Stability analysis and control synthesis for fuzzy‐observer‐based controller of nonlinear systems: a fuzzy‐model‐based control approach

Adaptive fuzzy fault-tolerant output feedback control of uncertain nonlinear systems with actuator faults based on dynamic surface technique

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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.

Artificial intelligence techniques for stability analysis and control in smart grids: Methodologies, applications, challenges and future directions

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.

In this paper, we investigate the stability of Takagi-Sugeno fuzzy-model-based (FMB) functional observer-control system. When system states are not measurable for state-feedback control, a fuzzy functional observer is designed to directly estimate the control input instead of the system states. Although the fuzzy functional observer can reduce the order of the observer, it leads to a number of observer gains to be determined. Therefore, a new form of fuzzy functional observer is proposed to facilitate the stability analysis such that the observer gains can be numerically obtained and the stability can be guaranteed simultaneously. The proposed form is also in favor of applying separation principle to separately design the fuzzy controller and the fuzzy functional observer. To design the fuzzy controller with the consideration of system stability, higher order derivatives of Lyapunov function (HODLF) are employed to reduce the conservativeness of stability conditions. The HODLF generalizes the commonly used first-order derivative. By exploiting the properties of membership functions and the dynamics of the FMB control system, convex and relaxed stability conditions can be derived. Simulation examples are provided to show the relaxation of the proposed stability conditions and the feasibility of designed fuzzy functional observer-controller.

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.

In this paper, we develop a systematic Lyapunov approach to the regional stability and performance analysis of saturated systems in a general feedback configuration. The only assumptions we make about the system are well-posedness of the algebraic loop and local stability. Problems to be considered include the estimation of the domain of attraction, the reachable set under a class of bounded energy disturbances and the nonlinear L2 gain. The regional analysis is established through an effective treatment of the algebraic loop and the saturation/deadzone function. This treatment yields two forms of differential inclusions, a polytopic differential inclusion (PDI) and a norm-bounded differential inclusion (NDI) that contain the original system. Adjustable parameters are incorporated into the differential inclusions to reflect the regional property. The main idea behind the regional analysis is to ensure that the state remain inside the level set of a certain Lyapunov function where the PDI or the NDI is valid. With quadratic Lyapunov functions, conditions for stability and performances are derived as linear matrix inequalities (LMIs). To obtain less conservative conditions, we use a pair of conjugate non-quadratic Lyapunov functions, the convex hull quadratic function and the max quadratic function. These functions yield bilinear matrix inequalities (BMIs) as conditions for stability and guaranteed performance level. The BMI conditions cover the corresponding LMI conditions as special cases, hence the BMI results are guaranteed to be as good as the LMI results. In most examples, the BMI results are significantly better than the LMI results

This paper investigates the relaxed non‐quadratic stability conditions, fuzzy observer designs and ℋ︁∞ controller designs for discrete‐time Takagi‐Sugeno fuzzy systems based on a relaxed approach in which fuzzy Lyapunov functions are used. First, a new relaxed condition of non‐quadratic stability is presented, which is shown to be useful in designing fuzzy controller and observer. Second, new fuzzy observers based on the relaxed non‐quadratic stability conditions have been proposed. Then, a sufficient linear matrix inequality (LMI)‐type condition is proposed to guarantee the existence of the ℋ︁∞ controllers based on the fuzzy observers designed. It is shown that the controller and observer parameters can be obtained by solving a set of LMIs that are numerically feasible with commercially available software. Finally, the effectiveness and less conservativeness of the proposed approach are demonstrated by two examples. Copyright © 2008 John Wiley & Sons, Ltd.

Relaxed conditions for stability of nonlinear, continuous and discrete-time systems given by fuzzy models are presented. A theoretical analysis shows that the proposed methods provide better or at least the same results of the methods presented in the literature. Numerical results exemplify this fact. These results are also used for fuzzy regulators and observers designs. The nonlinear systems are represented by fuzzy models proposed by Takagi and Sugeno (1985). The stability analysis and the design of controllers are described by linear matrix inequalities, that can be solved efficiently using convex programming techniques. The specification of the decay rate, constrains on control input and output are also discussed.

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

In this paper, a robust adaptive controller based on fuzzy observer for a class of unknown nonlinear systems with bounded external disturbances is presented. Firstly, a fuzzy observer based on fuzzy-basis-functions is designed to estimate the system state variables, and the fuzzy observer can guarantee the uniform ultimate bounds of the observer errors by SPR theory. Secondly, the adaptive controller with a fuzzy observer which requires only the system output variable is showed to control a class of unknown nonlinear systems with external disturbances, and the adaptive controller can guarantee the uniform ultimate bounds of the tracking errors. Finally, this simulation results presented adaptive controller applied to a single link robot control system demonstrate and confirm the effectiveness of the proposed method.

This study is concerned with the problem of observer-based output feedback control for T-S fuzzy systems with time-delay under imperfect premise matching. The objective is to design an observer-based controller such that the closed-loop control system is asymptotically stable. The proposed fuzzy controller and fuzzy observer do not need to share the same premise membership functions and number of rules with T-S fuzzy systems, which makes the controller and observer design more flexible. Using a recently developed integral inequality, a delay-dependent method for the controller and observer design is provided. With the help of some advanced matrix inequalities, the proposed conditions are presented in the form of linear matrix inequalities (LMIs). An illustrative example is given to show the effectiveness of the present result.

A model-based robust nonlinear control via second-order sliding mode for a class of nonlinear affine systems is presented in this paper. The proposed approach consists of the inverse dynamics of the system for a compensation of the nominal nonlinear terms and a second-order sliding mode control to deal with the given parameter uncertainty as well as the unknown disturbances. Within a stability analysis, the stability conditions are derived and proved by means of Lyapunov's second method. Moreover, the robust nonlinear control approach is applied to the tracking control of a hydrostatic transmission, which is a typical nonlinear system with parameter uncertainty and unknown disturbances. The simulation and experimental results highlight the applicability of the proposed method.

Almost sure stabilization of positive systems under asynchronous Markovian switching