Advanced Process Control
Tutorial Problem Set 3
Interaction Analysis and Multi-Loop Control
1. For the liquid storage system shown in the drawing, it is desired to control liquid levels
h1 and h2 by adjusting volumetric ‡ow rates q1 and q2 . Flow rate q6 is the major load
variable. The ‡ow-head relations are given by
p
p
q3 = cu1 h1
q5 = cu2 h2
q4 = k(h1 h2 )
where cu1; cu2 and k are constants.
Figure 1: Two tank system
2. For the liquid-level storage system in Exercise 1:
(a) Derive a transfer function model of the form,
C(s) = Gp (s)M (s) + GL (s)L(s)
where L is the load variable and GL is a 2
1 matrix of load transfer functions.
(b) Draw a block diagram for a multi-loop control system based on the following
pairing:h1 q1 =h2 q2 . Do not attempt to derive transfer functions for the
transmitters,control valves, or controllers.
3. For the ‡ow- pressure process shown in Fig. 2, desired to control both pressure P1 and
‡ow rate F . The manipulated variables are the stem positions of the control valves,
M1 and M 2 . For simplicity assume that the ‡ow-head relations for the two valves are
given by
F = 20M1 (P0 P1 )
F = 30M2 (P1
P2 )
The nominal steady-state conditions are F = 100 gal /min., P0 = 20 psi, P1 = 10 psi,
and P2 = 5 psi. Use the RGA approach to determine the best controller pairing.
4. A blending system is shown in the drawing. Liquid level h and exit composition c3 are
to be controlled by adjusting ‡ow rates q1 and q3 . Based on the information below, do
the following:
1
Figure 2: Flow-pressure process
(a) Derive the process transfer function matrix, GP (s).
(b) If a conventional multi-loop control system is used which controller pairing should
be used? Justify your answer.
(c) Obtain expression for the ideal decouplers T21 (s) and T12 (s) in the con…guration
of Fig. 3
Figure 3: Blending system process
AVAILABLE INFORMATION
i The tank is 3 ft. in diameter and is perfectly mixed.
ii Nominal steady-state values are:
h = 3 ft.
q 3 = 20 ft.3 /min.
c1 = 0:4 mole / ft.3
c2 = 0:1 mole / ft.3
q 1 = 10 ft.3 / min.
iii The density of each process stream remains constant at
= 60 lb./ ft.3 :
iv The primary load variable is ‡ow rate q2 :
v Inlet compositions c1 and c2 are constant.
vi The transmitter characteristics are approximated by the following transfer functions
2
with time constants in minutes:
4
Gm11 (s) =
(mA=f t:)
0:1S + 1
100
(mAf t:3 =mole)
0:2 s + 1
Gm22 (s) =
5. A multivariable system has the following state-space model
dx
3
=
1
dt
y = Ix
2
4
x+
2 0
0 1
u
(a) Find the eigenvalues of the matrix A, the corresponding matrix of eigenvectors
1
and con…rm that for this system A =
where is the diagonal matrix of
the eigenvalues of A.
(b) Obtain the corresponding transfer function matrix model for this system.
6. The mechanistic model for a certain novel bioreactor has been linearized around a
proposed steady-state operating point, resulting in the following approximate linear
state-space model:
dx
4
=
5
dt
y = Ix
1
2
x+
1 0
0 2
u
(a) Show that the bioreactor is open-loop unstable at this operating condition.
(b) Obtain the corresponding transfer function matrix G(s) for the bioreactor at this
operating point. …nd the system poles and zeros.
(c) Certain economic consideration dictate that the bioreactor is most pro…tably operated at this unstable steady state. Under proportional-only feedback control,
using two single-loop controllers with gains kc1 and kc2 , with:
KC1
0
0 KC2
u=
x
where KC2 = 2:Find the range of values which KC1 must take for the close-loop system
to be stable.
7. A 2
2 system is modeled approximately by:
y1
y2
=
2
S +3
1
S +7
1
S +1
1=2
S +4
m1
m2
(a) Find the poles and zeros of the transfer function matrix.
(b) obtain explicit expression outputs y1 (t); y2 (t) in response to inputs m1 (t) =
m2 (t) = 1 at t > 0:Plot these responses.
3
Figure 4: Closed-loop TF block diagram
8. The transfer function of a 2
2 system is given as:
G (s) =
1
5s+1
100
3s+1
0:0001
0:2 s + 1
1
0:5 s + 1
(a) Set s = 0 and obtain the steady-state gain matrix K = G(0).
(b) First …nd the eigenvalues of K, then …nd its singular values. Obtain the ratio
of the larger to the smaller singular values and compare it to the corresponding
ratio of the eigenvalues. From your knowledge of what does the value you have
obtained imply about the "conditioning" of this process?
(c) By examining the steady-state gain matrix for this system carefully, comment on
the relative e¤ectiveness of the two process input variables in a¤ecting the two
output variables
9. A two-input / two-output system whose transfer function matrix model is as shown
below:
1
2
y1
m1
= s +4 1 3s 1+ 1
m2
y2
s+1
2s + 1
is under feedback control, using the y1 m1 = y2 m2 pairing so that the process transfer
function matrix,G(s);is as indicated above; the controller, Gc has been chosen has two
pure proportional controllers,i.e.:
K c1 0
0 K c2
Gc =
and it is desired to use the following speci…c controller parameters: KC1 = 1; KC2 = 5:
(a) First show that with the given controller parameters, the close-loop system will
be unstable
(b) Shown that with the same controller parameters, if the loop pairing is switched.
(c) Find the appropriate transfer function matrix …rst, since the transfer function
matrix
(d) in this case is di¤erent from the one used in part (a).
4
Figure 5: Closed Loop Transfer Function
10. An approximate model identi…ed for an experimental, laboratory scale continuous
stirred-tank polymerization rector is:
y1
y2
=
0:81
2:6s + 1
0:55
4:3s + 1
0
0:013
3:5s + 1
m1
m2
where,in terms of deviations from their respective steady states, the input and output
variables are:
y1 = Polymer production rate(gm / min.)
y2 = Weight average molecular weight
m1 = Monomer ‡owrate (gm / min.)
m2 = Chain transfer agent ‡owrate (gm / min.)
Not included in the model is the e¤ect of the catalyst ‡owrate; for the experimental
system this was permanently set at …xed value.
(a) First study the given transfer function matrix; does it shown any evidence of
interaction between the process variable? Which output variable is expected to
be more susceptible to the e¤ect of interactions and why?
(b) Calculate the RGA for this process. What does this suggest in terms of input/
output pairing and the loop interactions resulting from such a pairing scheme?
Input your results in light of your initial assessment in part (a).
11. A certain multivariable system has three outputs y1 ; y2 and y3 which can be controlled
by any of four available inputs m1 ; m2 ; m3 ;and m4: Through pulse testing, the following
3 4 transfer function matrix model was obtained:
3 2 m 3
2
3 2 0:5e 0:2s
0:07e 0:3s
0:04e 0:3s
0:01e 0:5s
1
y1
3s + 1
2:5s + 1
2:5s + 1
10s + 1
6
7
m
0:1s
7
6
0:5e
6 2 7
4 y2 5 = 4
0
0
0
2:1s + 1 5 4 m 5
3
0:003e 0:2s
0:006e 0:4s
0:004e 0:5s
y3
0
1:5s + 1
s+1
1:6s + 1
m4
Which input / output pairing con…guration is expected to give the best results for a
multiple single loop strategy?
[Hint:one input/output pair is obvious and can be detected by inspection; eliminate
this to reduce the problem to a 2 3 loop pairing problem.]
5
12. The following is a transfer function model representing the dynamic behavior of certain
portions of a hot oil fractionator:
2
3
m1
"
#
4:05e 27s
1:77e 28s
5:88e 27s
1:44e 27s
6 m2 7
y1
50s
+
1
60s
+
1
50s
+
1
40s + 1
6
7
= 4:38e 20s 4:42e 22s
7:2
1:26
4 m3 5
y2
33s + 1
44s + 1
19s + 1
32s + 1
m4
where, in terms of deviation variables:
y1 = Top end point;
y2 = Bottom re‡ux temperature;
m1 = Top draw rate;
m2 = Side draw rate;
m3 = Bottoms re‡ux duty;
m4 = Upper re‡ux duty.
(a) Which of these four inputs should be paired with the two outputs?
(b) Compute Ziegler-Nichols’s PI controller tunings for the chosen pairing.
(c) Obtain expression for the dynamic decouplers T21 (s) and T12 (s). Are they realizable?
(d) Suppose you introduce steady state decouplers for the process. What are the
e¤ective transfer functions between the PI controller outputs and plant outputs?
13. The following incomplete RGA was determined experimentally(after four very long
and tedious experiments) for a 3 3 system with outputs y1 , y2 , y3 to be paired with
the inputs m1 , m2 , m3: Is there enough information to determine an input / output
pairing? If so what input / output pairing is recommended?
2
3
?
?
0:6
4 ?
0:8
? 5
0:2
0:2
?
6