Optimal Risk-Sensitive Filtering for System Stochastic of Second and Third Degree

ABSTRACT

The risk-sensitive filtering design problem with respect to the exponential mean-square cost criterion is con-sidered for stochastic Gaussian systems with polynomial of second and third degree drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed-form optimal fil-tering equations are obtained using quadratic value functions as solutions to the corresponding Focker- Plank-Kolmogorov equation. The performance of the obtained risk-sensitive filtering equations for stochastic polynomial systems of second and third degree is verified in a numerical example against the optimal po-lynomial filtering equations (and extended Kalman-Bucy for system polynomial of second degree), through comparing the exponential mean-square cost criterion values. The simulation results reveal strong advan-tages in favor of the designed risk-sensitive equations for some values of the intensity parameters.

The risk-sensitive filtering design problem with respect to the exponential mean-square cost criterion is con-sidered for stochastic Gaussian systems with polynomial of second and third degree drift terms and intensity parameters multiplying diffusion terms in the state and observations equations. The closed-form optimal fil-tering equations are obtained using quadratic value functions as solutions to the corresponding Focker- Plank-Kolmogorov equation. The performance of the obtained risk-sensitive filtering equations for stochastic polynomial systems of second and third degree is verified in a numerical example against the optimal po-lynomial filtering equations (and extended Kalman-Bucy for system polynomial of second degree), through comparing the exponential mean-square cost criterion values. The simulation results reveal strong advan-tages in favor of the designed risk-sensitive equations for some values of the intensity parameters.

Cite this paper

nullM. Alcorta-Garcia, S. Rostro and M. Torres, "Optimal Risk-Sensitive Filtering for System Stochastic of Second and Third Degree,"*Intelligent Control and Automation*, Vol. 2 No. 1, 2011, pp. 47-56. doi: 10.4236/ica.2011.21006.

nullM. Alcorta-Garcia, S. Rostro and M. Torres, "Optimal Risk-Sensitive Filtering for System Stochastic of Second and Third Degree,"

References

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[2] M. V. Basin and M. A. Alcorta-García, “Optimal Filtering for Bilinear Systems and Its Application to Terpo- lymerization Process State,” Proceedings of IFAC 13th Symposium on System Identification, 27-29 August 2003, Rotterdam, 2003, pp. 467-472.

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[8] M. A. Alcorta-García, M. V. Basin, S. G. Anguiano and J. J. Maldonado, “Sub-optimal Risk-Sensitive Filtering for Third Degree Polynomial Stochastic Systems,” IEEE Control Applications & Intelligent Control, St. Petersburg, 8-10 July 2009, pp. 285-289.

[9] H. K. Khalil, “Nonlinear Systems,” 3rd Edition, Prentice Hall, Upper Saddle River, 2002.

[10] W. H. Fleming and W. M. McEneaney, “Robust Limits of Risk Sensitive Nonlinear Filters,” Mathematics of Control, Signals, and Systems, Vol. 14, No. 1, 2001, pp. 109-142. doi:10.1007/PL00009879

[11] W. M. McEneaney, “Max-Plus Eigenvector Representations for Solution of Nonlinear H1 Problems: Error Ana- lysis,” SIAM Journal on Control and Optimization, Vol. 43, 2004, pp. 379-412. doi:10.1137/S0363012902414688

[12] D. L. Lukes, “Optimal Regulation of Nonlinear Dynamic Systems,” SIAM Journal on Control and Optimization, Vol. 7, No. 1, 1969, pp. 75-100. doi:10.1137/0307007

[13] T. Yoshida and K. Loparo, “Quadratic Regulatory Theory for Analitic Nonlinear Systems with Additive Controls,” Automatica, Vol. 25, No. 4, 1989, pp. 531-544. doi:10.10 16/0005-1098(89)90096-4

[14] N. N. Ural’ceva, O. A. Ladyzenskaja and V. A. Solonnikov, “Newblock Linear and Quasi-Linear Equations for Parabolic Type,” American Mathematical Society, Provi- dence, 1968.

[15] H. Sira-Ramírez, R. Márquez, F. Rivas-Echeverría and O. Llanes-Santiago, “Automática y Robótica: Control de Sistemas no Lineales,” Pearson, Madrid, 2005.

[16] A. H. Jazwinski, “Stochastic Processes and Filtering Theory,” Dover Publications, Inc., New York, 2007.

[1] M. V. Basin and M. A. Alcorta-García, “Optimal Filtering and Control for Third Degree Polynomial Systems,” Dynamics of Continuous Discrete and Impulsive Systems, Vol. 10, 2003, pp. 663-680.

[2] M. V. Basin and M. A. Alcorta-García, “Optimal Filtering for Bilinear Systems and Its Application to Terpo- lymerization Process State,” Proceedings of IFAC 13th Symposium on System Identification, 27-29 August 2003, Rotterdam, 2003, pp. 467-472.

[3] F. L. Lewis, “Applied Optimal Control and Estimation,” Prentice Hall PTR, Upper Saddle River, 1992.

[4] V. S. Pugachev and I. N. Sinitsyn, “Stochastic Systems Theory and Applications,” World Scientific, Singapore, 2001.

[5] S. S.-T. Yau, “Finite-Dimensional Filters with Nonlinear Drift. I: A Class of Filters including both Kalman-Bucy and Benes Filters,” Journal of Mathematical Systems, Estimation, and Control, Vol. 4, 1994, pp. 181-203.

[6] R. E. Mortensen, “Maximum Likelihood Recursive Non- linear Filtering,” Journal of Optimization Theory and Applications, Vol. 2, No. 6, 1968, pp. 386-394. doi:10.1007/ BF00925744

[7] W. M. McEneaney, “Robust H∞ Filtering for Nonlinear Systems,” Systems and Control Letters, Vol. 33, No. 5, 1998, pp. 315-325. doi:10.1016/S0167-6911(97)00124-2

[8] M. A. Alcorta-García, M. V. Basin, S. G. Anguiano and J. J. Maldonado, “Sub-optimal Risk-Sensitive Filtering for Third Degree Polynomial Stochastic Systems,” IEEE Control Applications & Intelligent Control, St. Petersburg, 8-10 July 2009, pp. 285-289.

[9] H. K. Khalil, “Nonlinear Systems,” 3rd Edition, Prentice Hall, Upper Saddle River, 2002.

[10] W. H. Fleming and W. M. McEneaney, “Robust Limits of Risk Sensitive Nonlinear Filters,” Mathematics of Control, Signals, and Systems, Vol. 14, No. 1, 2001, pp. 109-142. doi:10.1007/PL00009879

[11] W. M. McEneaney, “Max-Plus Eigenvector Representations for Solution of Nonlinear H1 Problems: Error Ana- lysis,” SIAM Journal on Control and Optimization, Vol. 43, 2004, pp. 379-412. doi:10.1137/S0363012902414688

[12] D. L. Lukes, “Optimal Regulation of Nonlinear Dynamic Systems,” SIAM Journal on Control and Optimization, Vol. 7, No. 1, 1969, pp. 75-100. doi:10.1137/0307007

[13] T. Yoshida and K. Loparo, “Quadratic Regulatory Theory for Analitic Nonlinear Systems with Additive Controls,” Automatica, Vol. 25, No. 4, 1989, pp. 531-544. doi:10.10 16/0005-1098(89)90096-4

[14] N. N. Ural’ceva, O. A. Ladyzenskaja and V. A. Solonnikov, “Newblock Linear and Quasi-Linear Equations for Parabolic Type,” American Mathematical Society, Provi- dence, 1968.

[15] H. Sira-Ramírez, R. Márquez, F. Rivas-Echeverría and O. Llanes-Santiago, “Automática y Robótica: Control de Sistemas no Lineales,” Pearson, Madrid, 2005.

[16] A. H. Jazwinski, “Stochastic Processes and Filtering Theory,” Dover Publications, Inc., New York, 2007.