Ultimate boundedness control for uncertain discrete-time systems via set-induced Lyapunov functions

Linear discrete-time systems affected by both parameter and input uncertainties are considered. The problem of the synthesis of a feedback control assuring that the system state is ultimately bounded within a given compact set containing the origin with an assigned speed of convergence is investigated. It is shown that the problem has a solution if and only if there exists a certain Lyapunov function which does not belong to a pre-assigned class of functions (i.e. the quadratic ones) but it is determined by the target set in which ultimate boundedness is desired. One of the advantages of this approach is that one can handle systems with control constraints. No matching assumptions are made. For systems with linearly constrained uncertainties, it is shown that such a function may be derived by numerically efficient algorithms involving polyhedral sets. An extension of the technique to continuous-time systems is presented