Nonlinear Control Systems II

The purpose of this Chapter is to describe some important tools for the design of feedback laws which globally asymptotically stabilize a nonlinear system in the presence of parameter uncertainties. We consider the case in which the mathematical model of the system to be controlled depends on a vector μ ∈ ℝ ^p of parameters, which are assumed to be constant, but whose actual values are unknown to the designer. The vector µ of unknown parameters could be any vector in some a priori given set \( \mathcal{P} \), and the goal of the design is to find a feedback law (obviously independent of μ) which globally asymptotically stabilizes the system for each value of \( \mu \in \mathcal{P} \). A problem of this type is usually referred to as a problem of robust stabilization.

In section 9.3 we have introduced the concept of semiglobal stabilizability, and we have shown (Theorem 9.3.1) how, using a linear feedback, it is possible to stabilize in a semiglobal sense (i.e. imposing that the domain of attraction of the equilibrium contains a prescribed compact set) a system of the form (9.23), under the hypothesis that the equilibrium z = 0 of its zero dynamics is globally asymptotically stable. In this section, in preparation to the subsequent study of the problem of robust semiglobal stabilization using output feedback, we extend the result of Theorem 9.3.1 to the case of a system modeled by equations of the form in which z ∈ ℝ ⁿ , ξ _i ∈ ℝ for i=1,…,r, u ∈ ℝ and \(\mu \in \mathcal{P} \subset {\mathbb{R}^p}\) is a vector of unknown parameters, ranging over a compact set \(\mathcal{P}\).

In this Chapter we will study problems of global stabilization of systems that can be modeled as feedback interconnection of two subsystems, one of which is accurately known while the other one is uncertain but has a finite L ₂ gain, for which an upper bound is available. More precisely, we consider systems modeled by equations of the form which describe the feedback interconnection of a system in which \({x_1} \in {\mathbb{R}^{{n_1}}}\), w ∈ ℝ, u ∈ ℝ, y ∈ ℝ and f ₁(0,0,0)=0, h ₁(0)=0, a system in which \({x_2} \in {\mathbb{R}^{{n_2}}}\) and f ₂(0,0)=0,h ₂(0)=0.

In this Chapter, we describe methods for global (robust) stabilization of nonlinear systems, by means of memoryless feedback, in cases in which the amplitude of the control input cannot exceed a fixed bound. Of course, if such a hard constraint is imposed on the amplitude of the control input, one cannot expect — in general — that global asymptotic stability is possible, unless the uncontrolled system already possesses this property to a certain extent. The simplest case in which so happens is when there exists a positive definite and proper function, whose derivative along the trajectories of the uncontrolled system is negative semi-definite but, possibly, not negative definite. In this case, in fact, under mild hypotheses, it is possible to find a smooth feedback law, whose amplitude does not exceed any (arbitrarily small) a priori fixed number, yielding global asymptotic stability. We discuss this special case first, as a point of departure for the analysis of more general structures that will be presented in the subsequent sections of the Chapter.