Minimal balanced realization of SISO systems using closed-form Gramians

This paper develops a simple approach to minimal balanced realizations, which enable balanced realizations for arbitrary linear discrete 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