Balanced truncation for linear switched systems
This addresses the need for reduced-order models of switched systems, which are common in control theory, but the approach is an extension of existing balanced truncation to a specific class of systems.
The paper proposes a balanced truncation method for model order reduction of linear switched systems, computing Gramians via coupled Lyapunov equations to eliminate hard-to-control and hard-to-observe states while preserving stability and enabling error bounds.
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.