The benefits of using corrected predictions will be illustrated by a simple example, a first-order-plus-time-delay model: Y(s) / U(s) = 5 e^-2s / 15s+1 Assume that the disturbance transfer function is identical to the process transfer function, Gd(s) = Gp(s). A unit change in u occurs at time t = 2

Question

The benefits of using corrected predictions will be illustrated by a simple example, a first-order-plus-time-delay model:
[latex]\frac{Y(s)}{U(s)}=\frac{5e^{-2s} }{15s+1}[/latex] (20-34)

Assume that the disturbance transfer function is identical to the process transfer function, [latex]G_{d} (s)=G_{p}(s)[/latex] . A unit change in [latex]u[/latex] occurs at time [latex]t = 2 min[/latex] , and a step disturbance, [latex]d = 0.15[/latex], occurs at [latex]t = 8 min[/latex]. The sampling period is [latex]\Delta t = 1 min[/latex].
(a) Compare the process response [latex]y(k)[/latex] with the predictions that were made 15 steps earlier based on a step-response model with [latex]N = 80[/latex]. Consider both the corrected prediction [latex] ilde{y} (k)[/latex] and the uncorrected prediction [latex]\hat{y} (k)[/latex] over a time period, [latex]0 ≤ k ≤ 90[/latex].
(b) Repeat (a) for the situation where the step-response coefficients are calculated using an incorrect model:
[latex]\frac{Y(s)}{U(s)}= \frac{4e^{-2s} }{20s+1}[/latex] (20-35)

Accepted Answer

The output response to the step changes in u and d can be derived from Eq. 20-34 using the analytical techniques developed in Chapters 3 and 4. Because [latex] heta = 2 min[/latex] and the step in u begins at [latex]t = 2 min, y(t)[/latex] first starts to respond at [latex]t = 5 min[/latex]. The disturbance at [latex]t = 8 min[/latex] begins to affect y at [latex]t = 11 min[/latex]. Thus, the response can be written as

[latex]y(t) =0 for t ≤ 4 min[/latex]

[latex]y(t) = 5(1)(1 − e^{-(t-4)/15} ) for 4 < t ≤ 10 min[/latex]

[latex]y(t) = 5(1)(1 − e^{-(t-4)/15}) for t > 10 min[/latex]

[latex] + 5(0.15)(1 − e^{−(t−10)∕15)})[/latex] (20-36)

The 15-step-ahead prediction, [latex]\hat{y}(k + 15)[/latex], can be obtained using Eq. 20-20 with [latex]j = 15[/latex] and [latex]N = 80[/latex]. The corrected prediction, [latex]\hat{y} (k + 15)[/latex], can be calculated from Eqs. 20-29 and 20-30 with [latex]j = 15[/latex]. But in order to compare actual and predicted responses, it is more convenient to write these equations in an equivalent form:

[latex] \hat{y}(k+j) = \sum\limits_{i=1}^{j}{S_{i}\Delta u (k+j-i)+\hat{y} ^{\circ } (k+j) }[/latex] (20-20)

[latex] ilde{y}(k+j) ≜ \hat{y} (k+j) +b(k+j)[/latex] (20-29)

[latex]b(k+j)=y(k)-\hat{y} (k)[/latex] (20-30)

[latex] ilde{y} (k) ≜ \hat{y} (k)+b(k)[/latex] (20-37)
[latex]b(k) = y (k-15)-\hat{y} (k-15)[/latex] (20-38)

(a) The actual and predicted responses are compared in Fig. 20.7. For convenience, the plots are shown as lines rather than as discrete points. After the step change in u at [latex]t = 2 min[/latex] (or equivalently, at [latex]k = 4[/latex]), the 15-step-ahead predictions are identical to the actual response until the step disturbance begins to affect [latex]y(k)[/latex] starting at [latex]k = 11[/latex]. For [latex]k > 10, \hat{y} (k) < y(k)[/latex] because \hat{y} (k) does not include the effect of the unknown disturbance. Note that [latex] ilde{y}(k) = \hat{y} (k)[/latex] for [latex]10 < k < 25[/latex] because [latex]b(k) = 0[/latex] during this period. For [latex]k > 25, b(k) ≠ 0[/latex] and [latex] ilde{y}(k)[/latex] converges to [latex]y(k)[/latex]. Thus, the corrected prediction[latex] ilde{y}(k)[/latex] is more accurate than the uncorrected prediction, [latex]\hat{y} (k)[/latex].

(b) Figure 20.8 compares the actual and predicted responses for the case of the plant-model mismatch in Eqs. 20-34 and 20-35. The responses in Fig. 20.8 are similar to those in Fig. 20.7, but there are a few significant differences. Both of the predicted responses in Fig. 20.8 differ from the actual response for [latex]t > 4[/latex], as a result of the model inaccuracy. Figure 20.8 demonstrates that the corrected predictions are much more accurate than the uncorrected predictions even when a significant plant-model mismatch occurs. This improvement occurs because new information is used as soon as it becomes available.

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