– 12S-1– SUPPLEMENT TO CHAPTER 12 FLOWSHEET CONTROLLABILITY ANALYSIS 12S.0 OBJECTIVES This chapter supplement introduces quantitative measures for controllability assessment to be used when developing the base-case design and in the detailed design stage (Stages 2 and 3, Table 12.1) and highlights how their integration into the design process can help to generate improved flowsheets that satisfy control performance criteria. At this point, the process creation stage has been completed and several promising process flowsheets exist. As they are evaluated, the control objectives are considered as constraints, the latter including: • Adequate disturbance resiliency, that is, the ability to reject disturbances quickly enough to meet specifications • Insensitivity to model uncertainty, that is, the ability to control easily, and to provide adequate closed-loop performance, with relatively insensitivity to model inaccuracies. An approach is introduced to screen the potential designs as early as possible, to identify the most promising designs for rigorous testing in stage 4, in which plantwide controllability assessment is completed. As demonstrated in this chapter supplement, it is important to verify the approximate analysis using rigorous dynamic simulation. Detailed multimedia instruction on the use of ASPEN HYSYS for dynamic simulation is available as part of the multimedia support that may be downloaded from the Wiley web site associated with this book. UNISIM and CHEMCAD could be used also for dynamic simulation. It is assumed that the reader is familiar with the basic concepts of linear systems theory. This material is covered typically in an introductory course on process dynamics and control at the undergraduate level. The subjects in that course that are prerequisite to understanding the concepts in this chapter supplement are: 1. Basic linear matrix theory, linearization, complex numbers, and Laplace and Fourier transforms. Note that Section 12S.1 provides some of this background material.
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– 12S-1–
SUPPLEMENT TO CHAPTER 12 FLOWSHEET CONTROLLABILITY ANALYSIS
12S.0 OBJECTIVES
This chapter supplement introduces quantitative measures for controllability assessment to
be used when developing the base-case design and in the detailed design stage (Stages 2 and 3,
Table 12.1) and highlights how their integration into the design process can help to generate
improved flowsheets that satisfy control performance criteria. At this point, the process creation
stage has been completed and several promising process flowsheets exist. As they are evaluated,
the control objectives are considered as constraints, the latter including:
• Adequate disturbance resiliency, that is, the ability to reject disturbances quickly enough to
meet specifications
• Insensitivity to model uncertainty, that is, the ability to control easily, and to provide
adequate closed-loop performance, with relatively insensitivity to model inaccuracies.
An approach is introduced to screen the potential designs as early as possible, to identify
the most promising designs for rigorous testing in stage 4, in which plantwide controllability
assessment is completed. As demonstrated in this chapter supplement, it is important to verify the
approximate analysis using rigorous dynamic simulation. Detailed multimedia instruction on the
use of ASPEN HYSYS for dynamic simulation is available as part of the multimedia support that
may be downloaded from the Wiley web site associated with this book. UNISIM and CHEMCAD
could be used also for dynamic simulation.
It is assumed that the reader is familiar with the basic concepts of linear systems theory.
This material is covered typically in an introductory course on process dynamics and control at the
undergraduate level. The subjects in that course that are prerequisite to understanding the concepts
in this chapter supplement are:
1. Basic linear matrix theory, linearization, complex numbers, and Laplace and Fourier
transforms. Note that Section 12S.1 provides some of this background material.
– 12S-2–
2. Pole and zero positions in the complex plane, and their impact on the time-domain
response of linear systems.
3. Linear stability theory and the impact of feedback.
4. Tuning of single-input, single-output controllers (P, PI, and PID controllers). Note that
Section 12S.4 provides instruction on model-based PI-controller tuning.
Key concepts relating to linear process models are reviewed in the first section of the
chapter. For deeper coverage, the reader is referred to the following undergraduate-level texts:
Bequette, B.W., Process Dynamics: Modeling, Analysis, and Simulation, Prentice Hall, Englewood
Cliffs, NJ, (2003).
Luyben, W.L., Process Modeling, Simulation and Control for Chemical Engineers, 2nd ed.,
McGraw-Hill, New York (1990).
Ogunnaike, B.A., and W.H. Ray, Process Dynamics, Modeling and Control, Oxford Univ. Press,
New York (1994).
Seborg, D.E., T.F. Edgar, and D.A. Mellichamp, Process Dynamics and Control, Wiley, New York
(1989).
Stephanopoulos, G., Chemical Process Control, Prentice-Hall, Englewood Cliffs, NJ (1984).
This supplement to Chapter 12,
1. Explains how to generate linear process models in their standard forms.
2. Defines quantitative measures that are used to analyze the controllability and resiliency
(C&R) of process flowsheets, and shows how to implement them using MATLAB.
3. Describes a method to carry out C&R analysis using the results of steady-state process
simulations.
4. Shows how to use quantitative analysis with steady-state and dynamic relative gain
arrays (RGA and DRGA) to reliably select control loop pairings and to use the IMC
model-based approach to provide preliminary tuning of single-loop PI controllers.
5. Analyzes, in Section 12S.5, selected case studies in Chapter 12 to demonstrate the
utility of the quantitative methods. For completeness, these analyses are verified with
dynamic simulations using single-loop PI controllers.
– 12S-3–
After reading this chapter, the student should
1. Be able to compute the frequency-dependent process transfer functions P , dP using
MATLAB, given a linear model in one of its standard forms.
2. Generate the C&R measures: relative-gain array (RGA), and disturbance cost (DC),
given the matrices P s and dP s, describing the effects of the manipulated
variables and disturbances on the process outputs, using MATLAB.
3. Select the appropriate pairings for a decentralized control system for a process using the
static and dynamic RGAs and appropriate resiliency measures, and provide preliminary
tuning using IMC-PI tuning rules.
4. Perform C&R analysis to select between alternative process configurations, given the
results of process simulations using linearized models.
12S.1 GENERATION OF LINEAR MODELS IN STANDARD FORMS
In this chapter supplement, several methods are described to assist the designer in rejecting
designs that do not provide acceptable closed-loop performance, using models linearized about a
steady state. These are generated by expressing the open-loop response of the process outputs, ys,
in terms of the variations of the inputs, us, and disturbances, ds:
sdsPsusPsy d+= (12S.1)
The procedure for deriving the linear state-space model and the input-output transfer
function model in Eq. (12S.1) involves the following steps:
Step 1. The nonlinear state and output equations are derived from the material and energy balances
that model the process. These are expressed in the form:
duxgy
duxfdt
xd
,,
,,
=
= (12S. 2)
– 12S-4–
where x is a vector of nx state variables, y is a vector of ny output (measured) variables,
u is a vector of nu manipulated variables, d is a vector of nd disturbances, and f and g
are vectors of nx and ny nonlinear functions, respectively.
Step 2. The state and output equations are solved at a stationary (steady) state that is defined either
in terms of the desired state variable values or those of the input variables:
***
***d,u,xgy
d,u,xf=
= 0 (12S.3)
where the stationary point is at *** dduu,xx === and . The solution of Eq. (12S.3)
requires that the degrees of freedom of the process be resolved through the specification of
nu+ nd values.
Step 3. The equations are linearized in the vicinity of the desired stationary point, by a Taylor
series expansion of Eq. (12S. 2):
( ) ( ) ( ) ( ) ( ) ( )
* * * * * *
* * * * * *
, , h.o.t.
, , h.o.t.U D
U D
d x f x u d A x x B u u B d ddty g x u d C x x D u u D d d
≅ + − + − + − +
≅ + − + − + − + (12S.4)
Note that the linear approximation is obtained by ignoring the higher order terms (h.o.t.)
of the Taylor series expansion. The matrices UDU D,C,B,B,A and DD are the Jacobian
matrices of appropriate dimension evaluated at the stationary point, defined as follows:
*** d,u,xj
ij,i x
faA∂∂
≡= *** d,u,xj
ij,i x
gcC∂∂
≡=
*** d,u,xj
ij,i,UU u
fbB∂∂
≡= *** d,u,xj
ij,i,UU u
gdD∂∂
≡=
*** d,u,xj
ij,i,DD d
fbB∂∂
≡= *** d,u,xj
ij,i,DD d
gdD∂∂
≡=
Step 4. The linearized equations are formulated in terms of perturbation variables that express the
deviation from the stationary point (or steady state): *xxx −= , *yyy −= , *uuu −= and
– 12S-5–
*ddd −= . Substituting the perturbation variables into Eqs. (12S.4) and ignoring higher-
order terms:
dDuDxCy
dBuBxAdt
xd
DU
DU
++≅
++≅ (12S.5)
Eqs. (12S.5) constitute the linear state-space representation of the system.
Step 5. The linearized equations are transformed into the Laplace domain:
sdsPsusPsy d+= , (12S.1)
where ( ) UU DBAIsCsP +−⋅= −1 and ( ) DDd DBAIsCsP +−⋅= −1 are matrices of the
appropriate dimension. Eq. (12S.1) constitutes the input-output transfer function
representation of the linear system.
As an example, the procedure for generating linear models in standard form is demonstrated
for an exothermic reactor, whose complete analysis is presented in Case Study 12S.1 of Section
12S.5.
Example 12S.1 Standard Linear Models for an Exothermic Reactor.
A continuous-stirred-tank reactor for the production of propylene glycol is analyzed in Case Study
12S.1, in Section 12S.5 below. Approximate linear models for the reactor are generated using the
five-step procedure as follows:
Step 1. Define the State and Output Equations. The hydrolysis of propylene oxide (PO) to
propylene glycol is an exothermic reaction catalyzed by H2SO4:
CH2-O-CH-CH3 + H2O → CH2OH-CH-OH-CH3
When water is supplied in excess, the reaction is second order with respect to the propylene oxide
concentration and zero order with respect to the water concentration. Its rate constant exhibits an
Arrhenius dependence on temperature, with k0 = 3.294×1026 m3/(kmol-h) and E = 1.556×105
kJ/kmol. Furthermore, it is customary to dilute the PO feed with methanol (MeOH), while the
H2SO4 catalyst enters the reactor with the feed. Operating conditions are sought for carrying out
– 12S-6–
this liquid-phase reaction in a 47-ft3 continuous-stirred-tank reactor (CSTR), with the liquid holdup
at 85% of its total volume (1.135 m3). The liquid feeds are fed at 23.9 oC, with one consisting of
18.712 kmol/h of PO and 32.73 kmol/h of MeOH. The water feed rate is from 160 – 500 kmol/h
(2.84 – 8.88 m3/h), selected to moderate the reactor temperature. To reduce the risk of vaporization,
the reactor is operated at a pressure of 3 bar. Under these conditions, the transients for the PO
concentration, CPO [kmol/m3], and temperature, T [oC], are determined by solving the following
species and enthalpy balances:
( ) 20PO
wPOin,POPO CTkV
qqCVdt
dC−
+−
ℑ= (12S.6)
( ) ( )( )V
TTqqHCTkCdt
dT wPO
P
0021 −+−∆−= (12S.7)
where, ( )2273.TR/EoekTk +−= m3/(kmol-h), R = 8.314 kJ/kmol-K, the molar flow rate of PO in
the feed, ℑPO,in = 18.712 kmol/h, V = 1.135 m3, ∆H = –9×104 kJ/kmol, the organic volumetric feed
rate, q0= 2.556 m3/h, the water volumetric feed rate is qw, T0 = 23.9 oC, and cP = 3,558 kJ/ m3 oC.
Implicit in the assumption of perfect level control is the pairing between the effluent volumetric
flow rate, F, and the liquid level, L. This leaves the temperature, T, as the output, to be controlled
by the water feed rate, qw, as the manipulated variable. The disturbances to the process are the
volumetric organic feed rate, q0, and the feed temperature, T0. Thus, [ ]TPO T,Cx = , [ ]Ty = ,
[ ]wqu = and [ ]Toin,PO T,d ℑ= .
Step 2. Solve at the Steady State. The state equations are solved at the steady state. The degrees of
freedom are resolved by fixing all of the input variable values and solving the two equations for the
two unknown state variables.
( ) 020 =−+
−ℑ
POwPOin,PO CTk
VqqC
V
(12S.8)
( ) ( )( ) 01 002 =−+
−∆−V
TTqqHCTkC
wPO
P
(12S.9)
Taking *u = qw* = 5.325 m3/h and [ ] [ ]TTo*in,PO* .,.T,d 923 71218=ℑ= and solving Eqs. (12S.8)
and (12S.9) gives [ ]T* ,.x 2.48 060= . Note that the fractional conversion of PO is X = 1 –
– 12S-7–
CPO/CPO,in = 1 – 0.06/2.374 = 0.975, where CPO,in = ℑPO,in/(q0 + qw). This solution is obtained
analytically, graphically (see Case Study 12S.1), or using a numerical method (e.g., the Newton-
Raphson method).
Steps 3 and 4. Linearize in the Vicinity of the Steady State. The Jacobian matrices for the linearized
approximation are:
( )
( ) ( ) ( )
∆−∂
∂+
+−
∆−∂
∂−−
+−
=
P
*PO**w*PO*
P
*PO*
*PO**w
CCH
TTk
VqqCTk
CH
CTTkCTk
Vqq
A 20
20
2
2
601 (12S.10)
( )
−−
−=
VTT
VC
B**
*PO
U 0601 , ( )
+=
Vqq
VB*wD 00
01
601 (12S.11)
[ ]10=C , [ ]0=UD , [ ]00=DD (12S.12)
The division by 60 in each matrix is to express time in minutes instead of hours. Note also that all
of the variables are expressed in physical units. Substituting numerical values into Eqs. (12S.10)
and (12S.11):
−−=
887008229039602039
....
A ,
−−
=8596000090..
BU ,
=
115700001470
..
BD
These matrices are scaled by assuming that all outputs and manipulated variables are nominally at
50% of their ranges, and the disturbance variable values are constrained to vary in the range ∆d =
[±50% ±5 oC]T. Thus:
=
−−== −
*
*POxxxs T
CS,
....
SASA0
08870016440
435520391
*wuuUxs,U qS,..
SBSB =
−−
== −
05550078201
ℑ=
== −
50050
007000033021 *in,PO
ddDxs,D.
S,.
.SBSB
(12S.13)
– 12S-8–
These matrices relate the input (manipulated and disturbance) variables to the output (controlled)
variables, with all of the variables scaled and in perturbation variable form.
Step 5. Generate Transfer Functions. These are computed using the scaled matrices in Eq.
(12S.13), for example
( ) [ ]
−−
−−
+=+−⋅=
−−
0555007820
887001644043552039
101
1
.
..s...s
DBAIsCsP s,Us,Us
Hence,
( )( )( )164811220
1106055209490328524005550
2 +++−
=++
−−=
s.s.s..
.s.s.s.sP (12S.14)
Note that sP is a scalar transfer function, since it relates perturbations in the single manipulated
variable, wq , to those in the single process output variable, T . Note that the process zero almost
cancels the fast process pole, meaning that the response to the manipulated variable is effectively
that of a first-order lag, with a time constant of approximately 9 min.
A similar computation yields the transfer function matrix, sPd :
[ ]
( )( )( )
( )( )
++
+++
=
−−
+=
−
164811220110900680
1648112204100
00700003302
887001644043552039
101
s.s.s..
s.s..
..
.s...s
sPd
(12S.15)
The columns of sPd define the responses of T to in,POℑ and oT , respectively. Note that the
temperature response to step changes in in,POℑ is of second order, while its response to changes in
the feed temperature is effectively of first order.
The reader is referred to the Wiley website that accompanies this text for useful MATLAB
functions and scripts for the generation of linear models in their standard forms.
– 12S-9–
21.2 QUANTITATIVE MEASURES FOR CONTROLLABILITY AND RESILIENCY
The quantitative assessment of the controllability and resiliency of chemical processes has
generated considerable interest. The term resiliency was introduced by Morari (1983), who also
pioneered qualitative measures for its assessment. Furthermore, Perkins (1989) presented an
approach for the simultaneous design of processes and their control systems that addresses
plantwide controllability directly.
All of the C&R measures use the linear approximations, sP and sPd , which describe
the effects of the control variables and disturbances, respectively, on the process outputs. A
commonly used controllability measure is the relative-gain array (RGA – Bristol, 1966), which
relies only on sP . The disturbance condition number (DCN; Skogestad and Morari, 1987) and
the disturbance cost (DC; Lewin, 1996) are resiliency measures that require a disturbance model,
sPd , in addition to sP . These C&R measures are especially useful in Stages 2 and 3 of the
design process (see Table 12.1) because they do not assume a controller structure or a specific
controller design and tuning.
It is assumed that each input variable is nominally at the midpoint of its range and is
expressed in perturbation variable form, and scaled by dividing by its nominal value. For example,
if Fi is an inlet flow rate, nominally at 500 lbmol/hr, its operating range is 0 ≤ Fi ≤ 1000 lbmol/h, in
perturbation variable form, −500 ≤ Fi ≤ 500, and in scaled form, −1 ≤ Fi ≤ 1. Thus, sP and
sPd are scaled by multiplying the gains in each column by the nominal value of the appropriate
input variable. As a result, all of the scaled inputs vary over the same range [−1, 1]. Note, however,
that the RGA is scale independent, whereas the DC is input scale dependent.
– 12S-10–
Relative-gain Array (RGA)
Steady-State RGA (Bristol, 1966)
Figure 12S.1 shows the block diagram for a multiple-input, multiple-output (MIMO)
process to be controlled by two single-loop controllers. Having closed one of the loops (y1 − u1), the
controller in the second loop, which manipulates u2 based on the feedback of y2, must be tuned. A
desirable feature of this controller is to have the effective process gain remain invariant, regardless
of the action of the other control loop.
c1 p11
p21
p12
p22
u1y1
y2u2
-
Figure 12S.1 MIMO process with one control loop.
When the controller c1 is put into manual operation, i.e., when it is turned off, the process
gain as seen by controller c2 is
yu
pc
2
222
1 ,OL= (12S.16)
where u2 and y2 are the deviations of the input and output from their nominal values in the steady
state. On the other hand, when c1 is put into automatic operation, the process gain as seen by
controller c2 is
yu
p p cp c
pc
2
222 12
1
11 121
11,CL
= −+
(12S.17)
In general, for MIMO systems, a useful measure is the ratio
process gain as seen by a given controller with all other loops open process gain as seen by a given controller with all other loops closed
– 12S-11–
When this ratio is close to unity, the given controller is relatively insensitive to interaction.
Computing this ratio for the MIMO process in Figure 12S.1:
21111
11222
22
2
2
2
2
11
1
pcp
cpp
p
uy
uy
CL,c
OL,c
+−
= (12S.18)
When the top loop is closed-loop stable, and when c1 has integral action,
lim
scp c p→ +
=0 1
11
11 1 11
Therefore, the ratio at steady state is
21122211
2211
21111
11222
22
2
2
2
2
10
lim0
lim
1
1
pppppp
pcp
cpp
ps
uy
uy
s
CL,c
OL,c
−=
+−
→=→ (12S.19)
Similarly,
lims
yu
yu
p pp p p p
c OL
c CL
→=
−−0
2
1
2
1
12 21
11 22 12 21
1
1
,
,
(12S.20)
Thus, for a two-input, two-output process, the RGA is defined as
( )
2 2
2 2
1 1
1 1
1 1
1 2, ,
1 1
1 2, , 111 12 11 22 12 21
21 22 12 21 11 222 2
1 2, ,
2 2
1 2, ,
det
c OL c OL
c CL c CL
c OL c OL
c CL c CL
y yu u
y yu u p p p p
Pp p p py y
u u
y yu u
−
λ λ − Λ = = = ⋅ λ λ −
(12S.21)
In general, the RGA can be computed using
( )TPP 1−⊗=Λ (12S.22)
where ⊗ denotes the element-by-element (Schur) product.
– 12S-12–
Theorem (2 × 2 Systems Only).
If λ11 (= λ22) is positive, there exists a pair of single-input, single-output (SISO) controllers, c1 and
c2, with integral action for the loops u1 − y1 and u2 − y2 such that the loops are stable by themselves
and together. If λ11 is negative, there are no controllers that can guarantee stability by themselves
and together. In other words, to guarantee closed-loop stability with either of the two SISO
controllers in automatic or manual, the controllers should be paired such that the RGA elements
corresponding to the pairings are positive. Negative RGA elements are an indication of the
presence of destabilizing positive feedback due to unfavorable process interactions. Similarly,
excessively large RGA elements are related to poorly conditioned processes; those in which the
effective process gain may be orders of magnitude different, depending on the input direction.
For systems of higher rank, a necessary condition for the stabilizability of a decentralized
control system is the selection of pairings such that λij > 0, and hence the RGA provides a useful
screening tool. The decentralized integral controllability (DIC) conditions (see Morari and
Zafiriou, 1989, pp. 359-367) provide additional necessary conditions for the stability of higher-
order systems, which depend only on the steady-state gain matrix, .P 0
Properties of the Steady-state RGA
The following properties are especially noteworthy when working with the RGA:
1. 1ij iji jλ = λ =∑ ∑ (rows and columns sum to unity)
2. If P is triangular (lower or upper), I=Λ
e.g.
=Λ⇒
−=
100010001
100530323
P
In such systems, the process interaction is in one direction only, and therefore, precludes the
possibility of the occurrence of destabilizing feedback.
3. For 2×2 systems only:
If P has an odd number of positive elements, 0 ≤ λij ≤ 1 If P has an even number of positive elements, either λij < 0 or λij > 1
– 12S-13–
Dynamic RGA (McAvoy, 1983)
Considering the same MIMO process in Figure 12S.1, y2 is expressed in terms of u1and u2:
2221212 upupy += (12S.23)
When c1 is in manual operation, u1 = 0 and
yu
pc
2
222
1 ,OL=
as for the steady-state analysis. When c1 is in automatic operation and it is assumed that the first
loop can be designed to give perfect control (i.e., the first loop's output is assumed to be held at its
set point),
y p u p u u pp
u1 11 1 12 2 112
1120= + = ⇒ = − (12S.24)
Substituting for u1 in Eq. (12S.23),
yu
p p ppc
2
222
12 21
111 ,CL= − (12S.25)
Hence, the dynamic RGA (DRGA) has precisely the same form as the steady-state array. Note that
the dynamic RGA assumes perfect control, which may not be an appropriate assumption, especially
at high frequencies. The computation of the DRGA requires care since it involves complex algebra.
Because columns and rows sum to unity only at the steady state, the DRGA should be computed
using:
( ) ωλ⋅λ=ω jsignDRGA ijijij 0 , (12S.26)
with λijjω computed conveniently for 2 × 2 systems using Eqs. (12S.19) and (12S.20), or using
Eq.(12S.22) in general.
An accepted rule of thumb is to avoid pairings between variables with negative RGA
elements and to select those with values close to unity, as illustrated in the following example.
– 12S-14–
Example 12S.2 LV Control of a Binary Distillation Column
Figure 12S.2 shows the LV configuration for the two-point composition control of a binary
distillation column discussed in Example 12.9. After assigning manipulated variables to regulate
the vapor and liquid inventories, the boilup rate, V, and the reflux flow rate, L, remain available to
control the distillate and bottoms product compositions, xD and xB, respectively. To assess the
controllability and resiliency of this configuration, the disturbances are taken to be the feed
composition, xF, and the flow rate, F. The column dynamics are approximated by a linear model in
transfer function form (Sandelin et al., 1990):
+
=
−
+−−
+−
−+
−+
−
−+
−+
−
−+
−+
−
Fs
s..s
s..
ss.
.ss.
s.s.s.
s..
s.s
.s.s..
B
D
xF
ee
eeVL
ee
eexx
129650
155160
1580040
1100010
50110
55051118
230
50111
048050118
0450
(12S.27)
To complete the process model definition, it is noted that the process input ranges are as follows: L
= 60 ± 60 kmol/h, V = 72 ± 72 kmol /h, F = Fnom ± 20 kmol /h, xF = xF, nom ± 6 %. In Eq. (12S.27),
the gain coefficients are in the appropriate units, and time is in minutes.
Figure 12S.2 Control of a binary distillation column using the LV configuration.
The qualitative guidelines presented in Chapter 12 are not sufficient to decide how to pair the two
manipulated variables with the two outputs. Without analysis, it is not clear whether this pairing
should be diagonal (i.e., xD−L, xB−V as shown in Figure 12S.2) or off-diagonal (i.e., xD−V,
xB−L ). However, using Eq. (12S.19), λ11 in the RGA is
– 12S-15–
11 2211
11 22 12 211.8p p
p p p pλ = =
− (12S.28)
Using the property that the RGA rows and columns add to unity,
−
−=Λ
8.18.08.08.1
,
and consequently, diagonal pairing is recommended, with the off-diagonal pairing resulting in
stability problems, either when both of the controllers are on automatic or when one of the
controllers is switched to manual operation. Although stable, significant interactions are anticipated
when both loops are closed, because of the large RGA element.
Figure 12S.3 Closed-loop response of the LV
configuration for binary distillation to the worst-
case disturbance, d = [20, 6]T, with decentralized
PI control - Outputs: xD (solid line), xB (dashed
line); Inputs: L (solid line), V (dashed line).
To verify this, Figure 12S.3 shows the closed-loop response for the process, diagonally paired with
IMC-tuned PI controllers (xD − L loop: Kc = -50, τI = 8 min; xB − V loop: Kc = 5, τI = 10 min). For
the IMC-PI tuning rules, the reader is referred to Section 12S.4. The simulation is computed for the
worst-case disturbance, d = [20, 6]T, identified using the disturbance cost analysis, to be discussed
shortly. As expected, the response is stable but shows significant interactions, with the bottoms
composition affected more significantly. The reader can try out this example under MATLAB,
using the interactive C&R Tutorial CRGUI available on the Wiley website that accompanies this
text. In the main menu, opt for the “Binary Column.”
In some cases, the RGA elements vary significantly with the frequency, which may indicate
bandwidth limitations on the diagonal dominance of the process. For this example, Figure 12S.4
– 12S-16–
shows λ11 and λ12 as a function of the frequency. Although there is considerable variation at high
frequencies, the diagonal dominance holds for the entire frequency range of interest. In this case,
the RGA and DRGA give the same pairing recommendations. For some processes, however, the
information furnished by the dynamic RGA can be crucial for the correct pairing selection. The
following example provides one such case.
Figure 12S.4 Dynamic RGA for the diagonal
(solid line) and off-diagonal (dotted line) pairings
for Example 12S.2.
Example 12S.3 Importance of the Dynamic RGA
Consider the process:
( )( )
+
=
−
+−−
+−
−+
−+
−
−+
−+
+−
++
2
15
11022
151
5110
32110
4
2
15
1204
131
1455
1211552
2
1
dd
ee
eeuu
e
e
yy
ss
ss
ss
ss
sss
ss
ss.
(12S.29)
Here the process inputs are limited to the ranges: u1 = 60 ± 60, u2 = 50 ± 50, d1 = d 1,nom ± 20, and
d 2 = d 2,nom ± 5, and time is in minutes.
For this system, λ11 = 2/3 in the steady-state RGA, suggesting that the variables be paired
diagonally. In the dynamic RGA, however, the diagonal dominance deteriorates at moderate
frequencies, as shown in Figure 12S.5. In fact, the process is off-diagonally dominant in the
frequency range of interest. The open-loop time constants are on the order of 10 min, and hence,
frequencies in the range 0.1 < ω < 1 rad/min are of particular interest. For this system, the off-
diagonal pairing (i.e., y1 − u2 and y2 − u1) is preferred, contrary to the pairing suggested by the
steady-state RGA. To verify the analysis in Figure 12S.5, the two pairings are simulated using
– 12S-17–
IMC-PI tuning rules (see Section 21.4). For the diagonal pairing, the controllers are tuned: y1 − u1
loop, Kc = 0.6, τI = 15 min; y2 − u2 loop, Kc = −0.37, τI = 20 min. In contrast, for the off-diagonal
The C&R analysis in the steady state predicts the superior performance of the modified
HEN, which allows all three target temperatures to be controlled at their setpoints in the face of
disturbances in the feed flow rate and temperature of the hot stream. More specifically, the steady-
state RGA indicates that a decentralized control system can be configured for the modified HEN in
which θ2 − F2, θ4 − F3 and T3− ϕ are paired, and in which the first loop is almost perfectly
decoupled, with moderate coupling between the other two loops. Finally, aided by DC analysis, the
nominal bypass fraction is selected to be 0.25, providing the best trade-off between increased plant
costs and adequate resiliency.
Given the design decision to use ϕ = 0.25, based upon the steady-state C&R analysis,
verification is performed by dynamic simulations with ASPEN HYSYS. The hot stream of n-octane at
2,350 lbmol/h is cooled from 500 to 300 oF using n-decane as the coolant, with F2 = 3,070 lbmol/h
and F3 = 1,200 lbmol/h. Note that these species and flow rates are chosen to match the heat-
capacity flow rates defined by McAvoy (1983), with F1 slightly increased to avoid temperature
crossovers in the heat exchangers due to temperature variations in the heat capacities. Additional
details of the ASPEN HYSYS simulation are:
– 12S - 65 –
(a) The tubes and shells for the heat exchangers provide 2 min residence times.
(b) The feed pressures of all three streams are set at 250 psi, with nominal pressure drops of
5 psi defined for the tubes, shells and for the bypass valve, V-3. Subsequently, these
pressure drops are computed based on the equipment and valve sizing and the pressure-
flow relationships.
(c) The bypass valve V-3 is sized carefully, ensuring that the nominal bypass fraction is
0.25, with the nominal valve position being 50% open (selecting a linear characteristic
curve).
(d) IMC-PI tuning parameters are presented in Table 12S.8.
Table 12S.8 IMC-PI tuning parameters for the alternative HENs.
HEN without bypass (Figures 12S.30 and 12.5)
Loop PV Range,oF Kc τI, min Action θ2 − F2 300-500 2 1.5 Direct θ4 − F3 300-500 1.5 2.5 Direct
HEN with bypass (Figures 12S.31 and 12.6)
Loop PV Range,oF Kc τI, min Action θ2 − F2 300-500 2 1 Direct θ4 − F3 300-500 1 2 Direct T3− ϕ 300-500 1 1 Reverse
The regulatory responses of the two configurations are discussed next. Figure 12S.32 shows
that, as predicted by the DC analysis, even the worst-case disturbance has little effect on the two
controlled variables, whose control loops are decoupled, as indicated by the RGA analysis.
Moreover, the uncontrolled output, T3, exhibits offsets of about ±4.5 oF, which compare well with
the value of ±4 oF predicted by the linear DC analysis. In comparison, Figure 12S.31 shows that,
for the HEN with bypass, the response also corroborates the results of the linear DC analysis. Most
importantly, the design with ϕ = 0.25 rejects the worst-case disturbance with no saturation,
indicating that the DC analysis is slightly conservative. In addition, the first control loop (θ2 − F2) is
perfectly decoupled, with slight interactions seen in the other two loops, again as predicted by the
static RGA analysis. For more details, the reader is referred to the section covering dynamic
simulation using ASPEN HYSYS on the multimedia CD-ROM that accompanies this text, where
– 12S - 66 –
the files HEN_1.hsc and HEN_2.hsc, are provided to enable the reproduction of the results in
Figures 12S.32 and 12S.33.
Figure 12S.32 Response of HEN without bypass to the worst-case disturbances: (a) Normalized changes in F1 (solid) and T0 (dashed); (b) Tracking errors (θ2 – solid; θ4 – dashed; T3 – dotted); (c) Manipulated variables (F2 – solid; F3 – dashed).
Figure 12S.33 Response of HEN with bypass to the worst-case disturbance: (a) Normalized changes in F1 (solid) and T0 (dashed); (b) Tracking errors (θ2 – solid; θ4 – dashed; T3 – dotted); (c) Manipulated variables (F2 – solid; F3 – dashed; V1 – dotted).
– 12S - 67 –
While steady-state C&R analysis often provides a good assessment of the controllability
and resiliency, dynamic analysis should be considered when the steady-state analysis is
inconclusive. The latter methods are discussed by Wolff et al. (1991) and Mathisen et al. (1993).
Case Study 12S.3 Interaction of Design and Control in the MCB Separation Process
Denn and Lavie (1982) show that recycles increase the process response time and static
gain. Furthermore, when the recycle loop contains a time delay, resonant peaks comparable in
magnitude to the steady-state gain may result. Since these phenomena are potentially destabilizing,
control systems for recycle processes should be designed carefully. In this regard, control systems
for recycle processes are designed using the nine-step design procedure of Luyben and coworkers
(1999), presented in Section 12.3, with particular emphasis on the need to impose flow control on
each recycle stream.
Figure 12S.34 Flowsheet for the MCB separation process.
– 12S - 68 –
Process Description.
Figure 12S.34 shows the Monochlorobenzene separation process introduced in Section 5.4.
The process involves a flash vessel, V-100, an absorption column, T-100, a distillation column, T-
101, a reflux drum, V-101, and three utility heat exchangers. As shown in Figure 5.23, most of the
HCl is removed at high purity (96 mol % by design) in the vapor effluent of T-100. However, in
contrast with the “treater” to remove the residual HCl in the design shown in Chapter 5, in Figure
12S.34 this is removed in the small vapor overhead purge in T-101. The Benzene and
Monochlorobenzene are obtained at high purity as distillate (99 mol % Benzene) and bottoms
liquid products (98 mol % MCB) in T-101. It is required to design a control system to ensure that
the process meets its quality specifications in the face of changes in the throughput demand (treated
as a disturbance) and feed composition changes, as listed in Table 12S.9. A preliminary control
system configuration is proposed, and then refined and checked using the C&R analysis. Finally,
the performance of the control system is verified using dynamic simulation.
Table 12S.9 Process disturbance scenarios.
Species Nominal d1 d2
Molar flow rates in kmol/h HCl 10 15 15
Benzene 40 60 50 MCB 50 75 35 Total 100 150 100
Preliminary Control System Configuration.
The nine-step control design procedure of Luyben and co-workers is applied to design the
preliminary control structure in Figure 12S.35:
Step 1. Set objectives. To achieve the primary control objective, the production level is maintained
by flow control of the feed stream using valve V-1.
Step 2. Define control degrees of freedom. As shown in Figure 12S.34, the process has twelve
degrees of freedom with four valves controlling the flow rates of the utility streams (V-2, V-5, V-9
– 12S - 69 –
and V-10), one controlling the feed flow rate (V-1), three controlling product stream flow rates (V-
6, V-8 and V-11), and the four remaining valves controlling internal process flow rates. Having
chosen constant feed flow in Step 1, the feed valve (V-1) is reserved for independent flow control.
Step 3. Establish energy management system. The steam valve, V-2 is used to control the flash
feed temperature. Furthermore, the temperature of the recycle and bottoms product streams is
contolled by adjusting the coolant valve, V-10.
Step 4. Set the production rate. As stated previously, the feed valve, V-1, is assigned to a flow
controller, whose setpoint regulates the production rate.
Step 5. Control product quality, and meet safety, environmental and operational constraints. The
pressure in V-100 is controlled by adjusting its vapor stream using valve, V-3. Pressure regulation
in the T-101 is carried out by adjusting V-5, the coolant valve to the condenser E-101. Since both
of the products from T-101 are required to meet specifications, the LV configuration is
implemented, noting that the reflux ratio in the column is less than five. Thus, the reflux valve, V-7,
is adjusted to control the distillate composition, and the reboiler steam valve, V-9, is used to
regulate the bottoms composition.
Step 6. Fix recycle flow rates and vapor and liquid inventories. The obvious choice for recycle
flow control is valve V-12. The liquid inventories in the flash drum, the reflux drum and the
column sump, are regulated using the valves V-4, V-8 and V-11, respectively. Note that the purge
stream that removes the residual HCl from the column overheads is less that 1% of the feed by
design. Thus, the valve V-6 is designed to fixed at 50% open and left uncontrolled. Regulations of
the vapor inventories in both V-100 and T-101 have been addressed, by installing pressure
controllers.
Steps 7 and 8. Check component balances and control individual process units. The HCl in the
feed is removed from the process, mostly in the T-100 overhead stream, with small traces removed
in the column purge stream. The benzene and MCB fed to the process are mostly removed in the
distillate and bottoms streams from T-101, respectively, with small traces removed with the HCl
product and in the purge.
Step 9. Optimize economics and improve dynamic controllability. It is noted that all of the control
valves have been assigned, but the HCl product quality is still uncontrolled. To correct this, a
– 12S - 70 –
cascade controller is installed to regulate the HCl product stream composition, which adjusts the set
point of either (a) the recycle flow controller, FC-2, or (b) the recycle temperature controller, TC-2.
Figure 12S.35 shows the first alternative, which manipulates the liquid feed rate to the absorber to
control the mass transfer of the organic species from the vapor stream. Clearly, quantitative
methods are required to enable the most appropriate configuration to be selected, as will be shown
next.
Figure 12S.35 Control system for the MCB separation process.
Control System Refinement using C&R Analysis.
Controllability and resiliency analysis has two roles in the improvement of the control
system in Figure 12S.35: (1) The RGA aids in defining the appropriate pairing between the
controlled outputs and manipulated variables where interaction is anticipated; (2) The DC assists in
checking that the operating ranges of key manipulated variables is sufficient to ensure adequate
– 12S - 71 –
disturbance rejection. To provide data for these two analytical methods, a dynamic simulation of
the MCB separation process is developed using ASPEN HYSYS.
The equipment items are sized as follows:
(a) The flash vessel, V-100, condenser, V-101 and reboiler, E-102, are installed assuming at
least 10 min liquid residence time, computed using the steady-state liquid feed rate as a
basis. Thus, for example, since the liquid feed to V-100 is nominally 137 ft3/hr, the
required vessel volume is 2×10×137/60 = 45.7 ft3, which is rounded up to 50 ft3. Similar
calculations give volumes of 120 ft3 for V-101 and 240 ft3
for E-102.
(b) The absorption column, T-100, is a 10-stage packed bed with a diameter of 1.5 ft.
(c) The distillation column, T-101, has 10 valve-trays with a diameter of 2.5 ft.
(d) The two heat exchangers are approximated as heat-requirement units, which assume that
the control variable is the heat transfer duty. Thus, E-100 is installed as a heater, with
volume of 20 ft3 and E-103 as a cooler, with a volume of 50 ft3. More detailed modeling
is possible by using heat exchangers, allowing the manipulation of steam and cooling
water flows. Pressure drops in these heat exchangers are defined by assigning a
pressure-flow relationship, established automatically by ASPEN HYSYS on the basis of
nominal flow rates.
(e) A number of valves are installed to enable flow and pressure regulation of the process.
Each valve is set to be 50% open, sized on the basis of nominal flow rates, and then
assigned to follow a pressure-flow relationship. When a valve is selected to provide
control, it is assigned to a controller, which manipulates the percentage valve opening.
One valve that is maintained at 50% open is V-6, which is intended to purge the residual
light gases in the feed to T-101.
Several of the control loops in Figure 12S.35 are required to ensure inventory control,
namely, all three level control loops and the two pressure control loops. Note that the pressure in V-
100 is assumed constant and the loop PC-1 is not simulated explicitly in the ASPEN HYSYS
simulation. In contrast, as pointed out repeatedly in the literature, pressure control in the column is
crucial to stabilize the internal flows in the column. Finally, the feed flow rate and temperature
controllers are clearly decoupled from the rest of the process, and therefore need not be included in
– 12S - 72 –
the C&R analysis. Thus, the interactions that need to be analyzed are the effects of the four valves:
V-7, V-9, V-10 and V-12 (or more precisely, the setpoint to FC-1), on four controlled variables:
xD,2, xB,3, xA,1 (the mole fractions of the benzene in the distillate, MCB in the bottoms and HCl in
the absorber overhead stream, respectively) and TR, the recycle temperature. Note that to improve
dynamic performance, the temperature of tray 4 is controlled instead of the distillate benzene
composition.
The interaction analysis is performed using the steady-state RGA. To generate information
to compute RGA, the loops under test in the simulated process are placed in “manual” mode, and
the process is simulated to “line-out” the outputs at open-loop steady-state values. Then, distinct
step changes in the four valve positions are imposed, and the new steady-state values of the outputs
recorded. Note that for consistency, the step direction is chosen such that its effect on AC-1 is in
the same direction. The results of these simulations are recorded in Table 12S.10. Thus, for
example, a 0.5% increase in the position of the reflux valve (V-7) leads to a decrease of 4.5 oF in
temperature in tray 4.
Table 12S.10 Simulation results for RGA calculations.
R (V-7) xD,2 (AC-1) xB,3 (AC-2) xA,1 (AC-3) TR(TC-2) Range 0-100 % 100-300 oF 0.5-1.0 0.5-1.0 50-250 oF Before 43.0 % 226.3 oF 0.9857 0.9596 121.2 oF After 43.5 % 221.8 oF 0.9638 0.9590 117.9 oF
Change +0.5% –4.5 oF –0.0219 –0.0006 –3.3 oF QR (V-9) xD,2 (AC-1) xB,3 (AC-2) xA,1 (AC-3) TR(TC-2)
Before 45.4 % 226.3 oF 0.9857 0.9596 121.2 oF After 44.9 % 221.5 oF 0.9576 0.9589 116.8 oF
Change –0.5% –4.8 oF –0.0281 –0.0007 –4.4 oF FR (FC-2) xD,2 (AC-1) xB,3 (AC-2) xA,1 (AC-3) TR(TC-2)
Before 45.0 % 226.3 oF 0.9857 0.9596 121.2 oF After 45.5 % 224.9 oF 0.9817 0.9582 122.1 oF
Change +0.5% –1.4 oF –0.0040 –0.0014 +1.1 oF QC (V-10) xD,2 (AC-1) xB,3 (AC-2) xA,1 (AC-3) TR(TC-2)
Before 76.0 % 226.3 oF 0.9857 0.9596 121.2 oF After 76.5 % 224.3 oF 0.9803 0.9605 119.0 oF
Change +0.5% –2.0 oF –0.0054 +0.0011 –2.2 oF
– 12S - 73 –
Dimensionless static gains are computed, accounting for the full range of each variable.
Thus, for example, the gain that relates variations of xD,2 to changes in R is:
5410050200540 2
11 ...
Rx
p ,D −=−
=∆
∆= (12S.93)
In the same way, the other 15 static gains are computed, given the overall steady-state transfer-
function matrix relationship:
−−−−
−−−−−−
=
C
R
R
R
,A
,B
,D
QFQR
...............
Txxx
221014043034405602802401626012118.76002401804504
1
3
2
(12S.94)
The RGA is computed from this linear model:
−−−−
−−−−
=Λ
019989859869112115811715
639447284100731351711815
........
.......
(12S.95)
The large RGA elements are indicative of significant sensitivity to model uncertainty, often related
to process nonlinearities. While the RGA indicates that the pairings: xD,2 – R, xB,3 – QR, xA,1 – QC,
and TR – FR, provide stable response, the large RGA elements are indicative of large interactions in
the process.
The above results, however, suggest a simpler control structure, in which FR is maintained
constant, and QC is adjusted to control xA,1, giving the steady-state transfer-function matrix
relationship:
−−−−−
=
C
R
,A
,B
,D
QQR
........
xxx
4402802401622118.76002804504
1
3
2 (12S.96)
In this case, the RGA is:
−−
−=Λ
83904990660005302963451070794696
...
...
... (12S.97)
– 12S - 74 –
This suggests performance superior to that obtained with the original confirmation using the
diagonal pairings: xD,2 – R, xB,3 – QR, and xA,1 – QC , leading to the modified control system shown
in Figure 12S.36. Note in particular, that the third loop is almost decoupled, with strong
interactions in the two distillation-column loops. The large RGA elements associated with the LV
configuration are significantly larger than those expected in a column operating independently
(compared with those computed for the SC configuration in Figure 12S.16), due to the additional
positive feedback contributed by the material recycle.
Figure 12S.36 Improved control system for the MCB separation process.
Next, the DC is computed for typical process load changes and disturbances, presented in
Table 12S.9. Two scenarios are considered: d1, a 50% increase in throughput, and d2, a composition
disturbance in which all three compositions are changed. Table 12S.11 shows the open-loop effect
each disturbance on the four outputs, indicating that the second disturbance has the greatest effect
on the top composition in T-101. The effect of the two disturbances on the three outputs controlled
by the control system in Figure 12S.36, expressed in scaled perturbation variable form, are:
– 12S - 75 –
For disturbance 1:
−−−
=⋅479601420001950
01...
dPd (12S.98)
For disturbance 2:
=⋅
033600398018350
02...
dPd (12S.99)
Note that the scaled perturbation variables are computed by dividing the changes on the output
variables in Table 12S.11 by their full-scale ranges. This allows the steady-state DC to be
computed directly:
[ ]
−−
=
−−−
−−−−−
=
−=
−
−
910307941025591
479601420001950
4402802401622118.76002804504
000 :1 edisturbancFor 1
11
...
.
.
.
........
dPPDC d
(12S.100)
[ ]
−=
−−−−−
=
−=
−
−
053302208029990
033600398018350
4402802401622118.76002804504
000 :2 edisturbancFor 1
11
...
.
.
.
........
dPPDC d
(12S.101)
The linear analysis suggests that the effect of the first disturbance cannot be rejected completely,
because it causes the first control variable, R, to saturate (the magnitude of the DC for this variable
is greater than unity). In contrast, the linear DC analysis predicts that the second disturbance is
rejected relatively easily. Table 12S.11 Data for DC calculations.
(a) Disturbance 1: Increased throughput by 50 %
xD,2 (AC-1) xB,3 (AC-2) xA,1 (AC-3) TR(TC-2) Range 100-300 oF 0.5-1.0 0.5-1.0 50-250 oF Before 226.3 oF 0.9857 0.9596 121.2 oF After 222.4 oF 0.9148 0.7200 141.1 oF
Change –3.9 oF –0.0709 –0.2396 +39.9 oF
(b) Disturbance 2: Composition change.
xD,2 (AC-1) xB,3 (AC-2) xA,1 (AC-3) TR(TC-2) Before 226.3 oF 0.9857 0.9596 121.2 oF After 263.0 oF 0.9976 0.9764 90.8 oF
Change 36.7 oF 0.0199 0.0168 –30.4 oF
– 12S - 76 –
Dynamic simulation using ASPEN HYSYS is used to verify the predictions of the linear
C&R analysis. The control loops shown in Figure 12S.36 are all PI controllers, with tuning
parameters tuned using the IMP-PI rules, given in Table 12S.12. Note that the level controllers are
loosely tuned, as in Case Study 12S.1. In contrast, the distillation column pressure controller, PC-2,
is tuned to ensure tight control of this key variable. The gains on the three composition controllers,
AC-1, AC-2 and AC-3, are tuned to ensure that the strong interaction between them does not lead
to loss of stability, while imparting acceptable regulatory performance.
Figure 12S.37 Response of the MCB separation
process to a 50% increase in throughput (d1):
(a) molar feed rates in kmol/hr – solid = MCB,
dashed = benzene, dotted = HCl; (b) changes in
product purities in % – solid = MCB, dashed =
benzene, dotted = HCl; (c) manipulated variables
– solid = V-9 (QR), dashed = V-7 (R), dotted =
V-10 (QC); (d) product flow rates in kmol/h– solid
= MCB, dashed = benzene, dotted = HCl.
– 12S - 77 –
Figure 12S.38 Response of the MCB separation
process to composition change disturbance (d2).
(a) Molar feed rates in kmol/hr, (b) changes in
product purities in %, (c) manipulated variables,
(d) product flow rates in kmol/h. Variables as in
Figure 12S.37.
The simulations shown in Figures 12S.37 and 12S.38 show that:
a) The 3×3 control system, paired as suggested by the RGA, provides stable performance
for both disturbances.
b) Both of the disturbances are step changes in the molar feed rates in the three species.
Note that the control system manipulates the draw rates needed while ensuring that the
product compositions stay on specification, by the action of the level controllers (see
Figures 12S.37d and 12S.38d).
– 12S - 78 –
c) The effects of both of the disturbances on the purities of the three products are rejected
successfully, despite the prediction of the linear DC analysis (see Figures 12S.37b and
12S.38b). The control action perturbations required to reject the first disturbance are
greater than for the second one, which is qualitatively in agreement with the DC
analysis (see Figures 12S.37c and 12S.38c).
Table 12S.12 IMC-PI tuning parameters for the MCB Separation Process (See Figure 12S.36).
Loop PV Range Set point Kc τi Action TC-1 150-350 oF 270 oF 3 2 min Reverse FC-2 0-200 lbmol/h 90 lbmol/h 1.4 0.5 min Reverse AC-1 200-300 oF 226.3 oF 5 25 min Direct AC-2 0.50-1.00 MCB 0.98 MCB 12 10 min Reverse AC-3 0.50-1.00 HCl 0.97 HCl 12 20 min Reverse PC-2 15-40 psia 26 psia 3 0.5 min Direct LC-1 0-100% 50% 2 30 min Direct LC-2 0-100% 50% 2 30 min Direct LC-3 0-100% 50% 2 30 min Direct
This case study has shown the advantages of employing C&R analysis to assist in the
design of a plant-wide control system using the procedure of Luyben and co-workers. The control
configuration pairing is determined using the steady-state RGA. The disturbance rejection afforded
by the process is predicted incorrectly by the linear DC analysis. This indicates that non-linear
approaches should be used in general. Nonlinear controllability and resiliency analysis is an area of
active research (e.g., Seferlis and Grievink, 1999; Solovyev and Lewin, 2001).
12S.6 MATLAB FOR C&R ANALYSIS
MATLAB and SIMULINK are invaluable tools for the frequency- and time-domain
calculations required for C&R analysis. In this section, several examples are carried out using
MATLAB, it being assumed that the reader is familiar with the MATLAB syntax. The reader is
referred to the multimedia CD-ROM that accompanies this text for sources of these and other
useful MATLAB functions and scripts for C&R analysis. In particular, the interactive C&R
– 12S - 79 –
Tutorial CRGUI can be used to test three example linear processes for controllability and resiliency
and simulate their closed-loop response under single-loop PI control.
Example 12S.11 Computing the Dynamic RGA
For the system given in Example 12S.3, the MATLAB script that generates the dynamic RGA in
Figure 12S.5 is:
% Example 12S.3 % This script file computes the dynamic RGA for Example 12S.3 % Define a vector of frequency values on a log scale wmin=-3;wmax=1;nw=30*fix(wmax-wmin); w=logspace(wmin,wmax,nw); s=i*w; % Data for process model kp=[2.5 5;1 -4]; %process gain matrix tp1=[15 4;3 20]; %process time constant tp2=[2 0;0 0]; %process time constant thp=[5 0;0 5]; %process delay % Compute the frequency response for each element of Pij p11=kp(1,1)./(tp1(1,1)*s+1)./(tp2(1,1)*s+1).*exp(-thp(1,1)*s); p12=kp(1,2)./(tp1(1,2)*s+1)./(tp2(1,1)*s+1).*exp(-thp(1,2)*s); p21=kp(2,1)./(tp1(2,1)*s+1)./(tp2(1,1)*s+1).*exp(-thp(2,1)*s); p22=kp(2,2)./(tp1(2,2)*s+1)./(tp2(1,1)*s+1).*exp(-thp(2,2)*s); % Compute lambda(1,1) and lambda(1,2) as functions of frequency. l11=p11.*p22./(p11.*p22-p12.*p21); lam11=sign(real(l11(1))).*abs(l11); l12=-p12.*p21./(p11.*p22-p12.*p21); lam12=sign(real(l12(1))).*abs(l12); %Plot the results figure semilogx(w,lam11,'-k',w,lam12,':k','LineWidth',2) xlabel('\omega [rad/min]','FontName','Times','FontSize',14) ylabel('DRGA','FontName','Times','FontSize',14)
– 12S - 80 –
As discussed in Example 12S.3, the steady-state RGA suggests diagonal pairings. However,
the dynamic RGA implies that these pairings are unstable for frequencies higher than about 0.5
rad/min. Thus, anti-diagonal pairings should be used.
Example 12S.12 Computing Disturbance Cost Maps
Consider the component parts in the LSF configuration represented by Eqs. (12S.56) and (12S.57).
In this example, the elements of the transfer function matrices are entered into MATLAB and used
to compute the DC contour maps for this configuration. P j P jdω ω and are computed for each
frequency, and used to compute DC for all of the disturbance directions. By looping over all
frequencies, the entire DC map is calculated, and repeated for each manipulated variable separately.
Note that, as mentioned in Example 12S.7, the inputs are nominally at 50% of the full range. Here,
the nominal inputs are taken as LH = LL = 11 kmol/min, QRH = 0.222×106 kcal/min, and the
maximum disturbance magnitudes are taken as F=18 kmol/min and xF = 0.2 (±20% of the full
range).
% LFS: This script computes P(s) and Pd(s) for the LSF configuration, given the % transfer-function matrices for the two component parts. It then uses the % matrices to compute DC contours % Definition of frequency and direction vectors. n=41; i=sqrt(-1); wmin=-3; wmax=0; dw=(wmax-wmin)/(n-1); tmin=0; tmax=180; dt=(tmax-tmin)/(n-1); w=logspace(wmin,wmax,n); % Frequency vector [rad/min] ome=wmin:dw:wmax; % Frequency vector in log scale. phi=tmin:dt:tmax; % Direction vector [degrees] a=pi*phi/180; % Direction vector [radians] s=w*i; % Vector complex s tt = exp(i*a); % Computing the direction in radian coordinates dd(1:n,1:2) = [real(tt'),imag(-tt')]; % tt in cartesian coordinates z=zeros(1:n,1:n); % matrix for storing computed DC values. % Gains and delay times for the high pressure column: KH=[0.017 -1.109 0.001 0.090; 0.011 -1.859 0.006 1.296; -0.33 59.0 -0.2 -41.05; 0.916 -123.7 1.127 -0.02; 4.0e-5 -0.994 0.001 0.003]; DH=[0.0 0.0 0.0 6.4; 1.3 0.0 0.1 0.1; 1.3 0.0 0.1 0.1;1.3 0.0 0.1 0.1;1.3 0.0 0.1 0.1]; % Gains and delay times for the low pressure column: KL=[ 0.792 -0.029 0.007 2.161 0.012; 0.790 -0.051 0.003 3.291 0.038];
– 12S - 81 –
% Note: The coefficients in the fourth column have been multiplied by -1 % since QRL = - QCH DL=[ 0.1 0.1 0.1 0.0 1.4;8.5 0.0 0.0 0.0 0.0];
for ku=1:3 % Looping over all manipulated variables (m=3) for k=1:n % Looping over all frequencies. % Computing the frequency response of each component part submatrix [see % Eqns. (12S.56) and (12S.57)]. ph=KH.*exp(-DH*s(k))./(13*s(k)+1); pl=KL.*exp(-DL*s(k))./(17*s(k)+1); % Computing P(s) and Pd(s) at the current frequency [See Eqns. (12S.60) % and (12S.61)] P=[ph(:,1:4) 0 ; pl(1,1:2)*ph(2:5,1:2) pl(:,:5)]; Pd=[ph(1,3:4) ; pl(:,1:4)*ph(2:5,3:4)]; % Scaling: P(:,1)=P(:,1)*11;P(:,2)=P(:,2)*0.222;P(:,3)=P(:,3)*11; Pd(:,1)=Pd(:,1)*18;Pd(:,2)=Pd(:,2)*0.2; u2 = inv(P)*Pd*dd'; % Computing DC for i_dir = 1:n % Looping over d direction 0 → 180 z(i_dir,k) = norm(u2(ku,i_dir)); end end % End of frequency loop. v = [0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0]; figure cs=contour(ome,phi,z); clabel(cs) title(['Disturbance Cost for Input ',num2str(ku)]); xlabel('log(w)'); ylabel('Direction [deg]'); end % End of manipulated-variable loop
This script generates the DC contour maps in Figure 12S.39 for each manipulated variable
separately. Note that there is no bandwidth limitation to perfect disturbance rejection in any of the
control variables.
– 12S - 82 –
Figure 12S.39 DC contour maps for the LSF
configuration to dehydrate methanol: (a) LH;
(b) QRH; (c) LL. The bounds on the disturbances
are ±20% from their nominal values. The DC
contour maps for each manipulated variable are
computed separately, with bold solid lines
indicating DC = 1. See Figure 12S.17 for the DC
contour maps for the SC, FS and LSR
configurations.
12S.7 SUMMARY
In this chapter, the methods for short-cut C&R analysis, using the results of steady-state
simulations, have been described. The methods require the use of software for the solution of
material and energy balances in process flowsheets (e.g., ASPEN PLUS, HYSYS.Plant) and for
controllability and resiliency analysis (i.e., MATLAB). The reader is now prepared to tackle small-
to medium-scale problems, and in particular, should be able to
– 12S - 83 –
1. Generate a linear model of a chemical process in one of its standard forms, using
either the equations expressed in a MATLAB function, or the solution of the
material and energy balances computed by a process simulator.
2. Compute the frequency-dependent process transfer functions using MATLAB, given
a linear model in one of its standard forms.
3. Generate the C&R measures of relative-gain array (RGA) and disturbance cost
(DC), given the process transfer functions, using MATLAB.
4. Select the appropriate pairings for a decentralized control system for the process
using the static and dynamic RGAs and appropriate resiliency measures.
5. Perform C&R analysis to select between alternative process configurations, given
the results of process simulations.
Several examples have been selected to show how the methods are used to screen
alternative flowsheets in stage 2 of the design process (Table 12.1). In the first example (Section
21.3), dynamic C&R analysis enables the most resilient heat-integrated distillation configuration to
be selected. In Case Study 12S.1, two designs for an exothermic reactor, involving either one or
two CSTR(s) in series, show that while the latter is more economical (assuming steady-state
operation), the former is more resilient to disturbances. In Case Study 12S.2, a steady-state analysis
of two heat exchanger network configurations leads to the conclusion that while a design equipped
with bypasses may be subject to significant constraints leading to poor resiliency, a design without
them may lead to poor dynamic performance. Here, dynamic C&R analysis is crucial. Finally, Case
Study 12S.3, involves a recycle processes and shows the benefits of C&R analysis in the detailed
design stage (Stage 3 in Table 12.1).
REFERENCES
Bequette, B.W., Process Dynamics: Modeling, Analysis, and Simulation, Prentice Hall, Englewood
Cliffs, NJ, (1998).
Bristol, E. H., On a New Measure of Interactions for Multivariable Process Control, IEEE Trans.
Auto. Control, AC-11, 133-134 (1966).
Chiang, T., and W. L. Luyben, Comparison of the Dynamic Performances of Three Heat-integrated