A digital controller is used to control the pressure in a tank by adjusting a purge stream. The control valve is air-to-open, and the process model has been identified as
$G= G_{v}G_{p}G_{m}= \frac {-20e^{-s}}{5s+1}$
The gain is dimensionless, and the time constant and time delay are in minutes. The sampling period is Δt=1 min.

Question

A digital controller is used to control the pressure in a tank by adjusting a purge stream. The control valve is air-to-open, and the process model has been identified as
[latex]G= G_{v}G_{p}G_{m}= \frac {-20e^{-s}}{5s+1} [/latex]
The gain is dimensionless, and the time constant and time delay are in minutes. The sampling period is Δt=1 min.

Compare the closed-loop performance of a discrete PI controller using the ITAE (disturbance) tuning rules in Table below. Approximate the sampler and ZOH by a time delay equal to Δt/2 in the model for the ITAE controller calculation. Use Simulink to check the effect of sampling period for different controllers, with Δt=0.05, 0.25, 0.5, and 1.0 min.

Type of Input	Type of Controller	Mode	A	B
Disturbance	PI	P	0.859	−0.977
		I	0.674	−0.680
Disturbance	PID	P	1.357	−0.947
		I	0.842	−0.738
		D	0.381	0.995
Set point	PI	P	0.586	−0.916
		I	1.03	−0.165
Set point	PID	P	0.965	−0.85
		I	0.796	−0.1465
		D	0.308	0.929

Accepted Answer

First adjust the process time delay for each controller calculation by adding Δt/2, which accounts for the time delay due to the sampler plus the ZOH. The controller settings calculated from Table (shown in the question) for different sampling periods are ( [latex] τ_{I} [/latex] in min):

PI
[latex]τ_{I} [/latex]	[latex]K_{c}[/latex]	Δt
2.48	-0.21	0.05
2.68	-0.19	0.25
2.89	-0.17	0.5
3.27	-0.14	1.0

These continuous controller settings are then substituted into Equation
[latex] G_{c}(z) = \frac{K_{c}(a_{0}+a_{1}z^{-1}+a_{2}z^{-2})}{1-z^{-1}} [/latex]
to obtain the corresponding settings for the digital controller. Figure shows that smaller sampling periods result in faster closed-loop responses for PI control.
There is no change in performance for Δt≤0.05 min. It is interesting to compare the results of this example with guidelines for choosing the sampling periods given in Equation
[latex] 0.01≤ \frac{\Delta t}{ au _{dom}} ≤ 0.05 [/latex]

[latex] \frac{t_{s}}{15}≤ \Delta t≤ \frac{t_{s}}{6} [/latex]

Question 17.5: A digital controller is used to control the pressure in a ta...

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