Error Bounds for Balanced Model Reduction of Linear Time-Varying Systems

Error-bounds are developed for balanced truncation of linear time-varying systems, leading to an extension of the `twice the sum of the tail' formulae, well-known in the time-invariant case. The approach relies on an operator-theoretic framework for analysis of linear time-varying systems. This provides a multivariable notion of frequency for such systems, which are thus characterized by rational functions of many complex variables, allowing the problem to be formulated in the linear-fractional framework. Using a time-varying version of standard necessary conditions for reduced-order modeling, based on convex operator inequalities, we show that these error-bounds for balanced truncation are related to the closest possible reduced-order modeling error in a sense which parallels the time-invariant case.