Geometric methods for output regulation in discrete-time switching systems with preview

This contribution is focused on a geometric methodology devised to achieve optimization, expressed as the minimization of the l2 norm of the tracking error, of regulation transients caused by instantaneous, wide parameter variations occurring in discrete-time, linear systems. The regulated system switching law is assumed to be completely known a priori in a given time interval. A set of feedback regulators, designed according to the internal model principle, guarantee closed-loop asymptotic stability and asymptotic tracking of the reference signal generated by an exosystem, for each regulated system (i.e., for each pair to-be-controlled system/exosystem). The compensation scheme for optimization of transients consists of feedforward actions on the regulation loop and switching policies for suitably setting the states of the feedback regulators and those of the exosystems at the switching times. The theoretical bases of this approach comprise (i) a geometric interpretation, specifically aimed at discrete-time stabilizable and detectable systems, of the multivariable autonomous regulator problem and (ii) a non-recursive solution, still aimed at discrete-time stabilizable systems, of the finite-horizon optimal control problem with final state wieghted by a generic quadratic function, based on a characterization of the structural invariant subspaces of the associated singular Hamiltonian system holding on the sole, fairly general, assumptions that guarantee the existence of the stabilizing solution of the corresponding discrete algebraic Riccati equation.