Discrete-time Nonlinear Filtering Research Articles

Unscented Kalman Filtering for Simultaneous Estimation of Attitude and Gyroscope Bias

We present an unscented Kalman filtering algorithm for simultaneously estimating attitude and gyroscope bias from an inertial measurement unit (IMU). The algorithm is formulated as a discrete-time stochastic nonlinear filter, with state space given by the direct product matrix Lie group $\text{SO(3)} \times \mathbb {R}^3$ , and observations in $\text{SO(3)}$ reconstructed from IMU measurements of gravity and the earth's magnetic field. Computationally efficient implementations of our filter are made possible by formulating the state space dynamics and measurement equations in a way that leads to closed-form equations for covariance propagation and update. The resulting attitude estimates are invariant with respect to choice of fixed and moving reference frames. The performance advantages of our filter vis-a-vis existing state-of-the-art IMU attitude estimation algorithms are validated via numerical and hardware experiments involving both synthetic and real data.

Explicit Fixed-Point Computation of Nonlinear Delay-Free Loop Filter Networks

An iterative method is proposed for the explicit computation of discrete-time nonlinear filter networks containing delay-free loops. The method relies on a fixed-point search of the signal values at every temporal step. The formal as well as numerical properties of fixed-point solvers delimit its applicability: On the one hand, the method allows for a reliable prediction of the frequency rates where the simulation is stable, while, on the other hand, its straightforward applicability is counterbalanced by low speed of convergence. Especially in presence of specific nonlinear characteristics, the use of a fixed-point search is limited if the real-time constraint holds. For this reason, the method becomes useful especially during the digital model prototyping stage, as exemplified while revisiting a previous discrete-time realization of the voltage-controlled filter aboard the EMS VCS3 analog synthesizer. Further tests conducted on a digital ring modulator model support the above considerations.

Filtering a nonlinear stochastic volatility model

We introduce a class of stochastic volatility models whose parameters are modulated by a hidden nonlinear dynamical system. Our aim is to incorporate the impact of economic cycles, or business cycles, into the long-term behavior of volatility dynamics. We develop a discrete-time nonlinear filter for the estimation of the hidden volatility and the nonlinear dynamical system based on return observations. By exploiting the technique of a reference probability measure we derive filters for the hidden volatility and the nonlinear dynamical system.

Nonlinear Filter Estimation of Volatility

A discrete time nonlinear filter is used to estimate the volatility in a financial model. New filters are derived for sums of unobserved quantities and the EM algorithm applied to determine the parameters of the model.

Discrete–time nonlinear filtering with marked point process observations: ii. risk–sensitive filters

We discuss in this article the risk–sensitive filtering problem of estimating a nonlinear signal process, with nonadditive non–Gaussian noise, via a marked point process observation. This extends to the risk sensitive case all the risk–neutral results studied in Dufour and Kannan [2].By going into a change of measure, we derive the unnormalized conditional density of the signal conditioned on the observation history. We also derive the unnormalized prediction density. Using these, we present two separate expressions for the optimal estimate of the signal. A similar analysis of the smoothing density of the signal is also studied under both the risk–sensitive and risk–neutral cases. We specialize the above optimal estimation to the linear signal dynamics and marked point process observation under some Gaussian assumptions. We obtain a Kalman type risk-sensitive filter. Due to the special nature of the observation process, the conditional mean and covariance estimates directly depend now on the point process

Information transmission using chaotic discrete-time filter

In this work an adaptive approach for transmitting information hidden in a chaotic carrier is presented. The proposed method uses as a chaotic generator a discrete time nonlinear filter with a sawtooth nonlinearity induced by the two's complement overflow in digital filters. The synchronization at the receiving end is ensured by an adaptive slave system. Two different methods for hiding the information in the chaotic carrier are analyzed: direct chaos modulation and parameter modulation.

Discrete time nonlinear filtering with marked point process observations

This work discusses the problem of estimating a nonlinear signal process with nonadditive non–Gaussian noise via a marked point process observation. We begin by introducing a modeling of discrete–time marked point processes. In order to derive the exact filter, namely the conditional density of the signal given the observation history, a reference probability is introduced; this probability measure renders the signal and the observation processes independent, as needed. We present a discretetime version of Zakai equation corresponding to our model. By specializing the signal to be linear, we moreover show, under suitable assumptions, that the filter is finite dimensional

Dynamic realizations of sufficient sequences

Let (U_{1},U_{2}, \cdots) be a sequence of observed random variables and (T_{1}(U_{1}),T_{2}(U_{1}, U_{2}), \cdots) be a corresponding sequence of sufficient statistics (a sufficient sequence). Under certain regularity conditions, the sufficient sequence defines the input/output map of a time-varying, discrete-time nonlinear system. This system provides a recursive way of updating the sufficient statistic as new observations are made. Conditions are provided assuring that such a system evolves in a state space of minimal dimension. Several examples are provided to illustrate how this notion of dimensional minimality is related to other properties of sufficient sequences. The results can be used to verify the form of the minimum dimension (discrete-time) nonlinear filter associated with the autoregressive parameter estimation problem.

A rate distortion theory lower bound on desired function filtering error (Corresp.)

A discrete-time nonlinear filtering lower bound algorithm is given for evaluating the error in a desired function of the state vector. The algorithm is based on a rate distortion bound derived previously by the author. The problem is formulated in terms of Monte Carlo analysis. The theory of backward Markovian models is used to evaluate the conditional expectation appearing in the Bucy representation for the ease of Gauss-Markov signal models. An approximation procedure is given for the case of nonlinear signal models. In comparison with the author's previous bound the bound algorithm obtained here is tighter and does not require the difficult computation of the entropy of the state vector.

Formal algorithms for continuous-time non-linear filtering and smoothing†

Kalman's formal limiting procedure is applied to some recent results in sequential discrete-time non-linear filtering and smoothing to obtain the corresponding estimation algorithms for continuous-time non-linear dynamic systems. The resulting filtering algorithm is found to agree with the well-known Detchmendy-Sridhar filter which was obtained via another method. The present smoothing algorithm is a new result. It is argued that the combined filter-smoothing results here lead to an estimation algorithm which is second order in both system dynamics and measurement function non-linearity.