Question 13.8: Consider a continuous time plant having a nominal model give...
Consider a continuous time plant having a nominal model given by
G_{o}(s) =\frac{1}{(s+1)(s+2)} (13.6.40)
Design a digital controller, C_{q}(z), which achieves a loop bandwidth of approximately 3[rad/s]. The loop must also yield zero steady state error for constant references.
We choose to carry out the design using the Delta transform and then to transform the resulting controller, C_{\delta }(\gamma ), into the Z form, C_{q}(z). The sampling period, Δ, is chosen to be equal to 0.1[s] (note that, with this choice, the sampling frequency is significantly higher than the required bandwidth). We first use the MATLAB program c2del.m (in the accompanying diskette) to obtain the discrete transfer function in delta form representing the combination of the continuous time plant and the zero order hold mechanism. This yields
D \left\{G_{ho}(s)G_{o}(s) \right\} = \frac{0.0453\gamma +0.863}{\gamma ^{2} + 2.764\gamma +1.725\gamma } (13.6.41)
We next choose the closed loop polynomial A_{cl\delta }(\gamma ) to be equal to
A_{cl\delta }(\gamma )= (γ + 2.5)^{2}(γ + 3)(γ + 4) (13.6.42)
Note that a fourth order polynomial was chosen, since we have to force integral action in the controller. The resulting pole assignment equation has the form
(\gamma ^{2} + 2.764\gamma + 1.725 )\gamma \bar{L}_{\delta } (\gamma )+ (0.0453\gamma + 0.863)P_{\delta } (\gamma ) =(γ + 2.5)^{2}(γ + 3)(γ + 4) (13.6.43)
The MATLAB program paq.m is then used to solve this equation, leading to C_{\delta }(\gamma ),which is finally transformed into C_{q}(z). The delta and shift controllers are given by
C_{\delta }(\gamma ) = \frac{29.1 \gamma ^{2} + 100.0\gamma + 87.0}{ \gamma ^{2} + 7.9\gamma } =\frac{P_{\delta }(\gamma )}{\gamma \bar{L}_{\delta } (\gamma )} and (13.6.44)
C_{q }(z) = \frac{29.1 z ^{2} + 48.3z + 20.0}{ (z-1)(z-0.21)} (13.6.45)
The design is evaluated by simulating a square wave reference using the SIMULINK file dcpa.mdl. The results are shown in Figure 13.8.
The reader is encouraged to use the SIMULINK file dcpa.mdl to check that the bandwidth of the loop (use the command dlinmod) exceeds the required value. It is also of interest to evaluate how the locations of the closed loop poles change when the sampling period is changed (without modifying C_{q}(z)).
