Parameterized frequency-dependent balanced truncation for model order reduction of linear systems

Balanced truncation is the most commonly used model order reduction scheme in control engineering. This is due to its favorable properties of automatic stability preservation and the existence of a computable error bound, enabling the adaption of the reduced model order to a specified tolerance. It aims at minimizing the worst case error of the frequency response over the full infinite frequency range. If a good approximation only over a finite frequency range is required, frequency-weighted or frequency-limited balanced truncation variants can be employed. In this paper, we study this finite-frequency model order reduction (FF-MOR) problem for linear time-invariant (LTI) continuous-time systems within the framework of balanced truncation. Firstly, we construct a family of parameterized frequency-dependent (PFD) mappings which transform the given LTI system to either a discrete-time or continuous-time PFD system. The relationships between the maximum singular value of the given LTI system over pre-specified frequency ranges and the maximum singular value of the PFD mapped systems over the entire frequency range are established. By exploiting the properties of the discrete-time PFD mapped systems, a new parameterized frequency-dependent balanced truncation (PFDBT) method providing a finite-frequency type error bound with respect to the maximum singular value of the error systems is developed. Examples are included for illustration.