An electronic PID temperature controller is at steady state with an output of 12 mA. The set point equals the nominal process temperature initially. At t = 0, the error signal is increased at the rate of 0.5 mA/min (equivalent to a rate of 2∘F/min).If the current settings are Kc = 2 (dimensionless)

Question

An electronic PID temperature controller is at steady state with an output of [latex]12 mA[/latex]. The set point equals the nominal process temperature initially. At [latex]t=0[/latex], the error signal is increased at the rate of [latex]0.5 mA / min[/latex] (equivalent to a rate of [latex]2^{\circ} F / min[/latex] ). If the current settings are

[latex]\begin{aligned}K_{c} &=2 ext { (dimensionless) } \ au_{I} &=1.5 min \ au_{D} &=0.5 min\end{aligned}[/latex]

(a) Derive an expression for the controller output [latex]p(t)[/latex].

(b) Repeat (a) for [latex] au_{D}=0[/latex] (i.e., a PI controller).

(c) Plot the two controller outputs and qualitatively discuss their differences.

Accepted Answer

(a) The error signal can be described by:

[latex]\begin{aligned}&e(t)=0.5 t \&E(s)=\frac{0.5}{s^{2}}\end{aligned}[/latex]

The PID controller transfer function is given by (Eq. 8-14):

[latex]\frac{P^{\prime}(s)}{E(s)}=K_{c}\left[1+\frac{1}{ au_{I} s}+ au_{D} s ight][/latex] (8-14)

[latex]\frac{P^{\prime}(s)}{E(s)}=K_{C}\left[1+\frac{1}{ au_{I} s}+ au_{D} s ight][/latex]

Substituting gives the controller output:

[latex]\begin{aligned}&P^{\prime}(s)=\frac{0.5 K_{C}}{s^{2}}\left[1+\frac{1}{ au_{I} s}+ au_{D} s ight] \&p^{\prime}(t)=0.5 K_{C}\left[t+\frac{1}{2 au_{I}} t^{2}+ au_{D} S(t) ight]\end{aligned}[/latex]

Substituting numerical values and adding [latex]\bar{p}=12 mA[/latex] gives:

[latex]p_{PID}(t)=12+\frac{1}{3} t^{2}+t+0.5 S(t)[/latex]

(b) The equation for a PI controller is obtained by setting [latex] au_{ D }[/latex] to zero.

[latex]p_{PI}(t)=12+\frac{1}{3} t^{2}+t[/latex]

(c) The plot of the controller response for both controllers is shown in Fig. S8.7.

The two controllers have similarly-shaped responses. The difference is the sudden jump at [latex]t=0[/latex] that occurs with the PID controller as a result of the derivative term. When the set point begins to change with a constant slope, there is a step change in the error derivative from 0 to [latex]0.5[/latex]. The derivative term in the controller gives it a jumpstart right when the setpoint begins to change that the PID controller does not have,

Question 8.7: An electronic PID temperature controller is at steady state ...

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