Delayed feedback tracking controller for single-input single-output nonlinear systems

Difficulties one faces in designing a satisfactory controller for systems involving stiff nonlinearities are well known. Stictions, Coulomb frictions and backlashes are but a few such examples. Unfortunately, there are few mechanical systems without such nonlinearities. The problem becomes even more complex if some of the state variables are not directly measurable, a situation which occurs not infrequently in practical systems. Unlike in linear systems where the missing state variables may be reconstructed [1], no corresponding methods are available for nonlinear systems. In fact, no general methods are currently available for analyzing and synthesizing controllers for nonlinear systems. At present, the describing function method is perhaps the best tool available for investigating stiff nonlinear systems [2]. A stable system is designed by adjusting the system gain or inserting a simple lead-lag compensation network. The parameter values are determined by graphically examining the Nyquist curve and the describing function. The process becomes very difficult, if not impossible, to apply when the order of the system is high and only the measurable variables are to be used in the feedback. When the reference input is a constant and only the output is available for feedback, the delayed feedback controller was given in [3]. In this paper, preliminary results are presented for the case when the reference input is a polynomial function and a multi-dimensional observable vector is available for feedback. It is based on augmenting the system by additional state variables and then feeding back delayed observable vectors as well as the augmented state variables.