How does one derive an optimal, continuous-time linear quadratic estimator from a Luenberger state observer? I am aware of a Kalman filter, but I would like to see a derivation of an observer without the stochastic elements for learning purposes. Specifically, given the system
\begin{align}
x^\prime &= Ax + Bu \\
x(0) &= x_0 \\
y &= Cx \\
\end{align}
we can design a Luenberger type of state observer with the equations
\begin{align}
\hat{x}^\prime &= A \hat{x} + Bu + L(y-C\hat{x})\\
\hat{x}(0) &= \hat{x}_0
\end{align}
where the gain $L$ must be chosen according to some criteria. If we let the error be $e=x-\hat{x}$, we can find that
\begin{align}
e^\prime &= x^\prime – \hat{x}^\prime\\
&=…\\
&=(A-LC)e
\end{align}
Hence, we need the matrix $A-LC$ to be Hurwitz or $\Re(\lambda(A-LC)) < 0$. This implies that the error goes to zero and $\hat{x}$ approaches $x$. Now, it seems like there should be an optimization formulation to find an optimal gain, $L$, that appears similar to the equations for a continuous-time Kalman filter. I don't know what that formulation is. If I try a formulation such as
$$\begin{array}{rcl}
\min\limits_{e,L} && \frac{1}{2}\int_0^\infty e^TQe + \frac{1}{2}\int_0^\infty \textrm{tr}(L^TRL)\\
\textrm{st} && e^\prime = (A-LC)e
\end{array}$$
the result seems off. Here, the constraint is bilinear, between $e$ and $L$, so we can't get a quadratic objective with a linear constraint and therefore a linear system to solve for optimality. What is the correct form of the optimization formulation?
Best Answer
It should be the same way as for LQ optimal controller design.
Consider $$\dot{e} = A^\top e + C^\top u$$ with $u = -L^\top e$. The spectrum of $A^\top-C^\top L^\top$ is the same as for $A-LC$. Then you minimize the quadratic cost $$J = \int_0^\infty \left(e^\top(\tau) Q e(\tau) + u^\top(\tau) R u(\tau) \right)d\tau.$$
Here $u$ is your estimation error injection into the model's dynamics of the observer. However, since this injection is somewhat virtual, the term $u^\top R u$ does not represent the energy cost minimization as it does for LQ control. The role of $R$ is thus less intuitive.