Fuzzy guaranteed cost sampled-data control of nonlinear systems coupled with a scalar reaction–diffusion process

Accepted Manuscript Fuzzy guaranteed cost sampled-data control of nonlinear systems coupled with a scalar reaction-diffusion process

Jun-Wei Wang, Han-Xiong Li, Huai-Ning Wu

PII: DOI: Reference:

S0165-0114(15)00487-X http://dx.doi.org/10.1016/j.fss.2015.09.027 FSS 6922

To appear in:

Fuzzy Sets and Systems

Received date: Revised date: Accepted date:

27 March 2014 31 July 2015 29 September 2015

Please cite this article in press as: J.-W. Wang et al., Fuzzy guaranteed cost sampled-data control of nonlinear systems coupled with a scalar reaction-diffusion process, Fuzzy Sets and Systems (2015), http://dx.doi.org/10.1016/j.fss.2015.09.027

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Fuzzy Guaranteed Cost Sampled-Data Control of Nonlinear Systems Coupled with a Scalar Reaction-Diffusion Process Jun-Wei Wang†, Han-Xiong Li‡, § and Huai-Ning Wu*, ¶ †

‡

§

School of Automation and Electrical Engineering University of Science and Technology Beijing Beijing 100083, P. R. China

Department of Systems Engineering and Engineering Management City University of Hong Kong Tat Chee Avenue, Kowloon, Hong Kong SAR.

State Key Laboratory of High Performance Complex Manufacturing, Central South University, Changsha 410083, Hunan, P. R. China ¶

Science and Technology on Aircraft Control Laboratory School of Automation Science and Electrical Engineering Beihang University (Beijing University of Aeronautics and Astronautics) Beijing 100191, P. R. China Abstract A fuzzy guaranteed cost sampled-data control problem is addressed in this paper for a class of nonlinear coupled systems of an n-dimensional lumped parameter system modeled by ordinary differential equation (ODE) with a scalar reaction-diffusion process represented by parabolic partial differential equation (PDE). A Takagi-Sugeno (T-S) fuzzy coupled ODEPDE model is initially proposed to accurately represent the nonlinear coupled ODE-PDE system. Based on the T-S fuzzy coupled ODE-PDE model, a fuzzy sampled-data controller is subsequently developed via the Lyapunov’s direct method to not only locally exponentially stabilize the nonlinear coupled system, but also provide an upper bound of the given cost function. Moreover, a suboptimal fuzzy sampled-data control design is also addressed in the sense of minimizing an upper bound of the cost function. The main contribution of this study lies in that a new parameterized linear matrix inequality (LMI) technique is proposed to reduce the conservativeness of the method, and an LMI-based fuzzy suboptimal sampled-data control design is developed for the nonlinear coupled ODE-PDE system based on this parameterized LMI technique. Simulation results on the sampled-data control of hypersonic rocket car are provided to illustrate the effectiveness and merit of the design method. Key words: Fuzzy control; Sampled-data control; Lumped parameter system; Distributed parameter system; Linear matrix inequalities (LMIs). *

Corresponding author. E-mail: [email protected]

1

1. Introduction Nowadays most real control systems are implemented using the digital technology since it is very cheap, fast, relatively easy to operate, flexible and reliable. This motivates investigation of the so called sampled-data systems that consist of a continuous-time plant or process controlled by a discrete-time controller as discussed in [1]. Over the past few decades, sampled-data control systems have been the focus of considerable research interest, see, e.g., [2]-[10] and references therein. However, these results [1]-[10] are only applicable for the systems modeled by ordinary differential equations (ODEs). Notice that there are some industrial processes whose behavior must be spatially distributed in nature such as heat conduction, fluid flow, and chemical engineering processes [11]-[13]. These processes are referred to as distributed parameter systems described by partial differential equations (PDEs). Sampled-data control techniques have been recently extended to develop the finitedimensional control design for linear parabolic PDE systems [14] and nonlinear parabolic PDE systems [15]. Based on the original PDE model, more recently, robust sampled-data controller has been proposed in [16] for semi-linear parabolic PDE systems. On the other hand, some industrial processes are more complex processes involve both ODE and PDE model description. For example, in a fluidized bed reactor, mass and energy balances are described by PDEs while vibrations of void fraction are represented by an ODE [17]. The models of lumped and distributed parameter systems may be coupled through their states or boundaries [18]-[21]. Thus, the study of control design for coupled ODE-PDE systems is of practical importance. For coupled ODE-PDE systems, we need to redefine the exponential stability in the sense of the appropriate norm. Some researchers have paid attention to the coupled ODE-PDE systems because of their applications in real world situations and significance for control theory, and proposed some approaches to solve the control synthesis problem of these systems, such as optimal control [17], [19], proportionalderivative (PD) control [20], backstepping state and output feedback control [22]. Furthermore, Krstic has developed a backstepping design method in his book [23] for linear ODE-PDE cascaded systems. However, these results only focus on linear ODE-PDE coupled systems. The investigation of guaranteed cost sampled-data control design is quite few for nonlinear ODE-PDE coupled systems. Using the standard Lyapunov stability theory and fuzzy control based on Takagi-Sugeno (T-S) fuzzy model [24], the fruitful results of stability analysis and control design have been reported for nonlinear ODE systems, see, e.g., [6], [8], [9], [25]-[35] and references therein. 2

For PDE systems, the state space is a Hilbert space. Applying the Galerkin method and the fuzzy control techniques, some finite-dimensional fuzzy control design methods have been developed in [36]-[40] for parabolic PDE systems, where the closed-loop stability of the original PDE systems is shown via the combination of singular perturbation method and the standard Lyapunov stability theory. On the other hand, by introducing the definition of exponential stability for PDE systems in the sense of L2 norm, the distributed fuzzy control methods have been proposed via Lyapunov techniques for PDE systems [41]-[44]. Despite these promising results, however, the fuzzy control techniques on the basis of the original PDE model are not mature for nonlinear PDE systems. Moreover, these results [25]-[44] cannot be directly used to solve the problem of control design for coupled ODE-PDE systems, since the state space of the coupled systems is the Euclidean space of ODE state along with the Hilbert space of PDE state. More recently, fuzzy control design methods have been reported for nonlinear ODE-PDE coupled systems, where the PDE models are described by first-order hyperbolic PDE [45] and parabolic PDE [46], respectively. To the best of authors’ knowledge, this study is the first paper to deal with the fuzzy guaranteed cost sampled-data control problem for nonlinear coupled ODE-PDE systems, which motivates this study. This paper will deal with the problem of fuzzy guaranteed cost sampled-data control for a class of nonlinear systems represented by the coupled model of ODE and parabolic PDE. We first introduce the state space and its norm of the coupled ODE-PDE systems. Then, the definition of locally exponential stability is presented in the sense of the chosen norm. The purpose of this study is to design a fuzzy sampled-data controller not only locally exponentially stabilizing the coupled ODE-PDE system but also minimizing the bound of the given quadratic cost function. In the proposed design procedure, a T-S fuzzy coupled ODE-PDE model is initially proposed to accurately represent the nonlinear coupled system. A new parameterized linear matrix inequality (LMI) technique is introduced to reduce the conservativeness of the design method. Based on the obtained T-S fuzzy coupled ODE-PDE model and this parameterized LMI technique, a suboptimal fuzzy guaranteed cost sampled-data controller only using the information of ODE state is subsequently developed via the Lyapunov’s direct method. The outcomes of this paper are presented in terms of standard LMIs, which can be directly solved via the polynomial-time interior-point method [47], [48]. Finally, the proposed design method is successfully applied to the suboptimal guaranteed cost sampled-data control of hypersonic rocket car. It must be pointed out that the present study is completely different from our previous results [41]-[46] and [49]. The main difference is that the system studied in this work is described by nonlinear ODE coupled with 3

a scalar parabolic PDE and the proposed controller is a sampled-data one, whereas the suggested controllers reported in [41]-[46] and [49] are continuous in time. The main contribution and novelty of this paper is summarized as follows: (i) A new parameterized LMI technique is introduced to reduce the conservativeness of the control design method; (ii) A less conservative LMI-based suboptimal fuzzy guaranteed cost sampled-data control design is developed for the coupled systems based on this parameterized LMI technique. The remainder of this paper is organized as follows. Section 2 introduces the problem formulation and preliminaries. Section 3 presents a suboptimal fuzzy guaranteed cost sampled-data control design based on the LMI optimization techniques. The simulation study on the sampled-data control of hypersonic rocket car is given in Section 4. Finally, Section 5 offers some concluding remarks. 

Notations: The following notations will be used throughout this paper. ƒ , ƒ , ƒ mun

and ƒ

n

denote the set of all real numbers, all positive scalars, n-dimensional Euclidean

space and the set of all m u n matrices, respectively.

˜

and ˜, ˜

ƒn

denote the Euclidean

norm and inner product for vectors, respectively. Identity matrix of appropriate dimension will be denoted by I . M ! (, d)0 means that it is symmetric and positive definite (negative definite, negative semi-definite, respectively). Omin ( A) and Omax ( A) stand for the minimum and maximum eigenvalues of a square matrix

A , respectively.

 $2 ([0, L ]; ƒ) is a Hilbert space of square integrable scalar functions

Y (x, t) ƒ ,

x [0, L]  ƒ with the inner product and norm: Y 1 (˜, t ),Y 2 (˜, t )

³

L

0

where Y 1 (˜, t ) , Y 2 (˜, t ) 

Y 1 ( x , t ),Y 2 ( x , t )

ƒ

dx and

Y1 (˜, t ) 2

Y1 (˜, t ),Y1 (˜, t )

1/ 2

. The superscript ‘T’ is used for the transpose of a vector or a

matrix. The symbol ‘ ’ is used as an ellipsis in matrix expressions that are induced by symmetry, e.g., ª S  [ M  N  ] X º ª S  [ M  N  M T  N T ] X º  ». « Y »¼ «¬

XT Y¼ ¬

2. Problem formulation and preliminaries 2.1 System description

4

This paper considers a class of nonlinear coupled systems of an n-dimensional nonlinear ODE system with a scalar parabolic PDE system of the following state-space form: x (t )

Tt ( z, t )

L

f ( x (t ))  g ( x (t )) ³ T ( z , t ) dz  B ( x (t )) u (t ) ,

(1)

0

aTzz ( z, t )  bT ( z, t )  M ( x (t ))

(2)

subject to the homogenous Robin boundary conditions aTz (0, t )  a0T (0, t )

0 , aTz ( L, t )  aLT ( L, t ) 0

(3)

and the initial conditions x (0)

x0 , T ( z , 0) T0 ( z )

(4)

n where x(t )  is the state of ODE subsystem (   ƒ is a given local domain

containing the origin x(t ) 0 ); u(t ) ƒm is the control input; T (˜, t )  parabolic PDE subsystem; the subscripts respect to

is the state of

z and t stand for the partial derivatives with

z , t , respectively, z [0, L] ƒ and t [0, f) are the position and time,

respectively, and L is an known constant; f ( x(t )) , g( x(t )) , B( x(t )) and M ( x(t )) are locally Lipschitz continuous functions in x(t ) with appropriate dimensions and satisfy

f (0) 0 and M(0) 0 ;

a ! 0 , b , a0 and aL are known parameters; T0 ( z ) and x0

denote the initial conditions. The coupled system is depicted in Fig. 1. For example, the model (1)-(4) can be used to describe the spatio-temporal dynamics in semiconductors [18]. On the other hand, this model can also capture the complex dynamics of hypersonic rocket car in [19], where the rigid motion and the surface temperature distribution of the car are modeled by ODE (1) and a scalar parabolic PDE (2)-(4), respectively. u(t )

n-dimesnional ODE

x (t )

³ T ( z, t )

scalar parabolic PDE

Fig. 1 Nonlinear coupled systems of an n-dimensional ODE with a scalar parabolic PDE

Remark 1. As shown in Fig. 1, the ODE state x(t ) is fed into PDE dynamics; in turn, the temperature T ( z, t ) is influenced by the ODE state x(t ) (see (2)), which can be referred to as a state couple. The control input u(t ) is directly added to the ODE system (1). The notation “ ³ ” denotes the integrator.

5

Let the state space of the system (1)-(4) be  u x (t )

2

 T ( ˜, t )

2 2

. The norm of such state space is

. For simplicity, when u(t ) { 0 , the coupled system (1)-(4) is referred to

as a free coupled system. We introduce the following definition of locally exponential stability in the sense of norm

x (t )

2

 T ( ˜, t )

2 2

for the free coupled system:

Definition 1. The free coupled system of (1)-(4) (i.e., u(t ) { 0 ) is said to be locally exponentially stable, if there exist constants G ƒ , V ƒ and J  ƒ  such that for any 

2



2

x(0)  T0 (˜) 2  G , the solutions x(t ) and T ( z, t ) of the free coupled system of (1)-(4) satisfy 2

x ( t )  T (˜, t )

2 2

dJ

x

This definition implies that

2 0

 T0 (˜) 2

2 2

exp( V t ) . 2

2

2

x(t )  T (˜, t ) 2 d JG for any x(0)  T0 (˜) 2  G .

2.2 T-S fuzzy model and problem formulation It has been pointed out that an exact T-S fuzzy ODE model construction from a given nonlinear dynamical ODE model can be obtained by the sector nonlinearity approach [25]. More recently, this exact T-S fuzzy ODE modeling approach has been extended to deal with the fuzzy model-based control problem for a class of semi-linear parabolic PDE systems [42], [43], a class of semi-linear hyperbolic PDE systems [41], [44], and a class of nonlinear ODE-PDE coupled (or cascaded) systems [45], [46]. In this section, a T-S fuzzy coupled ODE-PDE model is utilized to accurately represent the nonlinear coupled ODE-PDE system (1)-(4). This T-S fuzzy coupled ODE-PDE model is described by fuzzy IF-THEN rules which express local dynamics using a linear coupled ODE-PDE model. The overall fuzzy coupled ODE-PDE model of the system is achieved by fuzzy ‘‘blending’’ of the linear coupled ODE-PDE models. Hence, the nonlinear coupled system (1)-(4) can be exactly represented by the following T-S fuzzy coupled ODE-PDE model: Plant Rule i : IF [1 (t ) is Fi1 and

"

and [l (t ) is Fil

­ x (t ) A x (t )  g L T ( z, t )dz  B u(t ) ° i i ³0 i THEN ® , i   {1, 2," , r} °¯Tt ( z, t ) aTzz ( z , t )  bT ( z , t )  d iT x (t ) where F ij , i  , j 1,2,", l are fuzzy sets, and Ai , Bi , gi and d i , for i  known matrices and vectors with appropriate dimensions,

6

(5) are real

r is the number of IF-THEN

rules. [ j (t ) , j 1,2,", l are the premise variables, which are assumed to be functions of ODE state x(t ) in this paper. Furthermore, we assume that ( Ai , Bi ) is controllable pair for each i  . By applying the center-average defuzzifier, product interference and singleton fuzzifier, the overall dynamics of T-S fuzzy coupled model (5) can be expressed as: x ( t )

¦

L

r i 1

hi ([ ( t ))[ Ai x ( t )  g i ³ T ( z , t ) dz  Bi u ( t )] 0

L

A ([ ) x (t )  g ([ ) ³ T ( z , t ) dz  B ([ ) u (t ) ,

(6)

0

Tt ( z, t ) aTzz ( z, t )  bT ( z, t )  ¦i 1 hi ([ (t ))diT x(t ) r

aTzz ( z , t )  bT ( z , t )  d T ([ ) x (t )

(7)

where [ (t )  [[1 (t ) [ 2 (t ) " [ l (t )]T and

S ([ )  ¦i 1 hi ([ (t )) Si , Zi ([ (t ))

–

r

hi ([ (t )) Zi ([ (t ))

¦

r i 1

Zi ([ (t )) ,

i

l j 1

Fij ([ j (t )) ,

, S(˜) {A(˜), B(˜), g(˜), d (˜)} ,

(8)

in which Fij ([ j (t )) is the grade of the membership of [ j (t ) in F ij , for i  . In this paper, it is assumed that Zi ([ (t )) t 0 , i 

r

and

¦Z ([ (t)) ! 0 , for all i

t t 0 . Then the following

i 1

conditions can be obtained for all t t 0 hi ([ (t )) t 0 , i 

and

¦

r

h ([ (t )) 1 .

(9)

i 1 i

In the sampled-data control systems, the control signals are kept constant during the sampling period and are allowed to change only at the sampling instant. Hence, the sampled-data control signals are stepwise, which is shown in Fig. 2. u( t )

t tk 1

tk

tk 1

tk  2

Th

Fig. 2 The sampled-data control singal

7

Based on the fuzzy coupled model (6) and (7), hence, this study considers the following sampled-data fuzzy control law for the nonlinear coupled system (1)-(4): Control Rule

j:

IF [1 (tk ) is F j1 and THEN u (t )

and [l (tk ) is F jl

"

K j x (t k ) , tk d t  tk 1 , j 

(10)

where K j  ƒ m u n , j 

are real matrices to be determined; t k

the sampling instant; Th

tk 1  tk denotes the constant sampling period. The overall fuzzy

kTh , k

0,1,2,", f is

sampled-data controller is inferred as follows: u(t )

¦

r

h j ([ (tk )) K j x (tk )

j 1

(

K ([ k ) x (tk )

tk d t  tk 1

,

1

,

k 0,1,2,", f

1

)

r

where K ([ k )  ¦ h j ([ (tk )) K j . Since the conditions given in (9) hold for all t t 0 , we have j 1

hi ([ (tk )) t 0 , i 

and

¦

r

h ([ (tk )) 1 .

(12)

i 1 i

Set W (t ) t  tk . It is clear that

W(t) 1 and W (t )  [0, Th ) , tk d t  tk 1 .

(13)

Then, fuzzy sampled-data fuzzy controller (11) is written as the following form: u(t )

¦

r j 1

h j ([ (tk )) K j x (t  W (t ))

(

K ([ k ) x (t  W (t ))

1

4

)

r

where K ([ k )  ¦ h j ([ (tk )) K j . Substituting (14) into (6), the closed-loop sampled-data j 1

coupled system can be represented as (7) and x (t )

L

A ([ ) x (t )  B ([ ) K ([ k ) x ( t  W (t ))  g ([ ) ³ T ( z , t ) dz . 0

Fig. 3 shows the block diagram of a fuzzy model-based sampled-data control system.

8

(15)

u(t )

x (t )

Nonlinear PDE-ODE coupled system (1)-(4)

T-S fuzzy coupled model (6) and (7)

x ( tk )

u(tk )

Zero-order holder

Fuzzy controller (11)

Sampler with sampling period Th

Sampled-data fuzzy controller

Fig. 3 The block diagram of a fuzzy model-based sampled-data control system

Furthermore, the following quadratic cost function is utilized in this paper:

³ x f

J

0

T

(t )Qx (t )  u T (t ) Nu (t ) dt ,

(16)

where Q ! 0 and N ! 0 are known matrices with appropriate dimensions. Therefore, based on the fuzzy coupled ODE-PDE model (6) and (7), the objective of this paper is to find a fuzzy sampled-data control law (11) not only to locally exponentially stabilize the nonlinear coupled ODE-PDE system (1)-(4), but also to minimize the value of the cost function (16). To this end, the following lemmas and assumption are presented: mum

Lemma 1 [50]. For any constant symmetric matrix M ƒ

, scalar h ! 0 , vector

function w :[0, h] o ƒm such that the following integrations are well defined, then h

h ³ w T ( s ) Mw ( s ) ds t 0

³

h

0

h

w T ( s ) ds M ³ w ( s ) ds .

Lemma 2 [51]. Let T (˜, t ) 

0

be an absolutely continuous scalar function with square

integrable derivative Tz ( z , t ) . Then the following inequality is achieved: 2

2

T (˜, t ) 2 d 2LT 2 (0, t )  4L2 Tz (˜, t ) 2 .

(17)

Notice that the fuzzy sampled-data controller (11) only utilizes the information of membership functions hi ([ (t )) , i 

at some specified points t k , k

0,1,2,", f . This

characteristic prevents the use of the parameterized LMI techniques [28] to derive less conservative design methods. However, the difference between hi ([ (t )) and hi ([ (tk )) , i

can be modeled as a class of special uncertain grades of membership. Some inequality

relaxation techniques have been reported over the past few years [6], [52] and [53] to enhance 9

the stabilization ability of fuzzy control systems subject to uncertain grades of membership. However, the results in [52] and [53] are more computationally demanding. To reduce the conservativeness but not significantly increase the computational demand of the control design, we make the following technique assumption for the membership functions hi ([ (t )) , i  . Assumption 1. The membership functions hi ([ (t )) , i  any i 

satisfy hi ([ (t ))  [D min , D max ] , for

and t t 0 , where 0  D min  D max  1 .

It has been pointed out in [52] that the property assigned for the membership functions hi ([ (t )) , i 

by Assumption 1 can be achieved by properly building the fuzzy plant model

for the nonlinear plant. (For the procedure of obtaining the fuzzy plant model of the nonlinear plant, please refer to [25], [26] and [54].) For example, the operating domain ˆ u is a bit larger than the one  u

, i.e.,  u

the fuzzy modeling process. hi ([ (t )) , i 

 ˆ u

, which

where ˆ  ƒn , is employed in

will lie in the range of 0 and 1 but not reach the

boundary values due to the fact that the nonlinear plant operates in the actual operating domain which is slightly smaller than the operating domain considered in the fuzzy modeling process. In Section 4, we will illustrate the detailed fuzzy modeling process for the hyperbolic rocket car such that Assumption 1 is achieved. The following Lemma 3 gives a new parameterized LMI technique based on Assumption 1 and the parameterized LMI techniques reported in [28]. Lemma 3. Under Assumption 1, given symmetric matrices & ij  ƒ n u n , i, j  9

and a scalar

1 satisfying 0  9 d D minD max . If there exist symmetric matrices b j  ƒ nu n such that

°­&ii  0, i  ® 1 °¯(r  1) &ii  0.5(&ij  & ji )  0, i z j, i, j  (

1 & ij  b j  0 ,

8

)

i, j 

(19) r

r

where & ij  & ij  0.59 1 (1  9 )( b i  b j ) , i, j  , then &([ , [ k )  ¦¦ hi ([ (t ))h j ([ (tk )) &ij i 1 j 1

 0 is fulfilled for any t  [tk , tk 1 ) ,

k 0,1,2,", f .

1 Proof. From Assumption 1, one can find a scalar 0  9 d D minD max such that the membership

functions hi ([ (t )) and hi ([ (tk )) , i 

satisfy the following constraints:

10

hi ([ (t ))  9 hi ([ (tk )) t D min  9D max t 0 , i  , for all t t 0 . Using (20) and considering the property

¦

r j 1

h j ([ (tk ))

¦ ¦

(20)

r

r

i 1

j 1 i

h ([ (t ))h j ([ (tk )) 1 , we

have

¦ ¦

& ([ , [ k )

r

r

i 1

j 1

¦i

r 1

¦

¦ ¦

[hi ([ (t ))  9 hi ([ (tk ))  9 hi ([ (tk ))]h j ([ (tk )) & ij

r j 1

r

r

i 1

j 1

[hi ([ (t ))  9 hi ([ (tk ))]h j ([ (tk ))( b j  b j )

[hi ([ (t ))  9 hi ([ (tk ))]h j ([ (tk ))[& ij  b j ]  9 ¦ i

r 1

¦

r

h ([ (tk ))& ij

j 1 ij

(21) where hij ([ ( t k ))  hi ([ ( t k )) h j ([ ( t k )) . Obviously, the following inequality holds:

¦ ¦ r

r

i 1

j 1

[hi ([ (t ))  9 hi ([ (tk ))]h j ([ (tk ))( & ij  b j )  0 , t  [tk , tk 1 )

if the inequalities (19) and (20) are fulfilled. Applying Theorem 2.2 in [28], on the other hand, we can get that the inequality

¦ ¦ r

r

i 1

j 1 ij

h ([ (tk ))&ij  0 holds if the LMIs (18) are fulfilled.

Therefore, it can be derived from above analysis that if the LMIs (18) and (19) hold under Assumption 1, &([ , [ k )  0 is then satisfied for t  [tk , tk 1 ) ,

k 0,1,2,", f . ƶ

Remark 2. Notice that the constraints (20) derived from Assumption 1 for the membership functions hi ([ (t )) and hi ([ (tk )) , i 

are introduced in the Lemma 3 to obtain the less

conservative results, i.e., the inequalities (18) and (19). These constraints are also employed in [6] to deal with the fuzzy sampled-data control design of nonlinear time-delay systems and in [52] to address the stability analysis of fuzzy control systems subject to uncertain grades of membership. For some real applications, these constraints can be easily verified, see, e.g., the truck-trailer with time delay [6]. But it must be pointed out that Assumption 1 to some extent brings some conservativeness of the T-S fuzzy model caused by locality [55] since this assumption requires that the membership functions hi ([ (t )) , i 

lie in the range of 0 and

1 but not reach the boundary values.

3. Fuzzy guaranteed cost sampled-data control design In this section, an LMI-based design method of fuzzy guaranteed cost sampled-data controller will be presented for the nonlinear coupled ODE-PDE system (1)-(4) based on the

11

T-S fuzzy coupled ODE-PDE model (6) and (7). To this end, we consider the following Lyapunov candidate function for the fuzzy system (7) and (15): V (t ) V1 (t )  V2 (t ) ,

(22)

where

V1 (t ) xT (t ) Px(t )  ³

t

t W (t )

(

(s  t  Th ) x T (s) Rx (s)ds

,

2 nun

in which 0  P ƒ

L

D ³ T 2 ( z , t ) dz

V2 (t )

,

0

3 nun

and 0  Rƒ

)

are real matrices and D ! 0 is a real scalar to be L

x T (t k ) Px (t k )  D ³ T 2 ( z , t k ) dz

determined. It is continuous in time since V (t k )

0

V (t k ) .

From the Newton-Leibniz formula, we can get

x(t )  x(t W (t ))

t

³W

t  (t )

x (s)ds .

(24)

Substituting (24) into (15), derives

x (t ) Ac ([ , [k ) x(t )  B([ ) K ([k )³

t

L

t W (t )

x (s)ds  g([ )³ T ( z, t )dz ,

(25)

0

where Ac ([ , [ k )  A ([ )  B ([ ) K ([ k ) . Taking the time derivative of V1 (t ) along the solution of the fuzzy ODE subsystem (25), where t  [tk , tk 1 ) , yields

V1 (t )

xT (t )[ PAc ([ , [k )  ]x(t )  2 xT (t ) PB([ ) K ([k )³

t

t W ( t )

L

t

0

t W (t )

2 xT (t ) Pg([ )³ T ( z, t )dz  Th x T (t ) Rx (t )  ³

x ( s)ds

x T (s) Rx (s)ds .

(26)

Applying Lemma 1 and R ! 0 , we can get when W (t ) ! 0

³

t

t W (t )

x T (s) Rx (s)ds d W 1 (t )³

t

t W (t )

d Th1 ³

t

t W (t )

When

W (t ) 0 ,

lim W 1 (t )³

t

t W ( t )

W ( t )o0

the t

t W ( t )

t

t W (t )

(27)

x ( s)ds 0 .

It is clearly seen that [30] 0

t

t W ( t )

x T (s)dsR³

inequality

x T ( s)dsR³

x T (s)dsR³

2 x T (t ) Px (t )  2 x T (t ) Px (t )

12

is

x (s)ds

x (s)ds . also

(27) fulfilled

obviously

because

2 x T (t ) PAc ([ , [k ) x(t )  2 x T (t ) PB([ ) K ([k )³

t

t W ( t )

x (s)ds

L

 2 x T (t ) Pg ([ ) ³ T ( z , t ) dz  2 x T (t ) Px (t ) .

(28)

0

Substituting (27) and (28) into (26), gets t V1 (t ) d xT (t )[ PAc ([ , [k )  ] x(t )  2 xT (t ) PB([ ) K ([k )³

t W (t )

L

t

0

t W ( t )

2 xT (t ) Pg([ )³ T ( z, t )dz  Th x T (t ) Rx (t )  Th1 ³ 2 x T (t ) PAc ([ , [k ) x(t )  2 x T (t ) PB([ ) K ([k )³

t

t W (t )

x (s)ds

x T (s)dsR³

t

t W (t )

x (s)ds

x (s)ds

L

 2 x T (t ) Pg ([ ) ³ T ( z , t ) dz  2 x T (t ) Px (t ) .

(29)

0

The following equation can be obtained by integrating by parts and considering the boundary condition given in (3): L

a ³ T ( z , t )Tzz ( z , t ) dz 0

L

 a LT 2 ( L , t )  a 0T 2 (0, t )  a ³ T z2 ( z , t ) dz . 0

Considering a ! 0 and using the inequality (17) in Lemma 2, we can get L

 a ³ Tz2 ( z , t ) dz d 0

a 2 a T (0, t )  2 2L 4L

³

L

0

T 2 ( z , t ) dz

which implies L a · a L § a³ T ( z, t )Tzz ( z, t )dz d aLT 2 ( L, t )  ¨ a0  ¸ T 2 (0, t )  2 ³ T 2 ( z, t )dz . 0 2L ¹ 4L 0 ©

(30)

Using (30) and taking the time derivative of V2 ( t ) along the solution of the PDE subsystem (7), yield V2 ( t )

L

L

L

2D a ³ T ( z , t )Tzz ( z , t ) dz  2D b ³ T 2 ( z , t ) dz  2D ³ T ( z , t ) dzd T ([ ) x ( t ) 0

0

0

L § a · L d D ¨ 2  2b ¸ ³ T 2 ( z, t )dz  2D ³ T ( z, t )dzd T ([ ) x(t ) 0 © 2L ¹ 0

a· § 2D aLT 2 ( L, t )  D ¨ 2a0  ¸ T 2 (0, t ) . L¹ ©

(31)

Hence, from (29) and (31), when t  [tk , tk 1 ) , the time derivative of V (t ) given by (14) along the solution of the fuzzy coupled ODE-PDE model (7) and (11) is given by V (t ) V1 (t )  V2 (t )

d xT (t )[ PAc ([ , [k )  ] x(t )  2 xT (t ) PB([ ) K ([k )³

t

t W (t )

13

x (s)ds

L

t

0

t W ( t )

2 xT (t ) Pg([ )³ T ( z, t )dz  Th x T (t ) Rx (t )  Th1 ³ 2 x T (t ) PAc ([ , [k ) x(t )  2 x T (t ) PB([ ) K ([k )³

t

t W (t )

x T (s)dsR³

t

t W (t )

x (s)ds

x (s)ds

L

 2 x T (t ) Pg ([ ) ³ T ( z , t ) dz  2 x T (t ) Px (t ) 0

L § a · L D ¨ 2  2b ¸ ³ T 2 ( z, t )dz  2D ³ T ( z, t )dzd T ([ ) x(t ) 0 © 2L ¹ 0

2D aLT 2 (L, t )  D  2a0  aL1 T 2 (0, t ) .

(32)

On the other hand, it can be derived from (14), (24), and (32) that V (t )  x T (t )Qx (t )  uT (t ) Nu(t ) V (t )  xT (t )Qx (t )

[ x(t )  ³

t

t W (t )

x (s)ds]T K T ([k ) K ([k )[ x(t )  ³

t W (t )

L

d L1 ³ ] T ( z , t ) ; ([ , [ k )] ( z , t ) dz , t  [tk , tk 1 ) 0

where ] ( z , t )  ª x T (t ) « ¬

x T (t )

x T ( s ) ds T ( z , t ) T (0, t ) T ( L , t ) º t W ( t ) ¼»

³

t

ª[ PAc ([ , [ k )  ]  Q «

; ([ , [ k )  « «

«

«¬ [ K ([ k ) 0

AcT ([ , [ k ) P 2 P  Th R

 K ([ k )

t

T

x (s)ds] (33)

and

 PB ([ ) K ([ k )

;1 ([ ) º »  PB ([ ) K ([ k ) ; 2 ([ ) » Th1 R 0 » » D; 3 »¼

0]T N [ K ([ k )

0

 K ([ k )

0] ,

in which ; 1 ([ )  [ L ( Pg ([ )  d ([ )D )

0

0] , ; 2 ([ )  [ L Pg ([ )

0

0] ,

;3  diag ^0.5aL1  2Lb 2a0  aL1 2aL ` . Now, the following theorem is stated for the fuzzy guaranteed cost sampled-data control design of the nonlinear coupled ODE-PDE system (1)-(4): Theorem 1. For a given local domain  u

, consider the fuzzy coupled ODE-PDE model

(6) and (7) with the cost function (16) and Assumption 1. For given scalars Th ! 0 and

R ! 0 , Z j , and symmetric

1 , if there exist a scalar 0  9 d D minD max

E ! 0 , matrices

matrices b j  ƒ ( 4 n  m  3)u ( 4 n  m  3) , j 

such that the following LMIs hold:

14

X ! 0,

­ : ii  0, i  ° ® 1 1 °¯ r  1 : ii  2  : ij  : ji  0, i z j , i , j 

(

3

4

)

< ij  b j  0 , i, j 

(35)

where : ij  < ij  0.59  1 (1  9 )( b i  b j ) ,

ª[ Ai X  Bi Z j  ] «

« «

< ij  «

« «

«

¬«

i, j 

XAiT  Z Tj BiT 2 X  Th R

 Bi Z j  Bi Z j T 1 R

<1,i

X

< 2,i 0

0 0





E; 3

0  Q 1





h

Z Tj º » 0 »  Z Tj » » , i, j  0 » 0 » »  N 1 ¼»

in which < 1,i  [ L ( g i E  Xd i )

0] , < 2 ,i  [ L g i E

0

0

0] ,

then there exists a fuzzy sampled-data controller (11) such that the resulting closed-loop nonlinear coupled system is locally exponentially stable and the cost function (16) satisfies 2

J  x0T X 1 x0  E 1 T0 (˜) 2 .

(36)

In this case, the corresponding control gain matrices K i , i  Ki

Z i X 1

(

are given by

,

.

i

3

7

)

Proof. See Appendix. Based on Lemmas 1-3, Theorem 1 provides a less conservative LMI-based condition for the existence of a fuzzy guaranteed cost sampled-data controller (11) for the coupled system (1)-(4) under Assumption 1. If this assumption is fulfilled, the desired control gains are constructed as (37) via the feasible solutions of LMIs (34) and (35), which can be directly solved via the feasp solver in the MATLAB LMI Toolbox [48]. T 1 1 On the other hand, as shown in Theorem 1, Jb  x0 X x0  E T0 (˜)

2 2

gives an upper

bound of the cost function J . The optimal controller to be introduced is in the strict sense a “suboptimal” one since Jb will be minimized instead of J in the control design scheme. In the following theorem, the suboptimal fuzzy guaranteed cost sampled-data control problem is transformed into a standard LMI optimization one.

15

Theorem 2. Under Assumption 1, consider the coupled system (1)-(4) with the cost function (16) and assume that the initial states x0 and T0 ( z ) are known. For given scalars Th ! 0 1 , if the following LMI optimization problem: and 0  9 d D minD max

min

U , E ! 0, X ! 0, R ! 0, Z j , b j , j

U

(38)

subject to LMIs (34), (35), and

ªU « « « ¬

T0 (˜) 2 º » 0 »!0 E »¼

x0T X

(39)

 !0 , Z , b , has a set of solutions U , E ! 0 , X ! 0 , R j j

j  , then a fuzzy

sampled-data controller (11) with control gains given by (37) is a suboptimal one for the coupled system (1)-(4) in the sense of minimizing the upper bound of the cost function (16). Proof. Using the Schur complement, the inequality (39) in the optimization problem (38) is equivalent to 2

x0T X 1 x0  E 1 T0 (˜) 2  U .

(40)

Hence, it is easily obtained from the inequality (40) and the proof of Theorem 1 that if the solution to the optimization problem (38) is obtained, the designed fuzzy sampled-data controller (11) is a suboptimal one. The proof is complete. The optimization problem (38) can be directly solved via the mincx solver in the MATLAB LMI Toolbox [48]. The control gains K i , i 

can be constructed as (37) via

the solutions to the optimization problem (38). Remark 3. It should be pointed out that the above suboptimal fuzzy sampled-data control design requires that the initial values x0 and T0 ( z ) are known. When x0 and T0 ( z ) are unknown, assume

2

2

x0 d I1 and T0 (˜) 2  I2 , where I1 and I2 are some given positive

constants. In this case, it is easily verified that the inequality (39) holds if the following inequality is satisfied: ª UI1 I I1 I « X « «

¬

I1I2 I º

» 0 » !0. E I »¼

(41)

Therefore, a suboptimal fuzzy sampled-data control design for this case can be obtained by replacing (39) with (41) in the optimization problem (38).

16

Remark 4. According to the proof of Theorem 1, we can only obtain a conservative result without Lemma 3, i.e., LMIs < ij  0 , i, j  . In such case, similar to the proof of Theorem 2, the suboptimal fuzzy sampled-data control problem of the coupled system (1)-(4) will be summarized as the following LMI optimization problem: min

U , E ! 0, X ! 0, R ! 0, Z j , j

U subject to LMIs < ij  0 , i, j 

and (39) (or (41)).

(42)

The optimization problem (42) can also be directly solved via the mincx solver in the

 ! 0, Z , MATLAB LMI Toolbox [48]. Obviously, the decision variables E ! 0 , X ! 0 , R j j

of the problem (42) must satisfy all constraints given in the problem (42). This fact

may result in the same Z j , j  , i.e., Z1

Z2

"

Z r . In this situation, the fuzzy

controller is reduced to a linear one and the performance will be decreased, which will be verified in Section 4. Remark 5. Notice that backstepping state and output feedback boundary control design methods have been developed in [22] for linear ODE-PDE coupled system, where the control input affects the entire system through the boundary conditions of the PDE. Different from the system considered in [22], a state-constrained optimal control design has been reported in [19] for the nonlinear ODE-PDE coupled system, where the control input is added to the linear ODE subsystem. However, only a necessary condition on the existence of a continuous-time optimal controller is presented in terms of a retrograde integro-differential equation of Volterra type [19], which is difficult to be verified. This study proposes an LMI-based sufficient condition for the guaranteed cost sampled-data fuzzy controller for the nonlinear ODE-PDE coupled system, where the ODE subsystem is nonlinear. Although it is inevitably conservative, the LMI-based design method proposed in this study can be effectively solved via the existing LMI optimization techniques [47] and [48]. Remark 6. It is worth mentioning that a Lyapunov-based fuzzy control design has been developed in [45] for a class of nonlinear cascaded ODE-hyperbolic PDE systems, where a fuzzy and entropy-like (or strict) Lyapunov function has been constructed. In [46], a Lyapunov-based design of a constrained fuzzy state-feedback controller has been developed for a class of nonlinear coupled ODE-parabolic PDE systems. The suggested design methods in [45] and [46] are presented in terms of LMIs and the suggested controllers are continuous in time. Different from the continuous controllers [45] and [46], this study proposes an LMI-based sufficient condition for the guaranteed cost sampled-data fuzzy controller for a class of nonlinear coupled ODE-PDE systems. The suggested sampled-data controller is

17

easily implemented in practice via the digital technology since it is very cheap, fast, relatively easy to operate, flexible and reliable.

4. Application to hypersonic rocket car In this section, simulation results on the sampled-data control of hypersonic rocket car (as shown in Fig. 4) [19] are presented to illustrate the effectiveness and merit of the proposed design method. The following mathematical model describes the dynamics of motion and spatiotemporal temperature of the car: (t ) u (t )  P w 2 (t ) , w yt ( z , t )

(43)

y zz ( z , t )  w 2 (t ) ,

(44)

subject to the boundary condition y z (0, t )

y (0, t )  y0 (0) , yz (1, t )

 y (1, t )  y0 (1)

(45)

and the initial condition y ( z, 0)

y0 ( z ) , w(0) w0 , w (0)

w1

(46)

where the ODE state variable w(t ) denotes the one-dimensional position of the car; the PDE state variable y ( z, t ) stands for the temperature and depends on time t as well as the spatial coordinate

z describing the position within the car; u(t ) denotes the control input of the

car; P denotes the friction coefficient; y0 ( z ) , w 0 , and w1 are the initial values. The PDE

 (t) of the car. On the other hand, system is controlled only indirectly via the velocity w notice that the nonlinear coupled ODE-PDE model is borrowed from [19]. However, different from the model introduced in [19], the frictional force is considered in development of ODE model (43). Hence, the ODE model developed in this paper is a realistic model. The frictional

 (t) . This nonlinear force is assumed to be proportional to the square of the velocity w ODE-PDE model has also been used in [46]. The value of P is set to be 0.002, i.e.,

P 0.002 . Similar to [19], the initial temperature y0 ( z ) of the car is in the following set to zero, i.e., y0 ( z )

0 , z[0,1] .

18

Fig. 4 Hypersonic rocket car

Define the state x (t )  [ x1 (t )

x2 (t )]T

[ w (t )

w (t )]T and T ( z , t )  y ( z , t ) , z[0,1] ,

the nonlinear system (43)-(46) is represented as the form of (1)-(4) with the following parameters: f ( x ( t ))  [ x 2 ( t ) a

1, b

0 , a0

 P x22 ( t )]T , g ( x ( t ))  [0

1 , aL 1 , M ( x (t ))

0]T , B( x(t ))  [0 1]T ,

x22 (t ) , L

1.

(47)

y

z

t

Fig. 5 Open-loop temperature profile of the hypersonic rocket car

Using the parameters given in (47), letting the initial values in (46) be w0 1 and w1 1 , i.e., the initial condition in (4) is chosen as T0 ( z )

y0 ( z )

0 and x 0

[ w0

w1 ]T

[1 1]T ,

Fig. 5 shows the open-loop temperature profile of the hypersonic rocket car. Let

^

`

  x (t ) ƒ2 x2 (t ) d N1 .

Let [ ( x2 (t ))  x2 (t ) , the nonlinear term x 22 ( t ) in (47) can be rewritten as x22 (t )

[ ( x2 (t )) x2 (t ) .

19

The local sector nonlinearity approach in [25] is adopted for the fuzzy coupled ODE-PDE model construction of the nonlinear coupled system (1)-(4). To this end, set the operating

^

`

domain ˆ  x (t )  ƒ 2 x2 (t ) d N . In this domain, calculating the minimum and maximum values of [ ( x2 (t )) yields min [ ( x2 (t ))

x2 ( t ) N

N and max [ ( x2 (t )) N . x2 ( t ) N

Using above minimum and maximum values, [ ( x2 (t )) can be represented by

[ ( x2 (t )) h1 ([ ( x2 (t ))) ˜ N  h2 ([ ( x2 (t ))) ˜ (N )

(48)

where h1 ([ ( x2 (t ))) , h2 ([ ( x2 (t )))  [0,1] and h1 ([ ( x2 (t )))  h2 ([ ( x2 (t ))) 1 .

(49)

By solving equations (48) and (49), the following membership functions are obtained: h1 ([ ( x2 (t )))

N  [ ( x2 (t )) and h2 ([ ( x2 (t ))) 2N

N  [ ( x2 (t )) . 2N

(50)

Define the fuzzy sets as “Big” and “Small”. Then, the nonlinear coupled system (1)-(4) with the parameters given in (47) can be exactly represented by the following T-S fuzzy coupled ODE-PDE model of two rules: Plant rule 1: IF [ ( x2 (t )) is “Big” ­ x (t ) A x (t )  g L T ( z, t )dz  B u(t ) ° 1 1 ³0 1 THEN ® T °¯Tt ( z, t ) aTzz ( z, t )  bT ( z, t )  d1 x (t ) Plant rule 2: IF [ ( x2 (t )) is “Small” ­ x (t ) A x (t )  g L T ( z , t )dz  B u(t ) ° 2 2 ³0 2 THEN ® °¯Tt ( z , t ) aTzz ( z, t )  bT ( z , t )  d 2T x (t ) where A1 a

1 º ª0 « 0  PN » , A2 ¬ ¼ 1, b

0 , d1

ª0 1 º «0 PN » , g1 ¬ ¼

[0 N ]T , and d 2

g2

ª0º «0» , B1 ¬ ¼

B2

N ]T .

[0

Then, the overall fuzzy coupled ODE-PDE model is given by

20

ª0 º «1 » , ¬ ¼

L 2 ­ x (t ) ¦i 1 hi ([ ( x2 (t )))( Ai x(t)  gi ³0 T ( z, t )dz  Bi u(t )) ° ® 2 °¯Tt ( z, t ) aTzz ( z, t )  bT ( z, t )  ¦i 1 hi ([ ( x2 (t )))diT x(t ).

(51)

Set the actual domain be x 2 ( t ) d N 1 , where N 1  N . For the open-loop system (43)

u(t ) 0 ), we can get

(i.e., set

 P x22 (t ) d 0 .

x 2 (t )

hi ([ ( x2 (t )))

,

(t ) w

given

(since

N 1 ! x 2 (t ) o 0 ,

We thus derive

i {1,2}

 P w 2 (t ) d 0

in

(50)

belong

tt0

to

P 0.002 ), i.e., if

x 2 (0) d N 1

and

the

compact

set

ª¬0.5  0.5N 1N1,0.5  0.5N 1N1 º¼ . Obviously, these membership functions satisfy Assumption 1 D min  0.5  0.5N 1N 1

with

d i ,k ( t )  hi ([ ( x 2 ( t )))  9 hi ([ ( x 2 ( t k ))) ,

Let Th

4 , N1

1, N

D max  0.5  0.5N 1N 1

and

.

For

brevity,

let

i {1,2} , k 0,1,2,", f .

1, 9

0.6 , Q I and

N

0.1 . It is obvious that

1 . By resorting to the LMI toolbox in MATLAB [48], the solution to the 0  9 d D minD max

optimization problem (38) can be obtained as follows: 0.1873º

ª 0.4645

U 18.9520 , E 1.3777 u106 , X « », ¬ 0.1873 0.1925 ¼ Z1

> 0.0253

 0.1076 @ and Z 2

> 0.0253

 0.1073 @ .

Hence, the upper bound of the cost function J Jb

18.9517 , i.e.,

³

f

0

2

x (t ) dt  J  J b

³

f

0

 x(t)

(52) 2



 0.1u2 (t ) dt is obtained as

18.9517 . Using (37) and (52), the control gain

matrices can be derived as: K1

>  0.2792

 0.8399 @ and K 2

>  0.2781

 0.8373 @ .

(53)

Applying the fuzzy sampled-data control law (11) with the control gains given in (53) to the fuzzy coupled ODE-PDE system (1)-(4) with parameters given in (47), the closed-loop temperature profile and the closed-loop position and velocity of the hypersonic rocket car are indicated in Figs. 6 and 7, respectively. It is easily seen from Figs. 6 and 7 that the suggested fuzzy controller can stabilize the coupled ODE-PDE system and x 2 ( t )  N 1  N , for all t t 0 , which implies that the T-S fuzzy coupled ODE-PDE model (51) is effective. Fig. 8

shows the control input u (t ) . Obviously, the overshoots in Figs. 6 and 7 are large, this is because the surface temperature T ( z, t ) of the car depends on the square of the velocity

w (t) (see, the equation (44)) and the initial velocity w1 of the car is positive, i.e., 21

x0

[1 1]T . Due to the inertia, the car will continuous to move forward within a few

minutes at the beginning.

y

z

t

Fig. 6 Closed-loop temperature profile of the hypersonic rocket car 1.5

w(t)

1 0.5 0 -0.5

0

4

t (sec.)

8

12

Fig.7 Closed-loop position and velocity of the hypersonic rocket car 0.2 0 -0.4

u(t) -0.8 -1.2

0

4

t (sec.)

8

12

Fig. 8 Control input u ( t )

For the closed-loop system resulting from the nonlinear system (43)-(46) plus the sampled-data fuzzy controller (11), the membership functions hi ([ ( x2 (t ))) , i {1,2} given t of in (50) will converge to 0.5 as t o f , since [ ( x2 (t ))  o 0 . Hence, we can further get of d i , k ( t )  t o 0.2 ,

i {1,2} . What is more, calculating the actual value of the cost function,

we have 12

J |³

0

 x(t)

2



 0.1u2 (t) dt 5.8277  Jb 18.9517  18.9520 U ,

22

i.e., the optimized upper bound of the cost function is ensured. In order to show the less conservativeness of the proposed method, using the same values of parameters and the same initial conditions ( Q I , N and x 0

0.1 , Th

1 , T0 ( z )

0

[1 1]T ), we solve the LMI optimization problem (42) to obtain the optimized

upper bound U 18.9534 , which is larger than the optimized upper bound (i.e., U 18.9520 ) derived by solving the optimization problem (38). The corresponding control gain matrices are obtained as K 1

K2

>  1.1617

 2.2349 @ . In this case, the fuzzy sampled-data

controller is reduced to a linear one. This is why the performance of the optimization problem (42) is worse than that of the optimization problem (38) for this example. It is obvious that Assumption 1 can help reduce the conservativeness of the proposed design method. On the other hand, to illustrate the merit of proposed design method, we give a performance comparison among the fuzzy sampled-data controller presented in this paper, a fixed-gain sampled-data proportional-integral (PI) controller, and a fixed-gain sampled-data nonlinear controller. The fixed-gain sampled-data PI control law is first given as follows: tk

K p x ( t k )  K i ³ x ( s ) ds

u (t )

0

(

tk d t  tk 1

,

5 [0.5 0.8] and K i

where K p

k 0,1,2,", f

, 4

)

[0.05 0.1] , and the fixed-gain sampled-data nonlinear

controller is given as u (t )

K non x (t k )  P w 2 (t k )

tk d t  tk 1

,

,

k 0,1,2,", f

(55) where K non

[0.5 0.7] . This nonlinear controller is obtained by employing the feedback

linearization method. When Th

0 , the system (43) driven by the controller (55) can be

(t ) written as a stable second-order system w

Set Th

2, P

K non x (t )

0.002 , T0 ( z ) 0 and x 0

[4

0.5w(t )  0.7 w (t ) . 0]T . Applying the sampled-data PI

controller (54), the fixed-gain sampled-data nonlinear controller (55), and the fuzzy sampled-data K2

>  0.0479

controller

(11)

with

control

gains

K1

>  0.0487

 0.3033 @

and

 0.3005 @ to the system (43)-(46), respectively, Fig. 9 indicates the resulting

closed-loop trajectories of x ( t ) . Moreover, the actual value of the performance J for the closed-loop sampled-data system driven by the fuzzy sampled-data controller (11) can be

23

approximately obtained as J |

³

30

0

 x(t)

2



 0.1u2 (t) dt 67.1286 , which is smaller than the

actual value (i.e., J | 76.9772 ) of the performance J for the one driven by the fixed-gain sampled-data PI controller (54) and the actual value (i.e., J | 73.9142 ) of the performance J for the one driven by the fixed-gain sampled-data nonlinear controller (55). It can be

concluded from the above analysis that in comparison to the fixed-gain sampled-data PI controller (54) and the fixed-gain sampled-data nonlinear controller (55), better performance can be obtained using the fuzzy sampled-data controller (11). 4.5 Fuzzy sampled-data controller (11)

4

Sampled-data PI controller (54) Sampled-data nonlinear controller (55) Continuous-time fuzzy controller (56)

3

1.5

0

0

10

Fig. 9 Closed-loop trajectories of

t (sec.)

20

30

x(t) using the fuzzy sampled-data controller (11),

the sampled-data PI controller (54), nonlinear feedback controller (55), and continuous-time fuzzy controller (56).

Moreover, a performance comparison between the fuzzy sampled-data controller (11) and the following continuous-time fuzzy controller u(t )

¦

r j 1

h j ([ (t )) K j x (t ) ,

(56)

is also provided to further illustrate the merit of the proposed control design method. The fuzzy controller (56) is simplified from the fuzzy controller reported in [46] neglecting the PDE state feedback part. A T-S fuzzy model of the form (51) without Assumption 1 is employed to determine the control gains of the fuzzy controller (56). To this end, remove the constraint N 1  N and set N

N1 1 . Based on the fuzzy controller design method given in

Theorem 3.1 [46], without considering the H’ performance and the input constraint, we get K1

>  0.9002

 4.3105 @ and K 2

>  0.9002

 4.3145 @ .

Applying the fuzzy controller (56) with the above control gains to the nonlinear coupled system (43)-(46), the resulting closed-loop trajectory of x ( t )

24

is also given in Fig. 9. It is

clear from Fig. 9 that the fuzzy controller (56) can provide a better performance in comparison to the fuzzy sampled-data controller (11). This is because that the fuzzy controller (56) is continuous in time and utilizes the full state information of the coupled ODE-PDE system (43)-(46). However, the fuzzy sampled-data controller (11) is easily implemented in practice due to the fact that it is a sampled-data one and only requires the state information of the ODE subsystem. Furthermore, it has been verified through numerical simulation that the continuous-time fuzzy controller (56) with the sampling period Th

1

cannot stabilize the nonlinear coupled ODE-PDE system (43)-(46).

5. Conclusion In this paper, the problem of fuzzy guaranteed cost sampled-data control design has been investigated for a class of nonlinear coupled ODE-PDE systems. By using Lyapunov’s direct method and a new parameterized LMI technique, the suboptimal fuzzy guaranteed cost sampled-data control problem is formulated as an LMI optimization problem, which can be effectively solved by using the existing LMI optimization techniques. The advantage of the proposed design method that the controller proposed in this study is easily implemented since it only uses the state information of the ODE subsystem and its control gain matrices can be directly constructed through the solutions to the LMI optimization problem. Finally, the simulation results on the sampled-data control of hypersonic rocket car illustrate the effectiveness and advantage of the proposed design method. In this study, the mature fuzzy model based control technique for ODE systems is extended to solve the sampled-data control problem of nonlinear coupled ODE-PDE systems. On the other hand, the developed framework of analysis paves the way for further improvements for more complicated nonlinear coupled systems, such as nonlinear coupled ODE-beam systems and nonlinear ODE systems coupled with wave PDE, that will be addressed in future research activities.

Acknowledgments This work was supported in part by the National Natural Science Foundation of China under Grants 61403026, 61473011, and 61421063, in part by the China Postdoctoral Science Foundation under Grant 2014M550618, and in part by the General Research Fund project from Research Grants Council of Hong Kong, SAR, under Grant CityU: 116212. The authors

25

also gratefully acknowledge the helpful comments and suggestions of the Associate Editor and anonymous reviewers, which have improved the presentation of this paper.

Appendix. Proof of Theorem 1 From (33), it can be obtained that for any non-zero ] (˜, t ) and t  [tk , tk 1 ) , V (t )  xT (t )Qx (t )  uT (t ) Nu(t )  0 ,

(A1)

;([ , [ k )  0 .

(A2)

if

Using the Schur complement, the inequality (A2) is equivalent to 4([ , [ k )  0

(A3)

where ª[ PAc ([ , [ k )  ] «

« «

4([ , [ k )  «

« «

«

¬«

AcT ([ , [ k ) P

 PB ([ ) K ([ k ) ;1 ([ )  PB ([ ) K ([ k ) ; 2 ([ ) 0 Th1 R

2 P  Th R

I 0 0





D; 3

0 Q 1





K T ([ k ) º » 0 »  K T ([ k ) » ». 0 » » 0 »  N 1 ¼»

Then, using (8), (9) and (12), the inequality (A3) can be rewritten as 4([ , [ k )

¦ ¦ r

r

i 1

j 1 i

h ([ (t ))h j ([ (tk ))4ij  0 ,

(A4)

where ª[ PAc ,ij  ] AcT,ij P  PBi K j «

2 P  Th R  PBi K j « «

Th1 R 4ij  «

« «

«

¬«

41,i

I

4 2,i 0

0 0

D; 3

0  Q 1



K Tj º » 0 »  K Tj » » , i, j  0 » 0 » »  N 1 ¼»

with Ac , ij  Ai  B j K i , 4 1,i  [ L ( Pg i  d iD ) 0 0] , and 4 2 ,i  [ L Pg i

0

0] .

Let

X P1 , R XRX , E D 1 , and Zi Ki X , Pre-

and

0  diag ^ X

post-multiplying X

X

EI

the I

matrix

i

4([ , [ k )

I ` , respectively, yields

26

.

by

(A5) the

block-diagonal

matrix

¦ ¦ r

r

i 1

j 1 i

¦ ¦

h ([ (t ))h j ([ (tk )) < ij

(

r

r

i 1

j 1 i

h ([ (t ))h j ([ (tk ))04ij0

A

.

6

)

It can be concluded from Lemma 3 that if the membership functions hi ([ (t )) , i  Assumption

¦ ¦ r

r

i 1

j 1 i

1

and

LMIs

(34)

and

(35)

are

fulfilled,

then

the

satisfy

inequality

h ([ (t )) h j ([ (tk )) < ij  0 holds, which implies the inequality (A3) is fulfilled since

(A4) and 0 ! 0 . Thus, the inequality (A1) is satisfied because of the equivalence between (A2) and (A3). From (A2), an appropriate scalar p2 ! 0 can be found such that ; ([ , [W )  p2 I d 0 , t  [tk , tk 1 ) .

(A7)

Utilizing (33) and inequalities (A7), we have for any non-zero ] (˜, t ) and t  [tk , tk 1 ) 2 2 2 2 2 V (t ) d  p2 L1 ] (˜, t ) 2 d  p2 L1 T (˜, t ) 2  p2 x(t ) d  p1 T (˜, t ) 2  p1 x(t ) ,

(A8)

where p1  p2 min{L1 ,1} . Since P ! 0 and D ! 0 , it is easily observed that V (t ) given by (22) and (23) satisfies



2

N 2 T (˜, t ) 2  x (t )

2

d V (t ) d N  T (˜, t ) 3

2 2

 x (t )

2



(A9)

where N 2  min ^O min ( P ), D ` and N3 is a sufficiently larger positive scalar. Inequalities (A8) and (A9) imply V (t ) d  p1N 31V (t )

(

t  [tk , tk 1 )

,

A

1

By integrating the inequality (A10) from t

t k to t

0 t k1 , we get

V (t k1 ) d V (t k ) exp(  p1N 31 (t k1  t k )) .

Since V (t ) is continuous at instant t V (t k ) V (t k ) , k

)

(A11) t k , we have

0,1,2,", f ,

(A12)

where V (t k ) and V (t k ) denote the right limit and left limit of V (t ) at instant t

tk ,

respectively. Hence, from (A11), we can get V (t k 1 ) d V (t k ) exp(  p1N 31 (t k 1  t k )) .

(A13)

For any t t 0 , there exists a positive integer k such that t  [tk , tk 1 ) . Hence, we can find from (A13) that V (t ) d V (t k ) exp(  p1N 31 (t  t k ))

27

d V (t k 1 ) exp(  p1N 31 (t k  t k 1 )) exp(  p1N 31 (t  t k ))

d"d

d V (0) exp( p1N31 (t  tk ))–i 1 exp( p1N31 (ti  ti 1 )) V (0) exp( p1N31t ) k

(

A

1

.

4

)

Using (A9), the inequality (A14) can be rewritten as 2

x (t )  T (˜, t )

2 2

d N 3N 21

x

2 0

 T0 (˜)

2 2

exp(  p N 1

t) , t t 0 .

(A15)

1 3

From (A15), the inequality in Definition 1 is fulfilled for the case when J

V

N 3N 21 and

p1N 31 . Based on the Definition 1, the closed-loop fuzzy coupled system (7) and (11)

subject to the homogenous Robin boundary conditions (3) and the initial conditions (4) is thus exponentially stable. From (A5), we have (37). Given N  1 . Taking into account (A12), rearranging and integrating (A1) from t to t

0

t N , give J

³ x tN

T

0

(t )Qx (t )  u T (t ) Nu (t ) dt   V (t ) t

t tN 0

V (t N )  V (0) .

(A16)

Since the closed-loop fuzzy coupled system (7) and (11) subject to the homogenous Robin boundary conditions (3) and the initial conditions (4) is exponentially stable, we gets 2

2

x(t ) o 0 and T (˜, t ) 2 o 0 as t o f . Thus, for J

³ x f

0

(

T

N o f , we arrive to

(t )Qx (t )  u T (t ) Nu (t ) dt  x 0T Px 0  D T0 (˜)

A

2

.

2

1

7

)

From (A5) and (A17), (36) is derived. The proof is complete.

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