PDC and Non-PDC fuzzy control with relaxed stability conditions for contintuous-time multiplicative noised fuzzy systems

Abstract

This paper presents a relaxed scheme of fuzzy controller design for continuous-time nonlinear stochastic systems that are constructed by the Takagi–Sugeno (T–S) fuzzy models with multiplicative noises. Through Nonquadratic Lyapunov Functions (NQLF) and Non-Parallel Distributed Compensation (Non-PDC) control law, the less conservative Linear Matrix Inequality (LMI) stabilization conditions on solving fuzzy controllers are derived. Furthermore, in order to study the effects of stochastic behaviors on dynamic systems in real environments, the multiplicative noise term is introduced in the consequent part of fuzzy systems. For decreasing the conservatism of the conventional PDC-based fuzzy control, the NQLF stability synthesis approach is developed in this paper to obtain relaxed stability conditions for T–S fuzzy models with multiplicative noises. Finally, some simulation examples are provided to demonstrate the validity and applicability of the proposed fuzzy controller design approach.

Introduction

Based on the T–S fuzzy model approach [1], [2], nonlinear control systems have been received a great deal of attention over the last three decades. This is a successful and systematical approach for dealing with the stability of nonlinear control systems. Through the T–S fuzzy model, the trajectories of nonlinear system can be approximated by several linear subsystems with determined membership functions. This kind of model can provide an effective representation of complex nonlinear systems. In general, T–S type fuzzy model can be classified into homogenous T–S fuzzy model [3], [4], [5], [6] and affine T–S fuzzy model [7], [8], [9], [10]. Different from the homogenous T–S fuzzy model, the affine T–S fuzzy model has a nonzero bias on the linear subsystems. Through this reason, the affine T–S fuzzy models are difficult to study and analyze. Hence, the T–S fuzzy model considered in this paper is the homogenous one. In general, the PDC concept [3], [4], [5], [6], [7], [8], [9], [10], [11] is applied to design the fuzzy controller for T–S fuzzy models. According to PDC concept, the nonlinear controller can be blended by using linear feedback gains via fuzzy rules. Through the T–S fuzzy model and PDC design concept, the linear control criteria can be applied to analyze the complex nonlinear systems. However, a fuzzy control system with a large number of rules under PDC control concept requires higher computational demand solving the solution. Unlike the previous PDC-based fuzzy control [3], [4], [5], [6], [7], [8], [9], [10], [11], this paper intends to develop a Non-PDC fuzzy control scheme to eliminate the previous lack and to release the drawback of the conventional PDC stability conditions.

In the previously mentioned literature, many practical stochastic behaviors are inevitable and are described by random processes. In this situation, the systems to be controlled are always modeled by stochastic systems. Based on Itô's stochastic differential equation [12], the multiplicative noise term is used to represent the unpredictable stochastic behaviors of systems. The multiplicative noise term is structured as that states multiplied by the zero-mean white noise. In [6], [13], [14], [15], Itô's stochastic differential equation has been employed to represent nonlinear stochastic systems via T–S fuzzy model with multiplicative noise. Because of the above motivation, the control problem of a nonlinear continuous-time stochastic dynamic system modeled by the homogenous T–S fuzzy model with multiplicative noise is investigated in this paper.

Previously, most of the stability analysis bases on a single quadratic Lyapunov function, i.e.,V(x(t))=x^T(t)Px(t) and P=P^T>0. By the convex optimization algorithm, the stability and stabilization problems of fuzzy control can be converted into LMI problems [16]. Therefore, some transformation methods are used to derive the LMI stability conditions. The feasible solutions of the fuzzy controllers can be directly solved by using the convex optimization algorithm [16]. However, it is a hard task to find a common matrix P=P^T>0 such that Lyapunov stability inequalities are all satisfied if the number of rules of a fuzzy system is large. In order to solve this problem, many efforts [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] have focused on developing the relaxed stability conditions for T–S fuzzy models. In [17], [18], some relaxed quadratic stability conditions for fuzzy models were proposed. On the other hand, the nonquadratic Lyapunov function was also used to derive the relaxed stability conditions for the T–S fuzzy systems [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. This nonquadratic Lyapunov function is usually called as fuzzy Lyapunov function (FLF) [23], [24], [25], [26], [27] or parameter-dependent Lyapunov function (PDLF) [31]. The above relaxed quadratic and nonquadratic stabilization concepts are employed in this paper to develop novel relaxed stability conditions for the T–S fuzzy model with multiplicative noises. By solving these relaxed stability conditions, the stability of the closed-loop system can be achieved by the designed fuzzy controllers.

The relaxed stability conditions developed in [17] took into account the interactions among the fuzzy subsystems in an analytical manner by using the property of the quadratic form. Overcoming the conservatism of [17], the approach of [18] collects the interactions in a single matrix and represents it in terms of an LMI that can be solved in a numerical manner. Extending the approach of [18], a PDC-based fuzzy controller is designed in this paper by deriving the relaxed quadratic stability conditions for the T–S fuzzy models with multiplicative noises. In addition, a Non-PDC fuzzy controller design is also developed in this paper based on the nonquadratic Lyapunov function. Using the nonquadratic Lyapunov function, some scholars have derived the relaxed stability conditions for the discrete-time [19], [20], [21] and continuous-time [22], [23], [24], [25], [26], [27], [28], [29], [30], [31] T–S fuzzy systems. However, the multiplicative noise was not considered for the systems in [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31]. Considering the stochastic behaviors, a Non-PDC fuzzy control scheme is thus developed in this paper by using the nonquadratic Lyapunov function for the T–S fuzzy systems with multiplicative noises. Compared with the approaches of [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], the derivations of the proposed design scheme are more complex due to the difficulty in handling membership function's derivatives and dealing with the additional multiplicative noises, simultaneously. In this paper, the PDC fuzzy controller designed with relaxed quadratic stability conditions and the Non-PDC fuzzy controller designed with the relaxed nonquadratic stability conditions are both investigated. The proposed Non-PDC fuzzy controller is to provide an alternative solution to applications of which time derivatives of membership functions are available. In this paper, the stability conditions are directly derived of the LMI forms that can be used to solve stabilizing fuzzy controllers without using the completing square technique. In contrast to the PDC fuzzy controller, the Non-PDC fuzzy controller can provide a better methodology with regard to relax the conservativeness of stability and stabilization problems for the T–S fuzzy systems with multiplicative noises.

The organization of this paper is given as follows. Section 2 formulates the problem and presents preliminary results. Results of quadratic stability with PDC-based fuzzy control are discussed in Section 3. The Non-PDC fuzzy controller design via nonquadratic Lyapunov function is investigated in Section 4. In Section 5, two numerical examples are given to illustrate the effectiveness and applicability of the proposed design approaches. Finally, Section 6 provides some concluding remarks.

Notations: the symbol ⁎ in a symmetric matrix denotes the transposed element in the symmetric position. The E{Q(t)} denotes the expected value of Q(t). I denotes identity matrix with appropriate dimension.