An observer-based adaptive iterative learning controller for MIMO nonlinear systems with delayed output

An observer based adaptive iterative learning control (AILC) is proposed for MIMO nonlinear systems with delayed output in this paper. Since the system state vector is unavailable for measurement, we apply the state tracking error observer to solve the problem of unmeasurable system state vector for the design of AILC. By using the state tracking error observer, a mixed time-domain and s-domain technique is first applied to derive an output observation error model. The output observation error model will become a decoupled MIMO linear systems whose input vector is the system uncertain vector and each diagonal element is a stable transfer function with relative degree one. Then, the output observation error model is further transformed by introducing an averaging filter matrix and some auxiliary signal vectors so that the AILC can be implemented without using differentiators. Based on the derived output observation error model, an MIMO filtered fuzzy neural network using delayed state estimation vector and state estimation vector as the input vector is applied to approximate the unknown system nonlinear function vector. Besides, a normalization signal is applied as a bounding function to design a robust learning component for compensation of the lumped uncertainties vector caused by function approximation error vector, state estimation error vector and delayed system output vector. Finally, a stabilization learning component is used to guarantee the boundedness of internal signals. Based on Lyapunov-like analysis, it is shown that all the adjustable parameters as well as internal signals remain bounded for all iterations. The norm of output tracking error vector will asymptotically converge to a tunable residual set whose size depends on some design parameters of averaging filter.