(374c) Estimation of the Arrival Cost in MHE Using Particle Filters

Moving Horizon Estimation (MHE) is an efficient state estimation method used for nonlinear systems.¹Since MHE is optimization-based it provides a good framework to handle bounds and constraints when they are required to obtain good state and parameter estimates. Recently, research in this area has been directed to develop computationally efficient algorithms for on-line application.²However, an open issue in MHE is related to the approximation of the so called Arrival Cost.¹The arrival cost is very important since it provides a mean to incorporate information from the previous measurements to the current state estimate. It is difficult to calculate the true value of the arrival cost therefore approximation techniques are commonly applied. The conventional method is to use the Extended Kalman Filter (EKF) to approximate the covariance matrix at the beginning of the prediction horizon. This approximation method assumes that the state estimation error is Gaussian. However, when state estimates are bounded the distribution of the estimation error becomes non-Gaussian. This introduces errors in the arrival cost term which can be mitigated by using longer horizon lengths. This measure, however, significantly increases the size of the nonlinear optimization problem that needs to be solved on-line at each sampling time.
Particle filters, recently, have become popular filtering methods that are based on Monte-Carlo simulations.³In this way they implement an optimal recursive Bayesian Filter that takes advantage of particle statistics to determine the probability density properties of the states. Recently, Prakash et al. ⁴have developed a constrained version of the particle filter that systematically handles bounds on states using truncated distributions. Moreover, because particles are being propagated through the nonlinear system the assumption of Gaussianity of the state estimation error can be dropped. In the present work, we exploit these features of constrained particle filters to approximate the arrival cost in the MHE formulation. We compare the performance of the MHE when the approximation of the arrival cost is done via PF with that of the conventional method using EKF. Also, we show that a benefit of having a better approximation of the arrival cost is that the horizon length required for the MHE can be significantly smaller than when using the conventional MHE approach. This is illustrated by simulation studies done on some benchmark problems proposed in the state estimation literature.