Parametric Uncertainty of an Adaptive LQ Controller with Periodic Excitation

Abstract

We present an example to demonstrate the applicability of the theory of stochastic complexity (cf. (Rissanen, 1989)) to compute the effect of statistical parameter uncertainty onto control performance. We consider a first order, continuous-time, linear stochastic control system with mixed quadratic criterion. The identification of the unknown system’s parameters takes place under conditions that are standard in the literature on identification and control: we consider an off-line estimation method with forgetting, for which a fairly complete theory is available. Another standard element of our estimation method is that we inject an external known signal, in this case a sinusoid signal to ensure identifiability. The intensity of this signal is a crucial design parameter, as has been pointed out in the theory of dual control (cf. (Feldbaum, 1965)). An approximate value of the optimal intensity of the dither will be determined for large frequency dither. Our results complement earlier results on the interaction of identification and control presented in (Åström and Wittenmark, 1971; Åström, 1993; Gevers, 1995; Keviczky, 1995).