Kalman Filtering with Equality and Inequality State Constraints

Both constrained and unconstrained optimization problems regularly appear in recursive tracking problems engineers currently address – however, constraints are rarely exploited for these applications. We define the Kalman Filter and discuss two different approaches to incorporating constraints. Each of these approaches are first applied to equality constraints and then extended to inequality constraints. We discuss methods for dealing with nonlinear constraints and for constraining the state prediction. Finally, some experiments are provided to indicate the usefulness of such methods.

💡 Research Summary

The paper addresses a gap in modern recursive tracking applications: while many engineering problems involve state constraints, conventional Kalman filters rarely exploit them. The authors propose two systematic ways to embed both equality and inequality constraints directly into the Kalman filtering framework, then extend these methods to handle nonlinear constraints and to constrain the prediction step itself.

Method 1 – Projection‑Based Approach
After the standard Kalman update, the posterior state estimate (\hat{x}{k|k}) and covariance (P{k|k}) are projected onto the feasible subspace defined by the constraints. For linear equality constraints (A x = b), the projection yields closed‑form updates:
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