Two-Step Stability Analysis for General 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.

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.

In this paper, the stability analysis for interval type-2 (IT2) polynomial fuzzy-model-based (PFMB) control system using the information of membership functions is investigated. To tackle uncertainties, IT2 membership functions are used in the IT2 polynomial fuzzy model and IT2 polynomial fuzzy controller. The stability of IT2 PFMB control system is investigated based on the Lyapunov stability theory and both sets of membership function independent (MFI) and membership function dependent (MFD) stability conditions are derived on the basis of the sum-of-squares (SOS) approach. To make the stability conditions MFD, the boundary information of IT2 membership functions is used in the stability analysis. To extract richer information of IT2 membership functions, the operating domain is partitioned into sub-domains. In each sub-domain, the boundary information of IT2 membership functions and those of the upper and lower membership function are obtained. Furthermore, to further relax the conservativeness, a switching polynomial fuzzy controller, together with the informations obtained in each sub-domain, is employed in investigating the stability analysis. Numerical examples and simulation results are given to demonstrate the validity of MFD and MFD switching methods.

Tracking control design of interval type-2 polynomial-fuzzy-model-based systems with time-varying delay

This study investigates the stability of fuzzy‐model‐based (FMB) control system, which is formed by a Takagi–Sugeno (T–S) fuzzy model representing a non‐linear plant and a fuzzy controller connected in a closed loop. The proposed fuzzy controller considers a higher design flexibility that the premise membership functions and the number of fuzzy rules are allowed to be not the same as those of the T–S fuzzy model. By employing simple membership functions and a smaller number of fuzzy rules, the complexity of the fuzzy controller can be reduced to facilitate its implementation. Based on the Lyapunov stability theory, a membership‐function‐dependent (MFD) stability analysis is proposed to develop stability conditions in the form of linear matrix inequalities (LMIs) to determine the system stability and facilitate the control synthesis. The information of the membership functions is brought to the MFD LMI‐based stability conditions, which are dedicated to the FMB control system under consideration. It is thus the proposed MFD stability analysis result is more relaxed compared with the existing membership‐function‐independent stability analysis results. A simulation example is given to illustrate the effectiveness of the proposed approach.

This paper presents the stability analysis of fuzzy-model-based control systems. Stepwise membership functions are introduced to facilitate the stability analysis. Through the stepwise membership functions approximating those of the fuzzy model and fuzzy controller, the information of the membership functions can be brought into the stability analysis. Based on the Lyapunov stability theory, stability conditions in terms of linear matrix inequalities are derived in a simple and easy-to-understand manner to guarantee the system stability. The proposed stability analysis approach offers a nice property to include the membership functions of both fuzzy model and fuzzy controller in the LMI-based stability conditions for a dedicated fuzzy-model-based control system. Furthermore, the proposed stability analysis approach can be applied to the fuzzy-model-based control systems of which the membership functions of both fuzzy model and fuzzy controller are not necessarily the same. Greater design flexibility is allowed by choosing the membership functions during the design of fuzzy controllers. By employing membership functions with simple structure, it is possible to lower the structural complexity and the implementation cost. Simulation example is given to illustrate the merits of the proposed approach.

This chapter investigates the stability of polynomial fuzzy model-based control systems by treating the membership functions and system states as symbolic variables. The information of membership functions is considered in the stability analysis and brought to the SOS-based stability conditions. Techniques are proposed to introduce slack matrix variables carrying the information of membership functions, including the property of membership functions, boundary information of membership functions and boundary information of premise variables, to the SOS-based stability conditions without increasing much the computational demand. Details of mathematical derivation are shown to help readers follow easily the analysis. A simulation example is given to show how to apply the obtained stability conditions during the control design and demonstrate the merits of the proposed stability analysis results.

This paper presents the stability analysis of interval type-2 fuzzy-model-based (FMB) control systems. To investigate the system stability, an interval type-2 Takagi-Sugeno (T-S) fuzzy model, which can be regarded as a collection of a number of type-1 T-S fuzzy models, is proposed to represent the nonlinear plant subject to parameter uncertainties. With the lower and upper membership functions, the parameter uncertainties can be effectively captured. Based on the interval type-2 T-S fuzzy model, an interval type-2 fuzzy controller is proposed to close the feedback loop. To facilitate the stability analysis, the information of the footprint of uncertainty is used to develop some membership function conditions, which allow the introduction of slack matrices to handle the parameter uncertainties in the stability analysis. Stability conditions in terms of linear matrix inequalities are derived using a Lyapunov-based approach. Simulation examples are given to illustrate the effectiveness of the proposed interval type-2 FMB control approach.

A review on stability analysis of continuous-time fuzzy-model-based control systems: From membership-function-independent to membership-function-dependent analysis

More relaxed stability analysis and positivity analysis for positive polynomial fuzzy systems via membership functions dependent method

This article proposes a novel Lyapunov stability analysis for a class of switched nonlinear systems under the mode-dependent average dwell time (MDADT) switching signals. For the first time, a polynomial fuzzy model is used to describe the nonlinearity of the switched systems instead of a Takagi–Sugeno fuzzy model, which can represent a wider range of nonlinear plants. As existing advanced results are relatively conservative and lead to that tighter bounds on the dwell time are not easy to be obtained, therefore, the stability analysis for switched systems is still quite challenging. In this article, to tackle this problem, by developing a polynomial multiple Lyapunov function (MLF) approach and exploring the feature of the membership functions, relaxed stability conditions are derived for the switched polynomial fuzzy-model-based control systems. In particular, the polynomial MLF is guaranteed radially unbounded by a two-step procedure, making traditional quadratic MLF a specific case, thereby improved stability conditions can be established in the form of sum-of-squares. Moreover, the new algorithm for multivariate Chebyshev membership functions based on linear programming is proposed to introduce the information of membership functions and further relax the stability conditions to obtain tighter bounds on MDADT. Finally, a simulation example demonstrates the effectiveness of the developed techniques.

This paper presents a sum of squares (SOS, for brevity) approach to polynomial fuzzy observer design for polynomial fuzzy systems. First, we briefly summarize previous results with respect to a polynomial fuzzy system and controller that are more general representation of the well-known Takagi-Sugeno (T-S) fuzzy system and controller, respectively. Secondly, we propose a polynomial fuzzy observer to estimate states of the polynomial fuzzy system and derive an SOS condition to design a polynomial fuzzy controller and observer. A key feature of the SOS design condition is that it realizes the so-called separation principle, that is, that a polynomial fuzzy controller and a polynomial fuzzy observer can be separately designed without lack of guaranteeing the stability of the overall control system. The design approach discussed in this paper is more general than the existing LMI approaches (to T-S fuzzy controller and observer designs). In addition, the design condition in the proposed approach can be represented in terms of SOS and is symbolically and numerically solved via the recent developed SOSTOOLS and a semidefinite program (SDP) solver, respectively. To illustrate the validity of the design approach, a design example is provided. The example shows the utility of our SOS approach to the polynomial fuzzy observer-based control.

This study proposes a new category of fuzzy‐observer‐based controllers to stabilise non‐linear plants based on the fuzzy‐model‐based control approach. A fuzzy observer is proposed to estimate the system states of the non‐linear plant and a fuzzy‐observer‐based controller using the estimated system states for feedback compensation is proposed to close the feedback loop for the control process. It does not require that the fuzzy observer and fuzzy‐observer‐based controller have to share the same premise membership functions and the same number of rules of the Takagi–Sugeno fuzzy model. A membership‐function‐dependent stability analysis approach is proposed to investigate the stability of the fuzzy‐observer‐based control system based on the Lyapunov method. A set of bilinear matrix inequalities (BMIs) is obtained to guarantee the system stability and control synthesis. To find a feasible solution of the BMI‐based stability conditions, a solution‐searching algorithm, which combines global searching algorithm and convex programming techniques, is proposed. A simulation example is provide to demonstrate its capability of relaxing control design flexibility and effectiveness.

Sum-of-squares-based robust H∞ controller design for discrete-time polynomial fuzzy systems

Transporting a dismantled offshore platform has its challenges in the process. The stability and longitudinal strength of the decommissioning barge used are common problems in this process. Stability analysis will be carried out in intact and damaged conditions while longitudinal strength analysis will be carried out during still water, hogging, and sagging conditions. From the results of the stability analysis that occurred, the initial GM value was obtained in the intact condition of 52.29 m and damage of 47.72 m, 47.69 m, 47.83 m, and 47.81 m. Then, the intact range value that occurred in intact condition was 86° and damage of 82°, 82°, 83.04°, and 82.9°. Next, the ratio area value that occurred in intact condition was 11.22 and damage of 37.24, 43.09, 36.95, and 47.59. Then, from the results of the longitudinal strength analysis, the bending moment value was obtained in still water, hogging, and sagging conditions respectively of 16410 Ton.m, 52740 Ton.m, and 37210 Ton.m and the value of shear forces in still water, hogging, and sagging conditions respectively of 619.092 Ton, 1552.973 Ton, and 1391.362 Ton. The results of stability and strength analysis meet their respective criteria.