What Is the Distance between Two Autoregressive Systems?

In the recent control literature there has been a great interest in controller design techniques which are robust with respect to plant perturbations. Crucial for all robustness studies is of course the availability of a “good metric” defined on the set of all plants. A metric is considered good if it can be easily computed and if it gives a measure of numerical robustness. On the set of proper transfer functions several metrics have been proposed. Most prominently we want to mention the gap metric introduced by Zames and El-Sakkary [15], the graph metric introduced by Vidyasager [13] and the pointwise gap metric introduced recently by Qiu and Davison [7]. All those metrics have in common that they induce the same topology on the set of proper transfer functions with a fixed McMillan degree. For recent contributions to the subject of gap metric and graph metric we refer to [3, 7, 11] and in particular to the survey article of Glüsing-Lüerßen [4], where a comparative study between those metrics is provided. A natural generalization of the set of proper transfer functions is the set of autoregressive systems. The main contribution of this short paper is a new metric on the space of autoregressive systems. Because those systems are not so widely known and for the convenience of the reader we summarize in the following section the relevant notions. For simplicity we will develop the theory over the real numbers IR.