Addresses the issue of stability of a fuzzy system described by fuzzy rules with singleton consequents. It first presents two canonical forms of a fuzzy system: a parametric expression and a state-space expression. A fuzzy system with singleton consequents is found to be a piecewise-polytopic-affine system. Then the paper gives stability theorems for a discrete-time system as well as for a continuous time system. It also gives a brief survey on the stability issues of fuzzy control systems.

On stability of fuzzy systems expressed by fuzzy rules with singleton consequents

PID type fuzzy controller and parameters adaptive method

A decoupled fuzzy sliding-mode controller design is proposed. The decoupled method provides a simple way to achieve asymptotic stability for a class of fourth-order nonlinear systems with only five fuzzy control rules. The ideas behind the controller are as follows. First, decouple the whole system into two second-order systems such that each subsystem has a separate control target expressed in terms of a sliding surface. Then, information from the secondary target conditions the main target, which, in turn, generates a control action to make both subsystems move toward their sliding surface, respectively. A closely related fuzzy controller to the sliding-mode controller is also presented to show the theoretical aspect of the fuzzy approach in which the characteristics of fuzzy sets are determined analytically to ensure the stability and robustness of the fuzzy controller. Finally, the decoupled sliding-mode control (SMC) is used to control three highly nonlinear systems and confirms the validity of the proposed approach.

Decoupled fuzzy sliding-mode control

We investigated the analytical structure of the Takagi-Sugeno (TS) type of fuzzy controllers, which was unavailable in the literature. The TS fuzzy controllers we studied employ a new and simplified TS control rule scheme in which all the rule consequent use a common function and are proportional to one another, greatly reducing the number of parameters needed in the rules. Other components of the fuzzy controllers are general: arbitrary input fuzzy sets, any type of fuzzy logic, and the generalized defuzzifier, which contains the popular centroid defuzzifier as a special case. We proved that all these TS fuzzy controllers were nonlinear variable gain controllers and characteristics of the gain variation were parametrized and governed by the rule proportionality. We conducted an in-depth analysis on a class of nonlinear variable gain proportional-derivative (PD) controllers. We present the results to show: (1) how to analyze the characteristics of the variable gains in the context of control; (2) why the nonlinear variable gain PD controllers can outperform their linear counterpart; and (3) how to generate various gain variation characteristics through the manipulation of the rule proportionality.

Constructing nonlinear variable gain controllers via the Takagi-Sugeno fuzzy control

This paper focuses on the control problem of the quadruple inverted pendulum by variable universe adaptive fuzzy control. First, the mathematical model on the quadruple inverted pendulum is described and its controllability is versified. Then, an efficient controller on the quadruple inverted pendulum is designed by using variable universe adaptive fuzzy control theory. Finally the simulation of the quadruple inverted pendulum is shown in detail. Besides, the experimental results on the hardware systems, i.e. real object systems, on a single inverted pendulum, a double inverted pendulum and a triple inverted pendulum are briefly introduced.

Variable universe adaptive fuzzy control on the quadruple inverted pendulum

Pad-analysis of fuzzy control stability

Design fuzzy controllers for complex systems with an application to 3-stage inverted pendulums

The framework of the theory of Truth-valued-flow Inference (TVFI) is introduced. Even though there are dozens of papers presented on fuzzy reasoning, we think it is still needed to explore a rather unified fuzzy reasoning theory which has the following two features: (1) it is simplified enough to be executed feasibly and easily; and (2) it is well structural and well consistent enough that it can be built into a strict mathematical theory and is consistent with the theory proposed by L.A. Zadeh. TVFI is one of the fuzzy reasoning theories that satisfies the above two features. It presents inference by the form of networks, and naturally views inference as a process of truth values flowing among propositions.

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Wei Xu

Papers

Pad-analysis of fuzzy control stability

Design fuzzy controllers for complex systems with an application to 3-stage inverted pendulums

Truth-Valued-Flow Inference (TVFI) and its applications in approximate reasoning