UDU factored Lyapunov recursions solve optimal reduced-order LQG problems

A new algorithm is presented to solve both the finite-horizon time-varying and infinite-horizon time-invariant discrete-time optimal reduced-order LQG problem. In both cases the first order necessary optimality conditions can be represented by two non-linearly coupled discrete-time Lyapunov equations, which run forward and backward in discrete time. The algorithm iterates these two equations forward and backward in discrete time respectively, until they converge. In the finite horizon time varying case the iterations start from boundary conditions and the forward and backward in time recursions are repeated until they converge. The discrete-time recursions are suitable for UDU factorisation. It is shown how UDU factorisation increases both the numerical efficiency and accuracy of the recursions. By means of a numerical example, two computer experiments and the benchmark problem proposed by the European Journal of Control, the results obtained with the new algorithm are compared to results obtained with algorithms that iterate the strengthened discrete-time optimal projection equations forward and backward in time. The convergence of both these algorithms is highly improved by an automatic procedure, introduced in this paper, to adjust the numerical damping, present in each algorithm. Furthermore the convergence properties are shown to be comparable. Especially when the reduced compensator dimensions are significantly smaller than those of the controlled system, the algorithm presented in this paper is more efficient.