Kalman Filter for Bilinear Systems with Application

In this paper, we present a new kind of the bilinear systems in the form of state space model. The evolution of this system depends on the product of state vector by its self. The well known Lotak Volterra and Lorenz models are special cases of this new model. We also present here a generalization of Kalman filter which is suitable to work with the new bilinear model. An application to real measurements is introduced to illustrate the efficiency of the proposed algorithm.

• Abdullah E. Al-Mazrooei

• Bilinear systems
• state space model
• Kalman filter.

[1] B.D. Anderson and J.B. Moore, Optimal Filtering, Prentice-Hall, Englewood
Cliffs, NJ, 1979.
[2] M. S. Gerwal and A. P. Andrews, Kalman Filtering: Theory and Practice
Using MATLAB, John Wily and Sons Inc., 2001.
[3] N. S. Goel, et al., On The Volterra and Other Nonlinear Models of
Interacting Populations, Academic Press Inc., 1971.
[4] D. Roy , Z.E. Musielak, Generalized Lorenz models and their routes to
chaos. III, Energy-Conserving horizontal and vertical mode Truncations,
Chaos, Solitons and Fractals, Vol. 33, pp. 1064–1070, 2007
[5] P.M. Pardalos and V.Yatsenko, Optimization and control of bilinear
systems: theory, algorithms and applications, Springer, 2008
[6] S. H. Strgate, Nonlinear Dynamics and Chaos, with Applications to
Physics, Biology, Chemistry, and Engineering, Presure Books l.l.c.,1994.
[7] H. Tanizaki, Nonlinear Filtering: Estimation and Applications, Springer-
Verlag, 1996.
[8] N. Wiener, Nonlinear Problems in Random Theory, Boston, MA: The
M.I.T. Press, 1958.