Robust H∞ nonlinear modeling and control via uncertain fuzzy systems

Robust H∞ nonlinear modeling and control via uncertain fuzzy systems

Available online at www.sciencedirect.com Fuzzy Sets and Systems 143 (2004) 189 – 209 www.elsevier.com/locate/fss Robust H ∞ nonlinear modeling and ...

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Available online at www.sciencedirect.com

Fuzzy Sets and Systems 143 (2004) 189 – 209 www.elsevier.com/locate/fss

Robust H ∞ nonlinear modeling and control via uncertain fuzzy systems Ji-Chang Lo∗ , Min-Long Lin Department of Mechanical Engineering, National Central University, Chung-Li, 32054, Taiwan Received 18 June 2002; received in revised form 25 December 2002; accepted 20 January 2003

Abstract In theory, an Algebraic Riccati Equation (ARE) scheme applicable to robust H ∞ quadratic stabilization problems of a class of uncertain fuzzy systems representing a nonlinear control system is investigated. It is proved that existence of a set of solvable AREs su4ces to guarantee the quadratic stabilization of an uncertain fuzzy system while satisfying H ∞ -norm bound constraint. It is also shown that a stabilizing control law is reminiscent of an optimal control law found in linear quadratic regulator, and a linear control law can be immediately discerned from the stabilizing one. In practice, the minimal solution to a set of parameter dependent AREs is somewhat stringent and, instead, a linear matrix inequalities formulation is suggested to search for a feasible solution to the associated AREs. The proposed method is compared with the existing fuzzy literature from various aspects. c 2003 Elsevier B.V. All rights reserved.  Keywords: Quadratic stabilization; Algebraic Riccati equation; Norm-bounded uncertainty; Takagi–Sugeno fuzzy model; Linear matrix inequalities (LMI)

1. Introduction Despite many successful industrial applications of fuzzy control systems, it has been di4cult in developing a general methodology for stability analysis and design for such systems. One reason for this is that fuzzy systems are essentially nonlinear. Recently, an increasing amount of works in fuzzy systems literature has been devoted to stability analysis and design of Takagi–Sugeno [1,2,3,4,5] fuzzy models. Among them, a prevailing stability analysis method based on Lyapunov theory is considered in [7,29] in which a common P matrix must be found and su4cient condition exists for stability analysis and state feedback design. ∗

Corresponding author. Tel.: +886-3-426-7330; fax: +886-3-425-4501. E-mail address: [email protected] (J.-C. Lo).

c 2003 Elsevier B.V. All rights reserved. 0165-0114/$ - see front matter  doi:10.1016/S0165-0114(03)00023-X

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Stability equivalence of fuzzy systems and its corresponding switched and hybrid systems also prompts new directions toward fuzzy stability analysis. Indeed, [16,17,32] show that a fuzzy system is equivalent to a switched and hybrid system from stability view point. Based on the result given by Narendra and Balakrishnan [23], Thathachar and Viswanath [32] provides a su4cient condition for the existence of a common P solution to the Lyapunov function while [15] extends the result to discrete-time fuzzy systems. Johansson et al. [16,17] derive a su4cient condition suggesting a piecewise quadratic Lyapunov function for each operating region where continuation of energy decreasing across cell boundaries is parameterized and solved using Sprocedure. Drawing upon progress in uncertain linear systems for robust control [40,24,19,37,36,35] and references therein, a number of results on robust fuzzy control have been reported in [28,25] and su4cient condition exists for stability analysis and stable feedback design for fuzzy systems. Furthermore, to overcome the disadvantage of using Fxed P in Lyapunov function, Cao et al. [3,4,5,11,13,14,6,10,12] construct a piecewise smooth quadratic Lyapunov function for each dominant subsystem determined by Fring strengths and stability conditions are obtained using the robust control results in [19], where the unstructured uncertainties are characterized by a DFE formulation. However, these boundary conditions are restrictive and hard to check a priori. Linear matrix inequality (LMI) techniques are employed to solve an H ∞ control problem of a nonlinear control system via robust H ∞ fuzzy control [8,9,22,21,33]. Inspired by these works an algebraic Riccati equation approach is sought to analyze the quadratic stabilization problems for uncertain fuzzy systems subject to external disturbances. Although LMI-based approach is commonly seen in the literature for an H ∞ fuzzy control problem, some protruding diHerences, which are not seen up to the present, with respect to those results come from three perspectives: (1) An optimization technique for representing a nonlinear system by a T–S fuzzy model is shown. (2) Explicit solution is obtained for controller gain, establishing connection with linear quadratic regulator (LQR). (3) Existence of a solution to a set of r 2 LMIs implies a linear controller exists. This paper is organized as follows. Section 2 clariFes the connection between a nonlinear control system and an uncertain fuzzy control system with DFE structure denoting modeling errors. Then the underlying fuzzy model is introduced and connections to robust H ∞ control theory are established where the uncertainty is conFned to the standard DFE structure. Section 3 contains the main result of this paper where solvability of a set of parameter-dependent algebraic Riccati equations guarantees quadratic stabilizability of an uncertain fuzzy system when a mimic LQR control law is designed. An LMI approach is suggested to search for a feasible solution instead of the minimal solution to the set of parameter-dependent AREs. Section 4 contains two examples that serve to validate the theoretical development established in the previous sections. Some important aspects of the design method are also discussed. Finally, conclusions are drawn in Section 5. Notations: Throughout this paper the notations are quite standard. M ¿N (M ¿N ) for symmetric matrices means that M − N is nonnegative (positive) deFnite.  ·  denotes the standard Euclidean norm. M t refers to transpose of matrix M .

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2. Uncertain fuzzy system setup 2.1. Modeling error and DFE uncertainty structure It is well known that a Takagi–Sugeno fuzzy model can be a universal approximator of a smooth nonlinear dynamics [38,39]. Therefore, a smooth nonlinear control system of the following form: x(t) ˙ = f(x) + g(x)u(t) + Dw w(t) with f(0) = 0 can be approximated using a T–S fuzzy model x(t) ˙ =

r 

i [Ai x(t) + Bi u(t)] + Dw w(t):

i=1

To account for the modeling error, we introduce the following notation     r r   i Ai x(t) + g(x) − i Bi u(t) e(x; u) = f(x) − i=1

i=1

and assume that the modeling error term e(x; u) satisFes the following norm-bounded assumption. Assumption 1 (Sector-type assumption). There exist some 1 ¿0, 2 ¿0 such that e(x; u)61 x+ 2 u for all x(t)∈Rn and u(t)∈Rm . Moreover, denote the corresponding modeling error set by (x; u) = {e(x; u): e(x; u)61 x + 2 u}. To cope with the error which is nonlinear in nature, we need the following lemma stating that the nonlinear error set (x; u) can be represented by a linear uncertainty set. Lemma 1 (Equivalent uncertainty set, Wang and Zhan [35]). Let l be the following linear uncertainty set: l (x; u) = {1 M1 x + 2 M2 u: M1 ∈Rn×n ; M2 ∈Rn×m ; M1t M1 6 I; M2t M2 6 I }: Then (x; u) = l (x; u). The aforementioned result is then expressed in terms of DFE structure to characterize the modeling error. To this end, we observe that Lemma 1 implies there exist two norm-bounded uncertainty matrices F1 and F2 satisfying F1t F1 6I and F2t F2 6I , respectively, such that e(x; u) = 1 F1 x + 2 F2 u     2 1 I F1 (1 I )x + I F2 (2 I )u = 1 2

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       F1 0 1 I F1 0 0 1 2 1 2 = I I x+ I I u 1 2 1 2 0 0 F2 2 I

  0 F2 

   

D

F

D

E1

= DFE1 x + DFE2 u;

F

E2

(1)

where 1 ¿0; 2 ¿0. Remarks. 1. Eq. (1) states the fact that modeling errors, under a mild assumption, can be expressed in a DFE structure. 2. It should be emphasized that although only modeling error is addressed, the similar technique can be readily applied to nonlinear uncertainty structure. SpeciFcally, if uncertainties are involved in the nonlinear model as x˙ = [f(x) + Mf(x)] + [g(x) + Mg(x)]u, then under a similar assumption Mf(x) + Mg(x)u61 x + 2 u as that of Assumption 1, it is justiFable to designate a DFE structure for the corresponding nonlinear uncertainties. 3. The determination of norm bound coe4cients 1 and 2 can be solved using constrained optimization softwares (i.e., Lagrange multipliers or Kuhn–Tucker method). In fact, a routine, known as fmincon, is available in the optimization toolbox of Matlab. It is formulated as follows (here only the method to Fnd 1 is shown). Given

range of state vector x f(x) − ri=1 i (x)Ai x maximize ; x

x = 0:

Note that in using this optimization routine fmincon, several tries on diHerent initial conditions may be required to capture the global extremes for the fact that the objective function is nonlinear. 4. Our approach is diHerent from the works of [8,9]. The method assumes a bounding matrix that bounds the modeling error in two-norm sense and then determines the controller gains by solving the corresponding Riccati inequality which is expressed in terms of fuzzy system matrices and bounding matrices. In contrast, our modeling error is directly expressed in a DFE structure. Moreover, explicit expressions for controller gains and connections with LQG optimal control are investigated. 5. In fuzzy literature, the technique for representing nonlinear systems by T–S fuzzy models is based on exact fuzzy modeling technique [21,22,25,26] where the output of the constructed fuzzy model is mathematically identical to that of original nonlinear system. Note that only a certain class of nonlinear system such as mass–spring system, Lorenz system and systems with sector nonlinearity [31] can have exact fuzzy model representations. Most plants in the industry have severe nonlinearity and uncertainties, posing additional di4culties to the exact modeling techniques. But the proposed optimization approach can tackle more severe nonlinearity and this is one of the focus of this work. 2.2. Fuzzy systems with DFE uncertainty structure On the heels of Section 2.1, we, in this paper, assume that the nonlinear control system is represented by the Takagi–Sugeno fuzzy model in which both plant modeling uncertainty and exogenous

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signal uncertainty are considered for each individual rule Ri ; i = 1; : : : ; r. If z1 is F1i and · · · and zp is Fpi then x(t) ˙ = (Ai + Di FE1 )x(t) + (Bi + Di FE2 )u(t) + Dw w(t);

(2)

where x is state vector from x1 to xn . z is premise vector from z1 to zp . Fpi denotes a fuzzy set of zp for rule i. Ai ∈Rn×n is state matrix of subsystem i. Bi ∈Rn×m is input matrix of subsystem i. Di ; E1 ; E2 are known matrices of subsystem i. F is uncertain parameter matrix satisfying F t F6I . Dw ∈Rn×q is disturbance matrix. w(t)∈Rq×1 is unknown but bounded L2 (0; ∞) external disturbance. By using a standard fuzzy inference method—singleton fuzziFer, product fuzzy inference and weighted average defuzziFer—the fuzzy model is established [25,32]. The inferred fuzzy model is described by x(t) ˙ =

r 

i [(Ai + Di FE1 )x(t) + (Bi + Di FE2 )u(t)] + Dw w(t);

(3)

i=1

where  = [1 ; : : : ; r ]t ∈Rr is the normalized Fring strength satisfying i ¿0 and ri=1 i = 1. A linear state feedback controller for each rule of the fuzzy model, known as parallel distributed compensation (PDC) [27,34], is designed to stabilize the uncertain fuzzy system (3) with a given level of disturbance attenuation for all admissible uncertainty satisfying F t F6I . They are formulated as If z1 is F1j and · · · and zp is Fpj Then u(t) = Kj x(t);

j = 1; 2; : : : ; r

(4)

and the overall defuzziFed controller output becomes u(t) =

r 

j Kj x(t);

j=1

r 

j = 1:

(5)

j=1

Substituting (5) into (3), the closed-loop system becomes x(t) ˙ =

r r  

i j [Ai + Di FE1 + (Bi + Di FE2 )Kj ]x(t) + Dw w(t):

(6)

i=1 j=1

Associated with the closed-loop system (6) is the H ∞ control performance criterion when initial condition is considered [8].



∞ t t 2 z (t)z(t) dt ¡ x (0)Px(0) + ( wt (t)w(t) dt; (7) 0

0

where z(t) is the controlled output z(t) = Cx(t):

(8)

And C ∈Ro×n is a designed matrix, P is a positive-deFnite weighting matrix and (¿0 is a prescribed level of disturbance attenuation we wish to achieve for system (6). The objective of the next section

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is to exploit the relations between robust H ∞ control and quadratic stabilizability of an uncertain fuzzy system in which a DFE structure with norm-bounded uncertainty F t F6I is formulated. 3. Main result The approach adopted in this paper to tackle the uncertain fuzzy control problem involves solving a parameter-dependent algebraic Riccati equation associated with an H ∞ -norm bound constraint (. To Fx ideas, we deFne the fuzzy algebraic Riccati equations (FAREs) corresponding to the problem of quadratic stabilization with an H ∞ -norm bound (¿0 as follows. Denition 1 (FAREs). Let (¿0 be a design parameter. A positive-deFnite symmetric matrix P∈Rn×n and controller gains Yj ∈Rn×m and scalars ,j ¿0 are solutions to the following FAREs: [Ai − Bi R−1 E2t E1 ]t P + P[Ai − Bi R−1 E2t E1 ] + ,j PDi Dit P + ,j P[Yj R−1 Yjt − Bi R−1 Yjt − Yj R−1 Bit ]P +

1 t E [I − E2 R−1 E2t ]E1 + (−2 PDw Dwt P + C t C + -ij = 0 ,j 1

(9)

where R := E2t E2 ¿0 and -ij ¿0, i; j = 1; : : : ; r, serving as a connection between inequality constraint and equality constraint. Remarks. 1. There are a total of r 2 AREs for the uncertain fuzzy system (3) governed by the fuzzy controller (10). For convenience, we denote G as the set of FAREs comprising r 2 equalities, i; j = 1; : : : ; r. 2. Note that in the absence of uncertainty and external disturbance w(t), the FAREs (9) becomes (Ai + Bi Kj )t P + P(Ai + Bi Kj ) + PBj Bjt P + C t C + -ij = 0 if Kj = − Bjt P. This corresponds to the Riccati equation which arises when one solves the standard LQR problem with performance index

∞ J = (z t z + ut u) d/: 0

The stabilizability of the uncertain fuzzy system (3) with a prescribed H ∞ -norm bound constraint on disturbance attenuation for all admissible uncertainties will be established using a quadratic Lyapunov function. We have the following main result stating that existence of a common P solution and Yj to a set of parameter-dependent AREs implies quadratic stability of uncertain fuzzy systems. Theorem 1 (Solvability of FAREs G). Given a constant (¿0. If for some -ij ¿0 there exist constants (,1 ; : : : ; ,r ) such that the parameter-dependent FAREs (9) have a common positive-de8nite symmetric solution P and (Y1 ; : : : ; Yr ). Then the uncertain fuzzy system (3) is quadratically

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195

stabilizable with disturbance attenuation via a fuzzy controller. Furthermore, a suitable stabilizing control law is given by u(t) =

r 

j Kj x(t);

Kj = −(E2t E2 )−1 (,j Yjt P + E2t E1 ):

(10)

j=1

Proof. Suppose the FAREs (9) has a positive-deFnite symmetric P for some ,j ¿0. We will show that the uncertain fuzzy system (3) is quadratically stabilizable with disturbance attenuation via a fuzzy control law (10) with the Lyapunov function V (t) = xt Px. Indeed, the time derivative of V corresponding to the closed-loop system is given by V˙ =

r  r 

i j xt {(Ai + Bi Kj )t P + P(Ai + Bi Kj ) + 2PDi F(E1 + E2 Kj )}x

i=1 j=1

+ xt PDw w + wt Dwt Px: Using a matrix fact DFE + E t F t Dt 6,DDt + 1=,E t E, we have V˙ 6

r  r 

i j xt {(Ai + Bi Kj )t P + P(Ai + Bi Kj ) + ,j PDi Dit P

i=1 j=1

+

1 (E1 + E2 Kj )t (E1 + E2 Kj )}x + (−2 xt PDw Dwt Px + (2 wt w: ,j

(11)

Let Kj = −(E2t E2 )−1 (,j Yjt P+E2t E1 ). It is straightforward but tedious to verify the following equalities: 1 t {E E2 Kj + Kjt E2t E1 } = −E1t E2 (E2t E2 )−1 Yjt P − PYj (E2t E2 )−1 E2t E1 ,j 1 −

2 t E E2 (E2t E2 )−1 E2t E1 ,j 1

(12)

1 t t K E E2 Kj = E1t E2 (E2t E2 )−1 Yjt P + PYj (E2t E2 )−1 E2t E1 ,j j 2 +

1 t E E2 (E2t E2 )−1 E2t E1 + ,j PYj (E2t E2 )−1 Yjt P: ,j 1

Substituting (12)–(13) into (11) leads to V˙ 6

r r  

i j xt {[Ai − Bi (E2t E2 )−1 E2t E1 ]t P + P[Ai − Bi (E2t E2 )−1 E2t E1 ]

i=1 j=1

+ ,j PDi Dit P + ,j P[Yj (E2t E2 )−1 Yjt − Bi (E2t E2 )−1 Yjt − Yj (E2t E2 )−1 Bit ]P +

1 t E [I − E2 (E2t E2 )−1 E2t ]E1 + (−2 PDw Dwt P}x + (2 wt w: ,j 1

(13)

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Now using (9), it follows that V˙ + z t (t)z(t) − (2 wt (t)w(t) 6 −

r  r 

i j 2min (-ij )x22 ¡ 0

(14)

i=1 j=1

for all x = 0. Hence, we conclude that (10) is the required stabilizing fuzzy control law. Integrating (14) from t = 0 to ∞ yields



∞ t 2 V (∞) − V (0) ¡ − z (t)z(t) dt + ( wt (t)w(t) dt: 0

It follows that



t 2 z (t)z(t) dt ¡ V (0) + ( 0

and thus

H∞

0

∞ 0

wt (t)w(t) dt

requirement (7) is achieved.

Remarks. 1. The present paper addresses robust stabilization problem and H ∞ performance of a nonlinear control system represented by an uncertain fuzzy system where the unstructured DFE uncertainty can be modeling error or nonlinear uncertainty or both. When compared with the results on robust fuzzy control [25, Theorems 3.4, 4.1, 4.2], our main result, Theorem 1, is distinct from them. SpeciFcally, we have the following observations. (a) The uncertainty in [25] can be a lumped fuzzy subsystem when a nominal system is extracted or a convex combination of some known matrices or both. (b) Although no disturbance was involved in [25] (i.e., a pure robust stabilization problem, no performance consideration), the robust results were obtained based on the concept that an uncertain linear system has a corresponding standard H ∞ formulation without uncertainty and, therefore, the stabilization problem of an uncertain fuzzy system can be solved by the existing H ∞ techniques. (c) In their formulation, only one Riccati inequality or its equivalent is required because of the extracted nominal term A0 . (d) DiHerent choice of A0 aHects stability analysis. 2. Before proceeding with further analysis on the fuzzy (nonlinear) controller (10), note that when (Yj ; ,j ) are identical for all j = 1; : : : ; r, a linear controller is recovered. In the sequel, the analysis on fuzzy (nonlinear) controllers and linear controllers is relegated in the next two sections. 3.1. Fuzzy controller The control law (10) is directly analogous to the optimal controller in the standard LQR control theory. In theory, Theorem 1 has established a su4cient condition for quadratic stabilizability of an uncertain fuzzy system satisfying H ∞ disturbance attenuation: Solvability of a set of FAREs. In practice, there are no simple analytical solution Fnding a common P to the set of FAREs G. Fortunately, a publicly available interior-point LMI solver can be applied. To this end, introducing new variables S = P −1 and Xj = ,j Yj , the FAREs (9) can be reformulated as Riccati-like inequalities shown below: 1 S[Ai − Bi R−1 E2t E1 ]t + [Ai − Bi R−1 E2t E1 ]S + SE1t [I − E2 R−1 E2t ]E1 S + SC t CS ,j + (−2 Dw Dwt + ,j Di Dit + [Xj (,j R)−1 Xjt − Bi R−1 Xjt − Xj R−1 Bit ] ¡ 0:

(15)

J.-C. Lo, M.-L. Lin / Fuzzy Sets and Systems 143 (2004) 189 – 209

Using Schur complements [1], we have the following parameter dependent LMIs:    S[Ai − Bi R−1 E2t E1 ]t      +[Ai − Bi R−1 E t E1 ]S   2   t −1 t t X SE [I − E R E ] SC    j 2 1 2   −Bi R−1 X t − Xj R−1 Bt   j i      +(−2 Dw Dt + ,j Di Dt  ¡ 0; w i     t   X −, R 0 0 j j     −1 t   [I − E R E ]E S 0 −, I 0 2 1 j 2   CS

0

0

197

(16)

−I

where i; j = 1; : : : ; r. Since (16) is convex with respect to S; Xj and ,j , the (S; Xj ; ,j ) can be determined via LMI. In view of the result above, we have the following theorem stating a su4cient condition for quadratic stability in the context of LMI formulation. Theorem 2 (QS with disturbance attenuation—fuzzy controller). Given (¿0. The uncertain fuzzy system (3) is quadratically stabilizable via a fuzzy control law (10) while satisfying the disturbance attenuation requirement (7) whenever there exist positive scalars (,1 ; : : : ; ,r ), a positivede8nite symmetric matrix P = S −1 and controller gains (X1 ; : : : ; Xr ) such that (16) is satis8ed. Furthermore, the local controller gain is simpli8ed to Kj = − (E2t E2 )−1 (Xjt S −1 + E2t E1 ). Proof. Follow immediately from Theorem 1 and the equivalence between quadratic matrix inequality and LMI through Schur complements. Note that the problem of Fnding P = S −1 and (X1 ; ,1 ; : : : ; Xr ; ,r ) satisfying (16) is a Fnite-dimensional convex feasibility problem similar to the one described in [1]. It can be solved using the LMI control toolbox. In the sequel, we denote (S; X1 ; ,1 ; : : : ; Xr ; ,r ) as a feasible solution. Since the feasibility problem solver feasp often yields undesirable high gain solutions, we, instead, use LMI linear minimization solver mincx where the objective function is to minimize the controller gain. 3.2. Linear controller Theorem 1 gives a su4cient condition for quadratic stabilizability via a fuzzy controller. The next theorem states that the solvability of a set of LMIs consisting of r 2 inequalities guarantees a simple linear state feedback controller that can be found by solving a set of r LMIs. It is indicated in [30, Remark 11, p. 56] that for a non-PDC controller (i.e., plant and controller do not share the same membership functions), feasibility condition to a set of r 2 LMIs is suggested. This research shows that searching for a non-PDC controller via r 2 -LMIs can be avoided because a non-PDC fuzzy structure provides no advantage over a linear controller. Theorem 3 (Fuzzy controller ⇒ Linear controller). If the uncertain fuzzy system (3) is quadratically stabilizable via a fuzzy controller through the r 2 -LMIs method, then it is also quadratically stabilizable via a linear controller (i.e., Kj s are identical).

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Proof. Given the assumption on the uncertain fuzzy system (3) being quadratic stabilizable via r 2 -LMIs method, solving the resulting LMIs generate a feasible solution (S; X1 ; ,1 ; : : : ; Xr ; ,r ) and then any permutations of (Xj ; ,j ) are still a solution to the resulting LMIs. To see this, categorize the set of r 2 LMIs (16) into r groups: That is, for a Fxed j, each group comprises r inequalities with i = 1; : : : ; r. Observe that each group has exactly identical terms except the (Xj ; ,j ), meaning a solution, for example, (S; X1 ; ,1 ) to group 1 is also a solution to other groups as well. Consequently, (S; X1 ; ,1 ; : : : ; X1 ; ,1 ) is a solution to the LMIs (16) and hence, a linear controller results. In light of Theorem 3, the following remarks are immediate. Remarks. 1. It should be emphasized that Theorem 3 states the result that under a conservative solution (i.e., r 2 -LMIs approach), existence of a fuzzy controller implies existence of a linear one. In other words, if a linear controller determined by solving a set of r LMIs does not exist then no fuzzy controller can be found via solving a set of r 2 LMIs. As such, other common P methods such as the relaxed method [26], which requires a set of r(r + 1)=2 inequalities, and/or the improved relaxed method [20] should be considered for seeking a fuzzy controller. 2. The signiFcance of Theorem 3 is that a fuzzy controller obtained via the r 2 -LMIs approach oHers no advantage over a linear state feedback controller in the satisfaction of the H ∞ requirement. The claim coincides with the results in [18,35] in the robustness theory. To synthesize the linear controller, one can construct the LMIs which are a direct result of (16) by setting X1 = · · · = Xr = X and ,1 = · · · = ,r = ,. Therefore, (16) leads to 

S[Ai − Bi R−1 E2t E1 ]t





     +[Ai − Bi R−1 E t E1 ]S   2   t −1 t t   X SE [I − E R E ] SC  2 1 2   −Bi R−1 X t − XR−1 Bt   i     − 2 t t  +( Dw D + ,Di D  ¡ 0; w i     t  X −,R 0 0      − 1 t  0 −,I 0  [I − E2 R E2 ]E1 S   CS 0 0 −I

i = 1; : : : ; r:

(17)

The Fnding is summarized as follows. Theorem 4 (QS with disturbance attenuation—linear controller). Given (¿0. The uncertain fuzzy system (3) is quadratically stabilizable via a linear control law while satisfying the disturbance attenuation requirement (7) whenever there exist a positive scalar ,, a positive-de8nite symmetric matrix P = S −1 and a controller gain X such that (17) is satis8ed. Furthermore, the controller gain is simpli8ed to K = − (E2t E2 )−1 (X t S −1 + E2t E1 ). Proof. Follow immediately from Theorem 2 and 3.

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199

Before we end this section, it is worth summarizing some of the signiFcances we have tried to convey in this presentation. They are listed below: 1. If the LMIs (16) have feasible solutions, then any (Xi ; ,i ) constitutes a linear controller and is su4cient to stabilize the uncertain fuzzy system (and, correspondingly, the nonlinear control system) with the required performance. This is guaranteed by Theorems 2 and 3. 2. In concept, our approach is vaguely similar to the work [2] which investigates discrete-time robust H ∞ problems via a relaxed method to Fnd state feedback gains without an explicit expression. In contrast, our presentation studies an explicit expression for the controller gains, draws connection with the standard LQR control theory, and analyzes the implication between fuzzy controllers and linear controllers. Furthermore, connections between a nonlinear control system and an uncertain fuzzy control system with a DFE structure is based on rigorous machinery while the same concern was shown via an example. 3. JustiFcation of the DFE structure characterizing modeling error and/or nonlinear uncertainty in fuzzy systems is exploited. Hence, the gap between robust theories on uncertain (T–S) fuzzy systems and the existing robust theories on uncertain linear systems is bridged.

4. Examples The problems of controlling a mass–spring–damper system and balancing an inverted pendulum are considered in this section as illustrations to demonstrate the applicability of materials proposed in the previous sections such as synthesis of modeling errors, the stability analysis and disturbance attenuation. 4.1. Mass–spring–damper system by

The dynamic equation of a mass–spring–damper mechanical system borrowed from [21] is given M y(t) S + g(y(t); y(t)) ˙ + f(y(t)) = 6(y(t))u(t) ˙ + w(t);

where M = 1 is the mass and assume that y(t) ∈ [−1:5

1:5];

y(t) ˙ ∈ [−1:5

1:5];

g(y(t); y(t)) ˙ = c1 y(t) + c2 y(t) ˙ + c3 y˙ 3 (t) + 8d (t)(c4 y(t) + c5 y(t) ˙ + c6 y˙ 3 (t)); f(y(t)) = c7 y(t) + c(t)y(t); 6(y(t)) ˙ = 1 + c8 y˙ 3 (t); 8d (t) ∈ [−1

1];

c(t) = c9 + c10 cos(2:t):

(18)

200

J.-C. Lo, M.-L. Lin / Fuzzy Sets and Systems 143 (2004) 189 – 209

In the case where [c1

c2

= [0:01

c3

c4

c5

1

0

0

0

0

0:1

c6 0

c7 0

c8

0:13

c9

c10 ]

1:155

0:655]

or = [0:01

0:01

0

0:1

0:01 0

0

0]

it is a problem considered in [21, Examples 1 or 2]. In this example, we assume that the parameters are given as [c1

c2

= [0:01

c3 1

c4

0:1

c5

c6

c7

0:01

0

0:1

c8

c9

0:01

c10 ] 0:13

1:155

(19)

0:655];

which is a combination of examples in [21]. Note that the exact modeling techniques used in [21] requires four rules such that a T–S fuzzy model can represent system (18) exactly while our approach needs only two rules. ˙ y(t)]t , we can obtain the state space representation of system (18) Denoting x(t) = [x1 x2 ]t = [y(t) with parameters (19) x(t) ˙ = F(x; t)x(t) + G(x; t)u(t) + Dw w(t); where

 F(x; t) =  G(x; t) =

(20)

−1 − 0:1(1 + 8d (t))x12 −1:175 − 0:018d (t) − 0:655 cos(2:t) 1  3

1 + 0:13x1 0



0 ;

Dw =

  1 0

;

:

A two-rule T–S fuzzy model is designed to approximate the nonlinear system If x1 is about F1i then x(t) ˙ = (Ai + D1 -1 Ed )x(t) + Bi u(t) + Dw w(t); where i = 1; 2 and the local system matrices are       −1 −1:175 1:329 0:671 A1 = A2 = ; B1 = ; B2 = 1 0 0 0       0 −0:1 −0:655 8d (t) 0:1 ; Ed = ; -1 = : D1 = 0 cos(2:t) 0 0 1 Note that the uncertain term D1 -1 Ed with -t1 -1 6I comes from the uncertainty 8d (t) and c(t) in the nonlinear system (18). The Fring strength i s of fuzzy sets F1i are x1 1 = 0:5 + ; 2 = 1 − 1 : 3

J.-C. Lo, M.-L. Lin / Fuzzy Sets and Systems 143 (2004) 189 – 209

201

The overall defuzziFed approximated system becomes x(t) ˙ =

2 

i [Ai x(t) + Bi u(t)] + D1 M1 Ed x(t) + Dw w(t):

(21)

i=1

The modeling error e(x; t) becomes     −0:1(1 + 8d (t))x12 0 0:13x13 − 0:3291 + 0:3292 e(x; t) = u(t): x(t) + 0 0 0 By further calculation on e(x; t) with x1 ∈[−1:5 1:5] and 8d (t)∈[−1 1], we have e 6 max {0:11 + 8d (t)x12 }x x1 ;8d (t)

+ max{0:13x13 − 0:3291 + 0:3292 }u x1

= 1 x + 2 u; where 1 = 0:45, 2 = 0:1098. The modeling error e(x; t) can be turn into     1 f1 (t) 0 2 f2 (t) e(t) = x(t) + u(t) 0 0 0 = D2 -2 Ee1 x(t) + D2 -2 Ee2 u(t); where -t2 -2 6I and   1 =0:5 2 =0:2 D2 = ; 0 0

 -2 =

f1 (t)

0

0

f2 (t)



 ;

Ee1 =

0:5 0 0 0



 ;

Ee2 =

0 0:2

 :

Adding the modeling error e(x; t) into the approximated system (21), we have an uncertain T–S fuzzy model representing the nonlinear system (18) x(t) ˙ =

2 

[i (Ai + DFE1 )x(t) + (Bi + DFE2 )u(t)] + Dw w(t);

i=1

where the uncertain terms DFEi with F t F6I is deduced from the modeling error and the system uncertainties 8d (t) and c(t), i.e.,       -1 0 Ed 0 ; E1 = ; E2 = : D = [D1 D2 ]; F = Ee1 0 -2 Ee2 Consider a linear controller u(t) = Kx(t)

202

J.-C. Lo, M.-L. Lin / Fuzzy Sets and Systems 143 (2004) 189 – 209 1

0.5

x 1; x 2

0

-0.5

-1

-1.5 0

1

2

3

4

5 time (sec)

6

7

8

9

10

Fig. 1. State evolutions of example 1: x2 -solid line, x1 -dashed line.

and a controlled output z(t) = x(t). To determine the quadratic stabilizability of the above system with disturbance attenuation level ( = 1, a feasible solution to a set of 2 LMIs (17) does exist via LMI solver, stipulating a linear control law. Here are the Fndings   2:2341 −0:4667 S= −0:4667 0:4784 and

 X =

0:2866 0:0132

 ;

, = 0:6027;

K = [−4:2084

− 4:7948]:

Fig. 1 shows states of the nonlinear model (not fuzzy model) oscillate near equilibrium and then die out eventually. The initial condition is x(0) = [−1 − 1:25]t . Fig. 2 is the control eHort, indicating the control law can stabilize the nonlinear mass–spring–damper system with the required performance  achieved. Fig. 3 shows that H ∞ requirement on z t z¡xt (0)S −1 x(0) + (2 wt w is achieved, where xt (0)S −1 x(0) = 6:0357. The disturbance in the simulation is assumed to be the same as [21]. 4.2. Balancing an inverted pendulum A balancing problem of an inverted pendulum will be considered to demonstrate that robust H ∞ quadratic stabilization via a linear controller for an uncertain fuzzy system representing a nonlinear model given below is valid. x˙1 = x2 ; x˙2 =

(M + m)mgl sin x1 − f1 (M + m)x2 − (mlx2 )2 sin x1 cos x1 − ml cos x1 u + w; (M + m)(J + ml2 ) − (ml cos x1 )2

z = C[x1

x 2 ]t ;

J.-C. Lo, M.-L. Lin / Fuzzy Sets and Systems 143 (2004) 189 – 209

203

10

8

u(t)

6

4

2

0

-2 0

1

2

3

4

5 time (sec)

6

7

8

9

10

Fig. 2. Control eHorts of example 1. 2.5

t 2 t z (t)z(t) ; ρ ω (t)ω(t)

2

1.5

1

0.5

0

0

1

2

3

4

5 time (sec)

6

7

8

9

10

Fig. 3. H∞ performance of example 1: z t z-solid line, (2 wt w-dashed line.

where M = 10kg, m = 0:22kg, l = 0:304m, J = 0:004963kgm2 , f1 = 0:007056N=(rad=s), g = 9:8m=s2 . The nonlinear model is represented by four-rule Takagi–Sugeno uncertain fuzzy model as follows: If x1 is about F1i then x˙ = (Ai + Di FE1 )x + (Bi + Di FE2 )u + Dw w; where i = 1; 2; 3; 4 and the local system matrices are obtained by evaluating the nonlinear model at x1 = 0; :=9; 2:=9 and :=3, respectively. That is, for i = 1; 2; 3; 4 we have         0:1 0 0 1 0 0 Ai = ; Bi = ; C= ; Dw = 0 0:1 a1i a2i bi 1

204

J.-C. Lo, M.-L. Lin / Fuzzy Sets and Systems 143 (2004) 189 – 209

with den = (M + m)(J + ml2 ) − (ml cos x1 )2 ;  (M + m)mgl  ; a11 =  den x1 =0  (M + m)mgl sin x1  a1i = ; den x1 x1 =:=9;2:=9;:=3  −f1 (M + m)  a2i = ;  den x1 =0;:=9;2:=9;:=3  −ml cos x1  bi = :  den x1 =0;:=9;2:=9;:=3 The fuzzy sets F1i can be found in [8] and ( = 0:5. The external disturbance w is assumed to be L2 (0; ∞) square wave with amplitude 1 and period 1:5 s in the simulations. That is  amplitude = 1; t 6 3; w= amplitude = 1=t; t ¿ 3: Consider a linear controller u(t) = Kx(t): Using fmincon optimization procedure over the range x1 ∈[−1; 1] and x2 ∈[−3:5; 3:5] yields 1 = 0:1452, 2 = 0:0041. Then following Section 2 and example 1, let D1 = D2 = D3 = D4 = D and E1 , E2 be given below:   0:6000 0 0:5000 0 D= ; 0 0:6000 0 0:5000     0:2420 0 0        E1 =   0 0:2420  ; E2 =  0  : 0 0 0:0082 To determine the quadratic stabilizability of the above system, a feasible solution to a set of 4 LMIs (17) does exist via LMI solver, stipulating a linear control law. Here are the Fndings   29:3219 −130:7048 S= ; −130:7048 675:3687 and

 X =

−0:0397 −0:1506

 ;

, = 15:2636;

K = [254:2911

52:5301]:

J.-C. Lo, M.-L. Lin / Fuzzy Sets and Systems 143 (2004) 189 – 209

205

1 x1 (linear) x2 (linear) x1 (nonlinear) x (nonlinear) 2

0.5

0

x ;x 1 2

-0.5

-1

-1.5

-2

-2.5 0

1

2

3

4

5 time (sec)

6

7

8

9

10

Fig. 4. State evolutions of example 2: linear and nonlinear controllers.

350 linear nonlinear 300

250

u(t)

200

150

100

50

0

-50 0

1

2

3

4

5 time (sec)

6

7

8

9

10

Fig. 5. Control eHorts of example 2: linear and nonlinear controllers.

Fig. 4 shows states of the nonlinear model (not fuzzy model) oscillate near equilibrium and then die out eventually. The initial condition is x(0) = [1 0]t . Fig. 5 is the control eHort, indicating the control law can stabilize the nonlinear pendulum system with the required performance achieved. Fig. 6 shows that H ∞ requirement on z t z¡xt (0)S −1 x(0) + (2 wt w is achieved, where xt (0)S −1 x(0) = 0:2484. For illustrating purpose, it is interesting to know that a feasible solution also exists for a fuzzy controller shown below: u(t) =

4  j=1

j Kj x(t) = −

4  j=1

j (E2t E2 )−1 (Xjt S −1 + E2t E1 )x(t);

206

J.-C. Lo, M.-L. Lin / Fuzzy Sets and Systems 143 (2004) 189 – 209 0.25

z t (t)z(t)[linear] ρ2ω t(t)ω(t) z t(t)z(t)[nonlinear]

zt(t)z(t) ; ρ2ωt(t)ω(t)

0.2

0.15

0.1

0.05

0

0

1

2

3

4

5 time (sec)

6

7

8

9

10

Fig. 6. H∞ performance of example 2: linear and nonlinear controllers.

where ,1 = 10:6760; ,2 = 10:6958; ,3 = 10:7503;   26:4982 −117:7181 S= −117:7181 613:0500 and

 [X1

X2

X3

X4 ] =

,4 = 17:0638;

−0:0323 −0:0319 −0:0321 −0:1036 −0:1779 −0:1755 −0:1713 0:0412

 ;

whose controller gains are 

K1





253:7684 53:0452



     K2   250:5177 52:3618   =       K3   248:2112 51:8172  K4

365:4836 69:1811

and xt (0)S −1 x(0) = 0:2568: Lastly, when compared with [8,9], the proposed method requires no Fnal check on whether the modeling error is bounded by the DFE uncertainty because Lemma 1 guarantees that the modeling error can be represented by DFE uncertainty. In contrast, their works need a Fnal check on the assumptions of boundedness. See [8, step 7, p. 580]; [9, step 6, p. 258].

J.-C. Lo, M.-L. Lin / Fuzzy Sets and Systems 143 (2004) 189 – 209

207

5. Conclusion Under a sector-type assumption on modeling error, exact representation of a nonlinear control system by an uncertain fuzzy system with DFE structure is established. Based on robust control theory, stability analysis and design of uncertain fuzzy control systems are investigated for the case where both plant uncertainty and exogenous signal uncertainty are present. The structure for uncertainty considered in the uncertain fuzzy model is the standard norm-bounded time-varying formulation—[MA MB] = DF[E1 E2 ] structure. The existence of a common P matrix to a Fxed Lyapunov function is obtained by solving the fuzzy equivalent of a set of parameter-dependent AREs involving uncertainties and H ∞ constraint (. Also, construction of the desired stabilizing state feedback control law is given in terms of a feasible solution to the FAREs. Consequently, guarantee on H ∞ performance is oHered along with the stability using the algebraic Riccati approach. To solve the simultaneous FAREs, the FAREs are converted into their equivalent LMI formulations which become a convex feasibility problem and can be solved using an extremely e4cient and publicly available LMI software. Some salient features that are distinct from the existing literature are emphasized in this presentation too. Acknowledgements This work was supported in part by the National Science Council of the ROC under grant NSC89-2213-E-008-075. References [1] S. Boyd, L.E. Ghaoui, E. Feron, V. Balakrishnan, Linear Matrix Inequalities in System and Control Theory, SIAM, Philadelphia, PA, 1994. [2] Y.-Y. Cao, P. Frank, Robust H∞ disturbance attenuation for a class of uncertain discrete-time fuzzy systems, IEEE Trans. Fuzzy Systems 8 (4) (2000) 406–415. [3] S. Cao, N. Rees, G. Feng, H ∞ control of nonlinear continuous-time systems based on dynamical fuzzy models, Internat. J. Systems Sci. 27 (9) (1996) 821–830. [4] S. Cao, N. Rees, G. Feng, Analysis and design for a class of complex control systems part I: fuzzy modelling and identiFcation, Automatica 33 (6) (1997) 1017–1028. [5] S. Cao, N. Rees, G. Feng, Analysis and design for a class of complex control systems part II: fuzzy controller design, Automatica 13 (6) (1997) 1029–1039. [6] S. Cao, N. Rees, G. Feng, H∞ control of uncertain fuzzy continuous-time systems, Fuzzy Sets and Systems 115 (2000) 171–190. [7] C. Chen, P. Chen, C. Chen, Analysis and design of fuzzy control system, Fuzzy Sets and Systems 57 (1993) 125–140. [8] B. Chen, C. Tseng, H. Uang, Robustness design of nonlinear dynamic systems via fuzzy control, IEEE Trans. Fuzzy Systems 7 (5) (1999) 571–585. [9] B. Chen, C. Tseng, H. Uang, Mixed H2 =H∞ fuzzy output feedback control design for nonlinear dynamic systems: an LMI approach, IEEE Trans. Fuzzy Systems 8 (3) (2000) 249–265. [10] G. Feng, Approach to quadratic stabilization of uncertain fuzzy dynamic system, IEEE Trans. Circuits Systems-I: Fund. Theory Appl. 48 (2001) 760–769. [11] G. Feng, S. Cao, N. Rees, C. Cheng, J. Ma, H∞ control of continuous time fuzzy dynamic systems, in: Proc. 6th IEEE Int’l Conf. Fuzzy Systems, vol. 2, Barcelona, Spain, 1997, pp. 1141–1146.

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