Miroslav Šimandl

Brief paper: Derivative-free estimation methods: New results and performance analysis

The derivative-free nonlinear estimation methods exploiting the Stirlings interpolation and the unscented transformation for discrete-time nonlinear stochastic systems are treated. The divided difference and unscented filters, smoothers, and predictors based on the methods are introduced in the unified framework. The new relations among the first order Stirlings interpolation, the second order Stirlings interpolation, and the unscented transformation are derived and their impact on the covariance matrices of the state estimates of the corresponding filters is analysed. The theoretical results are illustrated and used for the explanation of the unexpected behaviour of the sigma point Gaussian sum filters given as a mixture of the derivative-free filters.

Filtering, predictive, and smoothing Cramér-Rao bounds for discrete-time nonlinear dynamic systems

Cramer-Rao lower bounds for the discrete-time nonlinear state estimation problem are treated. The Cramer-Rao bound for the mean-square error matrix of a state estimate is particularly important for quality evaluation of nonlinear state estimators as it represents a limit of cognizability of the state. Recursive relations for filtering, predictive, and smoothing Cramer-Rao bounds are derived to establish a unifying framework for several previously published derivation procedures and results. Lower bounds for systems with unknown parameters are newly provided. Computation of filtering, predictive, and smoothing Cramer-Rao bounds, their mutual comparison and utilization for quality evaluation of some nonlinear filters are shown in numerical examples.

Unscented Kalman Filter: Aspects and Adaptive Setting of Scaling Parameter

This technical note deals with the unscented Kalman filter for state estimation of nonlinear stochastic dynamic systems with a special focus on the scaling parameter of the filter. Its standard choice is analyzed and its impact on the estimation quality is discussed. On the basis of the analysis, a novel method for adaptive setting of the parameter in the unscented Kalman filter is proposed. The results are illustrated in a numerical example.

Advanced point-mass method for nonlinear state estimation

State estimation of discrete-time nonlinear non-Gaussian stochastic systems by point-mass approach, which is based on discretization of state space by a regular grid and numerical solution of Bayesian recursive relations, is treated. The stress is laid to grid design which is crucial for estimator quality and significantly affects the computational demands of the estimator. Boundary-based grid design, thrifty convolution, and multigrid design with grid splitting and merging are proposed. The main advantages of these techniques are nonnegligible support delimitation, time-saving computation of convolution, and effective processing of multimodal probability density functions, respectively. The techniques are involved into the basic point-mass approach and a new general-purpose, more sophisticated point-mass algorithm is designed. Computational demands and estimation quality of the designed algorithm are presented and compared with the particle filter in a numerical example.

Brief paper: Active fault detection and control: Unified formulation and optimal design

A new unified formulation of the active fault detection and control problem for discrete-time stochastic systems and its optimal solution are proposed. The problem formulation stems from the optimal stochastic control problem and includes important special cases: an active detector and controller, an active detector and input signal generator, and an active detector with a given input signal generator. The optimal solution is derived using the so-called closed loop information processing strategy. This strategy respects the influence of the current decision and/or input on the future behavior of the observed system, allows penalizing future wrong decisions, and improves the quality of fault detection. The proposed formulation and obtained solution also provide better understanding of the active fault detection and its relation to the optimal stochastic control. The results are illustrated in numerical examples.

Stochastic Integration Filter

The technical note deals with state estimation of nonlinear stochastic dynamic systems. Traditional filters providing local estimates of the state, such as the extended Kalman filter, unscented Kalman filter, or the cubature Kalman filter, are based on computationally efficient but approximate integral evaluations. On the other hand, the Monte Carlo based Kalman filter takes an advantage of asymptotically exact integral evaluations but at the expense of substantial computational demands. The aim of the technical note is to propose a new local filter that utilises stochastic integration methods providing the asymptotically exact integral evaluation with computational complexity similar to the traditional filters. The technical note will demonstrate that the unscented and cubature Kalman filters are special cases of the proposed stochastic integration filter. The proposed filter is illustrated by a numerical example.

Anticipative grid design in point-mass approach to nonlinear state estimation

Numerical solution of the Bayesian recursive relations by the point-mass approach is treated. The stress is laid on the grid design as the main user design problem in this approach. An anticipative approach for finding a minimum sufficient number of grid points is developed, yielding a new adaptive algorithm reducing the computational operations without a loss of estimation accuracy. A numerical example is presented to compare the new algorithm with the standard point-mass technique and with a sequential importance sampling nonlinear filtering algorithm.

Truncation nonlinear filters for state estimation with nonlinear inequality constraints

The paper focuses on the state estimation problem of nonlinear non-Gaussian systems with state subject to a nonlinear inequality constraint. Taking into account the available additional information about the state given by the constraint increases the estimate quality compared to classical state estimation methods which cannot utilize the information. Considering the constraint in the form of an inequality involving a nonlinear function of the state makes the state estimation problem difficult and hence treated only marginally. In this paper, a generic local filter for the inequality constrained estimation problem is proposed. It covers the extended Kalman filter, unscented Kalman filter, and divided difference filter as special cases and enforces the constraint by truncating the conditional density of the state. The truncation is computationally cheap, yet it provides high estimate quality of the constrained estimate. The same idea is then utilized in a truncation Gaussian mixture filter which is also proposed in the paper to increase the estimate quality further by providing a global constrained estimate. Superior estimate quality and computational efficiency of the proposed filters are illustrated in two numerical examples.

Rolling horizon for active fault detection

This paper presents a feasible design of suboptimal active fault detection system in multiple-model framework. The optimal solution for finite horizon is approximated by means of well known rolling horizon scheme which belongs to the class of limited look-ahead policies. The suboptimal input signal which is chosen from given discrete set is obtained by l-step closed loop optimization. It is shown that such input signal can improve fault detection.

Sigma point Gaussian sum filter design using square root unscented filters

Abstract Local and global estimation approaches are discussed, above all the Unscented Kalman Filter and the Gaussian Sum Filter. The square root modification of the Unscented Kalman Filter is derived and it is used in the Gaussian Sum Filter framework. The new Sigma Point Gaussian Sum Filter is designed and some aspects of the filter are presented. Estimation quality and computational demands of the filter are illustrated in a numerical example.