Regular Design Equations for the Reduced-Order Kalman Filter

Reduced-order Kalman filters yield an optimal state estimate for linear dynamical systems, where parts of the outputs are not corrupted by noise. The design of such filters can either be carried out in the time domain or in the frequency domain both for continuous-time and for discrete-time systems. Here, the continuous-time case is discussed. Different from the full-order case with all measurements corrupted by noise, the design equations of the reduced-order filter are not regular. This is due to the rank deficient measurement covariance matrix, which also gives rise to a rank deficient solution of the Riccati equation. This causes problems when using standard software. By an adequate transformation of the system equation, the Riccati equation can be subdivided into a regular part and a vanishing part, so that standard software becomes applicable. The regular (reduced-order) design problem also allows a formulation of the conditions for the stability of the filter and it is shown, what they mean for the original system. In the frequency domain spectral factorization of the non-regular polynomial matrix causes no problems. The known proof of optimality of the factorization result, however, also requires a regular measurement covariance matrix. Starting from the regular reduced-order design problem in the time domain also a reduced-order regular frequency domain design can be formulated. The regular design equations in the frequency domain then allow a formal proof of optimality of the result of the spectral factorization and it turns out, that this result coincides with the known one obtained from the non regular polynomial matrix equation. Presented are also the conditions on the matrix fraction description of the system guaranteeing a stable filter in the frequency domain.