Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems

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Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems Zi-Peng Wang, Huai-Ning Wu∗

Q1

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

a r t i c l e

i n f o

Article history: Received 25 December 2014 Revised 12 June 2015 Accepted 9 August 2015 Available online xxx Keywords: Guaranteed cost control Sampled-data control Fuzzy control Distributed parameter system Bilinear matrix inequality (BMI)

a b s t r a c t In this paper, a finite dimensional guaranteed cost sampled-data fuzzy control (GCSDFC) problem is addressed for a class of nonlinear parabolic partial differential equation (PDE) systems. Initially, applying the Galerkin’s method to the PDE system, a nonlinear ordinary differential equation (ODE) system is derived. The resulting nonlinear ODE system can accurately describe the dominant dynamics of the PDE system, which is subsequently expressed by the Takagi– Sugeno (T–S) fuzzy model. Then, a guaranteed cost sampled-data fuzzy controller is developed to stabilize exponentially the closed-loop slow fuzzy system while providing an upper bound for the quadratic cost function. A novel time-dependent functional is constructed to derive the condition for the existence of the proposed controller which is presented by bilinear matrix inequalities (BMIs). Moreover, a suboptimal GCSDFC problem to minimize the cost bound can be formulated as a BMI optimization problem. A local optimization algorithm that views the BMI as a double linear matrix inequality (LMI) is given to solve this BMI optimization problem, in which a Latin hypercube sampling (LHS) method is proposed to find an initially feasible solution for starting the algorithm. Furthermore, it is shown that the proposed controller can ensure the exponential stability of the closed-loop PDE system. Finally, simulation results on the Fisher equation and the temperature profile of a catalytic rod show that the proposed design strategy is effective. © 2015 Published by Elsevier Inc.

1

1. Introduction

2

Since it can take the advantages of both linear system theory and fuzzy logic theory, the Takagi–Sugeno (T–S) fuzzy control approach has been widely used for the controller synthesis of various nonlinear systems over the past few decades (see, e.g., [2–4,6,16,22,26,33,35,36]). However, the existing studies are mainly developed for nonlinear ordinary differential equation (ODE) systems. In practice, the behavior of most industrial processes must depend on time and spatial position, such as fluid flow, heat conduction, elastic wave, and chemical reactor processes. These spatially distributed processes can be described by nonlinear partial differential equations (PDEs) with homogeneous or mixed boundary conditions. Since PDE systems are infinite dimensional systems, the existing T–S fuzzy control approaches for nonlinear ODE systems may not be directly applicable to the control design of nonlinear PDE systems. Until now, the modeling and control of nonlinear PDE systems remains an open and challenging issue.

3 4 5 6 7 8 9 10

∗

Corresponding author. Tel.: +86 1082317301; fax: +86 1082317332. E-mail addresses: [email protected] (Z.-P. Wang), [email protected], [email protected] (H.-N. Wu).

http://dx.doi.org/10.1016/j.ins.2015.08.009 0020-0255/© 2015 Published by Elsevier Inc.

Please cite this article as: Z.-P. Wang, H.-N. Wu, Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.08.009

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As is well known, according to the properties of the spatial differential operator (SDO), PDE systems could be classified as elliptical equation, hyperbolic equation, and parabolic equation, etc. The control of parabolic PDE systems has been studied extensively in the past two decades (e.g., [1,9–11,13–15,25,32]). The results of [1,9–11,13–15,25,32] have involved linear parabolic PDE systems [13,14,32], quasi-linear parabolic PDE systems [1,10,15], and nonlinear parabolic PDE systems [9,11,25]. For parabolic PDE systems, the eigenspectrum of the SDO can be separated into an infinite dimensional stable fast complement and a finite dimensional slow one. Based on the application of the spatial discretization methods (predominantly Galerkin’s technique), the proposed finite dimensional ODE systems can exactly describe the dominant dynamic behavior of parabolic PDE systems, which is then used for the basis of the finite dimensional controller design. Recently, those mentioned above T–S fuzzy control technique has been successfully applied to the finite dimensional fuzzy controller synthesis of nonlinear parabolic PDE systems in [5,7,37,39,40,44]. A finite dimensional constrained fuzzy control with guaranteed cost and a multiobjective optimization based fuzzy control are introduced for a class of nonlinear distributed process systems in [37,40], respectively. In [7,39], the fuzzy observer-based control problems for nonlinear parabolic PDE systems are studied. Nonlinear parabolic PDE systems that involve time delay and stochastic via fuzzy control are proposed in [5,44], respectively. However, it is worth pointing out that the existing studies mainly concentrate on controlling continuous-time physical systems via continuous-time controllers. On the other hand, the digital controllers have many merits in speed, small size, accuracy, and low price, which are widely used to control various continuous-time systems in the modern control process. When a continuous-time plant is controlled by a digital control algorithm, the closed-loop system is known as a sampled-data system. Three main approaches have been developed to control sampled-data systems which are described by ODEs: the discrete-time system approach [18,31,34], the time-delay system approach [17,29], and the impulsive system approach [8,30]. Recently, these approaches have been extended to the T–S fuzzy ODE systems with sampled-data control in the past few years ([20,23,24,41,43,45], and references therein). For example, the time-delay system approach is introduced to sampled-data fuzzy control for nonlinear ODE systems in [20,24,43]. Then, an improved input delay approach to stabilization of fuzzy ODE systems under variable sampling is proposed based on a novel Lyapunov–Krasovskii functional (LKF) in [45]. These results of [20,24,43,45] are further improved in [41] and successfully applied to chaotic systems. More recently, the sampled-data fuzzy control design for T–S model-based fuzzy ODE systems using the discrete-time system approach is put forward in [23]. It is noted that most of the existing results on sampled-data fuzzy control focus on nonlinear ODE systems. As mentioned earlier, the behavior of most industrial processes is described by nonlinear PDE systems that play an important role in engineering applications. However, the existing studies for PDE systems mainly design a continuous-time controller to control the continuous-time plant. To fully enjoy the benefits of the digital technology in control engineering, it is of importance to develop sampled-data control methods for PDE systems. Unfortunately, there are very few results focusing on sampled-data control design for nonlinear parabolic PDE systems, which motivates the present study. In this study, a finite dimensional sampled-data fuzzy control approach will be developed for a class of nonlinear parabolic PDE systems. Galerkin’s technique is initially applied to the PDE system to obtain a nonlinear slow system of finite dimensional ODEs. Subsequently, the T–S fuzzy model is used to describe exactly the nonlinear slow system. Then, a guaranteed cost sampleddata fuzzy controller design is proposed such that the closed-loop slow system is exponentially stable and an upper bound of the quadratic cost function is provided. The condition for the existence of the proposed controller is derived based on a novel time-dependent functional and presented by bilinear matrix inequalities (BMIs). Moreover, a suboptimal GCSDFC problem to minimize the cost bound can be formulated as a BMI optimization problem. Based on the existing LMI optimization techniques [12,19], a local optimization algorithm that views the BMI as a double LMI is also given to solve this BMI optimization problem, where a Latin hypercube sampling (LHS) method is proposed to find an initially feasible solution for starting the algorithm. Furthermore, it is proven that the proposed controller can ensure the exponential stability of the closed-loop PDE system. Finally, the proposed design strategy is successfully employed to the control of the Fisher equation and the temperature profile of a catalytic rod. The main contribution and novelty of this paper are summarized as follows: (i) A novel time-dependent functional is constructed to finite dimensional guaranteed cost sampled-data fuzzy control design for a class of nonlinear parabolic PDE systems; (ii) Based on the time-dependent functional technique, LHS method, and the LMI technique, a suboptimal GCSDFC problem is formulated as a BMI optimization problem, which can be solved by a local optimization algorithm. Notations: N is the set of nonnegative integers. R+ , R denote the set of nonnegative real and real numbers, respectively. Rn , Rn×m denote the n-dimensional Euclidean space and the set of all real n × m matrices, respectively. l2 denotes the subset of R∞ consisting of all square summable infinite sequences of real numbers, such that l 2 = {x = [x1 · · · x∞ ]T ∈ R∞ | xl 2 < ∞}  ∞ 2 n 2 where xl 2  i xi . For given constants α , β ∈ R, L ([α , β ]; R ) is a space of piecewise continuous, square-integrable vec2 n tor functions defined on the interval [α , β ]. L ([α , β ]; R )  {ζ : [α , β ] → Rn and ζ 2,[αβ ] < ∞} with the norm ζ 2,[αβ ]   β T α ζ (z)ζ (z)dz.  ·  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-semidefinite (positive-definite, negative-semidefinite, negative-definite, respectively). σ ( · ), σ¯ ( · ) denote the minimum singular value and the maximum singular value of a matrix, respectively. diag{ · } stands for a diagonal block matrix. 0n and In denote the n × n zero matrix and n × n identity matrix, respectively. MT stands for the transpose

A

of the matrix M and a symmetric matrix by BT

A B = ∗ C

B . C

Please cite this article as: Z.-P. Wang, H.-N. Wu, Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.08.009

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3

2. Preliminaries and problem formulation Consider a class of nonlinear parabolic PDE systems of the following form:

70

∂ x¯(z, t ) ∂ x¯(z, t ) ∂ 2 x¯(z, t ) = E1 + E2 + f (x¯(z, t )) + ku b¯ (z)u(t ) ∂t ∂z ∂ z2 71

(1)

subject to the boundary conditions

∂ x¯(α1 , t ) = d1 ∂z ∂ x¯(α2 , t ) = d2 M2 x¯(α2 , t ) + N2 ∂z M1 x¯(α1 , t ) + N1

72

(2)

and the initial condition

x¯(z, 0) = x¯0 (z) 73

(3)

where x¯(z, t ) = [x¯1 (z, t ) · · · x¯n

(z, t )]T

is the state vector,   [α1 , α2 ] ⊂ R is the domain of definition of the process, z ∈  is the

78

) ) spatial coordinate, t ∈ [0, ∞) represents the time, and u(t ) ∈ R p is the vector of manipulated input. ∂ x¯∂(z,t and ∂ ∂x¯(zz,t denote the 2 z first- and second-order spatial derivatives of x¯(z, t ), respectively. f (x¯(z, t )) is a Lipschitz continuous nonlinear vector function satisfying f (0) = 0, ku is a constant vector, b¯ (z) = [b¯ 1 (z) · · · b¯ p (z)] is a known smooth vector function of z, where b¯ i (z) ∈ L2 (; R) represents how the control action ui (t) is distributed in the domain . E1 , E2 , M1 , N1 , M2 , and N2 are constant matrices, d1 and d2 are column constant vectors, and x¯0 (z) is the initial condition, which is a sufficiently smooth function of z.

79

2.1. Finite dimensional approximation for parabolic PDE systems

74 75 76 77

80 81 82



2

Let H  L2 (; Rn ) be a Hilbert space of one-dimensional functions defined on , with inner product ζ1 , ζ2  

 ζ1 (z), ζ2 (z)Rn dz and norm

inner product in

Rn .

1

ζ1 2  ζ1 , ζ1  2 , where ζ 1 and ζ 2 are two elements of H and ·, ·Rn denotes the standard

Define the operator A in H as

∂ x¯ ∂ 2 x¯ + E2 2 A x¯  E1 ∂z ∂z 83

with its domain



x¯ ∈ D(A ) 

x¯ ∈ H

∂ x¯ ∂ 2 x¯ ∂ x¯(αi , t ) are absolutely continuous, = di , i = 1, 2 . and ∈ H , Mi x¯(αi , t ) + Ni ∂z ∂z ∂ z2

86

For the spatial operator A , we define the eigenvalue problem as A φ j (z)  λ j φ j (z), j = 1, 2, . . . , ∞, where λj and φ j (z) denote the jth eigenvalue and the corresponding orthonormal eigenfunction, respectively. Then, define the eigenspectrum of A as σ (A )  {λ1 , λ2 , . . . , λ∞ } and make the following assumption for σ (A ):

87

Assumption 1. All eigenvalues of A are ordered such that λ j  λ j+1 , and there is a finite number m so that λm+1 < 0.

84 85

88 89 90 91

92

Based on Assumption 1 and applying Galerkin’s technique [40] to (1)–(3), we can obtain an approximate finite dimensional nonlinear ODE system. Without loss of generality, letting x¯(z, t ) = x¯1 (z, t ) ∈ R, f (x¯) = f (x¯1 ), x¯0,1 (z) = x¯1 (z, 0), and ku = ku,1 in which ku, 1 is a constant value. First, by the separation of spatial variables and time, the following infinite dimensional nonlinear ODE system is derived:

⎧ ⎨ x˙ s (t ) = As xs (t ) + fs (xs (t ), x f (t )) + Bs u(t ) x˙ f (t ) = A f x f (t ) + f f (xs (t ), x f (t )) + B f u(t ) ⎩ xs (0) = xs,0 , x f (0) = x f,0

(4)

where

xs (t ) = [x1 (t ) · · · xm (t )]T ∈ Rm x f (t ) = [xm+1 (t ) · · · x∞ (t )]T ∈ l 2 As (t ) = diag{λ1 (t ) · · · λm (t )} A f (t ) = diag{λm+1 (t ) · · · λ∞ (t )} fs (xs (t ), x f (t )) = fs (xs (t ), x f (t )) =

φs (z), f (x¯1 )

φ f (z), f (x¯1 )

φs (z), ku,1 b¯ u (z) B f = φ f (z), ku,1 b¯ u (z) xs,0 = φs (z), x¯0,1 (z) Bs =

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φ f (z), x¯0,1 (z) φs (z) = [φ1 (z) · · · φm (z)]T φ f (z) = [φm+1 (z) · · · φ∞ (z)]T . x f,0 =

93

Then, we can obtain the following finite dimensional nonlinear slow system by neglecting the fast modes:

x˙ s (t ) = As xs (t ) + fs (xs (t ), 0) + Bs u(t ), xs (0) = xs,0 .

(5)

100

Remark 1. Based on the Galerkin’s technique, the PDE system (1)–(3) is divided into a finite dimensional slow system and an infinite dimensional fast one in (4). Then, we derive the finite dimensional nonlinear slow system (5) by neglecting the fast modes. Since the finite dimensional nonlinear slow system (5) should describe the dominant dynamic behavior of PDE system (1)–(3). Thus, the eigenspectrum separation has greater impacts on the stability of PDE system (1)–(3). If the eigenspectrum separation is provided small, it may make the designed finite dimensional controller invalid and even lead to the PDE system instability. However, it is worth mentioning that although the Galerkin’s method is an effective technique to deal with the parabolic PDE systems, it cannot be applied to the elliptical and hyperbolic PDE systems.

101

2.2. Fuzzy modeling of finite dimensional ODE system

94 95 96 97 98 99

102 103 104

Assume that the nonlinear slow system (5) can be described by the following T–S fuzzy model: Plant rule i: IF θ 1 (t) is μi1 and  and θ g (t) is μig , THEN

x˙ s (t ) = Ai xs (t ) + Bs u(t ), i ∈ S  {1, 2, . . . , r} 105 106 107

(6)

where Ai = As + Fi , r is the number of IF-THEN rules, μij , j = 1, . . . , g are the fuzzy sets, θ 1 (t) ∼ θ g (t) are the premise variables, Fi ∈ Rm×m are known constant matrices. Then, the overall dynamics of system (6) is represented by

x˙ s (t ) =

r

ωi (θ (t ))Ai xs (t ) + Bs u(t )

(7)

i=1

108

where θ (t ) = [θ1 (t ) · · · θg (t )], and

 ϑ (θ (t )) ωi (θ (t )) = r i , ϑi (θ (t )) = μi j (θ j (t )) i=1 ϑi (θ (t )) j=1 g

109

in which the grade of membership of θ j (t) in μij is denoted by μij (θ j (t)). Moreover, we have r

ωi (θ (t )) = 1, ωi (θ (t ))  0, i ∈ S

(8)

i=1

110

for all t ≥ 0.

111

2.3. Problem formulation

112 113 114 115 116

In this paper, we assume that the state x¯(z, t ) of system (1) can be completely available as an output. Let 0 = t0 < t1 < · · · < tk < · · · < limt→tk tk = +∞ be sampling time instants and define hk  tk+1 − tk . The required assumption for the sampling time sequence {tk } is that {tk } ∈ S (h)  {tk , k ∈ N|  hk  h, 0 <  h}. Then, using the sampler, one can obtain x¯(z, tk ). In this case, we can immediately yield xs (tk ) by using the modal analyzer [32,38]. For example, x¯(z, t ) = x¯1 (z, t ), one can calculate jth element of xs (tk ) as follows:

x j (tk ) = x¯1 (z, tk ), φ j (z). 117 118 119 120

Remark 2. Notice that in practice, the complete state x¯1 (z, tk ) is impossible to obtain. However, the following information can provide: First, measure the state x¯1 (z, t ) at m spatial positions; then x¯1 (zi , t ) ≈ φsT (zi )xs (t ). By using the sampler, we have x¯1 (zi , tk ) ≈ φsT (zi )xs (tk ). Define x¯1z (tk )  [x¯1 (z1 , tk ) · · · x¯1 (zm , tk )]T and z [φ s (z1 )φ s (zm )]T , then we obtain x¯1z (tk ) ≈ z xs (tk ). If the sampling locations of spatial positions zi , i = 1, 2, · · · , m are well chosen, the matrix z will be invertible, and thus

xs (tk ) = −1 z x¯1z (tk ). 121 122

124 125

(10)

In this situation, one can use (10) to replace (9) in the modal analyzer. If the measurements are available at more than spatial positions, then a least square is suitable for xs (tk ), and then use

xs (tk ) = (Tz z )−1 Tz x¯1z (tk ). 123

(9)

(11)

to take the place of (9). Furthermore, if the state x¯1 (z, t ) is measured at only a few spatial positions (less m, possibly only one), one can use a state estimator to provide the estimates of x¯1 (z, t ). Then, a finite dimensional sampled-data observer can be designed to estimate xs (tk ), which will be left for future research activities. Please cite this article as: Z.-P. Wang, H.-N. Wu, Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.08.009

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Fig. 1. A distributed parameter model sampled-data fuzzy controller.

126 127 128 129 130

Assume that the control signal is generated by a zero-order hold (ZOH) function with the sequence of hold times {tk } ∈ S (h). Then, based on the parallel-distributed compensation (PDC) scheme [35], the following sampled-data fuzzy control law is adopted for the fuzzy system (7): Controller rule j: IF θ 1 (t) is μj1 and  and θ g (t) is μjg , THEN

u(t ) = K j xs (tk ), t ∈ [tk , tk+1 ), j ∈ S 131 132

(12)

where K j ∈ R p×m , j ∈ S are the control gain matrices to be determined. Therefore, the final fuzzy sampling controller can be inferred as follows:

u(t ) =

r

ω j (θ (tk ))K j xs (tk ), t ∈ [tk , tk+1 ), j ∈ S.

(13)

j=1

133 134

Fig. 1. shows the structure of the above sampled-data controller. Substituting (13) into (7), we obtain the following closed-loop fuzzy system:

x˙ s (t ) =

r r

ωi (θ (t ))ω j (θ (tk ))[Ai xs (t ) + Bs K j xs (tk )], t ∈ [tk , tk+1 ).

(14)

i=1 j=1

135

In this paper, the following quadratic cost function associated with the system (14) is considered:

 Js =

tf

t0

[xs (t )W1 xs (t )dt + uT (t )W2 u(t )]dt

(15)

141

T C m×m and 0  W = DT D where tf is the final time of control and t f ∈ S (h), 0  W1 = CW ∈ R p×p are the known weightW1 ∈ R 2 W2 W2 1 ing matrices. Thus, our objective of this paper is to find a sampled-data fuzzy controller (13) not only to stabilize exponentially the closedloop fuzzy system (14), but also to minimize the value of the cost function defined in (15). Moreover, the resulting sampled-data controller is desired to ensure the exponential stability of the closed-loop PDE system. To the end, the following two Lemmas will be used to deduce the main results.

142

Lemma 1 ([8]). For given scalars a and b satisfying

136 137 138 139 140



b > 0, a > −2 b 143

set c =

1 2



(16)

|4b − a2 | and define

v = v(a, b) 

⎧1 √ 2c ⎪ arctanh , if a > 2 b ⎪ ⎪ c a ⎪ ⎪ ⎪ √ ⎪ ⎨2 , a 2c 1 ⎪ ⎪ arctan , ⎪ ⎪ a ⎪ ⎪c

⎪ ⎩π , 2c

if a = 2 b

√ if 0 < a < 2 b √ if − 2 b < a  0.

(17)

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Given positive scalars α , T and σ , and τ ∈ (0, T], φ (t, φ 0 ) is the solution

φ˙ = − 145

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v (φ 2 + aφ + b), t  α , φ(α) = φ0 > 0. τ +σ

(18)

Then there exists a scalar ϱ > 0 which depends on σ , a, b, and T so that for some initial value φ0∗ ∈ (0, ), φ(t, φ0∗ ) has the property

˙ t ) < 0 for t ∈ [α , α + τ ]. φ(α + τ ) = 0, φ( 146

(19)

Lemma 2. Considering the closed-loop fuzzy system (14), we can get the following relation:

xs (t )  μ0 xs (tk ), t ∈ [tk , tk+1 ). 147

where μ0 is a positive scalar.

148

Proof. Select V0 (t ) = xs (t )2 . For t ∈ [tk , tk+1 ), the time derivative of V0 (t) along the solution of system (14) yields

V˙ 0 (t ) =

r r

ωi (θ (t ))ω j (θ (tk ))[2xTs (t )Ai xs (t ) + 2xTs (t )Bs K j xs (tk )]

i=1 j=1

 2Axs (t )2 + 2Bs K xs (t )xs (tk ) 

μ1V0 (t ) + μ2V0 (tk )

(20)

149

where μ1 = 2A + Bs K  and μ2 = Bs K  in which A = max Ai , K  = max K j , i, j ∈ S. Then, it follows from (20)

150

that

i





j



V0 (t )  eμ1 (t−tk ) + (μ2 /μ1 ) eμ1 (t−tk ) − 1 V0 (tk ) 151

for t ∈ [tk , tk+1 ). Considering tk+1 − tk  h, we obtain







V0 (t )  eμ1 h + (μ2 /μ1 ) eμ1 h − 1 V0 (tk ), t ∈ [tk , tk+1 ). 152

The proof is complete. 

153

3. Main results

154

In this section, a guaranteed cost sampled-data fuzzy controller will be designed for the fuzzy system (14). Next, we first introduce a novel time-dependent functional for the system (14). For any {tk } ∈ S (h), the following functions ρ , ψ : [t0 , ∞) → R+ are introduced:

155 156

ρ(t ) = t − tk , ψ(t ) = tk+1 − t, t ∈ [tk , tk+1 ), k ∈ N. 157 158

From Lemma 1, for any scalars a and b satisfying (16), and k > 0, there exists a positive scalar ϱ which depends on a, b, k , and ∗ ∈ (0, ), the unique solution φ (t, φ ∗ ) hk so that for some initial value φk,0 k k,0

φ˙ k = − 159

(21)

v

hk + k

∗ (φk2 + aφk + b), t  tk , φk (tk ) = φk,0

has the following property:

φk (tk+1 ) = 0, φ˙ k (t ) < 0 for t ∈ [tk , tk+1 ]. 160

(22)

Define a function φ : [t0 , ∞) → R+ associated with {tk } ∈ S (h) by ∗ φ(t ) = φk (t, φk,0 ), t ∈ [tk , tk+1 ), k ∈ N.

161

162

(23)

Notice that hk ≤ h, it is easily observed from (22) and (23) that φ (t) has the property as follows: − φ(tk+1 ) = 0 forall k ∈ N and 0  φ(t )   for t  t0 ; ˙ t )  − v (φ 2 (t ) + aφ(t ) + b) < 0, t = tk , k ∈ N. φ(

(24) (25)

h¯

163

where h¯ = h + and = max { k }. It is worth pointing out that the functions ψ (t) and φ (t) have analogous properties, which are k

164 165

− − continuously decreasing on (tk , tk+1 ), nonnegative, and ψ(tk+1 ) = φ(tk+1 ) = 0. With the functions ρ (t), ψ (t), and φ (t), the following time-dependent functional is chosen for the fuzzy system (14):

Vs (t ) = xTs (t )Pxs (t ) + ψ(t )



t

t−ρ(t )

x˙ Ts (t )Rx˙ s (t )dt + ψ(t )[x˜Ts (t )X1 x˜s (t ) + 2(xs (t ) − x˜s (t ))T X2 x˜s (t )]

+ φ(t )x˜Ts (t )(ψ(t )S1 + (h¯ − ψ(t ))S2 )x˜s (t ) 166

where x˜s (t ) = xs (t ) − xs (t − ρ(t )), P > 0 ∈

Rm×m , R

>0∈

Rm×m , and S

(26) w

>0∈

Rm×m , w

= 1, 2, X1 =

X1T

∈

Rm×m , and X2

∈

Rm×m .

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167 168 169

7

It should be mentioned that the above functional (26) has some important features. Due to two “not necessarily” positive definite terms x˜Ts (t )X1 x˜s (t ) and (xs (t ) − x˜s (t ))T X2 x˜s (t ) in (26), the functional Vs (t) may be non-positive definite. Considering − − ρ(t0 ) = x˜s (t0 ) = 0, ρ(tk+1 ) = ψ(tk+1 ) = φ(tk+1 ) = 0, and x˜s (tk+1 ) = 0, k ∈ N, we get

Vs (tk ) = xTs (tk )Pxs (tk ), k ∈ N 170 171 172 173

(27)

which means the functional (26) is positive definite at sampling times. Moreover, observing limt→tk+1 Vs (t ) = Vs (tk+1 ), k ∈ N and using the fact that ψ (t), ρ (t), φ (t), and x˜s (t ) are continuous on (tk , tk+1 ), k ∈ N, the functional Vs (t) is continuous on [t0 , +∞). Next, the functional (26) will be employed to design the guaranteed cost sampled-data fuzzy controller for the system (14). For brevity, we give the following notations:

I1 = [Im 0m 0m ], I2 = [0n Im 0m ], I3 = [0m 0m Im ] I4 = [Im − Im 0m ], Ai j = [Ai + Bs K j − Bs K j − Im ]. 174

For t ∈ [tk , tk+1 ), along the solution of the system (14), and using (24) and (25), we obtain

V˙ s (t )  2xTs (t )P x˙ s (t ) −



t

t−ρ(t )

x˙ Ts (s)Rx˙ s (s)ds + ψ(t )x˙ Ts (t )Rx˙ s (t ) − x˜Ts (t )X1 x˜s (t )





− 2(xs (t ) − x˜s (t ))T X2 x˜s (t ) + 2ψ(t ) x˜Ts (t )X1 + (xs (t ) − x˜s (t ))T X2 x˙ s (t )

v

(φ 2 + aφ + b)x˜Ts (t )(ψ(t )S1 + (h¯ − ψ(t ))S2 )x˜s (t ) h¯ + 2φ(t )x˜Ts (t )(ψ(t )S1 + (h¯ − ψ(t ))S2 )x˙ s (t ) + φ(t )x˜Ts (t )(S2 − S1 )x˜s (t ). −

175

Let ξ (t ) =

0=2

[xTs (t ) x˜Ts (t ) x˙ Ts (t )]T .

r r

Since x˜s (t ) = xs (t ) − xs (t − ρ(t )) =



ωi (θ (t ))ω j (θ (tk ))ξ T (t )Mi j x˜s (t ) −

i=1 j=1



r r



t

t−ρ(t )

t

t−ρ(t ) x˙ s (t )dt,



for any matrices Mi j ∈

(28) R3m×m ,

we get

x˙ s (t )dt

ωi (θ (t ))ω j (θ (tk ))ξ T (t )(Mi j I2 + I2T MiTj )ξ (t )

i=1 j=1

+ ρ(t )

r r

ωi (θ (t ))ω j (θ (tk ))ξ T (t )Mi j R−1 MiTj ξ (t ) +

i=1 j=1

176



t

t−ρ(t )

x˙ Ts (s)Rx˙ s (t )dt.

Using (14), for any matrices Pi j ∈ R3m×m , gives the following relationship:



0 = 2ξ

T

(t )Pi j

r r

(29)



ωi (θ (t ))ω j (θ (tk ))(Ai xs (t ) + Bs K j xs (t − ρ(t ))) − x˙ s (t )

i=1 j=1

=

r r

ωi (θ (t ))ω j (θ (tk ))ξ T (t )(Pi j Ai j + ATi j PiTj )ξ (t ).

(30)

i=1 j=1

177

Considering ρ(t )  h¯ − ψ(t ) and using (28)–(30), we have

V˙ s (t ) + xTs (t )W1 xs (t ) + uT (t )W2 u(t ) 

r r

ωi (θ (t ))ω j (θ (tk ))ηT (t )i j (t, h¯ )η(t )

(31)

i=1 j=1

178

where

η(t ) = [ξ T (t ) φ(t )x˜Ts (t )]T i j (t, h¯ ) = 1i j + ψ(t )2i j (h¯ ) + (h¯ − ψ(t ))3i j (h¯ )  ij  10 + ϒ j i20j 1i j = ∗

0

∗

− S1 h¯

⎡ ⎤ i11j (h¯ ) i21j (h¯ ) 2i j (h¯ ) = ⎣ v ⎦ ⎡ ij ⎤ 12 (h¯ ) + Mi j R−1 MiTj i22j (h¯ ) 3i j (h¯ ) = ⎣ v ⎦ ∗

− S2 h¯

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ϒ j = I1TW1 I1 + I4T K Tj W2 K j I4 i10j = I1T PI3 + I3T PI1 − I2T X1 I2 − I4T X2 I2 − I2T X2T I4 + Mi j I2 + I2T MiTj + Pi j Ai j + ATi j PiTj bv i11j (h¯ ) = I3T RI3 + I2T X1 I3 + I3T X1 I2 + I4T X2 I3 + I3T X2T I4 − I2T S1 I2 h¯

bv 1 i12j (h¯ ) = − I2T S2 I2 , i20j = I2T (S2 − S1 ) ¯h 2  av  ij 2w (h¯ ) = − I2T + I3T Sw , w = 1, 2. 2h¯ 179

Obviously, if the following inequality holds: r r

ωi (θ (t ))ω j (θ (tk ))i j (t, h¯ ) < 0

(32)

i=1 j=1

180

then we have

V˙ s (t ) + xTs (t )W1 xs (t ) + uT (t )W2 u(t )  0. 181 182 183

(33)

Thus, we obtain the following result: Theorem 1. Consider the fuzzy system (7). Given a sampled-data fuzzy controller (13) with {tk } ∈ S (h) and scalars a and b satisfying (16), if there exist matrices P > 0, R > 0, X1 = X1T , X2 , Sw > 0, Pi j , and Mij satisfying

wi j < 0, w = 1, 2, i, j ∈ S 184

where

1i j

⎡ ij ⎤ j T T 111 i112 I1TCW I4T K Tj DW 1 2 ⎢ ∗ −vS1 ⎥ 0 0 ⎥ =⎢ ⎣ ∗ ⎦ ∗ −I 0 ∗

2i j

185

∗

∗

∗

∗

−I

⎡ ij j T T ⎤ 211 i212 hMi j I1TCW I4T K Tj DW 1 2 ⎢ ∗ −vS2 0 ⎥ 0 0 ⎢ ⎥ ⎥ =⎢ ∗ ∗ −hR 0 0 ⎢ ⎥ ⎣ ∗ ⎦ ∗ ∗ −I 0 ∗



=



=

ij 111 ij 211

(34)

ij 10 ij 10

+ h + h

ij 11 ij 12

∗

(h), 

=

(h), 

=

ij 112 ij 212

−I

 (h)

ij +h 20 ij +h 20

ij 21 ij 22

 (h)

then the closed-loop fuzzy system (14) is exponentially stable and the cost function (15) satisfies

Js  Jb  xTs,0 Pxs,0 .

(35) 

186

Proof. See Appendix A.

187 189

Theorem 1 presents a less conservative condition for the existence of a guaranteed cost sampled-data controller, which can stabilize exponentially the closed-loop fuzzy system (14) and provide the relation (35) for the cost function (15). From Theorem 1, an upper bound of the cost function Js is given by Jb = xTs,0 Pxs,0 . To minimize the upper bound of the cost

190

function (15) in (35), one can seek to minimize an upper bound of Jb = xTs,0 Pxs,0 , i.e.,

188

Jb < ρ¯ 191

where ρ¯ > 0 needs to be minimized. Therefore, for some given scalars a and b, the following minimization problem is considered:

192

min ρ¯ subject to inequalities (34 ) and (36 ) U

193

195 196

(37)

where

U  194

(36)

{ρ¯ > 0, P > 0, R > 0, X1 = X1T , X2 , Sw > 0, w = 1, 2, K j , Pi j , Mi j , i, j ∈ S}

is a set of decision variables. Due to the nonlinear terms Pi j Ai j + ATi j PiTj in the set U, the inequalities (34) are BMIs. Thus, the minimization problem (37) is a BMI optimization problem. BMI problems are known to be NP-hard and nonconvex, which cannot be directly solved by LMI Please cite this article as: Z.-P. Wang, H.-N. Wu, Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.08.009

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Algorithm 1  Step 1: Set K j = [ j1 · · ·  jm ] and Tjn = [γ jn1 1 · · · γ jn1 p ]T , γ jn1 n2 are scalar parameters with n1 = 1, . . . , m and n2 = 1, . . . , p. 1 √  Step 2: Give the ranges 0 < b ∈ [θ1 , 1 ], −2 b < a ∈ [θ2 , 2 ], γ jn1 n2 ∈ [g jn1 n2 , G jn1 n2 ], and generate multiple sets of a, b, and γ jn1 n2 by utilizing the LHS method.  Step 3: With these sets of a, b, and γ jn1 n2 , search a set of the values a, b, and γ jn1 n2 such that the LMIs in the optimization problem (37) are feasible. Then, we can get K j , go to Step 4.  Step 4: Solve the following LMI optimization problem with K j obtained previously:

min ρ¯ subject to LMIs (34 ) and (36 ) U1

where

U1 

{ρ¯ > 0, P > 0, R > 0, X1 = X1T , X2 , Sw > 0, w = 1, 2, Pi j , Mi j , i, j ∈ S}.

Then, set k = k + 1 and ρ¯ k = ρ¯ . If |ρ¯ k − ρ¯ k−1 | < δρ¯ , go to Step 6; Otherwise, go to Step 5.  Step 5: Using Pi j obtained in Step 4, solve the LMI optimization problem as follows:

min ρ¯ subject to LMIs (34 ) and (36 ) U2

where

U2 

{ρ¯ > 0, P > 0, R > 0, X1 = X1T , X2 , Sw > 0, w = 1, 2, K j , Mi j , i, j ∈ S}.

Then, set k = k + 1 and ρ¯ k = ρ¯ . If |ρ¯ k − ρ¯ k−1 | < δρ¯ , go to Step 6; Otherwise, go to Step 4.  Step 6: A suboptimal solution of (37) is derived, then stop.

197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216

toolbox of Matlab [19]. How to deal with the nonlinear terms Pi j Ai j + ATi j PiTj arises for solving the minimization problem (37). Recently, solving the BMI problems has been proposed in [21] by using some local and global optimization techniques. In this study, we solve the BMI optimization problem (37) with a local optimization algorithm that views the BMI as a double LMI. As is clear, for some given a, b, and ρ¯ > 0, if Kj satisfying (34) are fixed, seeking the matrices P > 0, R > 0, X1 = X1T , X2 , Sw > 0, w = 1, 2, Pi j , and Mij so that the inequalities of (34) and (36) hold is an LMI feasibility problem. On the other hand, if Pi j satisfying (34) are fixed, then seeking the matrices P > 0, R > 0, X1 = X1T , X2 , Sw > 0, w = 1, 2, K j , and Mij is also an LMI feasibility problem. However, searching an initially feasible solution for the BMI optimization problem (37) to start the local optimization algorithm is still a difficult problem. To overcome this difficulty, we use LHS method to find the appropriate values of Kj such that the LMIs in the optimization problem (37) are feasible (see Steps 1–3). Next, the local optimization algorithm for the design of a suboptimal guaranteed cost sampled-data fuzzy controller is presented as follows: Remark 3. It is noted in Step 1 of Algorithm 1 that if the gain matrices Kj have high dimensions such that there exist many parameters γ jn1 n2 in K j = [ j1 · · ·  jm ], which may lead to more computing time. To reduce the computational cost, we can select some special structures of Kj (e.g., K j = [1 · · · m ] or K j = [ j1 0 p×(m−1) ]) to replace Kj in Step 1 of Algorithm 1. Remark 4. LHS is a uniform stratified sampling method that has a small variance. With this technique, the design space for each factor is uniformly divided. This method was first proposed in [28] and the effectiveness of this technique can be easily verified in [27,42]. Moreover, the LHS technique is low in cost and easy for implementation and can provide a good global parameter search. It is noted that the exponential stability of the closed-loop fuzzy system (14) is shown in Theorem 1. However, the stability of the closed-loop PDE system (1)–(3) is not discussed. Next, we will show that the sampled-data fuzzy controller (13) proposed in Theorem 1 can stabilize exponentially the closed-loop PDE system when the eigenspectrum separation is provided sufficiently large.

219

Theorem 2. Assume that for given scalars a and b satisfying (16) and matrices P > 0, R > 0, X1 = X1T , X2 , Sw > 0, w = 1, 2, Kj , Pi j , and Mi j , i, j ∈ S such that the LMIs (34) hold. Then, there exists a positive integer m∗ such that if m ≥ m∗ , the sampled-data fuzzy controller (13) can ensure that the closed-loop PDE system (1)–(3) is exponentially stable.

220

Proof. See Appendix B. 

221

4. Numerical illustration

222

In this section, the proposed design strategy is used to control the Fisher equation and the temperature profile of a catalytic rod to verify its effectiveness.

217 218

223

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Example 1. We consider the control problem of the Fisher equation, which is a scalar reaction-diffusion equation. The closedform description of the Fisher equation is given by

∂ x¯ ∂ 2 x¯ = + βT (x¯ − x¯2 ) + βU b¯ (z)u(t ) ∂ t ∂ z2 226

(38)

subject to the Dirichlet boundary conditions

x¯(0, t ) = 0, x¯(π , t ) = 0 227

where x¯ ∈ R, u(t ) ∈ R denote the state and the manipulated input, respectively. The process parameters are given by

βT = βU = 2.0, b¯ (z) = 228

(39)



2/π sin (z).

The eigenvalue problem for the operator of this process A x¯ =

∂ 2 x¯ ∂ z2

x¯ ∈ D(A ) = {x¯ ∈ H is absolutely continuous, 229

can be solved, and its solution has the following form:



λ j = − j , φ j (z) = 2

230 231

2

π

sin ( jz), j = 1, 2, . . . , ∞.

For this system, the first two eigenvalues are selected as the dominant ones. By Galerkin’s technique, we can obtain the following ODE system:



x˙ s (t ) = 232

∂ 2 x¯ ∈ H , x¯(0, t ) = 0, x¯(π , t ) = 0} ∂ z2



−1 0 x (t ) + 0 −4 s



f1 (xs (t ), 0) f2 (xs (t ), 0)



+ Bs u(t )

(40)

where

xs (t ) = [x1 (t ) x2 (t )]T , Bs = [βU 0]T f1 (xs (t ), 0) = βT x1 (t ) − βT f¯1 (xs (t ))  π f¯1 (xs (t )) = φ1 (z)(x1 (t )φ1 (z) + x2 (t )φ2 (z))2 dz 0

f2 (xs (t ), 0) = f¯2 (xs (t )) = 233

βT x2 (t ) − βT f¯2 (xs (t ))

 π 0

φ2 (z)(x1 (t )φ1 (z) + x2 (t )φ2 (z))2 dz.

Notice that f¯1 (xs (t )) and f¯2 (xs (t )) can be written as

f¯1 (xs (t )) = p1 x21 + p2 x1 x2 + p3 x22 f¯2 (xs (t )) = q1 x21 + q2 x1 x2 + q3 x22 234

where

p1 =

 π 0

 π 0

φ12 (z)φ2 (z)dz = 0

 π 16 , q1 = φ12 (z)φ2 (z)dz = 0 15 0 0  π  π 32 φ1 (z)φ2 (z)dz = , q3 = φ23 (z)dz = 0. q2 = 2 15 0 0

p3 =

235

 π

4 3

φ13 (z)dz = , p2 = 2

Thus, (40) can be expressed as follows:



x˙ s (t ) = 236 237

φ1 (z)φ22 (z)dz =

−1 + βT − βT p1 x1 −βT q2 x2



−βT p3 x2 x (t ) + Bs u(t ). −4 + βT s

(41)

Select θ1 (t ) = x1 (t ) and θ2 (t ) = x2 (t ) as the premise variables. Assume that θ1 (t ) ∈ [−1, 1] and θ2 (t ) ∈ [−0.5, 0.5]. Under these constraints, we have

θ¯1,1 = min θ1 (t ) = −1, θ¯1,2 = max θ1 (t ) = 1 x1 (t )

x1 (t )

θ¯2,1 = min θ2 (t ) = −0.5, θ¯2,2 = max θ2 (t ) = 0.5. x2 (t )

x2 (t )

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[m3Gsc;August 18, 2015;11:2] 11

Then, θ g (t) is straightforward represented as

θg (t ) = μg,1 (θg (t ))θ¯g,1 + μg,2 (θg (t ))θ¯g,2 2

=

μg,w (θg (t ))θ¯g,w

(42)

w=1

239

where 2

μg,w (θg (t ))  0, w = 1, 2,

μg,w (θg (t )) = 1.

(43)

w=1

240

241

Solving (42) and (43), the following membership functions are derived:

μg,1 (θg (t )) 

θg (t ) − θ¯g,1 θ¯g,2 − θ¯g,1

μg,2 (θg (t )) 

θ¯g,2 − θg (t ) . θ¯g,2 − θ¯g,1

(44)

From (44), we get the following T–S fuzzy model: 4

x˙ s (t ) =

ωi (θ (t ))Ai xs (t ) + Bs u(t )

(45)

i=1

242

where ωi (θ (t )) = μ1,i1 (θ1 (t ))μ1,i2 (θ2 (t )) and



Ai = 243 244

−1 + βT (1 − p1 θ¯1,i1 ) −βT q2 θ¯2,i2

−βT p3 θ¯2,i2 −4 + βT



with i = i2 + 2(i1 − 1), i1 , i2 = 1, 2. From (13), the sampled-data fuzzy controller can be represented as

u(t ) =

4

ω j (θ (tk ))K j xs (tk ).

(46)

j=1

245

Then, we obtain the closed-loop system as follows:

x˙ s (t ) =

4 4

ωi (θ (t ))ω j (θ (tk ))[Ai xs (t ) + Bs K j xs (tk )].

(47)

i=1 j=1

246

It is assumed that the initial condition of the PDE system (38) and (39) is given by



x¯(z, 0) = x¯0 (z) = 0.5 247 248 249 250 251 252 253



2

sin z − 0.1

π

2

π

sin 2z.

(48)

Then, we obtain xs,0 = [0.5 − 0.1]T . Select CW1 = diag{10, 10}, DW2 = 0.5, h = 0.5, and δρ¯ = 5 × 10−3 . We first run Steps 1–3 of Algorithm 1 in order to find an initially feasible solution. Letting K j = [γ j 0], γ j , j = 1, . . . , 4 are scalar parameters. Give the ranges a ∈ [0, 2], b ∈ [0.5, 2], and γ j ∈ [−2, 0]. Then, we solve the LMIs in the optimization problem (37) with 100 sets of a, b, and γ j generated by LHS technique and find a set of the values a = 0.1539, b = 0.7699, γ1 = −1.0139, γ2 = −0.4401, γ3 = −1.9632, and γ4 = −1.5028. Next, continue Algorithm 1 (i.e., run Steps 4–6, iteratively). The algorithm is terminated after 200 iterations and a suboptimal solution of problem (37) can be obtained. With this solution, we get ρ¯ = 58.07665978 and the following control gain matrices:





K1 = −1.0088 0.0086





K2 = −0.4402 −0.0007





K3 = −1.9573 0.0035





K4 = −1.5047 −0.0041 . 254 255 256

Now, with the preceding control gain matrices and the initial condition (48), we apply the sampled-data fuzzy controller (46) to the system (38) and (39). The simulation results are shown in Figs. 2–5. Furthermore, calculating the value of the cost function, we obtain



Js =

4 0

(100xs (t )2 + 0.25u(t )2 )dt

= 5.9002 < Jb = 58.07665462 < 58.07665978 = ρ¯ Please cite this article as: Z.-P. Wang, H.-N. Wu, Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.08.009

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Fig. 2. Open-loop profile of evolution of x¯(z, t ).

Fig. 3. State trajectories of the closed-loop slow system xs (t).

Fig. 4. Control action u(t).

Fig. 5. Closed-loop profile of evolution of x¯(z, t ).

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13

259

which indicates that the optimized upper bound of the cost function is ensured. These results show that the resulting guaranteed cost sampled-data fuzzy controller can achieve the design purposes for slow system (47) and regulate the state x¯(z, t ) at the expected steady state (i.e., x¯(z, t ) = 0).

260

Example 2. The temperature profile of a catalytic rod can be described by

257 258

∂ x¯ ∂ 2 x¯ ν = + βT (e− 1+x¯ − e−ν ) + βU (b¯ (z)u(t ) − x¯) ∂ t ∂ z2 261

subject to the Dirichlet boundary conditions

x¯(0, t ) = 0, x¯(π , t ) = 0 262

(50)

where x¯ ∈ R is the state and u(t ) ∈ R denotes the manipulated input. The following values are given to the process parameters:

βT = 50, βU = 2.0, ν = 4.0, b¯ (z) = 263



2/π sin (z).

The spatial differential operator of the parabolic PDE system (49) and (50) is represented as: A x¯ =

∂ 2 x¯ ∂ z2

x¯ ∈ D(A ) = {x¯ ∈ H is absolutely continuous, 264



λ j = − j , φ j (z) = 265

2

π

sin ( jz), j = 1, 2, . . . , ∞.

Now, we select the first two eigenvalues as the dominant ones for (49) and (50). We can obtain the ODE system by Galerkin’s method as follows:



x˙ s (t ) = 267

∂ 2 x¯ ∈ H , x¯(0, t ) = 0, x¯(π , t ) = 0} ∂ z2

Then, the eigenvalue problem for this operator can be solved analytically and its solution is of the form: 2

266

(49)

−1 0



0 x (t ) + −4 s





f1 (xs (t ), 0) + Bs u(t ) f2 (xs (t ), 0)

(51)

where

xs (t ) = [x1 (t ) x2 (t )]T , Bs = [βU 0]T f1 (xs (t ), 0) = −βU x1 (t ) + f¯1 (xs (t ))  π ν f¯1 (xs (t )) = βT φ1 (z)(e− 1+x1 (t )φ1 (z)+x2 (t )φ2 (z) − e−ν )dz 0

f2 (xs (t ), 0) = −βU x2 (t ) + f¯2 (xs (t ))  π ν f¯2 (xs (t )) = βT φ2 (z)(e− 1+x1 (t )φ1 (z)+x2 (t )φ2 (z) − e−ν )dz. 0

268

Then, (51) can be represented as



x˙ s (t ) = 269

where

θ1 (t ) =

−1 − βU + θ1 (t ) θ3 (t )

⎧ ⎨0,

 θ2 (t ) x (t ) + Bs u(t ). −4 − βU + θ4 (t ) s

x (t ) ⎩ 2 1 2 f¯1 (xs (t )), x1 (t ) + x2 (t ) ⎧ ⎨0, θ2 (t ) = x (t ) ⎩ 2 2 2 f¯1 (xs (t )), x1 (t ) + x2 (t ) ⎧ ⎨0, θ3 (t ) = x (t ) ⎩ 2 1 2 f¯2 (xs (t )), x1 (t ) + x2 (t ) ⎧ ⎨0, θ4 (t ) = x (t ) ⎩ 2 2 2 f¯2 (xs (t )), x1 (t ) + x2 (t )

(52)

x21 (t ) + x22 (t ) = 0 x21 (t ) + x22 (t ) = 0 x21 (t ) + x22 (t ) = 0 x21 (t ) + x22 (t ) = 0 x21 (t ) + x22 (t ) = 0 x21 (t ) + x22 (t ) = 0 x21 (t ) + x22 (t ) = 0 x21 (t ) + x22 (t ) = 0.

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To avoid the discontinuity of the functions f¯2 (xs (t )) and f¯2 (xs (t )), we assume that x1 (t ) ∈ [−0.3533, 20] and x2 (t ) ∈ [−0.9, 0.9], which ensures that 1 + x1 (t )φ1 (z) + x2 (t )φ2 (z)  0. Under these constraints, we have

θ¯1,1 = min θ1 (t ) = −0.1773, θ¯1,2 = max θ1 (t ) = 6.0885 x1 (t ),x2 (t )

x1 (t ),x2 (t )

θ¯2,1 = min θ2 (t ) = −3.0191, θ¯2,2 = max θ2 (t ) = 3.0131 x1 (t ),x2 (t )

x1 (t ),x2 (t )

θ¯3,1 = min θ3 (t ) = −2.8746, θ¯3,2 = max θ3 (t ) = 2.8717 x1 (t ),x2 (t )

θ¯4,1 = 272

x1 (t ),x2 (t )

min θ4 (t ) = 0, θ¯4,2 = max

x1 (t ),x2 (t )

x1 (t ),x2 (t )

θ4 (t ) = 3.8306.

With the above values, θ g (t) can be represented by

θg (t ) = μg,1 (θg (t ))θ¯g,1 + μg,2 (θg (t ))θ¯g,2 =

2

μg,w (θg (t ))θ¯g,w

(53)

w=1

273

where 2

μg,w (θg (t ))  0, w = 1, 2,

μg,w (θg (t )) = 1.

(54)

w=1

274

275

Solving (53) and (54), we obtain the membership functions as follows:

μg,1 (θg (t )) 

θg (t ) − θ¯g,1 θ¯g,2 − θ¯g,1

μg,2 (θg (t )) 

θ¯g,2 − θg (t ) . θ¯g,2 − θ¯g,1

(55)

From (44), the following T–S fuzzy model is derived: 16

x˙ s (t ) =

ωi (θ (t ))Ai xs (t ) + Bs u(t )

(56)

i=1

276

where ωi (θ (t )) = μ1,i1 (θ1 (t ))μ1,i2 (θ2 (t ))μ3,i3 (θ3 (t ))μ4,i4 (θ4 (t )) and



Ai =

−1 − βU + θ¯1,i1 θ¯3,i 3

277 278

θ¯2,i2 −4 − βU + θ¯4,i4



with i = i2 + 2(i3 − 1) + 4(i2 − 1) + 8(i1 − 1), i1 , i2 , i3 , i4 = 1, 2. From (13), the sampled-data fuzzy controller can be expressed as follows:

u(t ) =

16

ω j (θ (tk ))K j xs (tk ).

(57)

j=1

279

Then, we can get the following the closed-loop system:

x˙ s (t ) =

16 16

ωi (θ (t ))ω j (θ (tk ))[Ai xs (t ) + Bs K j xs (tk )].

(58)

i=1 j=1

280

Assume that the initial condition of the PDE system (49) and (50) to be



x¯(z, 0) = x¯0 (z) = 0.9 281 282

2

π

sin 2z.

(59)

Then, we get xs,0 = [0 0.9]T . Choose CW1 = diag{0.2, 0.2}, DW2 = 2, h = 0.2, and δρ¯ = 10−2 . Using Algorithm 1, we can obtain ρ¯ = 3.94756402 and the control gain matrices as follows:









K1 = −2.9706 −1.254 , K2 = −2.7771 −0.8804









K3 = −2.6221 −0.8656 , K4 = −2.6938 −0.8777

















K5 = −2.6192 0.8703 , K6 = −2.6902 0.8798 K7 = −2.9722 1.2577 , K8 = −2.7786 0.8824

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Fig. 6. State trajectories of the closed-loop slow system xs (t).

Fig. 7. Control action u(t).

Fig. 8. Closed-loop profile of evolution of x¯(z, t ).

















K9 = −0.5757 −1.1838 , K10 = −0.4304 −0.9155

K11 = −0.3241 −0.9677 , K12 = −0.3551 −0.8775

















K13 = −0.3223 0.9738 , K14 = −0.3529 0.8794

K15 = −0.5784 1.1889 , K16 = −0.4325 0.9176 . 283 284 285

Next, we apply the sampled-data fuzzy controller (57) with the preceding gain matrices to the system (49) and (50) under the initial condition (59). The simulation results are shown in Figs. 6–8. Furthermore, calculating the actual value of the cost function, we have



Js =

4 0

(0.04xs (t )2 + 4u(t )2 )dt

= 0.2244 < Jb = 3.94756401 < 3.94756402 = ρ¯ 286

which means that the optimized upper bound of the cost function is ensured.

287

5. Conclusion

288

In this paper, the guaranteed cost control problem of a class of parabolic PDE systems has been investigated via finite dimensional sampled-data fuzzy control approach. Galerkin’s technique is initially applied to the parabolic PDE system to obtain a

289

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299

low-order nonlinear slow system. Subsequently, the resulting nonlinear slow system can be expressed by the T–S fuzzy model. Then, a novel time-dependent functional is used to derive the BMI condition for the existence of the guaranteed cost sampleddata fuzzy controller such that the closed-loop slow fuzzy system is exponentially stable and an upper bound of quadratic cost function is provided. Moreover, to minimize the cost bound, a suboptimal guaranteed cost sampled-data fuzzy controller can be formulated as a BMI optimization problem, which is solved by the proposed algorithm. Furthermore, it is shown that the proposed controller can stabilize exponentially the closed-loop PDE system. The main contribution and novelty of this paper are that a novel time-dependent functional is introduced to finite dimensional guaranteed cost control design for nonlinear parabolic PDE systems and the BMI optimization problem can be solved via a local optimization algorithm based on the LHS method and the LMI technique. Finally, the proposed design strategy is applied to the control of the Fisher equation and the temperature profile of a catalytic rod to illustrate its effectiveness.

300

Acknowledgments

301

305

This work was supported in part by National Basic Research Program of China (973 Program) (2012CB720003); in part by the National Natural Science Foundations of China under grants 61473011 and 61421063; in part by the General Research Fund project from Science and Technology on Aircraft Control Laboratory of Beihang University under grant 9140C480301130C48001; and in part by the Innovation Foundation of BUAA for Ph.D. Graduates. The authors also gratefully acknowledge the helpful comments and suggestions of the Associate Editor and anonymous reviewers, which have improved the presentation.

306

Appendix A

307

Proof of Theorem 1:. By Schur complement and using (34), we have

290 291 292 293 294 295 296 297 298

302 303 304

r r

ωi (θ (t ))ω j (θ (tk ))i j (t, h) < 0.

(A.1)

i=1 j=1

308 309

Obviously, the inequality (A.1) means that there exists a small enough scalar ¯ so that for any ∈ (0, ) ¯ , the inequality (32) holds. Thus, we can get the inequality (33). From (33), it follows that



tf

t0

=

(xTs (t )W1 xs (t ) + uT (t )W2 u(t ))dt 

tf

t0

{V˙ s (t ) + xTs (t )W1 xs (t ) + uT (t )W2 u(t )}dt −



tf

t0

V˙ s (t )dt

 Vs (t0 ) = xTs,0 Pxs,0 310

which implies that

Js  xTs,0 Pxs,0 . 311

In addition, from (32), we have r r

ωi (θ (t ))ω j (θ (tk ))i j (t, h¯ ) < 0

i=1 j=1

312

which indicates that there exists a sufficiently small scalar κ > 0 so that the following inequality holds: r r

ωi (θ (t ))ω j (θ (tk ))i j (t, h¯ )  −κ I

(A.2)

i=1 j=1

313

where

i j (t, h¯ ) = 1i j

1 (ψ(t ) h¯



=



2i j

314

j i111

∗

1i j

+ (h¯ − ψ(t ))

j i112 −vS1



2i j

)

 j ¯ i j R−1 MT i j i211 + hM 212 ij = . ∗ −vS2

Thus, from (28)–(30), and considering (A.2), we have Please cite this article as: Z.-P. Wang, H.-N. Wu, Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.08.009

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V˙ s (t ) 

r r

17

ωi (θ (t ))ω j (θ (tk ))ηT (t )i j (t, h¯ )η(t )

i=1 j=1

 −κηT (t )η(t )  −κξ T (t )ξ (t ). 315

(A.3)

Next, let us define

¯ 0] ξ (t0 + ϑ)  ϕ(ϑ), ϑ ∈ [−h,

(A.4)

316

where ϕ (ϑ) is an artificial initial condition constructed by

317

Notice that ψ (t) and φ (t) are bounded on [t0 , +∞), it follows from (26) that there exist scalars α 0 > 0 and α 1 > 0 such that

 ξ (t0 ), ϑ = 0 ϕ(ϑ) = ¯ 0). 0, ϑ ∈ [−h,

Vs (t )  α0 ξ (t )2 + α1 318



t

t−h¯

ξ (ς )2 dς .

(A.5)

Choose κ 0 > 0 such that

¯ κ0 h¯ ) − κ  0. κ0 (α0 + α1 he 319

By (A.3) and (A.5), we obtain

d(eκ0 t Vs (t ))  e κ0 t dt 320

(A.6)

  (κ0 α0 − κ)ξ (t )2 + κ0 α1

ξ (ς ) dς . 2

t−h¯

Integrating the aforementioned equation from t0 to t yields

eκ0 t Vs (t )  eκ0 t0 Vs (t0 ) +



t

t0

eκ0 ς {(κ0 α0 − κ)ξ (ς )2 + κ0 α1

¯ κ0 h¯ ) − κ)  eκ0 t0 Vs (t0 ) + (κ0 (α0 + α1 he 321



t



t

t0

 ς ς −h¯

ξ (τ )2 dτ }dς

¯ κ0 h eκ0 ς ξ (ς )2 dς + κ0 α1 he ¯



t0

t0 −h¯

eκ0 ς ξ (ς )2 dς .

Then, using (A.4) and (A.6), it is immediate from (A.7) that eκ0 t Vs (t )  eκ0 t0 Vs (t0 ), i.e.,

Vs (t )  e−κ0 (t−t0 )Vs (t0 ). 322

324

(A.8)

By (27), we further obtain from (A.8) that

xs (tk )  323

(A.7)



κ0

σ2 /σ1 e− 2 (tk −t0 ) xs (t0 )

(A.9)

where σ1 = σ (P ) and σ2 = σ¯ (P ). Moreover, we notice that for any given t > t0 , there exists an integer k0 ∈ N such that t ∈ [tk0 , tk0 +1 ). Thus, by Lemma 2 and (A.9), we get

 κ0 xs (t )  μ0 xs (tk0 ) = μ0 σ2 /σ1 e− 2 (tk0 −t0 ) xs (t0 )  κ0 κ0  μ0 σ2 /σ1 e 2 h xs (t0 )e− 2 (t−t0 )

325

which means that the slow fuzzy system (14) is exponentially stable over S (h). The proof is complete. 

326

Appendix B

327

Proof of Theorem 2:. By substituting the sampling control law (13) into (4), we get the following closed-loop system:

⎧ r r

⎪ ⎪ ωi (θ (t ))ω j (θ (tk ))[Ai xs (t ) + Bs K j xs (t − ρ(t ))] +  fs x˙ s (t ) = ⎪ ⎨ i=1 j=1

r

⎪ ⎪ ⎪ ω j (θ (tk ))B f K j xs (t − ρ(t )) ⎩ x˙ f (t ) = A f x f + f f (xs , x f ) +

(B.1)

j=1

328 329

330

where  fs = fs (xs , x f ) − fs (xs , 0). Since fs and ff are Lipschitz continuous, there exist positive scalars κ 1 , κ 2 , and κ 3 so that



 fs   κ1 x f l2  f f (xs , x f )  κ2 xs  + κ3 x f l2

(B.2)

for all xs  ∈ Rm and xf  ∈ l2 . Please cite this article as: Z.-P. Wang, H.-N. Wu, Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.08.009

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Select a functional candidate for system (B.1) as

V (t ) = Vs (t ) + V f (t ) 332 333

(B.3)

where Vs (t) is defined in (26) and V f  qxTf (t )x f (t ), q > 0 is a given constant. Using (B.1), gives the following relationship:



0 = 2ξ T (t )Pi j

r r



ωi (θ (t ))ω j (θ (tk ))(Ai xs (t ) + Bs K j xs (t − ρ(t ))) − x˙ s (t ) +  fs

i=1 j=1

=

r r

ωi (θ (t ))ω j (θ (tk ))ξ T (t )(Pi j Ai j + ATi j PiTj )ξ (t ) + 2ξ T (t )Pi j  fs .

(B.4)

i=1 j=1

334

From (28)–(29), and (B.4), for t ∈ [tk , tk+1 ), we have

V˙ s (t ) 

r r

ωi (θ (t ))ω j (θ (tk ))ηT (t )i j (t, h¯ )η(t ) + 2ξ T (t )Pi j  fs

i=1 j=1

 −κξ (t )2 + 2κ1 κ4 ξ (t )x f (t )l 2 335

(B.5)

where κ4 = max σ¯ (PiTj ), i, j ∈ S. By using (B.1) and (B.2), we obtain i, j

V˙ f (t ) = 2qxTf (t )x˙ f (t ) = 2qxTf (t )A f x f (t ) + 2qxTf (t ) f f (xs , x f ) + 2q

r

ω j (θ (tk ))xTf (t )B f K j xs (t − ρ(t ))

j=1

 2qλm+1 xTf (t )x f (t ) + 2qx f (t )l 2  fs (xs , x f ) + 2qI4T K Tj BTf x f (t )ξ (t )  2qλm+1 x f (t )2l 2 + 2qκ2 x f (t )l 2 ξ (t ) + 2qκ3 x f (t )2l 2 + 2qκ5 x f (t )l 2 ξ (t ) 336

where κ5 =

σ¯ (I4T K Tj BTf ).

(B.6)

Thus, from (B.5) and (B.6), for t ∈ [tk , tk+1 ), we have

V˙ (t ) = V˙ s (t ) + V˙ f (t )  ξ˜T (t )ξ˜(t ) 337



= 338

(B.7)

where ξ˜(t ) = [ξ (t ) x f (t )l 2 ]T and

−κ ∗

 κ1 κ4 + q(κ2 + κ5 ) . 2q(λm+1 + κ3 )

Then, it can be verified that there exist an integer m > m∗ and a scalar κ¯ 0 > 0 such that   −κ¯ 0 I, and thus

V˙ (t )  −κ¯ 0 ξ˜(t )2l 2 339

(B.8)

Moreover, letting xˆ(t ) = [xTs (t ) xTf (t )]T , the closed-loop system (B.1) can be rewritten as

xˆ˙ (t ) = Aˆ xˆ(t ) + Bˆxˆ(tk ) + fˆ(xˆ) 340

where

⎡ Aˆ =

⎣ ⎡

r

(B.9)

⎤ ωi (θ (t ))Ai

i=1

0

0⎦ Af

⎤ ω (θ ( t )) B K 0 s j j k ⎢ ⎥ ⎢ j=1 ⎥ Bˆ = ⎢ r ⎥ ⎣ ⎦ ω j (θ (tk ))B f K j 0  fˆ(xˆ) = 341

r

j=1

  fs . f f (xs , x f )

Similarly in Lemma 2, we get that the system (B.9) has the following property:

xˆ(t )l2  μ¯ 0 xˆ(tk )l2 , t ∈ [tk , tk+1 ) 342 343

where μ ¯ 0 is a positive scalar. Then, by applying an analogous analysis in the proof of Theorem 1, it is easy to deduce that the closed-loop PDE system is exponentially stable. This completes the proof.  Please cite this article as: Z.-P. Wang, H.-N. Wu, Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.08.009

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344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 Q2 401 402 403 404 405 406 407 408 409 410 411 412 413 414

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Please cite this article as: Z.-P. Wang, H.-N. Wu, Finite dimensional guaranteed cost sampled-data fuzzy control for a class of nonlinear distributed parameter systems, Information Sciences (2015), http://dx.doi.org/10.1016/j.ins.2015.08.009