiator. That study was based on simulation. In this chapter,
we study the performance of a closed-100p nonlinear system under a sampled-data,
high-gain, observed-based controller.

Sampled-data control of nonlinear systems was investigated by several researchers.
A tapic of particular interest is how certain continuous-time prOperties are affected by
sampling [51]. Considerable work was done by Sontag in [56] to study the preservation
of controllability and observability under sampling. Monaco and Normand-Cyrot
[44, 46] gave an explicit power series representation of sampled systems, in terms
of the vector fields of the original continuous-timedata system. Grizzle at el. in
[12, 55, 47] showed that certain properties, like observer error linearizability, are not

inherited from continuous-time systems.

77

The Chapter is organized as follows: Section 3.2 describes the continuous-time
design. Section 3.3 describes the sampled-data system, where we discuss the dis-
cretization of the high-gain observer, the discrete-time closed—100p model, and the
development of the discrete-time singularly perturbed form. Section 3.4 shows per-
formance recovery by the output feedback sampled-data controller. Section 3.5 shows
example of recovery of the region of attraction. Finally, the conclusions are given in

Section 3.6.

3.2 Continuous-time Design

3.2.1 The Class of Nonlinear Systems

We consider a single-input—single—output nonlinear system represented by

:i: = Acx+Bc¢(z,z,u)éf(a:,z,u) (3.1)
:2 = I/2(:I:,z,u) (3.2)
y = 0.x (3.3)
c = e(z,z) (3.4)

where u E u Q R'" is the control input, y 6 y Q R‘9 and C E R‘ are measured
outputs, and :I: E X g R" and z e Z g R‘ constitute the state vector. The r x r

matrix A, the r x 1 matrix BC, and l x r matrix Cc, given by

'01 0‘ To”

0 o 1 o o
Ac= .Bc=

o 01 o

o .04 1

 

 

 

 

78

Cc=[1o ...... 0]

represent a chain of r integrators.

Assumption 3.1 The functions gt : X x Z X Ll —) R and 1b : X x Z x Ll —)R‘
are locally Lipschitz in their arguments over the domain of interest. In addition,

¢(o,o, 0) = 0, 5(0, 0, 0) = 0 and 9(0, 0) = 0.

3.2.2 Partial State Feedback Control

The task of feedback control is to stabilize the origin of the closed loop system using
only the measured outputs y and C. We follow a two-step approach to this problem.
First, we design a partial state feedback control that uses measurements of a: and C.
Then we use a high-gain observer to estimate :6 from y. The state feedback control

takes the form
u = 7(12, C) (3.5)

We allow any state feedback design that holds the following four pr0perties.
Assumption 3.2

1. 7(0,0) = 0;

2. 7 is a locally Lipschitz function in its argument over the domain of interest;

3. '7 is a globally bounded function of :L';

4. The origin (:1: = 0,2: = 0) is an asymptotically stable equilibrium point of the

closed-loop system.

For some of our results, the forth property will be strengthened to exponential stabil-
ity. The global boundedness requirement is typically achieved by saturation of 7(-),
or its :r-input, outside a compact region of interest. The boundedness of 7(-) may

also follow from design constrains.

79

3.2.3 High-Gain Observer (HGO)

To implement the control (3.5) using output feedback, we design the high-gain ob-

server

i; = Aci- + B,¢,(e, c, u) + H(y — 065:) (3.6)

where ¢o(:i:,C , u) is a nominal model of the nonlinear function ¢(:r, z, u). If ¢ is a
known function of 1:, C and u, we can take (to = «t. On the other hand, if such nominal
model is not available, we can take 45., = 0, which results in a linear observer. The

function (to is required to satisfy the following assumption.

Assumption 3.3 ¢a(:r,C,'y(:r,C)) is locally Lipschitz in its argument over the do-
main of interest and globally bounded in 1:. Moreover, ¢o(0, 0, 0) = 0.

The output feedback controller is taken as
u = 7(i. C)
The estimation error e = :I; — :5 satisfies the equation
e = (A6 — HCc)e + B¢A(z, z, e, t)

where
A(°) = [¢($r 2, u) - ¢o(£!Cru)l
It is shown in [54, 16] that the output feedback controller recovers the performance

achieved under state feedback if the observer gain H is chosen as

HT: 21. 22.
e e2

as
h———l

80

where e is a small positive parameter and the positive constants a.- are chosen such

that the roots of
s"+als"‘l +~-+a,_ls+a,. =0 (3.7)

have negative real parts. This choice of H assigns the eigenvalues of (A, -- H C.) at
1/6 times the roots of (3.7). Using singular perturbation analysis, it is shown in [16]
that the estimation error decays to 0(6) values after a short transient period of the
form [0,T1(e)] where T1(e) tends to zero as 6 tends to zero. During this transient
period, the estimate :5 may exhibit peaking behavior where the transient response
takes the impulsive-like form (a/e‘)e‘°‘/‘ for some positive constants 6 and a. It
is shown in [16] that this peaking phenomenon can be overcome by saturating the

control or the state estimates outside a compact region of interest.

3.3 Sampled-Data Control

In sampled-data control, the output variables y and C are sampled and the control
input u is applied at the sampling points. Let T denote the sampling period. We
assume that the control is implemented through a zero-order hold. Hence, a is
constant over the sampling period. The continuous-time observer is discretized and
implemented in discrete—time as a difference equation. Since the observer dynamics,
parameterized by e, are fast for small 5, we choose T = as for some a > 0. For
convenience we denote the signals at the kth sampling point by z(lc), :i:(k), u(lc), etc.

The control is implemented by

WC) = 763(k), ((10)

81

3.3.1 Discretization of the HGO

In this section, we investigate discrete-time implementation of the observer (3.6).
We start by scaling the observer variables to avoid inherent ill-conditioning of the
realization (3.6) when 6 is very small. Let

_ i—l~ -_

to obtain

(1 = D(Ac — H Cc)D“q + DH :1 + DBc¢o(D"q, C. u) (3-8)

D"(6)q (3-9)

H)
II

where

13(5) = diag [1, E, . . . ,ér-l]

The characteristic equation of €D(Ac — H C,,)D'l is (3.7); hence it is Hurwitz. The
scaled observer is

. 1 _

q = ‘gleq '1‘ Boy + ch¢o(D lg: C: 74)] (3'10)

where A, = eD(A¢ - HCJD“, B, = 6DH, and DBc = 6'43," Equation (3.10)
is a standard singularly perturbed system whose coefficients are of order 0(1/6)
irrespective of the order of the system. Note that in equation (3.6) some right-hand
side coefficients are of order 00/6”). The solution of (3.8) does not exhibit peaking
for small 6, which makes it easier to discretize the equation. The effect of peaking is
now contained in the output equation (3.9), and can be overcome by saturating the
estimates 5: outside a compact region of interest.

Depending on whether or not the nominal function «to is zero, the HGO (3.8) can

be linear or nonlinear. Linear observers can be discretized using different methods.

82

 

Forward Difl’. Backward Difl‘. Bilinear Transformation

 

A... (I + $21.) (I - £14.)“ ‘1—2‘ M (I + 62AM! - gifts-1 ‘1-°=‘ N2M2

3,... $3., §Ml B, §M230
Cdo Co = I)"1 D—lMl D—1M2

 

 

 

 

 

 

Table 3.1. Coefficient of the discrete time implementation of the linear HGO.

We consider three methods: Forward Difference, Backward Difference and Bilinear
Transformation. For the nonlinear case, we use the Forward Difference method.

Linear HGO The linear high-gain observer is a special case of the general form
(3.8) when ¢o(-) = 0. It is implemented in discrete-time by

q(k+l) = Aaaq(k)+Baay(k)

:i:(k) = CdOQ(k)+Dday(k)

where Ado, Baa. 0,10 and Dd, depend on the discretization methods, as shown in Table
3.1.
Nonlinear HGO When 45., aé 0, the observer (3.8) is nonlinear. We discretize it

using the Forward Difference method, to obtain

q(k + 1) = Asq(k) + Baez/(k) + Te'"‘B.¢.(D"q(k). C(k). u(k))

5W“) = Chad")

where A40, Bag, and 040 are given in the first column of Table 3.1

83

3.3.2 Discrete-Time Closed-LOOp Model

To analyze the closed-100p system, we derive a discrete-time model that describes the
state variables at the sampling points. We start by discretizeing the plant dynamics.
A key step in deriving our closed-100p model is the representation of the system in a
singularly perturbed form that reflects the fact that the observer dynamics are faster
than the plant dynamics. As in continuous-time analysis [8], arriving at the desired
singularly perturbed form requires replacing the observer states by scaled estimation
errors. Unlike the continuous-time case, however, this change of variable by itself
may not be sufficient and we may have to perform additional changes of variables to
weaken the slow input into the fast equation. To keep track of this slow input, our

discretization of the plant must have 0(T'“) accuracy.

Assumption 3.4 The nonlinear functions f(a:, z, u) and ¢(:r, z, u) are 0' +1 in the

domain of interest.

Plant Dynamics

An equidistant grid on the time axis with a fixed step size T = t“; —t,, > 0 (sampling
period) is considered and u(t,,) = constant over the time interval [tk,tk+1) (zero-order

hold). For convenience, let us rewrite (3.1) and (3.2) as

X = F(Xru) (3'11)

where

x= i ,Frx,u)= “2“)
z ¢($,z,u)

By expanding the solution of (3.11) in a Taylor series about x(k) and using the fact
that u(k) is constant during the sampling interval, the plant dynamics are described

84

in discrete-time by
x<k+1> = x(k)+i:?.-a(x(k).u(k»+r+*R(x(6).u(k),T) (3.12)
l=l '

where F;(x, u), for 1 g l 5 r, are determined recursively by

F1(X,'U) = F(Xtu) (3'13)
mam) = Ween) (3.14)

and the remainder term R(x,u, T) is bounded on compact sets of (x, u), uniformly

in T for all T S T‘ for any given T‘ > 0. Using (3.13)-(3.14), it can be verified that

A! @_ Al—ch , ,
F.(x,u)= ““2“ ‘ W") (3.15)

gl(X’ 11)

where h; and 9;, for 1 S l S r, are determined recursively by

0h

 

 

h.(x.u)=¢(x.z,u), h.(x,u)= 6:1(x.u)F(x,U)
91(x.u)= new), 9106") = 0;: (x,u)F(x.u)

Using (3.15) in (3.12), the equation for a: can be written as

r T! r Tl l .
50054-1) = w(k)+(2_21-,A,—A£z(k)+Z-,,—ZA£" ch,(x(k),u(k))

[:1 i=1

+T'“R¢(X(k), "(’01) (316)

where R;(-) is formed of the first r components of R(). The double summation term

of (3.16) can be written as
' Ti ,_,
?Ac BCGJ'

i=1

85

where
r-j-l-l (j!)TI-l

G-_ ___._,.
’ ,=, (j+i—1)!h

Hence, (3.16) can be rewritten as

r—l T! r Tl
$06 + 1) = $06) + X 74:30) + Z 745’13c0z(x(k). u(’0)
. (=1 .

(=1

+T' +1PAW“). WC), T) (3-17)

where we used the fact that A; = 0. Note that, because of the property A; = 0, the

Taylor series (3.12) is written with only r terms.

Observer Dynamics

To represent the observer dynamics, we perform the change of variables

mac) = :1— ngoc) - 6.061 1 s .- s r

r—i

For the linear HGO, 17(k) satisfies the equation

1
”(k + 1) = —€r-1 D(I - DdoCc)$(k + 1) + DCaAaCJoID-ITIUC)
1 _
“Er—_‘chdoleoCc + A4004: (1 - DdoC'cHzUc) (3.18)

Using the values of Ada, Baa, 0,1,, and Ddo from Table 3.1, we rewrite (3.18) for each
of the three discretization methods.
Forward Difference Method:

no: + 1) = Adon(k) + €r1_10[:r(k + 1) — :I:(k) — new» (3.19)

 

86

Backward Difference Method:

1
6r-l

 

17(k + 1) = AdonUc) + AdoD[$(k + 1) — :1:(k) — TAcx(k + 1)] (3.20)

Bilinear Transformation Method:

 

1 MgD{[:c(k + 1) — 1:06)] — g—ACW + 1) + 3(a)} (3.21)

er-l

1706 + 1) = Kean“) +

where Ado = MzAdoMg-l.
For the nonlinear HGO, n(k) satisfies the equation

1
Gr—l

-TBc¢o($(k) - N WC). C (k), ”(10) (3-22)

17(k + 1) = Adofl(k) +

 

D[:I:(k + 1) - :I:(k) - TAc:r(k)]

where N = e"1D‘1; N (e) is an analytic function of e. The estimate 52(k) is given by

£5505) = $505) - Gr—‘fli(l€), l S ‘l S 1'

3.3.3 The Discrete-time Singularly Perturbed Form

The closed-100p sampled—data system can be represented at the sampling points by
a discrete-time model in the singularly perturbed form:

x(k+l) = x(k)+6‘1'(x(k).U(k),6) (3-23)
€(k+l) = A1606)+6P(x(k),u(k),=i(k),e) (3-24)
WC) = 7(i(k),C(k)) (325)
W) = $06)-N(6)€(k)+6W($(k),€(k).6) (3-26)

87

where W(z(k),C(k), e) is a continuous function of e at e = 0, and A f is independent
of e and its eigenvalues satisfy IAI < l. The fast variable C replaces 17 through a
change of variables. To illustrate the idea of arriving at (3.23) and (3.24), let us note
that substitution of $(k + 1) using (3.17) in (3.19), (3.20), (3.21), or (3.22) yields an

equation of the form

We + 1) = AMUC) + 6h(x(k), “(11). 6) - éch¢o(5=(k). ((1011105))

r-1 1 .
+Z—€,_l_jH,-DAgx(k) (3.27)

i=2

where the function h(-) and the matrix A I depend on the observer equation; that is,
equations (3.19), (3.20), (3.21), or (3.22) yield different h, A f, and HJ’s, and (to = 0
for linear observers. For some values of r, equation (3.27) is in the form (3.24). In
particular, for r 5 2, the summation term on the right-hand side of (3.27) does
not exist. Therefore, the equation takes the form (3.24), and (3.26) simplifies to
:i:(k) = z(k) — Nn(lc). In the case of a linear HGO discretized using the bilinear
transformation method (3.21), the summation term on the right-hand side of (3.27)
takes the form 23210) = 0(6) which can be grouped with h(-). In this case, (3.27)
takes the form (3.24) for r S 3 and (3.26) simplifies to :i‘:(k) = 1:05) — Nn(k).

In the more general case where r > 2 (or r > 3 in the bilinear case), (3.27)
is not in the form (3.24) since the summation term is of the order l/e"3 (l/c"‘
respectively). We can perform a change of variables to weaken the :r-input to the
n—equation. Using the fact that, for sufficiently small 6, x is much slower than 11, we

consider the quasi-steady-state value of 1) given, up to an 0(6) error, by

r—1
1 .
17 = Afr] + :2 mHjDAiz
J:

88

OI'

r-l
_ 1 -
n = (I — Ar) ‘12:; £,_,_,.H,-DA:1-

The change of variables

1
67-1-1.

g = 1, _ (I — A ,)-l g HjDAix (3.28)

-l
1:2

 

transforms (3.27) into

Ar£(k) + 671(x(k), “(10.6) - 60331545“), C (k). u(11))

r-1 1 ~ .
+2711. amaze) (3.29)

i=3

((1: +1)

Inspection of (3.27) and (3.29) shows that (3.29) has the same form as (3.27) except
that the summation term is of order l/e"4 compared with 1/e"3 in (3.27). Repeated
application of this transformation yields (3.24). Noting that in the change of variables
(3.28) the coefficient of a: has negative powers of 6, it is not obvious why on the right-
hand side of (3.26) :1: appears as :I:+6W(z, C, c) where W(:z:, C, 6) has only nonnegative
powers of c. This crucial fact can be verified by tedious calculations which can be

found in Appendix A .

3.4 Performance Recovery

Now we show that the output feedback sampled-data controller

1101) = 701(k). ((1‘))

recovers the performance of the state feedback controller (3.5) for sufficiently small

6. The performance recovery manifests itself in three points. First, the origin

89

(:1: = 0, z = 0, u = 0, :i' = 0) of the closed-loop system under the sampled-data
output feedback controller is asymptotically stable. Second, the sampled—data out-
put feedback controller recovers the region of attraction of the continuous-time state
feedback controller in the sense that if ’R. is the region of attraction under state feed-
back, then for any compact set 8 in the interior of ’R. and any compact set Q C R',
the set S x Q is included in the region of attraction under sampled-data output feed-
back control. Third, the trajectory of (1:, z, u) under sampled-data output feedback
control approaches the trajectory under continuous-time state feedback control as
e —) 0.

The closed-100p sampled—data system is represented in the singularly perturbed

form

x03 + 1) = x06) + 67"(x(k). ((15), 6) (3-30)
506 + 1) 415(k) + 69(x(k), ((k), 6) (331)

where

f(X’ g? 6) = W(X’ u, 6), G(X’ £3 6) = F(X’ u, it e)

and u and :i‘: are given by (3.25) and (3.26), respectively. The functions .F(-) and Q'()
are continuously differentiable functions of their arguments and globally bounded
functions of C. At 6 = 0, (3.31) reduces to

5(k +1) = «415(k)

whose steady-state solution is C = 0. Substitution of C = 0 in (3.30) results in the

reduced system

90

1106 + 1) = x06) + 67"(x(k), 0, 6) (332)

From (3.12), we see that ‘Il(x,u, 6) can be written as

1'66 u. 6) = F (x, u) + 6106 u, 6) (333)

where <I>(x, u, e) is a continuously differentiable function of its arguments such that

<I>(0,0,e) = 0. Moreover, from (3.26) we see that

:t]5=o = :I: + £W(:I:, 0, e) (3.34)

Using (3.33) and (3.34) we can rewrite the reduced system (3.32) as

X“ + 1) = X“) + 6F (x(k).7(x(k), C (k))) + 621(x(k),7(x(k). C(16)). 6) (335)

where <I> is a continuously differentiable function of its arguments and

||‘1>(x, 7(x. C), 6)II S llxll

3.4.1 Boundedness and Ultimate Boundedness

The boundedness and ultimate boundedness of trajectories are established in the
following theorem, where we show that trajectories of equations (3.30) and (3.31)

starting in S x Q come arbitrarily close to the origin as time progress.

Theorem 3.1 Let Assumptions 3.1 to 3.3 hold; then there exists 6; > 0 such that
for every 0 < e _<_ e}, the trajectories (x05), C (16)) of the system (3.30)-(3.31) starting
at x(0) E S and (2(0) 6 Q are bounded for all k 2 0. Moreover, the trajectory x(t) is

bounded for all t 2 0. In addition, given any 7‘ > 0, there exist 6;,T1 and an integer

91

16‘ (all dependent on 7') such that, for every 0 < c g 6;, we have
”606)" S T, V k 2 16‘, ||x(t)|| S T, V t 2 T1

Proof: We divided the prove to two parts.

Proof of Boundedness

We start by proving boundedness of (x(k),C(k)) at the sampling points, then we
prove boundedness of x(t) for all t. Boundedness at the sampling points can be
shown in two steps:

First : we show the positive invariance of an appropriately chosen set A, which
is arbitrarily small in the direction of the fast variable C.

Second : we show that any closed-loop trajectory, starting in the compact set
S x Q, enters the positively invariant set A in finite time.

We know that the equilibrium point of the continuous-time system under state
feedback is asymptotically stable with a region of attraction R. By the converse
Lyapunov theorem of Kurzweil [39] there is a continuously differentiable Lyapunov
function 1’00 and three positive definite functions U1(x), U2(x) and U3(x), all de-

fined and continuous on R, such that

U1(x)SV(x) S 0100

x133311211100 = 00

231%, 7(3. 0) S. -U3(X)

Writing the forward difference of the Lyapunov function and using the mean value

92

theorem we have
Amos» 6 vow: + 1)) - vows» = g-(nemxs + 1) — x(k)l

where n(k) is a point on the line segment joining x(k) and x(k + l). The propemess
of V(x) in R guarantees that with any finite co > mazxestc), the set (I = {x E
R : V00 5 Co} is a compact subset of R, and S is in the interior of (2.

Due to assumptions 3.1 to 3.3, we have for all (x,C) 6 (I x R'

III-"(1.6 on s K1. ||<I>(x.7(x(k).C(k)).6)ll 5 K2. "900516)" 5 K3

where K1, K2, and K3 are positive constants independent of e. For C; > co and
63 > 0, let

{2: {x6 R:V(x) 3 c2}, A=D x {W(C) Sc362}

For any 0 < E < 1, there is L, independent of e, such that for all (x,C) E A and

every 0 < e < E we have

"$006, 6) - $00016)" S LIIIEII

For the boundary-layer system, we define the Lyapunov function W(£) = CTPC,
where P is the positive definite solution of the Lyapunov equation A'fPA, - P = -I.

This function satisfies

Am1n(P)||€ II2 S W(E) S lll’lllléll2

where "P" > 1.

We start by showing that there exist positive constants c; and 61 (dependent on

93

c3) such that the compact set A is positively invariant for 0 < 6 < 61.

|/\

|/\

IA

AV(x(k))
%¥(N(k))[x(k+ 1) -x(k)]

%(x(k))lx(k+l)-x(k)l+[—(~(k))-—(x(k)) [x(k+1)- x(k)l
%;(x(k))(es(x(k),z(o. e» + L2||x(k + 1) - 1(1)"?

—€U3(X(k)) 'l' 5 gzl

 

 

(x(k))[[ Ilfl 1(6) 6(1) 1) — f(x(k) o e)"

+12 —(x(k))" ||<1>(x(k). (mac) ((1)) e)" + L2||x(k + 1)— x(k)||’
—€U3(X(l€)) + €L2L1[[€(l€)" + €2L2K12 + 6211ng

 

 

where ||[6V/0x](x)|| g L; in Q. In A, we have

Thus

and

Am.-..(P)||€||2 S W(€) S 636’

usu< (A; We)

AV(x(k)) S 6l-Ua(x(k)) + 6K4] (335)

where K4 = L1L2V63/Amgncp) 'l‘ L2(K? '1' K2).

For any positive constant 6, < 02, trajectories starting inside {V(x) 5 5,} cannot

leave {V(x) S c2} provided 6 is sufficiently small. This is seen from

V(X(k +1)) S V(x(k)) - 6U3(X(k)) + 3K4 S 66 + 62K: < 02 (3-37)

whenever 62 < (f g (o; — Ea)/K4. Trajectories starting inside {V(x) 5 0;} but

94

outside {V(x) 5 6,} remains inside {V(x) _<_ c2}, provided 6 is sufficiently small.

This is seen from

V(x(k+1)) S V(x(k))-6U3(x(k))+6’K4

< c2 — cpl + €2K4 < 02 (3-38)

whenever e < 62 11.5: p1 /K4, where p; = min-5,514,050, U3(x).
Thus, we conclude thatfor 63 S min{61, 62}, the set {x 6 V(x) 5 ca} is positively
invariant for 0 < e < 63.

On the other hand, from (3.31) we have

AW(£ (11)) = 6% + 1)P€(k + 1) - ET(k)P£(k)
S {RICH/11154: - P1606) + 26||€(’€)||||ArllIIPII"90416).{(16).6)”
+6’ll9(x(k).£(k), 6)||’||Pll
s -llt(k)||2 + 6 (2||£(k)|||lArllllPllK3 + eKillPll)
Hence

W050: + 1)) s (1- 117M)" was» + 6KallPll(2II£(k)llllA;|| + ex.) (339)

Since C(k) e {W(C) 5 C362}, we have

W(c(k+1)) s (1- -"Ii—Tl)cac’+2e’KallPllllArllA": —(——,m,,,

+6’K3IIPII
Choosing c3 large enough ensures that

wow: + 1)) s sic, (3.40)

95

We conclude from (3.37), (3.38) and (3.40) that the set A is positively invariant.

Now we consider the initial state (x(0), 2(0)) 6 8 x Q. It can be verified that the
corresponding initial state {(0) satisfies ||C(0)|| 3 Us ’1, where 1 depends on S and
Q. Using the fact that x(0) is in the interior of (2, it can be shown from (3.30) that

:7(x(i).£(i).6)

i=0

||x(k) - x(0)|| = 6 _<_ ekK1

 

 

 

 

as long as x(k) 6 fl. Therefore, there exists a positive constant K5, independent
of e, such that x(k) 6 Q We 5 Ks/e. Let it, be the first time C(16) enters the set
{W(C) _<_ C362} and take It, s min{l}, — 1, Lil}. During the time interval [0,76,], we
have

5 ¢ {W(€) S C362} and We) 6 (2.

Since x(lc) 6 f2, inequality (3.39) is satisfied. We rewrite (3.39) as

1 W(£(k))
_2_”P_"W(§(k)) + eKallPll (ZIIAIII m 1‘ ‘K3)[ -

l
+ (1 — m) W(€(k)) (3.41)

W(E(k + 1)) S

 

where we used the fact that "CH2 5 W(£)/z\,,,~,,(P). Choosing c3 large enough ensures
that, for W(C(lc)) 2 C362, the bracketed term on the right-hand side of (3.41) is non-

positive. Hence

W(£(k + 1)) S AW(E(k))

where 0 < A = [1 — l/(2||P||)] < 1. Therefore W(C(k)) S z\"W(C (0)). Recalling that
W(€) S II1"IIII€(’6)||2 and ”6(0)" S U6”, we obtain

12
—2

€2r

W(1.106)) S A‘llPll

 

(3.42)

96

which shows that lim),_,,, W(C (16)) = 0. Hence, {(16) must enter the set {W(C) S ezca}
in finite time. From

12
AkllPll€2r_2 S 6263

 

we see that C(k) will be inside {W(C) S e’ca} for k satisfying

 

1n 12
k 2 (‘1 )
ln (1')
where K, = l2||P||/c3. To ensure that C(k) enters {W(C) g 6263} before x(k) leaves

0, we choose 64 small enough that for 0 < e g 64

Mitts...

1n 6) .

 

or equivalently

eln (%) + eln (g—f) < K5ln (%) (3.43)
Such a choice of 64 is possible since the left-hand side of (3.43) tends to zero as 6
tends to zero.

Taking e; = min(€, £1, 64, e3) guarantees that for every 0 < e _<_ e}, the trajectory
(x(k), C(k)) enters A during the interval [0, A(e)] and remains inside thereafter, where
A(e) is the integer part of Ks/e. Thus the trajectory is bounded for all k _>_ A(e)
and it is bounded for k 6 [0, A(e)] by virtue of

||x(k) - x(0)|| S 6kK1

and (3.42).
Now we show the boundedness of x(t) for all t. The continuous time system (3.11)

can be represented by

97

1(1) =x(k)+ [greets/mar (3.44)

for IcT 5 t S TU: + 1). Add and subtract 1):, F(x(k),u(k))dr to (3.44), to obtain

x(1) = x(k)+ [;F(x(k),u(k))dr

+1; IF(x(Ar).u(k)) — meme») «6 (1145)
item the Lipschitz property of F(x, u) we have
(woo). «1(1) - F(x(k). 21(1))" s Ls||x(t) — 100"
Using this inequality in (3.45) yields
"1(1) - won s TuFococ). 11(1))" + [T Lsuxo) — x(k)||dr
Apply Gronwall Lemma to obtain
||x(T) - 2(6)" 5 TeL’TIIF(x(k),u(k))II

Let F(x,u) 9- M(x,C). Since F(-, -) is continuous and F(0,0) = 0, M(-, -) is con-
tinuous and M(0, 0) = 0. Therefore

||x(t)-x(k)|| S TKrllM(x(k),£(k))ll (3-46)

where K7 > 0. Thus, boundedness of x(k) and 60:) implies boundedness of x(t) for
all t 2 0.

98

Proof of Ultimate Boundedness

Next we show that trajectories of equations (3.30) and (3.31) starting in S x Q come
arbitrarily close to the origin as time progress. We know that for every 0 < e _<_ e}
the trajectories of the closed 100p system, starting from (x(k),£(lc)) E S x Q are
inside the set A for all k 2 76. Thus, we can find 65 = 65(7’) 5 e} such that for every
0 < e S 65 we have ||C(k)|[ S r, V k 2 It. Since (I is bounded, there is r, > 0 such
that Q C B(0, r,) n R.

To prove ultimate boundedness we need to show the following steps

First : Tl'ajectories starting inside the set

A = {V(x(k)) S 62} f1 {W(€ (11)) S 6362}
reach the set
I") = {V(x(k)) _<_ 2(6)} 0 {W(£(k)) .<. c362}

in a finite number ‘of steps I: = k‘, where lim,_,o c5(e) = 0.

Second : There exists a positive constant c, > c, (dependent on s with
limHo c4(e) = 0) such that trajectories lying in A at time k = lc‘ must belong
to the set

A = {V(X(k)) S 64(6)} n {W(€(k)) S 0362}

for all k _>_ k‘ + 1.
Using (3.36), it can be shown that for all (x,£) e A we have

s 1
VlXU‘ + 1)) S V(X(k)) " §U3(X(k)) - ‘5 [5U3(X(k)) - 6K4] (3-47)
Since U3(x) is positive definite and continuous, the set {U3 (x) S 26K;} is a compact

99

set for sufficiently small 6. Let c5(e) = maxu,(x)52,K,V(x). Whenever V(x(k)) >
c5(e) we have U3(x(lc)) > 26K4, and

vow: + 1)) _<_ 142(k)) — §Us(x(k)) (1421)

“Starting with V(X(0)) > c, it can be seen from (3.48) and mathematical induction
that as long as V(X(k)) > C5(6)

I:
VlXU‘ + 1)) s V(x(0)) — 522.6(1))

i=0

< 141(0)) - $21114 = V(x(0)) - emu + 1)

i=0

This inequality shows that there is a finite time k' such that V(x(k‘)) S c5(e).
From now on we can use 16‘ as our starting point. It follows from (3.47) that if

U3(x(k')) S zero then
V(x(k‘ + 1)) .<. V(x(k‘)) + 8K4 5 nos) + 52K4 2 «(6)
Else if U3(x(k‘)) > 241,, then
V(x(k’ + 1)) s vows» - §Ua(x(k‘)) 5 one)
In general, if U3(x(k)) 2 2e11, for all k’ _<_ k _<_ It: + j, then from (3.47)

vote: + ,- + 1)) s vows +2) — e [gators +1» — do]

i
s V(x(’6’)) - ego [gees +1» — an]
S V(x(k‘))Sc5(6)

100

In the second and complementary’case U3(x(k)) > 2€K4 for all 0 < k‘ 5 k S
16‘ + j - l, but U3(x(k’ + j)) 5 26K4, it follows from (3.47) that

V(x(k‘ +1 + 1)) s V(x(k' +1» — «2 [éveodk' +1» — do]

|/\

vows» — ’2 [$036416 +1» - em]

‘6 [éU3(X(k:: 1)) " 5K4]
V(X(k.)) - e [$036202 +1» - em]

c5(e) + €2K4

l/\

|/\

Thus,
V(X(k + 1)) S 64(6). V11 2 16"

From now on, if c5(6) < V(x(k)) S c4(e) we have U3(x(lc)) > 26K4 which implies
that the Lyapunov function as seen in (3.47) is decreasing and for V(x(k)) S c5(e)
where U3 (x(k)) 5 2€K4 the Lyapunov function may increase but V(x(k + 1)) 5 c4 (6)
for all k > 15'. From (3.46) and the fact that limHo c4(e) = 0, we conclude that there
is a finite time T1 such that, for sufficiently small 6, ||x(t)|| S r for all t _>_ T1.

3.4.2 Exponential Stability of The Origin and 'II‘ajectory

Convergence

/

At this stage of the analysis we assume that the origin x = 0 of the continuous time
system
2 = 1"(x,”1(=v,C))'1g Fe(x) (3.49)

is exponentially stable and place ourselves in small ball of radius r > 0 around
the origin; the value of 1' will be determined later on. Theorem 3.1 guarantees that
trajectories of the system (3.30)-(3.31), starting in S x Q, enter this ball in finite time

101

and stay thereafter. This result allows us to study the trajectory convergence. Let
x, (t) be the solution of the continuous-time closed-le system under state feedback,

starting at x,(0) = x(0), i.e.,

Xr = FC(XT)) Xr(0) = X(0)

The following theorem shows that the origin is exponentially stable and x(t) = x(t, e),
the solution of the sampled-data closed-lOOp system under output feedback, converges

to x,(t) as e —) 0, uniformly in t, for all t 2 0.

Theorem 3.2 Let Assumptions 3.1 to 3.3 hold and the vector field F,(x,.) be continu-
ously differentiable around the origin. Moreover, assume the origin of the closed-loop
system under state feedback is exponentially stable. Then, there exists 6; > 0 such
that for every 0 < e S 6;, the origin of the system (3.30)-(3.31) is exponentially sta-
ble. Furthermore, given any 1’ > 0, there exists 6; > 0 such that for every 0 < e S 6;,

we have

le(t) - Xr(t)ll < 'r. V t 2 0

Proof: The proof consist of two parts. First, we show exponential stability of the

origin. Second, we show trajectory convergence.

Proof of Exponential Stability of The Origin

Theorem 3.1 guarantees that trajectories of the system (3.30)-(3.31), starting in
S x Q, enter this ball after a finite time and stay thereafter. Due to exponential
stability of (3.49), there exists a continuously differentiable Lyapunov function V(x)
[34] defined over B(0, r2) C R for some r2 > 0, and four constants al, a2, a3 and a4

such that for all x E B (0, r2) we have

aillxll’ S V(x) S callxll2 (3-50)

102

OVF
S -013||x||2 (3-51)

6X3”;— 3] s cenxn (3.52)

For the fast variables {(16), we use the Lyapunov function W(£(k)) = CT(k)PE(k).
The composite Lyapunov function v = V + W satisfies

Av = AV + AW
1 l - ..
< "560311X(k)ll2 " EIIEUCNP - G‘I’TAa‘I’

where

A3: ice-661 492—653 5,: le(k)ll

-/32 - 6163 +51; - r34 — 6135 l|€(k)l|

for some nonnegative constant fig. Hence

6 1
Au 3 jeans»): - ,utu’

for sufficiently small 6. Then, there exists 6; > 0 such that for every 0 < e S 6;, the

origin of the system (3.30)-(3.31) is exponentially stable.

Proof of Trajectory Convergence

The outline of the proof goes as follows. Let 2,06) be the solution of
22.01 + 1) = 12.01) + Teen». 2240) = x(0)

and show that for [IT 5 t 5 (k + 1)T, le,(t) — 2,06)” < r/2. Argue that for any
1' > 0, 3 e; > 0 such that ||x(k) - )2,(k)|| < r/4 V k 2 O and 0 < e 5 6;. Using
(3.46) argue that ||x(t) — x(k)|| = 0(6) < r/4. Finally, show that ||x(t) — x..(t)|| < r.

103

We proceed to prove the first part. The solution of the reduced system under state
feedback is given by

2(1) = xr(k) + [T1 Fe(xr(T))dT. x.(0) -—- x(0) (3.53)

where t 6 [Tk,T(k + 1)]. By adding and subtracting F,(x,(k)) to the integral in
(3.53), we have

erc + 1) = 111,06) + ”lbs (16)) + T29(Xr(k)) (3-54)
where ||g(x,.(k))|| s K... The 0(T2) approximation of (3.54) is
so: + 1) = 55-06) + TF.()2r(k)). 2(0) = x(0) (3-55)
Define e = x, — 2., where e(0) = 0. Then
6(16 + 1) = 6(16) + TFe(6(k)) + TAFe(xr(k)i e(k)) + T29(xr(k))

where AF,(x,, e) = F,(x,) - F,(xr — e) — F,(e). Using the mean value theorem with
the scalar function AFc,(x,, e), we get

as, or,
ax, (exr + (1 - (9)6) - 0xr (106)] 6

 

 

A1"‘e(xr,e) = [

where 0 < o < 1 and 0 < p <1. Therefore
llAFC(Xn 6)" _<_ QLrlerllllell '1' MI - a - PIIIBII2

IIAFc(xr.e)ll2 S 51 llxrll4 + 52||6||‘

104

llFe(6)|| S lel6||
"606 + 1) - 6(k)||"’ s m. [||F.(6(k))||2 + ((46.6.6). coo»? + r223]
The Lyapunov function V(e) satisfies inequalities similar to (3.50 )-(3.52) in e in the
neighborhood of e = 0. Then
6V
AV = V(6(’6 + 1))- V(6 (11)) — 732-061(16)) [e(k + l) - 6(6)]
—(e(k» (eve + 1) - 11(6))” + [— (111(k)) — —(e(k»] (ea. +1) - es»

-:r [-2--049L7|IX(’1)||[ l|e(k)||’+TB(-) (1156)

IA

where

so = {—929- + «1.1221 - 9 - p|||6(k)||] l|6(k)||’
+Ar[e.1<. + man: "6(6)" + TLe535elle(k)ll ] l|6(k)ll
+T [IIII¢5:i<llll)(r(k)ll4 '1' T2L953Kg]

Using the fact that the solution of the nominal system decays to zero exponentially
fast, we have ||x,(k)|| _<_ arm”. Also for all Hell 5 r1 < r,, with 0 < r1 5
a3/[2a4L7ll - g — pl], we obtain ||B(-)]| 3 T6,. Then from (3.56) we obtain

V(e(k+1» _<_ V(6(k))-T[%i-aieL7llx(k)ll]ll6(k)ll’+T251
= [1_T9;+Te.eLnIIx(k)u]V(e(k»+T’ar (3.57)

We can relate to the initial condition as follows:

V(e(k+1» .<. II[1-r—+Te.eL..ux(1»I]V(e(o»

j=i+l

105

+T’6,
i=0 j=i+l

k k 03
1+2 II [1-T—,—+creux(t)u]]

 

From e(0) = 0 and due to the exponentially decaying bound on llxll, there is a

number of samples p such that for
o i < p, a4ngope‘°3T‘ 2 921, l < l - T921 + TaAQmee’asT‘ S 65
o i > p, owl/mafia“ < 9,1, 0 < 1 — T? + Toz4ngpe"‘”"37’i 5 A1 < 1

Then
I:
V(6(k + 1)) S T264 [1 + Z agar-P] = T264 [1 + ant-P]

i=0
and we have

"606 + 1»): s air/(err + 1)) 5 r21, [1+ 56,1114]
l

”606 + 1)II S TJ

 

64(1 + 65A'f") <
01

rel-t

for sufficiently small T. Thus,
use) - 2(4)" < 3
Consider now the difference x, (t) — x,(lc).
use) - 2(6)" 3 TIIFe(x.-(k))ll + f; Lianne) - sooner
Using Gronwall lemma for t E [kT, (k +1)T], we obtain
"26(1) -— 36(6)" 3 TeL'°TIIF.(x.-(k))ll s TKe

Thus,
||Xr(t) - 55(6)" = ”260) - xr(k) + 260:) — 2.0:)" < 3

106

In the second step we show that ||x(lc) - )2, (16) H < r/4. We partition the time interval
[0, 00) into two intervals [0, A(e)] and [A(e), 00), where A(e) is the number of samples
needed to get the estimation error C (k) to 0(6), as defined after (3.43). In the first
interval [0,A(e)],

x01) = 2(0) + 6”i_:1-7’()c(1’)1€(1)115)

i=0

k-l

llXU‘) " X(0)ll S 6% K1: ekKl

where ||f(x, C, 6)” 5 K1 Vx E 52. Hence
||x(k) - x(0)|| S 6A(6)K1 V k E [0115(6)]
The same can be done for )2, and we conclude that
”25(6) - 25(0)" S 6A(6)Ke
where Fc (x) S K 10. Therefore
llx(k) - 2.6)) s mean + K10) < {- v k e (o. 4(6))

for sufficiently small 6. Notice that the effect of the estimated error C does not appear
in the picture because of the saturation of the control in f(-). At the end of this
period, as in Theorem 3.1, we have "C(15)” 3 «$56 = (IT; also ||x(lc)-)Z,(k)|| < r
and ||x(k)|| < r We 6 [0,A(e)].

In the second time interval [A(e), 00), we need to study the difference x(k)—X,(k).
Since

X(’6 + l) = x06) + T7:(x(k),£(k)i 6). x(0) = X0 (353)

107

where f (x. 6, 6) = 44666) + ”()6 706.016) and 114666) = F6511), let us con-
sider (3.58) as a perturbation of (3.55) and define the difference E = )2 - )2, where
||‘é(A(e))|| < r/4 at the end of the first interval.

6'(16 + 1) = 606) + TFc('6(16)) + T5(x(k). 6(6), 606)) + T21’(x(’6)i 706(6), ((16)), 6)

where £(x, C, E) = M(x, C) — F,()2,) — F,(é). By adding and subtracting F,()2(k)) to

£(-) we rewrite it as

5066, 6') = M(x,€) - Fe(x) + AM(X1 6)

where AM(x, E) has the same structure of AF,(x,e). Repeating with E the same
argument used with e, we can show that ||é(k)|| 5 r/4 We 2 A(e). Using (3.46),
we obtain

||x(t) - x(k)|| s mummies») s 1;. w. 2 o

for sufficiently small T. Then

||X(t) - 56(6)" S ||x(t) - x(k)|| + ||x(k) - Xr(lc)|| < g

Finally,
||x(t) - xr(t)ll S '||x(t) - $606)” + llxr(t) - iii-(16)"
r r
< — — =
- 2 + 2 T
which completes the proof.

108

3.5 Example on the Recovery of the Region of At-
traction

We apply the deve10ped technique to show how we recover the region of attraction

as 6 tends to zero.

Example 3.1 Consider the second order system

i, = :52 (3.59)
:52 = 2161+ 10 tanh(u) (3.60)

which has an exponentially unstable mode. The bounded state feedback control
a = — tanh(:rl + 2:2) stabilizes the origin and results in a finite region of attraction.
To implement the sampled-data controller, we consider the discrete-time linear full-

order HGO (discretized by the bilinear transformation method)

(106 + 1) = Area“) + 3.100(6)

5606) = Caeq(k)+Daey(k)

where
1 1 — a1 - 02 1 2 (11 + 02
A,, = —— , Bdo = _—
d8t() —202 1 + (11 — 02 det(-) (12
1 1 1 2 (11 '1' 02
0.. = —— , D... = —
det(-) _ez _1_+_6n det(-) 23

£ 6 E

and det(-) = 1 + a; + 012. We take the observer initial conditions as :i:(k = 0) = 0.

109

 

 

 

 

 

 

 

 

 

Figure 3.1. Recovery of region of attraction: 6‘ = 0 (solid), 6" = 0.007 (dotted),
e‘ = 0.057 (dashed), and e‘ = 0.082 (dash-dotted)

The observer parameter c = 0.5T and the observer poles are real due to the choice
a1 = 2, a2 = 1. This example shows how the sampled—data output feedback recovers
the region of attraction achieved under state feedback.

Figure 3.1 shows the region of attraction under state feedback control (the solid
line), and three compact subsets that are recovered using the output feedback
sampled-data controller. For each specified compact subset there is a design pa-
rameter 6' such that for every 6 _<_ e‘ the sampled-data output feedback controller
is able to recover the given subset. The upper bound 6‘ is tight in the sense that
for e > 6' a part of the set is not included in the region of attraction. The bounds
6' = 0.082, 0.057, and 0.007 correspond to the three sets, from smallest to the

110

largest. This show that as 6 gets smaller we recover the region of attraction under

state feedback control.

3.6 Conclusion

In this Chapter we study sampled-data control of nonlinear systems using high-gain
observers. The observer is designed in continuous time, then discretized using three
different discretization methods. To perform the closed—100p analysis we represent
the interconnected system in a singularly perturbed form. The reduction of the neg-
ative power of e is done in number of steps to reach the singularly perturbed form.
Closed-loop analysis shows that the sampled-data controller recovers the performance
of the continuous-time controller as the sampling frequency and the observer gain
become sufficiently large. Performance recovery is shown in two steps. First we
show boundedness of trajectories which come arbitrarily close to the origin as time
progress. Second, we assume that the origin of the continuous time system is expo-
nentially stable and show that trajectories enter some ball in the neighborhood of

the origin in finite time and stay thereafter.

lll

CHAPTER 4

Application to the Pendubot

4.1 Introduction

One of the usual concerns with high-gain observers is how they function in the pres-
ence of noise. This can be addressed either by performing additional analysis to
study the effect of noise or by carrying out simulations and experiments. In Chapter
2 we used simulations to study the effect of noise. In this chapter, we study the
behavior of a closed-loop system by performing experiments. The performance of
the closed-100p system will reveal the effect of noise and the choice of the sampling
rates.

This chapter applies the high-gain observer in the control of the Pendubot, an
electro-mechanical system consisting of two rigid links interconnected by revolute
joints. The pendulum motion is controlled by actuation of the first link. We perform
experiments to confirm the open-loop conclusions of Chapter 2. The experimental
results confirm all the conclusions of that chapter. Moreover, we perform experiments
to confirm the closed-100p results of Chapter 3 regarding ultimate boundedness and
trajectory convergence. We use the experiment to compare the use of high-gain
observers as a tool for calculating velocities from position sensors with the engineering

practice of using Euler’s formula (difference method). The algorithm provided by the

112

manufacturer of the pendubot use Euler’s formula. The chapter starts by introducing
the pendubot in Section 4.2. The mathematical model and the equilibrium manifold
are described in Sections 4.3 and 4.4, respectively. Because of the nature of the
pendubot, the control algorithm is divided into swing up and balancing controls.
Detailed description of that is given in Section 4.5. The hardware description and

the experimental results are in Sections 4.6 and 4.7.

4.2 What Is The Pendubot?

The pendubot (Pendulum Robot) is an electromechanical system consisting of two
rigid links interconnected by revolute joints. The first joint is actuated by a DC-
motor while the second joint is unactuated as a simple pendulum.

The Pendubot is, in some ways, similar to the inverted pendulum on a cart, where
the linear motion of the cart is used to balance the pendulum. The Pendulum uses
instead the rotational motion of the first link to balance the second (pendulum) link.
In this regard, the Pendubot is also similar to the more recent rotational inverted
pendulum, pioneered by Professor Furuta of The Tokyo Institute of Technology [22].
In the rotational inverted pendulum, the axis of rotation of the pendulum is perpen-
dicular to the axis of rotation of the first link. The Pendubot has both joint axes
parallel, which results in some additional rotational coupling between the degrees of
freedom. This additional coupling, which is not found in either the linear inverted
pendulum or the rotational inverted pendulum makes the Pendubot more interesting
and more challenging from both a kinematic and a dynamic standpoints. For ex-
ample, in both the linear inverted pendulum and the rotational inverted pendulum,
the Taylor series linearization computed around any operating point results in a con-
trollable linear system. Moreover, the linearized model (A,B,C) is the same at all

operating points. With the Pendubot, the linearization is operating point dependent;

113

 

 

 

 

 

 

 

 

 

 

 

)__—I
I Table ;
Lin“
when
(6) e=

 

 

 

 

L)

Figure 4.1. Front and side view drawing of the pendubot

in other words, the linearization (A,B,C) changes at each configuration and there are
even special configurations where the linearization is uncontrollable.

The Pendubot possesses many attractive features for control research and edu-
cation. It can be used to investigate system identification, linear control, nonlinear
control, Optimal control, learning control, robust and adaptive control, fuzzy logic
control, intelligent control, hybrid and switching control, gain scheduling, and other
control paradigms. One can program the Pendubot for swing up control, balancing,
regulation, tracking, identification, gain scheduling, disturbance rejection, and fric-
tion compensation to name just a few of the applications. Some of these applications
are described in [59, 60, 58]. The maker of the Pendubot is Mechatronic Systems,

Inc. Figure 4.1 shows front and side view of the pendubot.

114

4.3 Mathematical Model

The pendubot is a two link robot with one actuator. The (Euler-Lagrange) equation

of motion is [61]

D(q)<’1' + 001,102+ 9(1)) = '7' (41)
where
. . . . h
D(q) = dll d12 ’ C(q,(j)(j = M2 M2 ‘1' (“II (11 = 1
d2) 6122 —hq'1 0 ‘12 he
9(9) = (#1 r '7’ = T
432 0

d1, = mag, + mg (13 +13, + 2111., cos(qz)) + 11 + 12
(122 = "82122 + I;
d, = as, = m2 (If,2 +1.1c2 cos(q2)) + 12
h = —mglll,2 sin(q2)
hr = -mrlrler sin(112)(i§ - 271121113 81110102221
h2 = "121110 811101063
4’1 = ("11ch + "1211) 9 c09491) + "inlay C0801) + <12)
(#2 = malezg 608((11 + qr)
'r : Torque applied to the first link by the DC — motor
m1(m2) : Total mass of link one (two)
ll : Length of link one
l,1(l,2) : Distance from base of the link to the center of mass of link one (two)

I) (12) : Moment of inertia of link one (two) to its centroid

115

90°

I
l
l
(D-
\
\

(:2

l

1«-——--—————s1

 

 

 

 

 

 

Figure 4.2. The Pendubot model parameters are shown on a pendubot schematic.

g : Acceleration of gravity

Figure 4.2 shows the parameters of the pendubot model. It also shows the direction

of increasing the angles. For simplicity let us define the parameters

01 = m1l§l+m2lf+ll
02 = "121:2'1'12

6'3 = melilez

116

04 = mich-l-mell

e?”
ll

melee

We can rewrite D(q), C(q,rj) and g(q) as

91 + 02 + 203 CW(Q2) 02 "l' 03 COS(Q2)

0(6) =
02 ‘l' 03 COS(Q2) 02
. -93 811162)?) -93 8in(112)((iz +61)
0(1), q) = . .
03 3m(92)41 0
049608011) + 959 608((11 + or)
g(q) =

95.9 c03(‘11 + 92)
Finally using the fact that the matrix D(q) is invertible, the equation of motion
is given by

('1' = D(q)‘1 l“? - C(q, (Di - g(q)]

4.4 The Equilibrium Manifold

If the pendubot is mounted so that the joint axes are perpendicular to gravity, then
there will be a continuum of equilibrium configurations, each correSponding to a

constant value of 7. Since at equilibrium ('1'1 = (j, = ('1'; = (jg = 0, we have

049608(qi)+059606(q1 +612) = 7

959008011 +92) = 0

117

Then, the pendubot will balance at

ll

8

m

l.
§

b
sle
V

111

q2 = n——q1; n=1,3

such that ('r/(04g)) S 1.

4.5 Controlling The Pendubot

The pendubot control strategy devel0ped in [60, 58] is divided into two parts: a
balancing control which balances the pendubot about one of its equilibrium config-
urations and a swinging control that swings the pendubot up from the downward

configuration to the inverted configuration.

Balancing Control

The balancing problem may be solved by linearizing the equation of motion about an
Operating point using a Taylor series expansion and designing a linear state feedback
controller. As we saw before, the pendubot has an equilibrium manifold which is a
continuum of balancing positions. The linearized system becomes uncontrollable at
q1 = 0, 1r as illustrated in Figure 4.3 which shows controllable and uncontrollable

positions of the arms. Notice that the reference position for ql is the horizontal axis.

Swing Up Control

The problem of swinging the pendubot up from the downward position to the inverted
position is an interesting and challenging nonlinear control problem. This problem

illustrates the nonlinear control ideas of nonlinear relative degree, partial feedback

118

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.3. The pendubot arm at a: controllable position, b: uncontrollable position

linearization and zero dynamics.

Partial Feedback Linearization

We take the point ((11 = —1r/ 2, q; = 1r) on the equilibrium manifold q1 + q2 = in/Z
as our desired equilibrium position and linearize the equation of motion about it.

Let us consider

61110.1 + 6112112 + hl + 4’1 -"-' 7' (412)

61214.1 + 6122112 + he + $2 = 0 (4.3)

119

Solving for {1'2 from (4.3) and substituting it in (4.2), we obtain

d d
(71(6111 — (13:21) + (’11— (11:22) '1' ($1 — 22:2) = 7'

Taking the control variable 1’ as

. = (1,, _ me)... (1.- 1,134..- .22..)

 

 

 

 

 

 

results in

91:01

(122512 '1‘ he + ‘22 “(121111

To track some reference signal (reference position) r = q‘,‘ and r = q‘,‘ = 0, the control

variable v1 can be designed as

or = kp(¢1i"' (II) + 111(1)? '" 41)

where the positive constant k, and k, are the control gains. Now we can define the

tracking error as our new states

eI=q1—q‘f r 62:41‘11’1’=41

01:02 r 772:2?

and rewrite the system as

C2 = —kp81-k482

120

1'71 =02
, _ 1
"2 de

11:61

(h. + e.) — g (6.6): — ,,) + (no: - 2))

2

The surface e = [e1 e2]T = [0 0]T in the state space defines an invariant manifold

for the system. The tracking error part can be written as

0 1
e = e = Ae
—k, -ka

Choose k, and k, such that A is Hurwitz. On the sliding surface e = 0, the dynamics
are given by

7’11 = 772 (4.4)

1 he

' = —— — — 4.5

172 (£22452 d” ( )
where

h2 = malllcg sin(q2)q¥ = 0
4’2 = melc29005(Ql’+7h)

dzz = "121224.12

which represent the zero dynamics with respect to the output y = e1. We see from
(4.4)-(4.5) that the zero dynamics are just the dynamics of the unactuated arm,
which has a periodic orbit. While the error e(t) converges to zero, the steady state
behavior for the first link converges exponentially to q, = —1r/2 for the middle
position (the first link downward and the second link upward) or q, = 7r/ 2 for the

upper position (both links are upward), and the second link oscillates about the

121

equilibrium (+1r, 0) for the middle position or (—1r,0) for the upper position. The
swing up control job then is to excite the zero dynamics sufficiently by the motion
Of link 1 such that the pendulum swings close to its unstable equilibrium. When the
pendulum is close enough tO the desired equilibrium position, we switch from the
partial feedback linearization controller to the linear balancing control in order to

balance the pendulum about the middle or upper position.

4.6 Hardware Description

The pendubot consists Of two rigid aluminum links Of lengths 14 in. and 8 in.
Link one is directly coupled to the shaft Of a 90 V permanent magnet DC motor
mounted tO the end of a table. The motor mount and bearings support the entire
system. Link one includes the bearing housing for joint two. The shaft extends
out in both directions Of the housing, allowing coupling to the second link and to
an optical encoder mounted on link one. The design gives both links full 360“ of
rotational motion. Link two is constructed of a % inch thick length Of aluminum with
coupling that attaches to the shaft Of joint two. The optical encoder resolution is
1250 pulse/rev.

All the control computations are performed in Pentium PC with a D/ A card and
encoder interface card. Using the software routines supplied with the pendubot, the
control algorithms are programed in C. Figure 4.4 shows a pictorial description Of

the interface between the pendubot and the controller.

4.7 Experimental Results

Because we only measure the positions Of the pendubot links, i.e., the angles ql and Q2,

by Optical encoders, we estimate the speeds ()1 and (jg. With 1:, = q1, $2 = ()1, $3 = Q2

122

 

 

 

 

 

 

 

 

 

 

 

 

 

 

”About J

 

 

Axisl
lnIeIfaoe
AxisZ

 

 

 

 

 

 

 

Figure 4.4. Pictorial description of the Pendubot’s interface with its controller

123

and x4 = ()2 as state variables, the states space model is given by

1131
532
1'3

:64

$2

$4

lie

We use a high-gain Observer to estimate the states. The nonlinearity of the system

is dependent on all states. If it is known exactly we can include it in the Observer

design resulting in the nonlinear full-order HGO

where

l01
00
00

 

00

0 0
0 0
0 1

 

00]

55
II

The Observer gains are designed as

 

 

“I”

124

“IN

COO

 

 

2 = Ace + B,¢,(:I:, 2) + L(y - 0,2)

1000
0010

where e > 0 is small parameter to be chosen. In the case of a linear observer,
4), = 0; hence, no knowledge Of the nonlinearity is included in the Observer design.
The discretization of the Observer can be implemented by any one of the methods
described in Chapter 2. We use the same state feedback control parameters for

output feedback. The system parameters are

0T: [0.0308 0.0106 0.0095 0.2087 0.063]

The HGO will be discretized with sampling period T and the Observer parameters
are as in L. We relate T to c by a = T/e. The parameter 6 can be made small by
either fixing T and increasing a, or fixing a and decreasing T.

The Optical encoder resolution will add noise to the links positions. The noise
depends on the number of pulses per revolution; the higher the number Of pulses the
smaller the noise. In our case, the Optical encoder resolution is 1250 pal se/ revelation
which means the maximum deviation for each link is 360/1250 = 0.288”. The ac-
tual deviation depends on the synchronization between the sampling points and the
Optical encoder pulses.

The pendubot starts at its only stable equilibrium point p (q) = —§, q2 = 0) as
shown in figure 4.1. The state estimates can start from the same initial conditions
Of the pendubot, which are known in this case, or with the default zero initial con-
ditions. Choosing the initial conditions Of the Observer at the default values brings
in peaking during the transient period. This phenomenon could derive the pendubot
to instability and saturation is used to limit the effect Of peaking. The physical
saturation Of the system by the manufacturer is :l:10 volts. We had to reduce the
saturation level in the case Of peaking to be able to balance the pendubot. The
saturation level depends on the sampling period T in such way that if T is large the

default saturation level is enough; otherwise we need tO reduce the saturation level.

125

4.7.1 Confirmation of the Open-LOOp Study

In Chapter 2 we arrived at various conclusions regarding the discretization Of the

high-gain Observer. The simulation-based study Of Chapter 2 addressed the following

issues:

1. the discretization method,

2. the order Of the Observer,

3. the effect of the ratio T/e,

4. the assignment Of the observer poles, and

5. the comparison with other numerical differentiators.

We use the pendubot experiment to address these issues. We need to balance the
robot arm at an unstable equilibrium point. Since we have two links, we use two
Optical encoders to measure their positions and then use the high-gain Observer to
estimate their speeds. The Sampling period range for the pendubot is limited. The
minimum sampling period we can get is Tm,“ = 0.0016 sec. The pendubot initial
conditions are :1: = [ ‘7', 0, 0, 0]T. The second link angle is zero along with the
extension Of the first link, as illustrated in Figure 4.2.

For large sampling periods the peaking phenomena is not that significant because
the system is only second order. When the sampling period decreases, the need for
saturation increases rapidly to be able to balance the pendubot. Experimentally, we
see the importance Of using saturation, especially when the high-gain observer starts

from zero initial conditions. The control input is limited to |u| < 10 volts.

Comparison of the Discretization Methods

In this part we compare three discretization methods (Bilinear, Backward Difference

and Forward Difference) for the linear high-gain Observer, with T = 0.003 sec.

126

 

 

 

 

 

 

 

 

 

Bilinear Trans. 0 = 1.0 1.5 2.0 2.5
Backward Diff. a = 1.2 1.5 2.0 2.5
Forward Diff. a = 0.8 0.9 1.0 1.1

 

 

Table 4.1. The values Of a = T/e for the different discretization methods.

a = T/e taken as shown in Table 4.1.

The choices for the bilinear and backward difference (BD) method were made
to check Figure 2.2 Of Chapter 2, where we found that for the highest derivative Of
the output, the bilinear method has less r.m.s. error than the BD method. This
will explain why we could not balance the pendubot arm by using BD method when
a = 1.0 while we were able to do it with a larger (1. Figure 4.5 (for T = 0.003
and a = 2.5) shows the position and the speed for both links when the steady state
position is (01 = —1r/2,02 = +1r). The speed of both links goes to zero at steady
state but, because of the Optical encoder resolution, we notice some chattering.

On other hand, the forward difference (FD) method will map the eigenvalues
of the HGO inside the unit circle. But if 0: increases while keeping T fixed, then
the eigenvalues will start to map outside the unit circle which will bring instability,
especially if a > 1.6. This is the reason why in Figure 2.2 Of Chapter 2 the HGO
discretized by FD method at a 2 1.5 was unstable for T = 0.01. For T = 0.003, the
poles will go outside the unit-circle for a > 1.1. But the, FD method will have less
r.m.s. error for smaller a as in Figure 2.2 of Chapter 2.

Figures 4.6, 4.7 and 4.8 for the bilinear, BD and FD methods confirm that as a
increases the steady state error decreases and the transient period includes oscillation.

The chattering effect we see at steady state is due to the resolution Of the Optical

encoder. The installed Optical encoders have a resolution Of 1250 pulse/revolution,

127

link one T=0.003

 

0 3.3/2

 

I
N

Position In red. 9

 

 

 

 

 

10

15

 

93

ON-h10’

 

Speed In rad/sec.
It»

1?

 

#1

 

 

 

 

0 5
Iimein sec.

10

15

link TWO

 

 

 

 

 

 

 

 

 

 

 

tirne'nsec.

Figure 4.5. The output signals for both links and their derivatives, T = 0.003 and

a=2.5

128

which results in an error Of 360°/1250 = 0.288". Increasing a when T is fixed reduces
the transient time.

If we consider the range Of the Operation for increasing a while fixing T and
the transient period, we find that the bilinear transformation method is the prOper

choice, which is in agreement with our earlier conclusion in Chapter 2.

Bilinear Implementation

On the basis of the foregoing comparative study, we adopt the bilinear transformation
as our discretization method. Let us study the design of the high-gain observer
with the bilinear implementation. The design parameters are the sampling period
T, the Observer parameter c and the a; coefficients which assign the roots Of the
characteristic equation.

Regarding the choice Of poles, we note that the bilinear HGO can be made an FIR
(finite impulse response) filter or an IIR (infinite impulse response)filter. Figure 4.9
shows that with complex poles the transient response is oscillatory and the transient
period is longer than the case Of real poles, especially for 60° 2 0 2 40". If 9 > 60"
the oscillation prevents the pendubot from balancing at the required position. Since
there is no advantage for using complex poles, we use real poles as we did in Chapter
2.

Another degree Of freedom is the choice Of the order of the Observer. For example,
we use reduced (first), full (second) and extended (third) order Observers to estimate
the speed of both links Of the pendubot. Figure 4.10 shows that the extended-order
observer has less transient oscillations and fast convergence Of the tracking error to
zero; in this figure T = 0.01 and a = 2.1. The same figure show also that the control
magnitude for the extended-order Observer is less than the other two Observers.

The parameter 0 determines the bandwidth of the discrete-time Observer, the

bandwidth increases with a. Increasing a while fixing T makes the peaking more

129

T=0.003, Bifinear Full Order Observer
1).-1.5

 

0.4

 

02’

 

   

—02

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

—o.4 A A .04 A A
5 10 15 5 10 15
(Ir-2.0 @125
0.4 . - 0.4
02 . . 02
o i o
—02 . I i -02
4.4 A A -0.4 A A
5 10 15 5 10 15

Figure 4.6. Using the bilinear transformation to calculate the steady state tracking
error e” = 1:1 +:I:3 -1r/2, where T = 0.0033ec. and a = 1.0, 1.5, 2.0, 2.5 respectively.

130

T=0.0m, Backward Ful.Order Observer

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Ira-1.2 (la-1.5
0.4 . . 0.4
0.2» . 0.2
0 0
~02) 4 -02)
-0-4 ‘ . -o.4 A
5 10 15 5 10 15
11:20 (1.25
0.4 . . 0.4 . -
0.2) 1 02» 1
olHlll’I" lllHllllll 0* )
-02A 4 _02. (
-0.4 A A -0.4 A
5 10 15 5 10 15

Figure 4.7. Using the backward difference method to calculate the steady state
tracking error e,,, where T = 0.003sec. and a = 1.2, 1.5, 2.0, 2.5 respectively.

131

T=0.003, Forward Full Order Observer

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0:0.8 a=0.9
0.4 1 - 0.4 - .
o 2 A 1 0 2
0 A 0
—0.2 A J -0.2 A l.
—0 4 A .0 4 A
5 10 15 5 10 15
new (1.1.1
0.4 . . 0.4 . -
0.2 A . 02 .
0 0
-o.2 A . -0.2
—0 4 A -0 4 A A
5 10 15 5 10 15

Figure 4.8. Using the forward difference method to calculate the steady state tracking
error cu, where T = 0.003sec. and a = 0.8, 0.9, 1.0, 1.1 respectively.

132

T=0.005;a=3; Bilinea FOHGO

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6=0° 0=20°

0.4 I - - 0.4

02- A 02
e 1 0’

0 Ullllll l l l I ll’
-o.2- A -o.2A A
-0.4 A A -o.4 A A

5 10 15 5 10 15
o-.-4o° o-so°

0.4 . - 0.4 . -

02 A 0.2

l
~02L A —0.2A
-0.4 A -04 A
5 10 15 5 10 15

Figure 4.9. The effect Of different choices Of poles real and complex on the transient
response, T = 0.005 and a = 3.0 discretized by the bilinear transformation.

133

 

 

 

 

 

 

Tracking error

 

 

 

 

 

 

 

 

 

—0.43 A A A -0.4A A A A —o.4

 

 

 

 

 

ControI u

 

 

 

 

 

 

 

 

 

 

Figure 4.10. Tracking error e and control signal for the full, reduced and extended
order HGO using the bilinear transformation, T = 0.01 and a = 2.1.

134

aggressive, and we need tO lower the saturation level. Moreover, peaking saturates
the controller, which disturbs the swing up control and the controller fails tO balance
the arm. SO, we study the effect Of increasing a while the sampling period is fairly
large, such as fixing T at 0.01 sec. For a 5 2 e is not sufficiently small. For a = 2.1
the high-gain Observer (discretized by the bilinear transformation method) gives good
estimates of the speed and the controller balances the pendubot arms. Increasing a
improves the performance by decreasing the estimation error. Figure 4.11 confirms
the results of Chapter 2. It is important to draw attention to the need Of lower
saturation levels, for a > 3.0. We find out that we need to saturate the outer-loop
control v1 such that —130 5 v1 5 120 only for the swing up period. For a > 6

peaking makes it difficult even with saturation to make the swing up successful.

4.7.2 Confirmation of the Closed-LOOp Study

The high-gain Observer poles are located at s = —l/e. We use the bilinear trans-
formation method for the discretization Of the HGO. We are going to discuss the

following points:
1. saturation as a tool to overcome peaking;
2. effect of decreasing e on the steady state error;
3. linear vs. nonlinear HGO; and

4. how large the sampling period can be.

Peaking and Saturation

As the Observer parameter 6 decreases, the peaking effect increases. Since this reduc-

tion Of 6 can be done by more than one method, the study is divided into two parts.

135

T=0.01; Biinear FOHGO

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

01:2.1 a=3
0.4 . . . . 0,4 l)
0.2 A 0.2
0 l 0 I
-0.2 . —0.2 .
-0 4 m .o 4 . 4
5 10 15 20 5 10 15 20
#5 (1:6
0.4 - . . . 0.4
0.2 . 1 0.2 r
0 ’ o .
-0.2 . -0.2 r
-0.4 ‘ P ‘ ‘ —0.4 4 ‘ ‘ ‘
5 10 15 20 5 10 15 20

Figure 4.11. Effect of increasing the ratio a = T/e on the full order HGO with real
poles at s = —1/e, T = 0.01 and a = 2.1, 3.0, 5.0, ,6.0 respectively.

136

First, the sampling time T is fixed and e is decreased through increasing a = T/e.
Second, a is fixed and T is decreased.

Let us start by taking the initial conditions of the Observer to be the same as the
system (12(0) = a:(0)). This is a case Of no peaking since peaking is induced by the
difference 2(0) - 15(0). Figure 4.12 represents the first case (T is fixed) where we can
see that increasing 0: doesn’t change the starting level Of 2:2. Figure 4.13 shows the
control magnitude which, in all cases, is less than 3 volts at the starting point.

For the second case as we decrease T and keep a = 2.5 fixed. The Speed estimate
for link one has the same starting point as T decreases but reaches steady-state faster
as shown in Figure 4.14. The same thing happens for the control as shown in Figure
4.15.

Now let us initiate the Observer at 2(0) = 0, which is different than the system
initial condition a:(0). Figure 4.16 represents the first case where we can see that
increasing or increases the peaking level in 1:2. This affects the control magnitude by
bringing more oscillation during the transient period as shown in Figure 4.17. Notice
that the control is saturated at the device limit, i.e., [u] _<_ 10 volts, while in the
nO-peaking case the control signal is |u| 5 3 volts.

For the second case as we decrease T and keep a = 2.5 fixed, the speed estimate
for link has more peaking as T decreases, but reaches steady-state faster as shown in
Figure 4.18. The same thing happens for the control as shown in Figure 4.19. With
large peaking, the balancing control fails to bring the pendulum arm to the balancing
region. This puts a limit on how small 6 could be , is, how small T when a is fixed
or how large a when T is fixed. For example, if a = 4 we cannot reduce T beyond
0.006 sec. This is contrasted with the no peaking case (when 23(0) = :r:(0)) where T

can be reduced tO 0.003 sec. Smaller values Of 6 result in smaller steady-state errors.

137

Speed estimate using the Bilinear Transformation Method
5 I l l l l

 

4.5

to 8’. o: 3

Speed estimate of lInk1 . xh2
1';

 

 

 

 

 

 

 

 

0.5
0
_o.5 l l I l I
0 0.01 0.02 0.03 0.04 0.05 0.13

timeinwc

Figure 4.12. The transient Of estimating speed Of the first link as or increases while
T = 0.01

138

Control applied to ink 1 using the Bilinear Translomration Method for T=0.01
$2.5 023.5

 

'

 

N
or o:
o:

N

 

Control input in volts
_. or

 

 

 

 

 

 

0.5L 05.
0 A A A A o A L A A
0 0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
1).-4.5 a=50
3 . ~ 3 .

 

 

N
or

N

   

Control Input In volts
d
- '01

.9
or
c
or

 

 

 

 

 

 

O

0.02 0.04 0.06 0.08 0.1 0 0.02 0.04 0.06 0.08 0.1
timeinsec tirneinsec.

0

Figure 4.13. The control during the transient period as or increases while T = 0.01

139

4.5

.10 .61
N U'l 03 0|

Speed estimate of link1. xl'I2
8‘.

Speed estimate using the Bilinear Translormation Mathis

 

 

I I

 

 

 

 

 

 

 

0.5
0
_0'5 1 J L l l
0 0.01 0.02 0.03 0.04 0.05 0.06
time in sec

Figure 4.14. The transient speed of link one while T decreases

140

controI applied to link 1 081119 the Bllnear Iranslormallon Mountains

T=0.01

 

yo
N or or

Control input in volts
_. 'ot

 

 

 

 

0.5 ' 1
o n n n n
0 0.02 0.04 0% 0.08 0.1
T=0.(X)6
3 r .
2.5 .

N

 

Control input in volts
-b 01

.°
or

 

 

 

O

0.02 0.04 0.06 0.08 0.1
timelnsec.

O

T=0.m8

 

 

 

 

 

0.02

0.04 0.06 0.08 0.1

T=0.w4

 

 

 

 

 

0
0

0.02

0.04 0.06 0.08 0.1
tinein see.

Figure 4.15. The control during the transient period as T decreases

141

 

 

Speed estimate of llnk 1 x2h

 

 

 

 

 

 

 

 

_1 60 L I P I l
0 0.01 0.02 0.03 0.04 0.05 013
line in sec.

Figure 4.16. Peaking in the transient of estimating speed of the first link as a
increases while T = 0.01

142

in

Control lnput In volts
O

.L
o

0

10

Control Input ln volts
0

Ellect ol peaking in control u applied to llnk 1 using bilinear melhod tor T=0.01

(Ir-2.5

 

 

 

 

---------

 

’ ........ ' ........ I ........ ' ....... .4
O

..................................

 

 

 

0.02 0.04 0% 0.08 0.1

 

 

------------

 

 

 

 

 

 

 

 

 

_5 ............ . . . :J .................... .l
_10 ........ ' . . .: ........ f ........ . ....... .4
0 0.02 0.04 0.06 0.08 0.1

Figure 4.17. The effect of the peaking on the control during the transient period as

limeinsoc

0 increases while T = 0.01

143

 

 

 

 

 

 

 

 

a-Jfi
1o ........................ 1
5» ........
0L ........
_5.... .................
_10 ........... x ....... E ................ T
0 0.02 0.04 0.06 0.08 01
0,-5.0
,0 .........
5 ........ J..
0 ................
4, ...................... ........ g ........ l
L— E E
_10 ........... ........ . ........ .........
0 002 0.04 006 OR 0.1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

EfleclolpealfnginspeedestirnaleusirigBilimar.a-2.5

 

l
8

Speed estimate of link 1 x2h
l
8

 

 

 

 

 

 

 

 

450 . ' -
— T=0.01
I ........ T30.”
“200 ' I ----- - T=0.006 ‘
,’ - - - 1.0.004
I
I I I l I
o 0.01 0.02 0.03 0.04 0.05 0.06

lineinsec.

Figure 4.18. Peaking in the transient speed of link one while T decreases

144

Efloclolpeaking in controluappfiedtolinld usingbilinearmethodlorfifi

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

T=0.01 T=0.m8
1o , .............. 1o ,, ............. 4
3 3 5
g 5 ......................................... 4 5 ............. ............. ............. 4
£- 3 ‘ I
5 ___|___.—— g |_.——;
g or ....................................... l 0 ............. .. ............. : ..............
7.3
E _5 ........................................ J _5 .........................................
o a
_10 .............. l ............. l ............. J 40, ............. z ............. I‘ ............. 4
0 002 004 006 0 002 004 006

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

:2
o
>
E
S
a.
5
T3 . .
g _5. ..... . .i ..... ............. ............. 1 _5 t. . . ......................................
0 . .
_10 ...... _l ..... l ............. :. ............. r _10 .. . . ._ ....... l ............. l ............. l
0 0.02 0.04 0.06 0 0.02 0.04 0.06
limeinsee line'nsec

Figure 4.19. The effect of the peaking on the control during the transient period as
T decreases

145

Steady-State Tracking

In this part of the experiment we study the effect of decreasing e on the steady state
tracking error. The parameter 6 can be decreased as long as the linear observer
poles remain inside the unit circle. In the bilinear transformation and backward
difference methods the poles are always inside the unit circle. In the presence of
noise, increasing a increases the bandwidth of the HGO which leads to complete or
partial distortion of the estimates, depending on the signal to noise ratio. Figure
4.20 shows the effect of increasing a when T = 0.003 sec.. Increasing 0 reduces the
transient time. We can obtain the same result by fixing a = 3.0 and decreasing T as
shown in Figure 4.21. These two results confirm Theorem 3.1.

Linear versus Nonlinear HGO

In this part we study the use of linear versus nonlinear high-gain observers to estimate
the pendubot states. Many runs were made to detect if there was any difference
between the two observers. We did not find noticeable differences between the two
observers. Figure 4.22 shows the steady-state error in one run of the experiment for
both observers, while Figure 4.23 shows another run. The two observers act almost
the same and this could be because of uncertainty in the model parameters. Paper

[8] reports an improvement with the nonlinear observer when the model is known.

Comparison between HGO and Euler method

It is shown in Section 2.5 that the Euler method (the most common method for
estimating speed from optical encoder) is a special case of a reduced-order high-gain
observer discretized using the forward difference method with T/e = 1. This is just
one of many Options we have when we address the problem as a high-gain observer

design. We can choose between reduced-order and full-order observers. Within each

146

tracking error using Bilinear T=0.003

a=1.o
1.5 . . r 1.5

 

 

 

 

 

 

 

 

 

 

 

 

-0.5 ‘ g
2

 

 

 

 

 

 

 

 

 

 

 

 

 

 

033-0 «=34

1.5 . . . 1,5 .

1 1 .
0.5 ' ' 0.5 -

o L I l o l

I ".1—
-0.5 9 ‘ ‘ -0.5 ‘
2 4 6 8 10 2 4 6 8 10
time in sec time in sec

Figure 4.20. The sampling time T = 0.003 is fixed while a is increased

147

Tracking error in degrees

 

 

 

 

 

 

 

 

 

 

 

 

T=0.01
1.5 '
1 .
0.5 ' ‘
o .
.5 4 . -
.0 5 10 15 20
T=0.005
1.5 .
1 n
0.5 ' 4
0 ’ w I
I l
-0.5 ‘ '
5 10 15
time in sec.

1.5

0.5

—0.5

1.5

0.5

-0.5

 

 

T=0.007

 

 

 

 

 

 

 

 

5 10 15 20
T=0.003
l
5 10 15
timeinsec

Figure 4.21. Fixing a = 3.0 and decreasing the sampling time T

148

Unear HGO T=0.02; 0:1 .8

 

 

 

 

 

 

 

0.2 I I I I I I I I
0 ..
E
.01:
—0.2 -
_04 i I I I I I I I
5 10 15 20 25 30 35 40 45
0.2 l l
o .-

 

 

o
tracking

 

 

 

 

 

 

0
-0.2 - 4
—0.4 i L J I 1 r 1 I
0 5 10 15 20 25 (I) 35 40
limeinsec

Figure 4.22. Tracking error at steady-state using linear and nonlinear HGO.

149

Linear HGO T=0.02; a=1 .8

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

0.2 l l I I I I l
0 1-
I?
.3
0
~02 -
~0.4 1 I 1 I I I I
0 5 10 15 20 25 30 35 40
Nonlinear HGO T=0.02;a=l.8
0.2 I I I I I I i
o .-
.0:
-0.2 -
.0" I I I I I I I
0 5 10 15 20 25 30 35 40
time in sec.

Figure 4.23. Tracking error at the steady-state using the linear and nonlinear in
another run

150

type, we have the flexibility of choosing several design parameters, most important
of which is the ratio T/e. We also have the freedom to use different discretization
methods. We have explored many Of these options in Section 2.5. In this section we
explore some of these Options, as applied to the pendubot, and compare the results
with those obtained using the Euler method and a variant of it.

The pendubot manufacturer provided an averaged Euler formula to estimate the
links, speed, where the sum of the last three calculated speeds by the Euler formula
was divided by 3. Notice that, for this method to work it needs very small sampling
period. Figure 4.24 compares the Operation range for the different discretization
methods of the high-gain observers and the Euler method, with and without aver-
aging. By Operation range we mean how large the sampling period can be before
the controller fails to balance the pendulum. This figure shows that the use Of the
reduced order high-gain Observer can guarantee larger Operation range (larger sam-
pling period). The Euler formula fails for T > 0.04 sec. Figure 4.25 shows that for
larger sampling periods the high-gain Observer outperforms the Euler method for the
same sampling period T = 0.04 sec. Even if we discretize the high-gain observer
with T = 0.06 and compare its performance with the Euler formula when T = 0.04,
the reduced order HGO outperforms the Euler formula. Figure 4.24 shows also that

the Euler method with averaging can not work for T > 0.01.

4.8 Conclusion

The experimental results confirm our earlier simulation and analytical results. Com-
parison of the three discretization methods (forward, backward and bilinear transfor-
mation methods) showed that the bilinear transformation method provides a larger
range of Operation when a increases while T is fixed. This feature helps in reducing

the transient time. After adapting the bilinear transformation as the discretization

151

 

 

  
 
 
 
 
  
 

Euler Formula with Averaging

Euler Formula

ROHG using Forward Dill.

FOHG using Forward Dill. Melh.

ROHG using Backward Dill. Melh.

FOHG using Backward Dill. Moth.

ROHG using Bilinear

Different algorithems for estimating the Pendubot arms speed

FOHG using Bilnear

 

o l I I I I I I I I
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1
Sampling Time T in sec.

Figure 4.24. Comparison Of the different discretization methods of the full order
(FOHG) and reduced order (ROHG) with the Euler formula with and without aver-

aging

152

Euler method T=0.04

 

 

 

 

 

 

 

 

 

1 i I I I 1 I I I I
0.5 r ..
o I.
-0.5 ‘
_1 1 I I I I I I I I
2 4 6 8 10 12 14 16 18 20
R.O.Bilinear (1:2, T=0.04
I I I I I I
g ..
0
O _
.E ‘v
X
0
g ..
I I I I I I
8 10 12 14 16 18 20

 

R.O.Biinear H5. T=0.01;

 

 

 

 

 

 

1 . . I I I I I I
05 - -
0 ' .WV—w————‘
—o.5 - -
_, I I I I L I I I
2 4 6 8 1o 12 14 16 18 20

timein see.

Figure 4.25. Comparing the performance of the reduced order HGO discretized by
bilinear method for a = 2 and T = 0.04 and a = 6.5 and T = 0.06 with the Euler
method when T = 0.04 sec.

153

method we confirmed all the results of Chapter 2.

The closed-loop study targeted different issues, such as the effect of peaking and
how the global saturation of the control will preserve stability of the closed-100p
system. Regarding the effect of decreasing e on the steady-state tracking error,
it was shown how the steady-state error improves as 6 becomes sufficiently small,
which confirms Theorems 3.1 and 3.2. Comparison of the linear and nonlinear high-
gain observers showed no advantage for one over the other. Comparison of the
performance of the high-gain Observer versus the Euler method showed that there
is a definite advantage for approaching the problem as a digital high-gain observer
design rather than simply using the Euler method. This advantage can be seen in
two ways. First, the high-gain Observer can work for a larger range of sampling
periods compared with the Euler method. Second, for the same sampling period, the

high-gain observer produces better performance with less oscillation at steady state.

154

CHAPTER 5

Conclusions and Future Work

5.1 Conclusions

This thesis presents a separation principle for sampled-data control of a class of
nonlinear systems. The basic ingredients Of this technique are: a continuous-time
high-gain observer that robustly estimates the derivatives Of the output, a global
bounded state feedback control, and a discrete-time implementation of the high-gain
Observer. For sufficiently small sampling period and sufficiently high Observer gain,
the sampled-data controller recovers the performance of the continuous-time state
feedback controller. The high-gain Observer is discretized using different discretiza-
tion methods to achieve the best discretization algorithm and parameters choice.
Chapter 2 provides answers to a number of important questions such as: what
is the best discretization method? what are the best choices for the observer pa-
rameters (12.9 and e and how do they relate to the sampling period T? and what is
the effect of the order of the Observer on the estimation of the output derivatives.
This is done for noisy as well as noise-free measurements. We also show how other
numerical differentiators are special cases Of the high-gain observer discretized by
the bilinear discretization method. The study can be extended to multi-input multi-

Output systems. Horn the simulation-based study Of Chapter 2 we move into closed-

155

100p analysis of Chapter 3. We study sampled-data controllers which use discretized
high-gain observers. The closed-loop analysis shows that the sampled-data output
feedback controller recovers the performance of the continuous-time state feedback
controller as the sampling frequency and the Observer gain become sufficiently large.
Performance recovery is shown in two steps. First we show boundedness of trajecto-
ries which come arbitrarily close to the origin as time progress. Second, we assume
that the origin of the continuous time system is exponentially stable and show that
trajectories converge to the origin.

In Chapter 4, we take the theoretical results one step farther by experimentally
testing the use Of the discretized high-gain observer in controlling the pendubot.
We show how saturation is used to overcome peaking and the effect Of decreasing
e on the steady-state error. The experimental results agree with and confirm all
conclusions of Chapters 2 and 3. Moreover, we show that the discretized high-gain
observer outperforms the Euler formula as a means for calculating velocities from

optical encoder.

5.2 Future Work

There are two lines for future work: one is extending the mom Of studies done in the
thesis and the other is using the deve10ped tools and techniques to study sample-data

control of nonlinear systems even when no high-gain observer is used.

5.2.1 Extending the SCOpe of the Sampled-Data Observer-

Based Control

The assumptions used in achieving the results Of this thesis are based on exponential
stability of the equilibrium point and the use Of static feedback control for stabiliza-

tion. These assumptions were used in the early work on the continuous-time case,

156

but later on the work was extended to cover more cases. There is room for extending

our discrete-time results to include the following:

o the use of dynamic controllers as in the work of Atassi and Khalil [8];
0 replacing exponential stability with asymptotic stability;

0 deveIOping the results for other control problems such as tracking, adaptive

control, and servomechanisms, in the spirit Of [4, 33, 32, 42, 43, 27, 63, 15].

e experimentally applying the discrete-time high-gain Observer to applications
where higher derivatives of the output are needed such as the induction motor

controller of [31] which requires both velocity and acceleration estimates.

5.2.2 Sampled-Data Control

Using the tools and techniques developed in the thesis, we can address other sampled-
data control problems. In particular, we can address the question of what we can
prove about the behavior of sampled-data control under fast sampling. Some re-
searchers address this question either by obtaining local results using linearization
[l 1] or nonlocal results using some SOphisticated techniques as the work Of Tee] at at
[65] which models the zero-order hold as a time delay element. The techniques de
veloped in the thesis can be used to provide some nonlocal results for sampled-data
control of nonlinear systems with fast sampling, in a way that could be much simpler

than [65].

157

APPENDICES

158

APPENDIX A

Appendix

A.1 Developing The Singularly Perturbed Form
For The Forward Difference Method

The state equations of the closed-100p system are

x(k+1) = x(k)+€‘1’(x(k) u(10.6)

n(k+ 1) =

                                  

:r(k +1) = :I:(k) + :3: ifA’ch) + 21;": 1B cG;()((k), u(lc))

T' +1RIOCUC). u(MIT)

The negative power of 6 appear in the term

                      

 

_ 6:1[E31D 3.413(1) + z —A:;-IB GI(~) + ruler)

r—1 1 c
e l. ,_l

 

r-l
= Z 26” ,_ __, lDA'II: x(k)+TZFAL’IDBCGAJ+a"1T’DR,(-)

= XII—— . -.I IDAIII (awgi‘LAL-‘B GI(-)+T2a'-‘DR.(-)

159

'where we used DBc = e'"ch. Notice that all the summation terms will be limited
to r — l or 1' because A; = 0. Rewriting n(k + l) we Obtain
r al-l

r—l I
"(k + 1) = A ,n(k) + Z Egg—104x“) + T ['2 711513.04.) + Ta “DR¢(-)]
(=2 ' =1 '

Let H; = ‘7'; and

r

h(x(k)Iu(k)I15) = a L}: —-—1A'.-‘B.GI(-) + Ta"‘DR¢(-)]

=1 l!
to obtain

n(l= + 1) = Am(k) + )3 3.17:,HIDAIzUc) + anon), an), e)

which represents (27), when ¢,(-) = 0, i.e., for the linear observer. It is clear that
the negative power Of e is limited to the summation part. We need to apply a change
Of variables in a number of steps to reduce the negative power of c by one in each

step. At quasi steady state 17(k + l) = 17(k), which can be written as

17(k) = A ,n,(k) + E CHI-'HIDAidk) + Ich(x(k), u(k), 6)

(=2

-1

nos) = (I — AI)“ [26+1-rnInAizIk) + ehrxrk). an). n]

(=2

Define 171(k) as
mac) = In) - (I — AI)“ [2: «Mummies + 6h(x(k). ans). 5]

160

Then

r-l
m (k + 1) = 77(k + 1) — (I — AI)“1 2 el+1"HIDA'c:I:(k + 1)
1:2

+(I — AI)“ leh(x(k +1)Iu(k + 1). e)]

We have expressions for 17(k + 1) and :c(k + 1), but we need to find an expression
for h(x(lc + l), u(k + 1), 5). Since the control is globally bounded, u(lc) and u(k + 1)
have the same order of magnitude, and so are x(k+ l) and x(k). This mean that the
difference between h(x(k), u(k), e) and h(x(k + 1), u(k + 1), c) has the same order of
magnitude. Substitute for ”(k + 1) and :I:(k + 1), and let 9 = (I — A ,)-1. The new
state 171(lc + 1) satisfies follow

111(k+1) = 141171001

+11er

 

—_—-l —Ze’Hj DAszc) + eh(x(k), u(k), 6)]

+-<I AIII —.—):e’H DAi=v(k)+6h(x(k) new]

- 1 r-1 1 r—lr-l aje'i-l-I
-2 . IZejH DA”z(k)+ .. IZZ— ——IHDAi+‘x(k)

 

 

- 1 r—l r alej-H

_9 er—l Z Z

j=2 [=1

 

 

 

uH,DA{+"‘BIGI(x(k)I1:00)]

cg [0”er Z ej+lHjDAiR¢(x(k), u(k), e) -l- h(x(k + l), u(Ic + 1), 6)]

j=2

We did multiply one of the brackets by (I - A ,)g = I to help in canceling some of

the terms and simplify the whole expression. Phrther simplification results in

r-lr—l a!

71100 + 1): Affll (k)- a; g: ij DAJ-HIEUC)
—IE 2: ,ngfie DA{+"‘B GIoIIk) nae»

j=21= l

161

r-1
"'9 Z ar+16j+2HjDAZIL(X(k)I “(MI 6)
i=2

-rth(x(k + 1)I "(k + 1). 6) - 6h(x(k)I WC), 6)

The double summation terms can be written as

 

r—lr—l r—l

22 l! r-a‘_T__j-‘H. DA’+‘:r(k) = :1 ZajlejDAiflk)

j=2l=1 ‘5 5 j=3
r-l r a! r-l .
221I6r__1-j_rHJDAj+l lB 6% = EzzblechGl(°)
j=2t= 1 '=

r-l r—1
2 a'+l€j+2HjDA{R¢(-) = 6’ z ejchDAiR1(-)

,-=2 ,5.»

Define h1(-) as

h1(x(k)I H(k)I 6) = h(x(k + 1)I "(k + 1). 6)- h(x(k)I “(16), 6)

+2031A: Be G ”()+€Z€jCjIDAjR;(°)

i=2

Then

771(k+ 1) = Amllk) - 9H I- -4 20116401910“)+€h1(x(k)Iu(k)I€)]

j=3

Repeating the transformation to the“ I step, we Obtain

III-(k+1)= AIm(k)+(-1)‘2‘LE a.-.-— —,—_‘,_D, DAIzIk)+eh.-(x(k). u(k).e)]

=i+2

The quasi steady state 1),-(k + 1) = 1),-(k) of this equation is

nI(k)=(- 11‘2““ [E a.-. —_——1,_,DA’z(k)+eh momma]

=i+2

162

Hence, the change of variables is taken as

n.+1(k)= n.-(k)+(- -1)‘“D LE —,——_1,._ lDAzx aci(k)+eh(x(k). We) a]

and the new variable satisfies

-i+2
+(-1)‘+‘D‘+‘eh.-(x(k +1),u(k + l), 6)

1).-+1 (k + l) = 1).-(k +1) + (-1)"+lg"+1 [:2 a; 56 _—-l——.__ DA’zUc + 1)]

Substitute for mac + l) and x(k + l) to obtain

17i+1(k+1)"' - Affli+1(k)+(- -1)'+IQ ”1 [:2 aji+1£,__lj_——1'DAj-’5(k) + €hi+1(X(k):"(k):€)]

'-i+3

where

ha+1(x(k),u(k),€) = hi(x(k+l),u(k+1),e)-hi(x(k).u(k).6)
+ )2 bjs+1AchGj(x(k),u(k))

jzi+2

+e E ejcfi+1DAiR¢(x(k),u(k),e)

j=i+2

If 2 _>_ r — 3 then A'= 0 which eliminates the negative power of e and the change of
variables reach its end

"5+1“? + 1) = AITRH“) + ("DH-IQi+l l 0 + €hi+l(X(k): ”(1‘): 6)]

Let 1),-+1 déf 5, then

£0: + 1) = 14,500+ e(—1)‘+‘ ((1 - Afr-1)“hi+1(x(k>,u(k),e)

163

 

Which can be rewritten as
£(k + 1) = A1606) + 6F(x(k), We). ilk), 6)
where
5+1

D(x(k),u<k),:e(k),e) = <-1)‘+‘ ((1 — An“) h.-+1(x(k).U(k),6)

From this we can see that as e -+ 0, the system stability depends only on the

eigenvalues of A f which lie inside the unit circle.

164

'lb Show (3.26), we need to write 1, in terms of E. This can be achieved by
listing all the transformations and write them in one step. Let N (e) = 6' ”1D“1 and

N(e)‘l = e—qu then

£(k) = 17.-(k)+(-1)"+‘Di+l L33 ajiejN"(6)Ai$(k)+ehs(x(k),u(k).6)]

°=i+2

77:“) = 77:40“) + (“l)iQ‘ LE 011—16" N _1(5)Ai$(k) + eh;_1(x(k),u(k),c)]

'=i+ 1

77206) = 7Mk) + (-1)’D2 LE: “1'16” N "(6)Ai=v(k) + €h1(x(k), WC). 6)]

7106) + (-1)9 [E ajoéj N “1(6)Ai$(k) + 6ho(x(k), We), 6)]

i=2

010‘)

Now we write n(k) equations with respect to £(k)

7706) = Q[:ajoejN-I(G)Ai$(k)+€ho(X(k):u(k):5)J

+ (- 1)D2 [22: “1'16” N 'l(€)Ai$(k) + €h1(x(k), Mk), 6)]

+ ("Di-IQi ['21 aj5_1£jN_l(€)Ai$(k) + EhI'-1(X(k)2 "(1‘): 5)]

+ (-1)‘D‘+1 [Z ajeejN'l(€)Ai$(k)+€hs(x(k),u(k).6)] +£(k)

j=i+2

120:) = £(k)+:2:(-1)“‘D‘ LE a::e"N(e)“Aiz-(k) + ehz—1(x(k). We), 6)]

'=l+l

From

5506) = $(k)-N(€)n(k)

165

where

5+1

* = I+¢s¥(—l)'N(e)g'N(¢s)"l LE egg—1.41;“ $0“)

=l+l

-N(e)£(k) + e gar-WW [hi-104k). nos), en

[1 + €M(€)] $06) - N (6)6 06) + 63($(k), g(k), 6)

= 313(k) - N (6)15 (16) + €W($(k), £(k), 6)

M(e) =

Z($(k),€(k), 6) =
W($(k), £(k), 6) =

if?1)‘-’V('E)Q'N(e)‘l [:23 @844]
'=1 ’= +1
:(-1)i-1N(€)Q'hz—1(x(k), u(k), e)

M (6)1706) + 360(k). £(k). 6)

166

A.2 The Transformation Steps To The Singulme
perturbed Form For The Other Discretization
methods

We need to study

1
er-l

 

not) = 31:13 [2:(k + 1) — mu.) — TAc:c(k + 1)]

61-1 DQ(-) = 57130 $(k +1) ‘- $(k) " 221'Ac($(k + 1) + $(k))]

 

for the backward difference method and bilinear transformation method respectively.
Other calculation will follow the same as we did in the forward difference method.
Notice that this need to be done in each step to show the true order of the error term

to get the singularly perturbed form.

A.2.1 Using Backward Difference Method

1:(k + 1).. - :r(k) + :2: u —A':c(lc) + Z ——A“‘Bch(-) + Triage)
Let Q() = a:(k + 1) — $(k) —- TAczUc + 1), then

r—l r TI
0 = 2Azz<k)+§:—,.—A:-IB.G.<->
1:1 1:1 '

+T'“R.()— TAcx(k)- :2:T—;—1

-Z

AHIZUC)

TH-l
——-A'B Gz( )T+2ACR1()

167

—l( _unTl

__.__ 2L1..— ————A':r(k) +TB.G:(')
T’“(I—~ TA.)R.( ) +2221” 'c’B [0:0 - 191-10)]
(:21

= “[fonr'u— TA.)R.()]

 

+30 —'II)T All) (’6) + Z! —A‘ 13cm)
(=2 L 1:21
Where
600) = 0
C710) = Gl(°)
5:0) = Gz(°) -le-1(°)
Then
0(-)= Tch Gz()+T'(I- TAc)R1( )]
+20 —_Al x(lc)+Z—u-1:-:A' 1%)

(=2

Now multiply both side by 31:10. Also notice that
DBc = 6"ch

l
DAi: r l—IALD

 

We get

      

T[Bl G;(-) + a"lTD(I - TAc):c(’)]
.ng ,. :‘——f—).°:‘DA' a.) Tzfl-m

1:2

168

= T [:i $11!-ch G;( ) + TeX-[DU " TA¢)R;()]
+ 2 ,. . —.f——).‘f DAM

The order of

-0(Z(1—..—:-_?‘f'DA‘.z(k)),.

        

Let Hga'fll, then

        

0(5— 1 _3 Eu - l)e‘"H;DAczc(k))

(:2

which is equivalent to 0 (Fl-‘5)

A.2.2 Using Bilinear Transformation

Starting from the difference equation in ??. Let

in.) = $(k + 1) — z:(k) — £11430: + 1) + z(k))

or) = 21% A£z(k)+:5—.A‘-‘B.G«(-)+r+la(-)
r—l (+1
— Z {Warme— 2T 2—(,.)A‘B Gm—

         

_ -l)T' T
_ :9 2(1!) A‘ g(k)+T[BcGz(')+T'(I-'54c)34(')]

 

+2:— ,. ZcAz-‘B [Gm — gm]

= T:[B: G((‘) + To, "' ZAc)R<l:()]
I
+2.10%ng A—'—' (k) + 23 31,41" 13 0‘”

1:2 (:2

169

Where

Go“ = 0
61H = G1(')

6%) =_ 049—50140
Then

m) = T[3 G.(-)+r<I—T-AC)R:(-)]

+201/____!2__)TA:1: (k)+z_ u A1-lB cG1()

1:2

Now multiply both side by firD, we get

 

 

 

6:10:70) = T [3110,” + a"1TD(I — f-P—Ac)1I?.,(-)]
+2( I! —:T'——"ll—/:2)1alDA'$(k)+TZl—A' 13 £10
1:2
= 7122—" —A‘ ‘3 G.<)+Ta'-‘D(I- demo]
1:l
+2 ,. ——',——'_/..23“' was)
The order of
€._ mo =0(:2:(1,. ,——’_———/f)"' DAiz(k))
= 0( r1_ 4 Eu —l/2)c"3H;DA'czr(k))
:3
which is equivalent to 0 (51.1).

Horn this point on we can apply the change of variables the same way described

for the forward difference method.

170

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171

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