Bruno Otávio Soares Teixeira

State estimation for linear and non-linear equality-constrained systems

This article addresses the state-estimation problem for linear and non-linear systems for the case in which prior knowledge is available in the form of an equality constraint. The equality-constrained Kalman filter (KF) is derived as the maximum-a-posteriori solution to the equality-constrained state-estimation problem for linear and Gaussian systems and is compared to alternative algorithms. Then, four novel algorithms for non-linear equality-constrained state estimation based on the unscented KF are presented, namely, the equality-constrained unscented KF, the projected unscented KF, the measurement-augmentation unscented KF, and the constrained unscented KF. Finally, these methods are compared on linear and non-linear examples.

Spacecraft tracking using sampled-data Kalman filters

The goal of this article is to illustrate and compare two algorithms for nonlinear sampled-data state estimation. Under idealized assumptions on the astrodynamics of bodies orbiting the Earth, we apply SDEKF and SDUKF for range-only as well as range and angle observations provided by a constellation of six LEO satellites in circular, equatorial orbits. We study the ability of the filters to acquire and track a target satellite in geosynchronous orbit as a function of the sample interval, initial uncertainty, and type of available measurements. For target acquisition, SDUKF yields more accurate position and velocity estimates than SDEKF. Moreover, the convergence of SDEKF is sensitive to the initialization of the error covariance; in fact, a nondiagonal initial covariance is found to be more effective than a diagonal initial covariance. Like SDUKF, by properly setting a nondiagonal initial error covariance, SDEKF also exhibits global convergence, that is, convergence is attained for all initial true-anomaly errors.

Gain-Constrained Kalman Filtering for Linear and Nonlinear Systems

This paper considers the state-estimation problem with a constraint on the data-injection gain. Special cases of this problem include the enforcing of a linear equality constraint in the state vector, the enforcing of unbiased estimation for systems with unknown inputs, and simplification of the estimator structure for large-scale systems. Both the one-step gain-constrained Kalman predictor and the two-step gain-constrained Kalman filter are presented. The latter is extended to the nonlinear case, yielding the gain-constrained unscented Kalman filter. Two illustrative examples are presented.

State estimation for equality-constrained linear systems

We address the state-estimation problem for linear systems in a context where prior knowledge, in addition to the model and the measurements, is available in the form of an equality constraint. First, we investigate from where an equality constraint arises in a dynamic system. Then, the equality-constrained Kalman filter (ECKF) is derived as the solution to the equality-constrained state-estimation problem and compared to alternative algorithms. These methods are investigated in an example. In addition to exactly satisfying an equality constraint on the system, ECKF produce more accurate and more informative estimates than the unconstrained estimates.

Unscented filtering for interval-constrained nonlinear systems

This paper addresses the state-estimation problem for nonlinear systems with an interval constraint on the state vector. Approximate solutions to this problem are reviewed and compared with new algorithms, which are based on the unscented Kalman filter. An illustrative example is discussed.

Unscented filtering for equality-constrained nonlinear systems

This paper addresses the state-estimation problem for nonlinear systems in a context where prior knowledge, in addition to the model and the measurement data, is available in the form of an equality constraint. Three novel suboptimal algorithms based on the unscented Kalman filter are developed, namely, the equality-constrained unscented Kalman filter, the projected unscented Kalman filter, and the measurement-augmented unscented Kalman filter. These methods are compared on two examples: a quaternion-based attitude estimation problem and an idealized flow model involving conserved quantities.

Kalman Filters [Ask The Experts]

In this issue, we invite Bruno Teixeira to explain the difference between the Kalman filter and the Kalman predictor. Bruno, who is a Ph.D. student at Federal University of Minas Gerais in Brazil, provides a complete review of the relevant equations in a convenient, common format.

Semiparametric identification of Wiener systems using a single harmonic input and retrospective cost optimization

We present a two-step method for identifying SISO Wiener systems. First, using a single harmonic input, we estimate a nonparametric model of the static nonlinearity, which is assumed to be only piecewise continuous. Second, using the identified nonparametric map, we use retrospective cost optimization to identify a parametric model of the linear dynamic system. This method is demonstrated on several examples of increasing complexity.

Maximum a posteriori state path estimation: Discretization limits and their interpretation

Continuous–discrete models with dynamics described by stochastic differential equations are used in a wide variety of applications. For these systems, the maximum a posteriori (MAP) state path can be defined as the curves around which lie the infinitesimal tubes with greatest posterior probability, which can be found by maximizing a merit function built upon the Onsager–Machlup functional. A common approach used in the engineering literature to obtain the MAP state path is to discretize the dynamics and obtain the MAP state path for the discretized system. In this paper, we prove that if the trapezoidal scheme is used for discretization, then the discretized MAP state path estimation converges hypographically to the continuous–discrete MAP state path estimation as the discretization gets finer. However, if the stochastic Euler scheme is used instead, then the discretized estimation converges to the minimum energy estimation. The minimum energy estimates are, in turn, proved to be the state paths associated with the MAP noise paths, which in some cases differ from the MAP state paths. Therefore, the discretized MAP state paths can have different interpretations depending on the discretization scheme used.

Joint Maximum a Posteriori Smoother for State and Parameter Estimation in Nonlinear Dynamical Systems

Abstract In this paper, we propose and demonstrate the direct use of optimization to search for the mode of the joint posterior state distribution of stochastic nonlinear dynamical systems. That is accomplished by forming a very large but sparse nonlinear optimization problem with the states in all time instants as decision variables. The proposed method generalizes well for parameter estimation without the need for treating them as augmented states and the introduction of artificial dynamics. It is also possible to estimate parameters such as the noise variances, which are assumed known in traditional methods.