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LIBRARY ‘1

. . 3 This is to certify that the
“hang“! §tate dissertation entitled
Unlverslty ______l

fi, A__—-—i

 

UNIVERSAL INTEGRAL CONTROLLERS WITH NONLINEAR
GAINS

presented by
Hyon Sok Kay

has been accepted towards fulfillment
of the requirements for the

PhD. degree in Electrical Engineering

Major Professofs’STgnature
M ay 16s,, 2 0 0 3

Date

 

MSU is an Affinnative Action/Equal Opportunity Institution

-.-. -.--- ‘.—.--.-

PLACE IN RETURN Box to remove this checkout from your record.
TO AVOID FINES return on or before date due.
MAY BE RECALLED with earlier due date if requested.

 

DATE DUE DATE DUE DATE DUE

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6/01 cJClFIC/DateDuepBS-sz

UNIVERSAL INTEGRAL CONTROLLERS WITH NONLINEAR GAINS
By

Hyon Sok Kay

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY
Department of Electrical and Computer Engineering

2003

ABSTRACT
UNIVERSAL INTEGRAL CONTROLLERS WITH NONLINEAR GAINS
By

Hyon Sok Kay

Various robust nonlinear control techniques have been developed for the regu-
lation of nonlinear systems under uncertainties and disturbances. Among the con-
trollers proposed for single-input single-output, input-output linearizable, minimum
phase, nonlinear systems, the Universal Integral Controller has a simple structure
that can be viewed as a natural extension of the classical PID controller and re-
quires minimal information about the system. While robust nonlinear controllers
ensure asymptotic regulation, they do not address the problem of transient per-
formance. In this dissertation, we extend the structure of the Universal Integral
Controller to provide more freedom that can be utilized to improve the transient
performance. We allow the integral, proportional and derivative gains to be non-
linear functions of the tracking error and its derivatives. Two possible schemes
for nonlinear integration are investigated: a nonlinearity placed before or after the
integrator. Our analysis shows that the new Universal Integral Controller achieves
regional and semiglobal regulation. More specific results are provided for the non-
linear PID controller, which is a special form of the Universal Integral Controller.
By simulation, we demonstrate that the new freedom in designing the nonlinear

gains can be used to improve the transient performance.

ACKNOWLEDGMENTS

I am deeply indebted to my advisor, Professor Hassan K. Khalil, for his en-
couragement and invaluable advice. I would like to extend my sincere thanks and
appreciation to my parents, Lee Gil Kay and Yesil Kim, and my wife, Young Yi Yu,
for their support and patience. I would also like to thank my children, Suyeon and

Sumin, for their love and understanding.

iii

Table of Contents

LIST OF TABLES vii
LIST OF FIGURES viii
1 Introduction 1
1.1 Integral Control .............................. 2
1.2 Universal Integral Controller ...................... 3
1.3 High-gain Observers ........................... 5
1.4 Controllers with Variable Gains ..................... 6

1.5 Overview of the Thesis .......................... 7

2 Universal Integral Controllers with Nonlinear Integral Gains 9
2.1 System Description ............................ 9
2.2 Sliding Mode Control .......................... 14
2.3 Design of Nonlinear Integral Gains ................... 17
2.4 Closed-loop Analysis ........................... 26
2.4.1 Nonlinearity before the integrator ............... 26

iv

2.5

2.6

2.4.2 Nonlinearity after the integrator ................
Simulation Results ............................
Conclusions ................................
Stability of Nonlinear Integrators ....................

Derivation of (2.43) ...........................

3 Universal Integral Controllers with Nonlinear Gains

3.1

3.2

3.3

3.4

3.5

Introduction ................................
Design of Nonlinear Proportional and Derivative Gains ........
Closed-loop Analysis ...........................
Simulation Results ............................
Conclusion ................................

S-procedure for Lemma 1 ........................

4 Nonlinear PID Controllers

4.1

4.2

4.3

4.4

Introduction ................................

Closed-loop Analysis ...........................
Simulation Results ............................

Conclusions ................................

5 Conclusions

40

48

49

51

56

56

57

62

65

69

72

75

75

79

90

95

99

BIBLIOGRAPHY 102

vi

List of Tables

3.1 Examples of sector conditions for stable systems determined by the
LMI ....................................

vii

1.1

2.1

2.2

2.3

2.4

2.5

2.6

2.7

2.8

2.9

3.1

3.2

List of Figures

Universal Integral Controller for relative-degree—one and two systems 4

Simulation results of the continuous sliding mode controller and Uni-
versal Integral Controller for the field-controlled DC motor ..... 19

Nonlinearity 1/J(é.,) for the Nonlinearity Before Integration scheme . 24
Two possible schemes for nonlinear integrators ............ 25

Simulation results of Universal Integral Controllers with nonlinear
integrators for the field-controlled DC motor ............. 42

Nonlinearities 101(ea) and ¢2(0‘) for the simulation of the field-controlled

DC motor ................................. 43
Simulation results of motion on a horizontal surface ......... 44
Simulation results of the magnetic suspension system ......... 46

Nonlinearity Men) for the simulation of the magnetic suspension
system ................................... 47

Nonlinearity before integrator when 3 = 0 ............... 50

Simulation results for the synchronous generator connected to an
infinity bus ................................ 67

Nonlinear proportional gain v1(e1) : k1(e1)e1 for the simulation of
the synchronous generator connected to an infinite bus ........ 68

viii

3.3

3.4

4.1

4.2

4.3

4.4

4.5

4.6

4.7

Simulation results for the single link manipulator ........... 70

Nonlinear prOportional and derivative gains k1(e1), k2(e2), k3(e3) for
the simulation of the single link manipulator ............. 71

Nonlinear PI controller for relative-degree-one systems and nonlinear
PID controller for relative-degree-two systems ............. 78

Simulation results of the nonlinear PID controllers for the field-
controlled DC motor ........................... 91

Nonlinearities 11:1(ea) and z/J2(e1) for the simulation of the field-controlled

DC motor ................................. 92
Simulation results of the nonlinear PID controllers for motion on a
horizontal surface ............................. 93

Nonlinear proportional gain k1(e1) for the simulation of the unstable
second-order system ........................... 96

Simulation results for the second-order unstable system with nonlin-
ear and linear proportional gains .................... 97

Simulation results for the second-order unstable system with linear
and nonlinear integrators ........................ 98

Chapter 1

Introduction

For input-output linearizable, minimum phase, nonlinear systems, the univer-
sal integral controller achieves robust asymptotic tracking. The controller which
can be viewed as an extension of the classical PID controller, uses a feedback signal

of the form
tip—181 + dpel
1 cup-1 dtP

 

de
ko/e1+k181+k2j+”'+kp_

where e1 is the tracking error and the derivatives of the error are calculated using
a high-gain observer. While it achieves robust steady-state performance, it does
not address the problem of transient performance. In fact, most of the time, the
improvement in the steady-state performance comes at the expense of degradation

of the transient performance. The goals of this dissertation are

1. Extend the structure of the Universal Integral Controller by replacing the

constant gains kg to kp_1 by nonlinear functions of 61 and its derivatives

2. Study the design of these nonlinear functions to improve the transient perfor-

mance of the system

3. Prove the stability of the closed-loop system under Universal Integral Con-

trollers with variable gains

In the next sections, we briefly review the main background elements of this disser-
tation. In Section 1.1, we review integral control of nonlinear systems, which leads
to the review of the Universal Integral Controller in Section 1.2. In Section 1.3,
we review high-gain observers. The idea of using nonlinear gains as a tool for im-
proving transient performance is not new in the control literature. In Section 2.4,
we review the literature on this idea. Finally, we give an overview of the thesis in

Section 2.5.

1 . 1 Integral Control

Integral control is extensively used in control system design. It achieves robust
asymptotic tracking of a reference signal, which is constant or approaches a constant
limit asymptotically, in the presence of external disturbances and unmodeled system
dynamics. An integral controller is composed of two parts: the integrator and
the stabilizing controller. Integration of the tracking error creates an equilibrium
point, where the tracking error is zero, and the controller stabilizes the augmented
system. While external disturbances or uncertainties of the system model move
the equilibrium point, the integral action ensures that the tracking error is zero at
equilibrium, as long as the stabilizing controller maintains the equilibrium point
asymptotically stable. The control problem is to design a controller that stabilizes
the equilibrium point of the augmented system in the domain of interest in the
presence of unknown disturbances.

The theory of integral control for linear systems was developed in the seventies

by Davison [8], Francis [12], and Desoer and Wang [9], among others. In the early

nineties, the integral control theory was extended to nonlinear systems and local
results were provided. Huang and Hugh [17],[18] used the extended linearization
method and allowed slowly varying external signals. Isidori and Byrnes [21] derived
necessary and suficient conditions for local solutions. Regional and semiglobal
results for input-output linearizable systems were reported later on. Freeman
and Kokotovic [13] used backstepping to design a state feedback controller for
systems with no zero dynamics. Mahmoud and Khalil [31], Khalil [25],[27] and
Isidori [19] designed output feedback controllers, with high-gain observers, which

achieve semiglobal regulation.

1.2 Universal Integral Controller

Mahmoud and Khalil [31] used integral control to design a robust min-max
output feedback controller for a single-input single-output, input-output lineariz-
able system with asymptotically stable zero dynamics. The integrator creates an
equilibrium point at which the tracking error is zero. The robust controller brings
the trajectory of the system to a positively invariant set. The size of that set can
be made small by choice of some control parameters, and inside the set the con-
troller acts as a high—gain controller stabilizing the equilibrium point. Khalil [27]
carried the work of [31] further by designing Universal Integral Controllers, where
continuous sliding mode control was used for robust nonlinear control. The control

input has the form

 

koa + klel + - - - + kp_1ep_1 + 6p)

# (1.1)

u = —k sign(.) sat (

 

 

 

 

 

 

 

 

 

 

 

 

‘KP

 

 

 

 

 

 

 

 

(a) A PI controller with K, = kkl/p, K p = k / [i followed by saturation

—>l , am
Is

81 KP 74 __1£_>

~—>]HGO 9 Kb

(b) A PID controller with K I = kkl/p, K p = [Chg/fl, K D : k/u followed by
saturation

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 1.1: Universal Integral Controller for relative-degree-one and two systems

where p is the relative degree of the system, a is the integrator output, e1 to e, are
the tracking error and its derivatives, and sat(-) is the saturation function. Only the
information about the relative degree p of the plant and the sign of its high frequency
gain sign(-) is necessary to design the controller, and the structure of the controller
can be viewed as an extension of classical PID controller. In particular, for relative-
degree—one systems, it is the classical PI controller followed by saturation, while for
relative-degree-two systems, it is the classical PID controller followed by saturation.
Figures 1.1 shows the structure of the controller for systems with relative degree

one and two.

1.3 High-gain Observers

High-gain observers provide an important technique for the design of out-
put feedback controllers for nonlinear systems. A high-gain observer estimates
the derivatives of the output of a system. The observer gain depends on a small
parameter c, which can be adjusted to guarantee that the estimation error decays
to the level 0(6) in an arbitrarily small time interval. Esfandiari and Khalil [10]
studied the use of high-gain observers in the design of output feedback control of
nonlinear systems, which are minimum phase and input—output linearizable. A key
contribution of their study is the use of saturation nonlinearities to overcome the
peaking phenomenon associated with high-gain observers. The observer is designed
to assign the observer eigenvalues at 0(1/6) values in the open left-half complex
plane. This results in exponential modes of the form (1 / e)“ exp(—ta/e) for some
positive constant a and positive integer m. While the decay of the exponential
term exp(—ta/e) can be made faster by decreasing e, the amplitude term (1 /e)"‘
grows larger with such decrease. This peaking phenomenon was known for linear
systems, but Esfandiari and Khalil showed that its impact on nonlinear systems is
more serious because it could destabilize the system. They proposed a technique
to overcome the efiect of peaking in the observer on the state of the system under
control, i.e., the plant. The idea is to saturate the feedback control law outside a
compact region of interest such that during peaking period the control saturates
and protects the plant from the efiects of peaking. Because the peaking transients
decay rapidly, the saturation period is small. Since its introduction in 1992, the
Esfandiari and Khalil technique has received a lot of attention in the control field

and has been included in a few textbooks [33],[28],[20]. One of the important con-

sequences of this technique is the ability to separate the design of output feedback
control for nonlinear systems into a state feedback design followed by observer de-
sign, and to prove that the output feedback controller recovers the performance of
the state feedback controller. Teel and Pray [41] developed a generic separation
principle, which showed that a globally stabilizable system by state feedback with
uniform observability can be stabilized semiglobally by output feedback. Atassi and
Khalil [3] provided a more comprehensive separation principle and showed that the
output feedback controller recovers the performance of the state feedback controller
in the sense of recovering asymptotic stability of an equilibrium point, its region of
attraction, and its trajectories. Extensive survey of the use of high-gain observers

in nonlinear control can be found in [26].

1.4 Controllers with Variable Gains

Numerous researchers in various fields have proposed to use variable gains as
a tool to improve the performance of controllers. The idea has been applied to
aircraft control [39], traffic control [6], vibration control [1] and power systems [42],
among other areas. In particular, as PID controllers are widely employed in control
systems, controllers with nonlinear PID gains have been designed to improve the
performance. Experimental and simulation results showing superior performance
of nonlinear proportional and derivative gains are found in Fertik and Sharp [11],
Clark [7], and Ni [32]. While integral control achieves asymptotic regulation, it has
been observed that the integrator can degrade the transient performance. Krike-
lis [29] proposed ‘intelligent’ integrators where a deadzone nonlinearity was placed

in a feedback loop around the integrator to prevent the buildup. This idea was

applied to digital systems by Ghreichi and Farison [15]. The idea of decreasing the
integral gain when the error is large can be found in Luo, Jackson and Hill [30] and
F‘ukuda, Fujiyoshi, Arai and Matsuura [14]. Shahruz and Schwartz [38] developed
a computer aided design technique for tuning nonlinear PI controllers. Izuno [22]
designed integral gain as a function of desired speed.

Only a few analytical results are available for nonlinear PI and PID controllers.
Seraji [35],[36],[37] used Popov criterion to obtain the range of the nonlinear gains
that stabilize the linear systems. Huang [16] showed Lyapunov stability of a non-
linear PD controller for second-order stable linear systems. Armstrong, Neevel and
Kusik [2] showed the stability of a nonlinear PID controller for linear systems by
switching off the nonlinearity which contributed to the positive terms of the deriva-
tive of Lyapunov function. Xu, Ma and Hollerbach [43] showed Lyapunov stability

for second-order robotic systems.

1.5 Overview of the Thesis

In Chapter 2, we describe single-input single-output minimum phase systems
with a well-defined normal form. A brief review of ideal sliding mode control, con-
tinuous sliding mode control and Universal Integral Controllers is provided. We find
that the input-to-state stability of the dynamics of the system on the sliding surface
is the essential property for the analysis. To prevent the buildup in the integrator,
we propose to place a nonlinearity that satisfies a sector condition before or after
the integrator. The nonlinear integrator is driven by an augmented error, which is a
weighted sum of the tracking error and its derivatives. Our analysis shows that the

controller with nonlinear integration achieves regional and semiglobal regulation.

We compare the performance of the controllers for linear integrator, nonlinearity
before integration and nonlinearity after integration schemes.

In Chapter 3, we investigate the use of nonlinear pr0portional and derivative
gains. We consider a nonlinear gain of the form k,- = k,(e,-) where v(e,-) = k,(e,~)e,-
satisfies a sector condition. The dynamics of the system on the sliding surface can
be represented as a Lure system. Linear Matrix Inequalities are used to find the
sector condition for the nonlinear gains, so that a Lyapunov function on the sliding
surface can be obtained. Our analysis shows that if the LMI problem is feasible,
then the controller, with nonlinear proportional, integral and derivative gains, will
achieve regional and semiglobal regulation. The effect of nonlinear gains on the
performance is investigated by simulation.

In Chapter 4, we consider systems with relative degree two. A Universal In-
tegral Controller with nonlinear gains reduces to a PID controller with nonlinear
proportional and integral gains. In Chapters 2 and 3, the augmented error is used
to drive the integrator for systems with higher relative degree. For relative degree
two systems, we also consider nonlinear integrators driven by the tracking error,
which is the classical form of the nonlinear integration found in the literature. In
Chapter 3, LMIs were used to find sector conditions for proportional and deriva-
tive gains. For systems with relative degree two, a Lyapunov function is obtained
analytically. Our analysis shows that a PID controller with nonlinear gains, with
the nonlinear integrator driven by either the augmented error or the tracking error,
achieves regional and semiglobal regulation. By simulation, we investigate the efiect
of nonlinear gains on the performance of the controller.

Our findings and some remarks are summarized in Chapter 5.

Chapter 2

Universal Integral Controllers with

Nonlinear Integral Gains

2.1 System Description

Consider the single-input single-output nonlinear system
i = f(x.9) + g(z.9)u. y = hw) (2.1)

where a: E R" is the state, u E R is the control input, y E R is the measured out-
put, and 9 E R‘ represents unknown constant disturbances and parameters. The
functions f, g and h depend continuously on 9, which belongs to a compact set
6 E R‘. We assume that for all 0 E 9, f, g and h are suficiently smooth on U9, an
open connected subset of R" that could depend on 6.

In this work, we consider input-output linearizable, minimum phase nonlinear sys-

tems, which have well-defined normal forms.

Assumption 1 The system (2.1) has a uniform relative degree p g n for all

:r E U9 and 9 E O, i.e.,

Lgh(:r,9) = L,L,h(z,a) = m = Lng‘2h(x,9) = o

|L9L’}"1h(:c,9)| 2 co > o (2.2)
where co is independent of 9. Moreover, there exists a difieomomhism

1)
E

= T(:r., 9) (2.3)

of U9 onto its image that transforms (2.1) into the normal form

1'7 = 407.39)

é.=£.-+., forlS‘iSp—l

. (2.4)
ép = b(n.£.0) + a(n. £.6)u

31:61

Under the conditions given in Byrnes and Isidori [5, Proposition 3.2b, Corollary
5.6], Assumption 1 holds locally or globally when 9 = 90 is known. Global con-
ditions when 1') : q(n,£) is also given in [5, Corollary 5.7]. In our assumption,
the mapping in (2.3) is a diffeomorphism from U, onto its image. Necessary and
sufficient conditions for a mapping from U to V to be a diffeomorphism of U onto
V is provided in Sandberg [34]. A mapping M : U —) M (U) is a diffeomorphism
of U onto its image if and only if detJ, ;£ 0 for all p E U and M is a proper map
of U into M (U), where JP is the Jacobian matrix of M at p E U. The results that
guarantee the existence of a diffeomorphism for a given U9 is not available in the

literature. For most systems satisfying the local conditions, a region over which the

10

normal form exists is determined in the process of transforming the system into the
normal form. While the requirement of the existence of normal form uniformly in 9
is more restrictive, there are many examples of physical systems where the normal
form exists uniformly over a compact set of system parameters.

We consider the tracking problem where y(t) is to asymptotically track a ref-

erence signal r(t). The reference signal has the following properties:

0 r and its derivatives up to the pth derivative are bounded and 1"”) is piecewise

continuous, for all t 2 O;
o limHoor(t) = w and limtnoo r(‘)(t) = 0 for 1 S i S p.

Define V(t) by V = [r — w,r(1), - -- ,r‘P“1)]T. By construction, V(t) is bounded for
all t Z 0 and limtnoo u(t) = 0. Let W C R, A C R”, and I‘ C R be compact sets
such that w(t) E W, V(t) E A, and r(P)(t) E I‘ for all t Z 0. Set at = (II/,9) and

DszO.

Assumption 2 For each d E D, a unique equilibrium point a': : :‘I':(d) 6 U9 and

a unique control 11 : u(d) exist such that

0 = f(§=.0) + 9(5. 5011(4). w = h(a'=.9)

With the change of variables (2.3), the equilibrium point 5(d) maps into

(fi(d)!£-(d))’ Where 5(d) : [11), 0, ° ' ' t 0]?"

For the tracking problem, let

11

and rewrite the system (2.4) as

2 = qo(z,e + V, (1)
(5326141. forlgigp—l

(2.5)
ép : bo(z, e + V, d, rm) + ao(z, e + V, d)u

ym=€1

where ym is the measured tracking error.

Assumption 3 There exists positive constants r1 and r2, independent of d,

such that for all d E D and u E A,

He” < r1 and ”2” < r2 => x 6 U9

Define the balls 8 z {e E R" : Hell < r1} and Z = {z E Rn‘t’ : ||z|| < r2}.

Since A is compact, there exists r3 2 0 such that

”V“ 3 r3 for all u E A (2.6)

Therefore, He + V]] < r1 + r3 for all e E 8 and u E A.

Assumption 4 There exist a C'1 proper function V}, : Z ——> R+, possibly depen-
dent on d, and class [C functions a1,a2,a3 : [0,r2) ——> R+ and 'y : [0, r1 + r3) ——>

R+, independent of d, such that

01(IIZII) S Vo(t.z»d) _<. 02(HZH) (2-7)

6V 6V
5‘1 + —°qo(z e + v d)_ < —a.(nzu) v l|2||_ > 7(lle+ um (28)

12

for all e E 8, z E Z, V E A and d E D. Furthermore,

”r(t's) < 031(010'2» (29)

Moreover, the equilibrium point z = O of 21' = qo(z, 0,d) is exponentially stable
uniformly in d, i.e., there exist positive constants 00, I30 and r0, independent

of d, such that
IIZ(t)|I S fioe‘“°‘||2(0)l|. V IIZ(0)II S 1'0. W Z 0 (2-10)

The system 2 = qo(z, e + u, d) is said to be input-to—state stable, viewing (e + V)
as the input, if there are a class [CL function fi(-) and a class [C function a(-) such
that for any initial state z(0) and any bounded input e(t) + V(t), the solution z(t)

exists for all t Z 0 and satisfies

||2(t)|| s mum)“. t) + a (0:313, ”e(t) + mm) (2.11)

It is said to be locally input-to-stable if there exist positive constants c1 and c2 such
that the inequality (2.11) is satisfied for ||z(to)|] 3 c1 and supQ0 ||e(t) + V(t)]| <
c2 [40].

The inequalities (2.7) and (2.8) in Assumption 4 imply that, viewing (e + V) as
the input, the system 2' = 400:, e + V, d) is locally input-to—state state in the domain

of interest.

13

2.2 Sliding Mode Control

Robust asymptotic tracking can be achieved by sliding mode control. A slid-
ing surface is chosen such that asymptotic tracking is achieved when motion is
restricted to the surface. Then, a discontinuous control is designed to render the
surface attractive and guarantee that all trajectories will reach the surface in finite
time. Assumption 4 ensures that the system is minimum-phase and allows us to
concentrate on stabilizing the motion of the e,- variables. Therefore, it is typical to
choose the surface s = 0 such that s is a weighted sum of the tracking error and its
derivatives:

p—l
s = Z k,e,- + e p
i=1

where the positive constants k1 to kp_1 are chosen such that the roots of
AM + k,_1AP-2 + - - - + k2). + k1 = o (2.12)

have negative real parts. The dynamics on the surface s = 0 are described by

 

'0 1 O 0 i
0 0 1 0
c: s s cdé‘Ac
0 0 0 1
_—k, —k2 —k3 —k,_,d

 

14

where C 2 [e1, - -- ,e,,_1]T and A is Hurwitz. The variable 3 satisfies a first-order

diflerential equation of the form
3' = A(-) + (LgLf._1h)s
where A(-) is a continuous function of z, e, d, V and r(P). The sliding mode control
u = —ksign(LgL’}‘1h)sgn(s)

where

1, for s > O
sgn(s) =
—1, for s < 0

ensures that

33' g —po|s|, for all s 96 0

provided the control gain k is sufficiently large to satisfy

AC)

_ S k —Po
LgL’; 1h

 

 

over the domain of interest.

While ideal sliding mode control achieves zero steady-state error, it is well
known [28, pages 198, 215] that, in practice, sliding mode controllers suffer from
chattering due to nonideal effects such as switching delays and unmodeled dynamics.

One approach to avoid chattering is to approximate the signum nonlinearity sgn(s)

15

by the saturation nonlinearity sat(s/u), where

1, for p > 1
3‘1“?) 2 p. for Ipl _<_ 1
—1, for p < —1

and u is a positive constant. In the presence of nonvanishing disturbance, the

continuous sliding mode controller

u : —k sign(L9Lfr—1h) sat (I?)

can guarantee only ultimate boundness with respect to a compact set, which can
be made arbitrarily small by decreasing u. However, a too small value of u will
again induce chattering due to nonideal effects [28, page 215].

Zero steady-state error can be achieved by including integral action in the controller.
This was done in Khalil [27] by augmenting the system with an integrator driven

by the tracking error: a 2 e1. The sliding surface is taken as

p—l
s = [too + Z k,e,- + e, = o (2.13)

i=1
where the positive constants ho to kp_1 are chosen such that the roots of

Ap+kp_1Ap—1+"‘+k1A+ko:—‘0

have negative real parts. The augmentation of the integrator creates a closed-loop

equilibrium point where the tracking error is zero. Since 33' < -Po]s| for ls] > u, s

16

reaches the boundary layer {s 3 [al} in finite time. The dynamics of a and C are

described by

 

 

 

 

_ 1 F.
0 1 O O O
0 0 1 0 0
Ca: (0+ 5
(2.14)
0 O O 1 0
L—ko —-k1 —k2 -—k,,_1‘ 1
d-—"-‘A..<..+B.s

where c, = [0, e1, - .- ,e,,_,]T and A, is Hurwitz. The fact that c}, = A,(, + 3,3 is
input-to-state stable, viewing 3 as input, together with Assumption 4, ensures that
the trajectory of the system enters a positively invariant set. Inside the set, the
controller acts as a high-gain controller that stabilizes the closed-loop equilibrium
point. The controller of [27] is called “Universal Integral Controller” because it
works for a class of nonlinear systems that have the same relative degree and sign
of the high-frequency gain L, L’}"1 h. Only bounds on the uncertain terms are needed

to tune the controller parameters.

2.3 Design of Nonlinear Integral Gains

While integral control ensures asymptotic tracking, it has been observed that
the buildup in the integrator can cause poor transient performance. Simulation
results of a continuous sliding mode controller and a Universal Integral Controller

for a field-controlled DC motor are shown in Figure 2.1. A field-controlled DC

17

motor, described in a normalized form by [28, page 30]

. dz}, .
v, = clifw + era—t- + 2,,

dw-cii cw
dt—3fa 4

has relative degree two, viewing the field voltage v f as input and the angular velocity
w as output. For simulation, we use the numerical data c1 = 0.8484, c2 = 0.1,
c3 = 4.242, c, : 1.2. The control parameters are chosen as k = 1, u = 0.5, k0 = 1.5
and k1 = 5 and the reference signal is r = 0.9. While Universal Integral Controller
achieves zero steady-state regulation error, the buildup in the linear integrator
causes large overshoot and settling time.

To prevent the integrator buildup and improve the transient performance, we
propose to use a nonlinearity. We consider two possible design choices, where a

nonlinearity is placed before the integrator:

p—l
(721/)(e1), sza+Zk,-e,-+ep

i=1

or after the integrator:
p—l
{7: e1, s=¢(a)+Zk,e,-+ep
i=1

We would like to choose i/J(-) as a locally Lipschitz function that belongs to a sector

(c1, oz) for some positive constants c1 and c2 > c1, i.e.,
61102 S PIMP) S 62192 (2-15)

18

 

 

>‘ 0.5h -

 

 

 

 

 

 

 

 

 

 

 

 

 

o l l l I I l l l J
0 1 2 3 4 5 6 7 8 9 1 0
0 I I I I I I I I I
-O.2 r- ~
-0.4 — —
b
-0.6 - -
-0 8 1 4L I I I l I I
1 2 3 4 5 6 7 8 9 10
1 I I I I I I
- - Continuous mode sliding controller
— Universal intggral controller
3 0.5 - _

 

 

 

 

 

4 5 6 7 8 9 10

Figure 2.1: Simulation results of the continuous sliding mode controller and Uni-
versal Integral Controller for the field-controlled DC motor

19

The freedom in choosing the nonlinearity i/J(-) can be exploited to improve perfor-
mance. For example, choosing ¢(-) to be small when l61| is large reduces the effect
of the integrator during the transient period. The presence of the nonlinearity 1/2(-),
however, complicates the dynamics of (a. It is no longer true that the dynamics
can be represented as a stable linear system driven by s, as in (2.14). The diffi-
culties encountered in studying the stability of the system are discussed further in
Appendix A where it is shown that such dificulties may be overcome if we use an

augmented error, e, = 2:11 l,e,- + 52,. The two possible schemes are:

c a nonlinearity, driven by the augmented error, is placed before the integrator:

p—l
(I = 111(ea), s = a + 2 he + e,, (2.16)

t=1

o a nonlinearity is placed after the integrator, which is driven by the augmented

CHOI'Z

p—l
0" : ea, 3 : 112(0) + Z k,e,- + e, (2.17)
i=1

The foregoing integrators provide the desired integral action since, at steady state,
b=0=>(¢(ea):0):>ea:0:>e1=0

due to the fact that eg to e, are derivatives of e1. For the case of (2.16), the dynamics

of the integrator have the form

p—l p—l
0" : ’f/J(S— 0+ Zhe, — Zia-q)
i=1

i=1

20

and by choosing l,- = k,- for 1 g i g p — 1, we obtain

d=¢@—a) (2m)

Similarly, for (2.17), with the same choice of l,, the dynamics of a are given by

e=s—¢w) aim

For any I/z(.) in the sector (c1,c2), the origin of b : —'I/I(0) is asymptotically sta-
ble, and when 5 7t 0 equations (2.18) and (2.19) are locally input-to-state stable
viewing s as input. For the N onlinearity Before Integration scheme, we choose the

nonlinearity ¢() to satisfy the sector condition

01192 S M(P) S 02192. for all p E 9 (2-20)

where O is any compact set. When |s| g c, the solution a of (2.18) is ultimately

bounded with the ultimate bound (1 + 5)c where 6 is a positive constant, i.e.,
|o(t)| g (1 + (5)0, for all t _>_ T

for some T 2 0. For the Nonlinearity After Integration scheme, the ultimate bound

on III] for (2.19), when Is] 3 c, is given by

(1 +6)c

wUMg-————,fmant2T
CI

21

when we choose the nonlinearity I/J(-) to satisfy the sector condition

(31192 S PIMP) S 02172. for all P (2-21)

The dynamics of C are described by
C 2 AC + B(s — a)
and

4': AC + B(s — 12(0))

respectively, and they are input-to-state stable since the roots of (2.12) have nega-
tive real parts; hence A is Hurwitz.
To design the output feedback controller, we estimate the state C using high-gain

observer:

a = éi+1 + (fit/5ixei — éI)» for 1 S i S P __ 1

(2.22)
ép = (fie/GPXex - éx)
where the positive constants 91 to 6,, are chosen such that the roots of
A” + pap—1 + - - - + pp_1A + [3,, = 0 (2.23)

have negative real parts, while the positive constant e is chosen to be small enough.
To overcome the peaking phenomenon associated with high-gain observers, we want
the right side of the (7 equation to be a globally bounded function of of e,, [10]. This
can be achieved by saturating ea outside a compact set of interest. In particular, if

L is greater than or equal to the maximum of |e,,| over the domain of interest when

22

state feedback control is used, we can define the function [I by

p. for lpl S L
£(p) = (2-24)
Lsgn(p) for [p] > L

and replace E, by [.(éa). In the case when the nonlinearity is placed before the

integrator, (7 equation is given by

(.7 : w(£(éa))

We can combine Ib(-) and [.(-) in one nonlinearity defined by

61192 S WU?) S 62192. for lpl S L
(2.25)

01L S |¢(P)| S 0211, for '13] > L

A nonlinearity satisfying (2.25) will lie in the shaded area of Figure 2.2. In summary,

the Nonlinearity Before Integration scheme is described by

5 : a + é,
if = V(éa) (2.26)
p—l
e, = klel + Z k,é,~ + (2,
i=2

where ¢(-) satisfies (2.25). This scheme is shown in Figure 2.3(a). The Nonlinearity

23

¢(-)

 

 

 

Figure 2.2: Nonlinearity 1/2(é,,) for the Nonlinearity Before Integration scheme

After Integration scheme is described by

s = 1,0(0) + e,
('7 : £(éa) (2.27)
p—l
e, = klel + Z k,é,~ + e,
i=2

where ¢(-) satisfies the sector condition (2.21) and £(-) is defined by (2.24). This

scheme is shown in Figure 2.3(b).
To overcome chattering, we replace the signum nonlinearity sgn(s) by a con-
tinuous function ¢(s/u). We do not limit ourselves to the choice of ¢(p) = sat(p) as

in [27]. Instead, we allow any function ¢(p) that is locally Lipschitz, odd, strictly

24

 

 

 

 

 

(a) Nonlinearity Before Integration

Cm

 

 

 

 

 

(b) Nonlinearity After Integration

Figure 2.3: Two possible schemes for nonlinear integrators

increasing for all |p| < 1, increasing for all |p| 2 1, ¢(0) = 0, limp—too ¢(P) = 1, and

(p2 — P1)[¢(P2) - ¢(p1)l Z 63(192 - 1902. for lp1|.|p2| S 1 (2-28)

It follows that
|¢(p)| 2 45(1) 2 Ca. for p 2 1 (2.29)
Typical examples are ¢(p) = sat(p), ¢(p) : tanh(p), ¢(p) : (2/7r) arctan(1rp/2),

and ¢(p) = p/ (1 + |p|). The continuous sliding mode control law is taken as

u : —ksign(LgL’f"1h)¢(§) (2.30)

with 1:1 to kp_1 and B, to 6,, chosen such that (2.12) and (2.23) are Hurwitz, the
remaining design parameters are the positive constants k, u, and e. The analysis
of the next section will determine the conditions that should be satisfied by these

constants.

25

2.4 Closed—loop Analysis

2.4.1 Nonlinearity before the integrator

For the integrator of the form (2.26), where a nonlinearity is placed before the

integrator, the closed-loop system can be represented in the form

2 = qo(z.e + 14d)
C: AC + B(s — a)
(I = 1/2(s — N(e)<p — a) (2.31)

5' = A(-) - k]a,(.)]¢(s_:1_:fl€)

es?) = Aflp + 63a [bum — k|a°(')l¢(s_—_W)i

 

 

 

 

 

u
where
p—l
A(z, e, a, V, (1, 7(9)) : ’l/J(S — N(e)<p — U) + 2: he,“ + bo(z, e + V, d, 7(9)) (2.32)
i=1
i. I —a 1 ... o
. : : : 1 . ‘
w: .A;= .n: ,_,(eI—e:)for1SzSp
3 _fip_1 0 1 6
_‘pp‘ -_ _fip 0 . . . 0a
and
N(€) : 0 kQEP—z k36p—3 ' ' ° kp_1€ 1

26

With the parameter c of the high-gain observer chosen sufficiently small, the closed-
loop system has a singularly perturbed form where the scaled estimation error (p is
the fast variable, and z, C, a and s are slow variables. The matrices A and A, are

Hurwitz by design. Let P : PT > 0 be the solution of the Lyapunov equation
PA + ATP = —I
and take V(C) : CTPC. Consider
a, 4.? {(z,e,a) : [3| 3 c, |0] g (1 + 6)c, V(C) g e2p1, Vo(t,z,d) g a4(cp2 + r3)}

where the positive constants c, 5, p1, p2 and the class [C function a, are to be
specified. We require that our assumptions hold in the set 9,, i.e., (z,e,a) E Q,
implies that (z,e) E Z x 8. Since e, = s — a - KC where K 2 [k1, - -- ,kp_1], the

inequality

llell S ”(II + lepl S (1 + llKll)llC|| + ISI + IUI S 6/22 < 1'1

should be satisfied, where

P1

p2 > (1 + “KID Amin(P)

 

+ 2 + 6 (2.33)

me inequality (2.7) we require that

IIZII S afl(a4(cpz+rs)) < 7‘2

27

where the class K function a, is defined as on, déf a2 0 7, and from (2.9), 0:, satisfies
the inequality a;1(a1(r2)) > r3. Thus, for any 6 and p1, we can ensure that our
assumptions are valid in the set (2,, by choosing p2 to satisfy (2.33) and then choosing

c to satisfy

cpz g min{r1, a;1(a1(r2)) — r3} (2.34)

From (2.9), a4 satisfies the inequality a;1(a1(r2)) > r3.
We start the analysis by showing that, for sumciently small 6, the scaled estimation
error (p will decay to a level of the order of 0(6) after some arbitrarily small time.

Let Pf = PfT > 0 be the solution of the Lyapunov equation
PfAf + ATP, = —I
and take Vf(gp) : IpTPfcp. The derivative of V, satisfies the inequality:

. 1
Vi S --E-ll<p||2 + 2||<P|l||PxBa||71(C)

where

71(c) 2 max{|bo(z, e + V, d)| + k|a0(z, e + V, d)|}

and the maximum is taken over all (z,e,a) E (2,, d E D, V E A and r0”) 6 I‘. In
arriving at the preceding inequality, we have used the property |¢(p)| _<_ 1 for all p.
For V, 2 £2p3, where p;, = 16||PfBa||2|lel|7f(c), we have V, < —§1;||gp||2. Therefore,
<p(t) enters the set

2. c1.5.f{<p E R” : V,(<p) S eng}

28

within a finite time interval [0, To(e)], and stays therein. We note that lim,_,o T0(e) :
0.

Since the right side of the slow equation of (2.31) is bounded uniformly in e, for
all (z(0),e(0),a(0)) E (2;, with b < c, there is a finite time T1, independent of e,
such that (z(t), e(t), e(t)) 6 $2,. for 0 g t 3 T1. By choosing 6 small enough, we can
ensure that To(e) < T1. Therefore, there exists 6‘; > 0 such that, for each 0 < e < 6;,
every trajectory, starting at a bounded 6(0) with (e(O), 2(0), 0(0)) 6 (lb, enters the
set (I, x E, in finite time.

In the next step, we establish that the set (2,, x 2, is positively invariant. Let us
choose p. small enough that u(1 + 6) < c and 6 small enough that ]N(€)lp] < 6p.

For u(1 + 6) 3 Is] 3 c, we have

Using (2.29), we obtain

33' S IA(')||S| - c.zklaoIISI (235)

We require the controller gain I: to be large enough to overcome the disturbance.

If k is designed to satisfy the condition
It 2 c4 + 72(c) (2.36)

where c, > 0 and
A(z, e, a, V, d, rm)
c3a0(z, e -l— V, d)

 

72(c) = max

with the maximum taken over all (z,e,o) E 9,, d E D, V E A and rm 6 I",

then 35' < 0 on the boundary |s| : c. On the boundary [a] = (1 + 6)c, since

29

|s — N(e)<p| < c+6u < la] and

$311014s - N (6MJ - 0)) = - sigma)

we have

06 = aI/J(s— N(e)<p—a) < 0

Due to the inequality

V S -l|C|l2 + 2HCIIIIF’BIKISI + IUI) (237)

S -|lCl|2 + leCIIIIF’BHd2 + 6)
we can show that V < 0 on the boundary V = c2p1, by choosing p1 to satisfy
p1 > 4|lPB||2||Pl|(2 + a)? (2.38)

From (2.7) and (2.8) and by defining the class K: function a4 as at, déf a2 0 '7, we
can verify that VD < 0 on the boundary V0 2 a4(cp2 + r3). Therefore, there exists
#I > 0 and 6301) > 0 such that for each 0 < u < III and 0 < e < 6;, the set {2, x 2,
is positively invariant.

Next, we show that s(t) enters the boundary layer {|s| g #1} in finite time, where
u, is chosen as u, = u(1 ——6) to ensure that |(s— N (e)<p) / u] S 1 inside the boundary

layer. Let us consider s in the set {pl 3 Is] 3 c} and choose 6 to satisfy

C
6<—4<

l
— 2.39
4k 4 ( )

30

Notice that sign(¢(§/u)) = sign(é/u) = sign(s), since |N(e)<p| < 6y. < u(1—6) g |s|.

If |§| 2 u, inequality (2.35) holds and from (2.2) and (2.36) we can show that
33' g —coc3c4|s|
On the other hand, if |.§] < [1, using (2.28), (2.36), (2.39), and the fact that

4%)]

C C
2 flslls — N(e)«pl 2 flslu — 26in)

MG) = Ssgn(S)

 

we can show that

606304
—-——-— S

seg|A(-)ns|—klaols¢(§)s 2 II

The latter bound on 53' holds for both cases and we conclude that s(t) reaches the
boundary layer {Isl g #1} in finite time and stays therein. After s(t) reaches {|s] g
In}, consider a in the set {(1 + 6))u g [a] g (1 + 6)c}. Since ls — N(e)<p| < p. < lo],

we have

sig1r1(1/I(s - N (ew - 0)) = sign(s - N (6W - 0) = - sign(a)

|s - N(€)r - 0| 2 l0! - Is - N(€)<p| > 5n
Moreover,

|s—N(e)<p—a|gu(1—6)+u6+(1+6)c=u+(1+6)c

W8 - N(€)<P - 0)] Z fills - N(€)¢p — 0|

31

where

 

. L
Elzclmm{1, }>0
p

Using these facts, we show that

0'51 = asign(¢(s - N(e)r - 0))ll/’(3 - N(6)r - 0)| < Jamal

Thus, 0(t) reaches the set {lal S (1 + 6)].I} in finite time and stays therein. Then,
from inequality (2.37), we show that V _<_ —%||C||2 for V(C) 2 uzpl, by choosing p1
to satisfy

pl 2 64|lPB||2|IPlI (2.40)

Inequality (2.40) implies (2.38) and from now on, we fix p1 as chosen above. Thus
e(t) reaches the set {V(C) g pzpl} in finite time and stays therein. From the
inequality Hell 3 ”C” + |e,,|, it follows that He“ < upz. Since limp,” V(t) = 0, we
have ”V(t)” g u in finite time. Using ||e(t)|| + ||V(t)|| < up; + 11 together with (2.7)
and (2.8), we can show that for Vo(z) 2 a4(#p2 + II). Va 3 —a3(||z||). Thus, z(t)
reaches the set {V0 3 a4(up2 + u)} in finite time and stays therein. Therefore, every

trajectory in 9,. x Z}, enters ‘11,, x 23, in finite time and stays therein, where the set

\P,, is defined as

‘1’» ‘3‘? {(46.0) I IS! S #1. l0! S (1 + 5M. V(C) S #2121. Vo(t.2.d) S aim/22 + M}

Finally, we show that every trajectory in ‘15,, x )3, approaches an equilibrium point

as time tends to infinity. When V = 0 and H” = 0, the closed-loop system (2.31)

32

has a unique equilibrium point (z = 0, e = 0, a = 6, (p = 0), where

6‘ : -— sign(LgL§_1h)u¢‘1(%®-)

and ¢’1(p) is defined for all [p] g 1. Let s? = (‘7 be the corresponding equilibrium
value of s.
Inequality (2.10) of Assumption 4 implies that in some neighborhood of z = 0

there is a Lyapunov function such that

6V 6V
61IIZII2SVISA2IIZIIZ. 7190(z.0,d)S—A3|l2|l2. [[5,— sxtnzn (2.41)

6

 

 

for some positive constants A1 to A4, independent of d. Consider
1 1
V2 = V1(z, d) + AsCTPC + 5A,? + 552 + «Frye (2.42)

where A5 and A5 are positive constants to be chosen, 5 = a — 5 and ‘s' = s — S. The

derivative of V2 can be arranged in the form
V2 3 -—xTP2x + (Avllsoll + Aslé'l + Aellzll)llv(t)ll + (Atollwll + Anlfil)lr"”(t)| (2.43)

where x 2 [“2“ ”C” III] |§| [lcpll]T and A7 to All are positive constants. The sym-

33

metric matrix P; has the form

r -

 

A3 —A12 _A13 —A14 "A15
A5 -}\23 -)\24 —)\25
P2 = A5C1 ’A34 —A35
c c k
° 3 — A... 4.,
ll 1
— - Ass
. e _

 

 

where the nonnegative constants A15 to A55 are independent of 5, A14 to A34 are
independent of p, A13 and A23 are independent of A5, and A12 is independent of A5.
In arriving at (2.43), we have used (2.28). The detailed derivation of (2.43) and
expressions for all the constants in P2 are given in Appendix B. By choosing A5
large enough then As large enough then u small enough then 6 small enough, we
can make P2 positive definite. Since lim,_,°o V(t) = 0 and limtnoo r(P) (t) = 0, there
exists a ball 3 around the equilibrium point, whose radius is independent of u and
e, such that every trajectory in B approaches the equilibrium point as time tends
to infinity. We can choose u and e sufficiently small to ensure that ‘11,, x )3, C 3.
Hence, there exists It; > 0 and e;(u) > 0 such that for each 0 < [J < u; and
0 < e < 6;, all trajectories in ‘11,, x )3, approaches the equilibrium point as time
tends to infinity.

Our conclusions are summarized in the following theorem.

Theorem 1 Suppose that Assumptions 1 to 4 are satisfied and consider the
closed-loop system (2.31) formed of the system (2. 5), the observer (2.22), the
nonlinear integrator (2.26 ) with nonlinearity satisfying the condition (2.25),

and the output feedback controller (2.30). Suppose é(0) is bounded and

34

(z(0),e(0),a(0)) E 91,, where b < c and c satisfies (2.34) and (2.36). Then,
there exists It“ > 0 and for each 0 < u < p“, there exists 5* : 6*(11) > 0 such
that for each 0 < p. < u‘ and 0 < e < 6*(p), all the state variables of the

closed-loop system are bounded and limtnoo e(t) : 0.

The estimate of the region of attraction (2., is limited by two factors: the region
of the validity of our assumptions shown (2.34), and the requirement (2.36) on the
controller gain k. If all the assumptions hold globally and k can be chosen arbitrarily

large, the controller can achieve semiglobal regulation.

Corollary 1 Suppose that Assumptions 1 to 4 are satisfied globally, i.e., U9 2
R”, a1, a2, a3 and 7 are class [Coo functions. For any given compact sets
N E It“1 and M 6 RP, choose c > b > 0 and r3 _>_ 0 such that N 6 (lb, and
choose In large enough to satisfy (2. 36). Then, there exists [1." > 0 and for each
0 < u < If, there exists 5‘ = 6"(u) > 0 such that for each 0 < u < p." and
0 < e < e‘(u), and for all initial states (2(0), e(O),c(O)) E N and é(0) E M, the
state variables of the closed-loop system (2. 31) are bounded and limHoo e(t) =

0.

The control parameter k can be chosen as the maximum magnitude of the actuator.
The choice of small It is limited by the system and controller delays, since too
small u can cause chattering problem. Since high-gain observer is an approximate
difierentiator, in practice, the choice of small 6 is limited by measurement noise and

unmodeled high-frequency dynamics of the senor.

35

2.4.2 Nonlinearity after the integrator

For the integrator of the form (2.27), where a nonlinearity is placed after the

integrator, the closed-loop system can be represented in a form similar to (2.31):

2': = 40(2, e + V, d)
4': AC + B(s - 12(0))
4 = 5(8 - N (6W - 1140)) (2-44)

3' = A(.) ._ klao(')|¢(s "1::(ellp)

where
p—l
A(z, e, a, V, d, rm) : ¢’(c)(s — N(e)<p — ¢(a)) + Z k,e,+1 + bo(z, e + V, d, r‘”’)
i=1

Only the dynamics of C and a and the disturbance term A(~) are different from
(2.31) and the analysis follows the same steps. We will mention only the parts that

are difierent from the previous section. Consider

 

(1+6)
C

a. ‘i—i‘ {(z,e,a) : Isl s c. lal s c. V(C) s cm. Vo(t.z.d) S ado/)2 + r3)}

We can show that our assumptions are valid in the set O, by choosing p2 to satisfy
(2.45) and then choosing c to satisfy (2.34), since from ep : s — We) — KC, the

inequality

Hell S IICII + lepl S (1 + llKll)llCll + [8| + M(U)] S cm < r1

36

should be satisfied, where

P1

p2 > (1 + MK“) Amin(-P)

 

+ 1 + 23(1 + 6) (2.45)
1

As in the previous section, we show that every trajectory, starting at a bounded
é(0) with (e(O), 2(0), 0(0)) 6 (2,, with b < c, enters the set 0, x )3, in finite time.
After showing that 33 < 0 on the boundary |s| = c, consider 0 on the boundary

|0| : (1 + 6)c/c1. We have
00 : 0£(s — N(e)<p — If/(0)) < 0
since

ls — N(e)<pl < c+ 5n < cllal < I¢(0)I

si311Ws - N (€)<p — 42(0)» = sign(s - N (6)<p — 4(0)) = - si1310(0)
Due to the inequality

V S -||€||2 + 2llClllllr’l-‘3||(|S|+|1/J(0)|) (2.45)

c
s -l|C|l2 + zucuupBuc (1 + i“ + 6))
we can show that V < 0 on the boundary V = c’pl, by choosing p1 to satisfy

2
p1 > 4||PB||2||P|| (I + $0 + 5)) (2.47)
1

37

Then we verify that Vo < 0 on the boundary V}, = a4(cp2 + r3). Therefore, the set
(2,, x 2, is positively invariant.
After showing that s(t) reaches {|s] S [21} in finite time, consider 0 in the set

{(1 +6)u/c1 3 I0] 3 (1 + 6)c/c1}. Since |s — N(e)(,0| < u < c1]0| < |i/J(0)|, we have

sign(lXS - ”(6W - ¢(0))) = sign(s - N(€)¢P - 10(0)) = — sign(U)

ls - N(e)r - 1MUN _>_ cilal -- Is - N(6)<pl > 6n
Moreover,

Is — New — V(UN s M(1 — a) + #6 + 23(1 + 5)c = u + %(1 + 6)c
1 1

I£(s — New — an 2 fills — New — aI

where
L

>0
C
[1+ c—2(1+5)C
1

 

51 2 min 1,
Thus,
air = osien<£<s — New — ¢(0)))|£(s — New — «Mam < —516#|0|

Therefore, 0(t) reaches the set {|0] S (1 + 6),u/c1} in finite time and stays therein.

Then, from inequality (2.46), V g —%||C||2 for V(C) Z pzpl if
2
p1 > 16||PB]|2||P|| (I + :30 + 6)) (2.43)
1

By choosing p1 to satisfy Inequality (2.48), which implies (2.47), e(t) reaches the

set {V(C) _<_ uzpl} in finite time and stays therein. Thus, 2(t) reaches the set

38

{V0 3 a4(p.p2 + 1.1)} in finite time and stays therein. Therefore, every trajectory in
(I, x )3, enters ‘1'” x )3, in finite time and stays therein, where the set \P,, is defined
as

(1+6)

C1

 

‘11.. d=—°-‘ {(z,e.a) = Isl s #1. Iol s u. V(C) s 112A. vo(t.z.d) s 04(up2 + u)}

When V = 0 and r(P) : 0, the closed-loop system (2.44) has a unique equilibrium

point (2 : 0,e = 0,0 : 0, (,0 = 0), where

6 = III—1(5) = — sign(LgL;“1h)¢-1 (mp-1 (315642))

provided t/I‘1(-) exists in the neighborhood of 5. Considering the composite Lya-
punov function of the form (2.42), we can show that all trajectories in \II,, x )3,
approaches this equilibrium point as time tends to infinity.

Our analysis shows that the sector condition (2.15) is not suficient to guarantee
asymptotic tracking of Universal Integral Controller with nonlinear integral gain
(2.27). The nonlinearity t/J(o) should be invertible in the neighborhood of s. In
practice, due to unknown system dynamics and disturbances, we cannot predict 3',

thus the nonlinearity should be designed such that
¢‘1(p) exists for 0 g p g u (2,49)

Notice that the sets 9, and ‘1!” are dependent on the sector condition (c1, c2). The
conditions for p, and p2, in (2.48) and (2.45), include the term c2/c1, and the bound
of 0 is proportional to 1/c1.

The following theorem summarizes the conclusion of our analysis.

39

Theorem 2 Suppose that Assumptions 1 to 4 are satisfied and consider the
closed-loop system formed of the system (2.5), the observer (2.22), the non-
linear integrator (2.27 ), with nonlinearity satisfying the conditions (2.21 ) and
(2.49), and the output feedback controller (2.30). Suppose 6(0) is bounded and
(2(0),e(0),0(0)) 6 m, where b < c, the size of 9;, depends on the sector condi-
tion (2.21 ) and c satisfies (2.34) and (2.36). Then, there exists u" > 0 and
for each 0 < u < u“, there exists 5* = 5"(u) > 0 such that for each 0 < u < [1."
and 0 < 6 < 6‘01), all the state variables of the closed-loop system (2.44) are

bounded and limtnoo e(t) = 0.

When the controller gain 11: can be chosen arbitrarily large, the controller

achieves semiglobal regulation.

Corollary 2 Suppose that Assumptions 1 to 4 are satisfied globally, i.e., U9 :—
R", 01, 02, a3 and 7 are class ICOO functions, and k can be chosen arbitrarily
large. For any given compact sets N 6 1‘2"“1 and M E R”, choose c > b > 0
and r3 2 0 such that N E (21,, and choose 1: large enough to satisfy (2.36).
Then, there exists [1" > 0 and for each 0 < u. < If, there exists 6" = e“(u) > 0
such that for each 0 < u < u" and 0 < e < e“(u), and for all initial states
(2(0), e(0), 0(0)) 6 N and é(0) E M, the state variables of the closed-loop system

(2.44) are bounded and lim,_,,,o e(t) = 0.

2.5 Simulation Results

Example 1: In Figure 2.4, the simulation results for the field-controlled DC motor
is shown. We use the same parameters as the simulation of Figure 2.1, and take

L() with L = 5. The tracking error, integrator output and control effort are shown

40

for controllers with linear integral gain and the following nonlinear integral gains:

6 = ¢,(e,), s = 0 + e, (2.50)

0 = c(é,), s = 116(0) + é, (2.51)

We use the nonlinearities shown in Figure 2.5. The buildup in the linear integrator
causes overshoot and large settling time. The transient performance of the nonlin-
ear integrator (2.50), where the nonlinearity is placed before the integrator, shows
no overshoot and smallest settling time. The nonlinearity ¢1(-) is designed so that
the gain is small when the augmented error is large to prevent the buildup in the
integrator during the reaching phase. It behaves similar to a PD controller until
it reaches the boundary layer; then during the sliding phase the integrator drives
the tracking error to zero. The transient performance of the controller with the
nonlinear integrator (2.51), where the nonlinearity is placed after the integrator,
shows better settling time compared to the linear integrator, but it does not im-
prove the overshoot. To satisfy the conditions (2.21) and (2.49), we choose ¢2(-)
as a monotonically increasing function, which restricts the freedom of choosing a
nonlinearity that reduces the efiect of integration on the control input during the
transient period.

Example 2: The advantage of the Nonlinearity Before Integration Scheme is

demonstrated for the motion on a horizontal surface with friction and disturbance.

tj+0.1y2:u—1

The control gains are chosen as u = 0.5, k = 2.5, k0 = 0.98, k, : 1.4 and L = 5.

41

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 _ I __ __1_,‘__ _I _ _ 1:_ I I T I I ..
/ ‘ -- —- _ 2 m. _‘_ _ _ _ _ _ __ _ _
/ — _. _ _ _ _
>‘ 0.5 *- r
o l l l l l l l l_ l
0 1 2 3 4 5 6 7 8 9 10
0 \ w I I I , I I I 'I ______ I. _____
I - \ . ...... .— . .— . 71(— ———————————————————
-0.5 a ‘ — —————— , / ’ —
\ , ’
O -1 r \ I ’ — nonlinearity before integrator ~
\ I / ’ -—- nonlinearity atter integrator
.15 r \ / - - linear integral gain —
\ , ’
_2 7-4 l l 1 l l l l l
0 1 2 3 4 5 6 7 8 9 10
1 I I
0.5 - r
3
0 - -
_O.5 l l l l l l l I l
0 1 2 3 4 5 6 7 8 9 10
t

Figure 2.4: Simulation results of Universal Integral Controllers with nonlinear in-
tegrators for the field-controlled DC motor

42

 

 

 

w,(ea). 112(0)

 

 

 

 

 

 

 

 

Figure 2.5: Nonlinearities 1121(ea) and 192 (0) for the simulation of the field-controlled
DC motor

43

-2

—4

-6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

I I I I I I T I I
/ ' 'I. § \
./ . \ ' \ -‘
. / _ - — — \ \
_/,’ ““““ ‘7:“;—._‘:::."_ —————
”I 4
I I I I I I_ I l
1 2 3 4 5 6 7 8 9 10
I I I I I I I I I
\\ . I ’ —————————————
\' Q . '/_." ”””” T
\\ . \ ’ ’ ’I ’I ' /
I L I I l I I I
1 2 3 4 5 7 8 9 10
I I I I I I U I I
I _____ ‘ 2' —_ _
._ - linear integrator '1
\ /. — nonlinearity before integrator
\ . ,- — - nonlinearity after integrator ~
I I I I I I I I I
1 2 3 4 5 6 7 8 9 10
t

Figure 2.6: Simulation results of motion on a horizontal surface

44

We use the nonlinearities shown in Figure 2.5. The simulation results are shown
in Figure 2.6. The Nonlinearity Before Integration scheme maintain the integrator
output small and shows better performance than the N onlinearity After Integration
scheme. The settling time for the Nonlinear After Integration Scheme is increased
by 100%. The buildup in the linear integrator causes 68% overshoot.

Example 3: A magnetic suspension system, where a ball of magnetic material is
suspended by an electromagnet whose current is controlled by the ball position, is

described by [28, page 31]

 

 

.. . . . Lot2
2 —k F F = -
- . . L
v=¢+Rz. <II=L(y)z L(y)=L1+1+;/a

With the vertical position of the ball y as output and voltage v of a voltage source
which controls the electromagnet as input, the system has relative degree three.
For simulations, we use m = 0.01 kg, 1: = 0.001 N/m/sec, g = 9.81m/sec2, a = 0.05
m, L0 = 0.01 H, L1 = 0.02 H and R = 100. The controller gain is chosen as
a = 0.05, k : 40, k0 : 1, k1 : 2, k2 = 2 and L = 0.5 for nonlinear integration.
Simulation results for controllers with linear integration 0 2 e1 and nonlinear in-
tegration 0 = Men), with 1p(-) designed as in Figure 2.8, are shown in Figure 2.7.

While controllers with nonlinear integration improves transient performance,
the augmented error e, is small and the nonlinear integration is Operating in the
linear region: 0 = e,,. This suggests that we may gain the benefits of the nonlinear
integrator if we simply drive the linear integrator by the augmented error. This is
indeed the case since the input-to-state stable dynamics of the integrator, 0 = s — 0

keeps the integrator output small when 3 reaches the boundary layer fast. Linear

45

0.16

0.14

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 2.7: Simulation results of the magnetic suspension system

46

r— _-
/."- \ \ \
/' _______ .—
/
L 1 l I 1
10 12 14 16 18 20
I I I4 —--—.T_ ~ _ I
______ \ I " —
.\" ’ ,-
4 — nonlinearity before integration ——————
--- linear integration
- - augmented error
I I I I 1 l 4 l J
0 2 4 6 8 10 12 14 16 18 20
I I I I I T____._' I I
A:_ _ _ f “—
I I I I 1 1 1 l I
2 6 s 10 12 14 16 18 20
t

0.05

 

 

 

 

 

 

 

0.04 -
0.03 -
0.02 *-
0.01 -
e“ 0 ~
5+
-0.01 ~
-0.02 r
-0.03 '-
—0.04 -
-0.2 —0. 1 5 -0.1 —0.05 0 0.05 0.1 0.15 0.2
e
a

Figure 2.8: Nonlinearity ¢(e,,) for the simulation of the magnetic suspension system

47

integrators driven by the augmented error, 0 2 ea, can be used to prevent integra-
tor buildup if the maximum control level I: is large enough to force 5 to reach the

boundary layer fast.

2.6 Conclusions

We designed Universal Integral Controllers with nonlinear integrators driven
by the augmented error. We considered two possible schemes: Nonlinearity Before
Integration and Nonlinearity After Integration. Our analysis shows that the con-
troller achieves regional and semiglobal regulation. Simulation results show that the
Nonlinearity Before Integration scheme prevents the integrator buildup and achieves
better transient performance than the Nonlinearity After Integration scheme and
the linear integrator, when the maximum permissible control level I: is not large
enough to ensure fast reaching phase. In the case of fast reaching phase, the benefit
of using a nonlinearity may not be significant, but driving the linear integrator by
the augmented error rather than the tracking error may still improve the transient

performance.

48

Appendix A Stability of Nonlinear Integrators

Consider the case

p—l
0 = I/J(e1), s = 0 + k,e,~ + 6p

i=1
On the surface s = 0,
p—l
ep : —0 -— Z k,e,~
i=1
and the system can be represented by the feedback connection of Figure 2.9 with

1

G(s) : s(st"1 + k,,,_lst’—2 + - - - + k1)

 

The transfer function G(s) has all poles in the left half plane, except a simple pole
at the origin. Application of the Popov criterion [28, Exercise 7.8] shows that the
feedback connection will be absolutely stable for w in the sector (0, k] if we can
choose a positive constant 7; such that

1 . .
h + Re[(1 + jwn)G(]w)] > 0, V w

For this inequality to hold with arbitrarily large k, we need the Nyquist plot of

0+nfl
s(sP‘l + kp_lsP-2 + - - - + k1)

 

(1 + nS)G(S) =

to lie in the right-half plane, which is possible only if (1 +ns)G(s) has relative degree
zero or one. This will be the case if p = 1 or p = 2. For p 2 3, (1 + ns)G(s) will
have relative degree higher than one and its Nyquist plot must cross in the left-half

plane. To overcome this dimculty we can use 0 = 1,1)(ea) instead of 0 :2 111(e1), where

49

 

 

 

 

 

 

 

| [<_
IN)
I _

Figure 2.9: Nonlinearity before integrator when 3 = 0
e,3 = Eff: l,e,- + ep. In this case the system, on the sliding surface s = 0, will be
represented by the feedback connection of Figure 2.9 with

(lp_.1 — kp_1)SP_2 + ' ' ' +(l1 - k1) +

1
sP‘l + kp-lsP-2 + - - - + k1

 

G(s) = i

so that the transfer function (1 + ns)G(s) will have relative degree zero. In fact,
the choice

l,:k,, forlgigp-l

yields G(s) = 1/s and for any 17 > 0, the Nyquist plot of (1 + ns)/s will be in the

right half plane.

50

Appendix B Derivation of (2.43)

The derivative of V2 of (2.42) is given by

V
——-1-qo(2,e + V, d)

v, =
62
— A5|]C||2 + 2A5CTPB(s — 0)
+ with — N (6)19 — a) (2.52)
+ s [we — New — a) + 2 he... + bo(') — klao|(-)¢ (55—253)]

--llrll2+2rTPfBa [boo- klao-l()¢(———s—IZ(‘)¢)]

We arrange (2.52) in a quadratic form of x = [”2” “CH l0] ['3'] ||<p|]]T. From (2.42),

we have
av1 _ 6V1 BVI
5:400?» e + V, d) ._ 62 qo(Z, 0, d) + 62 [qo(z, 8 ‘t‘ V, d) _ 90(2) 0: (1)]

S -/\a||1’all2 + A4Lq||3|l(||€|| + Ill/ID)
where L, is a Lipschitz constant of qo(-). Since
llell = II3 - 0 - KCII = ”5 - 5 - KCH S IIKIIIICII + |§| + I51
where K : [k1, - -- ,kp_1], the first term of (2.52) satisfies
93-40(2 e + V d)- < -)\a.||2||2 + A4Lq||ZII(IIKIIIICl| + W + I'S'l + IIVH) (2-53)

The following inequality is satisfied for the second term of (2.52).

—A5||<|12 + 21.01336 — a) s —A.II<II2 + ZAsIIPBIHICIKIé'I + IaI) (2.54)

51

We choose u small enough that
L 2 (2+5)#> ls-N(6)<P—0|
then c1p2 _<_ pimp) g czp2 and the third term of (2.52) is arranged in the form

have - N (6)<p - U)

= Mir/)6 -— N (EN) - 5)

= —Ae(§ - N (6)<p — (3)1143 - N (are - 5) + 66(3' - N (6)r)¢(§ - N (elr - 5)
S _,\6e,(§ - N(6)<p - (3)2 + Ahazl‘S' - N(€)r||§ — N(6)<p — 6|

g —)\6c102 + A6(c2 — 1:1)32 + A5(c2 — c1)]|N(€)H2‘/’2 + A5(2c1 + c2)|§||0|

+ 66(261 + C2)||N(¢5)||||<Pl||5| + 266(61 + 02)||N(6)IIH<P|H§| (2-55)

Consider the fourth term of (2.52) and divide it into three parts. The first part

satisfies the inequality

are — N(6)<p — 0) -—- we — New — a)

S c2|§|(|§| + llN(6)|l|l<P|| + lél) (2-55)

The second part can be rewritten as

p—l pp—2
S Z k,e,+1 = 3 Z kiei-i-l + kp_1€p]
i=1

_izl

 

.p_2
3" Z kiei+1 + kp—1(S “ 0 — KO]

3:1

5 §(|IK1||||C||+ kp—Il‘s'l + kp-Ilél) (2.57)

52

Where K1 2 [—kp_1k1,k1 '— kp_1k2, ' ' ' ,kp_2 — k§_1]. Since

11((1) = —k sig11(1r.gI/;-1 ’04) (E)

we can arrange the remaining part of the fourth term of (2.52) as

M“ + V» d» r") - klao(z, e + v, d)|¢ (8 — Tm?)

= bo(2, e + V, d, r") — bo(0, 0, d, 0) + bo(0, 0, d, 0) + ao(0, 0, d)u(d)

+ klao(0, O,d)|¢ (i) — k|a0(0, o, d)|¢ (g + 5 ;N(‘)"’) (2.58)
8 - N (6W)

+ klao(0,0, d)|¢ (W) — klao(2, e + V, d)|¢( It

Recalling that at equilibrium,

bo(0, o, d, 0) + ao(0, o, d)u(d) = o

and inside the boundary layer ‘11,“

4, (8- NOW) < 1

u
we have
- s-N(€)<p
S bo(Z,e+l/,d,7'p)—k]ao(z,€+l/,d)]¢ —;—

 

S I'S'l [Lb(||2|l + llell + IIVII + lrpl) + lcLI(||ZII + ”8“ + ||V|l)] (2-59)

+ 5 klao(0.0.d)|¢ (E) — k|a0(o,o, d)|¢ (5 + 5 ;N(‘)‘P)]

53

where L, and L, are Lipschitz constants of ao(-) and bo(-), respectively. Moreover,

using property (2.29) of ¢(-), namely

(p1 - p2)(¢(p1) — ¢(p2)) 2 c3(p1 — 192)2

and the minimum co of ao(-), as in (2.2),

 

g [k|ao(0,0,d)|¢ (E) -— k|a0(0, o, ,1)”, (s + s- — N(€)<P)]

 

u

_ .. S + S" — N(e)<p §

— —kIao(o.o.d)I(s — New + New ]¢( # ) - ¢ (3)]
s —C—;’5Iao(o.o. due — New)? + 5%Iao(o.o.d)IIN(e)eII§ — N(6)<p|

s -“°ff"(§ — New): + Egglaowfi.d)|llN(e)llll<pll(|‘s'l + ||N(e)llllsoll) (2.60)

 

where L, is the Lipschitz constant of ¢(-). From (2.58), the last term of (2.52)

satisfies

— fine“? + 2633. [boo — klao|(-)¢ (—3 “ Z“"”)]

1
S -;ll<pll2 + ZH‘PHHPIBaH [Li(llz|| + Hell + llVll + ITPI) + kLa(llzll + ”6” + III/Ill]

+ 2||<pll|leBa|| [1c%lao(0.0.d)l(l§l + l|N(6)|ll|<p|l)] (2.61)

54

We choose 6 small enough that ||N(e)|| 3 N1 for some positive constant N1 > 1.

From (2.53) to (2.57) and (2.59) to (2.61), we set

1.. = $(AILIIIKII)

A13 2 %(A4Lq)

423 = A5HPB||

An = $0.41,, + L, + kLa)

A21 = §I215IIPBII+ IIK1|l+ IIKII(L1+ kLa)]
1,, = $41,121:, + c2) + C2 + 1,.-. + L. + 11...]
A4,, = A5(c2 — c1) + e2 + 1,.1 + L. + kL,

A15 : ||P,B,,]](Lb + kLa)

A25 2 ”PfBaHIIK]](Lb + kLa)

1
A35 : §[A6(2C1 + C2)N1 + 2]]PfBa]](Lb + [CI/4)]

2CoC3kN1 + kL¢|ao(0, 0, d)]N1
u u

 

1
A45 2 - [C2N1 + 2A6(C1 + C2)N1 +

2
kL 0,0,d
+2|]P,B,,|| (L. + IL, + “‘3; )')]

kL¢|ao(0,0,d)|N12 + 2k|]PfBa||L¢|a(0, o, d)|N1
u u

 

A55 = A6(C2 - C1)Ni2+

 

 

A7 = 2||P,B,,|](Lb + kLa)
A, = L. + kL,

A, = 1,1,,

A10 = 2]]P;B,,]|Lb

An = Lb

55

Chapter 3

Universal Integral Controllers with

Nonlinear Gains

3.1 Introduction

In Chapter 2, we designed Universal Integral Controllers with nonlinear inte-
gral gains. In this chapter, we investigate the use of nonlinear proportional and
derivative gains in addition to nonlinear integral gains. We consider only the Non-
linearity Before Integration scheme, which showed the best transient performance in
simulations. Nonlinear proportional and derivative gains provide us more freedom
in designing a controller and can be utilized to further improve transient perfor-
mance.

For a Universal Integral Controller with a nonlinearity placed before the inte-
grator, fiom the closed-loop equation (2.31), the error dynamics have the form of a

stable linear system driven by (s — 0):

CzAC+B(s—0)

56

If the nonlinearity is designed to hold the integrator output 0 small, the response
of the system depends on the eigenvalues of A, which are determined by the gains
k1 to kp_1. But, the design of the proportional and derivative gains k,- is limited by
the control level In. Large k,’s increase the disturbance term A(-) in (2.32), and a
higher controller gain k is required to overcome the disturbance (2.36), which may
violate the actuator limit.

We propose to replace the constant pr0portional and derivative gains k,- with
nonlinear gains k,(-), which can be functions of the tracking error and its derivatives
C. Our goal is to design a Universal Integral Controller with nonlinear proportional,
integral and derivative gains that improve the transient performance, while preserv-

ing the stability properties of the systems, both regional and semiglobal.

3.2 Design of Nonlinear Proportional and Deriva-

tive Gains

Taking nonlinear proportional and derivative gains is equivalent to designing

a nonlinear sliding surface:

p—l
s = 11:00 + z k,(-)e,- + e,,
i=1
d r P4
:3 11:00 + 2 v,(-) + ep

1:1

57

where k,(-) is a nonlinear function of e1 to ep_1 and v,(-) = k,(-)e,-.

 

 

 

 

 

 

 

 

The dynamics

of C 2 [e1,- - - ,e,,_1]T have the form
f . . . . 1
o 1 o o 0 o o o u,(-)
o o 1 o o o o 0 v2(.)
6 = C + ' (s — 0)
o o 0 1 o o o o v,_2(-)
Lo 0 0 OJ L—l —1 —1 —1_ .174“.
“l—i‘ AmC + Bmv(-) + B(s — 0) (3.1)
where v(-) : [v1(-), - - - ,vp_1(-)]T. Finding a class of nonlinear functions v,(~), which

ensure that the dynamics of C are asymptotically stable when (5 — 0) = 0, is a

challenging problem. In this work, we consider nonlinear gains of the form

where v,(e,~) satisfies the sector conditions

i.e., the nonlinear gains k,(e,-) that are bounded by

li S ki(ei) S mi:

2 2
lie,- S €IUI(€I) S mien

fori=1,~-

vi = 11103:) = (91(8081

forlgigp—l

:p—1

(3.2)

Then, the study of stability of the system (3.1) with the given sector conditions

(3.2) reduces to a Lure problem. For Lure systems, Lyapunov functions have been

found analytically only for systems of order two. When 5 — 0 = 0, the derivative of

58

the Lyapunov function

C1 .p 17
V = CTPC + 221/ 01(T)d7', P = PT 2 n 12 > o
0
P12 P22
with the choice of A1 : p22, satisfies the inequality
- 2P12 —P11 + P127712
V S -(T C = -CTQC

—P11 + P127712 2(P2212 — P12)

and Q = QT can be made positive definite by choosing p12 > 0 and p22 large. For
higher order systems, Linear Matrix Inequalities can be applied to determine the

stability for the given sector conditions v,(e,~) E (1,, m,) [4, Chapter 8].

Lemma 1 Consider

c’ = Amc + Bmv(<)

with v(C) satisfying the sector conditions ( 3. 2). There is a Lyapunov function
of the form

p—l Ci
V : CTPC + 2 Z A,/ v,(r)dr (3.3)
i=1 0

where P 2 PT > 0 and A,- are positive constants, and its derivative satisfies

the inequality
V = 2(ch + vT(<)A)(Amc + B..v(<)) s —c"c (3.4)
where

A Z diag()\1, ' ' ° ,Ap_1)

59

if the LMI problem

AfiP + PA... + I — 2TK, PB... + A3111 + TK,
< 0 (3.5)

33,110 + AA... + K,T AB... + B,’—',’,A -— 2T

is feasible, where

KP Z diag(llm1, ° ° ' ,lp_1mp_1)
K, = diag(11 + m1, . .- ,I,,_1 + m,_,)

T = diag(7'1, . .. ,7.-.)

and TI to rp_1 are nonnegative constants.

Lemma 1 results from the S-procedure [4, page 23] and detailed steps are shown in
Appendix C.

For relative-degree-p systems, we determine the desired range of the nonlinear
gains and solve the LMI problem. If the LMI problem is feasible, our analysis in
the next section shows that the controller achieves regional and semiglobal regula-
tion. Examples of sector conditions which ensure the stability for second, third and
fourth-order systems are computed by the MATLAB LMI toolbox and are shown
in Table 3.1. For example, in the second column of the table, we started solv-
ing the LMI problem for relative-degree-three systems with (ll, m1) = (39,41),
(l2, m2) = (5.9, 6.1), and (13, m3) = (39,41) and increased the range of the nonlin-

ear gains until the LMI problem was infeasible.

60

Table 3.1: Examples of sector conditions for stable systems determined by the LMI
2nd 3rd 3rd 4th 4th
II 0.1 2.5 16 13 180
m1 1000 5.5 48 21 332
Z2 0.1 3.5 12 18 134
mg 1000 7.5 36 30 250

 

 

 

 

 

 

 

 

 

 

 

 

L», 2.5 6 14 58
mg 5.5 10 22 86
l, 4 9
m., 7.5 15

 

In summary, a Universal Integral Controller with nonlinear proportional, inte-

gral and derivative gains is taken as

u = —k si@(LgL?—1h)¢(-E) (3.6)
where
§:0+&
é : ¢(éa) (3'7)

and the augmented error e, is given by

p—l
é, : v1(e1) + Z v,(é,-) + 63,,
i=2

The nonlinear integral gain ip(0) satisfies the condition (2.25), and the nonlinear
proportional and derivative gains v,- = v,(e,~) are continuous, piecewise difierentiable

and satisfy the sector conditions (3.2).

61

3.3 Closed-loop Analysis

The closed-loop system can be represented in the singularly perturbed form

2 : go(Z,e+V,d)
é: AmC + Bmv(C) + B(s - a)

0 = Ip(s — Nm(C, s, (p, e) — 0) (3.8)
s' = A(°) - klao(')|¢(s — NA: 8' (Ad)

 

 

.e 2 mm. [boo-kIao<->I¢(s’N'"(C’s’w’e))]

[l
where
p—l
AC7": 8: 0, VI d) 7.00)) : ¢(8 — Nm(<t S, ()0! 6) _ U) + Z vi(ei)ei+l + bo(Z, e + V: d) 7.09))

i=1

p—l

Nm(C. 5.4916) = 204031) — "1(éi))+ ép — en
i=2

Only the dynamics of C, the estimation error N...(C, s, p, e) = s — § and the distur-
bance term A(-) are different from (2.31). We present in detail only the part of the
analysis that is different from Section 2.4.1.

Suppose that v(C) is chosen such that the LMI problem (3.5) is feasible. Take
the Lyapunov function V of the form (3.3) whose derivative satisfies the inequality

(3.4). Consider
a... ‘1—1‘ {(2.20) : Isl s c. IaI s (1 + 5)c. V(C) s «2212.. vo(t.z.d) s «(cm + rsl}

We show that our assumptions hold in the set 0,, by choosing p; to satisfy (3.9)

62

and then choosing c to satisfy (2.34). Since e p = s — 0 — 2:13,“ v.(C.), the inequality

llell S llCll + lepl S (1 + lle||)||C|| + ISI + IUI S 692 < r1

should be satisfied, where K... 2 [m1, - -- ,mp_1] and

P1

 

+2+5 (am

In the first step, we show that every trajectory, starting at a bounded é(0) with
(e(0), 2(0), 0(0)) E m, and b < c, enters the set {2, x E, in finite time.

From the continuity of v.(-), we can choose 5 small enough so that
|N...(C,s,p,e)| < 6p. Then we show that 53' < 0 on the boundary Is] 2 c and
00 < 0 on the boundary |0| = (1 + 6)c. Consider C on the boundary V = czpl,

where p1 is to be specified. From (3.2) and (3.4),

v s —|ICH2 + 2|lC|l(||1"Bl|+ A._1m._.)(IsI + IaI) (3.10)

S -|lCl|2 + 2llC||(||P13|| + )‘P-lmp-l)c(2 + 5)
and we can show that V < 0 on the boundary V : czpl, by choosing p. to satisfy
p1 > 4(IIPBII + Ap—1mp-1)2(HPII+1Inwl{r\rmi})(2 + a): (3.11)

Then we verify that V.) < 0 on the boundary V0 2 a4(cp2 + r3). Therefore, the set
(2, x )3, is positively invariant.
In the next step, we show that s(t) reaches the boundary layer {|s] S #1} and

0(t) reaches the set {|0] S (1 + 6)u)} in finite time. Then, from (3.10), we show

63

that V s -—%II<II"’ for V(C) 2 112p. if
P1 > 54(HPBH + Ap_1mp_1)2(llPI| + 11935117713) (3-12)

Since (3.12) implies (3.11), by fixing p. to satisfy (3.12), we show that C (t) reaches
the set {V(C) _<_ uzpl} in finite time and stays therein. Then, 2(t) reaches the set
{V}. g a4(up2 + 11)} in finite time and stays therein. Therefore, every trajectory in
O, x )3, enters 43, x 2, in finite time and stays therein, where the set \II,, is defined

as

‘Pp d5" {(46.0) 1 ISI S #1. IO! S (1 + (5)11. V(C) S #2101, Vo(t.2.d) S 040412 + 11)}

When V = 0 and rm 2 0, the closed-loop system (3.8) has a unique equilibrium

point (2 = 0,e = 0,0 =2 b“, (p : 0), where

6 = — sign(LgLI‘1h)e¢-1 (ES—:9)

Following steps similar to Appendix B, we conclude that all trajectories in \II,. x )3,
approaches the equilibrium point as time tends to infinity.

Our conclusions are summarized in the following theorem and corollary.

Theorem 3 Suppose Assumptions 1 to 4 are satisfied and consider the closed-
loop system formed of the system (2. 5), the observer (2.22), the nonlinear inte-
gral gain ( 3. 7) satisfying condition (2.25), nonlinear proportional and deriva-
tive gain satisfying (3.2), and the output feedback controller (3. 6). Suppose
the LMI problem (3. 5) is feasible, é(0) is bounded, and (2(0),e(0),0(0)) E {25,

where b < c and c satisfies (2. 34 ) and (2.36). Then, there exists ,0“ > 0 and

64

for each 0 < u < If, there exists 6* : 601) > 0 such that for each 0 < ,u < If
and 0 < e < 6* (u), all the state variables of the closed-loop system are bounded

and limthoo e(t) = 0.

If all the assumptions hold globally and k can be chosen arbitrarily large, the

controller can achieve semiglobal regulation.

Corollary 3 Suppose Assumptions 1 to 4 are satisfied globally, i.e., U9 : R",
01, a2, a3 and 7 are class ICOO functions, the LMI problem (3.5) is feasible,
and k can be chosen arbitrarily large. For any given compact sets N E R"+1
and M 6 RP, choose c > b > 0 and r3 2 0 such that N e 05, and choose k large
enough to satisfy (2.36). Then, there exists It > 0 and for each 0 < u < [1“,
there exists 6* = 6* (u) > 0 such that for each 0 < p. < u“ and 0 < e < 6‘02), and
for all initial states (2(0), e(0),0(0)) E N and 6(0) 6 M, the state variables of

the closed-loop system (3. 8) are bounded and Hm...” e(t) = 0.

3.4 Simulation Results

Example 1: A synchronous generator connected to an infinite bus can be described

[28, page 25-26] by

M6 = P — D6 — mEqsin6

TEq Z -772Eq + 7’3 C086 + EFD

The system has relative degree three viewing the field voltage EpD as input and
the angle 6 as output, for 0 < 6 < 1r. For simulations, we use P = 0.815, 171 = 2.0,

172 = 2.7, 173 = 1.7, r :2 6.6, M = 0.0147 and D/M = 4. The simulation results

65

shown in Figure 3.1 are for the controllers with linear and nonlinear proportional
gains. Since 3 enters the boundary layer fast, we drive the integrator with the
augmented error 4e. without using nonlinearity. Controller parameters u = 0.5,

= 2.5 and limit L = 10 for the integrators are chosen for the simulation. The
error dynamics inside the boundary layer is given by

81 0
+ (s — 0)

—k1 —k2 82 1

C=AC+B(s—0):

When the linear gains are chosen as k. : 8 and k2 = 4, the system C 2 AC is
underdamped and shows fast response with overshoot, while the linear gains k1 = 6
and k2 = 5 remove the overshoot but increase the settling time. The nonlinear
proportional gain increases to k1(e1) = 12.5 when the tracking error is large for
fast rise time, and decreases to k1(el) = 6 as the error becomes small to prevent
overshoot. The nonlinear pr0portional is shown in Figure 3.2. The derivative gain
was fixed as k; = 5. The simulation results show that the nonlinear proportional
gain achieves comparable settling time with the linear gains k1 = 8 and k2 = 4,
without showing overshoot.

Example 2: The nonlinear model of a single-link manipulator with flexible joints

[28, page 25], damping ignored, is described by

[(11 + MgLsinq1 + (“(41 — 92) Z 0

JiI'z—KUII -<12) =u

Viewing the angular position q. as output and the torque u as input, the system

has relative degree four. For simulations, the numerical values I = 0.5 kg/mz,

66

 

 

 

 

 

 

 

 

 

 

12 I I I I I
A 10'— _
3
x 8"" _
6 1 L l l
0 0.5 1 1.5 2 25 3

 

 

 

._. k1=8, k2=4 _
_ _ k1=6, k2=5
_ k1=nonlmear, K2=5

 

 

 

 

p-

l

2 2.5 3

 

Figure 3.1: Simulation results for the synchronous generator connected to an infinity
bus

67

15 I

 

v1(e1), k1(e1)

 

 

 

‘ ______ v1(e1)

 

 

 

 

. . i - - “1““)
_,5 I I I I I I I I
-1 —0.8 -0.6 —0.4 -0.2 0 0.2 0.4 0.6 0.8
e

Figure 3.2: Nonlinear proportional gain 01(e1) = k1(e1)e1 for the simulation of the

synchronous generator connected to an infinite bus

68

 

J = 0.5 kg/mz, M = 1 kg, 9 = 9.8 m/secz, L = 1 m and k = 20 N/m are used.
In Figure 3.3, we compare the performance of the controllers with the linear and
nonlinear proportional and derivative gains. The maximum control level is set as
k = 12, and the linear gains are chosen as k1 = 31.25, kg = 25 and k3 = 7.5.
The nonlinear gains are designed to satisfy the conditions 31.25 g k1(e1) g 45,
25 g k2(62) _<_ 28 and 6 g k3(e3) _<_ 7.5. For the range of the nonlinear gains chosen,
the LMI problem is feasible, as shown in Table 1. The nonlinear gains are plotted
in Figure 3.4. The integrator is driven with 23,, with the limit L = 90. For the
reference 7‘ = 2 with the initial condition y(0) = 0, the nonlinear gains achieve
better settling time but show small damping. When the reference 1' = 1 is applied
at t = 4, the nonlinear gains reduce the settling time without overshoot. Since the
nonlinear gains k,(e,-) are functions of diflerent variables, it is difficult to design the

gains to change the poles of C 2 AC as the tracking error becomes small.

3.5 Conclusion

In this chapter, we designed Universal Integral Controller with nonlinear in-
tegral, proportional and derivative gains. We showed that, if the LMI problem in
Lemma 1 is feasible for the nonlinear gains, the controller stabilizes the system
regionally and semiglobally. The nonlinear gains provide us freedom to alter the
error dynamics inside the boundary layer as the state of the system changes. The
simulation results showed that the nonlinear gains can be chosen to improve the
transient performance for relative-degree-three systems. It is more challenging to

design the gains for higher relative degree.

69

 

I I I I I I I
- — linear gains
— nonlinear gains

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

\
l l l l l l
2 3 4 5 e 7
$060 I I I I I I I
Vco _k1(91)
:40- __k2(92)
m~ _\ I‘\ _ k3<e3>
VN ._ ____________________________________
:20—
o" ____,_______.._.____.\_________..___._
:F 0 l l l l l l l
o 1 2 s 4 5 e 7
20

 

 

 

 

 

 

.-
p-
n—
p—
b
u—

 

n¢

Figure 3.3: Simulation results for the single link manipulator

70

 

 

45 I I T I I I I

 

 

 

 

 

 

‘ ‘
25-.....................‘_,——-—--~—-—---——‘—-——>-«—v—-—---—.--vv-<r.--.---.--....d

I
I
I
I
I
I
I
I
\
I
I
I
I
I
I
I
I

20F ........... ; ............ g ............. : .......... , .g ............ f, ............ g ............ g ........... _
15- ........... ....... , ............ g ........ . ,_ .......... ............ g ........... g ........... _

'—-_.—.—.—~.—._.—.—..—.—-—.—--—-—.—.h

. .—

—._
.—-.— ‘_-'—.
__._.—— —.—.._

5 I I I I I I I
-2 -1.5 —1 —0.5 0 0.5 1 1.5 2
e

61’ 2’ es

 

 

 

Figure 3.4: Nonlinear proportional and derivative gains k1(e1), k2(e2), k3(ea) for
the simulation of the single link manipulator

71

Appendix C S-procedure for Lemma 1 [4]

A linear matrix inequality (LMI) has the form

F($):F0+inFi >0

i=1

where a: E R“ is the variable and F,- = F? E Rn“ are given. The LMI is equivalent
to a set of n polynomial inequalities in 2:. The convex constraint on 3:, i.e. {a2 :
F(:z:) > O} can represent a wide variety of convex constraint on 2:. The LMI problem
is to find :1: such that F(:z:) > 0 or determine that the LMI is infeasible. Eficient
algorithms have been developed for LMI problems, which often represent constraints
in control design.

For the constraint that some quadratic functions be nonnegative whenever some
other quadratic functions are nonnegative, the S-procedure can be applied to form
an LMI that is a conservative approximation of the constraints. Let Go, - - - ,G,D be

quadratic functions of the variable C E R":
Gi(€) = £TTI£ + 2w§ré + m. for i = 0. - -- .p
where T,- = T,T. Consider the following condition on Go, - - - ,Gp:
Go 2 0 for all C such that G,(§) Z O, for i = 1,--- ,p
The above constraint holds if

P
there exist T1 2 0, - -- ,TP 2 0 such that for all C, G0(C) — ZT,G,(C) Z 0

i=1

72

The sufficient condition can also be represented as

To wo P T,- w,-
— Z T, Z 0
w? no ‘=1 w? 17.

Note that in Lemma 1, we require inequality (3.4) be true if every nonlinear
gain v,(') satisfies the section condition (3.2); both inequalities can be arranged in
quadratic forms. We apply the S-procedure to find the suficient condition. From

(3.4), we have

2((”P + vT(C)A)(AmC + Bmv(C)) + (TC

= CT(2PAm + I )C + 2CT(PBm + ATIIA)v(C) + 2vT(C)ABmv(C)
T
( AgP+PAm+I me+A3;,A c

v(C) B,7,‘,P + AA", AB", + Ball v(C)

‘i—i‘ (film... 3 o

73

The sector conditions (3.2) can be arranged as

205(6) — liCi)(Vi(Ci) — miCz')

 

 

. T .
0 0
C Zlimi -(lI + mi) C
_ 0 0
— 0 0
V(C) —(l,- + m,) 2 1’(C)
I . .0 0
dé‘ cm... s o
for 2' = 1, - - - , p— 1, where the elements of T,- are zero except the ith diagonal element

of each submatrix. Thus, the inequality (3.4) is true, for v,-(C,) that satisfies the

 

 

sector condition (3.2), if

To—ZTI'H=T0—

 

 

p—l

i=1

which is the LMI problem of (3.5).

—2TKp

74

 

TK,

<0

—2T

 

 

 

Chapter 4

Nonlinear PID Controllers

4. 1 Introduction

For relative-degree one or two nonlinear systems, the controllers designed in
the previous two chapters specialize to nonlinear versions of the classical PI and
PID controllers. In this chapter we study the special case of relative degree one or
two systems for various reasons. First, there has been a lot of interest in the control
literature to develop nonlinear PID controllers, or more precisely PID controllers
with nonlinear gains, in order to improve the performance of the system. In Chap-
ter 1, we described the various ideas that have been proposed in the literature and
the analytical results that are available for some of those ideas. It is important to
emphasize that all the results available in the literature are for the case when the
plant is linear or for nonlinear robotic systems. Second, by specializing to the case
of relative degree one or two systems, we can obtain results sharper than those we
obtained in Chapter 3 for the general relative degree case. In Chapter 3, we could
not obtain an analytically verifiable stability condition. What we obtained was a

condition in the form of feasibility of an LMI problem, which could be checked only

75

numerically. In this chapter we derive analytical stability conditions. Third, in the
case of relative degree one or two systems we can consider new structures for the
nonlinear integrator that we cannot consider in the general relative degree case. We
saw in Appendix A of Chapter 1 that for relative degree higher than two, we need
to drive the integrator by an augmented error. For relative-degree—two systems, we
can also consider nonlinear integrators that are driven by the tracking error. For

relative-degree-one systems, we have the controller

 

=-k' (L.h)¢ ”“1
u sign ( p. ) (4.1)

F = M81)

which is a special case of the controller (2.26) when p = 1. For relative-degree—two

systems, we consider two different controllers:

 

 

u = —ksign(L,L,h)¢ (0 + Mel)“ + é?)
I‘ (4.2)
0” : ¢(k1(el)€1 + C2)
and
u = —Icsign(L9th)¢ (0 + Mel)“ + E2)
" (4.3)

('7 = 1,0(el)

The controller (4.2) is a special case of the controller (3.6) when p = 2. The
controller (4.3) is a new structure in which the integrator is driven by a nonlinear

function of the tracking error e1 instead of the augmented error

éa = k1(€1)+ éz

76

In both (4.2) and (4.3) the estimate éz is obtained by the high-gain observer

él éz ‘I' %(81 — él)

(4.4)

(1»-

fi .
2 = 272031 — 31)

where 61, fig and e are positive constants with e chosen sufficiently small. To
overcome the peaking phenomenon that can be induced by high-gain observers,
we require the nonlinearity 1/z(éa) for the controller (4.2) be a globally bounded
function of ea:

61192 S W09) S 62192. for |p| S L
(4.5)

C1L S i'I/1(p)| S C211, for |p| > L
while the nonlinearity 1/J(e1) for the controllers (4.1) and (4.3) satisfies the sector

condition

61192 S 191/410) 5 c2192 (4.6)

Schematic diagrams of the controllers (4.1), (4.2) and (4.3) are shown in Figure 4.1.
All the schemes are versions of classical PI or PID controllers with nonlinear gains
and with saturation nonlinearity ¢(-). It is worthwhile to note that the PI and PID
controllers considered in [27] as special cases of the Universal Integral Controller
when p = 1, and p = 2, respectively, are special cases of (4.1) and (4.3), with

¢(el) = ko’el, k1 = constant, and ¢(-) = sat(~).

77

Q

 

 

 

 

 

 

 

1M)

 

—>| f

 

 

 

 

vv
A“ q

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

61 ¢(-) —> k ——“—>
(a) Nonlinear PI controller
kI(-) >3; ‘3“ >',z/2(-) 9| f " >3; 5 ¢(-) 9
HGO Q

 

 

(b) Nonlinear PID controller with nonlinear integrator driven by the augmented

CHOI

&

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

¢(-) —> I —"
kid ‘jég
HGO éz

 

 

 

 

 

 

¢(°)

->[k

 

 

 

(c) Nonlinear PID controller with nonlinear integrator driven by the tracking

error

Figure 4.1: Nonlinear PI controller for relative-degree-one systems and nonlinear

PID controller for relative-degree-two systems

78

4.2 Closed-loop Analysis

We present the proof of regional and semiglobal regulation of the nonlinear PID
controllers (4.2) and (4.3) for relative-degree-two systems. The closed-loop system

for the controller (4.2) can be represented in the singularly perturbed form

2': = qo(z,e+u, d)
a = 1M8 - N90 - 0)

éI = —a — k1(e1)e1 + s (4,7)
, = A(-) — klao(-)l¢(s ' N‘p)

 

 

at) = Afcp + 63a [500) — kla°(')l¢(s _uNwH

where

s = 0' + k1(el)el + ez
A(z, e, a, V, d, 6‘) = 'I/J(S — N<p — a) + k1(el)e2 + k’l(€1)€1€2 + bo(z, e + u, d, f)
%(61 — él) —fl1 1 0

(P: ) Af: I Ba: 1 N:[O 1]
Eg—ég —fig 0 1

79

For the controller (4.3), the closed-loop system is given by

Z : QO(zIe+VId)
{7 = $031)
él = —a — k1(e1)e1 + s (4.3)

s' = A(-) — klao(-)I¢(s ‘ M”)

 

 

6‘? = Afcp + e3, [500) ._ k|a0(.)|¢(s —flNw)]

where
A(z, e, a, u, d, i‘) : W81) + k1(el)e2 + k’1(e1)elez + bo(z, e + V, (1,?)

The dynamics of a and the disturbance A(-) are different between (4.7) and (4.8).
We present the analysis for (4.7), and remark on the part that is difi'erent for (4.8).

For (4.7), consider
a. “‘é‘ {(z,e,a) : Isl s c, IUI s (1 + 6)c. Ian 5 cpl. mm) _<_ ado/’2 + ’13)}

where the positive constants A, 0, p1, p2 and the class IC function on, are to be
specified. Our assumptions should hold in the set QC, i.e., ”all < 1'1 and “z“ < T2
for all (3,2, 0') E De. From Assumption 4 of Chapter 2, upper bounds on e and z

are given by

“3“ S I81|+ I€2I = I€1I+ I — 0‘ — k1(31)€1 + SI
3 [0| + (1 + E1)|81|+|s| S cpg (4.9)

IIZII S afl(a4(cp2 + 7‘3))

80

where in is the maximum of k1(e1) over RC, and

p2 > (1 +l'cI)p1 + (2+ 6)

def _1
a4 2 0‘2 ° '7
Thus, c should be chosen to satisfy

Ch 3 min{r1, a;1(a1(r2)) — r3} (4.10)

where a;1(a1(r2)) > 1;», from (2.9).

For (4.8), we take QC as
:2. dz! {(2, 6,0) : Isl s c. V(C) s cm. Vo(t.z.d) S am» + 73)}

where

V=3<TP<+A/,e1w(r)dr. c: a . P: p“ p”

2
81 P12 P22
The matrix P 2 PT > 0 and the positive constant A are to be specified. The

inequality Hell 3 cpg holds with

2P1

p2 > (2 + Tel) Am?)

 

+ 1 (4.11)

and our assumptions are valid in QC if c satisfies (4.10) with p2 as in (4.11).

Let P, : P? > 0 be the solution of the Lyapunov equation

5.4, + A3219, 2 —I

81

and take Vf(<p) : IpTPfgo. In the first step, we show that (p enters the set
2, 43f {cp E R2 : Vf(<p) 3 62,03}

in a finite time To(e), with limHoo To(e) = 0, and 2, can be made arbitrarily small

by choosing 6 small. The derivative of V, satisfies

- 1
VI 3 7W + 2II‘PIIIIPfBaII71(C)

where 71(c) = max{|bo(z, e + V, d)| + k|a0(z, e + V, d)|} and the maximum is taken

over all (z,e,a) E QC, d E D, V E A and I" E 1". We have
' 1 2 2
Vf < —-2—6||<p|| , for V, 2 5 p3

where p3 : 16||PfB||2||Pf||712(c). Thus, <p(t) decays to )3, in a finite time To(e)
and stay therein. We choose 5 small enough that |NcpI < 6p, where 6 is a positive
constant to be specified. With 1/2(-) satisfying (4.5), the right-side of the slow
equation of (4.7) is bounded uniformly in 6. Hence, every trajectory in 05, with
b < c, is inside QC during [0, T1], where T1 is independent of e. We choose 6 small
enough that To(e) < T1. Thus, every trajectory starting at (e(0),z(0),a(0)) 6 $2,,
with bounded é(0) enters (2,, x E, in finite time.

Next, we show that the set (2,, x )3, is positively invariant. Choose [1 small

enough that ,u(1 + 6) < c. For ”(1 + 6) S |s| g c, we have

..(.(s—,W))-..(s—,~I)-....

 

 

82

and from (2.29), |¢(p)| 2 45(1) 3 c3 for |p| _>_ 1. Hence

sé S |A(°)||S| - CaklaoIISI (4-12)

With the controller gain In taken large enough to satisfy

’6 2 C4 + 72(C) (4.13)

where c.; is a positive constant, 72(c) = max{|A(z, e, a, V, d, I‘)| /|c3ao(z, 6 + VI ‘0”.

and the maximum is taken over all (z, e,a) 6 no, d E D, V E A and i‘ E 1", we have

ss<0 on|s|=c

On lal = (1 + 6)c, using

IS - N<p(6)<pl < 6+ M < IUI

sign(IMS - N ¢(€)<p — 0)) = - sign(a)

we can show that

06 2 azp(s — N<p(e)<p — a) < 0, on |a| = (1 + 6)c

From the inequality

elél < —k1(€1)ei+I31I(I5I+I0I) (4-14)

_<_ —&1ef + 81(2 + 6)c

83

where £1 is the minimum of k1(el) over (L, we have

elél < 0, 0n Iell : Cpl

if p1 is chosen to satisfy

2+6
k1

 

p1 > (4.15)

For (4.8), we show that V(C) < 0 on the boundary V(C) = c2p1. The derivative

of V is given by

V = "P1202 — P22k1(81)ei ‘- (”C(31) — P12)¢(31)31 + (P11 — ”010(31)

"' (P22 + P12k1(€1))0€1 + CTPBOS + A1/J(el)s

By taking A 2 p11 and choosing p11 large and 1212 > O, the derivative of V can be

arranged in the form

V S -Amin(Q)|lC||2 + (HPII + 662IIIC|||S| (4-15)
where Q = QT > O is given by

P12 %(P12’21 + P22)

%(P127C1 + P22) Alglcl + pggkl — p12C2

and 7:1 is the maximum of k1(el) over DC. The derivative of V satisfies the inequality

V<O, onV=c2p1

84

when p1 is chosen to satisfy

> (IIPII + 602?
2(/\min(Q))2

 

(4.17)

From (2.7) in Assumption 4, we have “2” 2 7(cp2 + r3) on the boundary V0 :

a4(cp2 + r3). By (2.8) and the bound of Hell in (4.9), we can show that
VI) < 0, on V0 = 014(sz + 7‘3)

Therefore, the set (2,, x E. is positively invariant.
In the third step, we show that every trajectory in (2,. x 23. enters the set ‘1'), x )3,

in finite time and stays therein, where ‘1’), is defined for (4.7) as

‘I’F‘ 4:! {(2,830) I l‘sl S #1: '0' S (1 + 6);,” lell S #Pl.%(t121d)s 04(flP2 + I‘d}

and for (4.8) as

‘11.. "=93 {(2,420) = ISI S #1. V(C) S #2101. Vo(t.z.d) S 04(IJP2 + 74)}

Consider s in {#1 g Isl g c}. With 6 chosen to satisfy

C4 1
5 _ _
<4k<4

we have sgn(¢(§/Iu)) = sgn(3/u) = 8211(8). since leI < 5# < ISI- If |§| 2 V.
inequality (4.12) holds. Using (4.13),

88 _<_ —C0C3C4|S|

85

where co is defined in (2.2) of Assumption 1. If |§| < u, we have |§| > 11/2 and

we)

600304
2

ss _<. IA(-)|Is| — klao(-)| [co (E)

C316
2

 

 

S |A(')||S| - Iao(-)||S| S - ISI

Thus, s(t) reaches {|s| 3 m} in finite time and stays therein. Consider a in
{(1 + 6),u 3 [0| 3 (1 + 6)c}. Since |§| < u < lol, we have sign(w/z(§ — 0')) =

sign(s — a) = — sign(a) and Is — 0| > 6p. Moreover, |§ — 0| 3 p. + (1 + 6)c and
|1/I(P)| Z E1|10|. for IPI S u + (1 + 6)C
with 51 = min{c1,c1L/(u + (1 + 6)c)}. Using these facts, we can show that
06 : asign(§ — o))|2/J(§ — 0)] < —516u|o|

Thus, 0(t) reaches the set {|o| g (1 + 6)u} in finite time and stays therein. From

inequality (4.14), we have
. E1 2
8181 S -—2—€1, for Iell 2 PM

by choosing p1 to satisfy

4

Inequality (4.18) implies (4.15). With p1 chosen as above, el(t) reaches the set

{lell g upl} in finite time and stays therein.

For (4.8), we consider that C = [0,e1]T in {V(C) 2 pzpl}. From inequality

86

(4.16), we show that

V < —A—““%9—)H<II2, for V(C) 2 4sz

by choosing p1 to satisfy
2(IIPII + AC2)3
p1 > (Amin(Q))2

 

(4.19)

Inequality (4.19) implies (4.17) and from now on, we fix p1 as chosen above. Thus
C (t) enters the set {V(C) g pzpl} in finite time and stays therein.

Consider 2(t) in {V}, > a4(#P2 + #4)}. Using the facts that ”2“ Z 7((up2 + u)
from (2.7), the bound Hell 3 up; (derived similar to (4.9)), limb,” V(t) = 0, and
(2.8), we can show that

Vo S -a3(||Z|l)

Therefore, every trajectory in QC x 2, enters \II,, x )3, in finite time and stays therein.
Finally, we show that every trajectory in ‘11,, x 2,, approaches an equilibrium

point

1101))

6 = — sign(ao(-))u¢-1(T

asymptotically. Let V1(z, d) be a converse Lyapunov function for assumption (2.10)

of exponential stability. Consider
_ 1 2 1 ~2 1 ~2 T
V2 _ VI(z,d) + 5A5e, + 52.60 + -2-s + (p Pfcp

where 6 = a — 6, 3" = s — 's', 5 = 6, and A5 and A6 are positive constants to be

87

chosen. The derivative of V2 can be arranged in the form

Vz S —xTP2x + (Avllwll + Aslé'l + Agllzll)llv(t)ll
(4.20)

+(A10IIPII + A11I§I)Ili(t)l
where x = [”2” |e1| |a| |§| ||<p||]T and A7 to All are positive constants, and P2 2 pg"
has a structure similar to P2 of (2.43). The matrix P2 can be made positive definite
by choosing A5 large enough, then A6 large enough, then 14 small enough then 6
small enough.

For (4.8), we consider the Lyapunov function
~ ~ 1
V2 = V1(z, d) + AsCTPC + 552 + prfcp

where C = C — C and C = [6, OP". The derivative of V2 has the form similar to (4.20)
with x = [||z|| ||C|| |§| ||<p||]T, and P2 can be made positive definite by choosing A5
large enough then u small enough then 6 small enough.

Regional and semiglobal results for the controller (4.2) is stated in Theorem 4
and Corollary 4, respectively, while Theorem 5 and Corollary 5 summarize the

results for the controller (4.3).

Theorem 4 Suppose that Assumptions 1 to 4 of Chapter 2 are satisfied and
consider the closed-loop system formed of the system (2. 5) with relative degree
two, the observer (4.4), the output feedback controller (4.2) with the nonlin-
ear integral gain 1M) satisfying (4.5), and continuous, bounded and piecewise
difi‘erentiable nonlinear proportional gain k1(-). Suppose 6(0) is bounded and
(z(0),e(0),a(0)) E 91,, where b < c and c satisfies (4.10) and (4.13). Then,

there exists ,u“ > 0 and for each 0 < a < u’, there exists 6* = e’(u) > 0 such

88

that for each 0 < u < u‘ and 0 < e < 6*(p), all the state variables of the

closed-loop system are bounded and limtnoo e(t) : 0.

Corollary 4 Suppose that Assumptions 1 to 4 are satisfied globally, i.e., U9 2
R", (11, a2, a3 and 7 are class [Coo functions, and k. can be chosen arbitrarily
large. For any given compact sets N E l?"+1 and M 6 RP, choose c > b > 0
and r3 2 0 such that N 6 (lb, and choose In large enough to satisfy (4.13).
Then, there exists u" > 0 and for each 0 < u < u", there exists 5‘ = 6"(u) > 0
such that for each 0 < ,u < u‘ and 0 < e < 6(a), and for all initial states
(z(0),e(0),o(0)) E N and 6(0) 6 M, the state variables of the closed-loop sys-
tem (4.7), with nonlinear integral gain 1p(-) satisfying (4.5), and continuous,
bounded and piecewise differentiable nonlinear proportional gain, are bounded

and lim¢_,0° e(t) = 0.

Theorem 5 Suppose that Assumptions 1 to 4 of Chapter 2 are satisfied and
consider the closed-loop system formed of the system (2. 5 ) with relative degree
two, the observer (4.4), the output feedback controller (4.3) with the nonlin-
ear integral gain 1/J(-) satisfying (4.6), and continuous, bounded and piecewise
differentiable nonlinear proportional gain k1(-). Suppose 6(0) is bounded and
(z(0),e(0),o(0)) 6 {25, where b < c and c satisfies (4.10) with (4.11), and (4.13).
Then, there exists pi“ > 0 and for each 0 < u < p.‘, there exists 6" = 6*(p) > 0
such that for each 0 < u < p‘ and 0 < e < 6*(p), all the state variables of the

closed-loop system are bounded and limtnoo e(t) = 0.

Corollary 5 Suppose that Assumptions 1 to 4 are satisfied globally, i.e., U9 2
R”, a1, a2, a3 and '7 are class [Coo functions, and k can be chosen arbitrarily

large. For any given compact sets N E R“1 and M E R", choose c > b > 0

89

and r3 2 0 such that N E Db, and choose 11: large enough to satisfy (4.13).
Then, there exists if > 0 and for each 0 < p. < u“, there exists 5* = 6*(p) > 0
such that for each 0 < p. < u“ and 0 < e < 6*(p), and for all initial states
(2(0), e(0), 6(0)) 6 N and 6(0) 6 M, the state variables of the closed-loop system
(4.8) with nonlinear integral gain ¢(-) satisfying (4.6) and continuous, bounded
and piecewise diflerential nonlinear proportional gain k1(e1), are bounded and

limt—mo C(t) Z 0.

4.3 Simulation Results

Example 1: Reconsider Example 1 of Section 2.5. The simulation results of Fig-
ure 2.4 demonstrated that the Universal Integral Controller with nonlinear integra-
tor driven by the augmented error improves the transient performance. In Figure
4.2, we compare two cases where the nonlinear integrator is driven by the augmented
error or the tracking error. Figure 4.2 shows the angular velocity, the integrator
output and control input for three controllers with the following linear or nonlinear
integral gains:

(7 = 31. (7 = ’l/Ir(éa). ('7 = IP2031)

We use the same system parameters and controller gains of Figure 2.1. The nonlin—
earities w1(-) and ‘l/J2(-) are shown in Figure 4.3. Both nonlinear integrators achieve
better transient performance than the linear integrator, showing no overshoot and
improved settling time.

Example 2: Reconsider Example 2 of Section 2.5. We compare the performance of
the nonlinear integrators, driven by the augmented error and the tracking error, for

the motion on the level surface. We use the same controller gains as in Figure 4.4.

90

 

 

 

 

 

 

 

 

 

1- I I — f_ - “-1- _ _1__ I I l I l d
/' ——"—————_..._._
>:05- 4
0 l l 1 1 l l L 1
° 1 2 3 4 5 6 7 s 9 1o
0‘ ‘I~___I___I_-__r 1 I I I I
-0-2—\. '—- ————————————————————————— :
\‘ _. _. ...——--
b -0.4" \ ’4’ ’ f -
\_ I ’
-0.6” -
—o.8 l l l l l 1 I 1
0 1 2 3 4 5 6 7 8 9 10
1 I I I I I I I

 

.—.—.—._

 

 

-—.—.._
‘—-—v—-—;S_A_A_ _-_

 

- - - linear interator _
_ nonlinear integrator dnven by 91

_ nonlinear integrator driven by ea

 

 

 

 

).
.—
I.
._
..

 

Figure 4.2: Simulation results of the
controlled DC motor

nonlinear PID controllers for the field-

91

 

 

 

 

 

4 v,(9,)
_ _ v2(eq)

 

 

 

 

 

 

_1 1 1 1 1 1

-1.5 —1 —0.5 0 0.5 1 1.5

Figure 4.3: N onlinearities ¢1(ea) and i/J2(e1) for the simulation of the field-controlled
DC motor

92

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

6 I I I I I I I r T
_ ./' '\ . \ _l
4 / \ \
>. I. ————————— 7 L _________ _
/.
2 — ' a
0 l I 1 l 4 l l l l
0 1 2 3 4 5 6 7 8 9 10
2 I I r I I I I I I
0\ ——__-_‘___r ............. _
‘\ / ’ . .
_2 _ '\ _/ --- linear Integrator g
b ‘\' / , x _ nonlinear Integrator driven by ea
\ .
_4 _ ‘ \ . ._ , __ , , . ’ _ _ nonlinear integrator driven by 61 -
-6 1 1 1 1 1 1 1 1 1
0 1 2 3 4 6 7 8 9 10
4 I 7 I I T I I I T
2 ~ _. ....... a
3 o _ .’. _
-2 _ _
_4 1 1 1 1 1 1 1 1 1
O 1 2 3 4 5 6 7 9 10

Figure 4.4: Simulation results of the nonlinear PID controllers for motion on a
horizontal surface

93

The nonlinearities ‘l/J1(-) and ¢2(-), which are driven by the augmented error and the
tracking error respectively, are shown in Figure 4.3. Simulation results are shown
in Figure 4.4. Both nonlinear integrators improve the transient performance and
show similar performance.

Example 3: An intuitive idea proposed for improving the performance using non-
linear gains is to use large proportional gain while the tracking error is large, and
decrease the gain as the error becomes small to reduce damping [11] [7]. This idea
is not suitable for our case because, inside the boundary layer, the dynamics of the

tracking error has the form of a first-order system driven by (s — a):

612—k161+s—U

If the nonlinear integrator keeps a small, the response of the system will be faster
with large 1:1. But k1 is limited by the control level 1:, since large k1 increases the un-
certainty A(-), which the controller should overcome (4.13). Nonlinear proportional
gain k1(e1) can be designed such that the gain is small when the tracking error is
large to keep the disturbance A(-) small, and increases as the error becomes small
to improve the settling time. For an unstable second-order system with constant
disturbance

y—y3—o.1y3=u+1

we designed the nonlinear proportional gain shown in Figure 4.5. A controller with
nonlinear integral gain 6 = ¢1(éa) with ¢1(-) shown in Figure 4.3, is used with
control parameters p. = 0.5 and k = 12 for simulation. The simulation results
are shown in Figure 4.6. The controller cannot stabilize the system with linear

pr0portional gain greater than [:1 = 0.5, due to the limited control level. The non-

94

linear proportional gain, shown in Figure 4.5, is designed such that it is small when
the tracking error is large and increases as the tracking error becomes small. The
nonlinear pr0portional gain improves the settling time by more than 50%. The
simulation results in Figure 4.7 compares the performance of the controllers with
the nonlinear integrators ¢1(ea), ¢2(e1) with the nonlinearities shown in Figure 4.3
and the linear integrator when the nonlinear proportional gain k1(el) is used. The
controllers with the nonlinear integrators show similar transient performance im-
provement. The controller with the linear integrator, driven by the tracking error,
cannot stabilize the system with the same control parameters, since the buildup in

the integrator increases the uncertainty term.

4.4 Conclusions

In this chapter, we designed a nonlinear PID controller with nonlinear integral
and proportional gains for relative-degree-two systems. The nonlinear integrator
is driven by the augmented error or the tracking error. We proved regional and
semiglobal regulation for both schemes. The simulation results show that nonlinear
integrators driven by the tracking error achieve performance improvement similar
to integrators driven by the augmented error. The nonlinear proportional gain can
be designed to reduce the uncertainty term when the tracking error is large and
obtain faster error dynamics inside the boundary layer. Simulation results show

that the nonlinear proportional gain can be designed to reduce the settling time.

95

 

 

 

 

6 I I I I I I I I
5... ............................................................................................. .—
4,_ ............................... .1
9: 3L -
xv-
2r _
1-...... .. ........ _
l 1 l l l l l J L
-5 —4 -3 —2 -1 0 1 2 3 4 5
61

Figure 4.5: Nonlinear proportional gain k1(e1) for the simulation of the unstable
second-order system

96

 

 

I I I l I I
— nonlinear proportional gain
- - linear proportional gain ‘

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 1‘—-1———+-__.._1____1
4 5 6 7 8 9 10
I I I r I I
1 1 1 1 1 1
4 5 6 7 8 9 10
I I I I I I
1 1 1 1 l 1
4 5 6 7 8 9 10

 

Figure 4.6: Simulation results for the second-order unstable system with nonlinear
and linear proportional gains

97

 

 

 

 

 

 

 

 

 

 

4 I I I I I I I I I
\ _ nonlinear integrator driven by ea
2 _ _ _ nonlinear integrator driven by 61 _
> - - . linear Integrator
o l. — _ _
_2 l l l l l l l l l
O 1 2 4 5 6 7 8 9 10
2 I I I I I T I I I
.l
1 - _’ -
o I

 

 

 

l.
h--
_-
_
|—
_
I—
.—
-
-

 

 

 

 

 

 

 

0 1 2 3 4 5 6 7 8 9 10
I I I I I I
.l
3 _
_20 1 1 1 1 1 1 1 1
1 2 3 4 5 6 7 8 9 10
t

Figure 4.7: Simulation results for the second-order unstable system with linear and
nonlinear integrators

98

Chapter 5

Conclusions

This dissertation provides more freedom in designing controllers which regu-
late single-input single-output, input-output linearizable, minimum phase nonlinear
systems. The structure of the Universal Integral Controller [27] is extended by re-
placing the controller gains with nonlinear functions of the tracking error and its
derivatives, while preserving the regional and semiglobal asymptotic stability. A
key idea in our design of the nonlinear gains and analysis of the closed-loop system
is that the uncertainty due to nonlinear gains can be overcome by the sliding mode
control as long as the system is input-to-state stable inside the boundary layer. The
effects of the nonlinear gains on the performance of the controller are investigated
by simulation.

In Chapter 2, two possible schemes for nonlinear integration are considered:
a nonlinearity placed before or after the integrator. The study of the nonlinear
integrator reveals that, when the integrator is driven by the augmented error, the
dynamics of the closed-loop system inside the boundary layer are input-to—state
stable, which is an essential property for the analysis of asymptotic stability. The

closed-loop analysis shows that the Universal Integral Controller with the nonlinear

99

integrator stabilizes the system regionally and semiglobally. The nonlinear integra-
tor is designed to prevent the buildup during the transient period. In simulations,
the Nonlinearity Before Integration scheme achieves superior transient performance
over the Nonlinearity After Integration and the linear integrator schemes.

Further extension of the structure of the controller is investigated in Chapter 3.
The proportional and derivative gains are replaced by nonlinear functions of the
tracking error and its derivatives. By choosing the proportional and derivative gains
as nonlinear functions of the form k,- = k,(e,~), the dynamics inside the boundary
layer have the form of a Lure system, and Linear Matrix Inequalities are used
to obtain bounds on the nonlinear gains k,(e,-). Our analysis proves that, if the
LMI problem is feasible, the controller with nonlinear proportional, integral and
derivative gains achieves regional and semiglobal regulation. Simulation results
demonstrate examples where nonlinear gains are utilized to improve the transient
performance.

The Universal Integral Controller with nonlinear gains specializes to a non-
linear PID controller for relative-degree-two systems. In Chapter 4, more specific
results are provided for the nonlinear PID controllers, which are not possible for
systems with higher relative degree. We consider the nonlinear integrator driven
by the tracking error, which is the classical form of nonlinear integrator found in
the literature. The proof of stability is completed analytically, and shows that PID
controllers with the nonlinear proportional and integral gains achieves regional and
semiglobal regulation for minimum phase nonlinear systems with relative degree
two. Simulation results show that the nonlinear integrators, driven by the tracking
error or the augmented error, show better transient performance than the linear

integrator. The nonlinear proportional gain is designed to improve the settling

100

 

time by decreasing the uncertainty that the control input should overcome when
the system is far from the equilibrium point.

In this work, we considered the nonlinear proportional and derivative gains of
the form v,(-) = k,(e,-)e,- among the possible choice of v,- : v,(e1, - - - ,ep_1). Finding
a more general class of nonlinear functions that preserve the property of input-
to-state stability is a challenging but rewarding problem, as it will provide more
freedom in the design of the controller.

Application of the Universal Integral Controller with nonlinear gains will be
an interesting problem. The structure of the controller could be extended further
for a specific problem. For example, the proportional gain of the nonlinear PID
controller cannot be taken as k1 = k1(e1, e2), since it requires that

61:
[£(81,€2)€1 + 1] u ?,£ 0

which is not true in general. But the desired proportional gain In; and the operating

range of a certain application might satisfy the condition.

101

BIBLIOGRAPHY

[1] M. Abe. Vibration control of structures with closely spaced frequencies by a
single actuator. J. Virbr. Acous. Trans. ASME, 120(1):117—124, Jan. 1998.

[2] B. Armstrong, D. Neevel, and T. Kusik. New results in npid control: tracking,
integral control, friction compensation and experimental results. IEEE Tran.
Contr. Sys. Tech., 9(2):399—406, 2001.

[3] AN. Atassi and H.K Khalil. A separation principle for the stabilization of a
class of nonlinear systems. IEEE Trans. of Automa. Contr., 44:1672-1687,
1999.

[4] S. Boyd, L.E. Ghaoui, E. Feron, and V. Balakrishnan. Linear Matrix Inequal-
ities in System and Control Theory, volume 15 of SIAM studies in applied
mathematics. SIAM, 1994.

[5] CI. Byrnes and A. Isidori. Asymptotic stabilization of minimum phase non-
linear systems. IEEE Trans. of Automa. Contr., 36(10):1122—1137, 1991.

[6] H.Y. Chiu, G.B. Stupp, and S.J.Jr. Brown. Vehicle-follower control with
variable-gains for short headway automated guideway transit systems. J. Dyn.
Syst. Meas. Contr., 99(3):183-189, 1977.

[7] D.W. Clark. Pid algorithm and their computer implementation. Trans. Inst.
Meas. Contr., 6(6):305—316, 1985.

[8] E.J. Davison. The robust control of a servomechanism problem for linear
time-invariant multivariable systems. IEEE Trans. of Automa. Contr., AC-
21(1):25—34, 1976.

[9] CA Desoer and Y.T. Wang. Linear time-invariant robust servomechanism
problems: A self-contained exposition. In C.T. Leondes, editor, Control and
Dynamic Systems, volume 16, pages 81—129. Academic Press, 1980.

[10] F. Esfandiari and H.K. Khalil. Output feedback stabilization of fully lineariz-
able systems. Int. J. Contr., 56:1007—1037, 1992.

102

[11] HA. Fertik and R. Sharpe. Optimizing the computer control of breakpoint
chlorination. ISA Trans., 19(3):3—17, 1980.

[12] BA. Francis. The linear multivariable regulator problem. SIAM J. Contr.
0pt., 15(3):486—505, 1977.

[13] RA. Freeman and RV Kokotovic. Optimal nonlinear controllers for feedback
linearizable systems. In Proc. American Control Conference, pages 2722—
2726, Seattle, WA, June 1995.

[14] T Fukuda, F. Fujiyoshi, F. Arai, and H. Matsuura. Design and dextrous control
of micromanipulator with 6 d.o.f. In Proc. IEEE Int. Conf. Robot. Auto.,
pages 1628-1633, Sacramento, CA, Arp. 1991.

[15] G.Y. Ghreichi and J .B. Farison. Sampled-data pi controller with nonlinear
integrator. In Proc. IECON, pages 451-456, Milwaukee, WI, Apr. 1986.

[16] H. Huang. Error driven controller for electric vehicle pr0pulsion system. In
Canadian Conf. Elec. Comp. Eng., volume 2, pages 744-747, St. John,
Canada, May 1997.

[17] J. Huang and W.J. Rugh. On a nonlinear multivariable servomechanism prob-
lem. Automatica, 26(6):963—972, 1990.

[18] J. Huang and W.J. Rugh. Stabilization on zero-error manifolds and the nonlin-
ear servomechanism problem. IEEE Trans. of Automa. Contr., 37(7):1009—
1013,1992.

[19] A. Isidori. A remark on the problem of semiglobal nonlinear output regulation.
IEEE Trans. of Automa. Contr., 42:1734—1738, 1997 .

[20] A. Isidori. Nonlinear Control, volume 2. Springer, London, 1999.

[21] A. Isidori and CI. Byrnes. Output regulation of nonlinear systems. IEEE
Trans. of Automa. Contr., 35(2):131-140, 1990.

[22] Y Izuno. Speed tracking servocontrol system with speed ripple reduction
scheme for traveling-wave-type ultrasonic motor. Elect. Comm. in Japan,
81(9):1728—-1735, 1998.

[23] HS. Kay and H.K. Khalil. Universal integral controllers with nonlinear inte-
grators. In Proc. ACC, pages 116—121, Anchorage, AK, May 2002.

[24] HS. Kay and H.K. Khalil. Universal integral controllers with nonlinear gains.
In Proc. ACC, Denver, CO, Jun. 2003. To appear.

103

 

[25] H.K. Khalil. Robust servomechanism output feedback controllers for feedback
linearizable systems. Automatica, 30(10):1587—1599, 1994.

[26] H.K. Khalil. High-gain observers in nonlinear feedback control. In H. Nijmeijer
and TI. Fossen, editors, New Directions in Nonlinear Observer Design,

volume 244 of Lecture Notes in Control and Information Sciences, pages
249—268. Springer, London, 1999.

[27] H.K. Khalil. Universal integral controller for minimum-phase nonlinear sys-
tems. IEEE Trans. of Automa. Contr., 45(3):490—494, 2000.

[28] H.K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, NJ, 3rd
edition, 2002.

[29] NJ. Krikelis. State feedback integral control with ’intelligent’ integrators. Int.
J. Contr., 32(3):465-473, 1980.

[30] RI. Luo, R.D. Jackson, and R.J. Hill. Digital controller for thyristor current
source. IEE Proc. B Elect. Pow. App., 132(1):46—52, 1985.

[31] NA. Mahmoud and H.K. Khalil. Asymptotic regulation of minimum phase
nonlinear systems using output feedback. IEEE Trans. of Automa. Contr.,
41(10):1402—1412, 1996.

[32] Y. Ni. Application of a nonlinear pid controller on starcom with a difierential
tracker. In Proc. 2nd Int. Conf. Ener. Pow. Deliv., volume 1, pages 29—34,
Singapore, Mar. 1998.

[33] Z. Qu. Robust Control of Nonlinear Uncertain Systems. Wiley series in
nonlinear science. Wiley, New York, 1998.

[34] I.W. Sandberg. Global inverse function theorems. IEEE Tran. Circ. Sys.,
CAS-27(11):998—1004, 1980.

[35] H. Seraj. A new class of nonlinear pid controllers with robotic applications. J.
Robot. Sys., 15(3):161—181, Mar. 1998.

[36] H. Seraji. Nonlinear and adaptive control of force and compliance in manipu-
lators. Int. J.Robot. 1265., 17(5):467—484, May 1998.

[37] H. Seraji. Nonlinear contact control for space station dextrous arms. In Proc.
IEEE Int. Conf. Robot. Auto., pages 899—906, Leuven, Belgium, May 1998.

[38] S Shahruz and AL. Schwartz. Nonlinear pi compensators that achieve high
performance. J. Dyn. Syst. Meas. Contr., 119:105—110, 1997.

104

[39] SA. Snell and P.W. Stout. Quantitative feedback theory with a scheduled gain
for full envelope. J. Guid. Contr. Dyn., 19(5):1095—1101, 1996.

[40] ED. Sontag and Y Wang. On characterizations of the input-to-state stability
property. Sys. Contr. Lett., 24(5):351—359, 1995.

[41] A. Teel and L Pray. Global stabilizability and observability imply semi-global
stabilizability by output feedback. Sys. Contr. Lett., 22(5):313—325, May
1994.

[42] HF. Wang, F.J. Swift, B.W. Hogg, H. Chen, and G. Tang. Rule-based variable-
gain power system stabiliser. IEE Proc. Gen. Trans. Distr, 142(1):29—32, Jan.
1995.

[43] Y. Xu, D. Ma, and J .M. Hollerbach. Nonlinear proportional and derivative
control for high disturbance rejection and high gain force control. In Proc.
IEEE Int. Conf. Robot. Auto., volume 1, pages 752-759, 1993.

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3 1293 02461 87

l

  

‘m