Quantitative Controllability Metric for Disturbance Rejection in Linear Unstable Systems

Haemin Lee and Jinseong Park ()
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Haemin Lee: Department of Mechanical and Automotive Engineering, Kongju National University, Cheonan 31080, Republic of Korea
Jinseong Park: Department of AI Machinery, Korea Institute of Machinery & Materials, Daejeon 34103, Republic of Korea

Mathematics, 2024, vol. 13, issue 1, 1-24

Abstract: This paper introduces a novel Gramian-based quantitative metric to evaluate the disturbance rejection capabilities of linear unstable systems. The proposed metric addresses key limitations of the previously introduced degree of disturbance rejection (DoDR) metrics, including their dependency on the final time and numerical problems arising from differential equation computations. Specifically, this study defines the steady-state solution of the DoDR metric, which avoids numerical issues by relying only on solving four algebraic equations, even when the Gramian matrices diverge. This study further strengthens its contributions by providing rigorous mathematical proofs supporting the proposed method, ensuring a strong theoretical foundation. The derived results demonstrate that the proposed metric represents the sum of the steady-state input energies required to reject the disturbances in the asymptotically stable and anti-stable subsystems. Numerical examples demonstrated that the proposed metric maintained the physical meaning of the original DoDR while offering practical computational advantages. This study represents a significant step toward the efficient and reliable assessment of disturbance rejection capabilities in unstable systems.

Keywords: Gramian-based controllability metric; degree of disturbance rejection (DoDR); unstable systems; steady-state solution; actuator placement (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2024
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