USING QUADRATICALLY CONSTRAINED QUADRATIC PROGRAMMING TO DESIGN REPETITIVE CONTROLLERS: APPLICATION TO NON-MINIMUM PHASE SYSTEMS

Repetitive Control (RC) aims for zero tracking error in the presence of a periodic disturbance. Non-minimum phase systems present a difficult design challenge to the sister field of Iterative Learning Control. This paper investigates to what extent the same challenges appear in RC. One challenge is that RC easily handles zeros outside the unit circle in the discrete time z-plane introduced by discretization, but the non-minimum phase zeros mapped from continuous time are normally much closer to the unit circle. A second challenge is the result of the small magnitude frequency response at zero frequency produced by the zero. A min-max cost function over the learning rate is presented along with the approach needed to easily compute the optimal solution as a Quadratically Constrained Linear Programming problem. This is shown to be an RC design approach that directly addresses the challenges of non-minimum phase systems. And it has the advantage that it can be designed based on frequency response data directly, without producing a pole-zero system model. But it is shown that this is not the preferred design approach for minimum phase systems. It is demonstrated that the most common approach to RC design, developed by Tomizuka, does not work on non-minimum phase systems. The design based on optimizing the learning rate that the authors advocate for minimum phase systems is seen to give good performance at most frequencies, but require a large number of gains to learn well at DC. One might still want to accept this tradeoff. A new design approach based on Taylor series expansion of the discrete time transfer function is given and shown to be competitive to the min-max approach under appropriate circumstance. The conclusion is that we now have effective methods to design repetitive control of non-minimum phase systems.