Abstract
Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville-von Neumann and Fokker--Planck type. Methods based on balancing generalized system Gramians and on minimizing an H2-type cost functional are considered. The focus is on the numerical implementation and a thorough comparison of the methods. Structure and stability preservation are investigated, and the competitiveness of the approaches is shown for practically relevant, large-scale examples.
Highlights
Model reduction methods for bilinear control systems are compared by means of practical examples of Liouville–von Neumann and Fokker– Planck type
While the computational effort of balanced model reduction is essentially determined by the solution of two generalized Lyapunov equations for controllability and observability Gramians, the effort of the H2-optimal interpolation method is mainly due to the solution of two generalized Sylvester equations in each step of the bilinear iterative rational Krylov algorithm (B-IRKA)
The idea of the averaging principle is to average the fast variables in the equation for z1 against their invariant measure, because whenever is sufficiently small, the fast variables relax to their invariant measure while the slow variables are effectively frozen, and the slow dynamics move under the average influence of the fast variables
Summary
Due to the growing ability to accurately manipulate single molecules by spectroscopic techniques, numerical methods for the control of molecular systems have recently attracted a lot of attention [4, 30, 41, 61]. The second approach is based on balancing the controllable and observable subspaces, and exploits the properties of the underlying dynamical system in that it uses the properties of the controllability and observability Gramians to identify suitable small parameters that are sent to 0 to yield a reduced-order system; for details, we refer to [29] Both methods require the solution of large-scale matrix Sylvester or Lyapunov equations. While the computational effort of balanced model reduction is essentially determined by the solution of two generalized Lyapunov equations for controllability and observability Gramians, the effort of the H2-optimal interpolation method is mainly due to the solution of two generalized Sylvester equations in each step of the bilinear iterative rational Krylov algorithm (B-IRKA) We stress that both generalized Lyapunov or Sylvester equations can be solved iteratively at comparable numerical cost (for a given accuracy), but they all require the dynamics of the uncontrolled system to be asymptotically stable [55]. The article contains an appendix, Appendix A, that records some technical lemmas related to the asymptotic stability of bilinear systems
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