Sufficient conditions for absolute stability and optimization using Genetic Algorithms of specific class Mamdani Fuzzy Logic State Variable Controllers

This paper has proposed a specific class of multiple input single output (MISO) Mamdani fuzzy logic state variable controllers (FLSVC) with triangle input membership functions, output membership functions with the same area, product fuzzy AND operator, sum fuzzy OR operator and centroid of area defuzzifier for control of stable Single Input Multiple Output (SIMO) linear time invariant systems. The proposed FLSVC in fact represents memoryless, nonlinear, locally Lipschitz mapping between its inputs and its output. According to the multivariable circle criterion for absolute stability, nonlinear part of the system must satisfy the sector condition. In this paper necessary and sufficient conditions have been determined, which must satisfy FLSVC parameters in order for the memoryless function represented by FLSVC to satisfy the sector condition. Subsequently, using the multivariable circle criterion, the sufficient conditions for absolute stability of feedback connected linear system and FLSVC have been determined. Based on the results achieved, an optimization method for both parameters and FLSVC decision rules has been proposed using genetic algorithms (GA).