1. Introduction

In practice, in order to realize a feedback stabilization, we must know the state of the system. This state can be obtained by several ways. One way consists to design an observer. The theory which combines the couple (observer, controller system) is variously called separation principle, dynamic output feedback controller or observer-based controller. In this paper, we are concerned with the separation approach to the design of stabilizing output feedback control using T-S fuzzy observer.

Recall that, T-S models [1] have been widely used for analysis and controller synthesis of nonlinear systems. The advantage of such approach relies on the fact that once the T-S fuzzy models are obtained, some analysis and design tools developed in the linear case can be used, which facilitates observer or/and controller synthesis for complex nonlinear systems see for example [2] and the references therein. However, many research on T-S fuzzy control and state observation for nonlinear systems described by ordinary differential equations (ODEs) exist in the literature. Indeed, the stability and the stabilization of such systems are mainly investigated through the direct Lyapunov method [3], [4]. Likewise, various works have been proposed to design observers [5-8]. The observer-based controller synthesis has been studied extensively with great success for such nonlinear systems. We may cite [9-13]. For nonlinear descriptor systems described by T-S models, the problems of control and observation design are mainly investigated in the literature, see for instance [14-17]. Notice that, generally an interesting way to solve the various problems raised previously (control and observation) is to write the convergence conditions on the LMI form.

In this paper, we deal with descriptor nonlinear systems. In fact, many chemical and physical processes can be described by nonlinear systems of differential and algebraic equations [18-20]. These systems are variously called descriptor systems, singular systems, or differential algebraic equations (DAEs). This formulation includes both dynamic and static relations. The numerical simulation of such descriptor models usually combines an ODE numerical method together with an optimization algorithm.

The main contribution of this paper is to give an observer-based controller design for a class of T-S descriptor models with measurable premise variables. A new design methodology through judicious use of the inequality of Young is proposed. The developed result is based on the separation between the dynamic and static relations in the descriptor model. The asymptotic stability of the closed-loop state system is studied by using the Lyapunov theory and the stability conditions are given in terms of LMIs. Besides, the proposed dynamic controller for a class of T-S descriptor systems can be synthesized by only an ODE structure.

The outline of the paper is as follows. The class of the fuzzy T-S structure of nonlinear descriptor systems is introduced in Section 2. The main result about fuzzy observer-based controller design for a class of T-S descriptor models with measurable premise variables is stated in sections 3. The control and observer gains are found directly from LMI formulation. An illustrative application is given in Section 4 to show the efficiency of the proposed approach.

In this paper, some notations used are fair standard. For example, means the matrix is symmetric and positive definite. denotes the transpose of . The symbol (or 0) represents the identity matrix (or zero matrix) with appropriate dimension, and .

2. System Description

In this paper, the following class of T-S descriptor systems is considered:

(1)

Where is the state vector with is the vector of differential variables, is the vector of algebraic variables, is the control input, is the measured output. are real known constant matrices with:

(2)

where constant matrices are supposed invertible

The premise variable is supposed to be real-time accessible and depended only on . is the number of sub-models. The are the weighting functions that ensure the transition between the contribution of each sub model:

(3)

They verify the so-called convex sum properties:

(4)

In what follows and to prove the convergence of the observer-based controller proposed in this work, we consider the following assumptions [20]:

Assumption A1: is regular, i.e.

Assumption A2: All sub-models (3) are impulse controllable and stabilisable.

Assumption A3: All sub-models (3) are impulse observable and detectable.

The main objective of this work is to design an observer-based controller for system (1) which assures asymptotic convergence to zero of the closed-loop system. The approach is based on the separation between dynamic and static relations in each sub-model (3) and the global fuzzy model is obtained by aggregation of the resulting sub-models. So, using (2), system (3) can be rewritten as follows:

(5)

The form (5) for system (3) is also known as the second equivalent form [20].

From (5) and using the fact that exist, the algebraic equations can be solved directly for algebraic variables, to obtain:

(6)

Substitution of the resulting expression for (equation (6) in equation (5) yields the following model:

(7)

Where

(8)

In descriptor form, subsystems (7) take the following equivalent form of sub-models (3)

(9)

Where

(10)

Then, the fuzzy descriptor system (1) can be rewritten in the following equivalent form:

(11)

So using (4) and (10), system (11) can be rewritten as follows:

(12)

3. Main Results

The objective is to design an observer-based controller for the T-S descriptor system (1). Therefore, first a new feedback controller design method is proposed. Secondly, an observer design approach permitting to estimate the unknown state is given.

3.1. Feedback Controller Design

In this section, the goal is to determine the gains of the following control law:

(13)

In order that the closed-loop system (1) is asymptotically stable.

Let the local linear feedback controller related with each submodel (7) given by:

(14)

and the expression of giving in (7) by:

(15)

So, in order to be able to state the result as LMIs, the key idea consists in substituting (15) in (14). Then, we obtain:

(16)

Where

(17)

Thus, the fuzzy state feedback controller (13) can be rewritten in the following equivalent form:

(18)

By substituting (18) into (12), the following closed-loop fuzzy system can be represented as:

(19)

Where

(20)

In order to prove that the closed-loop system (19) converges to zero, it suffice to prove that converges toward zero. Then, the following result can be stated.

Theorem 1.: The closed-loop system described by (19) is globally exponentially stable if there exist matrices , verifying the following LMIs:

(21)

The fuzzy local feedback gains are given by:

(22)

Note that, to determine the control gains and given in (13), according to the expression of given in (17), it is clear that we have:

(23)

Where

(24)

Then, a particular solution of (23) using the pseudo inverse matrix denoted is given by:

(25)

Proof: Let us consider the quadratic Lyapunov function as follows:

(26)

The time derivative of the Lyapunov function (26) along the trajectories of the system (19) is obtained as:

(27)

Therefore, we have the following stability conditions:

(28)

From (20), inequality (28) becomes:

(29)

Let . Premultiply and postmultiply (29) by , to obtain the LMI conditions given in (21). From the Lypunov stability theory, if the LMI conditions (21) are satisfied, the closed-loop system (19) is exponentially asymptotically stable. This completes the proof of Theorem 1.

3.2. Observer Design

As it was mentioned before, to apply state feedback control it is necessary to know all the states of the system. In real application, not all the states are available to be measured, thus the necessity to use an observer arises. Therefore, in this section, we propose an observer design for a class of T-S descriptor systems (1). Based on the form (12), the proposed observer takes the following form:

(30)

where and denote the estimated state vector and output vector respectively. The activation functions are the same than those used in the T-S model (15). are the gains of observer witch are to determined such that asymptotically converges to .

Denoting the state estimation error by:

(31)

It follows from (12) and (30) that the estimated error equation can be written as:

(32)

Where

(33)

To prove the convergence of the estimation error toward zero, it suffices to prove from (32), that converges toward zero. The result is stated in the following theorem.

Theorem 2.: There exists an observer (30) for (1) if there exist matrices verifying the following LMIs:

(34)

The fuzzy local observer gains , are given by:

(35)

Proof: Let us consider the following quadratic Lyapunov function as follows:

(36)

Estimation error convergence is ensured if the following condition is guaranteed:

(37)

By using (32), the condition (37) can be written as:

(38)

which is equivalent to the following stability conditions:

(39)

Letting , it follows that (39) is equivalent to (34). From the Lypunov stability theory, if the LMI conditions (34) are satisfied, the estimated error equation (32) is exponentially asymptotically stable.

3.3. Dynamic Output Feedback Controller Design

In this section, we propose a novel approach for the separation principle theory of a class of T-S descriptor systems (1). A fuzzy dynamic output feedback controller for T-S descriptor system (1) is defined by:

(40)

in order that the closed-loop system be asymptotically stable where and are the control gains to be determined.

As in subsection 3.1, in order to be able to state the result in the form of LMIs, the key idea lies in the use of the equivalent control law of (40) given by:

(41)

where is defined in (17) which can be determined by Theorem 3 below and the control gains and used in (40) can be determined by equation (25).

Let

(42)

Combining (30), (32) and (41), we obtain the following augmented closed-loop system:

(43)

Where

(44)

and and are defined in (20) and (33) respectively.

The following theorem provides the main result of this paper.

Theorem 3.: The closed-loop T-S model described by (43) is globally asymptotically stable if for a fixed scalar , there exist matrices verifying the following LMIs:

(45)

where

(46)

The dynamic controller gains and are given by:

(47)

Proof: To prove the convergence to zero of the state variable system (43), it suffices to prove that converges toward zero. Thus, let us consider the candidate Lyapunov function as follows:

(48)

With

(49)

The time derivative of the Lyapunov function (48) along the trajectories of the system (43) is obtained as:

(50)

Therefore, we have the following stability conditions:

(51)

which become from (44) and (49):

(52)

Now multiplying the two inequalities given in (52) on the left and right by and respectively and defining new variables we obtain:

(53)

By considering the following:

(54)

Matrices and can be rewritten again as:

(55)

Lemma 1: Young's Inequality

For any matrices and with appropriate dimensions, the following property holds for any invertible matrix and scalar

(56)

Applying Lemma 1 and taking , (55) becomes:

(57)

Hence, using the Schur complement [21], the inequalities and hold if the following two matrix inequalities are satisfied.

(58)

Where

(59)

Then, from (20), (33) and the use of the changes of variables:

(60)

we establish the LMI conditions (45) given in Theorem 3.

4. Application to an Inverted Pendulum System

To illustrate the effectiveness of the proposed method, we consider a nonlinear inverted pendulum system controlled by a separately excited direct-current (DC) motor [22]. Using the procedure of fuzzy model construction given in [23], the following T-S descriptor model with measurable premise variables can be derived:

(61)

where is the state vector, denotes the control input voltage, is the measured output. is the angle of the pendulum, is the angular velocity, is the current of the motor.

(62)

Note that, the application of the proposed dynamic controller (40) for inverted pendulum system requires that the above model (61) takes the form (12). To do so, let:

With:

This show that the model (61) is a particular case of the system (1). Then, the fuzzy descriptor system (61) can be rewritten in the following equivalent form:

(63)

With (see equation (8)):

Consequently, using Theorem with , the following dynamic controller gains and were obtained:

(64)

(65)

Simulation results with initial conditions

are presented in Figures 1, 2 and 3. These simulation results show the performance of the proposed observer-based controller designed with the parameters , given by (64) - (65). They show that the global asymptotic stability of the closed loop T-S system with the control law is satisfied. Moreover, Figures 2 and 3 show that the observer gives a good estimation of angular velocity and current of the motor.

Figure 1. and with fuzzy observer-based controller

Figure 2. and with fuzzy observer-based controller

Figure 3. and with fuzzy observer-based controller

On the other hand, using (25), we obtain the following control gains:

Notice that with these numerical values of and , we obtain exactly the same simulation results given in Figures 1, 2 and 3.

5. Conclusions

An observer-based fuzzy controller design approach for a class of T-S descriptor systems with measurable premise variables is proposed in this paper. The approach is based on the separation between dynamic and static relations in the descriptor model. The convergence of the closed-loop system is studied by using the Lyapunov theory and the stability conditions are given in terms of LMIs. Simulation results are given and demonstrate the good performance of the proposed dynamic controller.