Asymptotic quadratic matrix equations and stochastic state filtering



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The concept and use of the steady-state Kalman gain in Kalman filtering theory is presented in this dissertation. Use of the steady-state Kalman gain, in place of the real-time Kalman gain with initial covariance Po = 0, is shown to reduce both the error and the number of computations required in determining the optimal estimate of system states. The steady-state Kalman gain is computed directly from the system coefficient matrices using a recently derived algorithm. This method bypasses possible existence problems associated with the real-time Kalman gain matrix. Historical background, theoretical development, and examples are presented for both continuous and discrete systems. The results show conclusively the benefits gained by using the steady-state Kalman gain rather than the real-time Kalman gain. Included in the examples are both constant coefficient and time-varying coefficient systems. A proof is given which shows that the conclusions drawn from the constant coefficient examples presented are true in general. Further examples to illustrate basic points of the proof are provided. Program listings and descriptions are included in the appendices.