| A viable control theoretic technique was developed for assessing the performance of fuzzy logic control systems. Stability and robustness were considered as measures of the closed loop system's performance. Essentially, the fuzzy logic controller (FLC) in the closed loop system generated the control inputs necessary to control the process, whose mathematical model was readily available. The stability theories developed were of the input-output type. The fuzzy controller was required to drive the system from all initial conditions in the feasible state space, asymptotically to the origin, which is the unique zero-error point. In the first of the two stability theories developed, input-output mappings were formulated linear and nonlinear processes. Essentially, a Lyapunov-like function was assumed, which measures the energy of the physical process that was being fuzzily controlled. From this, the output variables were identified and included in the time derivative of the Lyapunov-like function. In the case of the linear process, on the other hand, the input-output mapping was readily formed in the standard way, using the system's time invariant matrices, and the same stability condition was imposed on the mapping. Alternatively, the Kalman-Yakubovich (Positive Real) lemma was proposed, when the linear system was open-loop stable. The second stability analysis proposed for the fuzzy controlled systems pertained to process set-point control systems. The growth of nonlinearity away from this set-point vicinity was required to diminish under fuzzy control. A set of controllability conditions of fuzzy control systems was proposed that using strictly, the fuzzy if-then rules and the discrete-time fuzzy dynamical system model. On the robustness side, a performance index was formulated based on the system's implicit states. The time derivative was expressed in terms of fuzzy sensitivity functions of the process variables with respect to the parameter perturbations. The first part of the theory was developed with the assumption that the system was decoupled, in the sense that one the parameter for a particular state could be perturbed with no effects on the other system's states. Finally, a set of robust bounds for the parameter perturbations, interactions, and the individual sensitivities were derived. (Abstract shortened by UMI.)... |
