From the Linear Quadratic Regulator (LQR) to the (Deterministic) Kalman Filter in Two Easy Steps
Abstract¶
This note is a tutorial on the deterministic version of the Kalman filter (state estimator), which is formulated as finding the state trajectory consistent with the system's equations with the minimal amount of L^(2) process and measurement uncertainty. As stated, this is an input signal design problem with linear dynamics and an objective that is affine-quadratic in the state and inputs. The first step is to convert this problem to one with a purely quadratic objective by embedding in a larger system using ``homogeneous coordinates''. This converts the problem to a purely quadratic (i.e. an LQR) problem, but with non-standard initial or final state constraints. This latter problem can then be solved using a version of the matrix Differential Riccati Equation (DRE) for the larger LQR problem. The second step is a partitioning of this larger problem, which then yields the optimal dynamic observer and the DRE of the traditional Kalman filter. For comparison, the solution of the traditional LQ-tracking (Servomechanism) problem is also treated using a similar construction.
Summary¶
Gives a tutorial derivation of the deterministic Kalman filter by embedding a trajectory-estimation problem into an enlarged LQR with homogeneous coordinates and nonstandard boundary constraints. The derivation then partitions the resulting Riccati equations to recover the dynamic observer, making the LQR-Kalman connection unusually explicit.