Determining the discrete equivalent Kalman steady state gain for the LQR problem

A method of mapping the continuous steady-state Kalman gain to a discrete equivalent is developed and presented. Using continuous and discrete system relationships makes it possible to derive a method for relating the continuous Kalman gain to a discrete equivalent. The method determines the weighting matrices necessary to locate the closed-loop eigenvalues in a specified region of the s-plane so only a continuous gain can be computed, but by making use of the relationship of the closed-loop continuous model and its discrete equivalent a relationship for the discrete Kalman gain can be determined based on computed quantities. The relationship is an overdetermined system of linear equations in the elements of the unknown Kalman discrete gain matrix. Using the least-squares method makes it possible to solve the system for the discrete Kalman steady-state gain. The eigenvalues of the continuous time closed-loop system can be mapped directly to the z-plane based on the sample rate and then compared with the z-plane closed-loop eigenvalues found on the basis of the determined discrete steady-state gain.<
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