Bayesian State Estimation of Nonlinear Systems Using Approximate Aggregate Markov Chains

Abstract

The conditional probability density function (pdf) is the most complete statistical representation of the state from which optimal inferences may be drawn. The transient pdf is usually infinite-dimensional and impossible to obtain except for linear Gaussian systems. In this paper, a novel density-based filter is proposed for nonlinear Bayesian estimation. The approach is fundamentally different from optimization- and linearization-based methods. Unlike typical density-based methods such as probability grid filters (PGF) and sequential Monte Carlo (SMC), the cell filter poses an off-line probabilistic modeling task and an on-line estimation task. The probabilistic behavior is described by the Foias or Frobenius−Perron operators. Monte Carlo simulations are used for computing these transition operators, which represent approximate aggregate Markov chains. The approach places no restrictions on system model or noise processes. The cell filter is shown to achieve the performance of PGF and SMC filters at a fraction of the computational cost for recursive state estimation.

Computing arrival cost parameters in moving horizon estimation using sampling based filters

“…On the average, the mean estimates are closer to the true state than the mode estimates [24] as indicated by the average MSE in the bar chart in Fig. 7.…”

“…The main idea in CF is the development of an aggregate Markov chain for describing the temporal dynamics of the state pdf over a discretized state space comprising a finite number of cells [24][25][26]. The transition probability matrix is com puted offline using propagation of samples from the cells.…”

“…The second illustrative example is a more challenging bench mark problem in nonlinear estimation research [10,19,24]. The dy- …”

The use of a cell filter for state estimation in closed-loop NMPC of low dimensional systems

“…In case of invertible and differentiable measurement function h, the lmv is readily computed by Eq. (11) at cell centers z z [12]. Otherwise a region of interest in the output space Y c R p is considered where measurements of x 2 X, i.e., y 2 Y are likely to be obtained.…”

“…Density based methods such as sequential Monte Carlo (SMC) filter has been shown to be more accurate and faster than optimization based methods [10,11]. Similar to SMC the cell filter (CF) is based on Monte Carlo integration but relies on aggregate Markov chains developed offline, which reduces the online computational burden [12]. Extension of both SMC and CF to constrained systems are also reported [13,14].…”