Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation
Complex time-dependent Lyapunov equation (CTDLE), as an important means of stability analysis of control systems, has been extensively employed in mathematics and engineering application fields. Recursive neural networks (RNNs) have been reported as an effective method for solving CTDLE. In the prev...
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MDPI AG
2022-08-01
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| author | Bolin Liao Cheng Hua Xinwei Cao Vasilios N. Katsikis Shuai Li |
| author_facet | Bolin Liao Cheng Hua Xinwei Cao Vasilios N. Katsikis Shuai Li |
| author_sort | Bolin Liao |
| collection | DOAJ |
| description | Complex time-dependent Lyapunov equation (CTDLE), as an important means of stability analysis of control systems, has been extensively employed in mathematics and engineering application fields. Recursive neural networks (RNNs) have been reported as an effective method for solving CTDLE. In the previous work, zeroing neural networks (ZNNs) have been established to find the accurate solution of time-dependent Lyapunov equation (TDLE) in the noise-free conditions. However, noises are inevitable in the actual implementation process. In order to suppress the interference of various noises in practical applications, in this paper, a complex noise-resistant ZNN (CNRZNN) model is proposed and employed for the CTDLE solution. Additionally, the convergence and robustness of the CNRZNN model are analyzed and proved theoretically. For verification and comparison, three experiments and the existing noise-tolerant ZNN (NTZNN) model are introduced to investigate the effectiveness, convergence and robustness of the CNRZNN model. Compared with the NTZNN model, the CNRZNN model has more generality and stronger robustness. Specifically, the NTZNN model is a special form of the CNRZNN model, and the residual error of CNRZNN can converge rapidly and stably to order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup></semantics></math></inline-formula> when solving CTDLE under complex linear noises, which is much lower than order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> of the NTZNN model. Analogously, under complex quadratic noises, the residual error of the CNRZNN model can converge to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mn>2</mn><mo>∥</mo><mi>A</mi><mo>∥</mo></mrow><mi>F</mi></msub><mo>/</mo><msup><mi>ζ</mi><mn>3</mn></msup></mrow></semantics></math></inline-formula> quickly and stably, while the residual error of the NTZNN model is divergent. |
| first_indexed | 2024-03-09T12:21:48Z |
| format | Article |
| id | doaj.art-6924df06e8f647aba6c584f7b6ad6058 |
| institution | Directory Open Access Journal |
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| language | English |
| last_indexed | 2024-03-09T12:21:48Z |
| publishDate | 2022-08-01 |
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| spelling | doaj.art-6924df06e8f647aba6c584f7b6ad60582023-11-30T22:39:12ZengMDPI AGMathematics2227-73902022-08-011015281710.3390/math10152817Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov EquationBolin Liao0Cheng Hua1Xinwei Cao2Vasilios N. Katsikis3Shuai Li4College of Computer Science and Engineering, Jishou University, Jishou 416000, ChinaCollege of Computer Science and Engineering, Jishou University, Jishou 416000, ChinaSchool of Business, Jiangnan University, Wuxi 214122, ChinaDepartment of Economics, Division of Mathematics and Informatics, National and Kapodistrian University of Athens, Sofokleous 1 Street, 10559 Athens, GreeceSchool of Engineering, Swansea University, Swansea SA2 8PP, UKComplex time-dependent Lyapunov equation (CTDLE), as an important means of stability analysis of control systems, has been extensively employed in mathematics and engineering application fields. Recursive neural networks (RNNs) have been reported as an effective method for solving CTDLE. In the previous work, zeroing neural networks (ZNNs) have been established to find the accurate solution of time-dependent Lyapunov equation (TDLE) in the noise-free conditions. However, noises are inevitable in the actual implementation process. In order to suppress the interference of various noises in practical applications, in this paper, a complex noise-resistant ZNN (CNRZNN) model is proposed and employed for the CTDLE solution. Additionally, the convergence and robustness of the CNRZNN model are analyzed and proved theoretically. For verification and comparison, three experiments and the existing noise-tolerant ZNN (NTZNN) model are introduced to investigate the effectiveness, convergence and robustness of the CNRZNN model. Compared with the NTZNN model, the CNRZNN model has more generality and stronger robustness. Specifically, the NTZNN model is a special form of the CNRZNN model, and the residual error of CNRZNN can converge rapidly and stably to order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>5</mn></mrow></msup></semantics></math></inline-formula> when solving CTDLE under complex linear noises, which is much lower than order <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>10</mn><mrow><mo>−</mo><mn>1</mn></mrow></msup></semantics></math></inline-formula> of the NTZNN model. Analogously, under complex quadratic noises, the residual error of the CNRZNN model can converge to <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mrow><mn>2</mn><mo>∥</mo><mi>A</mi><mo>∥</mo></mrow><mi>F</mi></msub><mo>/</mo><msup><mi>ζ</mi><mn>3</mn></msup></mrow></semantics></math></inline-formula> quickly and stably, while the residual error of the NTZNN model is divergent.https://www.mdpi.com/2227-7390/10/15/2817complex time-dependent Lyapunov equationzeroing neural network (ZNN)complex linear noisecomplex quadratic noisenoise-suppression |
| spellingShingle | Bolin Liao Cheng Hua Xinwei Cao Vasilios N. Katsikis Shuai Li Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation Mathematics complex time-dependent Lyapunov equation zeroing neural network (ZNN) complex linear noise complex quadratic noise noise-suppression |
| title | Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation |
| title_full | Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation |
| title_fullStr | Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation |
| title_full_unstemmed | Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation |
| title_short | Complex Noise-Resistant Zeroing Neural Network for Computing Complex Time-Dependent Lyapunov Equation |
| title_sort | complex noise resistant zeroing neural network for computing complex time dependent lyapunov equation |
| topic | complex time-dependent Lyapunov equation zeroing neural network (ZNN) complex linear noise complex quadratic noise noise-suppression |
| url | https://www.mdpi.com/2227-7390/10/15/2817 |
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