Fuzzy Functional Observer Based Controller Design and Fault Detection for Nonlinear Systems

Abstract

This thesis presents fuzzy model based stability analysis and controller design techniques for nonlinear systems using functional observer considering time-delay, external disturbances and model uncertainty. A novel fuzzy functional observer based robust fault detection technique for delayed nonlinear systems is also included in this thesis. Takagi-Sugeno (T-S) fuzzy model represents a nonlinear system as a fuzzy summation of linear models around the operating points that are expressed by respective fuzzy rules. As this approach uses singleton consequent parts, the defuzzification process of the whole system is straightforward: the overall system dynamics and output are determined by the fuzzy summations of the linear consequent parts. As a result, existing linear tools and techniques can be applied for analysing the stability of the system and designing controller accordingly. Parallel distributed compensation (PDC), a fuzzy blending of the linear compensators designed for the linear subsystems of the fuzzy model, is an effective technique for synthesising a fuzzy controller for a nonlinear system. In case the system states are not readily available, fuzzy observers are employed to obtain PDC controllers. Functional observer directly estimates a function of states in one step rather than doing it in two steps, i.e., estimating the states and computing the function of the estimated states. It reduces the real-time computational effort of the observer. Unknown input observer can decouple external disturbances from the observer error dynamics. This research investigates the existence and stability conditions for the functional observers for nonlinear systems represented by T-S fuzzy models. The main challenge of designing a functional observer for a T-S fuzzy system is obtaining the observer parameters so that the estimation error approaches zero not only for the individual linear subsystems but also for the whole observer after fuzzy blending. The fuzzy functional observer is employed to obtain a PDC controller. Apart from reducing the real-time computational burden, this controller design technique reduces the observer size to the dimension of the controller. Considering time-delays and model uncertainties, a new set of stability conditions are presented. Lyapunov-Krasovskii functionals are used to obtain the stability conditions for delayed systems. Free-weighting matrices are used for obtaining delay dependent stability conditions to increase the solution domain of the stability conditions. This technique is applied to design fuzzy power system stabilisers for single machine infinite bus system. The proposed fault detection technique uses a fuzzy functional observer to obtain a residual. The main advantage is that this technique does not require any calculation of a threshold for the real-time comparison with the residual. The concept of the unknown input observer is used to decouple external disturbances from the error dynamics of residual generation and fault estimation observers. The stability conditions obtained from the Lyapunov stability analysis approach appear in the form of convex inequality conditions. If the inequalities are not linear, the stability conditions are transformed as linear matrix inequalities so that observer parameters can be constructed by solving these inequalities.