Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems

P. Benner, T. Damm: Lyapunov equations, energy functionals, and model order reduction of bilinear and stochastic systems
SIAM Journal on Control and Optimization 49 (2), 686 - 711, 2011

DOI: 10.1137/09075041X
Zentralblattnummer: pre05931218
Keywords: Lyapunov equations; Gramians; energy functionals; balanced truncation; model order reduction; bilinear systems; stochastic systems
Mathematics Subject Classification Code: 93A15 (93A30 93C10 93E20 93B40)


Abstract:

We discuss the relation of a certain type of generalized Lyapunov equations to Gramians of stochastic and bilinear systems together with the corresponding energy functionals. While Gramians and energy functionals of stochastic linear systems show a strong correspondence to the analogous objects for deterministic linear systems, the relation of Gramians and energy functionals for bilinear systems is less obvious. We discuss results from the literature for the latter problem and provide new characterizations of input and output energies of bilinear systems in terms of algebraic Gramians satisfying generalized Lyapunov equations. In any of the considered cases, the definition of algebraic Gramians allows us to compute balancing transformations and implies model reduction methods analogous to balanced truncation for linear deterministic systems. We illustrate the performance of these model reduction methods by showing numerical experiments for different bilinear systems.

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