Stability analysis and design of Takagi–Sugeno fuzzy systems (2005)

8/12/2019 Stability analysis and design of TakagiSugeno fuzzy systems (2005)

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Stability analysis and design of TakagiSugeno fuzzy systems

Chen-Sheng Ting *Department of Electrical Engineering, National Huwei University of Science and Technology,

64 Wunhua Rd., Huwei, Yunlin 632, Taiwan, ROC

Received 29 November 2004; received in revised form 14 June 2005; accepted 28 June 2005

Abstract

This work presents stable composite control criteria for multivariable Takagi Sugeno (TS) fuzzy systems. On the basis of the linear matrix inequality (LMI) controlstrategy and parametric optimization, the composite fuzzy control algorithms arederived. Unlike earlier studies of fuzzy control systems on an LMI framework, thisinvestigation develops a supervisory control approach, such that a fuzzy controllercan be synthesized more efficiently. Moreover, a robust control scheme is applied tothe TS fuzzy model with parametric uncertainties. The sufficient conditions arededuced in the form of reduced LMIs and adaptive tuning rules. Finally, numeric sim-ulations are given to validate the proposed approach.

2005 Elsevier Inc. All rights reserved.

Keywords: TS fuzzy system; Parallel distributed compensation; Reduced stability conditions

0020-0255/$ - see front matter 2005 Elsevier Inc. All rights reserved.doi:10.1016/j.ins.2005.06.005

* Tel.: +886 5 6315617; fax: +886 5 6315609.E-mail address: [email protected]

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1. Introduction

The analysis and design of TakagiSugeno fuzzy system have been studiedfor decades since this system was introduced [14]. The main feature of the TSfuzzy model is that the consequents of the fuzzy rules are expressed as analyticfunctions. The choice of the function depends on its practical applications. Spe-cically, the TS fuzzy model can be utilized to describe a complex or nonlinearsystem that cannot be exactly modeled. Mathematically, the TS fuzzy model isan interpolation method. The physical complex system is assumed to exhibitexplicit linear or nonlinear dynamics around some operating points. Theselocal models are smoothly aggregated via fuzzy inferences, which lead to the

construction of complete system dynamics.During the past few years, TS fuzzy control systems have been extensivelydiscussed. Most works are based on Lyapunov s direct method to derive thesufficient conditions for the fuzzy systems. In [10], Ma et al. developed separa-tion theorems for designing fuzzy controller and fuzzy observer. Tong et al. [16]and Lee et al. [7] proposed robust control strategies for the MIMO TS fuzzymodel with uncertainties. Their results indicated that a common matrix P isrequired for each fuzzy local system, to guarantee the stabilization of theglobal fuzzy system (in the sense of Lyapunov). The exploration of the com-

mon matrix for the synthesis of the controller or observer can be recast as aconvex problem and solved by LMI optimization techniques [11]. In spite of the advantages of LMI, the existence of a solution that satises the sufficientconditions is not guaranteed. Specically, as the number of fuzzy rule isincreased or too many constraints are imposed, a solution may become infea-sible [9].

In an attempt to rectify this situation, Tanaka et al. [15] derived the relaxedstability conditions to eliminate the conservative constraints on LMIs. Ref. [19]also utilized matrix measures with a properly chosen of parameter p to

obtain the trade-off between conservativeness and computational convenience.Instead of searching for the common positive denite matrix P , a piecewisequadratic function cited in [2] is adopted as a Lyapunov candidate in each fuz-zy subspace. The Lyapunov function is modied to be piecewise differentiableto deal with discontinuity in the transition region [3]. This approach treats thedesign of a fuzzy system as the design of a set of linear-time invariant extremesystems. The method provides the advantages of applying linear system theoryto the design work of a fuzzy system. However, the results may tend to limit thedesign of controllers [13]. In contrast, conventional nonlinear control strate-

gies, such as sliding mode control and feedback linearization control, providealternative means of designing of fuzzy systems. The former approaches offerrobust control, as indicated in [6], while the latter are theoretically complete,as described in [5]. In spite of their success, their derivations have merelyfocused on SISO systems.

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This investigation presents a composite fuzzy control for stabilizing multi-variable fuzzy system with uncertain parameters. The proposed approach uti-

lizes the parameter optimization method to deal with the effects of coupling,which results in a parallel distributed compensation (PDC) framework. Fromdifferent viewpoints, the dynamics of the fuzzy system actually contain explicitinformation and can be analyzed in advance to facilitate the design. On theassumptions about the system s properties, a supervisory control is used to re-lax the stabilization constraints. Therefore, the LMI solution is more likely tobe obtained than by satisfying the basic stability condition. The presentedmethod is more exible than feedback linearization [5], and can be easily ap-plied to a MIMO system. Moreover, a robust fuzzy controller is developed

by incorporating the adaptive control strategy to extend this concept to a fuzzysystem with parameter uncertainty. The adaptation laws are adjusted on-line toguarantee asymptotic stability. Finally, numerical examples are presented toverify the effectiveness of the proposed fuzzy controllers.

The remainder of this study is organized as follows. Section 2 reviews theconventional TS fuzzy model and issues about its stability. Section 3 describesthe design of the composite fuzzy controller. Section 4 introduces a robust con-trol scheme. Section 5 presents the numerical examples. Finally, Section 6 offersconcluding remarks.

2. TakagiSugeno fuzzy control system

Exact mathematic models of most physical systems are difficult to obtain,because of the existence of complexities and uncertainties. However, thedynamics of these systems may include linear or nonlinear behaviors for smallrange motion. Lyapunov s linearization method is often implemented to dealwith the local dynamics of nonlinear systems and to formulate local linearized

approximation. That is, the complex system can be divided by a set of localmathematical models. Takagi and Sugeno have proposed an effective meansof aggregating these models by using the fuzzy inferences to construct the com-plete dynamics of the system [14].

Given the properly dened input variables and membership functions, theTS fuzzy rules for a multivariable system considered herein are in the form of

Rl : If z 1t is M l1 and . . . and z jt is M l j then

_ xt Al xt Bl ut ; l 1; 2; . . . ; r ; 1

where x(t) 2 Rn is the state vector; u(t) 2 Rm is the control input vector;Al 2 Rn

n and B l 2 R n m are the system matrix and the control input matrix,

respectively; M lk is the fuzzy set (k = 1,2, . . . , j ); z(t) = [z1(t), z2(t , . . . , z j (t))]T is

the premise variable vector associated with the system states and inputs, and

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r is the number of fuzzy rules. Center of gravity defuzzication yields the out-put of fuzzy system

_ xt Pr l 1wl z Al xt Bl ut

Pr l1wl z

; 2

where wl z Q ji1 M

li z i and M

li z i denotes the grade of the membership

function M li , corresponding to zi (t). Let l l (z) be dened as

l l z wl z

Pr l1wl z

. 3

Then the fuzzy system has the state-space form

_ xt Xr

l1l l z Al xt Bl ut . 4

Clearly, Pr l 1l l z 1 and l l (z) P 0 for l = 1,2, . . . , r.

Many published results, concerning the control of the fuzzy system, arebased on the parallel distributed compensation (PDC) principle [7,10,11].The fuzzy system is assumed to be locally controllable. The design of the fuzzycontroller shares the same antecedent as the fuzzy system and employs a linearstate feedback control in the consequent part. For each local dynamics the con-troller is dened as


ut K l xt ; l 1; 2; . . . ; r ; 5

where K l is the local state feedback gain. Consequently, the defuzzied result is

ut Xr

l 1l l z K l xt . 6

Substituting (6) into (4) yields_ xt X

r

i1 Xr

j1l i z l j z Ai Bi K j xt . 7

A sufficient condition for the stability of the control is deduced using Lyapu-nov s direct method. Suppose that a common positive denite matrix P exists,so that the following conditions are satised [15]:

Ai Bi K iT P P Ai Bi K i < 0 8

for i = 1,2, . . . , r, and

Ai Bi K j A j B j K i2

T

P P Ai Bi K j A j B j K i

2 < 0 9


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for 1 6 i < j 6 r. The above terms in Eqs. (8) and (9) are regarded as the basicstabilization conditions. When these conditions are satised, the fuzzy system

(7) is then asymptotically stable. The design work can be transformed into aconvex problem [11], which is efficiently solved by LMI optimization. If thesolution is feasible, meaning that the stabilization constraints are met, thenlocal state feedback gains are obtained. In spite of the advantages of LMI,the existence of the solution is not guaranteed. As the number of fuzzy rulesis increased or if too many constraints are imposed, a solution may becomeinfeasible [9].

Cao et al. [2] introduced the idea of piecewise smooth quadratic Lyapunovfunction to solve the problem of searching for the common matrix P over LMI.

Based on that approach, the fuzzy system is considered to be a set of lineartime-invariant extreme systems. In each fuzzy subspace, the local dynamicsstands for the nominal model and the effect of other rules is interpreted asthe uncertainty. Accordingly, the linear uncertain system theory can be appliedto the analysis and design of fuzzy systems. However, the Lyapunov function isdiscontinuous while the states of the system are maintained in the transitionregion. To overcome this deciency, the Lyapunov function is modied to bepiecewise differentiable [3]. Sun et al. [13] continued this idea to exploringthe adaptive control for estimating the bound of the uncertainty. Moreover,

Udawatta et al. [18] addressed a method to determine a large convex domainfor common P and to derive fuzzy-chaos controller for nonlinear systems withchaotic attractive features.

In the PDC scheme, each control rule is designed based on the corre-sponding rule of the TS fuzzy model. A stable feedback gain of the localfuzzy model is much easier to obtain specically as the subsystem is control-lable. Notably, the stabilization of each local model does not ensure thestability of the global system. However, the basic stabilization conditions (8)and (9) tend to be conservative, and further relaxation is often desired

[15,19]. This result motivates the exploration the composite fuzzy control inthis study. The proposed approach employs the nonlinear programmingmethod to solve the coupling problem (9) in the PDC framework. Whensome assumptions regarding the properties of the system hold, the supervisorycontrol overcomes the coupling effect. The basic stabilization constraintscan be relaxed. Therefore, the LMI solution can be easily obtained than byapplying the conservative constraints. The presented approach is derived asfollows.

3. Design of composite fuzzy control system

Before the derivation, the following assumptions are made regarding theTS fuzzy system (1).


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Assumption 1. The pairs (A l , B l ), l = 1, . . . , r, are controllable. That is, thefuzzy system is locally controllable.

Assumption 2. For each local model, the control input matrix B l is of full rankand has the same column space as the others: col (B 1) = col( B 2) = = col( B r ).

Based on these assumptions, a state feedback control gain K l can be ob-tained by pole placement design or Ackerman s formula, such that each localdynamics is stably controlled. Moreover, B l can be expressed as B l = B 0D l ,where D l 2 Rm

m is a full rank matrix and B 0 denotes the basis of the controlinput matrix. The representation of the global control input matrix, denoted by

B , is in the form

B Xr

l 1l l Bl B0 D; D X

r

l1l l Dl !. 10

The global control input matrix dominates the control performance, so thefollowing assumption is made regarding D.

Assumption 3. The matrix D in (10) is of full rank in the system operatingregion.

The column vectors of D should be linearly independent to conrm the fullrank property of D. Let D be expressed in the column vectors form:

D d 1 . . . d m ; d i Xr

l1l l qil ; 11

where qil denotes the i th column vector of Dl . Notably, each column vector of D is time-varying and is a linear combination of the corresponding column vec-

tors of Dl s. Proving the independence of the column vectors of D is compli-cated. However, from the geometric point of view, the designer can verifythis property, especially for the low dimension D. In the one-dimensional case,D l is simply scalar. The independence of the columns of D is easily determinedby checking the congruency of the signs of the scalars. In the two-dimensionalcase, the column space forms a plane. Eq. (11) implies that d i lies on the regionover the linear combination of the column vectors, qil s. A parallelogram isused to add the vectors, and the largest region in which each column lies uponcan be delimited. If the regions do not overlap, the matrix D is concluded to be

of full rank and invertible.Let the fuzzy control rules be dened as


ut K l xt u st ; l 1; 2; . . . ; r ; 12


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where us(t) 2 Rm . The premise part of the controller shares the same fuzzy setsof the TS fuzzy plant model, and the consequent part consists of a local state

feedback K l x(t) and a supervisory control us(t), which will be discussed later.The output of the fuzzy controller is

ut Xr

j1l j K j xt u st . 13

The closed-loop system is given by

_ xt Xr

i1 Xr

j1l i z l j z Ai Bi K j xt X

r

i1l i Biu st

Xr

i1l 2i G ii xt 2X

r

i< jl il j G ij xt Bu st ; 14

where G ii = A i + B i K i , G ij = (Ai + B i K j + A j + B j K i )/2. For convenience, thevariables in time t will be suppressed.

The controller synthesis initially considers the stabilization of the local fuzzydynamics. That is, the stable state feedback gains are determined for every sub-system. Suppose that there exists a symmetric and positive denite matrix P ,and some matrices K i , (i = 1, . . . , r), such that the following reduced stability

conditions hold: Ai Bi K i

T P P Ai Bi K i6 Q i; i 1; . . . ; r ; 15

where Qi is a positive denite matrix. Based on this assumption, each subsys-tem is locally controllable and a stable feedback gain is obtainable. Intuitively acommon matrix P that satises Eq. (15) can be obtained more easily than thatcan one that fullls the basic stabilization conditions. When the LMI method isapplied, the conditions (15) can be veried efficiently. If a feasible solution isobtained, the design proceeds to exploit the supervisory control to deal withthe coupling terms.

Choose the Lyapunov function candidate, V 1(x) = xT Px . The derivative of V 1(x) with respect to time is

_V 1 x Xr

i1l 2i x

TG Tii P PG ii x

2Xr

i< jl il j x

TG Tij P PG ij x 2 xT PBu s

6

Xr

i1 l2i x

T

Q i x 2Xr

i< j l il j xT

G Tij P PG ij x 2 x

T

PBu s. 16

Given the matrix property, clearly,

kminG Tij P PG ij k xk2 6 xTG Tij P PG ij x 6 kmaxG

Tij P PG ij k xk

2; 17


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where kmin(max) denotes the smallest (largest) eigenvalue of the matrix.Dene

a maxi; j

kmax G Tij P PG ij for 1 6 i < j 6 r . 18

A relaxed condition concerning the coupling effect is expressed as

Xr

i< jl il j x

TG Tij P PG ij x 6 j 1k xk2; and j 1

r r 12

a. 19

Finding the maximum value of

Pr i< j l il j x

TG Tij P PG ij x is equivalent to

determining the maximum value of Pr

i< j l il jkmaxG Tij P PG ij . It can be pre-sented as a nonlinear programming. The optimal algorithms are employed to

seek for the best solution. Moreover, the Matlab Optimization Toolbox [1]consists of functions that minimize or maximize general nonlinear functions.By using the toolbox, the nonlinear programming is expressed in the followingform:

maxi; j Xi< j l il j kmaxG Tij P PG ij ; 1 6 i < j 6 r

subject to : Xr

i1l i 1; l i P 0;

Xr

j1l j 1; l j P 0. 20

The largest eigenvalue of G Tij P PG ij can be obtained in advance, so the max-imum value is determined to be

j 2 maxi; j

Xi< j

l il jkmaxG Tij P PG ij . 21

The following supervisory control is chosen

u s BT Px

k xT PBk2 j k xk2; if k xT PBk 6 0;

0 if k xT PBk 0;8>:

22

where j > j j , j = 1 or 2. If kxT PB k 5 0, then substituting (22) into (16)gives

_V 1 x6 Xr

i1l 2i x

TQ i x 2j jk xk2 2j k xk2 6 X

r

i1l 2i x

TQ i x V 2 x;

23


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where V 2(x) is a positive denite function. As kxT PB k = 0, based on the fullrank assumption of the matrix D, it follows that xT PB 0 = 0. Therefore,

xT Ai Bi K iT P P Ai Bi K i x

xT Ai B0 Di K iT P P Ai B0 Di K i x

xT ATi P PAi x 6 xTQ i x; i 1; . . . ; r 24

and

Xr

i< jl il j x

TG Tij P PG ij x

Xi< j lil j x

T

Ai A jT

P P Ai A j x

Xi< j l il j xT ATi P PAi x xT AT j P PA j x6 Xi< j l il j xTQ i Q j x; 1 6 i < j 6 r . 25

The time derivative of V 1(x) becomes

_V 1 x6 Xr

i1l 2i xTQ i x Xi< j l il j x

TQ i Q j x V 3 x; 26

where V 3(x) is a positive denite function. Hence, the closed-loop fuzzy systemis asymptotically stable.

The results are summarized in the following theorems.

Theorem 1. If the Assumptions 13 regarding the fuzzy system (1) hold and thereexist a common positive denite matrix P and some feedback gain matrices K i ,

(i = 1, . . . ,r), such that the reduced stability conditions (15) are satised, then the fuzzy closed-loop system is guaranteed to be asymptotically stable as determined by control laws (12) and (22).

Corollary 1 [15]. Consider the special case in which B 1 = B 2 = = B r = B 0 , and B 0 is of full rank. Given the fuzzy controller (12), the equilibrium of the fuzzy con-trol system is asymptotically stable if a common positive denite matrix P and some matrices K i ,(i = 1, . . . ,r), exist such that conditions (15) are satised.

Based on Corollary 1, the designer simply veries that the common controlinput matrix is of full rank. The asymptotic stability of the fuzzy control systemis thus achieved. A more exible design methodology than fuzzy feedback lin-earization control [5,11], which can be easily applied to multiple input fuzzysystem, is provided.


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4. Control of uncertain fuzzy systems

Motivated by the results of Section 3, the design principle is extended to theTS fuzzy system with uncertainties. Controlling these systems is particularlychallenging [7,16]. Consider the TS fuzzy model,


_ xt Al D Al xt Bl D Bl ut ; l 1; 2; . . . ; r . 27

Notably, the model is almost the same as (1), except for the terms D Al and D B l ,which stand for the parametric uncertainties of each fuzzy subsystem and are

time-varying with appropriate dimensions. The fuzzy system is then inferredto be

_ xt Xr

l1l l z Al D Al xt Bl D Bl ut . 28

The fuzzy control rule is dened as


ut K l xt u st ua t ; ua t 2 Rm

; l 1; 2;. . .

; r ; 29which leads to

ut Xr

j1l j K j xt u st ua t . 30

The controller has the same structure as (12), except for the additionalterm ua which is related to the uncertainty. The rst two terms in (30) have al-ready been discussed in Section 3, so the development of ua is consideredherein.

Substituting (30) into (27) yields

_ x Xr

i1 Xr

j1l il j Ai D Ai x Bi D Bi K j x u s ua

Xr

i1 Xr

j1l il j Ai D Ai x Bi D Bi K j x X

r

i1l i Bi D Biu s ua

Xr

i1 Xr

j1l il j Ai Bi K j x X

r

i1 Xr

j1l il jD Ai D Bi K j x

Xr

i1l i Bi D Biu s ua . 31


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The result derived in the preceding section can be used by rearranging

_ x Xr

i1l 2i G ii x 2X

r

i< jl il jG ij x X

r

i1l i Biu s X

r

i1l iD Ai x X

r

i1

Xr

j1l il j D Bi K j x X

r

i1l iD Biu s X

r

i1l i Bi D Biua

Xr

i1l 2i G ii x 2X

r

i< jl il jG ij x X

r

i1l i Biu s D A F t ; x x

D B F t ; x x D BS t ; xu s B g D B g ua ; 32

where

D A F t ; x Xr

i1l iD Ai; D B F t ; x X

r

i1 Xr

j1l il j D Bi K j ;

D BS Xr

i1l iD Bi B g

1r X

r

i1 Bi; D B g X

r

i1l i Bi D Bi B g . 33

The following assumption is made regarding the uncertainties to derive theproposed robust control scheme.

Assumption 4. The parametric uncertainties D A i and D B i in (27) are matched.That is,

D Ai B g d Ai; D Bi B g d Bi; i 1; . . . ; r ; 34

where dA i and dB i are matrices of compatible dimensions.

Based on this assumption, there exist the matrices

E : R Rn ! Rm n ; F : R Rn ! Rm n; G : R Rn ! Rm m; H : R Rn ! Rm m

such that

D A F B g E t ; x; D B F B g F t ; x; D BS B g G t ; x; and D B g B g H t ; x.

Accordingly,

_ x Xr

i1l 2i G ii x 2X

r

i< jl il j G ij x X

r

i1l i Biu s B g ua Ex Fx Gu s Hua

Xr

i1l 2i G ii x 2X

r

i< jl il j G ij x Bu s B g ua nt ; x; u; 35


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where n(t, x , u) is the lumped uncertainty and is given by

nt ; x; u E x F x G u s H u

a. 36

This representation implies that the uncertainty vector n() does not inuencethe dynamics more than the control vectors ua and us [4].

The uncertainty n is unknown, so generally, the estimated n is required fordesigning the controller. That the fuzzy system is a universal approximationthat can approximate any real continuous function on a compact set to an arbi-trary accuracy is well known. A number of interesting results concerning adap-tive fuzzy control have been obtained [17]. Consequently, the fuzzy basisfunction is used herein to approximate the uncertainty n and deduce the adap-tive laws for the estimation of the parameter and the upper bound. The adap-tive control utilizes the r -modication [12] to avoid chattering caused byswitching.

Consider the Mamdani type fuzzy inference that approximates the i th ele-ment of n, ni , as follows:

Rl : If x1t is ~ M l1 and . . . and xnt is ~ M

ln then ni is

~ Dil ;l 1; 2; . . . ; r .

The output of the inference is

^ni xjhi Pr l 1hil Q

nh1l ~ M lh xh

Pr l 1Q

nh1l ~ M lh xh

hTi x x; 37

where hi = (hi 1, hi 2, . . . , hir )T is an adjustable parameter vector; hil is the center of ~ Dil for i = 1,2, . . . , r, and x (x) is called the fuzzy basis function [17]. Then, theestimation of n is given by ^n xjh hTx x; h 2 Rr m. The optimal parametermatrix is dened as

h argminh2 X h

sup x

k^n xjh nt ; xk 38such that

k^n xjh nt ; xk6 e1 e2k xk; 39

where e1 and e2 are unknown positive constants and estimated by the adaptivemechanism as e1 and e2.

Choose the adaptive control law asua ^n u1 u2; 40

where u1 and u2 are used to reduce the effect of estimation errors and will bedepicted later.


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Let the parameter errors be ~h h h; ~e1 e1 e1; ~e2 e2 e2; andchoose the candidate Lyapunov function,

V V 1 12gh

tr ~hT~h

12g1

~e21 12g2

~e22; 41

where V 1 is as dened in the previous section, and gh , g1 and g2 are positiveconstants that relate to the adaptation rate. The time derivative of V is

_V _V 1 1gh

tr ~hT _h

1g1

~e1 _e1 1g2

~e2 _e2 6 V 3 x 2 xT PB g n ^n u1 u2

1

ghtr ~h

T _h 1

g1~e1 _e1

1

g2~e2 _e2; 42

where V 3(x) is a positive function obtained from Theorem 1. Let Z 2 BT g Px;then,

_V 6 V 3 x Z Tn ^n Z Tu1 u2 1gh

tr ~hT _h

1g1

~e1 _e1 1g2

~e2 _e2

V 3 x Z Tn n Z T~n Z Tu1 u2 1gh

tr ~hT _h

1g1 ~e1

_e1

1g2 ~e2

_e2 6 V 3 x kZ k1e1 k Z k1e2k xk

Z T~n Z Tu1 u2 1gh

tr ~hT _h

1g1

~e1 _e1 1g2

~e2 _e2; 43

where n n xjh ; ~n ~hT

x x, and kk1 denotes 1-norm. Rearranging (43)yields

_V 6 V 3 x tr ~hT

x Z T 1gh

_h ~e1 kZ k1 1g1 _e1 k Z k1e1 ~e2 kZ k1k xk 1g2

_e2 k Z k1e2k xk Z Tu1 u2. 44

Let u1 and u2 be

u1i e1 tanh z i e1

l ; i 1; 2; . . . ; m;u2i e2k xk tanh

z i e2k xkl ; i 1; 2; . . . ; m;

45

where u1 i , u2 i , and zi denote the i th components of vectors u1, u2, and Z , respec-tively, and l is a designed small positive constant. The hyperbolic tangent func-tion in the control design is used to avoid chattering that would otherwise becaused by discontinuous control. Ref. [12] reveals that for any l > 0 and anyz 2 R , the following inequalities hold,


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0 6 j z j z tanh z

l 6 ml; z tanh z l P 0; 46where m is a constant given by m = e (m + 1) , such that m = 0.2785.

The adaptive laws regarding r -modication are_h ghx Z T r h h

0;_e1 g1kZ k1 r e1 e

01;

_e2 g2kZ k1k xk r e2 e02;

47

where h0; e01; e02 and r > 0 are design constants. Then,

tr ~hT x Z T 1gh_h tr r

~hTh h0h i

r2

tr h h0Th h0h i r2

tr ~hT~h

r2

tr h h0Th h0; 48

~e1 kZ k1 1g1

_e1 r ~e1 e1 e01 r2

e1 e01 2 r2

~e21 r2 e1 e01

2 49

~e2 kZ k1k xk 1g2

_e2 r ~e2 e2 e02

r2

e2 e02 2 r

2~e22

r2

e2 e02 2. 50

Based on (46), one yields

kZ k1e1 Z Tu1 Xm

i1j z i je1 z i e1 tanh

z i e1l 6 mml;

kZ k1e2k xk Z Tu2 X

m

i1j z i je2k xk z i e2k xk tanh

z i e2k xkl 6 mml

51

and

_V 6 V 3 r2

tr h h0Th h0h ir2

tr ~hT~h

r2

tr h h0Th h0r

2 e1 e0

1

2 r

2~e2

1

r

2 e1 e0

1

2 r

2 e2 e0

2

2 r

2~e2

2

r

2 e2 e0

2

2

2mml 6 V 3 r2

tr ~hT~h ~e21 ~e

22h i

r2

tr h h0Th h0

r2

e1 e01 2

r2

e2 e02 2

2mml. 52


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Suppose V 3(x) > a1V 1(x) for some positive real number a1. Based onc = max(2,1/ gh , 1/g1, 1/g2),

12

V 1 tr ~hT~h ~e21 ~e

22 P 1c V . 53

If r is chosen such that 2a1r P 1, and b r2 tr h h

0Th h0he1 e01 2

e2 e02 2 is dened, then

_V 6rc

V b 2mml qV d; 54

where q rc, and d b 2mml. Therefore,

V t 6 dq

V 0 dq e q t . 55

From the above discussion, V is obviously a bounded function, which impliesthat x; ~h; e1, and e2 are all bounded. Moreover, given any l 1 P ffiffiffiffid=kmin P qp there exists T (l 1) such that kxk 6 l 1 for all t P T . Accordingly, the state x(t)will eventually be conned within a certain range.

The following theorem summarizes the results.

Theorem 2. Consider the TS fuzzy system described in (27). If Assumptions14 hold, and there exist a common positive denite matrix P and some matricesK i , (i = 1, . . . ,r), such that the reduced stability conditions (15) are satised, thenthe fuzzy controller (29) with the adaptive laws (47) can stabilize the global system. Furthermore, given any l 1 P ffiffiffiffiffiffiffiffi ffid=kmin P qp there exists T( l 1 ) such thatkxk 6 l 1 for all t P T.

Based on the proposed adaptive control scheme, the design parametersh0; e01; e02; r ; l

must be chosen appropriately. The parameters h0; e01; e02

can

be regarded as initial estimates of the unknown h*, e1, and e2, respectively. Usingestimates that are closer to the optimal values yields a more accurate result. Thecontrol designer generally uses a priori knowledge or an off-line identication of the system to determine the best values. In the absence of any a priori informa-tion, these parameters may be set to be zero for simplicity. The parameter l re-ects the width of the boundary layer associated with the normal sign function.Its value should not to be too small to prevent chattering. The parameter r hasto be chosen such that the condition r 6 2a1 is satised.

According to the above analysis, the design procedure for TS fuzzy systems

is summarized as follows.

Step 1: Conrm that Assumptions 1 and 2 are satised for the designedsystem.

Step 2: Determine the basis B 0 of the control input matrix.


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Step 3: Verify the full rank assumption regarding the matrix D using a com-puter program.

Step 4: Solve the LMI problem, Eq. (15) and obtain P , K i , Q i , i = 1, . . . , r .Step 5: Execute the nonlinear program, based on Eq. (20), to determine j .Step 6: Construct the fuzzy controller (12).The following procedure may be

followed to control uncertain fuzzy systems.Step 7: Verify the matching conditions (34) imposed on D Ai and D B i ,

i = 1, . . . , r.Step 8: Determine the design parameters h0; e01; e02; r ; l based on a priori

knowledge or off-line identication of the system.Step 9: Construct the composite fuzzy controller (30) with adaptive laws

(47).

5. Numerical examples

This section concerns the computer simulation of the design procedure andveries the effectiveness of the proposed algorithms for both SISO and MIMOsystems.

Example 1. Consider an inverted pendulum system whose dynamics is [16]_ x1 x2;

_ x2 g sin x1 amlx 22 sin2 x1=2 a cos x1u

4l =3 aml cos2 x1 ;

where x1 is the angle of the pendulum from the equilibrium position; x2 is theangular velocity, and u is the force applied to the cart. The parameters are gi-ven as follows: g = 9.8 m/s 2, the gravity constant, m = 2.0 kg, the mass of thependulum; M = 8 kg, the mass of the cart; 2 l = 1.0 m, the length of the pendu-

lum, and a = 1/( m + M ).The nonlinear system is represented by the TS fuzzy model

R1 : If x1 is about 0; then _ x A1 x B1u;

R2 : If x1 is about p2

; then x A2 x B2u;

where the matrices A1, A2, B 1, and B 2 are given by [16]:

A1 0 1 g

4l =3 aml 0

24 35; A2

0 12 g

p4l =3 aml b20

24 35;

B1 0

a4l =3 aml

24 35; B2

0 ab

4l =3 aml b224 35


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and b = cos(88 ). The local linear models are in controllable canonical formand have the same column spaces for each control input matrix. Assumptions

1 and 2 hold. Moreover, if B 1 is chosen as the basis, D l is simply a scalar. Thesigns of Dl are the same so Assumption 3 holds. The LMI method yields thefollowing results,

P 1.9205 1.0361

1.0361 1.2653 ; Q1 Q2 0.3242 0.03260.0326 0.3108 ; K 1 117.9103 17.9807 ; K 2 2458.7 606

which guarantee the stability conditions (15). With the help of the Optimiza-tion Toolbox [1], the parameters are obtained as j 1 = 2.8064 and j 2 =0.7016. The initial state vector is x0 1 0 T and j is chosen to be 1. Fur-thermore, the parallel distributed compensation is compared with proposedcontroller. The feedback gains of PDC are designed to be ^ K 1 1106.7 304.2 and ^ K 2 2798.1 794.8 , which satisfy the basic stabiliza-tion conditions (8) and (9). Figs. 1 and 2 show the closed-loop system perfor-mance, where the solid lines stand for the control effect as determined by the

Fig. 1. Response of x 1(t).


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developed approach and the dashed lines represent the PDC behaviors. The g-ures indicate the effectiveness of proposed algorithm.

The system model used to simulate the perturbed condition is

R1 : If x1 is about 0; then _ x A1 D A1 x B1 D B1u;

R2 : If x1 is about p2 ; then _ x A2 D A2 x B2 D B2u.

If the parameters l , m and M are changed to 56.5% of their nominal values,then

D A1 0 011.1328 0 ; D B1 00.1136 ;

D A2 0 0

7.2051 0

; D B2

00.004

.

Essentially, the mean perturbations are about 30% of the nominal values, cor-responding to A1, B 1, A2, and B 2. The uncertainty is estimated by applying theve fuzzy rules and the following data are used in the simulation: h0 1 1 0 1 1 , r = 10, l = 0.1, e01 e02 0.01, and gh = g1 = g2 = 1.



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Figs. 3 and 4 plot the simulation results. If the effect of uncertainty is neglected,the PDC controller is not robust.

Example 2. Consider a two-link robot system whose dynamics is as follows[8].

M qq C q; _q_q G q s;

where

M q m1 m2l 21 m2l 1l 2 s1 s2 c1c2m2l 1l 2 s1 s2 c1c2 m2l 22" #;

C q; _q m2l 1l 2c1 s2 s1c20 _q2_q

1 0

; G q

m1 m2l 1 gs1m2l 2 gs

2

;

where q q1 q2 T, and q1,q2 are generalized coordinates; M (q) is the mo-

ment of inertia; C (q) includes coriolis and centripetal forces, and G (q) is thegravitational force. The other quantities are: link mass m1, m2 (kg), link lengthl 1, l 2(m), angular position q1,q2 (rad), applied torques s s1 s2

T N m,

Fig. 3. Response of x 1(t) under perturbed conditions.


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gravitational acceleration g = 9.8 m/s 2, and short-hand terms s1 = sin( q1),s2 = sin( q2), c1 = cos( q1), and c2 = cos( q2).

Let x1 = q1, x2 _q1, x3 = q2, x4 _q2, m1 = m2 = 1 kg, l 1 = l2 = 1 m, andangular positions q1 and q2 are constrained within p=2; p=2 . The TS fuz-

zy model for the system is given by the following nine-rule fuzzy model: R1 : If x1 is about

p2

and x3 is about p2

; then _ x A1 x B1u;


and x3 is about 0; then _ x A2 x B2u;


and x3 is about p2 ;

then _ x A3 x B3u;

R4 : If x1 is about 0 and x3 is about p2


R5

: If x1 is about 0 and x3 is about 0; then _ x A5 x B5u; R6 : If x1 is about 0 and x3 is about

p2



and x3 is about p2


Fig. 4. Response of x2(t)under perturbed conditions.


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and x3 is about 0; then _ x A8 x B8u;

R9 : If x1 is about p2 and x3 is about

p2 ; then _ x A9 x B9u;

where x x1 x2 x3 x4 T; u s1 s2

T, and

A1

0 1 0 0

5.927 0.001 0.315 8.4 10 6

0 0 0 1

6.859 0.002 3.155 6.2 10 6

266664377775

A2

0 1 0 0

3.0428 0.0011 0.1791 0.0002

0 0 0 1

3.5436 0.0313 2.5611 1.14 10 5

266664377775

A3

0 1 0 0

6.2728 0.003 0.4339 0.0001

0 0 0 1

9.1041 0.0158 1.0574 3.2 10 5

266664

377775

A4

0 1 0 0

6.4535 0.0017 1.2427 0.0002

0 0 0 1

3.1873 0.0306 5.1911 1.8 10 5

266664377775

A5

0 1 0 0

11.1336 0 1.8145 0

0 0 0 1

9.0918 0 9.1638 0

266664

377775

A6

0 1 0 0

6.1702 0.001 1.687 0.0002

0 0 0 1

2.3559 0.0314 4.5298 1.1 10 5

266664377775

A7

0 1 0 0

6.1206 0.0041 0.6205 0.0001

0 0 0 1

8.8794 0.0193 1.0119 4.4 10 5

266664377775


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K 4 252.9162 32.048 0.9893 0.23817.034 0.1841 173.7095 20.9899" #

K 5 257.698 33.2081 92.9015 10.347489.1484 11.749 172.9205 20.978" #

K 6 252.3267 32.0422 0.6541 0.34376.6782 0.1873 172.8575 20.9634" #

K 7 254.37 33.2737 94.5779 10.2385

91.0372 11.1737 170.5918 21.1106" # K 8

246.857 31.9635 6.9556 0.27050.5788 0.4908 170.9743 20.9125" #

K 9 248.9674 33.1026 90.3182 10.362386.0267 11.7596 167.7326 20.8511" #

P

0.8783 0.0038 0.0443 0.00020.0038 0.0057 0.0005 0.0002

0.0443 0.0005 1.2034 0.00260.0002 0.0002 0.0026 0.0055

266664

377775

.

Fig. 5. Largest lying regions of column vectors of D .


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The parameters j 1 and j 2 are determined to be 5.436 and 0.0377, respectively,by applying the Optimization Toolbox. The initial condition of the state vector




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is x0 0.5 0.3 0.5 0.3 T [8] and j = 0.04. In simulation, the proposedmethod is compared with a PD controller provided for each link with

Fig. 10. Response of x1(t) under perturbed conditions.



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K P = 100 and K D = 20. Figs. 69 show the performance of each link; the solidlines refer to the proposed approach and the dashed lines present the control

effect obtained by PD controller. With respect to the uncertainty of the fuzzysystem, the parameters of the robot, the link mass and the length, are assumedto be perturbed 30% from their nominal values. In the adaptive control scheme,the following values are adopted: h0 0; r 15; l 0.5; e01 e02 0.1;g1 g2 0.1; and gh = 1. Figs. 1013 plot the trajectories of the links. Thesimulation results reveal the system controlled by the proposed method per-forms well. The effectiveness of the proposed controller design is nallydemonstrated.

6. Conclusion

This study presents a systematic control design for multivariable TS fuzzysystem. The control algorithms are simple and easy to apply. Based on this ap-proach, the relaxed stabilization conditions are derived so that the solutions byLMI are more feasible. A composite fuzzy controller is designed to stabilize thecontrol performance if the TS fuzzy system satises the assumptions men-tioned above. Additionally, a robust control scheme that incorporates theadaptive tuning laws is investigated to deal with the problem of parametricuncertainties in the TS fuzzy model. The effectiveness of proposed approachis illustrated by computer simulations of the inverted pendulum and the two-link robot.

References

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stability, Fuzzy Sets Syst. 85 (1997) 110.[3] S.G. Cao, N.W. Rees, G. Feng, H 1 control of uncertain fuzzy continuous-time systems, Fuzzy

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delay, IEEE Trans. Autom. Control 34 (11) (1989) 11991203.[5] Y.W. Cho, C.W. Park, M. Park, An indirect model reference adaptive fuzzy control for SISO

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Documents

Stability analysis and design of Takagi–Sugeno fuzzy systems (2005)