A comparison between the pseudomeasurement and extended Kalman observers

The class of nonlinear measurements that can be transformed into a linear function of the state where the coefficient matrix is a nonlinear function of the observation are called pseudomeasurements. The dynamic system is assumed to be linear. If the system is observable, then by processing the pseudomeasurements by a linear observer structure (PMO), the estimation errors are shown to converge to zero. This structure is contrasted with the extended Kalman observer (EKO) which processes these same measurements but in their original nonlinear form. To compare the PMO with the EKO, the bearings-only problem associated with homing missile guidance problem (which is an important member of the pseudomeasurement class) is used. By algebraic manipulations the EKO is reduced to the PMO structure but with different resulting gains. New gains for the EKO are determined by modifying the update formula for the "error variance" propagation in accordance with the PMO algorithm. A simulation of the homing missile problem in two dimensions shows that the PMO and the EKO using the modified gains reduce all initial estimation errors rapidly toward zero and with very similar responses. In contrast, the original EKO has a far more sluggish and markedly different response.