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Control Theory for Automation – Advanced Techniques 10.9 Control of Nonlinear Systems 195

The importance of the feedback linearization lies in

the fact that the linearized structure allows us to ap-

ply the wide selection in the control design methods

available for linear systems. Before going into the de-

tails of such applications an engineering interpretation

of the design aspects of the feedback linearization will

be given. Speciﬁcally, the feedback linearization tech-

nique developed for nonlinear systems will be applied

for linear plants.

10.9.2 Feedback Linearization

Versus Linear Controller Design

Assume asingle-input single-outputlinear systemgiven

by the following state-space representation

x =Ax+Bu ,

y =Cx .

The derivatives of the output can sequentially be gener-

ated as

y =Cx ,

y =C

x =CAx ,

y =CA

x =CA

x ,

(r)

=CA

r−1

x =CA

x+CA

r−1

Bu ,

where CA

B =0fori < r −1andCA

r−1

B =0 with r

being the relative degree. Note thatr is invariant to sim-

ilarity transformations. Observe that this derivation is in

close harmony with the canonical form derived earlier.

The conditionsof the exact statecontrol (r =n) are that:

•

The system iscontrollable: rank(B, AB, A

B,...) =

•

The relative degree is equal to the system order

(r =n), which could be interpreted as the involutive

condition for the linear case.

A more direct relation between the feedback lin-

earization and the linear control design is seen if the

linear system is assumed to be given by an input–output

(transfer function) representation

n−r

(s)

where the transfer function of the process has appro-

priate B

n−r

(s)andA

(s) polynomials of the complex

frequency operator s. The subindices refer to the degree

of the polynomials. If the system has a relative degree

of r, the feedback generates

(r)

=v,

which is formally equivalent to a series compensator of

C(s) =

(s)

n−r

(s)

The above compensator is realizable in the sense

that the numerator and the denominator polynomials are

of identical order n. To satisfy practical needs when

realizing a controller in a stable manner it is required

that the zeros of the process must be in the left half

plane. This condition is equivalent to the requirement

that the zero dynamics remains exponentially stable as

the feedback linearization has been performed.

10.9.3 Sliding-Mode Control

Coming back to the nonlinear case, now a solution is

looked for to transform nonlinear dynamic systems via

state-variable feedback to dynamic systems with rel-

ative degree of 1. Achieving this goal the nonlinear

dynamic system with n state variables could be handled

as one single integrator, which is evidently easy to con-

trol [10.47]. The key point of the solution is to create

a ﬁctitious system with relative degree of 1. Addition-

ally, it will be explained that the internal dynamics of

the closed-loop system can be deﬁned by the designer.

For the sake of simplicity assume that the nonlinear

system is given in controllable form by

(n)

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

where the state variables are

x =(y,

y,...,y

(n−1)

) .

Create the ﬁctitious output signal as the linear combina-

tion of these state variables

z = h



x =h

y+h

y+...+y

(n−1)

Now use the method elaborated earlier for the feedback

linearization. Consider the derivative of the ﬁctitious

output signal and, taking y

(n)

= f (x)+g(x)u into ac-

count,

z = h



x =h

y+h

...

+...+ f (x)+g(x)u

is obtained. Observe that

z appears to be a function of

the input u, meaning that the relative degree of the ﬁcti-

tious system is 1. Now expressing u, and just as before,

introducing the new input signal v,

u =

g(x)



v −h

y−h

...

−...−h

n−2

(n−1)

− f (x)



Part B 10.9

196 Part B Automation Theory and Scientiﬁc Foundations

is obtained. Consequently, the complete system can be-

come realized as one single integrator by

z = v.

The internal dynamics of the closed-loop system is

governed by the h

coefﬁcients. Using Laplace trans-

forms and deﬁning

H(s) =h

s+...+h

n−2

n−1

z(s) can be expressed by

z(s) = H(s)y(s) .

Introduce the z

ref

reference signal for z

z(s) =z(s)−z

ref

(s) = H(s)[y(s)−y

ref

(s)]

= H(s)

y(s) ,

where

ref

(s) = H(s)y

ref

(s) .

The question is, what happens to

y(t) = L

−1

[

y(s)] if

z(t) = L

−1

[

z(s)] tends to zero in steady state. Clearly,

having all the roots of H(s) on the left half plane,

y(t)

will tend to zero together with

z(t) according to

y(s) =

H(s)

z(s) .

Both y(t) → y

ref

(t)and

z(t) → 0 is highly expected

from a well-designed control loop. In addition, the dy-

namic behavior of the y(t) → y

ref

(t) transient response

depends on the location of the roots of H(s). One pos-

sible setting for H(s)isH(s) =(s+λ)

n−1

Note that in practice, takingmodel uncertaintiesinto

account, the relation between v and z becomes

z ≈ v.

To see how to control an approximated integrator ef-

fectively recall the Lyapunov stability theory with the

Lyapunov function

z(t)) =

z(t)

Then

z(t) →0if

z) =

z < 0; moreover,

z(t) =0 can

be reached in ﬁnite time,if theabove Lyapunovfunction

satisﬁes

z) < −η|

z(t)|,

where η is some positive constant. By differentiating

z) the above stability condition reduces to

z(t) ≤−η sign[

z(t)].

The above control technique is called the reaching law.

Here η is to be tuned by the control engineer to com-

pensate the knowledge about the uncertainties involved.

According to the above considerations the control

loop discussed earlier should be extended with a dis-

continuous component given by

v(t) =−K

sign[

z(t)],

where K

>η. To reduce the switching gain and to

increase the speed of convergence, the discontinuous

part of the controller can be extended by a proportional

feedback

v(t) =−K

sign[

z(t)]−K

z(t) .

Note that the application of the above control law may

lead to chattering in the control signal. To avoid this

undesired phenomenon the saturation function is used

instead of the signum function in practice.

To conclude the discussion on sliding control it has

been shown that this concept consists of two phases. In

the ﬁrst phase the slidingsurface is to be reached (reach-

ing mode), while in the second the system is controlled

to movealong thesliding surface(sliding mode).In fact,

these two phases can be designed independently from

each other. Reaching the sliding surface can be realized

by appropriate switchingelements. The wayto reach the

sliding surface can be modiﬁed via the parameters in the

discontinuous part of the controller. Forcing the system

to move along the sliding surface can be effected by as-

signing various parameters in the H(s). The algorithm

shown here can be regarded as a special version of the

variable-structure controller design.

10.10 Summary

In this chapter advanced control system design meth-

ods have been discussed. Advanced methods reﬂect the

current state of the art of the related applied research ac-

tivity in the ﬁeld and a number of advanced methods are

available today for demanding controlapplications. One

of the driving forces behind browsing among various

advanced techniques has been the applicability of the

control algorithms for practical applications. Starting

with the stability issues, then covering performance and

robustness issues, MIMO techniques, optimal control

strategies, and predictive control algorithms have been

discussed. The concept of feedback linearization for

certain nonlinear systems has also been shown. Sliding

control as a representative of the class of variable-

Part B 10.10

Control Theory for Automation – Advanced Techniques References 197

structure controllers has been outlined. To check the

detailed operation of the presented control techniques

in numerical examples the reader is kindly asked to use

the various toolboxes [10.48].

References

10.1 T. Kailath: Linear Systems (Prentice Hall, Upper

Saddle River 1980)

10.2 G.C. Goodwin, S.F. Graebe, M.E. Salgado: Control

System Design (Prentice Hall, Upper Saddle River

2000)

10.3 W.S. Levine (Ed.): The Control Handbook (CRC, Boca

Raton 1996)

10.4 B.G. Lipták (Ed.): Instrument Engineers’ Handbook,

Process Control and Optimization, 4th edn. (CRC,

Boca Raton 2006)

10.5 P.K. Sinha: Multivariable Control (Marcel Dekker,

New York 1984)

10.6 S. Skogestad, I. Postlethwaite: Multivariable Feed-

back Control (Wiley, New York 2005)

10.7 A.F. D’Souza: Design of Control Systems (Prentice

Hall, Upper Saddle River 1988)

10.8 R. Bars, P. Colaneri, L. Dugard, F. Allgöwer,

A. Kleimenow, C. Scherer: Trends in theory of con-

trol system design, 17th IFAC World Congr., Seoul,

ed. by M.J. Chung, P. Misra (IFAC Coordinating Com-

mitteeonDesignMethods,SanFrancisco2008)

pp. 93–104

10.9 K.J. Åström, B. Wittenmark: Computer-Controlled

Systems: Theory and Design (Prentice Hall, Upper

Saddle River 1997)

10.10 G.C. Goodwin, R.H. Middleton: Digital Control and

Estimation: a Uniﬁed Approach (Prentice Hall, Up-

per Saddle River 1990)

10.11 R. Isermann: Digital Control Systems,Vol.I.Fun-

damentals, Deterministic Control (Springer, Berlin

Heidelberg 1989)

10.12 R. Isermann: Digital Control Systems, Vol. II.

Stochastic Control, Multivariable Control, Adaptive

Control, Applications (Springer, Berlin Heidelberg

1991)

10.13 J.M. Maciejowski: Multivariable Feedback Design

(Addison-Wesley, Indianapolis 1989)

10.14 J.D. Aplevich: The Essentials of Linear State-Space

Systems (Wiley, New York 2000)

10.15 R.A. Decarlo: Linear Systems. A State Variable Ap-

proach with Numerical Implementation (Prentice

Hall, Upper Saddle River 1989)

10.16 D.C. Youla, H.A. Jabs, J.J. Bongiorno: Modern

Wiener–Hopf design of optimal controller, IEEE

Trans. Autom. Control 21, 319–338 (1976)

10.17 M. Morari, E. Zaﬁriou: Robust Process Control (Pren-

tice Hall, Upper Saddle River 1989)

10.18 E.C. Garcia, M. Morari: Internal model control:

1. A unifying review and some new results, Ind.

Eng. Chem. Process Des. Dev. 21, 308–323 (1982)

10.19

O.J.M. Smith: Close control of loops with dead time,

Chem.Eng.Prog.53, 217–219 (1957)

10.20 V. Ku

cera: Diophantine equations in control –

a survey, Automatica 29, 1361–1375 (1993)

10.21 J.B. Burl: Linear Optimal Control: H

and H

∞

Meth-

ods (Addison-Wesley, Indianapolis 1999)

10.22 J.C. Doyle, B.A. Francis, A.R. Tannenbaum: Feed-

back Control Theory (Macmillan, London 1992)

10.23 K. Zhou, J.C. Doyle, K. Glover: Robust and Optimal

Control (Prentice Hall, Upper Saddle River 1996)

10.24 M. Vidyasagar, H. Kimura: Robust controllers for

uncertain linear multivariable systems, Automatica

22, 85–94 (1986)

10.25 B.D.O. Anderson, J.B. Moore: Optimal Control

(Prentice Hall, Upper Saddle River 1990)

10.26 A.E. Bryson, Y. Ho: Applied Optimal Control (Hemi-

sphere/Wiley, New York 1975)

10.27 A.E. Bryson: Dynamic Optimization (Addison-

Wesley, Indianapolis 1999)

10.28 H. Kwakernaak, R. Sivan: Linear Optimal Control

Systems (Wiley-Interscience, New York 1972)

10.29 F.L. Lewis, V.L. Syrmos: Optimal Control (Wiley, New

York 1995)

10.30 S.I. Lyashko: Generalized Optimal Control of Lin-

ear Systems with Distributed Parameters (Kluwer,

Dordrecht 2002)

10.31 D.S. Naidu: Optimal Control Systems (CRC, Boca Ra-

ton 2003)

10.32 D.P. Bertsekas: Dynamic Programming and Op-

timal Control, Vol. I,II (Athena Scientiﬁc, Nashua

2001)

10.33 E.F. Camacho, C. Bordons: Model Predictive Control

(Springer, Berlin Heidelberg 2004)

10.34 D.W. Clarke (Ed.): Advances in Model-Based Pre-

dictive Control (Oxford Univ. Press, Oxford 1994)

10.35 J.M. Maciejowski: Predictive Control with Con-

straints (Prentice Hall, Upper Saddle River 2002)

10.36 J.A. Rossiter: Model-Based Predictive Control –

a Practical Approach (CRC, Boca Raton 2003)

10.37 R. Soeterboek: Predictive Control – a Uniﬁed Ap-

proach (Prentice Hall, Upper Saddle River 1992)

10.38 D.W.Clarke,C.Mohtadi,P.S.Tuffs:Generalised

predictive control – Part 1. The basic algorithm,

Automatica 23,137(1987)

10.39 D.W.Clarke,C.Mohtadi,P.S.Tuffs:Generalised

predictive control – Part 2. Extensions and inter-

pretations, Automatica 23,149(1987)

10.40 C.R. Cutler, B.L. Ramaker: Dynamic matrix control

– a computer control algorithm, Proc. JACC (San

Francisco 1980)

Part B 10

198 Part B Automation Theory and Scientiﬁc Foundations

10.41 C.E. Garcia, D.M. Prett, M. Morari: Model predictive

control: theory and practice – a survey, Automatica

25, 335–348 (1989)

10.42 D.Q. Mayne, J.B. Rawlings, C.V. Rao, P.O.M. Scoka-

ert: Constrained model predictive control: stability

and optimality, Automatica 36, 789–814 (2000)

10.43 F. Borrelli: Constrained Optimal Control of Linear

andHybridSystems(Springer, Berlin Heidelberg

2003)

10.44 T.A. Badgewell, S.J. Qin: Nonlinear Predictive

Control Chapter: Review of Nonlinear Model Pre-

dictive Control Application, IEE Control Eng. Ser.,

Vol. 61, ed. by M.B. Cannon (Kouvaritabis, London

2001)

10.45 A. Isidori: Nonlinear Control Systems (Springer,

Berlin Heidelberg 1995)

10.46 J.J.E. Slotine, W. Li: Applied Nonlinear Control

(Prentice Hall, Upper Saddle River 1991)

10.47 V.I. Utkin: Sliding Modes in Control and Optimiza-

tion (Springer, Berlin Heidelberg 1992)

10.48 Control system toolbox for use with MATLAB. User’s

guide (The Math Works Inc. 1998)

Part B 10

199

Control of Unc

11. Control of Uncertain Systems

Jianming Lian, Stanislaw H.

Zak

Novel direct adaptive robust state and output

feedback controllers are presented for the out-

put tracking control of a class of nonlinear systems

with unknown system dynamics and disturbances.

Both controllers employ a variable-structure radial

basis function (RBF) network that can determine

its structure dynamically to approximate unknown

system dynamics. Radial basis functions are added

or removed online in order to achieve the de-

sired tracking accuracy and prevent to network

redundancy. The raised-cosine RBF is employed

to enable fast and efﬁcient training and output

evaluation of the RBF network. The direct adaptive

robust output feedback controller is constructed

by utilizing a high-gain observer to estimate

the tracking error for the controller implemen-

tation. The closed-loop systems driven by the

variable neural direct adaptive robust controllers

are actually switched systems.

11.1 Background and Overview..................... 200

11.2 Plant Model and Notation ..................... 203

11.3 Variable-Structure Neural Component .... 203

11.3.1 Center Grid .................................. 205

11.3.2 Adding RBFs ................................ 205

11.3.3 Removing RBFs ............................ 207

11.3.4 Uniform Grid Transformation ......... 208

11.3.5 Remarks...................................... 208

11.4 State Feedback Controller Development.. 209

11.4.1 Remarks...................................... 211

11.5 Output Feedback Controller Construction 211

11.6 Examples ............................................. 213

11.7 Summary ............................................. 216

References .................................................. 217

Automation is commonly understood as the replace-

ment of manual operations by computer-basedmethods.

Automation is also deﬁned as the condition of be-

ing automatically controlled or operated. Thus control

is an essential ingredient of automation. The goal of

control is to specify the controlled system inputs that

force the system outputs to behave in a prespeciﬁed

manner. This speciﬁcation of the appropriate system

inputs is realized by a controller being developed

by a control engineer. In any controller design prob-

lem, the ﬁrst step is to construct the so-called truth

model of the dynamics of the process to be controlled,

where the process is often referred to as the plant.

Because the truth model contains all the relevant char-

acteristics of the plant, it is too complicated to be

used for the controller design and is mainly used as

a simulation model to test the performance of the de-

veloped controller [11.1]. Thus, a simpliﬁed model

that contains the essential features of the plant has to

be derived to be used as design model for the con-

troller design. However, it is often infeasible in real

applications to obtain a quality mathematical model be-

cause the underlying dynamics of the plant may not

be understood well enough. Thus, the derived mathe-

matical model may contain uncertainties, which may

come from lack of parameter values, either constant

or time varying, or result from imperfect knowledge

of system inputs. In addition, the inaccurate model-

ing can introduce uncertainties to the mathematical

model as well. Examples of uncertain systems are

robotic manipulators or chemical reactors [11.2]. In

robotic manipulators, inertias as seen by the drive mo-

tors vary with the end-effector position and the load

mass so that the robot’s dynamical model varies with

the robot’s attitude. For chemical reactors, their trans-

fer functions vary according to the mix of reagents

Part B 11

200 Part B Automation Theory and Scientiﬁc Foundations

and catalysts in the vessel and change as the reaction

progresses [11.2]. Hence, effective approaches to the

control of uncertain systems with high performance are

in demand.

11.1 Background and Overview

One approach to the control of uncertain systems is so-

called deterministic robust control. Deterministic robust

controllers use ﬁxed nonlinear feedback control to en-

sure the stability of the closed-loop system over a spec-

iﬁed range of a class of parametric variations [11.3].

Deterministic control includes variable-structure con-

trol and Lyapunov min–max control. Variable structure

control was ﬁrst introduced by Emel’yanov et al. in the

early 1960s. It is a nonlinear switching feedback con-

trol, which has discontinuity on one or more manifolds

in the state space [11.4]. A particular type of variable-

structure controlis slidingmode control [11.5,6]. Under

sliding mode control, the system states are driven to

and are then constrained within a neighborhood of the

intersection of all the switching manifolds. The Lya-

punov min–max control was proposed in [11.7], where

nonlinear controllers are developed based on Lyapunov

functions and uncertainty bounds. Deterministic robust

controllers can guarantee transient performance and ﬁ-

nal tracking accuracy by compensating for parametric

uncertainties and input disturbances with robustifying

components. However, they usually involve high-gain

feedback or switching, which, in turn, results in high-

frequency chattering in the responses of the controlled

systems. On the other hand, high-frequency chattering

may also excite the unmodeled high-frequency dynam-

ics. Smoothing techniques to eliminate high-frequency

chattering were proposed in [11.8]. The resulting con-

trollers are continuous within a boundary layer in

the neighborhood of the switching manifold so that

the high-frequency chattering can be prevented. How-

ever, this is achieved at the price of degraded control

performance.

Another effective approach to the control of un-

certain systems is adaptive control [11.9–11]. Adaptive

controllers are different from deterministic robust con-

trollers because they have a learning mechanism that

adjusts the controller’s parameters automatically by

adaptation laws in order to reduce the effect of the un-

certainties. There are two kinds of adaptive controllers:

indirect and direct adaptive controllers. In indirect

adaptive control strategies, the plant’s parameters are

estimated online and the controller’s parameters are

adjusted based on these estimates, see [11.12,p.14]

and [11.13, p. 14]. In contrast, in direct adaptive con-

trol strategies, the controller’s parameters are directly

adjusted to improve a given performance index with-

out the effort to identify the plant’s parameters [11.12,

p. 14]. Several adaptive controller design methodolo-

gies for uncertain systems have been introduced such

as adaptive feedback linearization [11.14, 15], adap-

tive backstepping [11.11, 16–18], nonlinear damping

and swapping [11.19] and switching adaptive con-

trol [11.20–22]. Adaptive controllers are capable of

achieving asymptotic stabilization or tracking for sys-

tems subject to only parametric uncertainties without

high-gain feedback. However, the adaptation laws of

adaptive controllers may lead to instability even when

small disturbances appears [11.23]. When adaptive

controllers utilize function approximators to approxi-

mate unknown system dynamics, the robustness of the

adaptation laws to approximation errors needs to be

considered as well. The robustness issues make the ap-

plicability of adaptive controllers questionable, because

there are always disturbances, internal or external, in

real systems. To address this problem, robust adap-

tive controllers were developed to ensure the stability

of adaptive controllers [11.23–26]. However, there is

a disadvantage shared by adaptive controllersand robust

adaptive controllers: their transient performance cannot

be guaranteed and the ﬁnal tracking accuracy usually

depends on the approximation errors and disturbances.

Thus, adaptive robust controllers, which effectively

combine the design techniques of adaptive control and

deterministic robust control, were proposed [11.27–33].

In particular, various adaptive (robust) control

strategies for feedback linearizable uncertain systems

have been proposed. A feedback linearizable system

model can be transformed into equivalent linear models

by a change of coordinates and a static-state feedback

so that linear control design methods can be applied

to achieve the desired performance. This approach

has been successfully applied to the control of both

single-input single-output (SISO) systems [11.10, 27–

29, 32–40] and multi-input multi-output (MIMO)sys-

tems [11.41–46]. The above adaptive (robust) control

strategies have been developed under the assumption

that all system states are available in the controller im-

plementation. However, in practical applications, only

the system outputs are usually available. To overcome

Part B 11.1

Control of Uncertain Systems 11.1 Background and Overview 201

Table 11.1 Limitations of different tracking control

strategies

L1 Prior knowledge of f and/or g

L2 No disturbance

L3 Needs fuzzy rules describing the system operation

L4 Requires ofﬂine determination of the appropriate

network structure

L5 Availability of the plant states

L6 Restrictive assumptions on the controller architecture

L7 Tracking performance depends on function

approximation error

the problem of inaccessibility of the system states, out-

put feedback controllers that employ state observers

in feedback implementation were developed. In partic-

ular, a high-gain observer has been employed in the

design of output feedback-based control strategies for

nonlinear systems [11.38,46–51]. The advantage of us-

ing a high-gain observer is that the control problem can

be formulated in a standard singular perturbation for-

mat and then the singular perturbation theory can be

applied to analyze the closed-loop system stability. The

performance of the output feedback controller utilizing

a high-gain observer would asymptoticallyapproach the

performance of the state feedback controller [11.49].

To deal with dynamical uncertainties, adaptive (ro-

bust) control strategies often involve certain types of

function approximators to approximate unknown sys-

tem dynamics. The use of fuzzy-logic systems for

function approximation has been introduced [11.28,

29, 32, 33, 36, 39, 40, 42–44, 46]. However, the fuzzy

rules required by the fuzzy-logic systems may not be

available. On the other hand, one-layer neural-network-

based adaptive (robust) control approaches have been

reported [11.26, 27, 34, 38] that use radial basis func-

tion (RBF) networks to approximate unknown system

dynamics. However, ﬁxed-structure RBF networks re-

quire ofﬂine determination of the appropriate network

structure, which is not suitable for the online opera-

tion. In [11.10,35,41], multilayer neural-network-based

adaptive robust control strategies were proposed to

avoid some limitations associated with one-layer neu-

ral network such as deﬁning a basis function set or

choosing some centers and variations of radial basis

type of activation functions [11.35]. Although it is not

required to deﬁne a basis function set for multilayer

neural network, it is still necessary to predetermine the

number of hidden neurons. Moreover, compared with

multilayer neural networks, RBF networks are charac-

Table 11.2 Advantages of different tracking control

strategies

A1 Uses system outputs only

A2 Guaranteed transient performance

A3 Guaranteed ﬁnal tracking accuracy

A4 Avoids deﬁning basis functions

A5 No need for ofﬂine neural network structure

determination

A6 Removes the controller singularity problem completely

terized by simpler structures, faster computation time,

and superior adaptive performance. Variable-structure

neural-network-based adaptive (robust) controllershave

recently been proposed for SISO feedback linearizable

uncertain systems. In [11.52], a constructive wavelet-

network-based adaptive state feedback controller was

developed. In [11.53–56], variable-structure RBF net-

works are employed in the adaptive (robust) controller

design. Variable-structure RBF networks preserve the

advantages of the RBF network and, at the same time,

overcome the limitations of fuzzy-logic systems and

the ﬁxed-structure RBF network. In [11.53], a growing

RBF network was utilized for function approximation,

and in [11.54–57] self-organizing RBF networks that

can both grow and shrink were used. However, all

these variable-structure RBF networks are subject to

the problem of inﬁnitely fast switching between dif-

ferent structures because there is no time constraint

on two consecutive switchings. To overcome this prob-

lem, a dwelling-time requirement is introduced into the

structure variation of the RBF network in [11.58].

In this chapter, the problem of output tracking

control is considered for a class of SISO feedback lin-

earizable uncertain systems modeled by

⎧

⎪

⎨

⎪

⎩

= x

i+1

, i = 1,...,n −1

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

y = x

(11.1)

where x =(x

,...,x

)



∈R

is the state vector, u ∈ R

is the input, y ∈ R is the output, d models the distur-

bance, and f (x)andg(x) are unknown functions with

g(x) bounded away from zero. A number of adaptive

(robust) tracking control strategies have been reported

in the literature. In this chapter, novel direct adaptive ro-

bust state and output feedback controllers are presented.

Both controllers employ the variable-structure RBF

network presented in [11.58],whichisanimproved

version of the network considered in [11.56,57], for

Part B 11.1

202 Part B Automation Theory and Scientiﬁc Foundations

Table 11.3 Types of tracking control strategies

T1 Direct state feedback adaptive controller

T2 Direct output feedback adaptive controller

T3 Includes robustifying component

T4 Fuzzy logic system based function approximation

T5 Fixed-structure neural-network-based function

approximation

T6 Multilayer neural-network-based function

approximation

T7 Variable-structure neural-network-based function

approximation

function approximation. This variable-structure RBF

network avoids selecting basis functions ofﬂine by

determining its structure online dynamically. It can

add or remove RBFs according to the tracking per-

formance in order to ensure tracking accuracy and

prevent network redundancy simultaneously. Moreover,

a dwelling-time requirement is imposed on the structure

Table 11.4 Comparison of different tracking control strategies

Controller Reference Limitations Advantages

types L1 L2 L3 L4 L5 L6 L7 A1 A2 A3 A4 A5 A6

[11.50]

√√ √

√√√

T2 T3 [11.62]

√√

√√√

[11.49]

√√ √

√√√

[11.40]

√√ √

√

T1 T3 T4 [11.32]

√√ √

√

[11.29]

√√√ √

√

T1 T4 [11.36]

√√√

√

[11.28]

√√√√√

√

T1 T5 [11.37]

√√√√

√√

[11.63]

√√√√

√√

T1 T3 T5 [11.27]

√√ √√

√

[11.34]

√√ √√

√

T2 T3 T5 [11.38]

√√√

√√

T1 T6 [11.64]

√√√

T1 T3 T6 [11.10]

√√√

√√

[11.35]

√

√√ √

T1 T7 [11.54]

√√√

√√

[11.52]

√√ √ √

√√

T1 T3 T7 [11.53]

√√ √

√√

[11.55]

√√

T2 T3 T7 [11.56,57]

√√√ √

variation to avoid the problem of inﬁnitely fast switch-

ing between different structures as in [11.52, 59]. The

raised-cosine RBF presented in [11.60] is employed in-

stead of the commonly used Gaussian RBF because the

raised-cosine RBF has compact support, which can sig-

niﬁcantly reduce computations for the RBF network’s

training and output evaluation [11.61]. The direct adap-

tive robust output feedback controller is constructed

by incorporating a high-gain observer to estimate the

tracking error for the controller implementation. The

closed-loop systems driven by the direct adaptive robust

controllers are characterized by guaranteed transient

performance and ﬁnal tracking accuracy. The lists of

limitations and advantages of different tracking control

strategies found in the recent literature are given in Ta-

bles 11.1 and 11.2, respectively. In Table 11.3, different

types of tracking control strategies are listed. In Ta-

ble 11.4, these tracking control strategies are compared

with each other. The control strategy in this chapter

shares the same disadvantages and advantages as that

in [11.56,57].

Part B 11.1

Control of Uncertain Systems 11.3 Variable-Structure Neural Component 203

11.2 Plant Model and Notation

The system dynamics (11.1) can be represented in

a canonical controllable form as

⎧

⎨

⎩

x =Ax+b



f(x)+g(x)u +d



y =cx,

(11.2)

where

A =



n−1

0 0



n−1



, b =



n−1







n−1



and 0

n−1

denotes the (n −1)-dimensional zero vector.

For the above system model, it is assumed in this chap-

ter that f (x)andg(x) are unknown Lipschitz continuous

functions. Without loss of generality, g(x) is assumed to

be strictly positive such that 0 < g

≤ g(x) ≤ g, where

and g are lower and upper bounds of g(x). The

disturbance d could be in the form of d(t), d(x)or

d(x, t). It is assumed that d is Lipschitz-continuous in

x and piecewise-continuous in t. It is also assumed that

|d|≤d

, where d

is a known constant. The control ob-

jective is to develop a tracking control strategy such

that the system output y tracks a reference signal y

accurately as possible. It is assumed that the desired tra-

jectory y

has bounded derivatives up to the n-th order,

that is, y

(n)

∈Ω

, where Ω

is a compact subset of R.

The desired system state vector x

is then deﬁned as



,...,y

(n−1)





We have x

∈Ω

, where Ω

is a compact subset of

.Let

e = y −y

(11.3)

denote the output tracking error and let

e = x−x



e,...,e

(n−1)





(11.4)

denote the system tracking error. Then the tracking error

dynamics can be described as

e =Ae+b



(n)

−y

(n)



(11.5)

=Ae+b



f(x)+g(x)u −y

(n)



. (11.6)

Consider the following controller,

g(x)



−

f(x)+y

(n)

−ke



, (11.7)

where

f(x)and

g(x) are approximations of f (x)and

g(x), respectively, and k is selected such that A

=A−

bk is Hurwitz. The controller u

in (11.7) consists of

a feedforward term −

f(x)+y

(n)

for model compensa-

tion and a linear feedback term −ke for stabilization.

Substituting (11.7)into(11.6), the tracking error dy-

namics become

e =A

e+b

d , (11.8)

where

d =



f(x)−

f(x)





g(x)−

g(x)



+d . (11.9)

It follows from (11.8) that, if only u

is applied to the

plant, the tracking error does not converge to zero if

d is

present. Therefore, an additional robustifying compo-

nent is required to ensure the tracking performance in

the presence of approximation errors and disturbances.

11.3 Variable-Structure Neural Component

In this section, the variable-structure RBF network em-

ployed to approximate f (x)andg(x) over a compact

set Ω

⊂R

is ﬁrst introduced. This variable-structure

RBF network is an improved version of the self-

organizing RBF network considered in [11.56], which

in turn was adapted from [11.61]. The RBF adding

and removing operations are improved and a dwelling

time T

is introduced into the structure variation of

the network to prevent fast switching between different

structures.

The employed self-organizing RBF network has

N different admissible structures, where N is de-

termined by the design parameters discussed later.

For each admissible structure illustrated in Fig. 11.1,

the self-organizing RBF network consists of n input

neurons, M

hidden neurons, where v ∈{1,...,N}

Part B 11.3

204 Part B Automation Theory and Scientiﬁc Foundations

Input layer Output layerHidden layer

(x)

1,υ

2,υ

,υ

f1,υ

f2,υ

g1,υ

g2,υ

,υ

Fig. 11.1 Self-organizing radial basis function network

denotes the scalar index, and two output neurons cor-

responding to

(x)and

(x). For a given input

x =(x

, x

,...,x

)



, the output

(x) is represented as

(x) =



j=1

fj,v

j,v

(x) (11.10)



j=1

fj,v

i=1





−c

ij,v



ij,v



(11.11)

where ω

fj,v

is the adjustable weight from the j-th hid-

den neuron to the output neuron and ξ

j,v

(x)isthe

radial basis function for the j-th hidden neuron. The

parameter c

ij,v

is the i-th coordinate of the center of

j,v

(x), δ

ij,v

is the radius of ξ

j,v

(x)inthei-th coordi-

nate, and ψ :[0, ∞) →

is the activation function. In

the above, the symbol

denotes the set of nonnega-

tive real numbers. Usually, the activation function ψ is

constructed so that it is radially symmetric with respect

to its center. The largest value of ψ is obtained when

ij,v

, and the value of ψ vanishes or becomes very

small for large



−c

ij,v



.Let

f,v



f 1,v

,ω

f 2,v

,...,ω





(11.12)

be the weight vector and let

(x) =



1,v

(x),ξ

2,v

(x),...,ξ

(x)





. (11.13)

Then (11.11) can be rewritten as

(x) =ω



f,v

(x), and

the output

(x) can be similarly represented as

(x) =



g,v

(x).

One of the most popular types of radial basis func-

tions is the Gaussian RBF (GRBF)thathastheform

ξ(x) =exp



−

(x −c)

2δ



(11.14)

Fig. 11.2 Plot of one-dimensional (1-D) raised-cosine ra-

dial basis functions

The support of the GRBF is unbounded. The com-

pact support of the RBF plays an important role in

achieving fast and efﬁcient training and output eval-

uation of the RBF network, especially as the size of

the network and the dimensionality of the input space

increase. Therefore, the raised-cosine RBF (RCRBF)

presented in [11.60] which has compact support is em-

ployed herein. The one-dimensional raised-cosine RBF

showninFig.11.2 is described as

ξ(x) =

⎧

⎪

⎨

⎪

⎩



1+cos



π(x−c)



if |x −c|≤δ

0if|x −c|> δ ,

(11.15)

whose support is the compact set [c−δ, c+δ].Inthe

n-dimensional space, the raised-cosine RBF centered

at c =[c

, c

,...,c

] with δ =[δ

, δ

,...,δ

] can be

represented as the product of n one-dimensional raised-

cosine RBFs

ξ(x) =

i=1

ξ(x

) (11.16)

⎧

⎪

⎨

⎪

⎩

i=1



1+cos



π(x

−c

)



if |x

−c

|≤δ

for all i ,

0if|x

−c

|> δ

for some i .

(11.17)

A plot of a two-dimensional raised-cosine RBF is

showninFig.11.3.

Unlike ﬁxed-structure RBF networks that require

ofﬂine determination of the network structure, the

employed self-organizing RBF network is capable

of determining the parameters M

, c

ij,v

,andδ

ij,v

dynamically according to the tracking performance.

Detailed descriptions are given in the following sub-

sections.

Part B 11.3