06.2 Luenberger Observer - 2

MATLAB Implementation of Luenberger Observer and Why this observer problem looks like a 05.4 Full State Feedback - Part 1

From 06.1 Luenberger Observer (State Estimation) - Full State Feedback - 2 we know that our estimated state-space system will work if we can guarantee that our estimated states converged faster than the real states, right? And this convergence will depend 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 05.4 Full State Feedback - Part 1 problem. We will use the same place function of MATLAB. But instead of using place(A,B,desired_closed_loop_poles), we will be using place(A,C,desired_closed_loop_poles). Now, how do we place our poles? For that, we will need the techniques from the 03.6 Pole Placement (Const Damping and Const Freq Line) s-plane.

Always use transpose while calculating for the observer

K = place(A',C',obspoles) % obs = observer

But for that we need to know the poles of the “model” or the system, right?

Someone might ask, if we know the system, why do we need the observer to begin with?

By “model”, we mean $A,B,C,D$ matrices that come from equation of motion. And using $\hat{y} = \hat{C}\hat{x}+\hat{D}u$, we might obtain whatever’s my sensor is measuring, say, $x_2$. For more clarification, refer to 04.2 Intuition (Mapping) of State-Space Realization. So, we know $x_2$ but we don’t know the numerical values for $x_1$, $x_3$ and so on. And Luenberger Observer calculates the states themselves, meaning, $x_1, x_2$ and so on.

05.3 Draft - Why Observability and Controllability are Dual Problems