Constrained state estimation using the Unscented Kalman Filter

Abstract

Abstract-A simple procedure to include state inequality constraints in the Unscented Kalman Filter is proposed. With this procedure, the information of active state constraints influences the state covariance matrix, resulting in better estimates. In a numerical example, the approach outperforms the Extended Kalman Filter implemented with constraint handling via "clipping". I. INTRODUCTIONIn the process industries one of the main goals is to make the end product at the lowest possible cost while satisfying product quality constraints. State estimation often play an important role in accomplishing this goal in process control and performance monitoring applications. There are many uncertainties to deal with in process control; model uncertainties, measurement uncertainties and uncertainties in terms of different noise sources acting on the system. In this kind of environment, representing the model state by an (approximated) probability density function (pdf) has distinct advantages. State estimation is a means to propagate the pdf of the system states over time in some optimal way. It is most common to use the Gaussian pdf to represent the model state, process and measurement noises. The Gaussian pdf can be characterized by its mean and covariance. The Kalman Filter (KF) propagates the mean and covariance of the pdf of the model state in an optimal (minimum mean square error) way in case of linear dynamic systems [5].All practical systems possess some degree of nonlinearity. Depending on the type of process and the operating region of the process, some processes can be approximated with a linear model and the KF can be used for state estimation. In some cases the linear approximation may not be accurate enough, and state estimator designs using nonlinear process models are necessary. The most common way of applying the KF to a nonlinear system is in the form of the Extended Kalman Filter (EKF). In the EKF, the pdf is propagated through a linear approximation of the system around the operating point at each time instant. In doing so, the EKF needs the Jacobian matrices which may be difficult to obtain for higher order systems, especially in the case of timecritical applications. Further, the linear approximation of the system at a given time instant may introduce errors in the state which may lead the state to diverge over time. In other words, the linear approximation may not be appropriate for some systems. In order to overcome the drawbacks of the

Kalman filtering with state constraints: a survey of linear and nonlinear algorithms

“…URNDDR is called the sigma point interval UKF in [49]. A simplified version ofURNDDR is presented in [51].…”

Robust calibration of an ultralow-cost inertial measurement unit and a camera: Handling of severe system uncertainty

“…These inequality constraints are placed on the constrained region by the projection scheme [21]. Technically, the states outside the constrained region are moved to the boundary of the constrained region.…”

“…The inequality constraint is applied to states after sigma point generation of the prediction and update [21]. As a consequence, since we use the UKF, the state covariance is determined within the filtering process.…”

Gait analysis using 3D motion reconstruction with an activity-specific tracking protocol

“…We assume within a gait cycle, linear displacement of the body center is dominated by the component along the walking direction and the thigh and leg rotation is on the Sagittal Plane. These constraints are incorporated into an Unscented Kalman filter by projecting the sigma points which are outside the feasible region onto the boundary of the feasible region in the time-update step [10]. Another assumption to model walking is based on the observation that when one leg is in mid-swing, the other will be in mid-stance with the hip, knee and ankle joint in a line perpendicular to the ground.…”