Nonquadratic Lyapunov Functions For Nonlinear

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(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 7, No. 2, 2016Nonquadratic Lyapunov Functions for NonlinearTakagi-Sugeno Discrete Time Uncertain SystemsAnalysis and ControlAli BouyahyaYassine ManaiJoseph HaggègeUniversity of Tuins El Manar,National Engineering School ofTunis, Laboratory of research inAutomatic Control, BP 37,Belvédère, 1002Tunis, TunisiaUniversity of Tuins El Manar,National Engineering School ofTunis, Laboratory of research inAutomatic Control, BP 37,Belvédère, 1002Tunis TunisiaUniversity of Tuins El Manar,Departement of ElectricalEngineering, National EngineeringSchool of Tunis, Laboratory ofresearch in Automatic Control, BP37, Belvédère, 1002Tunis, TunisiaAbstract—This paper deals with the analysis and design of thestate feedback fuzzy controller for a class of discrete time Takagi-Sugeno (T-S) fuzzy uncertain systems. The adopted frameworkis based on the Lyapunov theory and uses the linear matrixinequality (LMI) formalism. The main goal is to reduce theconservatism of the stabilization conditions using some particularLyapunov functions. Four nonquadratic Lyapunov Functions areused in this paper. These Lyapunov functions represent anextention from two Lyapunov functions existing in the literature.Their influence in the stabilization region (feasible area ofstabilization) is shown through examples, the stabilizationconditions of controller for discrete time T-S parametricuncertain systems is demonstrated with the variation of thelyapunov functions between (k, k 1) and (k, k t) sample times.The controller gain can be obtained via solving several linearmatrix inequalities (LMIs). Through the examples andsimulations, we demonstrate their uses and their robustness.Comparative study verifies the effectiveness of the proposedmethods.Keywords—Nonquadratic Lyapunov functions; Non-PDC;Linear Matrix Inequality; Parametric Uncertain Systems; TakagiSugenoI.INTRODUCTIONFuzzy control systems have experienced a big growth ofindustrial applications in the recent decades, because of theirreliability and effectiveness.In recent years, there has been growing interest in thestudy of stability and stabilization of Takagi–Sugeno (T–S)fuzzy system[1, 2, 3,4, 5] due to the fact that it provides ageneral framework to represent a nonlinear plant by using aset of local linear models which are smoothly connectedthrough nonlinear fuzzy membership functions.Nonlinear systems are difficult to describe. Takagi-Sugenofuzzy model is a multimodel approach much used to modelisenon linear sytems by construction with identification of inputoutput data [6,7]. The merit of such fuzzy model-based controlmethodology is that it offers an effective and exactrepresentation of complex nonlinear systems in a compact setof state variables.With the powerful T–S fuzzy model, anatural, simple, and systematic design control approach can beprovided to complement other nonlinear control techniquesthat require special and rather involved knowledge.Nowadays, T–S fuzzy model-based control approaches havebeen applied successfully in a wide range of applications.One of the most important issues in the study of T–S fuzzysystems is the stability and stabilization analysis problems [8].Via various approachs, a great number of stability andstabilization results for T–S fuzzy systems in both thecontinuous and discrete time have been reported in theliterature [9,10].Two classes of Lyapunov functions are used to analyzethese systems: quadratic Lyapunov and nonquadraticLyapunov functions. The second class of function is lessconservative than the first. Many researches have investigatednonquadratic Lyapunov functions [11, 12, 13, 14, 15, 16].Many works try to reduce the conservatism of quadraticform. Several approaches have been developed to overcomethe above mentioned limitations. Piecewise quadraticLyapunov functions were employed to enrich the set ofpossible Lyapunov functions used to prove stability [11].Multiple Lyapunov functions have been paid a lot of attentiondue to avoiding conservatism of stability and stabilization.Some works try to enrich some properties of the membershipfunctions [17, 18], others introduce decisions variables (slackvariables) in order to provide additional degrees of freedom tothe LMI problem [19, 20].For every case, The Lyapunov function used to prove thestability has the most important effect to the results. To leavethe quadratic framework, some works have dealt withnonquadratic Lyapunov functions. In this case, some resultsare available in the continuous and the discrete cases[21],[22],[23]. In the discrete case, new improvements hasbeen developed in [24], by replacing the classical one sampletime variation of the Lyapunov function by its variation overseveral samples (k samples times variations). This conditionreduces the conservatism of quadratic form and give a largesets of solutions in terms of linear matrix inequality LMI. The299 P a g ewww.ijacsa.thesai.org

(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 7, No. 2, 2016relaxed conditions admitted more freedom in guaranteeing thestability and stabilization of the fuzzy control systems andwere found to be very valuable in designing the fuzzycontroller, especially when the design problem involves notonly stability, but also the other performance requirementssuch as the speed of response, constraints on control input andoutput .In this paper, a new stabilization conditions for discretetime Takagi Sugeno parametric uncertain fuzzy systems withthe use of [25,26, 27] new nonquadratic Lyapunov functionsare discussed. This condition was reformulated into LMI.[28,29,30,31,32,33,34,35], which can be efficiently solved byusing various convex optimization algorithms.The organization of the paper is as follows. First, T-Sfuzzy modeling is discussed. Second, we discuss the proposedapproachs to stabilize a T-S fuzzy system in closed loop withthe new lyapunov functions. Third, simulation results showthe robustness of this approachs and their influence in thestabilization region (feasible area of stabilization). We finishby a conclusion.II.SYSTEM DESCRIPTION AND PRELIMINAIRIESIn this section, we describe the concept of the TakagiSugeno parametric uncertain system. It’s based on the statespace representation.Consider the discrete time fuzzy model T-S parametricuncertain systems for nonlinear systems given as follows.If z1 (t ) is M i1 andand z p t is M ip then x(k 1) ( Ai Ai ) x(k ) ( Bi Bi )u (k ) y (k 1) (Ci Ci ) x(k )i 1.r(1)is the number of model rules, x k n is the states vector; u k m is the input vector; Ai n n ,the statesBi n mthecontrol(3),rThe term M ij z j k is the membership degree ofinM ijz j k .Since r wi z k 0 i 1 w z k 0 ii 1.r(4)we have 0 hi z k 1 r(5) hi z k 1 i 1The final output can be written under the following formr x(k 1) hi z k ( Ai Ai ) x(k ) ( Bi Bi )u (k )i 1 (6) ( Az ( k ) Az ( k ) ) x(k ) ( Bz ( k ) Bz ( k ) )u (k ) r y (k ) h z k (C C ) x(k ) (C C ) x(k )iiiz(k )z(k ) i 1 Ai , Bi represents parametric uncertainties matrices inWhere M ij (i 1, 2.r, j 1, 2. p) is the fuzzy set and rmatrix,r wi z k M ij z j k j 1 wi z k i 1, 2, hi z k r wzk i i 1matrixandz1 k ,., z p k are known premise variables.The T-S fuzzy model is written under the following form:r wi z k ( Ai Ai ) x(k ) ( Bi Bi )u (k ) x k 1 i 1r wi z k i 1 r wi z k (Ci Ci ) x k i 1 y k r wi z k i 1the state space representation. These uncertainties matrices arewritten under the following form. A H a FEa B H b FEb C H c FEc, T, T T FF 1 FF 1 FF 1 With H a , Hb , H c , Ea , Eb , Ec are constants matrices.(7)Lemma 1, 2 and 3 present the techniques and powerfultools used through the development of the next theorems.Lemma 1 (Schur Complement) [36,37]Consider A,G,L,P and Q matrices with appropriatesdimensions. The next properties are equivalent:(2)1. AT PA Q 0 , P 0 Q2. PAr : is the number of model rules.With(8)AT P 0 P (9) Q AT G3. G T 0, P 0T G A G G P (10)300 P a g ewww.ijacsa.thesai.org

(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 7, No. 2, 2016 Q AT LT LA L AT G 4. G, L 0, P 0TT G GT P L G A r V ( x(k )) xT (k ) hi z k Gi i 1 (11) T 1 r r hi z k Pi R hi z k Gi x (k ) (17) i 1 i 1 T T 1 x (k )Gz ( k ) ( Pz ( k ) )Gz ( k ) x(k )Lemma 2 [38]Relaxaion : Whatever the choise of the Lyapunov function,the analysis of the stabilization leads us to the inequality (12)with multiple sumrrrr h z k .h z 2k 1 h z k .i0 1 ik 1 1 i0 1 ik 1 1i0ik 1j0(12)The final equation of the Lyapunov function variationobtained by [40] which represent the stabilization condition ofdiscrete time T-S systems is written under the following form. i0i1 ,.ik 1 , j0 j1 ,. jk 1 . h jk 1 z 2k 1 i0 ,.ik 1 , j0 ,. jk 1 0Consider i0 ,.ik 1 , j0 ,. jk 1 P0 * TAG BF G G(18)i0 j0 i0 0 * 0Aik 1 G Bi j 1 Fjk 1 G GT P So [40] propose the following theorem: i 00, j0 . i kk 1 ,1j k 1 matricesand hi functions having the convex sum properties.The inequality (12) is verified if the next 0.5r r 1 kconditions are verified i0 , j0 ,., ik 1 , jk 1 1, 2,., r i0i1 ,.ik 1 , j0 j1 ,. jk 1 j0 j1 ,. jk 1 ,i0i1 ,.ik 1 0Consider the discrete Takagi-Sugeno (15), the control law(16) and the i0i1 ,.ik 1 , j0 j1 ,. jk 1 defined in (18). If it exist awheredefinite positive matrix P and matrices G, Fi , i {1.r} suchthat the conditions (12) and (13) of lemma 2 are verified thesystem is globally asymptotic stable in closed loop.i0 j0 ,., ik 1 jk 1Lemma 3 [39]Consider X and Y , Q QT 0 matrices of appropriatedimensions, the following inequality is verifiedXY T YX T XQX T YQ 1Y T(14)The use of these lemmas will be shown in the next section.III.Theorem [40](13)STABILIZATION ANALYSISThis section recalls the technique of the stabilizationanalysis of discrete T-S model based on a nonquadraticLyapunov function. In the discrete case, we consider thevariation of the Lyapunov function between two sample time.If the final equation of this variation is negative, we obtain asufficient condition of the T-S stabilization with the statefeedback controller.We propose a new Lyapunov function based on theLyapunov function in equation (17), by multiplying theLyapunov matrices Pz ( k ) by a scalar 0 . So the new formof the Lyapunov function is written under the following formin equation (19).(19)V ( x(k )) xT (k )Gz (Tk ) ( Pz ( k ) )Gz ( k ) 1 x(k )and the non-PDC control law is written under thefollowing form in equation (20).ru (k ) Fi G i 1 x k (20)i 1The variation of the Lyapunov function between k and k tsample times is given by the next equation (21) kV x k x k t Gz ( k ) T ( Pz ( k ) )Gz ( k ) 1 x k t Consider the discrete time fuzzy Takagi-Sugeno systemunder the following form.T x k Gz ( k ) T ( Pz ( k ) )Gz ( k ) 1 x k (21)Tr x(k 1) hi z k Ai x(k ) Bi u (k ) i 1 Az ( k ) x(k ) Bz ( k ) u (k ) (15) r y (k ) h z k (C ) x(k ) C x(k ) iiz(k ) i 1The non-PDC control law is described by the followingequation:rx k 1 ( Az ( k ) Bz ( k ) Fz ( k )Gz ( k ) 1 ) x k with Az ( k ) hi z (k ) Aii 1x k 2 ( Az ( k 1) Bz ( k 1) Fz ( k 1)Gz ( k ) )( Az ( k ) Bz ( k ) Fz ( k )Gz ( k ) 1 ) x k . 1.ru (k ) Fi G 1 x k The final output x k 1 is written between k and (k t)samples under the next form.(16)i 1The Lyapunov function used in [40] expressed in equation(17).x k t ( Az ( k t 1) Bz ( k t 1) Fz ( k t 1)Gz ( k ) 1 ) . ( Az ( k ) Bz ( k ) Fz ( k )Gz ( k ) 1 ) x k 301 P a g ewww.ijacsa.thesai.org

(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 7, No. 2, 2016The variation of the Lyapunov function for discrete systemshould be negative kV x k 0 . This equation isequivalent to T 1 1 T * Gz ( k ) ( Pz ( k ) )Gz ( k ) ( Az ( k t 1) Bz ( k t 1) Fz ( k t 1)Gz ( k ) ) .x k x k 0 1 T 1 . ( A B F G z(k )z ( k ) z ( k ) z ( k ) ) Gz ( k ) ( Pz ( k ) )Gz ( k ) Pz ( k ) * T Az ( k ) G Bz ( k ) Fz ( k ) Gz ( k ) Gz ( k ) 0 (22)The equation (22) is equivalent to equation (23) * Gz (Tk ) ( Pz ( k ) )Gz ( k ) 1 ( Az ( k t 1) Bz ( k t 1) Fz ( k t 1)Gz ( k 1) 1 ) . 0 . ( A B F G 1 ) G T ( P )G 1 z(k )z (k ) z (k ) z (k )z (k )z (k )z (k ) Az ( k t 1) Gz ( k t 1) Bz ( k t 1) Fz ( k t 1)(23)Consider the next modification(29)Az ( k t ) Bz ( k t ) Fz (k t )Gz (k t ) Az (k t )Gz (k t ) Bz (k t ) Fz (k t ) G 1 1z (k t )(24)Using the congruence with the full rank matrix G, weobtain * G ( Pz ( k ) ) Gz Tz 1( Az ( k t 1)Gz ( k t 1) Bz ( k t 1) Fz ( k t 1) ) . G ( Az ( k )Gz ( k ) Bz ( k ) Fz ( k ) ) ( Pz ( k ) ) 0(25)So the variation of the lyapunov function kV x k 0 1z(k )holds if the equation (25) is negative. The use of the SchurComplement (Lemma 1) with the equation (25) give the nextequation Pz ( k ) T Gz ( k ) Az ( k )Gz ( k ) Bz ( k ) Fz ( k ) * T (*)Gz (Tk ) Pz ( k )Gz ( k ) . 0 . Az ( k 1)Gz ( k 1) Bz ( k 1) Fz ( k 1) (26)(27)with i (*)Gz (Tk ) Pz ( k ) Gz ( k ) Az ( k t 1) Gz ( k t 1) Bz ( k t 1) Fz ( k t 1) . Az ( k i ) G Bz ( k 1) Fz ( k i ) (30)Therefore we state the following theorem for the discretetime Takagi-Sugeno fuzzy systems.Theorem 1These two theorems represent sufficient conditions of thediscrete time T-S stabilization with state feedback with ksample times variation of the Lyapunov function. In the nextsection, we present the analysis of the stabilization of thediscrete time T-S parametric uncertain systems.IV.The application of the lemma 1 with equation (27) with G give the next inequalityAik 1 G Bik 1 Fjk 10 Pi0 * T Ai0 G j0 Bi0 Fj0 Gi0 Gi0 0* 0Aik 1 G jk 1 Bik 1 Fjk 1 Gik 1 GiTk 1 Pik 1 and 0 such that the conditions (12) and (13) of lemma 2are verified the discrete time T-S system is globallyasymptotic stable in closed loop. Pz ( k )* 0 TT Gz ( k ) Az ( k ) Gz ( k ) Bz ( k ) Fz ( k ) i Gi0 G Ti0 i0i1 ,.ik 1 , j0 j1 ,. jk 1 definite positive matrices Pi and matrices G i , Fi , i {1.r}Let’s consider the following inequality: * The use of the lemma 2 with the equation (29), give thefinal condition of discrete time T-S systems stabilization. Thiscondition should be negative.Consider the discrete time Takagi-Sugeno (15), the controllaw (20) and the i0i1 ,.ik 1 , j0 j1 ,. jk 1 defined in (30). If exist aFor each iteration i {1 .r} Pi0 Ai0 G j0 Bi0 Fj0 0 0 * T Gz ( k t 1) Gz (k t 1) Pz (k t 1) 0 * 0 T G G 2 0PARAMETRIC UNCERTAIN SYSTEMS STABILIZATIONANALYSISA. New Lyapunov Function: First approachIn the next, we treat the case of the discrete time TakagiSugeno parametric uncertain systems.Consider the uncertain system described in equation (6)(28)Recursively by the use of Schur Complement we obtainthe inequality (29).In that case, the equation (30) becomes.302 P a g ewww.ijacsa.thesai.org

(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 7, No. 2, 2016 Pz ( k )0 * TAG BF G Gz(k )z(k ) z(k ) z(k ) z(k ) z(k ) 0 * T0AG BF G G Pz ( k t 1) z ( k )z ( k t 1) z ( k t 1)z(k )z(k )z ( k ) Az ( k ) Az ( k ) Az ( k ) , Bz ( k ) Bz ( k ) Bz ( k )Forthe Az ( k ) , Bz ( k )uncertainties,(31)termtheAz ( k ) G Bz ( k ) Fz ( k ) is transformed in the following form byintroducing two scalars 0 0, 0 0 . The use of lemma 3on uncertainties Az ( k ) , Bz ( k ) gives the next two inequalities: 0 H A Eaz Gz a z 0 1GzT EazT Eaz Gz 00 * T0 0 H a Az Az H aT (32) 0 F E Ebz Fz 0 H B Ebz Fz 0 * 0 b z 1TzTbz0 0 H b Bz Bz T H Tb(33)Taking in consideration the two inequality (32) and (33),the equation (31) become equation (34) * Pz ( k ) 10 Az ( k )Gz ( k ) Az ( k ) Fz ( k ) 0 Gz ( k ) GTz(k ) 200 0 * (34) 000 * * * k 1I00 0 k 1I0 00 GzT( k ) Gz ( k ) k2 1 Pz ( k ) 0 GzT( k ) Gz ( k ) k2 2Ebz ( k ) Fz ( k )Eaz ( k )Gz ( k )Az ( k )Gz ( k ) Bz ( k ) Fz ( k )(35)with i2 i H a H aT i Hb HbTAfter using lemma 2 the equation (35) become (37) i0 i1 ,.ik 1 , j0 j1 ,. jk 1 Pi0 Ebi0 F j0 Eai0 Gi0 A G B Fi0j0 i0 j0 0 0 * * Pz ( k ) Ebz ( k ) Fz ( k ) Eaz ( k )Gz ( k ) Az ( k )Gz ( k ) Bz ( k ) Fz ( k ) 0 0 * * * 0 I000 0 I000000 0 I000 GiT0 Gi0 02 G Gz ( k ) 0000 0 0 * 0 0 T2 Gk 1 Gk 1 k 1 Pi0 000 GiTk 1 Gik 1 k2 2 * * Ebik 1 Fjk 1 k 1 I0Eaik 1 G jk 10 k 1 IAik 1 G jk 1 Bik 1 F jk 100(37)So we state the next theorem for the stabilization of thediscrete time T-S parametric uncertain systems.Theorem 20Tz(k ) * 0 IWith G GT P 2k 1k 1k 1k 1 Az ( k t 1) Gz ( k t 1) Bz ( k t 1) Fz ( k t 1) i 1 T T 1 T T 0 i Gz Eaz ( k i ) Eaz ( k i ) Gz i Fz Ebz ( k i ) Ebz ( k i ) FzThe use of Schur Complement (lemma1) give the nextequation (35) which represents the final condition ofstabilization of T-S parametric uncertain systems with the useof the Lyapunov function (19) and the control law (20).(36)20Consider the discrete time uncertain Takagi-Sugenosystem (6), the control law (20) and the i0i1 ,.ik 1 , j0 j1 ,. jk 1defined in (37). If exist a definite positive matrices Pi ,000matrices Gi , Fi , i {1.r} and positives scalars i , i and 0 such that the conditions of lemma 2 are verified thesystem is globally asymptotic stable in closed loop.The next work deals with the addition of more variables inthe equation i0i1 ,.ik 1 , j0 j1 ,. jk 1 to give a large field of solutions.303 P a g ewww.ijacsa.thesai.org

(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 7, No. 2, 2016In this case a condition of stabilization is developed based onnew Lyapunov functions and a new non-PDC control law(20).B. New Lyapunov Function : Second approachConsider the new non quadratic lyapunov function inequation (38) and the non-PDC control law in equation (20).In this new function, we associate for each Lyapunov matricesPz ( k ) a scalar Tz (k )V ( x(k )) x (k )GT( i Pz ( k ) )Gz ( k ) x(k ) 1scalars i 0 such that conditions of lemma 2 are verified,the system is globally asymptotic stable in the closed loop.C. New Lyapunov Function : Third approachThe third Lyapunov function proposed in this paperrepresents an extention from the first Lyapunov function andthe next Lyapunov function described bellow.(38)where i 0 with i 1,.r Consider the same transformations and lemmas used toobtain equations (30), (37), theorems 1 and 2. The new formof equation for the stabilization of discrete time T-S fuzzyparametric uncertain systems with the use of the Lyapunovfunction (38) is under the following form in equation (39). i0i1 ,.ik 1 , j0 j1 ,. jk 1 i0 Pi0 * * 0 I 0 Ebi0 Fj0 0 0 I Eai0 G j0 A G B F00 GTj0 i0 j0 i0 j0 0 0 If it exists a definite positive matrices Pi , matricesGi , Fi , i {1.r} and positives scalars i , i and positivesThe following Lyapunov function is used by [16,17]. T r r V ( x(k )) xT (k ) hi z k Gi hi z k Pi R i 1 i 1 1 r hi z k Gi x(k ) i 1 (41)Which Pz is symmetric and definite positive matrix, andGz is full rank matrix. The nonlinearities are expressed by theterms hi z k 0 with the convex sum property hi z k 1r * 0i 10.0The Lyapunov function used by [13, 24], is written under thefollowing form. G j0 02rV ( x(t )) h k ( z (t ))Vk ( x(t ))(42)k 1000Vk ( x(t )) xT (t )( Pk R) x(t )(43)So the third proposed Lyapunov function is written underthe following form in equation (44) 0 0* 0 0 k2 1 ik 1 Pik 1 000 GTjk 1 G jk 1 2k 2 * * Ebik 1 Fjk 1 k 1 I0Eaik 1 G jk 10 k 1 IAik 1 G jk 1 Bik 1 Fjk 100 G Tjk 1 G jk 1(39)With 2k 1 k 1 H a H aT k 1 H b H bT 2TT k 2 k 2 H a H a k 2 H b H b(40)Using the equation (39) and the Lyapunov function (38)and the non-PDC controller (20), we propose the next theoremfor the stabilization of the T-S parametric uncertain systems.Theorem 3Consider the discrete uncertain Takagi-Sugeno system (6),the control law (20) and the i0i1 ,.ik 1 , j0 j1 ,. jk 1 defined in (39).V ( x(k )) xT (k )Gz T ( Pz R)Gz 1x(k )where 0 and 0 1(44)The new form of equation for the stabilization of discretetime T-S fuzzy parametric uncertain systems with the use ofthe Lyapunov function (44) is under the following form inequation (45). i0i1 ,.ik 1 , j0 j1 ,. jk 1 , R Pi R Ebi0 Fj0 E Gai0 j0 Ai G j Bi Fj00 0 00 0 * 0 I * * 000 0 I00 G G j0 0200Tj0000304 P a g ewww.ijacsa.thesai.org

(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 7, No. 2, 2016 0 0 * 0 0 2 k 1 Pi R 000 GTjk 1 G jk 1 2k 2 * * Ebik 1 Fjk 1 k 1 I0Eaik 1 G jk 10 k 1 IAik 1 G Bik 1 Fjk 100 G Tjk 1 G jk 1(45)With the equation (45) and the Lyapunov function (44)and the non-PDC controller (20), we propose the next theoremfor the stabilization of the T-S parametric uncertain systems.Theorem 4Consider the discrete uncertain Takagi-Sugeno system (6),the control law (20) and the i0i1 ,.ik 1 , j0 j1 ,. jk 1 , R defined in(45). If exist a definite positive matrices Pi , matricesR , Gi , Fi , i {1.r} and positives scalars i , i , positivescalar 0 and 0 1 such that the conditions of lemma2 are verified the system is globally asymptotic stable inclosed loop.In the next section, we add more values to the LMI inorder to demonstrate their influence in stabilization region byaffecting to each lyapunov matrices Pi , a positive scalar i .D. New Lyapunov Function : Fourth approachThe fourth Lyapunov function used in this paper is writtenunder the following form in equation (46)V ( x(k )) xT (k )Gz T ( i Pz R)Gz 1 x(k )(46)Under this Lyapunov function, the new condition ofstabilization obtained in the next equation. i0i1 ,.ik 1 , j0 j1 ,. jk 1 , R i Pi R Ebi0 Fj0 Eai0 G Ai G Bi Fj00 00 0 * 0 I * * 000 0 I00 G G 020T0 00 * 0 0 T2 G G k 1 i Pi R 00 * k 1 I0 * 0Eaik 1 G0 k 1 IAik 1 G Bik 1 Fjk 100 G G Ebik 1 Fjk 1T2k 2(47)We state the following theorem for the stabilization of thediscrete time T-S fuzzy parametric uncertain systems.Theorem 5Consider the discrete uncertain Takagi-Sugeno system (6),the control law (20) and the i0i1 ,.ik 1 , j0 j1 ,. jk 1 , R defined in(47). If it exist a definite positive set of matrices Pi , matricesR , Gi , Fi , i {1.r} and positives scalars i , i , i 0 and0 1 such that the conditions of lemma 2 are verified,then the system is globally asymptotic stable in closed loop.Four Lyapunov functions were proposed in this paper.They represent a direct extension from two other function inthe literature. In the next section, we present their robustnessby showing their influence on the stabilization region.V.SIMULATION AND VALIDATION OF RESULTSConsider TS discrete uncertain system with unstable openloop models. This system is modeled with two subsystems, sowe have r 2 .0.1 0.2 0.27 0.1 -0.3 0.5 A 1 0.4 -0.6 -0.3 , A 2 0.20.1 -0.9 0.1 0.5 0.1 -0.4 0.7 0.8 1.1 1 1 B 1 0.8 , B 2 0.5 H a 0.45 -0.9 0.73 0.45 1 0 0 1 H b 0 1 0 E a 0.2 1 -0.4 , E b 0 0 0 1 0.19 For the simulation the membership functions are choosenas follows:000h1 z k 1, h2 z k 1 h1 z k 1 0.9 x1 (k ) With the application of theorem 2, with 0.6 the resultsof LMI gives definite positive matrices P1 , P2 and matricesG1 ,G 2 , F1 and F2 :305 P a g ewww.ijacsa.thesai.org

(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 7, No. 2, 2016-0.0251 -0.0135 5.7746 0.0654 0.0332 0.3660 -0.0964)With the application of theorem 4, with 0.6 , the resultsof LMI gives other definite positive matrices P1 , P2 andmatrices G1 ,G 2 , R, F1 and F2 .0.90.80.70.60.5U(k) 5.4174 0.0220 -0.0128 5.5756 0.0131 P1 0.0220 5.6812 0.0189 , P2 0.0131 5.8226 -0.0128 0.0189 5.5808 -0.0251 -0.0135 0.6985 -0.0527 0.1130 0.7715 -0.0602 G1 -0.0160 0.3234 0.0015 G 2 -0.0224 0.3343 0.0384 0.0242 0.5142 0.0342 0.0193 F1 (0.0997 -0.1065 -0.1300)F2 (0.0777 -0.08420.40.30.20.10 2.6444 0.0149 -0.0295 2.6291 P1 0.0149 2.6722 -0.0537 , P2 -0.0149 -0.0295 -0.0537 2.6953 0.0295 0.3327 0.0229 -0.0870 0.3117 G1 0.0811 0.2159 -0.1457 G 2 0.0093 -0.0823 -0.1033 0.2099 0.0119 -4.7998 -0.0984 R 0.1461 -4.8765 -0.2455 -0.4761 F1 0.0399 -0.0191 -0.0272 ,-0.0149 0.0295 2.6012 0.0537 ,0.0537 2.5782 0.0126 -0.0188 0.1183 0.0265 0.0194 0.0491 -0.10510152025Samples3035404550Fig. 3. Evolution of the non PDC controller signalThe next figure 4, show the feasible areas of stabilizationfor proposed theorems 2 and 3 and the effect of the choice ofthe parameters and i to this areas. For the theorem 2 wechoose ( 0.6 ) and for theorem 3 we choose0.2377 0.4018 -4.9245 F2 (0.0248 -0.0103 -0.0176)( 1 1, 2 1.6 )7The figures 1, 2 and 3 show the convergence of statevariables and the control signal to the equilibrium point zerowith the application of the theorem 2.theorem 2theorem 354045500.150.20.254x1x2x33Fig. 4. Comparison between theorem 2 and 3Theorem 3 gives a larger stabilization region than theorem2. So by affecting for each Lyapunov matrices P a scalar we obtain a large stabilization region than a single scalar common to all Lyapunov matrices.Fig. 1. Evolution of the state variables of sub-system 1The next figures present the effect of increasing of numberof parameters with i 0 in the stabilization region.21x1,x2,x30.1a-20The figure 5 present the feasible area corresponding totheorem 3 with ( 1 0.01, 2 0.06 ) presented by the mark(o) and ( 1 1, 2 1.6 ) presented by the mark ( ). So evenwe choose i near to zero, we obtain a larger stabilizationregion (feasible area of 04550Fig. 2. Evolution of the state variables of sub-system 2306 P a g ewww.ijacsa.thesai.org

(IJACSA) International Journal of Advanced Computer Science and Applications,Vol. 7, No. 2, .10.120.140.160.180.20.220.24aFig. 5. Stabilization region of theorem 3Fig. 7. Stabilization region of theorem 5Figure 6 presents a comparison between theorem 4 and 5,it show the effect of the choice of parameters and iThe conclusion obtained throught these figuresdemonstrates that the choice of a large number of parameters i affected to each Lyapunov matrices give a best stabilizationregion then one parameter and a smaller (near to zero)number also have a great effect to the stabilization region.For the simulation, consider 0.6 for theorem 4 and 1 1, 2 1.6 for theorem 5. The use of theorem 5 give alarge stabilization region then theorem 4.The figure with the mark ( ) represent the stabilizationregion of theorem 5 and the figure with the mark (o) representthe stabilization region of theorem 0.220.24Fig. 6. Comparison between theorem 4 and 5Figure 7 presents the effect of choice of parameter i nearto zero with the use of theorem 5.The figure ( ) present the feasible area of stabilization forthe values of ( 1 0.01, 2 0.06 ) and (o) present thefeasible area for the values ( 1 1, 2 1.6 ). So to obtain alarge stabilization region, i should be near to zero.Figures 4,5,6 and 7 represent a comparison between theproposed theorems.CONCLUSIONVI.This paper has developed a new fuzzy controller with statefeedback for discrete time T-S parametric uncertain systems.The analysis of the stabilization problem is established by theuse of Lyapunov function technique. In this case, four newLyapunov functions are proposed. In This Lyapunov functionsmore parameters and slack matrix variables are introduced inorder to facilitate and enrich the stabilization analysis. In thefirst Lyapunov function, a multiplication with a commonscalar to each Lyapunov matrices is considered, In the secondeach Lyapunov matrices is multiplied with their own scalars.The use of the second function has a great influence to thestabilization region than the first. In the third and fourthfunctions, more parameters and slack matrix variables areintroduced with common and single scalars for each Lyapunovmatrices. Through the simulation results a singl

time Takagi Sugeno parametric uncertain fuzzy systems with the use of [25,26, 27] new nonquadratic Lyapunov functions are discussed. This condition ijwas reformulated into LMI. [28,29,30,31,32,33,34,35], which can be efficiently solved by using various convex optimization algorithms. The organization of the paper is as follows. First, T-S

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