Discrete Model Matching Adaptive Control for Potentially Inversely Non-Stable Continuous-Time Plants by Using Multirate Sampling

Abstract

Adaptive control theory has been widely applied for stabilizing linear time invariant plants of unknown parameters (Goodwin & Sin, 1984). One of the more used methods for such a purpose is based on the model reference adaptive control (MRAC) problem (Astrom & Wittenmark, 1997). Such a method requires some assumptions relative to the plant to be controlled in order to carry out the synthesis of a stable controller (Narendra & Annaswamy, 1989). One of them is that the plant has to be inversely stable, what means that its zeros have to be located within the stability domain. However, this information is not always available to the designer when the system under control contains unknown parameters. There are several alternatives to circumvent this drawback and carry out the stable adaptive control design. Some of them consist on relaxing the control performance from the model matching to that achievable from the closed-loop pole placement (AlonsoQuesada & De la Sen, 2004 and Arvanitis, 1999). In this way, the stabilization of the closedloop system can be ensured although its transient behaviour cannot be fixed to a predefined one. On one hand, the work (Alonso-Quesada & De la Sen, 2004) includes an estimates modification in the estimation algorithm to ensure the controllability of the estimated plant model without assuming any knowledge about the parameters of the plant to be controlled. This controllability property is crucial to avoid pole-zero cancellations between the estimated plant and the controller, which are both time-varying. In this context, a projection of the estimated plant parameters into a region in the parameter space where the closedloop system is free of pole-zero cancellations for all time can be alternatively used provided that the true plant is controllable and the knowledge of a region where the true plant parameters belong to (Goodwin & Mayne, 1987). On the other hand, the research (Arvanitis, 1999) proposes an adaptive pole-placement control for linear systems using generalized sampled-data hold functions. Following such a technique, gain controllers essentially need to be designed. Concretely, a periodic piecewise constant gain controller is added in the feedback chain. In the non-adaptive case, such constant gain values are those required so that the discretized closed-loop model under a fundamental sampling period and a zero-order hold (ZOH) be stabilized. For such a