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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.