Composite systems control by combined math-analytical and FS/NN computing approach

Copyright©IFAC Automatic Systems for Building the Infrastructure

in Developing Countries, Istanbul, Republic of Turkey, 2003

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IFAC

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Invited Plenary Paper 2

COMPOSITE SYSTEMS CONTROL BY COMBINED MATH-ANALYfICAL AND FSINN COMPUTING APPROACH

Georgi M. Dimirovski1,2,Yuan-Wei Jing3, Jun Zhao\ and Si-Ying Zhang3

IDogus University, Department of Computer Engineering

Faculty ofEngineering, Acibadem, Kadikoy, TR-34722 Istanbul, Turkey Fax# +90-216-327-9631; Email: gdimirovski@dogus.edu.tr

2SSCyril and Methodius University, Institute ofAutomation& Systems Eng.

Faculty ofElectrical Engineering, MK-1 000 Skopje, R. ofMacedonia 3Northeastern University, Institute of Control Science

School ofInformation Science& Eng., Shenyang, Liaoning, P.R. of China Fax# +90-24-2389-0912; Email: ywjjing@peoplemail.com.cn

Abstract: A control systems engineering approach, employing a two-level overall system architecture and different but compatible formalisms for system representation on the upper and lower levels, has been investigated in detail. One design alternative is based on employing fuzzy-system approximators and solving for the adaptive tracking of the given, arbitrary, desired system outputs. The other alternative is based on state equations of composite systems and the use of neural-network approximators to deal with uncertainties and control adaptation. In both alternatives similarity property of subsystems has been exploited. Both designs can be implemented within the standard computer process control technology, and are therefore believed to be promising in applied systems engineering.Copyright©2003 IFAC

Keywords: Complex composite systems, fuzzy systems, neural networks, non-linear systems.

1. INTRODUCTION

It may seem a paradox, but all the exact science is dominated by the idea ofapproximation - Bertrand Russell.

In this paper, we present an overview of our recent research endeavours to develop an advanced yet practically applicable systems engineering design methodology for composite systems control by means of compatible use of combined math-analytical and fuzzy-system or neural-network models. Although we made considerable reference to known research works by other authors, no attempt has been made for extensive citation because it not possible without a risk not to mention even the most important contributions that appeared recently in the literature due to paper size limits albeit those for an invited paper. Instead, we point out that rather

extensive research has been carried out world-wide in this topic, and a number of new and innovative designs have been proposed during the last decade. Hence the interested reader is strongly recommended to search the literature beginning with the monographs cited in here and references therein (e.g., see, Astroem and co-authors, 2001; Bemusou and Titli, 1982; Boyd and co-authors, 1994; Gao and Wang, 1994; Jing and co-authors, 1996, 1997; Jing, 1997; Krstic and co-authors, 1995; Narendra and Annaswamy, 1989; Polycarpou; 1996, 1998 a; Siljak., 1978; Spooner and co-authors 2002; Trentelman and Willems, 1993; Vidyasagar, 1981; Wang, 1994; Yen and Langary, 1999). In here, we confine merely to our own results that appeared to be comparable in terms of performance with those in the literature. In general, these have been obtained by synthesising suitable Lyapunov functions, an idea first used by Baily (1966), in order to ensure the

With reference to studies by Mesarovic and co-authors (1970) and Saridis (1989), we have devoted our research endeavours to an approach to integrated control and supervision for complex systems, typically a sort of partially hierarchical two-level scheme, by using both math-analytical and fuzzy-system and/or neural-net formalisms in a compatible setting as appropriate. The soft computing models typically play the roles of either approximators (Dirnirovski and ling, 2003; Liu and co-authors, 2001; Ge and co-authors, 1999; Polycarpou, 1996; 1998; Spooner and Pasino, 1996; Zhang and co-authors, 2003) regardless the level where employed, or of a coordinating supervisor alternatively. At this point we emphasise that both fuzzy-system and neural-net identification modelling heavily rely on time-domain sequences and series of measurable variables (e.g., see Box and co-authors, 1994; Haykin, 1999; Gupta and Rao, 1995; Spooner and co-authors, 2002; Werbos, 1990; Yen and Langary, 1997). Hence, assuming data sets of k-time process sequences are available, we have investigated two-layer and two-level control system architectures for complex systems that may comprise supervisory as well as local executive controls and interconnection emulators (Dirnirovski, 1998).

Further the paper is organised as follows: a brief discussion on the background and current research results; an overview analysis of pure math-analytical approach; the novel control design based on combined math-analytical and fuzzy-system approach; the novel control design based on combined math-analytical and neural-net approach; a couple of illustrative examples from the literature for the purpose of comparison; and the conclusion drawn from the entire research.

tremendous developments in signal processing and its cross-fertilisation with systems theory has taken place. In control science and engineering, this gave rise to the matured discipline of identification modelling of systems closely related to process state variable and parameter estimation (Ljung, 1999). In

this regard, ideas on combined handling of the identification tasks in both time and frequency domains pave the way towards deeper understanding of the underlying phenomena in real-world processes. Although both are equally valuable they are rewarding to the full when combined.

...

.

.,- '.'-'._'?(~ -

..

~:

.

; - "--

--

~-_._-

_.

- -- --....

--

_....-

- -_._-

._-_._.-..---. ---_...

--: 1 ~ j i stability of composite control system Not surprisingly, Lyapunov stability theory and his direct second method appeared to be indispensable over and over again.

As the above message by famous Bertrand Russell (one of the founders of modern mathematical science, Whitehead and Russell, 1927) states, systems and control sciences as well as systems engineering have undergone an incomparable development since Tsien's book on Engineering Cybernetics (1950) following Wiener's seminal work on Cybernetics (1948). Note, in addition, Rosenbrock's [1977, essay in Automatica 13, p. 390] words: "... My own conclusion is that engineering is an art rather than a science, and by saying this I imply a higher, not a lower status ... ". Within the framework of systems science, due its underlying philosophy and physics, indeed there is clear-cut understanding that original real-world systems and their conceptual models as well as their feasible mathematical representations (Figure 1) should be viewed in terms of the fact that dynamical processes in the real world (below the light speed) constitute a unique non-separable inter-play of the three fundamental natural quantities of energy, matter and information (Dirnirovski and co-authors, 1977; Gitt, 1989; MacFarlane, 1993). Moreover, energy and matter are information carriers, but solely information has the impact capacity as to change, direct and shape the flow of energy and matter, hence the impact of command, control, guidance, management and supervision in systems engineering (Dirnirovski and co-authors, 2000 a, b; 2001). Hence it is these naturally existing phenomena that have given rise to all developments in systems and control science and engineering up to nowadays and so it shall be in the future (for instance, see Fradkov and co-authors, 1999,2002; Vukobratovic, 2000).

Fig. 1. The general concept of controlled dynamical process in engineering terms, with non-separable interplay flow of energy, information and matter (Dirnirovski and co-authors, 1977; Dirnirovski and ling, 2003)

Next, let us recall that the main source of information in real-world systems is bound to be in the data sets of measurable signals and any other way gathered observation data implying some class of time series (Box and co-authors, 1994). Hence the

2. ON BACKGROUND AND CURRENT RESEARCH

Incontrol systems engineering, given the availability of operationally integrated distributed computer control systems, a pragmatic direction towards exploiting in parallel both system structure modelling and identification, and state variable and process parameter estimation exist for quite some time. Inturnthis has enhanced to investigate features such as similarity, symmetry, and partial similarity

3.1 The Case ofNon-linear Operating Regime

However, the somewhat more specific assumptions, given below, are also needed.

Here,

A,

Band

C

represent the linear parts of / '

g and

h,

respectively, while

F(x

j ) ,

G(x;)

and

N

xj=/(xJ+g(xJu j + IHij(x)

j=l.j~j (1) i

=

1,2,... ,N

Yj

=

h(x j ), N

.x

=

Ax +F(x)+Bu +G(x)u

+

"H(x.)

I I I I I I .~. 1J ) (2) j=l,j'" y;

=

Cx;

+

M(xJ i

=

1,2,.

",N

Assumption 3.1: The linear parts of system(l) are

completely controllable and observable.

Under Assumption 3.1, plant system representation (1) can be described as follows:

In here, traditionally, variables

x

j

ERn,

Yj E Rm, U; E Rmare the states, the outputs and the control inputs of the i-th subsystem

Sj'

respectively. Furthermore, the standard realistic assumptions

/(0)

=

0,

g(O)

=

0,

H ij

(0)

=

0,

h(O)

=

°

are also adopted.

In the more general non-linear system framework (ling, 1997; Dirnirovski and co-authors, 1977; Zhang, 1994), let consider the following class of non-linear composite system S

=

UN

S :

,=1 I

3. ON MATH-ANALYTICAL APPROACH TO CONTROL OF COMPOSITE SYMILARITY

SYSTEMS

For the purpose of exploring the problems of concern and clarifying the limitations of pure math-analytical approach, let us recollect some basic results on composite interconnected systems in math-analytical setting (Jing, 1997).Itis assumed the plant system is composed of

N

interconnected sub-systems

Sj

that may have some similarity property (Zhang, 1994). We have made a certain comparison analysis relative to some of our recent pragmatic control engineering case-studies (Dimirovski and co-authors, 1996; 2000 a, b; 2001), and also relative to known results on stability analysis (Bernusou and Titli, 1982; Narendra and Annaswamy, 1989) of composite systems.

(1994), Ossman (1994), Sandeep (1994) and others, and these were aimed at extending or improving the existing results. Hence we were able to compare well both approaches to composite systems control.

The insight to dynamical system structure and system properties gained via linearized models, of course, has demonstrated its constructive power in control design tasks. Nonetheless, in composite systems a clear-cut understanding of linear and non-linear models in this regard is crucial (e.g., see Astroem and authors, 2001; Dirnirovski and co-authors, 2000 a, b; 2001; Ge and co-co-authors, 1998; Khalil, 1996; Spooner and co-authors, 2002) given that soft-computing approximators (Dirnirovski and Andreeski, 2003; Polycarpou, 1998 a, b; Tong, 1999) are input-output type of models by and large.

We point out in addition, in parallel to this research we have worked on the control problem for composite systems by using math-analytical techniques only and gained considerable insights into restrictive assumptions, which had to be introduced in order to derive the respective results reported in (ling and authors, 1996, 1997, 1998; Liu and co-authors 2003; Zhang and co-co-authors, 2001; Zhao and Dimirovski, 2003). The background of these studies is found in the well-known works by Chen and co-authors (1991), Gao and Wang (1994), Mahmoud Inreality of systems engineering, the ultimate task to confront has been always: how to obtain most appropriate model representations of nonlinear, inter-acting and counter-inter-acting, dynamical processes in feasible operating regimes of the object system to be controlled (Astroem and co-authors, 2001). Various kinds of approximations have been and shall be used with a focus on one and neglecting another of process phenomena, of course.

It is therefore we clam here that in complex systems control both math-analytical and soft-computing representation models can be employed as long as these are being used in a compatible way according to the systems science framework. Hence, one way or another we face approximation models of complex object systems to be controlled, and thereupon deriving control designs desired. Moreover, we believe there are no obstacles to extend this combined approach to all kinds of complex controlled processes.

or symmetry (Zhang, 1994) while observing the structural properties of controllability, reachability, and obesrvability needed. The latter concepts for linear composite system models been well established (e.g., see Davison, 1977; Klamka, 1991; Rech and Peret, 1987; Rosenbrock, 1970; Siljak, 1977; Vidyasagar, 1981). Not surprisingly, within non-linear system models these turned out to be much more involved because controllability, reachability, and obesrvability in non-linear operating regimes have many facets (e.g., see Dirnirovski and Gough, 1990; Dirnirovski and co-authors, 2002; Fradkov and co-authors, 1999; Klamka, 2003; Zhang, 1994, Zhou and co-authors,

T(s)

=

FC(s! -

A)-I

B

(6)

In addition, it turned out necessary to make use of the defInition and the lemma cited below.

IIH

ij

(x/t),x j (t - ,),t)1

~

r

ijllx/t)11

+

(5)

+

r

ijllx

j

(t -

1")11

+

flij

Assumption 3.6: For every subsystem

(A, B).

there existsa matrix

F

E

R

mx1,such that

(8)

FC= B

T

P.

such that

is strictly feedback positive real. Then, from Kalman-Yakubovich Lemma (Boyd et al., 1994), it follows that there exist matrices

P

E

R

nxn

,

Q

E

R nxn ,

K

E

R mxn ,

p

=

pT

>

0,

Q

=

QT

>

0,

satisfying

(A

+

BK)

T

P

+

P( A

+

BK)

=

-Q,

(7a)

Re[

A(A

+

BK)]

<

0,

(7b)

Assumption 3.3: All functions

i =

1,2,···,

N ,

areintegrable.

The norms of vectors and matrices,

lit

and

Iltxm

respectively, are defmed in the usual way. These represent appropriate norms in

Rn

and

R nxm ,

respectively. With Assumption 3.2, the elements of

Hij and

G

satisfy the Lipschitz condition in

O.

Therefore, H ij and

G

also satisfy the Lipschitz

condition in

0

for some constants

a

ij

>

0,

f3

>

0.

M(x

_{i )} are the respective non-linear parts of each of them. For convenience, in what follows, terms

F

(x

J

are denoted as

Hi;

(x

J

for all

i

=

1,2,···,

N .

Furthermore, another two appropriate assumptions are also adopted.

Assumption 3.2: In some bounded closed

neighbourhood

0

of the origin in

Rn,

the elements ofHij and

G

do have continuous partial derivatives of first order with respect to each variable.

3.2 The Case with Operational Delay Interconnections

To clarify this case, let us adopt to consider a linear operating regime in the presence of delays into the interconnections, the representation model of which is as follows:

In this case, for system representation (4) we have adopted the following assumptions:

Assumption 3.4: For each sub-systems

S;,

pair

(A, B)

iscompletely controllable.

(9a)

x(t)

=

f(t,x(t))

Definition 3.1: Consider a system described by

x(t)

=

f(t,x(t))

(10)

x(t)

=

rp(t),

tE[to-"tol

(11)

Suppose functions r_l(-) and r₂(-) belong to class

Ka)'

and r₃

0

belong to class

K.

System(10)-( 11)

is to possess uniform ultimate boundedness with respect to a fInite constant

T(

S, r),

if the system has the following properties: there exists a

Cl

function

V(-) :

[to -

,,00)

x

Rn

~

R+

satisfying

(i) for all

t

E

[t

0 - "

00),

X

ERn,

Lemma 3.1: Consider a general delay function

differential equation

holds true. With regard to the issue of the uniform ultimate boundedness of dynamic systems, the following available, general result was needed. where

t ER

isthe time variable, and

x(t)

E

Rn

is the state. The system is said to be uniform ultimate bounded with respect to

S,

iffor any given set

S

and any

r

E (0,00),

there exists a

T(S,r)

E[0,00)

such that for every solution

x(·):

[to,oo)~Rn,x(to)=xo' (9b)

and for all

t

~

to

+

T(S,r),

Ilxoll

~

r

~

x(t)

E

S

(9c) N (4)

+

IBH

ij

(xj(t),x/t - ,),t)

j=l.j*;

y;

(t)

=

CX

_i

(t) .

S·

i'

The problem when delay interconnections are present turned out to be theoretically much more involved than appears to be (ling, 1997;lingand co-authors, 1998). Recently Liu and co-authors (2003) derived a stabilization result when both nonlinear uncertainties and multiple time-delays are present. However, typically the nonlinearity phenomenon is assumed to be negligible and abandoned.

Again traditionally, Xi E

Rn,

is the state vector,

u; E

R'i

is the input vector, and

y;

E

R

m, is the output vector.

A, B,

C

are the constant matrices of appropriate dimensions, and , denotes the delay in the interconnections.

Assumption 3.5: There exist non-negative scalar constants fl ij ,

r

ij'such that

(ii) there exists a constant

q

>

1

satisfying

Ilx(c;)11

<

qllx(t)lI, t -

r

~

c;

~ t, t

2:

to

(13)

such that

where E:

>

0

is a positive number, and satisfies

Itis apparent now in which sense the case with delay interconnections is much more involved.

4.COMBINED CONTROL OF COMPOSITE NONLINEAR SYSTEMS USING ON

FUZZY-SYSTEM ADAPTIVE EMULATION Adaptive control theory has attracted a significant attention of numerous researchers during past couple of decades, and has become a powerful methodology in resolving feedback control of non-linear systems with parameter uncertainties. There exist many math-analytical adaptive control algorithms for non-linear systems with unknown functions (e.g. see Khalil, 1996). On the other hand, fuzzy control is already broadly used in industry, and considerable development of fuzzy adaptive control has taken place (Wang, 1993; 1994). On the grounds of theory of fuzzy systems as universal approximators considerable many methods of fuzzy adaptive control have been derived (e.g., see Ge and co-authors, 1999; Polycarpou, 1998b; Tong, 1999; Dirnirovski and co-authors, 2000 a, b; Yeong-Chan, 2000). It is pointed out the class of composite systems is assumed to be characterized by unknown functions, and by presumption its model can be represented via using data sets of inputs and outputs.

4.1 Problem Statement and Assumptions

Consider a class of composite non-linear system represented by the following set of differential equations:

Y / ni-I)

=

fj ( Xi)

+

g i ( x

J

Ui

+

m i ( X) ,

i=I,···,p.. (16)

Here,

u

i is the control input of the i-th sub-system,

and x j= [Yj"

y:

ni -I

)] is the output of the i-th

sub-system. Functions f(xJ and gj(x;) are assumed to be known. The vector x =[x ... x ] is the output

l ' ' p

of the composite interconnected system, which is assumed to be measurable. However, function

m_i(x)is unknown. Further functions f.(xJ,

gi(X_i)and mi(x) are supposed to be smooth

functions, and

n

j are positive integers. The objective is to design a fuzzy adaptive controller for the

11

composite system, which guarantees output tracking a desired reference-output signal Y r

=

[y rl " Y rp ] .

By expressing the respective errors

e

j!

=

Yj - Yrj'

. . (n;-I) (ni-I) d

e

j2 = Y i - Y r j ' ,

e

jni = y j - y , an

taking

e

j

=

(e

j l'

,e

jn

)T ,

system equations can be

expressed as

~j

=

Aje_i

+b;{/;O+

_{gi(')U i +m,(x)- Y;jn}i)},

i

=

1, ... ,

p , (17) where

A.

, and

b.

I are system state and input matrix! system vector of appropriate dimensions.

Let us choose a matrix

K.

_l such that A ._m, = A -bK_{I I t}

is a Hurwitz matrix (all eigenvalues are in the left open plane of complex numbers). Then (17) can be re-written as

ei =Am;e+bj{/;O+g;Oui

+

(18)

+m;(x)

+

K;e_i-

Y;;)}

In the next sub-section, we present our design solution for the indirect adaptive controller of the considered class of non-linear composite systems and the relevant new theorem.

4.2 An Indirect Fuzzy Adaptive Control

A generic parameter non-linearized fuzzy logic system approximating the unknown functions is to be employed, because in practical applications the fuzzy system must not require so many fuzzy rules as the parameter linearized one. Suppose that there are available a few fuzzy IF-THEN rules about the unknown function

m

j

(x)

as follows:

R

(r) IF . Ari (n,-I). A r, m;: YI IS 1,1'"'' YI IS I,n,''''

A

r (np-I).

A

r

..., Ypis

;,1' ...,

Yp IS

;,n

p'

THEN mj(x) is

C;i

,rj=I, .. ·,

M

j . (19)

Because mi(x) IS unknown, we will use a synthesised fuzzy system

m

_i

(x /

()j) to approximate mj(x). The fuzzy system performs a mapping from

V

<;;;;;

R(p*n)

~

R

(Wang, 1994). Let consider

V

= VI,I X ..•x

V

n.p' i =

1,···

p ,

j

=

1, ...

n .

This fuzzy logic system can be used to design the adaptive controller.

We select the fuzzy model in which the parameters are not Iinearized that enables the reduced set of fuzzy rules. Its model form may be represented by means of the following formula

i

=

1, ...

,p , (20)

- I - I 1 .

where

y

,X;,j' ai,j are parameters. QuantItyXi,j

stands for y~j-l), and

B

i stands for the sum of

d· bl - I - I 1 hil

M

.

a Justa e parameters

y

, x i , j , a i , j ' w e i IS the number of fuzzy rules of i-th subsystem. Then we can take

where Q_i is the obligation aggregation of

B;.

Here, it is needed

B

i be bounded and

a:,

to be positive.

Hence, Qi is represented as follows:

Assumption 4.2.

gi

(xJ

*"

O.

Assumption 4.3.

Yi

is known, and all order

derivative of

Y

i are known.

Assumption 4.4. Equation

P;A

_m

+

A~P;

+

Qi

=

0

has the positive definitive matrix solution

P;

in which

Qi

is positive definite matrix.

Finally, we present the main result in terms of the following Theorem 4.1.

Theorem 4.1: Consider system (16) along with

Assumptions 4.1-4.4 satisfied, take the initial condition

B

_i

(0)

E Qi'and use the controller design given by means of (26)-(29) below, then overall system ensures asymptotically stable output tracking of the assigned desired output. The two-component control laws are constructed as follows:

(26)

and the adaptive law(i

=

1,' .. ,

p )is

A

where

B;

is the estimation of

8

_i' ,

Let us take

U

c be the obligation aggregation of x,

U =U

_{c e,l,!}

x"'xU .. x···xU

_{etl,} c,p,n}

U ..

is

j , etl.}

the obligation aggregation of Xi,j' Then by making use of the projection

Pro

_i

(8

<1»

=

<1> _ (11<1>11-

13)<1>

T

8

<1> . J , 811<1>112 ' (24)

<1>T

8

~

0

if

11<1>112

=

13;

8, otherwise,

(28) (29) where

In addition, following Wang (1994), we have derived

1\ 1\

ami (X/()i) =Yi-mJX/();)b .-2(xi,j-Xi,j)2,(30c)

aal

L

Mi bl M' (a . .)3 t,) I,} mi 1=1 (30a) A Omi(X/()i) b~i = - - , 1 M, aYi

~)~i

1=1 A A

the algorithm of

ami

(x /

8

i ) /

aB; :

I

Omi(x;'Bi)

=

Yi-m;(X/O;) b . -2(xi,j -Xi,j),(30b)

I M· ml ( )2

a- .

~bi (Ji,j

X,,) L.J ml

1=1

A - A

the estimation

B;

of

B

_i• and

8

i

=

8 ; -8

i , for the

system error dynamics can be derived as follows:

Now we can adopt the following realistic assumptions, which are not restrictive:

Assumption 4.1. Vi

(i

=

l,. ..

,p)

is bounded, and

4.3 Concluding Remark 5.1 Problem Statement and Assumptions

Consider now a class of composite, non-linear, uncertainty, similar systems described as follows:

Definition 5.1 (Qu, 1996): Suppose V be a

Lyapunov function of a given continuous-time system, and let

In addition, a non-restrictive assumption, and certain defmition and lemma were needed to derive the main result Theorem 5.1. These are given below.

Assumption 5.1: There is uncertain function

Pi (Xi'

t)

>

0

such that

(31)

Xi

=

Aixi +BJu;

+

/;(Xi,t)]+hi(x) ,

i

=

1,2,"',N

Here,

Xi

ERn,

Ui E

R

mare the state and the input of the i-th subsystem, respectively,

B;

E

R

nxm,

pair

(Ai,BJ

is controllable,

andh(xi,t)

is a

uncertain vector function,

hi (x)

is uncertain

.

(T

T

T)T

smoothing vector, andX

=

_{XI 'Xz , " ' XN} . For system (31), due to the characteristic features of similarity composite systems (Zhang, 1994), there is a similar parameter

1;

such that

r-

I

AT.

=

A

and

r-

I

B.

=

B

(32) I I I I I • . P

V

~

I -

t

e; Qiei

~ O.

i=1

5. COMBINED CONTROL OF COMPOSITE NONLINEAR UNCERTAINTY SYSTEMS USING

NEURAL-NETWORK ADAPTIVE EMULATION The robust adaptive control of non-linear composite systems with uncertainty is rather important topic in various applications. In allusion to the non-boundary uncertainties of the subsystems, the adaptive control approaches were all based on the model adaptive control (Ioanu, 1986; Krstic and co-authors, 1995; Narendra and Annaswamy, 1989). The stability of the entire system is assured by the condition of the

M

matrix, which related to the interconnections boundaries, without the need for knowing the parameters of the subsystems or the interconnection. The deficiency of these kinds of approaches is that it is difficult to validate the positive definiteness of the

M

matrix beforehand because its elements of depend on subsystem uncertain parameters.

Hence, the proposed fuzzy-logic control system guarantees both system stability and tracking of the required reference.

Theorem 4.1 has been proved in B. Liu and co-authors. (2001). The proof is based on synthesising an appropriate Lyapunov function for the tracking control error dynamics of the overall system so that

Investigations in Gao and Wang (1994) have shed additional light on the uncertainty composite large-scale systems. With regard to the time-varying linear uncertain composite systems, a non-linear controller was designed to make the system real stable by the Riccati approach. In Gao and Wang (1994) and in Qu (1996), the overall system has been made real stable through adaptive control design, however, no considerations about the uncertainties of the input gains were given. This issue was resolved in Zhou and co-authors (1999) and in Zhu and co-authors (1999) by using artificial neural networks, but their discussion is about the common large-scale, time non-varying, linear systems, and a corrective problem is solved in Liu (1999). The results about the uncertainties of the input gains are not satisfactory for practical applications. More recently, neural-networks structures with one hidden layer were successfully applied to various non-linear control systems. In Dirnirovski and co-authors (2000 a, b; 2001) thermal composite system control applications in presence of uncertainty gains, the bounds of which are handled learning neural-net structures, were solved. Two solutions to the problem of interconnections by using high-order neural nets were proposed in Y. W. Zhang (2000), and inY-XZhang and co-authors (2001).

YI

~lxll)~

V(x,t)

~

Yz

~lxll),

(34)

V(x, t)

~ -Y

3

(1Ixll)

+

<p(t) ,

(35) where

lim

Y .

(s)

=

00(1

~

j

~

3)

is a continuous

t~oo J

strictly positive definite function, and if

<p(t)

is a positive time-varying continuous function, which satisfies

<p(t)

~ rZ

<

00; then the system state converges to some neighbourhood of the origin and is referred toreal stable with

IIX(t)11

:S:r.

Lemma 5.1 (Zhu and co-authors, 1999): If

(A,B)

is controllable, then for a positive definite matrix Q

and a positive number

Y

>

0,

there exists a unique solutionPfor the following equation

5.2 Robust Adaptive Control Using Neural-Network Approximator

Note that a model of the non-linear connection

hi (x),

i =

1,2,' ..

,N

can be built to almost any degree of accuracy via high-order neural networks

and backpropagation algorithm (Gupta and Rao; 1995; Hopfeld, 1987; Werbos, 1990).

Let us suppose

x

be the input of the high-order neural network,

Y

j be the output. Let us construct

Yj

=

W;s(x),

where:

W;

E RnxLis weight matrix;

Sj(x),

i=1,2,···,L is the element of

s(x)

E

R

Lx\

N .

and given by

Sj(x)

=

IIII[s(x/ej

)]dj(/) ,

k=\ jel,

f

T.I

i

=

12 ...

L}

is a collection of

L

not-ordered

1.1I ' "

subsets of

{1,2, ... , n}; d

j(i) is nonnegative integer; and

s(x/ej)=

Po/oX

+A

_o,j=I,"',n,

1+

r

kj

k

=

1,···,N . For these models of high-order neural networks, we can find out that there exists an integer

L, a

d

/0

and an optimised matrix

W;.,

such that for anyG

>

0,

Ih

j

(x) - W;. s(x)1

~

G is satisfied. So, if the high-order term is large enough, there exists a weights matrix such that

W;. s(x)

can approximate

hj(x)

as desired with

11w;'11

~

Mw·

Then composite system (35) becomes:

_B

.

=

_{-YI ()}

₊

_{2ko. P T x .}

I

T -I

I

(46)

I " I .1 I I

5.3 Concluding Remark

with the adaptive law

where

p/

is the estimation of

p/

by using neural networks,

p/

(x, t)

=

Zj

(t)cr j

(xj), and Zj

(t)

is the weight vector. The network approximation error G_P,

(t)

is time-varying and bounded. Its boundary is uncertain (remark: this is a more rational assumption than in Qu, 1996). The quantities

0

j E

R

nxLand

e

=

[B 0 ... O]T i

=

1 2 ... N

p.,

y,

1.,

J . are

i " " , " " I I '~.l

constructed in the theorem proving process. Then the state

xj

is ultimately consistent bounded on the set

(43) { n . ( ) J1j

k;

Dj = xj ER .VOj

x

~--'=t

ko·a. k

,I I 1 i

=

1,2,···,N

i

=

1,2,··

·,N,

X

j

=

Axj

+BJu j

+

h(xj,t)]+

+

W;. s(x)

+

Gj(x),

Theorem 5.1: Consider composite system (31) and design the controller in the following form

X.

=

A.x. +B.[u.

+

h(x.,t)]-I -.. I I I I I I

i

=

1,2, ...

,N ,

W;s(x)

+

w;

s(x)

+

Gj(x),

(38) where the error

fiT;

satisfies

fiT;

=

W

j -

W;•.

(37) where Gj

(x)

= hj

(x) - W;. s(x)

is the weight estimate error. Moreover there exists Ei ~

0,

such that

IGj

(x)1

~

G, i

=

1,2,"', N.

If

W;

denotes the estimation of the uncertain weights matrix

W;',

then

a b c

Uj=Uj+Uj+U j

(39)

The process of constructive proof of Theorem 5.1. (Y.-X. Zhang and co-authors, 2001) proceeds in two steps:

Step (1). Firstly, by making use of Lemma 5.1, it has to be proved that there exist a nominal

a

b

Rn

R m

controlleruj = Uj

+

Uj : ~ and a Lyapunov function

VOj(xJ

for the nominal subsystem

x

j

=

Ajxj

+

BJu j

+

h(xj,t)]

such that conditions of Definition 5.1 are satisfied, i.e. that the nominal subsystems are real stable.

Step (2). Next, taking the result of step (1) as a known assumption, the proof of the theorem proceeds via designing the adaptive control law by synthesising a class of appropriate Laypunov functions so that each sub-system and the overall systems are made real stable.

with

U;

=

-(y

+

I)B T

pT;-I Xj ,

b ;:-1 ~ 2 ( )BTpT-1 Uj = -':>j p j

x,t

j Xj' T T c

B

j

W;S(X)

B

j

0

j Uj

=

2

+

2 '

A

j

[1

+

IIB

j

11]

~,j [1

+

IIB

j

11 ]

(40) (41) (42)

6. ON PERFORMANCE OF ANALYTICAL PLUS SOFT-COMPUTING CONTROL OF UNCERTAINTY COMPOSITE SYSTEMS Inthis section we illustrate the control performance which may be achieved by employing hybrid

'analytical + soft-computing' adaptive control of non-linear composite similarity systems with uncertainty in the case of the two alternative control designs in Section 4 and 5.

[

~21]

=

[0

X22

°

For the case of 'analytical

+

fuzzy-system' control design, let consider the following example of composite non-linear system having similarities:

X,~[~l ~},+

+

[~}u,

+

x,',e''''''

cos 101 +

x"e'"

sin 10,]+

+

5(:::

:::J.

[-1 1]

x

2

₌

° °

x

2

+

+ [

~

Ju,

+

xi, e''''''

cos 101 +

x" e '"

sin 101] +

(

COSX11

J

+5

,

cosx

₁₂

The overall system performance that may be achieved is illustrated by means of the responses of sub-system state variables depicted in Figure 2.

Fig. 2. Simulated state responses of both sub-systems in the composite plant under control of the proposed 'math-analytical + fuzzy-system approximator' adaptive control law.

Apparently, the stabilization effect is rather good, but some tracking remained albeit ultimate consistently bounded. The latter represents a weakness is of this design. Insofar, we could not find an explanation for this phenomenon and this a topic for further research.

For the case of 'analytical + neural-net' control design, let consider the mechatronic system of the axis-tray drive, the schematic of which is depicted in Figure 3. The equations of system dynamics have

been derived elsewhere (Han and Cbe, 1993). For comparison, we have adopted the same case-study parameters

m

₁

= 0.5, m

₂

= 1.0,

/1

=

/2

= 1,

1

= 1.0,

ill

= 0.1 sin(1

Ot),

with

g

= 0.0098 .

Then state equations can be transformed into similarity systems representation (Zang and co-authors, 2003) as follows:

[

~Il

]

=

[0 1

][Xll]

+

[0]U

₁

+

X I2

° °

x

12

1

+ [0.1

sin~

Ol)x"

1

~][::H~}u,

+

gl

sin

x"

)+

+ [ 0.1 sin(101

~x"

sine

x" )

J

Fig. 3. Mechatronic system of axis-tray drive (Zhang and co-authors, 2003)

By choosing

y

=

1,

Q

= / and solving the Riccati equation for matrix

P,

and then by using Theorem 5, the obtained adaptive control design is:

U;

=-(y+l)B

T

px

_i

=-2*[0

1]*Px

_i,

u~

=

-~i-lji

B

T

PX

_i

= -50*

j/

*[0

0.2]*

Px

j ,

BTWS(x)

AJl

+

IIBI1

2]

where the evaluated parameters are ~i

= 0.02,

Ai

= 0.001.

For evaluating the approximate

Pi

2,

we constructed a neural network with three levers each having six nodes. Similarly, for on-line emulation of

hi

(x),

made use of a network with three levers each having two nodes. For initially perturbed values of state variables [1.5 -1.0rand

[1.5

-1.oy,

and a short time interval

t

=

[0,10]

the computer simulation produced the sate evolution dynamics in closed loop given in Figure 4.

interconnections for which high-order neural networks and back-propagation are employed.

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In Section 4, a new constructive result on the output-tracking control for a class of non-linear composite systems by applying a novel synthesis of fuzzy adaptive control algorithm was presented. The algorithm has the advantage of reducing the number of fuzzy rules, which are needed within the fuzzy-logic control subsystem, which attenuates the rules' exploding problem usually present when plants are of high-order or rather complex.

In Section 5, a novel robust adaptive controller for a class of uncertain non-linear similar composite systems that employs neural networks to emulate on-line the non-linear subsystem dynamics and uncertainty interactions was presented. It was constructed via adopting two steps: in the first step, it makes use of low-order neural networks to design decentralized controllers in order to make the nominal subsystems real stable; in the second step, the concluding result of the fIrst step is taken as an assumption, and the overall control performance is further improved by handling the impact of We have presented two conceptual designs and the respective system-theoretical developments for constructing hybrid analytical and soft-computing control and supervision of complex systems having several similarity non-linear sub-systems and possessing uncertainties. One conceptual design is based on synergetic use of non-linear math-analytical and fuzzy-system models. The other oneisbased on synergetic use of non-linear math-analytical and neural-network models. Both capture well the uncertainties present within the plant via the respective self-adaptation soft-computing structures.

Th. 1101state of subsyst. I Th. 10 2 stete of subsyst. I

7. CONCLUSION

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