Functional Learning in Optimal Non-linear Control

This paper describes one approach to the application of functional learning techniques to assist in achieving near optimal control of nonlinear systems in the presence of disturbances and/or unmodelled dynamics. A standard approach to achieving robustness of open loop optimal control of nonlinear systems is to apply feedback control based on plant linearization and application of linear quadratic control methods. In earlier studies, it has been shown that such methods can be enhanced by augmenting with adaptive loops, achieving what is termed adaptive-Q control. Here, instead of the adaptive-Q filter being a linear system with coefficients adjusted by a least squares law, the filter's coefficients are functionally dependent on a subset of the optimal states associated with a nominal plant. The functional representation is updated by a least squares law in the case that `measurements' are linear in the function's unknown parameters, as when the function is represented by a sum of bisigmoids in the function input variable space. Such algorithms, and their convergence properties, have been previously studied in an identification context. A simulation study of the optimal quadratic regulation of the nonlinear 2-state Van der Pol equations is used to demonstrate improved performance in the presence of either a constant unknown disturbance of unmodelled dynamics, or stochastic disturbances. The approach could well have application in areas such as aircraft control or robot control where gain schedules are learnt on line. Such applications will be the subject of further study.