Find an expression for the amount of derivative kick that will be applied to the process for the position form of the PID digital algorithm (Eq. 8-25) if a set-point change of magnitude Δysp is made between the k − 1 and k sampling instants. (a) Repeat for the proportional kick, that is, the sudden

Question

Find an expression for the amount of derivative kick that will be applied to the process for the position form of the PID digital algorithm (Eq. 8-25) if a set-point change of magnitude [latex]\Delta y_{s p}[/latex] is made between the [latex]k-1[/latex] and [latex]k[/latex] sampling instants.

[latex]p_{k}=\bar{p}+K_{c}\left[e_{k}+\frac{\Delta t}{ au_{I}} \sum\limits_{j=1}^{k} e_{j}+\frac{ au_{D}}{\Delta t}\left(e_{k}-e_{k-1} ight) ight][/latex] (8-25)

(a) Repeat for the proportional kick, that is, the sudden change caused by the proportional mode.

(b) Plot the sequence of controller output values at the [latex]k-1[/latex], [latex]k, \ldots[/latex] sampling times for the case of a set-point change of [latex]\Delta y_{s p}[/latex] magnitude made just after the [latex]k-1[/latex] sampling time if the controller receives a constant measurement [latex]\bar{y}_{m}[/latex] and the initial set point is [latex]\bar{y}_{s p}=\bar{y}_{m}[/latex]. Assume that the controller output initially is [latex]\bar{p}[/latex].

(c) How can Eq. 8-25 be modified to eliminate derivative kick?

Accepted Answer

From inspection of Eq. [latex]8-25[/latex], the derivative kick [latex]=K_{c} \frac{ au_{D}}{\Delta t} \Delta r[/latex]

[latex]p_{k}=\bar{p}+K_{c}\left[e_{k}+\frac{\Delta t}{ au_{I}} \sum\limits_{j=1}^{k} e_{j}+\frac{ au_{D}}{\Delta t}\left(e_{k}-e_{k-1} ight) ight][/latex] (8-25)

a) Proportional kick [latex]=K_{c} \Delta r[/latex]

b)

[latex]\begin{aligned}&e_{1}=e_{2}=e_{3}=\ldots=e_{k-2}=e_{k-1}=0 \&e_{k}=e_{k+1}=e_{k+2}=\ldots=\Delta r \&p_{k-1}=\bar{p} \&p_{k}=\bar{p}+K_{c}\left[\Delta r+\frac{\Delta t}{ au_{I}} \Delta r+\frac{ au_{D}}{\Delta t} \Delta r ight]\end{aligned}[/latex]

[latex]p_{k+i}=\bar{p}+K_{c}\left[\Delta r+(1+i) \frac{\Delta t}{ au_{I}} \Delta r ight] \quad, \quad i=1,2, \ldots[/latex]

c) To eliminate derivative kick, replace [latex]\left(e_{k}-e_{k-1} ight)[/latex] in Eq. [latex]8-25[/latex] by [latex]-\left(y_{k}- y _{k-1} ight)[/latex]. (Note the minus sign.)

Question 8.8: Find an expression for the amount of derivative kick that wi...

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