This paper develops an input-output stability analysis on thebasis of the rules, and the control input values generated by the fuzzycontroller for each “cell partition” of the phase plane. Itis assumed that the rule-base has been designed for a linear plant whoseapproximate model is available. Furthermore, the rules may have beenderived using the operator manual-type if-then rules or based directlyon the approximate model. The method is based on formulating aninput-output dissipative map and then applying a Lyapunov-like analysisfor stability convergence. For a linear system, an input-output mappingmay always be formulated as an energy-like function. By proper choice ofoutputs, and the inputs generated by the fuzzy controller, theenergy-like function is forced to be dissipative fairly easily byapplying the Kalman-Yakubovich lemma. It is a known fact that thelinearized model of a stable nonlinear system should be stable in theneighborhood of the equilibrium point. Thus, this allows theKalman-Yakubovich lemma to be applied in this neighborhood, in theformation of a dissipative input-output map. Subsets of the rule baseare considered in each of the four quadrants of the state space
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