4.6 Constructing Robust Linear Controllers

In this section we use the Lyapunov theorem to construct robust state-feedback controllers. One of the basic issues in the control of dynamical systems is the effect of uncertainties, or neglected nonlinearities, on the stability of the closed-loop system. A controller is viewed as robust if it maintains stability for all uncertainties in an expected range. We consider a class of dynamical systems modeled by

where , and the functions h and f model uncertainties, or nonlinearities, in the system. We refer to h as the matched uncertainty because it affects the system dynamics via the input matrix B in the same fashion as the input u does. In other words, the uncertainty h matches the system input u. The vector f models the unmatched uncertainty. We assume that the uncertain elements h and f satisfy the following norm bounds:

1. ? h( t, x, u) ? ? ? _h ? u ? + ? _h ? x ?

2. ? f( t, x) ? ? ? _f ? x ?,

where ? _h, ? _h and ? _f are known nonnegative constants. We further assume that the matrix A is asymptotically stable. If this is not the case, we apply a preliminary state-feedback controller u = ? Kx such that A ? BK is asymptotically stable. Such a feedback...