Towards Time-Varying Balanced Realization via Riccati Equations

This paper presents a new approach for solving balanced realization problems with emphasis on the time varying case. Instead of calculating the exact solutions for balancing at each time instant, we estimate with arbitrary accuracy the balancing solutions by means of Riccati equations associated with the balancing problems. Under uniform boundedness conditions on the controllability grammians and their inverses, the solutions of the Riccati equations exist and converge exponentially as their initial time goes to -¿ to give what we term ¿-balancing solutions. The parameter ¿ has the interpretation of the gain of a differential equation. It determines the accuracy of the balancing transformation tracking and the exponential rate of convergence. Their exponentially convergent behaviour ensures numerical robustness.