**Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem**

Yassine Manai and Mohamed Benrejeb

Additional information is available at the end of the chapter

http://dx.doi.org/10.5772/48075

## **1. Introduction**

16 Will-be-set-by-IN-TECH

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[14] M. Teixeira and S. H. Zak, "Stabilizing controller design for uncertain nonlinear systems

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*Contr.*, vol. 41, pp. 1003–1008, 1996.

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Fuzzy control systems have experienced a big growth of industrial applications in the recent decades, because of their reliability and effectiveness. Many researches are investigated on the Takagi-Sugeno models [1], [2] and [3] last decades. Two classes of Lyapunov functions are used to analysis these systems: quadratic Lyapunov functions and non-quadratic Lyapunov ones which are less conservative than first class. Many researches are investigated with non-quadratic Lyapunov functions [4]-[6], [7].

Recently, Takagi–Sugeno fuzzy model approach has been used to examine nonlinear systems with time-delay, and different methodologies have been proposed for analysis and synthesis of this type of systems [1]-[11], [12]-[13]. Time delay often occurs in many dynamical systems such as biological systems, chemical system, metallurgical processing system and network system. Their existences are frequently a cause of infeasibility and poor performances.

The stability approaches are divided into two classes in term of delay. The fist one tries to develop delay independent stability criteria. The second class depends on the delay size of the time delay, and it called delay dependent stability criteria. Generally, delay dependent class gives less conservative stability criteria than independent ones.

Two classes of Lyapunov-Razumikhin function are used to analysis these systems: quadratic Lyapunov-Razumikhin function and non-quadratic Lyapunov- Razumikhin ones. The use of first class brings much conservativeness in the stability test. In order to reduce the conservatism entailed in the previous results using quadratic function.

As the information about the time derivatives of membership function is considered by the PDC fuzzy controller, it allows the introduction of slack matrices to facilitate the stability

analysis. The relationship between the membership function of the fuzzy model and the fuzzy controllers is used to introduce some slack matrix variables. The boundary information of the membership functions is brought to the stability condition and thus offers some relaxed stability conditions [5].

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 483

1

(3)

(4)

 

1

*l*

*h z t dx t dx t*

0

*zt xt dt dt*

for all *t*.

*p i ij j j w zt M z t* 

1 2 *<sup>p</sup> zt z tz t z t*

,

(2)

 , ,0 ,

*i i i i i ii xt h zt A A D D x t B B ut x t t*

*xt t t* 

The final outputs of the fuzzy systems are:

The term *Mi j* <sup>1</sup> *z t* is the grade of membership of *<sup>j</sup> z t* in *Mi*<sup>1</sup>

0, 1,2, ,

*ii i*

0

*w zt i r*

1

*h zt i r*

*h zt*

0, 1,2, ,

*h zt w zt w zt*

1

The time derivative of premise membership functions is given by:

for all *t*.

 

*<sup>s</sup> <sup>i</sup> i il il*

Consider a PDC fuzzy controller based on the derivative membership function and given by

 1 1

The fuzzy controller design consists to determine the local feedback gains ,and *i m F K* in the consequent parts. The state variables are determined by an observer which detailed in next

*u t h z t Fx t h z t K x t* 

*ii m m*

*h zt*

1

*r r*

*i m*

*r k k*

*i*

*r*

1 *r i i*

Since

*r i i i*

we have

the equation (4)

section.

*r i i i*

 

 1

We have the following property:

*h zt*

<sup>1</sup>

*w zt*

where

In this chapter, a new stability conditions for time-delay Takagi-Sugeno fuzzy systems by using fuzzy Lyapunov-Razumikhin function are presented. In addition, a new stabilization conditions for Takagi Sugeno time-delay uncertain fuzzy models based on the use of fuzzy Lyapunov function are presented. This criterion is expressed in terms of Linear Matrix Inequalities (LMIs) which can be efficiently solved by using various convex optimization algorithms [8],[9]. The presented methods are less conservative than existing results.

The organization of the chapter is as follows. In section 2, we present the system description and problem formulation and we give some preliminaries which are needed to derive results. Section 3 will be concerned to stability and stabilization analysis for T-S fuzzy systems with Parallel Distributed Controller (PDC). An observer approach design is derived to estimate state variables. Section 5 will be concerned to stabilization analysis for timedelay T-S fuzzy systems based on Razumikhin theorem. Next, a new robust stabilization condition for uncertain system with time delay is given in section 6. Illustrative examples are given in section 7 for a comparison of previous results to demonstrate the advantage of proposed method. Finally section 8 makes conclusion.

*Notation*: Throughout this chapter, a real symmetric matrix 0 *S* denotes *S* being a positive definite matrix. The superscript ''T'' is used for the transpose of a matrix.

## **2. System description and preliminaries**

Consider an uncertain T-S fuzzy continuous model with time-delay for a nonlinear system as follows:

$$\begin{aligned} \text{IF } z\_1(t) \text{ is } M\_{\vert \cdot \vert} \text{ and } \dots \text{ and } z\_p(t) \text{ is } M\_{\vert \cdot \vert}\\ \text{THEN} \quad \begin{cases} \dot{\boldsymbol{x}}\left(t\right) = \left(A\_i + \Delta A\_i\right) \mathbf{x}\left(t\right) + \left(D\_i + \Delta D\_i\right) \mathbf{x}\left(t - \tau\_i\left(t\right)\right) + \left(B\_i + \Delta B\_i\right) \boldsymbol{u}\left(t\right) \\ \mathbf{x}\left(t\right) = \boldsymbol{\phi}\left(t\right), t \in \left[-\tau, 0\right] \end{cases} \end{aligned} \tag{1}$$

where 1,2, , , 1,2, , *Mij i rj p* is the fuzzy set and *r* is the number of model rules; *<sup>n</sup> x t* is the state vector, *<sup>m</sup> u t* is the input vector, *n n Ai* , *n n Di* , *n m <sup>i</sup> <sup>B</sup>* , and <sup>1</sup> , , *<sup>p</sup> zt zt* are known premise variables, *t* is a continuous vector-valued initial function on ,0 ; the time-delay *t* may be unknown but is assumed to be smooth function of time.. , *Ai i D* and *<sup>i</sup> B* are time-varying matrices representing parametric uncertainties in the plant model. These uncertainties are admissibly norm-bounded and structured.

$$0 \le \mathfrak{r}\left(t\right) \le \mathfrak{r}\_{\prime\prime} \qquad \mathfrak{r}\left(t\right) \le d \prec 1\_{\prime\prime}$$

where 0 and *d* are two scalars. The final outputs of the fuzzy systems are:

$$\mathbf{x}\left(t\right) = \sum\_{i=1}^{r} h\_i \left(z\left(t\right)\right) \left\{ \left(A\_i + \Delta A\_i\right) \mathbf{x}\left(t\right) + \left(D\_i + \Delta D\_i\right) \mathbf{x}\left(t - \tau\_i\left(t\right)\right) + \left(B\_i + \Delta B\_i\right) u\left(t\right) \right\} \tag{2}$$

$$\mathbf{x}\left(t\right) = \phi\left(t\right), \quad t \in \left[-\tau, 0\right],$$

where

482 Fuzzy Controllers – Recent Advances in Theory and Applications

proposed method. Finally section 8 makes conclusion.

**2. System description and preliminaries** 

1 1

*THEN*

*IF z t is M and and z t is M*

and <sup>1</sup> , , *<sup>p</sup> zt zt* are known premise variables,

,0 ; the time-delay

and *d* are two scalars.

*xt t t*

, ,0

as follows:

function on

structured.

where 0 

some relaxed stability conditions [5].

analysis. The relationship between the membership function of the fuzzy model and the fuzzy controllers is used to introduce some slack matrix variables. The boundary information of the membership functions is brought to the stability condition and thus offers

In this chapter, a new stability conditions for time-delay Takagi-Sugeno fuzzy systems by using fuzzy Lyapunov-Razumikhin function are presented. In addition, a new stabilization conditions for Takagi Sugeno time-delay uncertain fuzzy models based on the use of fuzzy Lyapunov function are presented. This criterion is expressed in terms of Linear Matrix Inequalities (LMIs) which can be efficiently solved by using various convex optimization

The organization of the chapter is as follows. In section 2, we present the system description and problem formulation and we give some preliminaries which are needed to derive results. Section 3 will be concerned to stability and stabilization analysis for T-S fuzzy systems with Parallel Distributed Controller (PDC). An observer approach design is derived to estimate state variables. Section 5 will be concerned to stabilization analysis for timedelay T-S fuzzy systems based on Razumikhin theorem. Next, a new robust stabilization condition for uncertain system with time delay is given in section 6. Illustrative examples are given in section 7 for a comparison of previous results to demonstrate the advantage of

*Notation*: Throughout this chapter, a real symmetric matrix 0 *S* denotes *S* being a positive

Consider an uncertain T-S fuzzy continuous model with time-delay for a nonlinear system

where 1,2, , , 1,2, , *Mij i rj p* is the fuzzy set and *r* is the number of model rules;

function of time.. , *Ai i D* and *<sup>i</sup> B* are time-varying matrices representing parametric uncertainties in the plant model. These uncertainties are admissibly norm-bounded and

0 , 1,

 *t td*

*xt A xt D t B ut*

*ii i i ii i*

*A Dx t B*

, *n n Di*

*t* is a continuous vector-valued initial

*t* may be unknown but is assumed to be smooth

, *n m*

*<sup>i</sup> <sup>B</sup>* ,

(1)

definite matrix. The superscript ''T'' is used for the transpose of a matrix.

*i p ip*

*<sup>n</sup> x t* is the state vector, *<sup>m</sup> u t* is the input vector, *n n Ai*

 

algorithms [8],[9]. The presented methods are less conservative than existing results.

$$z\left(t\right) = \left[z\_1\left(t\right)z\_2\left(t\right)...z\_p\left(t\right)\right],$$

$$M\_i\left(z\left(t\right)\right) = w\_i\left(z\left(t\right)\right) \Big/ \sum\_{i=1}^r w\_i\left(z\left(t\right)\right), w\_i\left(z\left(t\right)\right) = \prod\_{j=1}^p M\_{ij}\left(z\_j\left(t\right)\right) \quad \text{for all } t.$$

The term *Mi j* <sup>1</sup> *z t* is the grade of membership of *<sup>j</sup> z t* in *Mi*<sup>1</sup>

$$\text{Since } \qquad \begin{cases} \sum\_{l=1}^{r} w\_l \left( z \left( t \right) \right) \succ 0 \\ w\_l \left( z \left( t \right) \right) \succeq 0, \qquad \qquad i = 1, 2, \dots, r \end{cases} $$

1

we have

0, 1,2, , *h zt i r* for all *t*.

The time derivative of premise membership functions is given by:

$$\dot{h}\_i \left( z \left( t \right) \right) = \frac{\partial h\_i}{\partial z \left( t \right)} \cdot \frac{\partial z \left( t \right)}{\partial x \left( t \right)} \cdot \frac{d\mathbf{x} \left( t \right)}{dt} = \sum\_{l=1}^s \nu\_{il} \dot{\xi}\_{il} \times \frac{d\mathbf{x} \left( t \right)}{dt}$$

We have the following property:

*i*

 1

*h zt*

 

*r i i i*

$$\sum\_{k=1}^{r} \dot{h}\_k \left( z \left( t \right) \right) = 0 \tag{3}$$

Consider a PDC fuzzy controller based on the derivative membership function and given by the equation (4)

$$\ln \left( t \right) = -\sum\_{i=1}^{r} h\_i \left( z \left( t \right) \right) \mathbb{F}\_i \mathbf{x} \left( t \right) - \sum\_{m=1}^{r} \dot{h}\_m \left( z \left( t \right) \right) \mathbb{K}\_m \mathbf{x} \left( t \right) \tag{4}$$

The fuzzy controller design consists to determine the local feedback gains ,and *i m F K* in the consequent parts. The state variables are determined by an observer which detailed in next section.

By substituting (4) into (2), the closed-loop fuzzy system without time-delay can be represented as:

$$\dot{\mathbf{x}}(t) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_i \left( \mathbf{z}(t) \right) h\_j \left( \mathbf{z}(t) \right) \left\{ \left[ \mathbf{A}\_{\text{Ai}} - \mathbf{B}\_{\text{Ai}} \mathbf{F}\_j - \sum\_{m=1}^{r} \dot{h}\_m \left( \mathbf{z}(t) \right) \mathbf{B}\_{\text{Ai}} K\_m \right] \mathbf{x}(t) + D\_{\text{Ai}} \mathbf{x} \left( t - \tau\_i(t) \right) \right\} \quad \text{(5)}$$

$$\mathbf{x}(t) = \phi(t), \quad t \in \left[ -\tau, 0 \right],$$

where ; *A i i i i i i ii i A A D D D and B B B*

The system without uncertainties is given by equation

$$\dot{\mathbf{x}}(t) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_i \{ \mathbf{z}(t) \} h\_j \{ \mathbf{z}(t) \} \left\{ \left[ \mathbf{A}\_i - \mathbf{B}\_i \mathbf{F}\_j - \sum\_{m=1}^{r} \dot{h}\_m \{ \mathbf{z}(t) \} \mathbf{B}\_i \mathbf{K}\_m \right] \mathbf{x}(t) + D\_i \mathbf{x} \{ t - \tau\_i(t) \} \right\} \tag{6}$$
 
$$\mathbf{x}(t) = \boldsymbol{\phi}(t), \quad t \in \left[ -\tau, 0 \right],$$

The open-loop system is given by the equation (7),

$$
\dot{\mathbf{x}}\left(t\right) = \sum\_{i=1}^{r} h\_i \left(\mathbf{z}\left(t\right)\right) \left(A\_{\text{Ai}}\mathbf{x}\left(t\right) + D\_{\text{Ai}}\mathbf{x}\left(t - \tau\_i\left(t\right)\right)\right) \tag{7}
$$

$$
\mathbf{x}\left(t\right) = \phi\left(t\right), \quad t \in \left[-\tau, 0\right],
$$

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 485

*<sup>T</sup>* and

**Lemma 1** (Boyd et al. Schur complement [16])

**Lemma 2** (Peterson and Hollot [2])

**Theorem 1** (Razumikhin Theorem)[5]

then the follow-ing inequality

2 2 *<sup>T</sup>* , then

if and only if

*s* 0 such that

**Lemma 3 [6]** 

Given constant matrices 12 3 , and with appropriate dimensions, where 1 1

1 1 32 3 <sup>0</sup> *<sup>T</sup>*

, , , , *<sup>n</sup> u x V tx v x t x* (8)

, , ,0 ,

 

   

1 3 2 3 2 1 0 or 0 \* \* *<sup>T</sup>*

Let , , and *<sup>T</sup> Q Q HE Ft* satisfying *<sup>T</sup> F tFt I* are appropriately dimensional matrices

0 *TT T Q HF t E E F t H*

<sup>1</sup> 0 *T T Q HH E E*

 Suppose , , : *uvw* are continuous, non-decreasing functions satisfying *u s* 0, *v s* 0 and 0 *w s* for 0 *s* , *u v* 0 0 0, and *v* strictly increasing. If there exist a continuous function : *<sup>n</sup> V* and a continuous non-decreasing function *ps s* for

> 

Assume that *<sup>a</sup> <sup>n</sup> a* , *<sup>b</sup> <sup>n</sup> b* , *a b n n N* are defined on the interval . Then, for any

 2 , *T*

*a Nb d d*

(9)

*T T a a X YN*

*b b YN Z*

 (10)

 

*V t x w x if V t x t p V t x* , , 

then the solution 0 *x* of (7) is uniformly asymptotically stable.

matrices *a a n n X* , *a b n n Y* and *b b n n Z* , the following holds:

 

*T*

is true, if and only if the following inequality holds for any 0

#### **Assumption 1**

The time derivative of the premises membership function is upper bounded such that *k k h* , for *k r* 1, , , where, , 1, , *<sup>k</sup> k r* are given positive constants.

#### **Assumption 2**

The matrices denote the uncertainties in the system and take the form of

$$\begin{cases} \Delta A\_i = D\_{a\_i} F\_{a\_i} \left( t \right) E\_{a\_i} \\ \Delta B\_i = D\_{b\_i} F\_{b\_i} \left( t \right) E\_{b\_i} \end{cases}$$

where , , and *i ii i DDE E a ba b* are known constant matrices and and *i i a b Ft Ft* are unknown matrix functions satisfying :

$$\begin{cases} F\_{a\_i}^T \left( t \right) F\_{a\_i} \left( t \right) \le I\_\prime \,\forall t \\ F\_{b\_i}^T \left( t \right) F\_{b\_i} \left( t \right) \le I\_\prime \,\forall t \end{cases}$$

where I is an appropriately dimensioned identity matrix.

#### **Lemma 1** (Boyd et al. Schur complement [16])

Given constant matrices 12 3 , and with appropriate dimensions, where 1 1 *<sup>T</sup>* and 2 2 *<sup>T</sup>* , then

$$
\boldsymbol{\Omega}\_1 + \boldsymbol{\Omega}\_3^T \boldsymbol{\Omega}\_2^{-1} \boldsymbol{\Omega}\_3 \prec 0
$$

if and only if

484 Fuzzy Controllers – Recent Advances in Theory and Applications

1 1 1

1 1 1

1

*i*

The matrices denote the uncertainties in the system and take the form of

 

*r*

*r r r*

*i j m*

*r r r*

*i j m*

where ; *A i i i i i i ii i A A D D D and B B B*

The system without uncertainties is given by equation

The open-loop system is given by the equation (7),

, for *k r* 1, , , where, , 1, , *<sup>k</sup>*

represented as:

**Assumption 1** 

**Assumption 2** 

where , , and *i ii i*

matrix functions satisfying :

*k k h* 

By substituting (4) into (2), the closed-loop fuzzy system without time-delay can be

*xt h zt h zt A B F h zt B K xt D xt t*

*xt t t* 

*x t h z t h z t A BF h z t BK x t Dx t t*

*xt t t* 

*xt h zt A xt D xt t*

*xt t t* 

The time derivative of the premises membership function is upper bounded such that

*i i ii*

*i j i ij m i m i i*

(6)

 , ,0 ,

 , ,0 ,

*k r* are given positive constants.

 *ii i ii i*

*i aa a i bb b*

*DDE E a ba b* are known constant matrices and and *i i a b Ft Ft* are unknown

, ,

*A DF tE B DF tE*

 

*F tF t I t F tF t I t* 

*i i i i*

*T a a T b b*

where I is an appropriately dimensioned identity matrix.

(7)

*i j i ij m i m i i*

 , ,0 ,

(5)

$$
\begin{bmatrix}
\boldsymbol{\Omega}\_1 & \boldsymbol{\Omega}\_3^T \\
\ast & -\boldsymbol{\Omega}\_2
\end{bmatrix} \prec 0 \text{ or } \begin{bmatrix}
\ast & \boldsymbol{\Omega}\_1
\end{bmatrix} \prec 0
$$

**Lemma 2** (Peterson and Hollot [2])

Let , , and *<sup>T</sup> Q Q HE Ft* satisfying *<sup>T</sup> F tFt I* are appropriately dimensional matrices then the follow-ing inequality

$$Q + HF(t)E + E^T F^T(t)H^T \prec 0$$

is true, if and only if the following inequality holds for any 0 

$$Q + \mathcal{X}^{-1}HH^T + \mathcal{X}E^TE \prec 0$$

**Theorem 1** (Razumikhin Theorem)[5]

 Suppose , , : *uvw* are continuous, non-decreasing functions satisfying *u s* 0, *v s* 0 and 0 *w s* for 0 *s* , *u v* 0 0 0, and *v* strictly increasing. If there exist a continuous function : *<sup>n</sup> V* and a continuous non-decreasing function *ps s* for *s* 0 such that

$$
u(\left|\mathbf{x}\right|) \le V\left(t, \mathbf{x}\right) \le \upsilon\left(\left|\mathbf{x}\right|\right), \qquad \forall t \in \mathfrak{R}, \ \mathbf{x} \in \mathfrak{R}^n,\tag{8}$$

$$\dot{V}\left(t, \mathbf{x}\right) \le -w\left(\left|\mathbf{x}\right|\right) \quad \text{if } V\left(t + \sigma, \mathbf{x}\left(t + \sigma\right)\right) \le p\left(V\left(t, \mathbf{x}\right)\right), \quad \forall \sigma \in \left[-\tau, 0\right],\tag{9}$$

then the solution 0 *x* of (7) is uniformly asymptotically stable.

#### **Lemma 3 [6]**

Assume that *<sup>a</sup> <sup>n</sup> a* , *<sup>b</sup> <sup>n</sup> b* , *a b n n N* are defined on the interval . Then, for any matrices *a a n n X* , *a b n n Y* and *b b n n Z* , the following holds:

$$-2\int\_{\Omega} a^T(a)Nb(a)da \le \int\_{\Omega} \begin{bmatrix} a(a) \\ b(a) \end{bmatrix}^T \begin{bmatrix} X & Y - N \\ Y^T - N^T & Z \end{bmatrix} \begin{bmatrix} a(a) \\ b(a) \end{bmatrix} da \tag{10}$$

$$\text{where } \begin{bmatrix} X & Y \\ Y^T & Z \end{bmatrix} \ge 0 \dots$$

#### **Lemma 4 [9]**

The unforced fuzzy time delay system described by **(7)** with u = 0 is uniformly asymptotically stable if there exist matrices *P* 0, 0, *<sup>i</sup> S* , *Xai* , *Xdi* , *Zaij* , *Zdij* and *Yi* , such that the following LMIs hold:

$$\begin{bmatrix} PA\_i + A\_i^T P + \tau \left( X\_{ai} + X\_{di} \right) + \left( 2\tau + 1 \right) P + Y\_i + Y\_i^T & -PD\_i \\ Y\_i^T - D\_i^T P & -S\_i \end{bmatrix} \prec 0 \tag{11}$$

$$S\_i \le P \tag{12}$$

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 487

(20)

*PR k r <sup>k</sup>* 0, 1, , (18)

2 ,

*PR k r <sup>k</sup>* 0, 1, , (22)

0,

(23)

2

, the Takagi-Sugeno system

(24)

(25)

*G G*

0, 1, , (19)

*PR j r <sup>j</sup>*

*P A P R P RA*

*ij j i*

*A P R P RA i j*

 

*G A BF ij i i j* and *G A BF ii i i i* .

In this section we define a fuzzy Lyapunov function and then consider stability conditions.

(21) is stable if there exist positive definite symmetric matrices , 1,2, , *<sup>k</sup> Pk r* , and *R*,

0, 1,2, , *<sup>j</sup> PR j r*

*ij ji ij ji k k G G G G P RP R*

0, 2 2

 

 

*ii k <sup>k</sup> ii P G P R P RG*

 

for , , 1,2, , such that

*ijk r i j*

*T*

, 1, ,

*ik r*

*T*

(21)

*xt h zt h zt Gxt h zt h zt x t*

 

*i i ii i j*

*r r ij ji*

 

 

0,

*ji i j*

and 1

1 2

where *ij r* , 1,2, , and

**Proof** 

where

**Theorem 2[18]** 

*T*

1

*r*

*k P PR*

 *T*

*k k*

The proof of this theorem is given in detailed in article published in [17].

The closed-loop system without time delay is given by equation (21)

1 1

*i i ij*

A sufficient stability condition, for ensuring stability is given follows.

Under assumption 1, and assumption 2 and for given 0 1

matrices 1 , , *<sup>r</sup> F F* such that the following LMIs hols.

$$\mathbf{A}\_{\circ}^{\mathrm{T}} \mathbf{Z}\_{\mathrm{adj}} \mathbf{A}\_{\circ} \le \mathbf{P} \tag{13}$$

$$\mathbf{D}\_{\dot{j}}^T \mathbf{Z}\_{\text{dij}} \mathbf{D}\_{\dot{j}} \le \mathbf{P} \tag{14}$$

$$
\begin{bmatrix} X\_{ai} & Y\_i \\ Y\_i^T & Z\_{aij} \end{bmatrix} \succeq \mathbf{0} \tag{15}
$$

$$
\begin{bmatrix}
\mathbf{X}\_{di} & \mathbf{Y}\_i \\
\mathbf{Y}\_i^T & \mathbf{Z}\_{dij}
\end{bmatrix} \succeq \mathbf{0} \tag{16}
$$

#### **3. Basic stability and stabilization conditions**

In order to design an observer for state variables, this section introduce two theorem developed for continuous TS fuzzy model for open-loop and closed-loop. First, consider the open-loop system without time-delay given by equation(17).

$$\dot{\mathbf{x}}\{t\} = \sum\_{i=1}^{r} h\_i\{z(t)\} A\_i \mathbf{x}(t) \tag{17}$$

The main approach for T-S fuzzy model stability is given in theorem follows. This approach is based on introduction of parameter which influences the stability region.

#### **Theorem 2 [17]**

Under assumption 1 and for 0 1 , the Takagi Sugeno fuzzy system (17) is stable if there exist positive definite symmetric matrices , 1,2, , *<sup>k</sup> Pk r* , matrix *<sup>T</sup> R R* such that the following LMIs hold.

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 487

$$P\_k + R \succ 0, \quad k \in \{1, \dots, r\} \tag{18}$$

$$P\_j + \mu \mathbb{R} \succ 0, \quad j \in \{1, \ldots, r\} \tag{19}$$

$$\begin{aligned} P\_{\phi} &+ \frac{1}{2} \Big\{ A\_i^T \left( P\_j + \mu R \right) + \left( P\_j + \mu R \right) A\_i \\ &+ A\_j^T \left( P\_i + \mu R \right) + \left( P\_i + \mu R \right) A\_j \Big\} \prec 0, \quad i \le j \end{aligned} \tag{20}$$

where *ij r* , 1,2, , and 1 *r k k k P PR* and 1 

#### **Proof**

486 Fuzzy Controllers – Recent Advances in Theory and Applications

The unforced fuzzy time delay system described by **(7)** with u = 0 is uniformly asymptotically stable if there exist matrices *P* 0, 0, *<sup>i</sup> S* , *Xai* , *Xdi* , *Zaij* , *Zdij* and *Yi* , such

> *T T i i ai di ii i*

 

0 *ai i*

0 *di i*

In order to design an observer for state variables, this section introduce two theorem developed for continuous TS fuzzy model for open-loop and closed-loop. First, consider the

> 1

The main approach for T-S fuzzy model stability is given in theorem follows. This approach

exist positive definite symmetric matrices , 1,2, , *<sup>k</sup> Pk r* , matrix *<sup>T</sup> R R* such that the

*x t h z t Ax t* 

*i i*

parameter which influences the stability region.

*r*

*i*

*T i aij*

*T i dij*

*X Y Y Z* 

*X Y Y Z* 

*PA A P X X P Y Y PD*

*T T*

**3. Basic stability and stabilization conditions** 

open-loop system without time-delay given by equation(17).

2 1 <sup>0</sup>

(11)

*<sup>i</sup> S P* (12)

*<sup>T</sup> Aj aij j ZA P* (13)

*<sup>T</sup> DZ D P j dij j* (14)

(17)

, the Takagi Sugeno fuzzy system (17) is stable if there

(15)

(16)

*i i i*

*Y DP S*

.

that the following LMIs hold:

is based on introduction of

Under assumption 1 and for 0 1

**Theorem 2 [17]** 

following LMIs hold.

where 0 *<sup>T</sup> X Y Y Z* 

**Lemma 4 [9]** 

The proof of this theorem is given in detailed in article published in [17]. The closed-loop system without time delay is given by equation (21)

$$\dot{\mathbf{x}}(t) = \sum\_{i=1}^{r} h\_i \left( z(t) \right) h\_i \left( z(t) \right) \mathbf{G}\_{\dot{\boldsymbol{\mu}}} \mathbf{x}(t) + 2 \sum\_{i=1}^{r} \sum\_{l \neq j} h\_i \left( z(t) \right) h\_j \left( z(t) \right) \left| \frac{\mathbf{G}\_{\dot{\boldsymbol{\mu}}} + \mathbf{G}\_{\dot{\boldsymbol{\mu}}}}{2} \right| \mathbf{x}(t), \tag{21}$$

where

$$G\_{ij} = A\_i - B\_i F\_j \text{ and } G\_{ii} = A\_i - B\_i F\_i \text{ .}$$

In this section we define a fuzzy Lyapunov function and then consider stability conditions. A sufficient stability condition, for ensuring stability is given follows.

#### **Theorem 2[18]**

Under assumption 1, and assumption 2 and for given 0 1 , the Takagi-Sugeno system (21) is stable if there exist positive definite symmetric matrices , 1,2, , *<sup>k</sup> Pk r* , and *R*, matrices 1 , , *<sup>r</sup> F F* such that the following LMIs hols.

$$P\_k + R \succ 0, \quad k \in \{1, \ldots, r\} \tag{22}$$

$$P\_j + \mu \mathbb{R} \ge 0, \quad j = 1, 2, \dots, r \tag{23}$$

$$\begin{aligned} P\_{\phi} + \left\{ G\_{ii}^{T} \left( P\_{k} + \mu \mathcal{R} \right) + \left( P\_{k} + \mu \mathcal{R} \right) G\_{ii} \right\} \prec \mathbf{0}, \\\ i, k \in \{ 1, \ldots, r \} \end{aligned} \tag{24}$$

$$\left\{\frac{G\_{ij} + G\_{ji}}{2}\right\}^T \left(P\_k + \mu \mathcal{R}\right) + \left(P\_k + \mu \mathcal{R}\right) \left\{\frac{G\_{ij} + G\_{ji}}{2}\right\} \prec 0,\tag{25}$$
  $\text{for } i, i, k = 1, 2, \dots, r \text{ such that } i \prec i$ 

for , , 1,2, , such that *ijk r i j* 

where

$$G\_{ij} = A\_i - B\_i F\_{j\prime\prime} \\ G\_{ii} = A\_i - B\_i F\_{i\prime}$$

$$\text{And }\ P\_{\phi} = \sum\_{k=1}^{r} \phi\_k \left( P\_k + R \right)^2$$

### **4. Observer design for T-S fuzzy continuous model**

In order to determine state variables of system, this section gives a solution by the mean of fuzzy observer design.

A stabilizing observer-based controller can be formulated as follow:

$$\begin{aligned} \hat{\boldsymbol{x}}\left(t\right) &= \sum\_{j=1}^{r} h\_{i}\left(\boldsymbol{z}\left(t\right)\right) \left\{ A\_{i}\hat{\boldsymbol{x}}\left(t\right) + B\_{i}\boldsymbol{u}\left(t\right) + L\_{j}\left(\boldsymbol{C}\_{i}\hat{\boldsymbol{x}}\left(t\right) - \boldsymbol{y}\left(t\right)\right) \right\} \\ \boldsymbol{u}\left(t\right) &= \sum\_{j=1}^{r} h\_{j}\left(\boldsymbol{z}\left(t\right)\right) \boldsymbol{F}\_{j}\hat{\boldsymbol{x}}\left(t\right) \end{aligned} \tag{26}$$

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 489

, the Takagi-Sugeno system (29) is stable if there

*PR k r <sup>k</sup>* 0, 1, , (30)

0,

 (32)

 

(31)

*i ij*

, the unforced fuzzy time delay system described

 

*A KC*

(33)

By applying Theorem 2[18] in the augmented system (29) we derive the following Theorem.

exist positive definite symmetric matrices , 1,2, , *<sup>k</sup> Pk r* , and *R*, matrices 1 , , *<sup>r</sup> F F* such

0, 1,2, , *<sup>j</sup> PR j r*

*ij ji ij ji k k G G G G P RP R*

 

0, 2 2

1 1

 

*A BF h B H R BF h B H R*

*r r i ij i ij i*

 

 

*ii k <sup>k</sup> ii P G P R P RG*

 

for , , 1,2, , such that

0

**5. Stabilization of continuous T-S Fuzzy model with time-delay** 

on the combination between Lyapunov theory and the Razumikhin theorem [5].

0, *<sup>i</sup> S* , *Xaij* , *Xdi* , *Zaij* , *Zdij Yi* , and *X* , such that the following LMIs hold:

The aim of this section is to prove the asymptotic stability of the time-delay system (6) based

by (7) with 0 *u* is uniformly asymptotically stable if there exist matrices 0, 1,2, , , *<sup>k</sup> Pk r*

*ijk r i j*

*T*

, 1, ,

The result follows immediately from the Theorem 2[18].

*ik r*

*T*

**Theorem 3** 

where

Under assumption 1 and for given 0 1

*ij*

Under assumption 1 and for given 0 1

*G*

And 1

 

*k P PR*

**Theorem 4** 

**Proof** 

*k k*

*r*

that the following LMIs hols.

The closed-loop fuzzy system can be represented as:

$$\begin{aligned} \mathbf{x}^{\prime}(t) &= \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_{i} \left( z(t) \right) h\_{j} \left( z(t) \right) \left| \left( A\_{i} - B\_{i} F\_{j} \right) - \sum\_{\rho=1}^{r} h\_{\rho} \left( z(t) \right) \left( H\_{\rho} + R \right) \right| \mathbf{x}(t) \\ &+ \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_{i} \left( z(t) \right) h\_{j} \left( z(t) \right) \left| \left( B\_{i} F\_{j} + \sum\_{\rho=1}^{r} h\_{\rho} \left( z(t) \right) \left( H\_{\rho} + R \right) \right| e(t) \right. \\\\ \mathbf{e}(t) &= \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_{i} \left( z(t) \right) h\_{j} \left( z(t) \right) \left| \left( A\_{i} - K\_{i} C\_{j} \right) e(t) \right| \end{aligned} \tag{28}$$

The augmented system is represented as follows:

$$\begin{split} \dot{\mathbf{x}}\_{a}(t) &= \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_{i} \big( \mathbf{z}(t) \big) h\_{j} \big( \mathbf{z}(t) \big) \mathbf{G}\_{ij} \mathbf{x}\_{a}(t) \\ &= \sum\_{j=1}^{r} h\_{i} \big( \mathbf{z}(t) \big) h\_{j} \big( \mathbf{z}(t) \big) \mathbf{G}\_{ii} \mathbf{x}\_{a}(t) + 2 \sum\_{i=1}^{r} \sum\_{i$$

where

$$\begin{aligned} \mathbf{x}\_a(t) &= \begin{bmatrix} \mathbf{x}(t) \\ e(t) \end{bmatrix} \\ G\_{ij} &= \begin{bmatrix} A\_i - \mathbf{B}\_i F\_j - \sum\_{\rho=1}^r \dot{h}\_\rho \mathbf{B}\_i \left( H\_\rho + \mathbf{R} \right) & \mathbf{B}\_i F\_j + \sum\_{\rho=1}^r \dot{h}\_\rho \mathbf{B}\_i \left( H\_\rho + \mathbf{R} \right) \\ \mathbf{0} & A\_i - \mathbf{K}\_i \mathbf{C}\_j \end{bmatrix} \end{aligned}$$

By applying Theorem 2[18] in the augmented system (29) we derive the following Theorem.

## **Theorem 3**

488 Fuzzy Controllers – Recent Advances in Theory and Applications

**4. Observer design for T-S fuzzy continuous model** 

A stabilizing observer-based controller can be formulated as follow:

*u t h z t Fx t*

*j j*

1

*j r*

*j*

The closed-loop fuzzy system can be represented as:

*i j*

*i j*

1 1

*i j*

*a*

*x t*

*ij*

*G*

where

*r r*

The augmented system is represented as follows:

*a i j ij a*

 

*x t*

 

*e t*

*x t h zt h zt Gx t*

*r*

1

, *G A BF ij i i j G A BF ii i i i*

In order to determine state variables of system, this section gives a solution by the mean of

*x t h z t h z t A BF h z t H R x t*

*i j i ij*

*h zt h zt Gx t h zt h zt x t*

1 1

 

*A BF h B H R BF h B H R*

*r r i ij i ij i*

   

(28)

 

*i ij*

 

*A KC*

 

2

*G G*

*h z t h z t BF h z t H R e t*

*e t h z t h z t A KC e t*

2

*<sup>r</sup> r r ij ji i j ii a i j a*

(26)

(27)

(29)

*i i i ji*

*x t h z t Ax t Bu t L Cx t y t*

ˆ ˆ ˆ

ˆ

1 1 1

*r r r i j i ij*

1 1 1

1 1 *r r*

*i j*

1 1

*j i ij*

0

*r r r i j ij*

where

And 1

 

fuzzy observer design.

*k P PR*

 *k k*

*r*

Under assumption 1 and for given 0 1 , the Takagi-Sugeno system (29) is stable if there exist positive definite symmetric matrices , 1,2, , *<sup>k</sup> Pk r* , and *R*, matrices 1 , , *<sup>r</sup> F F* such that the following LMIs hols.

$$P\_k + R \succ 0, \quad k \in \{1, \dots, r\} \tag{30}$$

$$P\_j + \mu \mathbb{R} \ge 0, \quad j = 1, 2, \dots, r \tag{31}$$

$$\begin{aligned} P\_{\phi} + \left\{ G\_{il}^{T} \left( P\_{k} + \mu \mathcal{R} \right) + \left( P\_{k} + \mu \mathcal{R} \right) G\_{il} \right\} \prec \mathbf{0}, \\\ i, k \in \{ 1, \ldots, r \} \end{aligned} \tag{32}$$

$$\left\{ \frac{G\_{ij} + G\_{ji}}{2} \right\}^T \left( P\_k + \mu R \right) + \left( P\_k + \mu R \right) \left| \frac{G\_{ij} + G\_{ji}}{2} \right\rangle \prec 0,\tag{33}$$
  $\text{for } i, j, k = 1, 2, \dots, r \text{ such that } i \prec j$ 

where

$$\mathbf{G}\_{ij} = \begin{bmatrix} A\_i - \mathbf{B}\_i \mathbf{F}\_j - \sum\_{\rho=1}^r \dot{h}\_\rho \mathbf{B}\_i \left( \mathbf{H}\_\rho + \mathbf{R} \right) & \mathbf{B}\_i \mathbf{F}\_j + \sum\_{\rho=1}^r \dot{h}\_\rho \mathbf{B}\_i \left( \mathbf{H}\_\rho + \mathbf{R} \right) \\\mathbf{0} & \mathbf{A}\_i - \mathbf{K}\_i \mathbf{C}\_j \end{bmatrix}$$
  $\text{And } \ P\_\phi = \sum\_{k=1}^r \phi\_k \left( P\_k + \mathbf{R} \right)$ 

## **Proof**

The result follows immediately from the Theorem 2[18].

### **5. Stabilization of continuous T-S Fuzzy model with time-delay**

The aim of this section is to prove the asymptotic stability of the time-delay system (6) based on the combination between Lyapunov theory and the Razumikhin theorem [5].

#### **Theorem 4**

Under assumption 1 and for given 0 1 , the unforced fuzzy time delay system described by (7) with 0 *u* is uniformly asymptotically stable if there exist matrices 0, 1,2, , , *<sup>k</sup> Pk r* 0, *<sup>i</sup> S* , *Xaij* , *Xdi* , *Zaij* , *Zdij Yi* , and *X* , such that the following LMIs hold:

$$\begin{bmatrix} P\_{\beta} + \left(P\_{k} + \varepsilon X\right)G\_{ij} + G\_{ij}^{T}\left(P\_{k} + \varepsilon X\right) \\\\ + \tau \left(X\_{aij} + X\_{di}\right) + \left(2\tau + 1\right)\left(P\_{k} + \varepsilon X\right) + Y\_{i} + Y\_{i}^{T} \\\\ Y\_{i}^{T} - D\_{i}^{T}\left(P\_{k} + \varepsilon X\right) & -S\_{i} \end{bmatrix} < 0 \tag{34}$$

where 1 *r k k k ij i i j P PX G A BF* .

$$S\_i \le \left(P\_k + \varepsilon X\right) \tag{35}$$

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 491

is equivalent to determining the

Finding the minimum value of <sup>0</sup>

The state equation of (7) with *u=0* can be rewritten as

on the Razumikhin theorem [5].

where *G A BF ij i i j*

1

,

2

*xt x t h P Xxt*

*k k*

*T T*

1

*k*

*<sup>r</sup> <sup>T</sup>*

*<sup>k</sup> k k h PX* 

.

minimum value of min <sup>0</sup> *r*

Define

Then,

Since

*r T*

*<sup>k</sup> k k <sup>k</sup>*

*<sup>k</sup> k k <sup>k</sup>*

1 *<sup>r</sup> <sup>T</sup>*

*k*

1 1

Then, based on assumption 1, an upper bound of <sup>1</sup> *x z*, obtained as:

,

*k i*

*k k i i*

1 1 1 1

1

*k*

*r r T T*

1 2

*<sup>k</sup> k k hx t P X xt*

<sup>1</sup> min <sup>0</sup> max for 0 , *<sup>r</sup>*

<sup>2</sup> max <sup>0</sup> min for 0 . *<sup>r</sup>*

2 2

 

,

2 , ,

*i*

 

 

(42)

*k*

In the following, we will prove the asymptotic stability of the time-delay system (7) based

 1 1 *<sup>i</sup>*

The derivative of *V* along the solutions of the unforced system **(7)** with 0 *u* is thus given by

*V x t h P Xxt x t h P Xxt xt xt*

<sup>1</sup>

*xz xt P X xt*

*<sup>r</sup> <sup>T</sup> k k*

, 2 2 .

*x t x t h P X x t x t h P X h h G D x t D x s ds*

*r r r r t*

*k k k k i j ij i i k k i j t t*

 

1 2

(41)

*r r t i j ij i i i j t t x t h h G D x t D x s ds* 

*x t x t t x s ds*

*i*

*i*

*t t*

*t*

 *xt x t P Xxt xt* 

*h P X kr*

*h P X kr*

$$\mathbf{G}\_{i\circ}^T \mathbf{Z}\_{\text{aij}} \mathbf{G}\_{i\circ} \le \left(\mathbf{P}\_k + \varepsilon \mathbf{X}\right) \tag{36}$$

$$\mathbf{D}\_{\dot{j}}^T \mathbf{Z}\_{d\dot{d}\dot{j}} \mathbf{D}\_{\dot{j}} \le \left(\mathbf{P}\_k + \varepsilon \mathbf{X}\right) \tag{37}$$

$$
\begin{bmatrix} X\_{aij} & Y\_i \\ Y\_i^T & Z\_{aij} \end{bmatrix} \succeq \mathbf{0} \tag{38}
$$

$$
\begin{bmatrix} X\_{di} & Y\_i \\ Y\_i^T & Z\_{dij} \end{bmatrix} \succeq \mathbf{0} \tag{39}
$$

*Proof* 

Let consider the fuzzy Lyapunov function as

$$\begin{aligned} V\left(\mathbf{x}\right) &= \mathbf{x}^T \left(t\right) V\_k\left(\mathbf{x}\right) \mathbf{x}\left(t\right) \\\\ V\_k\left(\mathbf{x}\right) &= \sum\_{k=1}^r h\_k \left(P\_k + \varepsilon \mathbf{X}\right) \end{aligned} \tag{40}$$

Given the matrix property, clearly,

$$\left\|\mathcal{A}\_{\min}\left(P\_k + \varepsilon X\right)\right\|\mathbf{x}\left(t\right)\right\|^2 \le \mathbf{x}^T\left(t\right)\left(P\_k + \varepsilon X\right)\mathbf{x}\left(t\right) \le \mathcal{A}\_{\max}\left(P\_k + \varepsilon X\right)\left\|\mathbf{x}\left(t\right)\right\|^2.$$

where min max denotes the smallest (largest) eigenvalue of the matrix.

Finding the maximum value of <sup>0</sup> *r T <sup>k</sup> k k hx t P X xt* is equivalent to determining the maximum value of max <sup>0</sup> *r <sup>k</sup> k k h PX* .

Finding the minimum value of <sup>0</sup> *r T <sup>k</sup> k k hx t P X xt* is equivalent to determining the minimum value of min <sup>0</sup> *r <sup>k</sup> k k h PX* .

Define

490 Fuzzy Controllers – Recent Advances in Theory and Applications

Let consider the fuzzy Lyapunov function as

 

maximum value of max <sup>0</sup> *r*

Finding the maximum value of <sup>0</sup>

*<sup>k</sup> k k h PX* 

.

Given the matrix property, clearly,

where min max 

where 1

 

*G A BF*

*k ij i i j*

*Proof* 

*r*

*k k*

.

*P PX*

 *aij di k i i*

*X X P X YY*

*k ij ij k*

*P P XG G P X*

2 1

*T T*

*i ik i*

 

*Y DP X S*

 *i k SPX* 

*<sup>T</sup> GZ G P X ij aij ij k*

*<sup>T</sup> DZ D P X j dij j k*

<sup>0</sup> *aij i*

0 *di i*

 1

 2 2 min max , *<sup>T</sup> k kk*

 *P X xt x t P Xxt P X xt* 

*<sup>k</sup> k k hx t P X xt*

denotes the smallest (largest) eigenvalue of the matrix.

*r T*

is equivalent to determining the

*r k kk k Vx hP X*

 

*T i aij*

*T i dij*

*X Y Y Z* 

*X Y Y Z* 

*T*

0

(35)

(36)

(37)

(38)

(39)

*<sup>T</sup> V x x tV xxt <sup>k</sup>* (40)

(34)

*T k i*

*P XD*

$$\begin{aligned} \kappa\_1 &= \min\_k \sum\_{k=0}^r h\_k \mathcal{A}\_{\text{max}} \left( P\_k + \varepsilon X \right) \text{ for } 0 \le k \le r, \\\\ \kappa\_2 &= \max\_k \sum\_{k=0}^r h\_k \mathcal{A}\_{\text{min}} \left( P\_k + \varepsilon X \right) \text{ for } 0 \le k \le r. \end{aligned}$$

Then,

$$\left\|\kappa\_1\right\|\mathbf{x}\left(t\right)\Big\|^2 \le \sum\_{k=1}^r \mathbf{x}^T\left(t\right)\left(P\_k + \varepsilon X\right)\mathbf{x}\left(t\right) \le \kappa\_2 \left\|\mathbf{x}\left(t\right)\right\|^2$$

In the following, we will prove the asymptotic stability of the time-delay system (7) based on the Razumikhin theorem [5].

Since

$$\|\mathbf{x}\left(t\right) - \mathbf{x}\left(t - \tau\_i\left(t\right)\right)\| = \int\_{t - \tau\_i\left(t\right)}^t \dot{\mathbf{x}}\left(s\right)ds\prime\_i$$

The state equation of (7) with *u=0* can be rewritten as

$$\dot{\mathbf{x}}\left(t\right) = \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_i h\_j \left[ \left(\mathbf{G}\_{ij} + \mathbf{D}\_i\right) \mathbf{x}\left(t\right) - \mathbf{D}\_i \int\_{t-\tau\_i\left(t\right)}^{t} \dot{\mathbf{x}}\left(s\right) ds\right],$$

where *G A BF ij i i j*

The derivative of *V* along the solutions of the unforced system **(7)** with 0 *u* is thus given by

$$\dot{V} = \mathbf{x}^T(t) \sum\_{k=1}^r \dot{h}\_k \left( P\_k + \varepsilon \mathbf{X} \right) \mathbf{x}(t) + 2\mathbf{x}^T(t) \sum\_{i=1}^r h\_i \left( P\_i + \varepsilon \mathbf{X} \right) \dot{\mathbf{x}}(t) = \mathbf{Y}\_1(\mathbf{x}, t) + \mathbf{Y}\_2(\mathbf{x}, t) \tag{41}$$

$$\begin{split} \mathbf{Y}\_{1}\left(\mathbf{x},t\right) &= \mathbf{x}^{T}\left(t\right) \sum\_{k=1}^{r} \dot{h}\_{k} \left(P\_{k} + \varepsilon \mathbf{X}\right) \mathbf{x}\left(t\right) \\ \mathbf{Y}\_{2}\left(\mathbf{x},t\right) &= 2\mathbf{x}^{T}\left(t\right) \sum\_{k=1}^{r} h\_{k} \left(P\_{k} + \varepsilon \mathbf{X}\right) \dot{\mathbf{x}}\left(t\right) = 2\mathbf{x}^{T}\left(t\right) \sum\_{k=1}^{r} h\_{k} \left(P\_{k} + \varepsilon \mathbf{X}\right) \times \sum\_{i=1}^{r} \sum\_{j=1}^{r} h\_{i} h\_{j} \left[\left(\mathbf{G}\_{ij} + \mathbf{D}\_{j}\right) \mathbf{x}\left(t\right) - D\_{i} \int\_{t-\tau\_{i}(t)}^{t} \dot{\mathbf{x}}\left(s\right) ds\right]. \end{split}$$

Then, based on assumption 1, an upper bound of <sup>1</sup> *x z*, obtained as:

$$\mathbb{E}\left(\mathbf{x},\mathbf{z}\right) \leq \sum\_{k=1}^{r} \beta\_k \cdot \mathbf{x}\left(t\right)^T \left(P\_k + \varepsilon \mathbf{X}\right) \mathbf{x}\left(t\right) \tag{42}$$

and for <sup>2</sup> *x t*, we can written as,

1 1

   

 2 11 1 1 1 1 1 1 , 2 2 *i rr r r r <sup>t</sup> <sup>r</sup> T T ij k k ij i i j kk i ij k i j t t k r r j xt hhx h P X G D x t hh x t h P X D h s h s G x s D x s s ds* 2 111 111 , 2 2 *i rrr rrr t T T ijk k ij i ijk k i ijk ijk t t r r j x t hhh x P X G D x t hhh x t P X D h s h s G x s D x s s ds* (43)

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 493

0 *di i*

 

*i*

 

*i i k ii i k i ii i*

*h x t Y P X D S Y P X D x t x t t Sx t t*

 

> 

<sup>1</sup> 21 1

*Y P XD S Y P XD xt P X*

*h t x t P X x t h x s P X x s ds*

*i i k ii i k i k*

*i k ii i k i k*

 

*T*

*T T*

     

 

 

 

*i*

*h x t Y P XD S Y P XD xt x t P Xxt*

for a sufficiently small

 

> 

 1 *V xt* for

   

(46)

 

*hh x t Y P X D h s h s G x s D x s s ds*

 

*T i dij*

*ijk k ij i ai di*

Hence, substituting (44) and (45) into (43), we have

1 1 1 1

*i k ai*

 

*r r T TT*

*h h s h s x s G Z G x s ds*

*T T*

Note that, by *Shur* complement, the LMI in (34) implies 0 *<sup>i</sup> L*

1 1

 

1 1

*r r t*

*i i t t*

*r r T TT*

 

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

*h h h s h s x s G Z G x s ds*

*i k j j j dij j j*

*h h h s x s s D Z D x s s ds*

 

*j j dij j j*

*i k ij ij k i i aij di*

*L P P XG G P X Y Y X X*

In order to use the *Razumikhin* Theorem, suppose *V xt*

,0 . Then, if the LMIs in (35)–(39) also hold, we have from (46) that

1

*i j k ij ij k i i aij di*

1

*T T i i k i k*

*<sup>r</sup> <sup>T</sup> T T*

*V hhx t P X G G P X Y Y X X x t*

*s s D Z D x s s ds*

1

*P hhx t P X G G P X Y Y X X x t*

*<sup>r</sup> <sup>T</sup> T T*

*i j k ij ij k i i aij di*

2

*V P hhh x t P X G D X X x t*

*r r <sup>t</sup> r r <sup>T</sup> ik i k i i k t t r r <sup>t</sup> r r T T*

 

1 1 1 1

1 1 1

 

*h h sx*

 

1 1 1

*i*

*i j*

1 1 *<sup>i</sup>*

*i j t t*

, where

*i t t*

*r r t*

*<sup>r</sup> <sup>t</sup> r r T T i ai*

*i k j t t*

1 1

*i j*

1

*i*

scalar 0 

> 

> > 1 1

*i j*

1

*i*

*i*

*rrr <sup>T</sup>*

*i*

111

*i k t t*

*ijk*

2

*X Y Y Z* 

Using the bounding method in(10), by setting *a xt* and *ij b Gxs ,* we have

 

$$\begin{split} & -\int\_{t-\tau\_{i}(t)}^{t} 2\mathbf{x}^{T}\Big(t\big) \big(P\_{k} + \varepsilon\mathbf{X}\big) \mathbf{D}\_{i} \times \sum\_{\nu=1}^{r} \sum\_{\varsigma=1}^{r} h\_{\nu}\big(s\big) h\_{\varsigma}\big(s\big) \mathbf{G}\_{\upsilon\varsigma}\mathbf{x}\big(s\big) \mathbf{ds} \\ & \leq \tau\_{i}\big(t\big) \mathbf{x}^{T}\big(t\big) \mathbf{X}\_{ai}\mathbf{x}\big(t\big) + 2\mathbf{x}^{T}\big(t\big) \big(Y\_{i} - \big(P\_{k} + \varepsilon\mathbf{X}\big)\mathbf{D}\_{i}\big) \times \int\_{t-\tau\_{i}(t)}^{t} \sum\_{\nu=1}^{r} \sum\_{\varsigma=1}^{r} h\_{\nu}\big(s\big) h\_{\varsigma}\big(s\big) \mathbf{G}\_{\upsilon\varsigma}\mathbf{x}\big(s\big) \mathbf{ds} \\ & + \int\_{t-\tau\_{i}(t)}^{t} \sum\_{\nu=1}^{r} \sum\_{\varsigma=1}^{r} h\_{\nu}\big(s\big) h\_{\varsigma}\big(s\big) \mathbf{x}^{T}\big(s\big) \mathbf{G}\_{\upsilon\varsigma}^{T}\mathbf{Z}\_{\alpha i\upsilon}\mathbf{G}\_{\upsilon\varsigma}\mathbf{x}\big(s\big) \mathbf{ds} \end{split} \tag{44}$$

For any matrices , and *XY Z <sup>a</sup> ai* satisfying

$$
\begin{bmatrix}
\mathbf{X}\_{\rm av} & \mathbf{Y}\_{\nu} \\
\mathbf{Y}\_{\nu}^{T} & \mathbf{Z}\_{\rm aiv}
\end{bmatrix} \succeq \mathbf{0}
$$

Similarly, it holds that

$$\begin{split} & -\int\_{t-\tau\_{i}(t)}^{t} 2\mathbf{x}^{T}(t) \big( \mathcal{P}\_{k} + \varepsilon \mathbf{X} \big) D\_{i} \sum\_{j=1}^{r} h\_{j}(s) D\_{j} \mathbf{x} \big( \mathbf{s} - \boldsymbol{\tau}\_{j}(s) \big) ds \\ & \leq \tau\_{i}(t) \mathbf{x}^{T}(t) \mathbf{X}\_{di} \mathbf{x}(t) + 2\mathbf{x}^{T} \big( t \big) \big( \mathcal{Y}\_{i} - \big( \mathcal{P}\_{k} + \varepsilon \mathbf{X} \big) D\_{i} \big) \int\_{t-\tau\_{i}(t)}^{t} \sum\_{j=1}^{r} h\_{j}(s) D\_{j} \mathbf{x} \big( \mathbf{s} - \boldsymbol{\tau}\_{j}(s) \big) ds \\ & + \int\_{t-\tau\_{i}(t)}^{t} \sum\_{j=1}^{r} h\_{j}(s) \mathbf{x}^{T} \big( \mathbf{s} - \boldsymbol{\tau}\_{j}(s) \big) D\_{j}^{T} \mathbf{Z}\_{dij} D\_{j} \mathbf{x} \big( \mathbf{s} - \boldsymbol{\tau}\_{j}(s) \big) ds \end{split} \tag{45}$$

For any matrices , and *XY Z di i dij* satisfying

$$
\begin{bmatrix}
\mathbf{X}\_{di} & \mathbf{Y}\_i \\
\mathbf{Y}\_i^T & \mathbf{Z}\_{dij}
\end{bmatrix} \ge \mathbf{0}
$$

Hence, substituting (44) and (45) into (43), we have

492 Fuzzy Controllers – Recent Advances in Theory and Applications

*xt hhx h P X G D x t hh x t h P X D*

*j*

*T T ijk k ij i ijk k i*

*ij k k ij i i j kk i*

11 1 1 1 1

*rr r r r <sup>t</sup> <sup>r</sup> T T*

*ij k i j t t k*

*x t hhh x P X G D x t hhh x t P X D*

*j*

 

 

*tx tXxt x t Y P XD h s h s G x s ds*

0 *<sup>a</sup>*

*ai*

*t x t X x t x t Y P X D h s D x s s ds*

 

 

*i di i k i jj j*

 

*T*

*X Y Y Z* 

 

 

*i*

*j t t*

1

*i*

*t t*

 

1 1

   

 

*h s h s G x s D x s s ds*

, 2 2

*rrr rrr t*

*ijk ijk t t*

Using the bounding method in(10), by setting *a xt* and *ij b Gxs ,* we have

*x t P X D h s h s G x s ds*

 

1 1

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

*ai*

 

*ai* satisfying

*x t P X D h s D x s s ds*

*k ij j j*

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

1

2

*h s x s s D Z D x s s ds*

*j j j dij j j*

*h s h s x s G Z G x s ds*

, 2 2

 

2

*i ai i k i*

 

*<sup>t</sup> r r <sup>T</sup> k i*

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

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

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

*j t t*

 

*h s h s G x s D x s s ds*

111 111

*i*

 

(43)

(44)

(45)

 

*i*

and for <sup>2</sup> *x t*, we can written as,

1 1

1 1

 

*r r*

 

*i*

*t t*

2

 

For any matrices , and *XY Z <sup>a</sup>*

*i*

*t t*

Similarly, it holds that

*i*

2

*i*

*j t t*

1

For any matrices , and *XY Z di i dij* satisfying

1 1

 

 

*r r*

 

2

2

 111 1 1 1 1 1 1 1 1 2 2 *i i rrr <sup>T</sup> ijk k ij i ai di ijk r r <sup>t</sup> r r <sup>T</sup> ik i k i i k t t r r <sup>t</sup> r r T T i k ai i k t t V P hhh x t P X G D X X x t hh x t Y P X D h s h s G x s D x s s ds h h h s h s x s G Z G x s ds* 1 1 1 *i r r <sup>t</sup> <sup>r</sup> T T i k j j j dij j j i k j t t h h h s x s s D Z D x s s ds* 1 1 1 1 1 1 1 *i r r T TT i j k ij ij k i i aij di i j <sup>r</sup> <sup>T</sup> T T i i k ii i k i ii i i <sup>r</sup> <sup>t</sup> r r T T i ai i t t i j P hhx t P X G G P X Y Y X X x t h x t Y P X D S Y P X D x t x t t Sx t t h h s h s x s G Z G x s ds h h sx* 1 1 *<sup>i</sup> r r t T T j j dij j j i j t t s s D Z D x s s ds* (46)

Note that, by *Shur* complement, the LMI in (34) implies 0 *<sup>i</sup> L* for a sufficiently small scalar 0 , where

$$\begin{split} \mathcal{L}\_{i}\left(\mathcal{S}\right) &= P\_{\beta} + \left(P\_{k} + \varepsilon \mathcal{X}\right) \mathbf{G}\_{i\bar{j}} + \mathbf{G}\_{i\bar{j}}^{T} \left(P\_{k} + \varepsilon \mathcal{X}\right) + \mathbf{Y}\_{i} + \mathbf{Y}\_{i}^{T} + \tau \left(\mathbf{X}\_{\operatorname{adj}} + \mathbf{X}\_{\operatorname{d}i}\right) \\ &+ \left(\mathbf{Y}\_{i} - \left(P\_{k} + \varepsilon \mathcal{X}\right) \mathbf{D}\_{i}\right) \mathbf{S}\_{i}^{-1} \left(\mathbf{Y}\_{i} - \left(P\_{k} + \varepsilon \mathcal{X}\right) \mathbf{D}\_{i}\right)^{T} \mathbf{x}\left(t\right) + \left(2\tau + 1 + \tau \mathcal{S}\right) \left(1 + \mathcal{S}\right) \left(P\_{k} + \varepsilon \mathcal{X}\right) \end{split}$$

In order to use the *Razumikhin* Theorem, suppose *V xt* 1 *V xt* for ,0 . Then, if the LMIs in (35)–(39) also hold, we have from (46) that

 1 1 1 1 1 1 1 1 1 *i r r T TT i j k ij ij k i i aij di i j <sup>r</sup> <sup>T</sup> T T i i k ii i k i k i r r t T T i i k i k i i t t V hhx t P X G G P X Y Y X X x t h x t Y P XD S Y P XD xt x t P Xxt h t x t P X x t h x s P X x s ds* 

$$\begin{split} \leq & \sum\_{i=1}^{r} h\_{i} \mathbf{x}^{T} \left( t \right) \left[ \left( P\_{k} + \varepsilon \mathbf{X} \right) A\_{i} + A\_{i}^{T} \left( P\_{k} + \varepsilon \mathbf{X} \right) + Y\_{i} + Y\_{i}^{T} + \tau \left( \mathbf{X}\_{ai} + \mathbf{X}\_{di} \right) \right] \mathbf{x} \left( t \right) \\ \quad + \sum\_{i=1}^{r} h\_{i} \left[ \mathbf{x}^{T} \left( t \right) \left( Y\_{i} - \left( P\_{k} + \varepsilon \mathbf{X} \right) D\_{i} \right) S\_{i}^{-1} \left( Y\_{i} - \left( P\_{k} + \varepsilon \mathbf{X} \right) D\_{i} \right)^{T} \mathbf{x} \left( t \right) + \mathbf{x}^{T} \left( t \right) \left( 1 + \delta \right) \left( P\_{k} + \varepsilon \mathbf{X} \right) \mathbf{x} \left( t \right) \right] \right] \mathbf{x} \left( t \right) \\ \quad + \tau \mathbf{x}^{T} \left( t \right) \left( 1 + \delta \right) \left( P\_{k} + \varepsilon \mathbf{X} \right) \mathbf{x} \left( t \right) + \tau \mathbf{x}^{T} \left( t \right) \left( 1 + \delta \right)^{2} \left( P\_{k} + \varepsilon \mathbf{X} \right) \mathbf{x} \left( t \right) \\ \quad = & \sum\_{i=1}^{r} h\_{i} \mathbf{x}^{T} \left( t \right) L\_{i} \left( \delta \right) \mathbf{x} \left( t \right) \\ \quad \prec 0 \end{split}$$

which shows the motion of the unforced system **(7)** with *u = 0* is uniformly asymptotically stable. This completes the proof.

#### **6. Robust stability condition with PDC controller**

Consider the closed-loop system (5). A sufficient robust stability condition for Time-delay system is given follow.

#### **Theorem 5**

Under assumption 1, and assumption 2 and for given 0 1 , the Takagi-Sugeno system (5) is stable if there exist positive definite symmetric matrices , 1,2, , *<sup>k</sup> Pk r* , and *R*, matrices 1 , , *<sup>r</sup> F F* such that the following LMIs hols.

$$P\_k + R \succ 0, \quad k \in \{1, \ldots, r\} \tag{47}$$

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 495

0 0 <sup>0</sup>

0 0  and

(50)

*P RD D P RD D P RD E*

*T ij ji ij ji*

 1

 , 1,2, , *<sup>T</sup> V xt x t P Rxt k r k k* 

, , 0 1, 1 , and 0, 1,2, , *T T*

 

The time derivative of *V xt* with respect to *t* along the trajectory of the system (21) is

 1 1

*kk kk*

*k k*

*r*

*k V xt h zt V xt* 

*k k <sup>k</sup> P PRR*

*r r*

*k k V xt h zt V xt h zt V xt* 

 

,

*k ai aj k bi bj k di di di*

*I*

*P RP R P RE E E E*

*k k k ai aj ai aj*

1

, 1 ,

(51)

*PR k r* .

 

(52)

*r ii i i i m i m m G A BF h z t BK* 

 

where

*k k*

1

 

*r*

*k P PR*

*Proof* 

with

where

given by:

for , , 1,2, , such that

*T*

*r ij i i j m i m m G A BF h z t BK* 

*EF EF EF EF*

*bi j bj i bi j bj i*

1

Let consider the Lyapunov function in the following form:

, 1, ,

with

*ik r*

\* 0 0

*I*

*I*

*ijk r i j*

<sup>2</sup> 2 2

*T T*

*G G G G*

*I*

 

2

2

\* \*

$$P\_j + \mu \mathbb{R} \ge 0, \quad j = 1, 2, \dots, r \tag{48}$$

$$
\begin{bmatrix}
\Phi\_1 & \begin{pmatrix} P\_k + \mu \mathbb{R} \end{pmatrix} D\_{ai} & \begin{pmatrix} P\_k + \mu \mathbb{R} \end{pmatrix} D\_{bi} & \begin{pmatrix} P\_k + \mu \mathbb{R} \end{pmatrix} \begin{pmatrix} P\_{di} \Delta\_{di} E\_{di} \end{pmatrix} \\ \ast & -\lambda I & 0 & 0 \\ \ast & \ast & -\lambda I & 0 \\ \ast & \ast & \ast & 0 \\ \ast, k \in \{1, \ldots, r\} \end{bmatrix} \tag{49}
$$

with

$$\Phi\_1 = P\_\phi + \overline{G}\_{ii}^T \left( P\_k + \mu \mathcal{R} \right) + \left( P\_k + \mu \mathcal{R} \right) \overline{G}\_{ii} + \mathcal{A} \left( P\_k + \mu \mathcal{R} \right) \left[ E\_{ai}^T E\_{ai} + \left( E\_{bi} F\_i \right)^T E\_{bi} F\_i \right]$$

$$
\begin{bmatrix}
\Phi\_2 \\
\* & -\lambda I & 0 \\
\* & \* & -\lambda I
\end{bmatrix} \prec 0
$$

$$
\begin{bmatrix}
\Phi\_2 & \begin{pmatrix} P\_k + \mu R \end{pmatrix} \begin{pmatrix} D\_{il} + D\_{aj} \end{pmatrix} & \begin{pmatrix} P\_k + \mu R \end{pmatrix} \begin{pmatrix} D\_{bi} + D\_{bj} \end{pmatrix} & \begin{pmatrix} P\_k + \mu R \end{pmatrix} \begin{pmatrix} D\_{di}\Delta\_{di}E\_{di} \end{pmatrix} \\
\* & 0 \\
\* & \* & -\lambda I
\end{pmatrix} \prec 0
\tag{50}
$$

$$
i, k \in \{1, \ldots, r\}
$$

$$
\text{for } i, j, k = 1, 2, \ldots, r \text{ such that } i \prec j
$$

with

494 Fuzzy Controllers – Recent Advances in Theory and Applications

11

 

stable. This completes the proof.

system is given follow.

**Theorem 5** 

with

*T T*

1

*i*

1

*i*

0

*x t P Xxt x t P Xxt*

 

*k k*

1

*i k i ik i i ai di*

*<sup>r</sup> <sup>T</sup> T T*

*<sup>r</sup> T TT*

**6. Robust stability condition with PDC controller** 

Under assumption 1, and assumption 2 and for given 0 1

matrices 1 , , *<sup>r</sup> F F* such that the following LMIs hols.

*I*

, 1, ,

*ik r*

1

1

1

*i*

 

*<sup>r</sup> <sup>T</sup>*

*hx t P X A A P X Y Y X X x t*

*hx t L x t* 

 

0 0 <sup>0</sup> 0 0

*PR k r <sup>k</sup>* 0, 1, , (47)

(48)

, the Takagi-Sugeno system

(49)

 

2

*h x t Y P XD S Y P XD xt x t P Xxt*

*i i k ii i k i k*

 

*i i*

which shows the motion of the unforced system **(7)** with *u = 0* is uniformly asymptotically

Consider the closed-loop system (5). A sufficient robust stability condition for Time-delay

(5) is stable if there exist positive definite symmetric matrices , 1,2, , *<sup>k</sup> Pk r* , and *R*,

0, 1,2, , *<sup>j</sup> PR j r*

*<sup>k</sup> ai k bi k di di di P RD P RD P R D E*

*I*

<sup>1</sup>

 

*T T T ii k <sup>k</sup> ii k ai ai bi i bi i P G P R P RG P R EE EF EF*

 

 

$$\begin{split} \boldsymbol{\Phi}\_{2} &= \left(\frac{\overline{\boldsymbol{G}}\_{ij} + \overline{\boldsymbol{G}}\_{ji}}{2}\right)^{T} \left(\boldsymbol{P}\_{k} + \mu \boldsymbol{R}\right) + \left(\boldsymbol{P}\_{k} + \mu \boldsymbol{R}\right) \left(\frac{\overline{\boldsymbol{G}}\_{ij} + \overline{\boldsymbol{G}}\_{ji}}{2}\right) + \lambda \left(\boldsymbol{P}\_{k} + \mu \boldsymbol{R}\right) \left[\left(\boldsymbol{E}\_{ai} + \boldsymbol{E}\_{aj}\right)^{T} \left(\boldsymbol{E}\_{ai} + \boldsymbol{E}\_{aj}\right) \right] \\ &+ \left(\boldsymbol{E}\_{ki}\boldsymbol{F}\_{j} + \boldsymbol{E}\_{bj}\boldsymbol{F}\_{i}\right)^{T} \left(\boldsymbol{E}\_{bi}\boldsymbol{F}\_{j} + \boldsymbol{E}\_{bj}\boldsymbol{F}\_{i}\right)^{T} \end{split}$$
 
$$\text{where } \ \overline{\boldsymbol{G}}\_{ij} = \left[\boldsymbol{A}\_{i} - \boldsymbol{B}\_{i}\boldsymbol{F}\_{j} - \sum\_{m=1}^{r} \dot{h}\_{m}\left(\boldsymbol{z}\left(t\right)\right) \boldsymbol{B}\_{i}\boldsymbol{K}\_{m}\right], \boldsymbol{G}\_{ii} = \left[\boldsymbol{A}\_{i} - \boldsymbol{B}\_{i}\boldsymbol{F}\_{i} - \sum\_{m=1}^{r} \dot{h}\_{m}\left(\boldsymbol{z}\left(t\right)\right) \boldsymbol{B}\_{i}\boldsymbol{K}\_{m}\right], \ \mu = 1 - \boldsymbol{\varepsilon}, \text{ and }$$
  $\boldsymbol{P}\_{\phi} = \sum\_{k=1}^{r} \phi\_{k}\left(\boldsymbol{P}\_{k} + \boldsymbol{R}\right)$ 

*Proof* 

Let consider the Lyapunov function in the following form:

$$V\left(\mathbf{x}\left(t\right)\right) = \sum\_{k=1}^{r} h\_k\left(z\left(t\right)\right) \cdot V\_k\left(\mathbf{x}\left(t\right)\right) \tag{51}$$

with

$$V\_k\left(\mathbf{x}\left(t\right)\right) = \mathbf{x}^T\left(t\right)\left(P\_k + \mu \mathcal{R}\right)\mathbf{x}\left(t\right), \; k = 1, 2, \dots, r$$

where

$$P\_k = P\_k^T, \ R = \mathbb{R}^T, \ 0 \le \varepsilon \le 1, \mu = 1 - \varepsilon, \text{ and } \left(P\_k + \mu R\right) \ge 0, \quad k = 1, 2, \dots, r \dots$$

The time derivative of *V xt* with respect to *t* along the trajectory of the system (21) is given by:

$$\dot{V}\left(\mathbf{x}\left(t\right)\right) = \sum\_{k=1}^{r} \dot{h}\_{k}\left(\mathbf{z}\left(t\right)\right) V\_{k}\left(\mathbf{x}\left(t\right)\right) + \sum\_{k=1}^{r} h\_{k}\left(\mathbf{z}\left(t\right)\right) \dot{V}\_{k}\left(\mathbf{x}\left(t\right)\right) \tag{52}$$

The equation (52) can be rewritten as,

$$\begin{split} \dot{V}\left(\mathbf{x}\left(t\right)\right) &= \mathbf{x}^T\left(t\right) \Big( \sum\_{k=1}^r h\_k\left(\mathbf{z}\left(t\right)\right) \left(P\_k + \mu \mathbf{R}\right) \Big) \mathbf{x}\left(t\right) + \dot{\mathbf{x}}^T\left(t\right) \Big( \sum\_{k=1}^r h\_k\left(\mathbf{z}\left(t\right)\right) \left(P\_k + \mu \mathbf{R}\right) \Big) \mathbf{x}\left(t\right) \\ &+ \mathbf{x}^T\left(t\right) \Big( \sum\_{k=1}^r h\_k\left(\mathbf{z}\left(t\right)\right) \left(P\_k + \mu \mathbf{R}\right) \Big) \dot{\mathbf{x}}\left(t\right) \end{split} \tag{53}$$

By substituting (5) into (53), we obtain,

$$\dot{V}\left(\mathbf{x}\left(t\right)\right) = \Upsilon\_1\left(\mathbf{x}, z\right) + \Upsilon\_2\left(\mathbf{x}, z\right) + \Upsilon\_3\left(\mathbf{x}, z\right) \tag{54}$$

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 497

> 

0

*ai ai*

*E*

0

 

(57)

*x t*

 

*G G G G*

*kij*

*xz xt h zt h zt h zt P RP R x t*

*<sup>T</sup> r r <sup>T</sup> ij ji ij ji kij k k*

0

*E*

0

1 1

*xt xt h zt h zt h zt E F*

, 2 2

*kij ai bi k*

+

*bi bi j k i ij*

0 0

*aj aj aj aj*

*bj bj i bj bj i*

 

> 

*P RD E*

 

 

*T*

*T*

*D D P R P RD D*

1

*h zt R R* 

*k*

*aj bj k k aj bj*

*k di di di*

0

 

<sup>1</sup>

*xz xt P Rxt*

*<sup>r</sup> <sup>T</sup> k k*

0 0

0 0

*aj aj aj aj*

 

0

*k k ai bi bi j bi bi j*

 

*P R P RD D E F E F*

*E F E F*

(58)

where R is any symmetric matrix of

*bj bj i bj bj i*

*E E*

 

*E F E F*

*E E*

 

0 0

*xt h zt h zt h zt D D P R E F*

*r r ai <sup>T</sup> ai*

*<sup>T</sup> r r <sup>T</sup> ai ai*

*k i ij bi bi j*

0

*T*

*D D P R P RD D*

*x t t h zt h zt D E P R xt*

*x t h zt h zt P R D E xt t*

*r r <sup>T</sup>*

*i*

 

*i k k di di di i*

*E*

Then, based on assumption 1, an upper bound of <sup>1</sup> *x z*, obtained as:

,

1

*r k k*

*ij ji ij ji k k*

2 2

*G G G G P RP R*

 

*kij ij*

0

*ai ai*

*bi*

0

1

2

*DE PR*

*di di di k T*

*xt xt t*

*xz h zt h zt h zt*

*r ij i i j m i m m*

 

*G A BF h z t BK*

*E*

*aj bj k k aj bj*

0

*r r <sup>T</sup> <sup>T</sup> i i k di di di k*

where

3

where

with

,

1 1

*k ai bi*

1 1

*k i*

1 1

*k i ij TT T*

*ij T*

*ai bi*

Based on (3), it follows that

*D D*

*P RD D*

1 1

2

proper dimension.

Adding *R* to (55), then

*k i*

*r r <sup>T</sup>*

1 1

*k i ij*

3

where

$$\Upsilon\_1(\mathbf{x}, \mathbf{z}) = \mathbf{x}^T(t) \Big( \sum\_{k=1}^r \dot{h}\_k \left( \mathbf{z}(t) \right) \cdot \left( P\_k + \mu \mathbf{R} \right) \Big) \mathbf{x}(t) \tag{55}$$

 2 2 1 1 2 1 1 , 0 0 0 0 *r r T T k i ii k k ii k i <sup>T</sup> r r <sup>T</sup> ai ai k i ai bi k k i bi bi i ai ai k ai bi bi bi i T i xz x t h zt h zt G P R P RG xt <sup>E</sup> x t h zt h zt D D P R E F E P RD D x t E F xt t h* 1 1 1 1 *r r <sup>T</sup> i k di di di k k i r r <sup>T</sup> i k k di di di i k i zt h zt D E P R xt x t h zt h zt P R D E xt t* (56)

 2 2 1 1 1 1 , where 0 with 0 0 *r r <sup>T</sup> k i ii k i TT T i k di di di ii T di di di k T ii k k ii T ai ai ai bi k k ai bi bi bi i xz h zt h zt xt xt t P RD E DE PR G P R P RG E D D P R P RD D E F* 1 0 0 where *ai ai bi bi i r ii i i i m i m m E x t E F G A BF h z t BK* 

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 497

 3 1 1 1 1 , 2 2 0 0 *<sup>T</sup> r r <sup>T</sup> ij ji ij ji kij k k k i ij <sup>T</sup> r r <sup>T</sup> ai ai kij ai bi k k i ij bi bi j k ai bi G G G G xz xt h zt h zt h zt P RP R x t E xt h zt h zt h zt D D P R E F P RD D* 1 1 0 + 0 0 0 0 0 *r r ai <sup>T</sup> ai kij bi bi j k i ij T aj aj aj aj aj bj k k aj bj bj bj i bj bj i E xt xt h zt h zt h zt E F E E D D P R P RD D E F E F* 1 1 1 1 1 where *r r <sup>T</sup> <sup>T</sup> i i k di di di k k i r r <sup>T</sup> i k k di di di i k i r ij i i j m i m m x t x t t h zt h zt D E P R xt x t h zt h zt P R D E xt t G A BF h z t BK* (57)

 3 1 1 2 2 , where 0 with 2 2 0 0 *r r <sup>T</sup> kij ij k i ij TT T i k di di di ij T di di di k T ij ji ij ji k k ai ai ai bi bi xz h zt h zt h zt xt xt t P RD E DE PR G G G G P RP R E D D* 0 0 0 0 0 0 *T ai ai k k ai bi bi j bi bi j T aj aj aj aj aj bj k k aj bj bj bj i bj bj i E P R P RD D E F E F E E D D P R P RD D E F E F* 

Then, based on assumption 1, an upper bound of <sup>1</sup> *x z*, obtained as:

$$\mathbf{Y}\_1\left(\mathbf{x},\mathbf{z}\right) \le \sum\_{k=1}^r \phi\_k \cdot \mathbf{x}\left(t\right)^T \left(P\_k + \mu \mathbf{R}\right) \mathbf{x}\left(t\right) \tag{58}$$

Based on (3), it follows that 1 0 *r k k h zt R R* where R is any symmetric matrix of proper dimension.

Adding *R* to (55), then

496 Fuzzy Controllers – Recent Advances in Theory and Applications

1

*k*

*<sup>r</sup> <sup>T</sup>*

By substituting (5) into (53), we obtain,

*r r <sup>T</sup>*

*i*

*xt t h*

1 1

*k i*

*xz h zt h zt*

*TT T*

*ii T*

*T*

1 1

*k i*

1

1 1

*k ai bi*

*V xt x t h zt P R xt x t h zt P R xt*

 <sup>1</sup> 1

*<sup>E</sup> x t h zt h zt D D P R*

*E F*

*xz x t h zt h zt G P R P RG xt*

*k i ai bi k*

*zt h zt D E P R xt*

*xz x t h zt P R xt*

*k k*

*k i ii k k ii*

*k k k k*

 

<sup>123</sup> *V xt xz xz xz* , , , (54)

0

 

0

(55)

*E F*

(56)

*x t*

0

*ai ai bi bi i*

*E*

*E F*

0

   

(53)

1 1

*k k*

*<sup>r</sup> <sup>T</sup>*

2

*r r T T*

0

*ai ai*

*r r <sup>T</sup>*

*x t h zt h zt P R D E xt t*

*k*

*<sup>T</sup> r r <sup>T</sup> ai ai*

*k i bi bi i*

*bi bi i*

*i k di di di k*

*i k k di di di i*

*P RD E*

*k di di di*

0

*T*

*D D P R P RD D*

*ai bi k k ai bi*

 

*E*

*r r T T*

 

*k k*

,

1 1

*k i*

1 1

*i*

   

1

2

*xt xt t*

*r r <sup>T</sup> k i ii*

*DE PR*

*di di di k*

*G P R P RG*

*ii k k ii*

0

1

 

*r ii i i i m i m m*

*G A BF h z t BK* 

0

*ai ai*

 

 

*bi bi i*

*E F*

*E*

*k i*

*P RD D x t*

0

2

*x t h zt P R xt*

The equation (52) can be rewritten as,

2

*T*

2

where

with

where

,

,

where

$$\Upsilon\_1\left(\mathbf{x}, \mathbf{z}\right) \le \sum\_{k=1}^r \phi\_k \cdot \mathbf{x}\left(t\right)^T \left(P\_k + \mathcal{R}\right) \mathbf{x}\left(t\right) \tag{59}$$

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 499

*bi bj*

*D D*

0

Then, based on Lemma 2, an upper bound of <sup>1</sup> *x z*, obtained as:

\* \*

*E E EF EF P R*

*T T ai aj ai aj bi j bj i k*

 

2 2

*EF EF EF EF*

*bi j bj i bi j bj i*

*T*

<sup>2</sup> 2 2

*T T*

*G G G G*

by Schur complement, we obtain,

with

complete the proof.

with: 2 *r*

1 1

*hxt*

1

It is assumed that <sup>1</sup> <sup>2</sup> *x t*

**7. Numerical examples** 

Consider the following T-S fuzzy system:

the premise functions are given by:

<sup>1</sup>

1 sin

; <sup>2</sup> *x t*

37.7864 26.8058 ; 26.8058 36.2722 *<sup>P</sup>*

 <sup>1</sup> 2 1

> . For 11 12

> > 2

 0, 0.5, <sup>21</sup>

1

*bi j bj i*

<sup>2</sup> 0 \* 0

*T ij ji ij ji*

If (49) and (50) holds, the time derivative of the fuzzy Lyapunov function is negative. Consequently, we have *V xt* 0 and the closed loop fuzzy system (5) is stable. This is

> 1

*x t h z t Ax t* 

1 sin

*hxt* <sup>1</sup>

*i i*

; <sup>2</sup> *x t*

98.5559 28.7577 ; 28.7577 22.9286 *<sup>P</sup>*

0.5, and <sup>22</sup>

(60)

5 4 ; 1 2 *<sup>A</sup>* <sup>2</sup>

0, we obtain

 -1.2760 -2.2632 -2.2632 -0.6389 *<sup>R</sup>*

2 4 20 2 *<sup>A</sup>*

;

*r*

*i*

 

 

*<sup>k</sup> ai aj k bi bj P RD D P RD D I*

*I*

 

*P RP R P RE E E E*

*k k k ai aj ai aj*

 

*EF EF*

*E E*

*G G G G D D P RP R P RD D D D*

*<sup>T</sup> T T ij ji ij ji ai aj*

*k k k ai aj bi bj T T*

 

Then,

$$\dot{V}\left(\mathbf{x}\left(t\right)\right) \leq \sum\_{k=1}^{r} \phi\_k \mathbf{x}^T \left(t\right) \left(P\_k + R\right) \mathbf{x}\left(t\right) + \Upsilon\_2\left(\mathbf{x}, z\right) + \Upsilon\_3\left(\mathbf{x}, z\right)$$

If

$$\begin{aligned} & \quad H\_{11} & \quad \left(P\_k + \mu \mathbb{R}\right) D\_{di} \Delta\_{di} E\_{di} \\ \geq & \underline{E}\_{di}^T \Delta\_{di}^T \left(P\_k + \mu \mathbb{R}\right) & \quad 0 \\ \text{where} \quad \quad H\_{11} &= \sum\_{k=1}^r \phi\_k \left(P\_k + \mathbb{R}\right) + \overline{G}\_{ii}^T \left(P\_k + \mu \mathbb{R}\right) + \left(P\_k + \mu \mathbb{R}\right) \overline{G}\_{ii} \\ & \quad \quad + \left[\begin{bmatrix} E\_{ai} \\ -E\_{bi} F\_i \end{bmatrix}\right]^T \left(\left[D\_{ai} \quad D\_{bi}\right]\right)^T \left(P\_k + \mu \mathbb{R}\right) + \left(P\_k + \mu \mathbb{R}\right) \left[D\_{ai} \quad D\_{bi}\right] \begin{bmatrix} \Delta\_{ai} & 0 \\ 0 & \Delta\_{bi} \end{bmatrix} \begin{bmatrix} E\_{ai} \\ -E\_{bi} F\_i \end{bmatrix} \end{aligned}$$

Then, based on Lemma 2, an upper bound of *H*11 obtained as:

$$\begin{aligned} &\sum\_{k=1}^{r} \phi\_k \left( P\_k + \mathcal{R} \right) + \overline{G}\_{ii}^T \left( P\_k + \mu \mathcal{R} \right) + \left( P\_k + \mu \mathcal{R} \right) \overline{G}\_{ii} + \mathcal{A}^{-1} \left( P\_k + \mu \mathcal{R} \right) \left[ D\_{ai} \quad D\_{bi} \right] \begin{bmatrix} D\_{ai}^T \\ D\_{bi}^T \end{bmatrix} \\ &+ \lambda \left[ E\_{ai}^T \quad - \left( E\_{bi} F\_i \right)^T \right] \begin{bmatrix} E\_{ai} \\ -E\_{bi} F\_i \end{bmatrix} \left( P\_k + \mu \mathcal{R} \right) \prec 0 \end{aligned}$$

by Schur complement, we obtain,

$$
\begin{bmatrix}
\Phi\_1 & \left(P\_k + \mu R\right)D\_{ai} & \left(P\_k + \mu R\right)D\_{bi} \\
\ast & -\lambda I & 0 \\
\ast & \ast & -\lambda I
\end{bmatrix} \prec 0
$$

with

$$\Phi\_1 = P\_\phi + \overline{G}\_{ii}^T \left( P\_k + \mu R \right) + \left( P\_k + \mu R \right) \overline{G}\_{ii} + \lambda \left( P\_k + \mu R \right) \left[ E\_{ai}^T E\_{ai} + \left( E\_{bi} F\_i \right)^T E\_{bi} F\_i \right]$$

$$\begin{aligned} & \left\| \begin{pmatrix} \overline{\mathbf{C}\_{ij}} + \overline{\mathbf{C}\_{ji}} \\ 2 \end{pmatrix}^{T} \begin{pmatrix} P\_{k} + \mu \mathbf{R} \\ \end{pmatrix} + \begin{pmatrix} D\_{k} + \mu \mathbf{R} \\ \end{pmatrix} \begin{pmatrix} \overline{\mathbf{C}\_{ij}} + \overline{\mathbf{C}\_{ji}} \\ 2 \end{pmatrix} \right\| + \left\| \begin{bmatrix} D\_{ai} + D\_{aj} & D\_{bi} + D\_{bj} \\ 0 & \Delta\_{bi} + \Delta\_{bj} \end{bmatrix} \right\| \\ & \times \begin{bmatrix} E\_{ai} + E\_{aj} \\ -E\_{bi}F\_{j} - E\_{bj}F\_{i} \end{bmatrix} \right\|^{T} \begin{pmatrix} P\_{k} + \mu \mathbf{R} \\ \end{pmatrix} + \left( P\_{k} + \mu \mathbf{R} \right) \times \left[ \begin{bmatrix} D\_{ai} + D\_{aj} & D\_{bi} + D\_{bj} \\ 0 & \Delta\_{bi} + \Delta\_{bj} \end{bmatrix} \begin{bmatrix} \Delta\_{ai} + \Delta\_{aj} & 0 \\ 0 & \Delta\_{bi} + \Delta\_{bj} \end{bmatrix} \begin{bmatrix} E\_{ai} + E\_{aj} \\ -E\_{bi}F\_{j} - E\_{bj}F\_{i} \end{bmatrix} \right] \\ & \prec 0 \end{aligned}$$

Then, based on Lemma 2, an upper bound of <sup>1</sup> *x z*, obtained as:

$$
\begin{split} & \left[ \frac{\overline{\mathbf{C}}\_{\overline{\boldsymbol{n}}} + \overline{\mathbf{C}}\_{\overline{\boldsymbol{n}}}}{2} \right]^{T} \left( \mathbf{P}\_{k} + \mu \mathbf{R} \right) + \left( \mathbf{P}\_{k} + \mu \mathbf{R} \right) \left[ \frac{\overline{\mathbf{C}}\_{\overline{\boldsymbol{n}}} + \overline{\mathbf{C}}\_{\overline{\boldsymbol{n}}}}{2} \right] + \lambda^{-1} \left( \mathbf{P}\_{k} + \mu \mathbf{R} \right) \left[ \mathbf{D}\_{\operatorname{ai}} + \mathbf{D}\_{\operatorname{aj}} \quad \mathbf{D}\_{\operatorname{bi}} + \mathbf{D}\_{\operatorname{bj}} \right] \begin{bmatrix} \mathbf{D}\_{\operatorname{ai}}^{T} + \mathbf{D}^{T}\_{\operatorname{aj}} \\ \mathbf{D}\_{\operatorname{bi}}^{T} + \mathbf{D}^{T}\_{\operatorname{bj}} \end{bmatrix} \\ & + \lambda \left[ \left( \mathbf{E}\_{\operatorname{ai}} + \mathbf{E}\_{\operatorname{aj}} \right)^{T} \quad \left( -\mathbf{E}\_{\operatorname{bi}} \mathbf{F}\_{j} - \mathbf{E}\_{\operatorname{bj}} \mathbf{F}\_{i} \right)^{T} \right] \times \begin{bmatrix} \mathbf{E}\_{\operatorname{ai}} + \mathbf{E}\_{\operatorname{aj}} \\ -\mathbf{E}\_{\operatorname{bi}} \mathbf{F}\_{j} - \mathbf{E}\_{\operatorname{bj}} \mathbf{F}\_{i} \end{bmatrix} \end{split} $$

by Schur complement, we obtain,

$$
\begin{bmatrix}
\Phi\_2 & \begin{pmatrix} P\_k + \mu R \end{pmatrix} \begin{pmatrix} D\_{ai} + D\_{aj} \end{pmatrix} & \begin{pmatrix} P\_k + \mu R \end{pmatrix} \begin{pmatrix} D\_{bi} + D\_{bj} \end{pmatrix} \\
\ast & -\lambda I & 0 \\
\ast & \ast & -\lambda I
\end{pmatrix} \prec 0
$$

with

498 Fuzzy Controllers – Recent Advances in Theory and Applications

Then,

If

<sup>1</sup>

*xz xt P Rxt* 

0

*r T*

*P R G P R P RG P R D D*

 

<sup>1</sup>

<sup>1</sup>

 

*ai aj ai aj a k k ai aj bi bj bi j bj i bi bj*

*E E E P R P R DDDD*

2 2 0

*P RP R DDDD*

*ij ji ij ji ai aj k k ai aj bi bj*

 

*k k ii k k ii k ai bi T k bi*

\* 0 0

*I*

*T T T ii k <sup>k</sup> ii k ai ai bi i bi i P G P R P RG P R EE EF EF*

 

 

0

*<sup>k</sup> ai k bi P RD P RD I*

2 3 , , *<sup>r</sup>*

*t P R t xz xz x*

*<sup>T</sup> ai ai ai ai bi k k ai bi bi i bi bi i*

 

*E F E F*

 

1

*T ai*

*E E D D P R P RD D*

0

(59)

0

*D*

*D*

0

*bi bj*

*i aj bi j bj i E EF EF* 

0

0

*<sup>r</sup> <sup>T</sup> k k*

1

*k*

,

1

0

*k di di di*

*H P R G P R P RG*

Then, based on Lemma 2, an upper bound of *H*11 obtained as:

*<sup>E</sup> E EF P R E F*

*bi i*

\* \*

*H P RD E*

 

*<sup>r</sup> <sup>T</sup>*

*T*

*T T ai ai bi i k*

1

*k*

11

*E DP R*

11

1

by Schur complement, we obtain,

*T*

*T*

*EF EF*

0

*G G G G*

 

 

where

with

*TT T di di di k*

*k k*

*T <sup>k</sup> V xt x* 

*k k ii k k ii*

$$\begin{split} \boldsymbol{\Phi}\_{2} &= \left(\frac{\overline{\mathbf{G}}\_{ij} + \overline{\mathbf{G}}\_{ji}}{2}\right)^{T} \left(\boldsymbol{P}\_{k} + \mu \boldsymbol{\mathcal{R}}\right) + \left(\boldsymbol{P}\_{k} + \mu \boldsymbol{\mathcal{R}}\right) \left(\frac{\overline{\mathbf{G}}\_{ij} + \overline{\mathbf{G}}\_{ji}}{2}\right) + \boldsymbol{\mathcal{A}} \left(\boldsymbol{P}\_{k} + \mu \boldsymbol{\mathcal{R}}\right) \left[\left(\boldsymbol{E}\_{ai} + \boldsymbol{E}\_{aj}\right)^{T} \left(\boldsymbol{E}\_{ai} + \boldsymbol{E}\_{aj}\right)\right] \\ &+ \left(\boldsymbol{E}\_{bi}\boldsymbol{F}\_{j} + \boldsymbol{E}\_{bj}\boldsymbol{F}\_{i}\right)^{T} \left(\boldsymbol{E}\_{bi}\boldsymbol{F}\_{j} + \boldsymbol{E}\_{bj}\boldsymbol{F}\_{i}\right)^{T} \end{split}$$

If (49) and (50) holds, the time derivative of the fuzzy Lyapunov function is negative. Consequently, we have *V xt* 0 and the closed loop fuzzy system (5) is stable. This is complete the proof.

## **7. Numerical examples**

Consider the following T-S fuzzy system:

$$\dot{\mathbf{x}}\{t\} = \sum\_{i=1}^{r} h\_i\{z(t)\} A\_i x(t) \tag{60}$$

with: 2 *r*

the premise functions are given by:

$$h\_1(\mathbf{x}\_1(t)) = \frac{1 + \sin x\_1(t)}{2};\ \ h\_2(\mathbf{x}\_1(t)) = \frac{1 - \sin x\_1(t)}{2};\ \ A\_1 = \begin{bmatrix} -5 & -4\\ -1 & -2 \end{bmatrix};\ \ A\_2 = \begin{bmatrix} -2 & -4\\ 20 & -2 \end{bmatrix};\ \mathbf{x}\_2(t) = \frac{1}{2};\ \mathbf{x}\_3(t) = \frac{1}{2};\ \mathbf{x}\_4(t) = \frac{1}{2}$$

It is assumed that <sup>1</sup> <sup>2</sup> *x t* . For 11 12 0, 0.5, <sup>21</sup> 0.5, and <sup>22</sup> 0, we obtain

$$P\_1 = \begin{bmatrix} 37.7864 & 26.8058\\ 26.8058 & 36.2722 \end{bmatrix}; \quad P\_2 = \begin{bmatrix} 98.5559 & 28.7577\\ 28.7577 & 22.9286 \end{bmatrix}; \quad R = \begin{bmatrix} -1.2760 & -2.2632\\ -2.2632 & -0.6389 \end{bmatrix}$$

**Figure 1.** State variables

Figure 3 shows the evolution of the state variables. As can be seen, the conservatism reduction leads to very interesting results regarding fast convergence of this Takagi-Sugeno fuzzy system.

In order to show the improvements of proposed approaches over some existing results, in this section, we present a numerical example, which concern the feasibility of a time delay T-S fuzzy system. Indeed, we compare our fuzzy Lyapunov-Razumikhin approach (Theorem 3.1) with the Lemma 2.2 in [9].

**Example 2.** Consider the following T-S fuzzy system with u=0:

$$\dot{\mathbf{x}}(t) = \sum\_{i=1}^{2} h\_i \left( \mathbf{z}(t) \right) \left\{ A\_i \mathbf{x}(t) + D\_i \mathbf{x} \left( t - \tau\_i(t) \right) \right\},\tag{61}$$

Robust Stabilization for Uncertain Takagi-Sugeno Fuzzy Continuous Model with Time-Delay Based on Razumikhin Theorem 501

> 1.021 0.064 , 0.064 0.664 *<sup>S</sup>*

> > *Za*

0.201 0.087 , 0.087 0.369

1.448 0.094 , 0.094 2.353

**Methods** max

Lemma 2.1 0.6308 Theorem 3.1 +

for Example 1

The LMIs in (34)-(39) are feasible by choosing , *X X ai a* , *X X di d* , *Y Y <sup>i</sup>* , *Z Z aij a* , *Z Z dij d*

a feasible solution is given by

*Xd*

This chapter provided new conditions for the stabilization with a PDC controller of Takagi-Sugeno fuzzy systems with time delay in terms of a combination of the Razumikhin theorem and the use of non-quadratic Lyapunov function as Fuzzy Lyapunov function. In addition, the time derivative of membership function is considered by the PDC fuzzy controller in order to facilitate the stability analysis. An approach to design an observer is derived in order to estimate variable states. In addition, a new condition of the stabilization

The stabilization condition proposed in this note is less conservative than some of those in

[1] T. Takagi, and M. Sugeno, "Fuzzy identification of systems and its application to modeling and control," *IEEE Trans. On System, Man and Cybernetics*, vol 15 (1), pp. 116–

[2] M.A.L. Thathachar, P. Viswanah, "On the Stability of Fuzzy Systems", *IEEE Transactions* 

0.849 0.227 , 0.227 0.246

1.451 0.178 , 0.178 0.883 *<sup>P</sup>*

2.523 0.707 , 0.707 2.155

*Zd*

1.5121 0.1801 , 0.1801 1.1057 *<sup>P</sup>* <sup>2</sup>

*Xa*

**Table 1.** Comparison results of maximum

and , *<sup>i</sup> S S i j* , 1,2, and for 0.5

0.611 0.169 , 0.243 0.421 *<sup>Y</sup>* 

of uncertain system is given in this chapter.

Yassine Manai and Mohamed Benrejeb

the literature, which has been illustrated via examples.

*National Engineering School of Tunis, LR-Automatique, Tunis, Tunisia* 

*on Fuzzy Systems,* Vol. 5, N°1, pp. 145 – 151, February 1997.

1

**8. Conclusion** 

**Author details** 

**9. References** 

132, 1985.

with:

$$A\_1 = \begin{bmatrix} -2.1 & 0.1 \\ -0.2 & -0.9 \end{bmatrix}, \ A\_2 = \begin{bmatrix} -1.9 & 0 \\ -0.2 & -1.1 \end{bmatrix}, \ D\_1 = \begin{bmatrix} -1.1 & 0.1 \\ -0.8 & -0.9 \end{bmatrix}, \ D\_2 = \begin{bmatrix} -0.9 & 0 \\ -1.1 & -1.2 \end{bmatrix}.$$

with the following membership functions :

$$h\_1 = \sin^2\left(\mathbf{x}\_1 + 0.5\right); \qquad h\_2 = \cos^2\left(\mathbf{x}\_1 + 0.5\right).$$

Assume that *<sup>i</sup>t xt xt* 0.5 sin 1 2 <sup>1</sup> where 1 2 , *<sup>T</sup> xt x t x t* . Then,

 0.5 *<sup>i</sup> t* . Table 1. shows that our approach is less conservative than Lemma 2.2. given in [9].


**Table 1.** Comparison results of maximumfor Example 1

The LMIs in (34)-(39) are feasible by choosing , *X X ai a* , *X X di d* , *Y Y <sup>i</sup>* , *Z Z aij a* , *Z Z dij d* and , *<sup>i</sup> S S i j* , 1,2, and for 0.5 a feasible solution is given by

$$P\_1 = \begin{bmatrix} 1.5121 & -0.1801 \\ -0.1801 & 1.1057 \end{bmatrix}, P\_2 = \begin{bmatrix} 1.451 & -0.178 \\ -0.178 & 0.883 \end{bmatrix}, S = \begin{bmatrix} 1.021 & -0.064 \\ -0.064 & 0.664 \end{bmatrix}'$$

$$Y = \begin{bmatrix} -0.611 & 0.169 \\ -0.243 & -0.421 \end{bmatrix}, X\_d = \begin{bmatrix} 2.523 & 0.707 \\ 0.707 & 2.155 \end{bmatrix}, X\_d = \begin{bmatrix} 1.448 & 0.094 \\ 0.094 & 2.353 \end{bmatrix}, Z\_d = \begin{bmatrix} 0.201 & -0.087 \\ -0.087 & 0.369 \end{bmatrix}'$$

$$Z\_d = \begin{bmatrix} 0.849 & -0.227 \\ -0.227 & 0.246 \end{bmatrix}'$$

## **8. Conclusion**

500 Fuzzy Controllers – Recent Advances in Theory and Applications

Figure 3 shows the evolution of the state variables. As can be seen, the conservatism reduction leads to very interesting results regarding fast convergence of this Takagi-Sugeno

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

In order to show the improvements of proposed approaches over some existing results, in this section, we present a numerical example, which concern the feasibility of a time delay T-S fuzzy system. Indeed, we compare our fuzzy Lyapunov-Razumikhin approach (Theorem

1 1 *h x* sin 0.5 ; <sup>2</sup>

*<sup>i</sup>t xt xt* 0.5 sin 1 2 <sup>1</sup> where 1 2 , *<sup>T</sup>*

*t* . Table 1. shows that our approach is less conservative than Lemma 2.2.

*x t h z t Ax t Dx t t*

1.9 0 , 0.2 1.1 *<sup>A</sup>* <sup>1</sup>

<sup>2</sup>

, *i i ii*

x1 x2

(61)

1.1 0.1 , 0.8 0.9 *<sup>D</sup>* <sup>2</sup>

2 1 *h x* cos 0.5 .

0.9 0 , 1.1 1.2 *<sup>D</sup>*

*xt x t x t* . Then,

**Figure 1.** State variables


3.1) with the Lemma 2.2 in [9].

**Example 2.** Consider the following T-S fuzzy system with u=0:

2

1

*i*

fuzzy system.

with:

1

Assume that

0.5 *<sup>i</sup>*

 

given in [9].

2.1 0.1 , 0.2 0.9 *<sup>A</sup>*

<sup>2</sup>

with the following membership functions :

This chapter provided new conditions for the stabilization with a PDC controller of Takagi-Sugeno fuzzy systems with time delay in terms of a combination of the Razumikhin theorem and the use of non-quadratic Lyapunov function as Fuzzy Lyapunov function. In addition, the time derivative of membership function is considered by the PDC fuzzy controller in order to facilitate the stability analysis. An approach to design an observer is derived in order to estimate variable states. In addition, a new condition of the stabilization of uncertain system is given in this chapter.

The stabilization condition proposed in this note is less conservative than some of those in the literature, which has been illustrated via examples.

## **Author details**

Yassine Manai and Mohamed Benrejeb *National Engineering School of Tunis, LR-Automatique, Tunis, Tunisia* 

#### **9. References**

	- [3] L. K. Wong, F.H.F. Leung, P.K.S. Tam, "Stability Design of TS Model Based Fuzzy Systems", *Proceedings of the Sixth IEEE International Conference on Fuzzy Systems*, Vol. 1, pp. 83–86, 1997.

**Chapter 21** 

© 2012 Kaddouri et al., licensee InTech. This is an open access chapter 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 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution,

**A Two-Layered Load and Frequency Controller** 

Automatic generation control (AGC) or called load frequency control (LFC) has gained a lot of interests in the past 30 decade (Benjamin &. Chan, 1982; Pan & Liaw, 1989; Kothari et al., 1989; Y. Wang et al., 1994; Indulkar & Raj, 1995; Karnavas & Papadopoulos, 2002; Moon et al., 2002; Sherbiny et al., 2003). LFC insures a sufficient and reliable supply of power with good quality. To ensure the quality of the power supply, it is necessary to deal with the control of the generator loads depending on the frequency with a proper LFC design. Therefore, the design of the controller is faced with nonlinear effects due to the physical components of the system, such as governor dead zone and generation rate constraints (GRC) and its complexity and the inherent characteristics of changing loads and parameters. Most actuators used in practice contain static (dead zone) or dynamic (backlash) nonsmooth nonlinearities. These actuators are present in most mechanical and hydraulic systems such as servo valves. Their mathematical models are poorly known and limit the static and dynamic performance of feedback control system (Corradini & Orlando,2002). Conventional PI controller has been often used to achieve zero steady state frequency deviation. However, because of the load changing, the operating point of a power system may change very much during a daily cycle (Pan & Liaw, 1989). Therefore, a PI controller which is fixed and optimal when considering one operating point may no longer be suitable with various statuses. On the other hand, it is known that the classical LFC does not yield adequate control performance with consideration of the speed – governor non-smooth

nonlinearities and GRC (Karnavas & Papadopoulos, 2002; Moon et al., 2002).

The problem of controlling systems with dead-zone nonlinearity has been addressed in the literature using various approaches some of which are dedicated to power systems. Reference (Tao & Kokotovic, 1994; X.-S. Wang et al., 2004) proposed adaptive schemes with and without dead zone inverse scheme, respectively, to track the error caused by the dead

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

Mavungu Masiala, Mohsen Ghribi and Azeddine Kaddouri

Additional information is available at the end of the chapter

**of a Power System** 

http://dx.doi.org/10.5772/48404

**1. Introduction** 

