Simple method for solving the constant gains of kalman filters with single output

A simple method is described for the calculation of the optimal gains for steady state Kalman filters in continuous- and discrete-time systems with single output. The procedure consists of comparing the optimal closed-loop characteristic polynomial with a determinant expansion containing the unknown constant gains. It is then shown that for the case when a computer program is not used for finding the roots of a polynomial, the optimal closed-loop characteristic polynomial in the discrete time problem is not as readily obtainable as that in the continuous-time problem, even though the system under consideration is of lower order. This drawback, however, is shown to be overcome by invoking the bilinear transformation.