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 Chapters 1, 2, 3 and 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 ℝ IRp, i.e. u = Φ(x, t)θ, where Φ(x, t) is a known matrix (called the regressor) function of measurable ¹ terms. The parameters θ_i, i ∈ {1, ⃛, p{, 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 θ in the sequel. In other words, u = Φ(x, t)θ is replaced by u = Φ(x, t)θ at the input of the system, and θ is estimated on-line 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.