Interval Analysis of the Eigenvalues of Closed-Loop Control Systems with Uncertain Parameters
Uncertainty caused by a parameter measurement error or a model error causes difficulties for the implementation of the control method. Experts can divide the uncertain system into a definite part and an uncertain part and solve each part using various methods. Two uncertainty problems of the control...
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MDPI AG
2020-04-01
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| Series: | Actuators |
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| author | Jing-Zhou Zhao Guo-Feng Yao Rui-Yao Liu Yuan-Cheng Zhu Kui-Yang Gao Min Wang |
| author_facet | Jing-Zhou Zhao Guo-Feng Yao Rui-Yao Liu Yuan-Cheng Zhu Kui-Yang Gao Min Wang |
| author_sort | Jing-Zhou Zhao |
| collection | DOAJ |
| description | Uncertainty caused by a parameter measurement error or a model error causes difficulties for the implementation of the control method. Experts can divide the uncertain system into a definite part and an uncertain part and solve each part using various methods. Two uncertainty problems of the control system arise: problem A for the definite part—how does one find out the optimal number and position of actuators when the actuating force of an actuator is smaller than the control force? Problem B for the uncertain part—how does one evaluate the effect of uncertainty on the eigenvalues of a closed-loop control system? This paper utilizes an interval to express the uncertain parameters and converts the control system into a definite part and an uncertain part using interval theory. The interval state matrix is constructed by physical parameters of the system for the definite part of the control system. For Problem A, the paper finds out the singular value element sensitivity of the modal control matrix and reorders the optimal location of the actuators. Then, the paper calculates the state feedback gain matrix for a single actuator using the receptance method of pole assignment and optimizes the number and position of the actuators using the recursive design method. For Problem B, which concerns the robustness of closed-loop systems, the paper obtains the effects of uncertain parameters on the real and imaginary parts of the eigenvalues of a closed-loop system using the matrix perturbation theory and interval expansion theory. Finally, a numerical example illustrates the recursive design method to optimize the number and location of actuators and it also shows that the change rate of eigenvalues increases with the increase in uncertainty. |
| first_indexed | 2024-03-10T20:19:01Z |
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| id | doaj.art-1ce37de0909e451f83c7dd724de36444 |
| institution | Directory Open Access Journal |
| issn | 2076-0825 |
| language | English |
| last_indexed | 2024-03-10T20:19:01Z |
| publishDate | 2020-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Actuators |
| spelling | doaj.art-1ce37de0909e451f83c7dd724de364442023-11-19T22:18:37ZengMDPI AGActuators2076-08252020-04-01923110.3390/act9020031Interval Analysis of the Eigenvalues of Closed-Loop Control Systems with Uncertain ParametersJing-Zhou Zhao0Guo-Feng Yao1Rui-Yao Liu2Yuan-Cheng Zhu3Kui-Yang Gao4Min Wang5Key Laboratory of CNC Equipment Reliability (Jilin University), Ministry of Education, School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130022, ChinaKey Laboratory of CNC Equipment Reliability (Jilin University), Ministry of Education, School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130022, ChinaKey Laboratory of CNC Equipment Reliability (Jilin University), Ministry of Education, School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130022, ChinaKey Laboratory of CNC Equipment Reliability (Jilin University), Ministry of Education, School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130022, ChinaKey Laboratory of CNC Equipment Reliability (Jilin University), Ministry of Education, School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130022, ChinaKey Laboratory of CNC Equipment Reliability (Jilin University), Ministry of Education, School of Mechanical and Aerospace Engineering, Jilin University, Changchun 130022, ChinaUncertainty caused by a parameter measurement error or a model error causes difficulties for the implementation of the control method. Experts can divide the uncertain system into a definite part and an uncertain part and solve each part using various methods. Two uncertainty problems of the control system arise: problem A for the definite part—how does one find out the optimal number and position of actuators when the actuating force of an actuator is smaller than the control force? Problem B for the uncertain part—how does one evaluate the effect of uncertainty on the eigenvalues of a closed-loop control system? This paper utilizes an interval to express the uncertain parameters and converts the control system into a definite part and an uncertain part using interval theory. The interval state matrix is constructed by physical parameters of the system for the definite part of the control system. For Problem A, the paper finds out the singular value element sensitivity of the modal control matrix and reorders the optimal location of the actuators. Then, the paper calculates the state feedback gain matrix for a single actuator using the receptance method of pole assignment and optimizes the number and position of the actuators using the recursive design method. For Problem B, which concerns the robustness of closed-loop systems, the paper obtains the effects of uncertain parameters on the real and imaginary parts of the eigenvalues of a closed-loop system using the matrix perturbation theory and interval expansion theory. Finally, a numerical example illustrates the recursive design method to optimize the number and location of actuators and it also shows that the change rate of eigenvalues increases with the increase in uncertainty.https://www.mdpi.com/2076-0825/9/2/31pole assignmentinterval theoryreceptance methodrecursive design method |
| spellingShingle | Jing-Zhou Zhao Guo-Feng Yao Rui-Yao Liu Yuan-Cheng Zhu Kui-Yang Gao Min Wang Interval Analysis of the Eigenvalues of Closed-Loop Control Systems with Uncertain Parameters Actuators pole assignment interval theory receptance method recursive design method |
| title | Interval Analysis of the Eigenvalues of Closed-Loop Control Systems with Uncertain Parameters |
| title_full | Interval Analysis of the Eigenvalues of Closed-Loop Control Systems with Uncertain Parameters |
| title_fullStr | Interval Analysis of the Eigenvalues of Closed-Loop Control Systems with Uncertain Parameters |
| title_full_unstemmed | Interval Analysis of the Eigenvalues of Closed-Loop Control Systems with Uncertain Parameters |
| title_short | Interval Analysis of the Eigenvalues of Closed-Loop Control Systems with Uncertain Parameters |
| title_sort | interval analysis of the eigenvalues of closed loop control systems with uncertain parameters |
| topic | pole assignment interval theory receptance method recursive design method |
| url | https://www.mdpi.com/2076-0825/9/2/31 |
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