Polytopic TSK Fuzzy Systems Analysis and Synthesis

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Polytopic TSK Fuzzy Systems: Analysis and SynthesisDiinitar P. FilevFord Motor Company24500 Glendale AV. . Detroit. MI 48239, USAe-mail: dfilev ord.com

AbstractIn this paper we discuss the problem of analvsis andsynthesis of TSK systems by using a polytopicrepresentation of the TSK fuzzy model. This allows usto derive the necessary and sufficient conditions fo rsynthesis of polytopic TSK com pensators . The closedloop TSK system under a polytopic TSK compensator ischaracterized w ith a stable characteristic polynomial that

s not dependent on the firing levels of the rules.Procedures fo r calculation of the compensators are alsopresented.

1 IntroductionMost of the theoretical and applied research on the so-called Takagi-Sugeno Kang (TSK) fuzzy systems [ ] isrelated to state space based methods for analysis andsynthesis. Following the pioneering work of Sugeno andTanaka [2], the problem of stability of TSK systems wasformulated and sufficient stability conditions were derivedby using Lyapun ov theorem. Lyapunov based approachwas applied to the analysis of fuzzy and neural systemsand to the design of state feedback compensators [3].Above mentioned results, however, used the state spacerepresentation of the TSK model carrying all advantages

and drawbacks of the state space technics.In this paper we discuss the issue of stabilization of aTSK fuzzy system based on its alternative polytopicinput-output representation [4, 51. We develo p apolynomial approach to the synthesis of stable TSKcontrol systems and derive t h e necessary and sufficientcondition for stabilization of a wide class of TSK fuzzysystems.

2 Necessary Conditions for Stability ofPolytopic TSK Fuz zy SystemsA dynamic SISO TSK fumy system P is described by arule-base of the following format:IF y(k-I) is Ail AND ... AND y(k-n) is Ain

T H E Ny(k) = bOu(k) ...+bnu(k-n) - a i ly ( k - l )- ... - aiiny(k-n), i = [ I , m] (1)

Reference fuzzy sets Ail 4 2 , ...,Ain represent llinguisticlabels defined over the output space; effectively, theselinguistic lables de termine a fuzzy partitioning of the statespace into rectuangular fuzzy regions Ri = Ail x Ai2 x ... xA i n , i = [ I m]. The output y(k) of the overall model isinferred from the individual linear subsy stems acco rding tothe TSK reasoning mechanism:y(k) = bo u(k) + ... + bn u(k-n)

mi = 1

- vi [ai1 y(k-I) - ... - ain y(k-n)]m

j=where the weights vi = 1ci /c j , i = [ l ,m] a r e th enormalized degrees of firing of the rules t i = Ai ~ ( y ( k - I )... A Ain(y(k-n)) [6]. An alternative representation [4, 51 ofthe system P and the compensator C is provided (for zeroinitial conditions) by the piolytopic (nonlinear) counterpartof (2):

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where vector v = [ v l v 2 ... vm]' is formed by the theweights vi's; the vi's are nonlinear functions of the pastoutput values y(k-I), y(k-2), ..., through the degrees offiring, the ri 's. We denote by V the set of all possibleweight vectors v asso ciated with a given TSK model; theelements vi's of a vector v E V are nonnegative and sumto one, i.e. vi 2 0 and C v i = 1 . We call (3) thepolytopic TSK model.

mi =

I t is easy to verify from expression (3) that the polytopicTSK model is a ratio of the numerator polynomial B(z)and the de nominator polytope A(z, v); polytope A(z,v) is aconvex sum of denom inator polynomials

. .m m= zn+ C v i a i l z n - l + ...+ C vi ai,;i= i = l

it is parame terized by the weights, the vi's.

In the following section we discuss the necessaryconditions for the stability of the TSK systems representedby the polytop ic model (3).An evident necessary condition for the stability of apolytopic TSK fuzzy system is the condition for itsstability in any region of the state space. This necessarycondition translates into a requirem ent for stability of thecharacteristic polytope A(z, v) for any vector v. Wedistinguish tw o main cases:i. State vector [y(k) y(k-I) ... y(k-n)]' resides in one of theregions Ri = A i l x Ai2 x ... x Ain, i = [ I , m]; in otherwords this is the special case vi = 1 and v. = O,j# i, i, j =J[I ;i i . State vector [y(k) y(k- I) ... y(k-n)]' resides in more thanone region Ri, i = [ I , m]; in other words this is thegeneral case where v =[vI v2 ... v,]', v E V, i.e. v is anyvector with non negative elemen ts that sum to one.

For the special case vi = I and v. = 0, j#i , i , j = [ I ,m]stability of the TSK fuzzy system is implied by the stabilityof the linear subsystem s. This case reflects situations inwhich the TSK system dynamics IS identical to thedynam ics of the i-th linear subsystem. Stability of theselinear subsystems is defined by the stability of de nominatopolynomials Ai(z), i=(l, m) and can be analyzed based onthe well-known criteria for stability of linear systems, e.gthe Schur criterion.

J

To deal with the general case we apply the ed ge theorem [7]According to it the zeroes of a polytope of po lynomials

lie in the open unit disc if and only if the edges eij(z) opolyto pe A(z, v):Ei,(z) = t Ai(z) + (1 - t) A,(z), i, j = [ 1, m]. t E [O, I ] (4)are Schur stable. Thus the conditions for stability ocharacteristic polytope A(z, v) can be transformed toconditions for stability of edges E..(z), i, j = [ 1, m]. A neasy check for stability of the convex edge polynomialsE..( z)'s follows from the Schur-Coh n stability criterion [8]stability of a pair of convex monic polynomials Ai(z) andA.(z) is implied by the lack of negative real eigenvalues othe matrix T(A i) T- (A.), i, j = [ I m], where

1J

J

1 I JI a1 I 1, 11-3 i i . n - 2 - a n0 1 a, n i i . n - 3 4 1 n- I

0 *I l*t4 at I*I 3-3n t ,n-1 -dl 3 la12

'(Ai

We summarize above discussion in the following theoremcovering the necessary conditions for the stability of apolytopic TSK fuzzy system.Theorem 1. Polytopic TSK system ( 3 ) is stable if lineasubsystem polynomials Ai(z ) are stable and the matriceT(Ai) T-I(A,), i, j = [ I , m] have n o negative eigenvalues.The exam ple bellow illustrates the application of T heoremfor analysing necessary conditions for stability of apolytopic TSK system.

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Example 1. Consider the TSK fuzzy system:

IF y(k-2) is . k- ) TH EN-13 a

y(k)=U(k) -1.32 y(k-1)-1.22 y(k-2) -0.44y(k-3)The nonlinear transfer function associated with this modelis:

Denominator polytope A(z, v) is formed by the convexsum of the stable polynomials:AI(z) = z3 - I .29 z2 + 1 . I9 z 0.41A2(z) = z3 + 1.32 z2 + 1.22 z + 0.44

Further we check the conditions for stability of thepolytope A(z, v). Matrices T(A I ) and T(A2) are:T ( A ~ ) = [ -0.88 w2)=[ 0.88

0.41 -0.19 -0.44 -0.22We form the matrix product:

-3.54 -10.19-1 -3.16T(A,) T4(A2) =[

Its eigenvalues are I = -0 .152 and h2 = -6.533;henceforth the polytope A(z, v) i s unstable and so is theTSK syste m. We can easily verify that for different valuesof parameter a this TSK system is unstable.If we co nsid er the vi's as independe nt uncertainties (as theyare considered in robust systems [9] then the necessaryconditions of Theorem I are sufficient conditions as well.However, as we pointed out above, the weights vi's arenonlinear functions of the state variables rather thanindependent parameters. On e important consequence of

Theorem 1 is that if the necessary cond itions are satisfiedand the state vector resides in a certain region of the statespace then the polytopic TSK system will be s table.Therefore, if we design a feedback that w ould force the statevector to remain in a certain region of the state space thenthe closed loop system will be stable. We discuss this issuein the next sections.3 Necessary and Sufficient Conditions forSynthesis of Polytopic TSK CompensatorsOur goal is to design a compensator C that would stabilizean object P described by a T SK fuzzy model. Due o thespecific structure of the plant model it is reasonable toconsider same type structure of the compen sator (same fuzzypartitioning of the antecedents):IF y(k-I) is A i l AND ... AND y(k-n) is Ain THIEN

u(k) = siOe(k)+ ... +si,e(k-c)- r i lu (k- l ) - ... - ricu(k-c), i=[l,nn] (6)

where e(k) = w(k) - y(k) arid w(k) is the desired output attime k. The output of the TSK compensator is:m

i =u(k) = vi [sioe(k) 4- ... +sice(k-n)- r i lu(k-l) - ... - qcu(k-c)] 7)The polytopic TSK compensator C is defined as a ratio oftwo polytopes:

We note that rules in both models have same antecedents,i.e. we deal with same degrees of firing ri's and weights vi'sin expres sions (1) - (3) and (6) - (8).For a particular vector v V , the polytopes A z, v), S z, v)and R (z, v) transforms into polynomialsAV@)= z n + c a i (v ) z i, S ,,(z) = c si(v) zi and RJz) =

ri(v) zi. For a certain condition given by the followingi=OLemma 1 we can calculate polynomials Sv(z ) and I (z)hatassign a polynomial D (z) to the closed loop system for any

n Ci=O i =OC

V .

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a 1 . Given a plant P with a TSK model 1 ) and adesired characteristic polynomial D(z). There exists a TSKcompensator that assigns the characteristic polynomialD(z) to the close d loop system if and only if the num eratorpolynomial B(z) and the denominator polytope A(z, v) ofthe TSK polytopic representation (3) are coprime for anyweight vector v E V. For any v E V polynomials S v(z)and Rv(z) of the compensator C, are the solutions of theDiop hantin e equation:

IF y(k-2) is . L Y k - Z ) T1 - 1

y(k)=l Ou(k-l)-u k-2)+y k-l)-OSy k-2)and a desired characteristic polynomialD(z) = z3+ 0.29 z2 0.0044 z - 0.0027.The polytopic TSK model of this plant is:

Proo f Follows from the known algebraic poleassignment theorem of linear control [4].In the above lemma we assumed that B(z) and A(z, v) arecoprime for any v E V . While condition of coprimenessof numerator and denominator polynomials in linearsystems is equivalent to controllability and observabilityof the system, in the case of a polytopic model it has adifferent formulation. The necessary and sufficientcondition for coprimeness of polynomial B(z) andpolytope A z, v) is given by the following theorem.Theorem 2. Let z b is the set of zeroes of numeratorpolynomial B(z). Polynomial B(z) and polytope A(z, v)= v i Ai(z) are coprime for any v E V if and only ifAi(z0), i = [ 1, m] are of a sam e sign for any zo E z b .Proof B(z) and A(z, v) have a common factor zo if andonly if B(z0) = 0 and A(zo,v) = 0 for some v E V. ButA(zo, v) = 2 vi Ai(z0) = 0 for some v E V if and only if0 [min(Ai(zO)),max(Ai(zo))], because of vi L 0 and

mi= 1

mi = 1

mi = 1c vi = 1 .Example 2. Consider a fuzzy system described by thefollowing TSK model:

T H E N- 1 1y(k)=lOu(k- I)-u(k-2)-y(k- I)-0.5y(k-2)

I O Z -V I (22 + z + 0.5 + ~2 (z2 - z + 0.5P(z, v) =

I O Z - 1z.2 + (vi- v2) z + 0.5-

We are looking for a polytopic TSK com pensator C by (8that assigns characteristic polynomial D(z) to the closeloop system. To apply Lemma 1 we first check thcoprimeness of polynomials B(z) = 10 z - 1 and A(z, v) =z2 + (V I- 2) z + 0.5. The only zero of B(z) is in zo = 0.1By substituting in A(z, v) we calcu late A (zo) = 0.6 anA ~ ( z o ) 0.41. According to Theorem 2 B(z) and A(z, vare coprime for any v E V. Therefore, we can calculateS,(z) and Rv(z) from Diophantine equation (9) for anweight vector v E V:(z2 + (V I- 2) z + 0.5)R(z, v) + (10 z - 1 ) S z, v) =

z3+ 0.29 z + 0.0044 z - 0.0027lmpulse responses of the plant and the closed loop systemare in Fig. la and Ib. Five consecutive values of thantecedent state variable y(k-2), vector v, and respectivepolynomials S z, v ) and R(z, v) are listed in Table 1.k y ( k - 2 ) ~ 1 ) 2 ) S, Z) V Z)4 0.22 0.31 0.69 0.0773 z - 0.0456 z - 0.09655 -0.06 0.80 0.20 -0.0221 z - 0.0459 z - 0.09716 0.015 0.39 0.61 0.0607 z - 0.0456 z - 0.09667 -0.003 0 53 0.47 0.0327 z - 0.0457 z - 0.09688 0.001 0.49 0 51 0.0402 z - 0.0457 z - 0.0968Table I . Five consecutive values of y(k-2), v, anpolynomials Sv(z) and R v(z).Comment. In Lemma we consid ered stabilization of theTSK system by assigning a des ired characteristic polynomiaof the closed loop system . Weak form s of this Lemma and

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Theorem 2 can be obtained if we exclude from the set z b(Theorem 2) all stable zeroes o f B (z) and include them inthe desired polynomial D(z).

a.I dI

b.ll

0 iFig. I . Impulse responses of the plant (a)and the closed loop TSK system (b) - Example 2

4 Analytic Form of the Polytopic TSKCompensatorLemma I does not provide an analytic form of thenonparametric TSK compensatorC hat assigns a desiredcharacteristic polynomial to the closed loop system; inaddition, the TSK com pens ator calculated accordingly maybe nonproper. For the more realistic case of a strictlyproper TSK polytopic model, however , we cananalytically specify the polytopes of a proper TSKcompensator that assigns a desired charactcr ist icpolynomial D(z) to the closed loop system.Lemma 2. Given a strictly proper po lytopic TSK model(3), i.e. bo = 0, with coprime B(z) and A(z, v). If desiredcharacteristic polynomial of the closed loop system is oforder 2n-I, i.e. D(z) = do z2n- ' + dl z2n-2 + ... d 2 n - l ,then there exists a proper polytopic TSK co mpensator oforder c = n - 1 . For any v E V the coefficients ofpolynomials Sv(z) and R v(z), of he TSK ompensatorare determined from the solution

of the matrix equation:_S C = D '

where joint vector contains the coefficients of thecom pen sator numerator and denom inator polytopes S,(z) andRv(z), and vector D is formed by the coefficients of thepolynomial D(z):

-C = [SO(V)D = [ I d l d2 ...d2n-l] ' ,(v) *..Sn- 1 (v) ro(v) 1 (v) ... rn- 1 )I

and matrix S is the modified Sylvester matrix of thepolytopic TSK model (3):

l0 . 0 1 0 .bl 0 . 091 1 . 0. bl . 0 . a i l . 0m 12)n . . 0 ain . 0s c i O b n . 0 09 , . Ii = 1 0 0 . bl 0 0

0 0 b n O O .ai,,ProoJ: The order (211-1) of desired polynomial D(z) andrequirement for a strictly proper polytopic TSK system Pfollow from the consideration of a proper TSK compen sator.Equation ( 1 1) can be easily derived by com bining the term swith equal powers of z in the Diophantine equation (9). Theinversion of modified Sylvester matrix S exists ffor anyweight vector v due to the assumed cop rimeness of B(z) andA z, v). The necessary arid sufficient conditions for thecopr ime ness of B(z) and A(x, v) are given by Theorem 2.By applying the abov e result to Example 2we calculate a nanalytic exp ression of the coefficients of polynomials Sv(z)and Rv(z). According to Lemma 2 the joint v ectorC = [so(v) s I (v) ro(v) rl(v)] isdetermined by the solution

1 0) of the matrix equation ( I 1 ) where s s the modifiedSylvester matrix of the polytopic TSK m odel:0 0 1 0 O O I OI IO 0.5 I 1 IO 0.5 I0 -1 0 0.5 0 1 0 0.5 -0.0027

C =

The next Example 3 provides an illustration of the procedurefor assigning a characteristiic polynomial to the closed loopTSK system.

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Example 3. Given a TSK fuzzy system:

IF y(k-3) is h y ( t - 3 ) H E N-ay(k)=u(k- 1 -0.25~(k-3)+3.37y (k- )-8.42y(k-2)+5.44y(k-3)

IF y(k-3) is -a ay(k)=u(k- 1)-0.25u(k-3)+4.1y(k- 1)-0.25~ k-2)+0.98~ k-3)The goal is to stabilize this system by assigning a stablepolynomial to the closed loop system.The polytopic TSK model of the system is:

z2 - 0.25P(z,v) = V I (23 - 3.37 Z2+ 8.42z - 5.4375) ~2 (23 4.1 z2t 0.25z - 0.98)In order to apply Lemma 1 or Lemma 2 we check thecoprimeness of the num erator polynomial

B z)= z2 - 0.25and denom inator polytopeA(z, v) = V I (z3 - 3.37 z2+ 8.42 z - 5.4375)+ v2 (z3 - 4.1 z2+ 0.25 z - 0.98)by using Theorem 2. The set of zeroes of the numeratorpolynomial is z b = [-0.5, O S ] . By substituting for z intothe polynomials AI = z3 - 3.37 z2+ 8.42 z - 5.4375 andA2(z) = z3 - 4.1 z2+ 0.25 z - 0.98 we calculate: AI(-0.5)= -10.6175 and A1(0.5) = -1.9425; A2(-0.5) = -2.255 andA2(0.5) = - 1.755. There fore, accord ing to Theorem 2 thenumerator and denom inator of P(z, v) are coprime for anyvector v E V. P(z, v) is strictly proper of order n=3 andby Lemm a 2 there exists a second order compensatorassigning a polynomial of order 5. We chose desiredcharacteristic polynomial of the closed loop system to bethe stable polynomial:D(z) = z5- 1.03 z4+ 0.03 13 z3 + 0. I397 z2

+ 0.002 z - 0.001.By substituting in I O ) we obtain an analytic expressionfor the joint vector C containing the compensatornumerator and denominator polynomials S v(z) and Rv(z):

-

where

s = V I

+ v2

0 0 0 1 0 01 0 0 -3.37 I 00 0 8.42 -3.37 1

0.25 0 I 5 .44 8.42 3.370 0.25 0 0 5 .44 8.42

- 0 0 0.25 0 0 -5.44

0 0 0 1 0 01 0 0 -4.1 1 00 1 0 0.25 -4.1 1

0.25 0 1 -0.98 0.25 -4.10 0.25 0 0 -0.98 0.25

- 0 0 0.25 0 0 -0.98

In Fig 2a and b are given the impu lse responsesof the plantand the closed loop system.

x 10 a. a 4 . 56r I

b. a.1.5

20 50 100

Fig. 2. Impulse responses of the plant a)and the closed loop TSK system (b) - Example 3.

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Theorem 2, Lemma I and Lemma 2 allow us to designnonpara metric TSK feedback compens ators that assign adesired polynomial to the closed loop system. This typeof control is characterized with compensator parametersthat vary with vector v. In essence, the compensator hasthe polytopic form (8)5 ConclusionNecessary and sufficient conditions for stabilizing TSKfuzzy systems by eliminating the nonlinearity of theclosed loop system were derived based on the polytopicrepresentation of the TSK model and procedures fordesigning pol ytopic TSK comp ensators were developed.References[ I ] Takagi, T. and Sugeno. M.. Fuzzy identification ofsystems and its application to modeling and control. IEEETransactions on Systems. Man and Cybernetics 15, 16-132,1985.[2] Tanaka, K. and Sugeno. M., Stability analysis of fuzzysystems using Lyapunov's direct method, Proc. of NAFIPS'90,133-136, 1990

[4] Filev, D., Control 01 nonlinear systems described byquasilinear fuzzy models . In: Lecture Notes in Control andInformation Sciences, v o l . 143. System Modelling andOptimization. Eds. H.-J. Sebastian and K. Tamer, Springer-Verlag, p. 591-598, 1989[ 5 ] Filev, D., Fuzzy modelling of complex systems, Int. J. ofApproximate Reasoning 4, 28 1-290, 1991[6] Yager, R. and Filev, D., E s s e n t l a l s o d e l i n g andControl , John Wiley Sorbs, New York, 1994[7] Bartlet, A.C., Hollot, C.V., and Huang, L., Root locationsof an entire polytope of polynomials: it suffices to check theedges , Math. Control Signals Systems I , 67-71, 1988[SI Ackermann, J.E. and Barmish, B.R., Robust Schurstability of a polytope of polynomials . IEEE Trans . onAutomatic Control, vol.AC-33. 984-986, 1988[9] Barmish, B.R.,Svstems , Macmillan Publ. Co., New York, 1994UNools for robustness of linear

[3] Tanaka. K. and Sano, M., A robust stabilization problemof fuzzy control systems and its application to backing upcontrol of a truck-trailer , IEEE Trans. Fuzzy Systems, v01.2,119-134, 1994

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Polytopic TSK Fuzzy Systems Analysis and Synthesis