Convergence of the Difference Equation for the Error Covariance Matrix Arising in Kalman Filter Theory

摘要

A detailed analysis of the stability properties of the error covariance equation in the time-invariant, discrete-time Kalman filter is presented. The scalar case is analyzed by means of a graphical construction. In the multidimensional case, it is shown that the equilibrium solution, P sub e, of the covariance equation satisfy an equation of the form A +BP(sub e) + P(sub e) C-P(sub e) DP(sub e) = 0, where the A, B, C, and D matrices are functions of the parameters of the random process. A technique for solving this equation is developed, and it is shown how the symmetry, definiteness, and local stability of a given equilibrium can be predicted. It is shown that the covariance equation has a stable, nonnegative definite equilibrium if the random process is not unobservable and either random walk or unstable, and it is shown that the stable equilibrium is the only nonnegative definite equilibrium if the random process is not undriven and either random walk or unstable. It is also shown in this case that any solution whose initial value is nonnegative definite converges to the stable equilibrium. (Author)