A nested computational approach for l2 optimization of regulation transients in discrete-time linear parameter varying systems

This work deals with the 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. A feedback regulator, designed according to the internal model principle, guarantees closed-loop asymptotic stability and zero error in the steady-state condition. The compensation scheme for the minimization of transients includes a feedforward action on both the plant and the feedback regulator and a switching policy for the states of the exosystem and the feedback regulator. The feedfoward action and the state switching policy are computed off-line, by means of a two-level nested algorithm. The lower level includes a sequence of finite-horizon optimal control problems (each one corresponding to a time interval between two subsequent switches), while the upper level combines relevant data from the lower-level problems into a global l₂ optimization problem. A substantial feature of this contribution is that the approach to the lower-level problem relies on an original procedure which provides the solution of a discrete-time finite-horizon optimal control problem in closed form as a function of time. Thus, discrete-time intervals with a large number of samples can easily be handled.