Model reduction of singular switched systems in discrete time

Based on our recently established solution characterization of switched singular descriptor systems in discrete time, we propose a time-varying balanced truncation method. For that we consider the switched system on a finite time interval and define corresponding time-varying reachability and observability Gramians. We then show that these capture essential quantitative information about reachable and observable state directions. Based on these Gramians we formulate a time-varying balanced truncation method resulting in a fully-time varying linear system with possible varying state dimensions. We illustrate this method with a small dynamic Leontief model, where we can reduce the size to one third without altering the input-output behavior significantly. We also show that the method is suitable for a medium size random descriptor system (100 × 100) resulting in a time-varying system of less then a tenth of the size where the outputs of the original and reduced system are indistinguishable.