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1M comma-p14

A SEPARATION PRINCIPLE FOR THE
CONTROL OF A CLASS OF NONLINEAR
SYSTEMS

By

AHMAD NAZIR ATASSI

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

A SEPARATION PRINCIPLE FOR THE
CONTROL OF A CLASS OF NONLINEAR
SYSTEMS

By

AHMAD NAZIR ATASSI

It is shown that the performance of a globally bounded partial state feedback
control of a certain class of nonlinear systems can be recovered by a sufficiently
fast high-gain observer. The performance recovery includes recovery of asymptotic
stability, the region of attraction, and trajectories. We deal with stability with respect
to an equilibrium point and stability with respect to a compact, positively invariant
set.

High-gain observers have been used in the design of output feedback controllers
due to their ability to robustly estimate the unmeasured states while asymptoti-
cally attenuating disturbances. The available techniques for the design of high-gain
observers can be classified into three groups: pole-placement algorithms, Riccati
equation-based algorithms, and Lyapunov equation-based algorithms. In this work,
we show that the abovementioned separation results hold for all these observer design

techniques.

To my mother, Hanaa, whose life and death are an inspiration
To my father, Fawaz, who showed me the way
To my uncle, Wa’el, my mathematics professor
To my grandmother, Thurayya, the mother of all
To my wife, Aivy, whose love gives my life a meaning
To Jeanne and Mary who repainted my memory

To myself for never giving up

iii

ACKNOWLEDGMENTS

I would like to thank my advisor, Professor Hassan Khalil, for his invaluable help
and for sharing his vast knowledge with me. I also would like to thank my beloved
wife, Aivy, for being my pillar of support and my shoulder of comfort. Finally, I thank

my brothers, Sammar and Faraj, for their continuous support and encouragement.

iv

TABLE OF CONTENTS

LIST OF FIGURES viii

1 Introduction 1

2 A Separation Principle for the Stabilization of a Class of Nonlinear

Systems 11
2.1 Introduction ................................ 11
2.2 The Class of Systems ........................... 13
2.3 Partial State Feedback Control ..................... 17
2.4 High-Gain Observer ............................ 18
2.5 Performance Recovery .......................... 20
2.5.1 Boundedness ........................... 21
2.5.2 Ultimate Boundedness ...................... 27
2.5.3 Trajectory Convergence ..................... 29
2.5.4 Recovery of asymptotic stability of the origin ......... 31
2.6 Examples ................................. 41
2.6.1 Example 1 ............................. 41
2.6.2 Example 2 - Inverted Pendulum ................. 42
2.6.3 Example 3 - VTOL aircraft ................... 45
2.7 Conclusion ................................. 48
3 A Separation Principle for the Control of a Class of Nonlinear Sys-
tems 60
3.1 Introduction ................................ 60
3.2 Definitions and Converse Lyapunov Results ............... 61
3.3 Problem Formulation ........................... 66
3.4 Performance Recovery - Semiglobal Separation Results ........ 68
3.4.1 Boundedness ........................... 70
3.4.2 Ultimate Boundedness ...................... 72
3.4.3 Trajectory Convergence ..................... 73

3.4.4 Uniform Asymptotic Stability .................. 75
3.5 Regional Separation results ....................... 82
3.6 Conclusion ................................. 85

A Separation Principle for the Control of a Class of Nonlinear Sys-

tems - Examples 86
4.1 Introduction ................................ 86
4.2 Robust Control I - Finite Time Convergence to a Set [10] ....... 88
4.3 Robust Control 11 - F inite-time Convergence to a Set [50, Section 3] . 94
4.4 Tracking [28] ............................... 103
4.5 Servomechanism [26, 37, 38] ....................... 108
4.6 Servomechanism [21] ........................... 118
4.7 Adaptive Control [27, 2, 1] ........................ 125
4.8 Conclusion ................................. 134

Separation Results for the Control of Nonlinear Systems Using Dif-

ferent High-gain Observer Designs 135
5.1 Introduction ................................ 135
5.2 High-gain Observers - A Comparative Study .............. 137
5.2.1 Pole Placement /Time—structure Assignment .......... 138
5.2.2 Riccati Equation-Based Algorithms ............... 138
5.2.3 Lyapunov Equation-Based Algorithm .............. 144
5.3 More Separation Results ......................... 145
5.4 Conclusion ................................. 147
A Smooth Converse Lyapunov Theorem for Robust Stability 149
6.1 Introduction ................................ 149
6.2 Definitions and the Main Results .................... 150
6.2.1 Strong Stability .......................... 150
6.2.2 Uniform Asymptotic Stability .................. 153
6.3 Proof of Theorem 6.1 ........................... 154
6.3.1 Case of a Complete System ................... 155
6.3.2 Case of a Forward Complete System .............. 164
6.3.3 Smoothing of Lyapunov functions ................ 165
6.4 Proofs ................................... 166
6.4.1 Proof of Lemma 6.1 ........................ 166
6.4.2 Proof of Theorem 6.2 ....................... 170

vi

7 Conclusions

A Technical Lemmas

B Cheap Control and H00 Disturbance attenuation

C Stability Results

C.1 Proof of Theorem 3.1 ...........................
C.2 Proof of Theorem 3.3 ...........................

BIBLIOGRAPHY

vii

180

185

190

194
194
196

201

2.1

2.2
2.3

2.4

2.5

2.6

2.7

2.8

2.9

2.10

LIST OF FIGURES

Recovery of region of attraction: 6* = 0 (solid), 6* = 0.007 (dashed),
6* = 0.057 (dash-dotted), and 6* = 0.082 (dotted) ........... 50
Peaking of the velocity: 2:2 (solid), 2‘72 (dashed) ............ 51
Trajectory convergence: state feedback (solid); output feedback with
e = 0.0015 (dashed), e = 0.001 (dash—dotted), and e = 0.0001 (dotted) 52
Effect of nonlinearity in the observer - without uncertainty: state feed-
back (solid), with linear observer (dashed), with nonlinear observer
(dotted) .................................. 53
Effect of nonlinearity in the observer - with uncertainty: state feedback
(solid), with linear observer (dashed), with nonlinear observer (dotted) 54
Trajectory convergence - without uncertainty: state feedback (solid);
output feedback with e = 0.02 (dashed), e = 0.008 (dash-dotted), and
e = 0.002 (dotted) ............................. 55
Trajectory convergence - with uncertainty: state feedback (solid); out-
put feedback with c = 0.02 (dashed), e = 0.008 (dash-dotted), and
e = 0.002 (dotted) ............................. 56
High-gain vs. nominal observer - without uncertainty: state feedback
(solid), nominal observer (dashed); high-gain observer with e = 0.01
(dash-dotted) and c = 0.002 (dotted) .................. 57
High-gain vs. nominal observer - with uncertainty: state feedback
(solid), nominal observer (dashed); high-gain observer with e = 0.01
(dash-dotted) and e = 0.002 (dotted) .................. 58
High-gain vs. nominal observer - importance of saturation: state
feedback (solid), nominal observer (dashed); high-gain observer with
e = 0.01 (dash-dotted) and e = 0.002 (dotted) ............. 59

viii

CHAPTER 1

Introduction

A separation principle is the property of a control system that allows the design of
a dynamic feedback controller in two separate steps. In the first step we design a
state feedback controller that achieves some desired properties such as asymptotic
convergence to an equilibrium point or asymptotic tracking by certain outputs of
some reference signals. This design assumes that all the state variables are available
for measurement. Then in the second step we design a state estimator (observer)
using measurements of some outputs (functions of the state variables). The state is
then replaced by its estimate in the state feedback controller to produce the dynamic
output feedback controller. The separation property facilitates the design of feedback
controllers in case we can only use measurements of outputs.

The literature provides several separation principles formulated for different
classes of systems. Hereafter we give a quick survey of the main ones.

For the case of linear time-invariant control systems

at = A$+Bu

y=Cx

if the system is stabilizable and detectable then a dynamical output feedback control

can be designed such that the closed-loop system has the origin as a globally asymp-
totically stable equilibrium point. The control input takes the form it = —K.i: where
K is designed to stabilize the closed-loop system under state feedback and 3‘: is the

estimate of the state 3: provided by the observer

:i:=A:i:+Bu+H(y—C:i;)

where H is the observer gain. This observer (Luenberger observer) replicates the
dynamics of the system and is driven by the output estimation error. The state es-
timation error x(t) — i(t) approaches zero asymptotically. This separation principle
means that the poles of the system and those of the observer can be assigned sepa-
rately and that the poles of the closed-loop system are the set of the state poles and
the observer poles. Thus, if global stabilization is achieved by some linear state feed-
back u = —K:1:, the corresponding output feedback it = —K:i: will globally recover
this property.

We understand from the above discussion that the statement of a separation
principle is a statement of recovery by an output feedback controller of performance
achieved by a state feedback controller. In the case of linear systems the recovery
includes recovery of asymptotic stability to an equilibrium point and asymptotic
tracking of a reference. But we don’t achieve closeness, in some sense, of trajecto—
ries under output feedback to trajectories under state feedback. This point will be
illustrated later on.

In what follows we always assume that the origin is an equilibrium point of
the system considered and all the vector fields and state functions mentioned are
at least continuous in some appropriate regions of interest. We also abbreviate
asymptotic stability (or asymptotically stable) by AS, local or locally by L and

global or globally by C, so we may have for example LAS or GAS.

Vidyasagar in [53] formulated a separation principle for a general nonlinear MIMO

(multiple-input multiple-output) control system

i7 : f(t,$,’lt)

y : TU, IL.)

This principle states that if f (t, ., .) is uniformly locally Lipschitz and if the system
is stabilizable and weakly detectable, then the closed-loop system composed of the
stabilizing state feedback and a weak detector has the origin as a LAS equilibrium
point. Stabilizability implies the existence of a state feedback law it = h(t, x) that
renders the origin of the system LAS. On the other hand, weak detectability implies
the existence of a dynamical system (weak detector) driven by the input u and the
output y and whose state is not very far from the state of the original system for
initial conditions close to the origin.

This result is local, i.e. valid only in a neighborhood of the origin, and not
constructive since it does not suggest a construction of the detector. The author
actually stated a global version of his separation principle for the case of exponential
stabilizability. This global version required, in addition to the above conditions, that
the state estimation error decays exponentially and that the vector fields involved be
globally Lipschitz.

It is noteworthy that the formulation of a separation principle requires the system
to have some stabilizability and detectability properties. In the examples hereafter
we highlight this idea. It is useful for the rest to define a feedback linearizable system.

It is well known [20] that if the nonlinear system

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

y = h(X)

has a vector relative degree (7‘1,...,rp), then it can be transformed into the form

t = At+BIf1(t,z)+gl(t.z)u1
2'; : f2(€,Z,’U.)

y=C€

where g1(., .) is invertible in the domain of interest and y is the only measured
output. If the sum of the relative degrees equals the dimension of the system it is
called fully linearizable, otherwise it is called input-output linearizable. The first
part of the system is a chain of integrators driven by a nonlinear function of the

states and the inputs. The second part is called the zero dynamics.

Khalil and Esfandiari in [11] used a separation approach for a fully linearizable
MIMO system with a high-gain observer that estimates the output and its derivatives.

This observer is a chain of integrators of the form 1

. a .
3/1 = y2+—€l(y1-y1)

. a ..
y2 = y3+-€-22(y1-y1)

yn = gal-(311-3}1)+f0(€,2)+90(€,2)u

driven by the output estimation error and by nonlinearities that reflect our partial
knowledge of the system at hand. The observer gain can be regulated through the

parameter 6 in such a way that the observer is fast enough that the state under

 

1For convenience, we give the observer equations for 8180 systems.

4

output feedback stays close to the state under state feedback and thus the stabilizing
property of the controller is not lost. This observer introduces peaking into the state
variables, see [11], which requires the disabling (saturation) of the controller outside
some region of interest. The disabling period is made short by an appropriate choice
of the observer gain. Notice that the stabilizability and detectability properties are

immediately satisfied due to the particular structure of the system studied.

Tornambe in [52] proposed a local separation principle for a class of input-output
linearizable non-minimum phase (unstable zeros dynamics) SISO nonlinear control
systems. The system should be observable in the region of interest meaning that the
state can be written as a function of the input, the output, and their derivatives.
Considering the derivatives of the input and the output as state variables transforms
the system into a double chain of integrators that can be linearized by state feed-
back. The fact that some of the states of the transformed system form a chain of
integrators composed of an output and its derivatives allows the use of a high-gain
observer similar to the one mentioned above to estimate these states. Two differences
exist between the two observers, one is that Tornambe’s observer did not use any
knowledge of the nonlinearities of the system, and the other is that it did not deal
with the peaking issue which makes the region of attraction of the system shrinks as
the observer gain grows.

Teel and Praly in [49] pr0posed an interesting and constructive global separation
principle for a wide class of autonomous SISO (single-input single-output) nonlinear
control systems. It is a combination and extension of the ideas used in Esfandiari
and Khalil [11] and Tornambe [52]. They stated that a globally stabilizable and
uniformly completely observable (UCO) system can be semi-globally stabilizable by
dynamic output feedback. The UCO property implies that the state of the system

can be written as a function of the input, the output, and a finite (not necessarily

equal) number of their derivatives. This function is called the observability map.
The semi-global stabilizability means that given any compact set in the state space,
then a dynamic output feedback controller can be designed such that this compact
set is contained in the region of attraction of the closed loop-system. The observer
used is the high-gain observer proposed by Esfandiari and Khalil in [11]. One
major limitation of this separation principle is that the exact knowledge of the

observability map implies a lack of robustness of the controller.

The last three results dealt with systems that can be transformed into a linear
system using a function of the input, the output and their derivatives and an appro-
priate state feedback. For the sake of completeness we mention two more separation
results that deal with systems that are affine in the inputs and whose unforced dy-
namics (for zero inputs) are Lyapunov stable (i.e. dissipative). The first [14] is a

global separation principle for a class of 8180 bilinear systems of the form

it = Ax+u(Ba:+b)

y=Cx

that are stabilizable and whose unforced dynamics are observable (the unforced
dynamics are linear). Structural conditions were provided to check the stabilizability
and observability properties and two globally stabilizing feedback controllers were
proposed. The two observers replicate the dynamics of the system and are driven
by the output estimation error; one requires the input to be small and the other

requires the input to be persistent.

The second [33] is a global separation result for a class of MIMO systems of the

form

m

i: = ALE—l- Z ”(313)212-
i:

y = C3:

that are stabilizable by bounded inputs ( a control input was suggested) and whose
unforced dynamics are observable (linear dynamics). Also, structural conditions
were given to check the stabilizability of the system. The observer replicates the
dynamics of the systems and is driven by the output estimation error. This [result

was shown to generalize the one for the bilinear system.

The above separation results were mainly concerned with AS to an equilibrium
point. The following separation results are for cases of adaptive control and ser-
vomechanism where not all of the states approach the origin and the system depends
on time-varying external signals (reference signals, disturbances, etc.).

In [27, 2, 1] Khalil and Aloliwi considered a 8180 minimum phase nonlinear

system which can be globally represented by an I/ O model of the form

P P
y(")=fo(-)+ z f.(.)6.+(go(.>+ z g.(.)0.-)u(m>+d(t,a,.)
2': 1 2': 1

where the different nonlinearities depend on the input, the output, and their
derivatives, 0 is an unknown vector of parameters that belong to a compact set,
and d(.) is a locally Lipschitz disturbance. Adaptive state feedback controllers
were designed so that the output can asymptotically track a bounded reference
signal with bounded derivatives. Later, a linear high-gain observer was used to
semi-globally recover what was achieved under state feedback. In some of the

scenarios studied the parameter estimation error 0 = 0 — 0 and some of the states

were only proven to be bounded. In other scenarios the tracking error as well as
the parameter estimation error were only ultimately bounded or were small in the

mean-square sense.

In [26] Khalil used the internal model principle to regionally (in a region of
interest) solve the servomechanism problem for a class of SISO uncertain systems
that can be transformed into a normal form with no zero dynamics. The proposed
dynamic state feedback controller achieved asymptotic convergence of the tracking
error but the overall state approached an invariant zero-error manifold. A similar
result was developed in [38] for a class of SISO systems represented by an I/O

equation which depends nonlinearly on a vector of time varying disturbances.

In [21] Isidori has shown that his previously proposed solution for the general
structurally stable regulation problem, see [22], can be coupled with the idea of high-
gain observer suggested by Khalil in [11] to solve a problem of robust semiglobal
output regulation in the presence of parameter uncertainties ranging over compact

sets. The class of systems considered is of the form

é = Z(M)Z+P0($1,w,#)
9': = Fx+Gu+P(z,z,w,u)

e = H27 - q(w,u)

where (F, G, H) is a chain of integrators, P(.) is a lower triangular vector of
nonlinearities, Z ()1) is Hurwitz in a compact set, and w is generated by a linear

neutrally stable exosystem.

It is noteworthy that other separation results can be formulated for a specific

application as is the case for polymerization reactors discussed in [54].

The purpose of this survey of literature is to show the classes of systems covered
by different separation results and the observers used to achieve the goal of successful
output feedback. Having done that we can exactly situate our version among the
others and point out its merits compared to them. In a concise way we can say the

following about our work:

(1) class of systems: it includes I / O linearizable systems, fully linearizable systems,
and observable systems (that can be represented by a higher order ODE in the

input and the output);
(2) observer: the robust high-gain observer of [11];

(3) state feedback: any globally bounded state feedback that stabilizes the system
with respect to an equilibrium point or a compact positively invariant set. No

Lyapunov function associated with the state feedback is needed; .

(4) recovery properties: it recovers the AS property, trajectories, as well as the

region of attraction (as opposed to local or global);

(5) it unifies and generalizes the cases of [49, 52, 11, 2, 1, 27, 26, 37, 38] which
encompass a wide class of systems and a wide class of control techniques and

objectives.

The goal of this work is to formulate, in a generic way, a separation principle
for a wide class of nonlinear systems. This principle is based on the idea of fast
estimation of the outputs and their derivatives cristalized in the high-gain observer
concept. The resulting output feedback controller will be shown to recover a wide
range of performance measures achieved by the state feedback controller. It is meant

by ”a generic way” that the statement of the separation principle does not depend

9

on a specific state feedback nor on the knowledge of a Lyapunov function associated
with this state feedback.

This work can be divided into three major sections. The first formulates a sepa-
ration principle for the case where the state feedback controller achieves asymptotic
stability to an equilibrium point and is given in Chapter 2. The second formulates a
separation principle for the case of asymptotic stability to a compact, positively in-
variant set and is given in Chapters 3 and 4. The third shows that different observer
design techniques yield separation results similar to those of Chapters 2 and 3, and
is given in Chapter 5. Chapter 6 gives converse Lyapunov results for stability with

respect to sets.

10

CHAPTER 2

A Separation Principle for the
Stabilization of a Class of

Nonlinear Systems

2.1 Introduction

A few years ago, Esfandiari and Khalil introduced in [11] a new technique in the
design of robust output feedback control for input-output linearizable systems [11].

The basic ingredients of this technique are
(1) A high-gain observer that robustly estimates the derivatives of the output;

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

used instead of the true states.

This technique has been the impetus for several results we have obtained over the

past few years. It was used in [11] and [30] to achieve stabilization and semiglobal

11

stabilization of fully-linearizable systems, in [26] to design robust servomechanisms
for fully linearizable systems, in [37] and [38] to extend the results of [26] to systems
having nontrivial zero dynamics. It was used also in adaptive control [27], variable
structure control [41], and speed control of induction motors [31].

As the results of [11] became known, other researchers adopted its technique in
their work. Teel and Praly [49, 50] and Lin and Saberi [35] used it in a few papers
to achieve semiglobal stabilization. Jankovic [23] used it in an adaptive control
problem. Isidori [21] used it to unify his pioneering work on servomechanisms [22]
with Khalil’s work [26]. Jiang, Hill, and Guo [24] used a reduced-order high-gain
observer to achieve semiglobal stabilization for a nonlinear benchmark example.

In most of these papers the controller is designed in two steps. First, a globally
bounded state feedback control is designed to meet the design objective. Second, a
high-gain observer, designed to be fast enough, recovers the performance achieved
under state feedback. This recovery is shown using asymptotic analysis of a singu-
larly perturbed closed-loop system. Our goal in the current chapter is to develop
this recovery property in a generic form that can be applied to any globally bounded
stabilizing state feedback control. In particular, we want a separation theorem that
is independent of the state feedback design and is derived under the least restric-
tive assumptions. To increase the utility of such a theorem, we want to allow for
model uncertainty. Finally, we want to demonstrate that the performance recovery
achieved with the high-gain observer is more than just asymptotic stability recovery.
It includes recovery of the region of attraction and trajectories achieved under state
feedback. These features of the theorem distinguish our work from the interesting
separation theorem proved by Teel and Praly [49], where it is shown that global
stabilizability by state feedback and observability imply semiglobal stabilizability
by output feedback. The result of [49] assumes perfect knowledge of the model and

shows only recovery of asymptotic stability and the semiglobal stabilization property.

12

The rest of the chapter is organized as follows. Section 2.2 introduces the class
of systems with which the chapter is concerned. Section 2.3 states the requirements
on the state feedback control. Section 2.4 introduces the high-gain observer used to
estimate the states. Section 2.5 discusses and proves the recovery of performance by

output feedback. Section 2.6 illustrates the previous results through simulations.

2.2 The Class of Systems

We consider a multivariable nonlinear system represented by

i: = Ax+Bq§(:r,z,u) (2.1)
2 = Maw) (2.2)
y = Ca: (2.3)
C = (102,2) (2.4)

where u E U _C_ Rm is the control input, y E y Q RP and C E R3 are measured
outputs, and a: E 26' Q RT and z E Z C; R6 constitute the state vector. The r x 1'

matrix A, the r x p matrix B, and the p x r matrix C, given by

P0 1 0.
0 0 1 0
A = block diag[A1,...,Ap], Az- =
0 0 1
b0 O-TiX’I‘z'

 

 

13

 

 

0
0
B=block diag[Bl,...,Bp], Bi:
0
L1.rz-x1
C=blockdiag[Cl,...,Cp], Ci: 1 0 0
IXTi

where 1 S i S p and r = r1 + + rp represent p chains of integrators. This system

satisfies the following assumption:

Assumption 2.1 The functions (f) : X x Z x L! ——> RP and w : X x Z x U -—) Re
are locally Lipschitz in their arguments over the domain of interest. In addition,

¢(0, 0,0) = 0, ¢(0,0, 0) = 0, and q(0, 0) = 0.

Assumption 2.1 guarantees that the origin is an equilibrium point of the Open-loop

system.

The main source of the system (2.1)—(2.4) is the normal form of a nonlinear
system having a vector relative degree (r1, ..., rp). It is well known [20] that if the

nonlinear system

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

t = A£+B[f1(t,z)+gl(t,z)u]

2° = f2(€,z,U)

14

31:06

In this case gl(., .) is nonsingular in the domain of interest. y is the only measured

output, and equation (2.4) is dropped.

Another source, where equation (2.4) is relevant, arises when the dynamics are
extended by augmenting a series of integrators at the input side [27, 49, 52]. Ref-
erence [27] considers a single-input single-output system modeled by the nth—order

differential equation

ym) = f0(-) + 90(-)#(n — p)

where a is the input, 3; is the output, f0 and 90 are functions of y, 31(1), ...,

y(n _ 1), a, ..., ”(n - p - 1). Augmenting (n — p) integrators at the input side,
denoting their states by z,- 2 pa — 1), setting a = #(n - p) as the control input of
the augmented system, and taking at,- = y“ ‘— 1) , results in a system of the form (2.1)—
(2.4) with r = n and Z = n - p. In this case all the components of z are measured;
hence q(:c, z) = z in (2.4). Another example of the use of extended dynamics can be
found in [49]. Reference [49] considers a single-input single-output nonlinear system
where complete uniform observability guarantees that the state x can be expressed
as x = h(y, ..., y(ny), p, ..., ”(nul) where p is the input, y is the output, and h(.) is
a known function. Furthermore, y(ny + 1) = d(x, u, ..., u(mul) where a is a known
function. The dynamics are extended by adding lu = max {nu, mu} integrators at

the input side. Taking x,- = y“), for 1 g i _<_ ny, z,- = #0:), for 1 S i g In, and

u 2 pa“ + 1), the system can be represented as

i: = Ax+Ba(h(:z:,z),z)

2 = Aoz-l- Bou

15

ysz

where (A, B, C) and (A0, BO) represent chains of ny and In integrators, respectively.
In this case (15 is independent of u and all the components of z are measured; hence

q(:r, z) = z in (2.4).

The model (2.1)—(2.4) may also arise in models of mechanical and electrome-
chanical systems where displacement variables are measured while their derivatives
(velocities, accelerations, etc.) are not measured. Examples of such models can be
found in [31, 15, 36, 19, 51, 24]. A model of induction motor [31] can be represented
in the form (2.1)—(2.4) with :1: = [(5, 5, (HT, where 6 = 0 — gref is the rotor position
error, and z constitutes the rotor flux and stator current. The measured variables y
and C are the rotor position error and stator current, respectively. Examples of mod-
els that can be put in the normal form are the models given in [15] and [36] for the.
inverted pendulum-on-a—cart system. These models, taking the cart displacement as
the measured output, have a relative degree two but are non—minimum phase. In [19]
and [51], the models given of the ball and beam system fit in the form of (2.1)—(2.4).
These systems can not be represented in the normal form because, taking the ball’s
position as one of the measured outputs, the relative degree is not well defined. A
last example of systems fitting the model (2.1)-(2.4) is the model of the benchmark
rotational/translational actuator given in [24] where the system has a well defined
relative degree with respect to the cart’s position but only locally. The design of the
globally stabilizing state feedback controller of [24] does not transform the system

into the normal form.

16

2.3 Partial State Feedback Control

Our goal is to design a feedback control to stabilize the origin of the closed-loop
system using only the measured outputs y and C. We follow a two-step approach
to this design problem. We first design a partial state feedback control that uses
measurements of :1: and C. Then we use a high-gain observer to estimate :1: from y.
We allow the state feedback control to be dynamic, which is the case, for example, in
the adaptive control of [27], the speed control of induction motors of [31] which uses
a flux observer, and the stabilizing control of [11] which includes a zero-dynamics

observer. The state feedback control is assumed to be in the form

19 = WM) (2.5)

u = 7(19,-’B,C) (2.6)

A non-dynamic state feedback control it = 7(ar, C) will be viewed as a special case of
(2.5)—(2.6) by dropping equation (2.5).

We allow any state feedback design that holds the following three properties:
Assumption 2.2

(1) F and 'y are locally Lipschitz functions in their arguments over the domain of

interest, I‘(0, 0, 0) = 0 and 7(0, 0, 0) = 0;
(2) F and y are globally bounded functions of x;

(3) The origin (:1: = 0, z = 0, 19 = 0) is an asymptotically stable equilibrium point

of the closed—loop system.

The state feedback control may have additional properties like conformity to certain
performance measures on the trajectories, and / or robustness to a certain class of un-

certainties. The global boundedness requirement is typically achieved by saturation

17

of P(.) and y(.), or saturation of their x-input, outside a compact region of interest.

The boundedness of y(.) may also follow from a design under control constraints.

2.4 High-Gain Observer

To implement the control (2.5)—(2.6) we use

it = 7(19, 5:, C) (2-8)
where the state estimate 5: is generated by the high-gain observer
2*: = Ar + 33003, 4, u) + H(y — 05:) (2.9)
The observer gain H is chosen as
0122/62

H = block diag[H1, . . . , Hp], Hz- : E (2.10)

. ._1
afi-l/frz

 

 

l

i r:
0.52
r,/ rz-xl

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

such that the roots of

r- ir-—1 i 1 i _
sl+alsl +---+a7.z,_ls +01”—

18

are in the Open left-half plane, for all i = 1,. . . ,p. The function do(;r,C,u) is a
nominal model of d(rr, z, u). If d is a known function of :r, C, and u, we can take
do = d. On the other hand, if such nominal model is not available, we can take
do = 0, which results in a linear observer. The function do is required to satisfy the

following assumption:

Assumption 2.3 do(:1:,C,*y(t9,;r,C)) is locally Lipschitz in its arguments over the

domain of interest and globally bounded in :r 1. Moreover, do(0, 0, 0) = 0.

The high-gain observer suggested above is a full-order one. It is possible to design

a reduced-order high-gain observer of order (r — p) [45]. We start by partitioning

y
the state vector :1: as x = . , where y = [y1, ..., yp]T, and rewriting the system

(2.1)—(2.2) as

y = Algv

v = A22v+822d(y,v,z,u)

where, A22, 822, A12 have the same structure as A, B, C, respectively, but with all

the ri’s replaced by (r,- — 1)’s. Then, the reduced-order observer will be

u) = (1422 - LA12)('w + Ly) + B22¢0(y, 17, C, u)

v = w+Ly

 

1 The need for this global boundedness property will be made clear in footnote
3 in Section 2.5.1. Moreover, global boundedness of do can always be achieved by
saturation outside a compact set of interest in the subspace of x.

19

where

fli/e
53/62
L = block diag[L1, . . . , Lp], L,- = '

5i, — 2/6” _ 2

 

 

~(Ti—1)X1

The positive constants B;- are chosen such that the roots of
sri-l+fiisri_2+-~+B,i.z,_231+fliz._1=0

are in the open left-half plane, for all i = 1, . . . , p.
Remark 2.1

(1 } In the case where one or more of the p channels that compose the system (2.1)—
(2.3) have relative degrees equal to one, then, there is no need to estimate their
states; they can be used as they are in the output feedback controller. This will

not modify the forthcoming analysis.

(2) We can use the measured y in the output feedback controller instead of its
estimate g = C5: even when we construct a full-order high-gain observer. This,

also, will not change the forthcoming analysis.

2.5 Performance Recovery

In this section we show that the output feedback controller (2.7)—(2.9) recovers the
performance of the state feedback controller (2.5)—(2.6) for sufficiently small 6. The

performance recovery manifests itself in three points. First, the origin (a: = 0, z =

20

0, 29 = 0, it = 0) of the closed-loop system under output feedback is asymptotically
stable. Second, The output feedback controller recovers the region of attraction of
the state feedback controller in the sense that if ’R is the region of attraction under
state feedback, then for any compact set S in the interior of 72 and any compact
set Q Q RT, the set S x Q is included in the region of attraction under output
feedback control. Third, the trajectory of (I, z,19) under output feedback approaches
the trajectory under state feedback as e —-> 0.

For the clarity of the proof we will follow the way used in [49] which establishes
asymptotic stability of the origin in three steps. The first step is to show boundedness
of trajectories, second, ultimate boundedness of these trajectories, and third, local
asymptotic stability. This allows us to deal with asymptotic stability as a local
property that will require some additional assumptions on the nonlinearities of the
closed-loop system, stated as a condition on the modeling error, which is not the
case for boundedness and ultimate boundedness of trajectories. Thus, the proof will
be divided into four sections. First we prove recovery of boundedness of trajectories;
second, we prove recovery of ultimate boundedness of trajectories; third, we prove
trajectory convergence; and fourth, we prove recovery of asymptotic stability of the
origin. In the latter case, we distinguish between asymptotic and exponential stability

in order to impose less conservative restrictions on the modeling error.

2.5.1 Boundedness

Let us first, for the purpose of analysis, replace the observer dynamics by the equiv-

alent dynamics of the scaled estimation error

_Iv-%j
nzj—W

21

for 1 S i S p and 1 g j 3 72" Hence, we have :i: = :1: -— D(e)n where

77 : [77111W1771r11177p1117lprplT
D(e) = block diag[Dl,...,Dp]

D,- = diag[eri — 1,...,1]rz. x r,

The closed-loop system can be represented by

j: = A2: + Bd(.r, 2, 7(19, 2: — man, 0) (2.11)
2 = to, 2,709,513 - Den, 4)) (2.12)
.9 = no, a: - 0(6),), 4) (2.13)

a; = A077 + eBg(:z:, 2,19, D(e)n) (2.14)

where

9($,z,19,D(€)77) = d(x,z,'r(19,i,C))-¢o(i,C,7(19,i,C))

and TAO = D'1(e)(A — HC)D(e) is an r x r Hurwitz matrix. The initial states are
(a:(0),z(0),19(0)) = (:ro,zo,t90) E S, and 12(0) 2 230 E Q, where S is any compact
set in the interior of ’R and Q is any compact subset of RT; thus, we have 77(0) 2
D—1(e)(.vo — 530) = no.

Remark 2.2 In the case of a reduced-order high-gain observer, the scaled error will

be
”a —2,-,-
Ti - 1 —j

772:7“:6

for 1 g i g p and 1 Sj S (r,- — 1). We also get the same structure of (211)—(214)
but with 7'; replaced by (Ti — 1), :1: replaced by (y, v), and it replaced by (y, a) where

A

v = v — D(e)n. The results of this paper will be the same for this reduced-order

22

observer.

The system (2.11)—(2.14) is a standard singularly perturbed one, and 77 = 0 is the
unique solution of (2.14) when 6 = 0. The reduced system, obtained by substituting
77 = 0 in (2.11)~(2.14), is nothing but the closed—loop system under state feedback.

For simplicity we write the system (2.11)-~(2.13) as

x=sa¢mn> 05)

where x = [$T, zT,19T]T and x(0) = [51%, 231,193]? Then, the reduced system is
given by

X=hflfl> mm)
The boundary-layer system, obtained by applying to (2.14) the change of time vari-
able 7' = 5 then setting 6 = 0, is given by

1’2_

(17' — A07) (2.17)

To fix the notation, let (x(t,e),n(t,e)) denote the trajectory of the system (2.11)—
(2.14) starting from (x(0),n(0)). The recovery of boundedness of trajectories is

summarized in the following theorem:

Theorem 2.1 Let Assumptions 2.1-2.3 hold; then, there exists e’i‘ > 0 such that for
every 0 < e 5 31‘, the trajectories (x, 7)) of the system (211)—(214) starting in S x Q

are bounded for allt Z 0.

Proof: The recovery of boundedness can be shown in two steps. First, we show the
positive invariance of an appropriately chosen set A. This set is arbitrarily small in
the direction of the error variable 1). Second, we show that, any closed-loop trajectory,

starting in the compact set 8 x Q, enters the positively invariant set A in finite time.

23

We know that the origin of (2.16) is asymptotically stable with a region of attrac-
tion ’R. Then, the converse Lyapunov theorem of Kurzweil [32, Theorem 7] 2 assures
the existence of a C1 Lyapunov function V(x) and three positive definite functions

U1(x), U2(x), and Ug(x), all defined and continuous on R, such that:

U1(X) S V(X) S U2(X) (2.18)
lim U1(x) = 00 (2.19)
X—)
%fr(x,0) s —U3(x) (2.20)

for all X E ’R. The properness of V(x) in R guarantees that with any finite c >
maxX e S V(x), the set D = {x E ’R : V(x) S c} is a compact subset of ’R. and Sis
in the interior of (2.

For the boundary-layer system we define the Lyapunov function W(17) = nTPon,
where P0 is the positive definite solution of the Lyapunov equation POAO + AgPO =

—I. This function satisfies

Amin(P0)||n||2 .<. W07) .<. Amax(PO)||n||2 (2.21)
aw 2
— < — 2.22
87} A077 _ H77” ( )

Let A = Q x {W(n) 3 p62}. Due to Assumptions 2.1, 2.2, and 2.3 (i.e., continuity
of the nonlinearities and global boundedness of P(.) and y(.) in x) we have, for all

x E 9 (continuous functions are bounded on compact sets) and all n 6 RT (global

 

2This theorem is built around a stability notion called Strong Stability in an open
set. The proof of Theorem 12 of [32] shows that asymptotic stability implies strong
stability in the region of attraction which is an open invariant set. Thus, we can
apply Theorem 7.

24

boundedness of the controller),

llfr(x, D(€)n)ll S k1 (2-23)

“900 D(6)77)|| S k2 (2.24)

where k1 and k2 are positive constants independent of 6. Moreover, for any 0 < E < 1,
there is L1, independent of e, such that, for all (x,17) E A and every 0 < 6 g '6', we

have

llfr(x, D(€)n) - fr(x,0)|| S L1|ln|| (2-25)

In the rest of the paper we always consider 6 _<_ E. We start by showing that there
exist positive constants p and £1 (dependent on p) such that the compact set A is

positively invariant for every 0 < 6 g 61. This can be done by verifying that

- 6V
V S EEC-Mao) + 5’93 (226)

for all (x, n) 6 W00 = C} X {W(n) S peg}, and

- 1
w s -;||n||2 + 2|ln|l||P0||llB|lk2 (2.27)

 

for all (x, n) E Q X {W(n) = p62}, where lC3 = L1L2\/P/’\min(P0): ||P0|l =
Amag;(P0), and L2 is an upper bound for ”-68%“ over 9. Taking p = 16k§||P0||3 and
61 = fl/k3, where 3 = minX E 30 U3(X), it can be shown that, for every 0 < e S 61,
we have

V g 0 (2.28)

for an (m) 6 {Von = c} x W07) 5 p9}, and

W s 0 (2.29)

25

for all (x,17) E {V(X) g c} x {W(7]) 2: p62}. From (2.28) and (2.29) we conclude
that the set A is positively invariant.

Now, we consider the initial state (x(0),:i:(0)) E S x Q. It can be verified that
the corresponding initial error 77(0) satisfies ”77(0)” 3 k/€(7'ma2: _ 1) for some non-
negative constant k dependent on S and Q, where 7'me = max {7‘1, ..., rp}. Since

the vector field fr(., .) is continuous, we can write
t
we) — x<0) = [O fr(x(r,6),D(€)n(T,6))dT (2.30)
Then, using (2.23) and the fact that x(0) is in the interior of D, we have
||X(t, 6) - X(0)|| S kit (2-31)

as long as x(t,e) E 9. Thus, there exists a finite time T0, independent of e, such

that x(t, 6) E Q for all t E [0, T0]. During this time interval we have 3
- 1
W S — 2—Cllnll2, for W07) 2 p62

Therefore,

 

W(n(t,e>) s gm: _ 1) exp (-01t/e) (2.32)

where 01 = 1/2||P0|| and 02 = k2||P0||. Choose 62 > 0 small enough that

def 6 02
_ _1 __ < _ ,
T(€) 1 n( 21” am) To (2 33)

for all 0 < e g 52. We note that 62 exists since the left-hand side of the preceding

inequality tends to zero as 6 tends to zero. It follows that W(n(T(e), 6)) S p62, for

 

3 Here we use an inequality similar to (2.27), obtained using (2.24), which is valid
for (x,17) E Q x RT. Inequality (2.24) requires global boundedness of 450 in 2:.

26

every 0 < 6 S 62. Taking 6’1‘ 2 min (6,61, 62) guarantees that, for every 0 < 6 S 61‘,
the trajectory (x(t, 6), 77(t, 6)) enters A during the interval [0, T(6)] and remains there
for all t Z T (6) Thus, the trajectory is bounded for all t Z T(6). On the other
hand, for t E [0, T(6)], the trajectory is bounded by virtue of inequalities (2.31) and

(2.32).<1

Remark 2.3 The constant 67; depends on the sets 8 and Q.

2.5.2 Ultimate Boundedness

Next, we show that trajectories of the system (2.11)—(2.14), starting in S x Q, come
arbitrarily close to the origin as time progresses. This is summarized in the following

Theorem:
Theorem 2.2 Under the conditions of Theorem 2.1, given any 6 > 0, there exist
65 = 65(6) > 0 and T1 = T1(§) such that, for every 0 < 6 S 65, we have

llx(t,6)ll + ||n(t,€)|| S E, Vt 2 T1 (234)

Proof. From the proof of Theorem 2.1 we know that, for every 0 < 6 S 6?, the
trajectory of the closed—loop system, starting from (x(0),:i:(0)) E S x Q, is inside

the set A for all t Z T(6), where A is 0(6) in the direction of the variable 17. Take

 

63 = min{6’1",§\/x\mm(P0/p)}. Then, 63 2 63(5) S (Y and for every 0 < 6 S 63 we

have

l|n(t, e)” s 5/2, w 2 mg) d=f Tc) (2.35)

In what follows we continue working with the Lyapunov function defined in the proof

of Theorem 2.1. It can be shown that, for all (x, 77) E A, we have

V g -U3(x) + k36 (2.36)

27

 

 

 

where k3- — L1\/p/’\min(P0) maxX E 9(‘9 0x H) Thus, we conclude that

- 1 d f

v s — §U3(x), forx i {x : U3(X) g 21.36 2 2(6)} (2.37)
Since U3(x) is positive definite and continuous, the set {X : U3(X) S [.1.(€)} is a
compact set for sufficrently small 6. Let c0(6) = maxU3(X) S ”(€){V(x x;)} c0(6) is
nondecreasing and lim,E _, 0 c0(6) = 0. Consider the compact set {X : V(X ) S 60(6 6.)}
We have {X : U3(x) S 72(6)} C {x : V(X) S c0(6)}. Choose 64 = 64(5) S 61 small
enough such that, for all 6 S 64, the set {X : U3(x) S p(6)} is compact, the set
{X : V(x) S c0(6)} is in the interior of Q, and

{X : V(X) S 00(6)} C {X3 IIXII S {/2} (2-38)

Then, for all x E 0 but X ¢ {x : V(X) S c0(6)}, we have an inequality similar to
(2.37).

Thus, we conclude that the set {X : V(X) S c0(6)} x {17 : W(17) S p62} is
positively invariant and every trajectory in Q x {17 : W(17) S p62} reaches {x :
V(X) S c0(6)} x {17 : W(77) S p62} in finite time. In other words, given (2.38), there

exists a finite time T = T(5) such that, for every 0 < 6 S 64
le(t, all g 6/2. Vt 2 T (239)

Take 65 = 65(6) 2 min (63,64) and T1 2 T1 (5) = max (T, T), then (2.34) follows
from (2.35) and (2.39).<1

In what follows we use the results of Theorem 2.2. Although it is understood
that different values of 5 give different values of 6’5, we use the same notation for

simplicity.

28

2.5.3 Trajectory Convergence

Let Xr(t) be the solution of (2.16) starting from x(0). The following theorem shows

that x(t, 6) converges to Xr (t) as 6 ——> 0, uniformly in t, for all t 2 0.

Theorem 2.3 Under the conditions of Theorem 2.1, given any 5 > 0, there exists

(5 > 0 such that, for every 0 < 6 S 63‘ we have

||X(t, 6) - Xr(t)|| S 6, W 2 0 (2-40)

Proof. We divide the interval [0,oo) into three intervals [0,T(6)], [T(6),T2], and
[T 2,00), where both T(6) and T2 are to be determined later, and show (2.40) for
each interval. This approach gives more insight into the factors that come into play
in each of these intervals.

0 From Theorem 2.2 we know that there exists a finite time T2 2 T(6), indepen-

dent of 6, such that, for every 0 < 6 S 6’2‘, we have
le(t, 6)“ s é/2, Vt 2 T2 (2.41)

From the asymptotic stability of the origin of the reduced system we know that there

exists a finite time T2, independent of 6, such that
ler(t)ll S 6/2, Vt 2 T2 (2.42)

Take T2 = max{T2, T2}. Then, using the triangular inequality along with (2.41) and

(2.42), we conclude that, for every 0 < 6 S 6’5, we have

llx(t, 6) - Xr(t)|l S E, Vt 2 T2 (2.43)

29

0 From the proof of Theorem 2.1 we know that
llX(ta€) - x(0)|l S W
during the interval [0, T(6)]. Similarly, it can be shown that
||Xr(t) - x(0)|l S kit
during the same interval. Hence,
le(t, 6) — Xr(t)|| S 2k1T(6), Vt E [0, T(6)] (2.44)

Since T(6) —> 0 as 6 —) 0, there exists 0 < 65 S 6’5 such that, for every 0 < 6 S 65,

we have

lleta‘E) - Xr(t)|| S E, W E [0,T(e)] (2-45)

0 Over the interval [T(6), T2], the trajectory x(t, 6) satisfies
X = f7~(x, D(6)77(t, 6)), ‘with initial condition X(T(6), 6)
Over the same time interval, the trajectory Xr (t) satisfies
)2 = fr(X, 0), with initial condition XT(T(€))

From (2.44), we know that

leme), e) — more)“ 5 2k1T<e> déf 6(e)

30

where (5 (6) —-> 0 as 6 —> 0+. By continuous dependence of the solutions of differential
equations on parameters over compact time intervals [28, Theorem 2.5], we conclude

that

||X(t, 6) - Xr(t)ll S 6(6) exp[L(T2 - T(6))l
+ %\/p/Amm<Po>e{exp[L(T2 — T(em — 1}

s [6(a)+%fi/Amin(Po)eJexpiL(T2—T(e))] (2.46)

 

 

where L is the Lipschitz constant of f7~(.,0) on 0. Thus, given (2.46), there exists

0 < 66 S 6’5 such that for every 0 < 6 S 66 we have

||X(t,6) — Xr(t)|| S E, W E [T(6),T2] (2-47)

Take 65 = min(65, 66), then, using (2.43), (2.45), and (2.47) we conclude (2.40).<1

2.5.4 Recovery of asymptotic stability of the origin

We treat first the case when there is no modeling error; then we proceed to the
more general case when modeling error is present. In order to avoid very restrictive
conditions on the modeling error, we separate the case where the origin of (2.16) is
asymptotically stable from the case where it is exponentially stable. At this stage
of the chapter we place ourselves in a small ball of radius 5 > 0 around the origin
(x, 17) = (0, 0); the value of 5 will be determined later on. Theorem 2.2 guarantees
that trajectories of the system (2.11)-—(2.14), starting in S x Q, enter this ball after

a finite time and stay thereafter.

Case 1: We deal with the case where the origin of (2.16) is asymptotically stable,

and there is no modeling error; i.e., we perfectly know 43 in (2.1)—(2.4) and use it as

31

450. We know [32, Theorem 7] that there exists a Cl Lyapunov function V and a
positive definite function U3, both defined on a ball B(0, r1) (_1 R for some r1 > 0,

such that for all x E B(0, r1)

6V
afdxfi) S -U3(x) (2-48)

Choose 5 < r1; then given Assumptions 2.1, 2.2, and 2.3, we can show that, for all

(Xfll) E BUM) X {”77” S 5} = A1, we have

||9(X,D(6)n)|| S L4.||77ll (2.49)

Consider the composite function 17(X, 17) = V(X)+(W(T]))1/2 and choose 0 < 6: S 6’5

such that 1/(4EZ‘NIPOH) — L2L3 -— [IPOIIL4/(/Amin(P0) > 0, where L2 is an upper

ov
73?

set. Then, we conclude that, for every 0 < 6 S 6:: and for all (x, 77) E A1, we have

bound for

 

 

 

 

over A1 and L3 is a Lipschitz constant of f1~(., .) in 7'] over the same

4 1
v s -U3(X) — —-——l|nll (2.50)
4e IIP0||

Thus, the origin of system (2.11)—(2.14) is asymptotically stable. We summarize the

above conclusion in the following theorem:

Theorem 2.4 Let Assumptions 2.1—2.3 hold and assume that (150 2 ¢- Then, there
exists 6: > 0 such that, for every 0 < 6 S 631‘, the origin of the system (2.11)--(2.14)

is asymptotically stable.

Remark 2.4 Theorem 2.4 covers the results obtained by Teel and Praly in [49]. It

also gives a nonlinear generalization of the linear separation principle.

During the proof of cases 2 and 3, we will use the following fact which is a special

case of Young’s Inequality [18, Theorem 156]:

32

Fact: Vx,y E R+, Vp > 1, V60 > 0 we have

1 _P_.
131/3 -f6p+ (60)p0y” 1
0

where p0 = p—l—T'

Case 2: We deal with the case where the origin of (2.16) is exponentially stable,
whether or not we know (15.

In this case, there exists a C1 Lyapunov function V2(x) [28, Theorem 3.13] defined
over B(0,r2) Q ’R, for some r2 > 0, and four positive constants a1, a2, a3, and 624

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

«11”an < v2<x ) s a2l|x|l2 (2.51)
6V
-—26fr(x,0 s -asllxll2 (2.52)
6V
723]] s a4llxll (2.53)

 

 

Let us consider V(X,n) = V2(x) + fiW(i7), where 3 > 0 is to be determined, as a
Lyapunov function candidate for the system (2.11)—(2.14). Choose 5 < r2; then,
given Assumptions 2.1, 2.2, and 2.3, we have, for all (x,17) E B(0, 5) x {”77” S 5} =
A2,

“906 D(6)77)|| S lelxll + lelnll (254)

Using (2.52), (2.53), (2.54), and Young’s inequality, it can be shown that for all

(x,77) E A2, we have

; (1 fl
v s «231W — 521m“? — blllxn’ — b2||n||2 (2.55)

33

where bl = 613/2 - a4L7/60 - _BL5HP0lla 02 = U/(QC) — (1412760 - (4L5 + 2L6)HHP0H,
L7 is a Lipschitz constant of fr(., .) in 77 over A2, and 60 > 0. Now, choose 6 small
enough and 60 large enough such that b1 > 0, then, it can be shown that there exists

0 < 63 S 6’5 such that, for every 0 < 6 S 63, we have

t/ s — min (a3/2,fi/(26)) [IIxH2 +l|nll21 (2.56)

Thus, we can conclude that the origin of (2.11)—(2.14) is exponentially stable.

The foregoing result is summarized in the following theorem:

Theorem 2.5 Let Assumptions 2.1—2.3 hold and suppose the vector field fr(X,0)
is continuously differentiable around the origin. Moreover, assume the origin of the
closed-loop system under state feedback is exponentially stable. Then, there exists
6’5" > 0 such that, for every 0 < 6 S 63, the origin of the system (2.11)—(2.14) is

exponentially stable.

Case 3: We deal with the case where the origin of (2.16) is asymptotically,
but not exponentially, stable. We start with an example that shows the need for
some conditions on the modeling error in order for the output feedback controller to
recover asymptotic stability of the origin. In addition, it gives an idea about how to

formulate these conditions.

Example: Consider the system

.131 = .132 (2.57)
.62 = f(x)+u (2.58)
y = 1131 (2.59)

34

Suppose we know a nominal model f0(x) = "‘11 of f (x), and that the state feedback
u = —x2 globally stabilizes system (2.57)w(2.59) for the actual function f.

To implement the output feedback controller we use the high-gain observer

, 2 -
x1 = x2+;(x1—x1),

. 1 .
$2 = —x1+u+ :2-(131—1‘1)

By passing to the error coordinates and applying the output feedback u = —§:2, we

get the closed-loop system

931 = x2 (2.60)

62 = f($) - $2 + 722 (2-61)

. 2 1

771 = 7771 + :72 (2.62)

. 1

772 = f(x) + x1 — (2 + 6)771 (2.63)
Suppose now that the actual nonlinearity f (x) = —x:13. By linearization of the system

(2.60)—(2.63) around the origin, we notice that, for any 6 E (0,1), the linearized
system has a positive eigenvalue, thus, the origin of the output feedback system is
unstable.

Clearly the Lyapunov analysis we performed in the exponentially stable case
fails in this example. To see the source of the problem, let us note that the state
feedback control u = —x2 stabilizes the origin of (2.57)—(2.59) when f = —x:1’, and

the Lyapunov function V(x) = %x% + x1332 + x3 + $511 satisfies

V = —x‘11 — mg (2.64)

35

Now, notice that, around the origin, we have If (x) — f0(x)[ ~ [231]. This modeling
error is bigger than the absolute value of the derivative of the Lyapunov function
V(x) along trajectories of (2.57)—(2.59). This observation motivates the upper

bound on the modeling error which will be stated in Assumption 2.4.

Realizing that the modeling error does not cause a problem in the exponentially
stable case, we want to focus our attention on the part of the dynamics that is
asymptotically, but not exponentially, stable. Towards that end we use the center
manifold Theorem [28]. We need the vector field fr(X,0) in (2.16) to be twice
continuously differentiable.

Let us write the system (2.11)—(2.14) in the form

56 = fr(X,0)+A1(X,77) (2-65)

677 = A077 + 6BA2(X, 77) + 68600 (2.66)
where

Albee) = fr(x,D(6)n)-fr(x,0)
A2007?) = l¢o($,C,7(t9,x,C))-¢o(i,C,7(69,i,C))l+
[65(33, 2, 7(19, in O) - 45(66, 3,709,130”
5(x) = ¢($:Z,’7(l9,$aC))—¢0($,C,’Y(l9,$,C))

Since the origin of (2.16) is asymptotically, but not exponentially, stable, Theorem
3.13 of [28] shows that %%(O, 0) has all its eigenvalues with either zero or negative

real parts. Thus, there is a change of variables
[67", 2T]T = TX (2.67)

36

such that (2.16) can be written as

x 2 A15: + 91(x,2)

Nl'

= A25 + 92(57, 2)

where A1 has all its eigenvalues with zero real parts and A2 is Hurwitz. In addition,
Theorem 4.1 of [28] guarantees the existence of a continuously differentiable center
manifold 2 :2 h(x), for all Hill S (11, for some (11 > 0. Shifting the center manifold

to the origin via the change of variable

a) = Z — h(x) (2.68)
puts (2.16) in the form
if: = A12 + 91(2, 77(2)) + N1(x,w) (2.69)
d) = Azw + N2(i‘,w) (2.70)
where
llNz'(i,w)ll S Aillwll, Vll(f=,w)ll S al2 (2-71)

for some d2 > 0. The positive constants A,, i = 1, 2, can be made arbitrarily
small by choosing d2 small enough. By inserting all these changes into the system

(2.11)—(2.14) we end up with

1'1“: — 90(27) + N1(2‘:,w) + M1(x,w, 77) (2.72)
d) 2 142(1) + N2(:T:, w) + M267, w, 77) (2.73)
677 = A077 + 6BM3(:Z‘, w, 77) + 6861(22) + 6862(x, w) (2.74)

37

where 61(1) d__ef 6(x)|w ___ 0 is the projection of the modeling error 6(.) onto the

center manifold, 62(x,w) d.—‘if 6(x) — 600),, = 0, [M1T,M2TJT = T231, M3 = A2,
and 90(23) = A157: + 91(2, h(x)).

Corollary 4.2 of [28] shows that the origin of the reduced system

57 = 90(5?) (2-75)

is asymptotically stable. The condition on the modeling error, needed to establish

the asymptotic stability of the origin of (2.11)-—(2.14), can be stated as follows:

Assumption 2.4 There exists a 01 function V3(:'i:) defined on B5;(0, r3), a ball
around :7: = 0 contained in the projection of 0 onto the subspace of 2:, that satisfies,

for all :7: E Bi“), 73),

590(6) 5 -05(||i||) (276)
“61(6)“ 3 coagunu) (2.77)
[%’a s clagmrzn) (278)

 

 

 

for some positive constants a, b < 1 such that a + b = 1, where (15 is a class K:

function defined on [0, r3], c0 2 0, and c1 > 0.

The existence of a Lyapunov function satisfying (2.76) is guaranteed by the converse
Lyapunov theorem [28, Theorem 3.14], but what we need here is for (2.77) and (2.78)

to be satisfied as well.

Remark 2.5 When the reduced system (2. 75) is one-dimensional, we can take
V3(:‘E) = - I62 g0(y)dy. Then 9523 = —g0(i') and Assumption 2.4 is satisfied, with
a = b = 1/2, if |61(:r)| S d3 [g0(x)[ for some d3 > 0; i.e., [(510%)] cannot approach

the origin faster than [g0(§:)|.

38

The recovery of asymptotic stability can now be stated as follows:

Theorem 2.6 Let Assumptions 2.1—2.4 hold. Let the origin of the closed—loop sys-
tem under state feedback be asymptotically, but not exponentially, stable, and let the
vector field fr(X,0) be twice continuously difierentiable around the origin. Then,
there exists (6 > 0 such that, for all 0 < 6 S 6’6, the origin of (211)—(214) is

asymptotically stable.

Proof. Consider the Lyapunov function candidate V(x,w, 77) = V3(2’:) + (wTPgw)0 +
(W(77))0, where P2 is the positive definite solution of the Lyapunov equation P2A2 +
A5132 = —1, and o =-. 1/(2a) > 1/2.

Let 5 < min (d1, d2, 73) and let 6 S £5, then according to Theorem 2.2 there exists
a finite time T3 after which we have llx(t, 6)” + I]77(t, 6)” S min (5/(2||T|[(1 + L)),5)
where L is a Lipschitz constant of h(.) over Bi“), r3). Then, using (2.67) and (2.68)

we can show that

llflt, 6)“ + llw(t,6)ll (1+ L)lli‘(t,6)ll + |l2(t,6)ll

|/\

< 2<1+ LineTv, e), 2T<t,e)>Tn

|/\

2(1+ L)||T|||lx(t,6)ll

|/\

(E, VtZT3

Due to Assumptions 2.1, 2.2 and 2.3 we have, for all (22,722, 77) E BE(O,5) X Bw(0, 5) x

{Hull S E} =43,

||52(i,w)|| S lelwll (2-79)
M1(i,w,77)
S L9||T||ll77|| (2.80)
M2(fi,w,n)
IIM3(i,w,n)ll S L10||77|| (2.81)

39

Using (2.71), (2.76)—(2.78), and (2.79)—(2.81), we can show that, for all (in), 77) E

A3, we have

V s -05(llill) + polluwnagumn + p1lln||ag(llill) - pawn?“ +
_ P
p3A2llw||20 + p4||n|lllwll20 1— éllnllQ" + pawn?” +

p7a‘5‘(llill)llnll20 -1+ psllwllllnll2" *1 (2.82)

where

Po = 01, P1 =61||T||L9, P2 =071
p3 = 2072HP2II, p4=2072||P2||||THL9, 95:073
705 = 20‘74llpollL1o, p7 = 2074||P0|l00, pg = 2074||P0||L8

are positive constants, and where

X W
’71 = ()‘min(P2))U ‘1 71 = ”1’2”" ’1
2'. P 0 _1 ___ A . P U "1
72 ”2” 1) 17021 72 (mm( 21)) )if1/2So<1
73 = (Amin(P0))U _ 73 = ”Poll" _
74 = upon" '1 , 74 = ammo)” -1 ,

 

 

The positive constants A1 and A2, defined by (2.71), can be made arbitrarily small
by choosing 5 small enough.

Next, we separate the five cross-product terms in (2.82) by repeatedly using
Young’s Inequality with p = 20 and 60 = q1, Q2, q3, q4, and q5, respectively. Con-
sequently, given that a = 1/(2a) and a + b = 1, we have all the terms in ”77“ or in
”call with the power 20 and all the terms in a5(||:'1':||) with the power 1.

Now, choose A1, A2 and q1, q2, Q3, q4, and q5 such that —1/2 + pOA1(q1)p1 +

701(612)p1 + P7/614 < 0 and —p2/2 + p0/\1/611+ 03A2 + 734013)”1 + 708/615 < 0, where

40

p1 = 1/(20 — 1). Then, it can be shown that there exists 0 < 6’6 S 6’5 such that, for
every 0 < 6 S 6’5 we have —p5/(26) + p1/Q3 + p6 + 707(q4)p1 + p8(q5)p1 < 0 which

implies that for every 0 < 6 S 6’5 and for all (in), 77) E A3, we have
- 1 _ P2 20 as 20
< —— — — , — —— .
v _ 205(H33ll) , lel ,6 llnll (2 83)

Thus, the origin of the system (2.11)~——(2.14) is asymptotically stable.<1

Remark 2.6 Theorems 2.4, 2.5 and 2. 6, along with Theorems 2.1 and 2.2, show the

recovery of the region of attraction.

2.6 Examples

We apply our technique to different systems in order to illustrate the theoretical
results and go beyond the theory to demonstrate some reasonable intuitions. We
also take advantage of these examples to show that the results obtained in this
chapter apply not only to input-output linearizable systems as in the previous work

[11, 27, 26, 30, 31, 38, 37, 41] but to any kind of system that fits the model (2.1)—(2.4).

2.6.1 Example 1

We consider a second order system having an exponentially unstable mode, together
with a bounded linear controller that achieves a finite region of attraction. The

system is

5131 = 1:2 (2.84)

x2 = 21131 + 10 tanh (u) (2.85)

where the control is u = —x1 - x2.

41

We consider a full-order high-gain linear observer (i.e., 450 = 0) with (11 = 02 = 1.
In this example we show how the output feedback controller recovers the region of
attraction achieved under state feedback.

Figure 2.1 shows the region of attraction under state feedback control, in addition
to three compact subsets that are recovered using the high-gain observer. In each
case the compact subset is specified, then a design parameter 6* is found through
multiple simulations at different points of the subset such that for every 6 S 6* the
output feedback controller is able to recover the given subset; i.e., it is a part of
the region of attraction of the new closed-loop system. The bound 6* is tight in
the sense that for 6 > 6* there is a part of the given set that is not included in the
region of attraction. The bounds 6*’s for these subsets are 0.082, 0.057, and 0.007,
respectively, starting from the smallest subset. Notice that the bigger the subset the

smaller the bound 6*. In all cases we take 52(0) = 0.

2.6.2 Example 2 - Inverted Pendulum

We consider the inverted pendulum-on-a-cart problem given in [15]. The system
Consists of an inverted pendulum mounted on a cart free to move on a horizontal

plane. The equations of motion are given by:

 

 

i'l = :62 (2.86)
, 1 . 2 .
x 2: —mgsrn x cos x +mlx Sln x —
2 M+msin2($3)l ( 3) ( 3) 4 ( 3)

bxg + u] (2.87)
(E3 = 274 (2.88)
, 1 . 2 .
x = M+m gsm .73 —mlx Sln it cos x +
4 [(M+msin2(x3))[( ) ( 3) 4 ( 3) ( 3)

bx2 cos (x3) — ucos (x3)] (2.89)

42

where M is the mass of the cart, m is the mass of the ball attached to the free end
of the pendulum, l is the length of the pendulum, g is the gravitational acceleration,
b is the coefficient of viscous friction opposing the cart’s motion, x1 is the cart’s
displacement, x2 is the cart’s velocity, x3 is the the angle that the pendulum makes
with the vertical, and x4 is the pendulum’s angular velocity. The values of the
different parameters of the model are M = 1.378Kg, m = 0.051Kg, g = 9.81m/sec2,
l = 0.325m, and b = 12.98Kg/sec. The nominal value of b is b0. It is shown in [15]

that the state feedback control

u = [mg sin (:63) cos (x3) — mlxi sin (x3) + boxg +
(M + msin2 (x3))v] (2.90)
v = —900x2 + 900[3.22x1 + 12273 + 7.44(lx4 + x2 cos (x3))] (2.91)
4

stabilizes the origin .

Let the measured outputs be (x1, 233). We use a full order high-gain observer to
estimate all the state variables, and use these estimates in the stabilizing control.
In order to avoid peaking in the state variables, induced by peaking of the observer
variables, we saturate the control input such that our region of interest is included in
the system’s region of attraction; we use u =2 100 tanh ( / 100). The nominal function
p0 is made globally bounded by saturating x2 and $4 at 15 and 20, respectively. We
design a full-order high-gain observer with multiple poles at —1/6.

Figure 2.2 shows how the velocity estimate peaks due to the difference in the

initial conditions between the position and its estimate. This is done for a linear

—‘

4This is a special case of the tracking control of [15] when the desired trajectory
is taken to be zero.

43

observer and the following choice of initial conditions and design parameters:

x(0) = (1,0, —0.5,0), 3(0) 2 0, b0 : 12.98, e = 0.001

Figure 2.3 shows how our output feedback controller recovers the trajectories
achieved under state feedback. We use a linear high-gain observer with three values

of 6. This is done for the following choice of initial conditions and design parameters:

x(0) = (1,—4,0.7,—9),:8(0) = (0,0,0,0),bO=12.98,

6 = 0.0015, 0.001, 0.0001

Intuitively we expect that a nonlinear observer that includes a model of the system’s
nonlinearities would outperform a linear one when the model is accurate. Figures
2.4 and 2.5 show that this intuition is justified. In Figure 2.4 we compare between
a linear observer and a nonlinear one with no modeling error; i.e., the nonlinearity
is perfectly known; thus b0 = b. In Figure 2.5 we do it with modeling error induced
by a nominal value b0 = 15.58 (20% error). In these two simulations we use the

following initial conditions and design parameters:

23(0) = (1,0,0.8, 0), 2(0) = (0,0,0, 0), e = 0.01

It is clear that when b0 2 b the nonlinear observer outperforms the linear one. But as
the model uncertainty increases, the performance of the nonlinear observer degrades

towards that of the linear observer.

44

2.6.3 Example 3 - VTOL aircraft

We consider the simplified PVTOL (Planar Vertical Take off and Landing) aircraft
modeled in [39] by

:81 = x2, x2 = —u1 sin(x5)+uu2cos(x5)
x3 = x4, x4 = ul cos(;1:5)+pu28in(x5)—g

(i5 = (176, 236 = Au?

where x1, x3, and x5 are the horizontal coordinate, the vertical coordinate, and the
inclination of the aircraft, respectively. We also consider, like [39], the linearizing

dynamic feedback

737 = x8, x8 = —V1 sin (x5) + V2 cos (x5) + x7xg
“1 = x7 + gxg

1
u2 = E(_V1 cos (x5) — V2 sin (x5) — 21138176)

The equilibrium point of the closed-loop system under state feedback is x =
(231,0,123,0,0,0,g,0), u = (9,0), and 17 = (0,0). The linearizing effect of this dy-

namic controller can be seen by applying the change of variables

X1 : $1—§Sin(x5), X2 = .752 — Lil-1‘6 COS ($5)
X3 = —x7 sin (x5), X4 = —x8 sin (x5) — x7226 cos (x5)
X5 2 (173+ ‘ECOS (1'5), X6 : $4 — §$6 sin ($5)

x7 = x7cos(x5)—g, X8 = x8cos(x5)—x7x68in(x5)

t0 obtain X4 2 V1 and X8 = V2. In the new coordinates the equilibrium point is

X .1— (51,0,§;3 + p/A,0, 0,0, 0,0). We choose the equilibrium point to be at xeq =

45

(2,0, 2,0, 0, 0, g, 0). To stabilize the aircraft at this equilibrium point, or to make
it track this constant trajectory, we define the change of variables 521 2 X1 — 2,
X3 = 3:3 - 2 — p/A, with the rest of the X variables unchanged. Then we take the

control

kph + 19222 + (63923 + k424 (2.92)

1’1

112 = [$121 + (C2522 + [€323 + (€424 (2.93)

where k1 to k4 are chosen to stabilize the origin )2 = 0. For the purpose of simulations
we take 9, p and /\ to be 1, 1, and 0.5, and k1, k2, k3, and k4 to be —24, -—50, —35,
and —10

Now suppose we only measure the position variables :31, x3, and x5, and set
y1 = $1, y2 = 2:3, and y3 = 2:5. We want to design an observer to estimate the
velocity variables 2:2, $4, and 2:6. Noting that the nonlinear functions sin (235) and

cos ($5) depend only on the measured output variable 2:5, the system takes the form
2': = A2: + B¢(u,y), y = C1:

If the nonlinear function ¢(u, y) is exactly known, we can design an observer that

yields linear error dynamics [20, Section 4.9]. In particular, the full-order observer
3%: = A5: + B¢(u, y) + L(y — C53)

results in the error equation

é: (A—LC)e

Where 6 = a: — :i'. Such observer design does not need the high-gain observer theory

presented in this paper. For the purpose of comparison with our method, we design

46

a reduced-order observer that yields linear error dynamics with eigenvalues located
at -—1, assuming perfect knowledge of the parameters p and A. We will refer to this
observer as the nominal reduced-order observer. On the other hand, when there is
uncertainty in modeling the nonlinearity ¢(u, y), the error dynamics will no longer be
linear. This case is covered by our high-gain observer which is designed to be robust
with respect to uncertainties in d). We design a reduced-order high-gain observer
with eigenvalues located at —1/6. We saturate the dynamic feedback controller as
follows: 2'37 = 20 tanh (./20), 2'38 = 200 tanh(./200), “1 = 40 tanh (./40), and 112 =
200 tanh (./200). These bounds have been figured Out from extensive simulations
done to see the maximal values that the state trajectories would take when the initial
state is in a region of interest around mag. The model nonlinearities are naturally
globally bounded so we do not need to saturate the nominal nonlinearities in the
observer.

To illustrate the capability of our output feedback controller to recover the state
trajectories we perform simulations with different values of the design parameter
6. Figure 2.6 shows that as e approaches 0 the trajectories under output feedback
approach the trajectories under state feedback.

To illustrate the robustness of the output feedback controller we perform sim-
ulations with a 15% error in A. Figure 2.7 shows that our design does its job in
recovering stability and trajectories for 5 small enough. The above simulations are

done with the following choice of initial conditions and parameters:
1(0) 2 (1,0, 1,0,0,0, 1,0), 33(0) 2 (0,0,0), 6 = 0.02, 0.008, 0.002

Figures 2.8 and 2.9 show that our observer outperforms the nominal observer in
terms of convergence rate and trajectory recovery. One may think that by increasing

the gain of the nominal observer we can have as good of a performance as the

47

high-gain observer. This is not true, and Figure 2.10 shows it. In the nominal
observer, the input and the nonlinearity are not globally bounded. Therefore, peaking
in the estimate passes to the state through the input. For this reason we saturate
our dynamic feedback. The above simulations are done with the following choice of

initial conditions and parameters:

37(0) = (1,0, 1,0, 0,0, 1,0), 1(0) = (0,0, 0), e = 0.01, 0.002,

nominal observer gain 2 100

Remark 2.7 Of course, the trajectories recovered are the entire state 3:1 to 11:8 but

Figures 2.6 to 2.10 only show a few of them.

2.7 Conclusion

We presented and proved a separation principle for a certain class of nonlinear sys-
tems. An output feedback controller using a sufficiently fast high-gain observer re-
covers the performance achieved under a state feedback controller. This includes
boundedness, ultimate boundedness, convergence of trajectories, and exponential
stability of the origin. We also found that we can recover asymptotic stability of the
origin when the modeling error is zero, but, when this error is not zero, we need to
impose some additional conditions. It is worthwhile to note that our results can only
Show semiglobal stabilization under output feedback even when the state feedback
control achieves global stabilization. Global separation results are more challeng-
ing as the discussions of [13] reveal. We performed simulations on various types of
Systems and controllers to illustrate the results obtained in this paper. These sim-
UIations showed the advantage of including a model of the system’s nonlinearity in

the observer, when a good model is available. Furthermore, they demonstrated the

48

effectiveness of the combination of high-gain observers and saturation in recovering

asymptotic stability and trajectories achieved under state feedback.

49

 

20

15"

-5—

 

 

 

-20

—20 20

 

Figure 2.1. Recovery of region of attraction: 6* = 0 (solid), 5* = 0.007 (dashed),

6* = 0.057 (dash-dotted), and 6* = 0.082 (dotted)

50

 

x . xz-estimate
/

 

 

 

 

_100 l 1 r 1 I l 1 l I
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

time

 

150 I I I I I I r I l

 

100 ‘

 

 

 

 

_150 l l l l l l l l I
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

time

Figure 2.2. Peaking of the velocity: 2:2 (solid), i2 (dashed)

51

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

’\
I
I \\
0 ' I « ,
’I
.A
x"-1 ' '
.II’/
_2 - I] J [
l I,
(I
_3 L
0 1 2 3 4 3 4
1
4
-1.5 m ‘
0 1 2 3 4 3 4
time

 

Figure 2.3. Trajectory convergence: state feedback (solid); output feedback with
= 0.0015 (dashed), e = 0.001 (dash-dotted), and e = 0.0001 (dotted)

52

 

 

 

 

 

 

 

 

 

O 0.5 1 1.5 2 2.5 3 3.5 4
time

Figure 2.4. Effect of nonlinearity in the observer - without uncertainty: state feedback
(solid), with linear observer (dashed), with nonlinear observer (dotted)

53

 

 

 

 

 

 

 

 

 

O 0.5 1 1.5 2 2.5 3 3.5 4
time

Figure 2.5. Effect of nonlinearity in the observer - with uncertainty: state feedback
(solid), with linear observer (dashed), with nonlinear observer (dotted)

54

 

 

 

 

-
p
_

 

 

.—
I—
p—

 

 

 

 

time

Figure 2.6. Trajectory convergence - without uncertainty: state feedback (solid);
output feedback with e = 0.02 (dashed), e = 0.008 (dash-dotted), and e = 0.002
(dotted)

55

 

2.5 l I I I I I I I I

 

 

 

p—
p—

 

 

 

 

 

.—
—

 

 

Figure 2.7. Trajectory convergence - with uncertainty: state feedback (solid); output
feedback with e = 0.02 (dashed), e = 0.008 (dash-dotted), and e = 0.002 (dotted)

56

 

 

 

 

 

 

 

 

 

 

8 10

 

 

 

 

 

 

 

 

 

 

 

Figure 2.8. High-gain vs. nominal observer - without uncertainty: state feedback
(solid), nominal observer (dashed); high-gain observer with e = 0.01 (dash-dotted)
and e = 0.002 (dotted)

57

2.5 . . . . 4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.9. High-gain vs. nominal observer - with uncertainty: state feedback (solid),
nominal observer (dashed); high-gain observer with e = 0.01 (dash-dotted) and e =
0 - 002 (dotted)

58

 

 

 

zol 1

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.10. High-gain vs. nominal observer - importance of saturation: state feed-
back (solid), nominal observer (dashed); high-gain observer with e = 0.01 (dash-
dotted) and e = 0.002 (dotted)

59

CHAPTER 3

A Separation Principle for the
Control of a Class of Nonlinear

Systems

3.1 Introduction

In Chapter 2 we introduce separation results for the stabilization of a class of systems
having a chain or more of integrators in their structure. Therein, we consider state
feedback controllers that make the origin of the closed-loop system an asymptotically
stable equilibrium point. In this chapter we are interested in the output feedback
implementation of controllers that achieve boundedness of trajectories under state
feedback control but not necessarily with convergence to an equilibrium point. Such
a situation can be encountered in adaptive tracking and regulation [2, 27] where
only the tracking error or both the tracking error and the parameter error converge
to zero. Another example is the convergence to a zero-error manifold as in the
servomechanism problem discussed in [26, 37, 38, 21]. Additional examples can be
found in stabilization problems in the presence of disturbances as in [50, 10] where

only finite-time convergence to a set can be achieved. In all these cases, it can be

60

shown that the trajectories approach an attractive, positively invariant, compact set.

In this chapter, we consider a class of systems similar to the one considered
in Chapter 2 and characterize the performance of the state feedback controller as
rendering a certain compact set positively invariant and asymptotically attractive.
Furthermore, as in Chapter 2, we require the control law to be globally bounded and
implement it using a high-gain observer. Then, we recover the same set of perfor-
mance measures that was recovered in Chapter 2. It includes recovery of the region
of asymptotic stability of the attractive set (i.e., recovery of arbitrary compact sub-
sets of this region), as well as the convergence of trajectories under output feedback
control to those under state feedback control as the observer gain approaches infinity.

We start with semiglobal separation results using results form [34]. Then, we give
similar separation results for a possibly finite region of attraction. For this task we
adapt the results of [34] to this case because [34] deals only with global convergence
to a set. In order to illustrate the theory developed hereafter, we present in the next
chapter several examples taken form [10, 50, 38, 21, 2].

This chapter is organized as follows: Section 3.2 states some definitions and recalls
results from [34], Section 3.3 formulates the problem. Section 3.4 discusses the set of
performance measures to be recovered by output feedback and proves this recovery
in a semiglobal setting. Section 3.5 adapts the results of [34] and Section 3.4 to the

finite region of attraction case.

3.2 Definitions and Converse Lyapunov Results

Consider the system

61

where for each t E R, :1:(t) E R” and d(t) E D, and where D is a compact subset of

Rd. The map f : R” x D -> R” is assumed to satisfy the following properties:

0 f is continuous in its arguments.

o f is locally Lipschitz in x uniformly in d. This means that for each compact

subset K of R” there is some constant c such that

||f(1‘,d) - f(z,d)|| S Clll‘ - 2|!

for all :r,z E K and all (1 E D.

Let MD be the set of all piecewise continuous functions from R to D. For each
d 6 MD, we denote by a:(t,a:0;d) the solution at time t of (3.1) with 22(0) = 2:0.

This solution exists and is defined on some maximal interval (T

+ .
$0,d’Tx0,d) With

— +
_OOSTmo,d<O<T£L‘0,dS+OO'

We say that a set A is a positively invariant set for (3.1) if
V1170 E A, Vd E MD, T26”! = +00 and :c(t,$0;d) E .A, W 2 0

Let A be a closed, non-empty subset of R”. The distance of g E R” with respect to

A IS defined as
4 —— inf — 7)

In the sequel we use the notation

62

where, for each d E D, fd(.) is the vector field defined by f(.,d). By ”smooth” we

always mean infinitely differentiable.

We define uniform asymptotic stability with respect to a set in the spirit of [4,

Definitions 4.1, 4.12] and [57, Section 1.10, Definition 1].

Definition 3.1 The system (3.1) is Uniformly Asymptotically Stable (UAS) with

respect to the compact positively invariant set A if the following two properties hold:

1. Uniform Stability: for any 6 > 0, there exists a constant 61 = 61(5) such that

|x(t,x0,d)|A g 'e for all d E MD, whenever IxOIA S 01(6) and t Z 0 (3.2)

2. Uniform Attraction: there is an oz > 0 and, for any 6 > 0, there is T = T(6) >

0, such that for every (1 E MD:

|x(t,x0, dllA < 6 whenever [xOIA < a and t 2 T (3.3)

Moreover, the system (3.1) is Uniformly Globally Asymptotically Stable (UGAS) with
respect to A, if (3.2) holds with a class (Coo function 6(6) and (3.3) holds for any
r > 0 with T = T(e,r). Finally, the system (3.1) is Uniformly Exponentially Stable
(UES) with respect to A, if there are three positive constants r, k, and 7 such that
the solution x(t,€,t0,d), starting from 5 at time to under the input d(t), exists for

all t 2 t0 and satisfies
lx(t,$0,t0, dllA S 1967“— t0)l$0|A, W Z 1*0

for all x0 6 {5: IEIA S r} dgf {20 and all d E MD-

63

Remark 3.1 Proposition 2.5 of [34] shows the equivalence between UGAS and

bounding |x(t,x0; dllA with a class IC£ function.

Herein, we state the definition of a Lyapunov function for (3.1) with respect to

the compact, positively invariant set A.

Definition 3.2 A Lyapunov function for the system {3.1) in the open set R with
respect to a compact, positively invariant set A (_I R is a function V : R —) R> 0

such that V is smooth on R/A and satisfies the following properties:

1. There exist two class [C functions 0‘1 and a2 such that for any as E R,
0105M) S V(f) S a2(|€|A) (3-4)

2. There exists a continuous, positive definite function (13 such that for any g E

R/A, and any d E MD,

L f dvu) s -a3(|€|A) (3.5)

A smooth Lyapunov function is one which is smooth on all of R.

In the case R = R”, we require 01 and 02 to be class ICOO.

In some situations the Lyapunov function candidate is time-dependent. In this case
we need a Lyapunov stability theorem for invariant sets where the Lyapunov function

could depend on time.

Theorem 3.1 Let A Q R be a compact, positively invariant subset of R" for the
system (3.1). Then, (3.1) is UAS with respect to A if there exist a C’1 positive defi-
nite, with respect to A, function V(t, {) : [0, 00) x U —> R, where U is a neighborhood

of A, two IC-functions 01,02, and a continuous positive definite function 03 such

64

that, for allt Z 0, we have

0106M) S V(té) S a205M) (3-6)
for any 5 E U, and

8V ,

E- + Lde (5) S -a3(|€|A) (3-7)

for any 5 E U/A and any (1 E D. In addition, ifU = R" and al is class (Coo, then

{3.1) is uniformly globally asymptotically stable with respect to A.

Proof. see Appendix C.1.<1

Remark 3.2 Theorem 3.1 along with [34, Proposition 2.5] constitute a powerful
machinery for showing U GAS. For example, we can show UGAS using two Lyapunov
functions, the first one shows ultimate boundedness and the second one shows local

UA S.

For GUAS, we state the converse Lyapunov theorem given in [34] as Theorem

2.9.

Theorem 3.2 Let A C R” be a compact, positively invariant set for the system
(3.1). Then, (3.1) is uniformly globally asymptotically stable with respect to A if and

only if there exists a smooth Lyapunov function V(x) with respect to A.

Moreover, we give a converse Lyapunov theorem for UES.

Theorem 3.3 Assume that the system (3.1) is uniformly exponentially stable with
respect to the compact, positively invariant set A. Then, there exists a function

V(t, x), defined and continuous an R2 0 x (20, where {20 = {leA 3 r0, r0 > 0}

65

and contains A, such that

 

lxlA s vex) s klxIA (3.8)
|V(t,$)-l“'(t,i‘)| S MILE—ill (3.9)
. V(t + h, x) — V(t, x)
1 < —AV t, , Vd e M 3.10
h 310 h ‘ ( x) D ( )

for all £5,573 6 $20, and allt _>_ to, where L and A < *y are positive constants and

5: = x(t + h, x, t0;d).

Proof. see Appendix C.2.<1

3.3 Problem Formulation

In many state feedback controller designs, the trajectories of the closed-loop system
do not converge to an equilibrium point. For example, in the servomechanism cases
discussed in [26, 38] the objective is to make the tracking error converge to zero while
keeping the states of the zero dynamics bounded. The same objective is achieved in
[2, 27] using an adaptive controller. Furthermore, in robust control, as in [50, 10],
we often achieve ultimate boundedness. In this section we formulate these design
objectives as steering the trajectories towards a compact, positively invariant set. Of
course, there are problems which can neither be cast as stabilization of an equilibrium
point nor as stabilization of a compact, positively invariant set. These problems,
such as [2] (when we only have partial persistence of excitation) and [29], will be the
subject of future research.

The class of systems considered can be represented by the multi-input multi-

output nonlinear model

it = Ax+Bq§(x,z,d(t),u) (3.11)

66

Z = u'.!(x,z,d(t),u) (3.12)
y = Cx (3.13)

C = q(r. z,d(t)) (3.14)

where u E Rm is the control input, C E R3 and y 6 RP are measured outputs,
x 6 RT and z E R6 constitute the state vector, and d(t) 6 Rd is a vector of signals
that belongs to MD- The matrices A, B, and C, represent p chains of integrators
as in Section 2.2.

The state feedback control is assumed to be in the form

{9 = F(19,x, (,d(t)) (3.15)

u 2 7(29, x, C,d(t)) (3.16)

We allow any state feedback design that satisfies:

Assumption 3.1 (1 ) I‘ and 'y are locally Lipschitz functions in 19, x, and C uniformly
in d over the domain of interest;

(2) I‘ and '7 are globally bounded functions of x;

(3) The closed—loop system is uniformly globally asymptotically stable with respect to

the compact positively invariant set A.
The system (3.11)—(3.14) with the controller (3.15)—(3.16) satisfies:

Assumption 3.2 The functions q, (f) and w are locally Lipschitz in x, z, and u uni-
formly in d over the domain of interest. Moreover, d(x, z,d,7(19,x,(,d)) is zero in

A uniformly in d.

To implement the control (3.15)—(3.16) we use the state estimates, it, generated

67

by the high-gain observer
in = A5: + B¢O(e, c, d(t), u) + H(y — Cr) (3.17)

The observer gain H is chosen as in 2.10.
The function ¢0(x,C,d(t),u) is a nominal model of d(x,z,d(t),u) which is re-

quired to satisfy the following assumption:

Assumption 3.3 (to is a locally Lipschitz function in x,(, and u uniformly in d
over the domain of interest. Furthermore, it is globally bounded in x and zero in A,

uniformly in d.

3.4 Performance Recovery - Semiglobal Separa-
tion Results

The main objective of this section is to show that the suggested output feedback
implementation of the control law recovers the performance of the state feedback
controller (3.15)—(3.16) for sufficiently small 6. The performance recovery manifests
itself in three points. First, the compact set A x {x — a": = 0} is a positively invariant
set of the closed-loop system under output feedback and the closed-loop system is
asymptotically stable with respect to A x {x - x = 0}. Second, the output feedback
controller achieves semiglobal stabilization; that is, for any compact set S which
contains A, and any compact set Q g RT, the set 8 x Q is included in the region
of attraction under output feedback control. Third, the trajectory of (x, 2,19) under
output feedback approaches the trajectory under state feedback as e —+ 0.

The analysis is done in three steps. First, we show the recovery of boundedness
of trajectories. Second, we show the recovery of ultimate boundedness of these

trajectories. Third, we show the recovery of local asymptotic stability with respect

68

to A. This allows us to deal with asymptotic stability as a local prOperty that could

require some additional assumptions on the modeling error.

Let us first, for the purpose of analysis, replace the observer dynamics by the
equivalent dynamics of the scaled estimation error (the scaling is similar to that of
Section 2.5.1). Then, as in Chapter 2, we have :1“: = x — D(e)n. Hence, the closed-loop

system can be represented by

:1; = Ax+B¢(x,z,d(t),y(.)) (3.18)
é==w@&fl®mWfl-Dmmcflm) mm)
.9 = I‘(19,x—D(c)n,C,d(t)) (3.20)
67') = A077+eBg(x,z,i9,D(e)77,d(t)) (3.21)

where g(.) = d(x,z,d(t),7(.)) — ¢0(x,C,d(t),y(.)) and A0 is a constant Hurwitz
matrix.

The system (3.18)-(3.21) is a standard singularly perturbed one, and 77 = 0 is the
unique solution of (3.21) when 6 = 0. Similar to Section 2.5.1, the reduced system
is the closed-loop system under state feedback. For simplicity we write the system
(3.18)—(3.20) as

X = fr(x, d(t), B(fln) (3-22)

where x = [xT, 2T,i9T]T and x(0) = [933,253,19ng- Then, the reduced system is

given by
X = fr(x, d(t), 0) (3-23)
The boundary-layer system is
dn

69

where r = f. Let (x(t,e),n(t,e)) denote the trajectory of the system (3.18)—(3.21)
starting from (x(0), 71(0)).

We know that (3.23) is uniformly globally asymptotically stable with respect to
the compact positively invariant set A. Then, Theorem 3.2 ensures the existence of
a smooth Lyapunov function V(x) in addition to two class Koo functions a1, a2 and

a continuous, positive definite function 03 such that, for all x E R", where n = r+€,

we have:

V(x) = 0 4:» x E A (3.25)
a1(|><|¢4) S V(x) S 02(leA) (326)

For the boundary-layer system we define the Lyapunov function W(n) = nTPOn,
where P0 is the positive definite solution of the Lyapunov equation POAO + AgPO =

-I. This function satisfies inequalities similar to (2.21)—(2.22).

3.4. 1 Boundedness

Let the initial states (x0, 20, 190) E S, and x0 6 Q, where S is any compact set in R"
which contains A and Q is any compact subset of RT. The recovery of boundedness

of trajectories is given in the following theorem:

Theorem 3.4 Let Assumptions 3.1—3.3 hold; then, there exists 6’] > 0 such that for
every 0 < e S 6’], the trajectories (x(t, e), 77(t, 6)) of the system (3.18)—(3.21) starting
in S x Q are bounded for allt 2 0 and all d E MD-

Proof. As in Section 2.5.1, we show the positive invariance of an appropriately chosen
set A, then we show that, any closed-loop trajectory, starting in the compact set

S x Q, enters the positively invariant set A in finite time.

70

The properness of V(x) guarantees that the set 9 = {x E R" : V(x) g c}, for
some c > 0, is a compact set and that S is in the interior of 9.

Let A : I) x {W(77) 3 p62}. Due to Assumptions 3.1—3.3 we have, for all x 6 Q,
all d E D, and all r) 6 RT,

llfT(Xa d1 D(€)7l)[l

“900 d, D(€)n)ll S k2 (3-29)

|/\

k1 (3.28)

where k1 and k2 are positive constants independent of 6. Moreover, for any 0 < E < 1,
there is L1, independent of e, such that, for all (x,17) E A, all (I 6 D, and every

0<eS€,wehave

||fr(X,d, D(€)77) - fr(X,d,0)|| S Llllnll (3-30)

In the rest of the chapter we always consider 6 g E. It can be shown that
- 0V

for an (m) 6 W00 = c} x {Won 3 p9}. and

- 1
w s 7177112 + 2llnllllPollllBllk2, va 6 v (3.32)

 

for all (x,77) E Q x {W(n) 2 p62}, where [:3 = L1L2\/p/)\mm(P0), ||P0|| =
Amax(P0), and L2 is an upper bound for [lg—EH over Q. As in Section 2.5.1, tak-
ing p = 16k§||P0||3 and 61 = fl/k3, where H = minX E 69 U3(x), it can be shown
that, for every 0 < e _<_ 61, the derivatives of V and W along the trajectories are
nonpositive on the boundaries of A. Thus, we conclude that the set A is positively

invariant .

71

Now, we consider the initial state (x(0), 1(0)) 6 S x Q. Using (3.28) and the fact

that x(O) is in the interior of Q, it can be shown that

”x(tm) — X(0)|| S kit (3-33)

as long as x(t,6) E Q. Thus, there exists a finite time T0, independent of 6, such
that X(t,€) E Q for all t E [0,T0]. As in Section 2.5.1, we can Show that there
exists 62 > 0 small enough and a time T (6) such that W(n(T (6), 6)) 3 pos2 for every
0 < 6 S 62. Taking 6’] = min (6,61,62) guarantees that, for every 0 < 6 S 61‘, the
trajectory (X(t,€),7](t,€)) enters A during the interval [0,T(6)] and remains there
for all t _>_ T (6) Thus, the trajectory is bounded for all t Z T (6) On the other
hand, for t E [0,T(6)], the trajectory is bounded by virtue of inequality (3.33) and

an inequality similar to (2.32).<1

3.4.2 Ultimate Boundedness

Hereafter, we show that trajectories of the system (3.18)—(3.21), starting in S x Q,
come arbitrarily close to the set A x {n = 0} as time progresses. This is summarized

in the following theorem:

Theorem 3.5 Under the conditions of Theorem 3.4, given any 5 > 0, there exist

6* = 6*(45 > 0 and T = T (6 such that, for every 0 < 6 S 6*, we have
2 2 1 1 2

|X(t, 6)IA + |ln(t,€)ll S E, Vt 2 T1 (3-34)

for all d E MD.

Proof. From the proof of Theorem 3.4 we know that, for every 0 < 6 3 61‘, the
trajectory of the closed—loop system, starting from (x(O),:ic(0)) E S x Q, is inside

the set A for all t 2 T(6), where A is 0(6) in the direction of the variable 17. Thus,

72

we can find 63 2 63(5) 3 61‘ such that for every 0 < 6 g 63 we have

Hna e)” g 6/2, w 2 my.) ‘1? Ta) (335)

for all d E MD: In what follows we continue working with the Lyapunov function
defined in the the proof of Theorem 3.4. It can be shown that, for all (x, n) E A, we
have

V g —a3(lxl_A) + k36, Vd E D
Without loss of generality a3 can be chosen to be class ICOO. Thus, we conclude that

- 1 — d f
v s — ,aguxa). forlxlA 2 33131.36) 2 Me), w e 72

Choose 64 : 64(5) 3 6’] such that u(64) < a§1(c) and ai—1(a2(u(64))) < 5/2.
Then, by a proof similar to that of [28, Theorem 5.1, Corollary 5.1], we conclude

that there exists a finite time T = T(5) such that, for every 0 < 6 g 64
Ix(t. «0L4 s é/2, Vt 2 T (3.36)

for all d E MD' Take 65 2 65(5) = min (63, 64) and T1 = T1 (5) = max (T, T), then
(3.34) follows from (3.35) and (3.36).<1

3.4.3 Trajectory Convergence

Let x7(t) be the solution of (3.23) starting from x(0). The following theorem shows

that x(t, 6) converges to x7»(t) as 6 -+ 0, uniformly in t, for all

73

Theorem 3.6 Under the conditions of Theorem 3.4, given any 5 > 0, there exists

63' > 0 such that, for every 0 < 6 S 6?; we have
”x(te) - Xr(t)|| S 6, Vt 2 0 (3-37)

for all d E MD‘

Proof. We divide the interval [0,00) into three intervals [0,T(6)], [T(6),T2], and
[T2,oo), where both T(6) and T 2 are to be determined later, and show (3.37) for
each interval.

0 From Theorem 3.5 and the uniform asymptotic stability with respect to A of the
reduced system we conclude that there exists a finite time T2 2 T(6), independent

of 6, such that, for every 0 < 6 3 6’5, we have

le(t)|A 5 6/2, Vt 2 T2 (3.33)

|X(t, 6)|A S 5/2, W 2 T2 (3.39)
for all d E MD' Then, we can write
”x(t, 6) - Xr(t)l| S ”x(t, 6) - dill + ”20““) - xll, Vt 2 T2 (340)

for all x E A and all d E MD- By taking the infimum of (3.40) over A and using
(3.38)-(3.39), we have

llX(ta€)—Xr(t)ll S lX(ta€)lA+er(t)lA

g 6, Vt 2 T2, Vd 6 MD (3.41)

74

0 As in Section 2.5.2, we can show that
llx(t,6) - Xr(t)|| S 2l€1T(6), W 6 [01(6)]

Since T(6) ——> 0 as 6 ——> 0, there exists 0 < 65 3 65 such that, for every 0 < 6 S 65,

we have

”x(té) - Xr(t)ll S 6, W E [0,T(6)l (3-42)
for all (l E MD'

0 Over the interval [T (6), T2], the trajectory x(t, 6) satisfies, for all 61 E MD,

>2 = fr(x, d(t), D(€)n(t, 6)), with ||X(T(€), 6) - Xr(T(6))l| S 5(6)

where D(6)n is 0(6) and 6(6) —> 0 as 6 —> 0+. Thus, as in Section 2.5.2, we conclude

that there exists 0 < 66 3 6’5 such that, for every 0 < 6 g 66, we have

||X(t,6) - 2000” S 5, W E [T(6),T2] (3-43)

for all (1 E MD: Take 65 = min(65,66), then, using (3.41), (3.42), and (3.43) we
conclude (3.37).<l

3.4.4 Uniform Asymptotic Stability

In this section we deal with local uniform asymptotic stability with respect to a
compact positively invariant set in the absence or presence of modeling errors. We
assume that the trajectory belongs to the ball B(A, 5) = {X : Ix] A g 5}. This is

justified given the result of Theorem 3.5.

75

Case 1: Herein, we deal with the case where the system (3.23) is uniformly
asymptotically stable with respect to the set A. Moreover, there is no modeling

error. We have the following theorem:

Theorem 3.7 Let Assumptions 3.1—3.3 hold and assume that (1)0 = d). Then, there
exists 62 > 0 such that, for every 0 < 6 g CZ, the system (3.18)—( 3.21 ) is uniformly

asymptotically stable with respect to the compact positively invariant set A x {77 = 0}.

Proof. We know from Theorem 3.2 that there exists a Cl Lyapunov function V and
a class (Coo function 623, both defined on a ball B(A, r1) g Q for some r1 > 0, such
that for all X E B(A, r1)

(9V

Choose 5 < r1; then given Assumptions 3.1—3.3 we can show that, for all (x,n) E
B(A, 5) x {”71” g 5} = A1 and all d, we have

||9(X,d,D(€)77)ll S L4||77||, d E D (3.45)

Consider the composite function V(x, n) = V()()-l-(W(17))1/2 and choose 0 < 62’ 3 6’5

such that 1/(4EZ‘HIPOII) — L2L3 — llPOllL4/1(’\min(P0) > 0, where L2 is an upper

 

 

 

 

bound for % over A1 and L3 is a Lipschitz constant of f7-(., ., .) in 17 over the
same set. Then, we conclude that, for every 0 < 6 g 6: and for all (x, n) 6 A1, we
have
4 1
V S —ag(|x|A) - ————-—||n||, d E D (346)
46 ”Poll

Thus, according to Theorem 3.1, the closed-loop system (3.18)—(3.21) is uniformly

asymptotically stable with respect to A x {77 = 0}. <1

76

Case 2: We deal with the case where the system (3.23) is uniformly exponentially
stable, whether or not we know a.

The result is summarized in the following theorem:

Theorem 3.8 Let Assumptions 3.1—~33 hold and assume that the closed-loop system
(3.23) is uniformly exponentially stable with respect to the set A. Then, there exists
6:; > 0 such that, for every 0 < 6 _<_ (’5, the system (3.18)—(3.21) is uniformly

exponentially stable with respect to the set A x {n = 0}.

Proof. Let X2(t) be the solution of (3.23) that starts from X at time t = 0. In this
case, according to Theorem 3.3, there exists a Lyapunov function V2(t, x) defined
over B(A,r2) g Q, for some r2 > 0, and three positive constants a1, a2, and a3

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

IXIA S V2(t, X) S allXIA (3-47)
hm V(t + h, X2(t + h)) - V(t, X)

h —> 0 h

||V2(t,X1) - V2(t,X2)|| S asllxi - X2l|.VX1, X2 6 B(Afl‘z) (3-49)

 

S -a2V(t, x), Vd E MD (3.48)

Let us consider V(t, X, 77) = V2(t, x) + fi‘/W(n), where S > 0 is to be determined, as
a Lyapunov function candidate for the system (3.18)-(3.21). Choose 5 < r2; then

we have

Claim: Given Assumptions 3.1—3.3, we have, for all (x,r)) E B(A, 5) x {”17” S
6} = A2,
”900 d, D(€)fl)|l S L5|XIA + L6l|77|| (3-50)

77

for all d 6 MD'

Proof. It suffices to show that

ll¢(X, d) - ¢0(X,d)ll S LlelA (3-51)

for all d E MD. Since both ()5 and (to are zero in A, we can write

ll¢(X, 01) - 450(X, d)|| S ||¢(X, d) - ¢(Xa, d)|| + ll¢0(X, d) - 450(Xa, d)ll (3-52)

for all Xa E A and all d E MD: Using the local Lipschitz property in (3.52) yields

||¢(X, d) - (2003 d)“ S L5||X — Xall (3-53)

for some L5 > 0. Taking the infimum over A yields (3.51).<1

Let x3(t) be the solution of (3.22) that starts from x at time t = 0. Then, using

(3.48), (3.49), and (3.50), it can be shown that, for all (x, n) 6 A2, we have

V(t + h, X3(t + h), n(t + h, 6)) — V(t, X, 7?)

 

 

lim < —a V t, +a L
h—+0 h _ 2 2( X) 3 7||n||
B B
- —-——||n|| + ||P0||(L5|XL4 + Lellnll) (3.54)
26 ”Poll \Amin( 0)

where L7 is a Lipschitz constant of fr(.,.,.) in 7) over A2. Then, there ex-

ist positive constants 5 such that —(a2/2) + fiIIPOIIL5/‘//\m,-n(PO)) < 0, and
0 < 63‘ 3 6’5 such that, for every 0 < 6 3 fig, we have —,B/(4€‘/][PO”) + a3L7 +

fillPOHLG/ilhmiflpo» < 0. This implies that, for every 0 < 6 3 6’5", we have

1m
h ——> 0 h

 

78

|/\

1 5
--a2V2(t, X) - ___—Hull
2 46 “Poll

-—}V (t X n) (3.55)

|/\

— min{—2-a2,———

1
46 ”PO ll

Thus, there exist positive constants b2 and b3 such that

“X“? din“? 6))IA X {0} S b2e—b3tl(X(0)in(0))lA x {0}, Vd E MD (356)

Thus, according to the definition of UES in Definition 3.1, the closed-loop system

(3.18)—(3.21) is uniformly exponentially stable with respect to A x {n = 0}. <1
During the proof of case 3, we will use Young’s Inequality given in Section 2.5.4.

Case 3: We deal with the case where the system (3.23) is uniformly asymptoti-
cally stable with respect to the compact, positively invariant set A. A condition on

the modeling error has to be imposed and is stated as

Assumption 3.4 There exists a 01 function V3(t, x) defined on [0, 00) x U, where
U = {x : M A 3 r3, r3 > 0} is a neighborhood of A in Q, three functions ibl and
i/Jg, and $3 defined and continuous on U which are positive definite with respect to

A (i.e., positive everywhere and zero only in A), such that, for all t Z 0, we have

w1(X) S V30, X) S $200 (357)
5%? + afit—(x, d. 0) s —w3(x) (3.58)
||¢( :6 z dew 1‘ C, d)) - ¢0($,C,d,v(19,w,é,d))llS 6010300 (3-59)
II%( (t x) 3 C1 «(2300 (3-60)

 

 

79

for all x E U and all (1 E D, for some positive constants co _>_ 0, Cl > 0 and a, b < 1,

such that a + b =1.

The existence of a Lyapunov function satisfying (3.57)-—(3.58) is guaranteed by The—

orem 3.2, but what we need here is for (3.59) and (3.60) to be satisfied as well.

Remark 3.3 Assumption 3.4 is similar to Assumption 2.4 of Chapter 2 in the sense
that it relates the modeling error magnitude and the rate of convergence of trajectories
near the attractor (which is a set in the case at hand). It can also be viewed as an
extension of Assumption 2.4 to the case of asymptotic stability with respect to a set.
However, in Assumption 2.4 we used the center manifold decomposition to reduce
the size of the set over which the assumption is satisfied and to reduce the structural
complexity of the modeling error by projecting it on the center manifold.

From different examples, presented later on, it seems reasonable to allow the Lya-

punov function candidate V3 to depend on time.

The recovery of asymptotic stability can now be stated as follows:

Theorem 3.9 Let Assumptions 3.1-3.4 hold. Then, there exists 6’5 > 0 such that,
for all 0 < 6 3 6’6, the system (3.18)—(3.21) is uniformly asymptotically stable with

respect to the compact positively invariant set A x {17 = 0}.
Proof. Consider the Lyapunov function candidate V(t, x,n) = V3(t, x) + (W(77))0
with a = 1/(2a) >1/2.

Let 5 < r3 and let 6 3 6’5, then according to Theorem 3.5 there exists a finite
time T3 > 0 after which we have |x(t, 6)]A + ||17(t, 6)“ S 5 for all t 2 T3.

Using Assumptions 3.1—3.4, we can show that, for all (x, 17) 6 A3 = B(A, 5) x

{“77” S E}, we have

llfr(X.d,D(€)77)-fr(X,d,0)|| S lelnll (3-61)
||9(X,d,D(€)77)ll S 00¢3(X)+L9||77|| (3-62)

80

for all (1 E D. and

. p ‘) __
v s —./.’3(x>+p1nnnwg(x> — 311nm?“ + pawn?” + pwgumnn-U 1 (3.63)

where

P1 = OILS: 92=071

p3 = 2Uvzlll’ollLe, p4 = 2Unlll’ollco
are positive constants and

z ,\ . P 0‘1 = P 0"]
’71 (min( 0)) “-021 71 H 0” ifl/ZSU<1

'72 = ”POHO —1 ’72 = (AminUDOllo _1

Next, we separate the two cross-product terms in (3.63) by repeatedly using Young’s
Inequality (see Section 2.5.4) with p = 20 and 60 = q1, Q2, respectively. Conse-
quently, given that o = 1/ (2a) and a + b = 1, we have all the terms in “77“ with the
power 20 and all the terms in w3(x) with the power 1.

Now, choose ql, Q2, such that —1/2+p1(q1)p1+p4/q2 < 0, where p1 = 1/(20—1).
Then, it can be shown that there exists 0 < 66 3 6’5 such that, for every 0 < 6 S 63
we have —p2 / (26) + p3 + p4(q2)p1 < 0 which implies that, for every 0 < 6 3 6’5 and

for all (x,n) 6 A3, we have
' 1 [’2 20
< __ __ 3. 4

for all (1 6 D. Thus, according to Theorem 3.1, the closed-loop system (3.18)-—(3.21)

is uniformly asymptotically stable with respect to A x {0}. <1

81

Remark 3.4 Theorems 3. 7, 3.8 and 3.9, along with Theorems 3.4 and 3.5, show

semiglobal stabilization.

3.5 Regional Separation results

In many cases, asymptotic stability with respect to a set, achieved under the state
feedback controller, is not global and the region of attraction is a subset of the state
space. This subset may be finite (bounded) or infinite. Examples of such cases can
be found in adaptive control [2, 27], and robust control [50].

In order to extend the previous separation results, we need, as we did in the
previous chapter, a converse Lyapunov theorem that yields a Lyapunov function
which goes to infinity at the boundary of an estimate of the region of attraction.
This can be done by extending the converse Lyapunov results of [34] to an estimate
of the region of attraction. In order to perform this task, we need to restrict the set of
time-varying parameters to the set, denoted by M’ , of all continuously differentiable
functions from R to D where the derivative (1’ (t) of d(t) belongs to a compact set
D1 C Rd.

First, we give a definition of the region of asymptotic stability of a closed set.

Definition 3.3 The region of uniform asymptotic stability of the system (3.1) with
respect to the closed positively invariant set A is the set of all points 2:0 E R" such

that
|:c(t,x0,d)|A —+ 0 as t —> +00

uniformly in d E MD‘

This region is not empty because it contains A.

82

Second, we give the needed converse Lyapunov theorem.

Theorem 3.10 Let d E MD' Let A C R" be a compact, positively invariant set for
the system (3.23). Assume that the system (3. 23) is UAS with respect to A. Let R be
an open and connected subset of the region of attraction that contains A. Then, there

exists a smooth Lyapunov function V in R and three positive definite, with respect

to A, functions U1, U2, and U3, all defined on R, such that

V(x) = 0<=>xeA (3.65)
11100 S V(x) S 11200 (3-66)
xgmaRUflX) = 00 (3.67)
ggffixfifi) g —U3(x),Vd€D (3.68)

Proof. see the proofs of Theorem 6.1 and Corollary 6.1.<1

Remark 3.5 The set R can be the region of attraction when the system is au-

tonomous.

In this case we can recover the same set of performance as in the previous section.

First, let us replace Assumption 3.1 with the following assumption:

Assumption 3.5 The time-varying parameter belongs to the set M5D. Items 1 and
2 of Assumption 3.1 hold. The closed-loop system under the state feedback controller

is uniformly asymptotically stable.
The separation result is summarized in the following theorem:

Theorem 3.11 Let Assumptions 3.2—3.4, and 3.5 hold. Let R be an open, connected
subset of the region of attraction that contains A and S be any compact subset of R
which contains A. Then, the conclusions of Theorems 3.4, 3. 5, 3. 6, 3. 7, 3. 8, and 3.9
hold.

83

Proof. Using Theorem 3.10, the proof of boundedness is similar to that of Theorem
3.4. The proof of ultimate boundedness is similar to that of Theorem 2.2. We show

that

- 1 d f
V S - 511300, forx 6? {X : U3(X) S 21636 3 #(6)} (3-59)
We define c0(6) = maxU3(X) S ”(€){V(x)}. Then, we have {X : U3(x) g u(6)} C
{X = V(x) S 60(6)}-
Finally, we choose 64 = 64(5) 3 6’f small enough such that, for all 6 g 64, the set
{X : U3(x) S ,u(6)} is compact, the set {X : V(x) g c0(6)} is in the interior of Q,

and

{x = V(x) S 60(6)} C {X = IXIA S 6/2} (370)

Then, we conclude that the set {X : V(x) S C0(€)} x {n : W(17) 3 p62} is
positively invariant and every trajectory in (2 x {r} : W(17) 3 p62} reaches

{X 3 V(x) S 60(6)} X {77 : W(n) S p62} in finite time.

The proofs of trajectory convergence and local uniform exponential stability are
similar to those of Theorems 3.6 and 3.8, respectively.

The proofs of local uniform asymptotic stability (with or without modeling errors)
are similar to those of Theorems 3.7 and 3.9. In this case we locally replace, using
Lemma A.3, the functions U4, i = 1,2,3, with class [C functions 62,-, i = 1, 2, 3, such

that alum) s U1(X),02(|xl,4) 2 U200, and a3(lxl,4) s U300 on a certain ball

around A.<l

84

3.6 Conclusion

We presented a separation principle for a class of nonlinear systems in cases where
trajectories do not necessarily go to the origin, namely, in cases where trajectories
go to a compact, positively invariant set A. An output feedback controller using a
sufficiently fast high-gain observer recovers the performance achieved under a state
feedback controller. This includes boundedness, ultimate boundedness, convergence
of trajectories, and exponential stability with respect A. We also found that we can
recover asymptotic stability with respect to A when the modeling error is zero, but,
when this error is not zero, we need to impose some additional conditions on it. It
is noteworthy that our results can only show semiglobal stabilization under output
feedback even when the state feedback control achieves global stabilization. As for
the case where the region of attraction is not the whole space we can only recover

compact subsets of an estimate of the region of attraction.

85

CHAPTER 4

A Separation Principle for the
Control of a Class of Nonlinear

Systems - Examples

4. 1 Introduction

In order to illustrate the theory developed in the previous chapter, we present
several examples taken form [10, 50, 38, 21, 2]. We present these examples as state
feedback control designs and apply the separation results of the previous chapter
to arrive at the output feedback control results proven in the respective references.
Moreover, we go beyond the results of those references and conclude the trajectory
convergence property mentioned earlier. This shows how we can use the results of

the previous chapter as a framework for the design of output feedback controllers.

Hereafter, we discuss several output feedback control problems that fit in the
generic form we discussed in Section 3.3. In each example we give the system con-
sidered and the control problem solved then we show that the system fits into the

model (3.11)—(3.14) and that the separation results previously obtained apply to the

86

problem at hand.

The forthcoming examples have been treated in several papers to be referenced
later on. In our presentation of these examples we conserve the notation used by the
respective papers for easy reference. The notation pertains only to the example at
hand and does not refer to any previous analysis unless it is explicitly mentioned.
These design cases, as presented in their corresponding references, have not been
treated as separation results; i.e., they were treated as output feedback design cases.
Thus, in order to apply our separation results, we reworked them as state feedback
design cases which did not require considerable changes to the original analysis given
in the corresponding references. Moreover, the high-gain observers suggested are the
ones used in those references.

Each example is divided into three main parts. The first part introduces the
system considered as well as any necessary technical assumptions. The second part
states the design objective and details the design process. The third part explains
how we can apply the separation results developed in the previous chapter to the
example at hand. The application of the separation results follows the following

points:
0 We give the system to which the theory applies.
0 We give the globally bounded state feedback controller.

0 We prove that the system with the controller is asymptotically stable with
respect to a certain compact positively invariant set that we specify. We also

give an estimate of the region of attraction.

0 We propose a high-gain observer for the output feedback implementation of the

controller.

87

0 We state the conclusions that can be made using the separation results and

show, if needed, that the conditions are satisfied.

We start with examples related to robust control. Then, we move to examples
related to tracking and servomechanism. Finally, we treat an example from adaptive

control.

Remark 4.1 To prepare the controller for output feedback implementation, we sat-
urate the control law outside a region of interest. It is implicit in this case that the
region of attraction recovered is the one achieved with the bounded controller.

In case the state feedback controller achieves global asymptotic stability, the out-
put feedback controller achieves semiglobal stabilization (i.e., the region of attraction
contains arbitrary compact sets fixed a priori by the designer). It is implicit that the
saturation level imposed on the control law depends on the compact set that we wish

to recover.

Remark 4.2 The design and analysis pertaining to each of the forthcoming examples
are extracted from their corresponding references. But for the simplicity of the pre-

sentation we used the pronoun ”we” to indicate the designers who are not necessarily

fl 7’

U3.

Remark 4.3 The assumptions and claims stated in each example are related only to

the example at hand and so are their numbers.

4.2 Robust Control I - Finite Time Convergence

to a Set [10]

Consider the perturbed linear time-invariant system

x(t) 2 Ax(t) + Bu(t) + E6(x, u, t) (4.1)

88

y(t) = 01130) (4.2)

where x 6 R", u E Rm, y 6 RP, and d(.) 6 RE. The following assumption imposes

growth and structural conditions on the nonlinearity:

Assumption 1: The disturbance vector d(.) satisfies the following inequalities:

||t5($. u, t)Il S 5x|l$l| + <5uIIUH + 5c

|l5($,u,t) - 5(th 11, t)|| < 161”” - fill

where 63;, (in, 60, and k1 are nonnegative constants and 6a < 1.

The objective is to design a dynamic state feedback control that ensures ultimate
boundedness of the state trajectories. The following assumption gives a stabilizing

static state feedback controller:

Assumption 2: There exists a state feedback control u = — f (x) and a Lyapunov

function W(x) such that

 

 

 

8W
5;le —- Bf(:t) + E6(x. —f(:t), t)] s -—fl4llrll2 + (6&3 + fizfllrll + 6cm
awntn2 s W(x) 3 nuts”?
6W
IE’ 3 kznstn
||f($) — x(t)“ s 1.3“. — tn
||f(:t)I| s t4ntn+k5

for all x and x, where fl4’7lw’ 77w, k2, k3 are positive constants, while fl3, fl2,fil, k4,

and k5 are nonnegative constants.

89

The first inequality ensures uniform ultimate boundedness of the closed-loop

system with respect to the set
Q = {x E R" : W(x) < 77ch} (4.3)

where

_l,
d§f5c53+52 + (6c53+fi2)2 +§c_fil_ 2
2A4 452 fl4

 

20

If the transfer function C (31 — A)_1E is minimum-phase and left-invertible, then
there exist nonsingular transformations F1,I‘2,F3, integer K, and integer indices

(12373, i=1,---,K such that

x = r15: = [235%,57? = [5:35:3ij

y = lefiftisTth u = F311, <5(-) = 1“350)

and such that the system (4.1)—(4.2) can be written as (all tildes will be dropped for

notational simplicity)

xa = Aaaxa + Aafyf + Aasys + Bau

xb = Abbxb + Abfyf + Bbu

xf = Afxf+Mfyf+Bfu+Ef[Daxa+Dbxb+Dfxf+6(.)]
yf = Cm

ys = ngb

where (A f’ B f) represents chains of integrators. The structure of the above matrices

90

is explained in [10] and the references therein.
The state component x f can be estimated using a high-gain observer. As for the

components xa and xb, they are estimated using the observer

23a 2: Aaai‘a + Aafyf + Aasys + Ba’u

H
c-
I

Abbib + Abfyf + Lb(ys — 0357b) + Bbu

where Lb is to be chosen. Thus, the implemented state feedback controller is u =

— f (F15?) dgf f1(x3, xf). The estimation error dynamics are

éa ‘2 Aaaea (4.4)

éb = (Abb_LbCS)eb (4.5)

Choose Lb such that (Abb — Lsz) is Hurwitz.
Set e3 = [cg-[11,651]? A3 = diag[Aaa, Abb — LbCS], and rewrite (4.4)—(4.5) as

és =- A383 (4.6)

Let P3 be the positive definite solution of the Lyapunov equation P3A3+AgP3 = —I.

The function V3 2 egPses satisfies

asllesll2 3 V8 3 nuts“? (4.7)

V3 < —d3V3 (4.8)

for some positive constants 118,773,613.

Next, we show that if Assumptions 1 and 2 hold globally, then the closed-loop

system (4.1), (4.6) with u = — f1 (£73,113 f) is globally uniformly ultimately bounded

91

with respect to the compact set
<I> = {(xT.e.T>T = vs s c? W(x) s m} (4.9)

for some c3 which can be made arbitrarily small and some c1 > c3 which depends
OI] Cs.

The derivative of W along the trajectories of the closed-loop system is

8W 8W

W = 337M313 — Bf(x) + E6(x, —f(x), t)] + EBlfflfcsflf‘) - f1(5ts,:tf)]
+ %V?VE[6(x, —f1(x3,xf),t) - (5(33, —f1(Is,$f)tt)l (4'10)

Using Assumptions 1 and 2 in (4.10) yields

W s —fl4||x||2 + (6433 + flz)llrll + 5cm

6W 6W
_ p _ _ “ ,
+ [II 6,, B” k3” 1n + H a. E” tltgnrlul ”at. as.” (4 11>
Using the bounds on 0W/6x and V3 into (4.11) yields
_1_
. 2 _ Vs 2
W S - 54W?“ ’ (5cfl3 + A2) + 2717}; 7 “513““ 5c51 (4-12)

where 271 = k2(HBH + IIEIIk1)k3 and ’7}, =HI’1”.
Let

—3

1
1 V 5
Cr (\fi/g) dgf 5E;- [(5c33 + 32) + 2717h (.773) ]

NIH

2
+ 4546081

1.
2

as

 

1 V
+ 5B; [(5cfi3 + 32) + 271?}; (i)

92

Notice that C17(0) : (3*. It is straightforward to see that

01’

W < 0 when “x” > cg; (fig) (4.13)
W < 0 when W'(x) > 77ng (VI/3) (4.14)

Using (4.14) and (4.8), it can be seen that the closed-loop system is globally

uniformly ultimately bounded with respect to the compact, positively invariant set

<I> defined in (4.9) where cl > 0%(c3).

The separation results apply to the system

ita
it
if
yf
ys

Aaaxa + Aafyf + Aasys + Ball.
Abbxb + Abfyf + Bbu
Afxf + Mfyf + Bfu+ EleaiEa + Dbxb + Dfxf + 6(x,u,t)]

Ckxf
0313b

This system fits into the model (3.11)—(3.14) with the vector of bounded disturbances

being 6(0,0, t) (bounded by Assumption 1), the x state component being x f’ the 2

state component being (xa, xb), and the C output being ys.

The state feedback controller considered is

(L'a = Aaaia + Aaf’yf + Aasys + Ball

:13b 2 Abbib + Abfyf + Lb(3/S “ 0353b) + Bbu

93

u : —f1(jaaib7$f)

Global boundedness is achieved by saturation outside a region of interest.
We have shown that the closed-loop system under this controller is globally
asymptotically stable with respect to the compact positively invariant set (I).

To implement the controller we use the high-gain observer
if = Afif + Mfyf +Bfu+Ef[Dai‘a + Dbib+Dfifl + Ld(yf - Cfi‘f)

The structure of the gain Ld is given in [10, Appendix B]. This structure is exactly
the one suggested in Section 3.3 with the difference that in [10] all channels of equal
relative degree are grouped together.

According to Theorems 3.4 and 3.5, trajectories of the closed-loop system under
output feedback come arbitrarily close to the set (I) x {e f = 0} in finite time, where
e f is the estimation error. Moreover, Theorem 3.6 ensures convergence of trajectories
under the output feedback controller to those under the state feedback controller as

the observer gain approaches infinity.

4.3 Robust Control 11 - Finite-time Convergence

to a Set [50, Section 3]

Consider the control system

2 = A(z, u,d(t)) (4.15)

y = C(z, d(t)) (4.16)

94

where the state z E R" and the input u E R. The functions A(.) and C(.) are
smooth in their arguments, and d(t) is a time-varying smooth disturbance signal

contained in a compact set D C Rd. For simplicity we denote d(t) and its time

derivatives d(t), d(t), - .. by the same symbol d, i.e., d = (d, d, d, - - ~).
We start with the following two definitions:

Definition 1: [50, Definition 2] (Uniform Complete Observability). Consider the

dynamical system

C=AKwLy=CK)

The above dynamical system is UCO if there exist two integers ny and nu and a C1

function ‘11 such that, for each solution of

C:A(C1u0)1u0:u1imaunu =‘U

we have, for all t where the solution makes sense,

at) = we), - - -,y("y)<t>,uo(t), - - -,un..(t)).

Remark 4.4 In [50] the authors used the more general notion of UCO of a function

u(() with respect to a dynamical system. In such a definition we have

mm» = we), - --.y("y)(t),uo(t),---,unu<t)).

In our case we need the function a to be equal to C because, later on, we need to

95

express the state variable 2 as function of the input, the output, and some of their

derivatives.

Definition 2: [50, Definition 4] (Semiglobal Practical Stabilizability). A point
z = 0 is said to be semiglobally practically stabilizable by dynamic state feedback
(respectively, output} feedback if, for each pair of compact sets (ICzs, Kzg), neighbor-
hoods of O with ngg C K38, there exist a locally Lipschitz dynamic state (respectively,
output) feedback u = u(z,C), C = 0(z,C) (respectively, u 2 21(3), C), C = 0(y,C)) and
a pair of compact sets (K43, ICU) such that all solutions of the closed-loop system,

with initial condition in 1C2); x ICU, enter the set lng x ICCs in finite time.

The objective is to design a state feedback controller that achieves semiglobal
practical stabilization of the point z = 0.
We assume that the point z = 0 is semiglobally practically stabilizable by a

static state feedback, as in the following assumption:

Assumption 1: [50, Assumption S-CPS]: There exists two integers Ny and Nu
so that the system (415)—(416) is UCO with 113, g Ny and nu 5 Nu. For each pair
of compact sets (K23, lng), neighborhoods of 0 and with lCzs C lng, we can find
1. a positive C1 function V, zero at 0, which is defined on D, an open set containing

Ing, and such that there exist three positive real numbers 195, c3, and cg satisfying
c3 < Cg, {z,V(z) g fig} C ICzs, ’Czl C {2 : V(z) 3 c3} (4.17)

and so that the set {z : V(z) 3 Ce} is compact and contained in Q.

2. a C2 function u(z) which is zero at 0, defined on (2 such that

2‘1

02 A(z,u(z),d(t)) S -<I>(z) (4.18)

96

where @(z) is continuous on Q and positive definite on {z : 193 g V(z) 3 cf} for

some positive real number 195 < W.

In order to use the state feedback {1(2) we need to know the output and ny of its
derivatives in addition to the input and nu of its derivatives. Thereafter, we need
to prepare to estimate the output and some of its derivatives, and design an input
whose derivatives (a number of them) are known.

It can be shown that there exist 77.3, + 1 smooth functions Ci and an integer

mu 3 ny such that, for each solution of

z' = A(z,u0,d(t))

it0:111

we have, for all t where the solution makes sense,

Let yo = y and y,- : yi, i : 1, - - -,ny. Then, if we write the system as

90 = 311
311 = 92
we use a high-gain observer with the nonlinearity Cny + 1(0, uO, . . - , umu, 0) to

97

estimate the output and its derivatives that are needed in 11(2).
Let [u = max{nu, mu} + 1. Thus, by adding in integrators to the system (4.15),
we can have u and its required derivatives (nu for a and mu for the Cny + 1) as

measured states of the system

z' = A(z,u0,d(t))

it0:111

iteu—l = ’0 (4.22)

For the moment, we are left with the task of designing a state feedback controller
for the system (4.22). Let {1 = uO — 11(2) and C,- = %-_TII for i = 1, - - -,£’u, with
K being a positive real number to be specified later on. Thus, the system (4.22) can

be written as

def

2 = A(z,t<z>+e,d(t)) = heads»
«51 = K52-Z—:(z)A<z,t(z)+tl,d(t))
£2 = K63
C5,” = K1_euv (4.23)

This system satisfies, with minor adjustments, the conditions of of Lemma 2.3 of [50]
which provides the desired state feedback controller. For the sake of completeness

we state the main condition of this lemma.

98

Assumption 2 [50, Assumption ULP]: For the system
z = h(z, 0,d(t))

there exists an open set 91 _C_ Rm, a nonnegative real number 19 < 1, a real number
c 2 1, and a C1 function V: {21 -> R) 0 such that the set {2 : V(z) g c+ 1} is a

compact subset of (21, and we have

%h<z.o,d(t>) s —<I>1<z)

where <I>1(z) is continuous on $21 and positive definite on the set {z : 19 < V(z) S

c+1}.

Claim: There exists a C1 function V1(z) that satisfies Assumption 2.
Proof. In order for the system (4.23) to satisfy Assumption 2 we need to adjust the
function V(z) and the various coefficients given is Assumption 1 as follows:
Pick 191 as an arbitrary number in (O, 1 / 8). Let K. be a C1 class [Coo function such

that
16(293) = 191, k(i9g) _>_ 8191, k(c5) Z 1, k(Cg) > 1 + [C(Cs) (4.24)

This function exists since Assumption 1 states that 0 < 193 < 19g 3 63 < cg. Now,

let V1(z) = k(V(z)) and c1 = k(c3) 2 1. Thus, we have

{z:Vl g 8191} C {z:V(z) 36g} C ICzs

lng C {2 : V(z) S c3} C {2 : V1(z) 3 Cl} (4.25)
The set {2 : V1 3 Cl +1} is a compact set and is contained in the set {2 : V(z) g

99

Cg}.
Finally, we have

an,

82 (zt0td(t)) S -‘1’1(Z)

where (1)1(z) = fi—fi(Vz)<I>(z) is continuous on Q and positive definite on the set {z :

191$ V1(z) 3 C1 +1} C {293 S V(z) S Cg}.
Hence, from Assumption 1 and the above adjustments, we conclude that the

system (4.23) satisfies Assumption 2 with the C1 function V1(z) defined on the

open set {21: Q and with '19 2291 <1 and c: c1214

Hereafter, we apply Lemma 2.3 of [50] to the system (4.23). Let the polynomial
p(s) =3€u+1+a€usgu +~~+a1

be Hurwitz and let Ac be the companion form matrix corresponding to p(s). Let PC
be the positive definite solution of Ach + PcAc = —I. Let ngg be an arbitrary
compact set where we choose to initialize C. Lemma 2.3 of [50] suggests the following

controller

’U = —K€u(a1€1+'”+agu§gu)

= —K€u(a1[u0 — 22(2)] + - . . + aguggu) (4.26)

and provides the Lyapunov function

tat/1(2) + M1€TPc€
01+1-V1 u1+1-€TPc€

 

 

W1(21€) :

100

where

T
#1 maX{ ,5 21315656 c8} 2 1

Furthermore, we have
[Cze X ICEg C {($45) : W1(z,§) S C? + It?)

0
and, by letting p = 31, there exists K* 2 1 such that, for all K 2 K*, the derivative
of W1 along trajectories of the closed-loop system (4.23), under the controller (4.26),

satisfies
W1 3 —<I>2(2.€) (4.27)

where (1)2(z,C) is a positive definite function on {(z,C) : 191 + p _<_ W1(z,C) S
c? + a? + 1}.

Thus, we have proved the following:
For any pair of compact sets (lCszng), neighborhoods of 0 with lCzs C ICZg, we
can find compact sets ICCs and Ing, gains az- ’s, a bound K*, integers Eu and ny, and

functions a and \I' such that for each K 2 K71; the dynamic state feedback

it = K62
52 = K63
Stu = [fl—6"”
v = —K€“(a1[u0—u(z)]+~~~+aguCgu) (4.28)

in closed-loop with the system (4.15)—(4.16} makes all solutions starting in lng x [ng

enter the set K23 >< K53 in finite time.

101

The theory developed in the previous chapter applies to the system

90

321

T $12 (4.29)

: 0713/ + 1(2,’U.,K€2,' ' °,Kmu€mu +11d(t))

: K€2
= K
53 (4.30)
= Kl—guv
(4.31)

where 2 can be expressed as function of the state variables given that the original

system is UCO. This system fits into the model (3.11)—(3.14) with (4.29) being the

x-dynamics and (4.30) being the z-dynamics. Furthermore, the vector of bounded

disturbance consists of d(t) and some of its derivatives, all denoted here by d(t). The

additional output C is represented here by the vector (u,C1, - . . ,Cgu). Since we are

dealing with a regional result we need to restrict the time-varying parameter d to

belong to M’D.

We consider the state feedback controller.

where

= —K€u(alfl + ° ° ° + agutfgu)

= —K€u(a1[u0 — x(z)] + - ~ - + agu€eu)

Global boundedness is achieved by saturating the 11(2) outside the set {2 : V(z) g
Cg}.

We have proved that trajectories of the closed-loop system under the state feed-
back controller, starting in the compact set IC 2 g x ngg (arbitrarily chosen), enter the
compact positively invariant set {(2, C) : W1 (2, C) s 191 + p} in finite time (positive
invariance is clear from (4.27)). An estimate of the region of attraction is the set
no = {(2,5) : W1(2,C) g c? + 11% + 1}.

To implement the controller we use a nonlinear high-gain observer with the non-
linearity Cny + 1(0, uO, - - - , umu, 0) as a nominal model for the nonlinearity
Cny + 1(2, uO, - - - , Umu, d(t)) (this is the same observer used in [50]).

Theorem 3.11 guarantees that the trajectories of the closed-loop system under
output feedback controller starting in S x Q, where S is a compact subset of (20 and

Q is a compact set in Rny, are bounded and will come arbitrarily close to the set

{(2,6) = W1(z.§) S 191+p} >< {6 = 0}

where e is the estimation error. Moreover, Theorem 3.11 shows that the trajectories
under output feedback control converge to those under state feedback control as the

observer gain approaches infinity.

4.4 Tracking [28]

We show that a tracking problem can be viewed as asymptotic stability with respect
to a positively invariant and compact set. Then we give an example where the

separation results of the previous chapter apply.

103

Consider the globally defined single-input single-output system

Tl : f0(7li 113)
x = Acx + Bab—(£53m — a(x)]
y = x1 (4.32)

with n E R" _ T, b(x) 7E 0 for all x 6 RT, and (Ac, BC) defines a chain of integrators.

Remark 4.5 The above system is not the most general input-output linearizable
system because the nonlinearities a(.) and b(.) depend only on a part of the state,

namely x, and does not depend on 1}.

Let (4.32) satisfy the following input-to—state stability assumption:

Assumption 1: There exists a C1 function V1(n) such that

01(Ilnll) S V107) S a2(ll77ll) (4.33)
8V
371400747) 5 —a3(nnn) + aunt“) (4.34)
for all 17 E R” — T and all x 6 RT, where 612-, i = 1, - . - ,4 are class Koo functions.

We need the output y to asymptotically track the reference signal y R' We assume

(1) (r - 1) (7‘)

that y R1 y R , - - -, y R are bounded and continuous and y R is bounded and
piecewise continuous.
Set YR = (yR, yg),-~, yg _ 1)) and e = x — YR. The system (4.32) can be

written in the error coordinates as

7') = mm(3 + YR)

104

-——)[u — a(x)] — ygl} d§f g(e + YR, u, t)

61 = y—yR (4.35)

The objective is to make e approach zero asymptotically. Consider the state feedback
u* = a(e + YR) + yg)b(e + YR) + Ke where K is such that Ac + BCK is Hurwitz.

Then, using Theorem 5.2 of [28], we conclude that
”60)” s 71e‘72tlle(0)ll (4.36)

for all t _>_ 0, for some positive constants 71 and '72.

Let r0 = supt Z 0 ||YR(t)|], p0(.) = a3_1(20z4(.)), and c = a2(p0(2r0)). Notice
that p0 thus defined is a class lCoo function. Since V1(n) is continuous and radially

unbounded, the set

4={n=V1(n)Sc}

is a compact subset of R" - T. Thus, we conclude that the set A = {e = 0} x A is
a compact subset of R".
We recall from Exercise 3.33 of [28] that a class [C function a, defined on [a, 0),

satisfies
a(sl + 32) g 0(281) + a(232).

Given this result and (4.34), we can write

6V

1
firemen/R) s jagtnnm + 44(21th for mm c (4.37)

105

for all t 2 0. Inequality (4.37) can be written as

8V

. 1 ' -
time... + YR) _<_ 743017;“) + attend», v 77 ¢ A (4-38>

Now, define the function

[ln(1+ s -— c)]2 for s 2 c
0 for s E [0,c]

The function A(.) is a C1 function on [0,oo). It is also nonnegative and strictly
increasing for s 2 c. Furthermore, using L’Hopital’s rule of differentiation (to show

X(s) -—> 0 as s —> 00), we have
/\'(s) g k, k > 0 (4.40)

for all s _>_ 0.
Consider the Lyapunov function candidate W(n) = /\(V1(n)). The following is a

technical result needed in the forthcoming analysis:

Claim 1: The function W(n) is C1 and radially unbounded. Furthermore, there

exist three class ICoo functions 61, 62,and 63 such that W(n) and %%(Vl(n))a3(llnll)

can be bounded as follows

51(I'Ill4) S W(n) S 520mg) (4-41)

5d—8(V1(n))03(|l77||) 2 5307M) (4-42)

for allne Rn—r.

Proof: The continuous differentiability of IV follows from that of V1 and x\. As for

106

(4.41) and (4.42), they follow from Lemma A.3.<1

Given the results of Claim 1, we can write

8W

E—f0(n,e + YR) S ‘63(l7l],§) + ka4(2llell)a V 77 ¢ A (4'43)

where k is defined by (4.40). Let p1(s) = 62(63_1(2ka4(2s))). Then, we can write

0W

Wf0(n,e + YR) 3 4641414), v n t 4, and Mn) 2 p1(||e||) (4.44)

for all t Z 0. Now, according to [48, Lemma 2.6], there exist a class ICE function 51

and a class KI function 7 such that, for every 77(0) E R” — r, we have

|n(t)|4 S 51(|n(0)|4.t) + 7(0 <Sup< ||€(T)|l) (4-45)

_r_t

By Lemma A.2 we conclude that there exists a class [CL function fl(., .) such that

|(€(t),n(t))lA S fi(|(6(0).77(0))|,4.t). W Z 0 (446)

Thus, using Proposition 2.5 of [34], we conclude that the closed-loop system is

asymptotically stable with respect to the compact set A.

The separation results apply to the system

7'7 = f0(77. e + YR)

. 1
8 = AC€+BC{WIE)

61 = y—yR (4.47)

[a — am] — 15:1} déf g(e + YR, t, t)

107

This systems fits into the model (3.11)—(3.14) with the vector of bounded exogenous
signals being (YR(t), yg)(t)).

The state feedback controller considered is u = u*. It achieves global asymp—
totic tracking uniformly in 1). Global boundedness of the control law is achieved by
saturation outside a region of interest.

We showed that the system (4.47 ) with the controller u = u* is globally asymp-
totically stable with respect to the compact positively invariant set A.

To implement the controller we use a nonlinear high-gain observer of the form
(3.17). Since we know the nonlinearities a(x) and b(x), we use them as the nominal
models.

Theorems 3.4 and 3.5 guarantee that the trajectories of the closed-loop system
under output feedback controller starting in 8 x Q, where S is a compact subset of
R" and Q is a compact set in RT, are bounded and will come arbitrarily close to
the set A x {e f = 0} where e f is the estimation error. Furthermore, Theorem 3.7
(no modeling error) guarantees local asymptotic stability of the compact positively
invariant set A x {e = 0}.

Finally, Theorem 3.6 shows that the trajectories under output feedback control
converge to those under state feedback control as the observer gain approaches in-

finity.

4.5 Servomechanism [26, 37 , 38]

The forthcoming example presents regional results on servomechanism. It is based
mainly on [38] which is a generalization of [37]. Both of these papers are continuation

of the work that was started in [26].

108

We consider a SISO system represented by the n-th order differential equation

y(71): f(y,...,y(n _1),u,...,u(m _1),w(t))
"—1),u,-~,u(m _1),w(t))u(m) (4.48)
where u is the control input, y is the measured output, um and ya) denote the i-th
derivatives of u and y, respectively, and w(t) is a continuous time-varying disturbance
signal which is assumed to belong to a compact set D C RI). The functions f and g
are (sufficiently) smooth nonlinearities with g(.) 76 0 for all values of its arguments in

the domain of interest U x D Q R" + m x RP. Let r(t) be a time-varying reference

(d(t)
signal and assume that V(t) = is generated by the pl-dimensional exosystem

1"(t)

b(t) = Sol/(t) (4.49)
where 50 has distinct eigenvalues on the imaginary axis. Clearly, V(t) belongs to a
compact set D1 C Rp 1, Vt Z 0.

We augment the system with a series of integrators at the input side and view

v = ”(m - 1) as the control input of the extended system. Taking

xi = g(z—1)—r(z—1),z:1,.n,n

CZ- = u(z—1),i=1,...’m
as state variables, the extended system can be represented as

i‘ = A$+Blf($,C,V(t))+9($,C,V(t))’0l
C = A2C+Bzv

e = Cx (4.50)

109

where e = y — r is the tracking error, and (A,B,C) and (.42,82) are chains of
integrators.
The following assumption ensures the existence of a change of variables such that

(4.50) can be written in the normal form

x = Ax + B[f(x,C,1/) +g(x,C,1/)v]

2 = w0(x,z,u)
e = Cx (4.51)

Assumption 1 [38, Assumption 1]: There exists a difieomorphism

a: = a: dgf T($1C1 V) : ‘IH/(Y, C) (4'52)

2 T1(27.C.V)

that maps (x, C) into (:13, z) for all V 6 D1 and transforms the last m state equations

of (4.50) into

2 = z/J0(x, z, z/(t))

which, together with the first it state equations of (4.50), define a normal form which

we assume to be defined in the domain D déf T(U x D) Q R" + m.

To specify the domains of validity of the above transformation, we assume ([38,
Assumption 3]) that there exists a domain N = N1 x N2 C R" x Rm, that contains

the origin, such that \Il;l(N) Q U for all I! 6 D1.

110

The objective is to design a state feedback controller that makes the output y
asymptotically track the reference signal r. We start by identifying the internal model
of the system, then we design a dynamic state feedback controller that incorporates
this model.

Of particular interest to us are the equations of motion on the zero-error manifold

(where e = 0 which implies x = 0). They are

 

f(0, at), m» + go, at). u(t>)v = 0 (4.53)
' _ -f(01 C1 ”U”
C ‘ ”42¢ + B2 l g(o. c. to») l “'54)

In the forthcoming development we will use a state feedback controller of the form

t = gale, c. tem-foe. c. V(t)) + t]

where gO(.) and f0(.) are nominal models of f () and g(.), respectively.

The following assumption identifies the internal model:

Assumption 2 [38, Assumption 2, Remark 3]: (1) There exists a unique mapping

C = A0(V) which solves the partial difi'erential equation

 

8A0 ]—f(01 <1 V(t))]

——SI/=A/\ u+B 4.55

61/ 0 2 0( ) 2 9(0.C.V(t)) ( )
(2) There exist q constants d1,---,dq such that, on the zero-error manifold, the

control component 17 is

to») 3:3 —gO(o. A000, v)g‘1(0.40(v).v)f(0./\o(v).v) + foe, 103), t)

111

and satisfies

Lgcu) = 4150/) + d2L36(z/) + - . . + 4,13 ‘ 1c(1/) (4.56)

for all V 6 D1, where L3C = $5012. Moreover, the polynomial equation
sq—dqsq_1—---—d2s—d1 =0

has distinct roots on the imaginary axis.
A routine calculation shows that there exist a q x q matrix S, q 2 p1, and a 1 x q

constant matrix F such that

6’52”) = SV(u) (4.57)

 

C(11) = FV(V) (4.58)

for all u 6 D1. In fact this happens for

 

 

 

( 0 1 0 0)
0 o 1 o
S =
0 o 0 1
(d1 d2 d3 dq]
( C(u) )
L3c(u)
1’0!) =
[ft—25(5)
(Lg-15(5) )

 

112

I‘=(100-~0)

Assumption 2 means that S has distinct eigenvalues on the imaginary axis.

Remark 4.6 If on the zero-error manifold we have C = /\0(V), then A(V) diff

T1(0, A0, V) is a well defined function of V.

Now, we proceed with the design of the dynamic state feedback controller. First,
we need a minimum-phase assumption. Before we state such an assumption, we give

the dynamics satisfied by x and 2 = 2 — /\(V) :

x : Ax + B[f(x,C, V) +g(x,C,V)v]

2 = 40(3, §,V) (4.59)
wheregb ()=1/2( 7 A _8A
0 . x, 2 + (V), V) 63.90%

Assumption 3 [38, Assumption 4]: There exists a C1 function W defined on Rm,

and four class IC functions 011,012, (131, and a, such that

l/\

a1(llill)

8W - ~ ~ ~
13—2'¢o(:t.z.v) s —¢1<Hz|l>tVIIZIIZGUIZ“)

W(E) S a2(||$|l)

for all (x,z,V) 6 N1 x N2 x D1.

Set Qa2 dgf {z : W(2) S a2}, a2 > 0.

Next, we consider a dynamic state feedback of the form

('7 = So+Je (4.60)

113

’U : (p(xiai Ca V) (4'61)

where p(x,o,C,V) is locally Lipschitz in (x,o) uniformly in (C,V). Then, [38, As-
sumptions 6,7,8, Lemma 1] ensure that the controller achieves convergence to the
zero-error manifold A = {x = 0} x {a = LV(V), 2 = A(V)} for some constant matrix
L. This manifold is compact, invariant, and asymptotically attractive uniformly in
V 6 D1. Hereafter, for the sake of completeness, we state these assumptions and
lemma.

The control strategy consists of two steps. First, the controller ensures ultimate
boundedness of trajectories. Then, it achieves asymptotic convergence to an equilib-

rium point.
a a . c
To state the next assumptlon let C = and C = h1(C, C, V), 1.e., part of the

dynamics of the closed-loop system.

Assumption 4 [38, Assumption 6]: There exists a C1 function V : R" + q —> R4.

which satisfies

510%”) S V(E) S 420%”)

W —¢2(ll€ll). we) 2 4(4)

55,11 (E1 Ca V)

|/\

for all C 6 X1, uniformly in (C, V) for all (C, V) 6 U2 x D1, where fi, 21, 52 and (152

are class [C functions and u > 0 is a design parameter.

Define X1 = Rq x N1 and Cal 2 {C : V(C) 3 a1} where a1 > 0 is chosen
such that Cal Q X1. It can be shown that the set 5201 x Qa2 is positively
invariant for some a2 > 0. Assumption 4 along with Assumption 2 imply that the

trajectory (C(t),z(t)), starting in Cal x Qa2 will, for all V 6 D1, eventually enter

114

the positively invariant set R7) dgf A” x Flt where A” déf {C 6 X1 : V(C) g (301)}

and r“ déf {z

: W(2) g 7(a)} for some class [C function 7. The set R” is a
neighborhood of (C, 2) = 0 whose size can be made arbitrarily small by choosing a

small enough.

Assumption 5 [38, Assumption 7]: There exists a compact, positively invariant
set S” Q A” such that C (t) enters S” in finite time. Furthermore, inside S”, the

control component 17 takes the form

77 = K06 + f2(C.l/)

where f2 satisfies f2(/\0(V),l/) = L2V(V).

Lemma 4.1 [38, Lemma 1] Suppose that Assumptions 2 and 5 hold. Then, there

exists a q x q matrix L such
I‘ = —(K01L + L2), LS = SL

Furthermore, the set {a = LV(V), x = 0, C = /\0(V)} is an integral manifold of the
closed-loop system (4.50) and (4.60)—(4.61).

Now, we need to design a state feedback controller (choose K0 and f2) such that
the zero-error manifold {o = LV(V), x = 0, C = A0(V)} is regionally attractive. To
a

study this attractiveness, we set 6 = o -— LV(V) and 17 = x . With this change of

~

2

 

 

variables, the zero-error manifold reduces to the origin 17 =- 0 which is an equilibrium
point of the closed-loop system (4.59) and (4.60)—(4.61). The next assumption is

needed to show attractiveness of the origin 7'] = 0:

115

Assumption 6 [38, Assumption 8]: The origin of the closed-loop system

~

17 = h1(77, V) is locally asymptotically stable.

The theory developed in the previous chapter can be applied to the autonomous

system

i: : A$+B[f($,C,V)+g($,C,I/)’U]

C = A2C+BQU

e = Cx (4.62)

This system fits the model (3.11)—(3.14) with Ci’s, being the additional measured
outputs C. The smoothness of f and g inherited from f and g implies a local Lipschitz
property in (x, C) uniformly in V 6 D1-

We consider the state feedback controller

(7 = So+Je

v = cp(x,o,C,V)

Global boundedness of the control law is achieved by saturation outside a region of
interest.

We have shown that the system with this controller is asymptotically stable with
respect to the zero error manifold A dgf {o = LV(V), x = 0, C = /\0(V)} with
Qal x Qag being an estimate of the region of attraction. Lemma 4.1 shows that A

is a center manifold for the system (4.62) with the controller. Since V E D1, then A

is a compact subset of R" + 2‘1 + m.

116

To implement the controller we use a linear high-gain observer to estimate the
tracking error and its derivatives (let the estimate be it). Let x be the scaled estima-
tion error. Then, the closed-loop system under the output feedback controller can

be written compactly as

Q:

77 = h3(77.X. V)

6X 2 A1X + €B[f1 (x, 2, V) + 91(x, 2, V)tp(o,x, 2, V)]

= A1X+elt2(n,x.V) , (4.63)

Theorem 3.11 guarantees that the trajectories of the closed-loop system under
output feedback control (starting in the compact set Qal x Qaz x Q, where Q is a
compact subset of R") are bounded and come arbitrarily close to the set A x {x—x =
0}. Moreover, Theorem 3.11 shows that the trajectories under output feedback
control converge to those under state feedback control as the observer gain approaches
infinity.

To establish asymptotic convergence of the closed-loop system (4.63) to the
equilibrium point (17, x) = (0,0) some conditions on the nonlinearities have to be

imposed. These conditions are given in the following assumption:

Assumption 7 [38, Assumption 9]: There exists a C1 function V(n)

R" + q + m —> R4. and a continuous positive definite function (133 such that

5117~

“71207.0. mu 5 4143(4). 412 o (4.65)
317 - ..
5114mm) -h3(n.0.V)l s 4243(6)”qu (4.66)

0<a51/2,0<b<1,c=1—;'—’ andqz20,forall(C,z,x,V)ESflxI‘HXQEXDl.

117

Note that inequality (4.64) of Assumption 7 follows from Assumption 6. Assump-
tion 7 provides a Lyapunov function for which Assumption 3.4 is satisfied. Both V
and (1)3 are positive definite with respect to A. Thus, by Lemma A.3 the bound
(3.57) is satisfied. Inequalities (4.64)—(4.65) are similar to (3.58)—(3.59). Finally,
inequality (4.66) is a variation of (3.60) and yields the same results. Then, according
to Theorem 3.11, the system (4.63) is asymptotically stable with respect to the set
A X {x — x = 0}.

Remark 4.7 Notice from (4.65) that the system’s nonlinearity h2(17,0,V) has to
be zero on the zero-error manifold. Therefore, it is very convenient to Choose the
nominal value of h2(r7,0, V) to be zero (which results in a linear observer), because
otherwise we have to know the manifold and we have to choose a nominal value which

is zero on this manifold.

4.6 Servomechanism [21]

The servomechanism example considered herein deals with a system that has a trian-
gular structure as opposed to the system considered in the previous example which
has a double chain of integrators. Moreover, no assumption of complete observability
(as defined in Section 4.3) is given, thus we can not proceed, by repeated differenti-
ation of the output, to find the input-output model and continue the design process
as in the previous example (complete observability is needed for the existence of an
input-output model as shown in Section 4.3). Even if the system is completely ob-
servable, the design presented in this section will, in general, be different from the
one presented in the previous section because the nonlinearities in the input—output
model could be different from the nonlinearities in the state model due to partial

differentiation while deriving the input—output model.

118

Consider the system

73 = Z(#)Z+P0($1,w,#)
x = Fx+Gu+P(z,x,w,/i)

e = Hx (4.67)

where

(010...0\ (0)
001-0 0
F: ,0:
000---1 0
000m0) (1)

 

 

 

 

0)

p1(z,$1.w. u) l

E
||
\A/
H
O
O

102(2. 4142,41, it)
P(2,x,w,u) =

pT—1(Z?x17$27°°°1xr _1,w,/J)

 

 

K pT(zax11$21”°1$T1w1#) )

with state x E R", control input u E Rm, and regulated output e E Rm. The
system (4.67) is subject to an exogenous input to 6 Rd in which 71 E ’P 6 Rp is a
vector of unknown parameters and ’P is compact set, p0(.) and P(.) are Ck functions
of their arguments (for some large k), and p0(0, 0, 71) = 0, P(O, 0, 0, u) = 0. Without

loss of generality we assume 0 E int(’P). The exosystem

d) 2 Std (4.68)

119

is assumed to be neutrally stable (the matrix S has distinct eigenvalues on the imag-
inary axis).
Assumption A: The eigenvalues of Z (11) have negative real part, for all u E P.

Moreover, the equation

06(w, u) 3

8w w = Z(u)€(w.u) +po(0.w. M) (4-69)

has a solution C(w, 41) defined for all a), 71.

Given Assumption A and the structure of F, G, H and P(.), a routine calculation

shows that the system

a((, na)(w,u)Sw = Z(#)C(w, It) +po(H7ra(w.u).w./t)
8w

 

F6610”: 11') + Gca(w, II) + P(C(w1.u)a7ra(w1#)1w1l1)
0 = H7ra(w, u) (4.70)

has a unique and globally defined solution «0(6), p), ca(w, 711) such that 7r“(0, u) = 0
and ca(0, u) = 0 for all 71. Hereafter, it is assumed that the function ca(w, 71) thus

determined satisfies the following:

Assumption B: For some set of real numbers a0, a1, - - ~ , aq __ 1, the identity
— 1
Lgcaw. H) = aoca(w. .u) + alLsca(w. M) + - - - + aq _ 1L3 ca(w, u) (4-71)
holds for all w, u, where L; = gag-ISM. Moreover, the polynomial equation

sq—aq_lsq_1—---—als—a0=0

120

has distinct roots on the imaginary axis.

Simple routine calculations Show that, under Assumption B, there exist a q x q

matrix (I), a 1 x q row vector I‘, and a globally defined mapping ra(w, a) such that

 

 

a
878(511‘1“) : (1),/410%”)
61(4),”) = Frattan) (4-72)
In fact, this happens for
(0 1 0 0 \
0 0 1 0
(I) :
0 0 0 1
{a0 a1 a2 aq_1]
{ ca(w,p) \
Lsca(w1#)
7110”,”) Z

— 2
L3 61(4),”)

\ List Icahn/1) ]

1“=(100~-0)

By Theorem 1 in [21] and using Assumptions A and B we conclude that the problem

 

 

of structurally stable output regulation for the system (4.67) is solvable.

Hereafter, we propose a feedback law for which we prove the existence of an

attractive zero-error invariant manifold. Furthermore, this manifold can be made

121

semiglobally attractive.

Set 2 = 2 — C(w,u), ii: = x — na(w,p), and

(.)(,,\( ., 1

(1) ~ ~ ~ ~
6 n x +19 ( .3: MM)
,7: . = .2 = 2 1 . 1 (4.73)

 

 

 

 

 

 

\e(T—1)/ \Ur} Kjar—FIST—1(21j11°”1j7‘—11w1#)f

Consider now a feedback law of the form

£1 = <I>€1 + Ne

u = M77+T€1 (4.74)

Then, it is possible to prove the following property:

Proposition 1 : Suppose Assumptions A and B hold. Suppose that (4.74)
asymptotically stabilizes the linear approximation of (4.67) at the equilibrium point

(C1, z,x) = (0,0,0), (w, u) = (0,0). Then, there exists a q x q matrix 11 satisfying
<I>II 2 11(1), TH = F (4.75)
where <1) and F are defined as in (4.72). As a consequence, the closed-loop system

61 = <I>C1+NHx

i = Z(u)2+p0($1.w.tt)
x = Fx+C(M17+TC1)+P(z,x,w,/i)

d) 2 SC) (4.76)

122

has a globally defined center manifold

Mo = {(61.Z.Ivtw) =61 = mam“), Z = ((0141). 3? = 7r“(66ml (4-77)

at (C1,2,x,w) = (0,0,0,0).
Proof. similar to that of Proposition 1 in [21], which applies to the output feedback

case.<1

Now, let us design the state feedback controller that makes MC semiglobally
attractive. The issue here is to choose N, T and M such that this goal is achieved.

Let C1 = C1 — Hra(w, 71). Then, in the coordinates (2, 2, C1), the closed-loop system

becomes
51 = 451 + NHx
3' = Z (142 +130(Hi.exp(3t)w0,u)
:if = F5: + C(Mn + TC~1) + P(z, x, exp(St)w0, a) (4.78)
where (.10 represents the value at time t = 0 of the state of the exosystem. System

0 are unknown. We

(4.78) is an uncertain system because the actual values of u and w
assume that the initial value 620 belongs to an a priori known compact set W 6 Rd.
The invariant manifold reduces to the origin (C1, 2, 2) = (0,0, 0) where the regulation
error e = 5:1 is zero. Thus, output regulation is achieved if the origin is attractive.

In order to be able to use the separation results of Chapter 3, Assumption C of

[21] has to be modified as follows:

123

Assumption C: There exists a positive definite smooth function V(2) satisfying

a1llil|2 5 WE) s 02||5||2 (4.79)
6V

g(zflfli+P0(Hi,eXp(St)thu)) < —C¥3]]2]]2+C]H.’L‘]2 (4.80)

for all 2, :Z‘,t and all ((120, a) E W x P, where 612- ’s are positive constants and c > 0.

For N choose any matrix such that the pair (Q, N) is controllable. Then, given
any compact set S of initial conditions (C1(0),2(0),:Z:(0)) 6 R4 x R" T T x RT,
find (via backstepping methods and high-gain feedback, for example) a pair of
matrices M and T such that the origin is locally exponentially stable with a basin

of attraction that includes the set S.

In order to apply our separation results, we consider the system

i = Z(V)Z+po($1.w.u)
7'7 = An+G(U+1fir(2.n.w.u))
(.2) = Sw

e=H17

This system fits the model (3.11)—(3.14) with a being the vector of bounded distur-
bances (constant in this case, thus it belongs to Mb).

We consider the state feedback controller.

51 = <I>€1 + Ne

11 = M77+TC1

124

This controller achieves semiglobal tracking uniformly in w and a. Global bounded-
ness is achieved by saturation outside a region of interest.

We showed, by construction, that the system with the controller is exponentially
stable with respect to the compact positively invariant zero-error manifold MC with
8 being an estimate of the region of attraction.

To implement the controller we use a linear high-gain observer. Boundedness,
ultimate boundedness, and convergence of trajectories under the output feedback
controller (starting in 8 x Q, where Q is a compact subset of R") are guaranteed
by Theorem 3.11. Moreover, Theorem 3.11 guarantees exponential stability with
respect to the compact positively invariant set MC x {17 — C0 = 0}, where C0 is the

estimate of 17.

4.7 Adaptive Control [27 , 2, 1]

We consider the system represented globally by the n-th order differential equation

g(nl = f0(.) + 2?: 1f,-(.).9,- + [gO(.) + 2;”: 1gi(.)02-]u(m) (4.81)

where u is the control input, y is the measured output, y(z) denotes the i-th derivative
of y, and m < n. The functions f2- and 91 are known smooth nonlinearities which
may depend on the output, the input, and their derivatives up to the (n —1)-th order
and (m — 1)-th order, respectively. The constant parameters are unknown, but the
vector 6 = [01, - - - , 9p]T belongs to Q, a known compact convex subset of RP.

The objective is to design a state feedback controller that renders all the signals
bounded and makes the output asymptotically track the bounded reference signal
y7-(t). We assume that all derivatives of yr up to the n-th order are bounded and

that y,(~n) (t) is piecewise continuous.

125

Letxi=y(i"1),ei=xi—y,(-z_1)fori=1,---,p;2i=u(i_1),i=1,---,m,
and v = W " 1). Let yr = (4.4),...4)" ‘ ”(017" and 342 = lyr,y(n)(t)lT-
To ensure that the system is input/output linearizable we make the following

assumption:

Assumption 1: [g0(x,2) + 0Tg(x,2)| Z k > 0, Vx E R", 2 E Rm and 0 6 521,

where 91 is a compact set that contains (2 in its interior.

We assume that there is a global change of variables C =
T1(y,-~,y("—1),u,~-,u(m_1)) such that the system (4.81) can be writ-

ten

5 = Ame + b(Ke + f0(e + yr, 2) + 6Tf(e + yr. 2)
+ [90(6 + yr, 2) + 0Tg(e + yr, 2)]v - y,(~n)} (4.82)

where Am 2 A — bK, (A,b) is a chain of integrators, and K is such that Am is
Hurwitz.
Before designing the controller, we have to make a crucial minimum-phase

assumption:

Assumption 2: The system C = F (g,y,~,0) has a unique bounded steady-state

solution C. Moreover, with C = C — C the system

C : F(€+C,€+yr,0)—F(C,y,0)

: F2(<~16+y’r1616)

126

has a continuously differentiable function V1(t, C), possibly dependent on 6, that sat-

is fies

Tulléll2 S V1085) S 772M“2

6V1 av, - 2 ~
3,: 35F“ ”nsllCH +774||C||||€l|

|/\

where 171,172, 173 > 0, and 174 2 0 are independent of 32,, C, and 6.

The state feedback controller is designed to adaptively cancel the nonlinearities

and stabilize the system (4.82). It consists of the dynamic controller

6 = Fp(6+6,e,z,yR) (4.84)

v = 1(1(6+ 6,6, 2,)7R) (4.85)
where 6 = 6 — 6 (6 is the estimate of 6) and

--KC + y7(~n) — f0(e +yr,Z) — 6Tf(e+yr,z)
g0(e + yr, 2) + 6Tg(e + Yr, 2)

 

«(4.) = (4.86)

The functions PP and w are locally Lipschitz in their arguments uniformly in y R
and 6. The adaptive law is a projection-type law (for more details see [27]).
Consider the Lyapunov function candidate V = eTPe + é-6F—16, where P is
the positive definite solution of the Lyapunov equation PAm + AgP = —Q, where
Q = QT > 0 and F—1 is a positive definite matrix. The derivative of V along the

trajectories of the closed-loop system (4.82)—(4.86) is

v = —eToe + 871—1739 — r3] (4.87)

127

where
(73(e, z, 3213,61) = 2eTPb[f(e + yr, 2) + g(e + yr, 2)w(e, 2, 32R, 6)]
The adaptive law is chosen to ensure that
éTr”1[é— 1‘6] 3 0 (4.88)

and 6(t) E (26 for all t 2 0 and all 6(0) 6 Q, where 96 is a compact set chosen
such that Q C {25 C S21. Inequality (4.88) ensures that V S —eTQe. Therefore,
all signals are bounded for all t Z 0. Since 32,. is bounded, we conclude that x(t) is
bounded, which implies, in view of assumption 2, that z(t) is bounded. By using [28,

Theorem 4.4] we can conclude that
e(t) —> 0 as t —1 oo (4.89)

To discuss parameter convergence let us define the regressor vector wr (t) as
W(t) = f(yr, 5) + g(yrt Ell/40121371210) (490)

where 2 is the steady-state solution of the zero dynamics, determined uniquely from

C- : T1(y7'1 Z)‘
Assumption 3: The regressor vector is persistently exciting.

Let
w(t) = f(e + yr, 2) + g(e + yr, z)1,b(e, 2, 3213, 6)

128

Then, the 62- and 6-dynamics can be written as

e = Ame—b6Twr+b6T(wr—w)

= PM)

@1-

Now let us work on (tar — 6)).

Claim: We can write

(9T

T9

wr-w=(f—f)+(§tl—gtl)+4_ . (491)
90-1-6

where

f(-) = f(yr. 2). s(-) = 906. 2)
f0 = foO’r. 2). tot) = 90m, 2). i4.) = 440. 2.32189)

Proof. Let "(h(.) = 1,12(0,2,31R,6). Then, we have

wr-w = (f+§t/3)-(f+gib)

(f—f)+(4t/3—gtt)+§(i—i)

The claim is proved if we show that

Q'T
= .. w
g0+0Tg T

— ~

(lb-111)

 

We have

it”) — 10 — 6T1" _ it”) - 10 - 17‘)?
£70 + 0T4 so + 6T4
6535—”) — f0 - 077) + 6117(4) + 0T5)
(90 + 9T§)(§0 + 6%)

$1
|
S1

 

 

 

129

Using the expression of w and car we can write

9T" ' 9T '
91/) + f

1/3—1/3 = -_—.——- ——.——
90+0Tg 90+0Tg

By using the above claim, the 62— and 6-dynamics can be written as

' A —b T A .
e = m go, 6 + 30 (4.92)

9 21:94)..pr 0 9 Ae(.)
where

400., 2) + 0740., 2)
4004, 2) + 9T6», 2)
4.4.) = (>610? —f)+(443—gt)1

Ae(.) = Fp(.)—2FQwaTPe

 

> Kg2

Since f, 901 g, and 1]) are Lipschitz functions in their arguments uniformly in 32. and

6 and since 6 is bounded, we have

IIAs(-)ll S 51||6|| +52||C~|| (4-93)
l|4e(-)ll S 53||€|| (4.94)
for some 6,- _>_ 0, i = 1, - --,3. Inequality (4.94) becomes clearer from the explicit

form of Fp(.) (see [27]).
From well known results in adaptive control theory (see for example [28, Section

1342]) and the fact that car is persistently exciting and Q is bounded, it can be

130

shown that the system

6 A —5ng e
L = m 7' ~ (4.95)
9 2Fgw7~bTP 0 9

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

Lyapunov function V2(t, e, 6) whose derivative along (4.92) satisfies

4119461)”? 5 V2(t.e.5) 5 82119.91? (496)

V2 < -64||e||2 - 6510112 + 651412 + 6714119)

 

+58||ellll5|l +59||5||||5ll (4.97)
(4.98)
6V -
“(W 2 ]] — /\3ll(e,9)ll2 (4.99)
a(e ,6)

for some positive constants 64, 65, and A7, i = 1,2,3, and some non-negative con-
stants 62', i = 7, - - -,9.
The derivative of V along the trajectories of the closed-100p system (4.82)—(4.86)

is
V _<_ —eTQe s —It1IIeII2 (4.100)

where k1 > 0. Consider the Lyapunov function candidate W(t, e, C, 6) = aV(e, 6) +
6V1 (t, C) + V2(t, e, 6). Then, using (4.92)—(4.94), (4.97), (4.100), and Assumption 2

it can be shown that the derivative of W along trajectories of the closed-loop system

131

(4.82)-- (4.86) satisfies

9 . T — -
Hell Hell
W s - uéu M uén <4-101>
_uc‘n , _ “in,

 

 

 

 

where M is given by

pak1+54—56 —§2,Z _18_774_2_+_‘§8 -

M = -222 55 _5

a a
_j—"Bam fig [3'73 ,

Choose 6 large enough to make

 

 

65 329

6
-129 6773
positive definite; then choose or large enough to make M positive definite. Therefore,

we conclude that (e = 0,6 = 0,6 = 0) is exponentially stable.

We consider the system

('2 = Ae+b{f0(e+yr,z)+6Tf(e+yr,z)
+ [90(6 + yr, 2) + 9T9(6 + yr, 2)]v - 9291)}

C : F2(€ay7'901€)

This system fits the model (3.11)—(3.14) with the vector of time-varying bounded
perturbations being d(t) = (yR(t),6,C—(t)) and the additional outputs C being the

input and (m - 1) of its derivatives; i.e., 2. Since we are dealing with a regional result

132

and in order to apply the separation results of Chapter 3, we need to restrict the

time-varying parameter d to belong to M5D. This can be guaranteed by assuming

that y)” +1)(t) exists and is continuous and bounded for all t 2 0 (notice that

((t) E M’D by Assumption 2).

We consider the state feedback controller.

Gr
||

Pp(é + 63 8, Z1 3’12)

’1) : ¢(5+9,8,2,yR)

This controller achieves exponential tracking of the reference signal with R" x Rm x
(25 being an estimate of the region of attraction. Global boundedness is achieved by
saturation of 1M.) and I‘p(.) outside a region of interest. 1

In the case at hand the compact positively invariant set A reduces to the origin
(e, 6, C) = 0 which is exponentially stable. Thus, we are dealing with a stabilization
problem of a time-varying system.

In [2] a case where only a part of the regressor vector is persistently exciting is
considered. This case does not fit into our formulation because we we do not have
stability with respect to a compact, positively invariant set.

Now, in order to implement the controller (4.84)—(4.85) and recover the perfor-
mance achieved under it, we use a linear high-gain observer to estimate the tracking
error and its derivatives.

Boundedness, ultimate boundednes, convergence of trajectories under output

feedback control, as well as local UES, are guaranteed by Theorem 3.11.

Remark 4.8 As in Remark 4.7 we choose the nominal value of the system ’3 non-

linearity 43 to be zero ( which results in a linear observer), because otherwise we have

 

1Actually, the saturation of l‘p(.) may not be needed since the projection-type
adaptive law guarantees boundedness of 6.

133

to know the steady state solution 2 and we have to choose a nominal value which is

zero at the steady state.

4.8 Conclusion

In this chapter we presented separation results for design cases Where the trajec-
tories converge to a compact, positively invariant set. For each of these cases, we
showed how the state feedback controller performance can be cast as convergence to
a compact, positively invariant set. Then, after suggesting an output feedback im-
plementation of the control law, we applied the previous chapter’s results and listed

the set of performance measures that can be recovered.

134

CHAPTER 5

Separation Results for the Control
of Nonlinear Systems Using
Different High-gain Observer

Designs

5.1 Introduction

In this chapter we are concerned with the separation approach to the design of stabi-
lizing output feedback control using high-gain observers. In the separation approach
the design is pursued in two steps. The first step focuses on the design of a stabiliz-
ing state feedback controller, while the second step is concerned with the design of
a high-gain observer that successfully provides state estimates such that the overall
closed-loop system is asymptotically stable.

High-gain observers are attractive because of their robustness, namely their ability
to estimate the unmeasured states while rejecting the effect of disturbances. The
available techniques for the design of high-gain observers can be classified into three

groups. First, pole-placement algorithms which lead to either a two-time scale

135

structure as in [11] or a multiple time-scale structure as in [46]. Second, Riccati
equation-based algorithms which lead to either an H2 algebraic Riccati equation
(ARE) as in [9] and [43, Section 4.4.1] or to an H00 ARE as in [42] and [43, Section
4.4.2]. Third, Lyapunov equation-based algorithm as in [17].

For linear time-invariant systems the recovery of asymptotic stability through
the use of high-gain observers is shown in [43], and references therein, in a Loop
Transfer Recover (LTR) context and in [42] in an H00 context. As for nonlinear
systems, Esfandiari and Khalil in [11] and [30] use pole placement/singular pertur-
bation to design a one-parameter observer gain and recover the robustness properties
of a controller designed to stabilize a fully linearizable system. This design results
in a standard two-time scale singularly perturbed system. In [46] Saberi and San-
nuti design a multiple-parameter observer to recover the global stabilizability of an
uncertain linear system. This design results in a standard multiple time-scale singu-
larly perturbed system. Tornambe in [52] uses a similar pole placement technique
to recover local asymptotic stability for a class of input-output linearizable systems.
Nicosia and Tornambe in [40] use singular perturbations to recover local asymptotic
stability for the case of a robot with elastic joints. Teel and Praly [49] combine ideas
from [11] and [52] to achieve semiglobal stabilization for a wide class of nonlinear
systems. Isidori in [21] shows that his previously proposed solution for the general
structurally stable regulation problem [22] can be coupled with ideas from [11] to
solve a problem of robust semiglobal output regulation. In [7] Busawon et at present
local and global separation results for a class of nonlinear systems using a high-gain
observer designed with a Lyapunov equation-based algorithm.

A characteristic of a high-gain observer is the peaking phenomenon of its transient
response. This phenomenon is examined carefully in [11]. Peaking occurs in the
observer variables and propagates to the state variables through the control law.

This peaking could be destabilizing in the case of a finite region of attraction, which

136

only allows recovery of local asymptotic stability as in [52], [40], and [7]. However,
despite peaking, global asymptotic stability results can be obtained at the expense
of imposing restrictive global Lipschitz conditions, as is the case of [46] and [7], and
tolerating a clearly unacceptable transient response. To remedy this problem the
idea of saturating the control law outside a region of interest was introduced in [11].
It is this saturation feature that leads to the semiglobal results of [11, 30, 49, 21] and
the regional results of Chapter 2 and 3.

Aside from some special cases like the case of relative degree-one systems, all
the different ideas to design high-gain observers boil down to various asymptotic
methods to approximate the derivatives of the outputs. In [25] Khalil illustrates this
observation through an example and in this chapter we prove it rigorously.

In this chapter we show that separation results, similar to those of Chapters 2
and 3, can be obtained if the other available algorithms for observer gain design
are applied in combination with global boundedness of the control law. Section 5.2
describes the different algorithms that can be used to design the gain of a high-gain
observer and shows that in all of them the gain asymptotically matches the gain
structure used in Chapters 2 and 3. Section 5.3 argues the separation results of

Chapters 2 and 3 when we use alternative observer gain designs.

5.2 High-gain Observers - A Comparative Study

We consider the system represented by (2.1)—(2.4). The estimation error dynamics

for the subsystem (2.1) can be written as

e = (A — HC)e + B(i(2:,:i:, z,u) (5.1)

137

where 6(.) = ¢(:z:,z,u) - ¢0(i:,C,u). We need to design an observer gain H that
stabilizes (A — H C) while rejecting the effect of the disturbance 6(..) This is achieved
if we could design H such that the transfer function between the disturbance input
and the estimation error (31 — A + H C')-lB is identically zero or arbitrarily close to
zero. We design H as a function of a parameter c or a set of parameters 62‘, i = 1, - - - ,p
such that this transfer function approaches zero as e or ei’s tend to zero. We present
three different approaches to the design of H and show that in all three cases the

gain H has approximately the structure (2.10).

5.2.1 Pole Placement / Time-structure Assignment

In pole placement we assign one or multiple time-scale eigenstructures to the ob-
server matrix and ultimately make the closed-loop system under output feedback
transformable into a standard singularly perturbed system. This is the approach
used in Chapter 2. A detailed exposé of this technique as applied to full or reduced

order observers can be found in [43, Section 4.3], [11], and Chapter 2.

5.2.2 Riccati Equation-Based Algorithms

Optimization-based techniques to design the high-gain observer can be reached
through two paths: Loop Transfer Recovery and disturbance attenuation. Both

can be applied to general stabilizable and detectable linear time-invariant systems

2': = A$+Bu (5.2)

y 2 Ca: (5.3)

The LTR algorithm consists of two steps. First, design a state feedback controller
u 2 F2: to shape the loop transfer function as desired. For example, by breaking

the feedback loop at the input side the 100p transfer function is given by T(s) =

138

F (31 — A)_lB. Second, design a Luenberger observer to estimate the state a: by :6
and implement the feedback control u = F 1%.

The error E between the target loop transfer function and the realized one is
given by E(s) = M(s)[I + M(s)]—1(I + F(sI — A)_lB), see [43, Lemma 2.2.1],
where M(s) : F(sI — A + HC)_lB. It is also shown that, for all 0 3 led] < 00,
E (jw) = 0 if and only if M ( jw) = 0. The observer gain can be designed to exactly
make E (s) = 0 or to depend on a small positive parameter p; i.e. H = H (,u), such
that the loop transfer function under output feedback control will approach that
under state feedback control asymptotically as u tends to zero; i.e. to asymptotically
make E(s) tend to zero.

The observer design for asymptotic LTR can be achieved by pole placement or
optimization-based methods. Pole placement is discussed in the previous section. In
optimization-based algorithms the objective is to find a gain H (u) that asymptoti-
cally minimizes either the H2 or the H00 norm of M (s). In other words, let M (s, ,u)
denote the matrix M (s) with H substituted by the designed H (u), then it can be
shown, [43, Theorem 4.4.2], that ||M(s,u)|| —+ inf||M(s)|| as it tends to zero. A
historical survey as well as clear explanation of this approach can be found in [43,
Section 4.4]. Basically, the idea is to solve a standard H2 or H00 control problem

for the following auxiliary system

:i: = ATz + CTu + FTw (5.4)
y = :1: (5.5)
z = BT22 (5.6)

The standard H2 or H00 control consists of determining the control gain H T to
minimize the H2 or H00 norm of the transfer function between the disturbance

input w and the controlled output z over the set of all possible gains while rendering

139

the matrix (A — HC) asymptotically stable. This transfer function is equal to
M T(s). The infinimum of its H2 or H00 norm over all possible gains is equal to zero
for a minimum-phase left-invertible system for any loop gain F. It is worth noting
that these LTR techniques work for general stabilizable and detectable systems; i.e,
not necessarily minimum-phase nor left-invertible, but “M(s, a)” is only guaranteed
to converge to some finite value, except for some cases where F satisfies certain

conditions.

It is shown in [43, Section 4.4.1] and references therein that the H2 optimization-
based technique yields, for a minimum-phase left-invertible system, a standard alge-

braic Riccati equation of the form

1
AP + PAT — —2PCTCP + BBT = 0 (5.7)
u

The observer gain is H2(u) = (1 / #2)P2 (,u)CT where P2(,u) is the unique positive
definite solution of (5.7) that makes the matrix [A — H2(u)C] asymptotically stable.

The H00 optimization yields an H00 algebraic Riccati equation of the form

AP + PAT — E12—PCTCP + $§PFTFP + BBT + [121 = o (5.8)
The observer gain is Hoo(u) = (1 / u2)Poo(u)CT where P0001) is the unique positive
definite solution of (5.8) that makes the matrix (A — Hoo(,u)C') asymptotically
stable. According to [56], this solution exists for an appropriately large 7 and
sufficiently small a (when the system (A, B, C) is stabilizable and detectable). Let
7* denote the infinimum of ||M(s)||oo over all possible gains. Then, equation (5.8) is
actually solvable for all 7 > 7* and every 0 < u < u* where p* depends on 7. More-

over, it can be shown, see [56], that ”M(s, 7)||oo < 7 for 7 > 7* (in our case 7* = 0).

140

The disturbance attenuation algorithm for observer design is detailed in [42],
where it is applied to a minimum-phase left-invertible linear system. It is based on a
parameterized H00 algebraic Riccati equation for a dual system. In our case it can

be applied to the system

:i: = A1: + 860(1):, C, u) + B6(a:, is, z, u) (5.9)
y = C1: (5-10)
z = a: (5.11)

considering 450 as the control input, 6(.) as the disturbance input, and 2 as the
controlled output. This methods yields, after some scaling, an H00 Algebraic

Riccati Equation similar to (5.8).

Now, let us go back to our problem of designing the gain H and apply the
optimization-based techniques suggested by the LTR theory. The observer design
problem formulated in (5.1), can fit into a Loop Transfer Recovery scheme if we
consider d>(.) to be the control input and F = I to be the controller gain. Thus, in
this case we have M (s) = (31 — A + H C)_1B. The problem is solvable since (A, B)
and (A, C) are controllable and observable pairs, respectively (left invertibility and
minimum-phase are implied by the structure of (A, B, C ))

The structure of the stabilizing solution of a general algebraic Riccati equation as
well as the eigenstructure of the observer matrix have been investigated in [47]. Here,
we apply these results to our case of interest; namely, when the triplet (A, B, C) rep-
resents a set of chains of integrators and F = I . In Appendix B we establish similar
results for the H00 algebraic Riccati equation (5.8). Basically the same structure

in (B.18)—(B.19), obtained in Appendix B, applies to both types of equations. The

141

observer gain is given by H = JZPOOCT = block diag[H1, - - - , Hp]. It can be shown
u

that, for i = 1, - - ~ ,p, we have

[ (az,+0(.,))/., .

of e- 62
(2+é( 2))/ z (5.12)

 

 

_ (at, + weave? ,

To apply the analysis of Chapter 2 we have to transform the closed-loop system
into a standard singularly perturbed form similar to (2.11)—(2.14) by scaling the

estimation error. The scaling turns out to be

:i: = $-S(€)77, where

S = diag[S'1,---,SK], S,- = diag[e:i _ 1,---,e,-, 1] (5.13)

This scaling, when applied to the observer (2.9) with the gain H just calculated,

yields
S_1(A — HC)S = 5‘1r + s—1A.R(s) (5.14)
where

5 = diag[6117~1, ' ' ' , érpI'rp] (5.15)

3(5) = diagloklllrlw“a0(€p)1rp)

142

 

 

 

 

_azl 1 0
-a22 0 1 0
F = block diag[l‘1,-~,I‘p], 11,:
—ai~,,-—1 0 1
b —a;.z. O-TZ'XTZ-
'1 0 0
1 O 0 0
A = block diag[A1,---,Ap], Ai=
1 0 0
_1 O‘Tz'XTi

and I‘ is nonsingular. This result yields a multiple time-scale singularly perturbed

closed loop system similar to (2.11)—(2.14) with the estimation error equation given

by

57'] 2 Fr) + 88601:, 2, 6,D(8)17) + A.R(£)17 (5.16)

In Section 5.4 we show how the results of Chapter 2 can be extended to the closed-

loop system formed of (2.11)—(2.13) and (5.16).

143

5.2.3 Lyapunov Equation-Based Algorithm

In [17] Gauthier, Hammouri, and Othman presented a class of single-input—single-

output nonlinear systems transformable into the triangular form

 

 

 

 

r r ' 7
$2 91(331)
$3 9261,1132)
j: = E + E u = F(x) + G'(:1:)u (5.17)
am 9n_1($1»°“a$n—1)
_ WE) . _ 972(17) .

To implement an output feedback control scheme, the following observer was sug-

gested
:i: = 17(5) + 6(5):; + so‘olcT(y — 05:) (5.19)
where the gain 500 is the solution of the following Lyapunov equation
(A + -21—6I)Ts00 + SOO(A + 2161) — CTC = 0 (5.20)

and A is an n x n matrix representing a chain of integrators. This solution exists
for sufficiently small 6. Through a Lyapunov analysis coupled with global Lipschitz
conditions, global exponential stability of the estimation error is established. These

results were extended to the single-input—multi-output case in [5].

The system (2.1)-(2.4), with the z dynamics dropped, becomes a special case of
(5.17)—(5.19) when 9i(-) = 0 for i = l,---,n — 1. Equation (5.20) is written for

a single chain of integrators. Let us examine its solution for a multiple chain of

144

integrators. Multiply (5.20) by 80—01 from the left and from the right to obtain
_ _ _ _ 1 _
A5001 + soolAT — SOOICTCSOOI + 25 1 = 0 (5.21)

It can be shown that the positive definite solution of (5.21) is a block—diagonal matrix

and is given by

5301(5) = block diag[SO—011(e),o--,Sg01,p(e)]

sgofne) = [(Sgo{,<e>>gj1r,xr,=[(s;0{,(1))g,g—$_—,1 (5.22)

Thus, the structure of the observer gain 30—01

CT is exactly the structure (2.10).
Consequently, if the scaling of Section 2.5.1 is used for the estimation error we obtain
a two time-scale singularly perturbed system similar to (2.11)—(2.14). Thus, if global
boundedness of the control law is implemented, we obtain the results of Chapter
2. It is noteworthy that in [17] all nonlinearities are required to be known which
undermines the robustness of the observer. However in the case where g,(.) = 0 for

i = 1, - - - , n — 1, the results of Chapter 2 show that imperfect knowledge of the two

remaining nonlinearities gn (cc) and (p(x) can be allowed.

5.3 More Separation Results

In Chapter 2 we decompose the closed-loop system into a reduced model correspond-
ing to the closed-loop system under state feedback controller and a boundary-layer
model corresponding to the Hurwitz matrix A0 of the error dynamics. Then, we re-
duce the parameter c to make the observer fast enough such that it brings the state
estimate close enough to its real value in short time and restores the stabilizing pow-
ers of the feedback controller. This is carried out using two Lyapunov functions: one,

V(x, z, 6), for the reduced model and another, W(n) = nTPon for the boundary-layer

145

model. It establishes boundedness of trajectories by proving that trajectories enter,
in a short time, the positively invariant set A 2' {V(r, z,6) S c} x {W(n) 3 p62}
where c and p are positive constants.

We notice that the observer gain of Chapter 2 depends only on one parameter c
in all of the p channels which results in a closed-loop system with a two time-scale
structure. Alternatively, one may have different parameters in the different channels
as we have seen. This eventually, after scaling, results in a closed-100p system with
a multiple time-scale structure.

For the case, where the observer gain is designed using one of the above-mentioned
algorithms, we can apply a multiple time-scale analysis to (5.16) and show that we
have p decoupled boundary-layer models. The i—th fast subsystem is an exponen-
tially stable linear time-invariant system with the Hurwitz matrix Fi° Thus we
can repeat the same steps of the previous analysis with the Lyapunov functions
V(x, z, 6), W1(n1), - - - , Wp(np) where W,- (172') = né-rPOZ-r), with P0,- being the unique
positive definite solution of the Lyapunov equation PI‘z-+I‘ZTP = —I. In this case the
positively invariant set is A = {V(x, z, 6) g c} x{W1(171) S pleg} x - - ~ x {Wp(np) 3
$612,}. All of the parameters ei’s should be simultaneously reduced to make the dif-
ferent boundary-layer models fast enough to bring the state estimate close to its real
value in short time. Ultimate boundedness and convergence of trajectories as well as
asymptotic stability analysis can be performed as in Chapter 2 using the above-given
Lyapunov functions. The foregoing analysis and discussion are summarized in the

following theorem

Theorem 5.1 Let the observer gain be designed using one of the above-given algo-
rithms, leading to the gain structure (5.12). Suppose that:

o The vector fields of the system (2.1)-(2.4), the dynamic controller (2.5)-(2.5), and
the observer (2.9) be locally Lipschitz in their arguments over the domain of interest

and vanish at the origin (at, z, 19) = X = 0.

146

o The functions F(6,:z:,(), 7(6,.r, C), and ¢0(x,C,u) are globally bounded in x.
o The origin X = 0 is an exponentially stable equilibrium point of the closed-loop
system under state feedback, with ’R, as its region of attraction.

Let the initial state x(O) be in a compact subset S of ’R and the initial observer
state 2(0) be in a compact set Q of RT.

Then, for sufficiently small 62‘, i = 1, - --,p, the origin (x,§:) = (0,0) is an en:-
ponentially stable equilibrium point of the closed-loop system under output feedback.
Moreover, for any compact subsets S C ’R and Q C RT, the set 8 x Q is'a sub—
set of the region of attraction. Furthermore, as the parameters e,- ’s tend to zero the
trajectory x(t,8) under output feedback approaches the trajectory Xr(t) under state

feedback control, uniformly in t, for t Z 0.

Remark 5.1 For simplicity we stated the separation results only for the case where
the origin x = 0 is exponentially stable. Similar results can be easily proved for
the case of asymptotic stability along the lines of Chapter 2, but we have to require

additional conditions on the local growth of the modeling errors.

Remark 5.2 Separation results similar to those of Theorem 5.1 and Remark 5.1 can
be stated for the case of stability with respect to a compact, positively invariant set

along the line of Chapter 3.

5.4 Conclusion

In this chapter we discussed the problem of output feedback control for a wide class
of nonlinear systems using high-gain observers designed using different algorithms.
These algorithms involved either pole placement, algebraic Riccati equation-based
techniques, or Lyapunov equation-based technique. We basically showed that all
these designs yield observer gains that asymptotically have the structure of the ob-

server gain used in Chapters 2 and 3. Consequently, if the idea of global boundedness

147

of the control law is implemented, the separation results of Chapters 2 and 3 apply

to the cases where these alternative techniques are used to design the observer gain.

148

CHAPTER 6

A Smooth Converse Lyapunov

Theorem for Robust Stability

6.1 Introduction

In [34] Lin, Sontag, and Wang present converse Lyapunov function theorems for
stability with respect to sets. Their work allows arbitrary bounded time-varying pa-
rameters in the system description, results in smooth Lyapunov functions, applies to
stability with respect to not necessarily compact sets, and deals with global asymp—
totic stability. In addition to the global results of [34], we need a converse Lyapunov
theorem that yields a smooth function defined on a possibly finite open set and that
approaches infinity at the boundary of this set. Thus our objective here is to modify
the results of [34] to make it suitable for our purposes.

The tools used to modify the results of [34] are inspired by Kurzweil [32]. It
consists of replacing the distance (with respect to a set in this case) by a continuous
positive definite function that has all the important properties of the distance (except,
may be, the triangular inequality) and that approaches infinity at the boundary of
the open set of interest.

In Section 6.2 we give basic definitions and the main results. In Sections 6.3 we

149

prove the converse Lyapunov theorem for uniform asymptotic stability.
The proof of the converse Lyapunov theorem closely follow that of [34] except in
places where some special discussions have to be carried. We will point out these

matters whenever they arise.

6.2 Definitions and the Main Results

Consider the system (3.1) (we repeat it here for easy reference)

13(1) = f($(t),d(t)) (6.1)

Let MD be as in Chapter 3. Let R be an open subset of R". The system is said
to be forward complete in R if Ta) (1 = +00 for all 2:0 6 R and all d E MD. It is
backward complete in R if CPI-0’ d = —00 for all 2:0 E R and all d E MD, and it is
complete in R if it is both forward and backward complete in R. In this chapter we

use notation and definitions given in Section 3.2.

6.2.1 Strong Stability

Let A be a compact subset of the open connected set R. Define the function “(A :

Rn —) R2 0U{+OO} by

)_ max(|£|A,]—€—[—E—]—F—.ZE) ifxER (62)
+00 if$¢R

00,4(5

where F is the complement of R in R", and [FIA z info7 u) E A x F ”77 — #ll

Remark 6.1 A similar function was first defined in K urzweil [32] for the case where

A reduces to an equilibrium point.

150

We can make some useful observations concerning wA(5); they are summarized

in the following Lemma:

Lemma 6.1 Let A be a compact set contained in the open connected set R g R".

The function “094(5) restricted to the set R satisfies the following properties:

I. It is positive definite with respect to A; i.e., “A(S) > 0 for all 6 E R/A and
6)./1(6) =0f01‘ allf E A. (1)./4(6) 2 MIA ifR= R”;

2. It is continuous in R;
3. It approaches infinity as 5 approaches the boundary of R;
4. The set {E E R : r1 S “A(5) 3 r2} for r1,r2 Z 0 is compact in R;
5. It is locally Lipschitz.
Proof. See Section 6.4.1.<1

Remark 6.2 The function “A“ shares some properties with HA, namely, positive
definiteness with respect to A, continuity, and local Lipschitz property. This fact
makes adapting many results of [34] to our case a straightforward process. Therefore,

we omit the proofs where replacing I] A by “A“ does not pose any challenge.

The following is a notion of stability in an open set inspired by Definition 1 of

[32] and Definition 2.2 of [34]:

Definition 6.1 Let R Q R" be an open connected set that contains A. We say that
the system (6.1) is Strongly Stable with respect to the compact positively invariant

set A if the following two properties hold:

1. Uniform Stability in R: there exists a class [Coo function 6(.) such that for any

6 Z 0 and every d E MD, we have

wA(:r(t,:c0; d)) S 5, whenever wA(:z:0) 3 6(6) and t 2 0 (6.3)

151

2. Uniform Attraction in R: for any r, e > 0, there is T = T(e, r) > 0, such that

for every d 6 MD,

wA(x(t,x0; d)) < 6 whenever wA(x0) < r and t 2 T. (6.4)

Remark 6.3 The definition of Strong Stability with respect to a set is an adapta-
tion of the notion of global UAS [34, Definition 2.2] to a possibly finite open set R
(wA(.) 2 HA ifR = R"). This notion is a generalization to a compact set of the
definition of strong stability given in [32].

The following definition generalizes [34, Definition 2.6] to the case of strong sta-
bility:
Definition 6.2 A Lyapunov function for the system (6.1) in the open set R with

respect to a compact, positively invariant set A C; R is a function V : R —> R> 0

such that V is smooth on R/A and satisfies the following properties:

I. There exist two class Koo functions a1 and a2 such that for any 6 E R,
C¥1(wA(lE)) S V(é) S 02(wA(€)) (65)

2. There exists a continuous, positive definite function 013 such that for any 5 E

R/A, and any d E MD,

L fdvc) _<. mums» (6.6)

A smooth Lyapunov function is one which is smooth on all of R.

It follows from the previous definition that V is continuous in all of R, zero if and

only if 6 E A, and onto.

152

The first main result of this chapter is:

Theorem 6.1 Let A C R" be a compact, positively invariant set for the system
(6.1). Let R be an open and connected set that contains A. Assume that the system
(6.1) is strongly stable in R with respect to A. Then, there exists a smooth Lyapunov

function V in R with respect to A.

Remark 6.4 It is possible to state Theorem 6.1 as a necessary and sufficient con-
dition. Moreover, a result similar to Proposition 2.5 of [34] for the case of strong
stability in an estimate of the region of attraction is also correct. The proofs of such

results follow from the corresponding ones in [34] by replacing || A with wA(..)
It is useful to restate the uniform attraction property as in the following lemma.

Lemma 6.2 The uniform attraction property defined in Definition 6.1 is equivalent
to the following:

There exists a family of mappings {Tr}, > 0 such that

o for each fixed r > 0, T7- : R> 0 —-> R> 0 is onto, continuous, and strictly

decreasing;
o for each fixed 6 > 0, T1- (6) is strictly increasing in r and limr _; 00 Tr (e) = 00

and, for each d E MD, we have

wA(x(t,x0; d)) < 6 whenever wA(x0) < r and t Z Tr(e)

Proof. straightforward from Lemma 3.1 in [34].<1

6.2.2 Uniform Asymptotic Stability

Let MCD be the set of all continuously differentiable functions from R to ’D where

the derivative (1’ (t) of d(t) belongs to a compact set ’D1.

153

The following lemma relates the notion of strong stability to that of uniform

asymptotic stability given in Chapter 3:

Theorem 6.2 Let d E M’D. Let the system (6.1) be Uniformly Asymptotically
Stable (UAS) with respect to the compact positively invariant set A. Let R be an
open, connected subset of the region of attraction that contains A. Then, the system

{6.1) is Strongly Stable in R with respect to A.

Proof. see Section 6.4.2.<1

The second main result of this chapter is:

Corollary 6.1 Let d E M'D. Let A C R" be a compact, positively invariant set for
the system {6.1). Assume that the system (6.1) is UAS with respect to A. Let R be
an open, connected subset of the region of attraction that contains A. Then, there

exists a smooth Lyapunov function V in R with respect to A.

Proof. if follows directly from Theorems 6.1 and 6.2.4

Remark 6.5 The open set R can be any time-independent estimate of the region of
attraction of (6.1) with respect to A. This region of attraction is not necessarily time-
independent nor positively invariant unless the system (6.1) is autonomous (see [4,
Chapter V,Proposition 415]). Thus, we can not take R to be the region of attraction

as in [32, Theorem 12] unless the system (6.1) is autonomous.

6.3 Proof of Theorem 6.1

The proof is divided into three main parts. In the first part we assume that the
system (6.1) is complete and we construct a Lyapunov function which is smooth in

R/ A. In the second part we drOp the completeness assumption and we construct an

154

auxiliary complete system for which a Lyapunov function is available from the first
part. Then, we prove that this function is also a Lyapunov function for the system

(6.1). Finally, we smooth the resulting Lyapunov function in R.

6.3.1 Case of a Complete System

Theorem 6.3 Let the assumptions of Theorem 6.1 hold. Moreover, assume that the
system (6.1) is complete in R. Then, there exists a Lyapunov function V in R with

respect to A which is smooth in R/ A.

Proof. First, we construct a useful locally Lipschitz function g({) and use it to con-
struct a not necessarily smooth Lyapunov function U (5) Then, we use a smoothing
result given in [34, Theorem B.1] to show the existence of a smooth Lyapunov func-
tion V({).

The system (6.1) is strongly stable in R with respect to A. Let 6 and Tr be as

in Definition 6.1 and Lemma 6.2.

Lemma 6.3 The function g : R ——> R, defined by

9(5) inf {wA($(t. 6; d))} (6-7)

= t S 0, d E MD
is well defined, continuous everywhere in R, locally Lipschitz on R/A, and satisfies
9($(t.€;d)) S 9(6). W > 0. W E Mp (6-8)
5654(6)) S 9(6) S wA(€) (6-9)
for allE E R.

Proof.

Some additional steps should be added to the proof of [34]. One is the proof that

155

g(.) is well defined on R, and the other is the proof of the local Lipschitz property
of g(..) The function g(C) is well defined because the trajectory x(t,€;d), t S 0
starting from C E R exists for any d 6 MD. Moreover, it stays in R for at least a
finite interval [T8, 0] in the negative time. Since the function w A is equal infinity on
the boundary and outside of R, then the infimum function always has a finite value
for every C in R. The proof of (6.8) and (6.9) is a straightforward extension of the

corresponding proof in [34].

Let us characterize g(.) in the set Kg, r dgf {C E R : e _<_ “54(5) < r} defined for

any 0 < e < r (KEJ is not necessarily compact). Note that, for all e and r with

0 < e < r, there exists qg, r g 0 such that:
CE K5,)”, d E MD, and t < (15,1‘ => wA(x(t,C;d)) 2 r
Therefore, for any C E K 5, r, we have
9(5) = inf{w,4(rr(t,€;d)) = t 6 [$5,130], d E MD, wA($(t,€;d)) S 7‘} (6-10)

Let us prove that g(.) is locally Lipschitz on R/A. Fix any CO 6 R/A, and let

_ . IéoLA |€0lF
s—m1n{ 2 , 2

Let B (C0, 3) denote the closed ball of radius 3 centered at {0. Then B (CO, 3) g K031",
for some 0 < o < r. Pick a constant C as in Proposition 5.5 of [34] with respect to
the closed ball B(CO, s) and T = ng', r]-

Pick any C, 77 E B (CO, 3). Then, from (6.10) (a property of the inf function), for any

156

6 > 0 there exist some d7), 6 E MD and some t1}, 6 E [gag r, 0] such that

9(7)) 2 wA(I(tn,e, 77; 0117, 6)) - 6 (5-11)

Moreover, from (6.10), we have

g(C) S WA($(tn,c,C;dn,e)) (6-12)

From (6.11) and (6.12), we conclude

9(C) — 9(7)) S wA($(tn, e, C; d7), 6)) - wA(-T(tn, e, v; dn, e)) + f (6-13)

Since B(CO, s) is compact, then, by Proposition 5.1 of [34] the points x(t”, 5, C; d7), 6)
and x(tn,€,n; (177,5) belong to a compact set E in R"; moreover, by (6.10), they
belong to {C : wA(C) S r}. Consider the set E = Efl{wA(§) g r}. We know from
Lemma 6.1 that {wA(C) g r} is compact, then the set E is compact and is contained
in R.

Using the Lipschitz property of w A in E , proved in Lemma 6.1, yields

9(() — 907) S Lll$(tn, e, C; do, 6) — x(t", 6,77; 6177,15)“ + ‘5 (614)

where L is a Lipschitz constant of w A in E. By Proposition 5.5 in [34], applied in
B (C0, 3), we have

9(() - 9(77) S CLIIC - 71H + 6 (6-15)

Note that (6.15) holds for all e > 0, then it follows that

9(C) - 9(9) S CLIIC — 77”

157

By symmetry we have g(n) — g(C) g CL||C — nll which proves that g(.) is locally
Lipschitz in R/A. The Lipschitz constant CL depends only on B (CO, 3) (although
L depends on E, E depends only on B(CO, 3) if we fix the time T).

Let us show that g(.) is continuous everywhere in R. From the Lipschitz property
we conclude that g(.) is continuous in R/A. Since, for C E A, wA(C) = 9(5) = 0,

then, using (6.9), we have

I907) - 9(€)| S [454(9) - wA(€)|

for C E A and 77 E R. Since ”A“ is continuous in R, the above inequality shows

that g(.) is continuous in R.<i

Lemma 6.4 The function U : R —-) R2 0, defined by

U(é) = sup {9(I(t,E;d))k(t)} (616)
t 2 0,d 6 MD

is well defined given that k : R> 0 —> R> 0 is any strictly increasing, smooth function

that satisfies:
0 there are two constants 0 < c1 < c2 < 00 such that k(t) 6 [c1, c2] for all t Z 0;

0 there is a bounded, positive, decreasing, and continuous function r(.) such that
k’(t) 2 r(t) for all t 2 0

(For instance, c :ct t is one example of k(t)).

158

Moreover the function U is continuous everywhere on R, locally Lipschitz on R/A,

and satisfies

015(wA(€)) SU(€) SCQwA(€) (6-17)

for all C E R. Finally, the function U(C) decreases along trajectories of the system
(6.1); i.e.,

L de (6) S -(3z(wA(C)), Vd e MD (6.18)

for all C E R/A, where 51 is a continuous positive definite function.

Proof

Some additional steps should be added to the proof of [34]. One is the proof that
U (.) is well defined on R, and the other is the proof of the local Lipschitz property
of U () The function U (C) is well defined for C E R because the value g(x(t,C; d))

exists for all t Z 0 and all d E MD given that
0 S 9(1‘(t.€; d)) S 9(6). V6 6 R, W > 0

and g(C) exists for C E R. The proof of (6.17) is a straightforward extension of the
corresponding proof in [34].
Let us characterize U(.) in any set of the form {C E R : 0 < wA(C) < r,r > 0}.

Note that, for any 0 < wA(C) < r,

U(E) = SUP {9($(t,€;d))k(t)}
0 S t S t€;d E MD

159

where t5 = Tr (%5(WA(§)))-

Let us prove the local Lipschitz property of U () on R/A. For any compact set
K g R/A, let

def
t : maxt <00
K CEKé

Finiteness follows from the above expression of U (C), since K Q {C E R : 0 <
wA(C) < r} for some r > 0, and from continuity of T7-(.),6(.), and wA(.).

For C0 9! A, pick a compact neighborhood K0 of C0 in R such that KOOA =
(0. By (6.17) we have (the continuous function wA() reaches its minimum on the

compact set K0)
U(5) Z 7‘0. V5 6 K0 (6.19)
for some constant r0 > 0. Let r1 = 2%; and let
K1= Kofl{v = “9-60” < L]
“ 2CL
where C is a constant such that (see Proposition 5.5 of [34])
llx(t.€; d) — x(t. n;d)ll _<_ CHE — nll. V5.17 6 K0. 0 .<_. t _<_ tKO. d 6 M1) (620)

and L is to be determined later on. The set K1 is a compact neighborhood of C0 in
‘R/ A. In the following we will show that there exists some constant L such that for

any C,17 6 K1, we have

|U(€) - U(77)| S l3H6 - 77“ (6-21)

160

We have to find a compact set in R/ A on which we can apply the Lipschitz prop-
erty of g(.) (this set will be K3). We know that (it is a property of the supremum),
for any C 6 K1 and any 6 E (0,r0/2), there exist tC,e E [0,tK0] and dC,e E MD,

such that

(1(5) 3 “1505,95; 45,999., a + .

Furthermore, using (6.9) yields

mo 3 5.4.4995, ..4; (15,.» + .

Thus, using (6.19), we obtain

wA(x(t€,€,C;d€,€)) _>_ Eg- —é
Since 6 E (0, r0/2)
WA($(t€,€,€;d£,€)) 2 T1 (6'22)

From Proposition 5.5 in [34] we know that there exists a compact set K 2 such that
x(t,C;d) 6 K2, VC 6 K1, Vt E [0,tK1], and Vd E MD

The set K 2 is contained in R, given the definition of strong stability (especially the
definition of stability in R). Let L be the Lipschitz constant of wA(.) in K2. Then,

for all 77 6 K1, we have (we apply the Lipschitz property of ”A” in K 2)

WA(IE(L€,€, Tl; (16’6” Z WA($(t€,€,§;d§-,€)) — Ll|$(t€,€,€; (15,5) _ $(t§,€.77;d§,g)ll

161

Therefore, using (6.20) and (6.22), for all 77 6 K1, we have (we apply the Lipschitz

property of x(t,C; d) in 1(1))

WA($(t§,€.TI; (15,.» 2 r1 — LCllé - nll

Thus, from the expression of K1, we have

wA(x(t€,€,n;d£,€)) 2 ’31 (6.23)

The compact set K3 dgf K2 fl{C E R : wA(C) 2 g} is a subset of R/A. The
inequalities (6.22) and (6.23) show that x(t€,£,C;d€,€) and x(t€,€,n; (15.6) belong
to K3. Then, we can apply the Lipschitz property of g(.) on the compact K3 and

obtain

lg($(t£,5:€i dé, 5)) _ g($(t€,€ani d€,g))l

S C1|l$(tg,€.€;dg,e) - $(t§,€.fl;d§,e)ll (624)

for some C1 > 0. Therefore, using the bounds on k(t) and (6.24), we have

I/\

U(E) — U(77) 9($(tg, 5. 6; Gig, 5))k(tg, 5) + 6 — 9617055077; (15, €))k(t€, 5)

S k(t€,€)[g($(t€’€,€;616,6» _ g($(t€,ei 7]; 616,5)” + 6

|/\

02|g($(t€,6,€;d€,6)) _ 9($(t€,5.773d§,5))l + 6

/\

_ 016114-66, ..6; 46,.) — x(tg, ..n; d5, .)1 + . (6.25)
Using Proposition 5.5 of [34], inequality (6.25) yields

U(E) - U(n) S I3|I€ - all + 6 (6-26)

162

for some constant L = C1c2 > 0 that only depends on the compact set K1. Since

(6.26) holds for any 6 E (0, 70/2),

~

U(E) - 11(7)) S Lllé — rill. V6.77 E K1

Thus, by symmetry, we can prove (6.21).

Let us prove continuity of U () everywhere in R. From the Lipschitz property of
U () we conclude that U () is continuous in R/A. For all C E A we have U (C) : 0
and wA(C) 2: 0. Then, for all 77 6 R, we have

|U(€) - U(77)| = U(77) S C2lwA(€) - w,4(77)|

Since wA(.) is continuous in R, so is U(.).

The proof of (6.18) is a straightforward extension of the corresponding proof in
[34]. Finally, let us smooth the function U (C) in R/A. By [34, Theorem B.1], there
exists a C00 function V : R/A ——> R2 0 such that for all C E R/A, we have

{_1_(52

“no — U(on < 2

and

[\Dt—J

L mm s —-c‘r(w,4(€)). Vd 6 Mo

Extend V to R by letting V = 0 on A and denote the extension by V. Note that V

is continuous on R and smooth on R/ A. Thus, V is the desired Lyapunov function

with 01(3) 2 3375(3), 02(8) = 2323, and a3 = %6. <1

163

6.3.2 Case of a Forward Complete System

Hereafter, we establish a Converse Lyapunov result without completeness of the
system. In order to use the previous result, we modify the vector field of (6.1) to
make it complete. Then we show that, if the system (6.1) is strongly stable, then
the new system with the modified vector field is also strongly stable. Finally, by
applying Theorem 6.3 we show the existence of a Lyapunov function for the new

system. The same function will be the desired Lyapunov function.

Let us define a new but complete vector field.

Lemma 6.5 Let f : R" x D —+ R” be continuous, where D is a compact subset of
Rm. Then there exists a smooth function a f : R" —+ R, with a f(x) 2 1 everywhere,

such that ||f(x,d)|| S af(x) for all x E R” and all (1 E D.

Proof. Let a(x) = maxd E D I] f (x, d)“. This function is continuous because of the
continuity of f (.,d). Choose any smooth function a f such that a f(x) 2 1 + a(x)

for all C E R". The function a f is the desired one.<1

For any given system
23 : x = f(x,d) (6.27)

not necessarily complete, the system

‘5
3?
5‘:
II
S...

 

2b I it = (:2), d) (6.28)

is complete since W s 1 for all x E R" and all d e D (see [16, Theorem 2.1,

page 17]).

164

Assume that f is continuous and locally Lipschitz in x uniformly in (1, then f

has these same two properties.

Lemma 6.6 Assume that A is a compact subset of R”. Let R be an open, connected
set that contains A. Suppose that the system 2 is strongly stable in R with respect

to A. Then, system 2b is strongly stable in R with respect to A as well.

Proof. straightforward from Lemma 7.2 of [34].<1

Now consider the system (6.1) and let 2b be the corresponding complete system.
By Lemma 6.6, we know that the system 2b is strongly stable in R with respect to
the set A. By applying Theorem 6.3 to the complete system 2b in R, there exists a

Lyapunov function V for 2b such that

|/\

V(é) S 92(wA(€)). V5 6 R
Lde(§) < —-a3(wA(C)), V5 6 R/A, and Vd e D

0195405))

for some class lCoo functions a1, 02 and some continuous positive definite function

013. Since a f (C) 2 1 in R, it follows that
Lde(€) S -a3(wA(C)), w: e n/A, and Vd e D
Thus, we conclude that V is also a Lyapunov function for 2. <1

6.3.3 Smoothing of Lyapunov functions

The Lyapunov function found previously is only smooth on R/A. Hereafter we
smooth this function in the domain of interest R. To do this, we first construct an

appropriate smooth function 6 then we prove that the function W = 60V is the

165

desired smooth Lyapunov function. This procedure is summarized in the following

lemma and proposition:

Lemma 6.7 Assume that V : R ——> R2 0 is C0, the restriction VlR/A is COO,
VI A = 0, and VIR / A > 0. Then, there exists a class lCoo functions ,6 which is
smooth on (0,00) such that 13(2)“) ——> 0 as t -—> 0+ for each i = 0,1,---; 6’ > 0,

Vt > 0, and W = ,6 0 V is a C00 function over R.

Proof. see [34, Lemma 4.3] we just need to restrict the domain of interest to R

instead of Rn.<l

Proposition 6.1 If there is a Lyapunov function for (6.1) in R with respect to A,

then there is also a smooth Lyapunov function.

Proof. straightforward from Proposition 4.2 in [34].<1

6.4 Proofs

6.4.1 Proof of Lemma 6.1

1) Positive definiteness with respect to A: From the definition of wA(.), we
have wA(C) _>_ ICIA 2 0. Now, if wA(C) = 0 then |C|A = 0. Thus, C belongs to the

closure of A, then, since A is closed, we have C E A. Next, if C E A, then [C | A = 0

 

and ICIF Z |F|A; i.e., Kl]? — lelA < 0. Thus, wA(C) 2 [CIA = 0, and we conclude

positive definiteness of “A” with respect to A. Finally, if R = R", then, the term

[fl—E — [7'21 is zero and, since [CIA Z 0, we have wA(§) = MIA-

2) Continuity in R: It follows from the fact that the maximum of two

continuous functions is continuous; see [6, Exercise 4, page 136].

166

3) Properness in R: As C approaches the boundary of R we have [C | F —> 0.

From the definition of wA(.) we always have

 

1 2
> _
WA“) - lap [FLA

. 2 . . .
Therefore, Since If]; 1S finite (A 1S compact), then wA(C) —> 00 as C approaches

the boundary of R.

4) Compact Level Sets: Since wA(.) is continuousin R, then the set {C E R :
r1 3 wA(C) g r2} is closed. Now we only need to show that this set is bounded.

Note that

{€ERIT1SwA(€)ST2}={€€R=7‘1SwA(€)}fl{€€R=wA(€)ST2}

Thus, it suffices to prove that the set 52 dgf {C E R : wA(C) 3 r2} is bounded.

In case R is bounded, the set (2 is necessarily bounded (of course this is possible
because A is bounded). Now, consider the case where R is not bounded. Suppose
that Q is not bounded. Then, as “C” ——> 00; i.e., as C approaches the boundary
of R, we have that wA(C) —+ 00 which is a contradiction because, in (2, we have
wA(C) 3 r2. The assumption that A be bounded is needed because otherwise the

set 0 is necessarily unbounded.

5) Local Lipschitz property: Let K be a compact subset of R/ A. Let us show

that there exists a positive constant k such that, for all C1, C2 6 K, we have

lwA(€1)- wA(€2)| S kl|€1 - €2|| (6-29)

First, consider the case where wA(C1) = |C1|A and wA(C2) = ICglA. In this case

167

we have

lwA(€1)* wA(€2)| = | lfllA — |€2|A|

Since || A is locally Lipschitz, we can write

I |€1|A - |€2|A] S ||€1 - E2”

Then, using (6.31), the inequality (6.29) is true.

Second, consider without loss of generality that

954051) = |€1|A
... (5) _ _1_..L
A 2 ‘ IolF IFIA

Then, from the definition of wA(.), we have

1 2
> ___——
(“(61) ‘— |€1|F IFIA

91,4(62) 2 |€2|A

Using (6.32) and (6.35) yields

“A(51)— wA(€2) S |€1|A — |€2|A

Then, using (6.31) in (6.36), we have

wA(€1)- wA(€2) S ||€1 - 62”

168

(6.30)

(6.31)

(6.32)

(6.33)

(6.34)

(6.35)

(6.36)

. (6.37)

Using (6.33) and (6.34) yields

1 2 1 2
_, < __————+_—
“91(9) ”4151) - (.21).. lFlA I€1IF IFIA
‘ 16le |51|F

 

1 ] (6.38)

s ] 1 —
[52]}? [61]]?

Now, if we can prove that, in K, there exists a positive constant k2 such that

1

1
—— — — < k — 6.39
[[62]]? léllFl - 2ll€1 £2” ( )

then, using (6.37), (6.38), and (6.39) in addition to (6.31) proves the local Lipschitz
property of wA(.) with k = max(1, k2). Hence, it suffices to prove (6.39). Since R
is an open set, F is a closed set. Thus, with a similar reasoning as the one that lead

to (6.31) we have

I |€1| F — I629] 3 ”£1 — €2ll (6.40)

Since K is compact, then there exists a positive constant k2 such that

1

—— < k 6.41
|€1|F|€2|F _ 2 ( )

Using (6.40) and (6.41) we can write

 

] 1 __ 1 ] < |€2|F-|€1|Fl
|€1|F |€2|F |€1|F|€2IF
S L2l|€1 -€2|| (6-42)

This concludes the proof of Lemma 6.1.<1

169

6.4.2 Proof of Theorem 6.2

Assume that the system (6.1) is UAS with respect to A with R as in the theorem.
Then, the system is forward complete in R and Definition 3.1 applies. Let us show
strong stability of (6.1) with respect to A. We divide the proof into two parts.
First, we show uniform stability in R. Then, we show uniform attraction in R. The

forthcoming proof is inspired by that of Theorem 12 of [32].

Since R is open, then there exists an open ball B0 = {C : |C | A < r0, r0 > 0}
around A such that BO is the largest ball in R that conatins A (i.e., BO is the first
ball whose boundary intersects with the boundary of R). It is important to notice

that r0 may be finite. Let us define the function

(15(3) = sup wA(C), for s E [0,r0)
lélA S s

the function (p(s) is continuous, positive definite, increasing (not necessarily strictly),

and goes to infinity as s —> r0. Moreover, we have

w,4(€) S ¢(|€|A)

for all C 6 80- Let 7(8) be a class [C function such that 7(5) 2 k¢(s) with k > 1.

Then, using the above inequality and the definition of wA(.), we have

IEIA S wA(€) S V(lflA) (6-43)

s-ll>mr07(s) = 00 (6.44)

Part I - Uniform Stability in R:

Fix 6 > 0. Consider the positive constant 17 = 7_1(e); 7‘1 exists since 7 : [0, r0] —)

170

[0, 00) is strictly increasing and onto. From Definition 3.1 we know that there exists

a positive constant 61(7)) such that, for all d E M'D and all t 2 0, we have
Imam... 614 s 77 (6.45)

whenever [xOIA s 61(7)).
Now, let wA(x0) 3 61(7)) which implies, given the definition of “A, that |x0| A _<_
61(77). This implies (6.45). Then, using (6.43), we have

Mama; «1» 6 7(4) = .

for all d 6 M51) and all t Z 0, whenever wA(x0) 3 6(6) = 61(7—1(6)).

Now, we have to find a class K00 function 6 to replace 6 in the above statement.

For a fixed 6, let 6(6) be the supremum of such 6. Then,

wA(x0) < 6(6) => wA(x(t,x0; d)) g 6 (6.46)

for all d E M’D and all t _>_ 0, and if 62 > 6, then there exists at least one initial

state 5:0 and one function (i E M’D such that

(“(60) S 62 and sup wA(x(t,:i':0,d)) > 6 (6.47)
t Z 0
Let 6(6) 2 %5(€). Then, (6.46) can be written as
wA(x0) 5 6(6) ==> wA(x(t,x0;d)) g 6 (6.48)

for all d E M’D and all t Z 0. The function 6(6) is positive and non-decreasing,

171

but not necessarily continuous. Furthermore, we have lim,E _, 06(6) 2 0 because

we can see from (6.48) that 6(6) 3 6, otherwise we would get a contradiction at t = 0.

Claim 1:
lim 8(6) = 00 (6.49)

Proof. Assume that lime _) 00 6 (6) = 600 < 00. According to (6.47), for every i 2 1
and for 62(i) = 26(i) + 1 : 6(i) + 1 > 6(i), there exists in E R and d,- 6 M51) such

that

wA(x0,-) S 62(i) and tsgp0(wA(x(t,x0,-; dill) > i (6.50)

This means that

lim sup{ sup (wA(x (t, inidillll = 00 (6.51)
i —> 00 t > 0

Let us find 0 < T* g 00 and a solution x(t,x0; d), where x0 6 R and wA(x0) S
600 + 1, such that limt _) T* x(t, x0; 61) = 00 and let us show that this constitutes a
contradiction to the fact that x0 belongs to R, a subset of the region of attraction.

def

Let xi(t) = x(t,x0,-;d,;), then {x,-(t)} is the sequence of solutions defined by

(6.50). Define

T* d__e_f sup{T_ > 0:1imsup(0

i——>000 SmtSTw'A

(332(0)) < 00} (6-52)

Note that 0 < T* S 00. Given (6.51), the quantity T* is the supremum of all T Z 0

such that all of the elements of the sequence are finite for all t E [0, T].

172

Case 1: First, suppose that
T* < 00
Consider a sequence {72-} of positive constants such that for every i Z 1 we have
if wA(x0) S i, then wA(x(t,x0;d)) S i + 1, 0 S t S 27,- (6.53)

for all d E M’D. This sequence is not empty because of the continuity of x(t, x0; d).
Without loss of generality we can take T* > T1 > 72 > - - - and limz- __) 00 r, = 0.

Let {le(t)} be the subsequence of {x,;(t)} such that
wA(x,1(T* — 71)) > 1, w 2 1 (6.54)

This subsequence is infinite. To show that, assume that it is finite and let 11 be the
set of indices such that xz-(t), i E I 1, does not belong to {le (t)} (i.e., 11 is infinite).
Take i 6 II, then wA(x,-(T* - r1)) S 1, then, from the definition of q, (6.53), and

uniqueness of solutions, we have
wA(x,(t)) _<_ 2, for T* — T1 St 3 T* + 71 (6.55)
Then, from the definition (6.52) of T*, we have
lim sup{ max (62 (x(t)))} < 00 (6.56)
i611 OStST*—r1 A 2

Then, using (6.55) and (6.56), we can write

limsup max w x't <00 6.57
2,611{Ostg,+71( A( .0») ( >

173

Since 11 is infinite, then

lim sup{ max (wA(x,-(t)))} < 00 (6.58)
i—>00 ogth*+r1

which is a contradiction with (6.52). Thus, 11 is finite and the subsequence {le(t)}
is infinite.

Moreover, let {x12(t)} be a subsequence of {le(t)} such that
wA(x,2(T* — 72)) > 2, Vi 2 1 (6.59)

As previously, we can show that the subsequence {x22(t)} is infinite. Similarly, we
construct a family of subsequences. Now, as we continue, we end up with the sequence

{xj(t)} such that
wA(:r.-(T* - 9-)) 2 j. W 2 1. w 2 1 (6.60)

Since wA(x’,-(0)) S 600 + 1 < 00, then, using Proposition 5.1 of [34], the set of
solutions starting from :77,- (0) is compact on [0, T] for any 0 < T < T*. Thus, the
sequence of solutions {53,-} is uniformly bounded (uniformly in i) on compact intervals
of time [0,T], 0 < T < T*. Let {62-} be the sequence of time-varying parameters

corresponding to the sequence of solutions {6,}. It can be shown that

t -
lliz-(t1)- 22.62)” s [,12 116.6). daemon (6.61)

Since the sequences {6,} and {Ji} are uniformly bounded and f (., .) is continuous,

then there exists a constant c513, independent of i, such that

”13601) - ji(t2)ll S Cxltl - t2| (6-62)

174

 

for all i Z 1. Then, using (6.62), for every 6 > 0, there exists a positive constant

ct > 6/ c1; such that

”173761) - 513602)” < 6 (6-63)

for all t1, t2 such that |t1——t2| < ct and all i _>_ 1. Then, see [6, Section 4.5, page 208],
the sequence {x,-} is equicontinuous. Thus, by Ascoli-Arzela’s Theorem [6, Theorem
4.5.8; Exercises 4.5.9, Probleml], this sequence has a subsequence (indexed by ij)
that converges (as i j —+ 00) for every t E [0, T], 0 < T < T*. Given the compactness
of [0, T], the previous subsequence converges uniformly in t (see [6, Section 4.5, page
206] for the definition of uniform convergence). By [12, Theorem 2.11], we see that
the limit is a continuous function in t on [0, T], 0 < T < T*.

Let {dz-j} be the subsequence corresponding to the subsequence {652-3.}. By the
Mean Value Theorem we can show that the sequence {Jij} is equicontinuous and

that it has a subsequence (indexed by i jk) that converges (as ij —> 00) uniformly

k
(in t) in every interval [0, T], 0 < T < T*. It can be shown that the set of functions
of MCD defined on [0, T] is a closed subset of the set of continuous functions defined
on [0, T] with values in Rd. Then, the limit of the convergent subsequence belongs
toM'DforallO<T<T*.

Let {5230,}. } be the subsequence corresponding to the subsequence {xijk }. Since
wA(:E,-(0)) S 600 + 1, we conclude, using Lemma 6.1, that the sequence of initial
conditions belongs to a compact set. Therefore, the sequence of initial states indexed
by ijk has a convergent subsequence in {C : wA(C) S 600 + 1}. Thus, the corre-
sponding subsequences (now indexed simply by i) of solutions {53,-} and time-varying
parameters {Ji} converge uniformly (in t) in every interval [0, T], 0 < T < T*. Let
x0, f(t), and d(t) be the respective limits.

Let us prove that x(t) is the solution of x = f (x, d) for x(0) = 5:0. Since f (x, d(t))

175

is continuous in x and t we can write
t - *
6,6) = 60,- + [0 f(x,-(r),d,-(r))dr, 0 _<_ t g T < T (6.64)

Since f (.,.) is continuous in its arguments and the sequences {53,-} and {Ji} are
uniformly bounded, then the sequence {f (x,, d,)} is uniformly bounded (in i) and
converges to f (x, d) for every t E [0, T], 0 < T < T* (see [6, Exercises 4.5.3, Problem
2]). Then, by the Lebesgue Dominated Convergence Theorem (see [12, Section 5.11,

page 232]), we can exchange limit and integration. Thus, we have
t - *
6(1) 2 5:0 + [O f(a‘:(r), d(r))dr, 0 St g r < :r (6.65)

Therefore, x(t) is the solution of x = f (x, d) for x(0) = 5:0; it is defined fort E [0, T*).

From (6.60) and the fact that limz- _, 00 r,- = 0 we conclude that

t ng* wA(x(t)) = 00 (6.66)

This is impossible since :70 belongs to the region of attraction.

Case 2: Now, let

Since T* = 00, then
lim sup{ ma); (wA(x,-(t)))} < oo (6.67)

i—>00 0St_T

for all finite T > 0. Since all x0,- belong to the region of attraction then, by Definition

176

3.3, for u > 0 there exist a finite T,(u) > 0 such that
livi(t)|,4 < u. t 2 T.- (6.68)
for all i Z 1. Choose u such that u < r0. Then, using (6.43), we have
wA(:v.-(t)) 6 7(2). for t2 T.- (6.69)
Thus, using (6.67) and (6.69), we have

lim sup{ sup (wA(x,-(t)))} < 00 (6.70)
i —+ 00 t Z 0

which contradicts (6.51).
Thus, we showed that if 600 is finite, we reach a contradiction. Therefore,

600 = 00.6

Since, by Claim 1, lime _) 00 6 (6) = 00, then we can choose a class lCoo function

6(.) such that 6(r) S 6(r). Hence, given 6 > 0, we have

wA($(t.xo; d)) S e

for all (1 6 M’D and all t 2 0, whenever wA(x0) S 6(6) S 6(6). Thus, we have

proved property 1.

Part II - Uniform Attraction in R:

Fix 6. Consider the positive constant 17 = 7_1(6). From Definition 3.1 we know that

177

there exists T = T(n) > 0 and a > 0 such that

|I(t.xo;d)|A < n (6.71)

for all d E M'D and all t _>_ T, whenever |C0| A < 0. Since R is a time-independent
subset of the region of attraction, without loss of generality, we can choose a such
that {C: IxOIA < a} C R.

Let wA(x0) < a which implies that |x0|A < a. Let T(6) = T(7—1(6)). Then, using
(6.71), we have

wA(x(t,x0; d)) < 7(17) < 6 (6.72)

for all (1 E M’D and all t Z T, whenever wA(x0) < 62.

Fix 6, r > 0 and let us prove that there exits T = T(6, r) > 0 such that
wA(x(t, 1120; d)) < 6 (6.73)

for all d E M’D and all t Z T, whenever wA(x0) < r.
Assume that the above is not true. Then, VT > 0 there exists an initial state
:20 = 5130(T) with wA(x0) < r such that wA(x(t,x0,d)) 2 6 for some t _>_ T and
some of E M’D. In other words, if we take a sequence {T z} of positive numbers such
that limz- _, ooTi = 00, then there exist sequences {ti} C R2 0, {x02} C R, and
{di} C M’D such that
lim ti = 00, wA(x0,-) < r, wA(x(t,-,$0i; dill Z 6 (6.74)

i——>oo

Using the uniform stability property (6.48) and uniqueness of solutions, we can write,

178

for any 7 2 0,
02A(x(t,x0; d)) < 6 (6.75)

for all d E M'D and all t 2 7, whenever wA(x(r,x0; d)) < 6(6) (note that 6 is fixed).

Hence,
6,400,160,; 3,» 2 8(6), Vt 6 [0,1,] (6.76)
Since wA(x0,-) < r, using the uniform stability property (6.48), we have
wA(x(t,$0iidz°)) < 6‘16), Vt 2 0 (6.77)

Thus the sequence of solutions {x(t, in; dill is uniformly bounded (uniformly in i).

As in Part I, we can find subsequences (indexed by i) {6,}, {530,-}, and {dz-(t)}
which converge on any interval [0,T], T > O to x(t), :20, and d(t), respectively.
Furthermore, 17:0 6 {C E R : wA(C) S r}, d 6 M’ , for any T > 0. Moreover, x(t) is
the solution of x = f(x, 6), x(0) = :20, defined for 0 S 0 < 00.

From (6.76) and the fact that limi _, 00 t,- = 00 we conclude that
9426(0) 2 8(6). t e [0. oo) (6.78)

This is impossible since 5:0 6 R that is, the solution x(t) must converge to A; i.e.

wA(x(t)) —+ 0 as t —+ 00. Thus T(6, r) exits such that (6.73) is satisfied.<1

179

CHAPTER 7

Conclusions

In order to implement a controller using output feedback, we can rely on a separation
principle which divides the design process into two steps. The first step consists
of designing a' state feedback controller to achieve the desired performance. The
second step consists of estimating the state using an observer, and then using these
estimates in the control law. This implementation should recover the performance
achieved under the state feedback control, at least asymptotically. This is the task
we have set to achieve in this work for a certain class of nonlinear systems. First,
we gave a comprehensive formulation for the state feedback controller performance
which describes the behavior of the entire state. Then, we recovered this behavior
using an observer, thus proving the possibility of an output feedback implementation
of the control law.

State feedback controllers achieve a wide range of performance. Sometimes we
need to steer the state to a point and have it stay thereafter. Other times we need a
part of the state to track a constant or a time-varying reference. We also may require
the input-output response of the system to satisfy certain criteria and limitations or
we may require the controller to'minimize some performance index apprOpriate to
the task at hand. We opted to a stabilization-type of performance description. We

found that this formulation covers a wide range of design objectives. We found

180

that numerous design objectives can be formulated as a stabilization problem of an
equilibrium point or a compact, positively invariant set. In both cases we considered a
system that contains one or several chains of integrators. In the case of an equilibrium
point we considered time-invariant systems. However, in the case of invariant set we
allowed a time-varying bounded parameter in the system’s nonlinearities. Moreover,
we used a high-gain observer to implement the control law and we required this law to
be globally bounded which can be achieved by saturation outside a region of interest.
It is noteworthy that this separation principle does not require any particular state
feedback structure as long as the control law is globally bounded.

The high-gain observer has an adjustable gain which allowed us to estimate the
state quickly enough to recover the full stabilizing power of the control law. In addi-
tion to recovering local stability, we recovered arbitrary compact subsets of the region
of attraction (or sometimes an estimate of it) achieved under the state feedback con-
troller. Moreover, we showed that trajectories under the output feedback controller
approach those achieved under the state feedback controller as the observer gain ap-
proaches infinity. Finally, for each stabilization case we presented several examples
to show some sources of the class of systems considered and to illustrate how we can
apply the separation results previously proved.

To prove the recovery of the aforementioned set of performance measures we
divided the task into several steps. First, we showed that trajectories starting in an
a priori given compact subset of the region of attraction are bounded. For this we
needed a Lyapunov function that approaches infinity at the boundary of this region.
This Lyapunov function was supplied by results due to Kurzweil [32] in the case of an
equilibrium point and by results due to Sontag et al [34] for the case of an invariant
set. Second, we showed that these trajectories come arbitrarily close to the attractor
(a point or a set). Before concluding asymptotic stability, we showed convergence of

these trajectories to the ones achieved under the state feedback controller. As for the

181

local property of asymptotic stability, we discussed three cases. First, we discussed
the case where the modeling error is zero; i.e., we perfectly know the nonlinearities
at the end of the chains of integrators and we use them in the design of the observer.
Second, we discussed the case where the convergence to the attractor under the state
feedback controller is of an exponential type. In this case we used a local Lyapunov
function supplied by the classic Lyapunov theory for the case of an equilibrium point
and by an adaptation of some results of Yoshizawa [55] to our purposes for the case
of a set. Third, we discussed the case where we have asymptotic stability under the
state feedback controller but where we have nonzero modeling error; i.e., we used
an imperfect knowledge of the system’s nonlinearities to build the observer. In this
last case, some conditions on the size of the modeling error had to be imposed in
order to recover local asymptotic stability. These conditions state that the modeling
error should be proportional to the rate of convergence of trajectories to the attractor
(raised to a power less than one) under the state feedback controller. Moreover, we
note that our results can only show semiglobal stabilization under output feedback
even when the state feedback control achieves global stabilization.

It is noteworthy that for the case where the attractor is a set we used some results
provided in [34] when the region of attraction is the whole state space. However, when
this region is a subset of the state space we had to adapt the results of [34] to our
purposes which required that we restrict the class of the time-varying parameters
and we only recovered an open estimate of the region of attraction.

Finally, we reviewed several techniques in designing high-gain observers. These
techniques are: pole-placement algorithms, Riccati equation-based algorithms, and
Lyapunov equation-based algorithms. In this work we showed that the observer gain
structures provided by these different techniques are asymptotically similar to the
structure used to show the separation results. Therefore, we concluded that the

abovementioned separation results hold for all these observer design techniques.

182

4..

sex-.1; _ ' '

.——-—'

 

 

 

 

The idea of this work (providing a formulation of controller performance and
establishing separation results based on this formulation) is still a fertile ground for
future research. Actually, we noticed that the controller performance of some design
examples can be described as achieving partial stability (stability of part of the state)
of the system as in [2], or as input-to-state stability (the magnitude of the trajectory
is function of the magnitude of the input) as in [29]. Moreover, an Optimization-
type of performance can be investigated (minimization of a performance index, for

example) as in [1].

183

APPENDICES

184

APPENDIX A

Technical Lemmas

Lemma A.1 Let A = A x {e = 0}, where A is a compact subset of R". Then, the

following holds:

|(e.n)|A = [HelanIfill/z (A.1)
Hell2 6 l(e.n)|,24 (A2)
Inlji, s 192.613, (A.3)

Proof: The definition of the distance with respect to a set implies

e, = e, — =_inf e, — ,—
|( 7))[A ( 77) bll bEA“( 7?) (0 b)||

inf [I
beA
2 - -21/2
[Hell +_mf-|ln—b|| ]
beA

To prove (A.1) , it suffices to show that

1613, = _inf- lln — 5112 (AA)
b e A

185

First, since A is compact, then, there exists b0 6 A such that
In! - = - inf- lln — 5H = lln - 50”
A b e A
Thus, from the definition of infimum, we have
36-0—665um—6m3=m& (40
b e A

Next, the definition of infimum implies that, for any 6 > 0, there exists bl 6 A such

that
IM-5w2—6S_mflln—HF
b 6 A
Thus, since (1)112, 3 Mn —51||2, we have
lug—es_mLMn—HF
b e A
This inequality is true for an arbitrary 6 > 0; then,
Im%s_mrin-az (46
b e A

From (A5) and (A6) we conclude (A.4).

Finally, using (A.1) , we have

IMP glvwwwmfi=uamfi,

lag slwW+wmfi=wemmi

Thus, we conclude (A2) and (A.3).<1

186

Lemma A.2 Let A = A x {e = 0}, where A is a compact subset of B". Let
(7)(t), e(t)) be the solution of the system (4.35) under the state feedback controller u*

and assume that

[U(tllg s (31(ln(0)|,.1.t)+7( sup |le(r)|l). W20 (Av
0<r<t

“e(t)“ s 92(||e(0)ll.t). W26 " (68)

where 61(.,.) and 62(.,.) are class ICE functions. Then, there exists a class IC£

function 6(., .) such that

|(6(t).77(t))|A S fi(|(€(0).n(0))|A.t). W 2 0 (A-9)

Proof. With a proof similar to that of Lemma 5.6 of [28], we show asymptotic stability
with respect to the set A.

First, using (A7) and (A8) we can write

|v(t)lg S 31(I77(8)l,4.t-8)+7( sup<t||6(T)||) (A-10)

8ST-

“66)“ S 62(||6(8)I|.t-8) (A-ll)

forOSsSt.

Next, let us rewrite (A10) for s = t/2

|77(t)lg S [31(l77(t/2)|,.1.t/2) + ’Y( SUP ||8(T)||) (Al?)
t/2 S 7 St

In order to estimate 77(t/2) we use (A.10) with s = 0

|n(t/2)l,1 s 6160(0)],1. t/2) + 7( sup new”) (4.13)
0 S r S t/2

187

Furthermore, using (All), we have

SUI) ”6(7)“ S 62(||€(0)||.0) (A-14)
OSrSt/2

SUP ”6(7)“ S 62(||€(0)H.t/2) (61.15)
fl2SrSt

Now, by substituting (A13), (A14) and (A15) into (A.12) we get

|77(t)|,.1 S 51(1’31(|77(0)|A.t/2)+7( SUP 2||€(T)||).Il/2)

O S r S t/
+ 7( sup ||6(T)||) (A-16)
t/2 S 7 St
|n(t)l,a S 51(fl1(|77(0)|,2;.t/2) + 7(fl2(lle(0)|l.0)).t/2)
+ 7(fi2(lle(0)||.t/2)) (A-17)
Notice that
|(6.n)|A = 51ng ”(6.77 - 5)“ S Hell + Inlg (A-18)

Define

6(7‘. 8) = 61(610‘. 3/2) + 7(520". 0)). 3/2) + 7(62(T. 3/2)) + 62(7". 8)

Notice that [3 is a class ICL function. Then, using (A.18) and (A.2)—(A.3), we have

|(€(t).n(t))|A S 51(51(|(€(0).n(0))|A.t/2)+7(62(|(e(0).77(0))|,4.0)).1/2)
+70320640).77(0))IA.t/2))+fl2(|(6(0).n(0))|A.t) (A119)

|(6(t).77(t))lA S fi(|(6(0).77(0))|A.t) (A20)

for all t Z 0.<1

188

Lemma A.3 Let A be a compact set contained in the domain D = {x : [xIA <
r0} C R”. Let V(t,x) : [0,00) x D —> R be a continuous function such that there
exist two continuous positive definite with respect to A functions V1 (x) and V2(x),

defined on D, such that

V16) 6 V(t,x) s V202)

for allt Z 0. Let Br 2 {x : [xIA S r} C D, for some 0 < r < r0. Then, there exist

a class IC functions 0‘1 and 02, both defined on [0, r], such that

01(IIIIIA) S V(M) S 012(lxlA)

for all x 6 Br and all t 2 0. Moreover, if D = R" and V1 (x) is radially unbounded,

then al and 02 can be chosen to be class ICOO.

Proof: similar to that of [28, Lemma 3.5].<i

189

APPENDIX B

Cheap Control and H00

Disturbance attenuation

The objective of this appendix is to find the structure of the stabilizing solution of
(5.8). For this purpose we perform some of the steps of [44]. The main idea is to
transfer the singularity from the Riccati equation to the system through appropriate
scaling of the state variables. In so doing the singular Optimal control problem is
transformed into a regular optimal control design for a singularly perturbed system.
For the purpose of clarity we perform the analysis directly on the auxiliary system
(5.4)—(5.6), with F = I.

Consider the cheap control problem of minimizing the performance index
1 00
J = -2-/0 (uszx + sz + u2uTu — 72de) dt (B.1)
subject to the worst case of disturbance and to the dynamic constraint

d: = ATa: + CTu + at (B2)

2 = BTx (3.3)

190

Since (A,C) is an observable pair, then, according to [56], this disturbance at-
tenuation problem admits, for sufficiently large 7, the unique stabilizing solution
21* = ——12CPoox, where P00 is the unique positive definite solution of (5.8) that
renders (AT — LIIZCTCPOO) asymptotically stable. This controller achieves a distur-

bance attenuation level of 7.

Now apply to both the system and the performance index the following scaling

xf = Sx
ii = ii—zzuu

where S is given in (5.13). We get the following problem

J = 1[000201sz _2xf + sz + 6T6 — 72de) dt (BA)
22' = BZTIL'if,’l=T1,°°-,p (B.6)

This can be written in the compact form

_ _1_ 0° T 2 T -T-_ 2T
J — 2/0 (fo xf+z z+u u 7 d d)dt (B7)
2,. = e-lATxf+e—1CT6+Sd (8.8)
2 = BTxf (B.9)

M = block diag[M1,---,Mp], Mi = diaglfiz‘."°.€:il

where E is given in (5.15).

Let us consider the subproblems corresponding to the boundary layer systems.

191

For the i-th boundary layer system we have the following problem

 

1 00 _ _
J,- = —2-/0 (zfzi+u$ui)dri (B.10)
dx;
d7? = Afar” +63%,- (BM)
2
Zi = BTJIif, ”l: 1,°--,p, Ti =t/CZ' (13.12)

For every i = 1, - - ~ , p, the i-the boundary layer problem (BLP) is a standard
quadratic regulator problem. Since (A,, Bi) and (AZ-, Ci) are controllable and observ-
able pairs, respectively, then the i-th BLP has the stabilizing solution it: = -—C,' Pix,- f

where P, is the unique positive definite solution of the algebraic Riccati equation
T T T
Aipi + PiAi — PiCi Clip; + BiBi = 0 (8.13)
that renders the matrix A? — Cng-P, asymptotically stable.

We know that the scaled complete problem (B.7)-(B.9) has a stabilizing solution
11* 2 —CK x f where K is the positive definite solution of the H00 algebraic Riccati

equation

6—1AK + KATE—1 — K8"ICTC£—1K + iZKszx + BBT + M2 = 0(B.14)
’7

By uniqueness of the solution, we can show that K is a block diagonal matrix. From

[3, Theorem 8.2] we borrow the solution structure

K(€) = block diag[61K1, - . - , 6pr] (B.15)

192

and substitute in (BM) we get

,, .T T , 12 2 T ,2_
:12:in + [12742- — KiCi C111; + 3-2-62- KiSi Ki + BiBi + All -— 0 (13.16)

for i = 1, - - - ,p. To study the asymptotic behavior of K we let (2' : 0, i = 1, - . - ,p

in (BIG) and we get
. . . T _ . T . . . T _
A,K,(0) + K,(0)A,~ K,(0)Cz CZK,(0) + BZBz — 0 (B.17)

By uniqueness of solution we conclude that Ki(0) 2 Pi, for i = 1,---,p. Going
along the lines of [8] we can show that for i = 1, - - - , p, there exist 63‘ such that, for

0 S 6,- S 6? we have
va=a+om) (RM)

The solution of the parameterized algebraic Riccati equation (5.8) of the original

problem (B.1)—(B.3) is then given by

P00 = SK(5)S (B19)

193

APPENDIX C

Stability Results

C.1 Proof of Theorem 3.1

Pick any d E MD- The derivative of V along the trajectories of (3.1) is given by

V(t,x) = 37;— + grew» 6 -a3(|$|A). w 2 0
Let us establish a set of initial conditions (independent of the initial time) for which
the solution of (3.1) is forward complete. Choose r > 0 and p > 0 such that
Br 2 {x E R" : lxlA S r} C U and p < minlxlA = ral(l$|A) (exists because A is
compact, thus {x : [xIA :2 r} is also compact). Then the set {x 6 Br : al(|x|A) S p}
is in the interior of Br.

Now, define the time-dependent set at, p by
“tap = {x 6 Br : V(t,x) S p}
Since

0209421) S 9 => V(tx) S p

194

the set Qt,p contains {x 6 Br : 02(lxlA) S p}. On the other hand, since
V(t,x) S p => 01(l55lA) S p

Qt,p is a subset of {x 6 B7: : 01(leA) S p}.
Thus,

{xEBr:02(|x|A) Sp} CthC {xEBrzal(|x|A) Sp}CB7~

for all t Z 0.

Since V(t, x) is negative on U /A, hence, V(t, x) is decreasing. Therefore, for any
to Z 0 and any x0 6 {x 6 Br : 02(lxlA) S p} C QtO: pa the solution starting at
(to, x0) stays in Qt, p, for all t 2 t0. Since Qt, p is bounded, the solution starting at
(t0,x0) is defined for all t 2 t0 and x(t) 6 Br.

Now, for any x0 6 {x 6 Br : 02(lxlA) S p}, we can write

179.4) = 95:1 + 859696)) 6 —ag(|x|A) s —a(V(x<t))), w 2 o

where a is the continuous positive definite function defined by

a(.) déf a3<a§1(.))

on [0,T]. Now, let 60 be the ICE-function defined in Lemma 4.4 of [34]. Thus, we

have
V(t,x(t)) g 60(V(t0,x0),t —- t0), VV(t0,x0) e [0,p]

195

Therefore, for any (1 E MD and any x0 6 {x 6 Br : (12(lxlA) S p}, the solution

x(t) of (3.1) satisfies

|~’l?(t)l,4 S 01—1((3a(a2(l$(t0)l/1).t-110)) d§f5(|113(lo)|,4.15-lo)

The function 6(.,.) is a class ICE function defined on [0,a2_1(p)] x [0,00). This
implies, by a mechanism similar to the first part of the proof of Proposition 2.5 of

[34], that the system (3.1) is UAS with respect to the set A.

If al(.) is class K300, then so is (12(.). Therefore, the set {x 6 Br : a2(|x|A) S p}
is bounded for any p > O. For any x0 E R", we can choose p large enough so that
x0 6 {x 6 Br : 012(l33lA) S p}. Thus, using an argument similar to the previous

one, we can conclude uniform global asymptotic stability of the system (3.1).<1

C.2 Proof of Theorem 3.3

This proof is an extension of the proof of Theorems 19.1 and 19.2 of [55] to the case
of exponential stability with respect to sets.

Define the function

V(t, C) = sup {|x(t + r,C, t; d)|Aeq'lT} (C1)
7‘ 2 0,d 6 MD

for all (t,C) E R2 0 X (20, where 0 < q < 1. Clearly, for r = 0 we have

IEIA S V0.6) (02)

196

Furthermore, using the definition of UES in Definition 3.1, we have

WM) 6 sup{/ce’lTlélAeqlT} (0.3)
720
S klflA (C4)

Inequalities (C.2) and (C4) prove (3.8).
Using the local Lipschitz property of [I A and Preposition 5.5 of [34], we have

(notice that, since A is compact, the set 90 is compact)

||$(t.€.t0; d)l,4 - [$6.77. to;d)lA| S CH6 - all (05)

for all C, 77 6 (20, all d E MD, and all t 6 [0,T], for any fixed T > 0.
There is no loss of generality in choosing k > 1. Hence, we can choose T > 0

such that k = e(1 — q)7T_ Then, for r 2 T, we have

|$(t+r.€.t;d)l,46q77 S 196—(1—qllTIEIA

S IEIA ((16)
which implies that

Now, using (C5) and (C7), we have

- lir(t + 7.77.15; d)|A leqlT}

cue — vlle‘nT

|/\

S Lllé — 77H (0.8)

197

for all C, 77 E (20, where L = Ceq'lT. Thus, we have proved (3.9). This local Lipschitz
property in x implies continuity of V(t, x) in x.
Let us prove the continuity of V(t, x) in t. Take 6 Z 0. Then, for C 6 (20, we can

write

|V(t + <16)- V(t.€)| S |V(t + 5.0 - V(t + 5.336 + 5.5.75; d))l

+ lV(t+6,x(t+6,C,t;d)) — V(t,C)| (C.9)

for d E MD. For 6 small enough, even if the trajectory x(t + 6,C,t; d)) may exit
from (20, the function is still defined and locally Lipschitz with a constant that we

denote by L. This implies

|V(t+5.€)—V(t.€)| S L||€-$(t+5.€.t;d)||

+ IV(t+6,x(t+6,C,t;d)) — V(t,C)| (C.10)

Let us discuss the second term of the right-hand side of (C.10). Let C = x(t+6, C, t; d).

By the uniqueness of solutions and a change of variable 7’ = r + 6, we can write

V(t +6,x(t+6,C,t;d)) = supT Z 0,d 6 MD{|x(t+6+r,C,t+6;d)|AeQ7T}
= supT _>_ 0,d 6 MD{|x(t+6+T,C,t;d)|Aeq’77-}

= supTI _>_ 5, d E MD{|x(t + r’,§, t; d)| Ae‘I’YTe-W}
= a(6)e-q76 (0.11)

where a(6) = supT Z 6,d 6 MD{|x(t + T,C,t;d)|Ae97T}. Given (C.11), (C.10) can

be written as

|V(t + 5.6) - V(t.€)| S L||€ - x(t + 5.6.1861)“ + lame—(175 - 9(0)] (012)

198

Given the fact that a(6) goes to a(O) as 6 goes to zero and the continuity of the
solutions of (3.1), we conclude from (C.12) that V(t, C) is continuous in t.
Finally, let us prove (3.10). Let C = x(t + h,C, t; d), h > 0. Again, by uniqueness

of solutions and a change of variable, we have

V(t + h,C) = supT 2 h, d E MD{|x(t + h + r, C, t + h; d)|_Aeq7T}e_q7h
< Wage-‘17" (013)

Using (C.13), we have

 

. V(t + h, 6') — V(t, 6) . e-Wh — 1
1 < v t, lm _—
h 510 h ‘ ( E) h L. 0 h
S ~97V(t.€) ' ((114)
Take /\ = q7.<1

199

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200

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