a study on adaptive robust control theory of uncertain...

A Study on Adaptive Robust Control Theory of

Uncertain Nonlinear Systems and

Its Applications to Environmental Systems

Yuchao Wang

March 2016

Academic Dissertation

A Study on Adaptive Robust Control Theory of

Uncertain Nonlinear Systems and

Its Applications to Environmental Systems

Yuchao Wang

March 2016

Presented to the Graduate School of

Comprehensive Scientific Research Program

in Partial Fulfillment of the Requirements

for the Degree of Doctor of Philosophy in

Biological System Sciences

at the

Prefectural University of Hiroshima

Contents

List of Notations 1

1 Introduction 2

1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Research Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

1.3 Organization of This Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Preliminaries 9

2.1 Stability of Nonlinear Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.1 Stability Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.1.2 Basic Stability Theorems of Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.3 Barbalat Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.4 A Modified Theorem of Lyapunov . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Function Approximation Using NNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2.1 Classification of NNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2.2 Radial Basis Function NNs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2.3 Backstepping Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3 Adaptive Robust Control of Uncertain Nonlinear Systems with NNs 22

3.1 Problem Formulation and Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2 Adaptive Robust Controller Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 Stability Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

3.4 Numerical Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.1 A Mass Spring Damper System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.4.2 A Chua’s Chaotic Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.5 Problem of Water Pollution Control of River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.5.1 Water Pollution Control System Background . . . . . . . . . . . . . . . . . . . . . 40

i

3.5.2 Modeling of Water Pollution Control Systems . . . . . . . . . . . . . . . . . . . . . 41

3.5.3 Controller Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

3.5.4 Simulation of Uncertain Water Pollution Control Systems . . . . . . . . . 46

4 Adaptive Robust Controller Redesign of Uncertain Nonlinear Systems with-

out NNs 47


4.2 Adaptive Robust Controller Redesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49



4.4.1 A Mass Spring Damper System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.4.2 A Chua’s Chaotic Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.5 Problem of Water Pollution Control of River . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.5.1 Modeling of Water Pollution Control Systems . . . . . . . . . . . . . . . . . . . . . 64

4.5.2 Controller Synthesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5.3 Simulation of Uncertain Water Pollution Control Systems . . . . . . . . . 65

5 Adaptive Robust Controller Redesign of Uncertain Nonlinear Systems with

Generalized Matched Conditions 68


5.2 Adaptive Robust Controller Redesign . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70



5.5 Optimal Long–Term Management for Uncertain Nonlinear Ecological Sys–

tems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

5.5.1 Significance of Ecological System Management . . . . . . . . . . . . . . . . . . . . 78

5.5.2 Modeling of Uncertain Nonlinear Ecological Systems . . . . . . . . . . . . . . 78

5.5.3 Simulation of Uncertain Nonlinear Ecological System Management 81

6 Adaptive Robust Backstepping Control of Uncertain Nonlinear Systems 86


6.2 Adaptive Robust Backstepping Controller Design . . . . . . . . . . . . . . . . . . . . . . . . 88



ii

6.4.1 An Inverted Pendulum System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.4.2 A 3rd Order Nonlinear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

7 Conclusions and Future Directions 110

7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

7.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

Appendix A Commonly Used Basis Functions 112

Appendix B List of Published Papers 115

Acknowledgements 116

References 117

iii

List of Notations

∈ element of

⊂ proper subset of

⇒ it follows

→ it tends to

t time

R set of real numbers

R+ set of non–negative real numbers

Rn n–dimensional Euclidean space

Rn×m set of n×m real vector

A⊤ transpose of the matrix or vector

A−1 inverse of the matrix or vector

I identity matrix

L∞ a function space and its elements are the essentially

bounded measurable functions

a := b a is defined by b

| · | absolute value of the real number

∥ · ∥1 matrix or vector 1–norm

∥ · ∥ Euclidean vector norm or its induced matrix 2–norm

λmin(·) minimum eigenvalue of the matrix

λmax(·) maximum eigenvalue of the matrix

Sign(·) sign function

f : A→ B f maps the domain A into the codomain B

1

Chapter 1

Introduction

In this chapter, we will make a brief review of the current status of research on uncertain

nonlinear control systems, and identify our research objectives.

1.1 Background

It is well known that in many practical dynamical systems, such as population dynamics,

economy systems, chemical processes, traffic systems, networks systems, and so on (see, e.g.

[1]–[5] and the references therein), there are always nonlinear uncertainties which are frequently

sources of instability and poor performance. Therefore, in the past decades, a lot of attention

has been given to the study of the nonlinear systems with uncertainties, and many important

results have been obtained. If the nonlinear uncertainties are with the matched conditions,

and partially known or have partially known bounded functions, robust control schemes [6]–

[10], adaptive control schemes [11], [12], and their combinations [13]–[22], and so on, have

been developed to guarantee the stability of such uncertain systems. The robust controller

is firstly proposed in [23] without using the word “robust” in the 1960’s, and designed to

function properly provided that the uncertainties are with acceptable changes in the system

dynamics. The aim of robust control schemes is to achieve stability or robustness of the

control systems, while assuming the worst–case conditions on the acceptable changes [24].

On the other hand, the word “adaptive control” has been developed in the 1950’s (see, e.g.

[25], [26]). Adaptive controller schemes would try to perform an online estimation of the

system parameter uncertainties, so that the designed control input can avoid, overcome, or

minimize the undesirable deviations from the prescribed system behavior [24]. Adaptive control

schemes different from robust control schemes in that it does not need the priori information

about the bounds on these system uncertainties. It is obvious that their combinations such

as robust adaptive controller, and adaptive robust controller have the advantages of both of

them. Robust adaptive controller is a modified adaptive controller in order to gain certain

robustness property. Adaptive robust controller is an adaptive version of robust controller

[16]. The key ideas of these types of controllers are to design an adaptation law to estimate

the unknown system parameters or control parameters, and then to synthesize the controllers

using the partially known functions. However, in industrial control environment, some systems

are always characterized by the completely unknown uncertainties which cannot be considered

to be partially known or bounded by known functions of some unknown parameters. Hence,

2

as the useful tools for modeling the complete unknown dynamical uncertainties, the neural

networks (NNs), and fuzzy logic, and so on, have been demonstrated in several studies (see,

e.g. [27]–[32] and the references therein) and widely used to construct the control schemes

in the control literature. For example, in [33], [34], the robust fuzzy controllers have been

reported to stabilize the uncertain nonlinear systems. The analyze robust stability of these

types of controllers can be completed via quadratic stabilization, linear matrix inequalities,

and H∞ control theory. The fuzzy logic was applied in [35]–[37] to synthesize controllers whose

control parameters can be on–line adjusted by the designed adaptation laws. An observer–based

adaptive controller has been developed in [38] where the unknown nonlinear uncertainties can

be modeled via NNs. In addition, for the helicopter altitude and yaw angle tracking with

unknown uncertainties, the NNs approximation–based adaptive controllers have been proposed

in [39], and the controllers of which can provide a better control performance than the model

based controllers. Furthermore, the model predictive control based on a neural network model

is attempted for air–fuel ratio in [40], where the model is adapted on–line to cope with nonlinear

dynamics and parameter uncertainties. Practically, if the NNs or fuzzy logic has a complicated

structure, it may lead to a heavy computational burden and longer learning time (see, e.g. [41],

[42]). On the other hand, if the NNs or fuzzy logic has a simple structure, it may result in

inaccurate approximation results. Thus, it is necessary to find an appropriate structure for the

NNs or fuzzy logic. In fact, this is a very complicated process. Therefore, to overcome the

completely unknown uncertainties, using simple structure and without actually using the NNs

or fuzzy logic, the problem of controller design is still one of the hot topics in recent years.

On the other hand, in many practical systems, there always exist the mismatched uncertain

problems which cannot be ignored. In order to overcome these effects on the systems, the

backstepping approach has been proposed (see, e.g. [43], [44] and the references therein). The

backstepping procedure control is a systematic and recursive design methodology for nonlinear

systems, and it can provide an effective control method to accommodate the uncertainties miss-

ing the matched conditions. In recent years, the backstepping control method has been well

developed, and various researches have achieved to combine the backstepping control method

with other control methods. For instance, in [45], [46], based upon backstepping approach and

sliding control method, a class of continuous output feedback controllers has been proposed. In

[47], a wavelet adaptive backstepping controller is designed for a class of second–order nonlinear

systems, in which the controller comprises a virtual backstepping controller and a robust con-

troller. In addition, with the aid of backstepping approach, the fuzzy adaptive control schemes

are extended to control nonlinear systems in strict–feedback form in [48], [49]. Furthermore,

the backstepping approach–based adaptive robust control law has been proposed in [50], and

improved by [51]. It is worth noting that in order to avoid the controller singularity problems,

the virtual control coefficients in [45]–[51] are considered to be known, and set equal to 1. How-

ever, the virtual control coefficients are always time–varying, and even unknown in many cases

(see, e.g. [52]–[57] and the references therein). Meanwhile, several backstepping approach–

based control methods have been proposed to guarantee the stability for the uncertain systems

3

which contain unknown time–varying virtual control coefficients. For examples, in [58]–[61],

backstepping approach–based adaptive control schemes are developed to such systems, where

the unknown time–varying virtual control coefficients are assumed to have the known upper

bounds or be upper bounded by the known smooth functions. Being different from [58]–[61], in

[62], [63], the signal compensation–based robust backstepping control schemes require the un-

known time–varying virtual control coefficients to have known lower bounds, and the designed

control laws can guarantee both the uniform boundedness of all the closed loop signals, and

uniform ultimate boundedness of the system states. Moreover the control conditions of virtual

control coefficients can be further relaxed in [64]–[68], in which the prior lower and upper bound

knowledge of virtual control coefficients are allowed to be unknown. However, the considered

systems in [64] are discrete–time (i.e. the virtual control coefficients are not continuous). The

backstepping approach–based adaptive fuzzy controllers in [65], [66], can guarantee the uni-

form ultimate boundedness of the resulting uncertain nonlinear systems, yet neither of them

can make the solutions of the above systems converge asymptotically to zero. When the dis-

turbances do not exist in the systems mentioned in [65], [66], the backstepping approach–based

adaptive control schemes which have been proposed respectively in [67] and [68], can guarantee

the control systems to achieve asymptotically stable. In fact, there are always disturbances,

which cannot be neglected in the practical systems.

Except for nonlinear uncertainties, in many practical dynamical systems, for example, pop-

ulation dynamics, combustion engines, chemical processes, rolling mills, surgery process, and

so on (see, e.g. [1], [2], and [69] and the references therein), time–delay is also inevitable, and

frequently source of instability and poor performance. Thus, in recent years, research on un-

certain systems with time delays is also a hot topic, and the problems of robust stability and

stabilization for such systems have received increasing attention. For example, for the uncer-

tain linear systems with input delays, the prediction based adaptive feedback control schemes

have been developed in [70]–[72], and these control schemes have been further improved in

[73], which can be applied to the uncertain nonlinear systems with input delays. Furthermore,

delay–dependent H∞ control schemes, and delay–independent adaptive robust control schemes

for uncertain systems with delayed states are respectively reported in [74]–[76], and [77]–[81],

and these schemes can achieve favorable control results with regard to the time delays. It

should be pointed out that the stability control schemes or criteria of [74]–[81] are with the

help of some types of appropriate Lyapunov–Krasovskii functionals to investigate the robust

stability for the control systems. It is obvious that there is a drawback when they use the

Lyapunov–Krasovskii functionals, that is the time delays in these systems must be assumed ei-

ther to be non–negative constants, or to be time–varying functions and the derivatives of which

must be less than 1. However, in some practical systems this assumption may not be satisfied.

More recently, the control condition of unknown time–varying delays can be further relaxed in

[82]–[84]. That is for the control systems mentioned in [74]–[81], the time delays can be allowed

to be time–varying functions and the derivatives of which need not require to must be less than

1. Although the adaptive robust state feedback controllers of [82]–[84] can guarantee that the

4

states of the control systems converge to be a ball which can be as small as desired, neither of

them can guarantee that the states of the control systems converge asymptotically to zero.

1.2 Research Objectives

In this paper, we also consider the problems of robust stabilisation and adaptive robust

controller design for uncertain dynamical systems with nonlinear uncertainties which are with

matched conditions or with mismatched conditions.

System with matched nonlinear

uncertaintiesAdaptive robust controller

Output

Distu

rba

nce

s

Input

Neural networks

Figure 1.1: Feedback Control System Block Diagram (Case 1)

1. For a class of uncertain dynamical systems with matched nonlinear uncertainties and

external disturbances, we suppose that nonlinear uncertainties are completely unknown, and

the upper bounds of external disturbances are unknown. Similar to [42], we introduce the

NNs with a simple structure to model the completely unknown nonlinear uncertainties. In

particular, we do not require the norm of the approximation error of the NNs to be assumed

smaller than a known constant such as [41]. For such a class of dynamical systems (Figure 1.1),

we propose an adaptation law to learn online the norm forms of the weight and approximate

error of the NNs, and the upper bound values of the external disturbances. With the help

of the learned values, we construct a state feedback controller to guarantee that the states

converge uniformly asymptotically to zero in the presence of matched nonlinear uncertainties

and external disturbances.

2. For the same systems which has been mentioned in 1, we suppose that nonlinear uncer-

tainties are completely unknown, and the upper bounds of external disturbances are unknown.

Based on the approximation property of NNs, we employ the virtual NNs to model the com-

pletely unknown nonlinear uncertainties. That is the NNs are not really used in the control

process. For such a class of dynamical systems (Figure 1.2), we propose two adaptation laws

to learn online the uncertain parameters which contain all the uncertain bound values of these

uncertainties. With the help of the learned values, we develop a state feedback controller with

a rather simple structure to guarantee that the states converge uniformly asymptotically to

zero in the presence of matched nonlinear uncertainties and external disturbances. It is obvious

5

that without using NNs, our control schemes can achieve a better control performance than

the control schemes of [42].

System with matched nonlinear

uncertaintiesAdaptive robust controller

Output

Distu

rba

nce

s

Input


System with matched delayed

nonlinear uncertaintiesAdaptive robust controller

Output

Distu

rba

nce

s

Input


3. For a class of uncertain dynamical systems with matched delayed nonlinear uncertainties

and external disturbances, we suppose that delayed nonlinear uncertainties are completely un-

known, and the upper bounds of external disturbances are unknown. In particular, the time

delays are not required must neither to be non–negative constants, nor to be time–varying

functions and the derivatives of which must be less than 1. Similar to 2, based on the approxi-

mation property of NNs, we employ the virtual delay–dependent NNs to model the completely

unknown nonlinear uncertainties. It is clear that if the delay–dependent NNs are used in the

actual control process, the time delays must be known. Fortunately, the NNs will not appear

in the actual control process, i.e. we do not require the time delays must be known. Similar

to 2, for such a class of dynamical systems (Figure 1.3), we propose two adaptation laws to

learn online the uncertain parameters which contain all the uncertain bound values of these

uncertainties. Then, we redesign the adaptive robust controller reported in 2, and synthesize a

new adaptive robust controller so that the uniform asymptotic stability of the matched delayed

uncertain nonlinear systems can be guaranteed. It should be pointed out that, similar to the

adaptive robust state feedback controllers of [82]–[84], our control schemes are also with simple

structure and without using NNs; different from the adaptive robust state feedback controllers

6

of [82]–[84], our control schemes can make a better control performance, that is the states of

the control systems can be guaranteed to converge asymptotically to zero.

4. We consider the problem of adaptive robust stabilization for a class of nonlinear systems

with mismatched structure uncertainties, external disturbances, and unknown time–varying

virtual control coefficients. Similar to 1, we introduce a simple structure NNs to model the

unknown structure uncertainties. For such a class of uncertain dynamical systems (Figure 1.4),

we also employ the adaptation laws to estimate the unknown parameters which contain the

upper bounds of the external disturbances and the virtual control coefficients, and the norm

forms of weights and approximate errors of the NNs. Then, by making use of the updated values

of these uncertain parameters, we propose a class of backstepping approach–based continuous

adaptive robust state feedback controllers for such uncertain dynamical systems. Here, we

show that the proposed control scheme can guarantee the uniform asymptotic stability of the

uncertain dynamical systems in the presence of structure uncertainties, external disturbances,

and unknown time–varying virtual control coefficients.

System with mismatched

nonlinear uncertaintiesAdaptive robust controller

Output

Distu

rba

nce

s

Input

Neural networks


1.3 Organization of This Paper

The remainder of the paper is organized as follows. In order to expand our studies, we

introduce some preliminaries in Chapter 2. Such as:

• the stability of nonlinear systems is discussed based upon Lyapunov Functional [85];

• the function approximation using NNs is detailed;

• the backstepping control design technique is briefly reviewed.

In Chapter 3, we develop a class of NNs–based adaptive robust controllers for the nonlinear

systems with matched uncertainties, and make the corresponding convergence analysis. To

demonstrate the validity of the results, we apply them to solve a practical problem of water

pollution control. For the same systems of Chapter 3, without using NNs, we propose a new class

of adaptive robust controllers with rather a simpler structure in Chapter 4. In the same place,

7

we also do the corresponding stability analysis and utilize the proposed control schemes to solve

the same problem of water pollution control of Chapter 3. In Chapter 5, we extend the control

schemes of Chapter 4 to stabilize a class of delayed nonlinear uncertain systems with generalized

matched conditions. As a numerical example, we utilize an uncertain exploited ecosystem with

two competing species to demonstrate the validity of the results. For the nonlinear systems with

mismatched structure uncertainties, external disturbances, and unknown time–varying virtual

control coefficients, we propose a class of NNs and backstepping approach–based continuous

adaptive robust state feedback controllers in Chapter 6, where we also show that the proposed

control schemes can guarantee the uniform asymptotic stability of such uncertain dynamical

systems, and apply the proposed control schemes to solve some practical problems. Finally, in

Chapter 7, we conclude this paper with some brief remarks and make some pointers to future

work directions.

8

Chapter 2

Preliminaries

In this chapter, we introduce some preliminaries which will be used in the follow chap-

ters. Firstly, we discuss the stability of nonlinear systems based upon Lyapunov Functional

[85]. Then, we briefly describe the function approximation using NNs. Finally, we review the

backstepping control design technique.

2.1 Stability of Nonlinear Systems

In this section, we discuss the stability of nonlinear systems in state space based upon

Lyapunov functional [85]. These knowledge will be used in the following chapters to analyze

the stability properties of our control systems.

-4 4x

4

-4

x

Figure 2.1: Stable in The Sense of Lyapunov

2.1.1 Stability Definitions

Consider an autonomous system which satisfies

dx(t)

dt= f x(t), t

∣(2.1a)

x(t0) = x0, t ≥ t0, x(t) ∈ Rn (2.1b)

We suppose that f(·) satisfies the standard conditions for the existence and uniqueness of

solutions.

9

Definition 2.1. Stability in The Sense of Lyapunov

Such as Figure 2.1, the equilibrium point x(t) = 0 of (2.1) is stable in the sense of Lyapunov

(see, e.g. [86]–[88]) at t = t0, if for any constant ε > 0 there exists a constant δ > 0 such that

‖x(t0)‖ < δ ⇒ limt→∞

‖x(t)‖ < ε (2.2)

Definition 2.2. Asymptotic Stability

Such as Figure 2.2, the equilibrium point x(t) = 0 of (2.1) is asymptotically stable (see, e.g.

[86]–[88]) at t = t0, if

1. x(t) = 0 is stable in the sense of Lyapunov.

2. And, there exists a constant δ > 0 such that

‖x(t0)‖ < δ ⇒ limt→∞

‖x(t)‖ = 0 (2.3)

4

-4

x

-4 4x

Figure 2.2: Asymptotically Stable

2.1.2 Basic Stability Theorems of Lyapunov

Theorem 2.1. Uniform Ultimate Boundedness

For the system (2.1), if there is a so–called Lyapunov positive function V x(t), t∣: Rn×R →

R+ which is satisfying that

1. For any x(t) ∈ Rn, and x(t) �= 0,

V x(t), t∣> 0, and V 0, t

∣= 0 (2.4)

2. For any x(t) ∈ Rn,

α1 ‖x(t)‖∣ ≤ V x(t), t∣ ≤ α2 ‖x(t)‖∣ (2.5)

10

3. And, for any x(t) ∈ Rn,

dV x(t), t∣

dt=

)∂V x(t), t

∣∂x(t)

⎡�f x(t), t

∣+

∂V x(t), t∣

∂t

≤ −α3 ‖x(t)‖∣ + ε (2.6)

Where α1(·), and α2(·) are K∞ class functions, α3(·) is a K class function, and ε is a positive

constant. Then, the equilibrium point x(t) = 0 of the system (2.1) is said to be uniformly

ultimately bounded (or stable in Lyapunov sense).

Theorem 2.2. Uniform Asymptotic Stability




V x(t), t∣> 0, and V 0, t

∣= 0 (2.7)


α1 ‖x(t)‖∣ ≤ V x(t), t∣ ≤ α2 ‖x(t)‖∣ (2.8)


dV x(t), t∣

dt=

)∂V x(t), t

∣∂x(t)

⎡�f x(t), t

∣+

∂V x(t), t∣

∂t

≤ −α3 ‖x(t)‖∣ (2.9)

Where α1(·), and α2(·) are K∞ class functions, and α3(·) is a K class function. Then, the

equilibrium point x(t) = 0 of the system (2.1) is said to be asymptotically stable.

2.1.3 Barbalat Lemmas

Lemma 2.1. Barbalat Lemmas [89]

1. Letting f(t) : R → R be uniformly continuous on [0,∞). Suppose that limt→∞∑t

0f(τ)dτ

exists and is finite. Then,

limt→∞

f(t) = 0 (2.10)

11

2. Suppose that f(t) ∈ L∞,df(t)dt

∈ L∞, and limt→∞∑t

0f(τ)dτ exists and is finite. Then,

limt→∞

f(t) = 0 (2.11)

3. If f(t) is differentiable and has finite limit limt→∞ f(t) < ∞, and df(t)dt

is uniformly con-

tinuous, then,

limt→∞

df(t)

dt= 0 (2.12)

Proof:

1. If the first statement is not true, then there is a constant ε > 0, such that for all T0 > 0,

there always exists T > T0, satisfying |f(T0)| ≥ ε. Because f(t) is uniformly continuous,

there is a positive constant δ > 0 to satisfy |f(t + τ) − f(t)| < ε2for all t > 0, and

0 < τ < δ. Thus, for all t ∈ [T1, T1 + δ]

|f(t)| = |f(t)− f(T1) + f(T1)| ≥ |f(T1)| − |f(t)− f(T1)| >ε

2(2.13)

which implies

(((([ T1+δ

T1

f(τ)dτ

((((=

[ T1+δ

T1

|f(τ)|dτ >εδ

2(2.14)

where the equality holds when f(t) does not change the sign for all t ∈ [T1, T1+ δ]. Thus,∑t

0f(τ)dτ cannot converge to a finite limit as t → ∞, which is a contradiction. Therefore,

the first statement is valid.

2. The second statement follows from the fact that if df(t)dt

∈ L∞ then f(t) is uniformly

continuous and the first statement can be applied.

3. The third statement follows from the fact that

f(∞)− f(0) =

[ ∞

0

df(τ)

dτdτ =

[ ∞

0

df(τ) (2.15)

the right side exists and is finite. Because df(t)dt

is uniformly continuous, we can apply the

first statement with df(t)dt

instead of f(t). ��

2.1.4 A Modified Theorem of Lyapunov

Base on Theorems 2.1, 2.2 and Lemma 2.1, we can also obtain the following theorem.

Theorem 2.3. Uniform Asymptotic Stability



12


V x(t), t∣> 0, and V 0, t

∣= 0 (2.16)


α1 ‖x(t)‖∣ ≤ V x(t), t∣ ≤ α2 ‖x(t)‖∣ (2.17)


dV x(t), t∣

dt=

)∂V x(t), t

∣∂x(t)

⎡�f x(t), t

∣+

∂V x(t), t∣

∂t

≤ −α3 ‖x(t)‖∣ + δ(t) (2.18)

where α1(·), and α2(·) are K∞ class functions, α3(·) is a K class function, and δ(t) is a positive

function which satisfies that

limt→∞

[ t

0

δ(τ)dτ ≤ δ < ∞ (2.19)

where δ is a positive constant. Then, the equilibrium point x(t) = 0 of the system (2.1) is said

to be asymptotically stable.

Proof:

From (2.17), and (2.18), we can have the following inequality.

0 ≤ α1(‖x(t)‖) ≤ V (x(t), t)

= V (x(0), 0)) +

[ t

0

V (x(τ), τ)dτ

≤ α2(‖x(0)‖) +[ t

0

V (x(τ), τ)dτ

≤ α2(‖x(0)‖)−[ t

0

α3(‖x(τ)‖)dτ +

[ t

0

δ(τ)dτ (2.20)

Taking the limit on the both sides of (2.20) as t approaches infinity, we can obtain that

0 ≤ α2(‖x(0)‖)− limt→∞

[ t

0

α3(‖x(τ)‖)dτ + limt→∞

[ t

0

δ(τ)dτ (2.21)

By this and (2.19), we further have

0 ≤ α2(‖x(0)‖)− limt→∞

[ t

0

α3(‖x(τ)‖)dτ + δ (2.22)

13

That is

limt→∞

[ t

0

α3(‖x(τ)‖)dτ ≤ α2(‖x(0)‖) + δ (2.23)

Then, from (2.20), we can also obtain

0 ≤ α1(‖x(t)‖) ≤ α2(‖x(0)‖) +[ t

0

δ(τ)dτ (2.24)

On the other hand, from (2.19), there also is

supt∈[0,∞)

[ t

0

δ(τ)dτ ≤ δ (2.25)

From (2.24) and (2.25), we deduce

0 ≤ α1(‖x(t)‖) ≤ α2(‖x(0)‖) + δ (2.26)

which means x(t) is uniformly ultimately bounded. On the other side, since x(t) is uniformly

continuous, α3(·) is also uniformly continuous. Hence, according to Lemma 2.1, there is

limt→∞

α3(‖x(t)‖) = 0 (2.27)

which implies

limt→∞

‖x(t)‖ = 0 (2.28)

Thus, we can say that the equilibrium point x(t) = 0 of the system (2.1) is asymptotically

stable. ��

2.2 Function Approximation Using NNs

In the past decades, there many approaches have been developed to approximate the un-

known functions. Among of them, artificial NNs are one of the most frequently employed

approximation methods due to the fact that artificial NNs are shown to be capable of uni-

versally approximating any unknown function to arbitrary precision [90]. Similar to biological

neural networks, artificial NNs consist of massive simple processing units that correspond to

biological neurons. With the highly parallel structure, artificial NNs own powerful computing

ability, and intelligence of learning and adaptation with respect to fresh and complete unknown

data. In this paper, we want to apply NNs to approximate the nonlinear structure uncertainties

such that we can construct some classes of adaptive robust controllers. In this section, we will

make an overview of function approximation using artificial NNs.

14

2.2.1 Classification of NNs

According to whether there is a feedback loop, the NNs can be divided into two classes:

feedforward NNs and recurrent (or feedback) NNs.

1. Feedforward NNs

A feedforward neural network is a nonlinear function of its inputs, which is the composition

of the functions of its neurons. Figures 2.3 and 2.4 show sketches of two feedforward NNs,

respectively.

1

2

n

Figure 2.3: Feedforward Neural Network with 1 Hidden Layer and n Neurons

1

2

n

1

2

m

Figure 2.4: Feedforward Neural Network with 2 Hidden Layers and n+m Neurons

Therefore, a feedforward neural network is the first and simplest types of artificial neural

network devised. In this network, the information moves in only one direction, forward from

the input neurons, through the hidden neurons (if any) and to the output neurons. There are

no cycles or loops in the network.

The neurons, who are the basic processing elements of NNs, have two main components:

(i) a weighted summer, and (ii) a nonlinear activation function (Figure 2.5). The neurons that

perform the final computation, i.e., whose outputs are the outputs of the network, are called

output neurons; the other neurons that perform intermediate computations, are termed hidden

neurons (see, e.g. [91], [92]).

15

Figure 2.5: Artificial Neuron Composition

2. Recurrent NNs

A recurrent neural network is a class of artificial neural network whose connection graph

exhibits cycles between neurons. In that graph, there exists at least one feedback connection

that, following the connections, leads back to the starting neuron; such a path is called a cycle.

Since the output of a neuron cannot be a function of itself, such an architecture requires that

time be explicitly taken into account, i.e. the output of a neuron cannot be a function of itself

at the same instant of time, but it can be a function of its past values [91]. Recurrent NNs

topologies range from partly recurrent to fully recurrent networks. Figures 2.6 and 2.7 show

block diagrams of two recurrent NNs, respectively. Figure 2.6 is an example of a partly recurrent

network with recurrence in the hidden layer. Figure 2.7 shows a fully recurrent network where

each node gets input information from all other nodes.

1

2

3

Figure 2.6: Partly Recurrent Network Block Diagram

1

2

Figure 2.7: Fully Recurrent Network Block Diagram

16

Thus, different from feedforward NNs, recurrent NNs can use their internal memory to

process arbitrary sequences of inputs. This makes them applicable to do temporal process-

ing and learn sequences, e.g., perform sequence recognition/reproduction or temporal associa-

tion/prediction, where they have achieved the best known results [93].

2.2.2 Radial Basis Function NNs

The radial basis function neural network (RBFNN) is a forward neural network with good

function approximate performance. Because RBFNN has a simple structure, fast learning rate

and no local minimum problem, it is widely used like function approximation, data mining and

image processing, and so on [94]. Therefore, this paper will also use RBFNN to model the

unknown system nonlinear uncertainties.

1. The Structure of RBFNN

Such as shown in Figure 2.8, RBFNN is a standard three layer neural network: one is an

input layer (made up of source nodes with k inputs); one is a hidden layer with L neurons

whose activation functions are the radial basis functions; one is an output layer who is a simple

linear function.

1

k

)(1h

)(2h

)(hL

dy

1h

2h

hL

2

Figure 2.8: Traditional RBFNN

2. Function Approximation Property of RBFNN

Here, we briefly mention the property of function approximation using RBFNN (see, e.g.

[39]–[42] and the references therein), as shown in Figure 2.8.

Property 2.1.

For a continuous smooth function h(X ) : Rk → Rs, there exists a RBFNN system such that

17

h(X ) = Ψ�hΦh(X ) + ε(X ), ∀ X ∈ S ⊂ Rk (2.29)

where X ∈ Rk is the input, Φh(·) ∈ RL is the neural network basis function with L denoting

the number of hidden–layer neurons, Ψh ∈ RL×s is the weight matrix of output layer, and

ε(X ) ∈ Rs is the approximation error which satisfies ‖ε(X )‖ ≤ ε < ∞, and where ε is an

unknown positive constant.

According to the function approximation property of the RBFNN described by (2.29), we

can have the following results.

Consider the nonlinear function Δ(·) : Rn → Rm×n, we denote the ith row of Δ(·) to be

that

Δi(x) :=]Δi1(x) Δi2(x) · · · Δin(x) , i = 1, 2, · · · ,m

Then, for all i ∈ {1, 2, . . . ,m}, and j ∈ {1, 2, . . . , n}, the scalar function Δij(x) can be

approximated by RBFNN as follows

Δij(x) = Ψ�ijφ(x) + εij(x)

= φ�(x)Ψij + εij(x) (2.30)

where φ(x) :=]φ1(x) φ2(x) · · · φl(x)

�∈ Rl is the radial basis function (Appendix A

shows a few of commonly used basis functions), Ψij ∈ Rl is the weight vector of the output layer,

εij(x) ∈ R is the approximation error, and l represents the number of hidden–layer neurons.

Further, Δi(x) can be expressed as

Δi(x) =]Δi1(x) Δi2(x) · · · Δin(x)

= φ�(x)]Ψi1 Ψi2 · · · Ψin

+]εi1(x) εi2(x) · · · εin(x)

= φ�(x)Ψi + εi(x), i = 1, 2, · · · ,m (2.31)

where Ψi ∈ Rl×n, and εi(x) ∈ R1×n.

By (2.31), Δ(x) can be rewritten into

Δ(x) =

⎤⎥⎥⎥⎥⎥⎥⎦

φ�(x) 0 . . . 0

0 φ�(x) . . . 0

......

. . ....

0 0 . . . φ�(x)

⎣∫∫∫∫∫∫⎢

⎤⎥⎥⎥⎥⎥⎥⎦

Ψ1

Ψ2

...

Ψm

⎣∫∫∫∫∫∫⎢+

⎤⎥⎥⎥⎥⎥⎥⎦

ε1(x)

ε2(x)

...

εm(x)

⎣∫∫∫∫∫∫⎢

(2.32)

18

where we define

Φ(x) :=

⎤⎥⎥⎥⎥⎥⎥⎦

φ(x) 0 . . . 0

0 φ(x) . . . 0

......

. . ....

0 0 . . . φ(x)

⎣∫∫∫∫∫∫⎢

∈ R lm×m

and

Ψ :=

⎤⎥⎥⎥⎥⎥⎥⎦

Ψ1

Ψ2

...

Ψm

⎣∫∫∫∫∫∫⎢

∈ R lm×n, ε(x) :=

⎤⎥⎥⎥⎥⎥⎥⎦

ε1(x)

ε2(x)

...

εm(x)

⎣∫∫∫∫∫∫⎢

∈ Rm×n

Finally, we can reduce Δ(x) to

Δ(x) = Φ�(x)Ψ + ε(x) (2.33)

2.3 Backstepping Approach

The backstepping approach is a systematic and recursive design methodology for nonlinear

systems (see, e.g. [43], [44] and the references therein), and it can provide an effective control

method to accommodate the uncertainties missing the matched conditions. Because of the

recursive design methodology, the designer can start the design process at the known–stable

system, and “back out” new controllers which progressively stabilize each entry system. The

process terminates when the final external control is reached. Therefore, this approach is called

backstepping [87]. The details of backstepping approach are as follows.

Consider a system of the strict–feedback form:

x1 = x2 + f1(x1)

x2 = x3 + f2(x1, x2)

...... (2.34)

xn−1 = xn + fn−1(x1, x2, · · · , xn−1)

xn = u(t) + fn(x1, x2, · · · , xn)

where x(t) = [x1, x2, · · · , xn]� ∈ Rn, is the state, u(t) ∈ R is the control input, the unknown

functions fi(·) : R × Ri → R, i = 1, 2, · · · , n, represent structure uncertainties which are

assumed to be continuous and differentiable.

Since there are structure uncertainties, it is clearly that the system (2.34) is unstable.

19

The aim is to design a controller u(t) for stabilizing (2.34) at the origin. When applying

backstepping, there contains the following n steps: at the previous n − 1 steps, the virtual

control functions αi(·), i = 1, 2, · · · , n− 1, are developed; at step n the actual controller u(t) is

proposed, such that the system stabilization can be guaranteed.

By employing the virtual control functions αi(·), the state variables can be transformed into

the following form:

z1(t) = x1(t)

zi(t) = xi(t)− αi−1(x1, x2, · · · , xi−1), i = 2, 3, · · · , n(2.35)

The details of recursive design are as follows.

Step 1:

Substituting the first equation of (2.35) into the the first equation of (2.34), it reduces that

z1 = z2 + f(x1) + α1(·) (2.36)

Then, the virtual controller α1(·) is designed for the subsystem (2.36). The control design

is based on the following Lyapunov function candidate

V1(z1) =1

2z21 (2.37)

The virtual controller α1(·) can be picked to make the derivative of V1(·) to be that

dV1(z1)

dt≤ −z21 + z1z2 (2.38)

Step i(2 ≤ i ≤ n− 1):

Similar to Step 1, substituting the ith equation of (2.35) into the ith equation of (2.34), it

reduces that

zi = zi+1 + f(x1, x2, · · · , xi) + αi(·)− αi−1(·) (2.39)

Then, the virtual controller αi(·) is designed for the subsystem (2.39). The control design

is based on the following Lyapunov function candidate

Vi(z1, z2, · · · , zi) =1

2z21 +

1

2z22 + · · ·+ 1

2z2i (2.40)

The virtual controller αi(·) can be chosen to make the derivative of Vi(·) to be that

dVi(z1, z2, · · · , zi)dt

≤ −i∫

j=1

z2i + zizi+1 (2.41)

20

Step n:

Similar to the above n−1 steps, substituting the nth equation of (2.35) into the nth equation

of (2.34), it reduces that

zn = u(t) + f(x1, x2, · · · , xn)− αn−1(·) (2.42)

Then, the actual controller u(t) is designed so that the system (2.34) is stable. The control

design is based on the following Lyapunov function candidate

Vn(z1, z2, · · · , zn) =1

2z21 +

1

2z22 + · · ·+ 1

2z2n (2.43)

The actual controller u(t) is designed to make the derivative of Vi(·) to be that

dVn(z1, z2, · · · , zn)dt

≤ −n∫

j=1

z2n (2.44)

In the light of Theorem 2.2, from (2.43), and (2.44), the control u(t) can ensure that the

system (2.34) is asymptotically stable.

Remark 2.1. From (2.43), and (2.44), it is obviously that limt→∞ ‖z(t)‖ = 0, which means

that limt→∞ x1(t) = 0, and limt→∞ xi(t) = αi−1(·), i = 2, 3, · · · , n. Thus, if we want all the

states are asymptotically stable, we must also design that limt→∞ αi−1(·) = 0, i = 2, 3, · · · , n.

21

Chapter 3

Adaptive Robust Control of UncertainNonlinear Systems with NNs

In this chapter, we consider the problem of robust stabilisation for a class of uncertain dynam-

ical systems with matched nonlinear uncertainties. We suppose that the nonlinear uncertainties

are completely unknown, and the upper bounds of external disturbances are unknown. For such

uncertain systems, similar to [42], we introduce a simple NNs here to model the nonlinear un-

certainties. Further, we propose an adaptation law with σ–modification to learn online the

norm forms of the weight and approximate error of the NNs, and the upper bound values of

the external disturbances. With the help of the learned values, we construct a state feedback

controller to guarantee that the states converge uniformly asymptotically to zero in the pres-

ence of matched nonlinear uncertainties and external disturbances. Finally, to demonstrate the

validity of the results, we apply it to solve a practical problem of water pollution.

3.1 Problem Formulation and Assumptions

We consider a class of uncertain nonlinear dynamical systems described by the following

differential equation.

dx(t)

dt= Ax(t) + Bu(t) + f x(t)

[+ d(t) (3.1)

where t ∈ R is the time, x(t) ∈ Rn is the state, and u(t) ∈ Rm is the control input. Moreover,

A ∈ Rn×n and B ∈ Rn×m are known constant matrices, f(·) : Rn × R → Rn is a bounded

unknown nonlinear continuous differentiable vector function satisfying f x(t)[= BΔ x(t)

[,

and represents the system uncertainties, and d(t) ∈ Rn is a bounded continuous vector function

which represents the external disturbances. The aim of this chapter is to construct a state

feedback controller u(t) such that some types of stability of closed–loop system (3.1) can be

guaranteed.

Before proposing our control schemes, we first introduce the following assumptions.

Assumption 3.1. The pare (A, B) given in (3.1) is completely controllable.

On the other hand, it follows from Assumption 3.1 that for any symmetric positive definite

matrix Q ∈ Rn×n, and any positive constant μ, there exists an unique symmetric positive

22

definite P ∈ Rn×n as the solution of the algebraic Riccati equation of the form

PA+ A�P − μPBB�P = −Q (3.2)

In this chapter, we want to make use of the method of NNs to approximate the nonlinear

structure uncertainties such that we can construct a class of adaptive robust controllers. Ac-

cording the function approximation property of the NNs in Chapter 2, we can further propose

the following assumption.

Assumption 3.2. On the Ωx ⊂ Rn, there exists the weight matrix θ∗ ∈ R lm×1 such that

for x ∈ Ωx, the following approximation equation

Δ(x) := Φ�(x)θ∗ + ε(x) (3.3)

holds with ‖ε(·)‖ ≤ ε < ∞, where ε is an unknown positive constant, and where

θ∗ :=

⎤⎥⎥⎥⎥⎥⎥⎦

θ∗1

θ∗2...

θ∗m

⎣∫∫∫∫∫∫⎢

∈ R lm×1, ε(x) :=

⎤⎥⎥⎥⎥⎥⎥⎦

ε1(x)

ε2(x)

...

εm(x)

⎣∫∫∫∫∫∫⎢

∈ Rm

Φ(x) :=

⎤⎥⎥⎥⎥⎥⎥⎦

φ(x) 0 . . . 0

0 φ(x) . . . 0

......

. . ....

0 0 . . . φ(x)

⎣∫∫∫∫∫∫⎢

∈ R lm×m

and

φ(x) :=

⎤⎥⎥⎥⎥⎥⎦

φ1(x)

φ2(x)

· · ·φl(x)

⎣∫∫∫∫∫⎢∈ Rl

with φj(x), j = 1, 2, · · · , l, being chosen as the commonly used Guassian functions such as:

φj(x) = exp

)−//x− cj

//2δ2j

[, j = 1, 2, · · · , l

and where cj =]cj1 cj2 · · · cjn

∥�is any given constant vector represents the center of the

receptive field, and δj is any given constant represents the width of the Gaussian functions.

23

Remark 3.1. Being different from [39]–[40], we do not need that the weight matrix in

Assumption 3.2 has the known bounds. Similar to [42], the norm of weight matrix can be

updated on–line independent the structure of the NNs via the designed adaptation laws. Fur-

thermore the given designed adaptation laws can also update the upper bound of the norm of

approximation errors, i.e. different from [39], the prior knowledge of the approximation errors

is not needed. Therefore, in this chapter, the unknown functions can be properly approximated

via a simple structure of RBFNN via using the designed adaptation laws. Since, the struc-

ture of RBFNN is simple which will make the NNs–based control schemes more practicable to

applications.

Assumption 3.3. There exists a continuous vector function d(t) of appropriate dimensions

such that

d(t) = Bd(t) (3.4)

and

‖d(t)‖ ≤ d (3.5)

where d is an unknown positive constant.

Without loss of generality, we also make the following definition:

Definition 3.1.

ψ∗ := ‖θ∗‖2 + 2ε+ 2d (3.6)

Since θ∗, ε, and d are unknown constants, it is obviously that ψ∗ is still an unknown positive

constant.

3.2 Adaptive Robust Controller Design

In this section, we propose an adaptive robust state feedback controller which can guarantee

the nonlinear uncertain systems described by (3.1) to some types of robust stability.

For system (3.1), we design a state feedback controller as follows.

u(t) = ϕ x(t), ψ(t)[

= −1

2μB�Px(t)− 1

2

B�Px(t)κ2 x(t), ψ(t)[

//B�Px(t)//κ x(t), ψ(t)

[+ σ(t)

(3.7)

where P ∈ Rn×n is the solution of the algebraic Riccati equation described by (3.2), μ is a

given positive constant, and σ(t) ∈ R+ is a give positive function which satisfies

limt→∞

[ t

0

σ(τ)dτ ≤ σ < ∞ (3.8)

24

where σ is an unknown positive constant.

And, where κ(x(t), ψ(t)) is defined as:

κ(x(t), ψ(t)) := ψ(t) +//Φ(x)//2 (3.9)

and ψ(t) is the estimate of the unknown ψ∗ which is updated by the following adaptation law:

dψ(t)

dt= −γσ(t)ψ(t) +

1

2γ//B�Px(t)

// (3.10)

where γ a given positive constant.

Then, applying (3.3), (3.5) and (3.7) to (3.1), we can obtain the follow closed–loop system.

dx(t)

dt= Ax(t)− 1

2μBB�Px(t)

−1

2

BB�Px(t)κ2 x(t), ψ(t)[

//B�Px(t)//κ x(t), ψ(t)

[+ σ(t)

+B)Φ�(x)θ∗ + ε(x)

{+Bd(t) (3.11)

On the other hand, defining

ψ(t) := ψ(t)− ψ∗

we can rewrite (3.10) to the following error system:

dψ(t)

dt= −γσ(t)ψ(t) +

1

2γ//B�Px(t)

//− γσ(t)ψ∗ (3.12)

3.3 Stability Analysis

According to the above descriptions, in this section, we can propose the the following theorem

which shows that the asymptotic stability of nonlinear uncertain system described by (3.11)

and the error system described by (3.12) can be guaranteed.

Theorem 3.1. Consider the adaptive closed–loop nonlinear uncertain system described by

(3.11) and the error system described by (3.12). Suppose that the Assumptions 3.1–3.3 are

satisfied. Then, the solution (x(t), ψ(t)) (t; 0, x(0), ψ(0)) of (3.11) and (3.12) are uniformly

ultimate bounded, and

limt→∞

x(t) = 0 (3.13)

Proof:

Consider closed–loop nonlinear uncertain systems described by (3.11) and (3.12), we choose

the following positive definite Lyapunov candidate function.

V (x(t), ψ(t)) = x�(t)Px(t) + γ−1ψ2(t) (3.14)

25

where P ∈ Rn×n is the solution of (3.2), and γ is a given positive constant.

Letting (x(t), ψ(t)) be the solution of (3.11) and (3.12) for t ≥ 0. Then, by taking the

derivative of V (·) along the trajectories of (3.11) and (3.12) it is obtained that for t ≥ 0,

dV (x(t), ψ(t))

dt= 2x�(t)P

dx(t)

dt+ 2γ−1ψ(t)

dψ(t)

dt

= x�(t)(PA+ A�P − μPBB�P )x(t)

−x�(t)PBB�Px(t)κ2 x(t), ψ(t)[

//B�Px(t)//κ x(t), ψ(t)

[+ σ(t)

+2x�(t)PB Φ�(x)θ∗ + ε(x)[

+2x�(t)PBd(t) + 2γ−1ψ(t)dψ(t)

dt(3.15)

It follows from (3.2) and (3.15) that

dV (x(t), ψ(t))

dt= −x�(t)Qx(t)

−//B�Px(t)

//2κ2 x(t), ψ(t)[

//B�Px(t)//κ x(t), ψ(t)

[+ σ(t)

+2x�(t)PB Φ�(x)θ∗ + ε(x)[

+2x�(t)PBd(t) + 2γ−1ψ(t)dψ(t)

dt(3.16)

Notice the fact that

X�Y ≤ ‖X‖‖Y ‖, ∀X, Y ∈ Rl

By this, and Assumptions 3.2, 3.3, from (3.16), we can further obtain

dV (x(t), ψ(t))

dt≤ −x�(t)Qx(t)

−//B�Px(t)

//2κ2 x(t), ψ(t)[

‖B�Px(t)‖κ x(t), ψ(t)[+ σ(t)

+2//B�Px(t)

// (‖Φ(x)‖‖θ∗‖+ ‖ε(x)‖)

+2//B�Px(t)

//‖d(t)‖+ 2γ−1ψ(t)dψ(t)

dt

≤ −x�(t)Qx(t)

−//B�Px(t)

//2κ2 x(t), ψ(t)[

‖B�Px(t)‖κ x(t), ψ(t)[+ σ(t)

+2//B�Px(t)

//‖Φ(x)‖‖θ∗‖+2ε

//B�Px(t)//+ 2d

//B�Px(t)//

26

+2γ−1ψ(t)dψ(t)

dt(3.17)

On the other hand, consider that for ∀a, b ∈ R, there is

2ab ≤ a2 + b2

By this, and (3.6), from (3.17), we can have

dV (x(t), ψ(t))

dt≤ −x�(t)Qx(t)

−//B�Px(t)

//2κ2 x(t), ψ(t)[

‖B�Px(t)‖κ x(t), ψ(t)[+ σ(t)

+‖Φ(x)‖2//B�Px(t)//

+ ‖θ∗‖2 + 2ε+ 2d[//B�Px(t)

//+2γ−1ψ(t)

dψ(t)

dt

= −x�(t)Qx(t)

−//B�Px(t)

//2κ2 x(t), ψ(t)[

//B�Px(t)//κ x(t), ψ(t)

[+ σ(t)

+‖Φ(x)‖2//B�Px(t)//+ ψ∗

//B�Px(t)//

+2γ−1ψ(t)dψ(t)

dt(3.18)

Applying (3.9), and (3.12) into above equation yields,

dV (x(t), ψ(t))

dt= −x�(t)Qx(t)

−//B�Px(t)

//2κ2 x(t), ψ(t)[

//B�Px(t)//κ x(t), ψ(t)

[+ σ(t)

+‖Φ(x)‖2//B�Px(t)//

+ ψ∗ + ψ(t)[//B�Px(t)

//−ψ2(t)σ(t)− ψ2(t) + 2ψ∗ψ(t)

[σ(t)

= −x�(t)Qx(t)

−//B�Px(t)

//2κ2 x(t), ψ(t)[

//B�Px(t)//κ x(t), ψ(t)

[+ σ(t)

+ ψ(t) + ‖Φ(x)‖2[//B�Px(t)//

− ψ2(t) + 2ψ∗ψ(t) + (ψ∗)2[σ(t)

27

−ψ2(t)σ(t) + (ψ∗)2σ(t)

= −x�(t)Qx(t)

−//B�Px(t)

//2κ2 x(t), ψ(t)[

//B�Px(t)//κ x(t), ψ(t)

[+ σ(t)

+κ x(t), ψ(t)[//B�Px(t)

//− ψ(t) + ψ∗

[2σ(t)

−ψ2(t)σ(t) + (ψ∗)2σ(t)

≤ −x�(t)Qx(t)

+

//B�Px(t)//κ x(t), ψ(t)

[ · σ(t)//B�Px(t)//κ x(t), ψ(t)

[+ σ(t)

+(ψ∗)2σ(t) (3.19)

Then, in the light of the following fact

0 <ab

a+ b≤ a, ∀a, b > 0

From (3.19), we can further reduce

dV (x(t), ψ(t))

dt≤ −x�(t)Qx(t) + σ(t) + (ψ∗)2σ(t)

= −x�(t)Qx(t) + ςσ(t) (3.20)

where

ς := 1 + (ψ∗)2

Moreover, for any positive definite matrix Q, there is

λmin(Q)‖x(t)‖2 ≤ x�(t)Qx(t) ≤ λmax(Q)‖x(t)‖2

According to this fact, (3.20) yields

dV (x(t), ψ(t))

dt≤ −λmin(Q)

//x(t)//2 + ςσ(t) (3.21)

Furthermore, letting

z(t) :=]x�(t) ψ�(t)

�(3.22)

We can obtain from (3.21) that

dV (z(t))

dt≤ −λmin(Q)

//x(t)//2 + ςσ(t) (3.23)

28

On the other hand, in light of the definition of Lyapunov function which is described by

(3.14), there always exist two positive constants αmin, and αmax such that

α1 ‖z(t)‖[ ≤ V (z(t)) ≤ α2 ‖z(t)‖[ (3.24)

where

α1 ‖z(t)‖[ := αmin‖z(t)‖2

α2 ‖z(t)‖[ := αmax‖z(t)‖2

Then, from (2.23) and (2.24), we want to show that the solution z(t) of the adaptive closed-

loop nonlinear system is uniformly bounded, and that the state x(t) converges asymptotically

to zero.

It is obviously that from (2.23) and (2.24), there is

0 ≤ α1(‖z(t)‖) ≤ V (z(t))

= V (z(0))) +

[ t

0

V (z(τ))dτ

≤ α2(‖z(0)‖) +[ t

0

V (z(τ))dτ

≤ α2(‖z(0)‖)−[ t

0

λmin(Q)//x(τ)//2dτ +

[ t

0

ςσ(τ)dτ (3.25)

Firstly, taking the limit as t approaches infinity on both sides of (3.25), we can obtain that

0 ≤ α2(‖z(0)‖)− limt→∞

[ t

0

λmin(Q)//x(τ)//2dτ + lim

t→∞

[ t

0

ςσ(τ)dτ

= α2(‖z(0)‖)− limt→∞

[ t

0

λmin(Q)//x(τ)//2dτ + ς lim

t→∞

[ t

0

σ(τ)dτ (3.26)

By this and (3.8), we can further have

limt→∞

[ t

0

λmin(Q)//x(τ)//2dτ

≤ α2(‖z(0)‖) + ς limt→∞

[ t

0

σ(τ)dτ

≤ α2(‖z(0)‖) + ςσ (3.27)

On the other hand, from (3.25), we can also have

α1(‖z(t)‖) ≤ α2(‖z(0)‖) +[ t

0

ςσ(τ)dτ

≤ α2(‖z(0)‖) + ςσ (3.28)

29

which implies that z(t) is uniformly bounded. Since z(t) has been shown to be continuous, x(t)

is also continuous. Hence, according to Lemma 2.1, from (3.27), there is

limt→∞

λmin(Q)//x(t)//2 = 0 (3.29)

That is

limt→∞

//x(t)//2 = 0 (3.30)

which means

limt→∞

x(t) = 0 (3.31)

Thus, we can complete the proof of Theorem 3.1. ��

3.4 Numerical Example

In this section, two numerical examples are used to demonstrate the applicability of the

proposed NNs–based adaptive robust control schemes.

3.4.1 A Mass Spring Damper System

Figure 3.1: A Mass Spring Damper System

Similar to [41] in Figure 3.1, we also consider a mass spring damper system in the presence

of structure uncertainties, external disturbances, and the behaviour of system is described by

the following equation:

my + b1y|y|+ k1y + k2y3 = u+ b2ω (3.32)

where y ∈ R is the displacement of the car away from equilibrium position, m is the link to the

spring, b1y|y| is the nonlinear dissipation or damping with b1 > 0, k1y + k2y3 is the nonlinear

spring term with k1, k2 > 0, u is the control input respecting an external force, ω(t) is the

external disturbance, b2 ∈ R+ is the disturbance gain. The purpose is determined a proper

30

controller u to guarantee mass spring damper system (3.32) stable on the unique equilibrium

position y = 0.

Under the transformations

x1(t) = y(t)

x2(t) = y(t)

The mass spring damper system (3.32) can be written into the state space form as follows.

dx(t)

dt= Ax(t) + f x(t)

[+Bu(t) + d(t) (3.33)

where

x(t) =

]x1(t)

x2(t)

⎡, A =

]0 1

0 0

⎡, B =

⎤⎥⎦ 0

1

m

⎣∫⎢

f(·) =

⎤⎥⎦ 0

− 1

m

)b1x2(t)|x2(t)|+ k1x1(t) + k2x

31(t)

{⎣∫⎢

=

⎤⎥⎦ 0

1

m

⎣∫⎢)− b1x2(t)|x2(t)| − k1x1(t)− k2x

31(t)

{

= BΔ x1(t), x2(t)[

d(t) =

⎤⎥⎦ 0

1

m

⎣∫⎢ b2ω(t) = Bd(t)

where we define

Δ · [ := −b1x2(t)|x2(t)| − k1x1(t)− k2x31(t)

d(t) := b2ω(t)

For simulation, similar to [41], the following parameters have been given for this system.

m = 8, b1 = 3, b2 = 0.5

k1 = 1, k2 = 1

And, the external disturbance is taken as

ω(t) = cos(2t)

31

Then, we will determine the structure of proximate NNs to model the nonlinear function Δ(·),and synthesize an adaptive robust controller so that the mass spring damper system (3.32) is

stable on the unique equilibrium position y = 0 (x1(t) = 0).

Step.1:

Similar to [42], we take a simple RBFNN, and give a Gaussian bias function for the RBFNN

as follows

Φ(x) = [Φ1(x), Φ2(x), · · · ,Φl(x)]�

Φi(x) = exp

}−‖x− ci‖2

δ2i

(, i = 1, 2, · · · , l

where the node number l = 7, the centers ci = [ci1, ci2]�, and ci1, and ci2 are evenly spaced in

[−3, 3], [−3, 3] with all δi = 0.5.

Step.2:

Then, taking

μ = 4, Q =

]4 0

0 6

⎡

under solving (3.2) we can obtain

P =

]7.4883 4

4 7.4883

⎡

Step.3:

For the controller (3.7) and adaptation law (3.10) choosing

γ = 1, σ(t) = 2 exp(−0.5t)

Finally, we choose the initial values as follows

x(0) =]3 −3

�, ψ(0) = 2

With chosen parameters setting as above, we can obtain the simulation results shown in

Figures 3.2 and 3.3 for this uncertain mass spring damper system.

It can be observed from Figure 3.2 that by employing the NNs–based adaptive robust control

schemes proposed in this chapter, one can guarantee the states of mass spring damper system

to converge asymptotically to zero. On the other hand, Figure 3.3 also shows that the adaptive

law ψ(t) is indeed uniformly bounded.

32

0 5 10 15-3

-2

-1

0

1

2

3

Time (Sec)

Res

ponse

ofSta

teV

ariable

x(t

)

x2(t)

x1(t)

Figure 3.2: Responses of States x1(t) and x2(t) (Mass Spring Damper System)

0 5 10 150

0.5

1

1.5

2

Time (Sec)

History

ofA

daptatio

nLaw

ψ(t)

Figure 3.3: History of Adaptation Law ψ(t) (Mass Spring Damper System)

33

3.4.2 A Chua’s Chaotic Circuit

In this subsection, such as Figure 3.4, we consider a typical Chua’s chaotic circuit consists

of one linear resistor (R), two capacitors (C1, C2), one inductor (L), and one nonlinear resistor

(λ).

Figure 3.4: A Chua’s Chaotic Circuit

Firstly, we define

z(t) :=

⎤⎥⎥⎦z1(t)

z2(t)

z3(t)

⎣∫∫⎢ :=

⎤⎥⎥⎦vc1(t)

vc2(t)

iL(t)

⎣∫∫⎢

Then, the dynamic equations of Chua’s chaotic circuit can be written into the following form

(see, e.g. [96], [97]).

dz(t)

dt= Gz(t) +Hλ z1(t)

[(3.34)

where

G =

⎤⎥⎥⎥⎥⎥⎥⎥⎦

− 1

C1R

1

C1R0

1

C2R− 1

C2R

1

C2

0 − 1

L−R0

L

⎣∫∫∫∫∫∫∫⎢, H =

⎤⎥⎥⎥⎦− 1

C1

0

0

⎣∫∫∫⎢

λ(·) = az1(t) + cz31(t), with a < 0, b > 0

Since (3.34) is not satisfying the form of (3.1), we need to perform a linear transformation

to transform it into the form of (3.1).

Similar to [96], we introduce the following definition.

z∗(t) := T−1z(t), or z(t) := Tz∗(t)

34

where T is a transformation matrix which is given as

T =

⎤⎥⎥⎥⎥⎥⎥⎥⎦

− R +R0

C1C2RL

RR0C2 + L

C1C2RL− 1

C1

− R0

C1C2RL− 1

C1C2R0

1

C1C2RL0 0

⎣∫∫∫∫∫∫∫⎢

Using the transformation in [96], the transformed system can be obtained as

dz∗(t)dt

= T−1GTz∗(t) + T−1Hλ z1(t)[

= G∗z∗(t) +H∗λ z1(t)[

(3.35)

where

G∗ = T−1GT

H∗ = T−1H

Here, similar to [96], choosing the parameters as follows:

R =10

7, R0 = 0, C1 = 1, C2 =

19

2

L =19

14, a = −4

5, c =

2

45

After computation, we can further obtain

dz∗(t)dt

= Az∗(t) +Bf z∗(t)[

(3.36)

where

A =

⎤⎥⎥⎦0 1 0

0 0 1

0 0 0

⎣∫∫⎢ , B =

⎤⎥⎥⎦0

0

1

⎣∫∫⎢

f(·) =14

1805z∗1(t)−

168

9025z∗2(t) +

1

38z∗3(t)

− 2

45

)28

361z∗1(t) +

7

95z∗2(t) + z∗3(t)

[ 3

The aim is to design a NNs–based adaptive robust controller which can make the transformed

system to track a reference signal. Therefore, the closed–loop configuration of (3.37) with

35

external disturbance can be represented by

dz∗(t)dt

= Az∗(t) +B)f z∗(t)

[+ u(t) + d(t)

{(3.37)

where u(t) is the control function which will be determined later, d(t) is the external disturbance

which will be chosen as 0.5 sin(2t).

The control objective is to control the state z∗(t) of the system to track the reference yd(t)

such as

limt→∞

z∗1(t) → yd(t) (3.38a)

limt→∞

z∗2(t) → yd(t) (3.38b)

limt→∞

z∗3(t) → yd(t) (3.38c)

Similar to [96], for simulation, we selected yd(t) = 1.5 sin(t). Then, we introduce the following

definition.

x(t) =

⎤⎥⎥⎦x1(t)

x2(t)

x3(t)

⎣∫∫⎢ :=

⎤⎥⎥⎦z∗1(t)− 1.5 sin(t)

z∗2(t)− 1.5 cos(t)

z∗3(t) + 1.5 sin(t)

⎣∫∫⎢

By this, and system (3.37), we can further have the following error system.

dx(t)

dt= Ax(t) +Bu(t) + BΔ x(t)

[+Bd(t) (3.39)

where

Δ(·) =14

1805x1(t) + 1.5 sin(t)

[ − 168

9025x2(t) + 1.5 cos(t)

[

+1

38x3(t)− 1.5 sin(t)

[ − 2

45

)28

361x1(t) + 1.5 sin(t)

[

+7

95x2(t) + 1.5 cos(t)

[+ x1(t)− 1.5 sin(t)

[[ 3

+ 1.5 cos(t)

It is obviously that the control objective can be transformed to control the transformed state

x(t) such that:

limt→∞

x(t) = 0 (3.40)

Then, we will determine the structure of proximate NNs to model the nonlinear function

Δ(·), and synthesize an adaptive robust controller so that (3.38) or (3.40) can be ensured.

According to the design procedure, the controller is given in the following steps.

36

Step.1:

Similar to [42], we take a simple RBFNN, and give a Gaussian bias function for the RBFNN

as follows

Φ(x) = [Φ1(x), Φ2(x), · · · ,Φl(x)]�

Φi(x) = exp

}−‖x− ci‖2

δ2i

(, i = 1, 2, · · · , l

where the node number l = 21, the centers ci = [ci1, ci2, ci3]�, and ci1, ci2, and ci3 are evenly

spaced in [−5, 5], [−5, 5], and [−5, 5] with all δi = 0.5.

Step.2:

Then, taking

μ = 2, Q =

⎤⎥⎥⎦24 21 5

21 22 3

5 3 2

⎣∫∫⎢


P =

⎤⎥⎥⎦16.9120 18.9442 6.9282

18.9442 27.8957 10.9443

6.9282 10.9443 6.9121

⎣∫∫⎢

Step.3:


γ = 20, σ = 0.3 exp(−0.5t)


x(0) =]5 −5 −2.5

�, ψ(0) = 4


Figures 3.5–3.10 for this Chua’s Chaotic Circuit.


schemes proposed in this chapter, one can guarantee the states of the error system (3.39) to

converge asymptotically to zero. Also, Figures 3.6–3.8 show that the states z∗1(t), z∗2(t), and

z∗3(t) can track their reference trajectories yd(t), yd(t) and yd(t), respectively. On the other

hand, Figure 3.10 also shows that the adaptation law ψ(t) is indeed uniformly bounded.

37

0 5 10 15-5

-4

-3

-2

-1

0

1

2

3

4

5

Time (Sec)

Res

ponse

ofSta

teV

ariable

x(t

)x3(t)

x2(t)

x1(t)

Figure 3.5: Responses of States x1(t), x2(t), and x3(t) (Chua’s Chaotic Circuit)

0 5 10 15-2

-1

0

1

2

3

4

5

Time (Sec)

Sta

teVariable

z∗ 1(t

)and

Its

Ref

eren

cey

d(t

)

yd(t)

z∗1(t)

Figure 3.6: State z∗1(t) and Its Reference Signal yd(t) (Chua’s Chaotic Circuit)

38

0 5 10 15-4

-3

-2

-1

0

1

2

3

Time (Sec)

Sta

teVariable

z∗ 2(t

)and

Its

Ref

eren

cey

d(t

)yd(t)

z∗2(t)


0 5 10 15-3

-2

-1

0

1

2

3

4

Time (Sec)

Sta

teVariable

z∗ 3(t

)and

Its

Ref

eren

cey

d(t

)

yd(t)

z∗3(t)


39

0 5 10 150

5

10

15

20

25

Time (Sec)

History

ofA

daptatio

nLaw

ψ(t)

Figure 3.9: History of Adaptation Law ψ(t) (Chua’s Chaotic Circuit)

3.5 Problem of Water Pollution Control of River

In this section, we will apply our NNs–based adaptive robust control schemes to solve a

problem of river pollution.

3.5.1 Water Pollution Control System Background

It is well known that fresh water is an exceptionally important resource, because life on earth

is ultimately dependent on it. However, the fresh water resources on earth are finite. Moreover,

because of water pollution, renewable fresh water resources are decreasing. Therefore, the

purification of polluted river has become one of the serious problems.

To solve this problem, many methods have been proposed, in the past decades. Among

them, biological treatment of organic contaminants has been widely used. Biological treatment

methods use microorganisms, which are mostly bacteria, in the biochemical decomposition of

polluted river to stable end products. More microorganisms are formed and a portion of the

waste is converted to carbon dioxide, water and other end products. Generally, biological

treatment methods can be divided into aerobic and anaerobic methods, based on availability

of dissolved oxygen (see, e.g. [98], [99] and the references therein).

Here, we briefly introduce the principle of biological treatment method. In the rivers, there are

two kinds of bacterium such as aerobic bacteria and anaerobic bacteria. For organic pollutants

in the river, when the oxygen is insufficient, the anaerobic bacteria will decompose; when the

oxygen is sufficient, aerobic bacteria will decompose. When anaerobic decomposition happens,

40

a strong foul odor generates, and the river begins to be contaminated. On the other hand,

when aerobic decomposition happens, the final product is a stable substance, and the river can

be automatically purified [98].

From above principle, it is obviously that in order to control the aerobic decomposition about

the organic contaminants, it is necessary to control the dissolved oxygen (DO) in the river. In

addition, aerobic decomposition causes oxygen consumption. Therefore, the biochemical oxygen

demand (BOD) of recycling aerobic decomposition is also needed to be considered. Then, based

on the DO and BOD in the river, we can model the river water quality control system.

3.5.2 Modeling of Water Pollution Control Systems

Similar to [3], [42], z(t) and q(t) will be used to denote the concentrations per unit volume of

BOD and DO, respectively, in the reach at time t. Here, by a reach one means a stretch of river,

of some convenient length, which has an industrial waste treatment facility at its beginning.

Suppose that the flow rate is constant and water is well–mixed in the reach. Then, by mass

balance concentrations one can obtain the following differential equations which describe the

dynamical behavior of BOD and DO in the reach.

dz(t)

dt= −k1(t)z(t) +

QE

m+ u1(t)[ −Q

Ez(t)

v

+υ1(t) (3.41a)

dq(t)

dt= −k3(t)z(t) + k2(t) qs − q(t)

[ − QEq(t)

v

+u2(t) + υ2(t) (3.41b)

where ki(t), i = 1, 2, 3, are respectively the BOD decay rate, the DO reaeration rate, and

the BOD deoxygenation rate. qs denotes the DO saturation concentration, QE is the effluent

flow rate, and v is the constant volume of water in the reach. Moreover, υ1(t) and υ2(t) are

respecticvely the uncertain disturbances that affect the rates of change of BOD and DO in the

reach. In addition, m is a constant whose optimal value is the BOD concentration of effluent

into the reach corresponding to steady state conditions at a desired level of DO concentration.

u1(t) and u2(t) represent respectively the additional controlled variation of BOD concentration

from its optimal value m∗, and the in–stream aeration rate in the reach, i.e. these are the

controls introduced to drive the river system response to and then maintain it inside a calculable

neighborhood of the desired steady state response in the presence of uncertainties.

In this chapter, for (3.41) we assume that the rate coefficients ki(t), i = 1, 2, 3, to be of the

form

ki(t) = hi +Δhi(t), i = 1, 2, 3 (3.42)

where hi, i = 1, 2, 3, are known positive constants, and Δhi(t), i = 1, 2, 3, are some bounded

41

functions which represent uncertain disturbances. Thus, from (3.41) and (3.42) the nominal

equation without controls, i.e. in the absence of uncertainties, can be written as follows.

dz(t)

dt= −h1z(t) +

QEm−Q

Ez(t)

v(3.43a)

dq(t)

dt= −h3z(t) + h2 qs − q(t)

[ − QEq(t)

v(3.43b)

If the desired steady state value of DO concentration in the reach has been given as q∗, then the

corresponding steady state value z∗ of BOD concentration and the effluent BOD concentration

m∗ should satisfy the following equilibrium equations.

0 = −h1z∗ +

QEm∗ −Q

Ez∗

v(3.44a)

0 = −h3z∗ + h2 qs − q∗

[ − QEq∗

v(3.44b)

It follows from (3.44) that one can obtain

z∗ =1

h3

)h2 qs − q∗

[ − γq∗{

(3.45a)

m∗ =1

γh3

h1 + γ[)

h2 qs − q∗[ − γq∗

{(3.45b)

where γ := QE

(v. Therefore, the optimal steady state z∗, q∗

[corresponding to m∗ should

satisfy the dynamical systems described by

dz(t)

dt= −h1z(t) +

QEm∗ −Q

Ez(t)

v(3.46a)

dq(t)

dt= −h3z(t) + h2 qs − q(t)

[ − QEq(t)

v(3.46b)

Thus, if the effluent BOD concentration is adjusted with respect tom∗, the dynamical systems

described be (3.41) can be modified into the following form.

dz(t)

dt= −k1(t)z(t) + γ m∗ + u1(t)

[ − γz(t) + υ1(t) (3.47a)

dq(t)

dt= −k3(t)z(t) + k2(t) qs − q(t)

[ − γq(t) + u2(t) + υ2(t) (3.47b)

Therefore, if one defines

x1(t) := z(t)− z∗

x2(t) := q(t)− q∗

Then, the uncertain water pollution control systems can be written to be the following

42

dynamical systems with nonlinear uncertainties.

dx(t)


[+ d(t) (3.48)

where

x(t) =]x1(t) x2(t)

�

u(t) =]u1(t) u2(t)

�

A =

]−(h1 + γ) 0

−h3 −(h2 + γ)

⎡

B =

]γ 0

0 1

⎡

f(·) =

⎤⎦ −Δh1(t) z∗ + x1(t)

[−Δh3(t) z∗ + x1(t)

[+Δh2(t) qs − q∗ − x2(t)

[⎣⎢

d(t) =]υ1(t) υ2(t)

�

It can be observed from the above definitions that

f x(t)[

= BΔ x(t)[

(3.49a)

d(t) = Bd(t) (3.49b)

where

Δ(·) =

⎤⎥⎦ −1

γΔh1(t) z∗ + x1(t)

[−Δh3(t) z∗ + x1(t)

[+Δh2(t) qs − q∗ − x2(t)

[⎣∫⎢

d(t) =

]1

γυ1(t) υ2(t)

⎡�

Moreover, for the simulation of this numerical example, similar to [3] and [42], we choose the

following parameters:

q∗ = 6.0, z∗ = 0.625, qs = 10.0

h1 = 0.32, h2 = 0.2, h3 = 0.32

γ = 0.1

43

For simulation, we give the uncertain time–varying parameters as follows.

υ1(t) = 0.15 sin(3t), υ2(t) = 0.15 cos(2t)

Δh1(t) = −0.08 cos(3t), Δh2(t) = 0.08 sin(3t)

Δh3(t) = 0.05 sin(5t)

Now, the problem is to determine a NNs–based adaptive robust control scheme which is

described by (3.7) with (3.10), such that the stability of (3.41) can be guaranteed in the presence

of uncertain disturbances.

3.5.3 Controller Synthesis

Then, we will determine the structure of proximate NNs to model the nonlinear function

Δ(·), and synthesis an adaptive robust controller so that the asymptotically stability of (3.38)

or (3.40) can be ensured. According to the design procedure, the controller synthesis is given

in the following steps.

Step.1:

Similar to [42], we take a simple RBFNN, and give the Gaussian bias functions for the RBFNN

as follows

Φ(x) = [Φ1(x), Φ2(x), · · · ,Φl(x)]�

Φi(x) = exp

}−‖x− ci‖2

δ2i

(, i = 1, 2, · · · , l

where the node number l = 21, the centers ci = [ci1, ci2]�, and ci1, and ci2 are evenly spaced in

[−10, 10], and [−10, 10] with all δi = 1.

Step.2:

Then, taking

μ = 2, Q =

]11.98 18.84

18.84 34.48

⎡


P =

]3 2

2 4

⎡

Step.3:


γ = 0.5, σ(t) = 2 exp(−0.5t)

44

0 5 10 15-10

-8

-6

-4

-2

0

2

4

6

8

10

Time (Days)

BO

D( x

1(t

)) and

DO

( x2(t

)) Res

ponse

s(m

g/l)

x2(t)

x1(t)

Figure 3.10: Responses of States x1(t) (BOD) and x2(t) (DO) (mg/l) (Water Pollution ControlSystem)

0 5 10 151

2

3

4

5

6

7

8

Time (Days)

History

ofA

datatio

nLaw

ψ(t)

Figure 3.11: History of Adaptation Law ψ(t) (Water Pollution Control System)

45

3.5.4 Simulation of Uncertain Water Pollution Control Systems

For simulation, we also choose the initial values as follows.

x(0) =]10 −10

�, ψ(0) = 8


Figures 3.10–3.11 for this river pollution control system.


schemes proposed in this chapter, one can guarantee the states of the pollution control systems

to be asymptotically stable in the presence of nonlinear state perturbations. On the other

hand, it can be known from Figure 3.11 that adaptation law ψ(t) is also uniformly ultimately

bounded.

46

Chapter 4

Adaptive Robust Controller Redesignof Uncertain Nonlinear Systems with-out NNs

In this chapter, we also consider the problem of robust stabilisation for a class of uncertain

dynamical systems with matched nonlinear uncertainties. We suppose that the nonlinear uncer-

tainties are completely unknown, and the upper bounds of external disturbances are unknown.

For such uncertain systems, we introduce the virtual NNs to model the nonlinear uncertainties,

i.e. the NNs are not really used in the actual control process. Further, we propose two adap-

tation laws with σ–modification to learn online the norm forms of the weight and approximate

error of the NNs, and the upper bound values of the external disturbances. With the help of

the learned values, we construct a state feedback controller with rather a simple structure to

guarantee that the states converge uniformly asymptotically to zero in the presence of matched

nonlinear uncertainties and external disturbances. Finally, to demonstrate the validity of the

result, we also apply it to solve a practical problem of water pollution.


Similar to Chapter 3, we also consider a class of uncertain nonlinear dynamical systems

described by the following differential equation.

dx(t)


[+ d(t) (4.1)

where t ∈ R is the time, x(t) ∈ Rn is the state, and u(t) ∈ Rm is the control input. Moreover,

A ∈ Rn×n and B ∈ Rn×m are known constant matrices, f(·) : Rn × R → Rn is a bounded

unknown nonlinear continuous differentiable vector function satisfying f(x) = BΔ(x), and

represents the system uncertainties, and d(t) ∈ Rn is a bounded continuous vector function

which represents the external disturbances. The aim of this chapter is to construct a state

feedback controller without NNs such that some types of stability of closed–loop system (4.1)

can be guaranteed.

Before proposing our control schemes, we also introduce the following assumptions.

Assumption 4.1. The pare (A, B) given in (4.1) is completely controllable.

On the other hand, it follows from Assumption 4.1 that for any symmetric positive definite

47

matrix Q ∈ Rn×n, and any positive constant μ, there exists an unique symmetric positive

definite P ∈ Rn×n as the solution of the algebraic Riccati equation of the form

PA+ A�P − μPBB�P = −Q (4.2)

Similar to Chapter 3, we want to make use of the NNs to approximate the nonlinear structure

uncertainties. But in this chapter, we do not really use the NNs to construct control schemes.

According the function approximation property of the NNs in Chapter 2, we can further propose

the following assumption.

Assumption 4.2. On the Ωx ⊂ Rn, there exists the weight matrix θ∗ ∈ R lm×1 such that

for x ∈ Ωx, the following approximation equation

Δ(x) := Φ�(x)θ∗ + ε(x) (4.3)

holds with ‖ε(·)‖ ≤ ε < ∞, where ε is an unknown positive constant, and where

θ∗ :=

⎤⎥⎥⎥⎥⎥⎥⎦

θ∗1

θ∗2...

θ∗m

⎣∫∫∫∫∫∫⎢

∈ R lm×1, ε(x) :=

⎤⎥⎥⎥⎥⎥⎥⎦

ε1(x)

ε2(x)

...

εm(x)

⎣∫∫∫∫∫∫⎢

∈ Rm

Φ(x) :=

⎤⎥⎥⎥⎥⎥⎥⎦

φ(x) 0 . . . 0

0 φ(x) . . . 0

......

. . ....

0 0 . . . φ(x)

⎣∫∫∫∫∫∫⎢

∈ R lm×m

and

φ(x) :=]φ1(x) φ2(x) · · · φl(x)

�∈ Rl

with φj(x), j = 1, 2, · · · , l, being chosen as the commonly used Guassian functions such as:

φj(x) = exp

)−((x− cj

((2δ2j

[, j = 1, 2, · · · , l

and where cj =]cj1 cj2 · · · cjn

∥�is any given constant vector represents the center of the

receptive field, and δj is any given constant represents the width of the Gaussian functions.

48

Remark 4.1. It is worth pointing out that, the vector function Φ(·) is used to construct

the control schemes in Chapter 3. Thus, the vector function Φ(·) of Chapter 3 needs to be

known and well designed. However, the adaptive robust controller of this chapter will not

involve the knowledge of Φ(·). Hence, the function Φ(·) is not required to be known for the

system designer. Therefore, the controller which will be given in this chapter is synthesized

without actually using the NNs.

Remark 4.2. Since φj(·), j = 1, 2, · · · , l are Guassian functions and −((x − cj((2/δ2j ≤ 0,

there is ‖φj(·)‖ ≤ 1, that is ‖φ(·)‖ ≤ √l or ‖Φ(·)‖ ≤ √

lm2. It is obviously that there are

always two positive constants p1, and p2 such that

‖Φ(·)‖ ≤ p1‖x(t)‖+ p2 (4.4)

In particular, we do not require p1, and p2 to be known.

Assumption 4.3. There exists a continuous vector function d(t) of appropriate dimensions

such that

d(t) = Bd(t) (4.5)

and

‖d(t)‖ ≤ d (4.6)

where d is an unknown positive constant.

Without loss of generality, we also make the following definition:

Definition 4.1.

ψ∗1 :=μ+ p21‖θ∗‖2

η1(4.7a)

ψ∗2 :=2 p2‖θ∗‖+ ε+ d

[η2

(4.7b)

It is obvious that ψ∗1, and ψ∗2 are still unknown positive constants.

4.2 Adaptive Robust Controller Redesign

The main purpose of this subsection is to redesign an adaptive robust controller without NNs

such that the state x(t) of nonlinear dynamical systems described by (4.1) can be guaranteed to

converge asymptotically to zero in the presence of nonlinear structure uncertainty, and external

disturbance.

Provided that the state is available for the systems described by (4.1) , we redesign the

49

adaptive robust controller as follows.

u(t) = ϕ x(t), ψ1(t), ψ2(t)[

= −1

2η1ψ1(t)B

�Px(t)− 1

2η2ψ2(t)Sign B�Px(t)

[(4.8)

where P is the solution of (4.2), η1, and η2 are given positive constants, and ψ1(t), and ψ2(t)

are the estimate of ψ∗1, and ψ∗2, respectively, which will be updated by the following laws:

dψ1(t)

dt= −γ1σ(t)ψ1(t) +

1

2γ1η1‖B�Px(t)‖2 (4.9a)

dψ2(t)

dt= −γ2σ(t)ψ2(t) +

1

2γ2η2‖B�Px(t)‖ (4.9b)

with the initial conditions ψ1(t0) ∈ (0, ∞), and ψ2(t0) ∈ (0, ∞), where γ1, and γ2 are given

positive constants, and σ(t) ∈ R+ is a give positive function which satisfies

limt→∞

[ t

0

σ(τ)dτ ≤ σ < ∞ (4.10)


Then, applying (4.3), (4.5) and (4.8) to (4.1), we can obtain the follow closed–loop system.

dx(t)

dt= Ax(t)− 1

2η1ψ1(t)BB�Px(t)− 1

2η2ψ2(t)BSign B�Px(t)

[+B

)Φ�(x)θ∗ + ε(x)

(+ Bd(t) (4.11)

Without loss of generality, we define

ψ1(t) := ψ1(t)− ψ∗1

ψ2(t) := ψ2(t)− ψ∗2

Then, we can also rewrite (4.9) into the following error systems.

dψ1(t)

dt= −γ1σ(t)ψ1(t) +

1

2γ1η1‖B�Px(t)‖2 − γ1σ(t)ψ

∗1 (4.12a)

dψ2(t)

dt= −γ2σ(t)ψ2(t) +

1

2γ2η2‖B�Px(t)‖ − γ2σ(t)ψ

∗2 (4.12b)


According to the descriptions as above, in this section, we can propose the the following

theorem which shows that the asymptotic stability of nonlinear uncertain system described by

(4.11) and the error systems described by (4.12) can be guaranteed.

50

Theorem 4.1. Consider the adaptive closed–loop nonlinear uncertain system described by

(4.11) and the error systems described by (4.12). Suppose that the Assumptions 4.1–4.3 are

satisfied. Then, the solution (x(t), ψ(t)) (t; 0, x(0), ψ(0)) of (4.11) and (4.12) are uniformly

ultimate bounded, and

limt→∞

x(t) = 0 (4.13)

where

ψ(t) :=]ψ1(t) ψ2(t)

∥�Proof:

Consider closed–loop nonlinear uncertain systems described by (4.11) and (4.12), we choose

the following positive definite Lyapunov candidate function.

V (x(t), ψ(t)) = x�(t)Px(t) + γ−11 ψ21(t) + γ−12 ψ2

2(t) (4.14)

where P ∈ Rn×n is the solution of (4.2), and γ1, and γ2 are given positive constants.

Letting (x(t), ψ(t)) be the solution of (4.11) and (4.12) for t ≥ 0. Then by taking the

derivative of V (·) along the trajectories of (4.11) and (4.12) it is obtained that for t ≥ 0,

dV (x(t), ψ(t))

dt= 2x�(t)P

dx(t)

dt+ 2γ−11 ψ1(t)

dψ1(t)

dt+ 2γ−12 ψ2(t)

dψ2(t)

dt

= x�(t)(PA+ A�P − μPBB�P )x(t)

+μ((B�Px(t)

((2 − η1ψ1(t)((B�Px(t)

((2−η2ψ2(t)

((B�Px(t)((1+ 2x�(t)PB Φ�(x)θ∗ + ε(x)

[+2x�(t)PBd(t) + 2γ−11 ψ1(t)

dψ1(t)

dt

+2γ−12 ψ2(t)dψ2(t)

dt(4.15)

It follows from (4.2) and (4.15) that

dV (x(t), ψ(t))

dt= −x�(t)Qx(t) + μ

((B�Px(t)((2

−η1ψ1(t)((B�Px(t)

((2 − η2ψ2(t)((B�Px(t)

((1

+2x�(t)PB Φ�(x)θ∗ + ε(x)[

+2x�(t)PBd(t) + 2γ−11 ψ1(t)dψ1(t)

dt

+2γ−12 ψ2(t)dψ2(t)

dt(4.16)

51

Further, notice the fact that

X�Y ≤ ‖X‖‖Y ‖, ∀X, Y ∈ Rl

By this, and Assumptions 4.2, 4.3, from (4.16), we can further obtain that

dV (x(t), ψ(t))

dt≤ −x�(t)Qx(t) + μ

((B�Px(t)((2


((2 − η2ψ2(t)((B�Px(t)

((1

+2((B�Px(t)

(( ‖Φ(x)‖‖θ∗‖+ ‖ε(x)‖[+2

((B�Px(t)((‖d(t)‖+ 2γ−11 ψ1(t)

dψ1(t)

dt

+2γ−12 ψ2(t)dψ2(t)

dt

≤ −x�(t)Qx(t) + μ((B�Px(t)

((2−η1ψ1(t)

((B�Px(t)((2 − η2ψ2(t)

((B�Px(t)((1

+2((B�Px(t)

((‖Φ(x)‖‖θ∗‖+ 2ε((B�Px(t)

((+2d

((B�Px(t)((+ 2γ−11 ψ1(t)

dψ1(t)

dt

+2γ−12 ψ2(t)dψ2(t)

dt(4.17)

Then, applying (4.4) to above equation yields

dV (x(t), ψ(t))


((B�Px(t)((2


((2 − η2ψ2(t)((B�Px(t)

((1

+2p1‖θ∗‖((B�Px(t)

((‖x(t)‖+2p2‖θ∗‖

((B�Px(t)((+ 2ε

((B�Px(t)((

+2d((B�Px(t)

((+ 2γ−11 ψ1(t)dψ1(t)

dt

+2γ−12 ψ2(t)dψ2(t)

dt(4.18)

On the other hand, consider that for ∀a, b ∈ R, there is

2ab ≤ a2 + b2

52

By this, and (4.7), (4.18) reduces

dV (x(t), ψ(t))


((B�Px(t)((2


((2 − η2ψ2(t)((B�Px(t)

((1

+p21‖θ∗‖2((B�Px(t)

((2 + ‖x(t)‖2

+2p2‖θ∗‖((B�Px(t)

((+ 2ε((B�Px(t)

((+2d

((B�Px(t)((+ 2γ−11 ψ1(t)

dψ1(t)

dt

+2γ−12 ψ2(t)dψ2(t)

dt

= −x�(t)(Q− I)x(t)− η1ψ1(t)((B�Px(t)

((2−η2ψ2(t)

((B�Px(t)((1+ μ+ p21‖θ∗‖2

[((B�Px(t)((2

+2 p2‖θ∗‖+ ε+ d[((B�Px(t)

((+2γ−11 ψ1(t)

dψ1(t)

dt+ 2γ−12 ψ2(t)

dψ2(t)

dt

= −x�(t)(Q− I)x(t)


((2 + η1ψ∗1

((B�Px(t)((2


((1+ η2ψ

∗2

((B�Px(t)((

+2γ−11 ψ1(t)dψ1(t)

dt+ 2γ−12 ψ2(t)

dψ2(t)

dt(4.19)

Furthermore, substituting (4.12) into above equation yields

dV (x(t), ψ(t))

dt≤ −x�(t)(Q− I)x(t)


((2 + η1ψ∗1

((B�Px(t)((2


((1+ η2ψ

∗2

((B�Px(t)((

−2σ(t)ψ21(t) + η1ψ1(t)‖B�Px(t)‖2 − 2σ(t)ψ∗1ψ1(t)

−2σ(t)ψ22(t) + η2ψ2(t)‖B�Px(t)‖ − 2σ(t)ψ∗2ψ2(t)

= −x�(t)(Q− I)x(t)


((2 + η1ψ1(t)((B�Px(t)

((2−η2ψ2(t)

((B�Px(t)((1+ η2ψ2(t)

((B�Px(t)((

−2σ(t)ψ21(t)− 2σ(t)ψ∗1ψ1(t)

−2σ(t)ψ22(t)− 2σ(t)ψ∗2ψ2(t)

53

= −x�(t)(Q− I)x(t)


((1+ η2ψ2(t)

((B�Px(t)((

−2σ(t)ψ21(t)− 2σ(t)ψ∗1ψ1(t)

−2σ(t)ψ22(t)− 2σ(t)ψ∗2ψ2(t) (4.20)

Then, turning our attention to (4.9), it is obvious that when the initial conditions ψ1(0),

and ψ2(0) are positive, for any t ∈ [0, ∞), the solutions ψ1(t), and ψ2(t) are always positive.

On the other hand, noting the fact that [100],

((B�Px(t)(( ≤ ((B�Px(t)

((1

By these, (4.20) can be rewritten into

dV (x(t), ψ(t))

dt≤ −x�(t)(Q− I)x(t)


((1+ η2ψ2(t)

((B�Px(t)((1

−2σ(t)ψ21(t)− 2σ(t)ψ∗ψ1(t)

−2σ(t)ψ22(t)− 2σ(t)ψ∗2ψ2(t)

= −x�(t)(Q− I)x(t)

−2σ(t)

)ψ1(t) +

1

2ψ∗1

(2

+1

2(ψ∗1)

2σ(t)

−2σ(t)

)ψ2(t) +

1

2ψ∗2

(2

+1

2(ψ∗2)

2σ(t)

≤ −x�(t)(Q− I)x(t)

+1

2(ψ∗1)

2σ(t) +1

2(ψ∗2)

2σ(t)

= −x�(t)Qx(t) + ςσ(t) (4.21)

where

Q := Q− I

ς :=1

2(ψ∗1)

2 +1

2(ψ∗2)

2

In addition, in this chapter, we will choose Q > I to satisfy Q > 0.

Then, in the light of (4.14), and (4.21), by employing the method which has been used in

Chapter 3, we can complete the proof of Theorem 4.1. ��

54


In this section, we also use the two numerical examples which have been used in Chapter 3

to demonstrate the applicability of the redesigned adaptive robust control schemes.

4.4.1 A Mass Spring Damper System

Under the transformations

x1(t) := y(t)

x2(t) := y(t)

After some mathematical transformations (see Chapter 3 for details), the mass spring

damper system of Figure 3.1 can be written into the state space form as follows.

dx(t)

dt= Ax(t) + f x(t)

[+Bu(t) + d(t) (4.22)

where

x(t) =

]x1(t)

x2(t)

⎡, A =

]0 1

0 0

⎡, B =

⎤⎥⎦ 0

1

m

⎣∫⎢

f(·) =

⎤⎥⎦ 0

− 1

m

)b1x2(t)|x2(t)|+ k1x1(t) + k2x

31(t)

(⎣∫⎢

=

⎤⎥⎦ 0

1

m

⎣∫⎢)− b1x2(t)|x2(t)| − k1x1(t)− k2x

31(t)

(

= BΔ x1(t), x2(t)[

d(t) =

⎤⎥⎦ 0

1

m

⎣∫⎢ b2ω(t) = Bd(t)

where we define

Δ · [ := −b1x2(t)|x2(t)| − k1x1(t)− k2x31(t)

d(t) := b2ω(t)

For simulation, similar to Chapter 3, the following parameters have been given for this

system.

55

m = 8, b1 = 3, b2 = 0.5

k1 = 1, k2 = 1

And, the external disturbance is taken as

ω(t) = cos(2t)

Then, we will determine a proper adaptive robust controller so that the mass spring damper

system (4.22) is stable on the unique equilibrium position y = 0 (x1(t) = 0).

Since the redesigned controller is simple, we just need to determine the control parame-

ters and control function of (4.8) with (4.9). According to the design procedure, the control

parameters and control function are given in the following steps.

Step.1:

Taking

μ = 4, Q =

]4 0

0 6

⎡>

]1 0

0 1

⎡


P =

]7.4883 4

4 7.4883

⎡

Step.2:


η1 = 0.5, η2 = 0.5

γ1 = 1, γ2 = 1

σ(t) = 2 exp(−0.5t)


x(0) =]3 −3

�

ψ1(0) = 2, ψ2(0) = 2


Figures 4.1–4.3 for this uncertain mass spring damper system.

56

0 5 10 15-3

-2

-1

0

1

2

3

Time (Sec)

Res

ponse

ofSta

teV

ariable

x(t

)x1(t)

x2(t)

Figure 4.1: Responses of States x1(t) and x2(t) (Mass Spring Damper System)

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4

Time (Sec)

Histroy

ofA

daptatio

nLaw

ψ1(t)

Figure 4.2: History of Adaptation Law ψ1(t) (Mass Spring Damper System)

57

0 5 10 15

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

Time (Sec)

Histroy

ofA

daptatio

nLaw

ψ2(t)

Figure 4.3: History of Adaptation Law ψ2(t) (Mass Spring Damper System)

It can be observed from Figure 4.1 that by employing the redesigned adaptive robust control

schemes of this chapter, one can guarantee the states of mass spring damper system to converge

asymptotically to zero. On the other hand, Figures 4.2, and 4.3 also show that the adaptation

laws ψ1(t), and ψ2(t) are indeed uniformly bounded, respectively.

4.4.2 A Chua’s Chaotic Circuit

Similar to Chapter 3, in this subsection, we also consider a typical Chua’s chaotic circuit

consists of one linear resistor (R), two capacitors (C1, C2), one inductor (L), and one nonlinear

resistor (λ) (Figure 3.4).

Using the transformation in Chapter 3, the transformed system can be obtained as

dz∗(t)dt

= Az∗(t) +Bf z∗(t)[

(4.23)

where

A =

⎤⎥⎥⎦0 1 0

0 0 1

0 0 0

⎣∫∫⎢ , B =

⎤⎥⎥⎦0

0

1

⎣∫∫⎢

f(·) =14

1805z∗1(t)−

168

9025z∗2(t) +

1

38z∗3(t)

58

− 2

45

)28

361z∗1(t) +

7

95z∗2(t) + z∗3(t)

[ 3

The aim is to determine an adaptive robust controller without NNs which can make the

transformed system to track a reference signal. Therefore, the closed–loop configuration of

(4.23) with external disturbance can be represented by

dz∗(t)dt

= Az∗(t) +B)f z∗(t)

[+ u(t) + d(t)

((4.24)

where u(t) is the control function which will be determined later, d(t) is the external disturbance

which will be chosen as 0.5 sin(2t).

The control objective is to control the state z∗(t) of the system to track the reference yd(t)

such as

limt→∞

z∗1(t) → yd(t) (4.25a)

limt→∞

z∗2(t) → yd(t) (4.25b)

limt→∞

z∗3(t) → yd(t) (4.25c)

Similar to Chapter 3, for simulation, we selected yd(t) = 1.5 sin(t). Then, we introduce the

following definition.

x(t) =

⎤⎥⎥⎦x1(t)

x2(t)

x3(t)

⎣∫∫⎢ :=

⎤⎥⎥⎦z∗1(t)− 1.5 sin(t)

z∗2(t)− 1.5 cos(t)

z∗3(t) + 1.5 sin(t)

⎣∫∫⎢

By this, and system (4.23), we can further have the following error system.

dx(t)

dt= Ax(t) +Bu(t) + BΔ x(t)

[+Bd(t) (4.26)

where

Δ(·) =14

1805x1(t) + 1.5 sin(t)

[ − 168

9025x2(t) + 1.5 cos(t)

[

+1

38x3(t)− 1.5 sin(t)

[ − 2

45

)28

361x1(t) + 1.5 sin(t)

[

+7

95x2(t) + 1.5 cos(t)

[+ x1(t)− 1.5 sin(t)

[[ 3

+1.5 cos(t)

Furthermore, it is clearly that the control objective can be transformed to control the

59

transformed state x(t) such that:

limt→∞

x(t) = 0 (4.27)

Then, the following work is to determine a proper adaptive robust control scheme in the

form of (4.8) with (4.9). According to the design procedure, the controller is determined in the

following steps.

Step.1:

Taking

μ = 2, Q =

⎤⎥⎥⎦24 21 5

21 22 3

5 3 2

⎣∫∫⎢ >

⎤⎥⎥⎦1 0 0

0 1 0

0 0 1

⎣∫∫⎢


P =

⎤⎥⎥⎦16.9120 18.9442 6.9282

18.9442 27.8957 10.9443

6.9282 10.9443 6.9121

⎣∫∫⎢

Step.2:


η1 = 5, η2 = 3, γ1 = 5

γ2 = 5, σ(t) = 2 exp(−0.5t)


x(0) =]5 −5 −2.5

�

ψ1(0) = 4, ψ2(0) = 4


Figures 4.4–4.9 for this Chua’s Chaotic Circuit.


schemes proposed in this chapter, one can guarantee the states of the error system (4.26) to

converge asymptotically to zero. Also, Figures 4.5–4.7 show that the states z∗1(t), z∗2(t), and

z∗3(t) can track their reference trajectories yd(t), yd(t) and yd(t), respectively. On the other hand,

Figures 4.8 and 4.9 also show that the adaptation laws ψ1(t), and ψ2(t) are indeed uniformly

bounded, respectively.

60

0 5 10 15-5

-4

-3

-2

-1

0

1

2

3

4

5

Time (Sec)

Res

ponse

ofSta

teV

ariable

x(t

)

x1(t)

x2(t)

x3(t)

Figure 4.4: Responses of States x1(t), x2(t), and x3(t) (Chua’s Chaotic Circuit)

0 5 10 15-2

-1

0

1

2

3

4

5

Time (Sec)

Sta

teV

ariable

z∗ 1(t

)and

Its

Ref

eren

cey

d(t

)

yd(t)

z∗1(t)


61

0 5 10 15-4

-3

-2

-1

0

1

2

Time (Sec)

Sta

teVariable

z∗ 2(t

)and

Its

Ref

eren

cey

d(t

)

z∗2(t)

yd(t)


0 5 10 15-3

-2

-1

0

1

2

3

Time (Sec)

Sta

teVariable

z∗ 3(t

)and

Its

Ref

eren

cey

d(t

)

z∗3(t)

yd(t)


62

0 5 10 150

5

10

15

20

25

Time (Sec)

History

ofA

daptatio

nLaw

ψ1(t)

Figure 4.8: History of Adaptation Law ψ1(t) (Chua’s Chaotic Circuit)

0 5 10 150

0.5

1

1.5

2

2.5

3

3.5

4

Time (Sec)

History

ofA

daptatio

nLaw

ψ2(t)

Figure 4.9: History of Adaptation Law ψ2(t) (Chua’s Chaotic Circuit)

63

4.5 Problem of Water Pollution Control of River

Similar to Chapter 3, in this section, we also apply our redesigned adaptive robust control

schemes to solve a problem of river pollution.

4.5.1 Modeling of Water Pollution Control Systems

Denoting z(t) and q(t) to be the concentrations per unit volume of BOD and DO, respec-

tively. And, the desired steady state values of BOD and DO concentration are denoted as z∗

and q∗, respectively.

With the following transformation

x1(t) := z(t)− z∗

x2(t) := q(t)− q∗

The uncertain water pollution control systems can be written to be the following dynamical

systems (see Chapter 3 for details).

dx(t)


[+ d(t) (4.28)

where

x(t) =]x1(t) x2(t)

�

u(t) =]u1(t) u2(t)

�

A =

]−(h1 + γ) 0

−h3 −(h2 + γ)

⎡

B =

]γ 0

0 1

⎡

f(·) =

⎤⎦ −Δh1(t) z∗ + x1(t)

[−Δh3(t) z∗ + x1(t)

[+Δh2(t) qs − q∗ − x2(t)

[⎣⎢

d(t) =]υ1(t) υ2(t)

�

Then, for the simulation of this numerical example, similar to Chapter 3, we give the

following parameters:

q∗ = 6.0, z∗ = 0.625, qs = 10.0

h1 = 0.32, h2 = 0.2, h3 = 0.32

64

γ = 0.1

For simulation, we give the uncertain time–varying parameters as follows.

υ1(t) = 0.5 sin(3t), υ2(t) = 1.0− 0.5 cos(2t)

Δh1(t) = −0.08 cos(3t), Δh2(t) = 0.08 sin(3t)

Δh3(t) = 0.05 sin(5t)

4.5.2 Controller Synthesis

Then, we will determine an adaptive robust controller so that the asymptotically stability of

(4.28) can be ensured. According to the design procedure, the control parameters and functions

are given in the following steps.

Step.1:

Taking

μ = 2, Q =

]40.32 44.96

44.96 53.32

⎡>

]1 0

0 1

⎡


P =

]6 4

4 5

⎡

Step.2:


η1 = 6, η2 = 3

γ1 = 0.5, γ2 = 1

σ(t) = 4 exp(−0.5t)

4.5.3 Simulation of Uncertain Water Pollution Control Systems

For this simulation, we also choose the initial values as follows.

x(0) =]10 −10

�

ψ1(0) = 8, ψ2(0) = 8


Figures 4.10–4.12 for this river pollution control system.

65

0 5 10 15-10

-8

-6

-4

-2

0

2

4

6

8

10

Time (Days)

BO

D( x

1(t

)) and

DO

( x2(t

)) Res

ponse

s(m

g/l)

x2(t)

x1(t)

Figure 4.10: Responses of States x1(t) (BOD) and x2(t) (DO) (mg/l) (Water Pollution ControlSystem)

0 5 10 150

1

2

3

4

5

6

7

8

9

Time (Days)

History

ofA

daptatio

nLaw

ψ1(t)

Figure 4.11: History of Adaptation Law ψ1(t) (Water Pollution Control System)

66

0 5 10 150

1

2

3

4

5

6

7

8

Time (Days)

History

ofA

daptatio

nLaw

ψ2(t)

Figure 4.12: History of Adaptation Law ψ2(t) (Water Pollution Control System)

It can be observed from Figure 4.10 that by employing the redesigned adaptive robust

control schemes proposed in this chapter, one can guarantee the states of the pollution control

systems to be asymptotically stable in the presence of nonlinear state perturbations. On the

other hand, it can be known from Figures 4.11 and 4.12 that adaptation laws ψ1(t), and ψ2(t)

are also uniformly ultimately bounded, respectively.

67

Chapter 5

Adaptive Robust Controller Redesignof Uncertain Nonlinear Systems withGeneralized Matched Conditions

In this chapter, we consider the problem of adaptive robust controller redesign for a class

of uncertain nonlinear systems with time–varying delays. We assume that the nonlinear uncer-

tainties with delayed states are with generalized matched conditions and can be approximated

by NNs which are not used in the actual control process, and that the time–varying delays are

bounded continuous functions. Particularly, we do not require that the derivatives of the time–

varying delays must be less than 1. Then, similar to Chapter 4, we introduce the adaptation

laws with σ–modification to estimate the unknown parameters which are containing the ideal

weight matrix and approximation error of the NNs. By making use of the adaptation laws, we

redesign the controllers of Chapter 4, and synthesize a new class of adaptive robust controllers.

By employing a designed Lyapunov function, we show that the uniform asymptotic stability of

such uncertain nonlinear time delay systems can be guaranteed by using the proposed control

schemes. Finally, the feasibility of applying the obtained results is demonstrated by considering

the problem of optimal long–management of uncertain exploited ecological systems with two

competing species.


Consider the following system

dx(t)

dt= F (x(t), t) +G(x(t), t)

)u(t) +

v∫i=1

ξi x(t− hi(t)), t− hi(t)[[

(5.1)

where t ∈ R+ is the time, x(t) ∈ Rn is the state, u(t) ∈ Rm is the control input, F (·) :

Rn × R → Rn, and G(·) : Rn × R → Rn×m are known functions, for any i ∈ {1, 2, · · · , v},ξi(·) : Rn × R → Rm represents the nonlinear system uncertainty, and hi(t) denotes the time–

delay which is assumed to be bounded continuous function satisfying

0 ≤ hi(t) ≤ hi < ∞, i = 1, 2, · · · , v (5.2)

where hi is unknown non–negative constant.

68

In addition, the initial condition of system (5.1) is given by

x(t) = χ(t), t ∈ [t0 − h, t0] (5.3)

where χ(t) is a known continuous bounded function, and h := max{h1, h2, · · · , hv}.Furthermore, the so–called unforced nominal system of (5.1) is given by

dx(t)

dt= F (x(t), t) (5.4)

The aim of the chapter is to design a state feedback controller u(t) that can guarantee the

asymptotic stabilization of nonlinear dynamical system (5.1) in the presence of time–varying

delays. The main result in this chapter relies on the following assumptions.

Assumption 5.1. For the unforced nominal system described by (5.4), the equilibrium

x(t) = 0 is asymptotically stable in the large. Further, there exists a differentiable Lyapunov

function V0(x, t) : Rn ×R → R satisfying

α1(‖x‖) ≤ V0(x, t) ≤ α2(‖x‖) (5.5a)

∂V0(x, t)

∂t+∇�

x V0(x, t)F (x, t) ≤ −α3(‖x‖) (5.5b)

where α1(·), and α2(·) are K∞ class functions, and α3(·) is a K class function.

Assumption 5.2. According to the function approximation property of NNs, on the Ωx ⊂Rn, for all i ∈ {1, 2, · · · , v}, there exists an ideal weight matrix θi ∈ Rlm×m such that for all

t ∈ [−h, t0], and x(t) ∈ Ωx, the following equation

ξi x(t− hi(t)), t− hi(t)[= θ�i ρi x(t− hi(t))

[+ εi x(t− hi(t))

[(5.6)

holds with ‖εi(·)‖ ≤ εi < ∞, where εi is an unknown positive constant. In addition,

ρi x(t− hi(t))[

=]ρi1 x(t− hi(t))

[ρi2 x(t− hi(t))

[· · · ρilm x(t− hi(t))

[{�(5.7)

which is a vector Guassian function and chosen as:

ρij(·) = exp

)−((x(t− hi(t))− cij

((2δ2ij

[, j = 1, 2, · · · , lm

It is clear

0 < ‖ρi(·)‖ ≤√lm (5.8)

And where cij =]cij1 cij2 · · · ciji

{�is any given constant vector represents the center of the

69

receptive field, and δij is any given constant represents the width of the Gaussian functions.

Remark 5.1. According to the function approximation property of NNs of Chapter 2, it

is not difficult to obtain (5.6). Thus, Assumption 5.2 is rather a standard assumption. In

particular, during the control process, the function approximation described by (5.6) will not

be really used. That is x(t − hi(t)), i = 1, 2, · · · , v are not needed during the control process,

which means hi(t), i = 1, 2, · · · , v can be unknown for the system designer.

Definition 5.1. For simplicity, we define

μ(t) := G�(x, t)∇xV0(x, t) (5.9)

Form (5.8), and (5.9), it is clear that for any i ∈ {1, 2, · · · , v}, there always exist two positivefunctions pi1(t), and pi2(t) such that

‖ρi(·)‖ ≤ pi1(t)‖μ(t)‖+ pi2(t) (5.10)

holds with

0 < pi1(t) ≤ pi1 < ∞ (5.11a)

0 < pi2(t) ≤ pi2 < ∞ (5.11b)

where pi1, and pi2 are unknown positive constants.

Definition 5.2. For convenience, we also define

ψ∗ =1

η1

v∫i=1

)pi1‖θi‖

[(5.12a)

φ∗ =1

η2

v∫i=1

)pi2‖θi‖+ εi

[(5.12b)

where η1, and η2 are given positive constants. From the previous description, it is obvious that

ψ∗, and φ∗ are still unknown positive constants.

5.2 Adaptive Robust Controller Redesign

In this section, we will redesign the adaptive robust control schemes which have been pro-

posed in Chapter 4, such that the uncertain nonlinear delayed system (5.1) can be uniformly

asymptotically stable.

According to Chapter 4, we propose a new adaptive robust controller such as follows.

u(t) = −η1ψ(t)μ(t)− η2φ(t)Sign(μ(t)) (5.13)

where η1, and η2 are given positive constants, and ψ(t), and φ(t) are the estimates of the

70

unknown positive constants ψ∗, and φ∗, respectively, which are updated by the following adap-

tation laws

dψ(t)

dt= −γ1σ(t)ψ(t) + γ1η1‖μ(t)‖2 (5.14a)

dφ(t)

dt= −γ2σ(t)φ(t) + γ2η2‖μ(t)‖1 (5.14b)

where γ1, and γ2 are given positive constants, and σ(t) ∈ R+ is a given function satisfying

limt→∞

[ t

t0

σ(s)ds ≤ σ < ∞ (5.15)


Further, letting

ψ(t) := ψ(t)− ψ∗ (5.16a)

φ(t) := φ(t)− φ∗ (5.16b)

Then, we can rewrite (5.14) into the following error systems.

dψ(t)

dt= −γ1σ(t)ψ(t) + γ1η1‖μ(t)‖2 − γ1σ(t)ψ

∗ (5.17a)

dφ(t)

dt= −γ2σ(t)φ(t) + γ2η2‖μ(t)‖1 − γ2σ(t)φ

∗ (5.17b)


According to the descriptions as above, in this section, we can obtain the following theorem

which shows that the asymptotic stability of nonlinear uncertain system described by (5.1) and

the error system described by (5.17) can be guaranteed.

Theorem 5.1. Consider the uncertain nonlinear delayed system (5.1). Suppose that the

Assumptions 2.1, 2.2 are satisfied. Under the controller (5.13) with (5.14), the state x(t) of

system (5.1) converges asymptotically to zero.

Proof:

For closed–loop systems (5.1), (5.13), and (5.17), we introduce the following Lyapunov function.

V (x, ψ, φ, t) = V0(x, t) +1

2γ−11 ψ2(t) +

1

2γ−12 φ2(t) (5.18)

Then, for any t ≥ t0, differentiating the function V (·) along the solutions of (5.1), (5.13),

and (5.17) yields

V (x, ψ, φ, t) =∂V0(x, t)

∂t+∇�

x V0(x, t)F (x, t)

71

+∇�x V0(x, t)G(x, t)

)u(t)

+v∫

i=1

ξ x(t− hi(t)), t− hi(t)[[

+γ−11 ψ(t)dψ(t)

dt+ γ−12 φ(t)

dφ(t)

dt(5.19)

According to Assumptions 2.1, 2.2, (5.9) and (5.13), from (5.19) becomes

V (x, ψ, φ, t) ≤ −α3(‖x‖)− η1ψ(t)‖μ(t)‖2 − η2φ(t)‖μ(t)‖1

+∇�x V0(x, t)G(x, t)

v∫i=1

)θ�i ρi x(t− hi(t))

[

+εi x(t− hi(t))[[


dt+ γ−12 φ(t)

dφ(t)

dt

≤ −α3(‖x‖)− η1ψ(t)‖μ(t)‖2 − η2φ(t)‖μ(t)‖1

+‖μ(t)‖v∫

i=1

)‖θi‖

((ρi x(t− hi(t))[((+ εi

[[


dt+ γ−12 φ(t)

dφ(t)

dt(5.20)

Further, by (5.10) and (5.11), (5.20) reduces


+‖μ(t)‖v∫

i=1

)pi1(t)‖θi‖‖μ(t)‖+ pi2(t)‖θi‖+ εi

[


dt+ γ−12 φ(t)

dφ(t)

dt

≤ −α3(‖x‖)− η1ψ(t)‖μ(t)‖2 − η2φ(t)‖μ(t)‖1

+‖μ(t)‖v∫

i=1

)pi1‖θi‖‖μ(t)‖+ pi2‖θi‖+ εi

[


dt+ γ−12 φ(t)

dφ(t)

dt(5.21)

Substituting (5.12) into (5.21), it can be further obtain that


+v∫

i=1

)pi1‖θi‖

(‖μ(t)‖2 +

v∫i=1

)pi2‖θi‖+ εi

(‖μ(t)‖

72


dt+ γ−12 φ(t)

dφ(t)

dt

≤ −α3(‖x‖)− η1ψ(t)‖μ(t)‖2 − η2φ(t)‖μ(t)‖1+η1ψ

∗‖μ(t)‖2 + η2φ∗‖μ(t)‖


dt+ γ−12 φ(t)

dφ(t)

dt(5.22)

On the other hand, noting the fact that (see, e.g., [100]),

‖μ(t)‖ ≤ ‖μ(t)‖1 (5.23)

By this, (5.16) and (5.17), (5.22) derives

V (x, ψ, φ, t) ≤ −α3(‖x‖)− η1ψ(t)‖μ(t)‖2 − η2φ(t)‖μ(t)‖1+η1ψ

∗‖μ(t)‖2 + η2φ∗‖μ(t)‖1

+η1ψ(t)‖μ(t)‖2 + η2φ(t)‖μ(t)‖1−σ(t) ψ2(t) + ψ∗ψ(t)

[−σ(t) φ2(t) + φ∗φ(t)

[≤ −α3(‖x‖) + 1

4(ψ∗)2 + (φ∗)2

[σ(t)

= −α3(‖x‖) + εσ(t) (5.24)

where

ε :=1

4(ψ∗)2 + (φ∗)2

[On the other hand, letting z(t) := [x(t) ψ(t) φ(t)]�, and following the Assumption 5.1 and

the definition of V (·) described by (5.18), for any t ≥ t0, there is

α1(‖z(t)‖) ≤ V (z, t) ≤ α2(‖z(t)‖) (5.25)

where

α1(‖z(t)‖) := α1(‖x(t)‖) + 1

2γ−11 ψ2(t) +

1

2γ−12 φ2(t) (5.26a)

α2(‖z(t)‖) := α2(‖x(t)‖) + 1

2γ−11 ψ2(t) +

1

2γ−12 φ2(t) (5.26b)

Integrating (5.24) with respect to time t ≥ t0, and applying into (5.25) reduces

0 ≤ α1(‖z(t)‖)

≤ V (z, t0) +

[ t

t0

]− α3(‖x(s)‖) + εσ(s){ds

73

≤ α2(‖z(t0)‖) +[ t

t0

]− α3(‖x(s)‖) + εσ(s){ds (5.27)

In the light of (5.15) and (5.27), it is obvious

α1(‖z(t)‖) ≤ α2(‖z(t0)‖) + supt∈[t0,∞)

[ t

t0

εσ(s)ds

≤ α2(‖z(t0)‖) + εσ (5.28)

which means z(t) = [x(t) ψ(t) φ(t)]� is uniformly bounded.

On the other hand, taking the limit as t → ∞ on both sides of (5.27), one has

0 ≤ α2(‖z(t0)‖) + limt→∞

[ t

t0

]− α3(‖x(s)‖) + εσ(s){ds (5.29)

It follows from (5.15) and (5.29)

limt→∞

[ t

t0

α3(‖x(s)‖)ds

≤ α2(‖z(t0)‖) + limt→∞

[ t

t0

εσ(s)ds

≤ α2(‖z(t0)‖) + εσ (5.30)

Since x(t) is uniformly continuous with respect to time t, α3(‖x(t)‖) is also uniformly

continuous with respect to time t. Thus, by making use of the Barbarlat lemma of Chapter 2,

(5.30) becomes

limt→∞

α3(‖x(t)‖) = 0 (5.31)

On the other side, according to Assumption 5.1, notice that α3(·) is a K class function, it

is clear from (5.31) that

limt→∞

‖x(t)‖ = 0 (5.32)

Therefore, the proof of Theorem 5.1 can be completed. ��


Similar to [103], to illustrate the utilisation of our approach, in this section, we consider the

following system.

dx(t)

dt= F (x(t), t) +G(x(t), t)

)u(t) + ξ1 x(t− h1(t)), t− h1(t)

[

74

+ξ2 x(t− h2(t)), t− h2(t)[[

(5.33)

where

F (·) =

]−x1(t) + 3x1(t)x

22(t)

−x2(t)− 2x21(t)x2(t)

⎡, G(·) =

]x1(t)x2(t) 0

0 2x2(t)

⎡

ξ1(·) =

]1.2 sin x1(t− h1(t))

[1.6 x1(t− h1(t))x2(t− h1(t))

⎡

ξ2(·) =

]x1(t− h2(t)) + x2(t− h2(t))

0.1 sin x2(t− h2(t))[

⎡

and

h1(t) = 1.5∥∥sin(2t)∥∥, h2(t) =

∥∥cos(3t)∥∥It is obviously that hi(t), i = 1, 2, can be bigger than 1.

Then, we will determine a proper adaptive robust controller so that nonlinear delayed system

(5.33) can be asymptotically stable.

Step.1:

For the nominal system of (5.33), we select the following Lyapunov function

V0(x(t), t) = 2x21(t) + 3x2

2(t) (5.34)

where

x(t) := [x1(t) x2(t)]�

It is obviously that

2‖x(t)‖2 ≤ V0(x(t), t) ≤ 3‖x(t)‖2 (5.35a)

∂V0(x(t), t)

∂t+∇�

x V0(x(t), t)F (x(t), t) ≤ −2V0(x(t), t) (5.35b)

which means Assumption 5.1 can be satisfied.

Step.2:

Then, we can give μ(t) as

μ(t) = G�(·)∇xV0(·) =

⎤⎦4x2

1(t)x2(t)

12x22(t)

⎣⎢

75

Step.3:

Further, for the control schemes (5.13) with (5.14), we choose

η1 = 5, η2 = 5, γ1 = 0.2

γ2 = 0.2, σ(t) = 5 exp(−0.5t)

Step.4:


x(t) = [0.5 cos(t) − 0.5 cos(t)]⊤, t ∈ [−1.5, 0]

ψ(0) = 0.3, ϕ(0) = 0.3


Figures 5.1–5.3 for this uncertain system.


schemes of this chapter, one can guarantee the states of the nonlinear delayed system to converge

asymptotically to zero. On the other hand, Figures 5.2, and 5.3 also show that the adaptation

laws ψ(t), and ϕ1(t) are indeed uniformly bounded, respectively.

0 2 4 6 8 10-0.5-0.4-0.3-0.2-0.1

00.10.20.30.40.5

Time (Sec)

Res

ponse

ofSta

teV

ariable

x(t

)

x1(t)

x2(t)

Figure 5.1: Responses of States x1(t) and x2(t)

76

0 2 4 6 8 100.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Time (Sec)

History

ofA

daptatio

nLaw

ψ(t)

Figure 5.2: History of Adaptation Law ψ(t)

0 2 4 6 8 100.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Time (Sec)

History

ofA

daptatio

nLaw

φ(t)

Figure 5.3: History of Adaptation Law ϕ(t)

77

5.5 Optimal Long–Term Management for Uncertain Nonlinear Eco-

logical Systems

In this section, we consider the problem of the optimal long-term management of uncertain

ecosystems with time–varying delays. We implement the control schemes which have been

developed in this chapter to manage such uncertain ecological systems so that their optimal

steady states can be guaranteed.

5.5.1 Significance of Ecological System Management

It is well known that since the 1940s, along with the increase of population, over–exploitation

of resources, changes in the environment and rapid economic growth, a lot of worldwide eco-

logical problems have emerged, such as environmental pollution, forest deterioration and land

desertification, and so on. These series of problems pose serious threat for continued human

survival and economic development. Therefore, restoration of degraded ecosystem and rational

management of existing natural resources have received widespread attention from the inter-

national community [104].

In the problem of the environmental management, there is a class of ecological systems which

are often employed to describe the dynamics of some species. It is quite important to consider

the optimal long–term management of exploited ecosystems, and to guarantee their optimal

steady states. It should be pointed out that a real ecosystem in nature will be usually subject to

some continually acting unpredictable disturbances due to diseases, migrating species, changing

in climatic conditions, and so on. Hence, it is necessary to develop some optimal management

schemes which can guarantee the convergence of the optimal steady state of a real ecosystem

in the presence of acting unpredictable disturbances [69], [84], and [105].

5.5.2 Modeling of Uncertain Nonlinear Ecological Systems

Similar to [69], [84], and [105], we also consider a class of exploited ecosystems. Their

nominal model can be described by the following nonlinear differential equations:

dx(t)

dt= f x(t)

[ −Hx(t) (5.36a)

x(t0) = x0 (5.36b)

where x(t) ∈ X is an n–dimensional biomass vector, its ith component xi(t) representing the

biomass of the ith species at time t ∈ R+, and X is defined by

X :=}x(t) ∈ Rn

∥∥∥xi(t) > 0, i = 1, 2, · · · , n, t ∈ R+√

Further, the function f(·) : X × R+ → Rn is known, and assumed to be continuous and

78

bounded. And, H ∈ Rn×n a diagonal matrix such as

H :=

⎤⎥⎥⎥⎥⎥⎦

h1 0 · · · 0

0 h2 · · · 0

......

. . ....

0 0 · · · hn

⎣∑∑∑∑∑⎢

We say that a constant harvest effort vector h = [h1 h2 · · · hn]� is admissible, if h ∈ H,

where H ∈ Rn is a prescribed constraint set. The corresponding non–trivial solution of

f x(t)[ −Hx(t) = 0 (5.37)

is assumed to be unique. Letting h∗ be the admissible constant harvest effort that maximizes

the quantity β�Hx(t) subject to (5.37), where β := [β1 β2 · · · βn]� is a prescribed constant

price vector, and letting x∗ := [x∗1 x∗2 · · · x∗n]� be the corresponding equilibrium state of (5.36).

Thus, under optimal steady state harvesting, the exploited ecosystem (5.36) becomes

dx(t)

dt= f x(t)

[ −H∗x(t) (5.38a)

x(t0) = x0 (5.38b)

It is obvious that if the exploited ecosystem (5.36) is not subjected to disturbances, the

harvest rate H∗x∗ is indeed optimal for the long–term management of the ecosystems. That is,

the equilibrium state x∗ is an optimal steady state. However, it is well known that similar to

any actual dynamical systems, a real ecosystem in nature should include some uncertainties.

In fact, any real ecosystem will be usually subject to some continually acting unpredictable

disturbances due to diseases, migrating species, changing in climatic conditions, and so on.

Therefore, a modified ecosystems may be described by

dx(t)

dt= f x(t)

[ −H∗x(t) +ν∫

j=1

Δfj x(t− hj(t))[

(5.39)

where Δfj(·) : X × R+ → Rn, j = 1, 2, · · · , ν, are unknown functions, and represent the

system perturbations which are assumed to be continuous and bounded. Furthermore, the

time–varying delays hj(t), j = 1, 2, · · · , ν, are unknown functions, and assumed to be any

continuous bounded function satisfying

0 ≤ hj(t) ≤ hj < ∞, j = 1, 2, · · · , ν

where hj is any unknown nonnegative constant.

On the other hand, it is clearly that the optimal exploited ecosystem may deviate from its

equilibrium state x∗, and the constant harvesting effort h∗ may be no longer optimal in the

79

presence of acting unpredictable disturbances. Therefore, for uncertain exploited time–delay

ecosystems, we have to introduce a control term which is designed to compensate the effect

of uncertainties on a real exploited ecosystem. Then, (5.39) can be further written into the

following form.

dx(t)

dt= f x(t)

[ −H∗x(t) +ν∫

j=1

Δfj x(t− hj(t))[+ u(t) (5.40)

where u(t) ∈ Rn is the control vector to be synthesized, its ith component ui(t) representing a

control on the growth or decay rate of the biomass of the ith species at t ∈ R+.

Furthermore, for this exploited time–delay ecosystem, we give the initial condition as follows.

x(t) = χ(t), t ∈ [t0 − hmax, t0] (5.41)

where χ(t) is a known continuous function, and

hmax := maxj

}hj(t), j = 1, 2, · · · , ν∣

Since this is an actual physical system, and x(t) represents the biomass, thus x(t) can be

always positive real number, i.e. x(t) > 0. On the other hand, x∗ represents the target biomass,

which is also always positive real number, that is x∗ > 0. Therefore, we can make the following

transformation

x(t) :=

]ln

)x1(t)

x∗1

[ln

)x2(t)

x∗2

[· · · ln

)xn(t)

x∗n

[ (�(5.42)

Then, the control objective can be translated to develop a proper controller which can

guarantee that limt→∞ x(t) = 0.

From (5.42), it is obvious that

x(t) = X∗ex(t) (5.43)

where

X∗ :=

⎤⎥⎥⎥⎥⎥⎦

x∗1 0 · · · 0

0 x∗2 · · · 0

......

. . ....

0 0 · · · x∗n

⎣∑∑∑∑∑⎢ , ex(t) :=

⎤⎥⎥⎥⎥⎥⎦

ex1(t)

ex2(t)

...

exn(t)

⎣∑∑∑∑∑⎢

Further, substituting (5.43) into (5.40) yields

dx(t)

dt= g x(t)

[+B x(t)

[u(t) +

ν∫j=1

Δgj x(t− hj(t))[

(5.44)

80

where

g(·) := E−x(t)(X∗)−1)f X∗ex(t)

[ −H∗X∗ex(t)(

Δgj(·) := B x(t)[Δfj X∗ex(t−hj(t))

[, j = 1, 2, · · · , ν

B(·) := E−x(t)(X∗)−1

E−x(t) :=

⎤⎥⎥⎥⎥⎥⎦

e−x1(t) 0 · · · 0

0 e−x2(t) · · · 0

......

. . ....

0 0 · · · e−xn(t)

⎣∑∑∑∑∑⎢

In the light of above definitions, the functions Δgj(·), j = 1, 2, · · · , ν, are still unknown,

and bounded in magnitude. In addition, the system (5.44) is with the form of (5.1). Therefore,

the control schemes which have been proposed in this chapter can be used to this uncertain

time-delay ecosystem.

5.5.3 Simulation of Uncertain Nonlinear Ecological System Management

For simulation, similar to [105], we consider the following system with the form of (5.40)

dx1(t)

dt= x1(t) 1− x1(t)− 0.8x2(t)

[ −m∗1x1(t)

+2∫

j=1

v1j(t)x1(t− hj(t)) + u1(t) (5.45a)

dx2(t)

dt= x2(t) 1− 0.25x1(t)− x2(t)

[ −m∗2x2(t)

+2∫

j=1

v2j(t)x2(t− hj(t)) + u2(t) (5.45b)

where x(t) = [x1(t) x2(t)]� is a biomass vector, u(t) = [u1(t) u2(t)]

� is the control input which

will be given to guarantee that the state x(t) reaches the optimal equilibrium state x∗, m∗i ,

i = 1, 2 are the constant harvest efforts, v1j(t), and v2j(t), j = 1, 2 are any uncertain functions,

and hj(t), j = 1, 2 are the time lags causing by the diseases, migrating species and changes in

climatic conditions, and so on.

Notice that (5.45) is a practical ecosystem system i.e. x(t) = [x1(t) x2(t)]� ∈ R+, and

x∗ = [x∗1 x∗2]� ∈ R+. In addition, for all i ∈ {1, 2}, the optimal equilibrium state x∗i , and the

harvest effort m∗i have the following relationship (see, e.g. [105]).

m∗1 = 0.25x∗1 + x∗2 (5.46a)

m∗2 = x∗1 + 0.8x∗2 (5.46b)

81

Further, based on the transformation x(t) = [ln(x1(t)/x∗1) ln(x2(t)/x

∗2)]�, the ecosystem

(5.45) can be rewritten into

dx1(t)

dt= x∗1 1− exp(x1(t))

[+ 0.8x∗2 1− exp(x2(t))

[

+ x∗1 exp(x1(t)[−1)

u1(t) +2∫

j=1

v1j(t)x∗1 exp x1(t− hj(t))

[[(5.47a)

dx2(t)

dt= 0.25x∗1 1− exp(x1(t))

[+ x∗2 1− exp(x2(t))

[

+ x∗2 exp(x2(t)[−1)

u2(t) +2∫

j=1

v2j(t)x∗2 exp x2(t− hj(t))

[[(5.47b)

Then, the following work is to determine a proper controller u(t) in the form of (5.13) to

ensure limt→∞ x(t) = 0 which means the state x(t) of the exploited ecosystem (5.45) converges

asymptotically to the optimal steady state of x∗.

It is clear that the unforced nominal system of (5.47) can be given by

dx1(t)

dt= x∗1 1− exp(x1(t))

[+ 0.8x∗2 1− exp(x2(t))

[(5.48a)

dx2(t)

dt= 0.25x∗1 1− exp(x1(t))

[+ x∗2 1− exp(x2(t))

[(5.48b)

For above unforced nominal system, there is following differential Lyapunov function

V0(x(t), t) = x∗1 exp(x1(t)− 1− x1(t))[+ x∗2 exp(x2(t)− 1− x2(t))

[(5.49)

which establishes global asymptotic stability of unforced nominal system (5.48) at the equilib-

rium x(t) = 0

Differentiating (5.49) with respect to the state x(t), ∇xV0(x(t), t) can be obtained as follows.

∇xV0(x(t), t) =]x∗1 exp(x1(t)− 1)

[x∗2 exp(x2(t)− 1)

[{�(5.50)

Then, it can be further reduced

μ(t) =]1− exp(−x1(t)) 1− exp(−x2(t))

{�(5.51)

Therefore, according to (5.13) with (5.14), for the nonlinear uncertain delayed system (5.47),

the input controller u(t) can be designed as follows.

u(t) = −η1ψ(t)]1− exp(−x1(t)) 1− exp(−x2(t))

{�

−η2φ(t)Sign

)]1− exp(−x1(t)) 1− exp(−x2(t))

{�[(5.52)

82

where

dψ(t)

dt= −γ1σ(t)ψ(t) + γ1η1

)1− exp(−x1(t))

[2+ 1− exp(−x2(t))

[2((5.53a)

dφ(t)

dt= −γ2σ(t)φ(t) + γ2η2

)∥∥1− exp(−x1(t))∥∥

+∥∥1− exp(−x2(t))

∥∥( (5.53b)

Thanks to Theorem 5.1, under the developed controller u(t) described by (5.40) with (5.41)

via properly choosing the control parameters η1, η2, γ1, and γ2, and function σ(t), the state

x(t) of (5.47) be can guaranteed to converge asymptotically to zero which implies the state x(t)

of the exploited ecosystem (5.45) converges asymptotically to the optimal steady state of x∗.

Furthermore, for simulation, the uncertain functions, and time delays are given as follows.

v1j(t) = 1 + 0.5 sin(t), v2j(t) = 1 + 0.5 sin(t), j = 1, 2

h1(t) = 1 + sin(πt), h2(t) = 1 + cos(πt)

It is clear that 0 ≤ h1(t) ≤ 2, and 0 ≤ h2(t) ≤ 2 which are satisfying (5.2), respectively. In

particular, the derivatives of h1(t), and h2(t) can be bigger than 1.

Moreover, for the system (5.45) with above uncertainties, the control parameters, and func-

tion of (5.40) with (5.41) are chosen as follows.

η1 = 1, η2 = 1

γ1 = 0.5, γ2 = 0.5

σ(t) = 8 exp(−0.5t)

Finally, the harvest effort m∗i , i = 1, 2, and initial conditions of x(t), ψ(t), and φ(t) are

given as follows.

m∗1 = 25/61, m∗

2 = 30/61

x1(t) = 5 + sin(πt), x2(t) = 5 + cos(πt), t ∈ [−2, 0]

ψ(0) = 4, φ(0) = 4

With the settings as above, the results of the simulation are shown in Figures 5.4–5.7.

From Figure 5.4, it can be seen that the state x(t) of (5.47) is uniformly bounded, and

converges asymptotically to zero. Thus, as shown in Figure 5.5, the state x(t) of (5.45) is also

uniformly bounded, and converges asymptotically to the optimal steady state of x∗. On the

other hand, it can be known from Figures 5.6, and 5.7 the adaptation laws ψ(t), and φ(t) are

indeed the estimate values of the unknown parameters decreasing.

83

0 5 10 15 200

0.5

1

1.5

2

2.5

3

Time (Months)

Response

ofState

Varia

ble

x(t)

x1(t)

x2(t)

Figure 5.4: State Variable x(t) of (5.47) (Harvested System with Two Competing Species)

0 5 10 15 200

1

2

3

4

5

6

Time (Months)

ResponseofStateVariable

x(t)

x1(t)

x2(t)

x∗

1= x

∗

2=

20

61

Figure 5.5: State Variable x(t) of (5.45) (Harvested System with Two Competing Species)

84

0 5 10 15 2000.51

1.52

2.53

3.54

Time (Months)

History

ofA

daptatio

nLaw

ψ(t)

Figure 5.6: History of Updating Parameter ψ(t) (Harvested System with Two CompetingSpecies)

0 5 10 15 2000.51

1.52

2.53

3.54

Time (Months)

History

ofA

daptatio

nLaw

φ(t)

Figure 5.7: History of Updating Parameter ϕ(t) (Harvested System with Two CompetingSpecies)

85

Chapter 6

Adaptive Robust Backstepping Controlof Uncertain Nonlinear Systems

In this chapter, we consider the problem of adaptive robust stabilization for a class of non-

linear systems with mismatched structure uncertainties, external disturbances, and unknown

time–varying virtual control coefficients. Similar to Chapter 3, we introduce a simple structure

NNs to model the unknown structure uncertainties. For such a class of uncertain dynamical

systems, we also employ the improved σ–modification adaptation laws to estimate the unknown

parameters which contain the upper bounds of the external disturbances and the virtual control

coefficients, and the norm forms of weights and approximate errors of the NNs. Then, by mak-

ing use of the updated values of these uncertain parameters, we design a class of backstepping

approach–based continuous adaptive robust state feedback controllers for such uncertain dy-

namical systems. Here, we show that the proposed control schemes can guarantee the uniform

asymptotic stability of the uncertain dynamical systems in the presence of structure uncertain-

ties, external disturbances, and unknown time–varying virtual control coefficients. Moreover,

we can also implement the proposed control schemes to solve some practical control problems.


We also consider a class of uncertain strict–feedback systems with structure uncertainties,

external disturbances, and unknown time–varying virtual control coefficients described by the

following nonlinear differential equations:

x1 = g1(x1)x2 + f1(x1) + d1(t)

x2 = g2(x2)x3 + f2(x2) + d2(t)

...... (6.1)

xn−1 = gn−1(xn−1)xn + fn−1(xn−1) + dn−1(t)

xn = gn(xn)u(t) + fn(xn) + dn(t)

where xi(t) = [x1, x2, · · · , xi]⊤ ∈ Ri, i = 1, 2, · · · , n are the states, u(t) ∈ R is the control input,

the unknown functions fi(·) : R × Ri → R, i = 1, 2, · · · , n, represent structure uncertainties

which are assumed to be continuous and differentiable, the unknown functions gi(·) ∈ R,

i = 1, 2, · · · , n, represent time–varying virtual control coefficients which are assumed to be

86

continuous and bounded, and the unknown functions di(t) ∈ R, i = 1, 2, · · · , n, represent

external disturbances which are also assumed to be continuous and bounded.

Provided that all the states are available, the state feedback controller can be represented

by a nonlinear function:

u(t) = ϕ xn(t), t[

(6.2)

Now, the work is how to synthesize a state feedback controller u(t) that can guarantee the

stabilization of nonlinear dynamical system (6.1) in the presence of structure uncertainties,

external disturbances, and unknown time–varying virtual control coefficients.

Before giving our control scheme, we also need to introduce the following preliminaries and

standard assumptions for nonlinear dynamical system (2.1).

In this chapter, we also want to make use of the method of NNs to approximate the non-

linear structure uncertainties such that we can construct a class of adaptive robust controllers.

According to the approximation properties of NNs in Chapter 2, we firstly make the following

assumption.

To synthesize the feedback controller u(t), we also require the following standard assump-

tions.

Assumption 6.1. For any i ∈ {1, 2, · · · , n}, over the compact region Ω(xi) ⊂ Ri, there

exists the ideal weight matrix ω∗i ∈ RLi×1 such that for xi(t) ∈ Ω(xi),

fi xi(t)[= (ω∗i )

�ρi xi(t)[+ εi xi(t)

[(6.3)

holds with ‖εi xi(t)[‖ ≤ εi < ∞, where εi is an unknown positive constant, and where

ρi xi(t)[=

]ρi1 xi(t)

[ρi2 xi(t)

[ · · · ρiLixi(t)

[{�with ρij xi(t)

[, j = 1, 2, · · · , Li, being chosen as the commonly used Guassian functions such

as the follows:

ρij xi(t)[= exp

)−((xi(t)− cij

((2δ2ij

⎧, j = 1, 2, · · · , Li

and where cij =]cj1 cj2 · · · cji

∣�is any given constant vector represent the center of the

receptive field, and δij is any given constant represent the width of the Gaussian functions.

Assumption 6.2. The sign of gi(·) are known with i = 1, 2, · · · , n. Without loss of

generality, we assume gi(·) > 0, i = 1, 2, · · · , n. Moreover, for any i ∈ {1, 2, · · · , n}, thereexist the known positive constants g

iand the unknown positive constants gi such as 0 < g

i≤

gi(·) ≤ gi < ∞Remark 6.2. Assumption 2.2 implies that, similar to [62] and [63], in this chapter, being

87

different from [58]–[61], the upper bound values of virtual control coefficients can be allowed

to be unknown, but the lower bound values of virtual control coefficients need to be known.

Moreover, as shown in [87], for any i ∈ {1, 2, · · · , n}, letting gi(·) = 0 is the necessary condition

for controllability of the uncertain nonlinear dynamical system (6.1).

In addition, following Assumption 6.2, we can easily get that:

gi(·)gi

≥ 1, i = 1, 2, · · · , n (6.4)

Assumption 6.3. The external disturbances di(t) ∈ R, i = 1, 2, · · · , n are continuous and

bounded with respect to time t. Furthermore for any i ∈ {1, 2, · · · , n}, there exist the unknownpositive constants di such that:

|di(t)| ≤ di < ∞, i = 1, 2, · · · , n (6.5)

Without loss of generality, we also introduce the following definitions.

ψ∗i :=

1

4∥ω∗

i ∥2 + εi + di, i = 1, 2, · · · , n (6.6a)

ϕ∗i :=

1

4g2i , i = 1, 2, · · · , n− 1 (6.6b)

κ∗i−1 :=1

4max{gj}2 +

1

4max{∥ω∗

j∥}2 +max{εj}+max{dj},

j = 1, 2, · · · , i, i = 2, 3, · · · , n (6.6c)

Here, it is obvious from the definitions given above that the parameters ψ∗i , ϕ

∗i , and κ

∗i−1 are

still some unknown positive constants.

6.2 Adaptive Robust Backstepping Controller Design

The main purpose of this section is to design an adaptive robust controller based on back-

stepping approach for the uncertain dynamical system (6.1). Similar to the conventional back-

stepping procedure, the control method is containing n steps. At the previous n− 1 steps, the

control functions αi(·), i = 1, 2, · · · , n − 1, will be introduced. Particularly, the control func-

tions αi(·), i = 1, 2, · · · , n−1, are virtual controllers, which will be determined by the adaptive

robust feedback control approach. Finally, at step n the actual controller u(t) is proposed, such

that the system stabilization can be guaranteed.

Moreover, by employing the virtual control functions αi(·), the state variables can be trans-

formed into the following form:

z1(t) = x1(t)

zi(t) = xi(t)− αi−1(·), i = 2, 3, · · · , n(6.7)

88

Considering Assumption 6.1 and system (6.1), the time derivative of (6.7) results in

z1 = g1(x1)z2 + (ω∗1)�ρ1(x1) + ε1(x1)

+d1(t) + g1(x1)α1(·)z2 = g2(x2)z3 + (ω∗2)

�ρ2(x2) + ε2(x2)

+d2(t) + g2(x2)α2(·)− α1(·)...

... (6.8)

zn−1 = gn−1(xn−1)zn + (ω∗n−1)�ρn−1(xn−1) + εn−1(xn−1)

+dn−1(t) + gn−1(xn−1)αn−1(·)− αn−2(·)zn = gn(xn)u(t) + (ω∗n)

�ρn(xn) + εn(xn)

+dn(t)− αn−1(·)

Before developing the control design procedure, for convenience, similar to [51], we also

introduce the following notations. For any i ∈ {2, 3, · · · , n},

μi−1(t) :=i−1∫j=1

∥∥∥∥∂αi−1∂xj

∥∥∥∥x2j+1 +

i−1∫j=1

∥∥∥∥∂αi−1∂xj

∥∥∥∥‖ρj(xj)‖2

+

∥∥∥∥∥∂αi−1∂ψi−1

˙ψi−1(t) +

∂αi−1∂φi−1

˙φi−1(t)

+∂αi−1∂κi−2

˙κi−2(t) +∂αi−1∂σi−1

σi−1(t)

∥∥∥∥∥ηi−1(t) :=

i−1∫j=1

∥∥∥∥∂αi−1∂xj

∥∥∥∥where ψi−1(t), φi−1(t), and κi−2(t) are respectively the estimate values of ψ∗i−1, φ

∗i−1, and κ∗i−2.

In addition, the update laws of ψi−1(t), φi−1(t), and κi−2(t) will be given later (see (6.10) and

(6.18) for details). Further we define that κ0(t) := 0, and ∂α1/∂κ0 := 0.

Now, we will give the recursive procedure for the adaptive robust backstepping controller

as follows.

Step 1:

Here, for the first equation of system (6.8), the virtual control function α1(·) is designed as

follows.

α1(x1; ψ1, φ1, σ1) = − 1

2g1

k1z1 − 1

g1

v11(·)− 1

g1

v12(·) (6.9)

89

where for any i ∈ {1, 2, · · · , n}, ki ∈ R+ are any given positive constants which satisfy

ki >

⎩∑⎨

0, i = 1

2, 2 ≤ i ≤ n

In addition, v11(·), and v12(·) are designed as

v11(z1; ψ1, σ1) :=π21(t)z1

π1(t)|z1|+ σ1(t)

v12(z1; φ1, σ1) :=φ21(t)z

31

φ1(t)z21 + σ1(t)

and where for any i ∈ {1, 2, · · · , n}, zi(t), and πi(t) are defined as:

zi(t) := [z1, z2, · · · , zi]�

πi(t) := ψi(t) + ‖ρi(xi)‖2

and for any i ∈ {1, 2, · · · , n}, σi(t) ∈ R+ are any positive uniform continuous and bounded

functions and which satisfy

limt→∞

[ t

t0

σi(τ)dτ ≤ σi < ∞

Furthermore, the functions ψ1(t), and φ1(t) are respectively the estimate values of the

unknown parameters ψ∗1, and φ∗1, which will be updated by the following adaptation laws:

dψ1(t)

dt= −γ11σ1(t)ψ1(t) + γ11|z1| (6.10a)

dφ1(t)

dt= −γ12σ1(t)φ1(t) + γ12z

21 (6.10b)

where γ11, and γ12 are any positive constants, and ψ1(t0), and φ1(t0) are finite.

On the other hand, letting ψ1(t) = ψ1(t)−ψ∗1, and φ1(t) = φ1(t)−φ∗1, then we can rewrite

equation (6.10) into the following error systems.

dψ1(t)

dt= −γ11σ1(t)ψ1(t) + γ11|z1| − γ11σ1(t)ψ

∗1 (6.11a)

dφ1(t)

dt= −γ12σ1(t)φ1(t) + γ12z

21 − γ12σ1(t)φ

∗1 (6.11b)

Consider the uncertain nonlinear subsystem described by the first equation of (6.8) and the

equation (6.9), we introduce the following Lyapunov function:

V1(t) =1

2z21 +

1

2γ−111 ψ

21(t) +

1

2γ−112 φ

21(t) (6.12)

90

Then, by taking the derivative of V1(t) along the trajectories of the first equation of (6.8)

and the equation (6.9), we can obtain that for any t ≥ t0:

dV1(t)

dt≤ ∥∥g1(x1)

∥∥|z1||z2|+|z1|

)((ω∗1((((ρ1(x1)((+

((ε1(x1)(((

+|z1||d1(t)|+ g1(x1)z1α1(·)

+γ−111 ψ1(t)dψ1(t)

dt+ γ−112 φ1(t)

dφ1(t)

dt(6.13)

By using Assumptions 6.2, 6.3, and definitions described by (6.6), (6.13) becomes:

dV1(t)

dt≤ φ∗1z

21 + z22 + ψ∗1|z1|

+((ρ1(x1)

((2|z1|+ g1(x1)z1α1(·)

+γ−111 ψ1(t)dψ1(t)

dt+ γ−112 φ1(t)

dφ1(t)

dt(6.14)

From (6.4), (6.9), (6.11) and (6.14), it can be further obtained that:

dV1(t)

dt≤ −1

2k1z

21 + z22

+π1(t)|z1| · σ1(t)

π1(t)|z1|+ σ1(t)+

φ1(t)z21 · σ1(t)

φ1(t)z21 + σ1(t)

+σ1(t)

)1

4(ψ∗1)

2 +1

4(φ∗1)

2

((6.15)

Notice the fact that for ∀a, b > 0, there exists

0 <ab

a+ b≤ b

Thus, we can rewrite (6.15) into

dV1(t)

dt≤ −1

2k1z

21 + z22 + ς1σ1(t) (6.16)

where

k1 := k1

ς1 := 2 +1

4(ψ∗1)

2 +1

4(φ∗1)

2

Step i (2 ≤ i ≤ n− 1):

Here, for the ith equation of system (6.8), the virtual control function αi(·) is designed as

91

follows.

αi(xi; ψi, ϕi, κi−1, σi) = − 1

2gi

kizi −1

gi

vi1(·)

− 1

gi

vi2(·)−1

gi

pi−1(·) (6.17)

where vi1(·), vi2(·), and pi−1(·) are designed as:

vi1(zi; ψi, σi) :=π2i (t)zi

πi(t)|zi|+ σi(t)

vi2(zi; ϕi, σi) :=ϕ2i (t)z

3i

ϕi(t)z2i + σi(t)

pi−1(zi; κi−1, σi) :=ϑ2i−1(t)zi

ϑi−1(t)|zi|+ σi(t)

and where for any i ∈ {2, 3, · · · , n}, ϑi−1(t) are defined as:

ϑi−1(t) := µi−1(t) + κi−1(t)ηi−1(t)

furthermore the function ψi(t), ϕi(t), and κi−1(t) are respectively the estimate values of the

unknown parameters ψ∗i , ϕ

∗i , and κ∗i−1, which will be updated by the following adaptation

laws:

dψi(t)

dt= −γi1σi(t)ψi(t) + γi1|zi| (6.18a)

dϕi(t)

dt= −γi2σi(t)ϕi(t) + γi2z

2i (6.18b)

dκi−1(t)

dt= −mi−1σi(t)κi−1(t) +mi−1ηi−1(t)|zi| (6.18c)

where γi1, γi2, and mi−1 are any positive constants, and ψi(t0), ϕi(t0), and κi−1(t0) are finite.

Here, letting ψi(t) = ψi(t)− ψ∗i , ϕi(t) = ϕi(t)− ϕ∗

i , and κi−1(t) = κi−1(t)− κ∗i−1, we can

rewrite adaptation laws (6.18) to be the following error systems.

dψi(t)

dt= −γi1σi(t)ψi(t) + γi1|zi| − γi1σi(t)ψ

∗i (6.19a)

dϕi(t)

dt= −γi2σi(t)ϕi(t) + γi2z

2i − γi2σi(t)ϕ

∗i (6.19b)

dκi−1(t)

dt= −mi−1σi(t)κi−1(t) +mi−1ηi−1(t)|zi| −mi−1σi(t)κ

∗i−1 (6.19c)

Then, for the uncertain nonlinear subsystem described by the ith equation of (6.8) and the

92

equation (6.19), we will introduce the following Lyapunov equation:

Vi(t) = Vi−1(t) +1

2z2i +

1

2γ−1i1 ψ2

i (t)

+1

2γ−1i2 φ2

i (t) +1

2m−1

i−1κ2i−1(t) (6.20)

Taking the derivative of Vi(t) along the trajectories the ith equation of (6.8) and the equation

(6.19), it can be obtained for any t ≥ t0 that:

dVi(t)

dt≤ −1

2

i−1∫j=1

kjz2j + z2i +

i−1∫j=1

ςjσj(t) + gi|zi||zi+1|

+|zi|)‖(ω∗i )‖‖ρi(xi)‖+ ‖εi(xi)‖

(+di|zi|+ |zi||αi−1(·)|+ gi(xi)ziαi(·)

+γ−1i1 ψi(t)dψi(t)

dt+ γ−1i2 φi(t)

dφi(t)

dt

+m−1i−1κi−1(t)

dκi−1(t)dt

(6.21)

On the other hand, the time derivative of αi−1(·) results in:

αi−1(xi−1; ψi−1, φi−1, κi−2, σi−1)

=i−1∫j=1

∂αi−1∂xj

gj(xj)xj+1 +i−1∫j=1

∂αi−1∂xj

fj(xj)

+i−1∫j=1

∂αi−1∂xj

dj(t) +∂αi−1∂ψi−1

˙ψi−1(t) +

∂αi−1∂φi−1

˙φi−1(t)

+∂αi−1∂κi−2

˙κi−2(t) +∂αi−1∂σi−1

σi−1(t) (6.22)

It is obvious that:

|αi−1(·)| ≤ μi−1(t) + κ∗i−1ηi−1(t) (6.23)

By using the Assumptions 6.2, 6.3, and the definitions given in (6.6), (6.21) becomes:

dVi(t)

dt≤ −1

2

i−1∫j=1

kjz2j + z2i + z2i+1 +

i−1∫j=1

ςjσj(t)

+ψ∗i z2i + ‖ρi(xi)‖z2i + φ∗i |zi|

+μi−1(t)|zi|+ κ∗i−1ηi−1(t)|zi|+gi(xi)ziαi(·)

93

+γ−1i1 ψi(t)dψi(t)

dt+ γ−1i2 φi(t)

dφi(t)

dt

+m−1i−1κi−1(t)

dκi−1(t)dt

(6.24)

Substituting (6.17) and (6.19) to (6.24), similar to the method used in Step 1, we can finally

obtained that:

dVi(t)

dt≤ −1

2

i−1∫j=1

kjz2j −

1

2(ki − 2)z2i + z2i+1 +

i−1∫j=1

ςjσj(t)

+πi(t)|zi| · σi(t)

πi(t)|zi|+ σi(t)+

φi(t)z2i · σi(t)

φi(t)z2i + σi(t)

+ϑi−1(t)|zi| · σi(t)

ϑi−1(t)|zi|+ σi(t)

−σi(t) ψ2i (t) + ψi(t)ψ

∗i

[ − σi(t) φ2i (t) + φi(t)φ

∗i

[−σi(t) κ2

i−1(t) + κi−1(t)κ∗i−1[

≤ −1

2

i∫j=1

kjz2j + z2i+1 +

i∫j=1

ςjσj(t) (6.25)

where

ki := ki − 2

ςi := 3 +1

4(ψ∗i )

2 +1

4(φ∗i )

2 +1

4(κ∗i−1)

2

Step n:

At the last step, for the last equation of (6.8), we will construct the actual controller u(t)

as follows.

u(t) = ϕ(xn; ψn, κn−1, σn)

= − 1

2gn

knzn − 1

gn

vn(·)− 1

gn

pn−1(·) (6.26)

where vn(·), and pn−1(·) are designed as:

vn(zn; ψn, σn) :=π2n(t)zn

πn(t)|zn|+ σn(t)

pn−1(zn; κn−1, σn) :=ϑ2n−1(t)zn

ϑn−1(t)|zn|+ σn(t)

and where the function ψn(t), and κn−1(t) are respectively the estimate values of the unknown

94

parameters ψ∗n, and κ∗n−1, which will be updated by the following adaptation laws:

dψn(t)

dt= −γnσn(t)ψn(t) + γn|zn| (6.27a)

dκn−1(t)dt

= −mn−1σn(t)κn−1(t) +mn−1ηn−1(t)|zn| (6.27b)

where γn, and mn−1 are any positive constants, and ψn(t0), and κn−1(t0) are finite.

Here, letting ψn(t) = ψn(t)−ψ∗n, and κn−1(t) = κn−1(t)−κ∗n−1, we can rewrite adaptation

laws (6.27) to be the following error systems.

dψn(t)

dt= −γnσn(t)ψn(t) + γn|zn| − γnσn(t)ψ

∗n (6.28a)

dκn−1(t)dt

= −mn−1σn(t)κn−1(t) +mn−1ηn−1(t)|zn|−mn−1σn(t)κ

∗n−1 (6.28b)

Then, for the uncertain nonlinear subsystem described by the last equation of (6.8) and the

equation (6.26), we will introduce the following Lyapunov equation:

Vn(t) = Vn−1(t) +1

2z2n +

1

2γ−1n ψ2

n(t) +1

2m−1

n−1κ2n−1(t) (6.29)

Taking the derivative of Vn(t) along the trajectories of the last equation of (6.8) and the

equation (6.26), and making the transformation similar to above n−1 steps, from (6.26), (6.28)

and (6.29), we can obtain that for any t ≥ t0:

dVn(t)

dt≤ −1

2

n∫j=1

kjz2j +

n∫j=1

ςjσj(t) (6.30)

where we define kn, and ςn as

kn := kn − 2

ςn := 2 +1

4(ψ∗n)

2 +1

4(κ∗n−1)

2

Here, similar to [51], we introduce the following definitions:

z(t) :=]z�n (t) ψ�(t) φ�(t) κ�(t)

{�k := min

}ki : i = 1, 2, · · · , n

ς := max {ςi : i = 1, 2, · · · , n}

95

where

ψ(t) :=]ψ1(t) ψ2(t) · · · ψn(t)

{�φ(t) :=

]φ1(t) φ2(t) · · · φn−1(t)

{�κ(t) := [κ1(t) κ2(t) · · · κn−1(t)]

�

Remark 6.2. For any i ∈ {1, 2, · · · , n}, according to the definitions of ki, and the

relationship between ki, and ki, it is obvious that k > 0. Moreover from the definitions of ςi,

i = 1, 2, · · · , n, it can also be obtained that ς > 0.


According to the above descriptions, we can propose the following theorem which shows

that the state xn(t) ∈ Rn of uncertain dynamical system (6.1) can be asymptotically stable.

Theorem 6.1. Consider the uncertain dynamical system described by (6.1) satisfying

Assumptions 6.1 – 6.3. Under the adaptive robust control scheme given by (6.7) and (6.26)

with (6.27), the state xn(t) is uniformly asymptotically stable in the presence of structure

uncertainties, external disturbances, and unknown time–varying virtual control coefficients.

Proof:

From (6.30) and the definitions of k, and ς, we can further obtain that for any t ≥ t0:

dVn z(t), t[

dt≤ −1

2k((zn(t)((2 + ς

n∫j=1

σj(t) (6.31)

On the other hand, according to the definition of Lyapunov function given in (6.29), there

always exist two positive constants δmin, and δmax such that for any t ≥ t0:

γ1 ‖z(t)‖[ ≤ Vn z(t), t[ ≤ γ2 ‖z(t)‖[ (6.32)

where

γ1 ‖z(t)‖[ := δmin‖z(t)‖2

γ2 ‖z(t)‖[ := δmax‖z(t)‖2

Now, by making use of the method which has been used in Chapter 2, we can complete the

rest proof of Theorem 6.1. The details are as follows.

Consider the continuity of the transformed control system (6.8), and error systems (6.11),

(6.19), and (6.28), it is obvious that any solution (zn, ψ, φ, κ) (t; t0, zn(t0), ψ(t0), φ(t0), κ(t0)) of

96

the system is continuous with respect to time t.

It follows from (6.31) and (6.32) that for any t ≥ t0

0 ≤ γ1 ‖z(t)‖[ ≤ Vn z(t), t[= Vn z(t0), t0

[+

[ t

t0

Vn z(τ), τ[dτ

≤ γ2 ‖z(t0)‖[ − [ t

t0

γ3 ‖z(τ)‖[dτ +

[ t

t0

ς

n∫j=1

σj(τ)dτ (6.33)

where

γ3 ‖z(t)‖[ :=1

2k((zn(t)((2

Then, taking the limit on the both sides of (6.33) as t approaches infinity, we can obtain

that

0 ≤ γ2 ‖z(t0)‖[ − lim

t→∞

[ t

t0

γ3 ‖z(τ)‖[dτ + limt→∞

[ t

t0

ςn∫

j=1

σj(τ)dτ (6.34)

Consider the definitions of σi(t), i = 1, 2, · · · , n, from (6.34) we can further obtain that

limt→∞

[ t

t0

γ3 ‖z(τ)‖[dτ ≤ γ2 ‖z(t0)‖[+ ς

n∫j=1

limt→∞

[ t

t0

σj(τ)dτ

≤ γ2 ‖z(t0)‖[+ ς

n∫j=1

σj (6.35)

On the other hand, from (6.33) we can also get that

0 ≤ γ1 ‖z(t)‖[ ≤ γ2 ‖z(t0)‖[+ ς

n∫j=1

[ t

t0

σj(τ)dτ (6.36)

For any i = 1, 2, · · · , n, it follows from the definitions of σi(t), there also exists that for any

t ≥ t0

supt∈[t0,∞)

[ t

t0

σi(τ)dτ ≤ σi (6.37)

Then it follows from (6.36) and (6.37) that

0 ≤ γ1 ‖z(t)‖[ ≤ γ2 ‖z(t0)‖[+ ς

n∫j=1

σj (6.38)

which implies that z(t) is uniformly bounded. On the other hand, consider that zn(t) is uni-

formly continuous with respect to time t, it is obviously that γ3 ‖z(t)‖[ is also uniformly

continuous with respect to time t. By making use of the Barbalat Lemma of Chapter 2, from

97

(6.35) we can easily obtain that:

limt→∞

γ3 ‖z(t)‖[ = 0 (6.39)

Moreover, consider the definition of γ3 ‖z(t)‖[, from (6.39) we can further obtain:

limt→∞

‖zn(t)‖ = 0 (6.40)

which implies that zn(t) converges asymptotically to zero.

In addition, because zn(t) converges asymptotically to zero, and noting the definitions of

αi(·), i = 1, 2, · · · , n − 1, we can easily obtain that limt→∞ αi(·) = 0, i = 1, 2, · · · , n − 1.

Then, from the relationship between xn(t), and zn(t) described by (6.7), we can further get that

limt→∞ xi(t) = 0, i = 1, 2, · · · , n. Therefore, we can say that the state xn(t) can converge

uniformly asymptotically to zero.

Thus, we complete the proof of Theorem 6.1. ∇∇∇Remark 6.3. In the design process of the actual controller u(t), we do not give an adapta-

tion law to estimate the upper bound value of the time–varying virtual control coefficient gn(·),and we also do not make use of the knowledge of the upper bound value of gn(·) to construct

the actual controller u(t). Thus, in this paper, it is not necessary for the time–varying virtual

control coefficient gn(·) to have upper bound.

Remark 6.4. In this chapter, the problem of adaptive robust stabilization is considered for

a class of dynamical systems in the presence of structure uncertainties, external disturbances,

and unknown time–varying virtual control coefficients. It is not difficult to consider the adaptive

robust tracking problem by making use of the method presented in this chapter. Moreover,

the result obtained in this chapter can also be extended to MIMO strict–feedback uncertain

systems, which may be our future investigation.


In this section, we will present two simulation examples to show the effectiveness of the

proposed control schemes.

6.4.1 An Inverted Pendulum System

Similar to [101], [102], as shown in Figure 6.1, we also choose an inverted pendulum system

(or a cart–pole system) with some modification considering the external disturbance.

Letting

x1(t) = θ(t)

x2(t) = θ(t)

98

Then, the mathematical description of the inverted pendulum system can be written as

x1 = g1(x1)x2 + f1(x1) + d1(t) (6.41a)

x2 = g2(x2)u(t) + f2(x2) + d2(t) (6.41b)

where

g1(·) = 1, g2(·) =cos(x1)

43(mc +m)l −ml cos2(x1)

f1(·) = 0, f2(·) =(mc +m)g sin(x1)−mlx22 cos(x1) sin(x1)

43(mc +m)l −ml cos2(x1)

d1(t) = 0, d2(t) = g2(·)ω(t)

and where θ(t) is the pole angular displacement, mc is the mass of cart, m is the mass of pole,

l is the half length of pole, g is the acceleration due to gravity, u(t) is the control input, and

ω(t) is the external disturbance. The purpose is to determine a proper controller u(t) such that

system (6.41) can be guaranteed stable on the position θ(t) = x1(t) = 0.

cm

1x

2x

sinmgl

u

Figure 6.1: An Inverted Pendulum System

For simulation, similar to [102], we choose the following parameters for this system.

mc = 1kg, m = 0.1kg

l = 0.5m, g = 9.8m/s2

To satisfy the control condition g2(·) = 0, we also need to determine the range of x1(t).

Similar to [102], we set |x1(t)| ≤ π/6.

99

It is obvious that the lower bound values giof gi(·), i = 1, 2 can be known as:

g1= 1, g

2=

120√3

167

Then, the following work is to determine a proper adaptive robust control in the form of

(6.26) with (6.27). According to the design procedure, the design is given in the following steps.

Step.1:

Similar to [42], we also take a simple RBFNN, and give the Gaussian bias functions for the

RBFNN as follows

ρ2j(x2) = exp

)−‖x2 − c2j‖2

δ22j

(, j = 1, 2, · · · , L2

where L2 = 9, c2j = [cj1 cj2]�, and the centers cj1, and cj2 are respectively spaced in

[−π/6, π/6], and [−π/6, π/6] with width δ2j = π/24.

Step.2:

The following work is to determine a proper adaptive robust control law in the form (6.7)

and (6.26) with (6.27). For the proposed control scheme and adaptation laws, we choose

k1 = 10, k2 = 12, γ11 = 0.2

γ12 = 0.2, γ2 = 0.2, m1 = 0.2

and

σ1(t) = 5 exp(−0.01t)

σ2(t) = 5 exp(−0.01t)

Step.3:

Finally, the initial conditions, and external disturbance are given as follows:

x(0) = [π/6 − π/6]�

ψ1(0) = π/12, φ1(0) = π/12

ψ2(0) = π/12, κ1(0) = π/12

and

ω(t) = 0.03 cos(t)N

100

0 1 2 3 4 5 6-2

-1.5

-1

-0.5

0

0.5

1

Time (Sec)

Res

ponse

ofSta

teV

ariable

x(t

)x1(t)

x2(t)

Figure 6.2: Responses of States x1(t) and x2(t)

0 1 2 3 4 5 60

0.10.20.30.40.50.60.70.8

Time (Sec)

History

ofA

daptatio

nLaw

ψ1(t)

Figure 6.3: History of Adaptation Law ψ1(t)

101

0 1 2 3 4 5 60

0.10.20.30.40.50.60.70.8

Time (Sec)

History

ofA

daptatio

nLaw

φ1(t)

Figure 6.4: History of Adaptation Law ϕ1(t)

0 1 2 3 4 5 60

0.10.20.30.40.50.60.70.8

Time (Sec)

History

ofA

daptatio

nLaw

ψ2(t)


102

0 1 2 3 4 5 60

0.10.20.30.40.50.60.70.80.9

Time (Sec)

History

ofA

daptatio

nLaw

κ1(t)

Figure 6.6: History of Adaptation Law κ1(t)

With the chosen parameter settings, the results of this simulation are shown in Figures

6.2–6.6.

From Figure 6.2, it is shown that by employing the control scheme proposed in this chapter,

the state x(t) of system (6.41) can be indeed guaranteed to converge asymptotically to zero.

Meanwhile, as shown in Figures 6.3–6.6, the adaptation laws ψ1(t), ϕ1(t), ψ2(t), and κ1(t) with

σ–modification are also uniformly ultimately bounded.

6.4.2 A 3rd Order Nonlinear System

In this subsection, we also consider the following third order nonlinear system.

x1 = g1(x1)x2 + f1(x1) + d1(t) (42a)

x2 = g2(x2)x3 + f2(x2) + d2(t) (42b)

x3 = g3(x3)u(t) + f3(x3) + d3(t) (42c)

where

g1(·) = 1 + cos(x1)2, g2(·) = 3 + cos(x1x2)

g3(·) = 3 + sin(x1x2x3), f1(·) = x1 exp(−0.5x1) + 2x21

f2(·) = x1x2 + 2x21 + 0.5x2, f3(·) = x1x2x3 + x2x3

d1(t) = d2(t) = d3(t) = 0.01 cos(2t)

103

It is clear that the lower bound values giof gi(·), i = 1, 2, 3 can be known as:

g1= 1, g

2= 2, g

3= 2

The purpose is to determine a proper controller u(t) (6.7) and (6.26) with (6.27) such that

system (6.42) is asymptotically stable.

Step.1:

Similar to [42], in order to approximate the structure uncertainties fi(·), i = 1, 2, 3, we take

the basis functions of RBFNN as follows.

ρij(xi) = exp

)−‖xi − cij‖2

δ2ij

(, i = 1, 2, 3, j = 1, 2, · · · , Li

where L1 = 11, L2 = 11, L3 = 11, c1j = cj1, c2j = [cj1 cj2]�, c3j = [cj1 cj2 cj3]

�, and the

centers cj1, cj2, and cj3 are respectively spaced in [−0.5, 0.5], [−0.5, 0.5], and [−0.5, 0.5] with

widths δ1j = 0.1, δ2j = 0.1, and δ3j = 0.1.

Step.2:

The following work is to determine a proper adaptive robust control law in the form (6.7)

and (6.26) with (6.27). For the proposed control scheme and adaptation laws, we choose

k1 = 0.5, k2 = 2.5, k3 = 2.5

γ11 = 1, γ12 = 1, γ21 = 1, γ22 = 1

γ3 = 1.5, m1 = 0.5, m2 = 2.5

and

σ1(t) = exp(−0.02t), σ2(t) = exp(−0.02t)

σ3(t) = exp(−0.05t)

Step.3:

Finally, the initial conditions are given as follows:

x(0) = [0.5 0.2 1]�, ψ1(0) = 1

φ1(0) = 1, ψ2(0) = 1, φ2(0) = 1

κ1(0) = 1, ψ3(0) = 1, κ2(0) = 1

104

0 0.5 1 1.5 2 2.5 3 3.5 4-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

Time (Sec)

Res

ponse

ofSta

teV

ariable

x(t

)

x1(t)x2(t)x3(t)

Figure 6.7: Responses of States x1(t), x2(t), and x3(t)

0 0.5 1 1.5 2 2.5 3 3.5 400.10.20.30.40.50.60.70.80.91

Time (Sec)

History

ofA

daptatio

nLaw

ψ1(t)


105

0 0.5 1 1.5 2 2.5 3 3.5 400.10.20.30.40.50.60.70.80.91

Time (Sec)

History

ofA

daptatio

nLaw

φ1(t)


0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Time (Sec)

History

ofA

daptatio

nLaw

ψ2(t)


106

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Time (Sec)

History

ofA

daptatio

nLaw

φ2(t)


0 0.5 1 1.5 2 2.5 3 3.5 40.5

0.6

0.7

0.8

0.9

1

Time (Sec)

History

ofA

daptatio

nLaw

κ1(t)


107

0 0.5 1 1.5 2 2.5 3 3.5 40

0.2

0.4

0.6

0.8

1

Time (Sec)

History

ofA

daptatio

nLaw

ψ3(t)


0 0.5 1 1.5 2 2.5 3 3.5 40

5

10

15

20

Time (Sec)

History

ofA

daptatio

nLaw

κ2(t)


108

With the chosen parameter settings, the results of this simulation are shown in Figures

6.7–6.14.

From Figure 6.7, it is shown that by employing the control scheme proposed in this chapter,

the state x(t) of system (6.42) can be indeed guaranteed to converge asymptotically to zero.

Meanwhile, as shown in Figures 6.8–6.14, the adaptation laws ψ1(t), ϕ1(t), ψ2(t), ϕ2(t), κ1(t)

ψ3(t), and κ2(t) with σ–modification are also uniformly ultimately bounded.

109

Chapter 7

Conclusions and Future Directions

7.1 Conclusions

In this paper, we have considered the problems of robust stabilisation and adaptive robust

controller design for uncertain dynamical systems with nonlinear uncertainties which are with

matched conditions or with mismatched conditions.

In Chapter 2, in order to expand our studies, we have introduced some preliminaries which

contain: 1. Lyapunov Functional–based stability of nonlinear systems; 2. function approxima-

tion using NNs; 3. backstepping control design technique.

In Chapter 3, for a class of uncertain dynamical systems with matched nonlinear uncertain-

ties and external disturbances, we have introduced the NNs with a simple structure to model

the completely unknown nonlinear uncertainties. Then, we employed an adaptation law to learn

online the norm forms of the weight and approximate error of the NNs, and the upper bound

values of the external disturbances. With the help of the learned values, we have proposed

a state feedback controller to guarantee that the states converge uniformly asymptotically to

zero in the presence of matched nonlinear uncertainties and external disturbances. Finally, to

demonstrate the validity of the results, we have applied them to solve a practical problem of

water pollution control.

In Chapter 4, for the same systems which have been mentioned in Chapter 3, we have used

the virtual NNs to model the completely unknown nonlinear uncertainties. That is the NNs

are not really used in the control process. For such a class of dynamical systems, we have

proposed two adaptation laws to learn online the uncertain parameters which contain all the

uncertain bound values of these uncertainties. With the help of the learned values, we have

developed a new class of adaptive robust controllers with a rather simple structure to guarantee

that the states converge uniformly asymptotically to zero in the presence of matched nonlinear

uncertainties and external disturbances. To demonstrate the validity of the results, we have

also utilized the proposed control schemes to solve the same problem of water pollution control

in Chapter 3.

In Chapter 5, we have extended the control schemes of Chapter 4 to stabilize a class of

delayed nonlinear uncertain systems with generalized matched conditions. In particular, the

time delays are not required must neither to be non–negative constants, nor to be time–varying

functions and the derivatives of which must be less than 1. As a numerical example, we have

employed an uncertain exploited ecosystem with two competing species to demonstrate the

110

validity of the results.

We consider the problem of adaptive robust stabilization for a class of nonlinear systems

with mismatched structure uncertainties, external disturbances, and unknown time–varying

virtual control coefficients in Chapter 6. Similar to Chapter 3, we have introduced a simple

structure NNs to model the unknown structure uncertainties. Then, we have developed a class

of NNs and backstepping approach–based continuous adaptive robust state feedback controllers,

and have also shown that the proposed control schemes can guarantee the uniform asymptotic

stability of such uncertain dynamical systems. Finally, we have applied the proposed control

schemes to solve some practical problems.

7.2 Future Directions

According to the contents and results of this paper, the research directions to which we

should pay more attention are presented as follows.

1. It is well known that various real–world systems such as chemical processing, communi-

cation networks, traffic control, automotive engine control and aircraft control, and so on,

can be modeled to switched systems, neutral systems, and their combinations [106]–[109],

and so on. The stability, robustness, and dynamical performances of these systems are

affected seriously by the existence of the unknown nonlinear uncertainties. On the other

hand, the structure of the control schemes in Chapter 4 is simple, which means our control

schemes are more practicable to application. Therefore, one of our future directions is to

expand our control schemes to switched systems, neutral systems, and their combinations,

and so on.

2. Similar to [80], [81], the adaptation laws of our control schemes are taking the following

classical form.dψ(t)

dt= −γσ(t)ψ(t) + γ∥ · ∥

where γ is a given positive constant, and σ(t) is a given positive function. It is obvious

that different control parameters will produce different control performances. Thus, it is

important to choose some appropriate control parameters for the control scheme to the

system designer. However, there is still no a method how to select the optimal parameters.

Thus, another of future directions is to achieve the optimal selection of parameters by

introducing the optimal control algorithm.

3. The control schemes of Chapter 6 (see, (6.26)) are relatively complex. If we simplify

the control schemes of Chapter 6 to the control scheme form of Chapter 4, there will

occur differential explosion which implies our control schemes of Chapter 4 cannot be

applied to the nonlinear systems with mismatched uncertainties. Therefore, the third

is to simplify the controller structure for the case of mismatched uncertain nonlinear

systems. In particular, the results obtained in control theory may be expected to have

more applications to environmental systems.

111

Appendix A

Commonly Used Basis Functions

A range of theoretical and empirical studies have indicated that many properties of the

interpolating function are relatively insensitive to the precise form of the basis functions.

Some of the most commonly used basis functions are as follows [95].

1. Gaussian Functions:

ϕ(x) = exp

(− x2

2σ2

), with σ > 0 (A.1)

2. Multi–Quadric Functions:

ϕ(x) = (x2 + σ2)1/2, with σ > 0 (A.2)

3. Generalized Multi–Quadric Functions:

ϕ(x) = (x2 + σ2)γ, with σ > 0, and 1 > γ > 0 (A.3)

4. Inverse Multi–Quadric Functions:

ϕ(x) = (x2 + σ2)−1/2, with σ > 0 (A.4)

-5 0 50

0.10.20.30.40.50.60.70.80.91

x

φ(x)

σ = 0.1

σ = 0.5

σ = 1

Figure A.1: Gaussian Functions

112

-20 -10 0 10 200

5

10

15

20

25

x

φ(x)

σ = 0.5

σ = 5

σ = 10

Figure A.2: Multi–Quadric Functions

-20 -10 0 10 2005

10152025303540

x

φ(x)

σ = 0.5, γ = 0.4

σ = 5, γ = 0.4

σ = 0.5, γ = 0.6

σ = 5, γ = 0.6

Figure A.3: Generalized Multi–Quadric Functions

-20 -10 0 10 200

0.5

1

1.5

2

x

φ(x)

σ = 0.5

σ = 5

σ = 10

Figure A.4: Inverse Multi–Quadric Functions

113

5. Generalized Inverse Multi–Quadric Functions:

ϕ(x) = (x2 + σ2)−γ, with σ > 0, and γ > 0 (A.5)

-20 -10 0 10 200

0.5

1

1.5

2

2.5

x

φ(x)

σ = 0.5, γ = −0.4

σ = 5, γ = −0.4

σ = 0.5, γ = −0.6

σ = 5, γ = −0.6

Figure A.5: Generalized Inverse Multi–Quadric Functions

6. Thin Plate Spline Function:

ϕ(x) = x2 ln(x) (A.6)

-20 -15 -10 -5 0 5 10 15 200

200

400

600

800

1000

1200

x

φ(x

)

Figure A.6: Thin Plate Spline Function

114

Appendix B

List of Published Papers

Published Papers in Journals:

1. Yuchao Wang, and Hansheng Wu: “Neural networks–based adaptive robust controllers

and its applications to water pollution control systems,” Int. J. Advanced Mechatronic

Systems, Vol. 5, No. 2, pp. 138-145, 2013.

2. Hansheng Wu, and Yuchao Wang: “Adaptive robust control schemes of optimal long–

term management for a class of uncertain ecological systems,” Journal of the Faculty of

Management and Information Systems, Prefectural University of Hiroshima, No. 5, pp.

51-63, 2013.

3. Yuchao Wang, and Hansheng Wu: “Adaptive robust backstepping control for a class of

uncertain dynamical systems using neural networks,” Nonlinear Dynamics, Vol. 81, No.

4, pp. 1597-1610, 2015.

4. Yuchao Wang, and Hansheng Wu: “A class of adaptive robust controllers for uncertain

dynamical systems with unknown virtual control coefficients,” Int. J. Advanced Mecha-

tronic Systems, Vol. 6, Nos. 2/3, pp. 65-74, 2015.

Published Papers in Conferences:

1. Yuchao Wang, and Hansheng Wu: “Adaptive robust control for a class of uncertain

neutral systems with time–delay,” 2012 Annual Conference of Electronics, Information

and Systems Society, I.E.E. of Japan, pp. 1391-1395, Sep. 2012.

2. Yuchao Wang, and Hansheng Wu: “Adaptive robust backstepping control for a class

of uncertain nonlinear systems using neural networks,” Proceedings of the 21st Annual

Conference of the SICE Chugoku Chapter, Japan, pp. 38-39, Nov. 2012.

3. Yuchao Wang, and Hansheng Wu: “Adaptive robust stabilization for a class of uncertain

nonlinear systems with unknown virtual control coefficients,” Proceedings of the 2014

International Conference on Advanced Mechatronic Systems, Japan, pp. 165-170, Aug.

2014.

115

Acknowledgements

How time flies. It has been another four years since I successfully finished my Master’s

degree in this University. I am a bit excited when writing these acknowledgements, not only

for my research, but also for the wonderful time that I have spent in Hiroshima City, Japan. In

all these years, many people have helped and supported me. Without them, this dissertation

would not have been completely so easily. I am very glad that I now have the opportunity to

express my gratitude to all of them.

First of all, I would like to express my deepest gratitude to my advisor, Hansheng Wu. I

have been amazingly fortunate to have an advisor who gave me the freedom to explore on my

own, and at the same time the guidance to recover when my steps faltered. Prof. Wu taught

me how to question thoughts and express ideas. His patience and support helped me overcome

many crisis situations and finish this dissertation.

Besides my advisor, I must also acknowledge Prof. Yegui Xiao for his suggestions for neural

network approximation technique in this study.

Last but not the least, I would like to thank my family: my parents Jingwu Wang and

Guifang Xia, for giving birth to me at the first place and supporting me spiritually throughout

my life; my wife Jie Li, without whose love, encouragement and editing assistance, I would not

have finished this dissertation.

116

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