Optimal control computation for cascade systems by structured Jacobi iterations

Abstract

A computational method is proposed for solving a structured finite-horizon optimal control problem. Specifically, a linear quadratic problem with discrete-time dynamics arising from a cascaded interconnection of N heterogeneous sub-systems is considered. This optimization problem is first formulated as a structured quadratic program whose size grows with extension of the time horizon T and/or cascade length N. An algorithm based on block Jacobi iterations is developed for solving the linear system of equations arising from the Karush-Kuhn-Tucker conditions for optimality. It is shown that the per-iteration complexity of the approach scales linearly in both TV and T. Moreover, the computations at each iteration are amenable to distributed implementation on a path graph structured network of TV parallel processors, with inter-iteration information exchange limited to adjacent nodes. It is shown that convergence of the block Jacobi iterations is guaranteed for the given problem formulation. Numerical experiments based on model data for an automated irrigation channel illustrates the merit of the approach compared to alternatives. It is observed that the number of iterations required for convergence scales favorably with problem size.