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

This article may be used for research, teaching and private study purposes. Any substantial or systematic reproduction, redistribution , reselling , loan or sub-licensing, systematic supply or distribution in any form to anyone is expressly forbidden. The publisher does not give any warranty express or implied or make any representation that the contents will be complete or accurate or up to date. The accuracy of any instructions, formulae and drug doses should be independently verified with primary sources. The publisher shall not be liable for any loss, actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directly or indirectly in connection with or arising out of the use of this material. 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. 1. Introduction The classical Kalman filter (KF) for linear systems provides optimal state estimates under standard noise and model assumptions (Jazwinski 1970). In practice, however, additional information about the system may be available, and this information may be useful for improving state estimates. A scenario we have in mind is the case in which the dynamics and the disturbances are such that the states of the system satisfy an equality or inequality constraint (Robertson and Lee 2002; Goodwin, Seron, and de Dona´2005). For example, in a chemical reaction, the species concentrations are non-negative (Massicotte, Morawski, and Barwicz 1995; Chaves and Sontag 2002), whereas in a compartmental model with zero net inflow (Bernstein and Hyland 1993), mass is conserved. Likewise, in undamped mechanical systems, such as a system with Hamiltonian dynamics, conservation laws hold, while, in the quaternion-based attitude estimation problem, the attitude vector must have unit norm