A Nested Computational Approach to l2-Optimization of Regulation Transients in Discrete-time LPV Systems

This article deals with the optimization, expressed as the minimization of the 2 norm of the tracking error, of the regulation transients caused by instantaneous, wide parameter variations occurring in the regulated systems. The parameter-varying regulated system is modeled by a set of discrete-time, linear, time-invariant systems and the regulated system switching law is assumed to be completely known a priori. For each LTI system, a feedback regulator, including the exosystem internal model, is designed in order to guarantee closed-loop asymptotic stability and zero tracking error in the steady-state condition. The compensation scheme for the minimization of the regulation transients consists of feedforward actions on the regulation loop and a state switching policy for suitably setting the state of the feedback regulators at the switching times. Both the state switching policy and the feedforward actions are computed off-line: the former by exploiting some geometric properties of the multivariable autonomous regulator problem, the latter by resorting to a two-level, nested algorithm. The lower level includes a sequence of discrete-time, finite-horizon optimal control problems, each defined in the time interval between two consecutive switches. The upper level combines relevant data from the lower-level problems into a global, 2-control problem. A significant feature of the approach to the lower-level problems is the original procedure providing the solution of the finitehorizon, optimal control problem stated for discrete-time stabilizable systems through the structural invariant subspaces of the associated singular Hamiltonian system.