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Robust Finite-time Stabilization of Linear Systems with Limited State Quantization (2403.17184v1)

Published 25 Mar 2024 in math.OC, cs.SY, and eess.SY

Abstract: This paper investigates the robust asymptotic stabilization of a linear time-invariant (LTI) system by a static feedback with a static state quantization. It is shown that the controllable LTI system can be stabilized to zero in a finite time by means of a nonlinear feedback with a quantizer having a limited (finite) number of values (quantization seeds) even when all parameters of the controller and the quantizer are time-invariant. The control design is based on generalized homogeneity. A homogeneous spherical quantizer is introduced. The static homogeneous feedback is shown to be local (or global) finite-time stabilizer for the linear system (dependently of the system matrix). The tuning rules for both the quantizer and the feedback law are obtained in the form of Linear Matrix Inequalities (LMIs). The closed-loop system is proven to be robust with respect to some bounded matched and vanishing mismatched perturbations. Theoretical results are supported by numerical simulations. \

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References (38)
  1. Z-P Jiang and L Teng-Fei. Quantized nonlinear control—a survey. Acta Automatica Sinica, 39(11):1820–1830, 2013.
  2. Encoding-decoding-based control and filtering of networked systems: Insights, developments and opportunities. IEEE/CAA Journal of Automatica Sinica, 5(1):3–18, 2017.
  3. N. Elia and S. K Mitter. Stabilization of linear systems with limited information. IEEE Transactions on Automatic Control, 46(9):1384–1400, 2001.
  4. R. W Brockett and D. Liberzon. Quantized feedback stabilization of linear systems. IEEE Transactions on Automatic Control, 45(7):1279–1289, 2000.
  5. F. Bullo and D. Liberzon. Quantized control via locational optimization. IEEE Transactions on Automatic Control, 51(1):2–13, 2006.
  6. M. Fu and L. Xie. The sector bound approach to quantized feedback control. IEEE Transactions on Automatic control, 50(11):1698–1711, 2005.
  7. G. Gu and L. Qiu. Networked control systems for multi-input plants based on polar logarithmic quantization. Systems & Control Letters, 69:16–22, 2014.
  8. J. Wang. Quantized feedback control based on spherical polar coordinate quantizer. IEEE Transactions on Automatic Control, 66(12):6077–6084, 2021.
  9. M. Fu and L. Xie. Finite-level quantized feedback control for linear systems. IEEE Transactions on Automatic Control, 54(5):1165–1170, 2009.
  10. Dynamic quantization of uncertain linear networked control systems. Automatica, 59:248–255, 2015.
  11. Quantized feedback sliding-mode control: An event-triggered approach. Automatica, 91:126–135, 2018.
  12. J. Wang. Quantized feedback control of discrete-time mimo linear systems with input and output quantization over finite data rate channels. IEEE Transactions on Automatic Control, 2022.
  13. D. F Delchamps. Stabilizing a linear system with quantized state feedback. IEEE Transactions on Automatic Control, 35(8):916–924, 1990.
  14. M L. Corradini and G. Orlando. Robust quantized feedback stabilization of linear systems. Automatica, 44(9):2458–2462, 2008.
  15. Stabilization of continuous-time linear systems subject to input quantization. Automatica, 58:167–172, 2015.
  16. J. Wang. Asymptotic stabilization of continuous-time linear systems with quantized state feedback. Automatica, 88:83–90, 2018.
  17. M. Kawski. Families of dilations and asymptotic stability. In Analysis of controlled dynamical systems, pages 285–294. Springer, 1991.
  18. V. I. Zubov. Systems of ordinary differential equations with generalized-homogeneous right-hand sides. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, (1):80–88, 1958.
  19. L. Rosier. Homogeneous lyapunov function for homogeneous continuous vector field. Systems & Control Letters, 19(6):467–473, 1992.
  20. L. Grüne. Homogeneous state feedback stabilization of homogenous systems. SIAM Journal on Control and Optimization, 38(4):1288–1308, 2000.
  21. Geometric homogeneity with applications to finite-time stability. Mathematics of Control, Signals and Systems, 17(2):101–127, 2005.
  22. Homogeneous approximation, recursive observer design, and output feedback. SIAM Journal on Control and Optimization, 47(4):1814–1850, 2008.
  23. Quantized super-twisting algorithm based sliding mode control. Automatica, 105:43–48, 2019.
  24. Global output feedback stabilization for a class of nonlinear systems with quantized input and output. International Journal of Robust and Nonlinear Control, 27(2):187–203, 2017.
  25. Finite/fixed-time stabilization of linear systems with states quantization. arXiv preprint arXiv:2309.01412, 2023a.
  26. Homogeneous finite/fixed-time stabilization with quantization. In 2023 62nd IEEE Conference on Decision and Control (CDC), pages 7643–7648. IEEE, 2023b.
  27. A. Polyakov. Sliding mode control design using canonical homogeneous norm. International Journal of Robust and Nonlinear Control, 29(3):682–701, 2019.
  28. A. F. Filippov. Differential equations with discontinuous righthand sides: control systems, volume 18. Springer Science & Business Media, 2013.
  29. E. Roxin. On finite stability in control systems. Rendiconti del Circolo Matematico di Palermo, 15(3):273–282, 1966.
  30. Finite-time stability of continuous autonomous systems. SIAM Journal on Control and optimization, 38(3):751–766, 2000.
  31. Y. Orlov. Finite time stability and robust control synthesis of uncertain switched systems. SIAM Journal on Control and Optimization, 43(4):1253–1271, 2004.
  32. Robust stabilization of mimo systems in finite/fixed time. International Journal of Robust and Nonlinear Control, 26(1):69–90, 2016.
  33. VV Khomenyuk. Systems of ordinary differential equations with generalized homogeneous right-hand sides. Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, (3):157–164, 1961.
  34. M. Kawski. Geometric homogeneity and stabilization. IFAC Proceedings Volumes, 28(14):147–152, 1995.
  35. A. Polyakov. Generalized homogeneity in systems and control. Springer, 2020.
  36. Robust feedback stabilization of linear mimo systems using generalized homogenization. IEEE Transactions on Automatic Control, 65(12):5429–5436, 2020.
  37. V. Utkin. Sliding modes in control and optimization. Springer Science & Business Media, 2013.
  38. S. P Boyd and L. Vandenberghe. Convex optimization. Cambridge university press, 2004.

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