Volume 16, Issue 2 (Journal of Control, V.16, N.2 Summer 2022)                   JoC 2022, 16(2): 41-54 | Back to browse issues page

XML Persian Abstract Print

Download citation:
BibTeX | RIS | EndNote | Medlars | ProCite | Reference Manager | RefWorks
Send citation to:
Saki S, Bolandi H. Definition of General Operator Space and The s-gap Metric for Measuring Robust Stability of Control Systems with Nonlinear Dynamics. JoC 2022; 16 (2) :41-54
URL: http://joc.kntu.ac.ir/article-1-849-en.html
Definition of General Operator Space and The s-gap Metric for Measuring Robust Stability of Control Systems with Nonlinear Dynamics
Saman Saki , Hossein Bolandi * 1
Abstract:   (2818 Views)
In the recent decades, metrics have been introduced as mathematical tools to determine the robust stability of the closed loop control systems. However, the metrics drawback is their limited applications in the closed loop control systems with nonlinear dynamics. As a solution in the literature, applying the metric theories to the linearized models is suggested. In this paper, we show that using the linear model is not adequate to analyze the robust stability. To this end, the definition of general operator space is proposed as the important novelty to determine the weakest topology between two nonlinear dynamic systems. In this space, all nonlinear operators (nonlinear dynamic systems) with differentiable manifold topology can be considered as isometric isomorphism. The result of this definition is possibility of the nonlinear gap metric solution which leads to definition of the s-gap metric. In fact, we show that the calculation of the nonlinear gap metric leads to the most conservative tangent spaces in the defined space. Also, based on the new results, the gain bound of the closed-loop system is determined, which together with the s-gap offers a new theory for robust stability analysis. Advanced operator theory and simulation results confirm the correctness of the claims.
Keywords: The Gain Bounds, The Generalized Stability Margin, Nonlinear Metrics, Operator Space and Nonlinearity
Full-Text [PDF 636 kb]   (506 Downloads)    
Type of Article: Research paper | Subject: Special
Received: 2021/04/2 | Accepted: 2021/12/2 | ePublished ahead of print: 2022/03/7
References
1. [1] T. T. Georgiou and M. C. Smith, "Optimal robustness in the gap metric," IEEE Trans. Automat. Contr., vol. 35, no. 6, pp. 673-686, 1990. [DOI:10.1109/9.53546]
2. [2] G. Vinnicombe, "Frequency domain uncertainty and the graph topology," IEEE Trans. Automat. Contr., vol. 38, no. 9, pp. 1371-1383, 1993. [DOI:10.1109/9.237648]
3. [3] G. Vinnicombe, Uncertainty and Feedback: H [infinity] Loop-shaping and the [nu]-gap Metric. World Scientific, 2001. [DOI:10.1142/p140]
4. [4] S. Kersting and M. Buss, "How to Systematically Distribute Candidate Models and Robust Controllers in Multiple-Model Adaptive Control: A Coverage Control Approach," IEEE Trans. Automat. Contr., vol. 63, no. 4, pp. 1075-1089, 2018. [DOI:10.1109/TAC.2017.2731946]
5. [5] M. Ahmadi and M. Haeri, "Multimodel control of nonlinear systems: An improved gap metric and stability margin-based method," J. Dyn. Syst. Meas. Control, vol. 140, no. 8, 2018. [DOI:10.1115/1.4039086]
6. [6] S. Saki, H. Bolandi, and S. K. Mousavi Mashhadi, "Optimal direct adaptive soft switching multi-model predictive control using the gap metric for spacecraft attitude control in a wide range of operating points," Aerosp. Sci. Technol., vol. 77, 2018, doi: 10.1016/j.ast.2018.03.001. [DOI:10.1016/j.ast.2018.03.001]
7. [7] S. X. Ding, "Application of factorization and gap metric techniques to fault detection and isolation part I: A factorization technique based FDI framework," IFAC-PapersOnLine, vol. 48, no. 21, pp. 113-118, 2015. [DOI:10.1016/j.ifacol.2015.09.513]
8. [8] L. Li and S. X. Ding, "Gap metric techniques and their application to fault detection performance analysis and fault isolation schemes," Automatica, vol. 118, p. 109029, 2020. [DOI:10.1016/j.automatica.2020.109029]
9. [9] T. Koenings, M. Krueger, H. Luo, and S. X. Ding, "A data-driven computation method for the gap metric and the optimal stability margin," IEEE Trans. Automat. Contr., vol. 63, no. 3, pp. 805-810, 2017. [DOI:10.1109/TAC.2017.2735023]
10. [10] H. Bolandi and S. Saki, "Design of adaptive model predictive control for a class of uncertain non-linear dynamic systems: stability, convergence, and robustness analysis," IET Control Theory Appl., vol. 13, no. 15, pp. 2376-2386, 2019. [DOI:10.1049/iet-cta.2019.0061]
11. [11] D. Shaghaghi, A. Fatehi, and A. Khaki-Sedigh, "Multi-linear model set design based on the nonlinearity measure and H-gap metric," ISA Trans., vol. 68, pp. 1-13, 2017. [DOI:10.1016/j.isatra.2017.01.021]
12. [12] M. Ahmadi and M. Haeri, "A systematic decomposition approach of nonlinear systems by combining gap metric and stability margin," Trans. Inst. Meas. Control, vol. 43, no. 9, pp. 2006-2017, 2021. [DOI:10.1177/0142331221989009]
13. [13] X. Tao, N. Li, and S. Li, "Multiple model predictive control for large envelope flight of hypersonic vehicle systems," Inf. Sci. (Ny)., vol. 328, pp. 115-126, 2016. [DOI:10.1016/j.ins.2015.08.033]
14. [14] T. T. Georgiou and M. C. Smith, "Robustness analysis of nonlinear feedback systems: An input-output approach," IEEE Trans. Automat. Contr., vol. 42, no. 9, pp. 1200-1221, 1997. [DOI:10.1109/9.623082]
15. [15] T. Gong and Y. Lu, "The relationship between gap metric and time-varying gap metric for linear time-varying systems," Int. J. Innov. Comput. Inform. Control, vol. 9, pp. 2125-2141, 2013.
16. [16] B. D. O. Anderson, T. S. Brinsmead, and F. De Bruyne, "The Vinnicombe metric for nonlinear operators," IEEE Trans. Automat. Contr., vol. 47, no. 9, pp. 1450-1465, 2002. [DOI:10.1109/TAC.2002.802767]
17. [17] S. Z. Khong and M. Cantoni, "On the metric property of an LTV generalisation of the $ν$-gap," in 2012 IEEE 51st IEEE Conference on Decision and Control (CDC), 2012, pp. 214-219. [DOI:10.1109/CDC.2012.6427107]
18. [18] S. Z. Khong and M. Cantoni, "Gap metrics for time-varying linear systems in a continuous-time setting," Syst. Control Lett., vol. 70, pp. 118-126, 2014. [DOI:10.1016/j.sysconle.2014.06.003]
19. [19] S. M. Djouadi, "On robustness in the gap metric and coprime factor uncertainty for LTV systems," Syst. Control Lett., vol. 80, pp. 16-22, 2015. [DOI:10.1016/j.sysconle.2015.03.006]
20. [20] M. S. Akram and M. Cantoni, "Gap metrics for linear time-varying systems," SIAM J. Control Optim., vol. 56, no. 2, pp. 782-800, 2018. [DOI:10.1137/16M1092775]
21. [21] M. Cantoni and H. Pfifer, "Gap metric computation for time-varying linear systems on finite horizons," IFAC-PapersOnLine, vol. 50, no. 1, pp. 14513-14518, 2017. [DOI:10.1016/j.ifacol.2017.08.2073]
22. [22] K. Glover and D. McFarlane, "Robust stabilization of normalized coprime factor plant descriptions with H/sub infinity/-bounded uncertainty," IEEE Trans. Automat. Contr., vol. 34, no. 8, pp. 821-830, 1989. [DOI:10.1109/9.29424]
23. [23] V. Zahedzadeh, H. J. Marquez, and T. Chen, "On the computation of an upper bound on the gap metric for a class of nonlinear systems," in 2008 American Control Conference, 2008, pp. 1917-1922. [DOI:10.1109/ACC.2008.4586772]
24. [24] V. Zahedzadeh, H. J. Marquez, and T. Chen, "Upper bounds for induced operator norms of nonlinear systems," IEEE Trans. Automat. Contr., vol. 54, no. 5, pp. 1159-1165, 2009. [DOI:10.1109/TAC.2009.2017813]
25. [25] V. Zahedzadeh, H. J. Marquez, and T. Chen, "On the robust stability of unforced nonlinear systems," in Proceedings of the 45th IEEE Conference on Decision and Control, 2006, pp. 343-348. [DOI:10.1109/CDC.2006.377066]
26. [26] V. Zahedzadeh, H. J. Marquez, and T. Chen, "On the input-output stability of nonlinear systems: Large gain theorem," in 2008 American Control Conference, 2008, pp. 3440-3445. [DOI:10.1109/ACC.2008.4587025]
27. [27] K. Zhou and J. C. Doyle, Essentials of robust control, vol. 104. Prentice hall Upper Saddle River, NJ, 1998.
28. [28] A. El-Sakkary, "The gap metric: Robustness of stabilization of feedback systems," IEEE Trans. Automat. Contr., vol. 30, no. 3, pp. 240-247, 1985. [DOI:10.1109/TAC.1985.1103926]
Add your comments about this article : Your username or Email:
CAPTCHA
Send email to the article author

Rights and permissions
Creative Commons License This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License.

© 2024 CC BY-NC 4.0 | Journal of Control

Designed & Developed by : Yektaweb