Feedback Control of a Discrete System with Memory within the Class of Zonal Control Actions

Abstract

The problem of optimal control for a discrete-time dynamical system
over a finite horizon is considered under uncertainty in the initial state values and
in the external parameters acting on the system. The control is generated according
to a feedback law that depends only on the observable part of the state vector
and its history. A functional form of the feedback law is introduced on the basis
of a family of zonal controllers: the space of the observable part of the state is
partitioned into a finite number of zones, and within each zone the control is defined
by fixed affine coefficients, including a memory term, that is, a dependence
on the previous measurement. Such a parametrization reduces the original optimal
control synthesis problem to a finite-dimensional optimization problem with
respect to the coefficients of the zonal law. For the cost functional averaged over
the set of admissible initial states and external parameter values, analytical formulas
for the gradient with respect to the feedback parameters have been derived.
First-order optimality conditions are formulated, and a projected gradient algorithm
is proposed for computing the optimal zonal coefficients in the presence of
constraints. Results of numerical experiments on test problems are presented;
they illustrate the effectiveness of the proposed approach and make it possible to
compare different variants of zonal controllers.