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Year 2018, Volume: 8 Issue: 1, 71 - 82, 01.06.2018

Abstract

References

  • Kalman, R. E., (1960), A new approach to linear filtering and prediction problems, Transactions ASME, Ser. D (Journal of Basic Engineering), 82, pp. 35-45.
  • Kalman, R. E., Bucy, R. S., (1961), New results in linear filtering and prediction theory, Transactions ASME, Ser. D (Journal of Basic Engineering), 83, pp. 95-108.
  • Bensoussan, A., (1992), Stochastic control of partially observable systems, Cambridge University Press, Cambridge.
  • Curtain, R. F., Pritchard, A. J., (1978), Infinite dimensional linear systems theory, Lecture Notes in Control and Information Sciences, Vol. 8, Springer-Verlag, Berlin. Wonham, W. M., (1968), On the separation theorem of stochastic control, SIAM J. Control, 6, pp. 326.
  • Grassides, J. L., Junkins, J. L., (2004), Optimal estimation of dynamic systems, Chapman and Hall/CRC, Boca Raton.
  • Fleming, W. M., Rishel, R. W., (1075), Deterministic and stochastic optimal control, Springer-Verlag, New York.
  • Ito, K., (1944), Stochastic integral, Proceedings of the Imperial Academy, Tokyo, 20, pp. 519-524.
  • Ito, K., (1951), On stochastic differential equations, Memoirs of the American Mathematical Siciety, , pp. 645-668.
  • Bucy, R. S., Joseph, P. D., (1968), Filtering for stochastic processes with application to guidance, Wiley, New York.
  • Kushner, H. J., (1990), Weak convergence methods and singularly perturbed control and filtering prob- lems, Birkh¨auser, Boston.
  • Kushner, H. J., Runggaldier, W. J., (1987), Nearly optimal state feedback controls for stochastic systems with wideband noise disturbances, SIAM Journal on Control and Optimization, 25, pp. 298
  • Kushner, H. J., Runggaldier, W. J., (1987), Filtering and control for wide bandwidth noise driven systems, IEEE Transactions on Automatic Control, 32AC, pp. 123-133.
  • Kushner, H. J., Ramachandran, K. M., (1988), Nearly optimal singular controls for sideband noise driven systems. SIAM Journal on Control and Optimization, 26, pp. 569-591.
  • Liptser, R. S., Runggaldier, W. J., Taksar M., (2000), Diffusion approximation and optimal stochastic control. Theory of Probability and Applications, 44, pp. 669-698.
  • Hu, H., (2000), Speech signal processing, Harbin Institute of Technology Press, Heilongjiang.
  • Wang, W., Liu, D., Wang, X., (2010), An improved wide band noise signal analysis method, Computer and Information Science, 3, pp. 76-80.
  • Bashirov, A. E., (1988), On linear filtering under dependent wide band noises, Stochastics, 23, pp. 437.
  • Bashirov, A. E., (1993), Control and filtering for wide band noise driven linear systems, AIAA J.Guidance Control and Dynamics, 16, pp. 983-985.
  • Bashirov, A. E., Eppelbaum, L. V., Mishne, L. R., (1992), Improving E¨otv¨os corrections by wide band noise Kalman filtering, Geophysical Journal International, 107, pp. 193-197.
  • Bashirov, A. E., Etikan, H., S¸emi, N., (1997), Filtering, smoothing and prediction for wide band noise driven systems, Journal of Franklin Institute, Engineering and Applied Mathematics, 334B, pp. 683.
  • Bashirov, A. E., (2015), Stochastic maximum principle in the Pontryagin’s form for wide band noise driven systems, International Journal of Control, 88, No.3, pp. 461-468.
  • Bashirov, A. E., Etikan, H., S¸emi, N., (2010), Partial controllability of stochastic linear systems, International Journal of Control, 83, pp. 2564-2572.
  • Bashirov, A. E., Mahmudov, N., S¸emi, N., Etikan, H., (2007), Partial controllability concepts, Inter- national Journal of Control, 80, pp. 1-7.
  • Bashirov, A. E., Ghahramanlou, N., (2015), On partial S-controllability of semilinear partially ob- servable systems, International Journal of Control, 88, pp. 969-982.
  • Bashirov, A. E., (2015), On weakening of controllability concepts, In: Proceedings of the 35th IEEE Conference on Decision and Control 1996, 11-13 December, Kobe, Japan, pp. 640-645.
  • Bashirov A. E., (2017), Linear filtering for wide band noise driven observation systems, Circuits Systems and Signal Processing, 36, pp. 1247–1263.
  • Bashirov, A. E., (2014), Wide band noises: invariant results, In: Proceedings of the World Congress on Engineering 2014, Vol. II, 2-4 July, London, UK, 5 p.
  • Bashirov, A. E., Mazhar, Z., Etikan, H., Ert¨urk, S., (2013), Delay structure of wide band noises with application to filtering problems, Optimal Control, Applications and Methods, 34, pp. 69-79.
  • Bashirov, A. E., Uˇgural, S., Ert¨urk, S., (2008), Wide band noise as a distributed delay of white noise, In: Proceedings of the World Congress on Engineering 2008, Vol. II, 2-4 July, London, UK, pp. 954.
  • Bashirov, A. E., Uˇgural, S., (2002), Representation of systems disturbed by wide band noises, Applied Mathematics Letters, 15, pp. 607-613.
  • Bashirov, A. E., Uˇgural, S., (2002), Analyzing wide band noise processes with application to control and filtering, IEEE Transactions on Automatic Control, 47AC, pp. 323-327.
  • Bashirov, A. E., (2003), Partially observable linear systems under dependent noises, Systems & Con- trol: Foundations & Applications, Birkh¨auser, Basel.
  • Bashirov, A. E., (2005), Filtering for linear systems with shifted noises, International Journal of Control, 78, pp. 521-529.
  • Bashirov, A. E., Mazhar, Z., (2007), On asymptotical behavior of solution of Riccati equation arising in linear filtering with shifted noises, In: K. Ta¸s, J.A. Tenreiro Machado and D. Baleanu (Eds.)
  • Mathematical Methods in Engineering, Springer-Verlag, Dordrecht, pp. 141-149. Bashirov, A. E., Mazhar, Z., Ert¨urk, S., (2008), Boundary value problems arising in Kalman filtering
  • Boundary Value Problems, 208, Doi: 10.1155/2008/279410.
  • Bashirov, A. E., Mazhar, Z., Ert¨urk, S., (2013), Kalman type filter for systems with delay in observa- tion noise, Applied and Computational Mathematics, 12, pp. 325-338. Agamirza E.
  • Bashirov, for the photograph and short biography, see TWMS J. Appl. and Eng. Math., V.1, No.1, 2011.

INVARIANT FILTERING RESULTS FOR WIDE BAND NOISE DRIVEN SIGNAL SYSTEMS

Year 2018, Volume: 8 Issue: 1, 71 - 82, 01.06.2018

Abstract

Filtering of wide band noise driven systems accounts the following problem. Given an autocovariance function, there are in nitely many wide band noise processes, which have this autocovariance function. Each of them produces its own best estimate. The problem is a selection of the best one of these best estimates. A similar problem arises in control theory as a selection of optimal one of the optimal controls. In this paper we investigate this problem for a wide class of wide band noises. It is proved that in the case of independent wide band and white noises corrupting, respectively, the signal and observations, the best estimates and the optimal controls in the linear ltering and LQG problems are independent of the respective wide band noises. We present a complete set of formulae for the best estimate and, respectively, for the optimal control in terms of the system parameters and autocovariance function of the wide band noise disturbing the signal system.

References

  • Kalman, R. E., (1960), A new approach to linear filtering and prediction problems, Transactions ASME, Ser. D (Journal of Basic Engineering), 82, pp. 35-45.
  • Kalman, R. E., Bucy, R. S., (1961), New results in linear filtering and prediction theory, Transactions ASME, Ser. D (Journal of Basic Engineering), 83, pp. 95-108.
  • Bensoussan, A., (1992), Stochastic control of partially observable systems, Cambridge University Press, Cambridge.
  • Curtain, R. F., Pritchard, A. J., (1978), Infinite dimensional linear systems theory, Lecture Notes in Control and Information Sciences, Vol. 8, Springer-Verlag, Berlin. Wonham, W. M., (1968), On the separation theorem of stochastic control, SIAM J. Control, 6, pp. 326.
  • Grassides, J. L., Junkins, J. L., (2004), Optimal estimation of dynamic systems, Chapman and Hall/CRC, Boca Raton.
  • Fleming, W. M., Rishel, R. W., (1075), Deterministic and stochastic optimal control, Springer-Verlag, New York.
  • Ito, K., (1944), Stochastic integral, Proceedings of the Imperial Academy, Tokyo, 20, pp. 519-524.
  • Ito, K., (1951), On stochastic differential equations, Memoirs of the American Mathematical Siciety, , pp. 645-668.
  • Bucy, R. S., Joseph, P. D., (1968), Filtering for stochastic processes with application to guidance, Wiley, New York.
  • Kushner, H. J., (1990), Weak convergence methods and singularly perturbed control and filtering prob- lems, Birkh¨auser, Boston.
  • Kushner, H. J., Runggaldier, W. J., (1987), Nearly optimal state feedback controls for stochastic systems with wideband noise disturbances, SIAM Journal on Control and Optimization, 25, pp. 298
  • Kushner, H. J., Runggaldier, W. J., (1987), Filtering and control for wide bandwidth noise driven systems, IEEE Transactions on Automatic Control, 32AC, pp. 123-133.
  • Kushner, H. J., Ramachandran, K. M., (1988), Nearly optimal singular controls for sideband noise driven systems. SIAM Journal on Control and Optimization, 26, pp. 569-591.
  • Liptser, R. S., Runggaldier, W. J., Taksar M., (2000), Diffusion approximation and optimal stochastic control. Theory of Probability and Applications, 44, pp. 669-698.
  • Hu, H., (2000), Speech signal processing, Harbin Institute of Technology Press, Heilongjiang.
  • Wang, W., Liu, D., Wang, X., (2010), An improved wide band noise signal analysis method, Computer and Information Science, 3, pp. 76-80.
  • Bashirov, A. E., (1988), On linear filtering under dependent wide band noises, Stochastics, 23, pp. 437.
  • Bashirov, A. E., (1993), Control and filtering for wide band noise driven linear systems, AIAA J.Guidance Control and Dynamics, 16, pp. 983-985.
  • Bashirov, A. E., Eppelbaum, L. V., Mishne, L. R., (1992), Improving E¨otv¨os corrections by wide band noise Kalman filtering, Geophysical Journal International, 107, pp. 193-197.
  • Bashirov, A. E., Etikan, H., S¸emi, N., (1997), Filtering, smoothing and prediction for wide band noise driven systems, Journal of Franklin Institute, Engineering and Applied Mathematics, 334B, pp. 683.
  • Bashirov, A. E., (2015), Stochastic maximum principle in the Pontryagin’s form for wide band noise driven systems, International Journal of Control, 88, No.3, pp. 461-468.
  • Bashirov, A. E., Etikan, H., S¸emi, N., (2010), Partial controllability of stochastic linear systems, International Journal of Control, 83, pp. 2564-2572.
  • Bashirov, A. E., Mahmudov, N., S¸emi, N., Etikan, H., (2007), Partial controllability concepts, Inter- national Journal of Control, 80, pp. 1-7.
  • Bashirov, A. E., Ghahramanlou, N., (2015), On partial S-controllability of semilinear partially ob- servable systems, International Journal of Control, 88, pp. 969-982.
  • Bashirov, A. E., (2015), On weakening of controllability concepts, In: Proceedings of the 35th IEEE Conference on Decision and Control 1996, 11-13 December, Kobe, Japan, pp. 640-645.
  • Bashirov A. E., (2017), Linear filtering for wide band noise driven observation systems, Circuits Systems and Signal Processing, 36, pp. 1247–1263.
  • Bashirov, A. E., (2014), Wide band noises: invariant results, In: Proceedings of the World Congress on Engineering 2014, Vol. II, 2-4 July, London, UK, 5 p.
  • Bashirov, A. E., Mazhar, Z., Etikan, H., Ert¨urk, S., (2013), Delay structure of wide band noises with application to filtering problems, Optimal Control, Applications and Methods, 34, pp. 69-79.
  • Bashirov, A. E., Uˇgural, S., Ert¨urk, S., (2008), Wide band noise as a distributed delay of white noise, In: Proceedings of the World Congress on Engineering 2008, Vol. II, 2-4 July, London, UK, pp. 954.
  • Bashirov, A. E., Uˇgural, S., (2002), Representation of systems disturbed by wide band noises, Applied Mathematics Letters, 15, pp. 607-613.
  • Bashirov, A. E., Uˇgural, S., (2002), Analyzing wide band noise processes with application to control and filtering, IEEE Transactions on Automatic Control, 47AC, pp. 323-327.
  • Bashirov, A. E., (2003), Partially observable linear systems under dependent noises, Systems & Con- trol: Foundations & Applications, Birkh¨auser, Basel.
  • Bashirov, A. E., (2005), Filtering for linear systems with shifted noises, International Journal of Control, 78, pp. 521-529.
  • Bashirov, A. E., Mazhar, Z., (2007), On asymptotical behavior of solution of Riccati equation arising in linear filtering with shifted noises, In: K. Ta¸s, J.A. Tenreiro Machado and D. Baleanu (Eds.)
  • Mathematical Methods in Engineering, Springer-Verlag, Dordrecht, pp. 141-149. Bashirov, A. E., Mazhar, Z., Ert¨urk, S., (2008), Boundary value problems arising in Kalman filtering
  • Boundary Value Problems, 208, Doi: 10.1155/2008/279410.
  • Bashirov, A. E., Mazhar, Z., Ert¨urk, S., (2013), Kalman type filter for systems with delay in observa- tion noise, Applied and Computational Mathematics, 12, pp. 325-338. Agamirza E.
  • Bashirov, for the photograph and short biography, see TWMS J. Appl. and Eng. Math., V.1, No.1, 2011.
There are 38 citations in total.

Details

Primary Language English
Journal Section Research Article
Authors

A. E. Bashırov This is me

K. Abuassba This is me

Publication Date June 1, 2018
Published in Issue Year 2018 Volume: 8 Issue: 1

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