Stability of Cascaded Takagi-Sugeno Fuzzy Systems

Transcription

1 Stability of Cascaded Takagi-Sugeno Fuzzy Systes Zs. Lendek R. Babuska B. De Schutter Abstract- A large class of nonlinear systes can be well approxiated by Takagi-Sugeno (TS) fuzzy odels, with local odels often chosen linear or affine. It is well-known that the stability of these local odels does not ensure the stability of the overall fuzzy syste. Therefore, several stability conditions have been developed for TS fuzzy systes. We study a special class of nonlinear dynaic systes, that can be decoposed into cascaded subsystes. These subsystes are represented as TS fuzzy odels. We analyze the stability of the overall TS syste based on the stability of the subsystes. For a general nonlinear, cascaded syste, global asyptotic stability of the individual subsystes is not sufficient for the stability of the cascade. However, for the case of TS fuzzy systes, we prove that the stability of the subsystes iplies the stability of the overall syste. The ain benefit of this approach is that it relaxes the conditions iposed when the syste is globally analyzed, therefore solving soe of the feasibility probles. Another benefit is, that by using this approach, the diension of the associated linear atrix inequality (LMI) proble can be reduced. Applications of such cascaded systes include ultiagent systes, distributed process control hierarchical large-scale systes. 1. INTRODUCTION Dynaic systes are often odeled in the state-space fraework, using a state-transition odel, which describes the evolution of states over tie a easureent (sensor) odel, which relates the easureents to the states. Traditionally, the class of linear, tie-invariant systes have doinated control theory. The linearity tieinvariance ake these types of systes easy to analyze. The disadvantage is that such odels usually fail to describe nonlinear systes globally. An accurate approxiation of a nonlinear syste can only be expected in the vicinity of an equilibriu point or trajectory. A large class of nonlinear systes can be well approxiated by TS fuzzy odels [1], which in theory can approxiate a general nonlinear syste to an arbitrary accuracy [2]. The TS fuzzy odel consists of a fuzzy rule base. The antecedents of the rules partition a given subspace of the odel variables into fuzzy regions. The consequent of each rule is usually a linear odel, valid locally in the region defined by the corresponding antecedent. Although the local odels are often chosen to be linear or affine, the stability of these odels does not ensure the stability of the fuzzy odel. Therefore, several stability conditions have been developed for TS fuzzy systes, ost of the relying on the feasibility of an associated syste of linear atrix inequalities (LMI) [3]-[6]. However, the The authors are with the Delft Center for Systes Control, Delft University of Technology, Mekelweg 2, 2628 CD Delft, The Netherls (eail: zs.lendek@tudelft.nl, r.babuska@tudelft.nl). B. De Schutter is also with the Marine Transport Technology Departent of the Delft University of Technology (eail: b@deschutter.info). coplexity of the syste grows exponentially with the nuber of antecedents the stability analysis proble eventually becoes intractable for a large nuber of rules. An iportant class of nonlinear dynaic systes can be represented as cascaded subsystes. In several cases, conclusions referring to the overall syste can be drawn based on the study of the individual subsystes. E. g., for linear systes, the stability of the individual subsystes iply the stability of the cascaded syste [7]. This property, however, in general does not hold for nonlinear or tievarying systes. Even global asyptotic stability of the individual subsystes does not necessarily iply stability of the cascade. In the literature, the stability of several types of cascaded systes has been studied. The ain otivation cae fro the linear-nonlinear cascade [8], resulting fro input-output linearization. Conditions to ensure the overall stability of ore general cascades, in which all subsystes are nonlinear, were derived in [7], [9], [1]. We study a special class of systes, represented as TS fuzzy systes, which can be decoposed into cascaded subsystes, analyze the stability of the whole syste based on the stability of the subsystes. This class of systes is very iportant, as any systes are naturally distributed (e.g., ulti-agent systes) or cascaded (e.g., hierarchical large-scale systes). Others ay be represented as cascaded subsystes, which are less coplex than the original syste. The ain benefit of this approach is, that it relaxes the conditions iposed by analyzing the global syste. A global analysis of the syste ay lead to infeasible LMI probles, even if the analyzed syste is stable. In this paper, we propose ore relaxed stability conditions, which ay render the associated LMI proble feasible. Moreover, for soe applications, the diension of the associated LMI syste is greatly reduced. The results presented can also be extended to observer design for the class of cascaded systes. The structure of the paper is as follows. Section 2 introduces the proposed cascaded setting for nonlinear systes presents stability conditions for cascaded systes for TS fuzzy odels. The proposed stability conditions for cascaded fuzzy systes are presented in Section 3. Exaples are given in Section 4. Finally, Section 5 concludes the paper /7/$ IEEE. Authorized licensed use liited to: Technische Universiteit Delft. Downloaded on June 9, 29 at 6:52 fro IEEE Xplore. Restrictions apply.

2 II. STABILITY OF CASCADED DYNAMIC SYSTEMS A. Preliinaries Consider the following nth order, general nonlinear syste with outputs: ±1 = fi (x, U) X2 = f2 (X, U) Yi = hi(x,u) Y2 = h2(x,u) ±n = fn (X, U) Y = h(x, u) Assue that this syste (both the states the outputs) can be partitioned into several subsystes. For the ease of notation, two subsystes are considered here (without loss of generality): (1) t1 = fl(tl, u) ~~~(2) Yi hi (xi, u) 2= f2(x, X2,U) (3) Y2 = h2(x, X2, u) with x = X1 UX2 1xn X2 = In general, such a partition of the odel ay not necessarily exist,, if it exists, it ight not be unique. Since stability is independent of the easureent odels h, h2, in the sequel, the odels h, h2 are not considered. Then, for two subsystes, the cascaded structure is depicted in Figure 1. Fig. 1. Cascaded subsystes. B. Stability of Cascaded Systes It is well-known that the cascade of stable linear systes is stable, since the eigenvalues of the joint syste are deterined only by the eigenvalues of the individual subsystes [7]. Therefore, the stability of the subsystes iplies the stability of the joint syste. However, the sae reasoning does not necessarily hold for nonlinear or tievarying systes. Even global asyptotic stability (GAS) of the individual subsystes does not necessarily iply the stability of the cascade. In the literature, the stability of several special cases is studied. The ain otivation coes fro a linear-nonlinear cascade resulting fro input-output linearization [8]. More general cascades in which both subsystes are nonlinear were studied conditions to ensure the overall stability were presented in [7]. Several relevant results are presented below. Definition 1: A continuous function a : >R+> R+ belongs to class IC if it is strictly increasing a?(o) =. If a(s) -> oo when s -> oo, then a is said to be of class /C. Consider the nonlinear, cascaded, autonoous syste It has been shown in tl = fi(xi) X2 = f2l, X2) [11] that if (4) (5). the functions fi f2 are sufficiently sooth in their arguents, * syste (5) is input-to-state-stable with regard to the input x1,. syste (4) X2 = f2(ix2) (6) are GAS, then the cascaded syste (4)-(5) is GAS. An equivalent sufficient stability condition is presented in [1]: the cascaded syste is GAS, if both subsystes are GAS all solutions are bounded. The ain difficulty with this approach is that in general, boundedness of all the solutions is not easy to deterine the conditions to ensure boundedness ay be very conservative. More relaxed sufficient stability conditions have been derived for systes of the for: i f (xi) ( x) (7) _t2 f2(x2) + 9(X1, X2) assuing that the individual subsystes are GAS, additionally, certain restrictions related to the continuity /or slope, apply for the interconnection ter g [8], [12], [13]. A theore for the unifor GAS (UGAS) of the cascaded syste (7) [7] is presented below. Assuptions: 1) Syste (6) is UGAS. 2) There exist constants cl, c2,,u > a Lyapunov function V(t, X2) for (6) such that V : R+ x R' --> R+ is positive definite, radially unbounded, V(t, X2) < o < ClV(t,Xi2) V'1X2 1 >, av OX2-121< 3) There exist two continuous functions 1, 2 :R+ R+ such that g(x) satisfies:..g(x).. <- 1QXfl.) +2QXfl.)..X2.. (9) 4) There exists a class IC function a(.) so that for all to >, the trajectories of the syste (4) satisfy r J 41 (t; to, xi (to)) dt < aq(x (to) ) (1) Theore 1: Let Assuption 1 hold suppose that the trajectories of (4) are uniforly globally bounded. If in addition, Assuptions 2-4 are satisfied, then the solutions ofthe Authorized licensed use liited to: Technische Universiteit Delft. Downloaded on June 9, 29 at 6:52 fro IEEE Xplore. Restrictions apply.

3 syste (7) are uniforly globally bounded. If furtherore, syste (4) is UGAS, then so is the cascaded syste (7). Proposition 1: If in addition to the above assuptions systes (4) (6) are exponentially stable, then the cascaded syste (7) is also exponentially stable. The proof of the Theore 1 the study of different cases of the interconnection ter can be found in [7], [8]. Stabilizability conditions for such cascaded systes were derived in [13], [14]. C. Stability of Fuzzy Systes Consider an autonoous fuzzy syste expressed as: IEwi(z)Aix, (1 where Ai, i = 1, 2,... is the state atrix of the ith local linear odel, wi is its corresponding noralized ebership function, z denotes the scheduling vector. Syste (11) can also be regarded as a linear paraeter varying (LPV) syste: =A(z) (12) with A(z) Z> Wi(z)Ai. For the syste (11), several stability conditions were derived. Aong the, a well-known frequently used condition is forulated below [3]. Theore 2: Syste (11) is exponentially stable if there exist p pt > so that AiP + PAi <, for alli 1, 2,...,. A siilar condition is used if the fuzzy syste is subject to vanishing disturbances (perturbations): z = 5 wi (z)aix + Df (t, x) (13) where D is a perturbation distribution atrix f is a vanishing disturbance, i.e., f (t, x) > when t > oo, f is Lipschitz, i.e., there exists,u > so that f(t, x) <p,ulxz for all t x. With these assuptions, a sufficient stability condition can be foralized by the following theore. Theore 3: Syste (13) is exponentially stable if there exist atrices p = pt, Q = QT, so that P>O Q>O < Ain (Q) +PDA<2 AfTp±+PA < -2Q i= 1, 2,..., Several variants of the above theore exist, together with algoriths to copute robustness easures [15]. However, these approaches are conservative, by disregarding the fact that the local odels are valid only in a region of the state-space. For fuzzy systes, the ebership functions often have a bounded support. Therefore, it is sufficient that xt (ATP + PA T)X < only where wi (z) >. Stability conditions for the case when the support of each ebership function is bounded were derived in [4]. Another approach, based on partitioning the state-space into operating interpolation regies, is described in [16]. Assuing that in (11), z can be expressed as soe function of x, the syste can be written as: = Ewi(K)Aix iekk X C Xk (15) where Kk is the index set of the linear subsystes active in the region Xk. Then, using a Lyapunov function of the for V(x) xtpx when x C Xi, the syste (15) is stable, under the conditions expressed by the following theore [16]. Theore 4: Syste (15) is stable, if there exist atrices Pi = PT, H HT>,F,i 1, 2,..., IKl, so that: Pi F THFi Pi > FAx F <x AkfP, + PtAk < Vx C Xi n X Vk C Ki (16) For ore relaxed conditions, the coputations of the corresponding atrices see [17], [18]. Siilar conditions for the discrete-tie case are described in [19]. Last, but not least, for a paraeter-dependent Lyapunov function of the for: V(x) = xtp(z)x = XT 5 wi(z)pi (17) the stability conditions can be forulated as follows [15]. Theore 5: The syste (11) is stable if there exist a >, P P >,P Pi >, A, = A T >, Q = QT > so that P -Ai <P P+Ai >Pi Gii < -Q (18) Gij + Gji <-Q 5I Ai < aq where Gij = Ai j + PjAi, j = 1, 2,...,. Note that all the above conditions rely on the feasibility of the corresponding linear atrix inequality proble. Since efficient algoriths exist for solving LMIs, they can be easily verified. However, two shortcoings of the above theores have to be entioned: 1) the conditions are conservative often lead to infeasible LMIs 2) the nuber of LMIs can grow exponentially with the nuber of local odels, depending on the support of the ebership functions (in particular for Theores 4-5). III. STABILITY OF CASCADED FuzzY SYSTEMS Consider the case when the syste atrices of the odel (1 1) for each rule i = 1, 2,..., can be written as: Ai (A1 O2\ (Ali ~A21 A2) A21i A2) Authorized licensed use liited to: Technische Universiteit Delft. Downloaded on June 9, 29 at 6:52 fro IEEE Xplore. Restrictions apply.

4 i.e., the syste can be expressed as the cascade of two fuzzy systes: 1=Ewij(zi)Aijxi (19) Xt2 = W2i (z) (A2liAl + A2iX2) or, equivalently: with =Ali(zl)xl (2) -t2 =A21 (Z) Xt1+A2 (Z) X2 A1(Zi) = Ewij(zi)Aixi, A21(z) = EW2i(z)A21iX1, etc. Below, we prove that if the subsystes i = Al(zl)xl (21) X2 =A2(z)2 (22) are stable, it is possible to apply Theore 1 to fuzzy systes of the for (19), i.e., Theore 6: If there exist two Lyapunov functions of the for Vi(xi) = xipxi V2(X2) P2X2 so that the subsystes (21) (22) are UGAS, then the cascaded syste (2) is also UGAS. Proof: Consider two Lyapunov functions of the for Vi(xi) = 4TP x V2(X2) = XT4P2X2 for the subsystes (21) (22). Note that such Lyapunov functions satisfy Assuptions 1 4 also ensure exponential stability. Furtherore,. Assuption 2 is satisfied as: V 1X211 >,u x242 =2 P2 X2 < 2Aax (P2) X2 2 < Cl V2 (X2) for any cl > 2Aax (P2) For the second condition Ai.n(P2) condition of Assuption 2, we have V x2 <, 2 = XT24P2 < 2 42H P2 < 2IiAax(P2) = C2. Assuption 3 is satisfied by choosing continuous functions aj(xf1 a2(qxl1) = - ( ) ax la21(z2) x1 Since the conditions of Theore 1 are satisfied, the cascaded syste is UGAS. Furtherore, if these Lyapunov functions ensure exponential stability of the subsystes, then, based on Proposition 1, the cascaded syste is also exponentially stable. LIi Z2 While it is true that the cascaded syste is stable under the above conditions, finding a Lyapunov function valid for the cascaded syste is not trivial. The construction of a crosster in the global Lyapunov function has been proposed in [12], under the condition that the cascaded syste satisfies Assuptions 2 3. Then, the global Lyapunov function is of the for: VO(Xl,X2) = Vl(Xl) + V2(X2) + T(Xl,X2) (23) where V1 V2 are Lyapunov functions for the (21) (22), respectively. systes For the case when the first subsyste is LTI, the authors of [12] proved that the cross-ter exists is continuous, Vo is positive definite radially unbounded. If (21) is globally exponentially stable, the result fro [12] can be extended to the syste (2). The cross-ter T is then given by: T(X1, X2)= V (-i2(s))a21(z(s)).tr-(s)ds where x1 x2 are the trajectories of the systes (21) (22), respectively. To verify the stability of the cascaded fuzzy syste (19) by the Theores 2-5 fro Section II-C, additional sufficient, but by no eans necessary conditions can be derived using a Lyapunov function of the for VO(X,X2) = Vl(Xi) + V2 (X2). For exaple, Theore 2 requires a coon atrix P = pt > so that A TP + PAi <. Let P = diag[pl P2] = pt >. Then, the additional condition to those of Theore 6, to satisfy the requireents of Theore 2 is the negative definiteness of Gi for all i = 1, 2,..., with G, = Al Tpl + PiAl Gi= V P2A21 A21TP2 A2TP2 + P2A2 In the sae way, for all the above theores, the stability conditions presented in Section II-C can be relaxed. The new conditions for Theores 3 4 are presented below. The conditions of Theore 3 can be replaced as follows. Theore 7: Consider the syste (19) expressed as:.c=zwi(z) (1 A2) (A21 (Z) ) This syste is stable, if there exist Pi = PfT >, P2 P2T >, so that: AfTP1 + P1Ajj < A2P2 + P2A2i < for all i = 1, 2,...,. The proof is siilar to that of Theore 6. Note that in order to satisfy all the conditions of Theore 3 an additional condition is needed: there exists Q =QT > so that: 1 < Ain (Q) in P2A21 (Z) i Authorized licensed use liited to: Technische Universiteit Delft. Downloaded on June 9, 29 at 6:52 fro IEEE Xplore. Restrictions apply.

5 However, this condition is necessary only when applying Theore 3, is no longer needed to prove the stability of the cascaded syste (2). In order to relax the conditions of Theore 4 let K1 K2 be the nuber of operating interpolation regies for the individual subsystes, with K{ K2 the index set corresponding to the local odels of the subsystes active in the atching region. Then, the conditions can be expressed as: Theore 8: The syste (15) is UGAS, if there exist atrices pl = (p1 )T >, Pi = (p21)t >, H1 HT >, H2 = H2T >, F1, 1, 2,...,K2, so that: F2j, i = 1, 2,...,K1, j PFI= (F¾ciH P= F2i)TH2F2 Xi = F i zcx1 n xl F3it = F2t2 VX2 C x-i n X2 ',I (24) Ax AfJkPL + P Alk<k A2¾kP2Y + P2A2k < Vk C K1 Vk C K2 The additional condition to satisfy Theore 4 is: Ain(Pi Al ) Ain (P2A2) < P2A21k, k C Ki Siilarly, this condition is necessary only to satisfy all the conditions of Theore 4 is no longer needed to prove stability of the cascaded syste (2). When applying Theore 5, the restrictions on Gii Gij have to be odified, which leads to conditions siilar to those of Theore 4. Note, that the proposed conditions are still only sufficient conditions for the stability of cascaded fuzzy systes. However, by taking the advantage of the special for of the syste, i.e., studying the subsystes instead of global fuzzy syste, the coplexity of the associated LMI proble is reduced with respect to the presented Theores 2-5. IV. EXAMPLES In this section, the benefits of the proposed approach are deonstrated on siulation exaples. A. Feasibility Consider the fuzzy syste: 2 + =,wi(z)aix (25) with w, (z) = w(t) t w2 (t) = 1- w (t), for all At t C [, At], i.e., the syste changes in tie fro one local odel to another. The state atrices of the local linear odels were roly generated are given as: A A2 = / The local odels are stable siulations indicate that the syste (25) is also stable. However, the LMI proble P > O ATP + PA1 < ATP + PA2 < is infeasible, so Theores 2 3 cannot be applied. The stability of this syste can be investigated using Theores 4-5. By exaining the for of the syste atrices, one can easily see that the syste can be cascaded, with x 1 [X1 X2 X3]1 X2 = [X4 X5 X6]T Based on Theore 6, the syste (25) is stable if the individual subsystes are stable. As such, in order to prove the stability of the syste (25), it is sufficient that the LMI probles (Theore 6) Pi > ATLP1 + P1All < ATP1 + P1A12 < A12P2 + P2A12 < A2fP2 + P2A22 < P2 > are feasible. Using Yalip's solvesdp [2] one can easily see that it is so. This exaple illustrates the ain benefit of the proposed stability conditions: while the conditions iposed by conventional ethods lead to an infeasible LMI syste, it is still possible to prove stability of the syste under study. B. Diension reduction Consider the nonlinear syste: 1 =X- X2 =-2x2 + 16x,z2sinz where z C [-w, w] is a easured variable. It can be proven that this syste is globally asyptotically stable, e.g., by using the Lyapunov function V = 64w4x2 +42 A fuzzy approxiation of this syste can be obtained by linearizing the syste around all z C {- w,j-r/2, -w/4,, w/4, w/2, w}. The obtained atrices Authorized licensed use liited to: Technische Universiteit Delft. Downloaded on June 9, 29 at 6:52 fro IEEE Xplore. Restrictions apply.

6 are: A(t-j) = A(O) = A()= At-gr2)= (47 2-2) A(- /4) =( 222 ) A(7/4) (= 2w2 2) A(7/2) =4r (412 2) i.e., there are 5 distinct local linear odels. Using Theore 2, this eans that 5 LMIs have to be solved, while in case of Theores 4 5, this nuber is uch larger (11 13, respectively). However, using the proposed approach, the proble is reduced to two one-diensional LMIs: -1P + P1 (-1) < -2P2 + P2(-2) < This exaple illustrates how, by analyzing the subsystes instead of the global fuzzy syste, both the nuber of LMIs their diension can be reduced. V. CONCLUSIONS In any real-life applications, a coplex process odel can be decoposed into sipler, cascaded subsystes. This partitioning of a process leads to increased odularity reduced coplexity of the proble, while also aking the analysis easier. In this paper, we have studied the stability of a cascaded fuzzy syste, based on its individual subsystes. We have proven that the stability of such a syste is deterined only by the stability of the individual subsystes. Furtherore, the proposed approach relaxes the conventional stability conditions ay reduce the diension of the proble to be solved. The benefits of studying stability based on subsystes have been deonstrated on siulation exaples. In our future research, we will investigate the theoretical conditions under which observers can be designed individually for such cascaded processes while aintaining the sae perforance (stability, convergence rate) as a centralized observer. [3] K. Tanaka, T. Ikeda, H. Wang, "Fuzzy regulators fuzzy observers: relaxed stability conditions LMI-based designs," IEEE Transactions on Fuzzy Systes, vol. 6, pp , [4] M. Johansson, A. Rantzer, K. Arzen, "Piecewise quadratic stability of fuzzy systes," IEEE Transactions on Fuzzy Systes, vol. 7, pp , [5] P. Bergsten, R. Pal, D. Driankov, "Fuzzy observers," in The 1th IEEE International Conference on Fuzzy Systes, 21, vol. 2, 21, pp [6] K. Tanaka H.. Wang, Fuzzy Control Systes Design Analysis: A Linear Matrix Inequality Approach. John Wiley & Sons, Inc., 21. [7] A. Loria E. Panteley, Advanced Topics in Control Systes Theory. Springer Berlin / Heidelberg, 25, ch. Cascaded nonlinear tievarying systes: Analysis design, pp [8] M. Arcak, D. Angeli, E. Sontag, "A unifying integral ISS fraework for stability of nonlinear cascades," SIAM J. Control Opti., vol. 4, pp , 22. [9] E. D. Sontag, "Sooth stabilization iplies coprie factorization," IEEE Transactions on Autoatic Control,, vol. 34, pp , [1] P. Seibert R. Suarez, "Global stabilization of nonlinear cascade systes," Systes Control Letters, vol. 14, pp , 199. [11] E. Sontag, "Rearks on stabilization input-to-state stability," in IEEE CDC, 1989, pp [12] M. Jankovic, R. Sepulchre, P. Kokotovic, "Constructive Lyapunov stabilization of nonlinear cascade systes," IEEE Transactions on Autoatic Control, vol. 41, pp , [13] A. Chaillet A. Loria, "Necessary sufficient conditions for unifor seiglobal practical asyptotic stability: Application to cascaded systes," Autoatica, vol. 42, pp , 26. [14] A. Bacciotti, P. Boieri, L. Mazzi, "Linear stabilization of nonlinear cascade systes," Math. Control, Signals, Syst., vol. 6, pp , [15] P. Bergsten, "Observers controllers for Takagi-Sugeno fuzzy systes," Ph.D. dissertation, Orebro University, 21. [16] A. Rantzer M. Johansson, "Piecewise linear quadratic optial control," IEEE Transactions on Autoatic Control, vol. 45, pp , 2. [17] M. Johansson A. Rantzer, "Coputation of piecewise quadratic Lyapunov functions for hybrid systes," IEEE Transactions on Autoatic Control, vol. 43, pp , [18] M. Johansson, "Piecewise linear control systes," Ph.D. dissertation, Departent of Autoatic Control, Lund Institute of Technology, [19] W.-J. Wang C.-H. Sun, "Relaxed stability stabilization conditions for a TS fuzzy discrete syste," Fuzzy Sets Systes, vol. 156, pp , 25. [2] J. Lofberg, " YALMIP: a toolbox for odeling optiization in MATLAB," in Proceedings of the CACSD Conference, Taipei, Taiwan, 24. [Online]. Available: joloef/yalip.php ACKNOWLEDGMENT This research is partly sponsored by Senter, Ministry of Econoic Affairs of the Netherls within the project Interactive Collaborative Inforation Systes (grant no. BSIK324). REFERENCES [1] T. Takagi M. Sugeno, "Fuzzy identification of systes its applications to odeling control," IEEE Transactions on Systes, Man, Cybernetics, vol. 15, pp , [2] C. Fantuzzi R. Rovatti, "On the approxiation capabilities of the hoogeneous takagi-sugeno odel," in Fifth IEEE International Conference on Fuzzy Systes (FUZZ-IEEE'96), 1996, pp Authorized licensed use liited to: Technische Universiteit Delft. Downloaded on June 9, 29 at 6:52 fro IEEE Xplore. Restrictions apply.