Balanced truncation for linear switched systems

TL;DR

This paper introduces a balanced truncation method for linear switched systems, enabling effective model order reduction while preserving stability and providing error bounds for the reduced models.

Contribution

It develops a novel approach to compute mode-specific Gramians via coupled Lyapunov equations and constructs reduced models with guaranteed stability and error bounds.

Findings

Reduced models preserve stability.

Error bounds relate outputs and inputs.

Method effectively eliminates hard-to-control states.

Abstract

We propose a model order reduction approach for balanced truncation of linear switched systems. Such systems switch among a finite number of linear subsystems or modes. We compute pairs of controllability and observability Gramians corresponding to each active discrete mode by solving systems of coupled Lyapunov equations. Depending on the type, each such Gramian corresponds to the energy associated to all possible switching scenarios that start or, respectively end, in a particular operational mode. In order to guarantee that hard to control and hard to observe states are simultaneously eliminated, we construct a transformed system, whose Gramians are equal and diagonal. Then, by truncation, directly construct reduced order models. One can show that these models preserve some properties of the original model, such as stability and that it is possible to obtain error bounds relating the observed output, the control input and the entries of the diagonal Gramians.