Adaptive Control

Abstract

This chapter is dedicated to present so-called direct adaptive controllers applied to mechanical and to linear invariant systems. We have already studied some applications of dissipativity theory in the stability of adaptive schemes in Chaps. 1 – 4. Direct adaptation means that one has been able to rewrite the fixed parameter input u, in a form that is linear with respect to some unknown parameters, usually written as a vector θ ∈ Rp, i.e., u = φ(x, t)θ, where φ(x, t) is a known matrix (called the regressor) function of measurable (In the technological sense, not in the mathematical one.) terms. The parameters (Formula presented), are generally nonlinear combinations of the physical parameters (for instance, in the case of mechanical systems, they will be nonlinear combinations of moments of inertia, masses). When the parameters are unknown, one cannot use them in the input. Therefore one replaces θ, in u, by an estimate, that we shall denote Formula presented) in the sequel. In other words, u = φ(x, t)θ is replaced by (Formula presented) at the input of the system, and (Formula presented) is estimated online with a suitable identification algorithm. As a consequence, one easily imagines that the closed-loop system stability analysis will become more complex. However, through the Passivity Theorem (or the application of Lemma 7.23 ), the complexity reduces to adding a passive block to the closed-loop system that corresponds to the estimation algorithm dynamics. The rest of the chapter is composed of several examples that show how this analysis mechanism work. It is always assumed that the parameter vector is constant: the case of time-varying parameters, although closer to the reality, is not treated here due to the difficulties in deriving stable adaptive controllers in this case. This is a topic in itself in adaptive control theory, and is clearly outside the scope of this book.