SOS-based Stability Conditions Research Articles

Optimally robust H∞ polynomial fuzzy controller design using quantum-inspired evolutionary algorithm

ABSTRACTThis paper proposes an optimally robust H∞ polynomial fuzzy controller design using quantum-inspired evolutionary algorithm (QEA) for continuous/discrete time polynomial fuzzy systems with model uncertainties and external disturbances. To improve control performance, QEA is adopted to evolve optimal control gains with a fitness function that is defined by performance requirements. The stability and robustness of the control system are then guaranteed by the proposed robust H∞ stability conditions, which are formed by the sum of squares (SOS) method. By using the principle of copositivity, novel relaxed SOS-based stability conditions are derived to reduce the conservativeness of solving SOS-based stability conditions, while the feasible solution space is broadened. Four numerical examples demonstrate the effectiveness of the proposed approaches.

Output-feedback tracking control for interval type-2 polynomial fuzzy-model-based control systems

In this paper, the output tracking control issues of polynomial-fuzzy-model-based (PFMB) systems equipped with mismatched interval type-2 (IT2) membership functions are investigated. The output-feedback IT2 polynomial fuzzy controller connected with the nonlinear plant in a closed loop drives the system states of the nonlinear plant to track those of a stable reference model. The system stability is investigated based on the Lyapunov stability theory under the sum-of-squares (SOS)-based analysis approach and the SOS-based stability conditions are derived subjecting to an H∞ performance. In addition, the fuzzy controller does not need to share the same membership functions with the plant. Moreover, the information of membership functions is included in the analysis to facilitate the analysis and relax the stability conditions. Numerical and experimental examples are presented to verify the effectiveness of the proposed tracking control approach.

Robust H ∞ controller design for polynomial fuzzy control systems by sum‐of‐squares approach

This study presents the robust H ∞ control of polynomial fuzzy systems when model uncertainties and external disturbances are simultaneously taken into account. Using a non-quadratic Lyapunov function, two stability conditions were derived in the form of sum-of-squares (SOS). The proposed polynomial fuzzy controller possesses H ∞ robust performance against norm-bounded disturbances and uncertainties. Simulation results demonstrated the effectiveness of the proposed SOS-based stability conditions, which are more general than those of existing approaches.

Two-Step Stability Analysis for General Polynomial-Fuzzy-Model-Based Control Systems

This paper investigates the stability of a polynomial-fuzzy-model-based (PFMB) control system formed by a nonlinear plant represented by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop. Three cases of polynomial fuzzy controllers are proposed for the control process with the consideration of a matched/mismatched number of rules and/or premise membership functions, which demonstrate different levels of controller complexity, design flexibility, and stability analysis results. A general polynomial Lyapunov function candidate is proposed to investigate the system stability. Unlike the published work, there is no constraint on the polynomial Lyapunov function candidate, which is independent of the form of the polynomial fuzzy model. Thus, it can be applied to a wider class of PFMB control systems and potentially produces more relaxed stability analysis results. Two-step stability conditions in terms of sum-of-squares (SOS) are obtained to numerically find a feasible solution. To facilitate the stability analysis and relax the stability analysis result, the boundary information of membership functions is taken into account in the stability analysis and incorporated into the SOS-based stability conditions. Simulation examples are given to illustrate the effectiveness of the proposed approach.

Synchronization of Chaotic Systems Using Sampled-Data Polynomial Controller

This paper presents the synchronization of two chaotic systems, namely the drive and response chaotic systems, using sampled-data polynomial controllers. The sampled-data polynomial controller is employed to drive the system states of the response chaotic system to follow those of the drive chaotic system. Because of the zero-order-hold unit complicating the system dynamics by introducing discontinuity to the system, it makes the stability analysis difficult. However, the sampled-data polynomial controller can be readily implemented by a digital computer or microcontroller to lower the implementation cost and time. With the sum-of-squares (SOS) approach, the system to be handled can be in the form of nonlinear state-space equations with the system matrix depending on system states. Based on the Lyapunov stability theory, SOS-based stability conditions are obtained to guarantee the system stability and realize the chaotic synchronization subject to an H∞ performance function. The solution to the SOS-based stability conditions can be found numerically using the third-party Matlab toolbox SOSTOOLS. Simulation examples are given to illustrate the merits of the proposed sampled-data polynomial control approach for chaotic synchronization problems.

Stability Analysis of Polynomial-Fuzzy-Model-Based Control Systems With Mismatched Premise Membership Functions

This paper investigates the stability of polynomial-fuzzy-model-based (PFMB) control system, which is formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop. To enhance the design flexibility, the number of rules and the shape of premise membership functions of the polynomial fuzzy controller are considered to be chosen freely and are different from those of the polynomial fuzzy model, however, which make the stability analysis more difficult and potentially lead to conservative stability analysis result. A sum-of-squares (SOS)-based stability analysis approach using the Lyapunov stability theory is proposed to investigate the stability of the PFMB control systems and synthesize the polynomial fuzzy controller. To facilitate the stability analysis and relax the stability analysis result, the property of the membership functions and the boundary information of the membership grades and premise variables are taken into account in the stability analysis and incorporated into the SOS-based stability conditions. A simulation example is given to illustrate the effectiveness of the proposed approach.

Output Regulation of Polynomial-Fuzzy-Model-Based Control Systems

This paper investigates the output regulation problem of polynomial-fuzzy-model-based (PFMB) control systems with the consideration of system stability and control synthesis using the sum-of-squares (SOS)-based approach. A PFMB control system is formed by a polynomial fuzzy model and a polynomial fuzzy controller with integral action connected in a closed loop. Three cases of PFMB control systems regarding the premise membership functions and number of fuzzy rules, which have their own properties and advantages, are considered. The stability of the PFMB control systems is investigated based on the Lyapunov stability theory. With consideration of the information of membership functions, SOS-based stability conditions corresponding to each case are obtained to guarantee system stability and to realize output regulation. Simulation examples are given to compare the three cases and illustrate the effectiveness of the proposed approach.

Improved Polynomial Fuzzy Modeling and Controller with Stability Analysis for Nonlinear Dynamical Systems

This study presents an improved model and controller for nonlinear plants using polynomial fuzzy model‐based (FMB) systems. To minimize mismatch between the polynomial fuzzy model and nonlinear plant, the suitable parameters of membership functions are determined in a systematic way. Defining an appropriate fitness function and utilizing Taylor series expansion, a genetic algorithm (GA) is used to form the shape of membership functions in polynomial forms, which are afterwards used in fuzzy modeling. To validate the model, a controller based on proposed polynomial fuzzy systems is designed and then applied to both original nonlinear plant and fuzzy model for comparison. Additionally, stability analysis for the proposed polynomial FMB control system is investigated employing Lyapunov theory and a sum of squares (SOS) approach. Moreover, the form of the membership functions is considered in stability analysis. The SOS‐based stability conditions are attained using SOSTOOLS. Simulation results are also given to demonstrate the effectiveness of the proposed method.

Fuzzy control of non-linear systems using parameter-dependent polynomial fuzzy model

This study investigates the stability of parameter-dependent polynomial-fuzzy-model-based (PDPFMB) control systems. A parameter-dependent polynomial fuzzy model is proposed to represent a non-linear plant. A parameter-dependent polynomial fuzzy controller is then employed to stabilise the non-linear plant. The stability of PDPFMB control systems is investigated using the sum-of-squares (SOS) technique based on a parameter-dependent Lyapunov function candidate. With the consideration of the information of system parameters, parameter-dependent SOS-based stability conditions are obtained to determine the system stability and polynomial feedback gains. A feasible solution to the stability conditions can be obtained numerically using the third-party Matlab toolbox SOSTOOLS. Unlike the membership functions, as the system parameters are not single signed functions, the traditional analysis approach obtaining the stability conditions cannot be applied. Instead, the membership functions and system parameters are considered as symbolic variables for the construction of the parameter-dependent SOS-based stability conditions. A simulation example is given to demonstrate the merits of the proposed approach.

Stability analysis of polynomial fuzzy-model-based control systems under perfect/imperfect premise matching

This study presents an improved sum-of-squares (SOS)-based stability analysis result for the polynomial fuzzy-model-based control system, formed by a polynomial fuzzy model and a polynomial fuzzy controller connected in a closed loop. Two cases, namely perfect and imperfect premise matching, are considered. Under the perfect premise matching, the polynomial fuzzy model and polynomial fuzzy controller share the same premise membership functions. While different sets of membership functions are employed, it falls into the case of imperfect premise matching. Based on the Lyapunov stability theory, improved SOS-based stability conditions are derived to determine the system stability and facilitate the controller synthesis. Simulation examples are given to verify the stability analysis results and demonstrate the effectiveness of the proposed approach.

Stabilization of Nonlinear Systems Using Sampled-Data Output-Feedback Fuzzy Controller Based on Polynomial-Fuzzy-Model-Based Control Approach

This paper investigates the stability of sampled-data output-feedback (SDOF) polynomial-fuzzy-model-based control systems. Representing the nonlinear plant using a polynomial fuzzy model, an SDOF fuzzy controller is proposed to perform the control process using the system output information. As only the system output is available for feedback compensation, it is more challenging for the controller design and system analysis compared to the full-state-feedback case. Furthermore, because of the sampling activity, the control signal is kept constant by the zero-order hold during the sampling period, which complicates the system dynamics and makes the stability analysis more difficult. In this paper, two cases of SDOF fuzzy controllers, which either share the same number of fuzzy rules or not, are considered. The system stability is investigated based on the Lyapunov stability theory using the sum-of-squares (SOS) approach. SOS-based stability conditions are obtained to guarantee the system stability and synthesize the SDOF fuzzy controller. Simulation examples are given to demonstrate the merits of the proposed SDOF fuzzy control approach.

Polynomial Fuzzy-Model-Based Control Systems: Stability Analysis Via Piecewise-Linear Membership Functions

This paper presents the stability analysis of polynomial fuzzy-model-based (PFMB) control systems using the sum-of-squares (SOS) approach. The PFMB control system under consideration requires that the polynomial fuzzy model and polynomial fuzzy controller share neither the same premise membership functions nor the same number of fuzzy rules. This class of PFMB control systems offers a greater design flexibility to the polynomial fuzzy controller. However, due to the imperfectly matched membership functions, it usually produces more conservative stability conditions by following the traditional stability-analysis approach for the FMB control systems. To facilitate the stability analysis, piecewise-linear membership functions (PLMFs) are proposed, which offer a nice property that the grades of membership are governed by a finite number of sample points. Thus, it allows the PLMFs to be brought to the SOS-based stability conditions, which are applied to the PFMB control systems with the specified PLMFs rather than any shapes. The system stability can be examined by checking only the PFMB control system at the sample points. It is worth mentioning that the PLMFs, which are not necessarily implemented physically, are a mathematical tool to carry out the stability analysis. To verify the stability-analysis result, a simulation example is given to demonstrate the effectiveness of the proposed approach.

Chaotic synchronisation using output/full state-feedback polynomial controller

This study presents the synchronisation of two chaotic systems, namely drive and response chaotic systems, using polynomial controllers. Both output and full state-feedback polynomial controllers are proposed, respectively, to drive the system states of the response system to approach those of the drive one. The system stability of the overall system is investigated by the Lyapunov stability theory. Stability conditions in terms of sum of squares (SOS) are derived to aid the design of the feedback gains of the polynomial controllers. By satisfying the SOS-based stability conditions, chaotic synchronisation is achieved with system performance guaranteed by an H∞ performance function. Simulation examples are given to illustrate the merits of the proposed polynomial control approach.

SOS-Based Stability Analysis of Polynomial Fuzzy-Model-Based Control Systems Via Polynomial Membership Functions

This paper presents stability analysis of polynomial fuzzy-model-based (FMB) control systems using the sum-of-squares (SOS) approach. Recently, stability analysis of the polynomial fuzzy-control systems, which is a generalized form of the well-known Takagi-Sugeno (T-S) FMB control systems, has been reported in the form of SOS-based stability conditions. Lack of information on the relations between membership functions and premise variables, in the existing stability analysis approaches, causes conservatism of their results. In this paper, to derive relaxed stability conditions for polynomial FMB control systems, membership functions which are approximated with polynomials and carrying relations between membership functions and premise variables are brought into the stability analysis. Considering a polynomial FMB control system and based on the Lyapunov stability theory, stability conditions in the form of fuzzy summations are derived, where each term contains product of membership functions of the polynomial fuzzy model and polynomial fuzzy controller. Each product term is approximated by a polynomial. In order to obtain better approximation, the operating domain of membership functions is partitioned to subregions. Then, SOS-based stability conditions for all subregions are derived. Unlike some published stability-analysis approaches, the proposed one can be employed for stability analysis of polynomial fuzzy-control systems under imperfect premise matching of which the fuzzy model and fuzzy controller do not share the same membership functions. The solution of the SOS-based stability conditions can be found numerically using SOSTOOLS, which is a free third-party MATLAB Toolbox. Numerical examples are given to illustrate the effectiveness of the proposed stability conditions.

Synchronization of generalized Hénon map using polynomial controller

This Letter presents the chaos synchronization of two discrete-time generalized Hénon map, namely the drive and response systems. A polynomial controller is proposed to drive the system states of the response system to follow those of the drive system. The system stability of the error system formed by the drive and response systems and the synthesis of the polynomial controller are investigated using the sum-of-squares (SOS) technique. Based on the Lyapunov stability theory, stability conditions in terms of SOS are derived to guarantee the system stability and facilitate the controller synthesis. By satisfying the SOS-based stability conditions, chaotic synchronization is achieved. The solution of the SOS-based stability conditions can be found numerically using the third-party Matlab toolbox SOSTOOLS. A simulation example is given to illustrate the merits of the proposed polynomial control approach.