Quantitative Controllability Metric for Disturbance Rejection in Linear Unstable Systems
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...
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MDPI AG
2024-12-01
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| Series: | Mathematics |
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| author | Haemin Lee Jinseong Park |
| author_facet | Haemin Lee Jinseong Park |
| author_sort | Haemin Lee |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-8ed7027ed95f4e9d83bac5796c617b33 |
| institution | Kabale University |
| issn | 2227-7390 |
| language | English |
| publishDate | 2024-12-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-8ed7027ed95f4e9d83bac5796c617b332025-01-10T13:17:56ZengMDPI AGMathematics2227-73902024-12-01131610.3390/math13010006Quantitative Controllability Metric for Disturbance Rejection in Linear Unstable SystemsHaemin Lee0Jinseong Park1Department of Mechanical and Automotive Engineering, Kongju National University, Cheonan 31080, Republic of KoreaDepartment of AI Machinery, Korea Institute of Machinery & Materials, Daejeon 34103, Republic of KoreaThis 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.https://www.mdpi.com/2227-7390/13/1/6Gramian-based controllability metricdegree of disturbance rejection (DoDR)unstable systemssteady-state solutionactuator placement |
| spellingShingle | Haemin Lee Jinseong Park Quantitative Controllability Metric for Disturbance Rejection in Linear Unstable Systems Mathematics Gramian-based controllability metric degree of disturbance rejection (DoDR) unstable systems steady-state solution actuator placement |
| title | Quantitative Controllability Metric for Disturbance Rejection in Linear Unstable Systems |
| title_full | Quantitative Controllability Metric for Disturbance Rejection in Linear Unstable Systems |
| title_fullStr | Quantitative Controllability Metric for Disturbance Rejection in Linear Unstable Systems |
| title_full_unstemmed | Quantitative Controllability Metric for Disturbance Rejection in Linear Unstable Systems |
| title_short | Quantitative Controllability Metric for Disturbance Rejection in Linear Unstable Systems |
| title_sort | quantitative controllability metric for disturbance rejection in linear unstable systems |
| topic | Gramian-based controllability metric degree of disturbance rejection (DoDR) unstable systems steady-state solution actuator placement |
| url | https://www.mdpi.com/2227-7390/13/1/6 |
| work_keys_str_mv | AT haeminlee quantitativecontrollabilitymetricfordisturbancerejectioninlinearunstablesystems AT jinseongpark quantitativecontrollabilitymetricfordisturbancerejectioninlinearunstablesystems |