02.4 Controller or Compensator Design
When we are adding a controller/compensator into our system (by system, I mean, the state-space with the observer and with the control gains), what is the transfer function of that controller? By the way, a compensator just means the controller block that sits between the measured plant output and the plant input.
But why would we want a transfer function for controller or compensator?
Because once the observer-based controller is written as its own dynamic system, we can study the controller itself in the frequency domain.
That lets us check things like controller gain, bandwidth, phase/magnitude behavior, and robustness margins. It is not for simulating the whole closed-loop response directly; it is for understanding what the controller block is doing.
We have derived substituting $\mathbf{\hat{y}}$ and $u = -Gx$ into the 06.1 Luenberger Observer (State Estimation) - Full State Feedback - 2 state equation, For observer-based feedback:
\[\mathbf{\dot{\hat{x}}} = \begin{bmatrix}\mathbf{\hat{A}-\hat{B}G-K\hat{C}+K\hat{D}G} \end{bmatrix} + Ky\]So we have the state equation but what about the output equation? we know the following: \(\text{plant output } y \rightarrow \boxed{\text{compensator/controller}} \rightarrow \text{plant input } u\) Using that information, we can develop a state-space formulation for the controller alone, which is as follows! $$ \begin{align} \mathbf{\dot{\hat{x}}} &= \begin{bmatrix}\mathbf{\hat{A}-\hat{B}G-K\hat{C}+K\hat{D}G} \end{bmatrix} + Ky
u &= -\mathbf{G\hat{x}+0y}
\end{align} $$
Now compare with generic state-space form: \(\dot{x}_c=A_cx_c+B_cr\) \(y_c=C_cx_c+D_cr\)
| For the compensator, the internal state is | the compensator input is the measured plant output | and the compensator output is the plant control input |
|---|---|---|
| $x_c=\hat{x}$ | $r=y$ | $y_c=u$ |
So: \(A_c=A-BG-KC+KDG\) \(B_c=K\) \(C_c=-G\) \(D_c=0\)
That is why your professor said (C) is like (-G) and (D=0). The output equation of the compensator is:
\[u=-G\hat{x}\]The key point: (u) is not “output (y)” of the plant. It is the output of the compensator. Then that same signal (u) becomes the input to the plant.
So there are two systems:
| System | Input | Output |
|---|---|---|
| Plant | (u) | (y) |
| Compensator/controller | (y) | (u) |
That is the loop.
And no, “compensator/controller” does not by itself mean closed-loop system. It is one block. When you connect it with the plant so that plant output (y) feeds the compensator and compensator output (u) feeds the plant, then the whole combined system becomes closed-loop.
This is exactly why L15/S10 describes the output feedback compensator as a dynamic state-space system whose input is (y) and whose output is (u).
And then you can derive a transfer function $H(s)$ from that! \(H_c(s)=-G(sI-A+BG+KC-KDG)^{-1}K\)
Once you have a transfer function for the controller alone, you can obtain Bode Plot, right?
So in the bode plot, when you are plotting magnitude of the $H(s)$ of the controller against a frequency sweep, what are these frequency correspond to?
Practically, it answers:
If the plant output/measurement (y) contains oscillation/noise/disturbance at frequency ($\omega$), how strongly will the controller convert that into control effort (u)?
So yes, it can correspond to excitation/disturbance frequency, but more generally it is the frequency of any signal entering the controller: motion, sensor noise, vibration, measurement fluctuation, etc.
High magnitude at high frequency means the controller may strongly amplify high-frequency measurement noise, causing large/fast control effort. That is why L15 warns that increasing observer convergence rate increases compensator gain and bandwidth.