CLOSED-FORM GRAMIANS AND MINIMAL BALANCED REALIZATION OF DISCRETE-TIME SISO SYSTEMS

This paper develops a simple approach to minimal balanced realizations, which enable balanced realizations for arbitrary linear SISO systems. First, the controllability, observability and cross Gramians are obtained from the Lyapunov matrix equations in a closed-form. Based on these Gramians, an algorithm is proposed to obtain a minimal balanced realization directly from the coefficients of the transfer function, using the Jury array. This algorithm is simpler and numerically more efficient in computing the Gramians than presently existing algorithms. Thus, it can be very useful for adaptive control or model reduction of time-varying discrete-time systems.