Stability analysis of Mamdani-type fuzzy controllers by calculating the Lyapunov exponent

Determining the stability of the controllers, either through simulations orthrough analytical techniques, is vital in their design and implementation.The analytical method of stability in the sense of Lyapunov requires findinga candidate function, as a sufficient but not necessary criterion for that pur-pose. This candidate function is elusive for fuzzy controllers. It is proposed,as a possible solution to this problem, to quantify the stability of the fuzzycontrollers using the Lyapunov exponent (EL) calculated numerically. Thetime series from which the Lyapunov exponents are calculated is obtainedfrom the output of various closed-loop Mamdani-type fuzzy controllers withthe nonlinear plant dynamics stabilized in an admissible region of operation.The experiments were carried out by implanting the numerical method on theplatform MATLAB, integrating it with data from the simulation of variousfuzzy controllers. The plant to be controlled is the inverted car-pendulumsystem modeled with the Euler-Lagrange formulation. In each experiment,the time series corresponding to the control signal was obtained and theLyapunov exponent was calculated. Although variations in magnitude areobserved, the calculated exponent is negative in all cases. This indicates thatthe Mamdani-type fuzzy controllers used are dissipative systems. As futurework, the use of EL in adaptive control is outlined.