06.1 Luenberger Observer (State Estimation) - Full State Feedback - 2
From 05.4 Full State Feedback - Part 1, we know that control gain, $G$, is applied to all the states, right? But in practice, we typically do not have all the states. So we estimate them. Now, the question is: how do we estimate them?
We know a typical state-space system looks like this:
\[\begin{aligned}\dot{x} = Ax + Bu \\ y = Cx +Du \end{aligned}\]But we do not know $x,A,B,C,D$ perfectly, right? So we start by saying that our estimates for them are $\hat{x},\hat{A},\hat{B},\hat{C},\hat{D}$. Using this, we can write our estimated state-space system as:
\[\begin{aligned}\dot{\hat{x}} = \hat{A}\hat{x} + \hat{B}u \\ \hat{y} = \hat{C}\hat{x} +\hat{D}u \end{aligned}\]So with our estimated quantities, we can get an estimated $\hat{y}$, right? But how do we correct our estimation? We use a predictor-corrector idea. We correct our prediction with:
\[\dot{\hat{x}} = \hat{A}\hat{x} + \hat{B}u + K(y-\hat{y})\]But now the question is how do we get $K$?
After a bit of mathematical manipulation with the formulation (for details, refer to L13), we see that this $K$ is determined by the $A$ and $C$ matrices (with the caveat that we assume a perfect plant model). But how do we get the numerical value of $K$?
Interestingly, this seems like a 05.4 Full State Feedback - Part 1 problem - something like determining $G$ values. How so?
See, our estimated state-space system works if we can guarantee that our estimated states converge faster than the real states, right? And this convergence depends on the poles of our estimated system. But can we make the poles of our estimated system go faster? Yes, we can. This is more like a 05.4 Full State Feedback - Part 1 problem.
We use the same place function in MATLAB. But instead of using place(A,B,desired_closed_loop_poles), we use place(A,C,desired_closed_loop_poles). Now, how do we place our poles? For that, we need the techniques from 03.6 Pole Placement (Const Damping and Const Freq Line) s-plane.