Stability Of Fuzzy Control System Research Articles

Stability Analysis of Systems with Fuzzy PI Controllers Applied to Electric Drives

This paper analyzes the stability of fuzzy control systems with applications for electric drives. Ensuring the stability of these systems is a necessity in practice. The purpose of the study is the analysis of the dynamic characteristics of the speed control systems of electric drives based on fuzzy PI controllers in the context of performing stability analyses, both internal and input–output, finding solutions to stabilize these systems and provide guidance on fuzzy regulator design. The main methods of treatment applied are as follows: framing the control system in the theory of stability of multivariable non-linear systems, application of Lyapunov’s theory, performing an input–output stability analysis, and verification of the stability domain. The article presents the conditions for correcting the fuzzy controller to ensure internal and external stability, determines the limits of the stability sector, and gives indications for choosing the parameters of the controller. The considerations presented can be applied to various structures for regulating the speed of electric drives which use various PI fuzzy controllers.

Fuzzy Adaptive Control Design Strategy of Nonlinear Switched Large-Scale Systems

The decentralized adaptive fuzzy control problem for switched nonstrict-feedback nonlinear large-scale systems is investigated in this paper. Fuzzy logic systems are adopted to identify the unknown nonlinear system functions and unknown interconnected nonlinearities. A linear state observer is designed to solve the problem of unmeasured states, and a new observer-based output-feedback decentralized control scheme is developed based on adaptive fuzzy control principle and bacsktepping design technique. The proposed control strategy not only solves the control design problem of more general form nonlinear switched systems in nonstrict-feedback form, but also does not require the switched signals to meet the restriction of average dwell time. The stability of fuzzy control systems under arbitrary switchings is proven based on the common Lyapunov function method. Simulation results of numerical examples are presented to further demonstrate the effectiveness of the proposed control strategy.

Dynamic switching based fuzzy control strategy for a class of distributed parameter system

In this work, a dynamic switching based fuzzy controller combined with spectral method is proposed to control a class of nonlinear distributed parameter systems (DPSs). Spectral method can transform infinite-dimensional DPS into finite ordinary differential equations (ODEs). A dynamic switching based fuzzy controller is constructed to track reference values for the multi-inputs multi-outputs (MIMO) ODEs. Only a traditional fuzzy logic system (FLS) and a rule base are used in the controller, and membership functions (MFs) for different ODEs are adjusted by scaling factors. Analytical models of the dynamic switching based fuzzy controller are deduced to design the scaling factors and analyze stability of the control system. In order to obtain a good control performance, particle swarm optimization (PSO) is adopted to design the scaling factors. Moreover, stability of fuzzy control system is analyzed by using the analytical models, definition of the stability and Lyapunov stability theory. Finally, a nonlinear rod catalytic reaction process is used as an illustrated example for demonstration. The simulation results show that performance of proposed dynamic switching based fuzzy control strategy is better than a multi-variable fuzzy logic controller.

Asymptotically Stable Fuzzy Control System with Modeling Error

Asymptotically Stable Fuzzy Control System with Modeling Error

Stability Analysis of a Type of Takagi-Sugeno PI Fuzzy Control Systems Using Circle Criterion

A type of Takagi-Sugeno (T-S) Proportional-Integral (PI) fuzzy controllers is studied. The T-S PI fuzzy controller is formed by a T-S Proportional-Derivative (PD) fuzzy controller connected with an integrator. In this particular structure, the T-S PD fuzzy controller is a weighted sum of some linear PD sub-controllers. The mathematical properties of our T-S PI fuzzy controller are also investigated. Based on these properties, the global asymptotic stability of the fuzzy control systems, in which the T-S PI fuzzy controllers are employed, are analyzed by using the well-known circle criterion. A sufficient condition with an elegant graphical interpretation in the frequency domain is further derived to guarantee the global asymptotic stability of the above fuzzy control systems. Finally, two numerical examples are provided to demonstrate how to deploy this method in analyzing the T-S PI fuzzy control systems in the frequency domain with the aid of some simple graphs.

New stability method applied to the stabilization of fuzzy control systems

This paper addresses stability analysis and stabilization for Takagi-Sugeno fuzzy systems via a new stability method previously developed and presented by the authors. The new stability method is here extended to fuzzy systems. In essence, the authors consider the stabilization of a fuzzy control system with parallel distributed compensation. This problem has been considered in detail previously by different authors using fuzzy Lyapunov function theory. However, in this paper the authors show that their new stability method presents a viable alternative to Lyapunov's method for such control systems. This statement is confirmed by a demonstration that the new stability method always provides necessary and sufficient conditions, the satisfaction of which guarantees that the system in question will be asymptotically stable.

Fuzzy Control System Design via Fuzzy Lyapunov Functions

This correspondence deals with the problems of analysis and design for a class of continuous-time Takagi-Sugeno fuzzy control systems. Sufficient conditions for the stability of fuzzy control systems are derived based on a fuzzy Lyapunov function. Both parallel and nonparallel distributed compensation controllers are considered. Sufficient conditions for the solvability of the controller design problem are given in the form of linear matrix inequalities. Unlike the fuzzy Lyapunov function approaches reported in the literature, the bound of the time derivatives of the fuzzy basis functions is not required in the proposed approaches. The effectiveness of the proposed approaches is shown through a numerical example.

Design of a mixed fuzzy controller for multiple-input multiple-output systems

Multiple-input multiple-output (MIMO) systems usually have characteristics of nonlinear dynamics coupling. Therefore, the difficulty in controlling MIMO systems is how to overcome the coupling effects between the degrees of freedom. The computational burden and dynamic uncertainty associated with MIMO systems make model-based decoupling impractical for real-time control. This work develops a mixed fuzzy controller (MFC) to solve this problem and improve control performance. This study first designs a traditional fuzzy controller (TFC) from the viewpoint of a single-input single-output (SISO) system for controlling each degree of freedom of a MIMO system. Then, an appropriate coupling fuzzy controller is also designed according to the characteristics of the system’s dynamics coupling and incorporated into a TFC to compensate for coupling effects between the degrees of freedom. This control strategy can not only simplify the implementation problem of fuzzy control, but also improve control performance. The state-space approach for analyzing the stability of fuzzy control systems is applied to evaluate the stability and robustness of this intelligent mixed fuzzy controller. To verify the applicability of the proposed mixed fuzzy controller, this work presents a two-link robotic manipulator with a complex dynamic model for a MIMO system to evaluate the stability and robustness of the MFC by numerical simulation, and to examine the control performance by comparing the simulation results of the MFC with those of a TFC for this MIMO system.

Stable controller design for T–S fuzzy systems based on Lie algebras

In this paper, we study the stability of fuzzy control systems of Takagi–Sugeno-(T–S) type based on the classical theory of Lie algebras. T–S fuzzy systems are used to model nonlinear systems as a set of rules with consequents of the type x ˙ ( t ) = A l x ( t ) + B l u ( t ) . We conduct the stability analysis of such T–S fuzzy models using the Lie algebra L A generated by the A l matrices of these subsystems for each rule in the rule base. We first develop our approach of stability analysis for a commuting algebra L A , where all the consequent state matrices A l 's commute. We then generalize our results to the noncommuting case. The basic idea here is to approximate the noncommuting Lie algebra with a commuting one, such that the approximation error is minimum. The results of this approximation are extended to the most general case using the Levi decomposition of Lie algebras. The theory is applied to the control of a flexible-joint robot arm, where we also present the decomposition procedure.

Genetic algorithm optimization of a model-free fuzzy control system

In this work, we present an approach, based on a genetic algorithm optimization search, for determining the values of the parameters of an adaptive stable fuzzy control system suitable to drive a given plant to a desired reference trajectory. The method is demonstrated on a naturally unstable system taken from the literature and then applied to the well known Chernick’s model of a nuclear reactor.

Time-Varying Characteristics Analysis and Fuzzy Controller Systematic Design Method for Pressurized Water Reactor Power Control

In this paper a systematic design method of fuzzy control systems is applied to the pressurized water reactor’s (PWR) power control. The paper includes three parts. In the first part, a simplified time-varying linear model of the PWR power system is constructed, and its inner structure and time-varying characteristics are analyzed. That provides a solid basis for study and design of the nuclear reactor power control system. In the second part, a systematic design method of fuzzy control systems is introduced and applied to control the nuclear reactor power process. The design procedures and parameters are given in detail. This systematic design method has some notable advantages. The control of a global fuzzy model can be decomposed into controlling a set of linear submodels. Each submodel controller can be independently designed by using a linear quadratic regulator approach. This systematic design method gives a sufficient and necessary condition to guarantee the stability of fuzzy control systems; thus, better control performance can be obtained due to the accurate control gains. In the third part, the control performance of the nuclear reactor fuzzy control system is examined by simulation experiments, including nuclear reactor power shutdown, start-up, and adjustment operations. The satisfactory experiment results have shown that the systematic design method for fuzzy control systems is effective and feasible.

Correction to "Comments on 'Stability of fuzzy control systems with bounded uncertain delays'"

The research in the above paper (X. Jiang and W. Xu, Comments on 'Stability of fuzzy control systems with bounded uncertain delays', IEEE Trans. Fuzzy Syst., vol. 12, pp. 285-286, Apr. 2004) was carried out while the first author was with the Department of Automation, Tsinghua University, Beijing 100084, China. He is now with the Faculty of Informatics and Communication, Central Queensland University, Rockhampton QLD 47012, Australia.

Comments on “Stability of Fuzzy Control Systems With Bounded Uncertain Delays”

In these comments, some misleading statements in the above paper have been pointed out and relation results are given.

Authors' Reply to Comment on “Stability of Fuzzy Control Systems With Bounded Uncertain Delays”

For comment see ibid., p.285-6 (2004). The authors reply to a comment on their paper (Z. Yi and P.A. Heng, see ibid., vol.10, p.92-7, 2002) by agreeing with the comment.

Stability analysis and synthesis of multivariable fuzzy systems using interval arithmetic

This paper deals with the design of stable rule-based fuzzy control systems. Interval analysis is applied to design a stable fuzzy Takagi–Sugeno–Kang controller using a robust condition to ensure the stability. The presented methodology starts with a state model of the plant, finds a candidate fuzzy controller and uses an interval arithmetic algorithm to verify the stability of closed-loop fuzzy model. It is important to emphasize the generality of the presented methodology for fuzzy controller synthesis since there are no constraints in the state vector nor in the control vector. This methodology can also be used with nonlinear plant models. In previous works we showed the applicability of the interval analysis to design a controller that ensures the stability of first order nonlinear system. In this paper, we extend the analysis and the synthesis of stable fuzzy control system to the multivariable case. An example with a fuzzy controller for a nonlinear system is presented to illustrate the design procedure.

A multiple lyapunov function approach to stabilization of fuzzy control systems

This paper addresses stability analysis and stabilization for Takagi-Sugeno fuzzy systems via a so-called fuzzy Lyapunov function which is a multiple Lyapunov function. The fuzzy Lyapunov function is defined by fuzzily blending quadratic Lyapunov functions. Based on the fuzzy Lyapunov function approach, we give stability conditions for open-loop fuzzy systems and stabilization conditions for closed-loop fuzzy systems. To take full advantage of a fuzzy Lyapunov function, we propose a new parallel distributed compensation (PDC) scheme that feedbacks the time derivatives of premise membership functions. The new PDC contains the ordinary PDC as a special case. A design example illustrates the utility of the fuzzy Lyapunov function approach and the new PDC stabilization method.

Stable fuzzy control system design with pole-placement constraint: an LMI approach

In this paper, the synthesis of an Linear Matrix Inequality (LMI)-based stable fuzzy control system with pole-placement constraint is presented. The requirements of stability and pole-placement region are formulated based on the Lyapunov direct method. By recasting these constraints into LMIs, we formulate an LMI feasibility problem for the design of the fuzzy state feedback control system that guarantees stability and satisfies desired transient responses. This theoretical approach is applied to a nonlinear magnetic bearing system concerning the issue of rotor position control. Simulation results show that the proposed LMI-based design methodology yields better performance than those of a linear local controller or single objective controller. In addition, it is observed that the proposed fuzzy state feedback controller provides superior stability robustness against parameter variations.

Design of stable fuzzy control systems using Lyapunov's method in fuzzy hypercubes

In this paper, a systematic procedure is presented to analyze and design a stable fuzzy controller for a class of nonlinear systems. Based on the Lyapunov's directive method, we account for the relationship between the stability of the control system and the I/O sets of the fuzzy controller. First, the state space is partitioned into fuzzy hypercubes according to the product space of the input fuzzy sets of the states. Then, all fuzzy hypercubes are considered and made to satisfy the Lyapunov's stability condition by choosing proper output centers of the associated fuzzy rules. Through the proposed design guideline, it is also possible to determine the performance and accuracy of fuzzy control systems. An example with a fuzzy controller for a chaotic system is presented to illustrate the design procedure.

Stability of fuzzy control systems with bounded uncertain delays

Global exponential stability of fuzzy control systems with delays is studied. These delays in the fuzzy control systems are assumed to be any uncertain bounded continuous functions. Stability of systems with uncertain delays is interesting since in practical applications it is not easy to know the delays exactly. Conditions for global exponential stability of free fuzzy systems with uncertain delays are derived. Criteria for design of nonlinear fuzzy controllers to feedback control the stability of global nonlinear fuzzy systems are given. Theorems are proved via the method of functional differential inequalities analysis.

EXTENDED CIRCLE CRITERION: STABLE FUZZY CONTROL OF A MILLING PROCESS.

In this work, the Extended Circle Criterion and fuzzy approaches are combined to develop a practical technique for checking stability of Fuzzy Control Systems (FCS). The controller is designed on the basis of skilled operator criteria and engineering knowledge about the process. Afterwards, stability is verified by means of the so-called Extended Circle Criterion, under the assumption that the static characteristic of the controller is sector-bounded (nonlinear portion) and the process (linear portion) can be rendered Strictly Positive Real (SPR). The method is tested in a FCS applied to the milling process. Theoretical and experimental results show that the closed loop system is stable.