Synthesis of nonlinear discrete control systems via time-delay affine Takagi-Sugeno fuzzy models

ISA TRANSACTIONS® ISA Transactions 44 (2005) 243-257

Synthesis of nonlinear discrete control systems via time-delay affine Takagi-Sugeno fuzzy models Wen-Jer Chang, * Wei Chang Depanment of Marine Engineering. National Taiwan Ocean University. Keelung 202. Taiwan. R. O. C.

(Received 3 March 2004; accepted 8 November 2004)

Abstract The affine Takagi-Sugeno (TS) fuzzy model played a more important role in nonlinear control because it can be used to approximate the nonlinear systems more than the homogeneous TS fuzzy models. Besides, it is known that the time delays exist in physical systems and the previous works did not consider the time delay effects in the analysis of affine TS fuzzy models. Hence a parallel distributed compensation based fuzzy controller design issue for discrete time-delay affine TS fuzzy models is considered in this paper. The time-delay effect is considered in the discrete affine TS fuzzy models and the stabilization issue is developed for the nonlinear time-delay systems. Finally, a numerical simulation for a time-delayed nonlinear truck-trailer system is given to show the applications of the present approach. © 2005 IS A-The Instrumentation, Systems, and Automation Society. Keywords: Takagi-Sugeno fuzzy model; Time-delay; S procedure; Iterative linear matrix inequality

1. Introduction It is well known. that the Takagi-Sugeno (TS) fuzzy models [1-20] have become one of the useful control approaches for complex nonlinear systems. It can provide an effective representation of complex nonlinear systems in terms of fuzzy sets, described by a set of IF-THEN rules, which can locally represent linear input-output relations of nonlinear systems. Moreover, its stability analysis and synthesis issue can be transformed to a linear matrix inequality (LMI) problem. The history of LMI in the analysis of dynamical systems goes back more than 110 years. The story begins in 1890, when Lyapunov published his seminal work introducing what we now call the Lyapunov theory and the first LMI presented to analyze the "'Corresponding author. Tel: +886-2-24622192, Ext. 7110; fax: +886-2-24633765. E-mail address: [email protected] 0019-0578/20051$ - see front matter

stability of a dynamical system was the Lyapunov inequality. It can be solved analytically by solving a set of linear equations. Today, the LMI techniques have emerged as powerful design tools in areas ranging from control engineering to system identification and structural design even in the TS fuzzy models. In control systems, recasting the stability analysis and control design problems as LMI problems is equivalent to finding solutions to the original problems. The recasting of stability analysis and design of fuzzy controllers to LMI problems was first made in Refs. [6,7]. The LMI problems can be solved with ease by a convex optimization algorithm. The time delays are natural components of dynamic processes. The considerable effort has been applied to different aspects of linear time-delay systems during recent years (see Ref. [21] and the references cited therein). The delay-independent stabilization is investigated in Refs. [1-5], and the delay-dependent stabilization problem was dealt

e 2005 ISA-The Instrumentation, Systems. and Automation Society.

244

Wen-Jer Chang, Wei Chang /ISA Transactions 44 (2005) 243-257

with in Refs. [22-24]. For the nonlinear timedelay systems, the global stability and stabilization issues were developed in Refs. [1,2] via TS fuzzy models. Considering uncertain nonlinear fuzzy systems with time delays, Refs. [3,4] discuss the robustness and H:)O control design for TS fuzzy models. Moreover, the conditions for global exponential stability of free fuzzy systems with the uncertain delays were presented in Ref. [5]. In those papers, the concept of parallel distributed compensation (POC) is proposed to d'esign the fuzzy controllers and the LMI technique is used to find the solutions of the POC-based fuzzy controllers. In the literature, the authors focused on the timedelay homogeneous TS fuzzy models. However, the TS fuzzy models can be divided into two categories: homogeneous TS fuzzy models [1-14] and affine TS fuzzy models [15-19]. In general, the controller design problem for the time-delay affine TS fuzzy models is a difficult one for the designers. The difficulty comes from the fact that the stability fonnulation is not a LMI fonn but a bilinear matrix inequalities (BMI) fonn. The BMI cannot be solved with ease by a convex optimization algorithm. Thus the iterative linear matrix inequality (ILMI) [16,17,25] algorithm was derived to solve the BMI fonnulations by using the LMI techniques. The fuzzy controller design problem for affine TS fuzzy models has been solved in Ref. [16]; however, it did not consider the time-delay effects. Therefore this paper concerns the open problem of stability analysis and stabilization for the discrete affine TS fuzzy models with timedelay effects. In this paper, the ILMI algorithm technology is used to develop a fuzzy controller design procedure for the discrete time-delay affine TS fuzzy models. The organization of this paper is presented as follows. In Section 2, the system descriptions and stability conditions of discrete time-delay affine TS fuzzy models is provided. In Section 3, a POCbased fuzzy controller design procedure is pre~ sented via ILMI algorithm for the discrete timedelay affine TS fuzzy models. A numerical example and simulated results are described in Section 4. Finally, concluding remarks are made in Section 5. Throughout this paper. the following notations are used. The Vtn denotes the n-dimensional Euclidean space. The /1./1 refers to the Euclidean vector nonn and I denotes an identity matrix. The'

notation W>O means the matrix W is a positive definite symmetric matrix.

2. System descriptions and stability conditions Global asymptotic stability analysis of the discrete affine TS fuzzy model with time delays is discussed in this section. It is shown that the asymptotic stability analysis issue to the discrete time-delay affine TS fuzzy models is considered based on the Lyapunov criterion. Besides, the S procedure [26,27] is used to ensure that the analysis results and the detail of the S procedure can be referred to [27]. The discrete time-delay affine TS fuzzy model used in this paper is of the following IF-THEN fonn. Plant part: Rule p i: IF xl(k) is Mil and x2(k) is M;2 and '" and xp(k) is M;p THEN x(k+ l)=A;x(k)+ A;(}X(k- 7)+ Biu(k) +ai' i= 1· "r, A

for x(k)eX;, ieI, x(k)=I/I(k), ke[-7,0], (I) nXn nxn mnxm where A·eVt B·e r • A·rd eVt 'r : J l , and a·r e VtnX I are constant matrices. x(k) e Vtn is the state vector, u(k) e Vtm is the input vector, M ip is the fuzzy set, p is the premise variable number, and r is the number of model rules. 7 is the constant time delay in the state and 7>0. I/I(k) is the initial condition of the state defined on - 7~ k ~ O. Besides, the region Xi ~ Vtn is assumed to be the n-dimensional convex polyhedron which is ~alled the cell. The set of cell indices is denoted as I and the union of all cells X = U i E jXi will be referred to as the partition. Let 10 ~ 1be the set of indices for the fuzzy rules that contain the equilibrium point and j I ~ 1be the set of indices for the fuzzy rules that do not ~ontain the equilibrium point. Here, the origin x(O)=O is the equilibrium point of the discrete time-delay affine TS fuzzy models and it is assumed that ai = 0 for i e io . Given a pair of (x(k).u(k», the final outputs of the fuzzy model (1) are inferred as follows:

Wen-Jer Chang, Wei Chang / [SA Transactions 44 (2005) 243-257

, ~ Wj[x(k)]{Ajx(k)

;=\

245

+AjaX(k- T) +B;u(k) +a;}

x(k+ 1)= - - - - - - , - - - - - - - -

~ wj[x(k)]

;=\

, =;=\ L

hj[x(k)]{A;x(k)

+AjaX(k- T)+ B;u(k) +a;},

where

fuzzy sets with the plant. It is based on the PDC concept [6-14,16-20]' Thus the TS fuzzy controller can be described as follows. Controller part: Ruleci: IF xl(k) is Mil and x2(k) is M;2 and ... and xp(k) is M,p

(3) p

w;[x(k)]=

n

Mdxi k )],

(2)

(4)

j=\

THEN u(k)= - F;x(k), i= 1· "r,

h;[x(k)]= ,w;[x(k)]

L ;=1

, for x(k)eX;,

w;[x(k)]

h;[x(k)]~O

iel.

(6)

The output of the PDC-type fuzzy controller is detennined by the summation

and

,

,

L

h;[x(k)]= 1.

u(k)=

(5)

;=1

L

;=1

h;[x(k)]{ - F;x(k)}.

(7)

Substituting Eq. (7) into Eq. (2), one can obtain the corresponding closed-loop system as follows:

For a nonlinear plant represented by Eq. (2), a fuzzy controller is designed to share the same

•

, , x(k+ 1)=

L L

;= 1 j= 1

h;[x(k)]hJx(k)]{(A;- B;Fj)x(k) + A;aX(k- T)+a;}

, , =~

~ h;[x(k)]hJx(k)]{Gijx(k)+A;aX(k- T)+a;}

;= 1 j= I

,

=~

h~[x(k)]{G/ix(k)+ AjaX(k- T)+a;}

i=1

(i= j)

+2

±±

1=

1 J= I

h;[X(k)]hJX(k)]{(Gij+Gj;)X(k)+ AiaX(k-T)+AjaX(k-T) +(a;+aj )} 2 2 2

(i
, j~1, h;[x(k)]hj[x(k)] {(G . +G . )x(k)+ A"r(k-T)+A .. r(k-T) (a.+a.)} 2 2 + T '

=;~I

I)

)1

Ill"

)
(8)

246

Wen-Jer Chang, Wei Chang lISA Transactions 44 (2005) 243-257

where Gij = A; - B;Fj . Now, the delayindependent stability conditions for the discrete time-delay affine TS fuzzy model (8) are described in the following theorem.

trices P>O, S>O, and scalars

[

Theorem 1 The discrete time-delay affine TS fuzzy model described by Eq. (8) is asymptotically stable in the large via the fuzzy controller (7) if there exist ma-

H~PHij- P+S T

PIJ,,PH,IJ,

H~Ppij T

g;jq~O

such that

1< 0

for i ,j E 10

pIJ.. ppIJ.. - S

,.

(9)

and

p

p

H~PHij- P+ s-:L g;jq T;jq

H~Ppij

HTJ,Pl'ij-:L g;jqD;jq q=1

p~PHij

P0PPij-S

.. PTp" I] '-IJ

q=1

,..

<0 for i,j E I" (10)

P

IITpp',IJ '-']

I'~PI';j -:L g;jqV;jq q=1

where Hij= (G;j+ G]I)/2, p;]=(A;d+Ajd)/2, l'ij=(a;+a)/2, Gij=Aj-BjFj E !)tn X I, and V;jq E !)t are defined such that

,

and the TjjqE!)tnxn, Djjq

for all x(k) which activates rule; (i.e., h;[x(k)]>O).

Proof' Select a discrete-type Lyapunov function as

k-I V[x(k)]=xT(k)Px(k)+

:L

m=k- T

x T(1iJ)Sx(1iJ).

(12)

By evaluating the first-forward difference of the Lyapunov function V[x(k)] along the trajectories of Eq. (8), one has

d V[x(k)]= V[x(k+ 1 )]- V[x(k)] =xT(k+ l)Px(k+ 1) - x T(k)Px(k) + xT(k)Sx(k) - xT(k- T)Sx(k- T) =

±± ±±

;= I j= I

h;[x(k)]h {X(k)]{ (Gij+Gj;')X(k) + a;+aj + A;(!X(k- T)+ Aj(!X(k- T)} TP ] 2 2 2

x

k= I

1= I

hk[X(k)]h/[X(k)]{(Gk/+Glk)X(k)+ ak+ a / + Ak(!X(k-T)+A/(!X(k-T)} 2 2 .2

- x T(k)Px(k) + xT(k)Sx(k) - xT(k- T)Sx(k- T) j

=;i jil kil IiI h;[X(k)]hJX(k)]hk[.¥(k)]h/[X(k)]{ (Gij;Gj;)X(k)+ a;;a l

247

Wen-Jer Chang, Wei Chang / [SA Transactions 44 (2005) 243-257

A;
+

2

P

2

ak+al Ak
-xT(k)Px(k)+xT(k)Sx(k)-xT(k- T)Sx(k- T) x(k) ]T[HLPHij-P+S ~ ~ ~ h;[x(k)]hJx(k)] x(k- T) pLPHij r

1=

r

I

j=

HLpp;j

[

I

1

.

PTpp Ij IJ.. -S

IL~PP;j

"TpH"Ij r'j

where Hij= (Gij+ G j ;)/2, Pij=(A;d+Ajd)/2, lLij=(a;+a)/2 and Gij=A;-B;Fj . Converting Eq. (13) to an LMI by applying the S procedure [26,27], one has r

r

[

x(k)

AV[x(k)]~ ~ ~ h;[x(k)]hJx(k)] x(k- T) 1= I j= I 1 .

]

T[ H~Pllij- P+ S p~Pllij "TplI"Ij r'j

p

<~

(14)

g;jqr;jq[x(k)],

q=1

where r;jq[x(k)] represented as

E

9t is defined in Eq. (11). Since g;jq~O and rijix(k)]~O, then Eq. (14) can be

r

AV[x(k)]~;~1

frl r

[ x(k) h;[x(k)]hJx(k)] x(k; T)

]T p

p

IILpHij - P+ S- ~

q=\

x

g;jq T;jq

HLppij-

~ ~;jqn;jq

q=\

P~PIL;j

PTPH" IJ Ij p

P

IL~Pllij- q=\ ~ g;jqnTJ'q

ILTj,PlLij-

~ ~;jqV;jq

q=\

x(k) ] X X(k;T)

[

<0.

(15)

Obviously, if p

H:j,PHij-P+S- ~ ~;jqT;jq q=1

p

HTpp Ij Ij..

p~PHij

HLPPij - ~ ~;jqn;jq q=1

pLppij p

p

p:j,PHij -

~ ~;jqnLq

q=\

"TppIj.. r'j

pLpPij -

~ ~;jqV ;jq

q=\

<0

(16)

248

Wen-Jer Chang, Wei Chang / [SA Transactions 44 (2005) 243-257

hold for all x(k)eX;, iell' then ~V[x(k)]
3. Fuzzy controller design via ILMI algorithm In this section, the ILMI [16,17,25] algorithm is used to develop a fuzzy controller design procedure for the discrete time-delay affine TS fuzzy model (8). The idea of the ILMI algorithm used in solving BMI problems is based on holding some matrix variables as a constant value and then converting it to a LMI problem. Therefore one can use the LMI toolbox of MATLAB to solve the fuzzy controller design problem. Theorem 2 The stability conditions described in Theorem 1 are satisfied if and only if there exist matrices P>O, E>O, S>O, F i , scalars ~ijq;:' 0, and a< 1 such that

1<0

H~ -E

H I).. I pI).. TE-

o

·u··-S PTE I) I)

o

A

(17)

for i,j e 10 ,

p

- aP+ S- ~ ~ijq T ijq

H~

1 HTEij Pi}

Hi}

-E

TE-IH;j Pi}

0

0 TE-I Pi}- S Pi}

0

1'"TE-I pI).. I)

q=1

o

<0

I I I .. PTEI) rl)

p

A

for i,j e II,

p

TE-IH;j - '" l: T I';j """ o"ijqn;)'q q=1

I'~E - I I'i} - ~ ~ijqV ijq q=1

ETpE-E~O

(18) where Hi) = (Gi)+ G ji )/2, Pi) = (Aid+ A jd )/2, I'i)= (ai+ aj)/2, Gi)=A i - BiFj , and the T ijq e !)lIlXII, nijq e !)lll x I, and v ijq e !)l are defined such that fijq[x(k)]=x T(k)Tijqx(k) + 2n~qx(k) + Vijq~O,

q= 1.. 'P, i= 1"'r and j= 1.. ·r.

(19)

Proof: Eq. (I8)=>Eq. (10): By Schur complement, the first inequality of Eq. (18) can be rewritten as p

H~E-IIIi}-P+S- ~ €ijqTijq q=1

p~E-IHi) P

I'~E-IHi}- ~ ~ijqn~q q=1

p

HTE-I p I).. I) TE- 1p .. p.. I)

I)

TE-I Pi}

I'ij

H~E- I I'i) - ~ ~;jqnijq q=1

S

TE- 11'" p.. I) I) P

I'~E- I I'i} - ~ ~ijqV ijq

.[(a-op o0] , 0

<

0

0

0

0 0

q=1

(20)

249

Wen-Jer Chang, Wei Chang / [SA Transactions 44 (2005) 243-257

where a<1. Multiplying this inequality from left and right by [xT(k)xT(k-r) 1] and [xT(k)xT(k - r) 1]T, respectively, and rearranging it gives

xT(k)H&E-IHijx(k) + xT(k)H&E-l p,ij+ p,&E-IHijx(k) + xT(k- r)p&E- 1pijx(k- r) + xT(k)H&E-1pijx(k- r) + xT(k- r)p&E-l Hijx(k) + p,&E-1pijx(k- r) +xT(k- r)pi;E- 1 p,ij+ p,i;E- 1 p,ij-xT(k)Px(k)+xT(k)Sx(k)-xT(k- r)Sx(k- r) p

- q=l L

gijqfijq[x(k)]«a-l)xT(k)Px(k)

(21)

or

p

+xT(k)Sx(k)-xT(k- r)Sx(k- r)-

L q=l

gijqfijq[x(k)]«a-l )xT(k)Px(k).

(22)

If there exist a matrix P>O such that ETpE- E:s;;O is held, then the following inequality is obvious:

p

+xT(k)Sx(k) - xT(k- r)Sx(k- r) -

L

q=l

gijqfijq[x(k)] < ( a-l)x T(k)Px(k).

(23)

The inequality (23) can be represented as p

p

H~.PHij- P+S- ~ gijq T ijq • q=l PTPH.· 'J 'J

HTpp .. 'J

'J

H&PP,ij - ~ gijqDijq q=l

.. PTp" 'J r'J p

P

p,i;PHij-

L

q=l

gijqn;J'q .

p,;J.Pp,ij-

L

q=l

gijqVijq

x(k) IT[(a-l)p

< x(k-r) [

0

1

(24)

0

Obviously, p

p

H&PHij-P+S- ~ gijqTijq H&P P,ij - ~ gijqDijq q=1 q=l pTPH .. P&pPjj-S P&pp,jj < 'J 'J HTpp .. 'J 'J

P

p,&PHij- q~1 gjjqni;q Since a< 1 and P>O, one has

P

p,Tpp .. 'J 'J

p,i;p P,ij - ~ gijqV ijq q=l

r °n a

-l)p

0 0

0 0

(25)

250

Wen-Jer Chang, Wei Chang / [SA Transactions 44 (2005) 243-257 p

P

H~PHij- P+ S-

L

q=!

~ijq T ijq

HTpp .. IJ IJ

HJJ,PJLij-

~ijqnijq

L

q=!

P~PJLij

P TPH" IJ IJ

p

<0.

(26)

P

JL~PHij - ~ gijqn~q

1'!.PpIJ.. '-IJ

q=1

JLTJ,p JLij -

~

q=!

gijqV ijq

The inequality (26) is equivalent to Eq. (10) and the proof is completed. Eq. (10)~Eq. (18): Now introduce a new matrix Y=P+AY, where AY is defined as (27)

where 1I·lb denotes a matrix induced two-norm and 8>0 has a small variation. Consider the following matrix: p

p

HTJ,YHij-P+S- ~

gijqTijq

q=1

PTYH.. IJ IJ

H~YPij

H~YJLij- ~ ~ijqnijq

pTYp .. -S IJ IJ

p~Y JLij

q=!

p

I,TYIIij '-iJ'

p

nr

~ <:.ijq I:. "'" ijq q=!

"TI'J'YPI'J' .-

JL~YJLij- q=! ~

gijqVijq

p

HTJ,PHij-

p

P+S- ~ ~ijqTijq q=1

p~PHij

=

H~PPij

H~PJLij- ~

q=1

P~PPij-S

gijqnijq

P TPII" IJ '-IJ

p

P

JL~PHij- q~1 gijqn~q

IITp .. '-IJ p IJ

JL~PJLij -

L

q=1

gijqV ijq

H~A YHij H~A Y Pi) H~A Y JLi)] + P~A Ylli) [

P~A YPi) P~A YJLij JL~A YHij JL~A Yllij P~A YPi) p

p HJJ,PHi) -

P+ S -

L

q=1

gijq T ijq

PTPH" IJ IJ

=

p

JL~PHij - q~! gijqn~q

IIT ppIJ.. IJ

HTpJLij-

J

L

q=1

~ijqnijq

.. PTp" IJ '-IJ

P

JL~PJLi)- q=1 L ~ijqVijq (28)

It is obvious that

Wen-Jer Chang. Wei Chang / [SA Transactions 44 (2005) 243-257 p

HiJ,PHij- P+S-

P

L

q=!

~ijqTijq

P!.PH IJ IJ..

Eq. (28):e;;

251

H!.Pp IJ IJ..

HiJ,pp,ij-

P!.pp,',-s IJ IJ ,,!.PpIJ.. r-IJ

0 0q

~ijqnijq +81.

P0 P p'ij p

p

P0PHij - q~1 gijq

L

q=!

p,TJ,pp,ij-

~

q=1

(29)

gijqVijq

Assume that there exist P>O, S>O, gijq~O, and Fi such that Eq. (10) holds. Since the inequality of Eq. (10) is strict and 8>0, one has p

P

HTJ,PHij - P+ S- ~

HTJ,pPij - ~ ~ijqOijq

gijq T ijq

q=!

q=1

PTpp IJ IJ.. -S

P0 PH ij

p !.p"" IJ r-IJ

p,TJ,PHij-

(30)

+01<0.

p

p

~

"!.PpIJ.. r-'J

gijq n 0q

q=!

P,0 P p,ij-

~

q=1

gijqVijq

Comparing Eqs. (29) and (30), one can obtain p

p

HTJ,YHij - P+ S- ~

q=!

gijq T ijq

H0 Yp,ij- ~ ~ijqnijq

HTyp .. IJ IJ

q=!

PTYII" IJ IJ

P0 Y p,ij

p

P,0 YH ij -

(31)

<0.

P

~ ~ijqnTJ'q

q=1

"Typ,IJ, r-IJ

p,TJ,Yp,ij-

~ ~ijqV;jq

q=!

By continuing, if Eq. (31) is held, there exists an a< I such that the following inequality is satisfied: p

H0 YHij-P+S-

L

q=!

p

~ijqTijq

p~YHij

HTyp" IJ IJ p!.Yp IJ IJ.. -S

H~Y p,ij-

q=!

~ijqnijq

P~YP,ij P

P

P,0 YH ij- ~ ~ijqn0q

L

q=!

p,Typ" IJ IJ

P,0 Yp,ij -

~ ~ijqV ijq q=1

[
0

0

0]

o.

0 0 (32)

Thus, if E- 1 is defined as Y, then Eq. (20) holds. As before, Eq. (20) is equivalent to Eq. (I8) by Schur complement. Since E-1=y, Y=P+~Y, and ~y~o, then P:e;;E- 1 and ETpE-E:e;;O are satisfied. The proof is completed. For the case of x(k) E Xi' where i E 1o, the proof of stability condition (17) can be directly obtained by setting the state bias term 8i =0 from 0 the proof of condition (I8). According to Theorem 2, a ILMI algorithm is used to find the feasible solutions for the stability conditions (17) and (18). The purpose of this alA

gorithm is to interactively, search for p, S, F i , ~ijq, a and to update the auxiliary variables E until a< 1. The detail of fuzzy controller design procedure is concluded as follows.

Fuzzy controller design procedure with ILMI algorithm: Step 1. Define the iterative auxiliary variables as follows:

(33)

252

Wen-Jer Chang, Wei Chang / [SA Transactions 44 (2005) 243-257

where t denotes an iteration index. Set t = I and the initial p(o) can be solved by the following Riccati equation: ATp(o)A- p(o)- (A TP(o)B)(1

+ BTp(o)B) -, (BTp(o)A) + Q= 0, (34)

4. A numerical example Consider a computer simulated truck-trailer system as shown in Fig. 1. The dynamic equations of a nonlinear truck-trailer system have been developed in Ref. [20]. It has the following form: vt) vt xl(k+ 1)= ( 1- Y xl(k)+ Tu(k),

(35a)

where

I

r

I

r

A=-~ Ai, B=-~ Bi and Q>O. ri=1

vt x2(k+ 1)= YXI(k)+X2(k),

(35b)

ri=1

The matrix Q is assigned by the designers. Then start the ILMI algorithm. Step 2. Using the auxiliary variables E(t) = PI to solve the optimization problem for (I-I) p(t), S(t), Fi(l) , and gijq(l) from E~s. (17) and (18) subject to minimizing a(t). That IS, Minimize act). Subject to p(t»O, S(t»O, E~)~(t)E(t) - E(t) ~ 0, and gii..q(l) ~ 0, Eq. (I7) for i e 10 , and Eq. (I8)foriell' If act) < 1, then P(I)' S(t), Fi(t), and gijq(l) obtained in Step 2 are the feasible solutions for the conditions (I 7), (I8) and thus stop the iterative manner; otherwise, go to Step 3. Step 3. Resolve the following optimization problem for P(t)' S(t), Filt), and gijq(l) using aft) determined in Step 2 and the auxiliary variables E(t) obtained in Step 2. Minimize trace(P(t». Subject to p(t»0, S(I»O, E~)Ijt)E(I)-E(t) ~ 0, and g;{q(t) ~ 0, Eq. (17) for i e 10 , and Eq. (I8)foriell' If II E u )' - p(1)11 < 'Y, where 'Y is a predetermined small value, then the conditions (17), (18) are not feasible and then stop the iterative manner. Otherwise, one set t = t + 1 and go back to Step 2 to. update the auxiliary variables E(I) using Pu~,)' where Pu ~ I) is determined in Step 3. 0 To carry on, a numerical example is presented to show the usefulness of the above fuzzy controller design procedure for the discrete time-delay affine TS fuzzy models. Through the controller design procedure and LMI toolbox of MATLAB, one can find suitable matrices P>O, E>O, S>O, F;, scalars gijq ~ 0, and a< 1 to satisfy conditions (I7) • and (18).

x3(k+ 1) =x3(k) + vt sin[x2(k) + (vtI2L)x, (k)], (35c)

where xl(k) is the angle difference between truck and trailer [note that x I (k) corresponds to two "jackknife" positions, 90° and -90°]; x2(k) is the angle of trailer; x 3(k) is the vertical position of the rear end of trailer. The sampling period t = 2 sec and the length of truck 1= 2.8 m, the length of trailer L = 5.5 m, the constant speed of the backward movement v = - 1 rn/ sec. Considering the time delay effects in this system, it is assumed that the angle x2(k) of the trailer is perturbed by time delay and the modified nonlinear truck-trailer system is given as

vt x2(k+ 1)= YXI(k)+ cp(k),

(36b)

x3(k + 1 ) =x3(k) + vt sin[ cp(k) + (vtI2L)x, (k)], (36c) cp(k)=PX2(k)+(1-P)x2(k-r),

(36d)

where cp(k) is a time-delay weighting function and P E [0, 1] is the weighting coefficient. The limits 1 and of p are corresponding to no delay term and to a completed delay term, respectively. For example, when p=1 then CP(k)=X2(k), i.e., the time delay term x2(k- r) is not considered. In this example, we assume that' p=0.7. Applying the above system parameters, Eq. (36) can be thus represented as

°

xl(k+ 1)= 1.3636x,(k)-0.7143u(k),

(37a)

Wen -Jer Chang, Wei Chang / [SA Transa ctions 44 (2005) 243-257

253

. '(Ic)

I.

L

o

Desired Path

RukZ

Rule I:

90' 20 I O' O' 0 I 0)

(x'. x; . •' )..., = (88" Rule 2:

(x. x, • • )...., = (0' 0' 0 I 0' 0' 0 10) Rule 3:

-90' -20 I O' 0' 0 I 0)

(x-. x; . •-)_ = (- 88"

Fig. I. A truck-traile r mode l.

x2( k + L)= 0.3636x I(k)+ 0.7x 2( k ) + 0.3X2(k - r),

(3 7b)

x3( k + I ) = x3( k ) - 2 sin[0.7x 2( k) + 0. 3X2(k - r )

- 1.8182x I (k)].

(37c)

To obtain the di screte time-delay affin e TS fu zzy model of the truck-trail er ystem (37), it is necessary to apply the lineari zation method. Let us choose three operating points:

operating points, in which (X ,Xd, U)oper2 i an equi librium point and the others are the offequilibrium points; the coordinate positions about them is shown in Fig. I . Through the above three linear subsystem and definin g membership functions as in Fig. 2, then one can obtain the di screte time-delay affin e TS fuzzy model, which is composed by three rules as follows. Plant part: Rul e p 1: IF X I (k) is M I I THEN

(x +,x: ,U+)oper l = ( 88 0 90 0 2010 0 0 0 010 ), (38a) (X ,Xd, U)oper2=( 0° 0 0 0 10

0

0

0

010 ), (38b)

M 31

and

= ( - 88 0 - 90 0 - 2010 0 0 ° 0 10 ),

(38c)

to construct three linear subsy tern for these three

11 (k)

o· Fig. 2. Me mbershi p functi o ns of x I (k).

Wen-Jer Chang. Wei Chang / [SA Transactions 44 (2005) 243-257

254

X(k+l)=[~:':!:6 0~7 ~]X(k) 0.2480

- 0.9548

1

+[O~ 0~3 ~o]X(k-T)

_[ -

-0.4092

+[ -0.~143]U(k)+[ o

o

o

0]0 ,

o

0

~ X[(60+90)1T1180J] 0

"111-

'

and

o

~

]. VIII

-0.3438

= (601T/180) x (901T/lS0).

(40)

(39a)

1 0 0] [

T 331 = 0 0 0 , 000

o

1.3636 x(k+ 1)= -0.3636 [ 0.3636

0.7 -1.4

o

0]

_[ -

~ X[( -60-90)1TI1S0J] 0

"331-

0.3 0 X(k-T) -0.6 0

o V 33\

0

1.3636 x(k+ 1)= -0.3636 [ 0.2480

+

0.7 - 0.9548

[~o 0~3 ~O]

x(k- T)

-0.4092

+[-0~1431U(k)+[ ~ 1 o

and

0.3438 (39c)

and the S procedure is presented by the following matrices. For Rule 11, 60°:OS;;X1(k):os;;90° which means

= (- 601T/lS0) x (- 901TI1S0),

(41)

where the notation Rule ij means the correlation between Ru Ie p i and Ru Ie p j of the plant part, e.g., Rule 11 means the S procedure to the range of Rule p 1 and Rule p 1 for the plant part. For the fuzzy controller design of truck-trailer system (37), the control purpose is to realize the backward movement of the truck-trailer system along the "desired path" (X3=0), i.e., x1(k) -+0, x2(k)-+0, and x3(k)-+0 without forward movements. Through the above discrete timedelay affine TS fuzzy model (39), the fuzzy controller can be designed by applying Theorem 2 and the fuzzy controller design procedure developed in Section 3. First, in Step 1 of the controller design procedure, let us assign the matrix Q as follows:

1 01 00] . [001

Q= 0

(42)

Then, the matrix Pro) can be obtained by solving Riccati equation (34),

Wen-Jer Chang, Wei Chang / [SA Transactions 44 (2005) 243-257 2~--~--~--~----~--~--~

255

new matrix P(1) from Step 3 and then substitute it into Step 1 to produce a new auxiliary parameter E(2). Repeat the fuzzy controller design procedure until a< 1. In this example, we get a feasible solution after four iterations of the fuzzy controller design procedure. The final a is 0.9531 and the feasible solution is obtained as follows:

0.9842 P= -2.2996 [ 0.6882

- 2.2996 8.2062 - 2.1495

0.6882] - 2.1495 , 0.9647 (45)

-20:------=:2O:----.co::=-----=eo!:----=eo----::,OO=----::!,:2O limlClle)

Fig. 3. Responses of the angle difference between truck and trailer x.(k).

8.2341 p(o)

= [ - 8.0758 3.1928

- 8.0758 15.432 - 6.3765

3.1928] - 6.3765 .

E()

[

0.6856

0.6856 -0.6681

0.4455

0.0230

s= -0.0265 [

4.3389 (43)

Thus the matrix tion (33),

E=

3.0693

0.0136

0.5024 -0.0265 1.9431 -0.0269

-0.6681] 0.5024 , (46) 2.6304 0.0136] -0.0269 , 0.0108 (47)

can be obtained from defini~)))=0.9489, ~33J=0.9489.

(48)

0.2503 0.1397 0.0212] E( 1) = PiOl = 0.1397 0.2430 0.2543 . [ 0.0212 0.2543 0.5886 (44)

Thus the PDe based fuzzy controller has the following form. Controller part: Rule c 1: IF x)(k) is Mil

Next, using the initial auxiliary parameter (44), the minimum a can be obtained in Step 2. If a< 1, then one can obtain the feasible solution for system (39). Otherwise, if a~ 1, one needs to get a

THEN u(k)=[3.4769 -4.4192 1.3954]x(k). (49a)

THEN u(k)=[4.2026 -5.5019 1.5527]x(k). (49b)

THEN u(k)=[3.4769 -4.4192 1.3954]x(k). (49c)

.to~--""':20:----.co.,&,.---..,.eo!:----=I11:-----:t~OO-----,-!,:20 limI CIIC)

Fig. 4. Response of the angle of trailer x2(k).

The simulation results are stated in Figs. 3-5 with the initial conditions x( 0) = [85° 90° 8 ]T, I/I( k) = 0 for - T=S:: k < O. The sampling time of the simulation is 2 sec. The time-delay term is assumed as T=2 in this simulation. From these simulated responses, one can find that the controlled modified truck-trailer model (37), which is driven by the fuzzy controller (49), is asymptotically stable.

Wen-Jer Chang. Wei Chang / [SA Transactions 44 (2005) 243-257

256

10~--~-----r----~----~--~----~

~o~--~~~---~~--~m~---ooL----l~OO~--~I~ tim, (lie)

Fig. 5. Response of the vertical position of rear end of trailer x3(k).

5. Conclusions In this paper, the Lyapunov criterion was applied to analyze the stability and stabilization problems for the discrete time-delay affine TS fuzzy models. Through variable-hold and S procedure, the stability condition has recast into an LMI problem from the BMI problem. Then, the fuzzy controller design procedure has been developed to implement a fuzzy controller for discrete timedelay affine TS fuzzy models. The main contribution of this paper is to extend the stability analysis and stabilization issues of discrete time-delay homogeneous TS fuzzy models to discrete timedelay affine TS fuzzy models.

Acknowledgments The authors wish to express their sincere gratitude to three anonymous reviewers who gave them some constructive comments, criticisms, and suggestions. This work was supported by the National Science Council of the Republic of China, under . Contract No. NSC93-221S-E-OI9-002.

References [1] Cao, Y. Y. and Frank, P. M., Stability analysis and synthesis of nonlinear time-delay systems via linear Takagi-Sugeno fuzzy models. Fuzzy Sets Syst. 124, 213-229 (200t). [2] Gu, Y., Wang, H. 0., and Tanaka, K., Fuzzy Control of Nonlinear Time-Delay System~: Stability and Design Issues. Proc. American Control Conf., Arlington, 6, ' 4771-4776 (200t).

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backing up control of a truck-trailer. IEEE Trans. Fu zzy SySI. 2, 119- 134 ( 1994). [2 1] Maun, M. Z. and Mohammad. J., Time-Delay Sy tems Ana lysis, Optimization and Applications. NorthHolland, New York, 1987. [22] Cao, Y. Y. , Sun , Y. X. , and Cheng, ., Delaydependent robust stabili zation of' uncertain systems with multiple tate delays. IEEE Trans. Autom. Control 43, 1608- 161 2 ( 1998). [23] Iva ne cu. D., Dion. J. M., Dugard , L. , and iculescu, . I.. Dynamic compensation for time-delay . ys tems: An LMI approac h. Inl. J. Robust Nonlinear Onlrol 10. 61 1- 628 (2000). [24] Tarbouriech, S., Peres. P. L. D., Garcia. G., and Queinnec, I., Delay-dependent stabili zation and disturbance tolerance for time-delay systems subject to actuator saturation . lEE Proc.- D: Control Theory App!. 149, 387- 393 (2002) . [25] Cao, Y. Y., Lam. J., and Sun , Y. X.. Static output feedback tabilization: An fLMI approach. utomatica 34, 1641 - 1645 (1998). [26] Has ibi , A. and Boyd, S., Quadratic Stabilization and Control of Piecewise- Linear Systems. Proc. of the American COnlrol Conf., Philadelphi a. Pennsylva ni a. 6, 3659- 3664 (1998). [27] Boyd. ., Ghaoui , L. EI , Feron, E. , and Bal akri hnan, Y., Linear Matrix Inequalities in System and ontrol Theory. SIAM , Philadelphia, 1994.

257

Wen-Jer Chang received the B.S. degree from ati onal Taiwan Ocean niver ity. Taiwan. R.O.C. in 1986. and the M.S. degree and the Ph.D. degree from the Nati onal Central University. Taiwa n. R.O. .. in 1990 and 1995. respec tivel y. He is currently a professor with the Depanment of Marine Engineeri ng of the ational Tai\ an Ocean Uni\'cr~i t y. Taiwan. R.O.C. He is now a member of Ihe CIEE. AC . CSFAT. and SAME. In 2003, Dr. Chan g was listed in the Marquis Who' Who in Science and Engineering of 2003-2()()4. and won the o UI ~ tand i n g young control engi neers award granted by the hinese Automation ontrol Society (CAC ). Hi recent research intereSL~ are fu zzy cOl1lrol. robust cOl1lrol. and perfonnance constrained control.

Wei Chang was born on February 6. 198 1 in Taiwan. R.O. . He received the B.S. degree from the Depanment of Mari ne Engineering of the National Taiwan ean ni versity. Taiwan. R.O.C. in 2003. In the ame depanment. he is currentl y working loward Ihe M.S. degree. His research interest focus on fuzzy control and dynamic system analysis.