A new approach to fuzzy control of interconnected systems.pdf

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SAMS, 2002, Vol. 42, pp. 16231637

A NEW APPROACH TO

FUZZY CONTROL OF

INTERCONNECTED SYSTEMS

MAGDI S. MAHMOUDa,*, MANAR M. SABRYb,y

and SALAH G. FODAc,z

aFaculty of Engineering, Arab Academy for Sciences and Technology,P.O. Box 2033 Al-Horriya, Cairo, Egypt

bProjects and Design Division, Saudi Arabian Texaco (J.O.),P.O. Box 9720, Ahmadi 61008, Kuwait

cElectrical Engineering Department, King Saud University,P.O. Box 800, Riyadh 11421, Saudi Arabia

(Received 29 May 2000)

This paper develops a new approach to the control of interconnected system using fuzzysystem theory. The approach is based on incorporating a group of local estimators onthe system level to generate the inputoutput database. An array of feedback fuzzy con-trollers is then designed to ensure the asymptotic stability of the closed loop system. Thedeveloped technique is applied to an unstable large-scale system, and extensive simula-tion studies are carried out to illustrate the potential of the new approach.

Keywords:

1. INTRODUCTION

In control engineering research, problems of decentralized control

and stabilization of interconnected systems are receiving considerable

interest in recent years [1,2] where most of the effort is focused on

*Corresponding author. E-mail: [email protected] mail: mmanar@ncc moc kw


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dealing with the interaction patterns. It is concluded that a systematic

approach to deal with the problems of interconnected systems is two-

fold: First is to base the analysis and design effort on the subsystem

level using conventional control methods and second is to deal with

interactions effectively. These methods are facilitated, in general, by

virtue of several mathematical tools including linearization, delay

approximation, decomposition and model reduction. This constitutes

the so-called model-based control system approach for which we

have seen numerous techniques [3]. Most of the available results

have so far overlooked the operational knowledge of the intercon-

nected system under consideration. In [4], a knowledge-basedcontrol system approach has been suggested to deal with the analysis

and design problems of interconnected systems by incorporating

both the simplest available model as well as the best available knowl-

edge about the system. For single physical systems, one of the earlier

efforts along this direction has been based on the development of

an expert learning system [56]. An alternative approach has been

to integrate elements of discrete event systems with differential equa-

tions [7]. A practically supported third approach has been the use offuzzy logic control by successfully applying fuzzy sets and systems

theory [9].

For interconnected systems, the foregoing approach motivates the

research into intelligent control by combining techniques of control

and systems theory with those from artificial intelligence. The main

focus should be on integrating a knowledge base, an approximate

(humanlike) reasoning and/or a learning process within a hierarchical

structure.

Fuzzy logic controllers [10] are generally considered applicable to

plants that are mathematically poorly understood (there is no accept-

able mathematical model for the plant) and where experienced

human operators are available for satisfactorily controlling the plant

and providing qualitative rules of thumb (qualitative control rules

in terms of vague and fuzzy sentences).

1. Hierarchical ordering of fuzzy rules is used to reduce the size of the

inference engine.2. Real-time implementation, or on-line simulation of fuzzy control-

1624 M.S. MAHMOUD et al.


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sensory data before imputing the systems output to the inference

engine.

A concerted effort has been made to formally reduce the size of the

fuzzy rule base to make fuzzy control attractive to interconnected

systems. Two of the difficulties with the design of any fuzzy control

system are:

. The shape of the membership functions.

. The choice of fuzzy rules.

The properties that a fuzzy membership function is used to charac-

terize are usually fuzzy. Therefore, we may use different membership

functions to characterize the same description. Conceptully, there are

two approaches to determine a membership function. The first

approach is to use the knowledge of human experts. Usually this

approach can only give a rough formula of the membership function;

fine-tuning is required. In the second approach, data are collected

from various sensors to determine the membership functions.

Specifically, the structures of the membership functions are specified

first. Then fine-tuning of the membership function parameters shouldbe implemented based on the collected data [8].

In this paper, we contribute to the further development of intelligent

control techniques of interconnected systems. It provides a new

approach to fuzzy control design for interconnected systems. The

approach consists of two stages: In the first stage, a group of local

state estimators is constructed to generate the data base of input-

output pairs. In the second stage, an array of feedback fuzzy controllers

is designed and implemented to ensure the asymptotic stability of theinterconnected system. Simulation studies on a large-scale system

with unstable eigenvalues are carried to illustrate the features and

capability of the new approach.

2. FUZZY SYSTEMS BACKGROUND

Fuzzy control is by far the most successful application of fuzzy sets

and systems theory to practical problems. Numerous applications offuzzy logic controllers to a variety of consumer products and industrial

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Fuzzy systems are linguistic knowledge based system. The heart of

a fuzzy system is what is so-called fuzzy IF-THEN rules. These rules

are statements in which some words are described by a continuous

membership function (Fig. 1). For example,

IF vessel temperature is high

THEN small opening of fuel value is required:

IF vessel temperature is low

THEN wide opening of fuel value is required: 1

In general, the starting point of constructing a fuzzy system is to

obtain a collection of fuzzy IF-THEN rules from human experts,

experiments or based on domain knowledge.

The next step is to combine these rules into a single system. There are

three types of fuzzy systems that are commonly used:

1. Pure fuzzy systems,

2. TakagiSugenoKang (TSK) fuzzy systems, and

3. Fuzzy systems with fuzzifier and defuzzifier.

The three systems are described briefly hereinafter.

The configuration of a pure fuzzy system is illustrated in Fig. 2. The

fuzzy rule base represents the collection of fuzzy IF-THEN rules. The

fuzzy inference engine combines these fuzzy IF-THEN rules into a

mapping from fuzzy set in the input space URn to fuzzy sets in the

output space VR based on fuzzy logic principles. If the dashed

FIGURE 1 a Temperature membership functions; b Valve opening membership



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feed back line in Fig. 2 is exists, the system becomes a fuzzy dynamic

system (FDS).

The main disadvantage in the pure fuzzy system is that its input and

output is fuzzy set, whereas in design and engineering the input and

output are real-valued variables.

Takagi, Sugeno and Kang [11,12] introduced another fuzzy system

whose input and outputs are real-valued variables. This system uses

rules in the following:

IF the input x is high then the output y cx: 2

Where the word high has the same meaning as in (1), and c is a

constant. Comparing (1) and (2) we can see that the THEN part of

the rule changes from logistic to into a simple mathematical formula

which leads to combine the rule easier. In fact, the TSK fuzzy

system is a weighted average of the value in the THEN parts of

the rules. Figure 3 shows the basic configuration of TSK fuzzy system.

The main problem with TSK fuzzy system is its THEN part,

which may not reflect a good framework to represent human knowl-

edge. To solve this problem, the third type of fuzzy systems is used.

Figure 4 illustrates the main structure of the fuzzy system with fuzzifier

and defuzzifier.

Comparing this system with a pure fuzzy system, we can see that the

only difference between the two systems is that are the fuzzifier thattransfer the real-valued variable into a fuzzy set, and the defuzzifier

FIGURE 2 Basic configuration of pure fuzzy systems.

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3. STATE ESTIMATION OF INTERCONNECTED SYSTEMS

In the sequel, the terms large-scale and interconnected are used inter-

changeably. The term large-scale system (LSS) does not have a unique

established meaning, but it covers systems that possess several particu-

lar features, such as multiple subsystem, multiple control agents,

multiple objectives, decentralized and/or hierarchical information

structures [1,4]. Any LSS includes many variables but their control

is faced by a well-known fact [3] that the states are not always

available for measurement and state must be estimated.

Many authors have considered the state estimation of LSSs in input

decentralized fashion. Here we summarize one convenient algorithm

[2]. Let the state model of the ith subsystem be described by

xit Aixit Biuit XN

i6jGijxj; 3

yit Cixit, i,j 1,2, . . . , N: 4

Where all vectors and matrices are appropriately defined and gi(.) is

FIGURE 3 Basic configuration of TakagiSergenoKang (TSK) fuzzy systems.

FIGURE 4 Basic configuration of fuzzy systems with fuzzifier and defuzzifier.



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the system. It is considered that (Ci, Ai) is completely observable for

i 1,2, . . . , N.

The following algorithm finds the optimal states of a LSS based on

decentralized estimation and control [4]:

Algorithm 1

Step 1: Read the matrices Ai, Bi and select Qi 0 and Ri> 0 a s

weighted matrices.

Step 2: Solve the following 2Nalgebraic Raccati equations for Hi, Ki

HiATi Ii Ai IiHi HiDiHi Qi 0; 5

KiATi Ii Ai IiKi KiSiKi Qi 0: 6

Where Di CTi Ci, Si BiR

1j B

T:

Step 3: Integrate the following set of N simultaneous equations for

ei(t), i 1, 2, . . . , N, using the initial condition ei(0)xi(0)

e1

.

.

.

eN

264

375

A1S1K1 . . . GIN

.

.

..

..

GNI ANHNDN

264

375

e1

.

.

.

eN

264

375

B1v1

.

.

.

BNvN

264

375 7

Step 4: Integrate the following set of n simultaneous equations for

x1(t), i 1,2, . . . , N

x1

.

.

.

xN

264

375

A1 S1K1 G1N S1K1 0

.

.

....

..

.

GN1 ANSNIN 0 SNKN

264

375

x1

.

.

.

xN

26

4

37

5

B1v1

.

.

.

BNvN

26

4

37

5

8

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4. INTERCONNECTED SYSTEM

Assume the following interconnected system of order 10:

A

1:5 0:3 0:25 0:1 0:5 r111 r112 r113 r114 r1150:1 0 0 0:2 0 r121 r122 r123 r124 r1250 0:2 1 0 0:4 r131 r132 r133 r134 r1350:6 0:1 0:25 2 0 r141 r142 r143 r144 r1450:4 0:2 1 0:5 0:1 r151 r152 r153 r154 r155

r211 r212 r213 r214 r215 1:5 0:3 0:25 0:1 0:5

r221 r222 r223 r224 r225 0:1 0 0 0:2 0r231 r232 r233 r234 r235 0 0:2 1 0 0:4

r241 r242 r243 r244 r245 0:6 0:1 0:25 2 0

r251 r252 r253 r254 r255 0:4 0:2 1 0:5 0:1

2666666666666664

3777777777777775

B

0 0 0 0

1 0 0 0

0 0 0 0

0 1 0 0

0 1 0 00 0 0 0

0 0 1 0

0 0 0 0

0 0 0 1

0 0 0 1

2666666666666664

3777777777777775

C

1 1 0 0 0 0 0 0 0 0

0 0 1 0 1 0 0 0 0 0

0 0 0 0 0 1 1 0 0 00 0 0 0 0 0 0 1 0 1

2

664

3

775

which is considered to be composed of two-coupled subsystems; each

of order 5. The coupling parameters are r1jk and r2jk where j and k

take values of 1, 2, 3, 4 and 5. In the sequel, we refer to the structure

of the interconnected system model as:

x A11

. . . G12 r1

.

.

..

..

264375x B1. . .

B2

24

35v 10



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Where G12(r1) and G21(r2) are the coupling matrices with r1

{r111, . . . , r155} and r2 {r211, . . . , r255}.

For typical values of r115 0.1, r124 0.1, r142 0.2, r222 0.1,

r242 0.15, r251 0.11 and all other values of coupling parameters

being zeros, we examined the stability of the system by computing

the eigenvalues of matrix A. They are { 1.0915, 1.0641,

0.477j0.0206, 0.477j0.00206, 0.022j0.0544, 0.022j0.0544,

1.8709j0.1713, 1.8709 j0.1713, 1.9306 j0.1413, 1.9306

j0.1413}, and it is quite clear that there are four eigenvalues lying

in the open right half of the complex plane, and thus the interconnected

system is unstable. Further, it is easy to check that the interconnectedsystem is both controllable and observable.

4.1. Estimation of the System State Variables and Outputs

A Matlab program is written to implement the computational algo-

rithm (1) of section 3 on the interconnected system. Different positive

and negative step inputs are applied to estimate the outputs. Theresults of two cases are illustrated in Figs. 5 and 6. It is observed

that the outputs tend to track conveniently the input signals.

FUZZY CONTROL 1631


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4.2. Design of an Array of Fuzzy Controller

We are going to treat the interconnected system at hand as being com-posed of two identical and coupled subsystems. The control system to

be designed is such that each subsystem has its own fuzzy negative

feedback controller array with its input being the output of the

respective subsystem (Fig. 7). The array of fuzzy controllers is built

from a collection of individual controllers based on single-input

single-output (SISO) design. A schematic of the subsystem fuzzy con-

troller is shown in Fig. 7.

In order to build each fuzzy controller, the following steps are imple-mented.

Step 1: The range of the inputs to each fuzzy controller [i, i] are

driven from the estimated value of the respective subsystem outputs,

where i 1,2,3,4.

Step 2:2N 1 fuzzy set MLi in [i, i] that are normal, consistent and

complete with triangular membership functions, as shown in Fig. 6 are

defined for each controller, where L 1, 2, . . . , 2Ni 1. That is we use

Ni fuzzy set M1i, . . . , MNii to cover the negative internal (i, 0), theother Ni fuzzy sets M

Ni2i , . . . , M

2Ni1i to cover the positive internal

FIGURE 6 Simulation results (case.2).



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Step 3: The following 2Ni 1 rules are considered:

IF yai is MLi or ybi is M

Li THEN u is K

Li : 11

Where L 1, 2,. . .

, 2Ni 1, and ai, biare the input to the fuzzy con-trolleri, and the center yLaiandyLb of the fuzzy setK

Li are chosen such

that

yLai and yLbi

0 for 1 1, . . . , Ni 0 for 1 Ni 1

0 for 1 Ni 2, . . . , 2Ni 1:

8

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A new approach to fuzzy control of interconnected systems.pdf