H_∞ Observer-Based Control Design for Switched Positive Systems Including Interval Uncertainties, Unstable Modes and Time-Varying Delay

Document Type : Research paper

Authors

1 Department of Electrical and Computer Engineering, University of Kashan, Kashan, Iran

2 Electrical and Electronic Engineering Department, Shahed University, Tehran, Iran

Abstract

The purpose of this study is to present a practical approach in which the effect of performance degradation and instability factors such as exogenous disturbances, parametric uncertainties, time-varying delay, and unstable modes can reduce to the minimum possible amount in linear switched positive systems. To reduce the effect of the mentioned destructive factors and to strengthen the robust design of switched positive systems, in this paper, instead of using the co-positive Lyapunov function along with the L1-gain, the quadratic Lyapunov-Krasovskii function utilized along with the L2-gain, which leads to the design of H_∞ performance. The latter method, especially when there is a requirement to estimate the parameters with the support of the output feedback approach by minimizing the interface parameters, provides the feasibility of designing a more convenient and efficient observer-based controller. The necessary and sufficient conditions for solving the problem concerning the positivity of the system, disturbance attenuation, and parametric uncertainties are expressed by two theories and implemented by the linear matrix inequality technique. The results of this technique's solution include the gains of the controller and observer. Considering that stable and unstable modes are in this system, it is necessary to guarantee the exponential stability of the whole system by the controllers and designing the average dwell-time switching regime. Finally, illustrative examples, including numerical, practical, and comparative, are presented to show the efficiency and performance of different aspects of the proposed approach. The smallness of the mean square error values in the example compared with the output feedback method in linear programming confirms the capabilities of the presented approach. For instance, the mean square error of the system output for the method of this paper is 0.008 and for the compared approach is 0.081.

Keywords

References

  1. Dejamkhooy, and A. Ahmadpour,. "Torque ripple reduction of the position sensor-less switched reluctance motors applied in the electrical vehicles," J. Oper. Autom. Power Eng., 2022.
  2. Bagheri, A. Rabiee, S. Galavani, H. Yassami, and A. Moeini, “A Practical Approach for Coordinated Transmission Switching and OLTCs’ Tap Adjustment: DIgSILENT-Based Improved PSO Algorithm,” J. Oper. Autom. Power Eng., vol. 9, no. 2, pp. 103–115, 2021.
  3. B. Anderson, J. A. Marshall, A. L’Afflitto , J. M. Dotterweich, “Model reference adaptive control of switched dynamical systems with applications to aerial robotics,” J. Intell. Robot. Syst., vol. 100, pp. 1265–1281, 2020.
  4. Zhao, Y. Niu, T. Jia, “Security control of cyber-physical switched systems under round-robin protocol: input-to-state stability in probability,” Inf. Sci., vol. 508, pp. 121–34, 2020.
  5. Niu, J. Zhai, H. Yin, E. Feng, C. Liu, Z. Xiu, “Multiobjective optimisation of nonlinear switched systems in uncoupled fed-batch fermentation,” Int. J. Syst. Sci., vol. 51, no. 10, pp. 1798–813, 2020.
  6. Du, H. R. Karimi, J. Qiao, D. Wu, C. Feng, “Stability analysis for a class of discrete-time switched systems with partial unstable subsystems,” IEEE Trans. Circuits Syst. II: Express Briefs., vol. 66, no. 12, pp. 2017–21, 2019.
  7. Zhang, L. Xiao, H. Li, “Robust control for switched systems with unmatched uncertainties based on switched robust integral sliding mode,” IEEE Access., vol. 8, pp. 138396–138405, 2020.
  8. Briat, “L1× `1-to-L1× `1 analysis of linear positive impulsive systems with application to the L1× `1-to-L1× `1 interval observation of linear impulsive and switched systems,” Nonlinear Anal.: Hybrid Syst., vol. 34, pp. 1–17, 2019.
  9. Du, B. Jiang, P. Shi , S. Zhou “H∞ filtering of discrete-time switched Systems with state delays via switched lyapunov function approach,” IEEE Trans. Autom. Control, vol. 52, pp. 1520–1525, 2007.
  10. Bergeling , R. Pates, A. Rantzer, “H-infinity optimal control for systems with a bottleneck frequency,” IEEE Trans. Autom. Control, vol. 66, no. 6, pp. 2732–2738, 2020.
  11. M. Haddad, V. Chellaboina, Q. Hui, Nonnegative and Compartmental Dynamical Systems. Princeton University Press, 2010
  12. Farina, S. Rinaldi, Positive Linear Systems: Theory and Applications. John Wiley & Sons, 2000.
  13. Sheikholeslami, F. B. Sheykholeslami, S. Khoshhal, H. Mola-Abasia, D. D. Ganji, H. B. Rokni, “Effect of magnetic field on Cu–water nanofluid heat transfer using GMDH-type neural network,” Neural Comput. Appl., vol. 25, pp. 171–178, 2014.
  14. Lv, B. De Schutter , W. Yu, S. Baldi, “Adaptive asymptotic tracking for a class of uncertain switched positive compartmental models with application to anesthesia,” IEEE Trans. Syst. Man Cybern. Syst., vol. 51, no. 8, pp. 4936–4942, 2019.
  15. E. Valcher, I. Zorzan, “On the stabilizability of continuoustime compartmental switched systems,” in proc. of the CDC, pp. 4246–4251, 2015.
  16. Shang, W. Qi, G. Zong, “Finite-time asynchronous control for positive discrete-time Markovian jump systems,” IET Control Theory Applic., vol. 13, pp. 935–942, 2019.
  17. E. Valcher, I. Zorzan, “On the stabilizability of continuoustime compartmental switched systems,” in proc. of the CDC, Osaka, Japan, 2015, pp. 4246–4251.
  18. Abbasi, I. Zamani, A. H. Mehra, M. Shafieirad, A. Ibeas, “Optimal control design of impulsive SQEIAR epidemic models with application to COVID-19,” Chaos, Solitons Fractals, vol. 139, p. 110054, 2020.
  19. Xiang, Z. Xiang, “Stability, L1-gain and control synthesis for positive switched systems with time-varying delay,” Nonlinear Anal.: Hybrid Syst., vol. 9, pp. 9–17, 2013.
  20. Xiang, Z. Xiang “Observer design of switched positive systems with time-varying delays,” Circuits Syst. Signal Process., vol. 32, pp. 2171–2184, 2013.
  21. Chen, M. Chen, J. Shen, “A novel approach to l1induced controller synthesis for positive systems with interval uncertainties,” J. Franklin Inst., vol. 354, no. 8, pp. 3364–77, 2017.
  22. Krokavec, A. Filasová “H∞ filter design for discrete-time linear positive systems,” In 2019 IEEE 58th Conf. Decision Control., pp. 2844–2849, 2019.
  23. Zhang, X. Zhao, R. Zhang, Y. Chen, “Improved controller design for uncertain positive systems and its extension to uncertain positive switched systems,” Asian J. Control, vol. 20, no. 1, pp. 159–73, 2018.
  24. Fridman, Introduction to Time-delay Systems: Analysis and Control. Springer, 2014..
  25. Li, Z. Xiang, “Positivity, exponential stability and disturbance attenuation performance for singular switched positive systems with time-varying distributed delays,” Appl. Math. Comput., vol. 372, p. 124981, 2020.
  26. Liu, W. Tao, L. Lee, J. Lu, “Finite-time boundedness and L2-gain analysis for switched positive linear systems with multiple time delays,” Int. J. Robust Nonlinear Control, vol. 27, no. 17, pp. 3508–23, 2017.
  27. Li, Z. Xiang, H. R. Karimi, “Stability and L1-gain controller design for positive switched systems with mixed time-varying delays,” Appl. Math. Comput., vol. 222, pp. 507–18, 2013.
  28. Abu-Khalaf , J. Huang, F. L. Lewis, Nonlinear H2/H∞ Constrained Feedback Control: A Practical Design Approach Using Neural Networks, Springer, London, 2006.
  29. Liu, X. Zhang, X. Lu, Q. Liu, “Stabilization of positive switched delay systems with all modes unstable,” Nonlinear Anal.: Hybrid Syst., vol. 29, pp. 110–20, 2018.
  30. F. Ansari, K. Benidis, R. Kurle, A. C. Turkmen, H. Soh, A. J. Smola, B. Wang, T. Januschowski, “Deep Explicit Duration Switching Models for Time Series,” Adv. Neural Int. Process. Syst., vol. 34, pp. 29949–61, 2021.
  31. Zhao, L. Zhang, P. Shi, M. Liu, “Stability of switched positive linear systems with average dwell time switching,” Automatica, vol. 48, no. 6, pp. 1132–7, 2012
  32. Zhang, X. Zhao, F. Zhu, Z. Han, “L1/`1-gain analysis and synthesis of markovian jump positive systems with time delay,” ISA trans., vol. 63, pp. 93–102, 2016.
  33. Volyanskyy, W. M. Haddad, J. M. Bailey, “Adaptive disturbance rejection control for compartmental systems with application to intraoperative anesthesia influenced by hemorrhage and hemodilution effects,” Int. J. Adapt. Control Signal Process., vol. 23, no. 1, pp. 1–29, 2009.
  34. Zaidi, M. Chaabane, F. Tadeo, A. Benzaouia, “Static state-feedback controller and observer design for interval positive systems with time delay,” IEEE Trans. Circuits Syst. II: Express Briefs, vol. 62, no 5, pp. 506–10, 2014.
  35. Liu, Y. Shen, X. Zhao, “Delay-dependent observer-based H∞ finite-time control for switched systems with timevarying delay,” Nonlinear Anal.: Hybrid Syst., vol. 6, no. 3, pp. 885–98, 2012.
  36. He, X. Gao,W. Qi, “Observer-based sliding mode control for switched positive nonlinear systems with asynchronous switching,” Nonlinear Dynamics, vol. 93, pp. 2433–2444, 2018.
  37. S. Sadabadi, D. Peaucelle, “From static output feedback to structured robust static output feedback: A survey,” Annu. Rev. Control, vol. 42, pp. 11–26, 2016.
  38. Zhang, R. Zhang, X. Cai, X. Jia, “A novel approach to control synthesis of positive switched systems,” IET Control Theory Appl., vol. 11, pp. 3396–33403, 2017.
  39. J. Hu, Y.W. Wang, J. W. Xiao, “Positive observer design for linear impulsive positive systems with interval uncertainties and time delay,” Int. J. Control, Autom. Syst., vol. 15, pp. 1032–1039, 2017.
  40. Yang, X. Li, J. Qiu, “ Output tracking control of delayed switched systems via state-dependent switching and dynamic output feedback,” Nonlinear Anal.: Hybrid Syst., vol. 1, no. 32, pp. 294–305, 2019.
  41. Liu, J. Lian, Y. Zhuang, “Output feedback L1 finite-time control of switched positive delayed systems with MDADT,” Nonlinear Anal.: Hybrid Syst. vol. 15, pp. 11–22, 2015.
  42. Berman, M. Neumann, R. J. Plemmons, R. J. Stern, Nonnegative Matrices in Dynamic Systems. WileyInterscience, 1989.
  43. Shu, J. Lam, H. Gao, B. Du, L. Wu, “Positive observers and dynamic output-feedback controllers for interval positive linear systems ,” EEE Trans. Circuits Syst. I, vol. 55, no. 10, pp. 3209–22.2008.
  44. J. DISTEFANO III, M. Jang, T. K. Malone, M. Broutman “Comprehensive kinetics of triiodothyronine production, distribution, and metabolism in blood and tissue pools of the rat using optimized blood-sampling protocols,” Endocrinology, vol. 110, no. 1, pp. 198–213, 1982.
  45. M. Haddad, T. Hayakawa, H. M. Bailey, “Adaptive control for non-negative and compartmental dynamical systems with applications to general anesthesia,” Int. J. Adapt. Contr. Signal Processing, vol. 17, no. 3, pp. 2093–5, 2003.
  46. Chellaboina, W. M. Haddad, J. Ramakrishnan, J. M. Bailey “On monotonicity of solutions of nonnegative and compartmental dynamical systems with time delays,” in Proc. of the Conf. Decis. Control, Maui, HI, USA, 2003, vol. 4, pp. 4008–4013.
  47. M. Haddad, V. Chellaboina, Q. Hui, Nonnegative and Compartmental Dynamical Systems, Princeton University Press, 2010.
  48. Volyanskyy, W. M. Haddad, J. M. Bailey, “Adaptive disturbance rejection control for compartmental systems with application to intraoperative anesthesia influenced by hemorrhage and hemodilution effects,” Int. J. Adapt. Contr. Signal Process., vol. 23, no. 1, pp. 1–29, 2009.
  49. Liu, N. Li, Z. Li, “Controller synthesis for switched positive linear systems by output feedback,” Int. J. Syst. Sci., vol. 49, pp. 3261–3271, 2018.
Journal of Operation and Automation in Power Engineering
Volume 12, Issue 3
August 2024
Pages 245-260

Files

History

  • Receive Date: 14 May 2022
  • Revise Date: 15 November 2022
  • Accept Date: 11 February 2023
  • First Publish Date: 18 April 2023

Share

How to cite

Statistics