Simple Proof of the Generalized Adaptive Law and Some Properties of the Constant-Trace Algorithm

A generalized adaptive law for model reference adaptive systems was recently proposed by Landau et al., which could easily yield socalled a constant gain algorithm, a decreasing gain algorithm and a variable gain algorithm by forcing some constraints to it. This law was derived based on the hyperstability theory and the concept of positive dynamic system. Unfortunately the proof was very complicated and difficult to understand. This paper shows an elegant proof of the law using the direct method of Liapunov and the positive real lemma. It may be useful to deeply understand the law and its properties.Next we examine an attractive algorithm obtained from the generalized adaptive law as a special case, which has a gain matrix keeping its trace constant. Attention has been called to it with experimental results showing its superiority to some well-known algorithms. No analysis of its properties, however, has been tried yet. The analysis is given here to make clear some interesting properties, which are demonstrated by simple numerical examples.