Real-Time Realization of a Family of Optimal Infinite-Memory Non-Causal Systems

Abstract

© 2018 In this paper, we consider a problem of designing discrete-time systems which are optimal in frequency-weighted least squares sense subject to a maximal output amplitude constraint. It can be shown for such problems, in general, that the optimality conditions do not provide an explicit way of generating the optimal output as a real-time implementable transformation of the input, due to instability of the resulting dynamical equations and sequential nature in which criterion function is revealed over time. In this paper, we show that, under some mild assumptions, the optimal system has exponentially fading memory. We then propose a causal and stable finite-dimensional nonlinear system which, under an L1 dominance assumption about the equation coefficients, returns high-quality approximations to the optimal solution. The fading memory of the optimal system justifies the receding horizon assumption and suggests that such approach can serve as a cheaper alternative to standard MPC-based algorithms. The result is illustrated on a problem of minimizing peak-to-average-power ratio of a communication signal, stemming from power-efficient transceiver design in modern digital communication systems.