Predictive Filtering for Nonlinear Systems

Abstract

A real-time predictive e lter is derived for nonlinear systems. The major advantage of this new e lter over conventional e lters is that it providesa method of determining optimalstate estimatesin the presenceof signie cant error in the assumed (nominal)model. The new real-time nonlinear e lter determines (predicts)the optimal model errortrajectorysothatthemeasurement-minus-estimatecovariancestatisticallymatchestheknownmeasurement- minus-truth covariance. The optimal model error is found by using a one-time step ahead control approach. Also, because the continuous model is used to determine state estimates, the e lter avoids discrete state jumps. The predictive e lter is used to estimate the position and velocity of nonlinear mass-damper-spring system. Results using this new algorithm indicate that the real-time predictive e lter provides accurate estimates in the presence of highly nonlinear dynamics and signie cant errors in the model parameters. ONVENTIONAL e lter methods, such as the Kalman e lter, 1 have proven to be extremely useful in a wide range of appli- cations, including noise reduction of signals, trajectory tracking of moving objects, and control of linear or nonlinear systems. The es- sential feature of the Kalman e lter is the utilization of state-space formulations for the system model. Errors in the dynamics system can be separated into process noise errors or modeling errors. Pro- cess noise errors are usually represented by a zero-mean Gaussian errorprocesswithknowncovariance (e.g.,agyro-errormodelcanbe represented by a random walk process ). Modeling errors are usu- ally not known explicitly, because system models are not usually improved or updated during the estimation process. The theoretical derivation of the expression for the estimate error covariance in the Kalman e lter is only available if one makes assumptions about the model error. The most common assumptions about the model error are that it is also a zero-mean Gaussian noise process. Therefore, in the e lter-type literature, most often process noise and model error are treated equally. The Kalman e lter satise es an optimality criterion, which min- imizes 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 e lter design. Therefore, model characteristics are combined with sequential measurements in order to obtain state estimates that are more accurate than both the measurements and model responses. As already stated, errors in the system model of the Kalman e l- ter are usually assumed to be represented by a zero-mean Gaussian noise process with known covariance. In actual practice the noise covariance is usually determined by an ad hoc and/or heuristic es- timation approach, which may result in suboptimal e lter designs. Otherapplicationsalsodetermineasteady-stategaindirectly,which may even produce unstable e lterdesigns. 2 Also, in many cases such as nonlinearities in the actual system responses or nonstationary processes, the assumption of a Gaussian model error process can lead to severely degraded state estimates.