# American Institute of Mathematical Sciences

January  2011, 15(1): 197-215. doi: 10.3934/dcdsb.2011.15.197

## Approximate tracking of periodic references in a class of bilinear systems via stable inversion

 1 Department of Applied Mathematics IV, Universitat Politècnica de Catalunya, Av. Víctor Balaguer, s/n, 08800 Vilanova i la Geltrú, Spain 2 Department of Applied Mathematics I, Universitat Politècnica de Catalunya, Av. Diagonal, 647, 08028 Barcelona, Spain

Received  December 2009 Revised  August 2010 Published  October 2010

This article deals with the tracking control of periodic references in single-input single-output bilinear systems using a stable inversion-based approach. Assuming solvability of the exact tracking problem and asymptotic stability of the nominal error system, the study focuses on the output behavior when the control scheme uses a periodic approximation of the nominal feedforward input signal $u_d$. The investigation shows that this results in a periodic, asymptotically stable output; moreover, a sequence of periodic control inputs $u_n$ uniformly convergent to $u_d$ produce a sequence of output responses that, in turn, converge uniformly to the output reference. It is also shown that, for a special class of bilinear systems, the internal dynamics equation can be approximately solved by an iterative procedure that provides of closed-form analytic expressions uniformly convergent to its exact solution. Then, robustness in the face of bounded parametric disturbances/uncertainties is achievable through dynamic compensation. The theoretical analysis is applied to nonminimum phase switched power converters.
Citation: Josep M. Olm, Xavier Ros-Oton. Approximate tracking of periodic references in a class of bilinear systems via stable inversion. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 197-215. doi: 10.3934/dcdsb.2011.15.197
##### References:
 [1] F. Amato, C. Cosentino, A. S. Fiorillo and A. Merola, Stabilization of bilinear systems via linear state-feedback control, IEEE Trans. Circ. Syst.-II, 56 (2009), 76-80. [2] R. O. Cáceres and I. Barbi, A boost DC-AC converter: Analysis, design and experimentation, IEEE Trans. Power Electronics, 14 (1999), 134-141. doi: doi:10.1109/63.737601. [3] M. Carpita and M. Marchesoni, Experimental study of a power conditioning system using sliding mode control, IEEE Trans. Power Electronics, 11 (1996), 731-742. doi: doi:10.1109/63.535405. [4] D. Cortés, Jq. Álvarez, J. Álvarez and A. Fradkov, Tracking control of the boost converter, IEEE Proceedings Control Theory Applications, 151 (2004), 218-224. doi: doi:10.1049/ip-cta:20040203. [5] S. Devasia, D. Chen and B. Paden, Nonlinear inversion-based output tracking, IEEE Trans. Automatic Control, 41 (1996), 930-942. doi: doi:10.1109/9.508898. [6] D. L. Elliott, "Bilinear Control Systems. Matrices in Action," Springer, 2009. [7] M. Farkas, "Periodic Motions," Springer-Verlag, 1994. [8] E. Fossas and J. M. Olm, Asymptotic tracking in DC-to-DC nonlinear power converters, Discrete Continuous Dynam. Systems - B, 2 (2002), 295-307. [9] E. Fossas and J. M. Olm, Galerkin method and approximate tracking in a nonminimum phase bilinear system, Discrete Continuous Dynam. Systems - B, 7 (2007), 53-76. [10] E. Fossas and J. M. Olm, A functional iterative approach to the tracking control of nonminimum phase switched power converters, Math. Control Signals Syst., 21 (2009), 203-227. doi: doi:10.1007/s00498-009-0044-5. [11] E. Fossas, J. M. Olm, A. Zinober and Y. Shtessel, Galerkin-based sliding mode tracking control of nonminimum phase DC-to-DC power converters, Internat. J. Robust Nonlinear Control, 17 (2007), 587-604. doi: doi:10.1002/rnc.1136. [12] H. K. Khalil, "Nonlinear Systems," 3rd edition, Prentice Hall, 2002. [13] R. R. Mohler, "Nonlinear Systems, vol. 2: Applications to Bilinear Control," Prentice Hall, 1991. [14] J. M. Olm, "Asymptotic Tracking with DC-to-DC Bilinear Power Converters," Ph. D thesis, Universitat Politècnica de Catalunya, 2004. [15] J. M. Olm, X. Ros-Oton and Y. B. Shtessel, Stable inversion of Abel equations: application to tracking control in DC-DC nonminimum phase boost converters, Automatica, (2010), in press. doi: doi:10.1109/CDC.2007.4434276. [16] A. Pavlov and K. Y. Pettersen, Stable inversion of non-minimum phase nonlinear systems: A convergent systems approach, in "Proceedings of the 46th IEEE Conference on Decision and Control," (2007), 3995-4000. [17] A. D. Polyanin, "Handbook of Exact Solutions for Ordinary Differential Equations," 2nd edition, Chapman & Hall/CRC, Boca Raton, 2003. [18] S. Sastry, "Nonlinear Systems. Analysis, Stability and Control," Springer-Verlag, 1999. [19] Y. Shtessel, A. Zinober and I. Shkolnikov, Sliding mode control of boost and buck-boost power converters control using method of stable system centre, Automatica, 39 (2003), 1061-1067. doi: doi:10.1016/S0005-1098(03)00068-2. [20] H. Sira-Ramírez, Sliding motions in bilinear switched networks, IEEE Trans. Circ. Syst., 34 (1987), 1359-1390. [21] H. Sira-Ramírez, DC-to-AC power conversion on a 'boost' converter, Internat. J. Robust Nonlinear Control, 11 (2001), 589-600. doi: doi:10.1002/rnc.575. [22] H. Sira-Ramírez, M. Spinetti-Rivera and E. Fossas, An algebraic parameter estimation approach to the adaptive observer-controller based regulation of the boost converter, in "Proceedings of the IEEE Int. Symp. Industrial Electronics," (2007), 3367-3372. [23] E. D. Sontag, Input to state stability: Basic concepts and results, in "Nonlinear and Optimal Control Theory" (eds. P. Nistri and G. Stefani), Springer-Verlag, (2007), 163-220. [24] E. D. Sontag and Y. Wang, New characterizations of input-to-state stability, IEEE Trans. Automatic Control, 41 (1996), 1283-1294. doi: doi:10.1109/9.536498.

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##### References:
 [1] F. Amato, C. Cosentino, A. S. Fiorillo and A. Merola, Stabilization of bilinear systems via linear state-feedback control, IEEE Trans. Circ. Syst.-II, 56 (2009), 76-80. [2] R. O. Cáceres and I. Barbi, A boost DC-AC converter: Analysis, design and experimentation, IEEE Trans. Power Electronics, 14 (1999), 134-141. doi: doi:10.1109/63.737601. [3] M. Carpita and M. Marchesoni, Experimental study of a power conditioning system using sliding mode control, IEEE Trans. Power Electronics, 11 (1996), 731-742. doi: doi:10.1109/63.535405. [4] D. Cortés, Jq. Álvarez, J. Álvarez and A. Fradkov, Tracking control of the boost converter, IEEE Proceedings Control Theory Applications, 151 (2004), 218-224. doi: doi:10.1049/ip-cta:20040203. [5] S. Devasia, D. Chen and B. Paden, Nonlinear inversion-based output tracking, IEEE Trans. Automatic Control, 41 (1996), 930-942. doi: doi:10.1109/9.508898. [6] D. L. Elliott, "Bilinear Control Systems. Matrices in Action," Springer, 2009. [7] M. Farkas, "Periodic Motions," Springer-Verlag, 1994. [8] E. Fossas and J. M. Olm, Asymptotic tracking in DC-to-DC nonlinear power converters, Discrete Continuous Dynam. Systems - B, 2 (2002), 295-307. [9] E. Fossas and J. M. Olm, Galerkin method and approximate tracking in a nonminimum phase bilinear system, Discrete Continuous Dynam. Systems - B, 7 (2007), 53-76. [10] E. Fossas and J. M. Olm, A functional iterative approach to the tracking control of nonminimum phase switched power converters, Math. Control Signals Syst., 21 (2009), 203-227. doi: doi:10.1007/s00498-009-0044-5. [11] E. Fossas, J. M. Olm, A. Zinober and Y. Shtessel, Galerkin-based sliding mode tracking control of nonminimum phase DC-to-DC power converters, Internat. J. Robust Nonlinear Control, 17 (2007), 587-604. doi: doi:10.1002/rnc.1136. [12] H. K. Khalil, "Nonlinear Systems," 3rd edition, Prentice Hall, 2002. [13] R. R. Mohler, "Nonlinear Systems, vol. 2: Applications to Bilinear Control," Prentice Hall, 1991. [14] J. M. Olm, "Asymptotic Tracking with DC-to-DC Bilinear Power Converters," Ph. D thesis, Universitat Politècnica de Catalunya, 2004. [15] J. M. Olm, X. Ros-Oton and Y. B. Shtessel, Stable inversion of Abel equations: application to tracking control in DC-DC nonminimum phase boost converters, Automatica, (2010), in press. doi: doi:10.1109/CDC.2007.4434276. [16] A. Pavlov and K. Y. Pettersen, Stable inversion of non-minimum phase nonlinear systems: A convergent systems approach, in "Proceedings of the 46th IEEE Conference on Decision and Control," (2007), 3995-4000. [17] A. D. Polyanin, "Handbook of Exact Solutions for Ordinary Differential Equations," 2nd edition, Chapman & Hall/CRC, Boca Raton, 2003. [18] S. Sastry, "Nonlinear Systems. Analysis, Stability and Control," Springer-Verlag, 1999. [19] Y. Shtessel, A. Zinober and I. Shkolnikov, Sliding mode control of boost and buck-boost power converters control using method of stable system centre, Automatica, 39 (2003), 1061-1067. doi: doi:10.1016/S0005-1098(03)00068-2. [20] H. Sira-Ramírez, Sliding motions in bilinear switched networks, IEEE Trans. Circ. Syst., 34 (1987), 1359-1390. [21] H. Sira-Ramírez, DC-to-AC power conversion on a 'boost' converter, Internat. J. Robust Nonlinear Control, 11 (2001), 589-600. doi: doi:10.1002/rnc.575. [22] H. Sira-Ramírez, M. Spinetti-Rivera and E. Fossas, An algebraic parameter estimation approach to the adaptive observer-controller based regulation of the boost converter, in "Proceedings of the IEEE Int. Symp. Industrial Electronics," (2007), 3367-3372. [23] E. D. Sontag, Input to state stability: Basic concepts and results, in "Nonlinear and Optimal Control Theory" (eds. P. Nistri and G. Stefani), Springer-Verlag, (2007), 163-220. [24] E. D. Sontag and Y. Wang, New characterizations of input-to-state stability, IEEE Trans. Automatic Control, 41 (1996), 1283-1294. doi: doi:10.1109/9.536498.
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