Use the DS design method to calculate PID controller settings for the process: G = 2e^-s/(10s+1)(5s+1) Consider three values of the desired closed-loop time constant:τc =1,3, and 10. Evaluate the controllers for unit step changes in both the set point and the disturbance, assuming that Gd =G.

Question

Use the DS design method to calculate PID controller settings for the process:
[latex] G = \frac{2e^{-s}}{(10s+1)(5s+1)} [/latex]
Consider three values of the desired closed-loop time constant: [latex]τ_{c}=1,3, [/latex] and 10. Evaluate the controllers for unit step changes in both the set point and the disturbance, assuming that [latex] G_{d} = G [/latex].
Repeat the evaluation for two cases:
(a) The process model is perfect [latex] (\widetilde{G} = G). [/latex]
(b) The model gain is incorrect, [latex] \widetilde{K} = 0.9, [/latex] instead of the actual value, K=2.
Thus,
[latex] \widetilde{G} = \frac{0.9e^{-s}}{(10s+1)(5s+1)} [/latex]

Accepted Answer

The controller settings from Equation
[latex] K_{c} = \frac{1}{K}\frac{ au _{1}+ au _{2}}{ au _{c}+ heta },\quad au _{I} = au _{1}+ au _{2},\quad au _{D} = \frac{ au _{1} au _{2}}{ au _{1}+ au _{2}} [/latex] (1)
are

	[latex] τ_{c} =1 [/latex]	[latex] τ_{c} =3 [/latex]	[latex] τ_{c} =10 [/latex]
[latex] K_{c}(\widetilde{K}=2 ) [/latex]	3.75	1.88	0.68
[latex] K_{c}(\widetilde{K}=0.9 ) [/latex]	8.33	4.17	1.51
[latex] τ_{I} [/latex]	15	15	15
[latex] τ_{D} [/latex]	3.33	3.33	3.33

The values of [latex] K_{c} [/latex] decrease as [latex] τ_{c} [/latex] increases, but the values of [latex]τ_{I} [/latex] and [latex] τ_{D} [/latex] do not change, as indicated by Equation (1).
Although some of these controller settings have been reported with three significant figures for purposes of comparison, calculating a third digit is not necessary in practice. For example, controllers with [latex] K_{c} [/latex] values of 8.33 and 8.3 would produce essentially the same closed-loop responses.
Figures 12.3 and 12.4 compare the closed-loop responses for the three values of [latex] τ_{c}. [/latex] As [latex]τ_{c} [/latex] increases, the responses become more sluggish, and the maximum deviation is larger after the disturbance occurs at τ=80. For case (b), when the model gain is 0.9, about 50% too low, the closed-loop response for [latex] τ_{c} =1 [/latex] in Fig. 12.4 is excessively oscillatory and would even become unstable if [latex] \widetilde{K} = 0.8 [/latex] had been considered. The disturbance responses for [latex] τ_{c} =3 \space and \space τ_{c} =10 [/latex] in Figure 12.4 are actually better than the corresponding responses in Fig. 12.3 because the former have shorter settling times and smaller maximum deviations.
This improvement is due to the larger values of [latex]K_{c} [/latex] for case (b).
The Simulink diagram for this example is quite simple, as shown in Fig. 12.5. (See Appendix C for a Simulink tutorial.) However, the simulation results for Figs. 12.3 and 12.4 were generated using a modified controller that eliminated derivative kick (see Chapter 8).

Question 12.1: Use the DS design method to calculate PID controller setting...

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