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July  2013, 9(3): 579-593. doi: 10.3934/jimo.2013.9.579

## On the robust control design for a class of nonlinearly affine control systems: The attractive ellipsoid approach

 1 Department of Control and Automation, CINVESTAV, Av, Instituto Politecnico Nacional 2508, Mexico D.F., Mexico, Mexico, Mexico

Received  September 2011 Revised  October 2012 Published  April 2013

This paper is devoted to a problem of robust control design for a class of continuous-time dynamic systems with bounded uncertainties. We study a family of nonlinearly affine control systems and develop a computational extension of the conventional invariant ellipsoid techniques. The obtained method can be considered as a powerful numerical approach that makes it possible to design a concrete stabilizing control strategies for the resulting closed-loop systems. The design procedure for this feedback-type control is based on the classic Lyapunov-type stability analysis of invariant sets for the given dynamic system. We study the necessary theoretic basis and propose a computational algorithm that guarantee some minimality properties of the stability/attractivity regions for dynamic systems under consideration. The complete solution procedure contains an auxiliary LMI-constrained optimization problem. The effectiveness of the proposed robust control design is illustrated by a numerical example.
Citation: Vadim Azhmyakov, Alex Poznyak, Omar Gonzalez. On the robust control design for a class of nonlinearly affine control systems: The attractive ellipsoid approach. Journal of Industrial and Management Optimization, 2013, 9 (3) : 579-593. doi: 10.3934/jimo.2013.9.579
##### References:
 [1] V. Azhmyakov, Stability of differential inclusions: A computational approach, Mathematical Problems in Engineering, 2006 (2006), 1-15. doi: 10.1155/MPE/2006/17837. [2] V. Azhmyakov, V. G. Boltyanski and A. Poznyak, Optimal control of impulsive hybrid systems, Nonlinear Analysis: Hybrid Systems, 2 (2008), 1089-1097. doi: 10.1016/j.nahs.2008.09.003. [3] V. Azhmyakov, An approach to controlled mechanical systems based on the multiobjective optimization technique, Journal of Industrial and Management Optimization, 4 (2008), 697-712. doi: 10.3934/jimo.2008.4.697. [4] V. Azhmyakov, A gradient-type algorithm for a class of optimal control processes governed by hybrid dynamical systems, IMA Journal of Mathematical Control and Information, 28 (2011), 291-307. doi: 10.1093/imamci/dnr010. [5] V. Azhmyakov, On the geometric aspects of the invariant ellipsoid method: Application to the robust control design, in "Proceedings of the 50th Conference on Decision and Control and European Control Conference," Orlando, USA, (2011), 1353-1358. doi: 10.1109/CDC.2011.6161180. [6] V. Azhmyakov, M. V. Basin and J. Raisch, Proximal point based approach to optimal control of affine switched systems, Discrete Event Dynamic Systems, 22 (2012), 61-81. doi: 10.1007/s10626-011-0109-8. [7] A. E. Barabanov and O. N. Granichin, Optimal controller for linear plants with bounded noise, Automation and Remote Control, 45 (1984), 39-46. [8] M. Basin and D. Calderon-Alvarez, Optimal filtering over linear observations with unknown parameters, Journal of The Franklin Institute, 347 (2010), 988-1000. doi: 10.1016/j.jfranklin.2010.01.006. [9] M. Basin, J. Rodriguez-Gonzalez and L. Fridman, Optimal and robust control for linear state-delay systems, Journal of The Franklin Institute, 344 (2007), 830-845. doi: 10.1016/j.jfranklin.2006.10.002. [10] F. Blanchini and S. Miani, "Set-Theoretic Methods in Control," Birkhäuser, Boston, 2008. [11] S. Boyd, E. Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory," SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777. [12] P. Chen, H. Qin and J. Huang, Local stabilization of a class of nonlinear systems by dynamic output feedback, Automatica, 37 (2001), 969-981. doi: 10.1016/S0005-1098(01)00047-4. [13] D. F. Coutinho, A. Trofino and K. A. Barbosa, Robust linear dynamic output feedback controllers for a class of nonlinear systems, in "Proceedings of the 42 IEEE Conference on Decision and Control," Maui, USA, (2003), 374-379. [14] M. A. Dahleh, J. B. Pearson and J. Boyd, Optimal rejection of persistent disturbances, robust stability, and mixed sensitivity minimization, IEEE Transactions on Automatic Control, 33 (1988), 722-731. doi: 10.1109/9.1288. [15] G. J. Duncan and F. C. Schweppe, Control of linear dynamic systems with set constrained disturbances, IEEE Transactions on Automatic Control, AC-16 (1971), 411-423. [16] E. Fridman and Y. Orlov, On stability of linear parabolic distributed parameter systems with time-varying delays, in "Proceedings of the 46th Conference on Decision and Control," New Orlean, USA, (2007), 1597-1602. doi: 10.1109/CDC.2007.4434196. [17] E. Fridman, A refined input delay approach to sampled-data control, Automatica, 46 (2010), 421-427. doi: 10.1016/j.automatica.2009.11.017. [18] L. El Ghaoui and S. Niculescu, "Advances in Linear Matrix Inequalities in Control," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719833. [19] W. Haddad and V. Chellaboina, "Nonlinear Dynamical Systems and Control," Princeton University Press, Princeton, 2008. [20] A. Isidori, A. R. Teel and L. Praly, A note on the problem of semiglobal practical stabilization of uncertain nonlinear systems via dynamic output feedback, System and Control Letters, 39 (2000), 165-171. doi: 10.1016/S0167-6911(99)00083-3. [21] I. Karafyllis and Z.-P. Jiang, "Stability and Stabilization of Nonlinear Systems," Springer, London, 2011. doi: 10.1007/978-0-85729-513-2. [22] C. T. Kelley, L. Qi, X. Tong and H. Yin, Finding a stable solution of a system of nonlinear equations arising from dynamic systems, Journal of Industrial and Management Optimization, 7 (2011), 497-521. doi: 10.3934/jimo.2011.7.497. [23] H. K. Khalil, "Nonlinear Systems," Prentice Hall, Upper Saddle River, 2002. doi: 10.1007/s11071-008-9349-z. [24] A. B. Kurzhanski and P. Varaiya, Ellipsoidal techniques for reachability under state constraints, SIAM Journal on Control and Optimization, 45 (2006), 1369-1394. doi: 10.1137/S0363012903437605. [25] A. B. Kurzhanski and V. M. Veliov, "Modeling Techniques and Uncertain Systems," Birkhäuser, New York, 1994. [26] W. Lin and C. Qian, Semi-global robust stabilization of MIMO nonlinear systems by partial state and dynamic output feedback, Automatica, 37 (2001), 1093-1101. doi: 10.1016/S0005-1098(01)00056-5. [27] M. Mera, A. Poznyak and V. Azhmyakov, On the robust control design for a class of continuous-time dynamical systems with a sample-data output, in "Proceedings of the 18th IFAC World Congress," Milano, Ialy, (2011), 5819-5824. [28] A. N. Michel, L. Hou and D. Liu, "Stability of Dynamical Systems," Birkhäuser, New York, 2007. [29] P. Naghshtabrizi, J. P. Hespanha, and A. R. Teel, Exponential stability of impulsive systems with application to uncertain sampled-data systems, Systems and Control Letters, 57 (2008), 378-385. doi: 10.1016/j.sysconle.2007.10.009. [30] E. Polak, "Optimization," Springer, New York, 1997. doi: 10.1007/978-1-4612-0663-7. [31] B. T. Polyak, S. A. Nazin, C. Durieu and E. Walter, Ellipsoidal parameter or state estimation under model uncertainty, Automatica, 40 (2004), 1171-1179. doi: 10.1016/j.automatica.2004.02.014. [32] B. T. Polyak and M. V. Topunov, Suppression of bounded exogeneous disturbances: Output control, Automation and Remote Control, 69 (2008), 801-818. doi: 10.1134/S000511790805007X. [33] A. Polyakov and A. Poznyak, Lyapunov function design for finite-time convergence analysis: Twisting controller for second-order sliding mode realization, Automatica, 45 (2009), 444-448. doi: 10.1016/j.automatica.2008.07.013. [34] A. Poznyak, "Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques," Elsevier, Amsterdam, 2008. [35] A. Poznyak, V. Azhmyakov and M. Mera, Practical output feedback stabilisation for a class of continuous-time dynamic systems under sample-data outputs, International Journal of Control, 84 (2011), 1408-1416. doi: 10.1080/00207179.2011.603097. [36] E. Sontag, "Mathematical Control Theory," Springer, New York, 1998. [37] E. D. Sontag, Further facts about input to state stabilization, IEEE Transactions on Automatic Control, AC-35 (1990), 473-476. doi: 10.1109/9.52307. [38] A. R. Teel, D. Nesic and P. V. Kokotovic, A note on input-to-state stability of sampled-data nonlinear systems, in "Proceedings of the 37th IEEE Conference on Decision and Control," Tampa, USA, (1998), 2473-2478. doi: 10.1109/CDC.1998.757793. [39] A. R. Teel, L. Moreau and D. Nesic, A note on the robustness of input-to-state stability, in "Proceedings of the 40th IEEE Conference on Decision and Control," Orlando, USA, (2001), 875-880. [40] E. D. Yakubovich, Solution of the optimal control problem for the linear discrete systems, Automation and Remote Control, 36 (1976), 1447-1453. [41] V. I. Zubov, "Mathematical Methods for the Study of Automatic Control Systems," Pergamon Press, New York, 1962.

show all references

##### References:
 [1] V. Azhmyakov, Stability of differential inclusions: A computational approach, Mathematical Problems in Engineering, 2006 (2006), 1-15. doi: 10.1155/MPE/2006/17837. [2] V. Azhmyakov, V. G. Boltyanski and A. Poznyak, Optimal control of impulsive hybrid systems, Nonlinear Analysis: Hybrid Systems, 2 (2008), 1089-1097. doi: 10.1016/j.nahs.2008.09.003. [3] V. Azhmyakov, An approach to controlled mechanical systems based on the multiobjective optimization technique, Journal of Industrial and Management Optimization, 4 (2008), 697-712. doi: 10.3934/jimo.2008.4.697. [4] V. Azhmyakov, A gradient-type algorithm for a class of optimal control processes governed by hybrid dynamical systems, IMA Journal of Mathematical Control and Information, 28 (2011), 291-307. doi: 10.1093/imamci/dnr010. [5] V. Azhmyakov, On the geometric aspects of the invariant ellipsoid method: Application to the robust control design, in "Proceedings of the 50th Conference on Decision and Control and European Control Conference," Orlando, USA, (2011), 1353-1358. doi: 10.1109/CDC.2011.6161180. [6] V. Azhmyakov, M. V. Basin and J. Raisch, Proximal point based approach to optimal control of affine switched systems, Discrete Event Dynamic Systems, 22 (2012), 61-81. doi: 10.1007/s10626-011-0109-8. [7] A. E. Barabanov and O. N. Granichin, Optimal controller for linear plants with bounded noise, Automation and Remote Control, 45 (1984), 39-46. [8] M. Basin and D. Calderon-Alvarez, Optimal filtering over linear observations with unknown parameters, Journal of The Franklin Institute, 347 (2010), 988-1000. doi: 10.1016/j.jfranklin.2010.01.006. [9] M. Basin, J. Rodriguez-Gonzalez and L. Fridman, Optimal and robust control for linear state-delay systems, Journal of The Franklin Institute, 344 (2007), 830-845. doi: 10.1016/j.jfranklin.2006.10.002. [10] F. Blanchini and S. Miani, "Set-Theoretic Methods in Control," Birkhäuser, Boston, 2008. [11] S. Boyd, E. Ghaoui, E. Feron and V. Balakrishnan, "Linear Matrix Inequalities in System and Control Theory," SIAM, Philadelphia, 1994. doi: 10.1137/1.9781611970777. [12] P. Chen, H. Qin and J. Huang, Local stabilization of a class of nonlinear systems by dynamic output feedback, Automatica, 37 (2001), 969-981. doi: 10.1016/S0005-1098(01)00047-4. [13] D. F. Coutinho, A. Trofino and K. A. Barbosa, Robust linear dynamic output feedback controllers for a class of nonlinear systems, in "Proceedings of the 42 IEEE Conference on Decision and Control," Maui, USA, (2003), 374-379. [14] M. A. Dahleh, J. B. Pearson and J. Boyd, Optimal rejection of persistent disturbances, robust stability, and mixed sensitivity minimization, IEEE Transactions on Automatic Control, 33 (1988), 722-731. doi: 10.1109/9.1288. [15] G. J. Duncan and F. C. Schweppe, Control of linear dynamic systems with set constrained disturbances, IEEE Transactions on Automatic Control, AC-16 (1971), 411-423. [16] E. Fridman and Y. Orlov, On stability of linear parabolic distributed parameter systems with time-varying delays, in "Proceedings of the 46th Conference on Decision and Control," New Orlean, USA, (2007), 1597-1602. doi: 10.1109/CDC.2007.4434196. [17] E. Fridman, A refined input delay approach to sampled-data control, Automatica, 46 (2010), 421-427. doi: 10.1016/j.automatica.2009.11.017. [18] L. El Ghaoui and S. Niculescu, "Advances in Linear Matrix Inequalities in Control," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719833. [19] W. Haddad and V. Chellaboina, "Nonlinear Dynamical Systems and Control," Princeton University Press, Princeton, 2008. [20] A. Isidori, A. R. Teel and L. Praly, A note on the problem of semiglobal practical stabilization of uncertain nonlinear systems via dynamic output feedback, System and Control Letters, 39 (2000), 165-171. doi: 10.1016/S0167-6911(99)00083-3. [21] I. Karafyllis and Z.-P. Jiang, "Stability and Stabilization of Nonlinear Systems," Springer, London, 2011. doi: 10.1007/978-0-85729-513-2. [22] C. T. Kelley, L. Qi, X. Tong and H. Yin, Finding a stable solution of a system of nonlinear equations arising from dynamic systems, Journal of Industrial and Management Optimization, 7 (2011), 497-521. doi: 10.3934/jimo.2011.7.497. [23] H. K. Khalil, "Nonlinear Systems," Prentice Hall, Upper Saddle River, 2002. doi: 10.1007/s11071-008-9349-z. [24] A. B. Kurzhanski and P. Varaiya, Ellipsoidal techniques for reachability under state constraints, SIAM Journal on Control and Optimization, 45 (2006), 1369-1394. doi: 10.1137/S0363012903437605. [25] A. B. Kurzhanski and V. M. Veliov, "Modeling Techniques and Uncertain Systems," Birkhäuser, New York, 1994. [26] W. Lin and C. Qian, Semi-global robust stabilization of MIMO nonlinear systems by partial state and dynamic output feedback, Automatica, 37 (2001), 1093-1101. doi: 10.1016/S0005-1098(01)00056-5. [27] M. Mera, A. Poznyak and V. Azhmyakov, On the robust control design for a class of continuous-time dynamical systems with a sample-data output, in "Proceedings of the 18th IFAC World Congress," Milano, Ialy, (2011), 5819-5824. [28] A. N. Michel, L. Hou and D. Liu, "Stability of Dynamical Systems," Birkhäuser, New York, 2007. [29] P. Naghshtabrizi, J. P. Hespanha, and A. R. Teel, Exponential stability of impulsive systems with application to uncertain sampled-data systems, Systems and Control Letters, 57 (2008), 378-385. doi: 10.1016/j.sysconle.2007.10.009. [30] E. Polak, "Optimization," Springer, New York, 1997. doi: 10.1007/978-1-4612-0663-7. [31] B. T. Polyak, S. A. Nazin, C. Durieu and E. Walter, Ellipsoidal parameter or state estimation under model uncertainty, Automatica, 40 (2004), 1171-1179. doi: 10.1016/j.automatica.2004.02.014. [32] B. T. Polyak and M. V. Topunov, Suppression of bounded exogeneous disturbances: Output control, Automation and Remote Control, 69 (2008), 801-818. doi: 10.1134/S000511790805007X. [33] A. Polyakov and A. Poznyak, Lyapunov function design for finite-time convergence analysis: Twisting controller for second-order sliding mode realization, Automatica, 45 (2009), 444-448. doi: 10.1016/j.automatica.2008.07.013. [34] A. Poznyak, "Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques," Elsevier, Amsterdam, 2008. [35] A. Poznyak, V. Azhmyakov and M. Mera, Practical output feedback stabilisation for a class of continuous-time dynamic systems under sample-data outputs, International Journal of Control, 84 (2011), 1408-1416. doi: 10.1080/00207179.2011.603097. [36] E. Sontag, "Mathematical Control Theory," Springer, New York, 1998. [37] E. D. Sontag, Further facts about input to state stabilization, IEEE Transactions on Automatic Control, AC-35 (1990), 473-476. doi: 10.1109/9.52307. [38] A. R. Teel, D. Nesic and P. V. Kokotovic, A note on input-to-state stability of sampled-data nonlinear systems, in "Proceedings of the 37th IEEE Conference on Decision and Control," Tampa, USA, (1998), 2473-2478. doi: 10.1109/CDC.1998.757793. [39] A. R. Teel, L. Moreau and D. Nesic, A note on the robustness of input-to-state stability, in "Proceedings of the 40th IEEE Conference on Decision and Control," Orlando, USA, (2001), 875-880. [40] E. D. Yakubovich, Solution of the optimal control problem for the linear discrete systems, Automation and Remote Control, 36 (1976), 1447-1453. [41] V. I. Zubov, "Mathematical Methods for the Study of Automatic Control Systems," Pergamon Press, New York, 1962.
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