The variance and passivity constrained fuzzy control problem for the nonlinear ship steering systems with state multiplicative noises is investigated. The continuous-time Takagi-Sugeno fuzzy model is used to represent the nonlinear ship steering systems with state multiplicative noises. In order to simultaneously achieve variance, passivity, and stability performances, some sufficient conditions are derived based on the Lyapunov theory. Employing the matrix transformation technique, these sufficient conditions can be expressed in terms of linear matrix inequalities. By solving the corresponding linear matrix inequality conditions, a parallel distributed compensation based fuzzy controller can be obtained to guarantee the stability of the closed-loop nonlinear ship steering systems subject to variance and passivity performance constraints. Finally, a numerical simulation example is provided to illustrate the usefulness and applicability of the proposed multiple performance constrained fuzzy control method.

In the literature, the problem of controlling surface ships in maneuvering situations has been receiving more and more attention from the operational safety and environmental viewpoints [

In control engineering, it is always required to develop some methodologies for designing controllers to achieve multiple performance requirements. In addition to the stability performance constraint, the individual state variance constraint and passivity constraint are usually considered for the control problem of linear and nonlinear systems. The individual state variance performance requirements of engineering systems are usually expressed as upper bounds on the steady-state variances for the stochastic control systems. Current control design techniques, such as Linear Quadratic Gaussian (LQG) and

The control problem for the systems with multiplicative noise has recently received a great deal of attention. Such models are found in many physical systems, such as aerospace engineering systems [

The contribution of this paper is to develop a methodology to design a fuzzy controller such that the stability constraint, individual variance constraint, and passivity constraint are simultaneously achieved for the nonlinear stochastic ship steering systems with multiplicative noise. The T-S fuzzy model is used to represent the nonlinear stochastic ship steering systems with multiplicative noise. According to the T-S fuzzy model with multiplicative noise, there are few approaches investigated to simultaneously achieve multiple performance requirements such as stability constraint, individual variance constraint, and passivity constraint. Therefore, the multiobjective fuzzy control methodology presented in this paper is worthy of the attention of control engineers. According to the T-S fuzzy model [

Consider a nonlinear stochastic system, which is constructed by a continuous-time T-S fuzzy model with multiplicative noise as follows.

Without loss of generality, it is assumed that the premise variables of the previous T-S fuzzy model are measurable. Given the pair

Applying the concept of PDC, the fuzzy controller is designed to share the same IF part of the T-S fuzzy model (

Considering each subsystem of the T-S fuzzy model (

The system (

The purpose of this paper is to find feedback gains

The sufficient conditions for guaranteeing the stability, individual variance constraint, and passivity constraint of closed-loop T-S fuzzy model with multiplicative noise are derived in this section. By assigning a common upper bound matrix of the state covariance matrices for all fuzzy rules, the sufficient conditions are derived based on the Lyapunov theory and passivity theory. According to the closed-loop T-S fuzzy model (

If there exist positive definite matrices

To analyze the stability of the closed-loop T-S fuzzy system (

It is obvious that if condition (

Subtracting (

For achieving the attenuating performance, the passivity theory provides a useful and effective tool to design the controller to achieve the energy constraints for the closed-loop systems. Considering the passivity constraint defined in Definition

Due to the fact

Theorem

If there exist positive definite matrices

Consider conditions (

Based on the conditions of Corollary

Consider a nonlinear ship steering system with state multiplicative noises. It is customary to write the dynamic equations using a coordinate frame fixed to the ship such as Figure

The coordinate of ship steering system.

In this paper, the previous nonlinear ship steering system is modeled by a T-S fuzzy model [

The membership function of state

where

In general, the PDC approach is a popular T-S fuzzy controller design method. In the PDC concept, one needs to first design a linear controller for each rule, after which the controller of the entire nonlinear system can be made by blending the linear controllers of all rules. The fuzzy controller produced using PDC is represented as follows.

For starting the fuzzy controller design, we select the supply rate

The simulation responses of states are shown in Figures

Responses of state

Responses of state

Responses of state

Responses of state

Responses of state

Responses of state

The performance-constrained fuzzy controller design problem for the nonlinear ship steering system has been studied in this paper. The nonlinear ship steering system was modeled by a stochastic T-S fuzzy model with state multiplicative noises. This paper also considered the individual state variance constraint and passivity constraint. Based on the PDC concept, the proposed fuzzy controller design approach was carried out by solving the LMI stability conditions. On the other hand, these LMI stability conditions can be solved by the convex optimal programming algorithm. In the numerical example, a nonlinear ship steering system has been introduced to illustrate the usefulness and effectiveness of the proposed fuzzy control methodology.

The authors would like to express their sincere gratitude to anonymous reviewers who gave them some constructive comments, criticisms, and suggestions. This work was supported by the National Science Council of the Republic of China under Contract NSC101-2221-E-019-036.