A process has the transfer function, Y(s) / U(s) = e^-s / (10s+1)(5s+1) (a) Use Eq. 20-65 to calculate the controller gain matrix, Kc, for Q = I, R = 0 two cases:(i) P = 3, M = 1 (ii) P = 4, M = 2 Assume that N = 70, Δt = 1 and that u is unconstrained for each case. (b) Compare the set-point respon

Question

A process has the transfer function,

[latex] \frac{Y(s) }{U(s)} =\frac{e^{-s} }{(10s+1)(5s+1)}[/latex]

(a) Use Eq. 20-65 to calculate the controller gain matrix, [latex]K_{c}[/latex], for Q = I, R = 0 two cases:

[latex]K_c riangleq (S^TQS+R)^{-1}S^TQ[/latex] (20-65)
(i) P = 3, M = 1
(ii) P = 4, M = 2

Assume that N = 70, Δt = 1 and that u is unconstrained for each case.

(b) Compare the set-point responses of two MPC controllers and a digital PID controller with Δt = 0.5 and ITAE set-point tuning (Chapter 12): [latex]K_{c}[/latex] = 2.27, [latex] au _{I}[/latex]= 16.6 and [latex] au _{D}[/latex] = 1.49. Compare both y and u

(c) Repeat (b) for a unit step disturbance and a PID controller with ITAE disturbance tuning: [latex]K_{c}[/latex] = 3.52, [latex] au _{I}[/latex] = 6.98, and [latex] au _{D}[/latex] = 1.73.

Accepted Answer

(a) The step-response coefficients are obtained by evaluating the step response at the sampling instants, t = iΔt = i (because Δt = 1):

[latex]S_{1}=0[/latex]

[latex]S_{i} =1-2e^{-0.1(i-1)} +e^{-0.2(i-1)} for i = 2, 3, . . . , 70[/latex]

The controller matrix K[latex]_{c}[/latex] for each case is shown in Table 20.2. Note that the dimensions of K are different for the two cases, because K[latex]_{c}[/latex] has dimensions of rM × mP, as noted earlier. For this SISO example, r = m = 1, and the values of M and P differ for the two cases.

Table 20.2 Feedback Matrices Kc for Example 20.5

For:P=3 and M =1:	[latex]K_{c} = \left [ \begin{matrix} 0 & 7.79 & 28.3 \end{matrix} ight ][/latex]
For P = 4 and M = 2:	[latex]K_{c} = \left [ \begin{matrix} 0 & 33.1 & 48.8 &-13.4 \ 0 & -71.4 & -97.4 & 57.3 \end{matrix} ight ][/latex]

(b) The unit step response can be derived analytically using Lapace transforms:

[latex]y(t) =0 for t ≤ 1[/latex]

[latex]y(t)=1-2e^{-0.1(t-1)} +e^{-0.2(t-1)} for t > 1[/latex]

Figure 20.12 compares the y and u responses for a unit set-point change. The two MPC controllers provide superior output responses with very small settling times, but their initial MV changes are larger than those for the PID controller. (Note the expanded time scale for u.)

(c) For the step disturbance, the output responses for the MPC controllers in Fig. 20.13 have relatively small maximum deviations and are nonoscillatory. By comparison, the PID controller results in the largest maximum deviation and an oscillatory response. Of the two MPC controllers, the one designed using P = 3 and M = 1 provides a slightly more conservative response.

Question 20.5: A process has the transfer function, Y(s) / U(s) = e^-s / (1...

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