Theoretical Justification of Brosilow 's Criteria
3.6 Numerical Example 2: High-Purity Distillation Col- umn
1 Meas. 2 Meas. 3 Meas. 4 Meas. 5 Meas
Measurement CandidatA
Figure 3.9. The 2-Norm of "Worst-Case" Steady-State Output Error for Various Measurement Sets Under Uncertainty B and "p-Optimal" Controller
3.6 Numerical Example 2: High-Purity Distillation Col-
as.
Measurement Set
Figure 3.10. The 2-Norm of "Worst-Case" Steady-State Output Error for Various Measurement Sets Under Uncertainty A and "p-Optimal" Controller
vated measurement error: uncompensated pressure variation. The following set of disturbances/noise is considered:
Feed flowrate (F) Feed composition ( z F )
Uncompensated pressure variation (P) Measured Variables
Measurements are usually not limited to a specific number although it is common to use two tray temperatures for two-point composition control. In this example, for the sake of simplicity, we restrict ourselves to two tray temperatures (T, and Tb). In addition, for brevity of presentation, we consider only the placements symmetric with respect t o the feedtray (such as tray #l/tray #41, tray #%/tray #40, and so on).
1 Meas. 2 Meas. 3 Meas. 4 Meas. 5
t
Measurement Set
Figure 3.11. The 2-Norm of "Worst-Case" Steady-State Output Error for Various Measurement Sets Under Uncertainty B and "Least-Square" Controller
This is logical since the column is symmetric with respect to the feedtray.
Uncertainty
We limit ourselves to uncertainty in the manipulated variables. They have been shown to be the dominant uncertainty for high-purtity distillation columns [60]. We choose the same uncertainty weight Wz that Skogestad & Morari [60] used in their study:
Performance and Disturbance Weight
The performance weight W p and the disturbance weight Wd are chosen as follows:
where
As usual, much tighter specifications are imposed in the low frequency region in order to ensure good steady-state response.
Steady-State Performance
We apply the Design-Dependent Screening Tool #1 for LQG/MPC of (3.89) to re- duce the number of measurements to consider. The plot of the left-hand side of the inequality (3.89) vs. measurement sets is shown in Figure 3.14(a). It represents the measure of the "worst-possible" performance when the controller is designed yielding no steady-state offsets in compositions nominally (in the absence of uncertainty and measurement error). The measurement set of T7 and T35 shows the best steady-state performance. In fact, it is the only measurement set that satisfies the condition [3.89).
This result can be interpreted physically. The temperatures measured close to the reboiler and the condenser have poor signal-to-noise ratio because the gains from dis- turbances to these measurements are "small." On the other hand, the measurements far away from the reboiler and the condenser are sensitive to model uncertainty since the relationships between the end-point compositions and the measurements become less direct. Hence, placement of the temperature sensors involves a compromise be- tween these two factors. This is apparent from the plots shown in Figure 3.14(a) that represent the values for the left-hand side of the inequality (3.89) when measurement error (uncompensated pressure variation)
/
model uncertainty are neglected. The measurement set T7/T35 is apparently the best compromise between the signal-to-noise ratio and the sensitivity to model uncertainty. Note that neglecting either the model uncertainty or the measurement error would have resulted in a wrong choice of measurements. Figure 3.14(b) represents the condition numbers of the s teady-state gain matrices from the disturbances to the measurements (Gsd(0)). Note that the con- dition number (Brosilow's criterion) does not reflect the measurements' sensitivity to the uncertainty correctly in this particular problem.
O u t p u t E s t i m a t i o n Based I M C Controller Design for R o b u s t Performance To verify the result, we design controllers for the following three candidates: Tl/T41, T7/T35 and Tl7/TZ5. For controllers, dynamic output estimators designed via Kalman filter design was combined with an IMC controller. IMC filters were designed separately for each candidate using the robust performance bounds derived for tf(FzMc(jw)) and @ ( I
-
FzMc(jw)). The design method is explained in detail in Chapter 4. The robust performance bounds on a ( F z ~ ~ ( j w ) ) and *(I - F z ~ c ( j w ) ) for the measurement set T7/T35 are shown in Figure 3.15(a). The bounds are "feasi- ble" since the following transfer function meets at least one of the bounds at every frequency as we can see from Figure 3.15(a):The bounds for the other two candidates were not "feasible" and the IMC filter was designed so that the bounds are satisfied for as wide a frequency range as possible.
The p-plot for robust performance (Figure 3.15(b)) shows that robust performance is achieved for the measurement set T7/T3.5. Although not shown, the SSVs for the other two candidates exceeded 1 in some frequency regions, implying robust performance is not achieved. Figure 3.15 shows the simulated responses of z~ and yr, to unit step disturbances in z~ and F and a measurement noise in the form of a pseudo-random binary signal of unit magnitude filtered through W,. The specific multiplicative
Figure 3.12. High-Purity Distillation Column Dlstilla
Y D
input uncertainty (i. e., WzAz) used for the simulation is
['r
-02]. he simulrtions confirm the physical interpretation given earlier.Feed (F,Z F ) C,
B o i l u p ( V ) B o t t o m (B)
(boiler heat i n B
a
af
v
\ .
CC
Figure 3.13. Control Problem In High-Purity Distillation Column
MEASUREMENT SET (TRAY*)
(a) Steady-State Robust Performance Measure
(b) Condition Number of G
ysdFigure 3.14. Robust Performance Measure at Steady St ate
- - a - - -
-
bound on a ( I-
F I M c ( j w ) ) bound on L S ( F Z M C ( ~ W ) )- - -
I1-
s ( j w ) l Is(jw>I
(a) Meeting the Norm-Bounds (b) SSV for Robust Performance Figure 3.15. Meeting the Robust Performance Norm-bounds on LS(I - F z ~ c ( j w ) ) and o ( ~ & C ( j w ) ) with FIMC = g ( s ) I for the Measurement Set T7/T&,
D A I I . 0 L I U . - - > *.
I O L I D L I . . - - > y Q
(a) Measurement Set Tray #1/Tray #41
o . . . . o L I . . - - > *.
S O L I D L X . 1 - - > y o
(b) Measurement Set Tray #7/Tray #35
O . B I . 0 L X . . --•
.,
. O L X D L I . . - - > y o
100 I# 3 0 0 4 0 0
? X U .
(c) Measurement Set Tray #17/Tray #25
Figure 3.16. Simulated Responses of xs and y~ to the Unit Step Disturbances in z~
and F with 20 % Input Uncertainty and Pseudo-Random Binary Measurement Noise