This paper develops an input-output stability analysis on the basis of the rules, and the control input values generated by the fuzzy controller for each "cell partition" of the phase plane. It is assumed that the rule-base has been designed for a linear plant whose approximate model is available. Furthermore, the rules may have been derived using the operator manual-type if-then rules or based directly on the approximate model. The method is based on formulating an input-output dissipative map and then applying a Lyapunov-like analysis for stability convergence. For a linear system, an input-output mapping may always be formulated as an energy-like function. By proper choice of outputs, and the inputs generated by the fuzzy controller, the energy-like function is forced to be dissipative fairly easily by applying the Kalman-Yakubovich lemma. It is a known fact that the linearized model of a stable nonlinear system should be stable in the neighborhood of the equilibrium point. Thus, this allows the Kalman-Yakubovich lemma to be applied in this neighborhood, in the formation of a dissipative input-output map. Subsets of the rule base are considered in each of the four quadrants of the state space.
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