Nonlinear Filtering Based On Sequential Model Error Determination

In this paper, a real-time predictive filter is derived for nonlinear systems. This provides a method of determining optimal state estimates in the presence of significant error in the assumed (nominal) model. The new real-time nonlinear filter determines (i.e., “predicts”) the optimal model error trajectory so that the measurement-minus-estimate covariance statistically matches the known measurement-minus-truth covariance. The optimal model error is found by using a one-time step ahead control approach. Also, since the continuous model is used to determine state estimates, the filter avoids discrete state jumps, as opposed to the extended Kalman filter. Introduction Many linear control systems require full state knowledge (such as the LQR). But most systems have sensors which cannot measure all states (e.g., spacecraft sensors measure body positions which must be converted to an attitude for control purposes). An essential feature of most control systems involves an algorithm which is used to both estimate unmeasured states and to filter noisy measurements. Since pointing errors are a combination of both control and estimation errors, the robustness of each component must be addressed to insure proper pointing. Conventional filter methods, such as the Kalman filter [1], have proven been to be extremely useful in a wide range of applications, including: noise reduction of signals, trajectory tracking of moving objects, and in the control of linear or nonlinear systems. The essential feature of the Kalman filter is the utilization of state-space formulations for the system model. Errors in the dynamic system model are treated as “process noise,” since system models are not usually improved or updated during the estimation process. The process noise is essentially used to “shift” the emphasis from the model to the measurements. The Kalman filter satisfies an optimality criterion which minimizes the trace of the covariance of the estimate error between the system model responses and actual measurements. Statistical properties of the process noise and measurement error are used to determine an “optimal” filter design. Therefore, model characteristics are combined with sequential measurements in order to obtain state estimates which meliorate both the measurements and model responses. In the Kalman filter, the errors in the system model are assumed to be represented by a zero-mean Gaussian noise process with known covariance. However, in actual practice the noise covariance is usually determined by an ad hoc and/or heuristic estimation approach which may result in suboptimal filter designs. Other applications also determine a steady-state gain directly, which may even produce unstable filter designs [2]. Also, in many cases such as nonlinearities in the actual system responses or non-stationary processes, the assumption of a Gaussian model error process can lead to severely degraded state estimates. In addition to nonlinear model errors, the actual assumed model may be nonlinear (e.g., threedimensional kinematic and dynamic equations [3]). The filtering problem for nonlinear systems is considerably more difficult and admits a wider variety of solutions than does the linear problem [4]. The extended Kalman filter is a widely used algorithm for nonlinear estimation and filtering [5]. The essential feature of this algorithm is the utilization of a first-order Taylor series expansion of the model and output system equations. The extended Kalman filter retains the linear calculation of the covariance and gain matrices, and it updates the state estimate using a linear function of the measurement residual; however, it uses the original nonlinear equations for state propagation and in the output system equation [5]. But, the model error statistics are still assumed to be represented by a zero-mean Gaussian noise process.