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dissertation entitled

OUTPUT FEEDBACK CONTROL FOR
NONLINEAR SYSTEMS USING VARIABLE
STRUCTURE CONTROL

presented by

Seungrohk Oh

has been accepted towards fulfillment
of the requirements for

 

Ph.D. degreein Electrical Engineering
H.K. Khalil It
Major professor

Date 6/21/1994

 

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OUTPUT FEEDBACK CONTOL FOR
NONLINEAR SYSTEMS USING VARIABLE
STRUCTURE CONTROL

By

Seungrohk 0h

A DISSERTATION
Submitted to
Michigan State University

in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering

1994

ABSTRACT

OUTPUT FEEDBACK CONTOL FOR
NONLINEAR SYSTEMS USING VARIABLE
STRUCTURE CONTROL

By

Seungrohk 0h

We study stabilization and output tracking for feedback linearizable systems using
output feedback variable structure control in the presence of modeling error and/or
external disturbance. In particular, a stabilization problem is studied for a nonlinear
system that can be transformed into a normal form with no zero dynamics. A tracking
problem is studied for a nonlinear system that is represented by an input-output
model, which can be transformed into the normal form with asymptotically stable
zero dynamics. In both cases a robust high-gain observer is used to estimate the state
variables while rejecting disturbances. A globally bounded discontinuous variable
structure controller is designed to compensate for modeling error and/or external
disturbance. For the stabilization problem, we show that the controller can stabilize
the closed-loop system and does not suffer from the peaking phenomenon which exists
in previous designs. In the tracking problem, we show that the controller ensures

tracking of the reference signal in the presence of unknown time-varying disturbances

and modeling errors. We give regional as well as semiglobal results, but we do not
require exponential stability of the zero dynamics nor global growth conditions. As
an application, we design a controller, with only position measurement, that ensures
tracking of a desired path for an 11 degree of freedom robot manipulator. A high-gain

observer is used to estimate the angular velocities.

To my parents and family

ACKNOWLEDGMENTS

I would like to express my gratitude to my academic advisor, Professor H. K.
Khalil, for his invaluable guidance, patience, help, and support during the course of
this work. I appreciate guidance committee members, Professors R. A. Schlueter, C.
R. MacCluer, and F. M. A. Salam, for their recommendation. I wish to thank Korea
Electric Power Cooperation for financial support. Finally, I would like to express my
appreciation to my wife In-Ja Park, my two lovely sons Yungeon and Jungeon. This
work would not have been possible without their sacrifice and support.

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

1

INTRODUCTION
1.1 Variable Structure Control ........................
1.2 Robust High-gain Observer and The Peaking Phenomenon ......

OUTPUT FEEDBACK STABILIZATION

2.1 Introduction ................................
2.2 System Description ............................
2.3 Controller Design .............................
2.4 Semiglobal Stabilization .........................
2.5 Example .................................
2.6 Conclusions ................................

OUTPUT FEEDBACK TRACKING

3.1 Introduction ................................
3.2 Problem Statement ............................
3.3 Controller Design .............................
3.4 Semiglobal Tracking ...........................
3.5 Conclusions ................................

ROBOT MANIPULATOR CONTROL

4.1 Introduction ................................

4.2 Manipulator Dynamic Model and Problem Statement .........
4.2.1 Physical Robot Properties ....................
4.2.2 Problem Statement ........................

4.3 Controller Design .............................

4.4 Example ..................................

4.5 Concluding Remark ............................

vi

viii

ix

1
6
9

12
12
13
16
29
31
37

38
38
39
45
55
58

59
59
60
61
61
61
71
82

5 Conclusions and thure Work

5.1 Conclusions ................................
5.2 Future Work ................................

A Proof of Lemma 2.1

B Proof of Lemma 2.3

BIBLIOGRAPHY

vii

83
83
84

85

88

90

LIST OF TABLES

4.1 Physical parameters of the manipulator.

viii

LIST OF FIGURES

1.1 The sliding mode .............................
2.1 The comparison of the state variable 2:1. ................
2.2 The comparison of the state variable 32. ................
2.3 The comparison of the estimates of state variables. ..........
2.4 The comparison of control inputs .....................
2.5 The phase portrait of each case ......................
2.6 The comparison of state variable $2 with e = 0.02 ............
4.1 Two Link Manipulator ..........................
4.2 Tracking of the desired path in Cartician coordinate with discontinuous

control input. ...............................
4.3 Tracking of the desired joint angles 0;. .................
4.4 Tracking of the desired angular velocities 0,. ..............
4.5 The phase portraits of estimates of errors é. ..............
4.6 Applied torques 7'1 and 7'2. ........................
4.7 Tracking of desired path in Cartician coordinate with continuous ap-

proximation, p = 0.05. ..........................
4.8 Tracking of the desired joint angles 0; with continuous approximation,

p = 0.05 ...................................
4.9 Tracking of the desired joint angles 9,- with continuous approximation,

p = 0.05 ...................................
4.10 Applied torque 7' with continuous approximation(p = 0.05) ......

4.11 The control input u with continuous approximation(p = 0.05)
4.12 The plot of control input u with the same initial condition(p = 0.05)

ix

71

76
76
77
77
78

79

79

80

80

81
82

CHAPTER 1

INTRODUCTION

Nonlinear control theory for feedback linearizable systems has been developed in
the last decade. The idea of feedback linearization is to cancel nonlinearities of a
nonlinear system. Since a nonlinearity can not be canceled in every nonlinear system,
there must be a certain structural property of the system that allow us to perform

such cancellation. For instance, if the nonlinear state equation is given by
:i: = Asc+BB'l(x)[u—a(a:)] (1.1)

where A and B are constant matrices, the pair (A, B) is controllable, and the matrix
[3(x) is assumed to be nonsingular for every 3:, then we can linearize equation (1.1)
via the state feedback

u = a(:r) + fl(:1:)v

to obtain the linear state equation
:r = A3: + Bv

Since the state model of a system is not unique, a nonlinear state equation might

be transformable into the form of (1.1) even if it is not originally in that form. If

a nonlinear system can be transformed to the form of (1.1), the nonlinear system
is called a feedback linearizable system [1, 2]. Feedback linearizable systems are

classified as follows [1, 2]:
0 State space linearizable system
0 Input-output linearizable system
0 State space and input-output linearizable system

To describe the classification of feedback linearizable system, consider a nonlinear

system

where f : U —-> R", G : U -—> RM”, and h 2 U —-> R” are sufficiently smooth
on a domain U C R". The nonlinear state equation (1.2) is said to be state space
linearizable if there exists a diffeomorphism T : U —> R" such that the change of
variables a: = T(z) transforms (1.2) into the form (1.1). The nonlinear system (1.2)-
(1.3) is said to be input-output linearizable if there exist a diffeomorphism T : U —+ R"
such that the change of variables :1: = T(z) transforms the system (1.2)—(1.3) into the

‘ form

:3: 2 Ax + BB-l($,9)lu ’ GUM!”

d = ¢(x,q)

y=Crr

where 1‘ E R’, q E R""', C is m x r, and d) : R" —+ R" is defined in a domain

T(U) = D C R”. Clearly, with u = cr(:r,q) + B(:r,q)v, the v—y behavior is linear.

If the function ¢(-) is linear with respect to its arguments, then the system (1.2)-
(1.3) is said to be state space and input-output linearizable. Hence a state space
and input-output linearizable system is a special case of an input-output linearizable
system. A comprehensive treatment of feedback linearizable systems using differential
geometry can be found in [2]. After linearization using feedback, we can use the
well developed linear control theory to achieve control tasks such as stabilization and
tracking. However a system can have modeling uncertainty and be subject to external
disturbance. Robust control schemes such as variable structure control(VSC) and
Lyapunov redesign have been widely used to compensate for modeling uncertainty
and external disturbance. Variable structure control‘ has been successfully used
in various applications [3, 4, 5] due to its robustness to modeling uncertainty and
disturbances [6]. The purpose of this dissertation is the design of control laws to
achieve stabilization and output tracking for input-output linearizable systems using
output feedback variable structure control in the presence of modeling error and/or
external disturbance. Most of the work on VSC assumes state feedback. There have
been, however, some efforts to develop output feedback VSC. These can be classified
into two classes of output feedback schemes. One scheme is direct output feedback
VSC which does not use an observer to estimate the states of the system [7, 8, 9] but
is limited to relative degree one systems. For instance, the paper [8] considers the

system

:i: = F(:r)+Bu

y=C:r

where the state a: E R", the input u E R”, the output y 6 Rm, B is the n x

m matrix, C is the m x 72 matrix. The paper [8] requires the matrix CB to be

 

’More description of VSC is given in Section 1.1.

nonsingular and designs the control input u as a function of the output. Under this
condition, the system has relative degree one. Relative degree one systems exclude
many applications. The other scheme is observer-based controllers which could be
used for relative degree higher than one systems [10, ll, 12, 13]. Papers [10, 11]
treated linear systems assuming perfect knowledge of the state model, leading to a

standard observer problem. For instance, the paper [10] considers the system

2': = Az+Bu

y=Cx

where the state x E R", the input u E R’”, the output 3; E R”, A is n x n matrix.
B is n x m matrix, C is m x 72 matrix, and (A,C) is observerble. It constructs the

observer

i=Ai+Bu+L(y—Ci~)

where the observer gain L is n x m matrix. The observer gain L is designed such that
(A— LC) is a Hurwitz matrix. Hence the error (e = :r —:ic) equation is e = (A— LC)e,
and e —» 0 as t —» 00. Using the variable structure controller and exponentially
decreasing error e, asymptotic stability is achieved. Papers[l2, 13] treated nonlinear
state models with uncertainty and/or external disturbance, and used robust high-
gain observers1 to reject the disturbance due to modeling uncertainties and imperfect
feedback cancellation of nonlinearities. However, the use of high-gain observers of
relative degree higher than one results in peaking in the state variables as well as
shrinking of the region of attraction [14]. One method, described in [14], to eliminate
the peaking phenomenon and the associated shrinking of the region of attraction is

combining a high-gain observer with a globally bounded controller. Since this idea

 

IMore description is given in Section 1.2.

was introduced in [14], it has been used in a few designs of continuous output feedback
control [15, l6, 17, 18, 19, 20, 21]. In particular, it is a very useful tool to achieve
semiglobal output feedback stabilization where the controller is designed to include
any given compact set in the region of attraction. In [14, 15, 16, 17, 18, 19. 20, 21]. a
Lipschitz continuous state feedback controller is designed to stabilize the origin. Then
a high-gain observer is constructed, and the closed-loop system is represented in the
singularly perturbed form. Using the Lipschitz property of the controller, it is shown
that the reduced model of the singularly perturbed system is the closed-loop system
under state feedback, and the closed-loop system under output feedback recovers the
performance of the closed-loop system under state feedback when the observer gain is
sufficiently high. This recovery approach can not be used in variable structure control
since it is discontinuous in the state variables. Therefore, the design procedure and
analysis in a variable structure control scheme has to be different from the approach
used in [14, 15, 16, 17, 18, 19, 20, 21]. We establish properties of robust high-gain
observers which are valid for any globally bounded control. The use of the high-gain
observer and the globally bounded control enables us to show that the estimation
error decays to arbitrarily small values during a short transient period. We use this
fact to design the variable structure control as a feedback function of the estimates
of the state variables. The sliding manifold is chosen in the observer space and a
globally bounded output feedback VSC is designed to satisfy the reaching condition
to the manifold. After showing that the trajectories reach the sliding manifold in
finite time, analysis is done on the sliding manifold to show stabilization or output
tracking. It should be noted that in [12] and [13] a scheme similar to ours is used,
namely, a high-gain observer is designed first and then the variable structure control
is designed as a function of the state estimates to ensure attractivity of the sliding
manifold. However, in both [12] and [13] the attractivity of the manifold is achieved by

designing the control to cancel or dominate ” peaking” terms. Hence the performance

of the closed loop system suffers from the peaking phenomenon. We design a globally
bounded control by requiring the attractivity of the manifold to hold for all t 2 T > 0
for arbitrarily small T, rather than for all t 2 0. In Chapter 2, we consider the
stabilization of a class of multivariable nonlinear systems, about an equilibrium point
at the origin. In particular, the system can be transformed into a normal form with
no zero dynamics. We show that a globally bounded VSC with a high-gain observer
can stabilize the origin regionally and semiglobally. In Chapter 3, we generalize the
results of Chapter 2 in two directions. First, we consider a class of nonlinear systems
which have zero dynamics. In particular, we consider a nonlinear system represented
by an input-output model which contains unknown external disturbances. Second, we
solve the more general tracking problem in the presence of disturbances, as opposed
to the stabilization problem of Chapter 2. In Chapter 4, as an application of results
of Chapter 2, we consider the trajectory tracking control problem for an n degree of
freedom(DOF) robot manipulator, using only position measurements. A robust high-
gain observer is used to estimate angular velocities. The controller design is different
from the design of Chapter 2. The controller in Chapter 4 satisfies a sliding mode
condition in the intersection of the sliding manifolds, while the controller in Chapter

2 satisfies a sliding mode condition in each sliding manifold.

1.1 Variable Structure Control

The idea behind variable structure control is to choose a suitable switching manifold
in the state space, and switch the control input on this manifold. The control input is
chosen to guarantee that trajectories near the switching manifold are directed toward
the surface. Therefore, trajectories of the system repeatedly cross and immediately
re-cross the switching manifold. Consequently, trajectories remain in the switching

manifold(a(t) = O) as illustrated in Figure 1.1, and it is called sliding mode. The

 

Figure 1.1. The sliding mode

dynamics of the system are then effectively constrained to lie within a certain manifold

of the full state space. For instance, consider the system

i=Ar+B[f(:c)+u] (1.4)

where x E R", the control input u E R, the function f : R" —-+ R, and (A. B) is
controllable canonical pair. Suppose that we choose the switching surface (or sliding

surface) as

0(a) = Ma:

where M = [m1,...,m,,_1,1]. and design the control input a = —f(x) — sgn(a(:r)).
Since %%(a(z)2) < 0, the trajectories of the system (1.4) always move toward sliding
manifold 0(a) = 0. It is shown [22] that if %%(a(z)2) < 0 when 0(x) ;£ 0, there is a
finite time t1 such that the trajectories of the system (1.4) reach the sliding manifold
0(3) = 0 and remain thereafter. Therefore the dynamics of the system on the sliding

manifold is given by

i=(A—BMA)::: (1.5)

Since we have the freedom to choose the sliding manifold, we can choose the matrix
M such that (1.5) has a desired behavior. Note that since the control input is discon-
tinuous, the right hand side of the differential equation does not satisfy the existence
and uniqueness conditions of the solution of differential equations in the conventional
way. Filippov [23] proved the existence and uniqueness of the solution in a generalized
sense. Various design methods for variable structure control are given in [3]. Variable
structure control schemes can reject disturbances due to modeling uncertainties and
external disturbances under the matching condition. To illustrate this point, consider

the system

i=Ax+B[f(:r)+d(t)+u] (1.6)

where d(t) is an bounded external disturbance. Let fo(:r) be a known nominal model
of f(:r). Suppose that we know the upper bound of modeling uncertainty and d(t),

i.e.,

|f($)-fo(:r)| S pt?)
|d(t)l < k

where p(:r) is a nonnegative locally Lipschitz function and k is a nonnegative constant.
Choosing the sliding surface 0(1) = Mar and u = —fo(:c) — (p(:r) + k)sgn(a(.r)), it can
be shown that %%(U(I)2) < 0. Hence the trajectories of the system (1.6) reach the
sliding manifold 0(3) = 0 and remain thereafter. Therefore dynamics on the manifold
are given by equation (1.5). We can observe that the dynamics on the manifold are
independent of modeling uncertainty and disturbance. Such a property of variable
structure control is called invariance property [6]. This property implies that variable
structure control can reject disturbances due to modeling uncertainties and external
disturbances. However the ideal behavior of sliding mode controllers is achieved in

the theoretical limit as the switching frequency becomes infinite. In practice the

small, but nonzero, delay in control switching will cause chattering around the sliding
manifold. This is not a desirable feature. One of the remedies for chattering is the
use of continuous approximations [24], e.g., the signum nonlinearity is replaced by

sat(:r) where

 

r

l :r>/1
Smhfi=<; -#SxS#

k—l :1:<—/1

However the continuous approximation results in uniform ultimate boundness around
the origin rather than asymptotic stability in the stabilization problem [25]. There-
fore we have to trade off performance precision vs chattering; a trade off which is

application dependent.

1.2 Robust High-gain Observer and The Peaking
Phenomenon

For simplicity, consider a single-input single-output system

2': = Am+B[f(:r)+g(:r)u] (1.7)

y = C2: (1.8)

where z E R", u is control input,

 

 

 

 

.0 1 .0. .0-
0 0 1 0 0
A: ,8: ,C= 10 01
0 0 1 0
_0 O-an bl-nxl

10

f(-) and g() are sufficiently smooth in a domain U C R", and g(m) is nonsingular.
Va: 6 U. Suppose nominal models of f(~) and g(-) are f0(-) and go(-), respectively.

Construct the observer

éer+Bmfin+ammwn+Lw—Cfl (rm

Let e = :r—:i:. Then

é=(A—LC)e+B(5(:r,e) (1.10)

where 6(x, e) = f(:c) — fo(.i:) + (g(x) — 90(2))u(i:). The 6 term consists of two terms:
a term due to the uncertainty in the state equation (1.7) and a term resulting from
imperfect cancellation of nonlinearity due to the use of the estimate of state rather
than the state itself. The observer structure should be able to reject the deriving term
6. We call such observer a robust observer. This can be achieved by designing the
L = L(e) such that (31 -- A + L(e)C)"1B —+ 0 as a design parameter c ——> 0. Singular
perturbation is a powerful tool to design such L(e). Let L = [0:1,- - - ,an]T, where a,

is chosen such that (A — LC) is Hurwitz. Choose the specially scaled observer gain

Let C,- = 533e,. We can rewrite equation (1.10) as

eé=(A—LCK+«Bé H1”

The perturbation term is multiplied by e. Hence its effect diminishes asymptotically

as c —+ 0. Since we want to allow the initial condition ”((0)” S 595-, where c, is

l At/c

positive constant, the solution of (1.11) contains terms of the form (M, e . Since

 

:r(t) is coupled with e through the control input, :r(t) exhibits an impulse-like behavior

11

as e -+ 0. We call this behavior the peaking phenomenon. This is not a desirable
feature. Elimination of peaking phenomenon in :r(t) is achieved in [14] by a globally

bounded continuous controller.

CHAPTER 2

OUTPUT FEEDBACK
STABILIZATION

2.1 Introduction

We consider the stabilization of a system that can be transformed into a normal
form with no zero dynamics, about an equilibrium point at the origin, using variable
structure output feedback control. A robust high gain observer is constructed to reject
the disturbance due to modeling uncertainties and imperfect feedback cancellation of
nonlinearities. We start by establishing properties of the robust high-gain observer
which are valid for any globally bounded control. In particular, we show that the
estimation error can be brought to arbitrarily small values in arbitrarily small time
intervals. Then we use this fact to design the variable structure control as a feedback
function of the estimates of the state variables. The variable structure control drives
the estimates of the states to a sliding surface in finite time. Once on that surface we
perform stability analysis of the closed-loop system to confirm that the trajectories
are ultimately bounded by a bound which can be made arbitrarily small by designing
the observer to be sufficiently fast. In the special case when there is no uncertainty

in the control input coefficient, we show asymptotic stability of the origin. Also,

12

13

we show ultimate boundness of trajectories and asymptotic stability for any given
compact set of initial conditions. We give an example to illustrate the performance

of our design and compare it with previous designs [12, 13].

2.2 System Description

Consider the multivariable nonlinear system

w = f<w>+Zg.-(w)ut (2.1)
i=1

y = h(w) 12.2)

where w 6 R" is the state, it E Rm is the control input, y E Rm is the measured
output. We assume that f, g,-, and h are sufficiently smooth in a domain U C R", and
f(0) = 0, h(0) = 0. Hence, the origin w = 0 is an equilibrium point of the open loop
system (2.1)-(2.2)(i.e., when u = 0). Before we state the class of nonlinear systems

we shall deal with, we recall the following definition.

Definition 2.1 [26] Let X and Y be two nonempty subspaces of R", and let T :

X -—i Y. The map T is said to be proper map ofX into Y if the inverse image under

T of each compact subset on is a compact subset ofX.

Assumption 2.1 For all to E U,

o The system (2.1)—(2.2) has a uniform vector relative degree {r1, . . . ,rm}, i.e.,
L.,h.-(w) = = L9,L}""’h.-(w) = 0
for all 1 S i,j S m, and the matrix

A(w) = {02100)}: {L..L}"‘h.-(w)}

14
is nonsingular.
o n=r1+---+rm.

o The mapping a: = T(w), defined by

and

_ 1 l i i m m
x— [:rl,...,:rrl,...,1‘l,...,;z:

is a proper map" of U into T(U).

It is well known, e.g., [2], that the uniform vector relative degree assumption is a
necessary and sufficient condition for the mapping a: = T(w) to be a local diffeomor-
phism in the neighborhood of every w E U. Moreover, the origin :1: = 0 in the new
coordinates belongs to D = T(U). The change of variables a: = T(w) transforms the

system (2.1)-(2.2) into the normal form

where C(z) = {G';(:::)} is nonsingular, Va: 6 D. The properness of the mapping
a: = T(w) ensures that it is a diffeomorphism of U onto T(U), c.f., [26]. Therefore,

the normal form is defined for all a: 6 T(U). The normal form can be rewritten in

 

‘For a semiglobal result in Section 2.4, the mapping T is required to be a proper map of R” into
R". This is equivalent to lim||z||-ooollT(-r)ll = 00 [26].

15

the compact form

2': = Ax+ B[F(:r)+G(:r)u] (2.4)
y = Cr (2.5)
where r _
0 1 0
0 0 1 0

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

 

 

 

 

 

0 0 1
0 0
0
0 F1
B=block diag[B1,...,Bm], B,= S , F:
0 Fm
l
- dr.xl
C=block diag[C1,...,Cm], Ci: 1 0 0
. lxr.

 

Let Fo(:c) and Go(:r) be known nominal models of F(:r) and (1(3), respectively. Sup-
pose that Fo(:r) and (30(3) are sufficiently smooth, 170(0) = 0, and 00(1') is nonsingu-
lar for all a: 6 D. We assume also that 170(3) and Go(.r) are globally bounded. This
can be always achieved by saturating the given nominal functions outside a bounded

domain of interest, as it will be illustrated later on.

Remark 2.1 The system equation (RU—(2.2} can be dependent on a vector 0 of
unknown constant parameters which belongs to a compact set 9 as long as Assumption
2.1 holds for all 0 E G. In this case, the mapping T() will depend on 0. This means

that the new state a: would not be accessible even if the original state w was available

16

for measurement. This point will not cause a problem in our design since the controller

will use only measurement of the output y.

The uncertainty in equation (2.4) satisfies the matching condition, which is a typical
assumption in the design of robust variable structure control [3, 4]. We make the

following assumption on the uncertainty.
Assumption 2.2 For all :1: E D,

0 there is a scalar nonnegative locally Lipschitz function po(x) such that

||F($) _' F0(a:)||oo S (00(35) (‘3-6)

[\—
‘1
v

IIG(x)Ga‘(r) — III... 3 k <1 1'

where k is nonnegative constant.

2.3 Controller Design

The output feedback controller is an observer-based one, which is designed to be
globally bounded to eliminate the peaking phenomenon. We construct the estimate

of the state at using the observer

*' *1 Oi .i ,

IE] : xJ+1+:(yi—xl)i 3:19 art—1 (25)

in z 6;: (31,41) + F0(i:) + [Gwen (2 9)
i=1

where i;- is the estimate of the state variable 13; and e is a positive constant to be

specified. The positive constants a; are chosen such that the roots of

s" + (1]3'"l + - - - + c1(:'.|_,s1 + 0:. = 0

17

are in the open left-half plane, for all i = 1, . . . ,m. We rewrite the observer equa-

tion (2.8)—(2.9) into the compact form
:2: At+B[F0(:t)+GO(i)u(:t)] + Dances—5:) (2.10)

where

L = block diag[L1, . . . , Lm], L,- =

 

 

r,x1

1 l l

 

D(e) = block diag[D,(c),,,,,Dm(e)], 0,-(6) = diage

Let ej- = x; — it; be the estimation error, and define the scaled variables

 

It can be shown that the closed-loop system is given by

:i: = Ax+B[F(:r)+C(:r)u(i')] (2.12)

:6 = (A—LC)<+CB(F(x)—Fo(i)+10(x)—Go(:e))u(a‘c)1 (2.131

where

C:[C11,...,Crlli...,C[,...,C:.,---aCinv°"va?an

It can be also verified that (A —- LC) is a Hurwitz matrix. For small 6, the closed-loop
system (2.12)—(2.13) is a singularly perturbed system with a: as the slow variable and
C as the fast one. We choose the sliding surface o'(§:) 2 [01(5), . . . ,a,,,(:i:)]T such that

o,(:i:) = :i' + m;'_1:i:','_1+...+ m‘lit'l, l S i S m, where m;- are chosen such that A,-

18

is Hurwitz where

 

 

F
0 1 0
0 0 1 0
,1,- =
0 0 1
_mi —m‘ —m‘ —m
1 2 "‘2 "'1 4 (r,—1)x(r,—1)
Rewrite C(53) as
o(:i:) = Mi‘ (2.14)
where
M=blOCk diag[M1,...,Mm], M5: [772'1 'rn‘2 m; —l 1
Define
. , T
z,=[:r'1,...,1:;._1] , z=[zl,...,2:m]T

5.: [q,..., ;._,]T, and z: [5,,...,§,,]T

Similarly define 2, and 2. Rewrite equation (2.8) as
3: A2+Ba(e)+[)(e)iéé (2.15)

where

A = block diag[A1,...,Am], B = block diag[B1, . . . , {3m}, 19.:

 

 

L - (r,—l)xl

19(6) = block diag[d1(e),...,dm(e)], d,(c) = diag[e"'2,e"'3,...,c,l]

19

L = block diag[L1, . . . ,Lim], L.- =

 

 

(r,—1)Xl

ézblOCk diagléla°'°7€m]s Ci: 1 0 ... 0

1X(r.—l)
We consider a control input of the form u, = (pJ-(i) + vj(:i:)sgn(oj(.ic)) where th‘(i‘)
and vj(:ic) are continuous and globally bounded functions. We will specify uJ- later on.

Since A is a Hurwitz matrix, for any positive definite matrix Q, there is a symmetric
positive definite matrix P such that PA + ATP = —Q [1]. Let V(z) = zTPz. Choose

the positive constants c,, and r such that l

a. ‘5—5‘ {x e R" | Ierrll s c...(/V(z) s c..1c D

where
(12 .
c2, = r—csr (2.16)
01
a1 = %, a2 = 2A FEW), and r > 1. The set 9,. is taken as the region of interest

in our analysis. The motivation for choosing Q, in .this form will become clear as we
proceed with the analysis. Achieving globally bounded functions Fo(:r) and Go(:r)
can be done by saturating these functions outside 9,. The following inequalities hold

almost everywhere for :1: 6 51,, C E R"

“17(3) + G($)U(i)ll _<_ k1||$||+ k2 (‘-

[\r
p—
‘1

||[F($) - F0(4)] + (C(13) - Go(i))U(5=)l

 

s kalle + k4IICll + 1., (2.18)

 

1The notation || - I] without a subscript denotes the Euclidean norm [I - ||2

20

where k1 to to; are nonnegative constants. Notice that the right-hand side of (2.17)
does not depend on the fast variable C, which follows from the use of a globally

bounded control input u(:i7). Let

00 = {(a: 6 R" | ”1142:“ S cso, (/V(z) S czo}, cso < c“ and cm < C”

.. n C
91={€€RlllCl|<€(,—_1;}
n = 9,, x11, ('3-19)
where “y = max,-{r,-} for all i = 1,...,m and c is an arbitrary positive constant.

Notice that 90 C 9,.

Lemma 2.1 Consider the singularly perturbed system (212)—(213) and suppose that
Assumption 2.2 is satisfied. Then, for all (:r(0),((0)) 6 9, there exist el and T1 =
T1(e) S T3 such that for all 0 < e < 61, ”(II < lee for allt E [T1,T4) where T3 is a

finite time and T; > T3 is the first time :r(t) exits from the set 0,.

Proof: Appendix A

Note that Lemma 2.1 was proved in [14] under slightly different conditions, but.
the proof is given in Appendix A for completeness. It is shown in the proof of
Lemma 2.1 that the fast variable C decays very rapidly during a short time period
[0, T1]. We design the control input u(;it) such that a sliding mode condition is sat-
isfied when ”(H < he and (1,513) 6 Q, X $1,. This will be done by showing that

0(§:)T0(:i:) S —az|[0(:i:)||1 as long as 0(2) 74 0 where 02 is a positive constant. We

0T(.i:)0(i:) = (#935;

as
UTMW + B(Fo(:i:) + Go(:ic)u(:it)) + D(e)LC(:r — fl]

= 0TM[A:i + B(Fo(:i) + Co(:i".)u(:i:)) + -:-B(e)LCC] (2.20)
where
B(e) = block diag[d1(e),...,dm(e)]
d,(e) = diag[e"'1, 6"‘2, . . . , e, 1]
'We rewrite equation (2.13) as
64' = (A — LC)C + eB[{F(i‘) — 13(2)} + {F(:r1— m1} +
{G(z) — G(i)}u(i1+{c(aca‘(i)— 1100012421] (2.21)

For simplicity, we set

an?) dé‘ two—Foo)
we) “é‘ Fen—Fen {mo—Genus)

an) ‘5 {G(i)Ga‘(:21—I}Go(:21u(i)
Let n = (+ e(A — LC)"B¢1(:i:). The last term on the right-hand side of equa-
tion (2.20) is

(.5)ch = [mourn—4A—Lcr‘Bwn

alt—-
b.

= imam” :1; B¢,(e) + 0(e), for a e n. (2.22)

where we used the fact that B(0)LC(A — LC)‘1B = :tB. Therefore, equation. (2.20)

is given by

0T(:i')0(i:) = 0TM[A:i: + B(Fo(§7) + Co(:i)u(:i:)) + %B(e)LCn

(\D
[O

:tBa'Mi‘) + 0(6)] (2.23)

We need an estimate of %MB(e)LCn to design the control input u(:i') such that the
sliding mode condition is satisfied. The forthcoming analysis is done for t 6 [T1, T4).
By inequality (A.4) of Appendix A, ||D(e)LC(:L‘ — 5:)” S b, for some positive constant
b. Hence, from equation (2.10), it is bounded uniformly in 6. Since F(-) and RH

are smooth enough on Q, 1 l1?“ and ”£351” are bounded for all 1" E QT. Hence.

there is a positive constant c1 such that
[[(A — LC)"IB<l>1(:i3)|] S CI for Vi‘ 6 Q, (2.24)
Using (2.21), we obtain

a) = eé+e2(A—LC)-‘Ba'>.(i)
= (A — LCx + eBtito) + no.2) + (130)}
+801 — LC)"B<2>1(:(‘1)
= (A — LC){n - .(A — LC)"B¢1(i~)} + eBtilti) + (to. 5c)
+¢>3(i)} + 61A — LCWBcisle)

= (A — LC)n + cB{¢2(x,:i:) + 03(2)} + e2(A — LC)'IB<l>1(i‘) (2.25)
It can be verified that the solution of equation (2.25) can be represented by

,7“) = C(A-LCW-TifltmTl)

+ T' eM-LCW-Ti/‘B[F(x) — Fat) + {C(x) — G(:iv)}'v(T)lal‘r

t
+ T e‘A‘LC)(“”/‘B{G(i)Gg‘(:t) — I}Go(:i)v(r)dr

t .
+C/T eM"LC)(“'l/‘(A — LC)"B¢1(:i:)dr

23

for (:r,:i:) 6 Q, x Q, where v(t) E I\'{u(:i'(t)} for almost all t and the convex hull
K{u(.i:(t))} is defined in [23]. Since F(—) and G(-) are locally Lipschitz on D and 11(2)

is globally bounded, there is a positive constant L1 such that
||F($) - F(:i:) + (C(17) - C(i))v(t)ll S L1HC|| S L26 (2.26)
for (:r,:i:) E Q, x Q, where L2 = ch1. Using (2.24), (2.26), and (A.4) of Appendix A.

illMl’?(c)LCn(t)lloo S -:-llMD(€)LC|loollle‘A""C""T"/‘9(T1)lloo
t .
+|| /T1 e‘A'LC)('-Tl/‘[e(A — LC)-IB¢1(.i‘)
+B{F(;r) — F(:t) + (0(2) - Gti‘))v(7)}1dTHa1
‘ t
+l||MD(e)L/ Ce‘A'LC)(""/‘B
C T]

x {C(:i‘)G'6l (:i‘) — I}Go(;i:)v(r)d7'|]oo

|/\

, t
%()M19(0)L/T Ce<A-LC)<'-*i/<B

x{G(:i)Gg‘(i~) — I}Go(i‘)v(r)d‘rlloo + kge (2.27)

for t 6 [T1 + 6 ln %, T4) where k3 is some positive constant. Using Assumption 2.2 and

the fact that u(i:) is globally bounded,

l|{G(i)GJI(i) - 1}Go(i)v(i)lloo S IIG(i‘)G61(i)- IlloollGO(i)v(tllloc

g (we, (2.23)

where ha is an upper bound on Go(i)u(;i:) to be specified. Since (A — LC) is Hurwitz,

it can be shown that

~ t
£((MD(0)L fT CeiA’LCH“’)/‘B{G(5:)Ggl(5:) -— I}Go(5:)v(r)drlloo ;<_ kkukt

24

where k, = maria; f0°° |h,-,-(t)]dt 2 1 and 11,-,(t) is the ith diagonal element of diagonal

matrix CeiA'Lcl‘B. Therefore,

1 . ‘ 1
-HMD(e)LCnHoo S k8: + kk..k.. Vt 6 [T1 + e ln —,T.,) (2.29)
e e
When It. > 1, we need to strengthen inequality (2.7) as follows
1
||G(x)Gg‘(x) — Inc. 3 k < T’ V2: 6 D (2.30)
1

Remark 2.2 When we choose all eigenvalues of (A — LC) to be real and negative.
ha“) 2 0 f0? allt _>_ 0 [27]. Then 0:. [:0 ]h;,'(t)]dl = a; [00° hi,‘(f)dl = OLH,‘,‘(O) = l
where H(s) = C(sI — A + LC)’1B. Thus k, = 1.

Consider the function

«2(2) ——— Ga‘tfll—Mmi‘ + Benin — (psi-1+ ansgnoom (2.31)

where
sgn(0(:i:)) : [sgn(01(:i:)), ' ' ° 9 Sgn(UM(i))lT

and 01 is any positive constant that satisfies

kktk, + 202

 

.) : .2
a‘ - l—kk, (”3)

for some 02 Z 0 and k, is defined by the inequality
ll - MW? + BF0(5')) - po(i)sgn(o)lla s k. (2.33)

for almost all :i: E 0,. Notice that k, can be calculated since Fo(o) and p0(-) are

known. We take the control input 21(2) as 111(2), saturated outside the set $1,. In

25

particular, let 1121(2) = —Ggl(:i‘).M(A.if + BF0(:i:)), 1112(2) = -C61(:i‘)(p0(:i')+ 01) and

5,1 = maxien, [1123(2)] and take

 

 

1,.1 ‘ - ’4’ ‘
111(2) 2 S} sat (kg-114) + 3.2 sat (wig-5)) sgn(01(.i‘)) (2.34)

where sat(-) is the saturation function and (113(2) denotes the ith component of the

vector W(:E). Inside the set 9,, we have

Hence, ku in inequality (2.28) can be taken by
lcu = k, + 01 (2.36)
Using Assumption 2.2, (2.23), (2.29) , (2.35), and (2.36), we obtain

0T(i)<'7(5=) S UTIMMi + B(Fair) + Go(i)U(i')) i B¢1(i)} + 0(6)}

+kk.(k, + a1)lla(:t:)|1

 

 

S —(Po(5?) + 01)l|0(5)11+(P0(i)+ 0(6))lla(i)lli
+4491“: + Ollllaiillll
S —02“0(i‘)lll

for sufficiently small e, where we used the fact that MB = I. We summarize our

findings in the following lemma.

Lemma 2.2 Consider the singularly perturbed system (2.12)—(2.I3} with the control
input 11(2) defined by (2.34). Suppose that [[(H < lite and (3),;i') 6 Q, x Q, fort E

26

[T1,T4), and e is small enough. Then as long as 0(2) 75 0, the sliding mode condition

ohm-(.2?) 5 41211012111]

is satisfied for all t 6 [T1 + e ln %, T4).

So far we showed that ”(H < he for t E [T1,T4) and the control input 11(2) (2.34)
satisfies the sliding mode condition when ”(H < he and (2,57) 6 Q, x Q, for all
t 6 [T1 + eln %,T4). It is shown in the following lemma that for sufficiently small e,

“(H < he and ($,i‘) 6 (I, X 51, for all t 2 T, for some T > 0.

Lemma 2.3 Consider the singularly perturbed system (2.12)-(2.13) , (2.34) and sup-
pose that Assumption 2.2 and (2.30) are satisfied. Then, for all (:r(0), ((0)) E 9, there
exist 62 > 0 and T = T(e) S T2 < 00 such that for all 0 < e < €2,(l’,.’l‘) E (I, x 9,.

and “(u < he for all 1 2 T.
Proof: Appendix B

Theorem 2.1 Consider the system (2.12)—(2.13). Suppose Assumption 2.1? and
(2.30) are satisfied. Let the observer gain be chosen as in (2.8)—(2..9) and the control
input be chosen as in (2.34). Then there is 63 > 0 such that for all 0 < c < e3,
the closed-loop system (2.12)—(2.13) and (2.34) is uniformly ultimately bounded with
respect to the set fl, = {($,C) E R" x R" I [[2]] S kp\/E,]]C|| < lee}, for some
nonnegative constants k,, and ls, and fl, defined by (2.1.9), is an estimate of the region

of attraction.

Proof: We have already proved in Lemma 2.2 that

1
T(:i:)0(:i:) S —02||0(:i:)||1, for all t E [T1+ eln :,T.,) (2.37)

q

as long as 0(i‘) # 0 provided that (r(t),:i‘(t)) E Q, x Q, and ”(H < he fort 6 [73.711).

Also, we proved in Lemma 2.3 that (2,53) never leave the set (I, x Q, and ”(Ill S he

[\D
‘1

for all t Z T. It was shown in the proof of Lemma 2.3 that T 2 T1 + e In % Hence.
inequality (2.37) holds for all t 2 T. Inequality (2.37) implies that 0(2) 2 0 holds
from some finite time. On the manifold 0(2) = 0, we have

5:" = x‘ (2.38)

r,—l r,
i

= —(m',2'] + 171.22; + + m','_12:'_1) + Cr]

: —{m'[(:r[ — ("—lCi) 'l' ' ' ' + mi.-1(1'i,—1 " €C;,—1)}+Ci. (2'39)

Substitution of equation (2.39) into the closed-loop equation (2.12)—-(2.13) yields

2' = A2+Ee
= Az+ED(e)C (2.40)
where
F 0 0 0
0 0 0 0

E = blOCk diag[E1, . . . , Em], E,“ =

0 0 O

r,-l

 

 

—m' —m‘ —m‘ 1
1 2
b .1 (Ti—1)XI"

Since A and (A - LC) are Hurwitz, for given positive definite matrices Q1 and Q2,
there are positive definite matrices P1 and P2 such that PIA + ATP, = -Q1 and
P2(A — LC) + (A — LC)TP2 = -Q2. LEI. W(Z,C) = ZTPIZ + CTP2C. The derivative

of W(z, C) along the trajectories of equations (2.13),(2.40) is given by

W(z,() = —zTQ12 + 22TP1ED(e)C — ECTQX

+2CTP231F(17) — F003) + (C(33) — Goiillufiill

28

Using (2.18), (2.38), and (2.39), we have

. 1
W(2,C) S -/\m'n(621)|l2|l2-;Am'n(622)llCll2
+r1|IC||2 + 2r21|<lll|2|| + 2ksl|PzBllllCll
T
S _ Ilzll Am.n(Ql) —r2 llzll
”(ll —r2 'I'Amin(Q2) ’ Tl llCll
+2k5||PzBllllCll
T
S _ IIZII )‘min(Ql) -7‘2 |l2l|
”(II -72 ilmin(Q2)-r1 llCll
+2k5hHP2BHe (2.41)

for some positive constants r1 and r2. Let

Am1n(Q1) -7‘2
'"7‘2 %Amin(Q2) '_ 7'1

P:

For sufficiently small 6, B is positive definite. It can be shown that

for some positive constants A; and r3 . Let r4 > r3, and define

0u = {(330 l W(21C) S 74‘}

It can be shown that there is a finite time t1 such that (z(t),((t)) E flu, Vt 2 t1.

Using (2.38)-(2.39), and ||C(t)|| < Ice for all t Z T, it can be shown that [[2]] S kpfi

for all t 2 t1, where hp is a nonnegative constant. 0

29

One can observe that k 31$ 0 in Assumption 2.2 when G(-) ¢ Go(-) on D. Conse-
quently, 01 in the control input (2.34) should satisfy the inequality (2.32). We can
verify that k5 = 0 in inequality (2.18) when k = 0. The presence of k5 ¢ 0 in inequal-
ity (2.18) results in uniform ultimate boundness. When k5 = 0, we obtain asymptotic

stability.

Corollary 2.1 Assume that all the assumptions of Theorem 2.1 are satisfied. Sup-
pose that G() = Go(-) on D. Then there is 64 > 0 such that for all 0 < e < e4, the
origin of the closed-loop system (212)-(213) and (2.34) is asymptotically stable and

Q, defined by (2.19,). is an estimate of the region of attraction.

proof: Since F(0) = 0, the additional assumptions implies that k = k5 = 0. (‘on-
sequently, from (2.41), W is negative definite for sufficiently small e. Therefore,

(2,0 -—2 0 as t—> 00. Using equation (2.38)-—(2.39), .T -—> 0 as t ——> 00. 0

Since k = 0 when G(-) = Go(~), inequality (2.32) reduces to on 2 202. So, we

can drop inequality (2.32) in this special case and take 011 = 202.

2.4 Semiglobal Stabilization

In this section we consider a globally defined nonlinear system (2.1)—(2.2). i.e., l' =
R". Since U = R", we require Assumption 2.1 to hold with T(U) = R”. The
properness of the mapping :1: = T(w) ensures that it is a global diffeomorphism of R"
onto R". Hence the normal form (2.4)—(2.5) is defined for Va: 6 R”. The objective of
this section is to design an output feedback controller that will stabilize the system
(2.4)-(2.5), and will include any given compact set in the region of attraction. We

modify Assumption 2.2 as follows.

Assumption 2.3 For every compact set D E R",

30

0 there is a scalar nonnegative locally Lipschitt~ function 230(1) such that

llthr) - Fo($)lloo S [30(1) (2.12)

IIG<x1Ga‘(:r) — III... s :2 <1 (2.43)

for all :r E D where ls: is nonnegative constant.

Note that [10(2) and It could depend on the set D. To Show semiglobal stabilization.
we need to show that region of attraction of the closed-loop system (2.12) (2.13)
can be made arbitrary large. We assume that 2(0) belongs to any given compact
sets flo. Since ”Mr” and V(z) are radially unbounded, i.e., limz_.oo||M2|| = 0c and

limz_.°oV(z) = 00, we can choose cw and cm large enough such that

00 = {(2 E R" | ”Mr“ S cw, (/V(z) S cm} and 520 C (20

Choose c,, and c2, such that
91" = {2 G R" I “Mil?” S c", V(z) S czr}

where c,, > C”, en. > czo, c,r = rife”, and r > 1. From Assumption 2.3, we can
find the nonnegative locally Lipschitz function p0(:r) and nonnegative constant Ir that
satisfy the inequalities (2.42) and (2.43) over the set (2,. Notice that the function
[10(2) and I: could depend on the set fl... With the same design procedure as in Section
2.3, it can be verified that

Q = 90 X 91 (2.44)

31

is in the region of attraction where Q] = {C 6 R" | ”C“ < 35.7,}, '7 = max,{r,} for

all i = 1, . . . ,m and c is an arbitrary positive constant. We summarize our finding in

the following theorem.

Theorem 2.2 Consider globally defined the system (2.12)—(2.18). Suppose that As-
sumption 2.3 are satisfied. Let the observer gain be chosen as in (RSV—(2.9) and the
control input be chosen as in (2.34). Then there is 65 > 0 such that for all 0 < e < es,
the closed-loop system (2.12)—(2.13) and (2.34) is uniformly ultimately bounded with
respect to the set a, = {(11-2) 6 an x an ) [[2]] g (cm/2, IICII < 12,6}, for some con-
stants hp, and lcl, and fl, defined by (2.44), is an estimate of the region of attraction.
Moreover, suppose that G(:r) = (30(2) V2 6 R". Then there is es > 0 such that
for all 0 < e < es, the origin of the closed-loop system (2.12)-(2.13) and (2.34) is

asymptotically stable and 0, defined by (2.44), is an estimate of region of attraction.

2 .5 Example

Consider the second order system

231 = .132 (2.45)
22 = alsinxl + u (2.46)
y = 551 (2.47)

Where a1 is an unknown coefficient that satisfies [01] S 0.4. Let D = {x E R2 I ”1‘” S
10.5} and take the nominal function 170(2) = 0. It can be verified that |F(:L‘)-Fo(l‘)l S

04]]2” for V2: 6 D. A robust observer is designed as

2 a 1 A 1
I1 = $2+;(7J’$1) (44b)
1

:12 = u+§(y—i‘1) (2.49)

32

We choose the sliding surface as 0(i') = :27] + £2. Let
n.={eeR2| —6§:ir136, —2.5ga:1+i232.5}
be the region of interest. It can be verified that Q, is a subset of D. Let
112(53): -552 - (0-5llill + 0~1)Sgn(0(i‘))

and take the control input

u(:i:) = —S1sat (2) — Sgsat (0&le + 0.1) (2.50)
2

 

5:2] = 8.5 and 52 2: maxieg,{0.5||5:|| + 0.1} = 6.1. Let

 

where 51 = maxi-:69,

00 = {xER2I —5.5_<_:r1‘§5.5, —2.1§x1+x2£2.1}

01

1.5
{<6 R2 I IICII s 7}

Q = 00XQ1

where (1 = %(:r1 — 5:1) and (2 = ($2 — 5:2). According to Corollary 2.1, the origin
of the closed-loop system (2.45)-(2.50) is asymptotically stable for sufficient small c
and Q is an estimate of region of attraction. The uncertainty of the system (2.45)—
(2.46) satisfies the assumptions of [13] and [12]. For comparison, we design controllers
according to the design schemes of [13] and [12], and refer to them as Case 1 and
Case 2, respectively. In both cases we use the same observer as in (2.48)—(2.49). The
sliding surfaces are taken as 01(i) = 5:1 + 532 in Case 1 and 02(y,:it) = y + 5:2 in Case

2. The control inputs are taken as

33

Comparison of x1

 

 

 

 

 

1 _ e . . . r

0.5 «
3';
0L _____________
|\/\ I””’

-0. ‘-1--"’ 1 4 1 1

50 1 2 3 4 5 6

Time

Figure 2.1. The comparison of the state variable 31 (the dotted line is globally
bounded control case, the dashed line is Case 1, and the solid line is Case 2 with
e = 0.04).

for Case 1, and

1.2
112 = -{-€-2-(y - ii)+1.1|lill+1-2p1(t)}Sgn(02(y.i)) (2.52)

for Case 2, where p1(t) is the solution of
fi1=(—'<1+ K2)P1+ Kzllilla 91(0) = ‘5

with x1 = 12.5, x2=11.53 when 6 = 0.04, and K1 = —25, n2=23.08 when 6 = 0.02.
For all three designs, we simulated the response for 2(0) = [1,0]T, 52(0) = [0, —1]T,
and a1 = -0.4. Notice that the initial conditions belong to (I, x (2,. Figures 2.1 to
2.5 show the simulation results for e = 0.04. Figure 2.1 and Figure 2.3 show that
xl(t) and i1(t) do not exhibit peaking for all cases. Figure 2.2 show that peaking
is evident in 2:2(t) for Case 1 and Case 2, but 32(t) dose not exhibit peaking in our
globally bounded control design. Meanwhile, Figure 2.3 shows that £2(t) exhibits

peaking in the globally bounded control case. The use of globally bounded controller

34

 

 

 

 

 

 

 

 

Comparison of x2 x2 For Globally Bounded Control
5 - o - , ......... -
05%
N E
X ; .
-1.5 *
O 2 4 6

 

Figure 2.2. The comparison of the state variable 9:; (the dotted line is globally
bounded control case, the dashed line is Case 1, and the solid line is Case ‘2 with

e = 0.04).

 

 

 

 

 

 

 

 

 

 

Comparison of £1 Comparison of 322
1.5 - - 15 - +
1% 1 1m
0 'I -------- 0 Av: ‘ " — h
‘\ z ’ ’ l':
-as “”A -
0 2 4 6 50 2 4 6
Time Time

Figure 2.3. The comparison of the estimates of state variables ( the dotted line is
globally bounded control case, the dashed line is Case 1, and the solid line is Case 2

with e = 0.04).

35

 

 

 

 

 

 

 

 

Comparison of Control Inputs Globally Bounded Control Input
5 -
g 0‘ ........................
Q.
_:_ :
r.» '55?
'5 E
o ‘.
0 -1O 1
.15 r
0 0 5 1

nine

 

Figure 2.4. The comparison of control inputs (the dotted line is globally bounded
control case, the dashed line is Case 1, and the solid line is Case 2 with e = 0.04).

 

 

 

 

 

 

 

 

 

 

 

 

Phase Portrait Phase Portrait Phase Portrait
15 0.5 3
. - . I I \
10» o» \ 2
x : \
.' ; \
‘g 5’ i ‘ (9: ‘0-5’ \\ ‘ ‘g 1 l
0 -1 \g 0
\\“
-5 -1.5 -
0 A1 2 040.2 9 0.2 0.4 1 0 0.5 1
x1 x1 . y

Figure 2.5. The phase portrait of each case (the dotted line is globally bounded
control case, the dashed line is Case 1, and the solid line is Case 2 with e = 0.04).

36

 

 

 

 

 

 

 

 

 

10 Comparison of x2 x2 For Globally Bounded Control
- 0 a ............
Orig ---------- 'O.2' 1
‘9: -1ol 8‘: -o 4: " 1
-20 0.6 ~ 1
-3o . ~ - z . .
0 1 2 3 0'80 2 4 6
Time Time

Figure 2.6. The comparison of state variable 9:2 with e = 0.02 (the dotted line is
globally bounded control case, the dashed line is Case 1, and the solid line is Case '2).

eliminates peaking in the state variables of the plant. The control inputs for Case
1 and Case 2 contain terms of the order of magnitude of the estimation error and
the observer gain in order to ensure attractivity to the sliding surfaces. These terms
act as impulse-like functions. Figure 2.4 shows that the maximum magnitude of
control inputs for Case 1 and Case 2 are very large, much larger than in the globally
bounded controller. The control input for globally bounded control is saturated over
(2,. Figure 2.5 shows that trajectories for Case 1 move toward the sliding surface for
Vt Z 0. The attractivity toward the sliding surface is achieved by canceling peaking
terms for Vt Z 0. But trajectories for Case 2 and the globally bounded controller
move toward sliding surfaces after short periods of time. The attractivity of the
sliding surface in Case 2 is achieved by overcoming peaking terms after some time
interval. The attractivity of the sliding surface in the globally bounded control case
is achieved after errors between states and their estimates have become small enough.
As we decrease or from c = 0.04 to e = 0.02, we observe in Figure 2.6 that peaking

increases.

37
2.6 Conclusions

We have designed a globally bounded discontinuous variable structure output feed-
back controller which stabilizes the closed-loop system (2.12) --(2.13), and (2.34). Our
design does not suffer from the peaking phenomenon present in the earlier work [12]
and [13]. This fact is illustrated by an Example. When G(-) 75 Go(-), the closed-
loop system is uniformly ultimately bounded with respect to 0., which can be made
arbitrary small by decreasing e. We can achieve asymptotic stability with arbitrary
positive constant 01 when G(-) = Go(-). The output feedback controller uses the
ideas advanced in [14] and [16]; namely, it uses a high-gain observer and a globally
bounded control. However, the design approach is quite different from [14] and [16]
due to the discontinuity of the VSC. In [14] and [16], the feedback control is designed
as a function of the state variables, then it was shown that the high-gain observer
recovers the performance under state feedback, using the fact that the controller is
Lipschitz continuous. In the current work we start by designing the high-gain observer
and then design the variable structure controller to control the state estimates. The
observer equation (2.10) must contain the control term BGo(5t)u(:i:), while in [14] and
[16] this control term can be dropped, as it was done in [16]. Suppose we start with a
state feedback VSC design and replace the signum nonlinearity sgn(o) by sat(o/u).
The control will be Lipschitz continuous with a Lipschitz constant proportional to
i. The results of [14] and [16] apply to this case but they require c to be sufficiently
small relative to quantities proportional to %. Hence they requires the ratio(e/p) to
be sufficiently small. This shows that those results can not recover the results of this

chapter by letting ,u -—> 0.

CHAPTER 3

OUTPUT FEEDBACK
TRACKING

3. 1 Introduction

In this chapter we generalize results of Chapter 2 in two directions. First, we con-
sider a class of nonlinear systems which have zero dynamics, while Chapter 2 was
limited to systems with trivial zero dynamics. Second, we solve the tracking problem
in the presence of disturbances, as opposed to the stabilization problem we studied
in Chapter 2. We consider a nonlinear system represented by an input-output model
which contains unknown external disturbances. We extend the dynamics of the sys-
tem by adding a series of integrators at the input side of the system and represent
the augmented system by a state-space model. The extension of the dynamics of the
system makes the derivatives of the input available for feedback. We construct a ro-
bust high-gain observer to estimate the tracking error and its derivatives. The sliding
surface is chosen in the observer space and a globally bounded output feedback VSC
is designed to satisfy the reaching condition to the surface. The use of the high-gain
observer and globally bounded control enable us to show that the estimation errors

decay to arbitrarily small values during a short transient period. After showing that

38

39

the trajectories reach the sliding surface in finite time, we show that the output tracks
the reference signal with arbitrarily small error. We assume that the zero dynamics of
the system are minimum phase, but we do not require exponential stability or global

growth conditions.

3.2 Problem Statement

We consider a single-input-single-output nonlinear system represented by the nth

order differential equation

y‘"’ = f(-) + g()u‘m’ (3.1)

where u is the control input, y is the measured output, u“) and y“) denote the
ith derivatives of u and y, respectively, and m < n. The functions f (~) and g(-)
could depend on y,ym,---,y("’1),u,um,-~,u(m“), and an external time-varying
disturbance d(t). We assume that f(-) and g(-) are sufficiently smooth and defined in
a region U, x U. x P where Uac C R", Uz C R“, I‘ is a known compact subset of R”.
d(t) E F, and d(t) is bounded. We set U = U; x Uz. The objective of this chapter
is to design an output feedback VSC which guarantees boundness of all variables
of the closed-loop system, and output tracking of a given reference signal y,.(t) in
the presence of modeling uncertainty and time-varying disturbance. We represent
an extended version of (3.1) by a state-space model. We augment a series of m
integrator at the input side of the system and denote the states of these integrators
by z, = u,22 = u“), up to 2m = um“) and set U 2 am) as the control input of the

augmented system. By taking 2:1 = y,2:2 = y“), up to 3,, = yin—1), we can represent

40

the augmented system by the state-space model

532' = $i+1s lSiSn—l

in = f(a‘, 2, d) + g(ar, z, d)v

 

z“. = 2..., ISjSm—ll (3.2)
em = v
y = 1’1 J
where .1: = [31,. - - ,1.an and z = [z], - - ~ ,zm]T. A similar model, called generalized

observerbility canonical form, was studied in [28] for state feedback VSC but no
integrators were augmented at the input side. The augmentation of integrators is not
needed in the state feedback case since all state variables can be measured. In the
output feedback case, high-gain observers can be used to robustly estimate a part
of the state vector which constitutes derivatives of the output. This restricts the
controller to use partial state feedback, e.g., [18]. By augmenting integrators at. the
input side of the system, we obtain the system (3.2) whose state (3:, 2) comprises 2
which is readily available for feedback and a: which constitutes derivatives of y; so it
can be robustly estimated by a high-gain observer. We make the following assumption

on the system (3.2)

Assumption 3.1 For all d E F and for all'(:r,z) E U, g(r,z,d) 75 0

Assumption 3.1 implies that the system (3.2) has a uniform relative degree n, which
is a necessary and sufficient condition for the existence of a local diffeomorphism that

transforms (3.2) into the normal form [2].

Assumption 3.2 0 There exist m smooth functions 7),,1 S i S m, defined on
:r

U x P such that the mapping = T(m, z,d) is a diffeomorphism ofl' onto
77

its image, for every d E F, that transforms the system (3.2) into the normal

41

form
it = 531+), ISiSn—l (33)
in = f(xinid)+g($17l1dlv (34)
7'7 = umd) (3.5)
y = In (3.6)
where 17 = [771, - . - ,qm]T.

o The origin (0,0) 6 T(U,F) and equation (3.5) has a unique equilibrium point
at 17 = 0 when (z,d) = (0,0), i.e., ¢(0,0,0) = 0.

Remark 3.1 In the special case when 9 is constant, one can verify that the change

of variables

11?. ' .
m=zt- "MU 1<ZSm

g

 

transforms the last m equation of (3.2) into

Tli : 77:41» lszsm

. f(rr,z,d)
77m : —— lzt =m+$n—m+i/9

9

thus transforming (3.2) into the normal form (3.3)—(3.6).

The mapping T is required to be a diffeomorphism of a domain U [onto its image. We
recognize that there are no results in the literature that will guarantee the existence
of the mapping T of Assumption 3.2. However, under Assumption 3.1, one can always
find a local diffeomorphism. As one solves for the local diffeomorphism, it is possible
to determine the region of validity of the mapping; see [1, Section 5.4] for examples
of determining the region of validity of the mapping in the case of state equation

linearization. The essence of feedback linearization strategies is to choose the input v

42

to control the behavior of x. Since equation (3.12) is driven by (x, d), it should have a
bounded-input-bounded-states stability property, with (z,d) as input and 7} as state.
Moreover, 17 should belong to the domain of o(-). We make the following assumption
for such a system. Before we state the assumption, we recall the definitions of class

1C function and IC£ function.

Definition 3.1 [1] A continuous function a : [0,a) —» [0,00) is said to belong to
class 1C if it is strictly increasing and 0(0) = 0. It is said to belong to class K... if

a=ooanda(b)—»ooasb—ioo.

Definition 3.2 [I] A continuous function 3 : [0,a) X [0, 00) —-> [0, 00) is said to belong
to class ICC iffor each fixed 3 the mapping B(b, s) belongs to class [C with respect to b.
and for each fixed b the mapping {3(b, 3) is decreasing with respect to s and 13(b. s) —+ 0

083-400.

Define the ball, U: x D,, = {n E Rm] ”77“ < d,,} where d,, are chosen such that
U; x D,, C T(U,d),Vd E F where F C F. We might have to restrict the disturbance
F C F to be able to choose the ball D,, such that D,, C T(U,d) Vd E F.

Assumption 3.3 There exists a Cl function W : D,7 —> R+ and class K functions,

a, : [0,dn) -—+ R+, i = 1,2,3, and 71:[0,d,)—r R+ such that

al(llnl|) S WM) 5 az(l|nll) (3.7)

8W 1'
—¢(a:.n.d) s —aallln|l), Vllnll 2 71 (55-8)
(912 d

_ x
V(x,17,d) E U; X D" X F and dr < ‘71-1 bag-1001M") where d, = SUP(.—.,d)eu,xf‘

This assumption implies that n = 0 is asymptotically stable when (x,d) = (0,0). It
can be seen [1, Theorem 4.10] that when (x,d) # (0,0) but bounded, the solution

43

17(1) satisfies the estimate

”77(1)” S 51(lln(0)ll~t) + “/(h). V12 0

x(t)

where h = sup ,Vt Z 0 , fl] is a class KLC function, and 7 is a class K
d(t)

function. We will consider the semiglobal case in Section 3.4. For semiglobal results.
Assumption 3.3 must hold globally. However, it is less restrictive than some of the
global assumptions used in [12]. In fact, [12] assumes that 7') = ¢(0,17,0) is globally
exponentially stable and (b is globally Lipschitz in (x,d). Under these conditions. it
can be shown that there exists a Lyapunov function W() such that Assumption 3.3

is satisfied.

Assumption 3.4 For every domain D" C D,,, there are nonempty domains D C ('3

and 1‘“ c r such that T(U,,D.,d) c U, x 13,, Vd e I“

Assumption 3.4 has the effect of restricting the class of disturbances. In the case of

no disturbance, for any D", we can always find Dz such that

zEDzCUz=>n€DnCDn

When the mapping T depends on unknown disturbance, existence of the set D, de-
pends on the class of disturbances. Note that this condition implies that the set F
depends on the choice of D,,.

Let fo(x, z) and go(x, 2) be known nominal models of f(x, z,d) and g(x, 2, d), respec-
tively. Suppose fo(x,z) and go(x,z) are sufficiently smooth and go(x, z) # 0 for all
(1,2) 6 U. We assume that f0(x, z) and 90(1', z) are globally bounded. This can be
always achieved by saturating the given nominal functions outside a bounded domain
of interest, as it will be illustrated later on. We make the following assumption on

the uncertainty.

44

Assumption 3.5 For all (x,z,d) E U x F

a there is a scalar nonnegative locally Lipschitz function p(:r, 2) such that

Iflar. z,d) — fo(:r,z)| s pus, 2) (3.9)

where lc is a nonnegative constant.

We assume that the reference signal y.(t) is smooth, bounded, has bounded derivatives

up to the nth order, and y1")(t ) 1S piecewise continuous. Let

34(1) = lyr(t) 311110)» 1311” 1)(tllT

3’30) = lyr(t) )y1”(i) -.y1”’”(i)y1"’(t)lT

We also assume that y. 6 1", and ye E 1"}; where Y, and Y3 are compact subsets of

R" and RM", respectively. By taking

61 = y - yr
62 = 31— yr
en : y(1'1—1)_y1(.n-1)
and e = [€1,62, ~ ~ - , en]T, we rewrite system (3.3)-(3.5) in error coordinates as
é = Ae+B{f(e+J/..n.d)+§(e+yr,n,d )v-y1"’} (311)

ii = ¢(e+y..n.d) (3.12)

45

where (A,B) is a controllable canonical pair of the form

101 .0. 10-

0 0 1 0 0
A: ,3:

0 01 0

_0 .0] .1.

 

 

 

 

Since our goal is to design a controller such that e —> 0, we make the following

assumption.

Assumption 3.6 There is a domain D6 = {e E [2"] He“ < (1.} such that V6 6 Dt

and V)’, 6 Yr, 6 + y. E U; where d. is a positive constant.

3.3 Controller Design

We design the augmented control input v for the system (3.2) to achieve our objective,
i.e., tracking of a given reference signal in the presence of time-varying disturbances
and modeling uncertainties. we use a globally bounded controller to eliminate the
peaking phenomenon. The actual control input u is the output of a series of m
integrators driven by the augmented control input v. Moreover, z is always available
since its components are obtained by integrating v. To estimate the derivatives of the

tracking error(e1 = y — y,), we construct the following observer

8,‘ = éi+1+$(€]—él), i=1,...,n—l (3.13)
6'
0i

C—n

(61-é1)+fo(é+yr,2)+go(é+ynz)v-y1") (3-14)

46

where e,- is the estimate of e,, e is a positive constant to be specified, and e =

[81,“ - ,én]T. The positive constants a, are chosen such that the roots of
s"+cr1s"’l +---+on_ls+an =0

are in the open left-half plane. We rewrite the observer equation (3.13)—(3.14) in the

compact form

é: Ae + B{f0(é + y.,z) + go(é + 32., 2):: — W} + D(e)LC(e — e) (3.15)
where L = [al,o--,an]T, C = [1,0,-~,0,0], and D(e) = diag [%,ci2,....(i,,,.(—1-,,[ Let
{,- = e,- — é, be the estimation error, and define the scaled variables

1 .
Ci: {1, i=1,~~-,n (3.16)

("‘1

 

The closed-loop equation can be rewritten as

e = Ae+B{f(e+y,,z,d)+g(e+y,,z,d)v—y1")} (3.17)
2. = A12+Blv (3.18)
at = (A—LC')c+eBlf(e+y..z,d)—fe(é+y..z)

+{g(e+y,.,z,d)-g0(é+yr,z)}v] (3.19)

where (A1,Bl) is a controllable canonical pair of the form defined in (3.11) and
C = [(1, (2, - - - , (,JT. It can be verified that (A — LC) is a Hurwitz matrix. We choose

the sliding surface

g(e) = Me (3.20)

47

where M = [7711, - - - .mn-1] is chosen such that.
f 0 l 0 i
0 0 I 0
A =
0 0 1
—m1 —m2 ——m,,_2 —m,,_1 ]

 

 

is Hurwitz. Define

qzlelt'.°aen—llT9 é=léls°'°~én-1]T9 C: [€11...‘Cn-1]T‘

and rewrite equation (3.13) as

£3: 21c; + Bow) + D(e)£(§(’ (3.21)
where F q _ 1
0 at
B = , I: =
1 an—l
. . n—lxl L a n—lxl

 

 

 

 

D(e) = diag[e"’2,c"'3, - - - ,e, 1], C = [1,0, - - - ,0,0]1x,,_1

We consider a control input of the form v = (p(é + y,,z) + V(é + y,,z)sgn(o(€-))
where (.p(-) and u(-) are continuous and globally bounded functions. We will specify
v later on. Since A is a Hurwitz matrix, for any positive definite matrix Q, there
is a symmetric positive definite matrix P such that PA + ATP = —Q [1]. Let

V(q) = qTPq. We choose cerand r such that

13.: {e e R") lMeI s V(q) s ce}

48

is in the interior of De where eq, = rglcer. a1 = M, a2 = Ni, and r > 1.
a) AmOI(P) ’\nli'l(P)
Since E, C De, we can choose on such that
02 0 711(1rl< Cn < Oildn) (33313)

Define the set 0,, = {77 E Rm] WM) 3 an}. From Assumption 3.3,

W(U) S 01) => 01(ll7llll S on
= HUH S amen)

= ||77|| < dn
Hence 9,, C D". From Assumption 3.4, we can, find Q; C U2 and F C F such that.
T(L/,,n.,d) c U, x 11,, va e 1‘
By inequality (3.22),

H17“ < 11(4) ==> IIUII < 0271(4))

=> W(n) < c,,

Hence the ball {77 E Rm] [[77]] < 71(d,)} C 9". Therefore, using inequality (3.8),

We 1"
e(t) e E,, Vt 2 0 and 2(0) 6 n. => 720) e amt/12 0 (3.23)

Even though the mapping T depends on the unknown disturbance d, with the bounds

on y. and (I known, we can find a compact set 51,, such that

7169,, and 66E, :5 zEQuCUz (3.24)

49

The set E, x $22, is taken as the region of interest in our analysis. Achieving globally

bounded functions fo(e + 34.2) and go(e + y..:) can be done by saturating these

functions outside the set E, x 9:, x I”; where
E, = {e E 3"] [Mel S cm, V(q) S cqs} C D6

and cc, < c”, cq, < cw. Note that E, is always defined since E, is in the interior
of D... It can be seen that the following inequalities hold almost everywhere for

(e,y..z,d) e E. x Y. x 0.. x 1", c e R“
|f(e+y.,z,d)+fo(e+y..z)v| s k1||e||+k2 (3.2.5)

lfle + yr. z,d) - fo(é. 2) + {9(6 + 12.241) - golé. 2)}vl S ksllell + kallCll + ks (3326)

where k1 to k5 are nonnegative constants. Notice that the right-hand side of (3.25)
does not depend on the fast variable C, which follows from the use of a globally

bounded control input v. Let

 

E0 = {e 6 R" I IIMeII S cm, (fl/(q) S cqo},cco < cc, and egg < cq,
c
Q] = {C E Rn l ”C” < €n_1}
Q = E0 X 91 X Q; (3.27)

where c is an arbitrary positive constant

Lemma 3.1 Consider the singularly perturbed system (3.17) (3.1.9) with any globally
bounded control v and suppose that Assumption 3.1 to 3.6 are satisfied. Then, for all
(e(0),((0),z(0)) e n and Vd e 1, there exist e, and T, = T1(e) 3 T3 such that for all
0 < e < e,, “g() < ice for allt€[T1,T4) and z(t) e 12,, for all t e [0,T4) where 1:, .5

a finite time and T, > T3 is the first time e(t) exits from the set E,.

Proof: The proof of ”C(t)” < he is the same as that of Appendix A; therefore we

omit it here. From Assumption 3.4, we can find (2,, such that Vd E f‘
2(0) 6 (L => 71(0) 6 0,,

Since e(t) E E, Vt E [0,T4), using Assumption 3.3, it can be shown that 17(t) E

(2,, Vt e [0,T4). From (3.24), z(t) e (L, C Uz Vt e [0,T4). 0

Lemma 3.1 implies that the fast variable C decays very rapidly during a short time
period [0, T1]. Lemma 3.1 also implies that z E Q" as long as e E E,. Next,
we design a globally bounded control input 22 such that a sliding mode condition is
satisfied when ”(H < he and (6,6) 6 E, x E,. This will be done by showing that

0(é)o(é) S —;i2|0(6)| as long as 0(6) # 0 where M is a positive constant. We have

a(é)c’7(é) .—_ aggé
= aM[Aé + B(f0(é + yr, 2) + go(é + yr,z)v + D(c)LC(e — 6)]

= aM[Aé + B(fo(é + yrs 2) + go(é + yrs 2)v + %D(€)LCCl

where

~

D(c) = diag[c”—1, ("-2, . . . , c, 1]

We need an estimate of %MD(€)LCC to design the control input 22 such that the
sliding mode condition is satisfied. Using that f(-), fo(-), g(-), are smooth on the
region of interest, derivatives of y,(t) and d(t) are bounded, and control input 1! is

globally bounded, it can be shown that for (e,é) E E, x E,

1 .
-€-|M0(€)LCC| S p(é + yuz) + khaki + less, We [T1+ e In 3T.) (3.2s)

01

Where ku is an upper bound on go(é + y,,z)v(é + Mus) to be specified, k, =

an [6” ICe(A’LC)‘BIdt Z 1, and k6 is a some positive constant. When k, > 1. we

need to strengthen inequality (3.10) as follow

1
I9(x.z.d)ga‘(:r.z) — 1| S k < 7, V(at,z,d) e U x r

t

Consider the function

$(é,yr,z) = 961(é+yr,z)[—M(Aé+Bf0(é+yr,z))+y(nl

-(p(é + yr. 2) + #ilsgn(0(é))l

where in is any positive constant that satisfies

kktks + 2’12
1 — kk,

#1 Z
for some in 2 0 and k, is defined by the inequality

l _ M(Aé + Bf0(é + yrazll + g(n) _ p(é + y,,z)sgn(o(é)| S ks

(3.29)

(3.30)

(3.31)

(3.32)

for almost every (é, ya, 2) E E, x YR x 92,. Notice that k, can be calculated since f0(-)

and p() are known. We take the control input v(é,yR,z) as zp(é,yn,z), saturated

outside the set E, x YRXQ”. In particular, let ¢1(é,yR, z) = —ggl(é+yr, Z){M(A€+
Bfo(é + 34.2)) + 2153”}. ¢2(é.y..z) = —ga‘(é + y.,z)(p(é + ynz) + m) . S.- =

max(é,yg,z)€E,xYRxQu l¢i(é,yR, le . and take

 

v(é,yR,z) = 51 sat (¢~IS'(1)) + 52 sat (if?) sgn(a(é))

(3.33)

5‘2

Where sat(-) is the saturation function. Inside the set E, x YR x 02,, we have
”(éayRJ) =1l"(é.yna~7) (334)
Hence, ha in inequality (3.28) can be taken by
ku = k, + )1] (3.35)
Using (3.28), (3.31), (3.34), and (3.35),

0(é)6(é) < l-{P(é+yr.:)+m}+kkz(1\'s+lti)-(/)(c+yr.:)+kotl|0(€)l

s —u2l0(é)| (3.36)

for sufficiently small c. We summarize our findings in the following lemma.

Lemma 3.2 Consider the singularly perturbed system (3.17)—(3.1.9)with the control
input v(é,y,,z) defined by (3.33). Suppose that Assumption 3.5 and (3.29) are sat-
isfied, and suppose ”(H < he and (e,é,z) E E, x E, x 9,, fort 6 [T1,T4). Then as

long as 0(é) # O the sliding mode condition
0(é)5’(é) S -#2|0(é)|

is satisfied for all t 6 [T1 + c In %, T4).

So far we showed that ”(H < lite for t 6 [T1, T4) and the control input 12 (3.33) satisfies
the sliding mode condition when ”(H < he and (e, e, z) E E, x E, x Q” for t 6 [T1, T4).
It is well known that Lemma 3.2 implies that sliding mode exists after a finite time.
Using the existence of sliding mode in the observer space, it is shown in the following

theorem that the output tracks the reference signal with arbitrarily small error.

53

Theorem 3.1 Suppose that Assumptions 3.] to 3.6 and (3.29) are satisfied. ('on-
sider the closed-loop system (RID—(3.19) formed of the plant (3.1) with control input
(3.33). Suppose that (e(O),C(0),z(0)) E 9. Then, Vd 6 I“, there is 63 > 0 such that
for all 0 < 6 < 63, all the state variables of closed-loop system are bounded and there
exists a finite time t1 > 0 such that ||e(t)|| S kM/E Vt > t1 where k, is a positive

constant.

Proof: The proof has a lot of similarity with that of Theorem 2.1 in Chapter 2.1: so
we do it briefly. To show z(t) E 9,, Vt Z 0 and ”(H < he Vt Z T1(e), it is sufficient

to show e(t) E E, Vt 2 0. Define

(22 = {e E R" | “Me“ S 6,2, (ll/(q) S 6,2}, and

$22 = {é E R" l llMéll S 0.32. W?) S Cq2}

where cw < 6,2 < cc, < c,,, cqo < 6,2 < ng < c,,, 6,, = gregéezmg > 1), and
ng = Efregc,2(r,2 > 1). Since {e(O),C(0)} E 0, there exists a finite time T2 < T.
such that e 6 ft; for all 0 S t S T2. For sufficiently small e, T1(e) + eln(%) S T2.
Note that é might be outside $22 for t < T1(e) + e ln(%), but since é]- : e] — en'JCJ. by
||C(t)|| < eli: Vt E [T1,T4) with sufficiently small e, e E Q; at t = T1(e) + cln(%). Using
0(é)o(é) < 0 and V((j) S O on the boundary of 02 , it can be verified that é E 92 for
t E [T1(e) + eln -:-,T4). Using a contradiction argument, it can be shown that e 6 Q,
for Vt Z T1(e) + e In '1'. Hence e E Q, for Vt Z 0. Therefore the sliding mode condition
is satisfied Vt 2 T where T = T1(e) + 6 ln %. This implies that o(é) = 0 holds from.

some finite time. On the manifold 0(é) = 0, we have

én—l = an (3.37)
= —(m1é1+ m2é2 + ° ° ° + mn—zén—il + Cu

: —{m1(ei — ("c-1(1) 'i' ' ' ' + mn—l(€n—l — 691-1)} + C71 (338)

54

Substitution of equation (3.38) into equation (3.17) yields

 

 

('1 = Ag + A5
= Ag + ADMC (3.39)
where
0 0 0
0 0 0 O
A =
0 0 0
—m1 —m2 ... —m,,_1 1
d (n-1)xn

Since A and (A — LC) are Hurwitz. for given positive definite matrices Q1 and Q2.
there are positive definite matrices P, and P; such that 19,/i + ATP, = —Q1 and
BM — LC) + (A - LC)TP2 = —Q2. Let W(q,() = qTqu + (Tar . Using (3.26).
(3.37), and (3.38), it can be shown that the derivative of W(q, C) along the trajectories

of equations (3.19) and (3.39) satisfies

W(q.<)s—A.W(q.<). for W(q.<)>r3e

for some positive constants A1 and r3. Let r4 > r3, and define

Q, = {(9.0 l W(qu) S 7‘46}

It can be shown that there is a finite time t, such that (q(t),((t)) E 9,, [Vt _>_ t].
Using (3.37)—(3.38), and “((1)” < lite for all z 2 T, it can be shown that Hell 3 (em/2

for all t 2 t, where k, is a positive constant. 0

55
3.4 Semiglobal Tracking

In this section we consider a globally defined input-output model (3.1), i.e.. l' =
Rn+m. For any given compact set of initial conditions of (x,z), for any ye 6 Y3.
and for any d E I‘, we will design output feedback controller that ensures tracking of
reference signal y,(t) where Y3 and F are any given compact set. Since U = RM”.

we need to modify the assumptions in Section 3.3.

Assumption 3.7 For all d E F and for all (x, z) E H"+m, g(x,z,d) ¢ 0

Assumption 3.8 0 There exist m smooth functions 17,,1 S i S m, defined on

:1:

BMW x P such that the mapping = T(x, z,d) is a global diffeomorphism
77

, for every d E F, that transforms the system {3.2) into the normal form (3.3)

to (3.6).

0 Equation (3.5) has a unique equilibrium point at 77 = 0 when (x,d) = (0,0),
i.e., ¢(0,0,0) = 0.

The class of systems that are an input-output linearizable and for which (3.13)—(3.6)
hold globally have been characterized in [29]. However the condition of [29] is a suf-
ficient condition for existence of a global diffeomorphism that transforms a nonlinear
system into the normal form. Hence we do not restrict only for such a system. For a
bounded-input-bounded-states stability, we modify Assumption 3.3 in the following

assumption.

Assumption 3.9 There exists a C1 function W : Rm —) 12+, a,(i = 1,2,3). and 71

are class ICC, functions such that

01(Hnll) S W(n) S 02(llnll) (3.40)

56

8W x

-.—¢(r.n.d) S —as(||77|l). Vllnll 2 71 (3.41)
do (I

for all (x,77,d) 6 12““ x I‘.

We also modify Assumption 3.5 for the uncertainty in the following assumption.

Assumption 3.10 For every compact set U E Rn+m, and for V(x, z,d) E U x F.

0 there is a scalar nonnegative locally Lipschitz function ,6(x, 2) such that

If(x,z,d) — f0(x,.:)| S p(x,:) (3.42)

Ig($.2.d)ga‘(sr.z) — 1| S 7c <1 (3.43)
where it is a nonnegative constant.

Note that fi(x, z) and lo could depend on the compact set U. To achieve our goal, we
need to show that the region of attraction of the closed-loop system (3.17)-(3.19) can
be made arbitrary large. We assume that all the initial condition are bounded. In
particular, e(O) and 2(0) belong to given compact sets E0 and 52,, respectively. Since
|Me| and V(q) are radially unbounded, we can choose cc, and egg large enough such

that

E0 = {e E R"| lMel S cc... y/V(q) S ego} and En C Eo

Choose cc, and r such that
Er: (8612") (Mel Sccra V(Q)chr}

c,, > cw, cq, > cqo, c,,, = rfifcfl, and r > 1. Since the mapping T is a global

diffeomorphism and d belongs to the compact set P, we can find the compact set 0,,

57

such that

2692:1769"

Choose c,, such that

QrI'—"{77€le Ill/(9)3617}, {277C017 and Cn>a2°71(l‘al

e + 3’,
where p, = max(,.yhd)egrxyrxr . Note that we can always find such a

d

constant c,7 since W(n) is radially unbounded. Since E, is a compact set, we can
take (18 such that E, C D, = {e 6 R"| ||e|| < d,}. We can also take d, such that
the set D, = {n G R”) “17” < d,} contains the set 0,, i.e., d,, > af1(c,,). One can
observe that we can choose the constants c,,, d,, and d, with no restriction on d since
the system (3.2) is defined globally. Hence we do not require Assumptions 3.4 and
3.6 in the semiglobal case. Using Assumption 3.9 and global diffeomorphism of the

mapping T, we can find a compact set 0,, such that

2(0) 6 Q, and e(t) E E, Vt Z 0
=> 17(t) E 9,, and e(t) E E, Vt Z 0

=> 2(t)€ 0., W20

Define
E, = {e e R"I lMel S cm V(q) S Q»)

where Ce,- < c,, and c,,, < c,,. From Assumption 3.10, we can find the nonnegative
locally Lipschitz function p(e + 3),, 2) and nonnegative constant [C over the set E, x
Y, x 0,, x I‘ . With the same design procedure as in the Section 3.3, it can be verified
that Q 2 E0 x Q] x Q, is in the region of attraction. We summarize our conclusion

in the following theorem.

58

Theorem 3.2 Suppose that Assumptions 3.7 to 3.10 and (3.2.9) are satisfied. ("on-
sider the closed-loop system (3.17)—(3.19) formed of the plant (3.1) with control input
(3.33). Suppose that (e(O),C(0),2(0)) E 0. Then there is e4 > 0 such that for all
0 < e < (.4, all the state variables of closed-loop system are bounded and there is a

finite time t1 > 0 such that ||e(t)|| S ch/e- as t 2 t, where It, is a positive constant.

3.5 Conclusions

The class of nonlinear systems considered in this chapter is more general than the
single-input single—output case of the class of nonlinear systems considered in Chap-
ter 2. It can be easily seen that the state-space model (2.1)—(2.2) of Chapter 2 will
give rise to an input-output model of the form (3.1), but with no zero dynamics. We
have designed a globally bounded output feedback VSC that ensures the tracking of
a reference signal with arbitrarily small error in the presence of bounded unknown
disturbances and modeling uncertainty. We can achieve the desired accuracy of track-
ing by choosing the design parameter e sufficiently small. Decreasing the parameter
e does not induce peaking of the system’s state variables due to the use of a globally

bounded control. When our assumptions hold globally, we derive a semiglobal result.

CHAPTER 4

ROBOT MANIPULATOR
CONTROL

4.1 Introduction

Robust state feedback control schemes for robot manipulators have been developed to
overcome modeling uncertainties and / or disturbances [30]. A number of these schemes
use variable structure control(VSC) to track the desired path for robot manipulators.
Most of the work on VSC assume that measurements of positions and velocities are
available for feedback [30]. However, the velocity, obtained by tachometers, is easily
contaminated by measurement noise [31]. One way to overcome this problem is to
estimate velocity from position measurement using observers [32, 33, 34, 35, 36]. The
papers [35, 36] used a discontinuous sliding observer with a continuous control scheme.
while [32, 33, 34] used a continuous observer with a continuous control scheme. The
paper [37] used a continuous observer with a VSC scheme. A common theme of
observer design is the use of high gain observers which reject disturbance due to
modeling uncertainty and imperfect feedback cancellation of nonlinearities. The VSC
used in [37] is similar to the one described in this chapter, but [37] used a circular

argument in the design of the observer. In fact, [37] required the observer gain to

59

60

be greater than a function of the estimates of the angular velocities, which itself is
dependent on the observer gain. In Chapter 2 we have established that the specially
scaled observer with globally bounded control can reject disturbances due to modeling
uncertainty and imperfect feedback cancellation of nonlinearity. We use the technique
of Chapter 2 to design a controller that achieves tracking of a desired path. Motivated
by the study of a 2 DOF manipulator control problem, the controller design is slightly
different than the design of Chapter 2. The controller in this chapter satisfies a sliding
mode condition in the intersection of the sliding manifolds, while the controller in
Chapter 2 satisfies a sliding mode condition in each sliding manifold. We show. via
an example, that the current controller is less conservative than that of Chapter 2.
To illustrate the performance of the controller, we consider the tracking control of
a 2 DOF manipulator with unknown payload, but with known payload range, and
design the discontinuous controller. To reduce chattering, a continuous approximate
controller replaces the discontinuous one. The continuous approximation reduces

chattering, but results in an increase in tracking error.

4.2 Manipulator Dynamic Model and Problem
Statement

Consider the equation describing the dynamics of an n degree of freedom(DOF) rigid

robot manipulator [38]

H(e)é+6(0,é)é+g(0) = T (4.1)

where 9 E R" is a vector of generalized coordinates(joint positions), 71(9) E R’""‘ is
the positive definite inertia matrix, C(B, 3W 6 R" is a vector of Coriolis and centripetal
torques, 9(0) 6 R" is a vector of gravitational torque, r is a vector of applied joint

torques.

61

4.2.1 Physical Robot Properties

The manipulator model (4.1) has the following inherent properties [38], which are

useful in control design.
(i) H(0) = 71(0)?“ > 0 v0 6 R".
(ii) II’H(0)|| s q. W e Rn.

(iii) ((60.00)) s anér v0 6 R".
(iv) new)” 3 q. v0 6 R".

where qh,qc, and q, are some positive constants. Property (i) implies that H'1(0)
always exists, and property (ii) implies that the elements of the inertia matrix H(d) are
bounded. Property (iii) means that the Coriolis and centripetal torques are quadratic

in the velocity 0. Property (iv) means that the gravitational torques are bounded.

4.2.2 Problem Statement

Let the desired path of 0 be 0, and assume that (0,0,) 5 o, and (0,0,0,) 6 00
where O, and 90 are compact subsets of 122" and R3“, respectively. The objective is
to design the applied torque 7' using only joint position 0 such that joint position 6
tracks the desired path 90 when the payload is unknown, but its range is known. We

use VSC to design the torque r.

4.3 Controller Design

Introducing state variables in error coordinates, e', = 0,- — 0,, e; = 0, — 0,1,, for

i = 1,...,n, and setting el = [e],e¥,...,e'f’l,e[‘]T, e2 = [e§,e§,...,e3’l,e§]T. the

6‘2

dynamic equation (4.1) has the following state-space representation

61 = 62
n

e; = —6ld,- + Fi(e1,e2,0d,dd) + Z G;(e'1,0d)rj 1S i S n

1:1

where F‘(e,0d,dd) denotes the ith component of the vector 'H'1(e1 + 0d){—('(e1 +
0d,e2 + dd)(e2 + 3,) — Q(e1+ 0(1)} and G;(e1,0d) denotes the ith row and jth column
entry of the matrix ’H’1 (e1+0d). Notice that from property (i), the matrix C(el, 0,) =
{G;(e1+0d)} exists for Ve1+0d E R". After setting e = [e], a], . . . , e], e}. . . . , e'f. tilT~

we rewrite the state equations in the compact form

e = Ae+B[—dd+F(e,6d,dd)+G(e1,9d)r] (4.2)
where q
0 1
A = block diag[A1, . . . , An], A, =
0 0
.l
. F! .
0
B = block diag[Bl,...,B,,], B,- = , F =
1
Fn

 

 

C=blockdiag[Cl,...,C,,], C,=]1 0]

Let Fo(-) and Co(-) be the known nominal models of F(-) and C(-), respectively. From
property (i), we see that G(-) is nonsingular V(e, +04) 6 R"; moreover, from property
(ii), G() is globally bounded. We choose Co(-) to have the same properties. We
assume that Fo(-) is globally bounded. This can be always achieved by saturating the
given nominal functions outside a bounded domain of interest, as it will be illustrated

later on. Before we state an assumption on the uncertainty, we recall the definition

63

of an M-matrix.

Definition 4.1 [3.9] A real n x 72 matrix M=[m,-,-] is said to be an M-matrix if m,,- S
0,i 95 j(i.e., all ofl diagonal elements of M are nonpositive), and if all principal

minors of M are positive.

Let N be an n x 11 matrix such that each component of N is an upper bound on
the absolute value of the corresponding component of the matrix [G(0)GE 1(0) — I ],
V0 6 R”. Notice that since G() and G; 1() are globally bounded, the matrix N is
well defined.

Assumption 4.1 The matrix [I — N] is an M-matrix.

Assumption 4.1 restricts the range of payload.

To estimate the derivative of the tracking error, e; = 0— 0.1, we construct the observer

1" at a i -t'

el = e2 + -£—1(el — 6,) (4.3)

1" " Oi t ~i i ~ ° n 3' ~ -

e2 = —0d,- + :3- (el — el) + Fo(e,0d,0d) + Z Goj(el,0d)7j, i = 1, . . . ,n (4.4)
i=1

where all a;- are positive constants, é;- is the estimate of the state variable e], and e
is a positive constant to be specified. We rewrite the observer equation (4.3),(4.4) in

the compact form
0 = A0 + B[—0, + Fo(é, 0, 0,) + Go(e., 0.07)] + D(e)LC(e — e) (4.5)

where

L = block diag[Ll, . . . , Ln], L.- =

D(e) = block diag[D1(e), . . . , Dn(e)], D,(e) = diag[- —

C = block diag[Cl, . . . , Cu], C, = [1 0]

64

Let C]- = e;- — e;- be the estimation error, and define the scaled variables Q,

for j = 1,2. The closed-loop equation can be rewritten as

e = Ae + B[—0d + F(€,0d,0d) + C(elaalel

(A — LC)C + eB[F(e,0,0,) — F0(é,0, 0,)

m
a.
ll

+(G(e,,0d) - Ga(ét.fla))rl

where
- - ~ -i -i ‘71 "T
Q: (tittis-~~€i~§2~“"§l9C2]

We choose the sliding surface 0(0) = [01(é), . . . ,o,,(é)]T such that
a,(e) = e; +m*,e;, 1S t g n
where m’, are positive constants. Rewrite 0(é) as
0(é) = Me

where

M = block diag[M1,. . ., Mn], M, = [m‘1 1]

Define C := [C], . . . .C,"]T, and rewrite equation (4.3) as
e', = 0e, + 0(é) + 125

where

A = diag[—m], . . . , —m';], L = diag[al. - - -.al‘]

1 .
= 72:5}

(4.9)

Consider a control input of the form r, = tp,(é, 0d, 0d, 0d)+vJ-(é, 04,0d)sgn(01(é)) where

coj(-) and vJ-(-) are continuous and globally bounded functions. We will specify TJ later

65

on. Let P = diag[fi, .. —1-], and take V(el) = elTPel. Notice that P is a positive

"2m?

definite matrix since all m] are positive constants. Choose the positive constants c,,

and r such that

a. ‘3—2‘ {6 6 R2" | ”Men s c... (Met) 5 a.)

i = 22 = .___l__. = 2 P . ' ' ‘
where c,, r01 c,,, a1 (MAP), a2 '\mln(P)‘ and r > 1. The set Q, IS taken as

the region of interest in our analysis. Define

a, dé‘ {e e 122" | ”Men 3 c...(/V(et) S c,,}

where c,, > c,,, and c,, > c,,. Achieving globally bounded functions Fo(e,0d, 0,) can
be done by saturating Fo(-) outside the set it, x (9,). It can be seen that the following

inequalities hold for almost every (e,0d,0d) E Q, x O , C E R2"
||F(e,0d,0d)+C(e1,0d)r|| g k1||e|| +k2 (4.10)

[If-(6,0,1, 6d) — F0(éa0daéd)+ {C(6110d)_ G0(é110d)}7ll S k3llell + k4llCll + k5 (4-11)

where 1:1 to k5 are nonnegative constants. Notice that the right-hand side of (4.10)
does not depend on the fast variable C, which follows from the use of a globally

bounded control input 7'. Let

00 = {(e E R2" | “Me” S Cam V(61)S cza}. Cso < Car and Czo < Czr
n C
01 = {C6 132 |||C|| < E}

Q = 90x91 (4.12)

66

where c is an arbitrary positive constant. Notice that (to C 0,. The following lemma.
states that the fast variables decays very rapidly during a short time period. The
proof of the lemma is same as the proof of Lemma 2.1 in Chapter 2, hence it is

omitted.

Lemma 4.1 Consider the singularly perturbed system {AW-(4.7) and suppose that
the torque r is globally bounded, Then, for all (e(0),C(0)) 6 0, there exist e, and
T1 = T1(e) S T3 such that for all 0 < e < e1, ”C“ < fee for all t 6 [T1,T4) where T3 is

a finite time and T, > T3 is the first time e(t) exits from the set 0,.

We design the control input T such that a sliding mode condition is satisfied when
“C“ < he and (e,é) E Q, x 0,. This will be done by showing that o(é)Té(é) < 0 as
long as 0(é) 79 0. We will show o(é)Tq(é) < 0 usingcontrol input 7' different from
the control input in Chapter 2. This is motivated by studying a 2 DOF manipulator

control problem which is studied in the Example section. We have

0T(é){7(é) = aTa—‘fé
8e
= eTM[Ae + B(F0(e,0,0,) + Go(e.,0,)a) + D(e)LC(e —- 0)]
. 1 -
= aTMlAé+B(Fo(é.0a.0.)+Go(ét.60)r)+E (e)LCC] (4.13)
where

\

D(e) = block diag[d1(e),...,dn((.)], d,(t) = diag[e, 1]

For simplicity, we set

0.(é.0..0.) 42‘ F(é,0a.0.) — Fa(é.0a0a)

67
and let 1] = C+e(A—LC)“B¢1(é, 0,), 0,). It is shown in Chapter 2 that equation (4.13)

is given by

0T(é)o"(é) = oTM[Aé+B(Fo(é,0d,0d)+Go(é,0d)r)+ D(e)LC17

Ali—-

—B¢.(é.0a.0a) + 0(a)] (4.14)

We need an estimate of %MD(e)LCr) to design the control input 1' such that the

sliding mode condition is satisfied. It is also shown in Chapter 2 that

~ ~ 1
iMD(e)LCn = iMD(0)L/T 06(A'LC"“”"B{G(ét.0065101.94)-I}
l

xGo(él, 0d)v(l)dl + 0(e)

for t 6 [T1 + eln%,T4) where v(t) 6 K:{r(t)} for almost all t and the convex hull
lC{r(t)} is defined in [23]. Define IC“ = a; If |h,-,-(t)|dt Z l where h,-,-(t) is the ith
diagonal element of diagonal matrix C e(A'Lcl‘B, for i = 1, . . . , n. Let kn, be an upper
bound of the absolute value of ith component for vector Go(-)r, to be specified. It

can be verified that

~ 1
][%MD(0)L/T Ce(A—LC)(t-l)/¢B{G(él, gd)G;1(é1, 0,) - I}Go(é1, 00)v(1)d’] ]

S [lama],- (4.15)

where [-],- denotes the ith component of a vector, K, = diag[lcn, . . . , km], and k, =
[huh . . . , k,,,]T. Choose the observer gain a; such that all eigenvalues of (A — LC) are

real and negative. Then all kt, = 1 [27]. Hence inequality (4.15) becomes

. t
[EMDWL [T Cc(A—LC)(t—l)/cB{G(él, 0d)G;1(él, 0d) — I}G0(é130d)v(l)dl]]

5 [Na], (4.16)

68

From properties (ii) and (iii), we can alway find a locally Lipschitz function p,(é, 0d, 0,)
such that

Want) — F.1(é.0..0.)l s p.(é.oa.0a)

for (é,0d,0d) E ft, x (9,. Define the constant k,,- by the inequality
([0, - M(Aé + BF0(é),0d,0d) — p(é,0d,0d)sgn(0)],-] g k,,- (4.17)

almost everywhere for (é,0d,0d,0d) 6 Q, x OD where

p), = diaglp.(-).....pn(-)t sgnta) = [sgn(a.).....sgn(an)lT

Notice that k,,- can be calculated since Fo(°) and p() are known. Define the vector
B=(1—N)“Nk,+') (4.18)

where 7 > 0 is a vector such that (I — N)') > 0 and k, = [k,1,...,lc,,,]T. Since
the matrix (I — N) is an M-matrix, such a vector '7 always exists [39]. Consider the

function

00.0.0.0.) = G;‘(é.0.)[0.—MAé—Fo(é.0a.0a)

—(p(é.0a.0a) + 0)sgn(o(é))l (4.19)

where 6 = diag[Bl, . . . , 6,,]T and 0,- is the ith component of the vector 6. We take the
control input 7' as w(é,0d,0d,0d), saturated outside the set 0, x OD. In particular,

let

0.(é,0..0a0.) = G;‘(é..0.)(0a-MAé—Fa(é,0a0.)]

0.(é.0..0.) = —Ga‘(é.,oa)(p(é.0..0a) + 0)

69

where p( =(-)[p1 ...,pn(-)]T. Define

52...... 21';(-)>S‘
0»:‘(é.0a.0a) = < w) s‘ 303-135"
55...... 0";(-)<5;mm
Sbmar ‘0“ l > Sbmar
Ihflésgdtédtédl = ( Wit) SimtnS¢i(')SSimaa
Simtn wi('l<5imtn

 

for i = l,...,n where w;() and W() denote the ith components of the vectors
wa(-) and W(), respectively, 53",,” = max], 00 0, 0d)eflrxeD 11):, ((3, 0,,0d,0d),9 Lam," =

min(é.94.0d.5d)en,xep w;(é,0d,0d,0d), Sam” and ngm are similarly defined. Take
Ti = 10:03. 90.00.50) '1' w§‘(é.00.éa)sgn(ot) (420)

Inside the set D, X OD, we have

where T() = [r1(-),...,r,,(-)]T. Hence, k, in inequality (4.15) can be taken by

k, = k, + 6 (4.22)
Using (4.14), (4.16) , (4.21), and (4 22) we obtain

eT(e)e(e) = aT[—p(e,0,0,)sgn(e)—0sgn(a )+— :MD(O)LC77
i¢1(é.00.0a) + 0(6)]

S -'T[0 - N(ks + 3)] + 619:10:]

i=1

z _aT((1— N10 — Nksl + 0.210.)
i=1

70

= —0T(I - Nl‘l + Ch: [0,]

i=1

< 4171

for sufficiently small e, where 0 = [[01|,..., |0,,|]T, ’7 > 0, and k is some positive

constant. We summarize our findings in the following lemma.

Lemma 4.2 Consider the singularly perturbed system (4.6)—(4.7) with the applied
torque 1 defined by (4.20). Suppose that Assumption 4.] is satisfied, “C“ < he, and
(e,é) E Q, x (2, fort E [T1,T4), and e is small enough. Then as long as 0(6) 75 O. the

sliding mode condition

is satisfied for sufficiently small e and for all t 6 [T1 + e In 4,71,).

So far we showed that “C” < he for t E [T1,T4) and the torque T (4.20) satisfies the
sliding mode condition when ”C“ < he and (e, 0) 6 ft, x Q, for all t 6 [T1 + e ln %, T4).
It is shown in Chapter 2[Lemma 2.3] that for sufficiently small e, "C” < he and
(e,é) 6 ft, x Q, for all t 2 T, where T = T1(e) + eln %. This allows us to arrive at

the following theorem.

Theorem 4.1 Consider the system (4.6)—(4.7). Suppose that Assumption 4.] is sat-
isfied. Let the observer gain be chosen as in (4.3)-(4.4), all eigenvalues of (A-LC)
be real and negative, and the applied torque be chosen as in (4.20). Then there is
e3 > 0 such that for all 0 < e < (53, the closed-loop system (4.6)-(4.7) and (4.20 ) is
uniformly ultimately bounded with respect to the set S), = {(e,C) 6 R2" x R?“ I [It I) g
k,,/e', ((4)) < its}, for some constants la, and h, and 9, defined by (4.12), is an estimate

of the region of attraction.

Proof: The same as the proof of Theorem 2.1.

71

4.4 Example

 

Figure 4.1. Two Link Manipulator

As an example we consider the problem of controlling the 2 DOF planar manipulator
shown in Figure 4.1, where m,- is the mass of each link, 01) is the mass of payload, l,-
is the length of each link, 1,, is the length of center of mass, and I,- is the moment of
inertia for each link. The matrices H (0), C(0, 0), and g (0) take the forms

71(0) = hu(9) h12(9) , C(0,0) = c92 092 + 091

[212(0) 422(0) —e0. 0

9(9) _ {mllcl + ("‘2 + ml)}llgC08(91) + (mg + ml)lczgco3(01 + 92)
(mg + m))lcggcos(01 + 92)

where hu(0) = mllf1 + (mg + m))[lf + lg, + lelcgcos(02)] + I, + [2, h12(0) = (mg +
m))(l§2+lllcgsin(02)) +12, h22(0) = (m2+m))lZ,+Ig, and c = —(mg+m))lllcgsin(0g).

RI
[0

Let the desired path of 01(t) and 02(t) be

0,1(t) = —90"’ + 52.5°(l — cosl.26t)

0,,(t) = 170° — 600(1— cosl.26t),

respectively. The control task is to design the applied torque, r,, such that 01(t) and
02(t) track the desired path, 0,1(t) and 0,2(t), respectively, with unknown payload.
We take the parameters values given in Table 4.1 from [40], and assume that the

range of payload is 0 kg—1.2 kg. We use the formulas, I,- = fimdf, 1,1 = 0.18, and

 

 

Table 4.1
' Parameters Lower Link Upper Link
Link Mass m,(kg) 12.3 10.9
Link Length l,(m) 0.36 0.25

 

 

 

 

 

Table 4.1. Physical parameters of the manipulator.

10.952447: 1 . . . .
1,2 = W where m2 = 10.9 + m). Defining state variables in error coordinates,

e‘, = 0, — 0,“, e; = 0,- — 0.1,, i = 1,2, and setting e, = [e] 6?]7, e; = [e] ti]?
e = [e] e; e? e§]T, the state-space model is
e = Ac + Bl—éd + 1703.90, 00) + G(e1,0d)r]

where C(el,0d) = 'H’l(e1 + 90) and

F(e,0d,0d) = ’H'l(ei + 00){-C(61+ 0(1762 + 0.0032 + 2d) " 9031+ 0:21)}

73

A = block diag[Al, A2], A,- =2

B : blOCk diag[Bl, 82], B, =
1

Take the nominal functions F0(.) and G0(-) when the payload is m) = 0.6 kg. The

observer is constructed as
2: A0 + B[—0, + Fo(e,0,0,) + Ga(e,,0d)7)] + D(e)LC(e — 0) (4.23)

where
L = block diag[Ll, L2], L,- =
D(e) = block diag[D1(e), 02(6)], D,(e) = diag[— —
e
C = blOClx' diag[C1,C2], C, = [I 0]

The sliding surfaces are chosen by 01(0) = 20] + 0.], and 02(0) = 20% + 0%. Let the

region of interest 9, be

0. = {e e R“| ”Mall 3 1, ,/V(e) g 0.5}

where M = block diag[M1,M2], M,- = [2 1], and V(e) = (e})2 + (e32. The globally
bounded nominal functions Fo(~) and Go(-) are taken by saturating Fo(-) and Co(-)

outside the set 9,, where

n, = {e 6 ml ||Mé|| S 1.2, ,/V(e) g 0.5}

74

0.06 0.02

It can be verified that N = is the matrix such that each component

0.57 0.19

of N is the upper bound of the corresponding component of the matrix [C(e1 +
04)G;1(61 + 03) — I], Vel + 94 E R"; moreover, the matrix [I — N] is an M-matrix. It

can be also verified that

IF‘(e.0a0.)-F..‘(e.0a0.)| s p1(e,9a,0a)

IF’(e.oa0.)-F:(e.0a0'.)| < ps(e,0a,0.)

for e E Q, where p1(-) = 0.06|(e§ + 0,1)(eg + 0.12)] + 0.05(e§ + 0.12)2 + 0.8 and p2(-) =
0.1](e; +0d1)(e§ +0d2)| +0.4(e§ +0.12)2 +1.6. Notice that since we chose all eigenvalues
of (A-LC) to be real and negative, let,- = 1. It can be verified that led = 52 and
kg = 103.5 in inequality (4.17). Hence using equation (4.18), we obtain 0 = [7 67]T
for 7 = [0.7 1.5]T. Define

at = 00s - 153453.94) - 102353.00, 00) - W03, 0,0,)sgn(a,-)

for i = 1,2, where F32(é,0d) = - [H‘1(é1 + 0d)9(é1 + 0.1)].- when m) = 0.6, and

Slime: 2&2 + F01() > Slimmi-
¢:‘(é’ 0d, 0d) = 2&2 + F01(°) Siiimin S 2&2 + F010) S Simon:
53min 2&2 + F01() S 50min

Slime: P:() + Bf > Slime:
$00,010:!) = pat) + B.- 53....-.. S pt(-) + B.- S State
S0min P:() + Bi < Slimin

75

for i = 1,2 where F51(é,0d,0d) = — [7'1"(61 + 0d)C(el + 0d,eg + 0d)(62 +éd)]1‘ and

52mm = min(é.9¢,éd)60rx9d{2é; 'l' F31(é, 0d» ail}
Slime: = maxwflméaemxodlgéé + F5103, 901,510}
55min = mi”(é,ad,éd)en.xod{Pi(é~ 9d, (9.1) + Bi}
Simon: 2 max(é.od.éd)en,xod{l’i(éa 9d» 6.dl + Bi}

Using the optimization toolbox of MATLAB, we determined these maxima and min-

= 11.34, 5‘ = —7.64, s2 = 5.1, S2 = —27.48, 5,}

- 1 _ . .
1ma to be S 0min cam” 0min — 9.22,

amaz:

53min = 7.64, 53m” = 74.19, and 53min = 68.6. Take the torque T = G61(€,0d)u

mar
where u = [u] u2]T. Let

90 = {(e E R2" I ||Me|| S 0.8, ‘/V(el) g 0.3},
1
01 = {46W l IICII<;}

Q = 00XQ1

where (i = %(e', — é'i), C; = (e'2 — ea), 1 = 1,2, and C = [(11 C; ([2 22]T. According
to Theorem 4.1, the tracking error e is uniformly ultimately bounded. We assume
a payload of m; = 1.2 kg and simulate the response for 6(0) 2 [0.2 0 0 0]T, (3(0) 2
[0 0 0 0]T with c = 0.005. Notice that the initial conditions belong to Q, x 9,.

Figures 4.2 to 4.4 show that the actual path tracks the desired path with small
error. The tracking error can be reduced as the design parameter c is decreased.
Figures 4.3 and 4.4 show that joint angles 0, and angular velocities 0,- do not exhibit
peaking. Figure 4.5 shows that attractivity of the sliding surfaces is achieved after
errors between tracking errors and their estimates become small enough. Figure 4.6
shows that chattering appears in the torque. One way to reduce chattering is the

use of continuous approximation of the discontinuous signum nonlinearity [‘24], i.e.,

76

Tracking of The Dosirod Path

0:
vv 1 T T ‘1' I V

 

 

 

 

n A l L A

'°'-8.4 -o.3 -o.2 -o.1 o 0.1 0.2 0.3
Horizontal Displacement

 

Figure 4.2. Tracking of the desired path in Cartician coordinate with discontinuous
control input (the solid line is desired path and the dashed line is the actual path. 8:
starting point, D: destination point).

 

 

 

 

 

 

 

 

 

0.5 1
On- ..... .1
365» u
< -1 '- .
3’ _____ -
.‘.51- - - .- ~ I . 4
A l J l A
‘o 0.5 1 1.5 2 2.5
Turn.
4 r fl 7
N
w
9'
<
H.
c l 1 L 1

Tim.

Figure 4.3. Tracking of the desired joint angles 9,- (the solid lines are the desired joint
angles and the dashed lines are the actual joint angles).

77

 

.‘

MUUR VELmlTY 1
o
is

 

 

 

I
J

 

 

 

 

meow: VELocmz
3

 

Figure 4.4. Tracking of the desired angular velocities 0,- (the solid line are the desired
angular velocities and the dashed are the actual velocities).

 

.e
0

Estimate 0! Enode2)
00

 

 

 

 

J 1

0.05

 

0.15 0.2 0.25

be
0
UI
Oh-

0.1
Estlmato o! Enorm)

Figure 4.5. The phase portraits of estimates of errors é (the solid line is for (13], (3;)
and the dashed line is for (éf, 33%)).

78

 

 

 

 

Tome 2

 

 

Figure 4.6. Applied torques T1 and 72.

sgn(:r) is replaced by sat(x) where

l I>p
sat”): f -uSzS/t
—1 :r<—p

However, the continuous approximation results in uniform ultimate boundness around
origin in the state feedback stabilization problem [25]. Hence tracking error in the
continuous approximation is larger than that of the discontinuous controller. We
simulate the system with 6(0) 2 [0.2 0 0 0]T, (3(0) = [0 0 0 0]T, and p = 0.05 for
continuous approximation. Comparing Figures 4.2 and 4.3 with Figures 4.7 and 4.8,
respectively, one can observe that tracking error in the continuous approximation is
larger than that of the discontinuous control case. Figures 4.9 and 4.10 show that
chattering does not appear in the angular velocities 0 and the torque T. Figure 4.11
shows that the control input u is saturated during a short transient period, which
is a consequence of saturating u outside the set 9, x 9,. Finally, we simulate the

system with e(O) = [0.2 0 0 0]T, é(0) = [0.2 0 0 0]T which is considered in [35].

79

 

0.6 v v 1 r 1 I

0.5 P

0.31--

0.2 '-

Voflcd Displacement

 

 

 

L A A

0.1 0.2 0.3

 

-O.3 ‘ ‘
-0.4 -O.3 —0.2 -O.1 O
Honzontel DiepIecement

Figure 4.7. Tracking of desired path in Cartician coordinate with continuous ap-
proximation, p = 0.05 (the solid line is desired path in Cartician coordinate and the
dashed line is the actual path.).

 

 

 

 

 

 

 

 

 

Time

Figure 4.8. Tracking of the desired joint angles 0,- with continuous approximation,
[1 = 0.05 (the solid lines are the desired the joint angles and the dashed lines are the

actual joint angles)

80

 

 

AMUMR VELOCITY 1

 

 

 

Time

 

O

 

ANGULAR VELOCITY 2
o o” :e
G
l

 

 

 

O,
O
U!
4
.A
U.
'0]-

2.5
Time

Figure 4.9. Tracking of the desired joint angles 0,- with continuous approximation,
11 = 0.05 (the solid lines are the desired velocities and the dashed lines are the actual
velocities)

 

 

 

 

 

 

 

 

p ..

 

 

Figure 4.10. Applied torque T with continuous approximationw = 0.05)

81

 

U1

 

 

 

 

 

 

 

 

 

 

.‘c 1 L 1 1 1
0 0.1 0.25 0.3 0.35 0.45 0.5
Time
40 T fi
20
O
9
-20 .. . . . ..
.40 r «
-m 1 l 1 1 1 A 1 l l
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5
Time

Figure 4.11. The control input u with continuous approximation( 2 0.05)
Since 81(0) 2 é1(0), the fast variable C(t) does not contain the peaking term. Hence
é2(t) does not contain the peaking term. Consequently, the control input 11 does not
saturate. This is shown in Figure 4.12.

If we use the control scheme in Chapter 2, it can be verified that

Hi

91,,- — Fg,(é,od,éd) — sat (2e; + Fg,(é, 9d, 6m)

—sat(p2(é.0d,6ld)+106)sgn(0,-),

for z' = 1,2. Recall that the corresponding discontinuous coefficient terms of the

control input in this chapter are

p1(-) = 0.06|(e; + (9d,)(eg + 9.12M + 0.05(e§ + 5.12)? + 0.8
p2(-) = 0.1|(e; + 6d,)(eg + owl + 0.4(63 + (9,2)? +1.6
31 = 7 and 82 = 67. One can observe that the magnitude of the discontinuous

coefficient terms are significantly increased if we use the control scheme in Chapter

82

 

 

 

 

 

 

 

 

 

 

 

Figure 4.12. The plot of control input u with the same initial condition(p = 0.05)

go

4.5 Concluding Remark

We have designed an output feedback variable structure controller that ensures track-
ing of a desired path, with arbitrarily small error, for an n DOF manipulator. The
desired accuracy of tracking can be achieved by choosing the design parameter c. We
require that the diagonal components of (G(0)GJI(0) — I) dominate the off diago-
nal components by requiring (I — N) to be an M-matrix. This is different from the
requirement ||G(0)G5'1(0) - III.>0 < 1 in Chapter 2. We show, via an example, that
the assumption on the input coefficient matrix uncertainty and the corresponding

controller in the current chapter are less conservative than those of Chapter 2.

CHAPTER 5

Conclusions and Future Work

5. 1 Conclusions

We have designed output feedback variable structure controllers to achieve the control
tasks of stabilization and tracking of a reference signal in the presence of modeling
uncertainties and / or external disturbances. Our design does not suffer from the peak-
ing phenomenon present in the earlier variable structure work [12] and [13]. This is
achieved by the use of robust high—gain observers and globally bounded control, which
is motivated by [14]. However, our design procedure and analysis are fundamentally
different from those of [14] due to the discontinuous nature of variable structure con-
trol. Another significant contribution over the earlier work [12] and [13] is the wider
applicability of our results. The papers [12] and [13] do not allow modeling uncer-
tainty in the input coefficient matrix, which is allowed in our work. Such uncertainty
is common in applications, like robot manipulators, and it complicates the controller
design and analysis. We assume a less conservative condition for the zero dynamics
of the system than that of [12]. In particular, we assume that the zero dynamics of
the system are minimum phase, but we do not require exponential stability or global
growth conditions as required in [13]. In the tracking control of robot manipulators.

we have designed a variable structure controller with only position measurement. The

83

84

controller uses a high-gain observer to estimate angular velocities. Although a similar
scheme is suggested in [37], that work is not complete since a circular argument is

used in the observer design.

5.2 Future Work

Since the variable structure control is discontinuous, it could excite high frequency
unmodeled dynamics. For instance, actuator dynamics in the robot manipulator
control problem can be ignored since their electrical time constants are much smaller
than the mechanical time constants associated with the robot motion. However if we
use a variable structure control scheme, the ignored dynamics could be excited. Hence
the controller design should be modified to compensate for the ignored dynamics. We
investigated reduction of chattering by the use of a continuous approximation of the
discontinuous signum nonlinearity via an example in Chapter 4. However analysis
has not been done in that chapter. A complete analysis for state feedback case is
given in [25]. Similar analysis for output feedback case is needed. Since adaptive
control schemes achieve good performance for parametric uncertainties [41, 42], it is
expected that combining a robust control scheme such as variable structure control
and an adaptive control scheme will give good performance for systems which contain
the parametric uncertainties, nonparametric uncertainties, and external disturbances.
The paper [43] studied the state feedback adaptive variable structure control in the
control of robot manipulators. In view of the results of this thesis, we foresee that an

output feedback version of [43] can be developed.

APPENDICES

APPENDIX A

Proof of Lemma 2.1

It can be verified that the right-hand side of equation (2.12) satisfies the conditions
of Theorem 4 of [23]. Using Theorems 4 and 5 of [23], the solution of equation (2.12

) can be represented by
:L'(t) = :L'(0) +/01[Ax(‘r) + B{F(:r(7')) + G(.’B(T))’U(T)}]d7‘

where v(t) E K{u(:i:(t)} for almost all t and the convex hull K{u(:i:(t))} is defined
in [23]. Note that we use Theorem 1 of [22] for the calculus of Filippov’s differential

inclusion. Using inequality (2.17) and the Gronwall—Bellman inequality, we obtain

 

k
:13 t < a: O et(ll’4ll+k1)+ ___2__ etUIAII-l-kil _ 1
II ()n _ n1)“ “AMI 1
k
= ‘(H’kil 2 t(l+k1l _1

for z(t) € 9,. Since cw < c” and am < c", there exists a finite time T3 such that
z(t) 6 (2,, Vi E [0, T3]. Let T, > T3 be the first time 22(t) exits from the set 0,. Since
z(t) e 9,, Vt e [0,T4),

ll$(t)l| S as, W E [01731) (A-ll

85

86

where 013 is a positive constant. It can be also verified that the right-hand side of
equation (2.13) satisfies the conditions of Theorem 4 of [23]. Using Theorems 4 and

5 of [23], the solution of equation (2.13) is represented by

at) = e‘A'LCW‘aO)

+/e 1" LC" 'V‘Bm )- Fo<i1+<01z1 - Gownvvndr
for Vt E [0,T4) where v(t) E I\'{u(:ic(t)} for almost all t. Since (A — LC) is Hurwitz,
“e(A-LC)t/(“ S kle—agt/t (A2)

for some positive constant a; and k1. Using inequalities (2.17),(2.18), (A.l). and
(A2), it can be verified that

t
||C(t)|| s hummus-W + / kze‘°*“-”"(kqu<(r)II + kadr, te [05m (A.3)
where k7 = 1:303 + k5. Let b(t) = ea“/‘||C(t)[|. Then inequality (A.3) is given by
kt i
M0:hM(0)H+’“(ewh—u+mafiwow

Let /\(t) = k1||C(O)II + £1£zg(ea,¢/c — 1). By the Gronwall-Bellman inequality,

“0

|/\

)1t)t/\(+/0( s)k,k4 ef: k‘k‘des

k1 [€76

= k -0 kikd (o;/c)t_ km:
zM(Hk +a_%hhk e 1

Thus
k1k7€

t < (klks‘ol/(lt __ _ (kiln-01h)!
||C()ll_kzllC(0)lle +m_€kl,€411 e 1

(I)
‘1

Choose 61 = alk4kl/2 so that

a!

a
k1k4——lS—3-, V0<e§el
c

1.6

Then

klc €—Ofl/2( + 9k1k7€

6"“ a,

 

 

”((1)” S

Set T1 = i—j ln(l,). Thus

C

”((1)“ < €12,sz [TIL/1.)

where k is some positive constant. Since c ln((i,) —> 0 as

small enough to ensure that T, < T3, V0 < c < (1.

9 Vt E [0,T4)

(A.4)

( —> 0, we can choose (1

APPENDIX B

Proof of Lemma 2.3
To show that "((t)“ < elf: for all t Z T1(e), it is sufficient to show that z(t) 6 Q, for
all t Z 0. Define

fl = {2: E R" | [Illlxll S 6.2, ‘/V(z) S 522}, and
{5' 6 R" l ”Mill S 5.2. \/V(5) S 522}

31
II

where c.o < 5.2 < 6.2 < c", czo < 5,: < 522 < c”, 6,; = fifflgégfia > 1), and

5,2 = affigéflfia > 1). Since {:r(0), C (0)} E 9, there exists a finite time T: < T, such

that a: 6 S2 for all 0 S t S T2, where T; is independent of c. For sufficiently small 6,

T1(£) + eln(%) 5 T2. Note that 2‘: might be outside (.2 for t < T1(e) + eln(%), but since
. = 2::-

forthcoming analysis is done for t 6 [T1(c) + cln(%), T4). By Lemma 2.2,

— {PK}, by (A.4) with sufficiently small 6, :i: 6 S2 at t = T1(e) + eln(-:-). The

aT(:E)&(:i:) S -02[|a(:i:)||1 for :i: 6 0

Recall that V(é) = 2TP2. The derivative of V(é) along the trajectories of equa-

tion (2.15) is given by

17(2) = —2TQ2 + 227P1‘30(:2) + 22TPD(C)LCC

88

89

S -/\mm(Q)ll5‘I|2+2||P3|l||5|||l0(53)||+€i€1f€ll5ll
S —a1l/+a2\/V||0(i‘)||+a3e\/l7 (8.1)

where ||PD(e)LC|[ S It], a3 = 4L

\/ AmmU’)

Since 522 = 211732652 With 7:52 > 1 and 522 = 'Z-f'T—‘sgésg Will] in > 1,

1‘2 “é‘ {:2 e R" | new! 3 6.2. MW) 3 a2}

where 6,2 < 632 < 6,2, max{ézg, fife”) < 632 < 532, and 7‘32 = (3:)(gfi) > 1 is always

defined and Q C (2. For it 6 S2 and sufficiently small 6,
17(2) 3 0 , for W(é) 2 (3.2 (8.2)

Since 0T(i:)o‘(:i:) < 0 and 17(2) _<_ 0 for ,/V(2) 2 6,2,1: 6 (’2 fort e [T1(c)+ cln 571,).
Using a contradiction argument, it can be shown that :c 6 Q, for Vt 2 T1(c) + c In 5
Hence a: E Q, for Vt Z 0. Therefore, we can conclude that (3,52) 6 Q, x Q, and

C< lire, Vt 2 T1(e) + eln%.

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