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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

Vadim Azhmyakov 1, , Alex Poznyak 1, and  Omar Gonzalez 1,

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.
Keywords: the invariant ellipsoid method, Affine dynamic systems, LMI-based optimization..
Mathematics Subject Classification: Primary: 93C10, 37N40; Secondary: 93C4.
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. E. Barabanov and O. N. Granichin, Optimal controller for linear plants with bounded noise, Automation and Remote Control, 45 (1984), 39-46.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

F. Blanchini and S. Miani, "Set-Theoretic Methods in Control," Birkhäuser, Boston, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

L. El Ghaoui and S. Niculescu, "Advances in Linear Matrix Inequalities in Control," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719833.  Google Scholar

[19]

W. Haddad and V. Chellaboina, "Nonlinear Dynamical Systems and Control," Princeton University Press, Princeton, 2008.  Google Scholar

[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.  Google Scholar

[21]

I. Karafyllis and Z.-P. Jiang, "Stability and Stabilization of Nonlinear Systems," Springer, London, 2011. doi: 10.1007/978-0-85729-513-2.  Google Scholar

[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.  Google Scholar

[23]

H. K. Khalil, "Nonlinear Systems," Prentice Hall, Upper Saddle River, 2002. doi: 10.1007/s11071-008-9349-z.  Google Scholar

[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.  Google Scholar

[25]

A. B. Kurzhanski and V. M. Veliov, "Modeling Techniques and Uncertain Systems," Birkhäuser, New York, 1994.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[28]

A. N. Michel, L. Hou and D. Liu, "Stability of Dynamical Systems," Birkhäuser, New York, 2007.  Google Scholar

[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.  Google Scholar

[30]

E. Polak, "Optimization," Springer, New York, 1997. doi: 10.1007/978-1-4612-0663-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

A. Poznyak, "Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques," Elsevier, Amsterdam, 2008.  Google Scholar

[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.  Google Scholar

[36]

E. Sontag, "Mathematical Control Theory," Springer, New York, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[40]

E. D. Yakubovich, Solution of the optimal control problem for the linear discrete systems, Automation and Remote Control, 36 (1976), 1447-1453.  Google Scholar

[41]

V. I. Zubov, "Mathematical Methods for the Study of Automatic Control Systems," Pergamon Press, New York, 1962.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

A. E. Barabanov and O. N. Granichin, Optimal controller for linear plants with bounded noise, Automation and Remote Control, 45 (1984), 39-46.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

F. Blanchini and S. Miani, "Set-Theoretic Methods in Control," Birkhäuser, Boston, 2008.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

L. El Ghaoui and S. Niculescu, "Advances in Linear Matrix Inequalities in Control," SIAM, Philadelphia, 2000. doi: 10.1137/1.9780898719833.  Google Scholar

[19]

W. Haddad and V. Chellaboina, "Nonlinear Dynamical Systems and Control," Princeton University Press, Princeton, 2008.  Google Scholar

[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.  Google Scholar

[21]

I. Karafyllis and Z.-P. Jiang, "Stability and Stabilization of Nonlinear Systems," Springer, London, 2011. doi: 10.1007/978-0-85729-513-2.  Google Scholar

[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.  Google Scholar

[23]

H. K. Khalil, "Nonlinear Systems," Prentice Hall, Upper Saddle River, 2002. doi: 10.1007/s11071-008-9349-z.  Google Scholar

[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.  Google Scholar

[25]

A. B. Kurzhanski and V. M. Veliov, "Modeling Techniques and Uncertain Systems," Birkhäuser, New York, 1994.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[28]

A. N. Michel, L. Hou and D. Liu, "Stability of Dynamical Systems," Birkhäuser, New York, 2007.  Google Scholar

[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.  Google Scholar

[30]

E. Polak, "Optimization," Springer, New York, 1997. doi: 10.1007/978-1-4612-0663-7.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

A. Poznyak, "Advanced Mathematical Tools for Automatic Control Engineers: Deterministic Techniques," Elsevier, Amsterdam, 2008.  Google Scholar

[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.  Google Scholar

[36]

E. Sontag, "Mathematical Control Theory," Springer, New York, 1998.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[40]

E. D. Yakubovich, Solution of the optimal control problem for the linear discrete systems, Automation and Remote Control, 36 (1976), 1447-1453.  Google Scholar

[41]

V. I. Zubov, "Mathematical Methods for the Study of Automatic Control Systems," Pergamon Press, New York, 1962.  Google Scholar

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