Square-Root Balancing and Computation of Minimal Realizations of Periodic Systems

We propose a numerically reliable approach for balancing and minimal realization of linear periodic systems with time-varying dimensions. The proposed approach to balancing belongs to the family of square-root methods with guaranteed enhanced computational accuracy and can be also used to compute balanced minimal order realizations from non-minimal ones. An alternative balancing-free square-root method for minimal realization has the advantage of a potentially better numerical accuracy in case of poorly balanced original systems. The key numerical computation in both methods is the solution of nonnegative periodic Lyapunov equations directly for the Cholesky factors of the solutions. For this purpose, a numerically reliable computational algorithm is proposed to solve nonnegative periodic Lyapunov equations with time-varying dimensions.