Question 13.11: A plant with nominal model Go(s) = 13/s²− 4s +13 is to be di...
A plant with nominal model G_{o}(s) = \frac{13}{s^{2} – 4s + 13} is to be digitally controlled. The sampling period is Δ and a zero order hold is used. Assume that the nominal sensitivity must satisfy
\left|S_{o}(e^{jω\Delta })\right| \leq \epsilon \lt 1 ∀ω\leq \frac{ω_{s}}{4} = \frac{\pi }{2\Delta } (13.8.12)
Use Lemma 13.3 on the preceding page to determine a lower bound for the sensitivity peak S_{max}.
We note that the nominal model has two unstable poles located at p_{1,2} = 2\pm j3 . When the discrete model for the plant is obtained, these unstable poles map into \zeta_{1,2}=e^{( 2\pm j3) \Delta }. We then apply Lemma 13.3,u sing normalized frequency. This leads to
\int_{0 }^{\pi}{\ln \left|S_{o}(e^{jω\Delta})\right| dω } = 4\pi \Delta (13.8.13)
If we split the integration interval in [0, π] = [0, \frac{π}{2} ] ∪ (\frac{π}{2} , π] then
ln S_{max} \gt 4\Delta – \ln (\epsilon ) (13.8.14)
This bound becomes smaller if the sampling frequency increases. The reader is invited to investigate the use of Theorem ?? on page ?? to obtain a tighter bound for S_{max}.