Stabilization of positive Takagi–Sugeno fuzzy discrete-time systems with multiple delays and bounded controls

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Abstract

This paper deals with the problem of stabilization by state feedback control of Takagi–Sugeno (T–S) fuzzy discrete-time systems with multiple fixed delays while imposing positivity in closed-loop. The obtained results are presented under linear programming (LP) form. In particular, the synthesis of state feedback controllers is first solved in terms of Linear programming for the unbounded controls case. This result is then extended to the stabilization problem by nonnegative controls, and stabilization by bounded controls. The stabilization conditions are derived using the single Lyapunov–Krasovskii functional (LKF). An example of a real plant is studied to show the advantages of the design procedure. A comparison between linear programming and LMI approaches is presented.

Introduction

The problem concerns a special class of nonlinear systems called Takagi–Sugeno models (T–S) [1]. From the history of the approach, this class can be interpreted as a collection of linear models interconnected by nonlinear functions, called membership functions, which are dependent variables. The most delicate problem is the choice of premise variables that partition the space [2], [3].

Many physical systems involve quantities that have intrinsically constant sign. For example, absolute temperatures, level of liquids in tanks and concentration of substances in chemical processes are always positive. These examples belong to the important class of systems which have the property that the state is nonnegative whenever the initial conditions are nonnegative. In the literature [4], [5], such systems are referred to be positive. Control of this kind of systems must take into account this positivity constraint; otherwise the mathematical model of these systems might move into infeasible regions, impossible to reach by the real system, causing loss of stability or performance when the controller is implemented in the real plant.

In recent years, positive systems have obtained great interest by researchers [4], [5], [6], [7], [8]. The class of positive T–S fuzzy systems has been considered for the first time in [9] and more developed in [10], [11]. The obtained results have been presented under LMIs [14], [15].

In this paper, we study the stabilization problem by using bounded state feedback control, while imposing positivity in closed-loop, for such class of discrete-time systems. The obtained results are new and presented under linear programming form. An application on the model of a real process is considered. A comparison between linear programming and LMI approaches is presented.

The rest of this paper is organized as follows: Section 2 gives some preliminary results, studying the stabilization problem by symmetrically bounded control by LMI approach by adding the positivity condition of closed-loop system. Conditions of stabilization and positivity by linear programming with unbounded control are also established. The state-feedback stabilization problem with non-symmetrically bounded and nonnegative control is considered in Section 3. In Section 4, two simulation examples are given to show the need of such controllers. Some concluding points are given in Section 5.

Notation:

  • MT denotes the transpose of a real matrix M.

  • For a square matrix Q>0, if QRn×n is positive definite.

  • A0 stands for a positive matrix A, that is with nonnegative elements: aij0.

Section snippets

Problem formulation and preliminary results

Specifically, the Takagi–Sugeno fuzzy system is described by fuzzy IF–THEN rules, which locally represent linear input–output relations of a system. The fuzzy system is of the following form:

Rule i: IF z1(k) is Fi1 and and zp(k) is Fip Then:x(k+1)=Aix(k)+q=1NAi1qx(kτq)+Biu(k)x(k)=Ψ(k)0,k[τmax,0]y(k)=Cix(k)

where τmax=max(τq), x(k)Rn is the state, u(k)Rm is the control input, y(k)Rp is the output, AiRn×n, BiRn×m, CiRp×n, τq is a fixed delay, with i=1,2,,r, r is the number of IF–THEN

Main results

This section concerns the study of conditions of positive stabilization with bounded controls by linear programming, where two different types of controls are considered: first sign-restricted controls is considered, then the result is extended to controls with non-symmetric bounds.

Numerical examples of simulation

To demonstrate the importance of the result of stabilization of linear programming, we consider the following numerical examples:

Conclusion

In this paper, we are concerned with the study of positive nonlinear systems. To obtain conditions of stabilization by state feedback control for nonlinear systems, while imposing positivity in closed-loop, the T–S fuzzy techniques are used. The study is performed by using a linear programming method. Finally, an application to the model of a real process with two tanks is presented together with a comparison between LMI and LP results based on a numerical example.

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