Model reductions of unstable SISO systems using closed-form Gramians

This paper develops a simple approach to balanced model reductions for unstable linear continuous-time SISO systems. First, the controllability, observability, and cross Gramians are directly obtained from the coefficients of the transfer function using the Routh array in a closed-form. Based on these Gramians, a simple algorithm is proposed to compute a reduced order model based on balancing and truncation technique for unstable systems. This algorithm is simpler and numerically more efficient in computing the Gramians than presently existing algorithms. If the unstable system can be balanced, then the proposed algorithm gives a reduced order model which has the same number of unstable poles as the original model