UDU Factored Discrete-Time Lyapunov Recursions Solve Optimal Reduced-Order LQG Problems

A new algorithm is presented to solve both the finitehorizon time-varying and infinite-horizon timeinvariant discrete-time optimal reduced-order linear quadratic Gaussian (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 illustrated how UDU factorisation may increase both the numerical efficiency and accuracy of the recursions. By means of several numerical examples 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 properties are illustrated 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.