Model Order Reduction for Stochastic Systems

Beschreibung

In this work we determine the stability preservation and check for an error bound after model order reduction (MOR) by balanced truncation (BT) for linear stochastic continuous systems, where the corresponding Gramians of the system are related with its input and output energy. We also study the stability preservation for a new type of Gramian that yields error bound for the same stochastic system. Furthermore, we research on the reachability space of stochastic systems. After that, we consider the associated Hankel operator of the system to compare the low-rank approximation of matrices with the BT method. For the stability preservation after BT, we extend the study to positive operators by considering Schneider's Theorem in the case that the Lyapunov inequalities share a common solution block. Then, we find that their projected common subsystem is stable. From that, we have the stability preservation after BT of the stochastic system for the standard Gramian as a special case. We define a new Gramian and study its stability preservation after BT, which is developed after a similar proof idea as for the standard Gramian. This turns out later to be different. Regarding about the error bound existence, we have a counter example for the standard Gramian. Instead, for the new Gramian there exists an error bound similar to the determinstic one, whose proof is based on the stochastic bounded real lemma. We also consider the reachability space of stochastic systems that we characterize with the image of the controllability Gramian. Finally, using the Hankel operator, we compare the low-rank approximation of matrices with the BT method. As a summary we have first that the stability after BT is preserved for both type of Gramians but the error bound is only guaranteed for the new Gramian system. Second, we observe different kind of behaviors in the reachability of stochastic systems. Finally, we have that the low-rank approximation performs better than BT.

Autor*in

Yolanda Rocío Rodríguez Cruz

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model order reduction by balanced truncation stochastic systems stability preservation and error bound

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