Question 13.3: The system nominal transfer function is given by Go(s) = 10/...
The system nominal transfer function is given by
G_{o}(s) =\frac{10}{s(s+1)} (13.5.12)
and the continuous time controller is
G (s) =\frac{0.416s+1}{0.139s+1} (13.5.13)
Replace this controller by a digital controller with Δ = 0.157[s] preceded by a sampler and followed by a ZOH using the three approximations outlined above. Test the unit step response for each such approximation.
- Replacing s by γ in C(s) we get
\bar{C}_{1}(\gamma ) =\frac{0.416 \gamma +1}{0.139 \gamma +1} (13.5.14)
- The ZOH equivalent of C(s) is
\bar{C}_{2}(\gamma ) =\frac{0.694 \gamma +1}{0.232 \gamma +1} (13.5.15)
- For the bilinear mapping with pre-warping we first look at the continuous time sensitivity function
S_{o}(jω) =\frac{1}{1 + C(jω)G(jω)} (13.5.16)
We find that |S_{o}(jω)| has a maximum at ω^{*} = 5.48 [rad/s]. Now using the formula 13.5.6 we obtain α = 0.9375. With this value of α we find the approximation
\bar{C}_{3}(\gamma ) = C (s) \mid _{s=\frac{\alpha \gamma }{\underset{\bar{2} }{\Delta }\gamma +1 } }\frac{0.4685 \gamma +1}{0.2088 \gamma +1} (13.5.17)
The closed loop unit step responses obtained with the continuous time controller C(s) and the three discrete-time controllers \bar{C}_{1}(\gamma ) , \bar{C}_{2}(\gamma ) and \bar{C}_{3}(\gamma ) are presented in Figure 13.4 on the facing page.
