closed loopcontrolfeedbacksystem
I have a question, maybe you can help me because I am not sure of the correct definition.
As an example I have a plant with a measured output y in a closed loop system. But as the feedback variable I use y' = y + 100 to substract this from the setpoint.
Does the calculation of y' belong to the plant and is y' the controlled variable then?
Or is y the actual controlled variable and how is y' then called?
Thank you in advance for your help!
What you have initially described is a 2nd order low pass filter then you've made a bit of a complication of things because the bandwidth is \$\omega_n\$ i.e. \$\omega_n\$ is the 3dB point when \$\zeta\$ is not causing the filter to peak i.e. has a value of \$\frac{1}{\sqrt2}\$: -
OK you may be trying to derive a general expression but that doesn't help much visualize the problem so for now I'm assuming \$\zeta\$ = \$\frac{1}{\sqrt2}\$.
Anyway, moving on and ignoring your expression for \$\omega_0\$, the bandwidth of the low-pass filter is from dc to \$\omega_n\$. Then you've modified the 2nd order low-pass filter expression with a 1st order high pass filter (\$s +z\$).
When s is very low, the low pass filter is unaffected other than having a "gain factor" of z. If z is unity then the low pass filter response is unaffected until jw approaches z in magnitude - this then marks a lower 3dB point and the net response climbs with increasing frequency until the original \$\omega_n\$ is reached then, because the low pass filter is a second order, the response starts to fall again.
What you have proposed is a band pass filter with finite gain at dc.
I'm not sure about the bandwidth
If the high pass filter (\$s +z\$) comes into play at significantly lower frequencies than \$\omega_n\$ then the net bandwidth reduces from \$\omega_n\$ to \$\omega_n - z\$.
That's how I see it anyway.
Lets rephrase:
When (if ever) is the system stabilizable?
If it is stabilizable, you can perform a controllable decomposition and then apply pole placement.
Best Answer
Assuming a state-space representation of your plant as follows: $$ \begin{cases} \dot{x}=Ax+Bu\\ y=Cx+Du\\ z\approx y+100 \end{cases} $$ These are usual signal names you would find:
In order to compose the feedback control loop, you are taking the plant output through an estimator system: $$ \hat{z}=y+100 $$
I avoided using your \$y'\$ notation, since I think it's misleading for the explanation, but it should be equivalent as the \$z\$ or \$\hat{z}\$ signals in this notation.