On Spectral Decomposition of States and Gramians of Bilinear Dynamical Systems

Alexey Iskakov and Igor Yadykin
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Alexey Iskakov: V.A. Trapeznikov Institute of Control Sciences of RAS, 117997 Moscow, Russia
Igor Yadykin: V.A. Trapeznikov Institute of Control Sciences of RAS, 117997 Moscow, Russia

Mathematics, 2021, vol. 9, issue 24, 1-19

Abstract: The article proves that the state of a bilinear control system can be split uniquely into generalized modes corresponding to the eigenvalues of the dynamics matrix. It is also shown that the Gramians of controllability and observability of a bilinear system can be divided into parts (sub-Gramians) that characterize the measure of these generalized modes and their interactions. Furthermore, the properties of sub-Gramians were investigated in relation to modal controllability and observability. We also propose an algorithm for computing the Gramians and sub-Gramians based on the element-wise computation of the solution matrix. Based on the proposed algorithm, a novel criterion for the existence of solutions to the generalized Lyapunov equation is proposed, which allows, in some cases, to expand the domain of guaranteed existence of a solution of bilinear equations. Examples are provided that illustrate the application and practical use of the considered spectral decompositions.

Keywords: bilinear systems; eigenmode decomposition; spectral expansions; generalized Lyapunov equation; Gramians; observability; controllability; small-signal analysis; numerical algorithm (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2021
References: View complete reference list from CitEc
Citations: View citations in EconPapers (1)

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