Control of parameter-varying systems

In this chapter the problem of feedback control of uncertain systems is considered, with a special attention to the control of polytopic systems. To properly introduce the results, let us reconsider the stability analysis problem for an uncertain system of the form $$\displaystyle{\dot{x}(t) = A(w(t))x(t)}$$ where \(w(t) \in \mathcal{W}\), with \(\mathcal{W}\) compact, and A(⋅ ) is continuous. In the case of a single stable linear system, stability is equivalent to the fact that the eigenvalues of A have negative real part (modulus less than one in the discrete-time case). The speed of convergence associated with the maximum real part (modulus) of the eigenvalues, precisely the maximum value of β > 0 such that ( 4.17) holds, is max{Re(λ), λ ∈ eig(A)}, where eig(A) is the set of the eigenvalues of A. In the case of an uncertain system, there is no analogous concept of eigenvalues. The eigenvalues of A(w), intended as functions of w, do not play a substantial role anymore, since they may all have negative real part bounded away from 0 (i.e. \(\max \{Re(\lambda ),\ \ \lambda \in eig(A(w))\} \leq \beta <0\)), and still the time-varying system be unstable. Indeed, the condition becomes necessary only.