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 l2 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, l2-control problem. A significant feature of the approach to the lower-level problems is the original procedure providing the solution of the finite-horizon, optimal control problem stated for discrete-time stabilizable systems through the structural invariant subspaces of the associated singular Hamiltonian system.