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.2005

This is to certify that the
dissertation entitled

LOGIC-BASED SWITCHING CONTROL OF NONLINEAR
SYSTEMS USING HIGH-GAIN OBSERVERS

presented by

LEONID B. FREIDOVICH

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Ph.D. degree in Mathematics

 

 

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LOGIC-BASED SWITCHING CONTROL OF NONLINEAR SYSTEMS
USING HIGH-GAIN OBSERVERS

By
Leonid B. Fr'ez'dovz'ch

A DISSERTATION

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

DOCTOR OF PHILOSOPHY
Department of Mathematics

2005

ABSTRACT
LOGIC-BASED SWITCHING CONTROL OF NONLINEAR SYSTEMS
USING HIGH-GAIN OBSERVERS
By
Leonid B. Freidovz'ch

The design of robust output and state feedback controllers for practical track-
ing and stabilization for nonlinear systems with large—scale parametric uncertainty
is considered. We consider a class of single-input single-output systems that can be
transformed into a special form, where unavailable states (if any) are derivatives of
the measured outputs. We consider a large class of nonlinear systems that could be
stabilized via a Lyapunov-based technique if the level of parametric uncertainty was
much lower and if the unmeasured states were available for feedback. We propose and
investigate a new approach based on subdividing the set of parameters into smaller
subsets, designing candidate controllers for each subset, and implementing a logic-
based switching between them. We use the output of an extended-order high-gain
observer to substitute the unavailable states in the candidate controllers, check the
inequality for the derivative of the Lyapunov function and switch if it is not satisfied,
and to approximately identify the parameters. We discuss the issues of digital im-
plementation and measurement noise and illustrate our design procedure on several

examples.

To the memory of my mother, Elena Moiseezma Al’tman, and

to the memory of my first advisor, Anatolii’ Arkad’em'ch Pervozvanskti’

iii

ACKNOWLEDGMENTS I

I would like to express my deepest and sincerest gratitude to my advisor, Professor
Hassan K. Khalil, for his guidance, constant encouragement, and support, without
which this thesis would not have been possible. It is hard to overrate his help, insight
into Control Theory, or boundless patience. He has taught me not only to learn and
obtain results but also to interpret, test, and present them.

I would like to thank the members of my Ph.D. committee, Professor Sheldon
E. Newhouse, Professor Vera M. Zeidan, Professor Charles R. MacCluer, and Pro-
fessor Ranjan Mukherjee for their willingness to serve on the committee as well as
for their assistance, encouragement, and valuable input. I am especially thankful to
my co-advisor, Dr. Newhouse, for his insight into the theory and methods of Dy-
namical Systems and Differential Equations and to Dr. Zeidan for a comprehensive
introduction to Optimal Control Theory.

My former advisors, co—authors, and teachers from St. Petersburg State Technical
University, especially Professor Anatoly A. Pervozvanski, Professor Ilya V. Burkov,
and Professor Sergej F. Burdakov, gave me a foundation as a researcher, encouraged
my ambition, and directed my, often times misplaced, energt To them, as well as to
my instructors at Michigan State University, go my most heartfelt thanks.

It is also pleasure to acknowledge various helpful discussions on switching control
with Dr. Daniel Liberzon and on different control problems with Dr. Sridhar Sesha-
giri, Professor Cevat Gokcek, Dr. Konstantin E. Avrachenkov, Professor Alexander L.
F radkov, Professor Anton Shiriaev, Dr. Ahmed Dabroom, and Dr. Ilya Kolmanovski.

To my family, relatives, and long-time friends, especially to Leonid Slavin, Zeynep
Altinsel, Boris Yakovlev, Alexander Levin, and Denis Moskvin, I owe the maximum
gratitude, for years of their unflinching support, patience, understanding, and not

giving up on me at a rough time.

Contents

List of Figures ............................................. viii
1 Introduction ............................................ 1
1.1 Feedback Control Design ......................... 1
1.2 Lyapunov Functions ........................... 7
1.3 Uncertainty and Robustness ....................... 9
1.4 High-Gain Observer-Based Design .................... 14
1.5 Literature Review for Switching Control ................ 19
1.6 Overview of the Thesis .......................... 21

2 Sliding-Mode Control-Based Tracking of Minimum-Phase Non-

linear Systems .......................................... 24
2.1 Description of the Idea .......................... 24
2.2 Motivating Example ........................... 26
2.2.1 Robust control [22] ........................ 26
2.2.2 Adaptive control [21] ....................... 29
2.2.3 Logic-based switching ...................... 30
2.2.4 Simulation results ......................... 33
2.3 Problem Formulation ........................... 38
2.3.1 Model and Main Assumptions .................. 38
2.3.2 State feedback control for p E ’P“) ............... 40

2.3.3 High-gain observer for p E 73“) ................. 43

2.3.4 Switching logic .......................... 45
2.4 Main Result ................................ 45
2.5 Remarks .................................. 52

Lyapunov-Based State and Output Feedback Stabilization for a

Class of Nonlinear Systems ................................ 54
3.1 Introduction ................................ 54
3.2 Class of Systems ............................. 57
3.3 Switching Logic .............................. 60
3.4 Main Result ................................ 65
3.5 Examples ................................. 71
3.5.1 Linear non-minimum phase system ............... 72
3.5.2 Nonlinear system and adaptive control ............. 77
3.6 Remarks .................................. 80
Nonlinear Example: Case Study ........................... 83
4.1 Model ................................... 83
4.2 Continuous Controller Design ...................... 84
4.2.1 Lyapunov-based switching output feedback ........... 84
4.2.2 Lyapunov-based switching state feedback ............ 86
4.2.3 Scale-independent hysteresis-based switching logic ....... 87
4.3 Practical Implementation ......................... 89
4.3.1 Digital implementation ...................... 89
4.3.2 Suppressing measurement noise ................. 91
4.4 Simulation Results ..... A ....................... 92
4.4.1 Noise-free case ........................... 93
4.4.2 Acceptable level of noise ..................... 96

vi

4.5 Remarks .................................. 99

5 Conclusions ............................................. 101
A Listings of the programs .................................. 104
Bibliography .............................................. 121

vii

List of Figures

1.1

2.1
2.2
2.3
2.4

3.1

3.2

3.3

4.1
4.2
4.3
4.4

4.5

Pendulum .................................

Simulation results for the case of p = 5 ................
Simulation results for the case of p = —10 ...............
Performance under unmodeled dynamics ................

Performance under unmodeled dynamics and measurement noise . . .

Simulation results for y, = 0.5 : the dashed line is for adaptive control
and the solid line is for switching control .................
Simulation results for y, = 0.1 sin(t) : the dashed line is for adaptive
control and the solid line is for switching control. ...........
Simulation results for y, = 0.5 sin(t) : the dashed line is for adaptive

control and the solid line is for switching control. ...........

Improving pre—routed search: the case of z" = 10 ...........
Improving pre-routed search: the case of z" = 40 ...........
Noise-free performance: the case of z" = 30 ..............
Performance of the state feedback controllers without and with filters:
the case of z" = 20 and on = 10‘4 ...................
Performance of the state feedback controller without filters in the pres-

ence of noise of different levels: the case of z" = 40 ..........

viii

35
36
37

79

97

4.6 Performance of the controller with hysteresis-based scale-independent

logic without filters and with filters, having II" = 0.01 and no 2 4 :

the case of 2" = 40 and a0 = 1 ..................... 99
A.1 state-d.mdl, page 1 ........................... 105
A2 state_d.mdl, page 2 ........................... 106
A3 state_d.mdl, page 3 ........................... 107
A.4 state_d.mdl, page 4 ........................... 108
A5 state_d.mdl, page 5 ........................... 109
A6 state-d.md1, page 6 ........................... 110
A.7 state_d.md1, page 7 ........................... 111
A8 state-d.mdl, page 8 ........................... 112
A9 hgo1d.m (High-gain observer), page 1 .................. 113
A.10 hgold.m (High-gain observer), page 2 .................. 114
A.11 1bsl.m (Lyapunov-based switching logic), page 1 ........... 115
A.12 lbsl .m (Lyapunov-based switching logic), page 2 ........... 116
A.13 lbsl .m (Lyapunov-based switching logic), page 3 ........... 117
A.14 main- sf.m, page 1 ............................ 118
A.15 main- sf .m, page 2 ............................ 119
A.16 main- sf .m, page 3 ............................ 120

ix

Chapter 1

Introduction

In this thesis we consider some results in the field of mathematical theory of
nonlinear feedback control design. This chapter is included in order to make the
presentation self-contained and easy to read for a non-specialist in control. We start
with a brief overview of some classical problem formulations, traditional and modern
techniques. We define terminology and provide some references for the results to be
used latter in the main part of the thesis. Finally, in the last part of this chapter we
give a brief preview of our results. In addition, we identify the contribution and show

the location of our results in the broader picture.

1.1 Feedback Control Design

Let us, following [43, Ch. 1], start with a simple physical motivating example.
Consider the pendulum of Fig. 1.1, to which one can apply a torque as an external
force. Assuming the rod is rigid and has zero mass, the system can be described by

the second-order nonlinear differential equation

mld(t) + mg sin(0(t)) = u, (1.1)

 

mg

Figure 1.1: Pendulum

where 9(t) is the value of the counterclockwise angle with respect to the vertical at
time t , 0(t) = %0(t), m is the mass at the tip, I is the length of the rod, 9 is
the acceleration due to gravity, and u is the value of the external torque (counter-
clockwise being positive) at time t, that we can assign arbitrarily. We call u the
input or control.

Suppose our goal is to design a control law that stabilizes the pendulum at
the upward vertical position ( 6 = 1r and d = 0 ), i.e. to find a formula for u, the
torque that moves the pendulum into the upward position and keeps it there.

It is clear that, assuming that the pendulum is initially at rest with

the goal can be achieved in many different ways appropriately defining the control
u = k(t) (1.2)

as a function of time (a solution in this form is called a feedforward control or
precomputed control [43, Sec. 1.4]). Moreover, the behavior of the solution of the
system (1.1), (1.2) can be optimized in order to minimize the transition time (obey—

ing certain physical restrictions on the control magnitude) or power consumption and

to satisfy certain specifications of the provoked motionl. However, it is clear that
since the upward equilibrium for the unforced or open-loop system (with u E 0 )
is unstable (physically and mathematically in the sense of Lyapunov), if the initial
conditions for the system are slightly (actually, arbitrarily!) different from the as-
sumed values, the feedforward control law would fail, since absolutely no function of
time can change stability.

The stabilization problem can be solved in the more realistic case when the initial

conditions are not known in advance, i.e. for
(6(0).6'(0)) e a,

where Q C R2 is a known (typically compact) set, using feedback control, provided
the pendulum is equipped with a set of sensors. The sensors measure certain set of

signals in the system

y = h (an), 9(a) (1.3)

on-line, called the outputs, and allow us to define u in the form of either static
output feedback
u = k(t.y) (1-4)

or dynamic output feedback
u = k1(t,y, Z), Z = k2(t,y,2). (1.5)

In a special case, when the output y is precisely the vector of state variables ( l9 and

9 ), (1.4) is called static state feedback while (1.5) dynamic state feedback.

 

1This kind of problems is in the field of the Theory of Optimal Control, which naturally
complements the problem we are interested in this thesis.

Introducing the notation (and omitting dependence of t )

(III 6 — 7T I2
:1: = = , and f(1:,'u) =
1:2 6 (tr/m — gsi1(1'1+ 7r)) /1

we can rewrite (1.1) and (1.3) in the general form of the vector differential equation

with input and output

1% = f(;1:,u.) and y = h(;r), (1.6)

where x E R" is a vector of state variables, :i: = iii—f is a vector of time derivatives
of :r, t is time, 3; 6 IR“ is a vector of measurements (outputs), and u 6 IR’"
is a vector of controls (inputs) to be determined. If k = m = 1 , the system is
called single-input single-output [$180]; if k > 1 and m > 1, it is called multi-
input multi-output [MIMO]. Typically, f () and h(-) are locally Lipschitz vector
functions and 33(0) 6 f2 , with Q C R" being a given (often compact) set of initial
conditions, containing the origin in its interior.

We are ready now to define the stabilization problem: design a control law in
the form of either dynamic or static feedback [23, 19, 37], that guarantees that the
solutions of the closed-loop system, (1.6) and (1.5) or (1.6) and (1.4), respectively,

exist and are asymptotically small in certain sense. In particular, one might require

0 either practical stabilization: all solutions are uniformly bounded, and for
a given sufficiently small positive number 60, there exists T > 0 such that

||;r(t)]| _<_ £0 for t 2 T,

0 or asymptotic stabilization: :r(t) = O is a solution of the closed-loop
system, which is stable in the sense of Lyapunov [12, 41, 23] (for every 6 > 0
there exists 6 > 0 such that ||:r(t)|| S 5 whenever ”1(0)” 3 6 ) and attractive,

i.e. tlirn |].'r(t)|| = 0 whenever 3(0) 6 f2.

It is easy to see that feedback linearization-based [23, Sec. 13.3] static state
feedback

it 2 ml (%sin(:r1 + 7r) — 2.172 — 1'1) (1.7)

or dynamic output, with y = r1 , feedback

21. : ml (% sin(y + 7r) + 82 — 3y) and 23 = y — 32: (1.8)

transform the system (1.1) into a linear exponentially stable system
5131 + 2171 + (131 = 0

Of

z‘3)+32+32+z=0,

respectively, and, therefore, solve the problem of global ( Q = R2 ) asymptotic stabi-
lization.

Suppose now that, instead of asymptotic stabilization of the upward position, we
would like to achieve a more challenging task: we need the pendulum to follow the
predefined reference signal 9d(t) as close as possible. Assuming that this signal is

twice continuously differentiable, redefining the state variables as
3:1 = 9 — 9d(t) and x2 = 9 — 9d(t)
it is not hard to see how to modify (1.7) and (1.8). We can take

u = ml (910) + %sin(x1 + 9d(t)) — 2:132 — 2:1)

CJ'I

01'

u = ml (9,,(t) + %sin(y + 9d(t)) + 82 — 3y) and 2 = y — 32:

in order to ensure that the solutions for each closed-loop system exist and that
lim [9(t) — 9d(t)| = 0.
t—+oo

In general, for the system (1.6), we define a controlled output

yc = hc(;r)

a sufficiently smooth (continuously differentiable with a certain number of uniformly

bounded derivatives) reference signal r(t) , and a tracking error

6(t) = hC(flv(t)) - T0)-

The goal now is to design a static or dynamic state or output feedback control law

to ensure existence of the solutions of the closed-loop system with 3(0) 6 fl and

0 either practical tracking: all solutions are uniformly bounded and the track-
ing error is uniformly ultimately bounded, i.e. for a given pre-specified (suffi-

ciently small) 50 > 0, there exists T > 0 such that ||e(t)|| S 60 for t Z T,

0 or asymptotic tracking: all solutions of the closed-loop system are uniformly

bounded and tlim ||e(t)|| = 0.

A special case of tracking with a constant reference signal is called regulation and
often can be reduced to a significantly simpler problem of stabilization. We would like
to notice also that for some practically important cases the measured and controlled
inputs are the same.

It is clear that neither stabilization nor tracking problem can be solved without
additional structural assumptions about the functions f (~) and h(-) . Depending

on the particular assumptions, different design strategies have been proposed in the

literature [23, 19]. A few of them are to be described below after we briefly recall one
of the techniques that is usually used while proving that the goal is achieved or even

during the design stage.

1 .2 Lyapunov Functions

As soon as an appropriate control law is designed, the closed-loop system can be

rewritten in the form

X = F(t,x), x(0) E Q. (1.9)

If the control law is locally Lipschitz and the reference signal is ‘nice’2, then F ()
is locally Lipschitz in x and piecewise continuous in t. It is known, that if this
is the case, solutions exist locally, i.e. there exist T 6 (0,00] such that the tra-
jectories are well-defined for t E [0, T) (see, e.g. [23, Th’m 3.1]). However, we
need to show that the solutions of (1.9) exist globally (extendable to [0, co) and
satisfy certain properties, in particular, we often need to proof stability or asymp-
totic stability. For nonlinear systems, the most universal and popular approach for
this kind of analysis is Lyapunov technique, see, e.g. [23, 12, 41] and especially [23,
Th’ms 4.18, 8.4, 4.8 and 4.9]).

For illustration purpose we assume that F(-) is time-independent, which is usu-
ally the case for the problem of stabilization. The idea is to use an auxiliary scalar-
valued continuously differentiable function V(x) , called Lyapunov function, with

the following pr0perties:

o V(0) = 0 and there exists C E (0, 00] such that every set in the nested family

{X 6 1R" : V(x) = c} with c E [0, C] is compact;

0 there exists 01 6 [0,0] such that Q C {3: 6 IR" : V00 2 c1} and, for

 

2For example, 9.1(t) in the tracking problem for the pendulum example could be taken twice
differentiable with bounded piecewise continuous second derivative.

practical stabilization, there exists c2 6 [0, C] such that V (x(t)) 3 C2 implies

”XU)” g 60;

o the time derivative of V(x(t)) along the trajectories of the system, i.e.

- (9V
V = B—XF(X)’ is strictly negative for x inside the set {x 6 1R" : V00 3 CI}
but outside the set {X E R" : V(x) 3 C2}, with C; = O for asymptotic

stabilization.

There are several standard control design techniques for nonlinear systems [19, 23]
in the form (1.6). Some of the approaches (not only analysis) are based on some
constructive applications of the theory of Lyapunov functions.

As an example, let us consider a problem of asymptotic tracking a bounded con-

tinuously differentiable signal 1rd(t) for the system

We can start with x = a: — ird(t) , and take V(X) = )8. Computing derivative of
V , we obtain:

V = 2x(—:i:d(t) + f1(:r:) + u).

Naturally, the choice
it = k(t,:r:) E 234(t) — f1(:r:) —§

ensures V S —x2 and solves the problem.

In a similar way, the stabilization and tracking control laws presented above for the
pendulum example could be obtained as well (if we start with a quadratic Lyapunov
function for the corresponding closed-loop systems).

However, for most practically important problems the situation is complicated by

the, so-called, presence of uncertainty.

1.3 Uncertainty and Robustness

Returning back to the problem of stabilizing the upward equilibrium of the pen-
dulum system (1.1) it is worth noting that for a ‘real’ physical system a control law
like ( 1.7) is not possible to realize if the value of either the mass, m , or the length,
l , is not known precisely. Moreover, if the pendulum is a model for a one-link ro-
botic manipulator, so that m mostly corresponds to the mass of the object, hold in
the grasp, it may vary significantly. A more realistic assumption is that the system,

instead of (1.6), should be described as

j: = f(p,13,’tt), y = h($)1 (110)

where p is a vector of unknown parameters of the model that are known to belong

to a given compact set ’P . In particular, for (1.1), we can take

1 1 ml 1:2
PM ll w l-

p2 —g/l plu + 132 sin(rl + 7r)

and, instead of (1.7), apply
u = (—]32 8111(1131 + 71’) _ 2172 - (ED/131a

where 13 is a vector of nominal values for the parameters. It is not hard to show,
that if the set of parameters 1’ is sufficiently small, this control law will still work,
however the corresponding modification for tracking of an oscillating signals would
lead to practical tracking (not asymptotic). In addition, if ’P is not small enough
or if practical tracking is not acceptable, the design must be done differently and the
whole mathematical problem should be reformulated.

In general, the function f () cannot be assumed known and therefore the designed

control law must work not for one specific model but for a certain class of models in

the form of (1.6). If this is done, we say: the control law is robust with respect to
the corresponding class of uncertainties. The main part of this thesis is devoted
to a problem of feedback control design in the presence of parametric uncertainty:
we will assume that the class of systems can be described by (1.10).

There are three basic techniques, dealing with uncertainty:

0 Adaptive control design, see e.g. [2, 18, 26], [42, Ch. 8], is based on the idea
of parameter identification. First, we design a control law for each possible
set of the parameters, e.g. in the form u = k1(t, p, y). Second, we design
an adaptation or identification law, e.g. 15 = k2(t,p, y), sometimes aiming to
achieve parameter convergence £11120 || 15(t) — pl] 2 0 and sometimes not. Finally,

the control is implemented as
’U, = kl(taf§iy) and IS: k2(tafivy)

o Robust control design, see e.g. [40, 47], [23, Sec. 14.1 and 14.2], [42, Ch. 7],
[30] and references therein, is based on the worst-case scenario. We do not
employ the control law, designed for the known-parameters case. We attempt
to find a control law that would work for all possible values of the parameters
simultaneously relying only on the knowledge of the whole set of parameters.
Inparticular, sometimes analyzing the derivative of a Lyapunov function, we
can choose the control law to dominate over all the uncertain terms instead of

canceling them as in feedback linearization-based approach.

0 Switching control design is similar to the adaptive design except for the
choice of the parameter identification law. Instead of adjusting the estimate of
the parameters continuously, according to a differential equation, we change it
abruptly at certain switching times if certain conditions are satisfied. Since a

contribution to this approach is the subject of this thesis, we give a more accu-

10

rate description, literature review and references below in a separate section.

To illustrate the ideas, let us consider the problem of asymptotic stabilization for

r=p+u. UJU

For adaptive design, we start pretending that p is known. In this case, the control

law
u _ _ :1:
— 2
would solve the problem since the closed-loop system would be :i: 2 —g— and asymp—

2

totic stability can be shown by the Lyapunov function V = :1: (applying, e.g. [23,

Th’m 4.9]). Since p is unknown, we use an estimate, p , for the parameter:

NIH

Let us define the estimation error: 13 = p — p. Augmenting the Lyapunov function

x2 , used above, with a quadratic term in p we obtain
V(:r, p) = $2 + p2 and v = —:r:2 — 215(1'5 — 1:).

It can be shown that the adaptation law

'6):
II
a.)

2 and ensures asymptotic stabilization. For a more general non-

results in V = —a:
linear system we typically employ a more advanced tool like Barbashin-Krasovskii’
Theorem [23, Th’m 4.2], or LaSalle’s Theorem [23, Th’m 4.4], or Barbalat’s Lemma-
based stability Theorem [42, Lemma 4.3], [23, Th’m 8.4].

To regulate the system ( 1.11) using robust design, we can use the same Lyapunov

11

function, once again, and compute the derivative

V(.'L‘) = r2 and V = —.’L‘2 + 2J5 (u + p + g) .

It is easy to see that since the set of parameters, ”P , is compact we can take3

a: .
u : —§ — Sign{I} SUPilPlia
p679

to obtain V S —:r:2, which ensures asymptotic stability“.

Known robust and adaptive techniques are not always applicable or practical in
the case when the set of possible values of the parameters is large. We will illustrate
what is meant by ‘not practical’ below, in the next chapter, via simulations. In order

to see some difficulties let us consider the system
i=pm

where pE’P with PD{—1,l} and P25{0}.

For the design in the case of known parameter we can proceed as above and obtain

that
a:
u = ——
2:0
stabilizes the system. Let us consider now
V = r2 + 152

 

3 . _ :r, if :r 2 0
Here srgn{:r:} _ { —:r:, otherwise
4It is easy to see that the trivial solution exists when 1(0) = 0 and the right-hand side of the
closed-loop system is locally Lipschitz otherwise.

12

and attempt to derive the adaptation law for

:r
u :- ——
213
We obtain
1) $9 p arl V 2(°+~L) .12 2~ f+$2
:r=—— —————:r: l( = 51:3: =—‘— —.
2p 2 2;; pp p p 213
We notice, that although the choice
'. _ 132
P — 215
would ensure V = —:r2 g 0, not only stability but even global existence of the

solutions cannot be argued since the right-hand is undefined at p = 0. Moreover, if
2(0) 79 0, 15(0) > 0, and p < 0, then p(t) must be monotonically decreasing as
t —+ 00 , is approaching the bad point and, clearly, p(t) 74» p .

Trying to apply robust design, we also run into some obstructions. Proceeding as

above, we obtain

V(:z:) = 3:2 and V = 2pxu.

It is clear that there is no choice for the control in a form u = k(t, at) that would

guarantee

v = (22rk(t, x))P S 0

simultaneously for all p E ’P, since the sign of p is undefined.

From an intuitive point of view, the complexity of the adaptive and robust designs
here is due to the unknown direction of control (whether positive u would result in
increasing or decreasing of :r(t) depends on the value of the parameter); this situation

is called unknown high frequency gain.

13

Let us assume that P = {—1,1}. We know that u = —a:/2 would work for
p = 1 and u = r/ 2 would work for p = —1. It is also obvious that if the wrong
control law is applied, a:(t) monotonically approaches either 00 or —oo. Assuming
that this ‘bad behavior’ can be identified (just check whether :r:(t) is positive and
increasing or negative and decreasing) it seems reasonable to suggest the following
control strategy. Put one of the control laws into the loop and wait. As soon as
escaping to too is identified, switch to the other control law. This control strategy
is an example of switching control design. In general, the most challenging part of
this approach is the design of the switching logic, i.e. identifying that the current

control law does not work and choosing the one to switch to.

1.4 High-Gain Observer-Based Design

Now we are going to present another technique that is useful for the stabilization
or tracking problems when some of the state variables are not available and therefore
must not be used in the control signal computation.

Consider again the stabilization of the upward position for the pendulum system
(1.1). We would like to derive a dynamic output] feedback control using the output
y = 3:1 as an alternative to (1.8). Our new control law will be built on a known static
state feedback control law5. Such a procedure is called observer-based design.
Had both states I, and 1:2 been available, we would have used (1.7). Since 3:; is

not available, we may obtain its estimate, 5:2, and use the feedback control law

u=ml(%sin(y+7r)—2ai:2—y).

 

Since 2:2(t) E y(t) = lim y(t) ’ y(t “ h)

h 0 h a a possible way to obtain an estimate for 2:2

 

5For simplicity of presentation we consider here only the case of known parameters.

14

is to use an Euler’s backward difference formula (assuming y(t) E 0 for -—e g t < 0 )

)2 W) -y(t -8)

:i:. t
.( E

 

7

where 5 > 0 is sufficiently small.

It is clear that for t 6 [0, e) the estimate obtained with this formula is not reliable
and huge (i2 = y/e for t < 5) since 8 is small. Such behavior, which is called
the peaking phenomenon, might provoke unacceptable transient and even finite
escape time in a more complicated system.

An alternative approach is to use a high-gain observer formula

; . 2 —:i?1 ; —i1
11:32+—(y————) and $22y
e 52

 

with arbitrary initial conditions and small 5 > 0. Defining the scaled estimation

[’71] [(y—iil/5]
’7: = . .
772 31-1172

we can rewrite the dynamics of the observer as

[”l l4 l [ml [0]
E = +5 and 532:3:2—172.
7'72 “1 0 772 y(t)

Now it is reasonable to believe6 that, if y(t) is uniformly bounded and e is suffi-

CI'I'OI'

ciently small, then 172(t) vanishes and 132(t) approaches 232(t), possibly up to an
error of an order of 0(5) 7. We note that 77(0) = 0(1/5) and hence peaking may
occur here as well. However, for either of this two schemes of estimating derivative,
the system can be protected by saturating the control signal outside of the region

of interest, based on the given compact set of initial conditions, as suggested in [7].

 

6See [23, Sec. 14.5.1] for further motivation.
7We write 5 = 0 a if lim f—(2 is finite.
f( l (9( )) 5—.0 9(5)

For a special case of the problem of output feedback control design, when the
missing (unavailable) state variables are derivatives of the outputs we can proceed in
a way similar to the presented above [3], [23, Sec. 14.5].

Consider the systemg

zitl Al‘l + B¢§(;rl,.r2, u) yl Cr]
2 and y = = , (1.12)
:32 ¢’(I1,$2,U) y2 h2‘(1131,~”l?:2)

where
"0 1 0‘ '0-
0 0 1 0 0
A: , B: , C=[1 0 0]
0 0 1 0
_0 0. _1‘

 

 

 

 

are matrices of appropriate dimensions, ¢(-), W'), and h2(-) are locally Lipschitz,
vanishing at the origin nonlinear functions.

The control design can be done in two steps. First, we derive a globally bounded
in $1 (saturated outside of the region of interact), locally Lipschitz dynamic partial

state feedback control law

u = k1(:r1,y2, z) and 23 = k2(:r1, y2,z). (1.13)

Then, we use a high-gain observer

13']: A571 + B¢0(i1,yg,U) + H(y1— C(31), (1.14)

 

8For simplicity, we present here only a simplified version of the result from [3].

16

where (but) is a globally bounded in i1 model for cf)() ,

H:

e 52 5"1

 

[0'1 02 (in, [T
7

e > 0 is a small parameter, and o ’s are chosen so that the roots of
s"1 + 013’”—1 + - - - + (in, = 0
are in the open left-half plane. Finally, we use the control law
u = k1 ($231,112, :5) and i = k2(:i:1, y-z, z). (1.15)

Under certain additional technical assumptions, it is shown in [3] that the dynamic
output feedback control law recovers the properties of the partial state feedback con-
trol law, provided 5 is sufficiently small. In particular, it is shown that the trajecto—
ries of the systems with these two control laws put into the loop are asymptotically
close to each other.

Following [7, 3], in order to analyze a system with a high-gain observer, we make

the change of variables

77 = ID(5)l-l($1- 131),

where

0(5) 2 diag[e""l 6"“? 1],
to transform the system (1.12), (1.15), (1.14) into the form

it = F(€.x,n) and 61'? = 4077 + €G(€,x.n), (1-16)

where F () and G() depend on e regularly and are globally bounded in 77, A0 isa

Hurwitz matrix, whose eigenvalues are exactly the roots of s"1 +len1-l+. - +01", 2 0,

17

x(0) = {1(5) = 0(1), and "0(0) 2 £2 (5) z 0 (871114) . This system is a singular

 

perturbation of the system (1.12), ( 1.13), called slow-subsystem, which could be

written as

XC : 17(0) XCan) (1.17)

with chO) 2 61(0). It is easy to see that the new variables X and 77 change with
different rates and so the motions'in the system can be divided into slow and fast.
Had {2(0) been 0(1) , the system (1.16) would have been in a standard sin-

gularly perturbed form [23, Ch. 11]. However, using

0 the Lyapunov function W = nTPOr) for the fast-subsystem

fie : A0770

where P0 = POT is a positive definite solution of the Lyapunov equation

AOPO + POAg‘ = —I,

o arguments, similar to the ones used in the proof of continuous dependence of
the solutions of a (regularly perturbed) system on parameters [23, Th’m 3.5],

and

o boundedness of F () and G() in 17

it can be shown that there exists T(5) > 0, such that [13(1) T (5) = O and x(t)
stays of the order 0(1) for t 6 [0, T (5)), while 17(t) decays to 0(5) -values during
this period. After this transient period, that corresponds to peaking, the system is,
basically, in the standard singularly perturbed form and could be analyzed using the
composite Lyapunov function V(x) + :WM), where V(XC) is a Lyapunov function9

for the slow subsystem (1.17).

 

9In [3] and [23, Ap. C.23] existence of this function is argued using a converse-Lyapunov-function
Theorem [23, Th’m 4.17]; note, however, that it is a natural assumption if the state feedback design
is done based on a Lyapunov function as above.

18

1.5 Literature Review for Switching Control

Very recently, switching control design attracted attention of many researches
in mathematical theory of control. The pioneering work by Morse [34] introduced
supervisory control design with dwell-time logic for regulation of uncertain linear

systems. We will follow the key starting point of this approach:

split the 'set of parameters into a finite (or infinite) family of smaller
compact subsets and design a family of control laws, each to work under

the assumption that parameters are in the corresponding small subset.

The idea has been developed later for the control of a fairly general class of linear
systems by many researches; see, in particular, [35, 17] and, especially, [28] and the
references therein. After the families have been designed we put one of the controllers
into the loop and wait to observe the behavior of the closed—loop system. It is obvious
that if switching of the control is not allowed faster than a certain fixed period, called
dwell-time, then the solutions are globally defined and that if the right controller is
put into the loop, then the tracking error must vanish exponentially and sufficiently
fast ensuring that the tracking problem is solved. What is not trivial to notice is that
if the control laws are designed appropriately then vanishing of the tracking error
ensures that all the other signals are bounded. It has also been suggested in [34] to
compare the output of the real system with the outputs of a family of models of the
system, corresponding to each subset of parameters. The switching in [34] is based on
using such output predictors or multi-estimators to compute (on-line) performance
indices for all models and then choosing a controller that corresponds to the smallest
index as soon as the dwell-time is over.

An alternative switching logic has been recently used in [4], refining an idea from
[11]. Locally robust candidate controllers are designed and arbitrarily ordered, then

pre—routed switching strategy is used based on checking whether the Lyapunov func-

19

tion has decreased by a certain percentage during a small pre—fixed period of time.
Extensions of the switching logic ideas to nonlinear supervisory control design is

still in an early stage. To the best of our knowledge, only a few results are known for

continuous-time nonlinear systems [14, 15, 16, 28, 36]. These papers, in particular,

make basic assumptions, similar to the following ones from [15]:
(i) the set of parameters is finite;

(ii) for each choice of parameters an input-to—state10 (or integral input-to—state)
stabilizing output feedback controller with a known gain function should be

available;

(iii) the multi-estimators are designed so that for each value of the parameters, the
interconnection of the real system and each model is integral input-to state (or

input-to-state) stable with respect to the corresponding estimation error.

It should be noticed also that in these papers, except for [36], the authors employ
the scale—independent hysteresis-based switching logic, originally introduced in [16].
In this logic the dwell-time period is not constant and is not specified in advance.
Instead of assuming that it is fixed and sufficiently small (so that, in particular, no
finite escape would be possible during the time when wrong controller is put into the
loop) the authors of [16] have suggested to switch when one of the performance indices
is smaller than all the other by a certain percentage. Several important nonlinear
examples have been worked out, but characterization of a class of nonlinear systems
satisfying the assumptions above is still an open problem.

A different switching strategy, which is similar to the one we are going to use, has

been used for the problem of state feedback regulation of an uncertain discrete-time

 

10The notion of input-to—state stability has been introduced by Sontag. Roughly, it means
that all the states are small whenever the input signal is small (see, e.g. [23, Sec. 4.8] for a precise
definition). Integral-input-to—state stability means that the states are small whenever input signal
is small not point-wise but in integral sense.

20

system in a recent paper [1]. One of the main assumptions in [1] is that the Lyapunov
function and its incremental difference over one period, for each candidate controller,
are independent of the unknown parameters and so are available for computation.
Moreover, the origin is an equilibrium point independent of which candidate controller
is in the loop. For the problem treated in this thesis neither the Lyapunov function nor
its derivative is assumed available and we use a high-gain observer to estimate several
derivatives of the output in order to obtain estimates of the Lyapunov function and
its derivative. When specialized to the regulation case, our problem does not require
the origin to be an equilibrium point under all controllers. Also, let us remark that
in our continuous-time formulation we have to deal with a possible finite escape time.

Lyapunov function and stability-based switching strategies were also used in non-
linear systems in a different situation [6, 27, 29, 31, 38]. There, the state space,
not the set of parameters, is partitioned into smaller subsets equipped with local or
regional candidate controllers and Lyapunov function candidates. We would like to
mention also [25], where switching logic is combined with a design that uses a robust

control Lyapunov function.

1.6 Overview of the Thesis

The goal of this thesis is to develop a systematic procedure for switching control
design and analysis, applicable for a large class of nonlinear uncertain systems.

In Chapter 2, published in [9], we consider a single-input single-output minimum-
phase nonlinear system with large parametric uncertainty. Roughly speaking, the
minimum phase property means that when both input and output are small, all
other signals are asymptotically small as well; see [23, p.517] for a precise definition.
We assume that the system can be represented globally in the normal form; i.e.,

after an appropriate global change of coordinates it can be written as a chain of inte-

21

grators followed by a nonlinear affine in control equation and that defines the so—called
zero dynamics; see [19, Sec. 4.1]. Our goal is to find a dynamic output feedback
control law to ensure that the output (practically) asymptotically tracks a bounded
smooth reference signal. Earlier work used high-gain observers with saturation to
derive adaptive as well as robust control laws for this problem. The-adaptive con-
trol law requires the nonlinear functions to be linearly parameterized in the unknown
parameters and could have unsatisfactory transient performance for a large parame-
ter set. The robust control law is based on a worst-case design and could be overly
conservative. High gain feedback is needed to implement both controllers in the case
when the set of parameters is large. As a result, the robust and adaptive controllers
may perform poorly in the presence of unmodeled dynamics and measurement noise.
In order to reduce the controller gain and improve performance we propose a new
approach based on partitioning the set of uncertain parameters into smaller subsets.
Robust control laws are designed for each subset and logic based switching is used
to choose the appropriate control law. The switching rule uses an estimate of the
derivative of a Lyapunov function, which is provided by a high-gain observer.

In Chapter 3, published in [8], we generalize the technique, presented in Chap-
ter 2, and apply it to a wider class of nonlinear systems and more general Lyapunov-
function—based state and output feedback control designs instead of the restriction
to sliding mode control and output feedback as in Chapter 2. It is worth noting,
that, in particular, we require here neither the sign of the high-frequency gain to
be known nor the system to be minimum-phase. The key idea is the same: split
the set of parameters into smaller subsets, design a controller for each of them, and
switch the controller if the derivative of the Lyapunov function does not satisfy a
certain inequality, after a dwell-time period. However, we do not order the candidate
controllers in advance, as earlier. Instead, we use estimates of the derivatives of the

states, provided by an extended order high-gain observer, to calculate instantaneous

22

performance indices. When the controller is falsified, we switch to a new controller
that corresponds to the smallest index among the controllers that have not been fal—
sified yet. This modification is important when the number of candidate controllers
is high and pre-routed search may lead to an unacceptable transient performance.

Chapter 4 is devoted to study some practical issues of the developed Lyapunov-
based switching control design strategy via numerical simulation. We follow the the-
ory, presented in the previous chapters, to design output and state feedback controllers
for a regulation problem for an example taken from [13]. The example is a second
order nonlinear system in strict feedback form with an unknown high-frequency gain
with finite set of uncertain parameters. Since most of the practical controllers are
implemented digitally, we develop a discrete, sampled-data versions of our dynamic
regulators and test the performance of the closed-loop systems in ideal (disturbance
free) conditions and in presence of measurement noise. For the state feedback case, the
regulation problem for this system has been solved previously using scale-independent
hysteresis-based logic, proposed in [16]. We review this alternative approach, follow-
ing [13, 15], and compare performance under different controllers in the loop. We show
how to reduce the influence of sensor noise with the help of a low-pass Butterworth
filter and analyze the acceptable level of noise.

Finally, we conclude with some remarks and provide directions for future research

in Chapter 5.

23

Chapter 2

Sliding-Mode Control-Based
Tracking of Minimum-Phase

Nonlinear Systems

We consider a single-input-single-output (SISO) minimum-phase nonlinear system
that can be represented globally in the normal form. In particular, we are interested in
an 8180 affine in control version of the model used in [3], with additional, sufliciently
smooth, bounded additive input and output disturbances. The model may depend
nonlinearly on a finite number of uncertain parameters. We assume that the uncertain
parameters, as well as the initial conditions, belong to known compact sets, which

could be large.

2.1 Description of the Idea

We are interested in the case when known methods of robust and adaptive control
fail to achieve satisfactory performance because available control efforts are restricted
while the compact set of the unknown parameters is very large. This set is to be

partitioned into a finite number of significantly smaller subsets. For the case when

24

the parameters are assumed to belong to one of these subsets, we design robust
output feedback control. Just to be concrete, the control law is based on continuous
sliding-mode state feedback [23, sec.14.1], implemented using a high-gain observer
[23, sec. 14.5]. A candidate controller is designed for each parameter subset such that
the derivative of a Lyapunov function satisfies a certain inequality. The idea is to
apply one of the controllers over a small pre-fixed period of time (called dwell time);
then check whether the estimated derivative of the Lyapunov function, calculated
using an extended order high-gain observer, satisfies the inequality. If the inequality
fails, we switch to the next controller. We make the observer’s dynamics sufficiently
fast so that the observer peaking occurs within the dwell time, whereas the latter is
shorter than any possible finite—escape time. By the above construction, there always
exists a control law that ensures that the corresponding inequality holds and thus,
as soon as it is chosen, no further switching of the control would be done; hence,
the switching period is finite. Due to the conservatism of the analysis, there is no
guarantee that the controller, designed for the actual small parameter subset, will be

used; however trajectories will still converge with the desired rate.

The rest of the chapter is organized as follows. We continue with further expla-
nation of the main ideas of the design for a motivating example. Then, the class
of systems of interest is formally introduced and state feedback control is designed
and partially analyzed. We proceed with the main result for the problem of output
feedback tracking, followed by a stability proof. We conclude with some remarks and

discussions.

25

2.2 Motivating Example

To show the idea and associated technical difficulties, we start with a simple moti-

vating example. Consider the parametrically-uncertain controlled Duffing’s equation

a + as + (pi/)3 = u, (2-1)

where p is the unknown parameter, y is the output, it is the control input, and
our goal is to asymptotically (practically) regulate y(t) to zero. We assume that
p belongs to a given compact set P = [—10,10] and the initial states (y(O),y(0))
belong to another known compact set 0 C R2. Had p been known and y been

available to measure, the simple feedback-linearizing regulator
u = py + (pg/)3 — ks/u, where s = 2y + y,

with positive k and p chosen by pole-placement techniques, would have guaranteed
that the origin of the closed-loop system is globally exponentially stable. Moreover,
performance issues would not be hard to address from the perspective of linear control
theory. To deal with parametric uncertainty, we use robust and adaptive control
designs from [22] and [21], respectively. Then, we introduce the new logic-based

switching design.

2.2.1 Robust control [22]

We use the continuous sliding mode control law

u = ,3, +(13y)3 — k Sat(8/rt)

26

where p is an estimate of p, Sat(-) is a smooth saturation function that is bounded,

continuously differentiable with bounded first derivative, and satisfies1

.3 - Sat(s) 2 s2 if [3] S1
(2.2)

s - Sat(s) Z [.9] if [9] > 1

and p, k are positive constants to be chosen to guarantee that the trajectories con-
verge to a positively invariant set where |s| S u. Towards that end, consider the
Lyapunov function candidate V(y,s) = 32 + 4y2. Suppose Q C {V(y,s) S 16} so
that V S 16 at t = 0. To guarantee a certain exponential convergence rate, we
require the inequality V + 2V S O to be satisfied along the closed-loop trajectories.

It is not hard to see that
V + 2V S — (216/11 - 6) 32 - 81/2 + 2|8y|(I13 — r2|+|153 — PHI/2),
provided |s| S u and
V + W S -|SI[2k — 618l- 2|y|(|15 - PI + I133 - P3|y2)l,

provided ls] > p. In the set {V S 16} we have Is] S 4 and [y] S 2, and it is not

hard to show that V + 2V S 0 if
k/u 2 3 + (SUP{|15 -10I+4l153 -p3|= m3 E 7’})2/16

and

k212+2sup{Ip—p|+4lp3—p3|: 1943519}.

With the choice 13 = 0, we need k/u 2 100500925 and k 2 8032. In this

design, a very high controller gain is needed to ensure convergence when the level of

 

1 Sat(s) = (l/tanh(1))tanh(s), Sat(s) = 2s/(1 + Isl), and Sat(s) = tan-1(2s) are examples
of such function.

27

uncertainty is high. A possible choice for the parameters is k = 8500 and u = 0.008.

Finally, we use the output feedback control law

u = [93/ + (13y)3 — isms/i.) with s = 2y + t, (2.3)

where y = 5:2 is the output of the second-order high-gain observer (HGO):

E1=i2+h1(y—Cfil), $32 =h2(y—.'f31), (2.4)

in which h1 = 2/5, h; = 1/52, and 5 > 0. The HGO parameter 5 is to be chosen
sufficiently small. Using simulation, we choose 5 = 10*. It is worth noting that
to make the performance under output feedback close to the performance under the
(unimplementable) state feedback, the HGO dynamics must be made sufficiently fast
relative to the dynamics under closed-loop state feedback. The latter dynamics are
not slow because of the high controller. gain. Consequently, a very small value of
5 > O is required when the uncertainty level is high.

It is well known that having too small values for u and 5 as well as too high
values for the gain It results in poor performance in the presence of measurement noise
and unmodeled high-frequency dynamics. It is important to notice that in the case
when there is no uncertainty (p z p) the preceding analysis would require k/u Z 3
(compared to 106 ) and k 2 12 (compared to 104. ) Moreover, as explained above,
smaller values of k/p _>_ 3 would allow us to use larger values for 5. Clearly, the
huge difference is due to the high uncertainty level and the worst-case design strategy.

We show next that adaptive control suffers from the same drawbacks.

28

2.2.2 Adaptive control [21]

We set p1 = p and p2 2 p3, so that the dynamics are linear in the parameters,

v+p1y+p2y3 =u

and (phpg) E P = [-10,10] x {—103,103] C R2. Following [21] we augment the
Lyapunov function as

W = V+l5i/’Yl “Hg/72,

where 71 and '72 are positive adaptation gains, p1 = 13, —p1, p2 = 132 —p2, and pl,
132 are estimates of the parameters. To guarantee that the inequality V + 2V S 0 is

satisfied when p, E 0 and 132 E 0, we require

W + W = —8y2 + 23(3s - ply — P2113 + u) + 2mm + 225252/72 s 0.

Assuming first that s is available, we take

u = 211(3, :9) = -2003 + 131.1! + 1523/3,

then W + 2V S -8y2 — 39432 + 2131(7133/ + 1230/71 + 2132(7gsy3 + jag/72.

We define the adaptive law with parameter projection:

7r¢i(')[1+(bi “El/5i if 152' > bi and ¢i(°) > 0
13.: mat-O [1 + (:3.- + bi)/5l if 23.- < -—b.- and at) < 0 (2.5)

7r¢i(') otherwise

where i=1 or i=2, b1=10, b2=103,

¢l(31y) : _Syr and ¢2(S,y) : —Sy3a

29

to ensure that W + 2V S 0 and the parameters do not leave the set

735 = [‘10 —(5, 10+ 5i X {—103 _ 6’103 +6]’

where 6 is a small positive number. To use the estimate s, provided by the second—
order HGO, we need to protect the system from the destabilizing effect of peaking
by saturating both the control and the adaptive law outside the region of interest.
Assuming that initially (at t = 0) pi = 0, p2 = 0, V S 16, and (p1,p2) E P,
we obtain that W S Mg, where ME > 16+ 102/71 + 106/72. Hence, |s| S M5,

lyl _<_ M3/2, and we redefine

¢1(') = *A’II Sat (éy/MI) , ¢2(') = —M2 Sat (593/942),
u = ’I/}() = Mu Sat([—200.§ + 1513/ + fi2y3]/Mu),

(2.6)

where M1 = Mg/z, M2 = Mg/8, and M1‘ = 205Ms +125Mg. It is worth not-
ing that the large values for the saturation levels is the result of the high level of
uncertainty (see [10] for more on this issue) and is an unwanted feature.

Using simulation, we choose 71 = 10, '72 = 103, and 5 = 10‘“. A too small
value of 5 is needed due to the same reasons as in the case of robust control and is
undesirable.

We are ready to present an alternative way to deal with this situation that allows
us to keep the controller and observer gains much smaller in order to achieve better
performance in the presence of measurement noise and unmodeled high-frequency

dynamics.

2.2.3 Logic-based switching

To reduce the level of uncertainty we adjust the parameter estimates on-line using

logic-based switching. Let us consider a covering of the compact set P by a finite

30

number of comparably small compact subsets {P(’f)},’: 1- We follow the analysis of

our robust control to design N control laws

a“) = 9% + (13“)yl3 - k.- Sat($'/u.-)a (2-7)

where p“) E P“), and 16,, u,- > 0 are chosen under the assumption that p E P“).
The next step is to define an algorithm to identify an appropriate regulator and
a corresponding rule to switch, among these N control laws on—line. Let r > 0
be a small fixed constant. Suppose that the control law u = u“) is applied for
to S t < to + 1'. For t 2 to + r we would like to check whether the inequality
V + 2V S 0 holds. If not, then p ¢ P“) and we should switch to u = u‘i“).
Again, we ‘dwell’ on this controller for some time (that is r) and then check if the
Lyapunov function is decreasing sufficiently fast to decide whether to switch to the
next candidate or not. This logic avoids problems with chattering and infinitely fast
switching. By choosing 7' small enough to ensure that the solution does not leave a
certain compact set, we are guaranteed that the solution is well defined for all t Z to.
In addition, it is obvious that switching has to stop in a finite number of steps.

However, to apply this logic, we need to estimate V and V on-line. Noting that
V + 2V = 2s2 + By2 + (8y + 43);; + 233;,

we estimate y and y by :32 and 53:3, respectively, which are provided by the third-

order HGO:
551: 532 + knit! ‘ 131), 1£2 = 533 + h2i(y — 531), 5373 = had?! - 531), (2-8)

where h1,- = 3/5,, h2, = 3/53, and h3,- = 1/5?, and 5, > O is sufficiently small. The

corresponding estimate for s is given by s = i2 + 2y. This observer ensures that the

31

differences (:52 — y) and (32:3 — y) decay exponentially to 0(5,) values within a short
peaking period T0 of the order of 5,- log(5,-) [3], following, possibly, each switching.
During this period, the system is protected from destabilizing effect of peaking since
the control law is globally bounded in :82.

For switching we check the inequality
V + 2x? = 2s? + 8;;2 + (8y + 49:22 + 25a, 3 a0 (2.9)

where a0 > 0 is a small parameter included to deal with possible non-vanishing
observation errors. Note that we have to wait for the peaking period To to use the
observer outputs in this inequality. Therefore, 7‘ must be greater than To. To finish
the design, we need to determine N, p“), In, fig, 5,, and r.

‘ Let us try to reduce k and increase u relative to the robust control design of
Section 2.1. For k.- = 200 and u,- = 0.36, we need to split the set of parameters into

several subsets {P(‘)}[’;0 such that if p, p“) 6 P“) then

Ip — 13|+ 4|p3 — 133| g min{k/2 — 6, Mic/p - 3} = 94.

45
A possible choice is N = 45, P = U P“);
i=1

 

p“) = \/3 222', 19“) = [f/22(i- 1),{/22(z'+1)],
for i odd;
13“) = ——€/22(z' — 1), P“) = [— V" 222', -,3/22(z° — 2)],

for i even. By simulation, we choose 7' = 2 x 10“4 and 5, = 10‘5. We notice that

the increase in the value of 5 is not impressive. The reason is a too high number of

32

subsets, which is due to the huge reduction in the value of the controller gains k and
Up. We suggest to follow this way if it is crucial to reduce the control effort and if
it is expected that the influence of unmodeled high-frequency dynamics will be more

significant than the influence of measurement noise.

2.2.4 Simulation results

Let us first compare performance of the three designs in the ideal situation when
there are neither measurement noise nor unmodeled dynamics. The simulation for
p = 5 and p = —10 are given in Fig. 2.1 and Fig. 2.2, correspondingly. In each figure,
the results of the adaptive controller are shown in the first row, those of the robust
controller in the second row, while the third row shows the results of the switching
controller. In all three cases, the first and second columns show the output y and the
control u, respectively. In the case of adaptive control we show also the parameter
estimates p1 and p2, in the robust control we show 3, and in the switching control
we show 3 and parameter estimate p.

The case when lpl is not too large is illustrated with p = 5 in Fig. 2.1. For the
adaptive and robust controllers, large control efforts, 6 x 106 and 10‘, respectively,
are needed during the short peaking period. For the switching controller, on the other
hand, the control signal does not exceed 2 x 102 in magnitude. Notice also, that the
switching logic gives an appropriate parameter estimate much faster than adaptation
and there are no fast control oscillations as in the case of robust control. Remarkably,
the output trajectories for all three designs are indistinguishable despite such a big
difference in the control effort.

The case when [pl is large is illustrated with p = ~10 in Fig. 2.2. The output
and control signals under adaptive and robust control are similar to the previous
case. For the switching controller, this is the worst possible case because the highest

level of the ‘equivalent control’ is needed to ‘overcome’ the system’s dynamics and

33

V(t)

 

 

5 W)
1
10x 0

 

p,<t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.5
9
g 1 5‘—*-—--\ 0 ”My-.- 0 ,
n, ,\ ,,,,,___...____2
n \N , ,/
ID 0 5 \ o _2 .1 . -100
\\
00 1 F 2 '5b 0 00001-40 0 52000 0 5
x104 ' ' Sit) ‘
1 2
73 . , 1 /‘\
u. \ I \\.._...______
\ 0
0 \m -2 —1
o 1 2 0 0.0002 00) 0 0.0002
1 500 4 2
m
.5 ...——-» 1 (
§ 0.5 0 - // . 2 .
i r/ 0 gm“
0 -500 0 —1 ‘
0 1 2 o 0.005 0.01 0 0.005 0.01 o 0.01 0.02

Figure 2.1: Simulation results for the case of p = 5

the switching logic has to go through all candidate controllers except the last one. As
a result, the output does not vanish during a short period of time and fast control
oscillation is needed. This kind of transient behavior is of course undesirable and
illustrates the main problem of the pre-routed search strategy (see Chapter 3 for
more on this issue). Nevertheless, the needed control effort of 103 is still smaller
than for the other designs and the performance is still comparable.

We remark that the simulation results above are typical for various initial condi-
tions satisfying the equation V(y(0), 3(0)) = 4. The only difference, that we have
observed, is in the steady-state values of the estimates for the parameters in the case
of adaptive and switching regulators. We have also observed similar transient per-
formance with the same difference in the control efforts2 independently of the initial

conditions that belong to the surface V = 4.

 

2By control effort we mean the highest level of the control signal.

34

 

 

 

 

Y“) 5 u(t) p (t) t
x 10 5 1 500 le )

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

   

 

 

 

 

 

 

 

 

 

 

 

 

1.5 10
G)
5 R _.__ - ._
g 1 \\ 5 \g Cl 0
\ \
m 05 \j 0 ‘2 -5 \“_S_____,—500 \\
\\,N C

0 W -5 -1 1000

0 1 2 x104 000001 0 0.5 0

1 2 2 5(1)

\\

W .I.
3 05 \ 0 i’lfl'ltw
12: \\ film 0 \ \--—-<

0 -1 l

0 pm 0 0.0002
2’ 1.5 0 10
'5' 1 .r, .l l l
'i oll 5 i .. .
m ll ll , . .

\ trill] ; ~
0 2000 —10 “ ”’ o '
0 1 2 0 0.005 0.01 0 0.005 0.01 0 0.01 0.02
Figure 2.2: Simulation results for the case of p = —10

We see that the proposed control strategy allows us to reduce the controller and
observer gains and a smaller control effort is achieved almost without sacrificing per-
formance in ideal conditions. Note, however, that our motivating example has a
special structure that allows us not to saturate the equivalent part of the control. For
a general case, when the later is not possible, we expect that there would be worst
case values of the unknown parameters and initial conditions for which the control
effort in the robust and logic-based switching designs would be the same.

We have also simulated the system in non ideal conditions to check the robustness
with respect to a small actuator delay that was not taken into the account at the
design stage.

We have observed that if (2.1) is replaced with

i)+py+(py)3=u, rdfi+u=u,

35

where rd = 10‘5, then the performance of adaptive and switching regulators stays
unimpacted but the robust regulator suffers from chattering, resulting in fast control
oscillation between the upper and lower limits and non vanishing s(t). The simulation

results for the case of p = 5 are shown in Fig. 2.3.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

y(t) x 10‘ Ult) s(t)
1 2 1.5
0.3
.6; 0.6 . ., ......
:3 .
§ 0.4 -
0‘2 . . ., . .....
o
o 1 2
1 400 15
g 0.6 . . ....... . \ :
3 ‘ 0 . o_5 .
"’ ' i
02 VVVVVV _200H . . 0.
o —400 _ ‘
0 1 2 0 1 2 05 1 2

Figure 2.3: Performance under unmodeled dynamics

We notice that although due to integration effect there is almost no visible degra-
dation of the output performance for the shown short period of time for the case of
robust controller, fast chattering in the control signal is unacceptable for practical
implementation.

Elimination of chattering effect through employing our switching logic is not sur-
prising since all the gains are reduced. However, the price for this is the need to
estimate an extra derivative and, correspondingly, necessity to use a third order ob-

server (2.4) instead of a second order observer (2.8). Let us investigate whether this

36

would lead to a closed-loop system that is more sensitive to measurement noise. Keep—
ing the unmodeled dynamics, we replace the input to the controller and observer with
y(t) + w(t), where w(t) is a band-limited white noise with magnitude of 10‘5, and
reset the initial conditions to zeros. The simulation results for the case of p = 5 are

shown in Fig. 2.4.

10'5 Y“)
5 x

 

 

 

 

robust
o

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1 2
1 -5
5x 0
.Z’
:3 0 0 0?“ :
5
1 -O.1
-5
0 1 2 0 1 2 0 1 2

Figure 2.4: Performance under unmodeled dynamics and measurement noise

We notice that the magnitude of the steady-state tracking error for the robust
control is a little smaller, however the noise signal is not amplified significantly more
at the output despite the higher order of the observer. It is also worth noting some
degradation of the control signal produced with the switching regulator, although this

signal is still much better than the one under robust control.

2.3 Problem Formulation

Now we present a generalization of the example above to the problem of tracking

for a class of minimum-phase nonlinear systems [19, 22].

37

2.3.1 Model and Main Assumptions

Consider a single—input-single—output nonlinear system that can be globally rep-

resented in the form

5: At + Blame 2) + r1025 z){u + w..(t)}l.

.3 : ¢’(pr€73)r (2.10)
y = {1+ we“)?
C : (f(ptgazlt

where the pair (A, B) represents a chain of n integrators, i.e.

'0 1 0 0* '0‘
0 0 1 0 0

A: ; ; I. -. -. ; 612"“ and B: : ER",
0 0 1 0 0
0 0 o 1 0
_0 0 0 0_ -1-

 

 

 

 

6 = [51, . . . ,fian 6 R" and z E lR’" are the state variables, u 6 R is the control
input, y E R and C 6 IR’ are the measured outputs, wu(t) 6 IR and wo(t) 6 IR are
smooth bounded input and output disturbances, q(‘), fo(-) and f1() are known
smooth nonlinear functions that depend on a vector p of unknown parameters, which
belongs to a known compact set P C IR", and 1,0() is an unknown function.

Classes of nonlinear systems that can be represented in the form (2.10) are dis-
cussed in [3]. They include input-output linearizable systems with global normal
forms [19], many mechanical and electromechanical systems, and the extended dy-
namics of systems represented by input-output models [21].

Assuming that the initial conditions, (6(0), 2(0)), are inside some given compact

set Qg C R("+m), our goal is to find a dynamic output feedback control law to

38

ensure that the output y (practically) asymptotically tracks a reference signal r(t),
i.e., for any given tolerance 50 > O, we need to design the controller to ensure that

ly(t) — r(t)| < 50 after some finite time. We assume that

(i) There exists a known positive constant 6 such that f1(-) 2 (5 \7’ 5 E R",

z E R", and p E P.
(ii) The reference signal and its derivatives, up to order n, are available on line3.

(iii) lwu(t)l is bounded by a priori known constant and the vectors
d(t) = [r(t),7"(t), ' ' ' ,7‘("’(t)lT and w(t) = [wo(t),wé(t), - ' ° ,wl"’(t)lT

belong to known compact sets D C Rn“ and W C RM“.

(iv) The system 2' = w(p,§, z), with 2 as the state and g as the input, is bounded-
input-bounded—state (BIBS) and there exist a continuously differentiable proper

function V0(t,p,z) and class [Coo functions ao(-) and 70(-) such that

V0(t,p,z) Z ao(llzll) and for all p E P and E 6 IR":

V0 2 6%? + %¢(p,£,z) S 0 whenever %(t,p,z) _>_ 70mg”).

It is well known that if the z-subsystem, with 2 as the state and 6 as the
input, is input-to—state stable (ISS) for all p E P, then assumption (iv) is satisfied.
Reference [22] makes the ISS assumption and requires the zero dynamics to be locally
exponentially stable when g = 0. Assumption (iv) relaxes both requirements. It is
possible to weaken this assumption even further and, in particular, require it to hold

on a certain compact set if we impose some restrictions on 95 and ’D. However, for

the sake of simplicity of the presentation we do not do this.

 

3It is possible to modify the analysis to deal with the situation when only r(t) is available by
means of treating the unavailable derivatives as additional bounded disturbances.

39

It is convenient to introduce a new state variable a: = g + w(t), so that

i: = A11: + Blfo(p. a: — w(t). z) + f1(p, :1: — w(t). z){u + w(t)} + 10900)],
(2.11)

2: If«’(p,$—U.’(t),2), yzajlt CZQ(p,$—IU(6),Z),

where (:1:(O),z(0)) E Q, a compact subset of R"+m such that (a: — w, z) 6 S25 and
w E W imply (3,25) 6 Q.
We partition the set of parameters P into N smaller subsets {P(f)},’~:1 and

design a regulator for each of them.

2.3.2 State feedback control for p E P“)

In sliding-mode control design [47], we use the error coordinates
e = [y — r(t), r2 — r'(t), . . . ,xn_1 — r("’2)(t)]T
and define the sliding surface s = 0 by
s = 2,, — r("‘1)(t) + Te,

where T = [t1,...,t,,_1]. Then, 5 = Ale + bm(2:,, — r("‘”) = Ale + bm(s — Te),
where the pair (A1,bm) represents a chain of (n — 1) integrators. Motion on
the sliding surface s = 0 is governed by e = (A1 — me)e. Therefore, T is de-
signed so that Am 3- A1 — me is Hurwitz. With this representation, our system
can be viewed as an interconnection of three subsystems: an exponentially input-
to-state stable linear e-subsystem e = Ame + bms, an 8180 scalar s-subsystem
s = f2(p,e,s,z,d,w,,,w) + f3(p,e,s,z,d,w)u, where f2(-) = f0(p,:r — w(t),z) —
Ma) + f1 (p, a: — w(t), z)w,,(t) + ulna) + T(Ame + bms), f3(-) = f1(p, :1: — w(t), z),

40

and a BIBS z-subsystem, satisfying assumption (iv), 23 = '1/21(t,p,e,s,z), where
2:1(t,p,e,s,z) ='1.i’(p,$ - w(t), 2)-
Let 1}“) E 73“) be a nominal value of p. For each 2' we design a continuous

sliding-mode state feedback controller:
u = u“) 2 11.42"", 6, s, c, d) — k, Sat(s/u,), (2.12)

where ueq(-) is an equivalent control component to be defined later and Sat() is a
smooth saturation function that satisfies (2.2). Each of these regulators is designed
to work in the case when p 6 pm as follows. We consider the Lyapunov function

candidate

V(e, s) = ueTPe + 32/2,

where u > O and P 2 PT > 0 is the solution of the Lyapunov equation PAm +
AiP = —I. For a fixed number 6,, 6 [0, 1/||P||), we compute V + 60V along the

trajectories of the closed-loop system with u = u“) :
V + 6,,V = -V(€T€ — (SveTPe) + sf3(v)[A,-(-) - k,- Sat(s/u,)],

where A,(p,e,s, z,d,w,,, w) = [f2(-) + 6,,3/2 + 2ueTPbm]/f3(-) + ueq(-) is an uncer-
tainty term. To make it smaller, we should choose ueq(-) to be the best available
model for —[f2(-) + 603/2 + 2ueTPbm]/f3(-).

We would like now to define a positively invariant compact set that contains the
trajectories of the closed-loop system provided V _<_ 0. Towards this end, let p(-) be

a strictly increasing unbounded function such that
p(R) 2 sup{||:r — w“ : V(e,s) S R,d E D,w E W},
where the “sup” is finite for any finite R 6 IR due to compactness of D and W.

41

Noting that, due to assumption (iv), V S R and V0 2 70[/)(R)] imply Va 3 0, we

define the set

U(R) = {(e,s,2): V(818) S R, Vo(t,P»3) S ”Vol/)(Rll W E R71) E 73},

which contains (6,8,2) whenever V S R. The set U (R) is compact because [eT,s]T

is linear in a: and d, and d E ’D. Moreover, if (6,8,2) 6 U(R), then ”1:“ S
p(R) + sup{||w|| : w E W} and ”2” S ag‘(70[p(R)]). We define R to be a fixed
positive number such that Q C U (R0) for some R0 6 (O, R). Next, we show how
to choose k,- and p,- in order to guarantee that U (R) is positively invariant. Let

p,(R), L1,(R), L2,(R), L3.,-(R) Z 0 be such that
|A1(')l S 101(3) and |f3(')A.-(°)| S L12(R)||€l|+ L21(R)|S| + 1431(3)

for all (6,3,2) 6 U(R), p e P“), d e D, and w e W.
Take 17 = V(l — 6,,||P||) and k,- > p,(R); then, inside U(R) and V19 6 73“),

using the inequalities above together with (2.2) we obtain
V + 6.,V S -l7||6||2 - We; — ps(R)l|SI.
provided Isl > 12,-, and
V + 5vV S -l7||€||2 + L11(R)|8|llell - [k16/us— L21(R)]|s|2 + L31(R)|Sl,
provided |s| S M. By choosing )2,- small enough, we can ensure that

plkfls/fli " L21(R)l > [LAID/212. (2-13)

42

Then there exist [3, > 0 and 7, > 0 such that
V + (SUV 3 — Ini11{p,'),,fi,V — uiL3,(R)}. (2.14)
If u,- is small enough to ensure that
#iL13i(R)/(6v + 52') S R (215)

then V = R and V}; S '70[p(R)] imply V g 0, due to assumption (iv). We conclude
now that if p 6 P“), then every trajectory of the closed-loop system (2.11), (2.12)

initiated inside the set 9 can not leave U (R) and, moreover, it will enter the set
U0 2 {(8) 8’2) I V(ev 3) S 60) ‘/0 S. ’70[p(R)]}9

where 60 Z comL3,(R)/(5v + 31) with co > 1, in finite time and stay thereafter. To

ensure that ly(t) - r(t)| S 60 inside this set, we require 60 g 1163/ IIP‘IH.

2.3.3 High-gain observer for p E 79“)

Because 6 and s are unavailable, instead of u = u“) we use the control

u = a“) 2 21.42%, 2, ed) — k.- Sate/2.), (2.16)
where
5 = 2,, — r<"-1> + Té,
(2.17)
é = [y -— 73562 — 1",. . . ,in_1 — r("‘2)]T

are estimates for s and e, and :i: is provided by the high—gain observer (HGO)

5: =Aoi+Hi(y—il)+b0h(fi(i)véa§and)1 (2'18)

43

in which (A0, b0) represents a chain of (n + 1) integrators and h() is a model for

the time derivative of [f()(p, :1: — w, 2) + f1 (p, a: — w, 2){u + 10"} + my] ; we can take

- . 01 0’2 01n+1
h E 0. The observer gamls takenas H,- = [—,—2—,...,n—+, , where a1,...,oz,,+1

are chosen so that the polynomial ("+1 + 018‘ + - - - + an“ is Hurwitz and e,- > 0
is a small parameter to be specified.

Note that the order of (2.18) is greater than the order of the original system
because we need to estimate the n -th derivative of the output in order to estimate
V. The latter is needed to implement switching, as explained in the next section. It
will be clear from the stability analysis that in order to estimate an ‘extra’ derivative

we need to impose the following additional assumptions

(v) f0(-), f1(-), u.,(.), Sat(), wu(t), 2115,")(1), and r(")(t) are continuously

differentiable; w;(t), wgn+l)(t), and r("+1)(t) are bounded,
(vi) ueq(-) is globally bounded in 5 and ég,...,én-1,

(vii) partial derivatives of ueq(-) with respect to (é, 5,0 are globally bounded in g

and 62,. . .,€n_1,

(viii) there exist c1,02,c3 > 0 such that" |lh(-)|| S C; + Cgllé” + c3|§|,

2.3.4 Switching logic

We start with z' = 1 at t = T1: 0 and for each 2'6 {1,...,N} we wait with
u = a“) in the feedback loop for t 6 (TILT.- + 7'), where 7' > 0 is to be specified,

and then check the inequality
2 + 6J7 _<. a,- — min{,u1%, £3117 - (aha-(RH. (2.19)

where V = §2/2 + uéTPé (2.20)

 

“This assumption is always satisfied when h(-) E 0.

44

and V = —1/éTé + 2uéTPbms + s(z2,,.., — A") + T[A,,,é + bm§]) (2.21)

are estimates of V and V, and a,- is a small positive constant that is included to
deal with possibly non-vanishing estimation errors. If at sometime T,“ 2 T,- + T
the inequality is not satisfied, we switch to the control law 21 = 12““).

The dwell-time T should be chosen sufficiently small not only to guarantee no
finite escape time but also to ensure that (e(t), s(t), 2(t)) does not leave U (R) It
will be justified below in the analysis that: the solutions, initiated inside U (R0),
exist for all t Z 0; (e(t),s(t),2(t)) stay in U(R); there are no more than (N — 1)
switching points; there exists M > 0 such that V S M.

Based on these properties we define 7" = as an upper bound for 1'.

T7112
2.4 Main Result

Let us assume that
(ix) 53(0) 6 {2, where Q C Rn“ is compact;

(x) with k,- > p,(R) and p,- > 0 chosen to satisfy (2.13) and (2.15), suppose

a,- > O is chosen small enough to satisfy
[01' + ML3i(R)]/[6v + fir] S 50/60 and 01 < #171
for every z'E {1,...,N}, where
co > 1 and 50 S min{ucg/||P‘III,R}.

Let U0 = {(e,s,z,7)) : V(€,S) S 50, “Z” S Vol/KR”, WU?) S 1"OE-2},

45

where r0 > 0, E > 0, W(77) = nTPmnn, and Pom > 0 is the solution of Lyapunov
equation 243,an + Pom/lam = —I.

If the trajectories are trapped inside 00 , then ly(t) — r(t)| < 60. We show next
that it is the case, after a transient period, and so the tracking error is asymptotically

sufficiently small.

Theorem 2.1 Consider the closed-loop system (2.11),(2.16),(2.17),(2.18) with the
preceding switching logic, initiated inside Q x (2. Suppose assumptions (i)-(x) are
satisfied. Then, there exists To 6 (QT), such that for every 7’ E (70,?) there exists
5,- 6 (0,5) such that for e,- 6 (0,5,) every trajectory is bounded,.enters in finite time

a compact set (70 , and stays thereafter.

To rewrite the closed-loop system in a singularly perturbed form, we introduce

the scaled variables

 

 

where

xn+l = f0(pa ZL‘ _ WU), Z) + fl(pa SL‘ _ IU(t), Z) [u + wU(t)] + whn)(t)

is the needed extension of the state. This extension is well-defined in view of assump-
tion (vi) and is continuous between the switching times, for t E [7},7}+1), as long as
the solution exists. At the switching times Ti, 7) could have a finite jump, while :i'
and 23,-, for j = 1,. .. ,n, are continuous. The jump in 1),,“ is clear in view of the
jump in In.” that results from control switching. A jump in 17,-, for j = 1,. .. ,n

will take place if 5,- 7E 51+1-

46

Following the derivations in [3], the closed-loop system (2.11),(2.16),(2.17),(2.18)

can be rewritten as

e = Ame + bms,

23 Z ¢’1(t1P161313),

s = f2(p,e, s, 2, d, w“, w) + f3(p,e,s, z, d,w)[u<"> + @103“), 62 51977100]:
517') = AM + 6.1209.- (13mm, 8, s, z, n, d, w. wu),

C = (II (tap, 6’ 312%

where Aom = A0 — [011, - - - ,an+1]T[1,0, - - - ,0] is a Hurwitz matrix;

94-) = — h(-) = d—d. (M) + f1(-){fl"" + wan} + w£"’<t>] — ho;
wi() z it“) _UU) = 216003“): é) g! C» d) _ 1184(1)“), 8, S, C: d) + ki [sat('§/#i) _ sat(8/#i)]i

(11(')= (100.16%);

and i E {1, - - - , N} is changed according to the switching rule described above. This
system is valid between the switching times.

Noting that
V g —6,,V(-) — DeTe + sf3(-)[A,-(-) + \II.(-) - k.- Sate/2.)]

and \Il(p(‘), e, s, (,1), d) is globally bounded in 17 due to assumption (vi), we define

M = sup{—6,,V(e, s) — DeTe + sf3(-)[A,-(') + \Il,() — k, Sat(s/p,)] :
(e,s,2) E U(R),i E {1,...,N},p 6 ’P,d 6 ’D,w E W,n E IR"+1},

so that as long as the solution exists and (e,s, 2) E U (R) the inequality V S M
will be satisfied uniformly in 17.
Due to finiteness of the dwell-time, there is no possibility for chattering (infinitely

fast switching). Hence, for any fixed 7' E (0, 'F) and any choice of e, > 0 the solution

47

of the hybrid closed—loop system does exist locally.

Let us consider first the case when the right controller is chosen. We show that
the solution exists globally, there will be no more switching, the trajectories con-
verge to a subset of (70, and stay thereafter. We notice that there exists to such
that p 6 ”PM”. Suppose the controller is switched at t = T,- to u = 2160) and
(e(T,-0),s(T,-o),2(T,-O)) 6 U(Rio), where Rio 3 R0 + (N —1)Mr. Note that since
(2 C U(Ro), at t: 0, (e,s,2) start in U(RO).

We show that 7' is small enough to guarantee that during the dwell-time the
‘slow states’ stay inside a compact set and the ‘fast states’ decay to small values. To
prove this claim we adopt the arguments presented in [3] (see also [7, Theorem 2]
and [23, pp. 616-617,713-718]) being careful in dealing with the need to estimate
an ‘extra’ derivative of the output. As a result of extending the system state, we
face a technical difficulty that is not typical for the analysis of peaking in high-gain
observers: the perturbation term on the right-hand side of the fast subsystem is not
uniformly bounded in 17. However, we can show that it can be bounded by a function
that grows linearly in the fast variables. To the best of our knowledge, [7] is the only
paper where a similar situation has been investigated. While Theorem 1 of [7] cannot
be applied directly to our current problem (even if all e.- are chosen equal to each
other) since switching between several control laws results in discontinuous right-hand
sides of the slow subsystem, we are going to use the technique presented in [7].

Let us consider the time-derivative of the Lyapunov function W(17) = nTPomn
along the system’s solutions: W = —17Tn/ e.- + nTbog,(). We would like to show that

if ”n“ is not small then W < 0. Hence, we need to argue that

_ 6$n+1 axn+l 1 6311+]
g1(.)_[-a—$1-—,..,—5‘x—n‘] [A$+b$n+1—1U]+ 62 ¢()
5&0). 3*(1').
+f1(') [We + 719‘s 5? + w;] + wf,"+1) — h(-)

48

satisfies a linear growth condition in r]. Imposing at this point assumptions (vi) and

(vii), it is left to notice that
s: s — (7),,“ — 071771) + T(e — e)

and

éj = 33) ‘ 1.0)“) - sll‘j(vlj+1 — 011771)

for j = 1,. . .,n - 1 are affine functions of r] uniformly in 5,, and so is h() due

to assumption (viii). Hence, there exist Go > 0 and 01 > 0 such that

 

‘ ”RIP 2 1 2 ”7)”
W<———+G +0 .. 2. —G , _ ——G' ,
_ 6‘0 0H77” 1”,)” 2&0 0 “7]” 25m 1 “7)“

provided (e,s,2) E U (R). Assume that 5,0 > O is sufficiently small to ensure
1/(26,-0) > Go. Noting that W 2 mega implies “77“ 2 5mm, we choose
r0 = (201)2||Pom||, so that “17“ _>_ 26313-0 and hence W S —6wW, where 6.0 =
[1/(251'0) —Go]/||Pom|| > 0. Since V S M in U(R), uniformly in 17, and Rio < R,

we have
V(e(t), so) 3 veer...» s01.» + M — T...) s R... + M < R

for t E [TimTio + T]. It is easy to see that n(t) is large but finite at t = Tia and
decays faster than an exponential mode of the form (1 /e§;)e‘(["'l/ 5‘0)‘ uniformly in
(e, s, 2) E U (R), so that for sufficiently small values of 5,0, 1)(t) enters the set
{W S r052} during a period of time To < T and is trapped inside.

For this controller, inequality (2.14) with i = i0 is satisfied under the state
feedback u = u(‘°). Under the output feedback it = 21“”), the right hand side of the
inequality is perturbed by the term sf3(-)\II,-o(-). Due to continuity of \II,0(-), the

identity \Il,o(p(‘°),e, s,(,0,d) E 0, and the fact that 1) is 0(5) for t 2 Tie + To, we

49

see that there exists r.) > 0 such that ||\II,0(-)|| S 1",), for t 2 ,0 + T0, and that n),
can be made arbitrarily small by choosing 5 small enough. Hence, V satisfies the
inequality

V + (SUV S — mild/110710, {310V - HioL3io(R)} + 5%

with ("1,0 E (0, (1,0). Since, during the same interval, V and V, given by (2.20) and
(2.21), are 0(5) close to V and V, respectively, we conclude that for sufficiently
small 5,

V + 60V S — Humming), [310V — MOI/310(3)} + aio

for t Z Tm + T0. Hence, there is 5,0 > 0 such that if 5,0 E (0,5,0) then as soon as
the right controller is chosen no more switching occurs and the trajectories converge
to the corresponding residual subset of (70.

If i0 = 1 we are done. Otherwise we have to consider the case of a wrong
controller in the loop. Suppose i 75 i0 and the controller is switched at t = T, to
u 2 it“) and (e(fl),s(fl",-),2(T,~)) E U(R,), where R.- S R0 + (i — 1)Mr. Following
exactly the same arguments as for the case of the right controller, it can be shown
that (e(t),s(t), 2(t)) E U(R; + Mr) for t E [T.-,Tg + 7'] and 17(t) E {W S r052} for
t Z T.- + To and they stay so until the next switching time.

At time t = T.- + T, we check the inequality (2.19). If it is not satisfied we take
71+] = T.- +T and switch to the next controller. Otherwise we stay with the controller
a = 21“) as long as (2.19) is satisfied. Once again, since V and V are 0(5) close

to V and V, respectively, we see that, for sufliciently small 5,,
V + 60V S - min{un.-, fi1V - #iL3i(R)} + Eli»
for some a,— E (a,, mm), which is close enough to a,- to ensure that

[512' + #iL31(Rll/[5v + fit] S 50/50

50

for some 50 > 1. Then, for V 2 50, V S —c for some c > 0. This shows that
V does not increase during the period [T,-, T,+1). Moreover, if the inequality (2.19)
is never violated, there will be no more switching and the trajectories reach the set
00 in finite time and stay thereafter. If at some time ta 2 T,- + r the inequality
is not satisfied, we set T,“ = ta and switch to the next controller. Obviously,
:1:(T,-+1) and 17(7111) are finite, therefore :i:(T,-+1) is finite and so is 1701.11). Hence,
if either T,“ = 00 or i + 1 = i0, the trajectories must enter a compact subset
of (70 in finite time and stay thereafter. Noting that i0 S N and using induction
in i = 1,... ,io we see that: (e(Tl),s(T1),2(T1)) -—- (e(O),s(0),z(0)) E Q C U(RO),
(e(T2), 8(T2),Z(T2)) E U(Ro + MT), (e(Ti), 8(T1), 2(Ti)) 6 U(Ro + (i - ler),
..., (e(T,O),s(T,-o), 2(T,-o)) E U(Ro+(io-1)Mr) C U(R,0). This completes the proof.
We notice also that it follows that there can be no more than (N - 1) switching

points.

2.5 Remarks

We have considered the design of output feedback control for a class of para-
metrically uncertain nonlinear systems. The main ingredients of our approach are
logic—based switching between several candidate controllers, designed for small sub-
sets of the set of parameters, and dynamic output feedback, based on a high-gain
observer. We have shown that the parameters of the controller can be tuned to
ensure semiglobal practical stabilization. The main technical difficulty is that the
existing theory of high-gain observers with saturation [23, sec. 14.5] does not cover
switching among controllers. We have shown that, provided the logic is chosen so
that there is no switching during the observer peaking time, the techniques of that
theory are applicable to the analysis of the closed-loop system between the switching

times. An additional technical difficulty is the need to estimate an extra derivative

51

of the output in order to approximate the Lyapunov function and its derivative. Our
derivations show that this difficulty can be resolved if, in addition to the control it-
self, its first partial derivatives are globally bounded with respect the estimated state.
Consequently, we have had to use a smooth saturation function.

It is worth noting that even in the case of no input and output disturbances our
switching controller ensures semiglobal ultimate boundedness only, whereas, under
certain additional assumptions, both adaptive and robust controllers from [21, 22]
guarantee asymptotically vanishing tracking errors. However, the adaptive control
design [21] is applicable to a much more restrictive class of nonlinear systems since
linearity in the unknown parameters, which is not needed for our design, is essential.
It is intuitively clear, and has been recently formally proved in [10], that in the
case of large parametric uncertainty the high-gain observer-based adaptive feedback
control technique [21] leads to very high saturation levels as well as high controller
and adaptation gains. The needed large values might beunimplementable in practice
and might provoke poor robustness with respect to unmodeled dynamics and high-
frequency measurement noise. The robust controller of [22] is based on worst-case
analysis and therefore it suffers from the same problem associated with high gains
when the level of uncertainty is large. In particular, when the level of uncertainty
is large, three parameters of the control law: the saturation level (k), the linear
feedback gain (k/u), and the observer gain (1/5) have to be high. A very small
value of p, which characterizes the thickness of the boundary layer, would result
in chattering and a very small vallie of e, which defines the observer bandwidth,
would result in high sensitivity to measurement noise. In our design, the set of
parameters is split into smaller subsets and the candidate controllers are designed
to work only on this subsets. As a result, the level of uncertainty is reduced and
we can increase both p and e. We presented simulation results confirming that our

switching design allows us to eliminate chattering observed under robust control in the

52

presence of small actuator delay. We remark that it is done when both controllers are
designed to ensure similar output performance in ideal (disturbance free) conditions
and without any restrictions on the values of the control parameters and the control
efforts. For the motivating example, we observe that the needed control effort for our
design is smaller than for the adaptive and robust ones. This leads us to believe that,
under severe restrictions on the available control effort, logic-based switching design
would be more practical; however we leave this issue for future investigation.

There are many possible avenues for extending our approach. An extension to the
multi-input—multi-output case seems to be straightforward. It is could be noticed from
the proof that the sliding-mode design for the candidate controllers is not essential.
Lyapunov redesign [23, sec. 14.2] or any other robust technique could have been
used instead and this is going to be a subject of our next chapter. In addition,
judging from the simulation results, the most important extension that is needed
is an implementation of a more intelligent switching logic to replace the pre-routed

search. This is done in the next chapter together with some further generalizations.

Chapter 3

Lyapunov-Based State and Output
Feedback Stabilization for a Class

of Nonlinear Systems

3.1 Introduction

In the previous chapter we have presented a solution for the tracking problem for
a class of minimum-phase nonlinear systems. The key idea of Chapter 2 is to check
whether a certain inequality for the derivative of the Lyapunov function is satisfied.
The inequality can be verified using estimates of the states and their derivatives, as
soon as the peaking time of the high-gain observer is over. If it is satisfied, then the
current controller ensures convergence of the trajectories to a set where the tracking
error is small. If not, then switching is necessary. It can be seen from the analysis
that the sliding-mode design for the candidate controllers, presented in the previous
chapter, is not essential and any Lyapunov-based design could have been used instead.
Moreover, it is worth noting that for state feedback control design, a similar switching

logic, with a high-gain observer providing derivatives of the states, could be used in

54

order to improve robustness and transient performance of the closed-loop system. We
present this generalization below with an important improvement in the switching
logic. We avoid following pre-routed search, that may result in an unacceptable
transient performance when the number of candidate controllers is high. We use the
available on-line information to identify the set to which parameters belong and to
choose a candidate controller to be put into the loop when the inequality for the
derivative of the Lyapunov function fails. However, the most important extension of
our previous result is the class of systems that can be handled with our approach.
This class, studied below, contains not only minimum-phase nonlinear systems with
a known sign of the high-frequency gain, studied in the previous chapter, but also
uniformly observable nonlinear systems with possibly unstable zero-dynamics and
uncertain sign of the high-frequency gain.

The motivation for the class of nonlinear systems, we introduce below, is the
well-known procedure of dynamic extension [46, 45] for output feedback control. The
idea is to augment thesystem with additional dynamics, typically by adding a chain
of integrators at the input. The original system can be parametrically uncertain,
non-minimum phase, and even with unknown high-frequency gain. The dynamic ex-
tension yields a system with additional outputs and no zero-dynamics. As was noticed
by Tomambé [46], implementing a high-gain observer to estimate derivatives of the
output, we end-up with a state feedback control problem for the extended system.
However, in the nonlinear case, designing state feedback law and then implementing
it employing the estimates of a high-gain observer could lead to a shrinking region
of attraction and unacceptable transient behavior due to the peaking phenomenon.
This difficulty can be overcome if the control law is saturated outside the region of
interest, as argued by Esfandiari and Khalil [7]. These ideas, combined, in particular,
with backstepping and robust high-gain feedback design, were used by Tee] and Praly

[45] to develop a powerful tool for semiglobal feedback stabilization of the class of uni-

55

formly observable systems. Their results were extended by many researchers; see e. g.
[3, 23]. The available techniques apply to parametrically uncertain systems. However,
when the parameters of the model belong to a known but comparably large compact
set, it is typically assumed that all the systems in the family are minimum-phase and
that at least the sign of high-frequency gain is fixed and known. The stability of the
zero dynamics (often in the form of input-to-state stability) is essential when high-
gain feedback is used to overcome uncertainty. Knowledge of the control direction
is important in the problem of smooth stabilization [44] and discontinuous switching
control seems to be the most efficient way to avoid it [28]. Advantages of allowing
discontinuity in control are well-known [32] and, in many cases, there is no need to
impose the structural assumptions, mentioned above, provided switching is allowed.
Hence, our goal in what follows is to develop a procedure of achieving stability via
switching between several candidate controllers, designed to ensure satisfactory per-
formance for a small subset of parameter uncertainty.

The rest of chapter is organized as follows. We start with a precise description
of the class of systems. Then, we discuss the design of extended high-gain observers
and the switching strategy. The formulation and proof of the main result conclude
the theoretical part of the chapter. The last part of the chapter presents two illus-
trative examples. The first one is a linear non-minimum phase system, where we
show that it is not possible to design a single stabilizing feedback controller for the
whole set of unknown parameters, and discuss the applicability of our design. The
second example is a nonlinear system from [20] that can be stabilized via a high-
gain observer-based adaptive control design [21]. We present simulation results and
compare the performance under our design with the alternative one. We note that
another example is treated in the next chapter where we consider a nonlinear systems
with unknown sign of the high-frequency gain, which can be stabilized using scale-

independent hysteresis-based logic [13, 16], and discuss some implementation issues

56

as well.

3.2 Class of Systems

We consider the regulation problem for a nonlinear system that can be represented

in the form

i=AI+BMMLCUL
(I: ¢'(p1$1C)u)a (3'1)
y=Ca

where the triple (A, B, C) E RM" x Rm” x 1R1” represents a chain of n integrators,

i.e.
'0 1 0 0 0' "0"
0 0 1 0 0 0
A: , B: , C=[1 0 0],
0 0 1 0 0
0 0 0 1 0
_0 0 O 0‘ -1-

 

 

 

 

(19(1), -) and w(p, -) are known continuously differentiable functions; C E IRS and
x = [$1, - - - ,xn]T E R" are vectors of state variables; y E IR and C are measured
outputs; p E 1R” is a vector of unknown parameters; and u E R is the control input.

We restrict ourselves to the single-input case only for simplicity of presentation.
Extension to the multi-input case is straightforward and can be done along the lines
of [3]. We assume that the differential equation for the a: subsystem is dropped if
n = 0 and that n 2 2 otherwise (since a: could be incorporated into C if n = 1).
Similarly, there is no C subsystem if s = 0. We do not assume any special struc-
ture of ¢() and therefore we consider a class of parametrically uncertain nonlinear

systems that contains in particular the following important subclasses.

57

0 Feedback linearizable systems with no zero-dynamics [s = 0] :

:i: = Aa: + Bo(p, SE, a) and y = C12.

0 Uniformly observable parametrically uncertain systems. Specifically, the model
(3.1) includes the case of an uncertain system representable by an n-th order
input-output model :i: 2 A11: + ng(p, 23,11) , augmented with a series of integra-
tors of v, [3, 46, 21] [cp() is independent of u and w() represents a chain of

s integrators of u] :

i‘ = Al‘ + B¢(P,$,C),
C: ASC + 3371,

y=CL

where v = CSC and the triple (A3, BS, 03) E RSXS x RS“ x 112‘” is similar
to the triple (A, B, 0). Here the C subsystem has to be solved on-line and all

its states are naturally available for feedback.

0 General nonlinear systems with all state variables available for feedback [n = 0] :
C=¢®£mi

In each case we assume that the vector p belongs to a known, relatively large,
compact set ’P. In order to simplify the control design, this set is partitioned into

smaller subsets as in the previous chapter:

N
p E ’P = UP“).

i=1

Our main assumption is as follows. For a given compact set Q C Rn”; of possible

58

initial conditions and every

teI:{1,o--,N}

there exists a continuously differentiable, bounded in :L' (saturated outside the region
of interest), state feedback control law with all partial derivatives bounded in :r as

well,

u = U") =3 9"’(1‘,C) (3-2)

such that for every p E P“) all the trajectories of the closed-loop system (3.1), (3.2)
initiated inside (2 are bounded and y(t) —+ 0 as t —» 00. Moreover, we assume that

a corresponding family of Lyapunov functions V(‘)(:1:, C) and auxiliary IC°° functions

llilll)

[AID + B¢ (15) $9 Cig(i)($’ C))]

a[i’(-), ag’(-), and a?’(-) are known such that

([1le

all/(”(17, C)
3x

 

 

) s V“’(:c.<) s aé” (

(3.3)
6V (”(230
+____6_C_.

 

 

 

[El

 

)

¢(I7a$,C,g“)(2,g)) S _agi)(

v13 6 P“) and V($,C) e U(‘)(R) with
U"’(R) = {(12.0 t V“) (MD 3 R}.

where R > 0 is chosen so that $2 is in the interior of n U “’(R).
561
Remarks.
1. It is not hard to modify the stability analysis of the closed-loop system for the
case when the static state feedback control law (3.2) is replaced by a dynamic

one. The dynamic part of the controller could be incorporated into the (-

59

dynamics of the system.

2. It is possible to extend the class of systems (3.1) to include additional dynamics

C

the input, and all the assumptions are uniform, in certain sense. However, we

:r
if they are input-to—state stable (ISS) in the region of interest, with l l as

shall not pursue this case in order to avoid some technicalities and refer the
interested reader to Chapter 2, where it was done for the special case where

there are no C-dynarnics.

Our goal is to design an output feedback control law to guarantee practical regu-

lation. The family of the control laws (3.2) cannot be implemented because
0 the vector :1: is not available for feedback and
e the value i“ E I, for which p E 73“.), is not known.

The first problem could be resolved with the help of an appropriate high-gain observer
(HGO). To deal with the second one we suggest to use ‘discontinuous adaptation’ in

the form of logic-based switching.

3.3 Switching Logic

First of all, we need to obtain a robust estimate :3: of the vector :1: or, equivalently,

of n — 1 derivatives of the main output y, in order to apply the control law
u = a“) s g(’)(:f:, c), (3.4)

which is close to (3.2), provided Ila: — all is small. However, it is also useful to
obtain estimates of y‘”) and C as well, to be used for parameter identification.
Estimating ‘extra’ derivatives is an alternative to the filter design of classical adaptive

control theory [2, Sec. 2.3] aiming to obtain a quasi-static regression model suitable

60

for parameter identification. In what follows we also use these extra derivatives in

1:1)

Differentiating the equations of the closed-loop system (3.1), (3.4) with a fixed i

order to check whether (3.3), which could be rewritten as

8V“)(:I:,C) . 8V“’(r.<) ' m
____+ , — < — '.
at .r + 3C C _ 0.3

 

 

 

 

is satisfied in order to decide whether or not to switch.

 

 

 

 

 

 

 

we obtain
y(n) =2i"(W1C’§")’ (3.6)
c: wi"(p,x.<, 2.2),
where
i 8 717A“) 11' 305,,,2(i) Ai
:1: C
(9(b(p,:z:,(,u) 89(i)(i‘$<) 1 6.9“)(1', C) “(2')
+ an “:11“, 81% 1: + 6C $(P,$,C,U ) 9
i 6 ,IL', all“) Ai 6/ ’33? 1 ’11“) *i
W(p,x.c,u) 69%,0; 09"“)(20 «a
+ a“ “:11“, 65% I + 6C w(P1$,C,u ) '
(i) ‘
It is crucial to notice that, since the functions it“) E g")(i:,(), W, and
(i) ‘ . .
a—gfig—C) are globally bounded in i, the functions 45‘1"” and wl')(-) satisfy a

linear growth bound in it, i.e.

Hat’w. cm)“ s M, + M2112“,

. . . (3.7)
”We”, 92.2)“ s M, + Man“.

for some nonnegative constants M1, Mg, M3,.and M4, provided (22,0 E U (’)(R).

61

We are ready now to design a high-gain observer for the extended closed-loop
system (3.1), (3.4), (3.6) following the standard procedure [3].

The variables :1: and y(") are estimated by 5: and in“, provided by

$5 = ASE + Bin-+1 + H(€i)(y — Ci),
(3.8)

I. 71' A .1 an 1 A
$n+l : 50(1)(y7€7$71‘)+ ———+—-(y_ Cf),

n+1
5i

where aili’(y, C ,it,:i:) is a nominal model of ¢li)(p,a', C, 2,25), which can be taken as

zero and is assumed to satisfy an inequality similar to (3.7),

e,’52’ ’5’?

1 I

y(gi):l31 % ffilT,

al, - - - ,an+1 are chosen such that the polynomial 8‘“ + an? + . . . + mg + an+1 is
Hurwitz and e,- > 0 is a small parameter to be specified.

Similarly, C is estimated by C2, provided by the following observer (we are going
to discuss later some simpler alternatives that are more suitable for special classes

mentioned in the introduction):

(3.9)

A)

C

 

<2 : ¢lj)(y1C1jai') + 52 v

where 113]” (y, C, :it, :5) is a nominal model for 1129’ (p, 11:, C, a}, :5), which can be taken as
zero and is also assumed to satisfy an inequality similar to (3.7).

We assume that the initial conditions for (3.8) and (3.9) belong to a given (arbi-
trary) compact set S2 C RHH”.

Using high-gain observers to estimate additional derivatives brings up the same

62

challenge in the analysis that we have faced in the previous chapter‘.

The next issue we are going to address is how to choose the Correct candidate
control law or, equivalently, the appropriate index i E I . It is intuitively clear that
if i = i" (p E 79(")) and the high-gain observers above are capable of providing good
estimates, then the inequality (3.3) is satisfied up to a small error, the trajectories of
the closed-loop system cannot leave the compact set U (")(R), and y(t) approaches
an invariant set where it is small, provided 5,. is sufficiently small. It is crucial
to notice that if (3.3) is satisfied for a value of i 75 i", we can still show ultimate
boundedness (practical regulation). Obviously, we cannot check the inequality (3.5)
on—line because it depends on the derivatives :i: and C. However, using the estimates

provided by (3.8) and (3.9), the following inequality is easy to verify:

av“) . av“) . .,- i‘
—ax—(2r)2+ 0C (2,()c25ao—a§’ (lllclll), (3.10)

 

where

2 ,. . ,2 .. T
I : Ax + BIn+1=l£2,' ' ' 7$n+ll

is an estimate for it and a0 > 0 is a small constant, introduced to deal with an
0(e,) estimation errors induced by the high-gain observers.

After a short peaking period, we can find out whether inequality (3.10) is satisfied.
As long as it is satisfied, we maintain the controller that is currently in the loop. When
it is violated, we switch to another controller and exclude the previous one from the
list of candidate controllers. How to choose the next controller is a crucial decision
that affects the performance of the system. We can switch systematically according to
a pre-sorted list, as in Chapter 2. The performance, however, might not be acceptable
if we have to switch through a long sequence of controllers before we settle at one

for which the inequality (3.10) is satisfied. A more intelligent way is to use on-line

 

1See the discussion on p. 48.

63

information to decide on the next controller. Noting that, after the peaking period,

the estimates in“ and C2 satisfy the equations

iii-+1 -' (V (pa j,C,g(i)(;EI, <)) Z 0(Ei)’

C2 _ w(pai7<29(l)(i1<)) : 0(Ei)’

for the true parameter p, we can use

m 2 line _ a (par, g“)(;i:,C))l + lléz — w (p.2.c,g<"(2.<)) ll

as a performance index to be minimized over all possible values of p. Invoking addi-
tional assumptions on how the functions d)(-) and 1/J() depend on p, we may use
gradient, normalized gradient, least square, or another standard estimation algorithm
[2]. Naturally, the algorithm choice will impact the performance, as discussed in [32]
for a similar problem. For computational simplicity, we adopt the following approach.
For each set 790), we choose a nominal parameter p(j). Assuming that the sets 130')
are small, it is reasonable to expect lJ (p) — J (pU))l to be small for all p E 190).
Hence we use J (pm) as an index for the set “Pm. If I is the set of indices to

choose from at some switching time, the next index is taken as
i = arg meip {J (p(j))}. (3.11)
J

Finally, we pick a small positive dwell-time constant T, which is greater than peaking

time, and proceed according to the following algorithm.

Step 1. Define initial time, say to := 0, the set of indices I 2:: {1,2, - - .,N}, and

an arbitrary initial value for i E I, say i := 1.

Step 2. Put the controller u = 12"), defined by (3.4) and (3.8), into the loop for

64

t6 ltn, to + T).

Step 3. For t Z to + T we continuously check the inequality (3.10) using current
estimates from (3.8) and (3.9). We keep the controller a 2 it“) in the loop

until the moment of time t,- Z to + T when the inequality fails.

Step 4. At t = t, we redefine := I \ {i} and choose a new value for i using

(3.11).

Step 5. Set to :2 t, and go back to Step 2.

3.4 Main Result

The following is the summary of our approach:

split the large set of parameters into a finite number of smaller subsets,

0 use any Lyapunov-function-based technique to design a smooth partial state

feedback candidate controller for each subset,

0 design an extended-order high-gain observer providing the estimates for the

states and their derivatives,

0 obtain the performance indices for the subsets, based on the algebraic equations
that must be instantaneously satisfied by the parameters, states and derivatives

of the states,

0 use the estimates, provided by the observer, to check whether the inequality for
the derivative of the Lyapunov function, corresponding to the current controller,
is satisfied and if it fails switch to the controller that corresponds to the smallest

performance index.

Theorem 3.1 Consider the closed loop—system (3.1), (3.4), (3.8), {3.9) under the
switching logic described above and with initial states in the compact set 9 x (2.
There exist positive numbers Tm,“ and Tmax, with Tm,“ < Tmax, and for every T E
(Tmimeax) there exists 5 E (0,1) such that if e,- E (0,5), for i = 1, - --,N, then
the trajectories will be bounded. Moreover, ly(t)] will be ultimately bounded by a
bound that can be made arbitrary small, provided a0 in (3.10) and e,- are chosen

sufiiciently small.

Proof. Assuming that, for a fixed value of i, the controller (3.4) is put into the

loop, define the scaled estimation errors

 

 

 

A x T

Tl = (FIE—flan ‘) ' ° '1xn;$n1¢(pv$7Cag(i)(i1C)) _ in+1:l 1

C _6 (i) A A £1
£1: . a £2 :¢(p,$7C!g ($,C)) _C21 S: '

5’ 52

Using (3.8) and (3.9), it can be shown that

it = Ax + B [2500,96, C, g“’(1‘,C)) + A¢‘"(p,$, 917,60],
C = 2 (19,33, (,9“’($,C)) + Aib“’(p,x,C,n,5.-),
(3.12)

51".? : AO‘mn + eianAfiéli) (p, I, C? T], 8i),

at = Air + s.BdA¢i"(p.x, c. n. 5.).

66

where

Ae"’(p,l‘, 971,81) = <15 (11,136 11‘”) — (25 (10,11, CM"),

Afr/)(i)(pa$7 €177,753) : ll} (p9$1<2fl(i)) — 11) (1313319150),

A (i) , _ (i), ,a _ “(1') - 5
$1 (1)11374177381) _ 1(p,$,C,I,L) $1 (yaC1$)I)9

Awl‘lp. 2. c. n, a.) = wi'lp, 2, c, 2. 2) — We, c. 2 :2),

2: = A2 + 182,.+1 + H(e.-)(y — 02).

in-l-l = ((90993: C) 11(0) — ”n+1:

H(e,~)(y — Ci) = H(€i)(l‘1 — 531)=l011€?-1, "n aanm,

-21 I 0
Aol = a Bol = 7
—I 0 I

 

 

 

 

5 —al 1 0 0 0
—022 0 1 0 0
Am, = , Bo", = ,
—a,, O 0 1 0
_ -—oz,,+1 0 0 ~ - 0 . _1_

and A0), Aom are Hurwitz matrices.

From the foregoing expressions for :3: and :i:, it can be seen that A¢(‘)(-),

Adm”, Aol’)(-), and Aibli’(-) depend continuously on e, and using (3.7) it can

be shown that there exist nonnegative constants L1, L2, L3, L4, L5, L6 such

67

that
llAel"(p2:v,C.n€i)ll S L1+ Lzllvll,

”At/Mm, cw” 3 L3 + Linn”,
IIAe“’(p,z,C,n,€z-)ll S lelvll,
llAi/J"’(p,a:,C,n,€¢-)Il S lelvll,

(3.13)

provided (:r,C) E U(‘)(R) and e,- E (0,1).
Let Pom = P3,; be the positive definite solution of the Lyapunov equation

A0,, 0 A3,, 0
+
0 A0) 0 AT

ol

Pom Pom=_la

 

 

and define the constants

av“) , _ . . 6V“) , _ - .
Ki =Sup {—_—a—(Q-L'£_C_) [A$+ B¢(p,$,€,g(z)($,<))] + _._a.(Ci_Cl w(p)x7gvg(z)(x7<))}1

where supremum is taken over all (:17, C) E U (”(R), :i: E Rn“, f) E P, and the
N

number R0 E (0, R) is such that Q C H U (‘)(R0). Let
i=1

T _ R_R0
m“_K1+K2+o--+KN

 

and assume that T < Tm”.

First of all, it is clear from the definition of K,- and the description of the switching
logic, that if at t = to the controller a“) is put into the loop and (r(to),C(to)) E
U(‘)(R.-) for some R,- E [R0,R — TKg), then for t E [t0,to + T) there will be no
switching and (x(t),C(t)) E U(‘)(R,- + TK,) C U (’)(R). We show that during this I
time the estimation errors must decay to 0(a) values. Using (3.13), it can be shown

that the derivative of the Lyapunov function

77 77

5

Mai) = Pom

 

 

 

 

68

 

 

satisfies 2

+C1

2

WS—1
51

+C2

 

 

 

 

 

 

 

 

l3]
_.,)

 

 

[U] [U]
C C
for some nonnegative constants c, and c2. Hence,
1 n2 1 0

VV _<_ — (— — Cg) - —
25; 6 28,- 6

Assuming that
1
e,- <5S min{ ,—}
4C2

- c
IVs-3m vwzfia
52

 

 

 

 

 

[Z] -

 

 

 

 

 

 

 

 

 

it can be shown that

for some positive constants c3 and c4. Therefore, as long as W(n(t),C(t)) 2 efc4,

we have

Wee). 4(2)) 3 wave). (050)) 2“t — tales/5" s 53%; Ht - Was/e
for some c5 > 0. After a short peaking period of the order of 0(5, log(e,)), the fast
variables 17 and C decay to, and stay at, 0(e,-) values. This situation may change
only with another control switching that could induce another peaking period.
Since 5log(5) —+ 0 as 5 —> 0, for an arbitrarily chosen Tm,“ E (O,Tm,,,,) there
exist 5 such that for every T E (Tmin, Tm“) and every 5,- E (0, 5), the fast variables
n(t) and C(t) decay to 0(e,-) values. Consequently, since lA¢(‘)(-)l S L5llnl| and

llAw(i’(')ll S lelnll for t _>_ to + T, we must have

(i) , (i) . O (i) (i) .
3;; (iiC) 5: + %_(ia C) (2 = %($,C) :13 + %—($,C) C + 0(5i)'

 

It is clear now that for any fixed a0 > 0, provided 5 > 0 is chosen small enough, if

i = i” then for t Z to + T, inequality (3.10) is satisfied, no more switching occurs,

69

 

 

 

and the trajectories must enter an invariant set of 0“.) (0(e,.)) size with 0".)(-)
being a class ICOO function. Similarly, if the inequality is satisfied for t Z to + T for
some i 79 i‘ ultimate boundedness follows. However, in this case the invariant set
is guaranteed to be of 0“) (0(5,-) + a0) size and, correspondingly, not only 5 but
also a0 must be chosen sufficiently small to ensure practical regulation (i.e. ultimate
boundedness of the solutions with an arbitrary pre-specified bound).

If at some time t,- 2 to + T (3.10) fails, another candidate controller would be put
into the loop immediately. It is left to notice that no more then N switching times
may occur and that the choice of Tm,x guarantees that independent of whether or
not (3.10) is violated, (r(t), C(t)) cannot leave the set F] U(’)(R). Q.E.D.

i=1

Remarks.

1. In the special case when the system is obtained via dynamic extension from
an n-th order input-output model there is no need to estimate C using the
high-gain observer (3.9) since 2“) in this case represents a chain of integrators
driven by the input, whose state is readily available. Also, in a slightly more
general case, when some of the equations in the C subsystem are independent
of the uncertain parameters, the estimation of the corresponding derivatives can

be done in a simpler way. Assume that for some j we have

(j = ¢j($i C? It),

then we can estimate the derivative C,- by

623- : ¢j(i) C) U),

though the high-gain observer estimate would work as well. It is not hard to see
that the forgoing analysis would still work with such minor modification because

as soon as lla: — at” = 0(e,-) we have [le — C2)” = 0(2)). Clearly, in this way

70

we may achieve certain reduction in the order of the observer. However, we
note that we pay a price since the contribution of the corresponding part in the

expression for the performance index J (p(‘)) must disappear.

2. For the case when the state feedback control law (3.2) ensures ultimate bound-
edness only and, correspondingly, there are small positive constants added to
the right-hand sides of (3.3), the main result still holds true and the change in

the foregoing analysis will be insignificant.

3. It is also possible to require that the control law and all its partial derivatives
be globally bounded in 2:2, - - - ,1" but not in :31 since 2:1 E y is readily
available for the feedback and does not need to be estimated. This observation
is especially useful when for each small subset of the set of parameters it is

_ possible to design an output (possibly dynamic, as noticed above) feedback
control law. In this case, saturating the estimates outside of the region of

interest is not necessary.

3.5 Examples

The main assumption used to develop our result for a class of uniformly observ-
able (with respect to control signals) uncertain systems is the existence of a family
of stabilizing candidate regulators for each small subset of the space of parameters.
We would like to show now how the whole procedure, including the design of the
candidate regulators and subdividing the initial large set of parameters, works for
some challenging situations. Specifically, as stated in the introduction, we are inter-
ested to see what are the advantages of switching for the problem of control design
of parametrically uncertain non-minimum phase systems and systems with unknown

high-frequency gainsz.

 

2See the next chapter for an example of this kind.

71

3.5.1 Linear non-minimum phase system

Our first example is a family of linear non-mininnnn phase systems. We shall not
formulate the procedure for the general class of linear control systems since our main
interest is not to develop a superior technique for linear system stabilization but to
develop a strategy that is easily applicable for a class of nonlinear systems as well. We
shall illustrate our approach on a simple one-parameter example in order to explain
what can go wrong when uncertainty is significant and why switching is essential.

Consider the family of one-parameter linear systems
231 = 22, 2.32 =‘U—Zl +22, y: 22 —p21, (3.14)

where [21, 2le is the state vector, v and y are the control and measured output
signals, and p is an unknown parameter that belongs to a known large compact set
”P C R. It is easy to see that all the systems in the family are of relative degree 1
and are non-minimum phase for p Z 0.

We are going to show that when p is known the system can be stabilized via
linear dynamic feedback control with an arbitrary guaranteed rate of decay of the
solutions. However, a single linear control law cannot ensure even stability for the .
whole family when the set 'P is sufficiently large. After that, we show how to resolve
this difficulty with the help of a logic-based switching design as proposed above.

Following the idea of [46] (see also [7, 45, 3], and [23, Section 145]) we will employ
dynamic extension and pole-placement techniques.

The system (3.14) can be represented by the second-order differential equation

v-v+y=v-pv-

72

Extending the dynamics by one integrator at the input

and introducing a new vector of state variables [x C lT = [2:1 1:2 C lT with

131:3!) $223]! (:7),

we obtain a linear system in the form (3.1):

 

:i: _ a: _ y _ :r
l,l=.4(,.)l +3., llzcl l, (3.15)
C C C C
where
0 1 0 0
_ _ _ 10 0
A(p)= —l 1 -p, B: 1, Czl l.
0 0 1
0 0 0 1

It is easy to check that the pair (A(p), R) is (uniformly with respect to p) control-
lable and the pair (0, A(p)) is (uniformly with respect to p) observable for every
p E IR.

We design first a state feedback control law for (3.15)

u = K(q)[2 ClT = k1(q)21 + k2(q)22 + k3(q)<, (3.16)

where q is a nominal value for the unknown parameter p, K (q) is designed so
that [24(q) + BK (q) + 01] is Hurwitz, and a > 0 is the desired rate of decay of the
solutions. This control law can be represented as a proper transfer function from y

to v. It is equivalent to the dynamic output feedback law

v = w + k2(r1)y. 21') = 121(2):) + k3(q)[w + k2(q)yl (3.17)

73

i

for the original system (3.14). It is worth noting that we are able to design this filter,
which allows us to avoid computation of derivatives of the output, essentially using
linearity. This trick would not work for a more general nonlinear system but we would
still proceed using a high-gain observer and saturating the control law outside of the
region of interest.

The necessary and sufficient conditions for stability of the closed-loop system

characteristic polynomial
Ap(3) = d8tlSI _ (AU?) +BK)] 2' 83+ (‘1 — k2 — k3)S2 + (1 - 191+ka +k3)s+pk1— R3,

are
a1=—1—k2—k3>0, a2(p)=1—k1+pk2+k3>0,

a3(p) = pkl — 1:3 > 0, a1 - a2(p) — a3(p) > 0,
where for convenience, we omitted the dependence of K on the fixed value q. Sup-
pose {0,1} C ’P. The conditions a3(1) > 0 and a2(p) > 0 imply a3(1) + a2(p) =

1 + pkg > 0 and, correspondingly,

1 1
——>a>-—a
In M

The conditions 03(p) > 0 and a2(0) > 0 imply a3(p) + a2(0) = 1 + (p — 1)k1 > 0

and, correspondingly,
1

—— > k > .
lp-H 1 u—u
Finally, the conditions a; > 0 and a2(0) > 0 imply —k2 > k3 + 1 > It; and,

correspondingly,
1

1
—1+—a>a>-1————2
u-u

M

74

Therefore,

2
(11 = —(k3+1)—k2 > ——, (12(0) = (k3+1)—k1 >

-a;,(()) > —1
Ipl

w—u’ w—u

and

al ' (12(0) — (13(0) > 0

lPl'lP—1l_1_lP_—1l>
cannot be satisfied with, say, p = 3. We conclude that any fixed choice of K in
(3.16) cannot guarantee stability of the closed-loop system for all possible values of
p when P is sufficiently large.

It is clear that had P been small we could have stabilized the system with a single
feedback gain K. Therefore, it is reasonable to subdivide P into smaller subsets,
design a regulator for each subset separately and use estimates of higher derivatives
for fast parameter identification. In order to organize switching, we need to construct
a family of Lyapunov functions satisfying a linear analog of (3.3). Toward this, let

P(q) be the solution of the Lyapunov equation

P(q)[4(q) + BK ((1) + 01] + [4(2) + BKM) + 01 lTP(<1) = -1-

El

as a Lyapunov function for the system (3.15), (3.16) and compute

V: l1 3:] = -20V- [:6] (I-Gp(<1)) [2:],
C C C C

where Gq(p) = P(q)[/l(p) — A(q)l + [A(p) — A(q)lTP(q). It is easy to see that the

We take
T

P(a)

1'

V0510:

 

 

 

T
P(<1)

 

 

matrix Gq(p) is affine in p and Gp(p) = 0. Consequently, there exists 6,,(p) > 0

75

such that

q — 6,,(p) < p < q + 6,,(p) implies V S —20V.

Since P is compact, there exist a number N, a finite set of points {p(ll, - - - , p‘Nl} C

P and positive numbers 61, - - - ,6N such that

N
P C UP“), where P“) 2 (pm — 5,,p‘i) — (5,) for i = 1, - - -,N.
i=1
Now, assuming that for each i we follow the foregoing procedure to design K (pm)
we must have that V S —20V along the solutions of the closed-loop system (3.15),
(3.16) with q = p“), provided p E P“).

We use the following linear high-gain observer to estimate 2: and C :
1 . 3 . ~. . 3 - ,
$1=$2+E(y—xl), $2=m3+-E—2(y—a:1), $3 = —(y—:cl) (3.18)
where e > 0 is sufficiently small so that
. i2 . .
53 = . = 33 + 0(5), C2 = k1(q)y + ($200532 + k3(P)C = C + 0(5)
1133

as soon as a short period of peaking is over. Note that, since (3.16) is implemented
dynamically in the form of (3.17), the signal u is not available and therefore we
cannot take C2 = u.

Finally, after the dwell-time period we need to check the inequality

T (I:
Sao-
.J

P(q)

I
A

T
.’L‘

P(q)

I

C

+0

A

2

 

 

 

 

 

 

 

C

with q = p“), where i is the index of the controller in the loop and a performance

76

index for (3.11) can be taken as
J(p(j)) : 153 + 151— 272 + p(j)C — C2 .

3.5.2 Nonlinear system and adaptive control

The purpose of the next example is to compare our logic-based design with a
more classical adaptive design. We borrow the example from [20, 21] and consider

the system:
0 2 . e
212224-621, 222u+z31 23=—23+21, 31:31,

where 21,22,23 are the states, 0 is an unknown parameter that belongs to the
interval [0, 2], u is the control input, y is the output, and el = 21 — y. is the
output error with y, being a reference signal.

We use an adaptive state feedback controller design from [21]:

a = v = Sat ($75) (—Ké - u — 3) +1}- 26(3)?) + £12 + 3233') + 3153)), (3.19)
3‘ . .. .. 1 22 A r.
0 = 7 Pro) (Sat (.25) (4(eTPb)(yy + y + yy))) , (3.20)
él 3? - yr 91 2
where e = 52 = y- y. = Q2/E , K = 4 , P is the solution of the
é3 if — yr Q3/52 3

Lyapunov equation for the closed-loop system with d = 0, i.e.,

0 1 0 0 0 —2 —1 0 0
0 0 1 P + P 1 0 —4 = 0 —1 0 ,
—2 —4 —3 0 1 —3 0 0 —1

77

Proj is the smoothed projection [39, 21]:

[1 + (2 — é)/0.1lp ifél > 2 and 1,9 > 0
Proj(,9) = [1+ (6— 0)/0.1l1p ifé < 0 and p < 0,

1,9 otherwise

the saturation levels are indicated as subscripts, and the scaled high-gain observer is

implemented as3
541 = (12 + 3(61 _ (11), 542 = (13 + 3(61— QI)1 5‘13 = 81- (11- (3-21)

As an alternative design, we propose to use (3.19) with the extended high-gain

observer

8‘ll: (12 + 4(61— C11), €fl2 = (13 + 6(81— 91), 542 = 94 + 4(61- (11), 544 = 81 — 91
(3.22)
and switch the value of 0 among the elements of the finite set {0, 0.01, . - - , 1.99, 2}

when the Lyapunov inequality

0 1 0 —1 0 0
227‘ 0 0 1 1022—27“ 0 —1 o éSao
—2 —4 —3 0 0 —1

with 5 = [52 53 q4/e3 lT is violated using the performance index
* ,. c. c . 1 c2 ,. 2.
«1(9) = lq4/€3+y§3’ -u-y+y-v-29(yy+y +111!)-

We take 0 = 1, 'y = 10, e = 0.001, T = 0.03, a0 = 1 and compare performance
under adaptive control (3.19), (3.20), and (3.21) (dashed lines) and our logic-based

 

3We use a high-gain observer that is different from the one in [21]. All the eigenvalues of the
observer characteristic polynomial are taken real to avoid unnecessary control oscillations during the
transient.

78

 

switching control (solid lines) for three different reference trajectories: y, = 0.5,

y. = 0.1sin(t), and yr = 0.5 sin(t). We show the tracking error e1, the control signal

u, the control signal for an extended system v, and the value of the parameter 9.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

e1(t) u(t)
1.5 1 -
\\ /'~
{I \‘ Ol ,’{A\ ......
1." \r\ l {ill \..___ -_.._,. 0
’l \‘\\\ _1 l . .4; . . .....
l. ,. l"~,\ . 1 .. .. . ,. - ll , .. , . .......
0.5 \\\ . I 2 l;
\\ ‘ l I]
\'\,‘ \ -3-l ”1',
0* ‘r l a“
-41, ........ .............. ,
lll!"
-O.5 4 -5 V/ 2 r
0 5 10 0 5 1O
V(t) 9“)
100 - . 2.5 r i
, , 2.1.. _ ............
50’ ...... .. ...... . |
: ‘ 1.5,..‘,(\. . _ ......
m’r-‘r—Tr-r-T-
olr/m ‘ 1‘],
l 0.5, . , ............. .......... 1
01 ................ . ............. ‘. l
'100 ‘ ‘ -o.5 , 4
0 5 1O 0 5 10

Figure 3.1: Simulation results for y, = 0.5 : the dashed line is for adaptive control
and the solid line is for switching control.

The performance under the logic-based controller is the same in all three cases
with only one switching time right after the dwell-time period to a controller that
corresponds to a value of d that is very close to 9. This is not surprising since
our identification procedure here results in approximating the solution of the linear
algebraic equation J (0) = 0. For the adaptive controller, we can see that since the
first reference signal is not persistently exciting é converges to a wrong value and

since the second one corresponds to a very small excitation level convergence of the

79

 

91(1) U(t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

2 1
’/.\ of If} \ \‘ ,N _- I ‘N‘; ,,,,,, / A \ -<
1.5 .,l \ l i ‘_ —.._-
. \ '
1": “\X -1 I all
1 ’ ‘ Z“ i l I)
\'1‘\ __ l .
o 5 X“ ’ 2 l i)“
1. \ ; -3 l l)
\ l 1'
o >-—- . 1 1
-4 . _
I;
’0.5 _5 ‘1
5 10 0 5 10
Wt) 9 (t)
100 . . 2.5 .
2 , .1 ..... , . .................. .,
50 L. I l’ 1
1 .5 ’ l l l A l
. ‘ l l I \ ‘— ____________
o L ’V’\:\ 4 i 1 '. L_l|f...l.ll.....‘.. .1 ..'..‘.. ..._..,h_ ...,-..m'. ,; _,",,,.'_.;,,,;__~,_',,., __;,__.5
l1 . .
l ; : 1' .
_ 0.5. .......
_50 . . . . . . , . . ...... -
- o ................ j ........... 1 ...........
-100 ‘ A -05 '. i
0 5 10 0 5 10

Figure 3.2: Simulation results for y, = 0.lsin(t) : the dashed line is for adaptive
control and the solid line is for switching control.

parameter is too slow; it also takes longer for the parameter estimate to converge
when the third reference signal is tracked. However, the closed-loop performance
under both strategies is close and the only advantages achieved by the switching

controller are a little better settling time and better identification in the case of not

persistently exciting reference signal.

3.6 Remarks

Extending the idea of the previous chapter, we have proposed a new design tech— ,

nique for output and state feedback control to practically stabilize a class of para-

80

e1“) u(t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.5 - . . 1 , 2‘ 2 ,,~\
/, -. _\ /, x, / \
I \\ 0 l I "\‘\,,/' / \\ "J/ / \1
1* 1 -1) l \/
‘9 1
:3.) l l,
‘.\\.\ j ‘ _2 l ll . . . . , . .......
0.5, \\.\ .. . l l ll
\ . -3 ’l I, .....
.\\ Fl ll
. \ i .
0 L‘- — ”4 W11];
-5 .’l‘ ........
-0.5 -6
0 5 10 0 5 10
V0) 0(1)
I 2,3‘. ......... .
. 1'5,|‘ ...... ...... ...le
[ll /\ l I
o l //, F‘"\__. #nfi-th-—M~~f~——~m—~~Li_4 1 -' - l 5 3.. rs ,- ...—.'....cd 2:: that-1.: _ 1" i
d . .
l i ‘
0.5 ,_ ......... , .......... , ..........
-50» i
o .. ............ . ............... ...........
‘1“) ‘ 4 -05 '. ;
0 5 10 0 5 10

Figure 3.3: Simulation results for y, = 0.5 sin(t) : the dashed line is for adaptive
control and the solid line is for switching control.
metrically uncertain uniformly observable nonlinear systems. Compared to our earlier
work and to many results available in the literature, we do not impose any kind of
minimum-phase assumption neither do we assume that the sign of the high-frequency
gain is parameter independent. Sliding-modebased design of the candidate controllers
has been replaced by a fairly general Lyapunov-based design. A new identification-
based switching logic has been developed instead of pre—routed search. As a result,
the transient time has been shortened significantly improving the overall performance
of the closed-loop system.

The developed procedure has been demonstrated by an example of uncertain non-

minimum phase linear system and an example of non-linear systems, previously re-

81

ported in the literature. However, what is not clear from the analysis presented in
Chapters 2 and 3 is robustness of the proposed procedure with respect to (non differen-
' tiable) measurement noise and sampling, unavoidable if the controller is implemented

digitally. We investigate this issue in the next chapter via simulation.

82

Chapter 4

Nonlinear Example: Case Study

In this chapter we consider an example of a nonlinear system with an unknown

high-frequency gain in the strict-feedback form [23, p. 595]. Our goals are:

0 to compare the performance achieved via our design and via a recently proposed

logic-based switching strategy [34, 14, 15, 28] and

o to study some practical issues related to digital implementation and measure-

ment noise.
4.1 Model
The following system has been investigated in [13, pp. 76—82] (see also [15])
21 = Plizi5 +P222, 5’32 = ”U, yc = 21 - 7‘, (4'1)

where 21 and 22 are the state variables, 1) = [p1, p2]T is a vector of unknown

parameters that belong to the finite set

42
p = U {pm} 2 {—1,—0.9,. . . ,0.9, 1} x {—1,1} C 1R2,
i=1

83

 

u is the control input, 7" is a constant reference, and yC is the controlled output.
The design procedure, presented in Chapter 3, is applicable for solving both output

and state feedback regulation problems.

4.2 Continuous Controller Design

4.2.1 Lyapunov-based switching output feedback

We start with the case when only 1:1 is available, i.e.
y = ya = 21 '— 7'.

To transform (4.1) into the form (3.1) let

a:

_ _ .,3 _ 1
$1 — 21 -— 7', 2:2 — p1~1 +p222, :1: — ,
332

so that

i: [3‘] = [ “”2 J. (4.2)
$2 3131(131 + T)2$2 + pzu

We use feedback linearization and pole-placement to derive the control law

u = —[w2y + ani2 + 3q1(y + r)25:2]/q2, (4.3)

84

where q = [(11, (12]T E 79 is a nominal value for p, w > 0, 17 > 0.25, and :32 is

an estimate for 1:2, provided by the high-gain observer

 

. . 3' —i:
E
. A 3 —:i:
m2=x3+#, (4.4)
2 9—331
x3_ 83 i

where E > 0 is sufficiently small. The parameters of the controller w and 77 are
chosen to ensure acceptable transient performance of the closed-loop system (4.2),

(4.3) with 532 = $2 and q = p, i.e.,
i1 = 33;, 1'22 = —w2x1 — anxg. (4.5)
The Lyapunov function candidate can be taken as
V($1,x2) = w(l + 77)::f + $132 + xg/w,
so that along the trajectories of (4.5)

V = —W($1,$2) = —w21:¥— (477 — 1):”;

Therefore, we start with i = 1, at which the first candidate controller (4.3), saturated

outside the region of interest, with q = p“), is given by

u = — Sat ([wle + (217w + 3p(li)(:r1 + r)2) 5:2] My) ,

85

.
‘7 A
_‘I

where Sat() is a smooth saturation function. We put this controller into the loop

and switch to another one as soon as the dwell-time period is over and the inequality

6V A , - A .
-—(:v1,:v2)1:2 + -—(:r1,:I:2)I3 + W(:I:1, 51:2) S a0,

84131 81132

fails, where the output of the high-gain observer (3.18) is used. It is possible to order
the. candidate controllers arbitrarily and to use pre—routed search, as in Chapter 2.

Alternatively, we can take
J (pm) : F3 _ 3plj)(131 + 702% _ pg)”

as the performance index for (3.11) and follow the switching logic from Chapter 3,

presented on p. 64.

4.2.2 Lyapunov-based switching state feedback

In the state feedback case,

31 =[z1—r, 2le,

we let C
. 3
(1:21—73 <2=Z2+fls (=[l],
92 C2
so that ,
. [(1] [P1 (C1 + T)3 + “(C2 " 91T3/Q2l]
= . = . (4.6)
(2 u
Following the idea of [13], we use the regulator
u = -[w2C1 + 277wcp(q, C) + 3ql(Cl + r)2<p(q, C )l/ (12, (4 7)

w(q, C) = (11(C1‘l‘ 7")3 + (1222,

86

such that the closed-loop system (4.6), (4.7) with q = p is equivalent to (4.5). We

use the family of Lyapunov functions

VmKhCQ) = V(C1,99(p"",€)) a

that are obtained from the Lyapunov function used in the output feedback, and,

correspondingly, the inequalities

v(i) . V(i) .
%§(Cl,<2)c2 + a)?“ can + W (<1, 99(1)“), 0) S “0’

with the high-gain observer

61:62 + 2(C1;C1) and 62 : (IE—2CI’ (4.8)

and the performance index
J (pm) = I62 - w(p‘”,C)
for the algorithm from p.64.

4.2.3 Scale-independent hysteresis-based switching logic

When all the state variables are available for feedback, the controller of [13] is
an alternative approach. It uses a scale—independent hysteresis-based switching logic

design. The first step of that approach is to design a multi-estimator for the closed-

) , (4.9)

loop system (4.6), (4.7). Following [13] we use the estimator

 

. (112? + (1222 21 — T
3Q13f(912f + (1222) + (12” (112? + (1222

87

where zq = [2,“, zq2 ]T and q = [q1, q2]T E P. It is easy to see from (4.2), that,

independent of u, the estimation error

Zl—T

 

 

(112? + CI232

for the true value of the parameters, q = p, satisfies the linear differential equation
ép + 6,, = 0

and therefore vanish exponentially, provided solutions of the closed-loop system exist.
In order to avoid solving all 42 systems (4.9) on-line, we use the state-sharing

approach, introduced in [16]. The system (4.9) can be rewritten as

[201 + 2‘11] [ (21 " T) + (112:1; + QZZ2

2",, + 29., qlzi‘ + q222 + 3quf + 3q1q22fz2 + qzu

The solutions of this system for every q E ’P are asymptotically close to the output

of the lower-order system

’f)0+’Uo=Zl—T, ’f)1+’Ul=Z:13, 1324-02222,
133+v3 = 2?, 134+v4 = zfzg, 135+v5 =u,

291 = v0 + (11711 + 4202, Zen = (11111 + 0202 + 3Qf'03 + 3910204 + 9205-

The hardest part of this approach is to verify that the so-called “2' j -injected system”

(4.9), (4.2) is strongly detectable [16] with respect to eq with the gain function

7d(eq) = ”eqll2 + lleqll4-

88

The second step, is to use this gain function in order to define “performance indeces”:

for every p”) E P, j E {1,---,42}.
Switching is organized as follows. Starting with initial value 2' = 1, we check the
inequality

(1') t < ' (j) t
u ( ) _. (1+ h)j€{r{}_{§42}{u ()},

where h > 0 is a fixed hysteresis constant. As soon as the inequality fails, we redefine

z' = argmin {p(j)(t)}
jE{l,---,42}

and switch to the corresponding candidate controller.

It is shown in [13] that, when there are no disturbances, the solutions of the
closed-loop hybrid system are well-defined, switching has to stop in finite time (with
some value 2' 9—- i0 6 {1, - - - ,42} and it is possible but not necessary that in = z" ),

all signals are bounded, and limtnoo [y(t)] = 0.

4.3 Practical Implementation

We discuss here some modifications of the controllers, designed aboVe, that are

needed for a practical implementation.

4.3.1 Digital implementation

Most controllers are implemented in practice using digital computers [5]. The
main reason is that some sensors provide the measurements with a certain sampling

period. In particular, instead of y(t) 2 z1 (t) — r for (4.3), (4.4), the only available

89

output signal for the feedback is

W] = y(krs) = 2mm) — r,

where 73 > O is a small sampling period of the sensor and k is a non-negative integer.
In addition, even when the signal from the sensor is not sampled, some actuators are
not capable to follow continuous signals and therefore, we use zero-order hold, i.e. do
not change the control signal during the sampling period. In particular, instead of

(4.3), we implement

u(t) = u[k] = -(w"’y[kl +2nwi2 +3q1(y[k] +r)25:2)/q2 for kn. s t < (k+1)r...

where 7,, is a sampling period of the controller, 5:2 is provided by a sampled-data
digital implementation for (4.4). Discretization for the high-gain observer can be done

in various ways [5]. Here we follow Euler’s forward formula and use

 

an: +1] = 5:1[k]+ To (mm + 3W] '5' 5’ 1W) ,

 

an. +1] = 552[k] + 7,, (2241.1 + 3%] ‘ film),

52

83

:83[k + 1] = i3[k] + 10(M) ,

where To is the discretization step. For simplicity, we assume that T5 = Tu = To.

In order to implement the Lyapunov-based logic, we also assume that the dwell—
time constant 1' is larger than Tu in a natural number times.

Similarly, we find discretizations for both dynamic state feedback controllers, de-

scribed in Section 4.2.

90

4.3.2 Suppressing measurement noise

Another issue that has to be analyzed is noise in the measurements provided by
sensors. To obtain a more realistic model, we have to take into account the fact that
the measured output is always contaminated by noise. We assume that the output
signal for each dynamic controller is subject to a random additive error. In particular,
the signal y, used in the dynamic output feedback controller (4.3), (4.4), is not equal
to 21 — 7‘ but

y = 2.1 — r + a,,w(t),

where w(t) is a measurable function of t with [w(t)] S 1 and an is a parameter
that defines the level of measurement noise. It is not hard to see that, since w(t)
is not necessarily differentiable, the change of variables used in Chapter 2 to deal
with smooth output disturbances is not applicable. It is also intuitively clear that,
since the high-gain observer (4.4) basically estimates two derivatives of the (formally
not differentiable) signal y(t) , certain level of noise (that is not suppressed due to
sampling) may cause i the control signal to saturate and the closed-loop system to
become unstable. A commonly used practical way to avoid this singularity is to
pass all the measurements through a low-pass stable linear filter of a sufficiently high
order. Following this practice, we are going to use a standard Butterworth filter,
provided by the ‘Analog Filter Design’ block from the Signal Processing Blockset
of Matlab/Sz'mulink, having 1'3 as a sampling time. In particular, for the dynamic

output feedback controller we will have
#:0y(na) + bno_1#:o-ly(no~l) + . . . + blfln?) + y : 21 .. 7‘ + flaw“),

where no is the order of the filter; bno_1,-- -,b1 are the positive coefficients, such
that 1/(s"° + bur 13%“ + - -- + bls + 1) is the standard transfer function of the

Butterworth filter, designed to suppress all the frequencies greater than 1; and pa > 0

91

is a small parameter (the cut-off frequency of the filter is l/pn rads/sec). We note
that this parameter has to be accurately tuned using extensive simulations; when [1,,
is too small the filter is useless, however, when it is too large, the filter may hurt the
transient and damage the overall performance of the closed-loop system even in the
noise-free case.

Similarly, we can modify the state signal used in the dynamic state feedback
controllers, i.e. (4.7), (4.8) and the one with hysteresis-based scale-independent logic,
implementing two filters. It is worth noting, however, that the stability proofs for the
later controller, presented in [13, 15], are not valid even in the presence of smooth
exponentially vanishing disturbances and it is not clear how to modify them.

Due to the presence of noise, we are going also to modify the Step 4 of the

algorithm, presented on p. 64 and use

Step 4. At t = t,, if I 94 {2'}, we redefine I := I \ {2}. Otherwise, we restore
the initial value of I, defining I := {1, - - - ,42}, and increase (10 in (3.10), so

that a0 2: 10ao. Then, we choose a new value for i using (3.11) as before.

The reason for this ad hoc modification is as follows. It is intuitively reasonable to
expect that the measurement noise, ignored at the design stage, must result in some
non-observer related additional errors in (3.10) even in the best case scenario. We
notice also that the initial value of the constant a0 might be crucial for the level of

acceptable noise.

4.4 Simulation Results

All the simulation results are done using M atlab/Sz'mulz'nk. The Simulink diagrams
and Level 1 Matlab files used to simulate the sampled-data state feedback controller
with the modified logic of Chapter 3 are given in the appendix. The programs for

all the other regulators are similar. For numerical integration, we have used ‘ode23s’

92

(stiff / Mod. Rosenbrock) and have checked some results with ‘ode45’ (Dormand—
Prince).

We present below simulation results for different sampled-data controllers, dis-
cussed above, discretized with the sampling period T5 2 7",, = To = 0.0001 and with
1' = 1.0, w = 1.0, and 77 = 0.7. We note that slightly larger values of the sampling
period would still results in the same performance as for the continuous implementa-
tion in a disturbance free case and slightly larger values of the sampling periods would
significantly increase the time of computation. We show the system’s regulated state
21(t) = 31(t) + r = (1(t) + 1' (column 1), the corresponding generated control input
u(t) (column 2), and the index, z'(t), of the controller put into the loop (column 3).

The controllers are numbered so that

( (i)

(n 04+Olfi—1L—D, Hi<22
p1 7P2 =

(—1+OJU—22J) Hi222'

4.4.1 Noise-free case

In Figures 4.1 and 4.2 we show results for two output feedback controllers. The
second rows correspond to the pre-routed search of Chapter 2, and the first rows
correspond to the identification-based logic of Chapter 3.

In the case when the index of the system’s parameters is small, i.e. close to the
initial value of the index, we note that, due to the longer transient (e.g. in the case of
z" = 10 in Fig. 4.1), pre-routed search results in a slightly larger overshoot. However,
we note that the control signal in the case of pre-routed search is smoother due to a
more graduate change in the control amplitude.

In the case when the initial value of the index, 2' = 1, is not close to the real one
(e.g. in the case of i‘ = 40 in Fig. 4.2), the difference in performance is significant.
We can see not only larger overshoot and longer settling time but also an undershoot.

The later is due to the fact that, when pre—routed search is used, several candidate

93

x1“) u(t) i(t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.5 10 1o
3‘3 5 ------- E 8
a, w
E ‘ 6 ......
m' OW‘
'. 4 . l
3 -5 . ......
3 2 .............
-o.5 ~ ~10 o
o 5 1o 5 10 o 5 1o

 

 

 

 

Lyap.-B. [Pre-Routed]

 

 

 

 

 

 

 

 

 

 

Figure 4.1: Improving pre—routed search: the case of z" = 10

controllers put into the loop are designed for the wrong sign of the high-frequency
gain.

For the rest of the chapter we use only Lyapunov—based design with the logic of
Chapter 3.

In Fig. 4.3, the first and the second rows show results for the Lyapunov-based
switching logic for the case of output and state feedback (1' = 0.03, 5 = 0.001, and
a0 = 0.1) correspondingly, and the third row shows results for the scale-independent
hysteresis-based logic (h = 0.1 and A = 0.5). We show here the results for z" = 30.

The parameters in this case are identified correctly and this is always the case
for the hysteresis-based logic and is the case for most values of the parameters when
Lyapunov-based switching logic is used. It is worth noting, however, that for all tried

values the shortest settling time is achieved with the output feedback controller and

94

x1“) U(t) i(t)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

1.5 . 10 50
"'3' 5 E 40 .
a) . .
.13. 2 30 ----- s +
- 0p”— :
a _5 ... . 2
_>Js 10, ........ .......
"'05 -10 0
5 10 0 5 10 0 5 10

 

 

 

Lyap.-B. [Pre-Routed]

 

 

 

 

 

 

 

 

 

 

Figure 4.2: Improving pre-routed search: the case of z" = 40

the longest with the hysteresis-based logic. In the hysteresis-based logic, we need
to allow a transient period for the multi-estimator to provide a reliable estimate for
the parameters. Such a period is typically much longer than the dwell time of the
Lyapunov-based design. During this transient period the system may operate under
a wrong controller, which may be one that is designed for the wrong sign of the high-
frequency gain. We have also observed, for all tried values of p, that (10 can be
chosen small enough to ensure exact identification in the no disturbance case for both

Lyapunov-based designs of Chapter 3.

95

Lyap.-B. [Out F.]

Lyap.-B. [St. F.]

S.-l. Hyst.-B.

In all our simulations, the signal w(t) is generated by the ‘Uniform Noise Gen-

Below we compare performance in the presence of noise for the state feedback

When the level of noise is sufficiently low, the performance of the two control laws

 

 

 

 

 

 

 

 

 

 

 

 

 

 

10

 

 

 

u(t)

 

 

 

 

 

 

 

 

 

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

 

 

 

 

 

 

 

10

 

 

 

 

 

 

 

 

 

Figure 4.3: Noise-free performance: the case of z" = 30

and initial seed 12345 or 12341.

4.4.2 Acceptable level of noise

96

erator’ block from ‘Communications Blockset’ of Simulink with sampling period 75

controller with the switching logic, proposed in Chapter 3, and for the state feedback
controller with hysteresis-based scale-independent logic, pr0posed in [13]. For sim-

plicity, we assume that both state variables are contaminated with noise of the same

is similar to the noise-free case. The low-noise case corresponds to an S 10‘5 without

low-pass filters and to an S 10‘4 with two filters, having no 2 4 and po = 0.01.

Performance of the closed-loop system is unacceptable when an reaches

0 10‘4 under the Lyapunov-based state feedback without any filters,

no .2 4 and [an = 0.01,

10’2 under the Lyapunov-based state feedback with two low-pass filters, having

10‘1 under the hysteresis-based logic without any filters, and

o 101 under the hysteresis-based logic with two low-pass filters, having no = 4

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

and [Ln = 0.01.
*1“) u(t) in)
50 . 10 so .
IL .
Q.
_l
2
3
2 4
u.
(L
_l
3' ...................
m'
'-
.J
5 10
u_ L
0. I
_l ,—\_-_.__._____
2 i
3 .
l; 5 10
A .
3' :
m‘ ....... g
n. .
I l S
T' -2 -1o 4 o ,
o,- o 5 10 o 5 10 o 5 10

Figure 4.4: Performance of the state feedback controllers without and with filters:

the case of 2"“ = 20 and an = 10’“

In Fig. 4.4 we show the simulation results for all four closed-loop systems when

97

 

on = 10“. The first and second rows corresponds to the Lyapunov-based state feed-
back without and with filters, the third and fourth rows corresponds to the hysteresis-
based logic without and with filters, respectively.

In order to illustrate our modification for the switching logic of Chapter 3, in
Fig. 4.5 we show performance under the Lyapunov—based state feedback without filters

for different levels of noise.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.5: Performance of the state feedback controller without filters in the presence
of noise of different levels: the case of z" = 40

When on = 10“ performance is almost the same as in the noise-free case. When
on = 10'3, no candidate controller ensure that the Lyapunov inequality is satisfied
with a0 = 0.1, however, after we switch through all of them and redefine: co = 1,
stabilization is successful although with a pretty large steady-state error. When

on = 10‘2 the system goes unstable.

98

 

 

 

S.-l. H.-B., no filter

 

 

 

 

 

 

 

 

 

 

 

S.-I. H.-B.. un=o.o1

 

 

 

 

 

 

 

 

 

 

Figure 4.6: Performance of the controller with hysteresis-based scale-independent
logic without filters and with filters, having po = 0.01 and no = 4: the case of
z" = 40 and co = l ‘

Finally, we show how the filters prevent the system from instability in the case of

the hysteresis-based switching.

In Fig, 4.5 we compare performance under the hysteresis-based logic without and
with filters in presence of noise with a significant noise level of an = 1. We see that
the filters save the closed-loop system from instability although the effect of noise is

not suppressed completely, as can be seen from the steady-state error.

4.5 Remarks

0 We have seen that the regulators based on the logic of Chapter 3 (summarized

on p. 64) outperform the ones based on the pre-routed search.

99

e We note that, when the sampling period is sufficiently small, the sampled-
data versions of the controllers, designed to work without sampling, perform as

expected.

0 It seems that under ideal (disturbance free) conditions the output feedback
controller outperforms both state feedback controllers. This is unusual though
we should note that the state feedback controllers in our case are designed based
on the output feedback design (for the known parameter case) and not the other

way as is done more traditionally for a less challenging control problems.

0 It is also clear from the simulation results that in the presence of measurement

noise, putting a low-pass filter does increase the acceptable level of noise.

0 It follows from our simulation results that a much higher level of noise may be
allowed in the system when the hysteresis—based scale-independent switching
logic design is used. However, we notice that no output feedback control design
is available with this approach. So, the Lyapunov-based switching output feed-
back control design, proposed in this thesis, is, to the best of our knowledge,
the only solution known for the problem when not all the state variables are

measured in the system (4.1).

100

Chapter 5

Conclusions

In this thesis, we have addressed the problems of robust tracking and stabilization
by output and state feedback for a class of single-input single-output nonlinear sys-
terns with large-scale parametric uncertainty. The main ingredients of our approach
are Lyapunov function-based dynamic feedback control design, high-gain observers,
and logic-based switching with dwell-time. The general summary of the developed

approach is as follows.

1. If needed, use the dynamic extension procedure and an appropriate change
of coordinates to transform the system into a special form, where unavailable

states (if any) are derivatives of the measured outputs.

2. To simplify design, split the large set of parameters into a finite number of

significantly smaller subsets.

3. Use a Lyapunov-function-based technique to design a smooth partial state feed-

back candidate controller for each subset of parameters.

4. Design an extended—order high-gain observer providing the estimates for all the

unmeasured states of the transformed system and their derivatives.

101

Obtain a performance index for each subset, based on the algebraic equations

CJ'I

that must be instantaneously satisfied by the parameters, states, derivatives of

the states, and the control signal.

6. Develop a dwell-time supervisor to orchestrate switching. We use the estimates,
provided by the observer, to check whether the inequality for the derivative of
the Lyapunov function, corresponding to the current controller, is satisfied and
if it fails switch to the controller that corresponds to the smallest performance

index.

7. Develop a sampled-data realization for the observer and each candidate con-

troller.

8. For each subset of parameters, use numerical simulations without logic to tune
the gains of the candidate controllers and the corresponding observers. Use
numerical simulations to tune the dwell-time constant, that has to be small
enough to avoid finite escape time and large enough to allow the transient in

the observer, provoked by peaking, to pass.

9. If sensor noise is expected to be an issue, use simulations to design fast low-pass

filters that almost do not influence the overall closed-loop system performance‘.

We have gradually introduced this approach and provided some mathematical results
and numerical examples to support it.

We have given some motivations and have solved the practical tracking problem
for a class of minimum-phase nonlinear systems in Chapter 2. There we have assumed
that the relative degree and sign of the high-frequency gain are known and have used

continuous sliding mode-based design with high-gain observers.

 

1This is possible since filter dynamics are equivalent to the fast sensor dynamics that has been
recently analyzed in [24] in the noise-free case.

102

In Chapter 3, we have shown that our design procedure is applicable for a much
larger class of nonlinear systems and for a general Lyapunov function-based design.
We have proved that practical stabilization is achieved.

A challenging nonlinear example, with a known solution for the state feedback
case, involving a recently proposed scale-independent hysteresis-based logic, has been
a subject of our investigation in Chapter 4. We have shown how to use our tech-
nique to design an alternative state feedback and a new output feedback controllers.
We have shown how to obtain a sampled-data realizations for our controllers (Mat-
lab/Simulink programs are shown in Appendix A) and how to design a low-pass
Butterworth filter in order to suppress the measurement noise. In addition, we have
compared performance of the regulators numerically.

Extensions of our approach to the multi-input multi-output case and extending
the class of systems is a possible direction of future research.

From practical point of view, it is of interest to investigate performance of our
technique for a class of electromechanical systems and design some hardware exper-
iments. .

However, the most promising and interesting direction of our future research is
providing theoretical support (stability proofs) for the discretization and low-pass

filter design procedures, investigated numerically in Chapter 4.

103

Appendix A

Listings of the programs

We show here the Simulink diagrams and Level 1 Matlab files used to simulate
the sampled-data state feedback controller with the modified logic of Chapter 3 that
have been used generating the simulation results presented in Chapter 4.

The main Simulink program (state_d.md1) is split into 8 parts as follows

Page System Name

 

1 state-d

state_d/Logic
state-d/Logic/performance indeces
state_d/Lyapunov Inequality
state-d/Sensor1
state_d/Sensor1/filter

state_d/Sensor2

0040501pr

state-d/Sensor2/filter1

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112

G:\WORK\sim_d\hgold.m Page 1
January 23, 2005 10:10:44 PM

function [sys,x0,str,ts] = hgold(t,x,u,flag,tau,vareps)
%

switch flag,

%%%%%%%%%%%%%%%%%%
% Initialization %
%%%%%%%%%%%%%%%%%%
case 0,
[sys,x0,str,ts] = mdlInitializeSizes(tau);

%%%%%%%%%%
% Update %
%%%%%%%%%%
case 2,
sys = mdlUpdate(x,u,tau,vareps):

%%%%%%%%%%
% Output %
%%%%%%%%%%
case 3,
sys - mleutputs(x);

%%%%%%%%%%%%%
% Terminate %
%%%%%%%%%%%%%
case 9,
sys - [1; % do nothing

%%%%%%%%%%%%%%%%%%%%
% Unexpected flags %
%%%%%%%%%%%%%%%%%%%%
otherwise
error(['unhandled flag - ',num23tr(flag)]);
end

%end hgold

%

%---=3:--sa-==833=====‘=-=====---:===========fl====

% mdlInitializeSizes

% Return the sizes, initial conditions and sample times for the S-function.

%=====xmaslsluan===========z====n=========uu======x--=============a====-
%

function [sys,x0,str,ts] - mdlInitializeSizes(tau)

%

sizes - simsizes;

Figure A.9: hgold.m (High-gain observer), page 1

113

G:\WORK\sim_d\hgold.m Page 2
January 23, 2005 10:10:44 PM

sizes.NumContStates
sizes.NumDiscStates

‘0

I
I ‘e

sizes.NumOutputs -
sizes.NumInputs =
sizes.DirFeedthrough
sizes.NumSampleTimes =

‘e

HOHHNO
\

‘0

ie

sys = simsizes(sizes);
x0 - [0; 0]:
str - I]:

ts = [tau 0]:

% end mdlInitializeSizes

 

 

 

mdlUpdate
Handle discrete state updates, sample time hits, and major time step
requirements.

wwwaoapdpao

function sys - mdlUpdate(x,u,tau,vareps)

sys=[x(1)+tau*(x(2)+2/vareps*(u-x(1)));
x(2)+tau/vareps“2*(u-x(1))l:

%end mdlUpdate

 

mleutputs
Return the output vector for the S-function

OPOFWWWW

function sys - mleutputs(x)
%

sys - x(2);

%end mleutputs

Figure A.10: hgold.m (High-gain observer), page 2

114

G:\WORK\sim_d\lbsl.m Page 1
January 23, 2005 10:11:37 PM

function [sys,x0,str,ts] = lbsl(t,x,u,f1ag,tau,a_0)

%

global P_curr N_curr aa_0 i_p
%

N_par=42;

%

for i_j=1:N_par,
P_init(i‘j,1)=i_j;
if i_j<22,
P_init(i_j,2)=-1+O.l*(i~j-1):
P_init(i_j,3)=-1;
else
P_init(i_j,2)=-1+O.1*(i_j-ZZ):
P_init(i_j,3)=1;
end
end
%

switch flag,

%%%%%%%%%%%%%%%%%%
% Initialization %
%%%%%%%%%%%%%%%%%%
case 0,
[sys,x0,str,ts] - mdlInitializeSizes(tau,a_O,N_par,P_init);

%%%%%%%%%%
% Update %
%%%%%%%%%%
case 2,
sys = mdlUpdate;

%%%%%%%%%%
% Output %
%%%%%%%%%%
case 3,
sys = mleutputs(t,x,u,N_par,P_init);

%%%%%%%%%%%%%
% Terminate %
%%%%%%%%%%%%%
case 9,
sys - l]; % do nothing

%%%%%%%%%%%%%%%%%%%%
% Unexpected flags %
%%%%%%%%%%%%%%%%%%%%

Figure A.11: lbsl .m (Lyapunov-based switching logic), page 1

115

G:\WORK\sim_d\lbsl.m Page 2

 

January 23, 2005 10:11:37 PM
otherwise
error([’unhandled flag = ',num28tr(flag)]);
end
%end lbsl
%
%=3===-33:==========I=============—=—— ..... ========================

 

% mdlInitializeSizes

% Return the sizes, initial conditions, and sample times for the S-function.
%ggaggggggg====================a================g======a ——————— =5

%

function [sys,x0,str,ts] = mdlInitializeSizes(tau,a_0,N_par,P_init)

%

global P_curr N_curr aa_0 i_p

%

P_curr-P_init;

N_curr=N_par;

aa_0=a_0;

i_p=1:
%

sizes - simsizes;

sizes.NumContStates = 0:
sizes.NumDiscStates = 1;
sizes.NumOutputs - 3:
sizes.NumInputs = N_par+1;
sizes.DirFeedthrough - l:
sizes.NumSampleTimes - 1;

sys = simsizes(sizes):

x0 - i_p;

str- H:

ts - [tau 0];

% end mdlInitializeSizes

%
%=-================--=‘88::3233=3:======8======88====:fl-===fl====------fl.

% mdlUpdate
% Handle discrete state updates, sample time hits, and major time step
% requirements.

 

%====:= ________ __‘_==B==--=====8'=====--Egfl-----fl=8===-========.

function sys = mdlUpdate

Figure A.12: 1bsl.rn (Lyapunov-based switching logic), page 2

116

G:\WORK\sim_d\lbsl.m
January 23, 2005

SYS=[];

%end mdlUpdate

%

z==2========a====a==============z=========a========== ----------- === -----

% mleutputs
% Return the output vector for the S-function

 

%__..

%

function sys a mleutputs(t,x,u,N_par,P_init)

%

global P_curr N_curr aa_0 i_p

%

if u(1)<aa_0,

%

elseif N_curr--l,

else

end

sys - [P_curr(i_p,1),P_curr(i_p,2),P_curr(i_p,3)]:

P_curr=P_init;

N_curr=N_par;

aa_0=aa_0*10, t, disp('too noisy?')

for ifimu=1:N_curr,
mu_curr(i_mu)=u(P_curr(i_mu,l)+1);

end

[min_mu,i_p]-min(mu_curr);

int_p-[l:i_p-1,i_p+1:N_curr]:

N_curr-N_curr-1:

P_int-P_curr(int_p,:):

clear global P_curr, global P_curr,

P_curr-P_int:

for i_mu-1:N_curr,
mu_curr(i_mu)=u(P_curr(i_mu,1)+1);

end

[min_mu,i_p]-min(mu_curr);

%end mleutputs

Page 3
10:11:37 PM

Figure A.13: 1bsl.m (Lyapunov-based switching logic), page 3

117

G:\WORK\sim_d\main_sf.m
January 23, 2005

clear,

%

o_n=4, % order of the Butterworth filter
mu_n=0.01, % cut-off frequency = 1/mu_n

%

am_n=0.000001; % am_n=0; % amplitude of noise
% .

for iiijjj=1:7,

%

am_n=am_n*10,
% mu_n=mu_n'10,
%
t_final=10.0;
%
w=1.0;
eta=0.7;
%
r=l.0; % desired output
%
%%%%%
%
h-0.1;
1ambda=0.5;
%
%%%%%
%
vareps=0.001;
tau=0.03:
a_0=0.1;
%
%%%%%
%
tau_u=0.0001: % sampling period of the controller
tau_s=tau_u; % sampling period of the sensor
tau_o=tau_u: % % sampling period of the high-gain observer
%
%%%%%
%
%%%%%
%
%
% ij-20;
ij=0;
%
for ijj=1:4,
ij=ij+10;
%

Figure A.14: main- sf .m, page 1

118

Page 1
10:12:28 PM

G:\WORK\sim_d\main_sf.m Page 2

 

January 23, 2005 10:12:28 PM
if ij<22,
pr1=-1+0.1*(ij—1);
pr2=-1:
else
prl=-1+0.1*(ij_22):
pr2=1;
end

disp('parameters:'), disp([pr1, pr2]),
disp(‘index:'), disp(ij),

%
%%%%
%
filter_off=l; % disp(‘the filter is off'),
tic, sim('state_d'), toc,
x12=y+r‘ones(size(y)); u2-u; i2=index; t2=t_ind; t2u=t_u;
%
figure,
%
subplot(431), plot(t,x12), grid, ylabel('L.-B./no filter'), title('x_l(t)'),
subplot(432), plot(t2u,u2), grid, title('u(t)'),
subplot(433), plot(t2,i2), grid, title('i(t)'),
clear x12 u2 i2 t t2,
%
%%%%
%
filter_off=-1; % disp(‘the filter is on'),
tic, sim('state_d'), toc,
x12=y+r'ones(size(y)); u2=u; iZ=index; t2=t_ind; t2u-t_u:
%
subplot(434), plot(t,x12), grid, ylabel('L.-B./\mu_n=0.01'), %title('x_1(t)'),
subplot(435), plot(t2u,u2), grid, %title('u(t)'),
subplot(436), plot(t2,i2), grid, %title('i(t)').
clear x12 u2 i2 t t2,
%
%%%%
%
filter_off=1; % disp('the filter is off'),
tic, sim('hespanha_d'), toc,
x13=y+r*ones(size(y)): u3-u; i3=index; t3-t_ind;
%
subplot(437), plot(t,x13), grid, ylabel('S.-I. H.-B./no filter'), %title('x_1¥
(tl'):
subplot(438), plot(t3,u3), grid, % title('u(t)'),
subplot(439), plot(t3,i3), grid, % title('i(t)'),
clear x13 u3 t 13,
%
%%%%

Figure A.15: main- sf .m, page 2

119

G:\WORK\sim_d\main_sf.m Page 3
January 23, 2005 10:12:28 PM

filtermoff=-l; % disp('the filter is on'),
tic, sim('hespanha_d'), toc,
x13=y+r*ones(size(y)); u3=u; i3=index; t3=t_ind;

subplot(4,3,10), plot(t,x13), grid, ylabel('S.-I. H.-B.,\mu_n=0.01'), %title('x_1¢
(tl'),
subplot(4,3,11), plot(t3,u3), grid, % title('u(t)'),
subplot(4,3,12), plot(t3,i3), grid, % title('i(t)'),
clear x13 u3 t i3,
%
end
end
return

Figure A.16: main- sf .m, page 3

120

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