Nonquadratic Lyapunov Functions for Nonlinear Takagi-Sugeno Discrete Time Uncertain Systems Analysis and Control

This paper deals with the analysis and design of the state feedback fuzzy controller for a class of discrete time Takagi -Sugeno (T-S) fuzzy uncertain systems. The adopted framework is based on the Lyapunov theory and uses the linear matrix inequality (LMI) formalism. The main goal is to reduce the conservatism of the stabilization conditions using some particular Lyapunov functions. Four nonquadratic Lyapunov Functions are used in this paper. These Lyapunov functions represent an extention from two Lyapunov functions existing in the literature. Their influence in the stabilization region (feasible area of stabilization) is shown through examples, the stabilization conditions of controller for discrete time T-S parametric uncertain systems is demonstrated with the variation of the lyapunov functions between (k, k+1) and (k, k+t) sample times. The controller gain can be obtained via solving several linear matrix inequalities (LMIs). Through the examples and simulations, we demonstrate their uses and their robustness. Comparative study verifies the effectiveness of the proposed methods. Keywords—Nonquadratic Lyapunov functions; Non-PDC; Linear Matrix Inequality; Parametric Uncertain Systems; TakagiSugeno


INTRODUCTION
Fuzzy control systems have experienced a big growth of industrial applications in the recent decades, because of their reliability and effectiveness.
In recent years, there has been growing interest in the study of stability and stabilization of Takagi-Sugeno (T-S) fuzzy system [1,2,3,4,5] due to the fact that it provides a general framework to represent a nonlinear plant by using a set of local linear models which are smoothly connected through nonlinear fuzzy membership functions.
Nonlinear systems are difficult to describe.Takagi-Sugeno fuzzy model is a multimodel approach much used to modelise non linear sytems by construction with identification of inputoutput data [6,7].The merit of such fuzzy model-based control methodology is that it offers an effective and exact representation of complex nonlinear systems in a compact set of state variables.With the powerful T-S fuzzy model, a natural, simple, and systematic design control approach can be provided to complement other nonlinear control techniques that require special and rather involved knowledge.Nowadays, T-S fuzzy model-based control approaches have been applied successfully in a wide range of applications.
One of the most important issues in the study of T-S fuzzy systems is the stability and stabilization analysis problems [8].Via various approachs, a great number of stability and stabilization results for T-S fuzzy systems in both the continuous and discrete time have been reported in the literature [9,10].
Two classes of Lyapunov functions are used to analyze these systems: quadratic Lyapunov and nonquadratic Lyapunov functions.The second class of function is less conservative than the first.Many researches have investigated nonquadratic Lyapunov functions [11,12,13,14,15,16].
Many works try to reduce the conservatism of quadratic form.Several approaches have been developed to overcome the above mentioned limitations.Piecewise quadratic Lyapunov functions were employed to enrich the set of possible Lyapunov functions used to prove stability [11].Multiple Lyapunov functions have been paid a lot of attention due to avoiding conservatism of stability and stabilization.Some works try to enrich some properties of the membership functions [17,18], others introduce decisions variables (slack variables) in order to provide additional degrees of freedom to the LMI problem [19,20].
For every case, The Lyapunov function used to prove the stability has the most important effect to the results.To leave the quadratic framework, some works have dealt with nonquadratic Lyapunov functions.In this case, some results are available in the continuous and the discrete cases [21], [22], [23].In the discrete case, new improvements has been developed in [24], by replacing the classical one sample time variation of the Lyapunov function by its variation over several samples (k samples times variations).This condition reduces the conservatism of quadratic form and give a large sets of solutions in terms of linear matrix inequality LMI.The www.ijacsa.thesai.orgrelaxed conditions admitted more freedom in guaranteeing the stability and stabilization of the fuzzy control systems and were found to be very valuable in designing the fuzzy controller, especially when the design problem involves not only stability, but also the other performance requirements such as the speed of response, constraints on control input and output .
In this paper, a new stabilization conditions for discrete time Takagi Sugeno parametric uncertain fuzzy systems with the use of [25,26,27] new nonquadratic Lyapunov functions are discussed.This condition was reformulated into LMI.[28,29,30,31,32,33,34,35], which can be efficiently solved by using various convex optimization algorithms.
The organization of the paper is as follows.First, T-S fuzzy modeling is discussed.Second, we discuss the proposed approachs to stabilize a T-S fuzzy system in closed loop with the new lyapunov functions.Third, simulation results show the robustness of this approachs and their influence in the stabilization region (feasible area of stabilization).We finish by a conclusion.

II. SYSTEM DESCRIPTION AND PRELIMINAIRIES
In this section, we describe the concept of the Takagi-Sugeno parametric uncertain system.It's based on the state space representation.
Consider the discrete time fuzzy model T-S parametric uncertain systems for nonlinear systems given as follows.
  If ( ) is and then Where (  The T-S fuzzy model is written under the following form: we have The final output can be written under the following form , i i AB represents parametric uncertainties matrices in the state space representation.These uncertainties matrices are written under the following form.
,, 1 H H H E E E are constants matrices.
Lemma 1, 2 and 3 present the techniques and powerful tools used through the development of the next theorems.
Lemma 1 (Schur Complement) [36,37] Consider A,G,L,P and Q matrices with appropriates dimensions.The next properties are equivalent: 1. 0 , 0 2. 0 , 0, 0 Lemma 2 [38] Relaxaion : Whatever the choise of the Lyapunov function, the analysis of the stabilization leads us to the inequality (12) with multiple sum Consider and i h functions having the convex sum properties.
The inequality (12) is verified if the next Consider X and ,0 T Y Q Q  matrices of appropriate dimensions, the following inequality is verified The use of these lemmas will be shown in the next section.

III. STABILIZATION ANALYSIS
This section recalls the technique of the stabilization analysis of discrete T-S model based on a nonquadratic Lyapunov function.In the discrete case, we consider the variation of the Lyapunov function between two sample time.If the final equation of this variation is negative, we obtain a sufficient condition of the T-S stabilization with the state feedback controller.
Consider the discrete time fuzzy Takagi-Sugeno system under the following form.
The non-PDC control law is described by the following equation: The Lyapunov function used in [40] expressed in equation ( 17) The final equation of the Lyapunov function variation obtained by [40] which represent the stabilization condition of discrete time T-S systems is written under the following form.
We propose a new Lyapunov function based on the Lyapunov function in equation (17), by multiplying the Lyapunov matrices () zk P by a scalar 0   .So the new form of the Lyapunov function is written under the following form in equation (19).
) and the non-PDC control law is written under the following form in equation (20).
The variation of the Lyapunov function between k and k+t sample times is given by the next equation ( 21) The final output   1 xk is written between k and (k+t) samples under the next form.
The variation of the Lyapunov function for discrete system should be negative The equation ( 22) is equivalent to equation ( 23) (24) Using the congruence with the full rank matrix G, we obtain So the variation of the lyapunov function 25) is negative.The use of the Schur Complement (Lemma 1) with the equation (25) give the next equation For each iteration i={1….r}Let's consider the following inequality: .....
The application of the lemma 1 with equation ( 27) with G  give the next inequality Recursively by the use of Schur Complement we obtain the inequality (29). * The use of the lemma 2 with the equation ( 29), give the final condition of discrete time T-S systems stabilization.This condition should be negative.
Therefore we state the following theorem for the discrete time Takagi-Sugeno fuzzy systems.

Theorem 1
Consider the discrete time Takagi-Sugeno (15), the control law (20) and the   such that the conditions ( 12) and ( 13) of lemma 2 are verified the discrete time T-S system is globally asymptotic stable in closed loop.
These two theorems represent sufficient conditions of the discrete time T-S stabilization with state feedback with k sample times variation of the Lyapunov function.In the next section, we present the analysis of the stabilization of the discrete time T-S parametric uncertain systems.

IV. PARAMETRIC UNCERTAIN SYSTEMS STABILIZATION ANALYSIS
A. New Lyapunov Function: First approach In the next, we treat the case of the discrete time Takagi-Sugeno parametric uncertain systems.Consider the uncertain system described in equation (6) In that case, the equation (30) becomes.www.ijacsa.thesai.org ) Taking in consideration the two inequality ( 32) and ( 33), the equation ( 31) become equation ( 34) The use of Schur Complement (lemma1) give the next equation (35) which represents the final condition of stabilization of T-S parametric uncertain systems with the use of the Lyapunov function (19) and the control law (20).
After using lemma 2 the equation ( 35) become ( 37) (37) So we state the next theorem for the stabilization of the discrete time T-S parametric uncertain systems.

B. New Lyapunov Function : Second approach
Consider the new non quadratic lyapunov function in equation ( 38) and the non-PDC control law in equation (20).In this new function, we associate for each Lyapunov matrices where 0 i   with   1,.. ir  Consider the same transformations and lemmas used to obtain equations ( 30), (37), theorems 1 and 2. The new form of equation for the stabilization of discrete time T-S fuzzy parametric uncertain systems with the use of the Lyapunov function ( 38) is under the following form in equation (39).

     
the equation ( 39) and the Lyapunov function (38) and the non-PDC controller (20), we propose the next theorem for the stabilization of the T-S parametric uncertain systems.

Theorem 3
Consider the discrete uncertain Takagi-Sugeno system (6), the control law (20)  i   such that conditions of lemma 2 are verified, the system is globally asymptotic stable in the closed loop.

C. New Lyapunov Function : Third approach
The third Lyapunov function proposed in this paper represents an extention from the first Lyapunov function and the next Lyapunov function described bellow The following Lyapunov function is used by [16,17].
P is symmetric and definite positive matrix, and z G is full rank matrix.The nonlinearities are expressed by the terms The Lyapunov function used by [13,24], is written under the following form.
So the third proposed Lyapunov function is written under the following form in equation (44 With the equation ( 45) and the Lyapunov function (44) and the non-PDC controller (20), we propose the next theorem for the stabilization of the T-S parametric uncertain systems.

Theorem 4
Consider the discrete uncertain Takagi-Sugeno system (6), the control law (20)   such that the conditions of lemma 2 are verified the system is globally asymptotic stable in closed loop.
In the next section, we add more values to the LMI in order to demonstrate their influence in stabilization region by affecting to each lyapunov matrices i P , a positive scalar i  .

D. New Lyapunov Function : Fourth approach
The fourth Lyapunov function used in this paper is written under the following form in equation ( 46) (47) We state the following theorem for the stabilization of the discrete time T-S fuzzy parametric uncertain systems.

Theorem 5
Consider the discrete uncertain Takagi-Sugeno system (6), the control law (20)   such that the conditions of lemma 2 are verified, then the system is globally asymptotic stable in closed loop.
Four Lyapunov functions were proposed in this paper.They represent a direct extension from two other function in the literature.In the next section, we present their robustness by showing their influence on the stabilization region.

V. SIMULATION AND VALIDATION OF RESULTS
Consider TS discrete uncertain system with unstable open loop models.This system is modeled with two subsystems, so we have 2 r  .

 
For the simulation the membership functions are choosen as follows: With the application of theorem 2, with =0.6  the results of LMI gives definite positive matrices 12 P , P and matrices G ,G , F and F : www.ijacsa.thesai.org With the application of theorem 4, with =0.6  , the results of LMI gives other definite positive matrices 12 P , P and matrices 1 2 G ,G , R, F and F .
The figures 1, 2 and 3 show the convergence of state variables and the control signal to the equilibrium point zero with the application of the theorem 2. Theorem 3 gives a larger stabilization region than theorem 2. So by affecting for each Lyapunov matrices P a scalar  we obtain a large stabilization region than a single scalar  common to all Lyapunov matrices.The next figures present the effect of increasing of number of parameters with i 0  in the stabilization region.
The figure 5 present the feasible area corresponding to theorem 3 with ( 12 0.01, 0.06

 
) presented by the mark (o) and ( 12      The conclusion obtained throught these figures demonstrates that the choice of a large number of parameters i  affected to each Lyapunov matrices give a best stabilization region then one parameter and a smaller (near to zero) number also have a great effect to the stabilization region.

VI. CONCLUSION
This paper has developed a new fuzzy controller with state feedback for discrete time T-S parametric uncertain systems.The analysis of the stabilization problem is established by the use of Lyapunov function technique.In this case, four new Lyapunov functions are proposed.In This Lyapunov functions more parameters and slack matrix variables are introduced in order to facilitate and enrich the stabilization analysis.In the first Lyapunov function, a multiplication with a common scalar to each Lyapunov matrices is considered, In the second each Lyapunov matrices is multiplied with their own scalars.The use of the second function has a great influence to the stabilization region than the first.In the third and fourth functions, more parameters and slack matrix variables are introduced with common and single scalars for each Lyapunov matrices.Through the simulation results a single scalar for each Lyapunov matrices give a better stabilization region.The proposed theorems of stabilization was generalized between k and k+t samples time variation.Future research includes the development of design methods using these Lyapunov functions with fuzzy controller and observer in discrete time non parametric and mixed T-S uncertain systems.

where 0 
 and 01   The new form of equation for the stabilization of discrete time T-S fuzzy parametric uncertain systems with the use of the Lyapunov function (44) is under the following form in equation (45).
46)Under this Lyapunov function, the new condition of stabilization obtained in the next equation.

Fig. 5 . 3 Figure 6 5 .
Fig. 5. Stabilization region of theorem 3 Figure 6 presents a comparison between theorem 4 and 5, it show the effect of the choice of parameters  and i  For the simulation, consider   =0.6  for theorem 4 and

Fig. 6 . 5 Figure 7 6 
Fig. 6.Comparison between theorem 4 and 5 Figure 7 presents the effect of choice of parameter i  near