State Estimation for Ellipsoidally Constrained Dynamic Systems with Set-membership Pseudo Measurements

In many dynamic systems, the evolution of the
state is subject to specific constraints. In general, constraints
cannot easily be integrated into the prediction-correction structure
of the Kalman filter algorithm. Linear equality constraints
are an exception to this rule and have been widely used and
studied as they allow for simple closed-form expressions. A
common approach is to reformulate equality constraints into
pseudo measurements of the state to be estimated. However,
equality constraints define deterministic relationships between
state components which is an undesirable property in Kalman
filtering as this leads to singular covariance matrices. A second
problem relates to the knowledge required to identify and
define precise constraints, which are met by the system state.
In this article, ellipsoidal constraints are introduced that can
be employed to model a bounded region, to which the system
state is constrained. This concept constitutes an easy-to-use
relaxation of equality constraints. In order to integrate ellipsoidal
constraints into the Kalman filter structure, a generalized filter
framework is utilized that relies on a combined stochastic and
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set-membership uncertainty representation.