Variance and Passivity Constrained Fuzzy Control for Nonlinear Ship Steering Systems with State Multiplicative Noises

Abstract

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.

1. Introduction

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 [1–8]. For ship steering, the stochastic disturbances are waves, wind gusts, and observation errors, and the deterministic disturbances are current, mean wind, and curvature of the prespecified track. In order to deal with the complex disturbances, the adaptive control approach was developed to solve the ship steering control problem [1–4]. In [5], an output feedback control problem was studied by using the vectorial observer backstepping technique. Applying the sliding mode control, a multivariable performance-constrained controller was developed in [6] for the dynamic ship positioning systems. In [1–6], the dynamic ship steering systems were approximated by the linearized models that may have sacrificed some nonlinear characteristics. In order to reserve the characteristics of original nonlinear systems, the authors of [7, 8] used the Takagi-Sugeno (T-S) fuzzy model to represent the nonlinear dynamic ship steering systems. Without loss of generality, the T-S fuzzy model [7, 8] can approximate a wide class of nonlinear systems by mixing several linear subsystems with IF-THEN rules.

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 𝐻_∞ design methods, do not seem to give a direct solution to this kind of design problem since they lack a convenient avenue for imposing design objectives stated in terms of upper bounds on the variance values. The covariance control theory developed in [9] provided a more direct methodology for achieving the individual variance constraints than the LQG control theory. The idea of covariance control theory has been widely applied in solving multiobjective control problems as well as filtering problems see [6, 10–13], for instance. In addition to the individual variance constraint, the passivity constraint is also considered in this paper. In the literature, the passivity concept [14–21] is presented to issue stability analysis and synthesis of linear or nonlinear systems. It is well known that the passive theory uses the behaviors of energy dissipative between input and output of systems. The most important definitions of the passive theory are storage function [16, 20] and power supply [16, 21] for decision the property of passivity. As for a storage function in passive theory, it measures the amount of energy stored in the internal of the system; hence it is naturally described by using internal variables of system state variables. To the authors’ knowledge, so far, the variance and passivity constrained stabilization problem of nonlinear stochastic ship steering systems with multiplicative noises has not been investigated. It is also an open problem to discuss the variance and passivity constrained fuzzy controller design for the stochastic T-S fuzzy models with multiplicative noises. For this reason, this paper tries to develop an applicable fuzzy control methodology for achieving the individual variance constraints and passivity constraints, simultaneously.

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 [22], signal processing systems [23], and biological movement systems [24, 25]. Different from the traditional additive noise, multiplicative noise is more practical, since it allows the statistical description of the noise to be unknown a prior but depend on the control and state solution. So far, many researchers have been working on various kinds of investigation of filtering and control for the control systems with multiplicative noise. Thus, there have been several approaches for dealing with this problem, including the linear matrix inequality (LMI) approach [23], the Riccati difference equation approach [25], the game theoretic approach [26], and the fuzzy control approach [19, 27–29]. Since this model reflects more realistic properties in engineering, a complete theory which includes control and estimation is worthy to be developed.

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 [30, 31], a parallel distributed compensation (PDC) [31] based fuzzy controller is designed via the Lyapunov theory in this paper. Subject to the individual variance constraint and passivity constraint, new sufficient conditions are derived by assigning a common upper bound matrix for the state covariance matrices of all fuzzy rules. By using the LMI technique [32] to solve the previous sufficient conditions, a PDC-based fuzzy controller can be obtained to guarantee the individual variance constraint and passivity constraint, simultaneously. Finally, a numerical example for controlling a nonlinear stochastic ship steering system with multiplicative noise is given to demonstrate the applicability and effectiveness of the proposed fuzzy controller design approach.

2. System Description and Problem Statement

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

Rule i: if𝑧₁(𝑡)is𝑀_𝑖1⋯and𝑧_𝑞(𝑡)is𝑀_𝑖𝑞then𝑥̇(𝑡)=𝐀_𝑖+^𝑚_𝑒=1𝐍_𝑒𝑖𝑣_𝑒𝑖(𝑡)𝑥(𝑡)+𝐁_𝑖𝑢(𝑡)+𝐃_𝑖𝑤(𝑡)𝑦(𝑡)=𝐂_𝑖𝑥(𝑡)+𝐇_𝑖𝑤(𝑡), (1a)(1b)where 𝑧₁(𝑡),…,𝑧_𝑞(𝑡) are the premise variables that may be functions of the state variables, 𝑀_𝑖𝑞 is the fuzzy set, 𝑞 is the premise variable number, 𝑟 is the number of fuzzy rules, 𝑥(𝑡)∈ℜ^𝑛𝑥 is the state vector, 𝑢(𝑡)∈ℜ^𝑛𝑢 is the input vector, 𝑦(𝑡)∈ℜ^𝑛𝑦 is the output vector, and the processes 𝑤(𝑡)∈ℜ^𝑛𝑤 and 𝑣_𝑒𝑖(𝑡)∈ℜ are mutually independent zero-mean Gaussian white noise with intensities 𝐖(𝐖>0) and 1, respectively. Referring to [33], it is assumed that 𝐸{𝑤(𝑡)}=0,𝐸{𝑣_𝑒𝑖(𝑡)}=0,𝐸{𝑥(𝑡)𝑤(𝑡)}=𝐸{𝑥(𝑡)}𝐸{𝑤(𝑡)}=0,𝐸{𝑥(𝑡)𝑣_𝑒𝑖(𝑡)}=𝐸{𝑥(𝑡)}𝐸{𝑣_𝑒𝑖(𝑡)}=0 and 𝐸{𝑤(𝑡)𝑣_𝑒𝑖(𝑡)}=𝐸{𝑤(𝑡)}𝐸{𝑣_𝑒𝑖(𝑡)}=0. The matrices 𝐀_𝑖∈ℜ^{𝑛𝑥×𝑛𝑥}, 𝐁_𝑖∈ℜ^{𝑛𝑥×𝑛𝑢}, 𝐃_𝑖∈ℜ^{𝑛𝑥×𝑛𝑤}, 𝐂_𝑖∈ℜ^{𝑛𝑦×𝑛𝑥}, 𝐇_𝑖∈ℜ^{𝑛𝑦×𝑛𝑤}, and 𝐍_𝑒𝑖∈ℜ^{𝑛𝑥×𝑛𝑥} are constant.

Without loss of generality, it is assumed that the premise variables of the previous T-S fuzzy model are measurable. Given the pair (𝑥(𝑡),𝑢(𝑡)), the overall fuzzy model can be described as follows:𝑥̇(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡×𝐀))_𝑖+^𝑚_𝑒=1𝐍_𝑒𝑖𝑣_𝑒𝑖(𝑡)𝑥(𝑡)+𝐁_𝑖𝑢(𝑡)+𝐃_𝑖𝑤(𝑡𝑦(𝑡)=_𝑟𝑖=1)ℎ_𝑖(𝑧(𝑡))𝐂_𝑖𝑥(𝑡)+𝐇_𝑖𝑤(𝑡), (2a)(2b) where ℎ_𝑖(𝑧(𝑡))=𝜔_𝑖(𝑧(𝑡))/∑^𝑟_𝑖=1𝜔_𝑖(𝑧(𝑡)), 𝜔_𝑖(𝑧(𝑡))=∏^𝑞_𝑗=1𝑀_𝑖𝑗(𝑧_𝑗(𝑡)), ℎ_𝑖(𝑧(𝑡))≥0, and ∑^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))=1.

Applying the concept of PDC, the fuzzy controller is designed to share the same IF part of the T-S fuzzy model (1a) and (1b). The proposed fuzzy controller can be represented as follows.

Rule i: if𝑧₁(𝑡)is𝑀_𝑖1and⋯and𝑧_𝑞(𝑡)is𝑀_𝑖𝑞then𝑢(𝑡)=−𝐅_𝑖𝑥(𝑡) (3) or 𝑢(𝑡)=^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))−𝐅_𝑖𝑥(𝑡). (4) Substituting (4) into (2a), the closed-loop T-S fuzzy model can be obtained as follows: 𝑥̇(𝑡)=^𝑟𝑟_𝑖=1𝑗=1ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡×𝐀))_𝑖−𝐁_𝑖𝐅_𝑗𝑥(𝑡)+𝐃_𝑖𝑤(𝑡)+^𝑚_𝑒=1𝐍_𝑒𝑖𝑣_𝑒𝑖(𝑡)𝑥(𝑡). (5)

Considering each subsystem of the T-S fuzzy model (5), the steady-state covariance matrix of the state vector 𝑥(𝑡) has the following form: 𝐗_𝑖=𝐗^T_𝑖>0, (6) where 𝐗_𝑖=lim_𝑡→∞𝐸[𝑥(𝑡)𝑥^T(𝑡)] and 𝐗_𝑖 is the unique solution of the following Lyapunov equation for each rule [34]: 𝐀_𝑖−𝐁_𝑖𝐅_𝑖𝐗_𝑖+𝐗_𝑖𝐀_𝑖−𝐁_𝑖𝐅_𝑖^T+𝐃_𝑖𝐖𝐃^T_𝑖+^𝑚_𝑒=1𝐍_𝑒𝑖𝐗_𝑖𝐍^T_𝑒𝑖=0,𝑖=1,2,…,𝑟. (7) The state variance constraint considered in this paper is defined as follows: 𝐗_𝑖𝑘𝑘≤𝜎²_𝑘, (8) where [⋅]_𝑘𝑘 denotes the kth diagonal element of matrix [⋅] and 𝜎_𝑘, 𝑘=1,2,…,𝑛_𝑥 denote the root-mean-squared constraints for the variance of system states. In addition to the previous individual state variance constraints (8), the passivity constraint introduced in the following definition is also considered in this paper.

Definition 1 (see [18]). The system (5) with external disturbance 𝑤(𝑡) and output 𝑦(𝑡) is called strictly input passive if there exists a positive scalar 𝛾 and symmetric positive definite matrix 𝐒=𝐒^T>0 such that 𝐸2^𝑡𝑝₀𝑦^T(𝑥)𝐒𝑤(𝑡)𝑑𝑡>𝐸^𝑡𝑝₀𝛾𝑤^T(𝑡)𝑤(𝑡)𝑑𝑡 (9) for all 𝑡_𝑝≥0 and 𝑤(𝑡)≠0.

The purpose of this paper is to find feedback gains 𝐅_𝑖 that can be used to construct the PDC-based fuzzy controller (3) or (4) such that the individual state variance constraint (8) and passivity constraint (9) are all satisfied, simultaneously.

3. Fuzzy Controller Design for T-S Fuzzy Models with Multiplicative Noises

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 (5), the sufficient conditions for achieving the stability, individual variance constraint, and passivity constraint are derived in the following theorem.

Theorem 2. If there exist positive definite matrices 𝐗>0 and 𝐒>0 and feedback gains 𝐅_𝑖 and dissipative rate 𝛾 satisfying the following sufficient conditions, then [𝐗_𝑖]_𝑘𝑘≤𝜎²_𝑘 and the closed-loop system (5) is strictly input passive and asymptotically stable: 𝐀_𝑖−𝐁_𝑖𝐅_𝑖𝐗+𝐗𝐀_𝑖−𝐁_𝑖𝐅_𝑖^T+𝐃_𝑖𝐖𝐃^T_𝑖+∑^𝑚_𝑒=1𝐍_𝑒𝑖𝐗𝐍^T_𝑒𝑖−𝐗𝐂^T_𝑖𝐒−𝐒^T𝐂_𝑖𝐗𝛾𝐈−𝐇^T_𝑖𝐒−𝐒^T𝐇_𝑖𝐆<0_𝑖𝑗+𝐆_𝑗𝑖2𝐗+𝐗𝐆_𝑖𝑗+𝐆_𝑗𝑖2^T<0,𝑓𝑜𝑟𝑖<𝑗𝐗−diag𝜎²₁,…,𝜎²_𝑛𝑥<0, (10)(11)(12)where 𝐆_𝑖𝑗=𝐀_𝑖−𝐁_𝑖𝐅_𝑗 and diag(𝜎²₁,…,𝜎²_𝑛𝑥) denotes a diagonal matrix with the diagonal elements 𝜎²₁,𝜎²₂,…,and𝜎²_𝑛𝑥.

Proof. To analyze the stability of the closed-loop T-S fuzzy system (5), a Lyapunov function is chosen as 𝑉(𝑥(𝑡))=𝑥^T(𝑡)𝐏𝑥(𝑡), where 𝐏=𝐏^T>0. The expected value of differential of the Lyapunov function 𝑉(𝑥(𝑡)) along the trajectories of (5) is given as follows: 𝐸𝑉̇(𝑥(𝑡𝑥̇))=𝐸^T(𝑡)𝐏𝑥(𝑡)+𝑥^T(𝑡)𝐏𝑥̇(𝑡)=𝐸^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))𝐆_𝑖𝑖𝑥(𝑡)+𝐃_𝑖𝑤(𝑡)+^𝑚_𝑒=1𝐍_𝑒𝑖𝑣_𝑒𝑖(𝑡)𝑥(𝑡)^T×𝐏𝑥(𝑡)+𝑥^T(𝑡)𝐏×𝐆_𝑖𝑖𝑥(𝑡)+𝐃_𝑖𝑤(𝑡)+^𝑚_𝑒=1𝐍_𝑒𝑖𝑣_𝑒𝑖(𝑡)𝑥(𝑡)+2^𝑟_𝑖<𝑗ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝑥^T(𝑡)×𝐆_𝑖𝑗+𝐆_𝑗𝑖2^T𝐆𝐏+𝐏^𝑖𝑗+𝐆_𝑗𝑖2𝑥(𝑡)=𝐸^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡×𝑥))_T(𝑡)𝐆^T_𝑖𝑖𝐏+𝐏𝐆_𝑖𝑖+^𝑚_𝑒=1𝑣^T_𝑒𝑖(𝑡)𝐍^T_𝑒𝑖𝐏+𝐏𝐍_𝑒𝑖𝑣_𝑒𝑖(𝑡)𝑥(𝑡)+𝑤^T(𝑡)𝐃^T_𝑖𝐏𝑥(𝑡)+𝑥^T(𝑡)𝐃_𝑖𝑤(𝑡)+2^𝑟_𝑖<𝑗ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝑥^T(𝑡)×𝐆_𝑖𝑗+𝐆_𝑗𝑖2^T𝐆𝐏+𝐏^𝑖𝑗+𝐆_𝑗𝑖2𝑥(𝑡). (13) Due to 𝐸{𝑥(𝑡)𝑤(𝑡)}=𝐸{𝑥(𝑡)}𝐸{𝑤(𝑡)}=0 and 𝐸{𝑥(𝑡)𝑣_𝑒𝑖(𝑡)}=𝐸{𝑥(𝑡)}𝐸{𝑣_𝑒𝑖(𝑡)}=0, one has 𝐸𝑉̇(𝑥(𝑡))=𝐸^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡))𝑥^T(𝑡)𝐆^T_𝑖𝑖𝐏+𝐏𝐆_𝑖𝑖𝑥(𝑡)+2^𝑟_𝑖<𝑗ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝑥^T(𝑡)×𝐆_𝑖𝑗+𝐆_𝑗𝑖2^T𝐆𝐏+𝐏^𝑖𝑗+𝐆_𝑗𝑖2𝑥(𝑡). (14)
It is obvious that if condition (10) is satisfied, then one has 𝐀_𝑖−𝐁_𝑖𝐅_𝑖𝐗+𝐗𝐀_𝑖−𝐁_𝑖𝐅_𝑖^T+𝐃_𝑖𝐖𝐃^T_𝑖+^𝑚_𝑒=1𝐍_𝑒𝑖𝐗𝐍^T_𝑒𝑖𝐀<0,_𝑖−𝐁_𝑖𝐅_𝑖𝐗+𝐗𝐀_𝑖−𝐁_𝑖𝐅_𝑖^T. (15)(16) Let 𝐏=𝐗⁻¹ and multiplying each item of the previous inequality from left-hand side and right-hand side by 𝐏, then one can obtain the following inequality: 𝐆^T_𝑖𝑖𝐏+𝐏𝐆_𝑖𝑖<0. (17) Besides, multiplying both sides of (11) by 𝐏, one has 𝐆_𝑖𝑗+𝐆_𝑗𝑖2^T𝐆𝐏+𝐏^𝑖𝑗+𝐆_𝑗𝑖2<0. (18) It is obvious that if the conditions (17) and (18) are satisfied, then 𝐸{𝑉̇(𝑥(𝑡))}<0 can be obtained from (14). Thus, one can conclude that if conditions (10) and (11) are satisfied, then the closed-loop system (5) is asymptotically stable.
Subtracting (7) from (15), one has 𝐀_𝑖−𝐁_𝑖𝐅_𝑖𝐗−𝐗_𝑖+𝐗−𝐗_𝑖𝐀_𝑖−𝐁_𝑖𝐅_𝑖^T+^𝑚_𝑒=1𝐍_𝑒𝑖𝐗−𝐗_𝑖𝐍^T_𝑒𝑖<0. (19) Due to the fact that conditions (10) and (11) are satisfied, one can obtain that the closed-loop system (5) is stable and the closed-loop system matrix (𝐀_𝑖−𝐁_𝑖𝐅_𝑖) is stable. In this case, it can be concluded that 𝐗−𝐗_𝑖≥0 via inequality (19). From the condition (12) and 𝐗≥𝐗_𝑖, one has 𝐗_𝑖𝑘𝑘≤𝐗_𝑘𝑘≤𝜎²_𝑘. (20)
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 1, one can define a performance function such as 𝐸^𝑡𝑝₀𝛾𝑤^T(𝑡)𝑤(𝑡)−2𝑦^T(𝑥)𝐒𝑤(𝑡)𝑑𝑡=𝐸^𝑡𝑝₀𝛾𝑤^T(𝑡)𝑤(𝑡)−2𝑦^T(𝑥)𝐒𝑤(𝑡)+𝑉̇(𝑥(𝑡))𝑑𝑡−𝑉(𝑥(𝑡))≤𝐸^𝑡𝑝₀𝛾𝑤^T(𝑡)𝑤(𝑡)−2𝑦^T(𝑥)(𝑡)+𝑉̇(𝑥(𝑡{𝐾(,𝐒𝑤))𝑑𝑡≜𝐸𝑥,𝑤,𝑡)} (21) where 𝐾()≜𝑥,𝑤,𝑡_𝑡𝑝0𝛾𝑤^T(𝑡)𝑤(𝑡)−2𝑦^T(𝑥)𝐒𝑤(𝑡)+𝑉̇(𝑥(𝑡))𝑑𝑡. (22) Using inequality (14) and Schur complement [32], one has 𝐸{𝐾(𝑥,𝑤,𝑡)}=𝐸^𝑟_𝑖=1ℎ_𝑖(𝑧(𝑡𝑥(𝑡)𝑤(𝑡)))^T×𝐆^T_𝑖𝑖𝐏+𝐏𝐆_𝑖𝑖−𝐂^T_𝑖𝐒−𝐒^T𝐂_𝑖𝛾𝐈−𝐇^T_𝑖𝐒−𝐒^T𝐇_𝑖𝑥(𝑡)𝑤(𝑡)+2^𝑟_𝑖<𝑗ℎ_𝑖(𝑧(𝑡))ℎ_𝑗(𝑧(𝑡))𝑥^T(𝑡)×𝐆_𝑖𝑗+𝐆_𝑗𝑖2^T𝐆𝐏+𝐏^𝑖𝑗+𝐆_𝑗𝑖2𝑥(𝑡). (23)
Due to the fact 𝐃_𝑖𝐖𝐃^T_𝑖≥0 and ∑^𝑚_𝑒=1𝐍_𝑒𝑖𝐗𝐍^T_𝑒𝑖>0, it is obvious that condition (10) implies 𝐀_𝑖−𝐁_𝑖𝐅_𝑖𝐗+𝐗𝐀_𝑖−𝐁_𝑖𝐅_𝑖^T−𝐗𝐂^T_𝑖𝐒−𝐒^T𝐂_𝑖𝐗𝛾𝐈−𝐇^T_𝑖𝐒−𝐒^T𝐇_𝑖<0. (24) Multiplying inequality (24) on the left-hand and right-hand sides by diag{𝐏,𝐈}, then one can obtain the following inequality via 𝐏=𝐗⁻¹: 𝐆^T_𝑖𝑖𝐏+𝐏𝐆_𝑖𝑖−𝐂^T_𝑖𝐒−𝐒^T𝐂_𝑖𝛾𝐈−𝐇^T_𝑖𝐒−𝐒^T𝐇_𝑖<0. (25) If conditions (18) and (25) hold, then one can obtain 𝐸{𝐾(𝑥,𝑤,𝑡)}<0. From (22), the inequality 𝐸{𝐾(𝑥,𝑤,𝑡)}<0 implies 𝐸2^𝑡𝑝₀𝑦^T(𝑥)𝐒𝑤(𝑡)𝑑𝑡>𝐸^𝑡𝑝₀𝛾𝑤^T(𝑡)𝑤(𝑡)𝑑𝑡 (26) for all nonzero external disturbance. From Definition 1, it can be thus concluded that if conditions (10) and (11) are satisfied, then the closed-loop system (5) is strictly input passive.

Theorem 2 provides the sufficient conditions for solving the state feedback gains such that the closed-loop system (5) is asymptotically stable subject to individual state variance constraints and passivity constraint. The conditions derived in Theorem 2 cannot be solved by the LMI technique [32] because they are of the bilinear matrix inequality forms. In order to apply the LMI technique to solve the present problem, the conditions of Theorem 2 can be rewritten in the following corollary by using the matrix transform technique.

Corollary 3. If there exist positive definite matrices 𝐗>0 and 𝐒>0 and dissipative rate 𝛾 and matrices 𝐊_𝑖 satisfying condition (12) and the following sufficient conditions, then [𝐗_𝑖]_𝑘𝑘≤𝜎²_𝑘 and the closed-loop system (5) is strictly input passive and asymptotically stable: 𝐀_𝑖𝐗−𝐁_𝑖𝐊_𝑖+𝐗𝐀_𝑖−𝐊^T_𝑖𝐁^T_𝑖+𝐃_𝑖𝐖𝐃^T_𝑖+∑^𝑚_𝑒=1𝐍_𝑒𝑖𝐗𝐍^T_𝑒𝑖−𝐗𝐂^T_𝑖𝐒−𝐒^T𝐂_𝑖𝐗𝛾𝐈−𝐇^T_𝑖𝐒−𝐒^T𝐇_𝑖𝐀<0_𝑖𝐗−𝐁_𝑖𝐊_𝑗+𝐀_𝑗𝐗−𝐁_𝑗𝐊_𝑖2+𝐀_𝑖𝐗−𝐁_𝑖𝐊_𝑗+𝐀_𝑗𝐗−𝐁_𝑗𝐊_𝑖2^T<0,𝑓𝑜𝑟𝑖<𝑗, (27)(28)where 𝐊_𝑖=𝐅_𝑖𝐗.

Proof. Consider conditions (10) and (11) of Theorem 2. By setting new variables 𝐊_𝑖=𝐅_𝑖𝐗, conditions (10) and (11) can be rewritten as (27) and (28), respectively. It is obvious that conditions (12) and (27)-(28) are of LMI forms for the desired parameters 𝐗 and 𝐊_𝑖.

Based on the conditions of Corollary 3, the fuzzy control gains can be solved via LMI technique [32] by using MATLAB LMI-Toolbox. In order to demonstrate the applicability and effectiveness of the proposed fuzzy controller design approach, a numerical example for the controller design problem of a nonlinear ship steering system is considered in the next section.

4. Constrained Fuzzy Controller Design for Nonlinear Ship Steering Systems

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 1. Its dynamic equation can be written as follows [7]: 𝑥̇₁(𝑡)=cos𝑥₃(𝑡)𝑥₄(𝑡)−sin𝑥₃(𝑡)𝑥₅(𝑡)𝑥̇₂(𝑡)=sin𝑥₃(𝑡)𝑥₄(𝑡)+cos𝑥₃(𝑡)𝑥₅(𝑡)𝑥̇₃(𝑡)=𝑥₆(𝑡)(𝑡)𝑥̇+0.3𝑤₄(𝑡)=−0.0358𝑥₁(𝑡)−0.0797𝑥₄(𝑡)+0.9215𝑢₁(𝑡)+0.01𝑣₁(𝑡)𝑥₄(𝑡)+0.005𝑣₂(𝑡)𝑥₄(𝑡)𝑥̇₅(𝑡)=−0.0208𝑥₂(𝑡)−0.0818𝑥₅(𝑡)−0.1224𝑥₆(𝑡)+0.7802𝑢₂(𝑡)+1.4811𝑢₃(𝑡)+0.01𝑣₁(𝑡)𝑥₅(𝑡)+0.007𝑣₁(𝑡)𝑥₆(𝑡)+0.01𝑣₂(𝑡)𝑥₅(𝑡)+0.005𝑣₂(𝑡)𝑥₆(𝑡)𝑥̇₆(𝑡)=−0.0394𝑥₂(𝑡)−0.2254𝑥₅(𝑡)−0.2468𝑥₆(𝑡)+1.4811𝑢₂(𝑡)+7.4562𝑢₃(𝑡)+0.008𝑣₁(𝑡)𝑥₅(𝑡)+0.005𝑣₁(𝑡)𝑥₆(𝑡)+0.006𝑣₂(𝑡)𝑥₅(𝑡)+0.01𝑣₂(𝑡)𝑥₆(𝑡), (29a)(29b)(29c)(29d)(29e)(29f)where 𝑥₁(𝑡) and 𝑥₂(𝑡) are the earth-fixed positions (𝑥,𝑦),𝑥₃(𝑡) is yaw angle of the ships,𝑥₄(𝑡) is surge, 𝑥₅(𝑡) is sway, 𝑥₆(𝑡) is yaw modes, and 𝑢(𝑡)=[𝑢₁(𝑡)𝑢₂(𝑡)𝑢₃(𝑡)]^T is the control and moment by the thruster system. Besides, the processes 𝑣₁(𝑡), 𝑣₂(𝑡), and 𝑤(𝑡) are all mutually independent zero-mean Gaussian white noise with intensity 1.

Figure 1 

The coordinate of ship steering system.

In this paper, the previous nonlinear ship steering system is modeled by a T-S fuzzy model [31] by assuming that the yaw angle changes between 𝑥₃(𝑡)∈(−𝜋/2,+𝜋/2). The membership function of state 𝑥₃(𝑡) is given in Figure 2. The T-S fuzzy model for nonlinear ship steering systems contains the following rules.

Figure 2 

The membership function of state 𝑥3(𝑡).

Rule 1:if𝑥₃(𝑡)𝜋isabout−2−𝜋2<𝑥₃𝑥̇(𝑡)=𝐀<0then₁+²_𝑒=1𝐍_𝑒1𝑣_𝑒1(𝑡(𝑡))×𝑥+𝐁₁𝑢(𝑡)+𝐃₁𝑤(𝑡)𝑦(𝑡)=𝐂₁𝑥(𝑡)+𝐇₁𝑤(𝑡), (30a)

Rule 2: if𝑥₃(𝑡)−𝜋isabout02<𝑥₃𝑥̇(𝑡)=𝐀<0then₂+²_𝑒=1𝐍_𝑒2𝑣_𝑒2(𝑡(𝑡))×𝑥+𝐁₂𝑢(𝑡)+𝐃₂𝑤(𝑡)𝑦(𝑡)=𝐂₂𝑥(𝑡)+𝐇₂𝑤(𝑡), (30b)

Rule 3: if𝑥₃(𝑡)isabout00<𝑥₃<𝜋2then𝑥̇(𝑡)=𝐀₃+²_𝑒=1𝐍_𝑒3𝑣_𝑒3(𝑡(𝑡))×𝑥+𝐁₃𝑢(𝑡)+𝐃₃𝑤(𝑡)𝑦(𝑡)=𝐂₃𝑥(𝑡)+𝐇₃𝑤(𝑡). (30c)

Rule 4: if𝑥₃(𝑡)𝜋isabout+20<𝑥₃<𝜋2then𝑥̇(𝑡)=𝐀₄+²_𝑒=1𝐍_𝑒4𝑣_𝑒4(𝑡(𝑡))×𝑥+𝐁₄𝑢(𝑡)+𝐃₄𝑤(𝑡)𝑦(𝑡)=𝐂₄𝑥(𝑡)+𝐇₄𝑤(𝑡), (30d)

where 𝐀₁=,𝐀000𝛽10000−1𝛽0000001−0.035800−0.0797000−0.020800−0.0818−0.12240−0.039400−0.2254−0.2468₂=,𝐀0001𝛼0000−𝛼10000001−0.035800−0.0797000−0.020800−0.0818−0.12240−0.039400−0.2254−0.2468₃=,𝐀0001−𝛼0000𝛼10000001−0.035800−0.0797000−0.020800−0.0818−0.12240−0.039400−0.2254−0.2468₄=,𝐁000𝛽−100001𝛽0000001−0.035800−0.0797000−0.020800−0.0818−0.12240−0.039400−0.2254−0.2468₁=𝐁₂=𝐁₃=𝐁₄=0000.92150000000.78021.481100001.48117.4562^T𝐃₁=𝐃₂=𝐃₃=𝐃₄=00000,𝐂0.3₁=𝐂₂=𝐂₃=𝐂₄=,𝐇001000₁=𝐇₂=𝐇₃=𝐇₄𝐍=1.₁₁=𝐍₁₂=𝐍₁₃=𝐍₁₄=,𝐍0000000000000000000000.010000000.010.00700000.0080.005₂₁=𝐍₂₂=𝐍₂₃=𝐍₂₄=,0000000000000000000000.0050000000.010.00500000.0060.01 (30e) and 𝛼=sin(2^∘) and 𝛽=cos(88^∘).

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.

Rule 1: if𝑥₃(𝑡)𝜋isabout−2−𝜋2<𝑥₃(𝑡)<0then𝑢=−𝐅₁𝑥(𝑡) (31a)

Rule 2: if𝑥₃(𝑡)−𝜋isabout02<𝑥₃(𝑡)<0then𝑢=−𝐅₂𝑥(𝑡) (31b)

Rule 3: if𝑥₃(𝑡)isabout00<𝑥₃<𝜋2then𝑢(𝑡)=−𝐅₃𝑥(𝑡) (31c)

Rule 4: if𝑥₃(𝑡)𝜋isabout20<𝑥₃<𝜋2then𝑢(𝑡)=−𝐅₄𝑥(𝑡), (31d) where 𝐅_𝑖, 𝑖=1,…,4, are the desired feedback gains for the fuzzy controller design.

For starting the fuzzy controller design, we select the supply rate 𝛾=1, 𝐒=𝐈 with compatible dimension and assign the individual state variance constraints as 𝜎²₁=3, 𝜎²₂=3, 𝜎²₃=0.02, 𝜎²₄=2, 𝜎²₅=1, and 𝜎²₆=0.8. Solving the sufficient conditions of Corollary 3 via LMI technique [32], the matrix 𝐗 and 𝐅_𝑖 can be obtained as follows: 𝐗=1.5365−1.6718×10⁻¹¹1.5083×10⁻¹⁰−0.7323−1.6718×10⁻¹¹1.50561.4886×10⁻⁸−1.7648×10⁻⁶1.5083×10⁻¹⁰1.4886×10⁻⁸0.00801.063×10⁻¹²−0.7323−1.7648×10⁻⁶1.063×10⁻¹²1.28981.7508×10⁻⁶−0.7292−1.4698×10⁻⁹−1.016×10⁻¹¹−2.2125×10⁻¹²1.4989×10⁻⁹−0.0551−2.7184×10⁻¹⁰1.7508×10⁻⁶−2.2125×10⁻¹²−0.72921.4989×10⁻⁹−1.4698×10⁻⁹−0.0551−1.016×10⁻¹¹−2.7184×10⁻¹⁰0.74048.5979×10⁻⁹8.5979×10⁻⁹,𝐅0.5004₁=111.3−2.3−0.7228−3.3−0.11958−529.30.61970.8−93.2−0.2−200.21126.1−0.1−411.9202.8,𝐅₂=,𝐅112.40.021−10.5228.80.1−1.9−0.1914.6−4446.3−0.21880−8040.0301−176.61904.20.1−362.9344₃=112.40.9−25.7228.81.9−4.7−0.5940−2271.7−1.11932.2−409.40.2−189.91472.20.3−390.5265.6,𝐅₄=.111.32.6−6.22283.8−1.1−1.1916.8−4058.7−0.91886733.70.2−178.51827.20.2−367.2330 (32)

The simulation responses of states are shown in Figures 3, 4, 5, 6, 7, 8 with initial condition 𝑥(0)=[10100000]^T. Besides, the external disturbances 𝑣₁(𝑡), 𝑣₂(𝑡), and 𝑤(𝑡) are chosen as zero-mean white noises with variance one. From the simulation results, the effect of the external disturbance 𝑤(𝑡) on the proposed system can be criticized as follows: 𝐸2∫^𝑡𝑝₀𝑦^T(𝑥)𝐒𝑤(𝑡)𝑑𝑡𝐸∫^𝑡𝑝₀𝛾𝑤^T(𝑡)𝑤(𝑡)𝑑𝑡=2.0004. (33) The ratio value of (33) is bigger than determined dissipation rate 𝛾=1; one can find that condition (9) of Definition 1 is satisfied. Besides, the variances of states 𝑥₁(𝑡), 𝑥₂(𝑡), 𝑥₃(𝑡), 𝑥₄(𝑡), 𝑥₅(𝑡), and 𝑥₆(𝑡)  are 2.4692, 2.4849, 8.8252×10⁻⁶, 0.5876, 0.5825, and 2.771×10⁻⁴, respectively. It is obvious that the individual state variance constraints are all satisfied. Therefore, the nonlinear ship steering system (29a), (29b), (29c), (29d), (29e), and (29f) controlled by the proposed fuzzy controller (31a), (31b), (31c), and (31d) is asymptotically stable subject to individual state variance constraints and strictly input passivity constraint.

Figure 3 

Responses of state 𝑥1(𝑡).

Figure 4 

Responses of state 𝑥2(𝑡).

Figure 5 

Responses of state 𝑥3(𝑡).

Figure 6 

Responses of state 𝑥4(𝑡).

Figure 7 

Responses of state 𝑥5(𝑡).

Figure 8 

Responses of state 𝑥6(𝑡).

5. Conclusions

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.

Acknowledgments

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.