Fuzzy impulsive control for uncertain nonlinear systems with guaranteed cost

Fuzzy impulsive control for uncertain nonlinear systems with guaranteed cost

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Fuzzy Sets and Systems ••• (••••) •••–•••

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Fuzzy impulsive control for uncertain nonlinear systems with guaranteed cost

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Zi-Peng Wang, Huai-Ning Wu

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Science and Technology on Aircraft Control Laboratory, School of Automation Science and Electrical Engineering, Beihang University, Beijing 100191, China

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Received 8 July 2014; received in revised form 10 May 2015; accepted 29 September 2015

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Abstract In this paper, a guaranteed cost fuzzy impulsive control (GCFIC) problem is addressed for uncertain continuous-time nonlinear systems which can be represented by the Takagi–Sugeno (T–S) fuzzy model with parametric uncertainties. Based on the T–S fuzzy model, a novel time-varying Lyapunov function is initially constructed to derive the existence condition of guaranteed cost fuzzy impulsive controllers, which cannot only exponentially stabilize the uncertain fuzzy system, but also provide an upper bound on the quadratic cost function. Then, two procedures for designing suboptimal guaranteed cost fuzzy impulsive controllers are given in the sense of minimizing an upper bound of the cost function: one casts the controller design into a parameter-dependent linear matrix inequality (LMI) optimization problem and the other casts the controller design into a sequential minimization problem subject to LMI constraints by using the cone complementary linearization (CCL) algorithm. Finally, an example is presented to illustrate the effectiveness of the proposed method. © 2015 Published by Elsevier B.V.

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The past few years have witnessed rapidly growing interest in fuzzy control of nonlinear systems. In particular, the so-called Takagi–Sugeno (T–S) fuzzy model has been widely employed for the control design of nonlinear systems (see, e.g., [1,2], and the references therein for a survey of recent development). Fuzzy logic theory enables us to utilize qualitative, linguistic information about a highly complex nonlinear system to decompose the task of modeling and control design into a group of easier local tasks. At the same time, it also provides the mechanism to blend these local tasks together to yield the overall model and control design. It can provide an effective solution to the modeling and control of plants that are complex, uncertain, and ill-defined.

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1. Introduction

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Keywords: Fuzzy control; Guaranteed cost control; Impulsive control; Uncertain nonlinear systems; Linear matrix inequality (LMI); Cone complementarity linearization (CCL)

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E-mail addresses: [email protected] (Z.-P. Wang), [email protected] (H.-N. Wu). http://dx.doi.org/10.1016/j.fss.2015.09.026 0165-0114/© 2015 Published by Elsevier B.V.

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Z.-P. Wang and H.-N. Wu / Fuzzy Sets and Systems ••• (••••) •••–•••

Impulsive dynamical systems are a class of hybrid systems composed of continuous ordinary differential equations with instantaneous state jumps at discrete instants, which provide a natural framework for mathematical modeling of many real world evolutionary processes where the states undergo abrupt changes at certain instants. During the past decades, impulsive control has been also gaining considerable attention in many science and engineering fields [3–7], because it can provide an efficient way to deal with the plants which cannot endure continuous control inputs. Using the impulsive control technique, the continuous-time systems can be stabilized from only small control impulses generated by samples of the state variables at discrete time instants. The stability properties of impulsive systems have been intensively studied in [8–12] and the references therein. Recently, the T–S fuzzy model based control technique combined with the impulsive control approach has been applied to deal with the stabilization and synchronization of chaotic dynamical systems, and many important results have been developed, e.g., [13–19]. However, it is noted that the existing fuzzy impulsive control methods are based on the use of the time-invariant Lyapunov function, which may neglect the hybrid structure characteristic of impulsive systems and thus the resulting stability and stabilization conditions may be conservative. Moreover, the knowledge of membership functions in the above mentioned results is not considered, which seems to have room for improvement. More recently, some nice results on impulsive control by the time-varying Lyapunov function have been reported in [20–22]. But, these works did not consider the performance of impulsive systems. Guaranteed cost control for uncertain continuous-time systems has been extensively studied in the past decade [23–31]. The underlying objective is to design a control system that is not only robust stable but also guarantees an upper bound of quadratic performance for all admissible parameter uncertainties. We note that all of the above works are based on the assumption that the control inputs of the system are continuous. However, because of the applications of digital actuators, the inputs of many control systems are only available at discrete instants. Recently, the analysis and design of guaranteed cost control for uncertain continuous-time systems via sampling information involve mainly two approaches. In the first approach, the guaranteed cost controller is designed by updating the inputs in a sample-and-hold fashion [32]. The second one is the so-called guaranteed cost impulsive control approach i.e., the controller is designed by updating the inputs only at discrete instants of time. Compared with continuous-time control, the advantage of impulsive control lies in that the transmission of the stabilization information from the plant to the impulsive controller at discrete time instants can drastically save the bandwidth of networks and communication cost. In [33], some sufficient conditions for the existence of a guaranteed cost control law for a class of uncertain linear impulsive switched systems are given. But, it is worth pointing out that the guaranteed cost controller design in [33] is still considered with the assumption that the control inputs are continuous. Furthermore, to the best of our knowledge, little attention has been paid towards guaranteed cost impulsive control for uncertain nonlinear systems. This paper deals with the problem of guaranteed cost fuzzy impulsive control (GCFIC) for uncertain nonlinear systems, which can be represented by a T–S fuzzy model with parameter uncertainties. A novel time-varying Lyapunov function is firstly constructed to explore the hybrid characteristic of the closed-loop fuzzy impulsive system. Then, using this Lyapunov function, a sufficient condition for the existence of guaranteed cost fuzzy impulsive controllers is derived, which cannot only guarantee that the closed-loop fuzzy system is exponentially stable, but also provide an upper bound of the given quadratic cost function. Furthermore, two procedures for designing suboptimal guaranteed cost fuzzy impulsive control laws are given in the sense of minimizing an upper bound of the guaranteed cost: one transforms the controller design into a parameter-dependent linear matrix inequality (LMI) optimization problem and the other utilizes the cone complementarity linearization (CCL) method to cast the controller design into a sequential minimization problem subject to LMI constraints. Finally, the proposed design method is successfully applied to the control of mass–spring–damper system. The main contribution and novelty of this paper are summarized as follows: (i) A novel time-varying Lyapunov function is introduced to the uncertain fuzzy impulsive systems, and the GCFIC problem is considered; (ii) Some less conservative results are obtained for the uncertain fuzzy impulsive systems based on the time-varying Lyapunov approach, and two LMI-based procedures are proposed for designing the desired guaranteed cost fuzzy impulsive controllers. The rest of this paper is organized as follows. The problem formulation is presented in Section 2. In Section 3, the GCFIC design is proposed for uncertain fuzzy systems. In Section 4, a simulation example is given to illustrate the effectiveness of the proposed method. Finally, concluding remarks are given in Section 5. Notations: Throughout this paper, if not explicitly stated, matrices are assumed to have compatible dimensions. N+ is the set of positive integers. R, R+ denote the set of real and nonnegative real numbers, respectively. Rn ,

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Rn×m denote the n-dimensional Euclidean space and the set of all real n × m matrices, respectively.  ·  denotes the Euclidean norm for vector or the spectral norm of matrices. For a symmetric matrix M, M > 0 ( 0, < 0,  0) means that it is positive-definite (positive-semidefinite, negative-definite, negative-semidefinite, respectively). λmin(·), λmax (·) represent the minimal and maximal eigenvalues   of a matrix, respectively. The transpose of a matrix M is A B T denoted by M and a symmetric matrix by B T C = A∗ B . C

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x(t) ˙ = (Ai + Ai (t))x(t) + (Bi + Bi (t))u(t) Rn

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Plant rule i: IF z1 (t) is μi1 and . . . and zs (t) is μis , THEN (1)

where x(t) ∈ is the state vector, u(t) ∈ is the control input vector, i ∈ S  {1, 2, . . . , r}, j = 1, . . . , s, are the fuzzy sets, r is the number of fuzzy rules, zj (t), j = 1, . . . , s, are the premise variables. Ai ∈ Rn×n and Bi ∈ Rn×m are known constant matrices. The uncertain matrices Ai (t) = H F (t)E1i , Bi (t) = H F (t)E2i , where H, E1i , and E2i are known constant matrices of appropriate dimensions, F (t) is an uncertain time-varying matrix satisfying F T (t)F (t)  I .

Rm

μij ,

Remark 1. As is well known, robust stability analysis and synthesis of systems with uncertainties is one of the most fundamental problems in system and control theory. Two main classes of uncertainties have been extensively investigated: time-varying structured uncertainties (see e.g., [23–26] and [29–33]) and polytopic type uncertainties in [34–36]. This paper considers the time-varying structured uncertainties of nonlinear systems by a novel time-varying Lyapunov function. The nonlinear systems with the polytopic type uncertainties will be left for future research activities.

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By the singleton fuzzifier, product inference and the center average defuzzifier, the uncertain fuzzy system (1) can be represented by x(t) ˙ =

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r 

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

(2)

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Consider an uncertain continuous-time nonlinear system which can be described by the following T–S fuzzy model with parameter uncertainties:

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2. Problem formulation

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where z(t) = [z1 (t) · · · zs (t)], hi (z(t)) =

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wi (z(t))

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μij (zj (t)), and μij (zj (t)) denotes the grade of

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membership of zj (t) in

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μij .

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Note that

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hi (z(t)) = 1, hi (z(t))  0, i ∈ S

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for all t, where hi (z(t)) can be regarded as the normalized membership function of the IF–THEN rules. Fuzzy impulsive controllers for stabilizing the fuzzy system (2) can be designed via parallel-distributed compensation (PDC) scheme. In the PDC scheme, fuzzy impulsive controllers share the same premise parts with (2). That is, the impulsive controller for rule j is given by

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Controller rule j : j j IF z1 (t) is μ1 and . . . zs (t) is μs , THEN

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u(t) =

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δ(t − tk )Kj x(t − ), j ∈ S

(4)

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where δ(t) is the Dirac delta function, Kj ∈ Rm×n are the impulsive feedback gain matrices, {tk } is an impulsive time sequence satisfying t1 < . . . < tk < tk+1 < . . . with lim tk = ∞ and t1 > t0 = 0, x(t − ) denotes the limit from the left k→∞

u(t) =

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tf J=

x T (t)Qx(t)dt +

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hi (z(t))hj (z(t))[(Ai + Ai (t))x(t)

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δ(t − tk )x(t )])dt.

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hj (z(tk ))Kj . By substituting (5) into (2),

Integrating (7) on both side from tk −  to tk +  yields

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For given positive matrices Q ∈ Rn×n and R ∈ Rm×m , we consider the following quadratic cost function for the system (2):

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δ(t − tk )Kj x(t − ).

where  is a sufficiently small positive constant.

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Remark 2. Strictly speaking, the Dirac delta function δ(t − tk ) used in Equations (4) and (5) is not a true function. While for many purposes this function can be manipulated as a function. One may use the following approximation to the Dirac delta function [37–39]   −1 if t ∈ [tk − 2 , tk + 2 ], k ∈ N+ δ(t − tk ) ≈ 0 otherwise

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hi (z(tk ))hj (z(tk ))(Bi + Bi (tk ))Kj x(tk− )

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where x(t) = x(t + ) − x(t − ), and x(t + ) denotes the limit from the right at time t . Here, we assume that x(t +) = x(t). Therefore, we can represent the closed-loop fuzzy system (7) as the following equivalent form: ⎧ r  ⎪ ⎪x(t) ⎪ hi (z(t))(Ai + Ai (t))x(t), t = tk ˙ = ⎪ ⎪ ⎨ i=1 (8) r r  ⎪  ⎪ ⎪ ⎪ hi (z(t))hj (z(t))(Bi + Bi (t))Kj x(t − ), t = tk , k ∈ N+ . ⎪ ⎩x(t) = i=1 j =1

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The objective of this paper is to design a fuzzy impulsive controller of the form (5) and determine a guaranteed cost Jb  +∞ as small as possible such that the closed-loop fuzzy system (8) is exponentially stable and J  Jb for all admissible parametric uncertainties. To the end, the following Lemmas are employed to derive the main results.

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Lemma 1. (See [40].) Let matrices H ∈ following inequality holds for any scalar ε > 0:

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Lemma 2. (See [41].) For symmetric matrices Qij ∈ Rn×n , the following inequality is fulfilled: r r  

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3. Main results

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In the section, we consider the GCFIC design problem of the uncertain fuzzy system (2). We first establish a sufficient condition such that the closed-loop system (8) is exponentially stable and an upper bound is provided for the cost function (6). Then two LMI-based design procedures are proposed to obtain suboptimal guaranteed cost fuzzy impulsive controllers for the uncertain fuzzy system.

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3.1. Guaranteed cost performance analysis

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In this subsection, we make an analysis for guaranteed cost performance of the system (8) by employing a novel Lyapunov function. For given positive scalars τ1 and τ2 , where τ1  τ2 , we use the notation S(τ1 , τ2 ) to denote the class of impulsive time sequences that satisfy τ1  tk − tk−1  τ2 , k ∈ N+ . For given impulsive time sequence {tk } ∈ S(τ1 , τ2 ), we introduce the following two piecewise linear functions ρ, ρ1 : [t0 , ∞) → R+ : t − tk−1 1 ρ(t) = , ρ1 (t) = , t ∈ [tk−1 , tk ), k ∈ N+ . tk − tk−1 tk − tk−1

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ρ1 (t) ∈ [ τ12 , τ11 ],

ρ1 (t) =

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Noting that

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it follows that there exists a function ρ2 (t) ∈ [0, 1] such that

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With the function ρ(t), we choose a time-varying Lyapunov function candidate for the system (8) as

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where P (t) = (1 − ρ(t))P1 + ρ(t)P2 , Pi > 0, i = 1, 2. For the sake of convenience, we set V (t) = V (t, x(t)). Using the property of ρ(t) by (10), one can obtain that + V (tk−1 ) = V (tk−1 ) = x T (tk−1 )P1 x(tk−1 ) and V (tk− ) = x T (tk− )P2 x(tk− ), k ∈ N+ .

In the sequel, the Lyapunov function (12) will be used to analyze the guaranteed cost performance of the system (8) under some given feedback gain matrices Kj , j ∈ S.

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Theorem 1. Consider the uncertain fuzzy impulsive system (8) with {tk } ∈ S(τ1 , τ2 ). For given scalar μ ∈ (0, 1) and control gain matrices Kj , j ∈ S, if there exist positive scalars αj , βj , γsl , j, s, l ∈ S, matrices P1 > 0 and P2 > 0 such that the following LMIs are satisfied: ⎧ ⎪ ⎨ pii < 0, p = 1, 2, 3, 4, i ∈ S (13) 1 1 ⎪ ⎩ pii + ( pij + pj i ) < 0, p = 1, 2, 3, 4, i = j, i, j ∈ S r −1 2 ⎧ ⎪ < 0, s, l, i ∈S

⎨ slii (14) 1 1 ⎪ ⎩

slii + ( slij + slj i ) < 0, i = j, s, l, i, j ∈ S r −1 2

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where

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1i + Q =⎣ ∗ ∗ ⎡ 2i + Q =⎣ ∗ ∗ ⎡ 3i + Q =⎣ ∗ ∗ ⎡ 4i + Q =⎣ ∗ ∗

P1 H −αj I ∗

T αj E1i



0 ⎦ −αj I ⎤ T P1 H αj E1i 2ij −αj I 0 ⎦ ∗ −αj I ⎤ T P2 H βj E1i 3ij −βj I 0 ⎦ ∗ −βj I ⎤ T P2 H βj E1i 4ij −βj I 0 ⎦ ∗ −βj I ln μ 1 P1 + (P2 − P1 ) 1i = ATi P1 + P1 Ai + τ2 τ1 ln μ 1 T P1 + (P2 − P1 ) 2i = Ai P1 + P1 Ai + τ2 τ2 ln μ 1 P2 + (P2 − P1 ) 3i = ATi P2 + P2 Ai + τ2 τ1 ln μ 1 P2 + (P2 − P1 ) 4i = ATi P2 + P2 Ai + τ2 τ2 ij = (I + Bi Kj )T P1 ⎤ ⎡ T −μP2 ij KjT R γsl KjT E2i 0 ⎢ ∗ 0 0 P1 H ⎥ −P1 ⎥ ⎢

slij = ⎢ ∗ ∗ −R 0 0 ⎥ ⎥ ⎢ ⎣ ∗ ∗ ∗ −γsl I 0 ⎦ ∗ ∗ ∗ ∗ −γsl I 1ij

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then the system (8) is exponentially stable and the cost function defined by (6) satisfies

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1 + 1)x T (t0 )P1 x(t0 ). μ

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pij < 0, p = 1, 2, 3, 4, i, j ∈ S.

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Npij < 0, p = 1, 2, 3, 4, i, j ∈ S

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1 T E + 1 P H H TP , N T T where N1ij = 1i + Q + αj E1i 1i 1 2ij = 2i + Q + αj E1i E1i + αj P1 H H P1 , N3ij = 3i + Q + αj 1

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TE βj E1i 1i

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+ and + For t ∈ [tk−1 , tk ), along the trajectories of the system (8), we have   D+ V (t) = x T (t) ATz P (t) + P (t)Az x(t) + x T (t) (ρ1 (t)(P2 − P1 )) x(t)

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2x (t)P1 Az (t)x(t)  x T

T

T (t)(αz E1z E1z

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1 + P1 H H T P1 )x(t) αz

ln μ 1 T P1 + αz E1z E1z + P1 H H T P1 ]x(t) τ2 αz ln μ 1 T + ρ(t)x T (t)[P2 Az + ATz P2 + P2 + βz E1z E1z + P2 H H T P2 ]x(t) τ2 βz 1 − ρ2 (t) ρ2 (t) T ln μ +( + )x (t)(P2 − P1 )x(t) − V (t). τ1 τ2 τ2

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which implies

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D+ V (t)  (1 − ρ(t))x T (t)[P1 Az + ATz P1 +

Then, based on the properties of convex combination, it follows from (3), (17), and (21) that   x T (t)Qx(t) + D+ V (t)  (1 − ρ2 (t))x T (t) (1 − ρ(t))N1ij + ρ(t)N3ij x(t)   ln μ + ρ2 (t)x T (t) (1 − ρ(t))N2ij + ρ(t)N4ij x(t) − V (t) τ2 ln μ V (t), t ∈ [tk−1 , tk ) <− τ2

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1 P2 H H T P2 )x(t), (20) βz    where E1z = ri=1 hi (z(t))E1i , αz = ri=1 hi (z(t))αi and βz = ri=1 hi (z(t))βi in which αi > 0 and βi > 0, i ∈ S. Substituting (19) and (20) into (18), we have:

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T 2x T (t)P2 Az (t)x(t)  x T (t)(βz E1z E1z +

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and

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ln μ P1 + 2P1 Az (t)]x(t) τ2 ln μ P2 + 2P2 Az (t)]x(t) + ρ(t)x T (t)[P2 Az + ATz P2 + τ2 ln μ 1 − ρ2 (t) ρ2 (t) T + )x (t)(P2 − P1 )x(t) − V (t) (18) +( τ1 τ2 τ2   where D+ V (t) denotes the upper right Dini derivative of V (t), Az = ri=1 hi (z(t))Ai , and Az (t) = ri=1 hi (z(t)) Ai (t). Using Lemma 1, we get

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= (1 − ρ(t))x T (t)[P1 Az + ATz P1 +

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+ 2x (t)P (t)Az (t)x(t)

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TE N4ij = 4i + Q + βj E1i 1i

T

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P2 H H T P2 ,

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Then, using Schur complement, it follows that the condition (16) is equivalent to

10

3 4

Proof. Assume that the LMIs (13) are feasible. Then, it follows from Lemma 2 that

7 8

7

38 39 40 41 42

(21)

43 44 45 46 47 48 49

(22)

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8

1 2 3 4 5 6 7 8 9

D+ V (t)  μV ¯ (t), t ∈ [tk−1 , tk ) where μ¯ = − lnτ2μ − η, η =

(23)

λmin (Q) max(λmax (P1 ),λmax (P2 )) .

2

Thus, from (23), we have

¯ k−1 ) , t ∈ [t V (t)  V (tk−1 )eμ(t−t k−1 , tk ).

3

(24)

On the other hand, it can be verified from Lemma 2 that if the conditions (14) are fulfilled then the following condition holds: r  r  r r  

10

13 14

hs (z(tk ))hl (z(tk ))hi (z(tk ))hj (z(tk )) slij < 0.

(25)

19

22

where

⎡ zk = ⎣ ϒzk =

25

γzk =

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

r r  

KzTk R

(26)

r r  

16



17 18 19 20 21

hi (z(tk ))hj (z(tk ))(I + Bi Kj )

22 23 24

hi (z(tk ))hj (z(tk ))γij , γij > 0 



25



26

Uzk = E2zk Kzk 0 0 , V = 0 H T P1 0 E2zk =

r 

27 28

hi (z(tk ))E2i .

29

i=1

Considering

14 15

0 ⎦ −R



i=1 j =1

28 29

−μP2 ∗ ∗

ϒzTk P1 −P1



26 27

9

12

i=1 j =1

23 24

7

13

1 T V V <0 γzk

zk + γzk UzTk Uzk +

20 21

6

11

Then, using Schur complement, it follows that the condition (25) is equivalent to

17 18

5

10

s=1 l=1 i=1 j =1

15 16

4

8

11 12

1

30

F T (t

UzTk F T (tk )V

k )F (tk )  I ,

+V

T

31

for scalar γzk > 0, we have

F (tk )Uzk  γzk UzTk Uzk

1 T + V V. γzk

32

(27)

36

+ V F (tk )Uzk < 0. T

(28)

−μP2 + ϒ¯ zTk P1 ϒ¯ zk + KzTk RKzk < 0 where ϒ¯ zk = ϒzk +

i=1 j =1

39

(29)

40 41 42

hi (z(tk ))hj (z(tk ))Bi (tk )Kj .

Noticing that P (tk ) = P1 and

P (tk− ) = P2 ,

43 44

it follows from (8) and (29) that

45

V (tk ) = x (tk )P1 x(tk ) = x T (t − )ϒ¯ zT P1 ϒ¯ zk x(t − ) T

k

37 38

Then, using Schur complement, it is immediate that condition (28) is equivalent to

r r  

34 35

Combining (26) and (27) together yields zk + UzTk F T (tk )V

33

k

46 47

k

48

< μx T (tk− )P2 x(tk− )

49

= μV (tk− ). For any t > 0, there exists an integer k1

(30) ∈ N+

such that t ∈ [tk1 −1 , tk1 ). Combining (24) and (30) together yields

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9

¯ k1 −1 ) V (t)  V (tk1 −1 )eμ(t−t

2

1

3

¯ k1 −1 )  μV (tk−1 −1 )eμ(t−t

4

μ(t−t ¯ k1 −2 )

 μV (tk1 −2 )e

5

2 3 4 5

 ...

6



7

6

k1 −1

V (t0 )e

μ(t−t ¯ 0)

7

.

8 9

8

Considering k1 

t−t0 τ2 ,

we have

9

10 11 12

10

1 V (t)  V (t0 )e−η(t−t0 ) μ

11 12

13 14 15 16 17 18

13

which implies that the system (8) is exponentially stable. Next, we will prove that the cost function (6) satisfies (15). For any tf > 0, there exists an integer k2 ∈ N+ such that tf ∈ (tk2 −1 , tk2 ]. Then by (30), we have tf

19 20

21

k 2 −1

(V (tk− ) − V (tk )) − V (t0 )

22 23

k=1

24



26

24

k 2 −1 k=1

27 28

19 20



25

1 ( − 1)V (tk ) − V (t0 ). μ

(31)

31 32

29

tk

36 37

41

32

tk

tf t0

D+ V (s)ds  −

k 2 −1 k=1

44 45 46

ln μ − τ2

ln μ τ2 tf

47

t0

48

tf

49 50 51 52

33 34 35

(32)

ln μ − τ2

37

39 40

tk+1 V (s)ds − V (t0 )

41 42

tk

ln μ V (s)ds − V (t0 ) + τ2

36

38

Substituting (32) into (31) yields

42 43

31

τ2 − ln μ (t −t ) [e τ2 k+1 k − 1] − ln μ τ2 1 − μ V (tk ).  − ln μ μ

35

40

30

 V (tk )

34

39

26

28

tk+1 tk+1 − ln μ (s−tk ) V (s)ds  V (tk )e τ2 ds

33

38

25

27

On the other hand, by (24), we have

29 30

16

18

D+ V (s)ds = V (t1− ) − V (t0 ) + V (t2− ) − V (t1 ) + · · · + V (tf− ) − V (tk2 −1 )

21

23

15

17

t0

22

14

43 44

t1

45

V (s)ds

46

t0

47 48

1 V (s)ds − V (t0 ). μ

t0

Then, from (22) and (33), we can obtain that

(33)

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10

tf

1 2

J tf 

3

t0

4

tf

5

=

6

1

x T (s)Qx(s)ds

2 3

(x T (s)Qx(s) + D+ V (s))ds −

t0

7 8



9

tf

4 5

D+ V (s)ds

6

t0

7

1 V (t0 ). μ

8

(34)

10 11 12 13

10

By (24), we can get

11

V (tk− )  V (tk−1 )e

− lnτ μ (tk −tk−1 ) 2



14 15

18

Jm 

19 20

=

21 22 23

=

24 25 26



27

32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

13 14

k=1 m 

15 16 17

uTzk (tk )Ruzk (tk )

18 19 20

x T (tk− )KzTk RKzk x(tk− )

21

k=1 m 

m 

k=1 m 

k=1 m 

[x T (tk− )KzTk RKzk x(tk− ) + V (tk ) − μV (tk− )] −

V (tk ) + μ

x T (tk− )[KzTk RKzk + ϒ¯ zTk P1 ϒ¯ zk − μP2 ]x(tk− ) −

k=1

V (tk ) +

m 

22 23

V (tk− )

24

k=1 m 

25 26

V (tk−1 )

27

k=1

 V (t0 )

29

31

m 

k=1

28

30

12

1 V (tk−1 ). μ

From the above inequality and (29), we conclude

16 17

28

(35)

where m is the maximum number of the impulsive control in the final time of control tf . Therefore, combing (34) and (35), we can obtain (15). This completes the proof. 2 When B(t) = 0, we can obtain the following result. Corollary 1. Consider the uncertain fuzzy impulsive system (8) with {tk } ∈ S(τ1 , τ2 ). For given scalar μ ∈ (0, 1) and control gain matrices Kj , j ∈ S, if there exist positive scalars αj , βj , j ∈ S, matrices P1 > 0 and P2 > 0 such that LMIs (13) and the following LMIs hold: ⎧ ¯ ii < 0, i ∈ S ⎪ ⎨ (36) 1 1 ⎪ ¯ j i ) < 0, i = j, i, j ∈ S ¯ ii + ( ¯ ij + ⎩

r −1 2

52

29 30 31 32 33 34 35 36 37 38 39 40 41 42 43

where ⎡

−μP2 ⎣ ¯

ij = ∗ ∗

ij −P1 ∗

⎤ KjT R 0 ⎦ −R

then the system (8) is exponentially stable and the cost function defined by (6) satisfies (15).

50 51

9

44 45 46 47 48 49 50

When P1 = P2 = P , the time-varying Lyapunov function reduces to the time-invariant one, we can obtain the following conservative result.

51 52

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11

Corollary 2. Consider the uncertain fuzzy impulsive system (8) with {tk } ∈ S(τ1 , τ2 ). For given scalar μ ∈ (0, 1) and control gain matrices Kj , j ∈ S, if there exist positive scalars αj ,γsl , j, s, l ∈ S, and a matrix P > 0 satisfying the following LMIs: ⎧ ˜ ii < 0, i ∈ S ⎪ ⎨ (37) 1 1 ⎪ ˜ j i ) < 0, i = j, i, j ∈ S ˜ ii + ( ˜ ij + ⎩ r −1 2 ⎧ ˜ ⎪ ⎨ slii < 0, s, l, i ∈ S (38) 1 1 ⎪ ˜ slj i ) < 0, i = j, s, l, i, j ∈ S ˜ slii + ( ˜ slij + ⎩

r −1 2 where



˜ i + Q PH  ⎣ ˜ ij = ∗ −αj I ∗ ∗

T αj E1i

0 −αj I

ln μ ˜ i = ATi P + P Ai + P  τ ⎡ ˜ ij K T R −μP  j ⎢ ∗ −P 0 ⎢ ˜ slij = ⎢ ∗

∗ −R ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗

36 37 38 39

44

47 48 49 50 51 52

6 7 8 9 10 11 12

15 16 17 18

T γsl KjT E2i 0 0 −γsl I ∗

⎤ 0 PH ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −γsl I

19 20 21 22 23 24

˜ ij = (I + Bi Kj ) P 

25 26

then the system (8) is exponentially stable and the cost function defined by (6) satisfies

27 28

(39)

Proof. Suppose that conditions (37) and (38) are feasible. Choosing P2 = P1 = P , βj = αj , and τ2 = τ , then one can verify that conditions (13) and (14) in Theorem 1 are also feasible for all scalar τ1 satisfying 0 < τ1  τ2 . 2

29 30 31 32 33 34

Remark 3. In comparison with the time-invariant Lyapunov function in [13–19], there are two important features of our time-varying Lyapunov function (12). First, the time-invariant Lyapunov function is the special case of the time-varying Lyapunov one, and the latter can lead to less conservative results; Second, observing that impulsive systems are a class of hybrid systems, the time-varying Lyapunov function can characterize the dynamic behavior of impulsive systems, which is more suitable than the time-invariant one.

35 36 37 38 39 40

3.2. Controller design

41 42

In the subsection, the design of guaranteed cost fuzzy impulsive controllers will be provided for the system (8). We have the following result:

45 46

5

14

42 43

4



40 41

3

13

34 35

2



T

1 J  Jb = ( + 1)x T (t0 )P x(t0 ). μ

1

43 44 45

Theorem 2. Consider the uncertain fuzzy impulsive system (8) with {tk } ∈ S(τ1 , τ2 ). For given scalar μ ∈ (0, 1), if there exist positive scalars α¯ j , β¯j , γ¯sl , j, s, l ∈ S, n × n matrices X1 > 0, X2 > 0, and m × n matrices K¯ j , j ∈ S satisfying the following inequalities: ⎧ ⎪ ⎨pii < 0, p = 1, 2, i ∈ S (40) 1 1 ⎪ ⎩ pii + (pij + pj i ) < 0, i = j, p = 1, 2, i, j ∈ S r −1 2

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Z.-P. Wang and H.-N. Wu / Fuzzy Sets and Systems ••• (••••) •••–•••

12

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49

⎧ ⎪ ⎨pii < 0, p = 3, 4, i ∈ S

1

1 1 ⎪ ⎩ pii + (pij + pj i ) < 0, i = j, p = 3, 4, i, j ∈ S r −1 2 ⎧ ⎪ ⎨slii < 0, s, l, i ∈ S

(41)

1 1 ⎪ ⎩ slii + (slij + slj i ) < 0, i = j, s, l, i, j ∈ S r −1 2 where ⎡ T⎤ 1i √1τ X1 X1 Q α¯ j H X1 E1i 1 ⎢ ∗ −X2 0 0 0 ⎥ ⎥ ⎢ ⎢ 1ij = ⎢ ∗ ∗ −Q 0 0 ⎥ ⎥ ⎣ ∗ 0 ⎦ ∗ ∗ −α¯ j I ∗ ∗ ∗ ∗ −α¯ j I ⎡ 1 T⎤ 2i √τ X1 X1 Q α¯ j H X1 E1i 2 ⎢ ∗ −X2 0 0 0 ⎥ ⎥ ⎢ 2ij = ⎢ ∗ −Q 0 0 ⎥ ⎥ ⎢ ∗ ⎣ ∗ ∗ ∗ −α¯ j I 0 ⎦ ∗ ∗ ∗ ∗ −α¯ j I ⎤ ⎡ T ¯ 3i X2 Q βj H X2 E1i ⎢ ∗ −Q 0 0 ⎥ ⎥ 3ij = ⎢ ⎣ ∗ ¯ 0 ⎦ ∗ −βj I ∗ ∗ ∗ −β¯j I ⎡ ⎤ T 4i X2 Q β¯j H X2 E1i ⎢ ∗ −Q 0 0 ⎥ ⎥ 4ij = ⎢ ⎣ ∗ 0 ⎦ ∗ −β¯j I ∗ ∗ ∗ −β¯j I ln μ 1 1i = X1 ATi + Ai X1 + X1 − X1 τ2 τ1 ln μ 1 2i = X1 ATi + Ai X1 + X1 − X1 τ2 τ2 ln μ 1 1 3i = X2 ATi + Ai X2 + X2 + X2 + (−X2 X1−1 X2 ) τ2 τ1 τ1 ln μ 1 1 T 4i = X2 Ai + Ai X2 + X2 + X2 + (−X2 X1−1 X2 ) τ2 τ2 τ2 ij = X2 + K¯ jT BiT ⎡ ⎤ T 0 −μX2 ij K¯ jT R K¯ jT E2i ⎢ ∗ 0 0 γ¯sl H ⎥ −X1 ⎢ ⎥ slij = ⎢ ∗ −R 0 0 ⎥ ⎢ ∗ ⎥ ⎣ ∗ ∗ ∗ −γ¯sl I 0 ⎦ ∗ ∗ ∗ ∗ −γ¯sl I

(42)

3 4 5

8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47

(43)

50 51 52

6 7

then the system with Kj = K¯ j X2−1 is exponentially stable and the cost function defined by (6) satisfies 1 J  Jb = ( + 1)x T (t0 )X1−1 x(t0 ). μ

2

48 49 50

Proof. Define Pi = = 1, 2, αj = βj = β¯j−1 , j ∈ S, γsl = γ¯sl−1 , s, l ∈ S, Kj = K¯ j P2 , j ∈ S. Preand post-multiplying both sides of the inequalities (40) with diag{P1 , P2 , P1 , αj I, αj I }, the inequalities (41) with Xi−1 , i

α¯ j−1 ,

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Z.-P. Wang and H.-N. Wu / Fuzzy Sets and Systems ••• (••••) •••–••• 1 2

13

diag{P2 , P2 , βj I, βj I }, the inequalities (42) with diag{P2 , P1 , I, γsl I, γsl I }, and using Schur complement, we can obtain (13) and (14). Since all the conditions of Theorem 1 are satisfied, we have the desired results. 2

3 4

7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

2 3

In the case of B(t) = 0, we can formulate Theorem 2 as follows.

4

5 6

1

5

Corollary 3. Consider the uncertain fuzzy impulsive system (8) with {tk } ∈ S(τ1 , τ2 ). For given scalar μ ∈ (0, 1), if there exist positive scalars α¯ j , β¯j , j ∈ S, matrices X1 > 0, X2 > 0, and K¯ j , j ∈ S such that inequalities (40), (41), and the following LMIs hold: ⎧ ¯ ii < 0, i ∈ S ⎪ ⎨ (44) 1 1 ⎪ ¯ j i ) < 0, i = j, i, j ∈ S ¯ ii + ( ¯ ij +  ⎩  r −1 2 where ⎡ ⎤ −μX2 ij K¯ jT R ¯ ij = ⎣ ∗  0 ⎦ −X1 ∗ ∗ −R

the upper bound on cost function (6) in (43), we seek to minimize an upper bound of Jb = ( μ1 + 1)x T (0)X1−1 x(0). We consider the following equation:

(45)

ρ,α¯ j ,β¯j ,γ¯sl ,K¯ j ,Xq

ρ subject to inequalities (40)–(42), and (46)

9 10 11 12 13 14 15 16 17

20 21 22 23 24 25 26 27

(46)

28 29

Therefore, we can consider the following minimization problem such that the upper bound Jb in (43) is made as small as possible for given scalars μ ∈ (0, 1), τ1 , and τ2 : min

8

19

As shown in Theorem 2, Jb = ( μ1 + 1)x T (t0 )X1−1 x(t0 ) gives an upper bound of the cost function J . To minimize

where we want to minimize ρ. By Schur complement, it follows that (45) is equivalent to the LMI    1 T (t ) −ρ + 1x 0 μ < 0. ∗ −X1

7

18

then the system with Kj = K¯ j X2−1 is exponentially stable and the cost function defined by (6) satisfies (43).

Jb < ρ

6

(47)

where ρ, α¯ j > 0, β¯j > 0, γ¯sl > 0, K¯ j , j, s, l ∈ S, Xq > 0, q = 1, 2. The corresponding fuzzy impulsive controller with control gains Kj = K¯ j X2−1 is a suboptimal one for system (8) in the sense of minimizing the upper bound of the cost function (6). Notice that minimization problem (47) involves the nonlinear term X2 X1−1 X2 in (41), which cannot be directly solved by LMI Toolbox of MATLAB. Next, we will give two design strategies to deal with the nonlinear term X2 X1−1 X2 . One way is to use the matrix inequalities X2 X1−1 X2  2εq X2 − εq2 X1 , q = 1, 2. In this case, one can transform the inequalities (41) into the following LMIs: ⎧ ˆ pii < 0, p = 3, 4, i ∈ S ⎪ ⎨ (48) 1 1 ⎪ ˆ pj i ) < 0, p = 3, 4, i = j, i, j ∈ S ˆ pii + ( ˆ pij +  ⎩  r −1 2 where ⎡ ⎤ ˆ 3i X2 Q β¯j H X2 E T  1i ⎢ −Q 0 0 ⎥ ⎥ ˆ 3ij = ⎢ ∗  ⎣ ∗ 0 ⎦ ∗ −β¯j I ∗ ∗ ∗ −β¯j I

30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

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14

⎤ T X2 Q β¯j H X2 E1i −Q 0 0 ⎥ ⎥ ˆ 4ij  ¯ 0 ⎦ ∗ −βj I ∗ ∗ −β¯j I ln μ 1 1 ˆ 3i = X2 ATi + Ai X2 + X2 + X2 + (−2ε1 X2 + ε12 X1 )  τ2 τ1 τ1 ln μ 1 1 ˆ 4i = X2 ATi + Ai X2 + X2 + X2 + (−2ε2 X2 + ε22 X1 ).  τ2 τ2 τ2 Then, we can obtain the minimization problem such that the upper bound Jb in (43) is made as small as possible for given positive scalars μ ∈ (0, 1), τq , εq , q = 1, 2 as follows. ⎡

1 2 3 4 5 6 7 8 9 10 11

ˆ 4i  ⎢ ∗ =⎢ ⎣ ∗ ∗

12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

min

ρ,α¯ j ,β¯j ,γ¯sl ,K¯ j ,Xq

ρ subject to LMIs (40), (42), (46), and (48)

(49)

where ρ, α¯ j > 0, β¯j > 0, γ¯sl > 0, K¯ j , j, s, l ∈ S, Xq > 0, q = 1, 2. The corresponding fuzzy impulsive controller with control gains Kj = K¯ j X2−1 is a suboptimal one for system (8) in the sense of minimizing the upper bound of the cost function (6). Remark 4. Observe that for given τ1 and τ2 , the optimization problem (49) with the selected values μ, ε1 , and ε2 can be solved by employing the Matlab’s LMI toolbox. Then, the problem is how to find the optimal values of μ, ε1 , and ε2 to minimize ρ. In this paper, this tuning issue is given as follows. Firstly, we seek a set of initial scaling parameters such that LMIs (40), (42), (46), and (48) are feasible. Then, applying a numerical optimization algorithm, such as the program fminsearch in the optimization toolbox of Matlab, a locally convergent solution to the optimization problem (49) can be obtained. The effectiveness of this optimization procedure can be easily verified in [43–46]. Moreover, for given a sufficiently small value τ1 and ρ, one can also get the maximum τ2 with the searched values of μ, ε1 , and ε2 by using a grid search method (see [47], [48], and [32]) or a Latin hypercube sampling (LHS) method in [48].

38 39

where

29 30 31 32 33 34 35 36 37

40 41 42 43 44 45 46 47 48 49 50 51 52

3 4 5 6 7 8 9 10 11 12

14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39



⎤ T X2 Q β¯j H X2 E1i −Q 0 0 ⎥ ⎥ ˜ 3ij  ¯ 0 ⎦ ∗ −βj I ∗ ∗ −β¯j I ⎤ T X2 Q β¯j H X2 E1i −Q 0 0 ⎥ ⎥ ˜ 4ij  0 ⎦ ∗ −β¯j I ∗ ∗ −β¯j I ln μ 1 ˜ 3i = X2 ATi + Ai X2 +  X2 + X2 + τ2 τ1 ln μ 1 ˜ 4i = X2 ATi + Ai X2 + X2 + X2 +  τ2 τ2

2

13

It is noted that the above design procedure presents a sufficient condition for the GCFIC problem, which can be efficiently solved by using standard numerical software. However, the conditions in the minimization problem (49) may be a conservative result by introducing the matrix inequality technique. In the following, we will give another controller design procedure, which a CCL algorithm is proposed to solve the nonlinear term X2X1−1 X2 in (41) in Theorem 2. First, we define a new variable W > 0 such that X2 X1−1 X2  W , and replace the inequalities (41) by ⎧ ˜ pii < 0, p = 3, 4, i ∈ S ⎨ (50) 1 ⎩ 1  ˜ pii + ( ˜ pj i ) < 0, p = 3, 4, i = j, i, j ∈ S ˜ pij +  r −1 2 −1 X 2 X 1 X2  W (51)

28

1

˜ 3i  ⎢ ∗ =⎢ ⎣ ∗ ∗ ⎡ ˜ 4i  ⎢ ∗ =⎢ ⎣ ∗ ∗

40 41 42 43 44 45 46 47 48

1 (−W ) τ1 1 (−W ). τ2

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Z.-P. Wang and H.-N. Wu / Fuzzy Sets and Systems ••• (••••) •••–••• 1 2 3 4 5 6 7 8 9 10 11 12

ˆ > 0, Xˆ 1 > 0, and Xˆ 2 > 0 satisfying Now, we introduce new matrix variables W   −Wˆ Xˆ 2 0 ∗ −Xˆ 1 W Wˆ = I, X1 Xˆ 1 = I, X2 Xˆ 2 = I.

15

1 2

(52) (53)

Obviously, the inequality (51) is equivalent to the conditions (52) and (53). Thus, the original inequalities (41) are satisfied if the conditions (50), (52), and (53) hold. It is worth pointing out that the CCL algorithm is one of the most commonly method to solve nonconvex feasibility problem with the constraints in (53). The CCL method is first proposed in [49] to design the static output-feedback controller. In the following, for given μ ∈ (0, 1), τ1 , and τ2 , the CCL algorithm is applied for the GCFIC problem to determine the minimum ρ:

13

16 17 18 19 20 21

30 31 32 33

8 9 10 11 12

17

(54)

18 19 20 21 22

min

W,Wˆ ,Xq ,Xˆ q ,K¯ j ,α¯ j ,β¯j ,γ¯sl

tr( ) subject to LMIs (40), (42), (46), (50), (52), and (54)

(55)

23 24

where

25

= W Wˆ k + W k Wˆ + X1 Xˆ 1k + X1k Xˆ 1 + X2 Xˆ 2k + X2k Xˆ 2 .

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7

16

Set ρmin = ρ.  Step 2: Find a feasible set of W 0 , Wˆ 0 , X10 , Xˆ 10 , X20 , Xˆ 20 satisfying (40), (42), (46), (50), (52), and (54). Set k = 0.  Step 3: Solve the following LMI problem for the variables W , Wˆ , X1 , Xˆ 1 , X2 , Xˆ 2 , K¯ j , α¯ j , β¯j , and γ¯sl :

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15

 Step 1: Choose a sufficiently large initial value ρ such that there exists a feasible solution to (40), (42), (46), (50), (52), and       W I X1 I X2 I  0,  0,  0. ∗ Wˆ ∗ Xˆ 1 ∗ Xˆ 2

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5

14

Algorithm 1:

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Set W k+1 = W , Wˆ k+1 = Wˆ , X1k+1 = X1 , Xˆ 1k+1 = Xˆ 1 , X2k+1 = X2 , Xˆ 2k+1 = Xˆ 2 .

 Step 4: Substitute the obtained matrix variables W , Wˆ , X1 , Xˆ 1 , X2 , and Xˆ 2 into (40)–(42) and (46). For given an error bound σ , if these conditions (40)–(42) and (46) are satisfied with |tr( ) − 6n|  σ

28 29 30

(56)

then set ρmin = ρ, decrease ρ by a small amount, and return to Step 2. If LMIs (40)–(42) and (46) are infeasible within a specified number of iteration N , then stop. Otherwise, set k = k + 1 and go to Step 3.

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Remark 5. In Algorithm 1, for given τ1 and τ2 , we can adjust the values of μ ∈ (0, 1) to get the minimum ρ. Moreover, by an approach similar to Algorithm 1, one can also obtain the maximum τ2 by adjusting the values of μ for given a sufficiently small value τ1 and ρ.

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48

51 52

38 39

41 42 43 44 45 46

Next, based on Corollary 2, the guaranteed cost controller design by the time-invariant Lyapunov function (i.e., V (x(t)) = x T (t)P x(t)) is given as follows:

49 50

37

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Remark 6. It is worth pointing out that the aim of the two proposed design procedures is to design the desired guaranteed cost fuzzy impulsive controllers. The first design procedure can be efficiently solved by LMI Toolbox. But, more parameters need to tune in order to obtain the minimum ρ. The latter design procedure is based on the CCL Algorithm, which is an iterative procedure and may obtain some less conservative results. However, this design procedure needs high computational cost by much iteration.

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Corollary 4. Consider the uncertain fuzzy impulsive system (8) with {tk } ∈ S(τ1 , τ2 ). For given scalar μ ∈ (0, 1), if there exist positive scalars α¯ j , γ¯sl , j, s, l ∈ S, n × n matrix X > 0 and m × n matrices K¯ j , j ∈ S satisfying the following LMIs:

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⎧ ˜ ii < 0, i ∈ S ⎪ ⎨ 1 1 ⎪ ˜ j i ) < 0, i = j, i, j ∈ S ˜ ii + ( ˜ ij +  ⎩  r −1 2 ⎧ ˜ slii < 0, s, l, i ∈ S ⎪ ⎨ 1 1 ⎪ ˜ slj i ) < 0, i = j, s, l, i, j ∈ S ˜ slii + ( ˜ slij +  ⎩  r −1 2

(58)



˜i  ⎢ ∗ ˜ ij = ⎢  ⎣ ∗ ∗

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5

˜ slij 

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T K¯ jT E2i

0 0 −γ¯sl I ∗

16



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0 γ¯sl H ⎥ ⎥ 0 ⎥ ⎥ 0 ⎦ −γ¯sl I

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˜ ij = X + K¯ jT BiT 

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then the system with Kj = K¯ j X −1 is exponentially stable and the cost function defined by (6) satisfies J  Jb = (

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1 + 1)x T (t0 )X −1 x(t0 ). μ

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(59)

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Thus, based on Corollary 4, we can get the following minimization problem such that the upper bound Jb in (59) is made as small as possible for given scalars μ ∈ (0, 1), τ1 , and τ2 :

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min

ρ,α¯ j ,γ¯sl ,K¯ j ,X

ρ subject to LMIs (57), (58), and (46) in which X1 is replaced by X.

(60)

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In this section, a numerical example is presented to verify the effectiveness of the proposed method. Consider an uncertain nonlinear mass–spring–damper mechanical system in [25] and [50] whose dynamic equation is given by

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ϕ(t) ¨ = c(t)ϕ(t) ˙ − 0.02ϕ(t) − 0.67ϕ 3 (t) + u(t)

(61)

where ϕ(t) is the displacement of the mass, and u(t) is the control input. Assume that ϕ(t) ∈ [−1.5, 1.5], ϕ(t) ˙ ∈ [−1.5, 1.5], and c(t)ϕ(t) ˙ = −0.1ϕ˙ 3 (t) where c(t) is the uncertain term and c(t) ∈ [−0.225, 0]. Using the exact T–S fuzzy modeling method (see [1]), system (61) can be represented as Plant Rule 1: IF ϕ(t) is μ1 , THEN x(t) ˙ = (A1 + A1 (t))x(t) + (B1 + B1 (t))u(t)

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4. Numerical illustration

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T XE1i 0 ⎥ ⎥ 0 ⎦ −α¯ j I

ln μ + Ai X + X τ2 ⎡ ˜ ij K¯ T R −μX  j ⎢ ∗ −X 0 ⎢ =⎢ ∗ ∗ −R ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗

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˜ i = XATi 

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XQ α¯ j H −Q 0 ∗ −α¯ j I ∗ ∗

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where

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(57)

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Plant Rule 2: IF ϕ(t) is μ2 , THEN x(t) ˙ = (A2 + A2 (t))x(t) + (B2 + B2 (t))u(t) where

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x(t) = [ϕ(t) ˙ ϕ(t)]T , H T = [−0.1125 0]     −0.1125 −0.02 1 , B1 = A1 = 1 0 0     −0.1125 −1.527 1 , B2 = A2 = 1 0 0 [A1 B1 ] = H F (t)[E11 E21 ]

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[A2 B2 ] = H F (t)[E12 E22 ]

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E11 = E12 = [1 0], E21 = E22 = 0

10 11

and the membership functions are

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12

ϕ 2 (t) ϕ 2 (t) h1 (ϕ(t)) = 1 − , h2 (ϕ(t)) = . 2.25 2.25 From (5), the fuzzy impulsive controller can be represented as

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16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

16

u(t) =

2 

hj (ϕ(t))

j =1

∞ 

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Then, the uncertain fuzzy system can be rewritten as ⎧ 2 ⎪  ⎪ ⎪ ⎪ hi (ϕ(t))(Ai + Ai (t))x(t), t = tk x(t) ˙ = ⎪ ⎪ ⎨ i=1 2 2  ⎪  ⎪ ⎪ ⎪ ⎪ x(t) = hi (ϕ(t))hj (ϕ(t))(Bi + Bi (t))Kj x(t − ), t = tk , k ∈ N+ . ⎪ ⎩

γ¯11 = 1.2857 × 10−4 , γ¯12 = 1.2857 × 10−4

38 39 40 41 42 43 44 45 46 47 48 49 50 51 52

21 22 23

(62)

Without loss of generality, we assume that the impulses are equidistant, i.e., {tk } ∈ S(τ, τ ). For system (62) with the initial condition x(t0 ) = [1 − 1]T , choosing Q = 5I , R = 9, τ = 0.28, and using the fminsearch in Remark 4, the optimal scaling parameters are obtained as ε1 = 1.1358, ε2 = 1.1359, and μ = 0.8167 with the initial value [ε1 ε2 μ] = [1 1 0.7]. The corresponding optimization problem (49) can be solved, whose solution is

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i=1 j =1

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k=1

α¯ 1 = 2.4864, α¯ 2 = 0.4513 β¯1 = 0.3289, β¯2 = 7.5199

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δ(t − tk )Kj x(t − ).

−4

γ¯21 = 1.2857 × 10 , γ¯22 = 1.2857 × 10   0.0467 −0.0221 X1 = −0.0221 0.0272   0.0918 −0.0170 X2 = −0.0170 0.0222   ¯ K1 = −0.0658 −0.0007   K¯ 2 = −0.0582 −0.0009 .

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−4

The two impulsive gain matrices are obtained as   K1 = −0.8410 −0.6738   K2 = −0.7474 −0.6119 and the min ρ is 84.451478. The simulation results are shown in Figs. 1–3 with {tk } ∈ S(0.28, 0.28). Furthermore, calculating the actual value of the cost function, we have

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Fig. 1. Responses of the positive ϕ(t) (dotted lines) and the velocity ϕ(t) ˙ (solid lines) without impulsive control.

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Fig. 2. Responses of the positive ϕ(t) (dotted lines) and the velocity ϕ(t) ˙ (solid lines) under impulsive control.

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Fig. 3. The impulsive control signal u(t).

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8 J ≈5

x(t) dt + 9 2

0



38

u(ϕk )

2

t0
= 7.6460 < Jb = 84.451477 < 84.451478 = min ρ which means that the optimized upper bound of the cost function is ensured. For comparison purpose, we select the same values with x(t0 ) = [1 − 1]T , Q = 5I , R = 9, τ = 0.28. The LMI conditions in the optimization problem (60) were found infeasible when the range of μ ∈ [10−5 , 1] with a step size 10−5 . This shows that our result is less conservative than the one by the existing time-invariant function in [13–19].

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5. Conclusion

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In this paper, based on the T–S fuzzy model, we have studied the GCFIC design for nonlinear systems with parameter uncertainties. Initially, a sufficient condition for the existence of guaranteed cost fuzzy impulsive controllers is derived by using a novel time-varying Lyapunov function, which can enable us to exploit more information on

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the impulsive intervals that is neglected by time-invariant one. Subsequently, two procedures have been proposed for designing the desired guaranteed cost fuzzy impulsive controllers, which can be effectively solved by using the existing LMI optimization techniques. Finally, the numerical simulation results on the control of mass-spring-damper system illustrate the effectiveness and advantage of the proposed design method. Our future work will be to extend the developed approach to fuzzy impulsive observer design problem.

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