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THESIS
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This is to certify that the

dissertation entitled

ROBUST ADAPTIVE OUTPUT FEEDBACK CONTROL
OF NONLINEAR SYSTEMS

presented by
Bader Nm Aioiiwi

has been accepted towards fulfillment
of the requirements for

Ph. D degree in Electrical Eng

 

     

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Robust Adaptive Output Feedback Control of

Nonlinear Systems
By

Bader N m Aloliwi

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

1999

ABSTRACT

Robust Adaptive Output Feedback Control of

Nonlinear Systems
By

Bader Nm Aloliwi

In this thesis we design a robust adaptive output feedback control to solve the
tracking problem for a class on nonlinear systems. We consider a single-input-single—
output minimum phase system represented globally by an nth order differential equa-
tion. We start by designing a state feedback control to achieve tracking error conver-
gence. The control uses a Lyapunov based adaptation to estimate uncertain parame—
ters. Then, we saturate the control over a compact set of interest to prevent peaking
and design a high-gain observer to estimate the unmeasured states. We show that
this control guarantees the boundedness of all the state variables of the closed-loop
system and achieves tracking of a given smooth reference signal without requiring
persistence of excitation. We show robustness to small bounded disturbances. If
the bound on the disturbances is not small but known, we go one step further by
designing a robust control component that ensures the boundedness of all signals and
makes the mean-square tracking error of the order 0(6 + u) where c and p are design
parameters. We pick the induction motor as an application candidate to demonstrate
the applicability of our technique. We design a robust control that uses an adaptive

observer to estimate the rotor resistance. The design guarantees the boundedness of

all closed-loop signals and makes the mean-square speed tracking error of the order
0(p) where p is design parameter. We show some experimental results. The design
is tested experimentally and the experimental results are in good agreement with the

theory.

To my parents

iv

ACKNOWLEDGMENTS

In the name of Allah, the most merciful and the most beneficent. All praise and
thanks are due to God the Almighty. I would like to express my deep and sincere
appreciation to my thesis advisor Professor Hassan K. Khalil. I would like to thank
him for his excellent guidance, friendly manner and professional example. I would
like also to thank my committee members: Professors Elias Strangas, Fathi Salam
and Sheldon N ewhouse. In particular, I would like to thank Professor Strangas who
graciously made his laboratory available for me. I would like to thank Andres Diaz,
for his help with the programming of Intel, F ida Khan for the valuable discussions
that we had together, and Abdullah Alswiayan for his help with LaTex.

My deepest thanks go to my parents whose love, support and encouragement have
made it possible for me to pursue my graduate studies. My wife Modi Alhamad has
contributed to this dissertation an enormous amount of love, support and sacrifice
for which I will always be grateful. Last but not least, I would like to acknowledge
the patience of may daughters, Beshayer, Bayader, Raghad and my son Belal, who

always have prayed for my graduation.

TABLE OF CONTENTS

LIST OF FIGURES viii
LIST OF TABLES x
1 Introduction 1
2 Tracking 6
2.1 Introduction ................................ 6
2.2 Problem Statement ............................ 7
2.3 Control Design .............................. 10
2.3.1 State Feedback .......................... 11

2.3.2 High-gain observer ........................ 16

2.4 Tracking Error Convergence ....................... 18
2.5 Examples ................................. 28
2.5.1 Linear Plant [14] ......................... 28

2.5.2 Nonlinear Plant .......................... 31

2.6 Appendix ................................. 33
2.6.1 Proof of( 2.32) .......................... 33

2.6.2 Proof of( 2.35) and ( 2.36) ................... 35

2.6.3 Proof of( 2.44) .......................... 35

2.6.4 Imbedded Convex Sets Assumption ............... 36

2.7 Conclusions ................................ 37

3 Robustness to Bounded Disturbance 48
3.1 Introduction ................................ 48
3.2 Robustness Property ........................... 49
3.3 Robust Output Tracking ......................... 57
3.4 Example: Nonlinear Plant ........................ 63
3.5 Conclusions ................................ 63

vi

4 Application to Induction Motors

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8

Introduction ................................
Induction Motor Model ..........................
Controller Design .............................
Adaptive Observer ............................
Closed—loop analysis ...........................
Experimental Setup ............................
Experimental Results ...........................
Conclusions ................................

5 Conclusions and Future work

BIBLIOGRAPHY

vii

68
68
7O
77
81
83
89
91
93

103

106

2.1
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
2.10

3.1

3.2

3.3

4.1
4.2
4.3
4.4
4.5
4.6
4.7
4.8
4.9

LIST OF FIGURES

Tracking error e. ............................. 38
Parameter errors,(a)- 51 ,(b)- 672,(c)- O3, (d)-§4 .............. 39
Projected Parameter errors: (a)- 91 — aé3,(b)- 92,(c)- 53, (d)¢6~4. . . . 40
Tracking error e. ............................. 41
Parameter errors 9. ............................ 42
Tracking error 6 using MRAC controller. ................ 43
Parameter errors: (a)- 91, (b)-92, (c)-93 , (d)-6l4. ........... 44
Tracking error 6 using MRAC controller. ................ 45
Parameter error 5,. ............................ 46
The x-axis is time. (a) Tracking error e; (b) Control u; (C) W(-) ; ((1)

Projected Parameter error: 91 + 52 (solid), 62, 93 (dotted). ...... 47

The case at = 0 (solid), d = sin(t) (dotted-dashed), and d = 53in(t)
(dashed) and no robustifying control. The x-axis is time. (3) Tracking
error e; (b) Projected parameters error: 91 + 92 ............ 64
The case when d = 53in(t) when a robustifying control is used with
17 = 0 (solid) and n = 5.2, ,u = 0.9 (dotted-dashed), and u = 0.3
(dashed). The x-axis is time. (a) Tracking error e; (b) 91 + 92. . . . . 65
The case when d(-) = O, 17 = 5.2, and p = 0.9. The x-axis is time. (a)
Tracking error e; (b) Projected parameters error: 91 + 52 (solid), 672,

93 (dotted) ................................. 66
Three phase winding of induction motor ................ 70
Three phase equivalent circuit of induction motor ........... 71
Experimental setup ............................ 95
Flow chart of the assembly program ................... 96
Flow chart of the assembly program (cont.) ............... 97
(a)- Reference and actual speed (b)- Speed estimation error ...... 98
(a)- Flux norms, estimated and reference (b)- Estimated rotor resistance 99
(a)- Reference and actual speed (b)- Estimated rotor resistance . . . . 100
(a)- Reference and actual speed (b)— Estimated rotor resistance . . . . 101

viii

4.10 (a)- Reference and actual speed (b)- Load Torque and (c)- Estimated
rotor resistance ..............................

ix

LIST OF TABLES

4.1 Induction motor parameters ....................... 91

CHAPTER 1

Introduction

Nonlinear adaptive control has been a subject of interest for many researchers over
the past decade. The motivation behind this was feedback linearization techniques
where one first cancels nonlinearities then designs the control to meet the design
specifications. By canceling the nonlinearities, one can use well developed linear

techniques to satisfy the design specification. For example, consider the nonlinear

system
:i:=a:z:2+u, yza: (1.1)
Using the control u = —a:r:2 + u1, the closed-loop system becomes

which can be seen as a linear system. One can use a fully developed linear control
design to make a: behave in a desired manner. This is true under the assumption that
the parameter a is known. However, claiming perfect knowledge of the nonlinearities
is not always possible. In some cases, only a nominal value no of a is known. In this
case the term (11:2 will not be completely canceled. There have been results in the
literature to solve the problem in this case. It is solved mainly by two directions.

Adaptive nonlinear control and robust nonlinear control. A combination of the two

has been lately addressed; that is, the robust adaptive control of nonlinear systems.
The first approach, can be seen by going back to (1.1). If a = a0 + Aa where Aa is
a variation of the parameter a from its nominal value, then an identifier is designed
to estimate the parameter a and the estimate a is used in feedback. The estimate is
obtained using a Lyapunov based design in most cases. More details of this method
can be found in Chapter 2. The robust control on the other hand uses a0 in the
feedback and uses another control component to overcome the effect of the error Aa.
An example combining the two approaches is given in Chapter 3.

Sastry and Isidori [43] were the first to address the adaptive control of nonlin-
ear systems. They achieved global adaptive control of a class of feedback linearizable
systems. However, global Lipschitz assumptions were imposed and overparameteriza-
tion was needed. Kanellakopoulos, Kokotovic, and Morse [20, 21] solved the tracking
problem for nonlinear systems that are of the parametric — pure - feedback form us-
ing the backstepping procedure without global growth restrictions on nonlinearities.
However, overparameterization was also needed. The work of Jiang and Praly [17]
was able to achieve the above results with half the number of adaptation laws. Fi-
nally, Krstic, Kokotovic and Kanellakopoulos [29] solved the adaptive nonlinear con-
trol without overparameterization but required the system to be transferable globally
to the strict feedback form. It is worth mentioning that all the above work requires
the use of full state measurement.

Marino and Tomei’s result [38] was the first on output feedback adaptive con-
trol without requiring output matching conditions and sector-type nonlinearities,
which were required in [18] and [19]. [38] detailed out the necessary geometric con-
ditions which characterize the class of nonlinear systems for which they were able
to design output feedback tracking control. They used augmented filters. Krstic,
Kokotovic and Kanellakopoulos [30] solved the output feedback tracking problem

using backstepping with observer based identifier. The class which was treated is

called Output — feedback — canonical form. Khalil [25] dealt with a single—input-
single—output (SISO), minimum-phase nonlinear systems which can be represented
by an n-th order differential equation. The class of systems includes those treated
in [38] and [30] as special cases. He extended the dynamics of the system by adding
integrators at the input side then transformed it into the normal form. The uncertain
nonlinear functions of the model depend linearly on constant unknown parameters.
By combining results from [11, 48, 47] with Lyapunov-based adaptive design [39, 14],
he designed a controller that achieves semiglobal asymptotic output tracking for ref-
erence signals which are bounded and have bounded derivatives up to the nth order.
The new adaptive controller is simpler than traditional ones since it does not use
filtering or error augmentation ideas. It is simply a state feedback controller with
a linear observer. An important drawback of the result of [25] is the requirement
of persistence of excitation not only for parameter convergence but even for track-
ing error convergence. This is unusual in adaptive control results where tracking
error convergence is shown without persistence of excitation. Jankovic [16] achieved
similar results starting from the normal form with no zero dynamics, using state
feedback and a high-gain observer. The objectives of this thesis are to overcome the
drawback of [25], that is, to show asymptotic tracking without requiring persistence
of excitation, to show robustness of the design to small bounded disturbances, and to
design a robust part of the control to overcome large bounded disturbances. Finally,
we apply nonlinear control with adaptive observer to induction motors.

In Chapter 2 we solve the tracking problem without persistence of excitation.
This is made possible by analyzing the closed-100p system under output feedback.
Unlike [25], we do not rely on singular perturbations for recovering what was achieved
under state feedback. We combine various Lyapunov functions to form a composite
Lyapunov function that shows convergence of the tracking error and partial parame-

ter error. By partial parameter convergence we mean convergence of a projection of

the parameter vector on a lower-dimensional subspace. The control procedure takes
two steps. First, we design a state feedback control that uses the estimate of the
parameters to cancel the nonlinearities and stabilize the resulting system. Second,
we design a high-gain observer, to estimate the state of the error system, together
with saturation of the control to prevent peaking [11]. A number of conclusions are
drawn at the end of the chapter. The results of [25] and [1] are shown to be two
special cases of Chapter 2. Chapter 3 deals with two issues: robustness in the usual
form of robust adaptive control results [14], and robust control. We show that, for
sufficiently small bounded disturbance, all signals in the closed-loop system will be
bounded and the mean square tracking error will be of order 0(6 + (11), where all is
an upper bound on the disturbance. Since 6 is a design parameter, we can choose
it small enough to make the mean square tracking error of order 0(d1). Second, we
present a robustness result that goes beyond the traditional robust adaptive control
results. We exploit the fact that our design is developed for a system represented in
the normal form, where the disturbance satisfies the matching condition, to design an
additional robust control component that ensures that for any bounded disturbance,
with a known upper bound, all signals in the closed-loop system will be bounded
and the mean square tracking error will be of order 0(6 + p), where both 6 and ,u
are design parameters. Our design uses the Lyapunov redesign technique, e.g., [7]
and [26, Section 13.1], and does not require the disturbance to be small.The idea of
combining adaptive control tools with robust control tools used in Section 5 has also
appeared in the state feedback designs of [31], [41], and [51].

In Chapter 4, we consider the induction motor as an application of our work on
nonlinear robust adaptive control. Many researchers became interested in induction
motors after the introduction of field orientation by Blaschke [5] in 1972. The idea
of field orientation is to transform the motor equations into coordinates that are

rotating with the rotor flux, which is used to compute the transformation. During

the late 80’s and early 90’s, a number of results that assume measurement of the rotor
flux were derived; see for example [10, 32, 34, 12]. However, rotor flux measurement
is not practical. Flux estimators, e.g. [49], were used for field orientation. We quote
a number of references that use flux observer for field orientation [23, 24, 40, 27].
The flux estimate is sensitive to the rotor resistance. To overcome the changes of the
value of the rotor resistance, three direction have emerged: robust control; see for
example [27, 44], adaptive observer with regular control; see for example [9, 46, 36],
and field orientation using the stator flux [13, 50].

In [27] the field-orientation transformation is done using the estimate of the rotor
flux rather than the flux itself, which in turn results in transformed variables which
are available for feedback. In [35], an adaptive observer for induction motors with
unknown rotor resistance is introduced. It is based on rotor speed and stator current
measurements. The adaptation is with respect to the rotor resistance. The design
is a Lyapunov based design. In Chapter 4 we carry the controller of [27] one step
further by adapting the rotor resistance on line using the adaptive observer of [35].
We analyze the closed-loop system under output feedback (we do not assume speed
measurement) and show experimental results. It should be noticed that the appli-
cation to induction motor in Chapter 4 is not a straightforward application of the
results in Chapters 2 and 3 because the motor equations do not fit the mathemat-
ical model used in those chapters. Instead, the techniques of Chapter 2 and 3 are
adapted to fit the induction motor case.

Finally, in Chapter 5 we give our conclusions and possible future research direc-

tions.

CHAPTER 2

Tracking

2.1 Introduction

The word tracking in the control literature means (y — y, = 0) where y is the output
of a given system (plant) and y, is a desired reference signal to be followed. TYacking
of uncertain nonlinear systems takes, in most cases, one of two methods: adaptive
nonlinear control or robust nonlinear control. By uncertain, it is meant that the
plant has some unknown parameters. In the adaptive case, an on-line identifier is
used to estimate the parameters. In the robust case, a robust control is designed
to make sure the system will maintain stability in case of the mismatch. In [25],
Khalil studied adaptive output feedback control for a class of nonlinear systems. The
system under consideration is single—input—single-output, input-output linearizable,
minimum phase, and modeled by an input-output model of the form of an nth-order
differential equation. The uncertain nonlinear functions of the model depend linearly
on constant unknown parameters. As mentioned in Chapter 1, [25] showed tracking
error convergence under output feedback only if a persistence of excitation condition
is satisfied.

In this chapter, we prove tracking error convergence without persistence of exci-

tation. This major improvement over [25] has been made possible by changing the

analysis approach. In [25] convergence is proved by showing that, under state feed-
back and the persistence of excitation condition, the set of zero tracking error and
zero parameter error is an exponentially stable invariant set. Then, singular pertur-
bation analysis is used to show that this same property is recovered under output
feedback for sufficiently small 6. This idea does not work in the lack of persistence
of excitation because the set of zero tracking error and zero parameter (or partial
parameter) error is not exponentially stable. Here, we analyze the closed-loop system
under output feedback directly and combine various Lyapunov functions to form a
composite Lyapunov function that shows tracking error and partial parameter con-
vergence. By partial parameter convergence we mean convergence of a projection of
the parameter vector on a lower-dimensional subspace.

Section 2.2 defines the class of nonlinear systems considered. It is followed by
the design of the control. We first, design a state feedback control which uses 6 an
estimate of the uncertain parameter 0. The control cancels all nonlinearities and
stabilizes the overall system. Second, we design a high-gain observer to estimate the
error state e and use the estimate é in the feedback control. As in [11], we saturate

the control to avoid peaking. We show the tracking error convergence in Section 2.4.

2.2 Problem Statement

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

the nth-order differential equation

310‘) 2 f0(') + 23:1 fi(')0i + [900 + Zf=19i(‘)6ilu(m) (2'1)

where u is the control input, 3; is the measured output, y“) denotes the ith derivative

of y, and m < n. The functions f,- and g, are known smooth nonlinearities which

may depend on y, y“), ...,y(”‘1),u,u(1), ..., um”); e.g.,

f0(.) : f0(y1 31(1)? ' ' ' 7y(n—l)7 11" ”(1), ' ' ' ’ u(m—.l))

The constant parameters 01 to 6,, are unknown, but the vector 0 = [01,...,9p]T
belongs to Q, a known compact convex subset of R”. We augment a series of m
integrators at the input side of the system and represent the extended system by
a state space model. The states of these integrators are 21 = u, 22 = a“), up to
zm = u("“1) and we set U = 21"") as the control input of the extended system.

Taking 3:1 2 y, 172 = y“), up to at" = y("‘1) yields the extended system model

 

,
it = 1171'“, ISiSn—l
in 2' f0(:r, z) + 6Tf(:1:, z) + [90(17, 2) + 0Tg(;r, z)]v
2.; = 2314.1, 1 S 2 S m — 1 f (22)
Z", = v
y = 5171 J
where
a: = [$1,...,:I:,,]T, z=[zl,...,zm]T

f : [f1,...,fp]T1 g=l91:"'agP]T

Assumption 2.1 |go(:c,z) + 0Tg(x,z)| Z k > O V x E R", z E Rm and 0 6 (21,

where 521 is a compact set that contains 9 in its interior.

Assumption 2.1 ensures that (2.2) is input-output linearizable by full state feedback

for every 0 E 0. Using the results of [6], it can be shown that there exists a global

diffeomorphism, possibly dependent on 9,

1‘ (I?

C TIL/1:72)

“2‘ To, 2)

with T1(0, 0) = O, which transforms the last m state equations of (2.2) into

C = “6.39) (2-3)

This, together with the first n state equations of (2.2), defines a global normal form.
As discussed in [25], the input—output model (2.1) has linear dependence on the
constant parameters 6, which is a restriction. But, in some cases, redefinition of
physical parameters may be needed to arrive at (2.2). The following example shows
how it could be done. Consider a single link manipulator with flexible joints and

negligible damping which can be represented by [45]

Iijl + MgLsinq1+ k(q1 — q2) = 0

J92 — “(11— (12) = 0

where ql and Q2 are angular position, and u is a torque input. The physical parameters

9, I, J, k, L, and M are all positive. Taking y = q1 as output, y then, satisfies

 

 

(4>=Q_L_"£-2- _-- _E £«_9kLM- i
y I (y smy ycosy) (1+J)y IJ Siny+IJu
Taking
_gLM _ k k gkLM _ k
9“ 1’02—(I+J)’63_ 1.1 ’ 0“ IJ
yields
31(4)=01(yzsiny—ycosy)-92y—63 sin y+04u (2.4)

equation (2.4) is of the form of (2.1). The class of systems includes as a special
case the nonlinear systems treated [22] and [37] for output feedback adaptive control
and the linear systems treated in the traditional adaptive control literature, e.g., [4]
and [39]. For example [25], the class of state—space models treated in [39] has an

input—output model of the form

p

~

y‘") = 3(D)[0(y)U] + My, - - . , y("‘”) + Z My, - - - , y("‘”)0.- (2-5)

i=1

where D = (d/dt) and B(D) = me’" + + be (m < n) is Hurwitz polynomial
with unknown coefficients b,- and 0(y) and 1b,- are smooth known nonlinearities with
o(y) sé 0 V y E R. bm(0) 75 O by assumption. Redefining the control input as
if = o(y)u, (2.5) is a special case of (2.1) and Assumption 2.1 and 2.2 are satisfied.
Objectives

The objective of this chapter is to design an adaptive output feedback controller
which guarantees boundedness of all state variables and tracking of a given reference
signal y,, where y, is bounded, has bounded derivatives up to the nth-order, and yfl")

is piecewise continuous.

2.3 Control Design

In this section, we first design a state feedback controller that ensures boundedness
of all signals and yields zero steady-state tracking error. This same controller is
used in the output feedback case with the states replaced by estimates provided by
a high-gain observer. We saturate the control outside a compact region of interest

to protect the system from peaking induced by the high-gain observer.

lO

2.3.1 State Feedback

In this section we assume that the state x are available for feedback. The state 2,
which are the derivatives of the control it, are always available for feedback. We
design an adaptive state feedback controller so that the output y tracks the given

reference signal yr. Define

e- = y““’—y“‘” = sci-31““) 13.2572

r r 1

and

e: [61, e2, ..., e,,]T

Let

3’“) = [310), 31(1)“), ---, Elm-”(WT
340) = [yr(t), y§1)(t), y$"“’(t)lT

3’30) = [it-(t), 951W), y$"‘”(t), y§"’(t)lT

and Y and YR be any given compact subsets of R" and Rn“, respectively, such that

31(0) 6 Y and 3230:) 6 YR V t Z 0. We rewrite (2.2) as

e = Ame + b{Ke + f0(e + yr, 2)
+ 6Tf(€ '1' yr, Z) + [90(6 + yr, 2)
+ 6Tg(e + y.,z>]v — W} (2.6)

z' = Agz+b2v (2.7)

11

where (A, b) and (A2, b2) are controllable canonical pairs of the form

 

 

 

 

010 0‘ "of
001 0 0
A: ,b=
00 10 o
00 01 0
_oo o 04 L1_

and K is chosen such that Am 2 A — bK is Hurwitz. We choose the initial states
of the integrators such that z(0) E Z0, a compact subset of R”, and define the set
of initial conditions for the error states as E0, a compact subset of R". To proceed
with the analysis we require that the zero dynamics to be exponentially stable. In

particular,

Assumption 2.2 The system C = F(C,y,.,0) has a unique steady-state solution 5.

Moreover, with 5 = C — (- the system

c = F(<’+é,e+y.,9) — F(§,yr,9)

~ _ (2.8)
: F2(Caeiyr7C)0)

has a continuously differentiable function V1(t, 5), possibly dependent on 0, that sat-

isfies 1
mllillz S. V1(t,5) S Uzlléllz (2-9)

6V1 6V1

79? + 35%“. e,y.. 6.6) s min? + muéunen (2.10)

where 171,7)2, 173 > O, and 174 2 O are independent of y. and 6.

 

1Throughout the thesis, [I ' ]| denotes the Euclidean norm.

12

The steady-state response of a nonlinear system is introduced in [15, Section 8.1].
Basically, it is a particular solution towards which any other solution of the system
converges, as time increases. The inequalities satisfied by V1 imply that such conver-
gence is exponential. They also imply that (2.8), with e as input, is input-to—state
stable [26, Theorem 5.2]. Consequently, the zero dynamics of (2.2) are exponentially
stable and (2.2) is minimum phase.

Let P = PT > 0 be the solution of the Lyapunov equation PAm + ALP = —Q

where Q = QT > O, and consider the Lyapunov function candidate
v = eTPe +géTr-1é (2.11)

where I‘ = FT > 0, d = d — 0, and d is an estimate of 9 to be determined by the
parameter adaptation law. The derivative of V along the trajectories of the system

is given by

V = —eTQe + ETF’15+ ZeTPb{f0(e + yr, z)6Tf(e + yr, z) + [90(6 + yr, 2)

+ 0Tg(e + yr, z)]v + Ke — 31101)}
Taking
—Ke + yin) — f0(e + 34. z) — de(e + yr, 2)

90(8 '1' yr, Z) + éTg(e '1' yr; 2)
déf tb(e, z,yg,é) (2.12)

 

we can rewrite the expression for V as

v _—. —eTQe +éTr—1[é- rt]

13

where

a = 26"?be (e + yr, 2) + g(e + 32,, awe, 2. ya, 61)]

_—. ¢(e, z,yR, 0') (2.13)
The parameter adaptation law is chosen as
(i = Proj(9, as) (2.14)
where Proj(é, ¢)2F¢ for d E Q and is modified outside Q to ensure that

éTF‘1[§— w] s 0 (2.15)

and d(t) belongs to a compact set (25 for all t Z 0, where 91 D Q); 3 9. This can be

achieved by standard adaptation laws with smoothed parameter projection to ensure

A

that Proj(9, (b) is locally Lipschitz; cf. [42]. As an example, consider the case when

it is the convex hypercube
Q={9|ai39i3bi}, ISiSP}

Let

Qa={6la.—650.:b.+6},1:23p}

14

where 6 > O is chosen such that 95 C 01, and choose I‘ to be a positive diagonal

A

matrix. In this case the projection Proj(6, 45) is taken as

'

”1114515 if a,- _<_ 9,- S b,- or
ifé,>b,and¢,~§00r
ifd,<a,-and¢,-20

[Proj(é, m], = < (2.16)

’Yz'i [1 + (bi — HON] (151-, if bl,- > b,- and d),- > O

 

\ Vii [1+(é,—a,)/d] (151', If 61' <04 and 451' <0

Inequality (2.15) ensures that V S 0. Therefore, e(t) and d are bounded for all
t 2 0. Since y, is bounded, we conclude that :1:(t) is bounded, which implies, in view
of Assumption 2.2, that z(t) is bounded. With all signals bounded, we conclude that
e(t) —) O as t —> 00.

In preparation for output feedback, we saturate the control outside a compact
region of interest. We assume that all initial conditions are in a given compact set;
in particular, 0(0) 6 Q, e(O) E E0, and 2(0) 6 Z0, where E0 and Z0 are compact sets.
The sets E0 and Z0 can be chosen large enough to cover any given bounded initial
conditions, but once they are chosen we cannot allow initial conditions outside them.

Let

c1 = maxeeg0 eTPe

C2 = maxoenienl %(é — OFF—IQ - 9)

and c3 > c1 + c2. Then e(t) E E (g {eTPe 3 c3} for all t Z 0. Let Z be a compact

subset of R’" such that Z0 is in the interior of Z and

z(O)EZoande(t)EEVt_>_O=>z(t)EZVtZO (2.17)

15

The set Z can be determined using the Lyapunov function V1 of Assumption 2.2.
The basic idea is to choose cz large enough that the set {V1 3 oz} is positively
invariant2and then determine the corresponding set in the z-coordinate using the
global diffeomorphism that maps 2 into C and vice-versa.

def

Let S 2 max |d7(e, z,yR,6l)| where the maximization is taken over all e E E1 =

{eTPe 3 c4}, 2 E Z, 3);; 6 YR, d E {26, where c., > c3. Define the saturated function

1123 by

 

U48, Z,yR,é))

Wk, Z,yn,é) = S sat ( S

where sat(-) is the saturation function defined as

1 :r>1
sat(;r)= :1: -1333]

—1 :I:<—1

Although the function ¢(-) depends on e, there is no need for saturation, since pro-

jection is used to bound 0. Hence, 9 will not exhibit peaking.

2.3.2 High-gain observer

To implement the state feedback adaptive controller using output feedback, we need
to estimate e; there is no need for estimating 2 since it is already available (the state
of the integrators at the input side). With the goal of recovering the performance
achieved under state feedback, we use the same high-gain observer used in [25],

namely,
6i = éi+1+(ai/€i)(el — é1), 15 i S n -1

(2.18)

8,, = (an/c")(€1—é1)

 

2The choice of C2 is shown in Section 2.4, where it is called c5.

16

where 6 is a small positive parameter to be specified. The positive constants a,- are

chosen such that the roots of

sn+ozls"_l +---+an-ls+an =0 (2.19)

have negative real parts. To implement the control using output feedback, the state

8 in w“ and d) is replaced by its estimate é. By taking

{Gel-‘6‘, 132312 (2.20)

671‘“:

 

and é = [(1, . . . , €n]T, the closed-loop system is represented by the standard singularly

perturbed form

e = Ame+b{Ke+f0(e+yr,z) +0Tf(e+yr,z)

+ [90(6 + yr: Z) + 0Tg(e + yr) 2)]ws(éi Z,yR,é) _ yfin)}

Z 2 A22 +02w8(é, 2,373,199)
> (2.21)

Q)’
II

Proj(é, ¢(é, 2,3712, é»

65 = (A — HC)E+ eb{f0(e +y.,z) + 0Tf(e + y,,z)

+ [go(e + y... z) + 6Tg(e + y... awe. 2,313.6) — yr} ,

 

where C = [1,0,...,O], H = [a1,...,an]T, (A -— HC) is Hurwitz, and e = e — 05
where D is a diagonal matrix with 6"“ as the ith diagonal element. To eliminate

peaking in the implementation of the observer, define q,- : 6‘"1é,, 1 S i _<_ n then,

17

the observer equation becomes

“it = (12' +ai€ —q,1£iSn—1
+1 (1 1) (2.22)

“in = 071831—611),

The system (2.22) will not exhibit peaking if el and q,(0) are bounded function of
5 since it is in the standard singularly perturbed form. In summary, the adaptive

output feedback controller is given by

,3 = ¢3(é,z,yn,é)

" = proj(é,¢(é,z,yn,é)) (2.23)
g = A22+b211

u = 21

2.4 Tracking Error Convergence

The first step in showing tracking error convergence is to confirm that for any initial
conditions in the given compact set, all signals of the closed-loop system (under
output feedback) are bounded. This property is shown in two steps. First we show

that there exist constants c5, c6 > 0 such that the set 3
R. = {{v 3 c3} n {0 e 0,}} x {v1 _<_ as} x {V5 3 c662} (2.24)

is positively invariant for sufficiently small 6, where V; = {TPé and P = PT > 0 is

the solution of the Lyapunov equation P(A — H C) + (A — H C)TP = —I. For this

 

3 Note that the set {V1 _<_ c5} could be time-dependent. See [26, Section 3.4] for
the use of time-dependent sets in the analysis of nonautonomous systems.

18

part we use the fact that f is 0(6) and consequently the derivative of
T 1‘7‘ —1 ‘
V = 6 PC + '2—9 F 6

along ( 2.21) satisfies

6
|/\

—eTQe + kc

|/\

—coeTPe + he

3 —c0V + glé'Tr-lé + he

where c0 = Am,n(Q)//\m,n(P). For all 0 E 95, §6~TF‘19 3 c2. Hence,
V S —coV + c0c2 + k6 (2.25)

On the boundary V = c3, the derivative of V is strictly less than zero V c3 > c2+ek/co.
Since c3 > c1 + c2, for sufficiently small 6, the set {V S c3} fl {0 6 Q5} is a positively

invariant set. For all e E E, it can be seen from (2.10) that

V1 S —773l|5||2 +714 fill?“
m:n( )

Using (2.9), we obtain

0 "3
< ...,—V /___£3_._.‘/
l/l _ 772 l + 774 Amin(P) V1/771

Therefore, choosing
122711 2 _Ca.._--
C5 > ( Tia ) UiAmin(P))

we can ensure that the set {V1 3 c5} is positively invariant. Finally, the derivative

0f V5 is given by

V5 = {-51% + 2€pr{fo(°) + 6%) + l90(') + Nam/2%)}

19

Since all state variables are bounded in Rs, we obtain

~. 1 , ,
u s 75% — ml: + WV; (226)
for some positive constant k1, independent of 6. Choosing
C6 > [2k1AmaI(P)]2

ensures that the set {15 g 6662} is positively invariant. This completes the proof that
R, is positively invariant for the chosen values of c5 and c6. The second part of the
argument, is to show boundedness of the signals under output feedback. For that we
need to show that the fast variable 5 decays rapidly to 0(6). Since V(e(0), 0(0)) < c3
and 112’ is bounded uniformly in 6, there exist a finite time T1 independent of e such

that Vt E [0, T1], 2(t) E Z and V(e(t),d(t)) 3 c3. During this time interval we have,
‘ 1 2 2
V: s 3101, for V: 2 cse

Using the fact that “5(0)” 3 leg/6"“) for some he > 0, we obtain

V450» < ——fi‘ e-‘W‘

— 62(11—1)

where 51 = kgllPll and 62 = Ell—115W" Choose 6" small enough that

1
T 4.3! i1 fll < —T
(6) ,82 n(C6€2n) — 2 I

 

for all 0 < c < 6". Hence,

Vt “(TD S 6662

20

for all 0 < 6 < 6“. By choosing 6" _<_ (c3 — c2)cO/k, we are guaranteed that the
trajectory enters the set R, during the interval [0, T(6)] and remains inside thereafter.
It follows that V t > T(6), e E E and since é = e + 0(6) we conclude that é E E1.
Since the saturation level was taken over all e E E1, 2 E Z, 311; 6 YR and d E {25,
the saturation function will not be effective, i.e., dvszw V t > T(6). Therefore the

closed-loop system is given by

e = Ame — 0672220) + A(-)
é~ : FP(€3 Q5) - (227)
C : F2(Cte)y1‘7C70)
65' = (A — HC)§ — 6b[éTw(t) + Ke] + 6A(-)
where
PM“, ¢) -—- Prom), ué, 2. Me, (9))
1W) = f(-) + 9(-)w(é(t), 2(t),yR(t),é(t)) (228)
M“) = b{K(8 — é) + (f0 '— f0) + 9T(f — f) + (90 — 90)“
+9719 — W} (229)
f() = f(é+yraz)
EN) = g(é + yr: 2)
Define w, as
wr(t) = f (32,, Z) + 90%, 5W0, 2,30%) (230)

where Z is the steady state solution of the zero dynamics, determined uniquely from

C = T1(yry 2)-

21

Definition 2.1 [26, Definition 13.1] A vector signal V(t) is said to be persistently

exciting if there are positive constants 621, 6312 and (i such that
t+5 T
021 Z/ V(T)l/ (T) dT 2 all
t

Assumption 2.3 w, satisfies one of the three following conditions:
0 w, is persistently exciting;
0 wr = 0;

0 There exists a constant nonsingular matrix 5', possibly dependent on 0, such

that

sw.(t) = "mm (2.31)

0

where wrl is persistently exciting.

The first case is treated in [25] and the second one is the regulation case of [1]. The
analysis of either one of the first two cases is a trivial specialization of the analysis
of the third case. Therefore, we concentrate our attention on the third case. Using

the transformation .5"1 to transform 9 into
”T —1 "T ~T
0 S = [01,02]
the equations for e and 0 can be rewritten as

é : Ame — béTS—IS'LUT '1' béT(wr — 22]) + A()

s—Ti = s-Tr,

22

01'

It is shown in [42] that if the set it satisfies the Imbedded Convex Sets assumption4

then F”, and I‘gp are Lipschitz in 6. Using the fact 5

 

w. — w = (f —f) +0721) — 0v) + 0 wr (232)

where

f(°) = f0?“ 2), at) = 9(3)" 2), 00f) = 9004, 2)

2lL() : 11!}(01 ZayR26)1 ALA) : 11)“), ZiyRi 6), 12K) : w(éa Z7yR1é)

it can be shown that e and 01 satisfy

e Am —bgw;fl e As(-)
.: 2 ~ + (2.33)
01 2F1gwr1bTP O 01 Ae(')
where
s-Trs-1 — P1 F2
rg‘ r.
A.() = A( ) + b6T[(f' — f) + (in/2 - 022)]
Ae(') : [Flp — 2P1g’wrlePC]

¥

4see Appendix 2.6.4
5See Appendix 2.6.1

23

and

. = 90:02
Kgl > 9() 90+0Tg > Kgg

for some positive constants K91 and Kgg independent of 6. Since f, go, 9 and w are

Lipschitz functions in their arguments, we have 6

”As(')”

|/\

”9w " 9’1?” S kCIHC” + kelllell + kélllgll

|/\

llf - ill kezllell + k£2|l€|| + kollill

and 7

HM)” S dolléll
”As(')” S 61|lell+62|l€l|+53||5||
”Ae(')“ S 54l|6||

for some 6,- 2 O, i = O, .., 4. Consider the system

é Am —bgw;.r1 e
51 2F1gwrleP O 61
Defining
éln = F— 61

(2.38) can be rewritten as

 

6See Appendix 2.6.2

||A(-)|l + llbéTW - f) + (91/3 - 010]”

(2.34)

(2.35)

(2.36)

(2.37)

(2.38)

7Note that F”, is Lipschitz in e since the set it satisfies the Imbedded Convex Sets

assumption [42].

24

e .4," —bng‘,r’ e
-. = ] ~ (2.39)
61” 2GP UlrleP 0 61"

Using well known results from adaptive control theory (see for example [26, Section
13.4]) and the fact that wrl is persistently exciting and Q() is bounded from below,
it can be shown that (2.39) has an exponentially stable equilibrium point at the
origin. Then, from the converse Lyapunov theorem, there exists a Lyapunov function

V2(t, e, 01) whose derivative along (2.33) satisfies

V2 S -65||€|l2 - 66ilélll2 + 67l|8|l|l€ll + 68llélllll€|l + <59I|6H2

~ ~ ~ __ (2.40)
+ 610||6||||91|| + 611l|€||l|<l| + 512H91||HC||

for some positive constants 65 and 66 and some non—negative constants 6,- ,7 _<_ i S 12.
Noticing the similarity of the right-hand side of e and 65 in (2.27) one can easily show
that

65 = (A — HC)6 — cbgé’fwn — 6K8 + 6A3 (2.41)

The derivative of V5 with respect to (2.41) satisfies

., 1 ~ -
V: S :Iléll2 + 73l|91llll€|| + millelllléll + ’rsIIClllléll + 76H€H2 (2-42)

for some non-negative constants 7,- ,3 g i g 6. Construct the Lyapunov function
candidate

W = 0V + av, + v2 + V5 (2.43)

where a > O and ,6 > 0 will be chosen later. Using the inequality8

V S -kvlllell2 + kv2llel|ll€|| + lCusllfllz + kmllflllléll + kv5 élllléill (2-44)

 

 

 

8See Appendix 2.6.3

25

 

together with (2.10), (2.40), and (2.42), it can be shown that the derivative of W
with respect to (2.27) satisfies

 

 

 

 

 

 

 

.- q T P _.
Hell [[6]]
- “All ”9',”
W S _ ~ M .. (2.45)
IICII ||c||
_ ”5H _ [ H611 _
where M is given by
akv1+65 — 69 :gm - 42—6“ _,._.s_,_
.1310 56 1311 W
M =
jag—4T1- 1:” fins 35—;‘5—1
_ ——‘m i— 7. — as.
Choose 6 large enough to make
P a
‘56 13”
_ 2%)": 3723 .

 

 

26

positive definite; then choose a large enough that

l' . ']
akvl + 65 — 69 mg —L’;—611

 

 

15m :212
2 66 2
_g -5 —6 .

is positive definite. Finally, choosing 6 small enough we can make M positive definite

. Hence, by [26, Theorem 4.4], we conclude that

 

 

' 1
Hell
H91“
~ —> O, as t —2 oo
IICH
_ ||€|l ,

It should be noted that the foregoing analysis does not imply exponential conver-
gence since the right-hand side of (2.45) is only negative semidefinite. This is a key
point in the analysis because considering (2.33) together with the E and 5 equations
from (2.27) one cannot show exponential stability of the set {(e, 01, 6,5) = O}. The
difficulty arises from the perturbation terms on the right-hand side of (2.33). While
those terms satisfy the growth condition (2.37), the constants 61 to 64 are not nec-
essarily small. Consequently, we see in (2.40) that the right-hand side contains the
positive term 69||el|2 which could dominate the negative term —65]|e||2. We overcome
this difficulty by including 0V in the composite Lyapunov function W and choosing
a to ensure that the negative term —ak1||e||2 dominates 69||e||2 and other cross prod-
uct terms. The function V, however, is positive definite in (e, 0), not only (e,01),

and that is why the right-hand side of (2.45) is only negative semidefinite.

27

2.5 Examples

2.5.1 Linear Plant [14]

Consider Example 6.4.1 of [14]. The system is represented by the transfer function

_ kp(S + 00)
_ .2 u
(5 ‘l' 018 + (10)

 

where [6,, > O and be > O. The input-output model is

3;; = -—a1y — day + k,,(0 + bou) (2.46)

The goal is to design an adaptive feedback controller that renders tracking of a

reference y,. In [14] the reference model is given by

 

and r is a command signal. In our case, the controller requires y,, y}, and 3),. We

generate them using the second order filter

2
can

2 , r
yr 32 + 2anns + wfi

where 10,, = 40 and (n = 1. Its input is r and its states are the smoothed y, and 9,.

Equation (2.46) can be rewritten as
if] =1 —91y — 02y + 6311, + 9411,

which takes the form (2.1) with n = 2 and m = 1. Assumption 2.1 is satisfied

for hp at 0. We augment an integrator at the input side, set x1 = y, x2 = y,

28

 

z = u, and treat v = a as the control input. Let el = y — yr, e2 = y — g. and
6T 2 [a0, a1, kpbo, kp]. The change of variables C = z — i272 transforms the system

into the normal form

6 = Ae+b{6Tf+0Tgv}

4' = —%§C+%§(el+yr)+(%§—%§)(e2+vr)
where _ q _ q
—.’L‘1 0
0 1 0 —.’L'2 0
A: ,b= ,f= ,9:
O O 1 z 0
0 1

 

 

 

 

Assumption 2.2 is satisfied since 03 > 0, 94 > 0, and V1 can be taken as V1 : %62.

Choose the matrix K = [6 5] to assign the eigenvalues of Am = (A — bK) at -2 and
-3. We obtain the matrix P by solving the Lyapunov equation PAm + AiP = —I.
The function it of (2.12) is given by

_ —6€1 — 562 + 61(61 "l" 31,) + 62(82 + yr) — 632 + y,-

w— é4

 

and (b :2 2eTPb[ f + gw]. We use the scaled state observer

€41 = (12+(81—(11)

642 = 6(61—01)

where él = q1 and 622 = Q2/6. The variable e is replaced by its estimate (3 in the

control and adaptive laws for the output feedback case. When y, is constant, the

29

I

vector w, of (2.30) is

w, =

 

 

b

Hence, the third case of Assumption 3 is satisfied with the transformation

 

 

 

1 0 0 0
O 1 0 O
S :-
g; 0 1 0
[ 0 O 0 1
rendering
. . r ~ .
_yr 01 _ 33 93

0 - 0
5w, 2 and S‘TH = ~2
0 03
0 0'4

 

 

 

 

It is expected that the term 01 — gidg will converge to zero. On the other hand 62, 03
and 04 are not expected to converge to zero. The plant was simulated Using Matlab
with 6 = 0.01, 0T = {—10 3 3 1], Q = [—15,—5] x [1,5] x [1.5, 5] x [0.75, 2], and
the adaptive law (2.15) is used with 6 = 0.01 and I‘ = diag[300, 10, 10, 0.05]. Two
types of command signals r were considered. First, a step of amplitude 2, Figures 2.1
to 2.3 are for the case when the output feedback controller developed in this chapter
is used. Figure 2.1 shows the tracking error c, Figure 2.2 shows parameter errors 67,
and Figure 2.3 shows the parameter error 9 in the new coordinate i.e., S ‘Té where the
first component converges to zero as predicted by the theory. Second, the command

signal r is taken as r = O.5sin(0.7t) + 2cos(5.9t). Figure 2.4 shows tracking error

30

e, Figure 2.5 shows the parameter errors 0. Note that d converge to zero since the
reference signal y, is persistently exciting. Figures 2.6 and 2.7 show simulation of
the model reference adaptive controller (MRAC) [14]. Figure 2.6 shows the tracking
error and Figure 2.7 shows parameter errors 0, when r is a step input of amplitude
2. Figures 2.8 and 2.9 show the same quantities when r = 0.5sin(0.7t) + 2cos(5.9t),
0, is a function of the plant’s parameters that is different from our 0 . In conclusion,
in tracking a step command signal our controller has shown better steady state error
over the MRAC one. There is no noticeable difference in the convergence of the
parameter errors between the two methods. However, in our approach we were able
to transform the parameter errors into another space where we were able to draw
some conclusions. In particular, we were able to predict that 01 - adv“. approaches
zero. In the case of a persistently exciting command signal, our method shows better
considerably faster tracking error convergence. The parameter errors are comparable

in their convergence rate.

2.5.2 Nonlinear Plant

Consider the nonlinear system
3'): a1y+a2(y+uy2)+b1u+u (2.47)

which takes the form (2.1) with n = 2 and m = 1. Suppose the reference signal y, is

a step input. Assumption 2.1 is satisfied for b1 34 0. We augment an integrators at

the input side, set x1 = y, 232 = y, z = u, and treat v = a as the control input. Let
1

el = y — yr, e2 2 y -— y, and 6T 2 [a1 a2 b1]. The change of variables C = z - 5x2

31

 

transforms the system into the normal form

é : Ae+b{f0+t9Tf+9Tgv}
(0 = —%{01(61+yr)+C+%(82+3;/T)+62(€1+yr)

92K + 52032 + llrllle2 + 902}

 

 

where
.731 O
01 O
A: 7b: ifOZZi : {171+ng 39: 0
O O 1
0 1

 

 

Assumption 2 is satisfied when 63 > O, and V1 can be taken as V1 = $2. Choose the
matrix K = [6, 5] to assign the eigenvalues of Am = (A —- bK) in the Open left-half
plane. We obtain the matrix P by solving the Lyapunov equation PAm+A£P = —I.
The function w of (2.12) is given by

—z — 661- 562 - 41(61 + yr) - 92[(el + v.) + 2(62 + 3202] + i).
673

 

1p:

and (b = 2eTPb[ f + 91/2]. We use the scaled state observer

641 = (12+(81-(11)

642 = 6(61—41)

where él = q1 and ég = q2/6. The variable e is replaced by its estimate 63 in the

control and adaptive laws for the output feedback case. The vector w, of (2.30) is

yr

wr yr

 

 

32

Hence, the third case of Assumption 3 is satisfied with the transformation

 

 

1 O O
S = —1 1 0
_ O O 1]
which yields _
yr 61 + 52
5w, = 0 and S'Tbl~ = 6.2
O _ _ 03 J

 

 

 

 

Simulation results for the output feedback case when y, = 1, 6 = 0.01, 0T = [5 2 3],
Q = [4, 6] x [1,3] x [2,4], and the adaptive law (2.15) is used with 6 = 0.01 and
I‘ = diag[3, 3,1.5], are shown in Figure 2.10. Figure 2.10-a shows the tracking error
e, Figure 2.10-b is the control u while Figure 2.10-c shows the control after saturation

1123, and 2.10-d shows projected parameter error 01 + 02 converge to zero.

2.6 Appendix

2.6.1 Proof of( 2.32)

33

 

The term (ll-J - 2b) can be written as

(00+6T0)(0o+01‘g)

+ —y1(-n)§0-yin)9T§+f0§0+f09TA§+élj§0+éTf9TQ
(§0+9T§)(§0+9T§)

where f0 = f0(y,,2). Then

Wail")—foéT0+éngo—0Tf’éT§+éTflIT0+6Tf’0T9-6Tf6T0

 

 

w _ 10 _ (00+9T0)(0o+éT0)

flat/l")—fo—6Tf)+5Tf(0o+0Ti)

 

_ (00+0T0)(0o+éT0)
: (90+ng) __1—-(f + 91p)
2 __9'7."
(00+0T0)w
Hence,
- — . _ ~ .~ _ 6‘7“
w.—w—(f—f)+(gw— Hymn).

34

 

2.6.2

 

 

 

2.6.

ll? l1;

Sill ((1 [I’ll

2.6.2 Proof of( 2.35) and ( 2.36)

9113 - fill) =

l/\

K'u
I

"rn
II

S

g(y..z)v(o.z,y1.,é)— 90% + 32., ave, 2:, y... 0“)

got, 2mm, 2, y... 0“) — got. aim, 2, yR, 0)

+g(yr,2)1/)(0, ZiyRaé) '- 9(8 + yrv z)z/’,(e7 Z, yRa 6)

+9(8 + yr, Z)I/}(€, Z, 3237 é) - g(é + yr, Z)w(éa Z) yR) 9)

kClllé“ ‘l‘ kelllell + kélllgll

fO’n 2') - f(é + M. 2)

f(yr72) _f(e+yra2)

+f(e+y,,Z)—f(é+y,,2)

+f(é+y,,z) - f(é+y.,z)

ke2l|€ll + kt2ll€|l + k<2llC~ll

2433 Proof of( 2.44)

We have

e = Ame—dew(t)+/\(o)
i = P(2éTwa)

since llA()|] S tong“, the derivative of V along the trajectories of the system satisfies

35

 

V < —k,,1||e||2 — 2eTPbéTa + kQIIeIHIEH + éTr-lb
s —k..1llell2 MVP—1Ui — 1‘2eTwa) + k2lle||||€|l
S —kvl“ell2 + éT(2éTPb’tD - 26Tl’lnb) + kzllellllfll
S -kv1l|€ll2 + 26~’T(éT - 80wa + k2|l6llll€ll
S —k.1 Hell2 + 25%wa + k2llellll€ll

g —kv1||e||2 + 26PbéTw. + 2éPbéT(w — w.) + k2||e||||6||

g —k,,1||e||2 + 26PbéTs-lsw. + 26PtéT(o — w.) + kgllellllfll
S —Ic..llell2 + kaolléllllélll + 2éPbéT([f — f+ 043 — 0151+ afia—gw.) + k2ll€llll€ll
V < —kv1||ell"’ + kv2llell||€l| + wilt“2 + kallé~ llllé ll + kvslléllllélll
where e = e — e

2.6.4 Imbedded Convex Sets Assumption

To have a smooth projection, fl is required to satisfy the Imbedded Convex Set
Assumption; that is,

There exists a known 02 function ’P from Q to R such that the following hold.

36

1. For each real number A in [0, 1], the set

{6079(9) S A}

is convex and contained in Q.
2. The row vector (8P/86’)(9) in nonzero for all 6 such that ”P(d) E [0, 1]

3. The parameter vector 0* of the particular system to be actually controlled

satisfies

P(d') S O

2.7 Conclusions

For the nonlinear output feedback adaptive controller, we have successfully shown
tracking error convergence without requiring the persistence of excitation as in [25].
This is a major improvement over [25]. Also, we have removed an unnecessary feature
of the controller of [25], namely, saturation of the right-hand side of the adaptive
law. It is not needed due to parameter projection. We have also allowed go and g,-
of (2.1) to be nonlinear functions instead of constants as in [25]. On the other hand,
we have required exponential stability of the zero dynamics which is stronger than
the bounded-input—bounded-state stability requirement used in [25]. It should be
noted that in [16], Jankovic used ideas similar to ours to design an adaptive output
feedback controller for nonlinear feedback linearizable systems. He proved tracking
error convergence without persistence of excitation. However, to do that he required
the parameter adaptation gain (I‘ in our case) to be sufficiently large. Our result

does not impose any such restrictions on 1".

37

0.5

 

 

 

 

 

 

5 10
Time (see)

Figure 2.1. Tracking error e.

38

15

(a)

 

 

 

 

 

-6 - .
0 5 10
Time (see)

(c)

15

 

 

-1)

 

 

 

0 5 10
Time (sec)

Figure 2.2. Parameter errors,(a)- 01 ,(b)- 02,(c)- 613, (d)-04.

15

(b)

 

—2

 

-3

1.5

0.5 .

 

 

 

15

 

 

 

 

5 10
Time (360.)

~

15

(a)

 

1O

 

-5.

 

 

 

-10 A .
O 5 10
Time (sec.)

(C)

 

 

-1

 

 

 

_2 A A
0 5 10
Time (sec)

Figure 2.3. Projected Parameter errors:

15

15'

40

(b)

 

-1

_2JL

 

 

 

 

_3 e .
O 5 10
Time (sec.)

(d)

15

 

0.5 '

 

 

 

5 10
Time (sec)

~ ~

(a)- 0, —- 3:19.01)- 02,0.)— 03, (0)4}...

15

 

0.5...“ .....

 

 

 

 

 

0 5 10 15 20 25
Time (see)

Figure 2.4. Tracking error e.

41

3O

35

 

 

 

 

 

 

 

 

.]
-2 — -
_3 1 1 1 1 1 1 1
0 100 200 300 400 500 600 700 800
Time (sec)

Figure 2.5. Parameter errors 0.

42

1.5

0.5

 

 

 

 

 

.........

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

 

 

 

 

"1
i i 1' 1 1 L 1' 1 i
2 4 6 8 10 12 14 16 18 20
Time(sec.)

Figure 2.6. Tracking error 8 using MRAC controller.

43

(a) (b)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 r 2 N r +
1 5 ’ i 1.8 . J
j i 4 1.6-
05'
1.4»
O .
1.2-
-O.5-
..1 1 [ U
-1.5 ‘ L ‘ 0.8 ‘ ‘ ‘
0 5 1O 15 20 0 5 1O 15 20
Time (sec.) Time (sec.)
(c) (d)
2 v v 2 . , e
1 .
1.5-
0 b
-1 . 1

 

I
N

P

01

 

 

 

-3 . .
W 0 - 1

_4 .

O 5 10 15 20 0 5 1O 15 20

Time (sec.) Time (sec.)

 

 

 

 

 

 

4' ~ ~

Figure 2.7. Parameter errors: (a)-é1, (b)-92, (c)-03 , (d)-04.

44

 

 

 

 

 

 

2 T I 1 l 1 i l
15_.. ,,,,,,, i, ...i
1... ............................................................................................. .—
0051 ........... 5 ............ 3 ............. g .................. , ..... ............ g ............ g .............
1 1 4' 1 1
100 200 300 400 500 600 700 800
Time(sec.)

Figure 2.8. Tracking error 6 using MRAC controller.

45

 

 

 

 

 

l l l l l

 

 

1 00 200 300 400 500
Time (sec.)

Figure 2.9. Parameter error 5,.

46

600

700

 

800

 

 

 

 

 

0 50 100
...(a)..-

150

 

 

—1o-1

 

 

 

—15 i 4
O 50 1 00
..(c)..-

150

 

 

 

 

 

 

 

 

 

 

 

0
i, ...... 1
_8 z ;
0 50 100 150
--(b)—-
2 7
o\/ g
\. ,. _ _ __ _._ _f_._ _._..
-2 i
0 50 100 150
—-(d)--

Figure 2.10. The :r-axis is time. ~(a) Tracking error e; (b) Control 21; (C) W(-) ; ((1)
Projected Parameter error: 91 + 02 (solid), 02, 03 (dotted).

47

 

 

 

CHAPTER 3

Robustness to Bounded

Disturbance

3.1 Introduction

Robustness of adaptive controllers to bounded disturbance is of utmost importance
for its practical use [14]. In the previous chapter we have achieved tracking using
adaptive output feedback control. In this chapter, we study the robustness of that
controller to bounded disturbance. We present two results in that direction. First,
we present a robustness result in the usual form of robust adaptive control results
[14]. We show that, for sufficiently small bounded disturbance, all signals in the
closed-loop system will be bounded and the mean square tracking error will be of
the order 0(6 + d1), where d1 is an upper bound on the disturbance. Second, if the
bound on the disturbances is large we go one step further to introduce a new control
component to ensure that for any bounded disturbance, with a known upper bound,
all signals in the closed-loop system will be bounded and the mean square tracking
error will be of the order 0(6 + p), where both 6 and II are design parameters. In the
design we use a Lyapunov redesign technique, and we do not require the disturbance

to be small.

48

3.2 Robustness PrOperty

Our goal is to prove that the adaptive output feedback controller of Chapter 2 is
robust with respect to small bounded disturbance. To simplify the presentation, we
rely heavily on definitions, assumptions, and proofs from Chapter 2.

Consider a perturbation of (2.1), given by

If") = f0(') + ifJ'Wi + [90(') + 259400.111“) + d(') C“)

i=1

where d() is a disturbance term of the form
d(t, :c, 2, v, 6) = df(t, 2:, z, 6) + (19(33, 2:, 0)?)

The error equation (2.6) becomes

é = Ame + b{K6 + fo(-) + 9Tf(°) + (90(') + 0Tg(-))v

+ d(t, e + y,, .2, 11,0) — 319)}

Assumption 3.1 The disturbance d() satisfies
“d(t, e + yr, 2.1/130), 9)” S d1

VtZO,e€E,y,—€Y,zEZ,andéER".

Suppose further that for sufficiently small d1 Assumptions 2.1 and 2.2 hold uniformly
in d and the set Z has the property (2.17) for all (1. Recall the set R, defined by
(2.24):

R, = {{V 3 c3} fl {9 6 95}} x {V1 3 c5} x {Vg S 6662}

with the same values for c3, c5, and c5 as determined in Chapter 2. We show that

49

 

the set R, is positively invariant for sufficiently small 6 and (1,. To show that, we
conduct our analysis assuming all signals are inside the set. Later on, we show that
the fast variable 5 enters the set in finite time. Hence, all variables will be trapped
inside the positively invariant set R,. Inside R,, (e — é) is 0(6), hence é E E1. Since
the saturation level was calculated by maximization over all e E E1, 2 E Z, 32, 6 YR
and R E (25, the saturation function will not be effective inside R,, i.e., ’l/jszw. Hence,

inside R, the output feedback controller of Chapter 2 is given by

_ —Ké + 215.") — me + yr, 2) — 0Tf(é + yr, z)

 

v — - - . (3.3)
g0(e + yr12') + 0Tg(e + yr: 2)
The error equation can be written as
e = Ame — béTuvu) + A(-) + bd(-) (3.4)

The derivative of

V = $191: + §éTr-1é

along (3.4) satisfies
V = —eTQe + 2eTPbA(-) + Zede(-)

We use an argument similar to the one used in the previous chapter to show bound-
edness of the state variables. First, to show that R, is positively invariant, for

sufficiently small 6 and ([1, we use the fact 5 is 0(6) to arrive at

V g —eTQe + kc + kddl (3.5)

50

 

where k > 0 is the same constant appearing (2.25) and kd > 0. Furthermore,

V S —C06TP€ + k6 + kddl
= —(:0V + gnéTr-lé + k6 + kdd,

S —COV + COC2 + k6 + kddl

Therefore, on the boundary V 2 c3, V < 0 for all 03 > 02 + W. In the ideal case,
when d = 0, V < 0 on V = 03 for all 6 < 6" _<_ co(c3 —— €2)/k. In the presence of the
disturbance d, V < O on V = 03 for all 6 < 6" and d1 < 31 (6) = [(63 — C2)Co — k6]/kd.
Hence V 6 < 6* and (11 < (II, the set {V 3 c3} n {F E {25} is a positively invariant set.
We can show, as in Chapter 2, that {V1 S c5} is a positively invariant set. Finally,

the derivative of V5 is given by
v, = {5T6 + 2€TPb{fo(-) + 07m + [goo + (Fm-We + do}
Since all variable are bounded in R,, and so is d, we obtain
V, s —2—16€TE — 271—7514 +(k1+ kddm/VE
for some positive constant fed, independent of 6, and k1 is the same constant appearing
in (2.26). Choosing

(i :_ J66“ 2k1AmaI(P)
1 2dema,(P)

 

it can be shown that for all (1 < (21 the set {V5 g 6662} is positively invariant. This
completes the proof that R, is positively invariant for all (1 < min{dl,d1}. The
second part of the argument is to show boundedness of the signals under output
feedback. For that we need to show that the fast variable 5 decays rapidly to 0(6).
Since V(e(0), 5(0)) < c3 and 1/2‘ and d are bounded uniformly in 6, there exists a finite

time T2 independent of 6 such that V t 6 [QB], z(t) E Z and V(e(t), 5(0) 3 c3. the

51

time T2 depends on d and equals T, when d = 0. Since the right—hand side depends
continuously on d, for sufficiently small 611 we can ensure that T2 2 %T1. During the

interval [0,T2] we have,
- 1
V6 3 -2—€||€||2, for V: 2 c662

Therefore
. fl _ 2 6
V€(€(t)) S m8 B t/
where £1 = kgllf’ll and 52 = TIFF From Chapter 2 we know that

fl? ) s 1
0662" 2

 

T(6) déf iln(

52 T1 S T2

for all 0 < 6 _<_ 6*. Hence,

Vc(€(T)) S 6662

Thus, the trajectories are guaranteed to enter the set R, during the interval [0, T(6)]

and will remain inside thereafter. For all t 2 T(6), the inequality
V g —eTQe + k6 + kdd,
is satisfied and since all signals are bounded we conclude that

T
lim 1/ eTQe dt 3 k6 + kddl
T o

T—>oo

which shows that the mean square tracking error is of order 0(6 + d1).
In summary, we have shown that the adaptive output feedback controller of Chap-
ter 2 is robust with respect to small bounded disturbance in the sense that for each

0 < 6 < 6*, there is 61’; == d’f(6) such that for all d(c) satisfying |d| g d1 < 611', the tra-

52

 

19th
is 0|

inC

811

p0

L5

jectories of the closed—loop system are bounded and the mean square tracking error
is of order 0(6 + d1). It is important to note that 6‘ is the same bound established
in Chapter 2.

In Chapter 2 we saw that sharper results can be obtained in the ideal case, d 2 0,
when Assumption 2.3 is satisfied with either partial or full persistence of excitation.
In the rest of this section we investigate the effect of persistence of excitation when

d ¢ 0.The closed-loop system is given by

6': _ Ame — :1?th A(.) + d(-)

5 = r (9,

-. p(~ <25) _ (3.6)
C : F2(C’e)yr’<70)

66' = (A — HC)6 — eb[éTuv(t) + Ke] + 6A(-)

Suppose Assumption 2.3 is satisfied with partial persistence of excitation, decom-

posed as in (2.31). Then the derivatives of V and V5 satisfy
V S *ka “8H2 + kv2|lelllléll + kv3|l€||2 + kmlléllllf“ + kaHEHHél” + kdldl (3-7)

. 1 - -
V6 S glléll2 + 73|l01|ll|€ll + 74l|6|l||€|| + 75l|C|||l€|| + valléllz + kd2d1 (38)

° _ w'r . .
e = Am ()9 ,1 e + A,()+bd() (3.9)

~

9'} 2I‘lgwrleP 0 o, Ae(')

As in Chapter 2, it can be shown that the system

('3 Am —bgw:.r e
= 1 (3.10)

51 2F1gwr1bTP 0 91

has an exponentially stable equilibrium point at the origin. Then, from the converse

Lyapunov theorem, there exists a Lyapunov function V2(t, e, 5) whose derivative along

53

 

(3.9) satisfies

V2 S —5s||6||2 - 66HéIH2 + érllelllléll + 58H51||H€|l+ 59|l6||2

+ (SIOHBHHéI H + 611H6||||5||+ 612llé1||||5|| + 6d1d1||ell + 6d2d1 H91 ||
01‘
V2 S ~5s||6l|2 - 66Hélll2 + 67|lellll€|| + 6s||51||||€|| + 59|l6||2

+amwman+mummm+ameM1+am (an)

From (2.10) together with (3.7), (3.8), and (3.11), it can be shown that the derivative
of

W=av+z3V1+V2+wE

along the trajectories of the system, satisfies

 

 

 

 

 

 

- 7 T - -
Hell Ilell
. llélll ~ llérll 6 5 ~
ivs— ~ M ~ -§kW—§WM+wfi em)
IICII IICII
_ nan , _ ”an .
where M is given by
'01.... s — 69 __,.. - __.._1.. ”
—_6m a 2m _-__1__68— s-akvs
2 2 2 2
M = (3.13)
- an m _1.__.
_ -akv2;57-14 -58-7;—0kvs —15-20kv4 _:_ _ 76 _ akv3 d

 

54

 

1V! is similar to M of Chapter 2, except for changes in the (1, 1) and (2, 2) elements.

Choose fl large enough to make

- H

 

 

1:. ._1.z—5'
2 2
~85”

2 (3773

positive definite; then choose a large enough that
111

p

akvl+é2a_59 :m Ml

 

 

 

flu 25 .1512
2 2 2
-l3fl4-511 -512
2 2 18773

is positive definite. Finally, choosing 6 small enough we can make M positive definite

for some C, > 0. Since all signals are bounded, the mean-square tracking error

lim 1/T||e||2dt
T—moT 0

and the mean-square partial parameter error

lim 1 [Tné “2 dt
T—rooT o 1

are of order 0(d1).

On the other hand, if w, is persistently exciting, then the derivative of V satisfies

V _<_ -lcv1|l€||2 + kv2|l8||||£|| + lvvalléll2 + kmllilllléll + kvsllélllléll + kdldl (3-14)

55

and the system
é Am —bger e
L = ~ (3.15)
0 2FlgwrbTP 0 0

has an exponentially stable equilibrium point at the origin. We can repeat the preced-

ing argument to show that there is a Lyapunov function V2(t, 6, f5) whose derivative

along (3.9) satisfies

V2 S. —<55|le||2 - (Sislléll2 + 57||6l|||€|l + ésllélllléll + 59||8||2

~ ~ ~ ~ (3.16)
+ 6w|lellll9ll + 6nl|€|l|l<ll + 612ll9lll|Cll + kddl

Moreover,

. 1 - -
V: S glléll2 + 73||9||||€|| + 14||8||||€l| + ’YsllCllllEll + 76||€||2 + kd2dl (3.17)

Inequality (2.10) together with (3.14), (3.16), and (3.17) show that the derivative of

W along the trajectories of the system, satisfies

- . T - -
Ilell llell
. é .. (5 5 6 .-
w s — ”~” M ”-” — 55-11212 - 5191+ Cddr (3.18)
ucu ncu
_ urn, _ nan,

 

 

 

 

where [VI is given by (3.13) and can be made positive definite. Hence, W satisfies
W s —ka + add, (3.19)

for some kw > O, which shows that all variables, including the parameter error 5,
converge to a ball centered at the origin, whose size is of the order of 0 (Mai). We

note that the additional properties we have shown under Assumption 2.3 may require

56

 

6‘ to be less than the bound established in Chapter 2.

3.3 Robust Output Tracking

If the bound d1 is not small enough, we introduce an additional robustifying control
component to make the mean-square tracking error arbitrarily small, irrespective of
the bound on the disturbance d, provided this bound is known. Once again, we
consider the perturbed system (3.1) and assume that Assumptions 2.1 and 2.2 are
satisfied uniformly in d(-). Moreover, we assume that the set Z has the property

(2.17) in the presence of d. The control is taken as

v = ¢8(é12,y12,é)
- S sat (—L——M e’z’SyR’é )

where

A — A (n) __ e _ AT ..
¢(é:Z,yR,6) : K€+yr “f0(e+yraz;) ? f(e+yraz)+'Ul
90(8 + yT1Z)+ 6Tg(e “’1‘ yr, Z)

 

and the robustifying control component U1 is to be designed using the Lyapunov
redesign technique, e.g., [26, Section 13.1]. The saturation level 5 is determined as
in Section 2.3.1, except for a new constant c3 to be determined. The constants c.;

and 0, used in calculating S are chosen in terms of the new 03. consider the set

R, = {{V g c3} n {é e m} x {v1 3 05} x (v, g 0662}

where the constants c3, c5, and c6 are yet to be determined. We limit our analysis

to this set to show that it is a positively invariant set. Inside R,, the saturation will

57

not be effective. Hence, the control is given by

__ —Ké + 1);") -— f0(é + yr, 2) — éTf(é + 31%) + 111

 

— . A . (3.20)
go(e + yr, 2) + 0Tg(e + yr, 2)
The error equation under (3.20) becomes
e = Ame — béTuvu) + b[v1 + d(-)] + A(-) (3.21)

where II) = f + 91!), and A() is defined in terms of the new 1,1). The disturbance d()

is required to satisfy the following assumption.

Assumption 3.2

||d(t,a:,z,v,0)|| S p(e,z) + kvl’U1l, O S k, <1

where p and k, are known.

Take 77(e, z) 2 p(e, z) and define 3 = 2éTPb,

- (13111-2.- for n(é, 211$: 21
”1(a) : (3.22)
2 A ‘ A A
’ 44271—323) 'i for n(e,z)lsl < it

As in the previous section, we start by showing that the set R, is positively invari-
ant. Then, we show that the trajectories enter R, in finite time and remain inside

thereafter. For the first part, consider the derivative of

V = eTPe + géTr-lé

58

 

along (3.20), which satisfies

V g —eTQe — 2é7‘Pb0‘Tu‘2 + éTPIé + 2éTPb[v1 + d()]
+ 2eTPbA(~) + 2(e7‘ - éT)Pb[v1 + d(-) — 9712;]

Using the adaptation law

5: Prom}, <25)

where

¢(é. 2., ya, (9) = 2éTPbif + 31)]

and defining

A,(-) = 2(eT — éT)Pb[v1+ d(-) — 9%]
+ I2éTPb|(n(e, z) — 77(é, 2)) + 2eTPbA(-)

the derivative of V satisfies
V g —eTQe + 3v, + |s|[77(é, z) + kvlv1|]+ At(-)

Outside the boundary layer, i.e., (n(é, z)|.§|) 2 11, the robustifying control is

77(é, 2) 3

(1 - kv) |§|

 

U1:—

and the derivative of V satisfies

V g —eTQe + [—$_‘+;}Ji—:JT’ + 17(é, z)|.§| + k$_TlLlJl—] + A.(-)
S ‘CTQB + At(')
3 —eTQe + kcf

59

 

 

where A,(-) S kcé inside the set R,. Inside the boundary layer, i.e., 17(é, z)|s| < 11,

the robustifying control is
1)2(é,z) 3

v1: —(1—kv)/-‘

 

Hence,

<.
/\

2 . , ~2 ,. A 2 . «2
—eTQe +1—"72‘2—;}%L + Isln(e, 2) + kaéagfléli + M)
.. 2 A
< —eTQe +[-n21i1- + nISIl + At(-)
Since enziafi + nlél) s g,

V S —eTQe + [£06 + g (3.23)

Therefore, on the boundary V = c3, V < O for all c3 > 62 + W. Choose 6* > 0
and 11" > 0 such that c3 > c2 + k—‘J—ém. Then, V < O on V : c3 for all 6 < 6*
and p < )1“. We can choose c5 and c6 large enough that the sets {V1 5 c5} and
{V5 5 6662} become positively invariant. Hence, the set R, is positively invariant.
The second part is to show that the fast variable 5 decays rapidly to 0(6). This can
be shown using an argument similar to the one used in Section 3.2. It can be shown
that there exists E and T (6) such that for all 0 < 6 < E, V¢(€(T)) g 6662. Hence,
the trajectories are guaranteed to enter the set R, within the time interval [0,T (6)]
and remain inside thereafter. Hence, inequality (3.23) is satisfies for all t 2 T(6).
Therefore, the mean-square tracking error is of order 0(11 + 6) where the design
parameters )1 and 6 can be made arbitrarily small.

If Assumption 2.3 is satisfied in the ideal case (1 = 0, inequalities similar to (3.12)
and (3.19) can be shown when d 79 O. The right-hand side of such inequalities will

have a term proportional to the disturbance upper bound despite the presence of the

60

 

 

robustifying control component. Thus, such analysis does not reveal an advantage
for the robustifying control. The only advantage we can Show is the fact that the
mean square tracking error can be made of the order 0(11 + 6).

Finally, in the ideal case d = 0, the controller with the robustifying component
recovers the tracking-error-convergence property of Section 2.4, provided Assump-

tion 2.3 is satisfied. First, notice that the control component v1 always satisfies

2

7] A

I’Ull S —|3|
11

Hence

IN S ke1||€||+ kélllfill

Therefore, the effect of U, can be seen on some terms of the bound on V. In particular,

-kvillel|2 + kvzllelllléll + knalléll2 + kvallflllléll + kvsllélllléill + 2leTPbllvll

<.
l/\

S 4131 H8“2 +12v2|le|lll€l| + kvslléll2 + kmllilllléll + kv5||€|llléln (324)

Similarly two terms of the bound on V, will be affected.

. 1 ~ ~ ~ ..
V: S gllEll2 + 13||01l|I|€l|+ 74l|6||l|€|l + 15||C||||€|| + 76K“? (325)

Since wrl is persistently exciting, we can repeat the argument used in Chapter 2 to

show that the homogeneous part of the system

é Am 4)ng e A,(~) + bvl
+
01 2FlgwrleP 0 61 Ae(')

(3.26)

61

 

 

 

is exponentially stable. Therefore, from the converse Lyapunov theorem, there exists

a Lyapunov function V2(t, e, 6) whose derivative along (3.26) satisfies

V2

+510||e||||61|l + 611||e||llC~H+ élzllélllllfll

S -<55||6||2 - 5s||51||2 + 57l|€||l|€|l +58||51|H|€|l + 59llell'2

(3.27)

Note that the effect of robustifying control component 121 on (3.27) can be seen in

the constants 57, 58, 59, and 510. From (2.10) together with (3.24), and (3.25), and

(3.27), it can be shown that the derivative of W along the trajectories of the system

satisfies

where M is given by

 

(ll-61,1 + 65 — 39

 

 

 

 

— q T , 1
“CH ”CH
“91H “91“
”CH ”CH

_ Il€|l . _ 1in .
—_5m_ 4—511

‘2 2
56 1%":
1;” 5773
_gs—Ts-Gkvs “75—0kv4
‘2 2

 

62

(3.28)

 

1 ~
g _ 76 — akv3
.l

It can be shown, as before, that NI can be made positive definite. Hence, by [26,

Theorem 4.4], we conclude that

Hell
”an
”in

_ lléll .

—+0, ast—>oo

 

 

3.4 Example: Nonlinear Plant

Consider the disturbed nonlinear system (2.47)
3'] = mg + a2(y + 11.7)?) + blii + u + d(t)

The disturbance d(t) is piecewise continuous and bounded. Figures 3.1 shows simula-
tion results when d = 0 (solid), d :2 sin(t) (dotted-dashed), and d = Ssin(t) (dashed)
; again without robustifying control. Figures 3.2 shows results for d = 533n(t) when a
robustifying control is used with 1) = 0 (solid) and 17 = 5.2, )1 = 0.9 (dotted-dashed),
and )1 = 0.3 (dashed). Notice the reduction in the tracking error and the projected
parameter error 61 + 52 as 11 decreases. Finally, Figure 3.3 demonstrates tracking
error convergence in the idea case d(-) = 0 while the robustifying component 1,0, is

used.

3.5 Conclusions

In this chapter we have shown that, for sufficiently small upper bound on the bounded
disturbance, all signals in the closed-loop system are bounded and the mean square
tracking error is of order 0(6 + d1). We have also shown that, for a large bound on

the disturbance, we can design a robustifying control component such that all signals

63

 

0.5 I 1

 

 

 

 

 

 

  

 

 

 

 

0 50 100 150
__(a)__
2 1 1
1.5.. .......... _(
1.... ................................... . . .. . .. . fi
1 \ ‘ '1 ‘ ’i 1‘ " ,A " 1‘ r1. " 1‘1 I‘ ’\ I‘ ’\ 1" “
11'1'\i"\"'i?"’1i"‘ \"i\'11"iil"i"‘ 1‘“
1' 1 z' \1 \ I ‘I -I \ ' \ ’ \I \I
~\1I'.‘u'.1'.\"//- v."."-\\ r- ’10.“ ,/‘1\';\.\"I[1\'\ /- ”1'1 3'1; -\
o 'J \.- .., v. \.’ \., .v~ \_7 l \, \.j .J \ ‘,I -v. \., _/ \ ‘.I
\ ' ‘ .
-0.5 L ‘
0 50 100 150

—-(b)—-

Figure 3.1. The case (1 = 0 (solid), 61 = sin(t) (dotted-dashed), and d = 5sin(t)
(dashed) and no robustifying~cont~rol. The x-axis is time. (a) Tracking error e; (b)
Projected parameters error: 01 + 02

64

 

 

 

 

 

 

 

 

 

0.5 1 1
‘ l . ,1 1
O \ \ \. i x. I \ 1
v v \ \ a \ .b 'v \ I \ \ v 1
_05 .............................................................................................. ..
_1 l 1
O 50 100 150
--(a)_-
2 1 r
1.5.. ...................... .1
1- ..
05};’ .\;:.I\ ...,.i’; (in I/\".,\ $1.12. 5‘. 'I.‘
\ ' I" A1,) [I‘ I I“ \
. '\ _\ ‘1',“\ 1 )’1 \H i 9
_. . ..\.. .. .. [\[u \\I. .\..' .. I\I‘/ ...... 1.. ..1
0 ' J “\I v ‘1“, \~ ‘9' \‘I \I \34“/ I \ 11/1)
-O.5 ‘ '
0 50 100 150
-—(b)-—

Figure 3.2. The case when d = 5sz’n(t) when a robustifying control is used with n = 0
(solid) and 7) = 5.2, p = 0.9 (dotted-dashed), and p = 0.3 (dashed). The :c-axis is
time. (a) Tracking error e; (b) 01 + 02.

65

 

 

 

 

 

 

 

 

 

 

 

 

-1
O 50 100 150
__(a)__
2 r 1
,_ ‘‘‘‘‘ -
_1M:'::ZZIZ':‘Z'Z Z'Z’Z'Z':'ZIZ'Z'Z'SiZ'Z'Z'Z'I'Z'22:12.;
_2 . 1
0 50 100 150
__(b)--

Figure 3.3. The case when d(-) = 0, 17 = 5.2, and_11 =~0.9. The :r-axis is time. (a)
Tracking error e; (b) Projected parameters error: 01 + 92 (solid), 02, 03 (dotted)

66

in the closed-loop system are bounded and the mean square tracking error is of order
0(6 + 11.), where both 6 and )1 are design parameters.

The robustness results of Sections 3.2 and 3.3 have potential application to adap-
tive control of nonlinear systems using neural networks or other nonlinear function

approximators. Consider a system whose input-output model is of the form
y(n) : F(~) + G(~)u("‘)

Using neural networks, the nonlinear functions F () and G () can be approximated,
to any desired tolerance, by neural networks. In the special case of linear-in-the—
weights neural networks, as in radial-basis-function networks, the functions F and G

can be represented by

_—.§_jh-() )V,+61(), )G(-=Zh( )W+62()

for some weights V,- and W,. It follows that the system can be represented in the
form (3.1) with d = 61 + 6211“"). Therefore, the results of this chapter show that
our adaptive controller can be used in this case. Moreover, the robust controller of
Section 3.3 shows that we can trade off a larger approximation error with the use of

the robustifying control component, leading to lower-order networks.

67

CHAPTER 4

Application to Induction Motors

4. 1 Introduction

Nonlinear and adaptive control of induction motors is becoming more realizable re-
cently with the advances in power electronics and fast digital signal processors. Khalil
and Strangas [27] introduced a robust nonlinear control approach to the speed track-
ing problem in induction motors. It differs from the previous approaches in a number
of key points. First, it does not use speed measurement. Motivated by the practical
consideration that position measurement by optical encoders is much more reliable
than the noisy speed measurement by tachometers, it uses position measurement.
Second, it does not require rotor flux measurement. It adopts a novel idea of per-
forming the field orientation change of variables using the flux estimate rather than
the flux itself. This makes all the new variables available for feedback. Third, it
allows uncertainty in the rotor resistance, the stator resistance, and the load torque.
It uses robust control techniques to overcome the effect of this uncertainty on the
tracking accuracy. The use of robust control is based on another change of variables
that brings the acceleration as one of the state variables. This change of variables,
which is dependent or! the uncertain quantities, results in a state equation where

the uncertain terms satisfy the matching condition. The controller is designed using

68

continuous approximations of variable structure control. The uncertain change of
variables is not used in the implementation of the controller, as both the speed and
acceleration are estimated from the position using a robust high-gain observer [11].
It is shown in [27] that the speed tracking error will be asymptotically bounded by
a bound that can be made arbitrarily small by choice of certain design parameters.
In [35], an adaptive observer for induction motors with unknown rotor resistance
was introduced. It is based on rotor speed and stator current measurements. The
adaptation is with respect to the rotor resistance. The design is a Lyapunov based
design. It was shown that the states of the adaptive observer are bounded and if, in
addition, a persistence of excitation condition is satisfied, then all error signals tend
exponentially to zero.

In this Chapter we carry the controller of [27] one step further by adapting the
rotor resistance on line. For the on-line adaptation we use the adaptive observer of
[35]. We prove that the robust controller retains the properties shown in [27] for any
bounded time-varying estimate of the rotor resistance. The boundedness of the rotor
resistance is guaranteed by using parameter projection. The closed—loop analysis
is given in Section 4.5. The experimental results, given in Section 4.7, are in good
agreement with the theory.

It should be noted that [36] has a similar adaptive speed control scheme. It uses
an adaptive observer to estimate the load torque, rotor flux and rotor resistance under
the assumption that the rotor speed and stator current are measured. Asymptotic
convergence of the load torque and rotor resistance errors is shown under a persistence
of excitation condition. The speed control is designed assuming measurement of
rotor flux. However, in the simulation, the rotor flux is replaced by its estimate. The

closed-loop system is not analyzed.

69

4.2 Induction Motor Model

09‘

 

br
1
a,
............. {.3 ‘2...-
b, ar

53

as

Figure 4.1. Three phase winding of induction motor

An induction motor consists of three stator and three rotor winding, as illustrated
in Figure 4.1 and Figure 4.2, where R is resistance, L inductance and the subscripts
s and 1' denote stator and rotor quantities respectively. This three phase represen-
tation can be transformed into two phase equivalent representation [28] using the

transformation matrices

 

 

top—-
NIH

70

ibs

 

 

 

 

 

 

 

 

 

 

Figure 4.2. Three phase equivalent circuit of induction motor

ll
103
l
U
b ,UC
va
2'01::
and

where 0 is the rotor flux angle with respect to

p

 

c080
~—sin0

l
2

c0301

"-Slllgl

l
2

c0562

—sin02

 

l
2

d

the rotor, 61 = 0 + 2375 and 62 =

9— 2f. The transformation matrix K 3 transforms the three phase stator equations into

equivalent two phase equations, and K, transforms the three phase rotor equations

into equivalent two phase equations, in the rotor frame of reference. Hence, the

71

dynamics of the induction motor in the two phase representation is given by

,' (ft/J, _
Rs‘sa + dto — 1130

where z' is the current, (I) is the flux linkage, R, is the stator resistance and R, is
the rotor resistance. The subscripts a and b refer to the two orthogonal axes of the
new two phase representation. Note that the rotor equations are in the rotor frame
of reference. The voltages UT, and 21,, equal zero since the rotor’s winding are short
circuited.

Let Z—f = pw, and 6(0) = 0. where p is the number of pair of poles, and 6 and
w are the angle and speed of the rotor, respectively. The rotor equations can be

transformed into the stator frame of reference using the transformation

cos 6 —sin 6

sin 6 cos 6

where F, and F, are 2 x l vectors representing quantity in the stator and rotor frames
of reference, respectively. Hence the motor equations can be written in the stator

frame of reference as

' dflsa _
R8280 + dt —' va
Raisb + d—th : vb

Rrira + 1‘11;ij +W¢rb : O

Rrirb‘l' (1E1? -mtpra : 0

Under the assumption that the magnetic circuits are linear and the iron loss is zero,

72

we replace the stator flux and rotor current by stator current and rotor flux using

ab...
I/Jsb
112m
#2...

Lsisa + AIlira
Lsisb + Mirb
Alisa + Lrira

Misb + Lrirb

where M is mutual inductance, and L, and L, are stator and rotor total inductances,

respectively. Combining the resulting equations with the mechanical equation, the

induction motor is represented by the fifth order differential equation model [33]

(i = w (4.1)
w = —p,\3'Jz', — TL/m (4.2)
A, = (—%I + pwJ)A, + gm, (4.3)
i. = ((3%! - 5M)» — (0377 + grunt + 7v. (4.4)

where A, = [wwwblfl 2'. = [imz'bFl vs = [um]?

The variables 0 and w denote the angular position and speed of the rotor, A, denotes

the rotor flux in the stator frame of reference, and 2', and v, denote the stator

current and voltage. The constants a,, B, 7, n and p are defined by a, = R, / L,,

[3 = M/OLer, 7 = 1/0L87 77 =1/0’, and [J = pM/er) Where 0’ =1— MZ/Ler

and m is the rotor’s moment of inertia. The resistances R, and R, will be treated as

uncertain parameters with R,, as the nominal value for R, and R, as a time-varying

bounded estimate of R, where we assume that R, E (2,, a compact interval. Let

73

0,, = R,,/L,, 61 = (R, —- R,)/R,, and 62 = (R, - R,,)/R,,. The load torque TL will

be treated as a bounded time-varying disturbance with bounded derivative.

Problem Statement

It is desired to design a feedback controller that solves the speed tracking problem
w(t) —w,,,(t) —-) 0 as t -+ 00 in the presence of the disturbance TL and the uncertain
parameters R, and R,, where the reference speed w,,f(t) and its derivatives d2,,,(t)
and d),,;(t) are bounded functions of t. The controller can only use feedback from 0

and i,.

Flux Observer

We use an open-loop observer [49] to estimate /\,.

3‘, :- (—%—I +pwJ);\, + iRJ—Mi, (4.5)

The estimation error e, = 3‘, — A, satisfies the equation

A

e. = (—%—I +plee. — 61%(Mz'. — it) (4.6)

The origin of é, = (—%f1 + pwJ)e, is exponentially stable. This property ensures
that, as long as 2', and :\, are bounded, the estimation error e,(t) will have an

ultimate bound of order 0(61), i.e., the steady-state error in e,(t) will be 0(61).

Augmented System
We augment the observer equation (4.5) with the motor equations (4.1)—(4.4) to
obtain an eighth-order model with (awn/)0, 21),, i,,ib,z/3,, 212,) as the state variables,

where 5‘, = [2,130, 12),]? We perform a change of variables to bring the equations

74

into coordinates that will be easier to work with. First, we replace 11), and 1,0,,

with the flux estimation errors 6, = 1,5, — 2/2, and ab = 113,, — 11),. Next, we replace

(ll/)0,wbaiaaibaeaaebavalvb) by (wdv fiv idaimedaeq, 0d: '01)) Where

2213 = 1132 + 1233, 13 = tan-‘(zfib/tb.)

A

idziacosfl+ibsinfi, z'q: ~iasinfi+ibcosfi
ed=eacosfi+ebsinfi, eq= —e,sin,6+ebcosfi
fidzvacosfi+vbsinfi, 27,: —v,sin;3+vbcos,6

(4.7)

(4.8)
(4.9)

(4.10)

This change of variables resembles the one used in the traditional field orientation

control, except that the new variables are defined in terms of the flux estimates 1,5,

and 1,51, instead of the actual flux components 11), and 11),. Consequently, the new

variables #2,, [2, id, i,, 13,), and 1“), can be calculated in terms of signals which are

available on line. The variables 21,, fi, id, 1,, ed, and e, satisfy

A A

2‘ Rf “ Rr ’2
wd = “fl/M + L—M’ld
-. it, ._ .
P = p“) + L—MZq/l/Jd
:: RF " Rr '7 ’t
2at = L—fll/Jd — (03077 + E-flMfid + pun,
Rr ’:2 " A
+ szq/wd + m + f1+ 6191+ 6292
5 _ A ’3 Rr 1
2q — -flpwwd — pond - (05077 + L—fiMfiq
EM. . . . 6 6
_ L—, ‘d‘q/wd + 7% + f2 + 193 + 294
' RT RT ’5 " Ry- “ I:
- RT Rr ’: " Rt 1
811 : _Eeq _ Equed/wd — 611:qu

75

(4.11)

(4.12)

(4.13)

(4.14)

(4.15)

(4.16)

where

f1(°) = -5%f6d—Pl3w€q
gl(-) = fi%(«/31—Mid)
92(-) = -Rso“fld
f2(') = —16%feq+pflwed
93(') = _Q%2q
94(') = —Rs07;q

To tackle the speed tracking problem, we introduce the state variables
231:0—fwre1, x2 =w-wref, $3 =u'J-d2re;
to replace 0, w, and 3,, respectively. Noting that
122 = hide, + piqwd — ed) — TL/m

we can see that the change of variables from (0,w,iq) to ($1,232,153) is invertible
provided (113,) — ed) gé O. The conditions (15,) — ed) 76 0 and 113, 75 0 can be ensured by
allowing some delay between the time the motor and the flux observer are switched
on, and the time the load and speed reference are applied. This will allow the flux to
build up so that when the control law becomes effective the system will be far from

singularity. For the flux tracking problem, we introduce the state variables

A A

[art/3d + fi“lid _ Fibre]

Zl=¢d—¢refa Z2=il:—‘I:— L
1' 1'

where the flux reference w,,f(t) and its derivative z/l,,;(t) are bounded function of t.

76

4.3 Controller Design

Let 1: = [I1,$2,JI3]T, z = [21,22]T, e = [8d,6q11‘, X : [1E,Z,6]T, and rewrite the

augmented system as

 

 

 

 

if 2 A11? + BI[F1+ Glilq] (4.17)
Z. 2 A22 + Bg[F2 + Ggi’d] (4.18)
é = A38 + 61%91 (4.19)
where _ q _ 1
O 1 0 0
A1 = 0 O l , Bl = 0
O 0 O l
0 1 0
A2 = a B2 =
0 0 1
Fl = #%(—Eq¢d — eqid + edgq) “ uprdz/h - 643.1 — quq)

“ #1160102}; + ‘33 + e: — 2earth!) + #(Rs’Y + gfflAMM'Jqlfid “ qud + edgq)

+ [1’76qu — TL/m - 123,8,

F2 = (dad—MidMIBMIZEL

r

r150

+ (iii-)2 - g5] _ "vbref + £15?qu + (IfszMzgg/tpd

_ %Rs’7nM’id —' fl(%ed + pigweq)

L.
~ R.
GI = H’YWa — ed), 02 = L—M7
‘ . 213 — ME
A, = —%—I— (iMiq) J, g, = ( d . d)
r mp, —Mz'q

The third-order channel from the 1‘), to 1:1 is feedback linearizable, the term F1 satisfies

A

the matching condition, and the control coefficient 01 = 117%,) — 6,) is positive

77

whenever 213d — ed is positive. Hence, for any bounded F1, it is possible to design
a robust feedback control function of ($,z,iq) that brings ||;r|| arbitrarily close to
zero. This is a typical task in nonlinear robust control theory [26]. Similarly, the
second-order channel from 13,, to zl is feedback linearizable and the term F2 satisfies
the matching condition. So, for any bounded F2, we can design a robust feedback
control function of (w, z, 3,) that brings ||z|| arbitrarily close to zero.

We use sliding mode control. The sliding surfaces for (4.18) and (4.17) are taken

as 31 = 0 and 32 = 0, where
81 = alzl + 74, S2 = a2$1 + (23:62 + $3 (420)

for some positive constants al to (13. Let D be a compact subset of R8, that contains
the origin, with the property that Ilfidl and Ilia - edl are bounded away from zero for

all X E D. We design the control as

i), = —k1 sat (31-) (4.21)
#1

a, = —k, sat (3) (4.22)

where #1 and #2 are small positive constants, and k1 and k2 are chosen such that

> ’63 +|a122 + FQI

4.2
1.- G2 ( 3)

 

k+a$+asr+F
19224122033 1|
1

 

(4.24)

for all X E D, R, E Q, and |R,| 3 kn (some constant), for some arbitrary k3 > O

and k4 > 0. With this control, it can be verified that

QZQIX92XQ3XQ4X95CD

78

is a positively invariant set,1 where 91 = {xIP,:r, g 4c§||P,||3}, (22 = {(32I 3 c2},
Q3 = {Izll S 61/01}, Q4 ={|31| S Cl}: Q5 = {6T6 S 03}, 1‘s = [1171. 12V, and P3 is

the positive definite solution of the Lyapunov equation P,A, + AZP, = —I with

o 1
A,=

-—a2 —a3
The positive constants c, to c3 are chosen such that c) 2 )1), c2 _>_ #2, ([03 > 116117,
and Q C D, where kg and 1177 are nonnegative constants such that “91” _<_ kg for all
X E D and k7 is an upper bound on IR, — R,|/R, for all R, E 9,. Moreover, all
trajectories starting in Q will reach the boundary layers {'31) _<_ [11} and {lszl 3 112}
in finite time and remain inside thereafter. Consequently, X reaches a residual set 00
where the speed tracking error 222 is 0(112). Thus, the error can be made arbitrarily
small by choosing 112 small enough. The flux observer (4.5) and the control law
(4.22) require the signals 11) and d) which are not measured. The last step in the
controller design is to estimate these signals from the measured rotor position 0.
This estimation problem can be addressed using the third-order model (4.17). We

use the high-gain observer [11]

€41 = 42 + 011(331 - 41) (4-25)

642 = 43 + 012($1 — 41) (4-26)

643 = 013(151 — 41) (4-27)

:82 = 52 sat (5:72) , 5:3 = 53 sat (_gg; ) (4.28)
3

L?) = {i2 + w,,f, 82 = 021'1 + 0.3.132 + 1333 (4.29)

 

1The proof is similar to the one shown in Section 4.5 for the output feedback case.
Hence, it is omitted here.

79

where a, to (13 are chosen such that polynomial

s3 + 0132 + 0123 + (13 (4.30)

is Hurwitz and c is a small positive constant. Define £1 = £15711, {2 = 92—192, {3 =

2:3 - 2:3, then the high-gain observer dynamics are transformed into by

66 = (A1 — HCI)€ + €b1i‘3 (4.31)

where the eigenvalues of .4 = (A1 - H Cl) are determined by (4.30). The saturation
level 52 and $3 satisfy 52 2 max '172', .93 2 max lx3l, where the maximization,
performed under state feedback, is taken over the set :1: 6 91 x {22 x 93. The
estimates (1) and 5‘2, as determined by (4.29), replace w and 32, respectively, in (4.5)

and (4.22). The actual controls 1), and 1), are given by

v, = 1“), cos [2 — 1?, sin )5, U, = 13,, sin p" + '1“), cos ,6 (4.32)

The feedback control design is now completely defined by (4.5), (4.21), (4.22), and
(4.25)-(4.32). By invoking singular perturbation analysis similar to [11], it can be
shown that, for sufficiently small 6, the set 0 is positively invariant, and the closed-
loop trajectory (X ,5) reaches a residual set 60 x {“5” g kgc} where 272 is 0(112).
The controller designed in this section is essentially the same as the controller of [27].

The differences are the following:

1. The flux controller is designed for a time varying reference 1/2,,,(t), while in

2

[27] it was designed for a constant reference. This change is added to include

field-weakening.

 

2Due to this change, the definitions of 2:, F2 and G; are different than those of
[27].

80

2. Instead of using a constant nominal value R,,, the controller uses a bounded

time varying estimate of R, which is obtained from an adaptive observer.

3. The controller is simplified in two aspects. First, the control laws contain only
sliding-mode components. There is no feedback linearization as in [27]. Second,
the high—gain observer is a linear one. It does not include driving nonlinear
terms as in [27]. These two changes are adopted to simplify the controller and
reduce on-line computations. In experimental testing of the controller [3], there

was a need to reduce the on-line computations.

4. In [27], the definition of the tracking errors 2:1, :122, and 2:3 included a scaling
factor a, e.g., 2:2 = (1(4) — 1.2,”). It was included to improve the performance of
the nonlinear high-gain observer by reducing the nonlinear driving term. There

is no need for such scaling in the current her since we use a linear observer.

4.4 Adaptive Observer

To motivate the equations of the adaptive observer, we derive it assuming measure-

ment of speed, as in[36]. The observer is taken as

A A

A, = (—%—I +pwJ)A, + %Mi, (4.33)
= (4%! — flpwJ)5\r — (0.77 + £54101, + 722. + u (4.34)

where u is to be chosen later. Note that (4.33) is the same flux observer we used in

(4.5). The estimation errors 6, = A, — A, and f, = 1', — 2', satisfy

('3, = (-——%—I +pwJ)e, — LEO), — Mz',) (4.35)
f, = fi%A, + fiIgr—e, — pfiwJe, — iRIfiMz', + u (4.36)

81

where R = R, — R,. Define z = f, + fie,, then
2' = u (4.37)

Choose

where k,- > 0. Then

2, = — k,+fln1— J2,+Enz— — sz—
[( -.L') Row} ~L,( C) p ( n) (4.38)
+18%Ar_,6%fMis_%—EC

Let

—§[2.T;.+ER3+BI;-’E(z —<) (z-<)+,1n(z -n>T(z-n>l

for some fig, 5,, and fl, > 0. The derivative of V, along (4.37) and (4.38) satisfies

V, = —(k,-+%t)z’7'§,
+J—R,(R +33 ETA —fl Tn—MTC)
+2 rL. ‘8 r L. s
+'fil—<%f(Z-C)T(,BC;3+u-CO)

+ 35%(2 - 77)T(u - 7'1 + finpwfls)

Based on this inequality, we choose,

‘ '1 d f
R _IBrrLrgs 2TAT + IBTIB%_:zTis+ +2: 2: 73' (br
’ d f

C : 18(23 + u _e_ ¢C
7?

= u + finpin, dzezf 1b,,

82

Hence,

'1. 7T?
‘0 S "‘st 23

where k 2 (k,- -+- 1‘35) > 0, from which we conclude that all adaptive observer states

are bounded.

4.5 Closed—loop analysis

In this section we address the stability of the induction motor if the feedback loop
is closed using estimates of the speed and rotor resistance and if R, is replaced by
its nominal value R,, in the adaptive observer. We show the boundedness of all
signals and we also show that the mean-square speed tracking error is of the order
of 0(112), where 112 is design parameter. The adaptive observer will not have V, _<_ 0
in the lack of speed measurements. Hence, to ensure boundedness of all states of the
adaptive observer we project the states 17, C and R,. The projection is done following
Chapter 2. Noting the way the terms (2 — 1)) and (z — C) appear in the Lyapunov
function V,, we can view 17 and C as estimates of 2. Therefore we project 17 and C
so that they belong to the same set that 2 belongs to. The set that contains 2 can
be determined from its definition, i.e., z = f, + fie, Therefore, a bound on 2 can be
obtained from bounds on f, and 6,. The adaptation for R,, n, and C using projection

is given by

A A

R. = Proj(¢.,R.) (4.39)
c’ = Proj(¢(,C) (4.40)
w) = Pr0j(¢mn) (4.41)

83

where the projection operator Proj(-, -) is defined in Chapter 2. We analyze the

closed~loop system by considering the set (2,, defined by
QnIQlX92X93XQ4XQ5X06XQ7=QX§26XQ7

where (21 to 95 are defined in Section 4.3, Q, = (V, 5 c6} and 527 = {V, 3 C762}
for some positive constant c, and c7, to be determined. We will show that Q, is

positively invariant for sufficiently small 6. Consider the set
97 = {V, 3 C762}
where V, : {TI-’5. The derivative of V, satisfies
1", = —§1€—ng + 2.5713312,

Since

233 2 F1 +G1’f)q

where F1 and G, are function of the states that are bounded in Q, and so is 13,, it,

satisfies |:i'3| g c,, for some constant 0,. It follows that

. 1 1
v<_—T _——————.v , v
‘- 26‘ ‘ 26Am,X(P) “LC [‘—

Choosing c7 > [2c,.A,,,,,(I—’)]2 ensures that

V, S 0 on V, = C762

84

Hence {V, 3 C762} is positively invariant for all states inside 9,. Consider next the

set
96 = {Va S Ce}
The derivative of V, satisfies
1", g 4,532, + 1,,
< —2k,V, + 11:45—er + 317%(2 - C)T(z - C) + i(z — n)T(z - 77)] + 117,16 + k,1|R,|

_ 5"

Using the fact that, in 9,, He,“ 3 ya; and ”1',” g «21:6, while ”77” g k, (due to
projection), we obtain

Ilz — nn 3 «‘24. + (Me—3+ k.

Choosing the adaptation gain 5,, 2 fine, yields

 

 

Hz — nll2 < 4..., + 2([3\/c—3 + 1,)? < 4c, + 2(q,/c—3 + 1,)2
fl.) ‘ fl, ‘ fines

Similarly,

Hz - CH2 < 4a.; + 2(fi,/'c§ + kc)2 < 41:6 + 2431/53 + k<)2

54 _ 54 _ BCCG

 

 

where the adaptation gain B, satisfies 5, Z B,c,. Let E, be any positive constant.
Then
V, g —2k,v, + I} + 14,16 + k,1|R,|

where It is a positive constant, independent of c, for all c6 2 56. Choosing c, >

max{65, W}, where k,, is an upper bound on [R,,], ensures that V, < 0 on
V, = 0,. Hence Q, = {V, 3 c6} is positively invariant for all states in (2,. For the

set

05 = {8T8 _<_ C3}

85

we consider the equation

~

é = A36 — “fr—91 “‘ (W — ”[3244 - J]
Inside (2, we have

eTAse + ||6|H|§ff|ll91|| + [C(26)

co
m
|/\

|/\

—‘E:llel|2 + Ilell(ksl%:-I + 1,.)

6”2 — llemksk? + %fkc2€)l

|/\

_&
L.

 

 

0 on etc :03

|/\

where ,/c;, > k5k7 + L,k,2c/R,. Hence (2,, is positively invariant. For the set
04 ={IS1I S 01)
we consider
S1: (1122 + F2 + GQDd

where ”F,” g k, + |R,.| and

O
A A

R1- = Pr0j(¢raRr)

We will show that R, is bounded in Q, by a bound independent of the bounds on
the control inputs. Note that f, is bounded in Q, and C is bounded by projection.

Boundedness of f, and 1/3, follow from boundedness of 21 and 22. Using

a) = hide, + 112,,(1/3, — ed) — TL/m

86

and the fact (if), — 6,) is bounded away from zero in 95, we can conclude that f, is

bounded. Since 1', and i, are bounded, 1', is bounded. It follows from

A, = (—%’—I + pwJ)A, + 119—1111, (4.42)

that A, is bounded. Since A, and e are bounded, A, is bounded. Therefore R, is
bounded for all states in S2,. Hence, F2 is bounded by constant that is dependent on

the 52,, but independent of the control level. Now
SISI S 31(0122 + F2) "" G2k1|81|

Choosing
k F
’41 Z 3 +IGC1;ZQ + 2|
2

 

yields 315', S —k3|sll on the boundary of {|31| 5 cl} provided c1 2 p1 and all states

are in 52,. Therefore {Isll g cl} is positively invariant. For the set

03 = “31' S 01/41}

we consider the equation

21: —0121+ 81

which yields

212'1 S —alzi2 +61|21| _<_ 0 V |21| Z cl/al

Hence Q3 is positively invariant for all states in (2,. For the set

Q2 = “32' S 62)

87

we consider the equation
S2 I: (12132 + (131‘3 + F1 — 01kg Sflt(S2/#2)

Then
8232 S —k201|82| + ISQHGQICQ + £13.13 + F1| + k2|32||82 - S2

< —)s.,|(k,G. — |a2$2 + 03173 + Fll - km)

Therefore Sgsg g —k4|32| for

[C4 + [(121152 '1” 035173 + F1|+ k,3€
[C2 2 G
l

 

Hence {|32| S 62} is positively invariant provided c2 2 p2 and all states are in (2,.

Finally, Consider the set

01 = {1133113311}, S. Cl}

Using V, = affix, as a Lyapunov function candidate for

(I), = A3138 + 8282

we obtain

5.
||

—xf:r, + 2I,P,8232
S ‘llxsll2 + 2llflvsHllPsllC'z

_<_ 0 for H2,” 22l|P3|l02

Hence V, S O on V, 2 cl where ,/c1 2 2c2||P,||%. This concludes the proof that Q, is
positively invariant. The second step is to show that the fast variable C enters the set
9, in finite time. The details of this step are omitted here since they are similar to

the ones shown in Chapter 2. Hence, we have shown that all states are bounded for

88

all t Z 0. Inside (2,, the inequalities 313'] S —k3|s)| and 323'2 g —k4|32| are satisfied
as long as lsll _>_ 111 and lsgl Z [12, respectively. Therefore, the trajectories reach the
boundary layers {Isll S 111} and {lsgl S )112} in finite time. From that time on, V,

satisfies

Vs S —|l$sH2 + kn“?

Since all signals are bounded, we conclude that the speed tracking error is of order

0(112). We summarize our conclusions in the following theorem

Theorem 4.1 Consider the induction motor given by (4.17)—{4.19) with
the observers {4.25)—(4.27) and (4.33)—{4.34) the output feedback control
(4.21), (4.22} and (4.5), and the adaptive laws (4.39)—(4.41). Sup-
Pose (21(0).$3(0).81(0).82(0).€(0).53(0).C(0).77(0).11(0)) 1'3 in Q X 96. and
(21(0),:%2(0),:f:3(0)) is bounded. Then, there exists 6" such that V c 6 (0,?) all state
variables of the closed-loop system are bounded and the mean-square speed tracking

error is of order 0012).

4.6 Experimental Setup

The experimental drive setup consists of a DSP board, an induction motor fed by a
voltage regulated PWM-inverter, a DC load machine fed by an AC-to—DC converter
and some periphery. The drive setup is shown in Figuer 4.3 and explained below.
The experimental test object is a 3-phase squirrel cage induction motor. The data
of this machine are given in Table 4.1. The supply of the induction motor is a
pulse width modulation (PWM) voltage-source inverter with MOSFETS. In order
to obtain the desired voltage source, the inverter is controlled by a microcontroller
(Intel 80C196NU) for PWM. Two 8-bit signals are sent to the Intel to control the

magnitude and phase angle of the voltage used for PWM. The six outputs of the Intel

89

are fed, via isolated drive control board, to the gate signals of the six MOSFETS of
the PWM-inverter.

The load of the Induction machine under test is a DC machine. The supply of the
DC machine is a current controlled thyristor-based rectifier. An 8-bit BEI encoder
is used to measure the rotor position.

The DSP is suitable for real-time calculations with high sampling rate. The DSP
used in the control system of this thesis is the AT&T (DSP32c). It is a 32 bit
floating point unit, a 16—/24—bit fixed-point unit, on-chip memory, and flexible serial
and parallel input/output ports. It has the capability of supporting a wide variety
of applications with computation-intensive, repetitive mathematical operations. The
arithmetic unit allows the device to perform up to 25 million floating point operations
per second (with clock rate of 50 MHz). This performance was sufficient to satisfy
many of the real time algorithms used in this thesis.

In order to perform real-time control in an accurate way, it is necessary to do
the control computations within a small time step. Therefore, programming the
DSPs is done in assembly language because programming in high-level languages
(the language C was an option) would result in non-optimized assembly code that
would result in a program that might be too slow. Code writing in assembly language
was time consuming but not a big problem. A library of low-level programs were
written, representing elementary blocks of the flow chart of Figure 4.4 and Figure 4.5.
Another library contains assembly language programs for analog and digital inputs
and outputs, as well as a C-language program to generate MATLAB data files. Some
standard configuration files were created to perform initialization of the DSP system
software (timers, interrupt vectors, I/O cards, etc).

The nominal value of R, was adjusted on line using the measured stator temper-
ature. The induction motor has a thermo—coupler implanted in the stator winding

which gives an accurate measurement of the stator temperature. Assuming linearity,

90

 

Table 4.1. Induction motor parameters

 

 

 

 

 

 

 

 

Parameter Value
L, 608 mH
L, 595 mH
0 0.22352
R, 0.04!)
R, 0.0259
m 0.05
p 2

 

 

 

 

the stator resistance is adjusted using the formula

T1 + 235

ST = s—
Rh) R0T0+235

where T, is the current temperature in °C and R,,, is the value of the stator resis-
tance at room temperature To. Hence, a more accurate stator resistance is used in the
implementation. The stator current is measured using LEM hall-effect current trans-
ducers. In order to implement the PWM, an Intel 196NU micro-controller board is
used to carry out the computation of the timing and to provide the switching signals

for the inverter.

4.7 Experimental Results

The induction motor used has nominal parameters that are given in Table 4.1

The flux norm is 11), = 0.02 Wb. The control parameters are chosen as a1 = 200,
a2 = 400, a3 = 40, [11 = 0.001, [1.2 = 200, 5'2 = 32, S3 = 320, (11 = 6, a2 =11, a3 = 6,
and e = 0.02. Some of the control parameters were chosen such that the two 100ps,
flux and speed, are stable. Others are determined based on simulation results using

MATLAB. At the time of implementation, some of the parameters were adjusted to

91

get the setup to work. In particular, we needed to decrease the saturation slope 1/H2
for the speed control more than the value that was used in simulation. The larger
value was causing chattering which ,in turn, created more noise. By increasing the
value of 112 the problem was eliminated. Since our controller assumes smooth speed
and flux references, we used two linear filters for smoothing them. The second order

speed filter is of the form

2
can

32 + 2C,w,s + 122,2,

 

X,(s) =

where w, = 10 and C, = 1. Its input is the reference speed and its states are the
smoothed speed and acceleration references. The reference position is calculated
using 6,,f = f w,,,. On the other hand, the flux reference is smoothed using a

first-order filter of the form
(‘00
s + w,

 

X109) =

where w, = 100. In order to implement the continues control using the DSP, we
needed to discretize all filters, observers, and integrations for the adaptive laws. The
forward difference method was used. If the system equation is 2': = f (2:, u) then at

step It the state is computed using

:1:(k) = :1:(k — 1) + hf(2:(k — 1),u(k —— 1))

where h is sampling period. In the implementation a sampling frequency of 5 kHz is
used. The frequency of the switching signals, generated by Intel, is 7.5 kHz.

Two experiments were conducted. Both tests were done when the motor was
under some load all the time. In the first test, the controller was given a flux reference
of 1,0,, I = 0.02 Wb at time 3.5 seconds and a period of 1.5 seconds to build the

flux, then a speed reference of 111,, f = 25 r/s was applied. Figure 4.6(a) shows the

92

speed reference (solid) and the actual speed3 (dashed). Figure 4.6(b) shows the
speed estimation error (11) — 1Z2). Figure 4.7(a) shows the flux reference 1,0,” and
the estimated flux 11),, and Figure 4.7(b) shows the estimate of the rotor resistance
R,. It is noted that during the first 30 seconds R, was going to the lower limit.
Then, when the speed increased, R, started to converge to the right value which is
higher than the nominal one since the motor started to warm up and hence the rotor
resistance increased. Figure 4.8(a) shows the speed reference (solid) and the actual
Speed (dashed) while reversing the Speed direction. Figure 4.8(b) shows the estimate
of the rotor resistance R,. Upon switching the speed from negative to positive, the
estimate of the rotor resistance went to its lower limit and stayed there. However the
speed controller was still able to ensure tracking. Figure 4.9 is similar to Figure 4.8
but before switching speed, the adaptation was turned off for a period of 8.5 seconds
to avoid the disturbances that caused R, not to converge in the previous case. Finally,
Figure 4.10(a) shows the reference and the actual speeds during load changes, shown
in Figure 4.10(b), and Figure 4.10(c) give the rotor resistance estimate R,. Note
that speed tracking was achieved in the presence of the varying load torque. At time
t=60 sec. the speed was not reaching its reference because 12, hit its upper limit and

stayed there.

4.8 Conclusions

In this Chapter, we have demonstrated via experimental results the plausibility of
incorporating the adaptive observer of [36] into the nonlinear robust controller of [27].
Furthermore, we have experimentally demonstrated the speed tracking convergence

and the convergence of the rotor resistance to its actual value . The results are in

 

3 The actual speed is obtained directly from the measured position using a second
order high-gain observer.

93

full agreement with the theory. It should be noted that this adaptive observer is
sensitive to R, and the stator current i,. If there is a measurement error, which is
usually the case at low load, then the convergence of the observer states will not
occur. Hence, no advantage is gained by having adaptation. In this case, using a
nominal value for R, as in [27] is as good as the adaptive but with less computations.
However, if the motor is run at high speed or torque then an accurate estimate of
R, is needed so no saturation of the flux would result. Figure 4.10(b) demonstrate
this fact. Before time t = 30 the estimate of the rotor resistance was not converging
to the right value. However, when the speed increased by 50% , with the same load,

R, started to converge to its actual value.

94

 

 

 

 

 

 

 

 

 

 

/ l
l
l

l

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

PC serial Port
, Magnitude and Angle
DSP 1°)” ”3'08 "‘9‘“ From 05? digital output
1
¢ V
e __
>Patel 196NU ]
<1
— Temperature . .
\ s- 51122
l |
Current Comm \/
Transdusers
DSP digital v0 F
I L‘ Inverter
U
9 Signal , ,
Counter Conditioner 1 I
DC supply
, m DC
... Induction
Motor MOW
Optical encoder Torque
Sensor , AC-DC
3, AC —_s Convener

 

 

 

Figure 4.3. Experimental setup

95

 

 

 

 

I lnitalization and clearing up UO’s I
I Load Control Parameters I

 

# Wait for interrupt I

l

I Measure currents. temperature, and position I

3/ 2 Transformation of the currents I
A l
R.

I Update

 

 

 

 

 

 

 

 

 

Speed estimation

Du-.-------o---a-o-o-~-o-c---

Flax observer

2‘, i,

l

Adaptive Observer

Figure 4.4. Flow chart of the assembly program

 

 

 

 

 

 

 

96

 

 

 

 

 

 

 

 

 

I Flux Control I
I Speed Control I
vd . va
U, '01,
va Mag.
12, I Angle

 

 

 

 

l

Write out the magnitude and angle
to Intel

l

I Store data in the memory

Figure 4.5. Flow chart of the assembly program (cont.)

 

 

 

 

97

40

 

 

 

 

 

 

 

 

 

20

30

_(a)_

40 50 60

 

 

 

 

10

20

30

..(b)-

Figure 4.6. (a)- Reference and actual speed (b)- Speed estimation error

98

 

 

 

 

 

 

 

 

 

 

 

 

 

0.04 1 1 r 1 1
" ‘l’
0.03 - Wha‘ 4
ref
‘3 o 02 - r . . -
3 ° I Flt"? 231,1" "'1‘“ "of" filly-DIRT!“ 71W 19.11"“, ‘1‘.’ .\F‘.’ .\va V1")! \ r‘ 1‘1 ' CI ("JV 4"! \-'-/' V1".
,'
0.01 — 1 -
I
l
o l l l l l
0 10 20 30 40 50
_(a)_
0.04 1 1 1 1 1
0.03 ~ 1
C: 0.02 '- .1
0.01 " _.
0 l l L l l
0 10 20 30 40 50
-(b)-

Figure 4.7. (a)- Flux norms, estimated and reference (b)- Estimated rotor resistance

99

 

 

 

40'- ' -

 

[rad/s]
o

'20’ (”rel ”
-40- .—._..- m u

 

 

 

 

-60- .1

 

 

 

_(a)-

 

0.04 T 1 r r r 1 1 1

I
L

0.035

0.03 ,.. .1

3 0.025

I
l

0.02 - .1

0.015 - J

 

 

 

0.01 L l 1 l l l l l
0

Figure 4.8. (a)- Reference and actual speed (b)- Estimated rotor resistance

100

 

[rad/s]
o

0.04

0.035

0.03

C: 0.025

0.02

0.015

0.01
0

Figure 4.9. (a)- Reference and actual speed (b)- Estimated rotor resistance

 

 

 

 

 

(”ref

 

 

 

10 20 30 40 50 60 70

 

I

I

I

 

 

10 20 30 40 50 60 70

101

80

 

 

 

 

 

 

 

[rad/s]
2° i
l
t
1
l
l
l
1
l
1
l
l
l
I

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

ref
-50 — - — ' 0.) -
L l l 1 l l l L
0 10 20 30 40 50 60 70 80
_(a)-
0.8 1 r I r 1 1 1 n
0.6 " ..
5 0.4 - . _ _
Ol2 . d
0 l l l l l L l l
0 10 20 30 40 50 60 70 80 90
-(b)-
0.04 r 1 r —1 1 f 1 T
0 03 I" "
c \f
0.02 " "
001 1 l l l l l l 1
0 10 20 30 40 50 60 70 80

Figure 4.10. (a)- Reference and actual speed (b)- Load Torque and (c)- Estimated
rotor resistance

102

 

CHAPTER 5

Conclusions and Future work

Conclusions

In this thesis, we have advanced the state of the art of robust adaptive output
feedback control of nonlinear systems. We have shown that the tracking error con-
vergence can be achieved without persistence of excitation, studied robustness of the
controller to bounded disturbance even when the bounded on the disturbance is not
small, and applied our techniques to speed tracking control of induction motors.

In Chapter 2 we have designed an adaptive output feedback controller to solve the
tracking problem for a class of nonlinear systems. We have not required a persistence
of excitation condition like [25]. We have introduced a transformation that projects
the parameter error on a lower-dimensional subspace. The convergence has been
shown by constructing a composite Lyapunov function and taking its derivative along
the trajectories of the closed-100p system. In Chapter 3 we have shown that, in the
presence of small bounded disturbance, all signals under the control of Chapter 2
are bounded and the mean-square tracking error is of the order 0(d1) where d1 is
a small bound on the disturbance. We have combined robust and adaptive control
to force the mean-square tracking error to be of the order 0(6 + u) where e and ,u
are design parameters. In Chapter 4 we have designed, and experimentally tested, a

robust nonlinear controller for speed tracking control of induction motors which uses

103

an adaptive observer to estimate the rotor resistance.

Future work

A number of research problems remain open and can be pursued in future work.
First, requiring exponential stability of the zero dynamics as in Chapter 2 is stronger
than the bounded-input-bounded-state assumption used in [25]. Future research
may attempt to relax this exponential stability assumption. Second, the analysis and
robust control design of Chapter 3, which are presented for bounded disturbance, can
be extended to unmodeled dynamics. Finally we need to investigate the convergence
of the rotor resistance estimate of Chapter 4 and develop rules for turning off the

adaptation when the conditions for convergence are not satisfied.

104

 

BIBLIOGRAPHY

105

 

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