Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control
Standard adaptive control is the preferred approach for stabilization and synchronization of chaotic systems when the structure of such systems is a priori known but the parameters are unknown. However, in the presence of unmodeled dynamics and/or disturbance, this approach is not effective anymore...
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| Format: | Article |
| Language: | English |
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Wiley
2018-01-01
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| Series: | Complexity |
| Online Access: | http://dx.doi.org/10.1155/2018/4989520 |
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| author | J. Humberto Pérez-Cruz |
| author_facet | J. Humberto Pérez-Cruz |
| author_sort | J. Humberto Pérez-Cruz |
| collection | DOAJ |
| description | Standard adaptive control is the preferred approach for stabilization and synchronization of chaotic systems when the structure of such systems is a priori known but the parameters are unknown. However, in the presence of unmodeled dynamics and/or disturbance, this approach is not effective anymore due to the drift of the parameter estimations, which eventually causes the instability of the closed-loop system. In this paper, a robustifying term, which consists of a saturation function, is used to avoid this problem. The robustifying term is added to the adaptive control law. Consequently, the learning law is also modified. The boundedness of the states and the parameter estimations is rigorously and thoroughly proven by means of Lyapunov like analysis based on Barbalat’s lemma. On these new conditions, the convergence to zero cannot be achieved due to the presence of unmodeled dynamics and/or disturbance. However, it is still possible to guarantee the asymptotic convergence to a bounded zone around zero. The width of this zone can be adjusted by the designer. The performance of this robust approach is verified by numerical simulations. Although, for simplicity, this strategy is only applied to the stabilization and synchronization of Zhang system, the procedure can easily be generalized to a broad class of chaotic systems. |
| format | Article |
| id | doaj-art-70d24ce80ca14a6993fdb886904b6809 |
| institution | Kabale University |
| issn | 1076-2787 1099-0526 |
| language | English |
| publishDate | 2018-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Complexity |
| spelling | doaj-art-70d24ce80ca14a6993fdb886904b68092025-02-03T06:10:53ZengWileyComplexity1076-27871099-05262018-01-01201810.1155/2018/49895204989520Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive ControlJ. Humberto Pérez-Cruz0SEPI-ESIME-Azcapotzalco, Instituto Politécnico Nacional, Ciudad de México 02250, MexicoStandard adaptive control is the preferred approach for stabilization and synchronization of chaotic systems when the structure of such systems is a priori known but the parameters are unknown. However, in the presence of unmodeled dynamics and/or disturbance, this approach is not effective anymore due to the drift of the parameter estimations, which eventually causes the instability of the closed-loop system. In this paper, a robustifying term, which consists of a saturation function, is used to avoid this problem. The robustifying term is added to the adaptive control law. Consequently, the learning law is also modified. The boundedness of the states and the parameter estimations is rigorously and thoroughly proven by means of Lyapunov like analysis based on Barbalat’s lemma. On these new conditions, the convergence to zero cannot be achieved due to the presence of unmodeled dynamics and/or disturbance. However, it is still possible to guarantee the asymptotic convergence to a bounded zone around zero. The width of this zone can be adjusted by the designer. The performance of this robust approach is verified by numerical simulations. Although, for simplicity, this strategy is only applied to the stabilization and synchronization of Zhang system, the procedure can easily be generalized to a broad class of chaotic systems.http://dx.doi.org/10.1155/2018/4989520 |
| spellingShingle | J. Humberto Pérez-Cruz Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control Complexity |
| title | Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control |
| title_full | Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control |
| title_fullStr | Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control |
| title_full_unstemmed | Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control |
| title_short | Stabilization and Synchronization of Uncertain Zhang System by Means of Robust Adaptive Control |
| title_sort | stabilization and synchronization of uncertain zhang system by means of robust adaptive control |
| url | http://dx.doi.org/10.1155/2018/4989520 |
| work_keys_str_mv | AT jhumbertoperezcruz stabilizationandsynchronizationofuncertainzhangsystembymeansofrobustadaptivecontrol |