Optimal Luenberger Observer

Finite horizon

Let us consider a LTI system with additive Gaussian noise $x_{k} y_{k} = A x_{k - 1} + B u_{k - 1} + w_{k - 1}, = C x_{k} + v_{k}, Q_{k - 1} R_{k - 1} = E [w_{k} w_{k}^{⊤}] = E [v_{k} v_{k}^{⊤}] .$ and a Luenberger observer $\overset{x}{^}_{k} = A \overset{x}{^}_{k - 1} + B u_{k - 1} - L_{k} (C \overset{x}{^}_{k - 1} - y_{k - 1}) . (1)$ The state estimate's error $e_{k} = \overset{x}{^}_{k} - x_{k}$ and its covariance $P_{k} = E [e_{k} e_{k}^{⊤}]$ can be expressed recursively as $e_{k} P_{k} = w_{k - 1} + (A - L_{k} C) e_{k - 1} + L_{k} v_{k - 1} = Q_{k - 1} + (A - L_{k} C) P_{k - 1} (A - L_{k} C)^{⊤} + L_{k} R_{k - 1} L_{k}^{⊤} . (2)$ The gain $L_{k}$ can then be designed such that $P_{k}$ is minimized. This can be achieved by finding the stationary point $\frac{\partial P _{k}}{\partial L _{k}} = 0$ , where $L_{k} = P_{k - 1} C^{⊤} (C P_{k - 1} C^{⊤} + R_{k - 1})^{- 1} . (3)$ Substituting (3) back into (2b) we attain the DDRE $P_{k} = Q_{k - 1} + A P_{k - 1} A^{⊤} - A P_{k - 1} C^{⊤} (C P_{k - 1} C^{⊤} + R_{k - 1})^{- 1} C P_{k - 1} A^{⊤}, (4)$ which governs the evolution of the state error's covariance.

The state of the observer then consists of $\overset{x}{^}_{k}$ and $P_{k}$ , its dynamics described by (1) and (4). This observer can also be used in the case where the dynamics of the system are time variant.

Infinite horizon

In the case where the covariances of the process and measurement noise are time-invariant, i.e. $Q = E [w_{k} w_{k}^{⊤}], R = E [v_{k} v_{k}^{⊤}],$ we can also design the gain of the observer based on the steady-state error covariance $P_{k} \to P$ at $k \to \infty$ . The covariance $P$ can be obtained by solving the DARE $P = Q + A P A^{⊤} - A P C^{⊤} (CP C^{⊤} + R)^{- 1} CP A^{⊤}, (5)$ based on which we evaluate the gain $L = P C^{⊤} (CP C^{⊤} + R)^{- 1} .$

Duality with LQR

It is of note that the form of (4) and (5) is identical to those of the LQR with different matrices:

LQR	Luenberger
$S$	$P$
$A$	$A^{⊤}$
$B$	$C^{⊤}$

Optimal and Predictive Control

Optimal Luenberger Observer

Finite horizon

Infinite horizon

Duality with LQR