**1. Introduction**

24 Advances in Discrete Time Systems

52 Advances in Discrete Time Systems

[37] Zhang, X.-M. & Han, Q.-L. A new finite sum inequality approach to delay-dependent H∞ control of discrete-time systems with time-varying delay, International Journal of

Robust and Nonlinear Control, 2008;18 630-647.

In recent years, there has been significant interest in the study of stability analysis and con‐ troller synthesis for Takagi-Sugeno(T-S) fuzzy systems, which has been used to approximate certain complex nonlinear systems [1]. Hence it is important to study their stability analysis and controller synthesis. A rich body of literature has appeared on the stability analysis and synthesis problems for T-S fuzzy systems [2-6]. However, these results rely on the existence of a common quadratic Lyapunov function (CQLF) for all the local models. In fact, such a CQLF might not exist for many fuzzy systems, especially for highly nonlinear complex sys‐ tems. Therefore, stability analysis and controller synthesis based on CQLF tend to be more conservative. At the same time, a number of methods based on piecewise quadratic Lyapu‐ nov function (PQLF) for T-S fuzzy systems have been proposed in [7-14]. The basic idea of these methods is to design a controller for each local model and to construct a global piece‐ wise controller from closed-loop fuzzy control system is established with a PQLF. The au‐ thors in [7,13] considered the information of membership function, a novel piecewise continuous quadratic Lyapunov function method has been proposed for stability analysis of T-S fuzzy systems. It is shown that the PQLF is a much richer class of Lyapunov function candidates than CQLF, it is able to deal with a large class of fuzzy systems and obtained re‐ sults are less conservative.

On the other hand, it is well known that time delay is a main source of instability and bad performance of the dynamic systems. Recently, a number of important analysis and synthe‐ sis results have been derived for T-S fuzzy delay systems [4-7, 11, 13]. However, it should be pointed out that most of the time-delay results for T-S fuzzy systems are constant delay or

© 2012 Li et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Li et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

time-varying delay [4-5, 7, 11, and 13]. In fact, Distributed delay occurs very often in reality and it has been drawing increasing attention. However, almost all existing works on distrib‐ uted delays have focused on continuous-time systems that are described in the form of ei‐ ther finite or infinite integral and delay-independent. It is well known that the discrete-time system is in a better position to model digitally transmitted signals in a dynamic way than its continuous-time analogue. Generalized H2 control is an important branch of modern con‐ trol theories, it is useful for handling stochastic aspects such as measurement noise and ran‐ dom disturbances [10]. Therefore, it becomes desirable to study the generalized H2 control problem for the discrete-time systems with distributed delays. The authors in [6] have de‐ rived the delay-independent robust H∞ stability criteria for discrete-time T-S fuzzy systems with infinite-distributed delays. Recently, many robust fuzzy control strategies have been proposed a class of nonlinear discrete-time systems with time-varying delay and disturb‐ ance [15-33]. These results rely on the existence CLKF for all local models, which lead to be conservative. It is observed, based on the PLKF, the delay-dependent generalized H2 control problem for discrete-time T-S fuzzy systems with infinite-distributed delays has not been addressed yet and remains to be challenging.

1 12 2

*j j*

ence rules. *F* ji (*i*=1, 2,…, *g*) are the fuzzy sets, *s*(*t*)=[*s* 1(t), *s* 2(*t*),…, *s* g(*t*)]∈*R* <sup>s</sup>

, *C* <sup>j</sup> *, B* 2j) represent the *j-*th local model of the fuzzy system (1).

mm

The constants *μ* <sup>d</sup> ≥0 (*d* =1,2, …) satisfy the following convergence conditions:

1 1 : *d d d d*

*d* +¥ +¥

() () ()

= +

j

*zt Cxt B ut*

*j*

*Remark* **1.** The delay term ∑*<sup>d</sup>* =1

paper, which is different from one in [6].

pressed by the following global model:

(*s*(*t*))= *<sup>ω</sup><sup>j</sup>*

(*t*)in*Fij*

∑ *<sup>j</sup>*=1 *<sup>r</sup> <sup>ω</sup><sup>j</sup>*

,*ωj*

*r*

*r*

å

(*s*(*t*))

=

(*s*(*t*))

of infinite delays as well as the DDPLKF defined later.

1j, *D* <sup>j</sup>

where *hj*

bership of *si*

1 2

*d*

¥ =

å

() () 1,2

: () () () ,

*R if s t is F and s t is F and and s t isF then*

( 1) ( ) ( ) ( ) ( )

*xt t t Z j r*

= "Î =

control input vector, v(t)∈*l* <sup>2</sup>[0 ∞) the disturbance input, φ(*t*) the initial state, and (*A* <sup>j</sup>

*xt Axt A xt d B ut Dvt*

m

+= + - + +

*j j g jg*

L

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays

*j dj d j j*


where *R* <sup>j</sup> *, j*∈*N:=*{1,2,…, *r*} denotes the *j-*th fuzzy inference rule, *r* the number of the infer‐

ble vector, *x*(*t*)∈*R* n the state vector, *z*(*t*)∈*R* q the controlled output vector, *u*(*t*)∈*R* <sup>m</sup> the

 m

distributed delay in the discrete-time setting. The description of the discrete-time-distribut‐ ed delays has been firstly proposed in the [6], and we aim to study the generalized H2 con‐ trol problem for discrete-time fuzzy systems with such kind of distributed delays in this

*Remark* **2.** In this paper, similar to the convergence restriction on the delay kernels of infin‐ ite-distributed delays for continuous-time systems, the constants *μ* <sup>d</sup> (*d* =1,2, …)are assumed to satisfy the convergence condition (2), which can guarantee the convergence of the terms

By using a standard fuzzy inference method, that is using a center-average defuzzifiers product fuzzy inference, and singleton fuzzifier, the dynamic fuzzy model (1) can be ex‐

<sup>1</sup> <sup>1</sup> <sup>1</sup>

*xt h st Axt A xt d B ut Dvt*

+ = + -+ +

( 1) ( ( ))[ ( ) ( ) ( ) ( )]

*<sup>g</sup> <sup>F</sup> ji*

(*s*(*t*))≥0 has the following basic property:

*<sup>j</sup> <sup>j</sup> dj d <sup>j</sup> <sup>j</sup> <sup>j</sup> <sup>d</sup>*

¥

m

(*s*(*t*)), with *F ji*

2 1

(*s*(*t*))=∏*<sup>i</sup>*=1

<sup>=</sup> <sup>=</sup>

*jj j j*

å å

( ) ( ( ))[ ( ) ( )]

*zt h st Cxt B ut*

= +

, *ω<sup>j</sup>*

1

L

= = = £ < +¥ å å (2)

<sup>+</sup>*<sup>∞</sup> μd <sup>x</sup>*(*<sup>t</sup>* <sup>−</sup>*d*) in the fuzzy system (1), is the so-called infinitely

(1)

55

the premise varia‐

http://dx.doi.org/10.5772/51778

, *A* dj *, B*

(3)

(*s*(*t*)) being the grade of mem‐

Motivated by the above concerns, this paper deals with the generalized H2 control prob‐ lem for a class of discrete time T-S fuzzy systems with infinite-distributed delays. Based on the proposed Delay-dependent PLKF(DDPLKF), the stabilization condition and control‐ ler design method are derived for discrete time T-S fuzzy systems with infinite-distribut‐ ed delays. It is shown that the control laws can be obtained by solving a set of LMIs. A simulation example is presented to illustrate the effectiveness of the proposed de‐ sign procedures.

*Notation***:** The superscript "*T*" stands for matrix transposition, *R* n denotes the *n-*dimension‐ al Euclidean space, *R* n×m is the set of all *n×m* real matrices, *I* is an identity matrix, the nota‐ tion *P*>0*(P*≥0*)* means that *P* is symmetric and positive(nonnegative) definite, *diag*{…} stands for a block diagonal matrix. *Z* - denotes the set of negative integers. For symmetric block ma‐ trices, the notation \* is used as an ellipsis for the terms that are induced by symmetry. In addition, matrices, if not explicitly stated, are assumed to have compatible dimensions.

### **2. Problem Formulation**

The following discrete-time T-S fuzzy dynamic systems with infinite-distributed delays [6] can be used to represent a class of complex nonlinear time-delay systems with both local an‐ alytic linear models and fuzzy inference rules:

$$\begin{aligned} \mathbf{R}^{\top}: \text{if } \mathbf{s}\_{\downarrow}(t) \text{ is } F\_{j\uparrow} \text{and } \mathbf{s}\_{\downarrow}(t) \text{ is } F\_{j\downarrow} \text{ and } \cdots \text{ and } \mathbf{s}\_{\downarrow}(t) \text{ is } F\_{j\mathbf{s}}, \text{then} \\ \mathbf{x}(t+1) = A\_{j}\mathbf{x}(t) + A\_{\neq} \sum\_{d=1}^{n} \mu\_{d} \mathbf{x}(t-d) + B\_{1\neq} \mu(t) + D\_{j} \mathbf{v}(t) \\ \mathbf{z}(t) = C\_{j} \mathbf{x}(t) + B\_{z\_{j}} \mu(t) \\ \mathbf{x}(t) = \phi(t) & \qquad \forall t \in \mathbb{Z}^{-} \qquad \qquad j = 1, 2 \cdots r \end{aligned} \tag{1}$$

where *R* <sup>j</sup> *, j*∈*N:=*{1,2,…, *r*} denotes the *j-*th fuzzy inference rule, *r* the number of the infer‐ ence rules. *F* ji (*i*=1, 2,…, *g*) are the fuzzy sets, *s*(*t*)=[*s* 1(t), *s* 2(*t*),…, *s* g(*t*)]∈*R* <sup>s</sup> the premise varia‐ ble vector, *x*(*t*)∈*R* n the state vector, *z*(*t*)∈*R* q the controlled output vector, *u*(*t*)∈*R* <sup>m</sup> the control input vector, v(t)∈*l* <sup>2</sup>[0 ∞) the disturbance input, φ(*t*) the initial state, and (*A* <sup>j</sup> , *A* dj *, B* 1j, *D* <sup>j</sup> , *C* <sup>j</sup> *, B* 2j) represent the *j-*th local model of the fuzzy system (1).

The constants *μ* <sup>d</sup> ≥0 (*d* =1,2, …) satisfy the following convergence conditions:

time-varying delay [4-5, 7, 11, and 13]. In fact, Distributed delay occurs very often in reality and it has been drawing increasing attention. However, almost all existing works on distrib‐ uted delays have focused on continuous-time systems that are described in the form of ei‐ ther finite or infinite integral and delay-independent. It is well known that the discrete-time system is in a better position to model digitally transmitted signals in a dynamic way than its continuous-time analogue. Generalized H2 control is an important branch of modern con‐ trol theories, it is useful for handling stochastic aspects such as measurement noise and ran‐ dom disturbances [10]. Therefore, it becomes desirable to study the generalized H2 control problem for the discrete-time systems with distributed delays. The authors in [6] have de‐ rived the delay-independent robust H∞ stability criteria for discrete-time T-S fuzzy systems with infinite-distributed delays. Recently, many robust fuzzy control strategies have been proposed a class of nonlinear discrete-time systems with time-varying delay and disturb‐ ance [15-33]. These results rely on the existence CLKF for all local models, which lead to be conservative. It is observed, based on the PLKF, the delay-dependent generalized H2 control problem for discrete-time T-S fuzzy systems with infinite-distributed delays has not been

Motivated by the above concerns, this paper deals with the generalized H2 control prob‐ lem for a class of discrete time T-S fuzzy systems with infinite-distributed delays. Based on the proposed Delay-dependent PLKF(DDPLKF), the stabilization condition and control‐ ler design method are derived for discrete time T-S fuzzy systems with infinite-distribut‐ ed delays. It is shown that the control laws can be obtained by solving a set of LMIs. A simulation example is presented to illustrate the effectiveness of the proposed de‐

*Notation***:** The superscript "*T*" stands for matrix transposition, *R* n denotes the *n-*dimension‐ al Euclidean space, *R* n×m is the set of all *n×m* real matrices, *I* is an identity matrix, the nota‐ tion *P*>0*(P*≥0*)* means that *P* is symmetric and positive(nonnegative) definite, *diag*{…} stands for a block diagonal matrix. *Z* - denotes the set of negative integers. For symmetric block ma‐ trices, the notation \* is used as an ellipsis for the terms that are induced by symmetry. In addition, matrices, if not explicitly stated, are assumed to have compatible dimensions.

The following discrete-time T-S fuzzy dynamic systems with infinite-distributed delays [6] can be used to represent a class of complex nonlinear time-delay systems with both local an‐

addressed yet and remains to be challenging.

sign procedures.

54 Advances in Discrete Time Systems

**2. Problem Formulation**

alytic linear models and fuzzy inference rules:

$$\overline{\mu} := \sum\_{d=1}^{\ast \circ \circ} \mu\_d \le \sum\_{d=1}^{\ast \circ} d \, \mu\_d < +\infty \tag{2}$$

*Remark* **1.** The delay term ∑*<sup>d</sup>* =1 <sup>+</sup>*<sup>∞</sup> μd <sup>x</sup>*(*<sup>t</sup>* <sup>−</sup>*d*) in the fuzzy system (1), is the so-called infinitely distributed delay in the discrete-time setting. The description of the discrete-time-distribut‐ ed delays has been firstly proposed in the [6], and we aim to study the generalized H2 con‐ trol problem for discrete-time fuzzy systems with such kind of distributed delays in this paper, which is different from one in [6].

*Remark* **2.** In this paper, similar to the convergence restriction on the delay kernels of infin‐ ite-distributed delays for continuous-time systems, the constants *μ* <sup>d</sup> (*d* =1,2, …)are assumed to satisfy the convergence condition (2), which can guarantee the convergence of the terms of infinite delays as well as the DDPLKF defined later.

By using a standard fuzzy inference method, that is using a center-average defuzzifiers product fuzzy inference, and singleton fuzzifier, the dynamic fuzzy model (1) can be ex‐ pressed by the following global model:

$$\begin{aligned} \mathbf{x}(t+1) &= \sum\_{j=1}^{\prime} h\_j(\mathbf{s}(t)) [A\_j \mathbf{x}(t) + A\_{4j} \sum\_{d=1}^{n} \mu\_d \mathbf{x}(t-d) + B\_{1j} \mu(t) + D\_j \mathbf{v}(t)] \\ \mathbf{z}(t) &= \sum\_{j=1}^{\prime} h\_j(\mathbf{s}(t)) [C\_j \mathbf{x}(t) + B\_{2j} \mu(t)] \end{aligned} \tag{3}$$

where *hj* (*s*(*t*))= *<sup>ω</sup><sup>j</sup>* (*s*(*t*)) ∑ *<sup>j</sup>*=1 *<sup>r</sup> <sup>ω</sup><sup>j</sup>* (*s*(*t*)) , *ω<sup>j</sup>* (*s*(*t*))=∏*<sup>i</sup>*=1 *<sup>g</sup> <sup>F</sup> ji* (*s*(*t*)), with *F ji* (*s*(*t*)) being the grade of mem‐ bership of *si* (*t*)in*Fij* ,*ωj* (*s*(*t*))≥0 has the following basic property:

$$\alpha o\_{j}(\mathbf{s}(t)) \ge 0, \sum\_{j=1}^{r} \alpha\_{j}(\mathbf{s}(t)) > 0, j \in N \quad \forall t \tag{4}$$

*Remark* **3.** According to the definition of (8), the polyhedral regions can be divided into two folds: operating and interpolation regions. For an operating region, the set *M(l)* contains on‐ ly one element, and then, the system dynamic is governed by the *s-*th local model of the fuz‐ zy system. For an interpolation region, the system dynamic is governed by a convex

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays

In this paper, we consider the generalized H2 controller design problem for the fuzzy system

For future use, we define a set*Θ*that represents all possible transitions from one region to

Considering the fuzzy system (10), choose the following non-fragile piecewise state feed‐

1 2 () , () () , () *<sup>T</sup> l l K EU t H U t U t I U t R l ll l l l l*

1 ( 1) ( ) ( ) ( )

¥ = += + - +

*x t A x t A x t d Dv t*

*cl dl d l d*

m

{( , ) | ( ) , ( 1) , } *l j* Q = ÎW + ÎW " Î *l j st st l j L* (11)

() ( ) () () *l l <sup>l</sup> ut K K xt st l L* =- +D ÎW Î (12)

´ D = £ Î (13)

(*t*)∈*R <sup>l</sup>* 1×*l*

are unknown real matrix functions representing time varying parametric uncer‐

the dynamics of the system is governed by the dynamics of the region model of Ω<sup>l</sup>

to Ω<sup>j</sup>

http://dx.doi.org/10.5772/51778

, and *l* ≠ *j*, when the system transits

<sup>2</sup> are unknown real time varying

<sup>å</sup> (14)

at the time *t,*

at that

57

combination of several local models.

itself or another regions, that is

from the region Ω<sup>l</sup>

back controller

here Δ*K* <sup>l</sup>

matrix satisfying*ΔUl*

for *s*(*t*)∈*Ω<sup>l</sup>*

time *t.*

(1) or equivalently (10), give the following assumptions.

Here *l* = *j*, when the system stays in the same region Ω<sup>l</sup>

to another one Ω<sup>j</sup>

tainties, which are assumed to be of the form

where *E* <sup>l</sup> *, H* <sup>l</sup> are known constant matrices, and *Ul*

Then, the closed-loop T-S system is governed by

, *l* ∈ *L* where

*<sup>T</sup>* (*t*)*ΔUl* <sup>≤</sup> *<sup>I</sup>*.

() ()

*zt C xt*

=

*cl*

*Assumption* **1.** When the state of the system transits from the region Ω<sup>l</sup>

.

and therefore

$$h\_j(\mathbf{s}(t)) \ge 0, \sum\_{j=1}^{\prime} h\_j(\mathbf{s}(t)) = 1, j \in N \quad \forall t \tag{5}$$

In order to facilitate the design of less conservative H2 controller, we partition the premise variable space *Ω* ⊆*R <sup>s</sup>* into *m* polyhedral regions Ω<sup>i</sup> by the boundaries [7]

$$\hat{\varepsilon}\Omega\_{i}^{\mathbb{V}} = \{ \mathbf{s}(t) \mid h\_{i}(\mathbf{s}(t)) = 1, 0 \le h\_{i}(\mathbf{s}(t + \mathcal{S})) < 1, i \in N \}\tag{6}$$

where *v* is the set of the face indexes of the polyhedral hull with satisfying ∂*Ω<sup>i</sup>* =∪*<sup>v</sup>* (∂*Ω<sup>i</sup> v* )

Based on the boundaries (6), *m* independent polyhedral regions*Ω<sup>l</sup>* , *l* ∈ *L* ={1,2⋯*m*} can be obtained satisfying

$$
\Omega \Omega\_l \cap \Omega\_j = \hat{\varepsilon} \Omega\_l^\vee, l \neq j, l, j \in L \tag{7}
$$

where *L* denotes the set of polyhedral region indexes.

In each region Ω<sup>l</sup> , we define the set

$$M(I) \coloneqq \langle i \mid h\_j(\mathbf{s}(t)) > 0, \mathbf{s}(t) \in \Omega\_\gamma, i \in N \rangle, l \in L \tag{8}$$

Considering (5) and (8), in each region Ω<sup>l</sup> , we have

$$\sum\_{l \in M(l)} h\_l(s(t)) = 1\tag{9}$$

and then, the fuzzy infinite-distributed delays system (1) can be expressed as follows:

$$\begin{aligned} \mathbf{x}(t+1) &= \sum\_{i \in M(l)} h\_i(\mathbf{s}(t)) [A\_i \mathbf{x}(t) + A\_{il} \sum\_{d=1}^n \mu\_d \mathbf{x}(t-d) + B\_{il} \boldsymbol{\mu}(t) + D\_l \mathbf{y}(t)] \\ \mathbf{z}(t) &= \sum\_{i \in M(l)} h\_i(\mathbf{s}(t)) [C\_i \mathbf{x}(t) + B\_{il} \boldsymbol{\mu}(t)] \end{aligned} \tag{10}$$

*Remark* **3.** According to the definition of (8), the polyhedral regions can be divided into two folds: operating and interpolation regions. For an operating region, the set *M(l)* contains on‐ ly one element, and then, the system dynamic is governed by the *s-*th local model of the fuz‐ zy system. For an interpolation region, the system dynamic is governed by a convex combination of several local models.

In this paper, we consider the generalized H2 controller design problem for the fuzzy system (1) or equivalently (10), give the following assumptions.

*Assumption* **1.** When the state of the system transits from the region Ω<sup>l</sup> to Ω<sup>j</sup> at the time *t,* the dynamics of the system is governed by the dynamics of the region model of Ω<sup>l</sup> at that time *t.*

For future use, we define a set*Θ*that represents all possible transitions from one region to itself or another regions, that is

$$\Theta = \{(l, j) \mid \mathbf{s}(t) \in \Omega\_{\,\,\,l}, \mathbf{s}(t+l) \in \Omega\_{\,\,\,j} \,\forall \, l, j \in L\} \tag{11}$$

Here *l* = *j*, when the system stays in the same region Ω<sup>l</sup> , and *l* ≠ *j*, when the system transits from the region Ω<sup>l</sup> to another one Ω<sup>j</sup> .

Considering the fuzzy system (10), choose the following non-fragile piecewise state feed‐ back controller

$$\mathbf{u}(t) = -(K\_l + \Delta K\_l)\mathbf{x}(t) \qquad \mathbf{s}(t) \in \Omega\_l \quad l \in L \tag{12}$$

here Δ*K* <sup>l</sup> are unknown real matrix functions representing time varying parametric uncer‐ tainties, which are assumed to be of the form

$$
\Delta K\_l = E\_l U\_l(t) H\_l,\\
U\_l^\top(t) U\_l(t) \le I,\\
U\_l(t) \in \mathbb{R}^{l\_1 \times l\_2} \tag{13}
$$

where *E* <sup>l</sup> *, H* <sup>l</sup> are known constant matrices, and *Ul* (*t*)∈*R <sup>l</sup>* 1×*l* <sup>2</sup> are unknown real time varying matrix satisfying*ΔUl <sup>T</sup>* (*t*)*ΔUl* <sup>≤</sup> *<sup>I</sup>*.

Then, the closed-loop T-S system is governed by

$$\begin{aligned} \mathbf{x}(t+l) &= \overline{A}\_{cl}\mathbf{x}(t) + A\_{cl} \sum\_{d=1}^{n} \mu\_{d} \mathbf{x}(t-d) + D\_{l} \mathbf{v}(t) \\ \mathbf{z}(t) &= \overline{C}\_{cl} \mathbf{x}(t) \end{aligned} \tag{14}$$

for *s*(*t*)∈*Ω<sup>l</sup>* , *l* ∈ *L* where

<sup>1</sup> ( ( )) 0, ( ( )) 0, *<sup>r</sup>*

<sup>1</sup> ( ( )) 0, ( ( )) 1, *<sup>r</sup>*

into *m* polyhedral regions Ω<sup>i</sup>

where *v* is the set of the face indexes of the polyhedral hull with satisfying

Based on the boundaries (6), *m* independent polyhedral regions*Ω<sup>l</sup>*

where *L* denotes the set of polyhedral region indexes.

, we define the set

Considering (5) and (8), in each region Ω<sup>l</sup>

( )

*iMl*

Î

In order to facilitate the design of less conservative H2 controller, we partition the premise

{ ( ) | ( ( )) 1,0 ( ( )) 1, } *<sup>v</sup> ii i st h st h st i N*

0||1

, ,, *v*

, we have

and then, the fuzzy infinite-distributed delays system (1) can be expressed as follows:

*i i di d i i*

m

*ii i l*

( ) 1

*iMl d*

Î =

å å

2

( ) ( ( ))[ ( ) ( )] ( )

*zt h st Cxt B ut s t*

*x t h s t Ax t A x t d B u t Dv t*

+ = + -+ +

( 1) ( ( ))[ ( ) ( ) ( ) ( )]

¥

= + ÎW

d

d < <

¶W = = £ + <Î (6)

 w

*s t st j N t* <sup>=</sup> ³ >Î " å (4)

*j j <sup>j</sup> h st h st j N t* <sup>=</sup> ³ =Î " å (5)

by the boundaries [7]

*lj i* W Ç W = ¶W ¹ Î *l jl j L* (7)

( ) : { | ( ( )) 0, ( ) , }, *Ml i h st st i N l L* = > ÎW Î Î *i l* (8)

1

<sup>å</sup> (10)

( ) ( ( )) 1 *<sup>i</sup> iMl h st* <sup>Î</sup> å <sup>=</sup> (9)

, *l* ∈ *L* ={1,2⋯*m*} can be

*j j j*

w

and therefore

∂*Ω<sup>i</sup>* =∪*<sup>v</sup>* (∂*Ω<sup>i</sup>*

obtained satisfying

In each region Ω<sup>l</sup>

variable space *Ω* ⊆*R <sup>s</sup>*

56 Advances in Discrete Time Systems

*v* )

$$\begin{aligned} \overline{A}\_{cl} &= \sum\_{i \in \mathcal{M}(l)} h\_i A\_{il'} & A\_{dl} &= \sum\_{i \in \mathcal{M}(l)} h\_i A\_{il'} & D\_l &= \sum\_{i \in \mathcal{M}(l)} h\_i D\_{i'} & \overline{\mathcal{C}}\_{cl} &= \sum\_{i \in \mathcal{M}(l)} h\_i \mathcal{C}\_{il'} \\ A\_{il} &= A\_i - \mathcal{B}\_{1i} \overline{\mathcal{K}}\_{l'} & \mathcal{C}\_{il} &= \mathcal{C}\_i - \mathcal{B}\_{2i} \overline{\mathcal{K}}\_{l} \end{aligned}$$

Before formulation the problem to be investigated, we first introduce the following concept for the system (14).

*Definition* **1.** [10] Let a constant *γ* >0 be given. The closed-loop fuzzy system (14) is said to be stable with generalized H2 performance if both of the following conditions are satisfied:


$$\|\|z\|\|\_{\varkappa} \leqslant \mathcal{Y} \|\|\nu\|\|\_{2} \tag{15}$$

1 22 3 4

é ù - ++ + ê ú

*TT T He U P U A P U A P P U B <sup>d</sup>*

\* \* ( ) <sup>0</sup> \*\* \*

ë û

Based on the proposed partition method, the following DDPLKF is proposed to develop the

1

Then, we are ready to present the generalized H2 stability condition of (14) in terms of LMIs

*Theorem* **1.** Given a constant*γ* >0, the closed-loop fuzzy system (14) with infinite distributed delays is stable with generalized H2 performance*γ*, if there exists a set of positive definite

<sup>2</sup> 0 ( ), *<sup>T</sup> CC P i Ml l L il il l* -<Î Î

*d ktd*

m

¥ - = = -

å å

( ) 2 ( ) ( ), ( ) () ()

= Î

*V t xt Pxt V t x k Qx k*

*V t lZl l L*

*<sup>t</sup> T T <sup>l</sup> <sup>d</sup>*

1

\* \* \* \*\*

*D DDX <sup>b</sup>*

\* \* \*\*

\*

stability condition for the closed-loop system of (14).

3

123

() () () ()

*Vt Vt V t V t*

=++

1 1

= =

*<sup>t</sup> <sup>T</sup>*

 hh

, *Q*, *Z* >0, the nonsingular matrix *F* and matrices *Xli*

g

( ) () ()

1 2

¥ -- = =- = +

å åå

*d d i dl t i*

m

1

*<sup>l</sup>* <sup>=</sup> *<sup>F</sup>* <sup>−</sup>*<sup>T</sup> PlF* , *<sup>Q</sup>*¯ <sup>=</sup> *<sup>F</sup>* <sup>−</sup>*<sup>T</sup> QF* , *<sup>Z</sup>*¯ <sup>=</sup> *<sup>F</sup>* <sup>−</sup>*TZF* , and*Pl*

where *He*{∗} stands for∗ + ∗*<sup>T</sup>* .

**3. Main Results**

where*<sup>P</sup>*¯

as follows

matrices*Pl*

where

*η*(*t*)= *x*(*t* + 1)− *x*(*t*).

ing the following LMIs:

{ } 0

1 2 34 1

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays

*D D DDX*

5 67 2

89 3

*DDX*

10 4

*D X*

5

<

http://dx.doi.org/10.5772/51778

, *Q*, *Z* >0,*F* is nonsingular matrix, and

, *l* ∈ *L* , *i* =1,⋯,4 satisfy‐

, *Yli*

(20)

0 ( ), *ill* P< Î Î *i Ml l L* (21)

0 ( ),( , ) *ilj* P < Î ÎQ *i Ml lj* (22)

(18)

59

(19)

*X*

for all non-zero v ∈ *I2*.

Now, we introduce the following lemmas that will be used in the development of our main result.

*Lemma* **1***.* [6] Let *<sup>M</sup>* <sup>∈</sup>*<sup>R</sup> <sup>n</sup>*×*n* be a positive semi-definite matrix, *xi* (*t*)∈*R <sup>n</sup>*and constant

*ai* >0(*i* =1, 2, ⋯), if the series concerned is convergent, then we have

$$\left(\sum\_{i=1}^{o} a\_i \mathbf{x}\_i\right)^r M \left(\sum\_{i=1}^{o} a\_i \mathbf{x}\_i\right) \le \left(\sum\_{i=1}^{o} a\_i\right) \sum\_{i=1}^{o} a\_i \mathbf{x}\_i M \mathbf{x}\_i \tag{16}$$

*Lemma* **2.** [14] For the real matrices *P*1, *P*2, *P*3, *P*4, *A*, *Ad* , *B*, *Xj* ( *j* =1,⋯,5) and *Di* (*i* =1,⋯,10) with compatible dimensions, the inequalities show in (17) and (18) at the fol‐ lowing are equivalent, where *U* is an extra slack nonsingular matrix.

$$(a) \begin{bmatrix} He\{P\_1^\top A\} + D\_1 & P\_1^\top A\_d + A^\top P\_2 + D\_2 & A^\top P\_3 + D\_3 & A^\top P\_4 + P\_1^\top B + D\_4 & X\_1 \\ \* & He\{P\_2^\top A\_d\} + D\_5 & A\_d^\top P\_3 + D\_6 & A\_d^\top P\_4 + P\_2^\top B + D\_7 & X\_2 \\ \* & \* & D\_8 & P\_3^\top B + D\_9 & X\_3 \\ \* & \* & \* & He\{B^\top P\_4\} + D\_{10} & X\_4 \\ \* & \* & \* & \* & X\_5 \end{bmatrix} < 0 \tag{17}$$

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays http://dx.doi.org/10.5772/51778 59

$$(b) \quad \begin{bmatrix} -He\langle U\rangle & P\_1 + U^T A\_2 & P\_2 + U^T A\_4 & P\_3 & P\_4 + U^T B & 0\\ \ast & D\_1 & D\_2 & D\_3 & D\_4 & X\_1\\ \ast & \ast & D\_3 & D\_6 & D\_7 & X\_2\\ \ast & \ast & \ast & D\_8 & D\_9 & X\_3\\ \ast & \ast & \ast & \ast & D\_{10} & X\_4\\ \ast & \ast & \ast & \ast & \ast & X\_5 \end{bmatrix} < 0 \tag{18}$$

where *He*{∗} stands for∗ + ∗*<sup>T</sup>* .

#### **3. Main Results**

*A*¯ *cl* <sup>=</sup> ∑ *i*∈*M* (*l*)

*hiAil*

58 Advances in Discrete Time Systems

*l*

*Ail* <sup>=</sup> *Ai* <sup>−</sup> *<sup>B</sup>*1*i<sup>K</sup>*¯

for the system (14).

for all non-zero v ∈ *I2*.

{ }

result.

*Di*

*Lemma* **1***.*

, *Adl* <sup>=</sup> ∑ *i*∈*M* (*l*)

, *Cil* <sup>=</sup>*Ci* <sup>−</sup> *<sup>B</sup>*2*i<sup>K</sup>*¯

*hiAdi*

*l*

, *Dl* <sup>=</sup> ∑ *i*∈*M* (*l*) *hi Di* , *C*¯

**•** The disturbance-free fuzzy system is globally asymptotically stable.

[6] Let *<sup>M</sup>* <sup>∈</sup>*<sup>R</sup> <sup>n</sup>*×*n* be a positive semi-definite matrix, *xi*

1 1 11 ( ) ( )( ) *<sup>T</sup>*

¥ ¥ ¥¥ = = ==

*i i ii*

*Lemma* **2.** [14] For the real matrices *P*1, *P*2, *P*3, *P*4, *A*, *Ad* , *B*, *Xj*

*i i i i i ii i*

(*i* =1,⋯,10) with compatible dimensions, the inequalities show in (17) and (18) at the fol‐

1 11 2 2 3 3 41 4 1

*T T TT ddd*

*He P A D A P D A P P B D X*

( ) \* \* 0 \* \* \* {} \* \* \*\*

ë û

*He P A D P A A P D A P D A P P B D X*

é ù + + + + ++ ê ú

*T T T T TT*

*a D PB D X*

2 5 3 6 42 7 2

+ + ++

8 39 3

*He B P D X*

+

*T T*

4 10 4

+ <

5

*X*

å å åå £ (16)

*a x M a x a a x Mx*

*ai* >0(*i* =1, 2, ⋯), if the series concerned is convergent, then we have

lowing are equivalent, where *U* is an extra slack nonsingular matrix.

\* {}

*d*

*cl* <sup>=</sup> ∑ *i*∈*M* (*l*) *hi Cil*

¥ < (15)

(*t*)∈*R <sup>n</sup>*and constant

( *j* =1,⋯,5) and

(17)

Before formulation the problem to be investigated, we first introduce the following concept

*Definition* **1.** [10] Let a constant *γ* >0 be given. The closed-loop fuzzy system (14) is said to be

stable with generalized H2 performance if both of the following conditions are satisfied:

**•** Subject to assumption of zero initial conditions, the controlled output satisfies

<sup>2</sup> || || || || *z v* g

Now, we introduce the following lemmas that will be used in the development of our main

Based on the proposed partition method, the following DDPLKF is proposed to develop the stability condition for the closed-loop system of (14).

$$\begin{aligned} V(t) &= V\_1(t) + V\_2(t) + V\_3(t) \\ V\_1(t) &= 2\mathbf{x}(t)^\top \overline{P}\_l \mathbf{x}(t), \; V\_2(t) = \sum\_{d=1}^\alpha \mu\_d \sum\_{k=t-d}^{l-1} \mathbf{x}(k)^\top \overline{Q} \mathbf{x}(k) \\ V\_3(t) &= \sum\_{d=1}^\alpha \mu\_d \sum\_{i=-d}^{-1} \sum\_{l=t+i}^{l-1} \eta(l)^\top \overline{Z} \eta(l) \qquad &l \in L \end{aligned} \tag{19}$$

where*<sup>P</sup>*¯ *<sup>l</sup>* <sup>=</sup> *<sup>F</sup>* <sup>−</sup>*<sup>T</sup> PlF* , *<sup>Q</sup>*¯ <sup>=</sup> *<sup>F</sup>* <sup>−</sup>*<sup>T</sup> QF* , *<sup>Z</sup>*¯ <sup>=</sup> *<sup>F</sup>* <sup>−</sup>*TZF* , and*Pl* , *Q*, *Z* >0,*F* is nonsingular matrix, and *η*(*t*)= *x*(*t* + 1)− *x*(*t*).

Then, we are ready to present the generalized H2 stability condition of (14) in terms of LMIs as follows

*Theorem* **1.** Given a constant*γ* >0, the closed-loop fuzzy system (14) with infinite distributed delays is stable with generalized H2 performance*γ*, if there exists a set of positive definite matrices*Pl* , *Q*, *Z* >0, the nonsingular matrix *F* and matrices *Xli* , *Yli* , *l* ∈ *L* , *i* =1,⋯,4 satisfy‐ ing the following LMIs:

$$C\_{il}^{\mathcal{T}}C\_{il} - \gamma^2 P\_l < 0 \quad i \in M(l), l \in L \tag{20}$$

$$\Pi\_{\text{all}} < 0 \quad i \in M(l), l \in L \tag{21}$$

$$\Pi\_{\,\,\,ll} < 0 \quad \text{ $i \in M$ } (l), (l, j) \in \Theta \tag{22}$$

where

$$
\boldsymbol{\Pi}\_{\text{ij}} = \begin{bmatrix}
\ast & \boldsymbol{\Sigma}\_{\text{I}1} & \boldsymbol{\Sigma}\_{\text{I}2} & \boldsymbol{\Sigma}\_{\text{I}3} & \boldsymbol{\Sigma}\_{\text{I}4} & \boldsymbol{X}\_{\text{i}\boldsymbol{I}} \\
\ast & \ast & \boldsymbol{\Sigma}\_{\text{I}5} & \boldsymbol{\Sigma}\_{\text{I}6} & \boldsymbol{\Sigma}\_{\text{I}7} & \boldsymbol{X}\_{\text{i}\boldsymbol{I}} \\
\ast & \ast & \ast & \boldsymbol{\Sigma}\_{\text{I}8} & \boldsymbol{\Sigma}\_{\text{I}9} & \boldsymbol{X}\_{\text{i}\boldsymbol{j}} \\
\ast & \ast & \ast & \ast & \boldsymbol{\Sigma}\_{\text{I}10} & \boldsymbol{X}\_{\text{i}\boldsymbol{k}} \\
\ast & \ast & \ast & \ast & \ast & \boldsymbol{\left(-\sum\_{d=1}^{\omega} d\,\mu\_{d}\right)} \boldsymbol{Z}\_{\text{i}}
\end{bmatrix},
$$

with

$$\begin{split} & \Lambda\_{lij} = P\_j + Y\_{I1} + A\_l F - B\_{1i} \overline{X}\_l F\_{.} \\ & \Sigma\_{l1} = \overline{\mu} Q - 2P\_l + \text{He} \{ \overline{\mu} X\_{l1} - Y\_{l1} \}, \ \Sigma\_{l2} = -X\_{I1} + X\_{I2}^T - Y\_{I2}, \ \Sigma\_{l3} = \mathbf{X}\_{l3} - Y\_{l1}^T - Y\_{l3}, \\ & \Sigma\_{l4} = \mathbf{X}\_{l4}^T + Y\_{I4}, \ \Sigma\_{l5} = \frac{1}{\mu} \mathbf{Q} - \text{He} \{ \mathbf{X}\_{l2} \}, \ \Sigma\_{l6} = -\mathbf{X}\_{l3}^T - Y\_{l2}, \ \Sigma\_{l7} = \mathbf{X}\_{l4}, \\ & \Sigma\_{l8} = \sum\_{d=1}^{\infty} \mu\_d d \mathbf{Z} - \text{He} \{ Y\_{l3} \}, \ \Sigma\_{l9} = Y\_{I4}^T, \ \Sigma\_{l10} = -I. \end{split}$$

**Proof.** Taking the forward difference of (19) along the solution of the system (14), we have *ΔV* (*t*)=*V* (*t* + 1)−*V* (*t*)=*ΔV*<sup>1</sup> + *ΔV*<sup>2</sup> + *ΔV*<sup>3</sup>

Assuming that*s*(*t*)∈*Ω<sup>l</sup>* , *s*(*t* + 1)∈*Ω<sup>j</sup>* . The difference of *Vi* (*t*), *i* =1,2,3can be calculated, re‐ spectively, showing at the following

$$
\Delta V\_1(t) = 2\overline{\mathbb{L}}\overline{A}\_d \ge \left(t\right) + A\_{dl}\sum\_{d=1}^{\alpha} \mu\_d \ge \left(t - d\right) + D\_l v(t)\overline{\mathbb{L}}^T \overline{P}\_j \left[\eta(t) + \mathbf{x}(t)\right] - 2\mathbf{x}^T(t)\overline{P}\_l \mathbf{x}(t) \tag{23}
$$

$$\begin{split} \Delta V\_{2}(t) &= \sum\_{d=1}^{\alpha} \mu\_{d} \sum\_{\varepsilon=\iota+1-d}^{\iota} \mathbf{x}^{\top}(\boldsymbol{\tau}) \overline{\boldsymbol{Q}} \mathbf{x}(\boldsymbol{\tau}) - \sum\_{d=1}^{\alpha} \mu\_{d} \sum\_{\varepsilon=\iota-d}^{\iota-1} \mathbf{x}^{\top}(\boldsymbol{\tau}) \overline{\boldsymbol{Q}} \mathbf{x}(\boldsymbol{\tau}) \\ &= \overline{\mu} \mathbf{x}^{\top}(t) \overline{\boldsymbol{Q}} \mathbf{x}(t) - \sum\_{d=1}^{\alpha} \mu\_{d} \mathbf{x}^{\top}(t-d) \overline{\boldsymbol{Q}} \mathbf{x}(t-d) \end{split} \tag{24}$$

1

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays

 h

D = å åå - (27)

 h*lZl*

<sup>å</sup> (29)

*b*holds for compatible vectors*a*and*b*, and any compatible matrix

1

 h

m

(28)

61

http://dx.doi.org/10.5772/51778

(30)

1 1

*Vt d t Z t* mh

1

1

´ - ++ -

*d li di i*

¥ =

[( ) ( ) ( ) ( )]=0

m

*A I x t A Dv t t*

1 2 3 4

<sup>1</sup> 2 2 <sup>1</sup>

*X X dt Z <sup>t</sup> lZ l*

*T*

( ) ( ) () ()

x

1 2

*Vt v tvt v tvt h t t v tvt*

() () () () () () () () () *T T T T*

1 1

¥ ¥ - - = = = -

å å å

*<sup>t</sup> <sup>T</sup> l l d d d l l d ltd*


*l l*

é ùé ù ê úê ú ê úê ú £ + ê úê ú ê úê ú ê úê ú ë ûë û

*X X*

m

1 1 3 3 4 4

*X X XU XU*

*l l*

*μd <sup>x</sup> <sup>T</sup>* (*<sup>t</sup>* <sup>−</sup>*d*), *<sup>η</sup> <sup>T</sup>* (*t*), *<sup>v</sup> <sup>T</sup>* (*t*)

1 1

*T*

*d d ltd*

¥ ¥ - = = = -

å å å

 m

( )

*iMl*

Î D - + +X +X £ Y + å (31)

*i ilj*

 x

x

 hh

2[ ( ) ( ) () () ] ()

*x tX x t dX tX v tXU l*

*<sup>t</sup> T T TT <sup>l</sup> l l <sup>d</sup> l d*

 h

X= + - + +

m

*d*

¥ =

å

 m

´ - --

*d d d d ltd*

Observing of the definition of *η*(*t*)and system (14), we can get the following equations:

1 2 3 1 4

*<sup>t</sup> <sup>T</sup> d d d d ltd*

1 2 3 2 4

2[ ( ) ( ) () () ]

*x tY x t dY tY v tYU*

*T T TT <sup>l</sup> l l d l*

2[ ( ) ( ) () () ]

*x tX x t dX tX v tXU*

 mh

*T T TT <sup>l</sup> l l d l*

1 1

*xt x t d l*

¥ ¥ - = = = -

å åå

[ ( ) ( ) ( )]=0

X= + - + +

¥ ¥ - = = = -

 h

() () () () () *<sup>t</sup> T T*

 m

1

 h

h

 h

3

m

where *<sup>X</sup>*¯

*M* >0, we have

with*ξ*(*t*)= *x <sup>T</sup>* (*t*), ∑

where

*li* <sup>=</sup> *<sup>F</sup>* <sup>−</sup>*<sup>T</sup> Xli*

Since ±2*a <sup>T</sup> b* ≤*a <sup>T</sup> Ma* + *b <sup>T</sup> M* <sup>−</sup><sup>1</sup>

m x

> *d*=1 *∞*

Then, from (23-30) and considering (14), we have

*F* <sup>−</sup><sup>1</sup>

(*i* =1, 2, 3)

From Lemma1, we have

$$-\sum\_{d=1}^{n} \mu\_d \mathbf{x}^r(t-d)\overline{\mathbf{Q}}\mathbf{x}(t-d) \le -\frac{1}{\mu}(\sum\_{d=1}^{n} \mu\_d \mathbf{x}(t-d))^r \overline{\mathbf{Q}}(\sum\_{d=1}^{n} \mu\_d \mathbf{x}(t-d))\tag{25}$$

Substituting (25) into (24), we have

$$
\Delta V\_2(t) \le \overline{\mu} \mathbf{x}^T(t) \overline{Q} \mathbf{x}(t) - \frac{1}{\mu} (\sum\_{d=1}^{\alpha} \mu\_d \mathbf{x}(t-d))^T \overline{Q} (\sum\_{d=1}^{\alpha} \mu\_d \mathbf{x}(t-d)) \tag{26}
$$

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays http://dx.doi.org/10.5772/51778 61

$$
\Delta V\_{\mathfrak{z}}(t) = \sum\_{d=1}^{\alpha} \mu\_d d \eta(t)^{\mathbb{T}} \overline{Z} \eta(t) - \sum\_{d=1}^{\alpha} \mu\_d \sum\_{l=t-d}^{t-1} \eta(l)^{\mathbb{T}} \overline{Z} \eta(l) \tag{27}
$$

Observing of the definition of *η*(*t*)and system (14), we can get the following equations:

$$\begin{aligned} \overline{\boldsymbol{\Delta}}\_{1} &= 2[\boldsymbol{\boldsymbol{x}}^{\top}(t)\overline{\boldsymbol{X}}\boldsymbol{n} + \sum\_{d=1}^{\circ} \boldsymbol{\mu}\_{d}\boldsymbol{\boldsymbol{x}}^{\top}(t-d)\overline{\boldsymbol{X}}\boldsymbol{\iota} + \boldsymbol{\eta}^{\top}(t)\overline{\boldsymbol{X}}\boldsymbol{\iota} + \boldsymbol{\nu}^{\top}(t)\boldsymbol{X}\_{\boldsymbol{\iota}\boldsymbol{\iota}}\boldsymbol{U}] \\ &\times [\overline{\boldsymbol{\mu}}\boldsymbol{\kappa}(t) - \sum\_{d=1}^{\circ} \boldsymbol{\mu}\_{d}\boldsymbol{\boldsymbol{x}}^{\top}(t-d) - \sum\_{d=1}^{\circ} \boldsymbol{\mu}\_{d}\sum\_{l=t-d}^{t-1} \boldsymbol{\eta}(l)] \boldsymbol{=} \boldsymbol{0} \end{aligned} \tag{28}$$

$$\begin{aligned} \Xi\_2 &= 2[\mathbf{x}^\top(t)\overline{Y}\_{l1} + \sum\_{d=1}^\upsilon \mu\_d \mathbf{x}^\top(t-d)\overline{Y}\_{l2} + \eta^\top(t)\overline{Y}\_{l3} + \mathbf{v}^\top(t)Y\_{l4}U] \\ &\times [(\overline{A}\iota - I)\mathbf{x}(t) + A\_{d1} + D\_l\mathbf{v}(t) - \eta(t)] \mathbf{=0} \end{aligned} \tag{29}$$

where *<sup>X</sup>*¯ *li* <sup>=</sup> *<sup>F</sup>* <sup>−</sup>*<sup>T</sup> Xli F* <sup>−</sup><sup>1</sup> (*i* =1, 2, 3)

*Πilj* =

60 Advances in Discrete Time Systems

with

<sup>−</sup>*He*{*F*} *<sup>Λ</sup>ilj Yl* <sup>2</sup>

*Λilj* = *Pj* + *Yl*<sup>1</sup> + *Ai*

*Σl*<sup>4</sup> = *Xl*<sup>4</sup>

*<sup>Σ</sup>l*<sup>8</sup> <sup>=</sup>∑ *d*=1 *∞*

Assuming that*s*(*t*)∈*Ω<sup>l</sup>*

*<sup>Δ</sup>V*1(*t*)=2 *<sup>A</sup>*¯

From Lemma1, we have

+ *Adi*

*<sup>F</sup>* <sup>−</sup> *<sup>B</sup>*1*i<sup>K</sup>*¯

*μd dZ* −H*e*{*Yl*3}, *Σl*<sup>9</sup> =*Yl*<sup>4</sup>

, *s*(*t* + 1)∈*Ω<sup>j</sup>*

*d*=1 *∞*

m

m

t

D = -

*μd x*(*t* −*d*) + *Dl*

1 1 1

 tt

*d d d td d td*

å å åå

¥ ¥ - = =+- = = - ¥ =

*V t x Qx x Qx*

*d d*

*x t Qx t x t d Qx t d*

= () () ( ) ( )

å

1 1 1

m

¥ ¥ ¥ = = =

*d d d d d d*

m

 *x t d Qx t d x t d Q x t d* m

<sup>1</sup> ( ) ( ) ( ) ( ( )) ( ( )) *T T*

*V t x t Qx t x t d Q x t d*

*T T*

1

 m

*cl <sup>x</sup>*(*t*) <sup>+</sup> *Adl*∑

2

m

Substituting (25) into (24), we have

2

m

*<sup>T</sup>* <sup>+</sup> *Yl*4, *<sup>Σ</sup>l*<sup>5</sup> <sup>=</sup> <sup>1</sup>

*ΔV* (*t*)=*V* (*t* + 1)−*V* (*t*)=*ΔV*<sup>1</sup> + *ΔV*<sup>2</sup> + *ΔV*<sup>3</sup>

spectively, showing at the following

*<sup>l</sup>F*,

*<sup>Σ</sup>l*<sup>1</sup> <sup>=</sup>*μ*¯*<sup>Q</sup>* <sup>−</sup>2*Pl* <sup>+</sup> <sup>H</sup>*e*{*μ*¯*Xl*<sup>1</sup> <sup>−</sup>*Yl*1}, *<sup>Σ</sup>l*<sup>2</sup> <sup>=</sup> <sup>−</sup> *Xl*<sup>1</sup> <sup>+</sup> *Xl*<sup>2</sup>

*<sup>F</sup> Pj* <sup>+</sup> *Yl* <sup>3</sup> *Yl* <sup>4</sup>

\* *Σl*<sup>1</sup> *Σl*<sup>2</sup> *Σl*<sup>3</sup> *Σl*<sup>4</sup> *Xl*<sup>1</sup> \* \* *Σl*<sup>5</sup> *Σl*<sup>6</sup> *Σl*<sup>7</sup> *Xl*<sup>2</sup> \* \* \* *Σl*<sup>8</sup> *Σl*<sup>9</sup> *Xl*<sup>3</sup> \* \* \* \* *Σl*<sup>10</sup> *Xl*<sup>4</sup>

\* \* \* \* \* (−∑

*<sup>μ</sup>*¯ *<sup>Q</sup>* <sup>−</sup>H*e*{*Xl*2}, *<sup>Σ</sup>l*<sup>6</sup> <sup>=</sup> <sup>−</sup> *Xl*<sup>3</sup>

*<sup>T</sup>* , *Σl*<sup>10</sup> = − *I*.

**Proof.** Taking the forward difference of (19) along the solution of the system (14), we have

. The difference of *Vi*

*v*(*t*) *T P*¯

( ) () () () ()

*t t T T*


<sup>1</sup> ( ) ( ) ( ( )) ( ( )) *T T*

+ *Di* 0

*d*=1 *∞ dμd* ) −1 *Z*

*<sup>T</sup>* <sup>−</sup>*Yl*2, *<sup>Σ</sup>l*<sup>3</sup> <sup>=</sup> *Xl*<sup>3</sup> <sup>−</sup>*Yl*<sup>1</sup>

*<sup>j</sup> <sup>η</sup>*(*t*) <sup>+</sup> *<sup>x</sup>*(*t*) <sup>−</sup>2*<sup>x</sup> <sup>T</sup>* (*t*)*<sup>P</sup>*¯

1

 tt

> m

 t


1 1

¥ ¥ = =

*d d d d*

D£ - - - å å (26)

mm

 m

*<sup>T</sup>* <sup>−</sup>*Yl*2, *<sup>Σ</sup>l*<sup>7</sup> <sup>=</sup> *Xl*4,

*<sup>T</sup>* <sup>−</sup>*Yl*3,

(*t*), *i* =1,2,3can be calculated, re‐

*<sup>l</sup> x*(*t*) (23)

(24)

Since ±2*a <sup>T</sup> b* ≤*a <sup>T</sup> Ma* + *b <sup>T</sup> M* <sup>−</sup><sup>1</sup> *b*holds for compatible vectors*a*and*b*, and any compatible matrix *M* >0, we have

$$-2\|\mathbf{x}^{\top}(t)\overline{X}\_{\parallel} + \sum\_{d=1}^{a} \mu\_{d}\mathbf{x}^{\top}(t-d)\overline{X}\_{\parallel} + \eta^{\top}(t)\overline{X}\_{\parallel} + \boldsymbol{\nu}^{\top}(t)X\_{\parallel}U \| \times \sum\_{d=1}^{a} \mu\_{d}\sum\_{l=t-d}^{l-1} \eta(l)$$

$$\leq \sum\_{d=1}^{a} d\,\mu\_{d}\boldsymbol{\xi}^{\top}(t) \begin{bmatrix} \overline{X}\_{\parallel} \\ \overline{X}\_{\parallel} \\ \overline{X}\_{\parallel} \\ X\_{\parallel}U \end{bmatrix} \overline{Z}^{-1} \begin{bmatrix} \overline{X}\_{\parallel} \\ \overline{X}\_{\parallel} \\ \overline{X}\_{\parallel} \\ X\_{\parallel}U \end{bmatrix}^{\top} \boldsymbol{\xi}(t) + \sum\_{d=1}^{a} \mu\_{d}\sum\_{l=t-d}^{l-1} \eta(l)\overline{Z}\eta(l)$$

with*ξ*(*t*)= *x <sup>T</sup>* (*t*), ∑ *d*=1 *∞ μd <sup>x</sup> <sup>T</sup>* (*<sup>t</sup>* <sup>−</sup>*d*), *<sup>η</sup> <sup>T</sup>* (*t*), *<sup>v</sup> <sup>T</sup>* (*t*) *T*

Then, from (23-30) and considering (14), we have

$$
\Delta V(t) - \boldsymbol{\nu}^{\boldsymbol{\tau}}(t)\boldsymbol{\nu}(t) + \boldsymbol{\nu}^{\boldsymbol{\tau}}(t)\boldsymbol{\nu}(t) + \boldsymbol{\Xi}\_1 + \boldsymbol{\Xi}\_2 \le \sum\_{i \in \mathcal{M}(l)} h\_i \boldsymbol{\xi}^{\boldsymbol{\tau}}(t) \boldsymbol{\Psi}\_{i\boldsymbol{\eta}} \boldsymbol{\xi}(t) + \boldsymbol{\nu}^{\boldsymbol{\tau}}(t)\boldsymbol{\nu}(t) \tag{31}$$

where

$$
\Psi\_{\underline{\boldsymbol{w}}\boldsymbol{\upbeta}} = \begin{bmatrix}
\Phi\_{\boldsymbol{\upbeta}}^{1} & \Phi\_{\boldsymbol{\upbeta}}^{2} & \Phi\_{\boldsymbol{\upbeta}}^{3} & \Phi\_{\boldsymbol{\upbeta}}^{4} \\
\* & \Phi\_{\boldsymbol{\upbeta}}^{5} & \Phi\_{\boldsymbol{\upbeta}}^{6} & \Phi\_{\boldsymbol{\upbeta}}^{7} \\
\* & \* & \Phi\_{\boldsymbol{\upbeta}}^{8} & \Phi\_{\boldsymbol{\upbeta}}^{9} \\
\* & \* & \* & \Phi\_{\boldsymbol{\upbeta}}^{10}
\end{bmatrix} + \sum\_{d=1}^{o} d\,\boldsymbol{\upmu}\_{d} \left[ \begin{array}{c} \overline{X}\_{\boldsymbol{I}\boldsymbol{\varbeta}} \\ \overline{X}\_{\boldsymbol{I}\boldsymbol{\upbeta}} \\ \overline{X}\_{\boldsymbol{I}\boldsymbol{\upbeta}} \\ X\_{\boldsymbol{I}\boldsymbol{\upbeta}} \boldsymbol{U} \end{array} \bigg| \overline{Z}^{-1} \begin{array}{c} \overline{X}\_{\boldsymbol{I}\boldsymbol{\upbeta}} \\ \overline{X}\_{\boldsymbol{I}\boldsymbol{\upbeta}} \\ \overline{X}\_{\boldsymbol{I}\boldsymbol{\upbeta}} \\ X\_{\boldsymbol{I}\boldsymbol{\upbeta}} \boldsymbol{U} \end{array} \right] \tag{32}$$

where*<sup>Σ</sup>*¯

Let *U* = *F* <sup>−</sup><sup>1</sup>

*li* <sup>=</sup> *<sup>F</sup>* <sup>−</sup>*TΣli*

respectively, then *Ξilj*

with *ζ*(*t*)= *x <sup>T</sup>* (*t*), ∑

It follows from (20) that

*d*=1 *∞*

*F* <sup>−</sup><sup>1</sup>

(*i* =1, 2, 3, 5, 6, 8, 10)

is equivalent to*Πilj*

Thus, if (21) and (22) holds, (32) is satisfied, which implies that

*μd <sup>x</sup> <sup>T</sup>* (*<sup>t</sup>* <sup>−</sup>*d*), *<sup>η</sup> <sup>T</sup>* (*t*)

FFF F F

\* \* \*

system (14) with v(*t*) = 0 is globally asymptotically stable.

2 2

é ù ê ú < = ê ú ê ú ë û

0 0

*P*

*T*

g l

0 0 () 0 0 () ()

*t Q t Vt Z*

l

*ilj*

It is noted that if the disturbance term*v*(*t*)=0, it follows from (31) that

, *G* =*diag*(*F* , *F* , *F*, *F*, *I*, *F* ), pre- and post multiplying (35) by *G <sup>T</sup>* , *G*

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays

() () () *<sup>T</sup>* D < *Vt v tvt* (36)

http://dx.doi.org/10.5772/51778

63

D< W å (37)

T

+ < å (39)

0 0

 l

(40)

0 00

ê ú ë û

é ù

*T*

*C C*

å (38)

3 3

*X X d XZX X X*

.

( ) () () () *<sup>T</sup> i ilj*

 z

*iMl Vt h t t* z

*T*

1 23 1 1 <sup>1</sup> 5 6 2 2

*ilj l l*

¥ -

m

*l l ilj ilj ilj l l ilj ilj d d*

= éùéù é ù êúêú ê ú W = <sup>+</sup> êúêú ê ú <sup>ê</sup>

<sup>F</sup> úêú ê ú ë û êúêú ëûëû

By Schur's complement, LMI (32) implies*Ωilj* <0, then*ΔV* (*t*)<0. Therefore, the closed-loop

Now, to establish the generalized H2 performance for the closed-loop system (14), under

T 0 ( (T 1)) ( ) ( ) *<sup>T</sup> t V x v tvt* =

( )

ê ú = = ê ú

l

() () () () () 0 0 0 ()

å

*z t zt x tC C xt h t t*

Î

 g

*iMl*

*il il <sup>T</sup> T T <sup>T</sup> cl cl <sup>i</sup>*

8 1

zero-initial condition, and v(*t*)≠0, taking summation for the both sides of (36) leads to

Î

with

Then

$$
\Delta V(t) - v^T(t)v(t) \le 0 \tag{33}
$$

if

$$
\Psi\_{\text{ll}} < 0 \tag{34}
$$

Using lemma 2, (32) is equivalent to (33)

$$
\boldsymbol{\Xi}\_{ij} = \begin{bmatrix}
\* & \overline{\Sigma}\_{1i} & \overline{\Sigma}\_{12} & \overline{\Sigma}\_{13} & U \Sigma\_{14} & \overline{X}\_{1i} \\
\* & \* & \overline{\Sigma}\_{15} & \overline{\Sigma}\_{16} & U \Sigma\_{17} & \overline{X}\_{1i} \\
\* & \* & \* & \overline{\Sigma}\_{18} & U \Sigma\_{19} & \overline{X}\_{1i} \\
\* & \* & \* & \* & \Sigma\_{10} & X\_{1i}U \\
\* & \* & \* & \* & \* & (-\sum\_{d=1}^{s} \mu\_{d}d)^{-1} \overline{Z}
\end{bmatrix} \tag{35}
$$

$$\text{where}\\
\bar{\Sigma}\_{li} = F^{-T} \Sigma\_{li} F^{-1} (i = 1, \ 2, \ 3, \ 5, \ 6, \ 8, \ 10).$$

<sup>T</sup> 1 234 1 1

=

<sup>F</sup> ê úê ú ë û ë ûë û

12 3 4

*TT T i l li l j l di l i*

*ilj l l <sup>l</sup>*

X = <sup>ê</sup> <sup>ú</sup> S S <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>S</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> - <sup>ê</sup> <sup>ú</sup> <sup>ë</sup> <sup>û</sup> <sup>å</sup>

*He U P Y U A Y U A P Y U Y D*

{ } () 0

é- ++ + + + <sup>ù</sup> <sup>ê</sup> <sup>ú</sup> <sup>ê</sup> S SS S <sup>ú</sup> <sup>ê</sup> <sup>ú</sup> SS S <sup>ê</sup> <sup>ú</sup>

\* \* \* \* \*( )

1 23 1 4

*l ll l l*

5 6 1 7

*l l l l*

8 1 9

10 4

*l l*

*U X U X U X*

1

=

*d d*

m *d Z* ¥ -

*X U*

1

(35)

*ilj ilj d l l ilj l l*

F F ê úê ú

m

*l l ilj ilj ilj ilj l l ilj ilj ilj*

é ù FFFF é ùé ù ê ú ê úê ú FFF ê úê ú Y = <sup>+</sup>

2 2 1

å (32)

*X X*

4 4

*ΔV* (*t*)−*v <sup>T</sup>* (*t*)*v*(*t*)<0 (33)

0 Y < *ilj* (34)

ê úê ú

1 3 3

*X X d Z X X XU XU*

¥ -

567

*ilj d*

\* \* \* \*\*\*

with 

62 Advances in Discrete Time Systems

Then

if

Using lemma 2, (32) is equivalent to (33)

\* \*

\*\* \*

\* \* \*\*

\*

8 9

10

Let *U* = *F* <sup>−</sup><sup>1</sup> , *G* =*diag*(*F*, *F*, *F* , *F*, *I*, *F* ), pre- and post multiplying (35) by *G <sup>T</sup>* , *G* respectively, then *Ξilj* is equivalent to*Πilj* .

Thus, if (21) and (22) holds, (32) is satisfied, which implies that

$$
\Delta V(t) < \mathbf{v}^{\mathrm{T}}(t)\mathbf{v}(t) \tag{36}
$$

It is noted that if the disturbance term*v*(*t*)=0, it follows from (31) that

$$
\Delta V(t) < \sum\_{i \in M(l)} h\_i \zeta^{r^T}(t) \Omega\_{i\emptyset} \zeta(t) \tag{37}
$$

with *ζ*(*t*)= *x <sup>T</sup>* (*t*), ∑ *d*=1 *∞ μd <sup>x</sup> <sup>T</sup>* (*<sup>t</sup>* <sup>−</sup>*d*), *<sup>η</sup> <sup>T</sup>* (*t*) *T*

$$
\boldsymbol{\Omega}\_{\boldsymbol{\upbeta}\boldsymbol{\upbeta}} = \begin{bmatrix}
\boldsymbol{\Phi}^{1}\_{\boldsymbol{\upbeta}\boldsymbol{\upbeta}} & \boldsymbol{\Phi}^{2}\_{\boldsymbol{\upbeta}\boldsymbol{\upbeta}} & \boldsymbol{\Phi}^{3}\_{\boldsymbol{\upbeta}\boldsymbol{\upbeta}} \\
\boldsymbol{\h}^{\star} & \boldsymbol{\Phi}^{5}\_{\boldsymbol{\upbeta}\boldsymbol{\upbeta}} & \boldsymbol{\Phi}^{6}\_{\boldsymbol{\upbeta}\boldsymbol{\upbeta}} \\
\boldsymbol{\h}^{\star} & \boldsymbol{\h}^{\star} & \boldsymbol{\Phi}^{5}\_{\boldsymbol{\upup}\boldsymbol{\upbeta}}
\end{bmatrix} + \sum\_{l=1}^{o} \boldsymbol{d} \, \boldsymbol{\mu}\_{d} \begin{bmatrix}
\overline{X}\_{\boldsymbol{\upiota}} \\
\overline{X}\_{\boldsymbol{\upiota}} \\
\overline{X}\_{\boldsymbol{\upiota}}
\end{bmatrix} \overline{Z}^{-1} \begin{bmatrix}
\overline{X}\_{\boldsymbol{\upiota}} \\
\overline{X}\_{\boldsymbol{\upiota}} \\
\overline{X}\_{\boldsymbol{\upiota}}
\end{bmatrix}^{\top} \tag{38}
$$

By Schur's complement, LMI (32) implies*Ωilj* <0, then*ΔV* (*t*)<0. Therefore, the closed-loop system (14) with v(*t*) = 0 is globally asymptotically stable.

Now, to establish the generalized H2 performance for the closed-loop system (14), under zero-initial condition, and v(*t*)≠0, taking summation for the both sides of (36) leads to

$$V(\mathbf{x}(\mathbf{T}+\mathbf{l})) < \sum\_{\iota=0}^{\mathbf{T}} \mathbf{v}^{\sf T}(t)\mathbf{v}(t) \tag{39}$$

It follows from (20) that

$$\begin{aligned} \mathbf{z}^T(t)\mathbf{z}(t) &= \mathbf{x}^T(t)\overline{C}\_{cl}^T\overline{C}\_{cl}\mathbf{x}(t) = \sum\_{i \in M(l)} h\_i \mathbf{\mathcal{A}}^T(t) \begin{bmatrix} C\_{il}^T C\_{il} & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \mathbf{\mathcal{A}}(t) \\ &< \gamma^2 \mathbf{\mathcal{A}}^T(t) \begin{bmatrix} P & 0 & 0 \\ 0 & Q & 0 \\ 0 & 0 & Z \end{bmatrix} \mathbf{\mathcal{A}}(t) = \gamma^2 V(t) \end{aligned} \tag{40}$$

with

$$\lambda\left(t\right) = \left[\mathfrak{x}\left(t\right)\_{\prime}\sum\_{d=1}^{\omega}\mu\_{d}\sum\_{\tau=t-d}^{t-1}\mathfrak{x}\left(\tau\right)\_{\prime}\sum\_{d=1}^{\omega}\mu\_{d}\sum\_{i=-d}^{-1}\sum\_{l=t-d}^{t-1}\eta\left(l\right)\right]$$

From (39) and (40), we have

$$\left\| z(t) \right\|\_{\ast}^{2} < \gamma^{2} \left\| \nu(t) \right\|\_{2}^{2} \tag{41}$$

*Τilj* = *Pj* + *Yl*<sup>1</sup> + *Ai*

trices *Xli*

where

*il*<sup>=</sup>

with 

, *Yli* , *Ml* *F* − *B*1*<sup>i</sup> Ml*

**Proof.** In (20) and (21), replace *Kl* ¯ with*Kl* <sup>+</sup> *<sup>Δ</sup>Kl*

there exists a set of positive definite matrices*Pl*

\* \*

<sup>−</sup>*He*{*F*} *<sup>Τ</sup>il Yl* <sup>2</sup>

g e

tain the results of this theorem, and the details are thus omitted.

*Remark* **4.** If the global state space replace the transitions*Θ*and all*Pl*

common*P*, Theorem 2 is regressed to Corollary 1, shown in the following.

, *l* ∈ *L* , *i* =1,2,3,4satisfying the following LMIs:

*l*

*<sup>F</sup> Pj* <sup>+</sup> *Yl* <sup>3</sup> *Yl* <sup>4</sup>

\* *Σl*<sup>1</sup> *Σl*<sup>2</sup> *Σl*<sup>3</sup> *Σl*<sup>4</sup> *Xl*<sup>1</sup> − *B*1*<sup>i</sup>*

\* \* *Σl*<sup>5</sup> *Σl*<sup>6</sup> *Σl*<sup>7</sup> *Xl*<sup>2</sup> 0 \* \* \* *Σl*<sup>8</sup> *Σl*<sup>9</sup> *Xl*<sup>3</sup> 0 \* \* \* \* *Σl*<sup>10</sup> *Xl*<sup>4</sup> 0 \* \* \* \* \* *Γ<sup>l</sup>* 0 \* \* \* \* \* \* −*εlI*

e

ê ú - + <Î Î

*I EE i Ml l L I*

2 2 <sup>2</sup> \* 0 0 ( ),

*i i l il T ll l*

*P CF B M B HF*

é ù -- -

ë û -

+ *Adi*

Furthermore, the control law is given by

, *<sup>Γ</sup><sup>l</sup>* =(−∑

*d*=1 *∞ dμd* ) −1

<sup>1</sup> *K MF l l*

*Corollary* **1.** Consider the uncertain terms (12). Given a constant*γ* >0, the closed-loop fuzzy system (14) with infinite-distributed delays is stable with generalized H2 performance*γ*, if

*Z* + *εlEl*

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays

*<sup>T</sup> El* .


, and then by S-procedure, we can easily ob‐

, *Q*, *Z* >0, the nonsingular matrix *F* and ma‐

*HlF*

0 ( ), *il* ¡< Î Î *i Ml l L* (47)

+ *Di* 0 0

s in Theorem 2 become a

http://dx.doi.org/10.5772/51778

65

(46)

The proof is completed.

The following theorem shows that the desired controller parameters and considered control‐ ler uncertain can be determined based on the results of Theorem 1.This can be easily proved along the lines of Theorem 1, and we, therefore, only keep necessary details in order to avoid unnecessary duplication.

*Theorem* **2.** Consider the uncertain terms (12). Given a constant*γ* >0, the closed-loop fuzzy system (14) with infinite-distributed delays is stable with generalized H2 performance*γ*, if there exists a set of positive definite matrices*Pl* , *Q*, *Z* >0, the nonsingular matrix *F* and ma‐ trices *Xli* , *Yli* , *Ml* , *l* ∈ *L* , *i* =1,2,3,4satisfying the following LMIs:

$$
\begin{bmatrix}
\* & -\gamma^2 I + \mathfrak{s}\_l E\_l^T E\_l & 0 \\
\* & \* & -\mathfrak{s}\_l I
\end{bmatrix} < 0 \quad \text{if } \mathfrak{s} \in M(I), l \in L \tag{42}
$$

$$\Upsilon\_{\text{all}} < 0 \quad i \in M(I), l \in L \tag{43}$$

$$\Upsilon\_{ilj} < 0 \quad i \in M(l), (l, j) \in \Theta \tag{44}$$

where

$$
\begin{bmatrix}
\* & \Sigma\_{\mathrm{I}1} & \Sigma\_{\mathrm{I}2} & \Sigma\_{\mathrm{I}3} & \Sigma\_{\mathrm{I}4} & X\_{\mathrm{I}1} & -B\_{\mathrm{I}1}H\_{l}F \\
\* & \* & \Sigma\_{\mathrm{I}5} & \Sigma\_{\mathrm{I}6} & \Sigma\_{\mathrm{I}7} & X\_{\mathrm{I}2} & 0 \\
\* & \* & \* & \Sigma\_{\mathrm{I}8} & \Sigma\_{\mathrm{I}9} & X\_{\mathrm{I}3} & 0 \\
\* & \* & \* & \* & \Sigma\_{\mathrm{I}10} & X\_{\mathrm{I}4} & 0 \\
\* & \* & \* & \* & \* & \Gamma\_{\mathrm{I}} & 0 \\
\* & \* & \* & \* & \* & \* & -\varepsilon\_{\mathrm{I}}I
\end{bmatrix}
$$

with

$$T\_{i l \dot{j}} = P\_j + Y\_{I1} + A\_i F - B\_{1i} M\_{l \prime} \qquad \Gamma\_l = \left( -\sum\_{d=1}^{\omega} d \, \mu\_d \right)^{-1} Z + \varepsilon\_l E\_l^T E\_{l \prime}.$$

Furthermore, the control law is given by

$$K\_l = M\_l F^{-l} \tag{45}$$

**Proof.** In (20) and (21), replace *Kl* ¯ with*Kl* <sup>+</sup> *<sup>Δ</sup>Kl* , and then by S-procedure, we can easily ob‐ tain the results of this theorem, and the details are thus omitted.

*Remark* **4.** If the global state space replace the transitions*Θ*and all*Pl* s in Theorem 2 become a common*P*, Theorem 2 is regressed to Corollary 1, shown in the following.

*Corollary* **1.** Consider the uncertain terms (12). Given a constant*γ* >0, the closed-loop fuzzy system (14) with infinite-distributed delays is stable with generalized H2 performance*γ*, if there exists a set of positive definite matrices*Pl* , *Q*, *Z* >0, the nonsingular matrix *F* and ma‐ trices *Xli* , *Yli* , *Ml* , *l* ∈ *L* , *i* =1,2,3,4satisfying the following LMIs:

$$
\begin{bmatrix}
\* & -\gamma^2 I + \mathfrak{c}\_l E\_l^T E\_l & 0 \\
\* & \* & -\mathfrak{c}\_l I
\end{bmatrix} < 0 \quad i \in M(l), l \in L \tag{46}
$$

$$\Upsilon\_{\mu} < 0 \quad i \in M(l), l \in L \tag{47}$$

where

with

*λ*(*t*)= *x*(*t*), ∑

trices *Xli*

where

*ilj*<sup>=</sup>

with

, *Yli* , *Ml*

*d*=1 *∞ μd* ∑ *τ*=*t*−*d t*−1

64 Advances in Discrete Time Systems

The proof is completed.

avoid unnecessary duplication.

there exists a set of positive definite matrices*Pl*

\* \*

+ *Adi*

<sup>−</sup>*He*{*<sup>F</sup>* } *<sup>Τ</sup>ilj Yl* <sup>2</sup>

g e

From (39) and (40), we have

*x*(*τ*), ∑ *d*=1 *∞ μd* ∑ *i*=−*d* −1 ∑ *l*=*t*−*d t*−1 *η*(*l*)

> 2 2 2 <sup>2</sup> *zt vt* () () g

The following theorem shows that the desired controller parameters and considered control‐ ler uncertain can be determined based on the results of Theorem 1.This can be easily proved along the lines of Theorem 1, and we, therefore, only keep necessary details in order to

*Theorem* **2.** Consider the uncertain terms (12). Given a constant*γ* >0, the closed-loop fuzzy system (14) with infinite-distributed delays is stable with generalized H2 performance*γ*, if

, *l* ∈ *L* , *i* =1,2,3,4satisfying the following LMIs:

*l*

e

ê ú - + <Î Î

*I EE i Ml l L I*

2 2 <sup>2</sup> \* 0 0 ( ),

*<sup>F</sup> Pj* <sup>+</sup> *Yl* <sup>3</sup> *Yl* <sup>4</sup>

\* *Σl*<sup>1</sup> *Σl*<sup>2</sup> *Σl*<sup>3</sup> *Σl*<sup>4</sup> *Xl*<sup>1</sup> − *B*1*<sup>i</sup>*

\* \* *Σl*<sup>5</sup> *Σl*<sup>6</sup> *Σl*<sup>7</sup> *Xl*<sup>2</sup> 0 \* \* \* *Σl*<sup>8</sup> *Σl*<sup>9</sup> *Xl*<sup>3</sup> 0 \* \* \* \* *Σl*<sup>10</sup> *Xl*<sup>4</sup> 0 \* \* \* \* \* *Γ<sup>l</sup>* 0 \* \* \* \* \* \* −*εlI*

*l i i l il T ll l*

é ù ---

ë û -

*P CF B M B HF*

¥ < (41)

, *Q*, *Z* >0, the nonsingular matrix *F* and ma‐

0 ( ), *ill* ¡< Î Î *i Ml l L* (43)

0 ( ),( , ) *ilj* ¡ < Î ÎQ *i Ml lj* (44)

*HlF*

+ *Di* 0 0

(42)

$$
\begin{bmatrix}
\* & \Sigma\_{l1} & \Sigma\_{l2} & \Sigma\_{l3} & \Sigma\_{l4} & X\_{l1} & -B\_{1i}H\_{l}F \\
\* & \* & \Sigma\_{l5} & \Sigma\_{l6} & \Sigma\_{l7} & X\_{l2} & 0 \\
\* & \* & \* & \Sigma\_{l8} & \Sigma\_{l9} & X\_{l3} & 0 \\
\* & \* & \* & \* & \Sigma\_{l10} & X\_{l4} & 0 \\
\* & \* & \* & \* & \* & \Gamma\_{l} & 0 \\
\* & \* & \* & \* & \* & \* & -\varepsilon\_{l}I
\end{bmatrix}
$$

with

$$\begin{aligned} \mathcal{T}\_{i\uparrow} &= P + Y\_{\mathcal{I}1} + A\_{i}F - B\_{i\uparrow}M\_{\mathcal{I}\uparrow}, \; \mathcal{T}\_{\mathcal{I}} = (-\overset{\sim}{\sum}d\mu\_{d})^{-1}Z + c\_{i}E\_{i\uparrow}^{\top}E\_{\mathcal{I}},\\ \Sigma\_{i\downarrow} &= \overline{\mu}Q - 2P + \text{He}\{\overline{\mu}X\_{\mathcal{I}1} - Y\_{\mathcal{I}1}\}, \; \Sigma\_{\mathcal{I}2} = -X\_{\mathcal{I}1} + X\_{\mathcal{I}2}^{\top} - Y\_{\mathcal{I}2}, \; \Sigma\_{\mathcal{I}3} = X\_{\mathcal{I}3} - Y\_{\mathcal{I}1}^{\top} - Y\_{\mathcal{I}3},\\ \Sigma\_{\mathcal{I}4} &= X\_{\mathcal{I}4}^{\top} + Y\_{\mathcal{I}4}, \; \Sigma\_{\mathcal{I}5} = \frac{1}{\mu}Q - \text{He}\{X\_{\mathcal{I}2}\}, \; \Sigma\_{\mathcal{I}6} = -X\_{\mathcal{I}3}^{\top} - Y\_{\mathcal{I}2}, \; \Sigma\_{\mathcal{I}7} = X\_{\mathcal{I}4},\\ \Sigma\_{\mathcal{I}3} &= \sum\_{d=1}^{a}\mu\_{d}d\underline{Z} - \text{He}\{Y\_{\mathcal{I}1}\}, \; \Sigma\_{\mathcal{I}9} = Y\_{\mathcal{I}4}^{\top}, \; \Sigma\_{\mathcal{I}10} = -I. \end{aligned}$$

where

*<sup>μ</sup>*¯=∑ *d*=1 *∞*

and Fig.2

*μd* =2−<sup>3</sup> <sup>&</sup>lt;∑

*d*=1 *∞*

*R* <sup>1</sup> :*if s*(*t*) *is* −1, *then*

*R* <sup>2</sup> :*if s*(*t*) *is* 1, *then*

*<sup>x</sup>*(*<sup>t</sup>* <sup>+</sup> 1)= *<sup>A</sup>*1*x*(*t*) <sup>+</sup> *Ad* <sup>1</sup>∑

*<sup>x</sup>*(*<sup>t</sup>* <sup>+</sup> 1)= *<sup>A</sup>*2*x*(*t*) <sup>+</sup> *Ad* <sup>2</sup>∑

*<sup>A</sup>*<sup>1</sup> <sup>=</sup> 0.9 0.1

*<sup>A</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>0.9 0.1

*<sup>C</sup>*<sup>1</sup> <sup>=</sup>*C*<sup>2</sup> <sup>=</sup> <sup>−</sup>0.1 <sup>0</sup>

*H*<sup>1</sup> =*H*<sup>2</sup> = 0.1 0 ,

*Ω*<sup>1</sup> ={*s*(*t*)| −1≤*s*(*t*)≤0}, *Ω*<sup>2</sup> ={*s*(*t*)|0≤*s*(*t*)≤1}

*<sup>P</sup>*<sup>1</sup> <sup>=</sup> 0.1944 0.0248

*<sup>Z</sup>* <sup>=</sup> 0.0048 0.0019

*K*<sup>2</sup> = −0.0171 0.1685 .

*V* (*t*)=0.1cos(*t*)×exp( - 0.05*t*).

*e*<sup>1</sup> =10,*e*<sup>2</sup> =11,

The subspaces can be described by

*z*(*t*)=*C*1*x*(*t*) + *B*21*u*(*t*)

*z*(*t*)=*C*2*x*(*t*) + *B*22*u*(*t*)

*d*=1 *∞*

*d*=1 *∞*

<sup>−</sup>0.5 <sup>1</sup> , *<sup>A</sup>*1*<sup>d</sup>* <sup>=</sup> 0.1 <sup>−</sup>0.5

<sup>−</sup>0.5 <sup>1</sup> , *<sup>A</sup>*2*<sup>d</sup>* <sup>=</sup> <sup>−</sup>0.1 <sup>−</sup>0.5

<sup>0</sup> <sup>−</sup>0.2 , *<sup>B</sup>*<sup>21</sup> <sup>=</sup> *<sup>B</sup>*<sup>22</sup> <sup>=</sup> <sup>1</sup>

with the H2 performance index *γ*min =0.11, we solve (42)-(44) and obtain

0.0248 0.3342 , *<sup>P</sup>*<sup>2</sup> <sup>=</sup> 0.1951 0.0252

0.0019 0.1275 , *<sup>F</sup>* <sup>=</sup> 0.3939 0.1516

*μd x*(*t* −*d*) + *B*11*u*(*t*) + *D*1*V* (*t*)

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays

*μd x*(*t* −*d*) + *B*12*u*(*t*) + *D*2*v*(*t*)

<sup>0</sup> <sup>0</sup> , *<sup>B</sup>*<sup>11</sup> <sup>=</sup> *<sup>B</sup>*<sup>12</sup> <sup>=</sup> <sup>1</sup>

<sup>0</sup> <sup>0</sup> , *<sup>D</sup>*<sup>1</sup> <sup>=</sup>*D*<sup>2</sup> <sup>=</sup> 0.1

Choosing the constants *<sup>c</sup>* =0.9, *μd* =2−3−*<sup>d</sup>* , *<sup>d</sup>* =10 we easily find that

*dμd* =2< + *∞*, which satisfies the convergence condition (2).

Simulation results with the above solutions for the H2 controller designs are shown Fig.1

0 ,

<sup>0</sup> , *<sup>E</sup>*<sup>1</sup> <sup>=</sup> *<sup>E</sup>*<sup>2</sup> <sup>=</sup> 0.05 <sup>0</sup> ,

0.0252 0.3358 , *<sup>Q</sup>* <sup>=</sup> 0.2876 0.0746

0.0476 0.6285 , *<sup>K</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>0.0223 0.1702 ,

0.0746 0.1636 ,

http://dx.doi.org/10.5772/51778

67

0 ,

### **4. Numerical Examples**

In this section, we will present two simulation examples to illustrate the controller design method developed in this paper.

*Example* **1.** Consider the following modified Henon system with infinite distributed delays and external disturbance

$$\begin{aligned} \mathbf{x}\_1(t+1) &= -\langle \mathbf{x}\_1(t) + (\mathbf{l} - \mathbf{c}) \sum\_{d=1}^{\ast \alpha} \mu\_d \mathbf{x}\_1(t - d) \rangle^2 + 0.1 \mathbf{x}\_2(t) - 0.5 \sum\_{d=1}^{\ast \alpha} \mu\_d \mathbf{x}\_2(t - d) + u(t) + 0.1 \nu(t) \\ \mathbf{x}\_2(t+1) &= \mathbf{x}\_2(t) - 0.5 \mathbf{x}\_1(t) \\ \mathbf{z}\_1(t) &= (\mathbf{l} - \mathbf{c}) \mathbf{x}\_1(t) + u(t) \\ \mathbf{z}\_2(t) &= 0.2 \mathbf{x}\_2(t) \end{aligned} \tag{48}$$

where the constant *c* ∈ 0,1 is the retarded coefficient.

Let*s*(*t*)=*cx*1(*t*) <sup>+</sup> (1−*c*)∑ *d*=1 +*∞ μd <sup>x</sup>*1(*<sup>t</sup>* <sup>−</sup>*d*). Assume that*s*(*t*)<sup>∈</sup> <sup>−</sup>1,1 . The nonlinear term *<sup>s</sup>* <sup>2</sup> (*t*) can be exactly represented as

*s* 2 (*t*)=*h*1(*s*(*t*))(−1)*s*(*t*) + *h*2(*s*(*t*))(1)*s*(*t*)

where the*h*1(*s*(*t*)), *h*2(*s*(*t*))∈ 0,1 , and*h*1(*s*(*t*)) + *h*2(*s*(*t*))=1. By solving the equations, the membership functions *h*1(*s*(*t*))and *h*2(*s*(*t*))are obtained as

$$h\_1(\mathbf{s}(t)) = \frac{1}{2}(1 - \mathbf{s}(t)), \qquad h\_2(\mathbf{s}(t)) = \frac{1}{2}(1 + \mathbf{s}(t))$$

It can be seen from the aforementioned expressions that *h*1(*s*(*t*))=1 and *h*2(*s*(*t*))=0 when *s*(*t*)= −1, and that *h*1(*s*(*t*))=0 and *h*2(*s*(*t*))=1 when*s*(*t*)=1. Then the nonlinear system in (48) can be approximately represented by the following T-S fuzzy model:

$$\begin{aligned} \text{R}^{1}: & \text{if } \text{s}(t) \text{ is } -1, \text{ then} \\ & \quad \text{x}(t+1) = A\_{1}\text{x}(t) + A\_{d1} \sum\_{d=1}^{\omega} \mu\_{d} \text{x}(t-d) + B\_{11}\mu(t) + D\_{1}V(t) \\ & \quad \text{z}(t) = C\_{1}\text{x}(t) + B\_{21}\mu(t) \\ & \quad \text{R}^{2}: \text{if } \text{s}(t) \text{ is } 1, \text{ then} \\ & \quad \text{x}(t+1) = A\_{2}\text{x}(t) + A\_{d2} \sum\_{d=1}^{\omega} \mu\_{d} \text{x}(t-d) + B\_{12}\mu(t) + D\_{2}v(t) \\ & \quad \text{z}(t) = C\_{2}\text{x}(t) + B\_{22}\mu(t) \end{aligned}$$

where

**4. Numerical Examples**

66 Advances in Discrete Time Systems

method developed in this paper.

and external disturbance

2 21 1 1 2 2

+= -

*xt xt xt z t cx t ut*

( 1) ( ) 0.5 ( ) ( ) (1 ) ( ) ( )

=- +

( ) 0.2 ( )

*zt xt*

=

Let*s*(*t*)=*cx*1(*t*) <sup>+</sup> (1−*c*)∑

exactly represented as

*s* 2

*<sup>h</sup>*1(*s*(*t*))= <sup>1</sup>

In this section, we will present two simulation examples to illustrate the controller design

*Example* **1.** Consider the following modified Henon system with infinite distributed delays

2

1 1

( 1) { ( ) (1 ) ( )} 0.1 ( ) 0.5 ( ) ( ) 0.1 ( )

+¥ +¥ = = + =- + - - + - - + +

å å

*d d d d x t cx t c x t d x t x t d u t v t*

where the*h*1(*s*(*t*)), *h*2(*s*(*t*))∈ 0,1 , and*h*1(*s*(*t*)) + *h*2(*s*(*t*))=1. By solving the equations, the

It can be seen from the aforementioned expressions that *h*1(*s*(*t*))=1 and *h*2(*s*(*t*))=0 when *s*(*t*)= −1, and that *h*1(*s*(*t*))=0 and *h*2(*s*(*t*))=1 when*s*(*t*)=1. Then the nonlinear system in (48)

 m

*μd <sup>x</sup>*1(*<sup>t</sup>* <sup>−</sup>*d*). Assume that*s*(*t*)<sup>∈</sup> <sup>−</sup>1,1 . The nonlinear term *<sup>s</sup>* <sup>2</sup>

(48)

(*t*) can be

11 1 2 2

m

where the constant *c* ∈ 0,1 is the retarded coefficient.

membership functions *h*1(*s*(*t*))and *h*2(*s*(*t*))are obtained as

<sup>2</sup> (1 <sup>+</sup> *<sup>s</sup>*(*t*))

can be approximately represented by the following T-S fuzzy model:

*d*=1 +*∞*

(*t*)=*h*1(*s*(*t*))(−1)*s*(*t*) + *h*2(*s*(*t*))(1)*s*(*t*)

<sup>2</sup> (1−*s*(*t*)), *<sup>h</sup>*2(*s*(*t*))= <sup>1</sup>

$$\begin{aligned} \label{eq:10} A\_1 &= \begin{bmatrix} 0.9 & 0.1 \\ -0.5 & 1 \end{bmatrix} \begin{array}{c} A\_{1d} = \begin{bmatrix} 0.1 & -0.5 \\ 0 & 0 \end{bmatrix} \begin{array}{c} B\_{11} = B\_{12} = \begin{bmatrix} 1 \\ 0 \end{bmatrix} \\\ A\_2 &= \begin{bmatrix} -0.9 & 0.1 \\ -0.5 & 1 \end{bmatrix} \begin{array}{c} A\_{2d} = \begin{bmatrix} -0.1 & -0.5 \\ 0 & 0 \end{bmatrix} \begin{array}{c} D\_1 = D\_2 = \begin{bmatrix} 0.1 \\ 0 \end{bmatrix} \end{array} \\\hline \begin{array}{c} C\_1 = C\_2 = \begin{bmatrix} -0.1 & 0 \\ 0 & -0.2 \end{bmatrix} \end{array} \\\hline \begin{array}{c} C\_1 = C\_2 = \begin{bmatrix} -0.1 & 0 \\ 0 & -0.2 \end{bmatrix} \end{array} \\\hline \begin{array}{c} H\_1 = H\_2 = \begin{bmatrix} 0.1 & 0 \end{bmatrix} \end{array} \\\hline \end{aligned} \\\hline \begin{array}{c} e\_1 = 10, e\_2 = 11, \\\ V(t) = 0.1 \cos(t) \times \exp(-0.05t). \end{array} \end{aligned}$$

The subspaces can be described by

*Ω*<sup>1</sup> ={*s*(*t*)| −1≤*s*(*t*)≤0}, *Ω*<sup>2</sup> ={*s*(*t*)|0≤*s*(*t*)≤1}

Choosing the constants *<sup>c</sup>* =0.9, *μd* =2−3−*<sup>d</sup>* , *<sup>d</sup>* =10 we easily find that *<sup>μ</sup>*¯=∑ *d*=1 *∞ μd* =2−<sup>3</sup> <sup>&</sup>lt;∑ *d*=1 *∞ dμd* =2< + *∞*, which satisfies the convergence condition (2).

with the H2 performance index *γ*min =0.11, we solve (42)-(44) and obtain

$$\begin{aligned} &P\_1 = \begin{bmatrix} 0.1944 & 0.0248 \\ 0.0248 & 0.3342 \end{bmatrix}, \ P\_2 = \begin{bmatrix} 0.1951 & 0.0252 \\ 0.0252 & 0.3358 \end{bmatrix}, \ Q = \begin{bmatrix} 0.2876 & 0.0746 \\ 0.0746 & 0.1636 \end{bmatrix} \\ &Z = \begin{bmatrix} 0.0048 & 0.0019 \\ 0.0019 & 0.1275 \end{bmatrix}, \ F = \begin{bmatrix} 0.3939 & 0.1516 \\ 0.0476 & 0.6285 \end{bmatrix} \quad K\_1 = \begin{bmatrix} -0.0223 & 0.1702 \end{bmatrix} \\ &K\_2 = \begin{bmatrix} -0.0171 & 0.1685 \end{bmatrix} \end{aligned}$$

Simulation results with the above solutions for the H2 controller designs are shown Fig.1 and Fig.2

*h*2(*s*(*t*))={

The subspaces are given as shown in Fig.3

**Figure 3.** Membership functions and partition of subspaces.

*<sup>A</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>0.986 0.1

*<sup>A</sup>*<sup>2</sup> <sup>=</sup> 0.5 <sup>−</sup>0.6

*<sup>C</sup>*<sup>1</sup> <sup>=</sup> <sup>−</sup>0.02 <sup>0</sup>

<sup>−</sup>0.5 <sup>1</sup> , *<sup>A</sup>*1*<sup>d</sup>* <sup>=</sup> <sup>−</sup>0.1 <sup>−</sup>0.5

0.6 0.5 , *<sup>A</sup>*2*<sup>d</sup>* <sup>=</sup> <sup>−</sup>0.05 <sup>−</sup>0.6

<sup>0</sup> <sup>−</sup>0.1 , *<sup>C</sup>*<sup>2</sup> <sup>=</sup> <sup>−</sup>0.1 <sup>0</sup>

*v*(*t*)=0.1cos(*t*)×exp( - 0.05*t*).

*<sup>h</sup>*1(*s*(*t*))={ <sup>1</sup> *<sup>s</sup>*(*t*)<sup>∈</sup> <sup>−</sup>3, <sup>−</sup><sup>1</sup> ,

−0.5*s*(*t*) + 0.5 *s*(*t*)∈ −1, 1 .

0.5*s*(*t*) + 0.5 *s*(*t*)∈ −1, 1 , 1 *s*(*t*)∈ 1, 3 .

*E*<sup>1</sup> = *E*<sup>2</sup> = 0.05 0 , *H*<sup>1</sup> =*H*<sup>2</sup> = 0.1 0 ,*e*<sup>1</sup> =10,*e*<sup>2</sup> =11, *e*<sup>3</sup> =12,

We expanded the state space from [-1,1] to [− 3,3], the membership functions are given as

Using the Theorem 2 and Corollary 1, respectively, the achievable minimum performance

index for the H2 controller can be obtained and is summarized in Table 1.

**Approach Performance**

<sup>0</sup> <sup>0</sup> , *<sup>B</sup>*<sup>11</sup> <sup>=</sup> <sup>0</sup>

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays

<sup>0</sup> <sup>0</sup> , *<sup>D</sup>*1= *<sup>D</sup>*<sup>2</sup> <sup>=</sup> 0.1

<sup>0</sup> <sup>−</sup>0.3 , *<sup>B</sup>*<sup>21</sup> <sup>=</sup> *<sup>B</sup>*<sup>22</sup> <sup>=</sup> <sup>1</sup>

0.5 , *<sup>B</sup>*<sup>12</sup> <sup>=</sup> <sup>1</sup>

0 ,

0 ,

0 ,

http://dx.doi.org/10.5772/51778

69

**Figure 1.** The state evolution x1(t) of controlled system.

**Figure 2.** The state evolution *x*2(*t*) of controlled systems.

*Example* **2.** Consider a fuzzy discrete time system with the same form as in Example, but with different system matrices given by

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays http://dx.doi.org/10.5772/51778 69

$$\begin{aligned} &A\_{1} = \begin{bmatrix} -0.986 & 0.1 \\ -0.5 & 1 \end{bmatrix} & A\_{1d} = \begin{bmatrix} -0.1 & -0.5 \\ 0 & 0 \end{bmatrix}, \ B\_{11} = \begin{bmatrix} 0 \\ 0.5 \end{bmatrix}, \ B\_{12} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \\\ &A\_{2} = \begin{bmatrix} 0.5 & -0.6 \\ 0.6 & 0.5 \end{bmatrix}, \ A\_{2d} = \begin{bmatrix} -0.05 & -0.6 \\ 0 & 0 \end{bmatrix}, \ D\_{1} = D\_{2} = \begin{bmatrix} 0.1 \\ 0 \end{bmatrix}, \\\ &C\_{1} = \begin{bmatrix} -0.02 & 0 \\ 0 & -0.1 \end{bmatrix}, \ C\_{2} = \begin{bmatrix} -0.1 & 0 \\ 0 & -0.3 \end{bmatrix}, \ B\_{21} = B\_{22} = \begin{bmatrix} 1 \\ 0 \end{bmatrix}, \\\ &E\_{1} = E\_{2} = \begin{bmatrix} 0.05 & 0 \end{bmatrix}, \ H\_{1} = H\_{2} = \begin{bmatrix} 0.1 & 0 \end{bmatrix}, e\_{1} = 10.e\_{2} = 11, \ e\_{3} = 12, \\\ v(t) = 0.1\cos(t) \times \exp(-0.05t). \end{aligned}$$

We expanded the state space from [-1,1] to [− 3,3], the membership functions are given as

$$h\_1(\mathbf{s}(t)) = \begin{vmatrix} 1 & s(t) \in \mathbb{I} - \mathbf{3} \ -\mathbf{1} \ \mathbf{J} \end{vmatrix}$$

$$h\_2(\mathbf{s}(t)) = \begin{vmatrix} 0.5 \mathbf{s}(t) + 0.5 & s(t) \in \mathbb{I} - \mathbf{1} \ \mathbf{1} \ \mathbf{J} \end{vmatrix}$$

$$h\_2(\mathbf{s}(t)) = \begin{vmatrix} 0.5 \mathbf{s}(t) + 0.5 & s(t) \in \mathbb{I} - \mathbf{1} \ \mathbf{1} \ \mathbf{J} \\ 1 & s(t) \in \mathbb{I} \ \mathbf{1} \ \mathbf{3} \ \mathbf{J} \end{vmatrix}$$

The subspaces are given as shown in Fig.3

**Figure 3.** Membership functions and partition of subspaces.

Using the Theorem 2 and Corollary 1, respectively, the achievable minimum performance index for the H2 controller can be obtained and is summarized in Table 1.

**Figure 1.** The state evolution x1(t) of controlled system.

68 Advances in Discrete Time Systems

**Figure 2.** The state evolution *x*2(*t*) of controlled systems.

with different system matrices given by

*Example* **2.** Consider a fuzzy discrete time system with the same form as in Example, but


**Table 1.** Comparison for generalized H2 performance.

By using the LMI toolbox, we have

$$\begin{aligned} \;\_{P\_1=\begin{bmatrix} 1.5359 & 0.5771 \\ 0.5771 & 1.4293 \end{bmatrix}} \quad \;\_{P\_2=\begin{bmatrix} 1.5254 & 0.6540 \\ 0.6540 & 1.5478 \end{bmatrix}} \quad \;\_{P\_3=\begin{bmatrix} 1.2754 & 0.5634 \\ 0.5634 & 1.4983 \end{bmatrix}} \\\;\_{Q}=\begin{bmatrix} 1.8101 & 0.1568 \\ 0.1568 & 0.5915 \end{bmatrix} \; \_{P} &= \begin{bmatrix} 0.0399 & 0.0285 \\ 0.0285 & 0.4640 \end{bmatrix} \; \_{P} = \begin{bmatrix} 3.1076 & 0.7119 \\ 0.8671 & 2.5352 \end{bmatrix} \\\;\_{P\_4=\begin{bmatrix} 0.0003 & -0.22971 \end{bmatrix} \; \_{P\_2=\begin{bmatrix} 0.1311 & -0.0371 \end{bmatrix} \; \_{P\_3=\begin{bmatrix} -0.1125 & -0.0005 \end{bmatrix} \end{aligned} $$

**Figure 5.** Trajectories from two initial conditions

Jun-min Li1\*, Jiang-rong Li1,2 and Zhi-le Xia1,3

\*Address all correspondence to: jmli@mail.xidian.edu.cn

1 Department of Applied Mathematics, Xidian University, China

2 College of mathematics & Computer Science, Yanan University, China

3 School of Mathematics and Information Engineering, Taizhou University, China

This paper presents delay-dependent analysis and synthesis method for discrete-time T-S fuzzy systems with infinite-distributed delays. Based on a novel DDPLKF, the proposed sta‐ bility and stabilization results are less conservative than the existing results based on the CLKF and delay independent method. The non-fragile stated feedback controller law has been developed so that the closed-loop fuzzy system is generalized H2 stable. It is also shown that the controller gains can be determined by solving a set of LMIs. A simulation

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays

http://dx.doi.org/10.5772/51778

71

example was presented to demonstrate the advantages of the proposed approach.

**5. Conclusions**

**Author details**

The simulation results with the initial conditions are shown Fig.4 and Fig.5

**Figure 4.** Trajectories from two initial conditions

Delay-Dependent Generalized H2 Control for Discrete-Time Fuzzy Systems with Infinite-Distributed Delays http://dx.doi.org/10.5772/51778 71

**Figure 5.** Trajectories from two initial conditions

#### **5. Conclusions**

Common Lyapunov function based generalized H2 performance

Piecewise Lyapunov function based generalized H2 performance

**Table 1.** Comparison for generalized H2 performance.

*<sup>P</sup>*<sup>1</sup> <sup>=</sup> 1.5359 0.5771

*<sup>Q</sup>* <sup>=</sup> 1.8101 0.1568

**Figure 4.** Trajectories from two initial conditions

0.5771 1.4293 , *<sup>P</sup>*<sup>2</sup> <sup>=</sup> 1.5254 0.6540

0.1568 0.5915 , *<sup>Z</sup>* <sup>=</sup> 0.0399 0.0285

The simulation results with the initial conditions are shown Fig.4 and Fig.5

0.6540 1.5478 , *<sup>P</sup>*<sup>3</sup> <sup>=</sup> 1.2754 0.5634

0.0285 0.4640 , *<sup>F</sup>* <sup>=</sup> 3.1076 0.7119

*K*<sup>1</sup> = 0.0003 −0.2297 , *K*<sup>2</sup> = 0.1311 −0.0371 , *K*<sup>3</sup> = −0.1125 −0.0005 .

0.5634 1.4983 ,

0.8671 2.5352 ,

By using the LMI toolbox, we have

γmin=0,4586

γmin=0,3975

(Theorem 2)

70 Advances in Discrete Time Systems

( Corollary1)

This paper presents delay-dependent analysis and synthesis method for discrete-time T-S fuzzy systems with infinite-distributed delays. Based on a novel DDPLKF, the proposed sta‐ bility and stabilization results are less conservative than the existing results based on the CLKF and delay independent method. The non-fragile stated feedback controller law has been developed so that the closed-loop fuzzy system is generalized H2 stable. It is also shown that the controller gains can be determined by solving a set of LMIs. A simulation example was presented to demonstrate the advantages of the proposed approach.

#### **Author details**

Jun-min Li1\*, Jiang-rong Li1,2 and Zhi-le Xia1,3

\*Address all correspondence to: jmli@mail.xidian.edu.cn


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**Section 2**

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**Chapter 4**

**Discrete-Time Model Predictive Control**

More than 25 years after model predictive control (MPC) or receding horizon control (RHC) appeared in industry as an effective tool to deal with multivariable constrained control problems, a theoretical basis for this technique has started to emerge, see [1-3] for reviews of

The focus of this chapter is on MPC of constrained dynamic systems, both linear and nonlin‐ ear, to illuminate the ability of MPC to handle constraints that makes it so attractive to in‐ dustry. We first give an overview of the origin of MPC and introduce the definitions, characteristics, mathematical formulation and properties underlying the MPC. Furthermore, MPC methods for linear or nonlinear systems are developed by assuming that the plant un‐ der control is described by a discrete-time one. Although continuous-time representation would be more natural, since the plant model is usually derived by resorting to first princi‐ ples equations, it results in a more difficult development of the MPC control law, since it in principle calls for the solution of a functional optimization problem. As a matter of fact, the performance index to be minimized is defined in a continuous-time setting and the overall optimization procedure is assumed to be continuously repeated after any vanishingly small sampling time, which often turns out to be a computationally intractable task. On the con‐ trary, MPC algorithms based on discrete-time system representation are computationally simpler. The system to be controlled which usually described, or approximated by an ordi‐ nary differential equation is usually modeled by a difference equation in the MPC literature since the control is normally piecewise constant. Hence, we concentrate our attention from

By and large, the main disadvantage of the MPC is that it cannot be able of explicitly dealing with plant model uncertainties. For confronting such problems, several robust model pre‐

> © 2012 Dai et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use,

© 2012 Dai et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

distribution, and reproduction in any medium, provided the original work is properly cited.

Li Dai, Yuanqing Xia, Mengyin Fu and

Additional information is available at the end of the chapter

now onwards on results related to discrete-time systems.

Magdi S. Mahmoud

**1. Introduction**

results in this area.

http://dx.doi.org/10.5772/51122
