Robust stabilizing controller design for Takagi–Sugeno fuzzy descriptor systems under state constraints and actuator saturation

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Robust stabilizing controller design for Takagi–Sugeno fuzzy descriptor systems under state constraints and actuator saturation Quoc Viet Dang, Laurent Vermeiren ∗ , Antoine Dequidt, Michel Dambrine LAMIH, UMR CNRS 8201, University of Valenciennes et du Hainaut-Cambresis, Le Mont Houy, 59313 Valenciennes, France Received 25 February 2016; received in revised form 13 February 2017; accepted 17 February 2017

Abstract The aim of this paper is to present a new framework for designing robust controllers for uncertain nonlinear systems described by Takagi–Sugeno fuzzy descriptor systems and subject to input and state constraints. For this, a novel structure of modified parallel distributed compensation control laws is proposed. The controller parameters can be obtained through the numerical resolution of an optimization problem with linear matrix inequality constraints derived from Lyapunov’s stability theory and the choice of an original Lyapunov function. This ensures the stabilization of the closed-loop system and the respect of the state and input constraints regionally. The technique is illustrated through three numerical examples. © 2017 Elsevier B.V. All rights reserved. Keywords: Takagi–Sugeno fuzzy descriptor systems; Actuator saturation; Parallel distributed compensation; Stability; Linear matrix inequality

1. Introduction Since their definition in [1] Takagi–Sugeno (T–S) fuzzy models have been widely used for the control or observation of nonlinear systems. Their universal approximation property makes them excellent candidates for nonlinear black box modeling. Alternatively, using the sector nonlinearity approach presented in [2], a Takagi–Sugeno representation of a given knowledge-based model describing the same dynamics can be obtained by possibly limiting the state space. Lyapunov’s direct method applied with quadratic Lyapunov functions leads to a systematic framework for the design of stabilizing controllers with the so-called parallel distributed compensation (PDC) structure [2–7] relying on the numerical resolution of optimization problems with constraints expressed on the form of linear matrix inequalities (LMIs) [8]. Improvements of this approach have been proposed by considering other types of Lyapunov functions such as piecewise quadratic functions [9,10], non-quadratic Lyapunov functions [11–15] or line-integral Lyapunov functions [16,17]. Extensions to polynomial TS models have also been proposed based on the sum-of-squares approach [18,19]. * Corresponding author. Fax: +33 3 27 51 13 16.

E-mail address: [email protected] (L. Vermeiren). http://dx.doi.org/10.1016/j.fss.2017.02.006 0165-0114/© 2017 Elsevier B.V. All rights reserved.

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Another way to reduce the conservatism of the results is to use the descriptor system approach for controller synthesis [20–25]. In practice, many physical systems can be naturally represented by regular descriptor form [26] such as, for example, mechanical systems. By inversion, it is always possible to rewrite such a model in a standard T–S model. However, the inversion procedure results in increasing the number of nonlinear terms. As with the sector nonlinearity approach the obtained standard T–S model has a number of local linear models which grows exponentially according to the number of nonlinearities in the original model; this may lead to complex systems untractable with current solvers, and this even for small order systems. Since no inversion procedure is needed for obtaining T–S fuzzy descriptor models as introduced in [20], an equivalent representation of an initial first-principles model can be obtained with a smaller number of local linear models. Another interest of T–S fuzzy descriptor models is that the original structure of the nonlinear systems is preserved. Due to physical/technological limitations or safety reasons, actuator saturation is a very common nonlinear phenomenon encountered in practice. Their effects can result in significant performance degradation of a controlled system and even make it unstable [27]. Therefore, the design of stabilizing controllers for dynamical systems under input saturation has been attracting the attention of several researchers (see the monographs [28–31] and references therein). Several methods have been developed to deal with the input saturation problem; among them the anti-windup strategy is certainly the most popular used in practice. The main idea of this strategy is to design a nominal controller that meets the desired performance requirements regardless of the saturation effects, and then an additional antiwindup compensator is introduced in order to minimize performance degradation of the closed-loop system [32–36]. However, most of these works deal with linear systems through a two-step method in which the nominal controller and the anti-windup compensator are designed separately. In recent years, some researchers have extended the antiwindup design to nonlinear systems. This has been done for systems in special forms such as feedback linearizable systems [37], quadratic systems [38], or rational differential systems [39,40]. In T–S fuzzy model-based control framework, there are also some works dealing with the design of anti-windup controllers. Among results existing in the literature, it can be cited [41] in which is given a simultaneous design method of an anti-windup based dynamic output feedback controller for constrained T–S fuzzy systems with external disturbances considering robust performance of the closed-loop system. This result is the extension of the anti-windup approach proposed in [42] combined with the dynamic output feedback controller for continuous-time T–S fuzzy systems given in [43]. Another common approach to deal with the saturation effect is to use a polytopic representation of the saturation as in [44]. Recently, for T–S fuzzy systems with stable local linear models, another way to take the saturation effects into account relying on the sector nonlinearity approach has been proposed in [45,46]. In [47,48], the same approach has been revisited resulting in the reduction of decision variables with respect to [45] and enlarging the procedure to unstable systems. Other extension to unstable systems with the design of dynamic output feedback has been proposed in [49]. Few papers concern the controller synthesis for T–S fuzzy descriptor systems subject to input saturation, one may refer to [50,47] where the first one used the polytopic decomposition of the saturation for the control of singular systems, whereas the second consider regular description systems and use the sector nonlinearity approach for the modeling of the saturation effect. Another result [51] deals with the anti-windup design for T–S fuzzy singular descriptor systems with time-delay. In this reference, it is essentially a two-step method that is given since the knowledge of a stabilizing dynamic output controller is assumed. In this paper, a new systematic framework is proposed for the design of a robust controller stabilizing an uncertain, regular T–S fuzzy descriptor system under state and input constraints. In order to get a simple controller allowing to cope with input saturation, a novel structure of modified PDC control law is proposed. Using a new structure for the Lyapunov function, conditions for the controller design are given which may be solved using LMI convex optimization. The rest of this paper is organized as follows: In section 2 a description of the considered class of systems is given together with the control problem statement; the main results are then presented in Section 3 with the proposed control law and the LMI-based design conditions; in section 5 some simulation results are reported, and conclusions are finally given in section 6.

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2. Notations The following notations will be used in this paper: • • • • • • • • • • •

n×m is the set of n × m real matrices. n denotes the set of n × 1 real vectors. Nn denotes the set of integer numbers {1, ..., n}.  Sn denotes the subset of n defined by: Sn = {x ∈ n : xi ≥ 0 ∀i ∈ Nn and ni=1 xi = 1}. AT stands for the transpose of the matrix A. He(A) = A + AT . A(i) stands for the i-th row of the matrix A. I, 0 are the identity and zero matrices of appropriate dimensions. P > 0 means that P is a symmetric, positive-definite matrix. diag (A1 , ..., An ) is the block diagonal matrix with square matrices as diagonal elements A1, ..., An . An asterisk * in a symmetric partitioned matrix denotes elements that can be deduced by symmetry. n  • Ah represents the convex combination hi Ai associated with a vector h ∈ Sn and a family of matrii=1

ces (A1 , ..., An ) having the same dimensions. Similarly, Ahh will denote a convex combination of the form n n   hi hj Aij , . . . i=1 j =1  • Given a matrix K ∈ m×n and a vector u ∈ m , P (K, u) is the polyhedral set defined by x ∈ n : |K(i) x| ≤ ui ,  ∀i ∈ Nm .   • Given a matrix P > 0, E (P ) denotes the ellipsoidal set defined by x ∈ n : x T P −1 x ≤ 1 . 3. Problem statement 3.1. System description In this paper, we consider a class of systems that can be described by the following uncertain, regular T–S fuzzy descriptor model with saturated inputs1 : nw 

wk (z) (Ek + Ek ) x˙ =

k=1

nh 

hi (z) [(Ai + Ai ) x + (Bi + Bi ) sat (u)]

(1)

i=1

where x(t) ∈ nx is the state vector constrained for technological or modeling reasons to remain in the polyhedral set x with 2nc faces defined by   x = P N, [1 . . . 1]T , (2) for a given matrix N ∈ nc ×nx . u(t) ∈ nu is the non-saturated control input; Ai , Ai , Ek , Ek are real n-by-n matrices; Bi , Bi are real n-by-nu matrices. It is assumed that the premise vector z(t) ∈ nz depends only on the measured variables; the membership functions wk and hi are such that w(t) = [w1 (t) . . . wnw (t)]T has values in Snw , and h = [h1 (t) . . . hnh (t)]T has value in Snh ; the saturated input vector sat(u) ∈ nu has for i-th component: sat(u)i = sign(ui ) min(u0,i , |ui |),

(3)

where u0,i > 0 is the saturation bound of the i-th control input. It is assumed that all the matrices Ek are nonsingular. The admissible uncertainties Ek , Ai , and Bi are assumed to be of the following form: Ek = Hke e Wke ;

Ai = Hia a Wia ;

1 Time is omitted when there is no ambiguity.

Bi = Hib b Wib

(4)

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with given constant matrices Hke , Hia , Hib , Wke , Wia , Wib and unknown time-varying matrix satisfying: Te e ≤ I ;

Ta a ≤ I ;

Tb b ≤ I

(5)

Remark 1. As already stated in the introductory section, descriptor fuzzy systems have been introduced in [20,21]. The systems considered in this paper are a simple extension of these ones by taking into account parametric uncertainties and input saturation effect. They can be obtained through a physical model of the form (E(x) + E(x))x(t) ˙ = (A(x) + A(x))x(t) + (B(x) + B(x))u(t) via the standard sector nonlinearity approach described in [2]. 3.2. Control law The class of controllers considered in this paper consists in dynamic state-feedback laws described by the following expressions: ⎧ nw nh

  ⎪ ⎪ wk (z)Eka x˙a = hi (z) Aai xa + Bia ψ(u) ⎪ ⎪ ⎪ i=1 ⎪   ⎨ k=1  nh  nw u= hj (z)wk (z) Kj k x + Cjak xa (6) ⎪ j =1 k=1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ ψ(u) = u − sat(u) xa (0) = 0 where xa ∈ nx is the state variable of the proposed controller, and Eka , Aai , Bia , Cjak , Kj k are matrices to be determined. Remark 2. The structure of the proposed controllers is new, it is reminiscent of dynamical anti-windup controllers for linear systems (see [31]). Note that it consists in a classical PDC law for descriptor TS systems combined with a filtered saturation error. T

By introducing the augmented state variable x¯ = [x T xaT ] and using notations of section 2, the closed-loop system (1)–(6) is described by: (E¯ w + E¯ w )x˙¯ = (A¯ h + A¯ h + (B¯ h + B¯ h )K¯ hw )x¯ + (D¯ h − B¯ h )ψ(u)

(7)

and the controller output is rewritten as: u = K¯ hw x, ¯

(8)

where a ) E¯ w = diag(Ew , 0) E¯ w = diag(Ew , Ew A¯ h = diag(Ah , Aah ) A¯ h = diag(Ah , 0)     Bh Bh B¯ h = B¯ h = 0 0     −Bh a K¯ hw = Khw Chw D¯ h = . a Bh

3.3. Control problem definition The main goal of this paper is to propose a new method for the design of a robust controller stabilizing the regular T–S fuzzy descriptor system (1) under state constraints (2), and saturated control input (3). More precisely, the following properties will have to be satisfied:

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P 1. (Regional exponential stability): There exists an ellipsoidal set e ⊂ x such that any solution of the closedloop system with initial state in this ellipsoid will exponentially converge to the origin with a decay rate less than a given positive real number α. P 2. (State constraints): The ellipsoidal set e is positively invariant with respect to the closed-loop system. This implies that any solution of the closed-loop system with initial state in e remains in the admissible set x . 4. Main results The proof of the main result of this paper is based on the use of the sector bound condition from [43]. It is recalled here in a formulation adapted to the studied problem: Lemma 1 (Sector bound condition). Consider two families of (nu ×2nx ) matrices L¯ j k and K¯ j k for (j, k) ∈ Nnh ×Nnw . Let u be the polyhedral set associated with these families and defined by    u = P K¯ j k − L¯ j k , u0 . (9) (j,k)∈Nnh ×Nnw

Then, if x¯ ∈ u , the inequality

ψ(K¯ hw x) ¯ T Q−1 ψ(K¯ hw x) ¯ − L¯ hw x¯ ≤ 0

(10)

is verified for any vectors h ∈ Snh and w ∈ Snw , and any matrix Q that is diagonal and definite positive. The following theorem constituting the main result of the paper provides conditions which allow the synthesis of a controller satisfying the properties described in the previous section. Theorem 1. Given a system described by (1)–(5) whose validity domain (2) is characterized by a matrix N ∈ nc ×nx and a positive real number α, assume there exist matrices P [1] > 0, P [2] > 0, Xij[11] , Xij[12] , X [22] , Yij[11] , Yij[12] , E˜ k , A˜ ∈ nx ×nx , K˜ , C˜ , L˜ [1] , L˜ [2] , B˜ ∈ nu ×nx , a diagonal matrix Q ∈ nu ×nu with Q > 0, and positive real numbers i

jk

jk

jk

jk

i

τija k , τijb k , τije k for (i, j, k) ∈ Nnh × Nnh × Nnw such that: ⎡ ⎣

−P [1] 0 − L˜ [1]

K˜ j k (l)

j k (l)

∗ −P [2] − L˜ [2]

C˜ j k (l)

j k (l)

⎤ ∗ ∗ ⎦≤0 −u20,l

(11)

∀(j, k, l) ∈ Nnh × Nnw × Nnu , ⎡ ⎤ ∗ ∗ −P [1] ⎣ 0 −P [2] ∗ ⎦ ≤ 0 [1] 0 −1 N(q) P ∀q ∈ Nnc , and    i∈Nh j ∈Nh k∈Nw

 hi hj wk

ij k Wij k

(12)

 ∗ <0 −Sij k

(13)

∀(h, w) ∈ Snh × Snw , where ij k is the symmetric partitioned matrix ij k = {ij[mn] k }

defined by: ij[11] k =

m,n∈N5

[11] [11] T [22] +X [12] T ,  [22] = He(X [22] ) +2αP [2] ,  [31] = A P [1] +B K He(Xij[11] ) +2αP [1] , ij[21] , i i ˜ j k −Ek Xij +Yij k =X ij ij k ij k T T [32] [12] [33] [11] [0] [0] T = Bh C˜ − Ek X + X [22] ,  = He(−Ew Y ) + H , with H = τ a H a H a + τ b H b H b +  ij k ij ij k ij ij k jk T [41] [12] T [42] T e e e [22] ˜ ˜ ˜ , ij k = Ai − Ek + X , ij[43] τij k Hk Hk , ij k = −Ek + Yij k ˜ [1] ,  [52] = L˜ [2] ,  [53] = −QB T ,  [54] = B˜ T ,  [55] = −2Q. = L ij[51] k jk ij k jk ij k ij k ij k i i

ij k

=

ij k [12] T −(Ek Yij )

i

i

ij k

i

i

˜ − E˜ k , ij[44] k = He(−Ek ),

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Wij k is the following partitioned matrix ⎡ Wia P [1] 0 ⎢ W b K˜ b Wi C˜ j k Wij k = ⎣ i jk −Wke Xij[11]

−Wke Xij[12]

0 0

0 0

−Wke Yij[11]

−Wke Yij[12]

⎤ 0 −Wib Q ⎥ ⎦ 0

and Sij k = diag(τija k I, τijb k I, τije k I ). Then, the modified PDC control law (6) with the gain matrices defined by Eka = E˜ k (X [22] )−1

Aai = A˜ i (P [2] )−1

Bia = B˜ i Q−1

Cjak = C˜ j k (P [2] )−1

Kj k = K˜ j k (P [1] )−1

(14)

is such that the two properties P 1 and P 2 hold for the closed-loop system (1)–(6). Proof. First note that from inequality (13), it can be deduced that He(X [22] ) < 0, and therefore X [22] is a nonsingular matrix. So, the expression of Eka in (14) is meaningful. Let P¯ = diag(P [1] , P [2] ). By congruence transformation of (11) with matrix diag(P¯ −1 , I ) and applying the Schur ˜ [2] ˜ −1 . complement lemma, it follows that the set E (P¯ ) is contained in the set u defined in (9) with L¯ j k = [L˜ [1] j k Lj k ]P Similarly, from inequality (12), it can be deduced that if x¯ ∈ E (P¯ ) then x ∈ x . T  By introducing the variable ξ = x¯ T x˙¯ T , another representation of the closed-loop system can be obtained: CL E ξ˙ = (ACL hhw + Ahhw )ξ + (Dh − Bh )ψ(u)

(15)

where the following notations have been used:   E = diag(I, 0) Kj k = K¯ j k 0     0 I 0 0 = A Aik = ¯ ik Ai −E¯ k A¯ i −E¯ k     0 0 Di = ¯ Bi = Di B¯ i ACL ij k = Aik + Bi Kj k

 Bi =

0 B¯ i



ACL ij k = Aik + Bi Kj k

The notation Lhw = [L¯ hw 0] will also be used. Let the partitioned matrices Pij for (i, j ) ∈ Nnh × Nnh be defined as follows: 

P¯ Pij = ¯ X

ij

where X¯ ij =



Xij[11] X [22]



0 Y¯

(16)

ij

Xij[12] X [22]



 Y¯ij =

Yij[11]

X [22]

Yij[12]

X [22]

 .

Since Hij0 k ≥ 0 for all (i, j, k) ∈ Nnh × Nnh × Nnw , it follows that ∀(h, w) ∈ Snh × Snw  [33]  hhw ∗ − He(E¯ w Y¯hh ) ≤ . [43] [44] hhw hhw Inequality (13) yields that He(E¯ w Y¯hh ) > 0, from which it can be concluded that the matrices Y¯hh — and thus Phh — are nonsingular for any h ∈ Sh .

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Consider now the function V defined as: V (ξ ) = ξ T EP−1 hh ξ.

(17)

Simple computations lead to V (ξ ) = x¯ T P¯ −1 x¯ which is a positive definite function of x. ¯ Let V˙ (ξ ) be the derivative of V along the trajectories of (15). For x ∈ x , it comes easily that   T 

P−1 P−1 hh ξ hh ξ +  V˙ (ξ ) + 2αV (ξ ) − 2ψ(u)T Q−1 (ψ(u) − Lhw ξ ) = hhw hhw Q−1 ψ(u) Q−1 ψ(u) where hhw =



He(ACL hhw Phh ) + 2αEPhh

∗

QDTh + Lhw Phh

−2Q



 ,

 hhw =

He(ACL hhw Phh )

∗

−QBTh

0

(18)

 .

The last term can be expressed as:  ij k = He(Hik T Wij k ) with T = diag(a , b , e ) and ⎡ ⎤ 0 0 0 ⎢ 0 0 0 ⎥ ⎢ a ⎥ b ⎢ Hik = ⎢ Hi Hi Hke ⎥ ⎥. ⎣ 0 0 0 ⎦ 0 0 0 By completing the square, it yields: T  ij k ≤ Hik Sij k Hik + WijT k Sij−1k Wij k

and thus: ij k +  ij k ≤ ij k + WijT k Sij−1k Wij k . Applying Schur complement lemma to (13) leads to hhw +  hhw < 0. So, for x¯ ∈ E (P¯ ) ⊂ u , inequality (10) holds. It comes then that V˙ (x) ¯ + 2αV (x) ¯ < 0. (19) [1] −1 n T Denoting e = {x ∈  x : x P x ≤ 1} and considering zero initial values for xa , positive invariance of e can be deduced straightforwardly as well as properties P 1 and P 2. 2 Remark 3. Condition (12) of the previous theorem is expressed on the form of a parameterized LMI. There exist conditions formulated as LMI constraints which are sufficient for inequality (12) to hold ([52,4,53,54]). The relaxation scheme given in [53] requires no slack variables and allows a good trade-off between the computational cost and the conservatism of the result. It will be retained in the numerical applications given in section 5. Remark 4. Previous results allow the synthesis of a robust controller for the constrained system (1). Using bisection method, it is possible to find the controller leading to the greatest estimated rate of exponential convergence α. However, the stabilization ensured by this controller is only regional. Such an optimization design will certainly lead to a closed-loop system with a too small domain of attraction. In order to characterize the size of the estimated domain of attraction e , the reference set method will be considered: given an nx × nx matrix R > 0, let β be the greatest positive real number such that βR is contained in e . This number β is a measure of the size of the domain e . Finding the controller such that the estimated domain of attraction e has the maximum measure β with respect to the choice of a given matrix R > 0 and for a specified minimum exponential rate α can be achieved by considering γ = 1/β 2 as an additional decision variable and searching for its minimal value while LMI constraints of Theorem 1 are satisfied together with the following additional one:   γR ∗ ≥0 (20) R P [1] which is equivalent to the inclusion of βR in e .

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5. Illustrative examples In this section, some numerical examples are included to illustrate the effectiveness of the proposed solution. A first example will show the practical applicability of the method and the trade-off resulting in the choice of the tuning parameter α. The goal of the second example is to compare the presented results with respect to those given in [47]. Note that all computations were done under Matlab with the LMI solver of the Robust Control Toolbox. 5.1. Example 1 Consider the open-loop unstable system described by:  

0.11θ˙ cos θ + 1 + 0.12θ˙ sin θ θ¨ = 0.3 sin θ + −1 + 0.1θ˙ 2 cos θ θ˙ + u

(21)     where the position θ and velocity θ˙ are assumed to remain in polyhedral region defined by x = |θ | ≤ π2 , θ˙  ≤ 1.5 , and u is the control input under saturation limit |u| ≤ 0.5. Taking as state vector x = [θ θ˙ ]T , the model can be rewritten in descriptor form as follows: E(x)x˙ = A(x) x + Bu where B = [0

1]T



(22)

and

 1 0 0.11 cos x1 1 + 0.12x2 sin x1 ⎡ ⎤ 0 1 ⎦ A(x) = ⎣ sin x1 0.3 −1 + 0.1x22 cos x1 x1

E(x) =

In order to simplify the model, the terms weighted by δ1 (x) = 0.12x2 sin x1 and δ2 (x) = 0.1x22 cos x1 will be considered as uncertainties. Their maximal magnitudes taken on the domain x are respectively δ1max = 0.18 and δ2max = 0.225. The premise variables are chosen as follows: z1 = 0.11 cos x1 z2 = 0.3

sin x1 x1

Therefore, the membership functions are defined as: z1 − z1 min w1 = w2 = 1 − w1 z1 max − z1 min z2 − z2 min h1 = h2 = 1 − h1 z2 max − z2 min

(23)

(24)

where z1 max = 0.11, z1 min = 0, z2 max = 0.3, and z2 min = 0.6/π . By using the sector nonlinearity approach [2], the nonlinear system (22) can be expressed on the uncertain T–S fuzzy descriptor form (1) with nw = nh = 2, nu = 1 and       1 0 1 0 0 1 E1 = E2 = A1 = z1 max 1 z1 min 1 z2 max −1      T 0 1 0 0 H1e = H2e = W1e = W2e = max A2 = z2 min −1 1 δ1      T 0 0 0 H1a = H2a = W1a = W2a = max B1 = B2 = 1 δ2 1     N1 = 2/π 0 N2 = 0 2/3 u0 = 0.5. In this section, it will be shown that the control law obtained from Theorem 1 satisfies all predefined properties stated in section 3.3. First, the LMI conditions with α = 0 were found to be feasible with:

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Fig. 1. Domain e with (in solid line) and without optimization (in dashed line) and closed-loop trajectories for Example 1.

Fig. 2. Control input and closed-loop system response.

P [1] =



0.7203 −0.1424 −0.1424 0.2258

 (25)

Then, extending previous LMI conditions by considering also constraint (20) with R = I , the minimal value of γ were obtained for   2.4362 −1.6884 [1]∗ P = (26) −1.6884 2.2051 The estimated domains of attraction for both solutions are plotted in Fig. 1. As expected, the second one is larger than the first one. It can also be observed in Fig. 1 that the simulated closed-loop trajectories with initial states taken on the boundary of e converge to the origin and never escape this domain. Fig. 2 shows the time evolutions of the input signal and the closed-loop system response for the optimized control law. Initial state is x(0) = [−1.5 0.8]T belonging to the optimal domain e . It can be seen that despite the input signal is initially saturated the system response converges asymptotically towards zero. The time evolution of the Lyapunov function is plotted in Fig. 3 (left).

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Fig. 3. Left: Time evolution of the Lyapunov function; Right: Projection of the optimal DOAs corresponding to different values of decay rate.

The optimal estimated domains of attraction corresponding to different values of the decay rate α are depicted in Fig. 3 (right). It can be observed that a desired fast dynamics results in a reduced domain of attraction. This leads to a trade-off in the choice of α. 5.2. Example 2 Consider the uncertain T–S fuzzy descriptor system (1) under state constraints defined in (2) subject to the saturated control input (3) with nw = nh = 2, nu = 1 and     1 a 0 E1 = H1e = H2e = −2.5 + b 1 1    T 1 1.5 0.15 W1e = W2e = E2 = −2 + a 0.75 0     −1.5 + a −4 0 A1 = H1a = H2a = −1 3+b 1    T −2.5 + a −4 0 W1a = W2a = A2 = −1 1+b 0.2     0 B1 = B2 = N1 = 1 0 1   u0 = 2 (27) N2 = 0 1 where the parameters are such that (a, b) ∈ [−2.5, 2.5] × [−5, 5]. For values of parameters (a, b) taken in a 51 × 51 square grid with resolution of 0.1 × 0.2, the existence of a stabilizing feedback for (1)–(15) will be tested according to the conditions of Theorem 1 presented in this paper and, for the sake of comparison, with those of Theorem 1 in [47]. Fig. 4 shows the feasibility sets of LMI obtained for the two cases. As can be seen, the feasibility set employing the results of this paper contains strictly the one obtained through application of Theorem 1 in [47]. This suggests that the conditions presented in this paper are less restrictive than previous ones. 5.3. Example 3 Consider the inverted pendulum–cart system described in [47]. By using Euler–Lagrange formulation, the following descriptor model is obtained: E(x)x˙ = A(x)x + Bu

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Fig. 4. Feasible solutions for Example 2.

Table 1 Physical parameters of the cart–pendulum system. Mass of the cart (kg) Mass of the pendulum (kg) Pendulum length (m) Viscous friction coefficient of the pendulum (N s/rad) Viscous friction coefficient of the cart (N s/m) Gravity (m/s2 )

with:

⎛

1 ⎜0 E(x) = ⎜ ⎝0 0 ⎛

⎞ 0 ⎟ 0 ⎟ mL cos θ ⎠ m+M ⎞ 0 1 0 0 0 1 ⎟ ⎟ 0 ⎠ 0 −bp 0 0 −bc

0 0 1 0 0 4/3mL2 0 mL cos θ

0 ⎜ 0 A(x) = ⎜ ⎝ mgL sinc θ mLθ˙ 2 sinc θ

M m L bp bc g

0.5 0.7 to 1.5 0.1 to 0.5 0.01 0.1 9.81

⎛ ⎞ 0 ⎜0⎟ ⎟ B =⎜ ⎝0⎠ 1

and where x = [θ q θ˙ q] ˙ T ; q being the cart position, θ the pendulum angle, and u the force applied to the cart. sinc is the unnormalized cardinal sine function (sinc x = sin(x)/x for x = 0, 1 else). Other parameters are described in Table 1. We will consider several values for the pendulum parameters: mi = 0.7 + 0.1i with i = 0, 1, . . . , 8, and Lk = 0.1 + 0.1k for k = 0, 1, . . . , 4. The aim for this example is to compare our results dealing with a TS descriptor representation of the system with previous results in the literature dealing with a standard TS representation. For this, we will consider the following bounds: |θ | ≤ 41◦ , |θ˙ | ≤ 5 rad/s, |u| ≤ 10. No uncertainties in the model will be considered here for the sake of uniformity with existing conditions. We will also consider only asymptotic stability, meaning that we will take α = 0. Note that, for this example, the LMI feasibility will be checked using the SDPT3 solver (version 4.0). Firstly, the model is rewritten on a descriptor TS form considering as premise variables z1 = cos θ (for E(x)), z2 = sinc θ , and z3 = θ˙ 2 (for A(x)). Applying Theorem 1, it has been obtained a stabilizing controller for any couple of parameters (mi , Lk ). Then, we transform the model (28) into standard state-space representation by multiplying every term with the inverse of E(x) leading to the equivalent model x˙ = A1 (x)x + B1 (x)u

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Fig. 5. Feasible solutions for [41].

with

⎛

0 ⎜ 0 ⎜  A1 (x) = E(x)−1 A(x) = ⎜ ⎜ z1 (x) ⎝ z2 (x) ⎛

0 1 0 0 1 0 −bp (m + M) D(θ) 0

⎞ 0 0 ⎜ ⎟ ⎜ cos θ ⎟ B1 (x) = E(x)−1 B = ⎜ −mL D(θ) ⎟ ⎝ ⎠ 4 1 mL2 D(θ) 3

cos θ mLbp D(θ)

⎞ 0 ⎟ 1 ⎟ cos θ mLbc D(θ) ⎟ ⎟ ⎠ 4 1 2 − mL bc D(θ) 3

θ  where D(θ ) = 43 mL2 (m + M) − m2 L2 cos2 θ , z1 (x) = [m(m + M)gL − m2 L2 θ˙ 2 cos θ ] sinc D(θ) , and z2 (x) = m2 L2 ( 4 Lθ˙ 2 − g cos θ ) sinc θ Using the sector nonlinearity approach with premise variables chosen as z (x), z (x), 3

D(θ)

1

2

cos θ 1 , and z4 (x) = D(θ) , a standard TS model with 16 rules is obtained. z3 (x) = D(θ) Applying Theorem 8 of [44], no feasible solutions were found due to numerical problems. Fig. 5 shows the feasible solutions obtained through Theorem 1 of [41] (taking y = x). Finally, we apply Theorem 3 of [49] to find a dynamic state controller (we fix again y = x), all but one point (for mi = 1.5 and Lk = 0.4) were found feasible. As the goal is different for this paper — it is not asymptotic stability that is insured from the given conditions but ultimate boundedness of the solutions — we then try to find the maximal ultimate bounds denoted β d in [49]: due to numerical problems, no solutions were found. This example illustrates the interest of keeping a descriptor representation of the system avoiding numerical problems inherent to the complexity arising in the transformation in a standard form.

6. Conclusion This paper presents a novel LMI-based framework for the controller synthesis of physical systems. Based on Lyapunov’s stability theory and the choice of modified PDC controllers, new conditions have been proposed for the design of control laws stabilizing systems that can be described by a T–S fuzzy descriptor model with state and input constraints. As the controller design is expressed on the form of an LMI problem, the available semi-definite solvers allow an easy and efficient computation. Through three numerical examples, the efficiency of the approach has been demonstrated.

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