I am studying control systems, and I have seen that a pole in the right half plane gives a limitation in bandwidth, imposing that the bandwidth of the system has to be high enough.
My question is : why does this happen?
I have seen for example that for zeros in the right half plane, the limitation is given by the fact that the phase decreases after a certain frequency, but why for a pole in the right half plane do I have a limitation as well?
Consider for example the following plant and controller:
$P(s)=\frac{10}{(s+2)(s-1)}$
$C(s)=\frac{4(s+1)}{s}$
where $P(s)$ is the plant and $C(s)$ is the controller, if I plot the Bode plot I have:
but to me it looks like that if I decrease the bandwidth the performances increases, but it happens exacly the opposite.
Am I doing something wrong or am I missing some concepts?
Best Answer
Let's denote the close loop transfer function as $T(s)=\frac{P(s)C(s)}{1+P(s)C(s)}$. For a RHP-pole $p_i$, the cut-off frequency (where $\left\vert T \right \vert $ drops below $0\,\mathrm{dB}$), is bounded by the following inequality $$ \omega_T\geq \Re(p_i)\tan \left( \frac{\pi}{2+2\frac{\left\vert M_T \right \vert}{3\, \mathrm{dB}}} \right) ,$$ where $\left\vert M_T \right \vert$ is the peak gain, given in $\mathrm{dB}$. If we want this peak to be less than $3 \, \mathrm{dB}$ across all frequencies, then the bandwidth is bounded by $\omega_T \geq \Re(p_i)$.
For more information, see Skogestad & Postlehtwaite. Multivariable feedback control and Seron et al.. Fundamental Limitations in Filtering and Control.