(383g) On the Utility of the Iterated Kalman Filter for Constrained Importance Sampling in Sequential Monte Carlo State Estimation

Efficient operation of engineering dynamic systems such as chemical and manufacturing processes relies on a knowledge of the state of the system at any given time. Based on the current state, strategic input decisions are made, intended to guide the dynamic system towards its performance targets. Measurement and/or  estimation of the state thus forms the foundation of routine plant operations such as process control, optimization and fault detection. The task of state estimation or filtering from noisy and often limited measurements is generally posed in the realm of Bayesian statistics. The underlying theory and equations have been known for over half a century. Unfortunately, for general nonlinear dynamic systems with uncertainties characterized by non-Gaussian probability density functions (pdf), closed-form equations are not possible to implement the state estimation task. Practical implementations are typically achieved by incorporating simplifying assumptions such as Gaussianity of pdfs, functional and statistical linearization of nonlinear functions, piecewise constant pdfs on discretized grids and Monte Carlo methods. Specifically, recent advances in Monte Carlo sampling methods coupled with decreasing cost and increasing speed of computation have fueled a resurgence of interest in numerically solving the Bayesian state estimation problem with as few simplifying approximations as possible.

Sequential Monte Carlo state estimators, widely known as particle filters, are suboptimal numerical implementations of the optimal recursive Bayesian state estimation methodology. These numerical methods are attractive in practice because of their applicability to any type of nonlinear dynamic system involving uncertainties that are characterized by non-Gaussian distributions. The central objective of the Bayesian method is to construct a conditional pdf of the current state subject to all available measurements. Since it is impossible to determine a time-varying, infinite dimensional conditional pdf for arbitrary nonlinear systems, particle filters employ a finite set of random samples or particles and associated weights to form a discrete approximation of the conditional pdf. Statistical moments of the conditional pdf are useful for state estimation, for example, the mean, median and mode may be used for point estimates and a measure of accuracy expressed as covariance, skewness and kurtosis. The Monte Carlo approach is practically appealing because the statistical moments are readily computed from the weighted samples.

Ideally, the conditional pdf is the best probability measure to generate the samples from. Since the pdf is generally not available, samples belonging to its support are drawn from a conveniently chosen pdf, known as importance or proposal density, which is amenable to sampling. Appropriate weights are then assigned to the samples to reflect their relative importance according to the conditional density. The importance density is a critical design choice of the particle filter, which determines the quality of the discrete representation of conditional pdf. An importance density that contains a full knowledge of the state transition probability and all available measurements is an optimal importance density in the sense that it minimizes the variance of the importance weights. However, it is generally not possible to define the optimal importance density for arbitrary nonlinear systems. All other choices of importance density cause the variance of weights to increase with time, rendering them as suboptimal choices. The most popular suboptimal choice has been the transition probability density itself, which is a probabilistic model for the state transition sequence in state space. Predictive samples of the state drawn from the transition density conditioned on past measurements represent the transition prior density as the importance density. Although it is straightforward to sample from the transition prior, the drawback is that it has no knowledge of the most recent measurement. The transition prior as an importance density can prove to be ineffective in at least three situations, (1) the latest measurement is located in a low probability region of the transition prior, (2) a highly accurate measurement causes narrow and peaked likelihood that invalidates much of the high probability region of the transition prior and (3) the use of switching functions in measurements such as calibration shifts, relays, thresholds and saturation. In these cases, the overlap between the supports of likelihood and transition prior becomes inadequate. In summary, the transition prior is a poor choice for importance density when the measurement noise variance is small or the system noise variance is large.

In this context of problems involving a significantly limited agreement between the transition prior (prediction) and the measurement, several methods are available to generate better importance densities that are aware of the latest measurement. Therein, the conditional density is effectively sampled by separate importance densities for the particles, where the importance densities are based on measurement updated transition priors locally. This approach is local in the sense that the update is provided by a suboptimal nonlinear filter in the vicinity of the local filtered state. For instance, a bank of N extended Kalman filters (EKF) have been used in the literature to pose N local Gaussian importance densities generating N samples of the particle filter. It is known as local linearization particle filter based on functional linearization by truncated Taylor series of nonlinear functions. Similarly, the unscented particle filter employed a bank of unscented Kalman filters (UKF) based on local statistical linearization by the unscented transform. Recently, the same approach is demonstrated with a bank of ensemble Kalman filters (EnKF) for local Gaussian importance densities in the particle filter. Statistical linearization is more accurate than truncated Taylor series, and it is preferable when Jacobians are difficult or impossible to calculate. These local linearization particle filters are based on local Kalman update derived under two assumptions, (1) the prior pdf is a Gaussian and (2) the state and measurement are jointly Gaussian. More accurate and problem specific local filters can improve the performance by designing problem specific importance densities, such as in the case where constraints on state variables must be respected.

This paper is focused on the utility of the iterated Kalman filter to pose an importance density that effectively samples over a constraint surface for state estimation of constrained systems. The proposed approach rests on the fact that the transition prior density can be updated with the latest measurement by using the mean and covariance of the transition density. The commonly used assumption of a Gaussian prior is applicable here, which leads to an updated density that is non-Gaussian due to the measurement nonlinear function. One advantage here is that we have an analytical form of the updated density up to a constant of proportionality. The iterated Kalman filter essentially estimates the mode of this updated density by solving an optimization problem on the updated density. The optimization problem is typically solved using a Newton-Raphson iterative numerical method. In this paper, we propose that the updated density can be used as the importance density. Note that a single importance density is utilized to generate all N samples. However, the support of the importance density must be established prior to drawing samples. Since its analytical form is known, it is possible to solve for the support points where the density essentially vanishes. This involves a second Newton-Raphson iterative numerical method. The solutions, while finding the mode and the support, can be readily limited to linear and nonlinear constraint surfaces in state space. Having a knowledge of the mode and support of the the importance density the samples can be drawn using the Metropolis-Hastings algorithm.

The proposed approach has several important differences and advantages compared to the local linearization particle filters available in the literature. First, a single iterated Kalman filter is utilized instead of a bank of N local linearization filters, thereby achieving significant computational savings. Second, although the iterated Kalman filter formulation is used, only the state update part is utilized while the covariance update part is discarded. Thus, the need for linearization is eliminated, which can improve the performance of the particle filter. Third, by limiting the support of the importance density to applicable equality and inequality constraints, state estimation of constrained systems is easily handled. This approach is computationally simpler and more efficient than imposing constraints on each of the N particles separately by local linearization particle filters. Lastly, the suggested importance density from the iterated Kalman filter is a non-Gaussian density, while the local linearization particle filters approximate all the local importance desities as Gaussian densities.

Several numerical examples are provided to demonstrate the particle filter for state estimation on benchmark problems from literature. Comparison is made with local linearization particle filters using banks of extended Kalman filter and the unscented Kalman filter. Average results from multiple realizations are shown for the mean squared error of estimation and the computational time.