For the same process model used in Example 17.6, plot the response and the controller output for a unit set-point change (τc= Δt= 1.0) for the nonringing Dahlin’s controller and compare it with the Direct Synthesis PID (BD conversion) and Vogel–Edgar approaches.

Question

For the same process model used in Example 17.6, plot the response and the controller output for a unit set-point change [latex](τ_{c}= Δt= 1.0) [/latex] for the nonringing Dahlin’s controller and compare it with the Direct Synthesis PID (BD conversion) and Vogel–Edgar approaches.

Accepted Answer

Applying Equation
[latex] (\frac{Y}{Y_{sp}})_{d}=\frac{(1-A)z^{-N-1}}{1-Az^{-1}} [/latex]
for N=0, the controller transfer function for the nonringing version of Dahlin’s algorithm is
[latex] \overline{G}_{DC} = (\frac{0.6321}{1-z^{-1}} )(\frac{1-1.5353z^{-1}+0.5866z^{-2}}{0.0514} ) [/latex]     (1)
For this controller the response y(t) and the controller output p(t) are shown in Fig. 17.12a. Next we derive a PID controller using the IMC approach presented in Chapter 12. Starting with the continuous-time second-order transfer function [latex] (τ_{1} =5,τ_{2} =3, K=1,θ=0), τ_{c} [/latex] in Table 12.1 is set equal to one to provide the best response for a set-point change. When the BD approximation [latex] s ≈ (1−z^{−1})/Δt [/latex] is substituted into the PID controller transfer function,
[latex] G_{BD} = 4.1111\frac{3.1486−5.0541z^{−1} +2.0270z^{−2} }{1.7272−2.4444z^{−1} +0.7222z^{−2}} [/latex]      (2)
The closed-loop response is shown in Fig. 17.12b. When the BD-PID digital controller is designed for this system, there is no need to correct for ringing such as is required for Dahlin’s controller. This is true for wide ranges of [latex]τ_{c} [/latex] and Δt that have been investigated.
Figure 17.12 shows the closed-loop response for the Vogel–Edgar controller [latex](G_{VE}) [/latex] for the same second-order model. This controller is
[latex] G_{VE} = 0.6321\frac{ 1−1.5353z^{−1} +0.5866z^{−2}}{0.0514−0.0366z^{−1} −0.0148z^{−2}} [/latex]     (3)
The tuning parameter A is selected to be [latex] 0.368(τ_{c} =1). [/latex] For this second-order system, the controlled variable response for [latex]G_{VE}[/latex] is superior to [latex] \overline{G}_{DC} . [/latex]
If a time delay is added to the model, the comparative performance of [latex]G_{VE} \space and \space \overline{G}_{DC} [/latex] is still the same, because both controllers utilize the same form of the Smith predictor.
Note that the controller parameters in Equations (1),(2) and (3) have been reported with four decimal points in order to avoid roundoff errors in the control calculations.

Question 17.7: For the same process model used in Example 17.6, plot the re...

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