Stabilization of Switched Systems Using Only A Single Fractional Order PI Controller

Abstract

The infinite root boundary does not exist because of m < n . Finally, the complex root boundary (CRB) is obtained by equating the real and imaginary parts of Pi(jw) to zero. In this case, Equation (4) is returned to two equations

Keywords

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References

1. D. Liberzon, Switching in Systems and Control, Birkäuser, 2003.
2. J. Daafouz, P. Riedinger, and C. Lung, “Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach,” IEEE Trans. on Automatic Control, vol. 47, no. 11, pp. 1883–1887, 2002.
3. S.L. Chen, Y.Yao, and X. Di, “Robust Stabilization for a Class of Uncertain Discrete-time Switched Linear Systems,” In: Discrete Time Systems, edited by M.A. Jordán, InTech-Open Access Company.
4. D.J. Leith, R.N. Shorten, W.E. Leithead, O. Mason, and P. Curran, “Issues in the design of switched linear control systems: A benchmark study,” Int. J. Adapt. Control Signal Process, vol. 17, no. 2, pp. 103– 118, 2003.
5. S. Kim, S.A. Campbell, and X. Liu, “Stability of a Class of Linear Switching Systems With Time Delay,” IEEE Trans. on Circuits and Systems-I: Regular Papers, vol. 53, no. 2, pp. 384-393, 2006.
6. Z. Sun, and S.S. Ge, “Analysis and synthesis of switched linear control systems,” Automatica, vol. 41, no. 2, pp. 181-195, 2005.
7. K. Wulff, Quadratic and Non-Quadratic Stability Criteria for Switched Linear Systems, Ph.D Thesis, National University of Ireland, 2004.
8. M.S. Branicky, “Stability of switched and hybrid systems,” in: Proc. the 33rd IEEE Conf. on Decision and Control, pp. 3498-3503, 1994.

9. Z. Sun, S. S. Ge, and T. H. Lee, “Controllability and reachability criteria for switched linear systems,” Automatica, vol. 38, no. 5, pp. 775–786, 2002.
10. J.P. Hespanha, D. Liberzon, D. Angeli, and E.D. Sontag, “Nonlinear norm-observability notions and stability of switched systems,” IEEE Trans. Automat. Control, vol. 52, no. 2, pp. 154–168, 2005.
11. S. Solmaz, R. Shorten, K. Wulff ve F.Ó. Cairbre, “A design methodology for switched discrete time linear systems with applications to automotive roll dynamics control,” Automatica, vol. 44, no. 9, pp. 2358-2363, 2008.
12. X. Xu and P. J. Antsaklis, “Optimal control of switched systems based on parameterization of the switching instants,” IEEE Trans. Automat. Control, vol. 49, no. 1, pp. 2–16, 2004.
13. K. Wulff, F. Wirth, and R. Shorten, “A control design method for a class of switched linear systems,” Automatica, vol. 45, no. 11, pp. 2592-2596, 2009.
14. C. Chen, S. Fei, K. Zhang, and Y. Lu, “Control of switched linear systems with actuator saturation and its applications,” Mathematical and Computer Modelling, vol. 56, no. 1-2, pp. 14-26, 2009.
15. S.H. HosseinNia, I. Tejado, B.M. Vinagre, “Robust Fractional order PI Controller for Switching Systems,” in: Proc. the 5th Symp. on. Fractional Differentiation and its Applications (FDA'2012), Hohai University, 2012.
16. Y.I. Neimark, “D-decomposition of the space of quasi-polynomials (on the stability of linearized distributive systems),” American Mathematical Society Translations, vol. 102, pp. 95-131, 1973.
17. D. Liberzon, ve A.S. Morse, “Basic problems in stability and design of switched systems,” IEEE Control Systems Magazine, vol. 19, pp. 59–70, 1999.
18. S.H. HosseinNia, I. Tejado, B.M. Vinagre, “A method for the design of robust controllers ensuring the quadratic stability for switching systems,” J. of Vibration and Control, vol. 20, no. 7, pp. 1085-1098, 2014.
19. I. Podlubny, “Fractional Order Systems and PID Controllers,” IEEE  Trans. on Automatic Control, vol. 44, no. 1, pp. 208-214, 1999.
20. J. Hwang, J.-F. Leu, and S.-Y. Tsay, “A note on time-domain simulation of feedback fractional-order systems,” IEEE Trans. on Automatic Control, vol. 47, no. 4, pp. 625-631, 2002.
21. J. Ackermann, D. Kaesbauer, “Design of robust PID controllers,” in: Proc. the European Control Conference, pp. 522-527, 2001.
22. M.-T. Ho, A. Datta, and S.P. Bhattacharyya, “A new approach to feedback stabilization,” in: Proc. The 35th Conf. on Decision and Control, Kobe, Japan, 1996.
23. İ. Işık, and S.E. Hamamci, "Anahtarlamalı Sistemleri Kararlı Yapan PI Kontrolör Setinin Hesabı," in: Proc. the TOK 2013 Turkish Automatic Control National Meeting, Malatya, Turkey, 2013, (in Turkish).
24. Y.C. Cheng, and C. Hwang, “Stabilization of unstable Şrst-order time- delay systems using fractional-order PD controllers,” J. of the Chinese Inst. of Engineers, vol. 29, pp. 241-249, 2006.