An innovation approach to optimize a Kalman filter with an H_∞error bound by secant method

Designing a Kalman filter with a constraint on the H_∞norm of the estimation error was first developed by Bernstein and Haddad in 1989. The main result is a sufficient condition for characterizing the Kalman filter. In this paper, similar to the standard Kalman filter, the properties of orthogonal principles are also shown to be preserved. Furthermore, the uniqueness, as opposed to an H_∞ filter, of the filter is implied by the orthogonal principles. An innovative approach to obtaining the minimum energy with a constraint on the H_∞ norm of the estimation error is proposed since the original work of Bernstein and Haddad does not, in general, reach the minimum energy of the estimation error. By means of the Secant method, the energy of the estimation error can be reduced as much as possible, under the condition that the H_∞error bound is still satisfied.