Comparison results of maximum

## 1. Introduction

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].

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 time-delay 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

## 2. System description and preliminaries

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

where

where

The final outputs of the fuzzy systems are:

where

*t*.

The term

Since

we have *t*.

The time derivative of premise membership functions is given by:

We have the following property:

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

The fuzzy controller design consists to determine the local feedback gains

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

where

The system without uncertainties is given by equation

The open-loop system is given by the equation ,

## Assumption. 1

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

## Assumption. 2

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

where

where I is an appropriately dimensioned identity matrix.

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

Given constant matrices

if and only if

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

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

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

Suppose

then the solution

**Lemma. 3** [6]

Assume that

where

**Lemma. 4** [9]

The unforced fuzzy time delay system described by with u = 0 is uniformly asymptotically stable if there exist matrices

## 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.

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

## Theorem. 2 [17]

Under assumption 1 and for

where

## Proof.

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

where

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

where

And

## 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:

The closed-loop fuzzy system can be represented as:

The augmented system is represented as follows:

where

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

## Theorem. 3

Under assumption 1 and for given

where

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

## Theorem. 4

Under assumption 1 and for given

where

## Proof.

Let consider the fuzzy Lyapunov function as

Given the matrix property, clearly,

where

Finding the maximum value of

Finding the minimum value of

Define

Then,

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

Since

The state equation of with u=0 can be rewritten as

where

The derivative of V along the solutions of the unforced system with

Then, based on assumption 1, an upper bound of

and for

Using the bounding method in, by setting

For any matrices

Similarly, it holds that

For any matrices

Hence, substituting and into , we have

Note that, by Shur complement, the LMI in implies

In order to use the Razumikhin Theorem, suppose

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

## 6. Robust stability condition with PDC controller

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

## Theorem. 5

Under assumption 1, and assumption 2 and for given

with

where

Proof

Let consider the Lyapunov function in the following form:

with

where

The time derivative of

The equation can be rewritten as,

By substituting into , we obtain,

where

Then, based on assumption 1, an upper bound of

Based on , it follows that

Adding

Then,

If

Then, based on Lemma 2, an upper bound of

by Schur complement, we obtain,

with

Then, based on Lemma 2, an upper bound of

by Schur complement, we obtain,

with

If and holds, the time derivative of the fuzzy Lyapunov function is negative. Consequently, we have

## 7. Numerical examples

Consider the following T-S fuzzy system:

with:

the premise functions are given by:

It is assumed that

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:

with:

with the following membership functions :

Assume that

Methods | τmax |

Lemma 2.1 | 0.6308 |

Theorem 3.1 | + |

The LMIs in - are feasible by choosing

## 8. Conclusion

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.