Advertisement

Application of the Attractive Ellipsoid Methodology to Robust Control Design of a Class of Switched Systems

  • V. Azhmyakov
  • J. H. Carvajal Rojas
Part of the Lecture Notes in Mechanical Engineering book series (LNME)

Abstract

Our contribution is devoted to an application of the newly elaborated robust feedback-type control methodology to a class of industrial robotic systems. We consider a formal prototype of an automated Continuous Stirred Tank Reactor (CSTR) in the presence of bounded (operating) uncertainties and external disturbances. The nonlinear model of the CSTR has a switched nature and implies a sophisticated dynamical behaviour. Moreover, the resulting control design is supposed to be the defined only by the given system output. The robustness property of the closed-loop automated system is determined here in the sense of a “practical stability” concept and is based on the Attractive Ellipsoid (AE) approach. The implementable control design scheme we propose involves the Bilinear Matrix Inequalities (BMIs) techniques in combination with the Multiple Lyapunov functions analysis.

Keywords

Continuous Stir Tank Reactor Output Feedback Control Practical Stability Arbitrary Switching Multiple Lyapunov Function 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. Antsaklis, P. (2000). Special issue on hybrid systems: Theory and applications—a brief introduction to the theory and applications of hybrid systems. Proceedings of IEEE, 88, 879–887.CrossRefGoogle Scholar
  2. Azhmyakov, V., Boltyanski, V., & Poznyak, A. (2008). Optimal control of impulsive hybrid systems. Nonlinear Analysis: Hybrid Systems, 2, 1089–1097.MathSciNetMATHGoogle Scholar
  3. Azhmyakov, V., Galvan-Guerra, R., & Egerstedt, M. (2010). On the LQ-based optimization technique for impulsive hybrid control systems. In Proceedings of the 2010 American Control Conference (pp. 129–135), Baltimor, USA, 2010.Google Scholar
  4. Azhmyakov, V., Basin, M., & Reincke-Collon, C. (2014). Optimal LQ-type switched control design for a class of linear systems with piecewise constant inputs. In Proceedings of the 19th IFAC World Congress (pp. 6976–6981), Cape Town, South Africa.Google Scholar
  5. Azhmyakov, V. (2011). On the geometric aspects of the invariant ellipsoid method: Application to the robust control design. In Proceedings of the 50th IEEE Conference on Decision and Control and European Control Conference (pp. 1353–1358), Orlando, USA.Google Scholar
  6. Azhmyakov, V., Cabrera Martinez, J., & Poznyak, A. (2015). Optimization of a class of nonlinear switched systems with fixed-levels control inputs. In Proceedings of the 2015 American Control Conference, Chicago, USA, 2015.Google Scholar
  7. Barkhordari, Y. M., & Jahed-Motlagh, M. R. (2009). Stabilization of a CSTR with two arbitrarily switching modes using modal state feedback linearization. Chemical Engineering Journal, 155, 838–843.CrossRefGoogle Scholar
  8. Blanchini, F., & Miani, S. (2008). Set-theoretic methods in control. Boston: Birkhäuser.MATHGoogle Scholar
  9. Boskovic, J. D., & Mehra, R. K. (2000). Multi-mode switching in flight control. In Proceedings of the 19th Digital Avionics Systems Conference (Vol. 2, pp. 6F2/1–6F2/8).Google Scholar
  10. Branicky, M. S. (1998). Multiple Lyapunov functions and other analysis tools for switched and hybrid systems. IEEE Transactions on Automatic Control, 43, 475–482.MathSciNetCrossRefMATHGoogle Scholar
  11. Daafouz, J., Riedinger, P., & Iung, C. (2002). Stability analysis and control synthesis for switched systems: A switched Lyapunov function approach. IEEE Transactions on Automatic Control, 47, 1883–1887.MathSciNetCrossRefGoogle Scholar
  12. Dahleh, M. A., & Pearson, J. B, Jr. (1988). Optimal rejection of persistent disturbances, robust stability and mixed sensitivity minimization. IEEE Transactions on Automatic Control, 33, 722–731.MathSciNetCrossRefMATHGoogle Scholar
  13. Davila, J., & Poznyak, A. (2011). Dynamic sliding mode control design using attracting ellipsoid medthod. Automatica, 47, 1467–1472.MathSciNetCrossRefMATHGoogle Scholar
  14. Doyle, J. C. (1983). Synthesis of robust controllers and filters. In Proceedings of the 22nd Conference on Decision and Control (Vol. 22, pp. 109–114).Google Scholar
  15. Duncan, G. J., & Schweppe, F. C. (1971). Control of linear dynamic systems with set constrained disturbances. IEEE Transactions on Automatic Control, 16, 411–423.MathSciNetCrossRefGoogle Scholar
  16. DeCarlo, R. A., Branicky, M. S., Petterson, S., & Lennartson, S. (2000). Perspectives and results on the stability and stabilizability of hybrid systems. Proceedings of IEEE: Special Issue Hybrid Systems, 88, 1069–1082.CrossRefGoogle Scholar
  17. Egerstedt, M., Wardi, Y., & Axelsson, H. (2006). Transition-time optimization for switched-mode dynamical systems. IEEE Transactions on Automatic Control, 51, 110–115.MathSciNetCrossRefGoogle Scholar
  18. Haddad, W., & Chellaboina, V. (2008). Nonlinear Dynamics Systems and Control. Princeton: Princeton University Press.MATHGoogle Scholar
  19. Hespanha, J. P., Morse, A. S. (1999). Stability of switched systems with average dwell-tim. In Proceedings of the 38th IEEE Conference on Decision and Control (pp. 2655–2660), Phoenix, USA.Google Scholar
  20. Khalil, H. (2002). Nonlinear systems. Upper Saddle River: Prentice Hall.MATHGoogle Scholar
  21. Kouhi, Y., Bajcinca, N., Raisch, J., & Shorten, R. (2014). On the quadratic stability of switched linear systems associated with symmetric transfer function matrices. Automatica, 50, 2872–2879.MathSciNetCrossRefMATHGoogle Scholar
  22. Liberzon, D. (2003). Switching in systems and control. Boston: Birkhäuser.CrossRefMATHGoogle Scholar
  23. Lunze, J., & Lamnabhi-Lagarrigue, F. (2009). Handbook of hybrid systems control: Theory, tools and applications. New York: Cambridge University Press.CrossRefMATHGoogle Scholar
  24. Lygeros, J. (2003). Lecture notes on hyrid systems. Cambridge, UK: Cambridge University Press.Google Scholar
  25. Martinoli, A., Mondada, F., Mermoud, G., Correll, N., Egerstedt, M., Hsieh, A., et al. (Eds.). (2013). Distributed autonomous robotic systems., Springer tracts in advanced robotics New York: Springer.Google Scholar
  26. Poznyak, A., Azhmyakov, V., & Mera, M. (2011). Practical output feedback stabilization for a class of continuous-time dynamic system under sample-data outputs. International Journal of Control, 84, 1408–1416.Google Scholar
  27. Poznyak, A., Polyakov, A., & Azhmyakov, V. (2014). Attractive ellipsoid method in robust control. New York: Birkhäuser.MATHGoogle Scholar
  28. Sajja, S., Corless, M., Zeheb, E., & Shorten, R. (2013). Stability of a class of switched descriptor systems. In Proceedings of the 2013 American Control Conference (pp. 54–58), Washington DC, USA.Google Scholar
  29. Shaikh, M. S., & Caines, P. E. (2007). On the hybrid optimal control problem: Theory and algorithms. IEEE Transactions on Automatic Control, 52, 1587–1603.MathSciNetCrossRefGoogle Scholar
  30. Shucker, B., Murphey, T., & Bennett, J. K. (2007). Switching rules for decentralized control with simple control laws. In Proceedings of the 2007 American Control Conference (pp. 1485–1492), New York, USA.Google Scholar
  31. Wicks, M. A., Peleties, P., & DeCarlo, R. A. (1994). Construction of piecewise Lyapunov functions for stabilizing switched systems. In Proceedings of the 33rd Conference on Decision and Control (pp. 3492–3497), Lake Buena Vista, USA.Google Scholar
  32. Zhai, G. S., Hu, B., Yasuda, K., & Michel, A. N. (2001). Disturbance attenuation properties of time-controlled switched systems. Journal of the Franklin Institute, 338, 765–779.MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer India 2016

Authors and Affiliations

  1. 1.Faculty of Biomedical Engineering, Electronics and MechatronicsAntonio Nariño UniversityNeivaRepublic of Colombia

Personalised recommendations