- Email: [email protected]
Robust control of distillation columns: μ- vs H∞-synthesis
J Proc C,mt Vol 7, No 1, pp. 19 30, 1997
~ ]
~
Cop}right (~ 1996 Elsevier Science Ltd Printed in Great Britain All rights reserved 0939-1524797 $ I 7.0(I + 0.00
ELSEVIER
Pll: S0959--1524(96)00008-X
Robust control of distillation columns: /l- vs H -synthesis Urs Christen, Hans E. Musch and Max Steiner* Measurement and Control Laboratory, Swiss Federal Institute of Technology (ETH) CH-8092 ZSrich, Switzerland Received 5 October 1994; in revised form 11 December 1995
High-purity distillation columns are known to be difficult to control due to their ill-conditioned and nonlinear behaviour. Two approaches for the design of robust controllers yielding high performance are presented. For the first approach, first principles are used to develop an uncertainty model describing the nonlinear dynamics within the entire operating range of an industrial distillation column. This structured uncertainty model is used for/l-synthesis. In a second approach which is based on loop shaping ideas, an H -controller is designed. This controller performs as well as the y-controller. The H~-approach offers the advantage that the burden for uncertainty modelling and computation is greatly reduced. However, the GS/T augmentation scheme, which is developed in this paper, must be used for the design of the H -controller to avoid the inversion of the plant in the controller. The paper concludes with a comparison of the H=- and H-synthesis methods. Copyright %2"1996 Elsevier Science Ltd.
Keywords: distillation column, ill-conditioned plants, g-synthesis, H~-synthesis
Distillation is one of the most important unit operations in the process industry. Almost all chemical plants use distillation columns for the separation of substances according to their relative volatility, and distillation processes are among the largest energy consumers in chemical plants. Tight control of the product compositions is fundamental for an economically and ecologically optimal operation of the entire plant. However, especially in the case of high product purities, distillation columns are known to be difficult to control because their dynamics are ill-conditioned. Furthermore, many distillation columns in the chemical industry operate over a wide range of feed flow rates and compositions, Many robust control design techniques have been successfully applied to this control problem ~ and, at first
trays, a total condenser and a steam-heated reboiler (Figure 1), serves as an example for a typical high-purity distillation process. The feed F is a mixture of mainly two components which is split into the two product streams D and B. The most important operating data including the operating range are summarized in Table l. Due to high investment and maintenance costs of composition analysers, the pressure-compensated ten> peratures on trays 10 and 44 are controlled instead of the product compositions. The manipulated variables chosen to control these two tray temperatures are the reflux k and the boilup V (indirectly controlled by reboiler heat duty). This L V control structure is very common in the chemical industry.
glance, this design task could be considered as solved. Upon closer consideration, however, some drawbacks of the design approaches proposed in the literature become apparent. Since a discussion of these drawbacks necessitates a basic knowledge concerning distillation dynamics, a short summary of the basic principles is given next.
IIl-conditionedplaJlt dynamics As mentioned above, the dynamics of high-purity distillation columns are ill-conditioned, which leads to high sensitivity to uncertainties in the manipulated variables.: Even small errors in these manipulated variables can cause a significant deterioration of the product qualities, 3 a fact which explains why open-loop control of high-purity distillation columns is hardly ever satisfactory. As shown in Figure 2, the column considered here has a condition number
The distillation process An industrial distillation column equipped with 50 sieve
* A u t h o r to w h o m c o r r e s p o n d e n c e s h o u l d be a d d r e s s e d
19
Robust control of distillation columns: U. Christen et al.
20
Vacuum
/~1
•/@I¢' 1 &
Distillate
i~
L[~
Reflux accumulator
102
Condenser
VT
.
.
'" ~ .. .~10 o
.
.
9
Reflux
10 10-5
-Feed - ~
_2.o._
F, x F
Vi / Yi
Figure 2
Yi Gas phase composition
ni Holdup Liquid phase composition
47 -~,8-~t95i3
~
Reboiler B, x B ..._
Bottom product Figure 1 Binary distillation column
Table 1 Steady-state data Column data No. of trays Column diameter (m) Feed tray number Murphree tray efficiency Relative volatility a
50 0.8 20 =0.4
Feed flow rate F (mol/min) Top pressure(mbar)
Nominal operating point
Feed composition (mol/mol)
Feed flow rate (mol/min) Reflux L (mol/min) Boilup V(mol/min)
~¢(G(jo))) = ~ ( G ( j o g ) ) cr(G(jw))
Singular values ( - - ) and condition number ( - -) of the
L i n e a r plant models
1.61
All controller designs within the context of this report are based on linear process models which were calculated by numerical linearization of a rigorous dynamic model. This rigorous model includes the dynamics of the tray compositions and holdups, as well as the
Operating data
Top composition .x',~(tool/tool) Bottom composition xA (moI/mol) Feed composition xF(mol/mol)
10 l
for inputs in the least amplified direction. If the gains of the actuators and therefore the directions of the maximum and minimum gains of the column are uncertain, large amplifications by the controller may be misaligned which causes the controlled outputs to deviate from the intended values. These observations are further clarified by Freudenberg. 4 He has shown that a controller for an ill-conditioned plant must not invert the plant. With plant inversion, the control system becomes very sensitive to uncertainty due to the strong directionality of illconditioned plants. Robustness properties may be good at the output of the plant, but very poor at its input, or vice versa. Freudenberg concludes that the controller should be well-conditioned and consequently, the openloop transfer function of the control system should be as ill-conditioned as the plant itself. This means that there are strong limitations to the achievable closedloop performance.
Ti Temperature
_.., Boilup,, i -" v ~ ~-Ij @
10-3 10-l Frequency (rad/min)
distillation column (nominal model)
xi
t
_
0.99 0.015 0.7-0.9 20~,6 60
0.8
33 65 104
(1)
of several hundred (~(.) and <7(.) denote the maximum and minimum singular values). In other words, the gain for inputs in the direction in which the plant amplifies most can be several hundred times larger than the gain
algebraic equations for the liquid and vapour flow rates, pressure drops and boiling points. Altogether, this model consists of a system of several hundred differential and algebraic equations. 5 In the case of this industrial column, the level of the reflux accumulator is controlled by the top product flow rate D while the reboiler level is controlled by the bottom product flow rate B. Since these levels have tight control, perfect level control and constant holdups in column top and bottom can be assumed. With these assumptions, a linear state-space model of order 102 (52 state variables representing compositions and 50 variables representing holdups) with the following structure can be calculated.
,,~ = A x + Bu + E d y = Cx
(2)
Robust control of distillation columns: U. Christen et al. • &'~D Axi : x = zk,c8 Dn~
[ AL ] u =
AV
V•X"F 1 d =
LAFJ
VATI0 1 y =
LAT,~]
-Ansc (3) The symbol A stands for the deviation from the operating point. The other symbols are as defined in Figure 1. Since the computation time for controller design strongly depends on the model order, an order reduction of these linear models is necessary. Using a balanced truncation, the models of order 102 are easily reduced to an order of 10-15 without significant loss of accuracy. The identification of linear plant models based on recorded step response is an alternative approach to obtain linear models. As pointed out by Skogestad, ~ it is very difficult to identify the dynamics of ill-conditioned plants. Furthermore, the recording of step responses should start at operating points where the product compositions are close to their setpoints and the dynamics are almost at steady-state. While this can be easily achieved in simulations, it is almost impossible in a real high-purity distillation column operated in open-loop (i.e. without composition control)• Therefore, a linearization of the rigorous dynamic model is used in this study,
Control objectives The main objective of the control system is to keep the controlled (pressure compensated) tray temperatures close to their setpoints despite the presence of disturbances. The most significant disturbances are changes in the feed flow rate and in the feed composition, The second design objective is a sufficiently fast setpoint tracking• Direct composition measurements are not installed; rather, as mentioned above, pressure compensated temperatures on trays 10 and 44 are controlled• Therefore, significant changes in the feed composition cause a slight worsening of the product compositions. The plant operator counteracts this effect by changing the setpoints for the temperatures on trays 10 and 44. These two objectives - a best possible disturbance compensation and a reasonable setpoint tracking - must be achieved for the entire operating range of the distillation column•
Design approaches
Basically, two different approaches in robust controller design can be chosen. With a thorough knowledge of the process, the mismatch between the dynamics of the linear model and the real plant dynamics can be modelled from a physical point of view. This includes the operating range of the column, inaccuracies of the sensors and
21
actuators, as well as the frequency content of the disturbances and setpoint signals. Based on a structured uncertainty model which describes the column dynamics within the entire °perating range' a c°ntr°ller is calculated using the ~-synthesis technique. This approach is discussed in the next section. In the second approach, the controller is designed based on loop-shaping ideas. A new augmentation scheme for the design of Ha-controllers is introduced which explicitly avoids the inversion of the plant in the controller by including the plant in the weight of the sensitivity. This ensures that Freudenberg's condition discussed above is satisfied.
N-synthesis Since high-purity distillation columns can be very sensitive to uncertainties in the manipulated variables, it is important for successful implementation that a controller guarantees its performance in the presence of input uncertainties. This particular design task is tYequentty solved by modelling a multiplicative input uncertainty for a nominal plant model and subsequently calculating the controller using p-synthesis. > ') If a distillation column is operated in a rather small range of feed flow rates and feed composition, such a controller can be expected to perform well. However. in the chemical industry the operating conditions of a distillation column often change over a (vide range depending on the actual load of the entire plant. The distillation column described here is a typical example of such a plant, where a controller should perform as well as possible not only, for the nominal operating conditions, but for the whole range of flow rates and compositions fed to the column. One way to handle this task is to design a controller for the nominal plant model (including a multiplicative input uncertainty) and subsequently test its robustness for other operating conditions. 6 Another more straightforward approach is the representation of the varying column dynamics by an extended uncertainty model as proposed by Skogestad et al. 2.* In these papers, the nominal dynamics of a distillation column are approximated by a crude first order model, The nonlinearity of the cotumn is described by time constant and gain uncertainty,. which are represented by divisive uncertainty at plant output and additive uncertainty, respectively• The authors explicitly state that the variations of the time constants and gains are clearly correlated, a fact which is not captured by the uncertainty description they propose. Consequently, this uncertainty, model describes a range of column dynamics which is wider than that of the real column. Therefore, a controller design based on this uncertainty model is always conservative. A similar plant and uncertainty model (using gain and delay uncertainty) has been used, e.g. by Lundstr6m et al2 for a study of a two degree of freedom controller design within the /~-framework. This gain and delay uncertainty, however, fails to represent the correlations in the
Robust control of distillation columns: U. Christen et al.
22
column dynamics, necessitating a more structured model of uncertainty. In the following section, an uncertainty model is developed based on physical considerations which takes the correlation between time constant and gain uncertainty into account and thus avoids this conservatism,
An
uncertainty model for the distillation
As a general guideline, Skogestad et al. 2~ recommend modelling uncertainty where it physically occurs, According to this principle, three types of uncertainty are identified in this report: •
Uncertainty of the manipulated variables (input
•
uncertainty) Uncertainty due to nonlinearity of the process
•
dynamics Unmodelled high-frequency dynamics and uncertainty of the temperature measurements (output certainty),
Input uncertainty As discussed above, ill-conditioned plants can be very sensitive to errors in the manipulated variables. Since measurement errors, parameter variations (e.g. changing heat of evaporation), and unmodelled high frequency dynamics are always present, errors between the setpoints and real flow rates of the reflux and the boilup must be expected? The bounds for the relative errors of the column inputs u are modelled in the frequency domain by a multiplicative uncertainty with two frequency-dependent error bounds w,,. These two bounds are combined in the diagonal matrix IV,, = %1. with (4)
Due to flow dynamics and energy balance, any change e.g. in the reflux may cause a change of the vapour flow rates within the column. Therefore, the errors in the manipulated variables can be interdependent, as represented by the 2 × 2 complex matrix A,(j'to). This matrix is limited in magnitude and only shapes the spatial direction of the error. If all flow measurements are carefully calibrated, an error bound of 10% for the low frequency range is reasonable. Because much higher errors must be assumed in the higher frequency range, the error bounds for input uncertainties are modelled using a first-order lead/lag transfer function with the bound exceeding 100% of the nominal value for frequencies above 0.5 rad/min: 1+ 20s w~(s) = 0.1 - -
1 + 0.02s
For the derivation of an adequate uncertainty model, it must be considered that the column is operated in closed-loop once a controller is designed and implemerited. In the low-frequency range, a reasonable controller will exhibit large gains ('integrating behaviour') in order to keep the controlled variables close to their setpoints. Thus, the composition profile as well as the internal flow rates are maintained very close to their steady-state values. Consequently, the column dynamics
column
~(jto) = [I +A (jto)W,(jto)]u(jto) II A , ( j t o ) ] ] ~ 1
Process nonlinearity
(5)
at low frequencies become a function of feed flow rate and feed composition only. Within the range of these feed conditions, the controller sets the largest internal flow rates for the smallest feed composition and the largest feed flow rate. The smallest internal flow rates, on the other hand, are set for the largest feed composition and smallest feed flow rate. Together, the composition profiles at these two steady-state operating points represent the bounds of the domain of all steady-state profiles.S Therefore, the low-frequency dynamics of a binary high-purity distillation column are bounded by two models representing the maximum and minimum column load. As a basis for further discussions the following three linear models are introduced: Model G~ column at nominal load (F = 33 tool/rain, X F = 0 . 8 tool/tool) Model G~ column at maximum feed flow rate and
minimum feed composition (increased load, F = 46 mol/min, XF = 0.7 tool/tool) Model GR column at minimum feed flow rate and maximum feed composition (reduced load, F = 20 tool/rain, XF = 0.9 mol/mol) The Nyquist plots (Figure 3) for the individual transfer functions between the two manipulated inputs and two controlled outputs demonstrate that a variation in the column load causes a simultaneous increase or decrease of the plant gains. Thus, we can assume that the dynamics must lie somewhere between model G~ and model GR. This can be represented by a linear combination of the two column models G~ (/to) and GR(/'(.o). ~(jto) = Gi(jto) + GR(Jto)+ 6a(jto ) Gx(jto)-GR(jto) 2 2 with II 6a(Jto)I[= < 1,6c ~ C~X~ (6) Since the points at fixed frequencies in the Nyquist plots are not on a straight line, the uncertain parameter 8a is chosen to be complex. This uncertainty model provides a reasonable approximation of the low-frequency dynamics of the controlled column for the whole range of feed conditions. It describes the correlation of gain and time constant uncertainty and avoids circular uncertainty bounds for the nominal model which are known to be very conservative. ~°
Robust control of distillation columns: U. Christen et al.
GTxoL
GTmV
/ .~
,
= .~ ea E
I
.
I
I
x
~+ x
\
/ N
/
I
~
"
= .~ ~ ~ )=,=(
Xx
"/
/
/ \
\
/
/
~.
-10G 6
200
-100~ ()
200 Real axis
GT44L
"--
GT44V ,
t:)
c~
•t:~ =E
,,.
~. . ~
0
I'
m
I ~
~<
,
t:l
I'
/
I
I
\
i
/ "
.=~rj
E
"
x
~
?
\ \
-40( 6
/
/
Real axis 0
23
700
/
-300 6
Real axis
600 Real axis
Figure 3 N y q u i s t p l o t s o f the i n d i v i d u a l t r a n s f e r f u n c t i o n s for m o d e l s r e p r e s e n t i n g d i f f e r e n t c o l u m n l o a d s a n d linear i n t e r p o l a t i o n s -: M o d e l GM . - . : M o d e l G~: - - : M o d e l GR; × : C0= I X I 0 ~ r a d / m i n : +: Co= 1 x 1 0 a r a d / m i n
It is important to note that such an uncertainty model in the frequency domain can only describe uncertain, but time-invariant plants. Therefore, the description is restricted to a description of all fixed or arbitrarily slowly varying ~ operating points within tile operating range of the distillation column. Output uncertainty
Three different sources of uncertainty are lumped together as an output uncertainty: transient effects during disturbance rejection, unmodelled high-frequency dynamics and uncertainty in the temperature measurements. This prevents an overly complex uncertainty model and significantly simplifies the numerical analysis, While disturbances can be compensated very well in the low-frequency range, for higher frequencies the composition on trays 10 and 44 will deviate from their setpoints. Due to the nonlinear vapour/liquid equilibrium, the gains of the individual transfer functions between the two manipulated inputs and controlled outputs may change in opposite directions, e.g. becoming higher for the transfer functions between the inputs and Tl0, and lower for the transfer functions between the inputs and T44.1° This behaviour can be described with independent multiplicative uncertainties for the two outputs of the models and a diagonal weighting matrix W,, = w,.l.
[ [6,~(.jo)) i:,(jco) = I + 0
with
0
]i, ] 4,(.jco) y(jco)
6,:(jco)
I[ 6,,(jco)[I < 1, 6, • C " ' (7)
The uncertainty bound w,(s)=0.1--1+167s 1 + 1.67s
(8)
is chosen to have large gains in the high frequency range to account for transient effects of disturbances and for neglected flow dynamics. At low frequencies, an uncertainty of 10% is assumed for the description of uncertainties in the temperature measurements. PelJbrmance
specification
The uncertainty model developed so far represents the set of all plants for which the performance of a controller should be guaranteed. F i g u r e 4 illustrates the uncertainty model including the performance specification in the frequency domain. The external inputs d and r are shaped by the weights W a = diag [w,.F, wr] and W, = w,l which are chosen based on physical considerations. As mentioned earlier, the most important disturbances entering this distillation column are slow
Robust control of distillation columns: U. Christen et al.
24
T
., I aP'
,I
[
" -
Performance ,specification 4
--..
' 2-21 6 t r------~" ~ ' - - - - "
Figure
____
'
,,
,11
'ILl
I~"_-'~!
[_2
1
,,
II
It ii
l
l
Jl
n
y
/+"
I
8 I j
9 Yl 0]!
ii
N
Input uncertainty ~Nonlinearity
II
jrOutput uncertainty
The structured uncertainty model
changes of the feed composition and step changes of the feed flow rate. The frequency content of these disturbances is shaped by first-order lags. As a worst-case scenario, the following transfer functions are chosen: 0.1 I~'..¢F ( S ) -- - (9) 1 + 180s 6 WF(S ) -- - (10) 1 + 120s It is not necessary for the gains of these transfer functions to correspond to the operating range of the column which is already taken into account by the uncertainty modelling of the nonlinearity. The gains of Wa rather reflect the scaling of the two signals x F and F to each other and with respect to the reference signal r. The changes of the setpoints for the controlled temperatures are small with a magnitude of _+0.2 K. Thus, W, is chosen as W, = 0.2I
(11)
The performance of the control system now is specified as a norm bound on the transfer function from the scaled inputs to the control error e, 2 which is weighted by W,~ = w,,(s)I. The weighting function w~(s) is chosen as a first-order lag with a large static gain of 100 to statically diminish the errors to 1% of the input signals. The selection of the crossover frequency for the first-order lag is a matter of optimization. If the crossover frequency is too high, the performance cannot be achieved for all the plants described by the uncertainty model. If the crossover frequency is too small, the performance of the closed-loop system is not maximized. The best results were obtained using the performance weight 100 I
W~,(s)-
(12)
1 + 27800s
/~-synthesis results The controller design based on the uncertainty model and the performance specification discussed above
require the framework of the structured singular value (SSV or/1). The SSV was introduced some 10 years ago by Doyle? 2 However, only the recent launch of the '/.tSynthesis and Analysis Toolbox '13 has provided a direct and easy access to the analysis and synthesis algorithms. Since a review of the mathematical properties 14 and the design procedures l-~ would go far beyond the scope of this report, any reader not familiar with the theoretical background is asked to consult the references. The numerical solution of the design task K2 HpA(M)fp= < 1
(13)
where the plant M incorporates the process models, the weighting functions and the controller, is difficult. Two iteration algorithms proposed - the DK-iteration ~5 and the more recently developed/,tK-iteration ~6- use an H=minimization alternating with a frequency dependent scaling of the plant. Both algorithms have two considerable drawbacks: firstly, neither algorithm can guarantee convergence. In our case, it was not possible to achieve convergence of the DK-iteration, and the solution of the design task using the/~K-iteration was very difficult. Secondly, both algorithms require a scaling of the plant at each iteration step, thus increasing the order of the plant. Consequently, the computation times become extraordinarily high (typically two hours on a SPARCstation 2 for six iteration steps using yK-iteration). The order of the resulting controller is very high (more than 80 state variables), but by a balanced truncation it is easily reduced to an order of 20. The good performance of this controller is demonstrated by a plot of the structured singular value (Figure 5). Since the SSV is below one for all frequencies except for a small and insignificant peak in the higher frequency range, this controller guarantees the specified performance, as well as stability for the set of all plants described by the uncertainty model. The tracking properties of this controller are adequate, which is illustrated by a plot of the singular values of the transfer function between the reference signals r and the plant output y based on the nominal closed-loop system with the process model G~ (Figure 6a). Up to the mid-frequency
Robust control of distillation columns: U. Christen et al.
_g
I
_ A ~~a,/~/
~ R P
~ = -~'~ 0.5 '~
o0-5
plant such that the H=-norm of the closed-loop transfer function is smaller than a fixed value. Doyle et al. ~ have elegantly solved this latter task for a general augmented plant. The engineering task, however, cannot be solved for the general case, but must be carried out individually for each plant. It is possible, though, to find weighting structures suited for special classes of problems, e.g. for ill-conditioned plants. It has often been suggested (e.g. refs 2,7,~) to use an input uncertainty to account for the effects of the bad condition of the plant. This can be done with the socalled S / K S / T (or mixed sensitivity) scheme weighting the sensitivity S, = (I + KG) -~ and its complement T,, = I - S, at the plant input. However, this leads to a controller which contains the inverted plant. 2° The disastrous consequences for the robustness of such a controller are demonstrated below. The same problem arises with the S / K S / T scheme in which the sensitivity S and its complement T,, at the output of the plant are weighted? That scheme has been used by Raisch et al. 2~ for the control of a binary and ternary distillation column. However, they have not shown the robustness of their controllers against input uncertainty. Moreover, their column is much better conditioned, the steadystate model of the binary column having a condition number of roughly 15. To avoid the inversion of the plant in the controller, an alternative weighting scheme, the G S / T scheme, is introduced. Using the G S / T scheme, the sensitivity is automatically weighted by the plant, and therefore, the controller does not invert it. 2° Hence, this scheme is especially suited for ill-conditioned plants.
/
RS-/ ~ ~
/ ~
10-3 10-~ Frequency (rad/min)
25
10j
Figure 5 Robustperformance (RP) and robust stability (RS) of the p-optimal controller range, the singular values are close to one and the maximum of the upper singular value is sufficiently small, The plots of the singular values of the sensitivity functions S, = [l + GK] ~ demonstrate the good disturbance rejection properties, which are insensitive to input uncertainty (Figure 6b). The two sensitivity functions with the nominal plant and with an input error of +10% in AL and -10% in AV are almost identical, The simulation of step responses using the nonlinear model shows the disturbance rejection capability in the time domain (Figure 7). At minimum as well as at maximum feed flow rates, the maximum control errors are small (i.e. not much higher than both the measurement noise and the resolution of an industrial temperature sensor), and the product compositions are kept close to their setpoints. The steady-state offsets of the product compositions arise from controlling tray temperatures at some distance from the column ends. From an industrial point of view, the low sensitivity of the performance to errors in the manipulated variables is very important. Based on this uncertainty model, multivariable controllers were designed and implemented for two industrial distillation columns with excellent results. 17. ~
H=-eontroller design
S/KS/T
By introducing the auxiliary input signal v in the S / K S / T scheme shown in Figure 8, the sensitivity S,, and the complementary sensitivity 7",, are the transfer functions from v to u and ~, respectively. The sensitivity and its complement are weighted by 14/,, and W,, which are chosen as
H=-synthesis The design of an H=-controller consists of two stages. In the engineering stage, the plant has to be augmented by weightings reflecting the specifications. In the mathematical stage, a controller is derived for the augmented
W, =
100 I 2000s + 1
a
b
10
'r.
r. 0.01 --
//. ,
0.0001
0.01 10_5
10_3
10_1
Frequency (racL/min)
101
10_5
10_3
nominal plant 10% input, error 10_l
Frequency (rad/min)
Figure 6 Singularvalues of the closed-loopsystem. (a) Complementarysensitivity:(b) sensitivityfunctions S,.
10 I
26
R o b u s t c o n t r o l o f distillation c o l u m n s : U. Christen ,_ 0.01S
etal.
~-, 0.018 [
0.016,
"~' .
.
.
.
.
.
.
.
~ .
"\ .._ s:.~ -
"~ O.Ol
.
.
.
.
.
.
.
.
.
.
.
.
.
r..) 0"0060
10
20
I':;~ .
.
.
.
. . . . .
|t .
.
.
.
.
.
.
.
.
.
.
.
................
.1""""
F = 20 mol/min I [ . . . . . 10% input error I
"
~ .
...........
0.008
0'016tt
.
]
30
"~
O.Ol
o~
0.008 [-q 0"0060
~
40
10
1/
L'."" 10% input error 20 30 40
4/;' i.
.0.2 [
e
e 0
10
20
30
-O4
40
70
0
10
20
30
40
160
•
g
50
g
[ F = 20 raol/min
120
I F = 46 tool/rain
, 20
o
lO
20 Time (h)
30
40
150
lO
20 Time (h)
30
40
Figure 7 Simulationresults with the ~t-optimalcontroller for an increase in feed composition (0.8-0.9 mol/mol)at t = 0 h and an increase of feed flow rate (+ 3.6 mol/min) at t = 20 h W~ = 0 . 1 . ! 8 0 S + l l 2.5s + 1 Wj -- 0.0021 (cf. Figure 9a). W, prescribes a maximum gain of 0.01 for the sensitivity S, at low frequencies up to about 10-4 rad/min, thus attenuating disturbances to 1% in that frequency range. The weight W,7 for the complementary sensitivity can be used to specify a multiplicative uncertainty at the input of the plant. In our case, we assign 10% uncertainty in the low-frequency range and more than 100% above 10-I rad/min, thus limiting the bandwidth of the control system. The weight W,/ is only
~~} rV t
, [_ ~
w
Figure 8 S/K,STTscheme weightingat the plant input
z
used to avoid a singular H~-problem and can be small. The plant used for the control design is the nominal plant GN. The resulting controller is as ill-conditioned as the plant (Figure 9b). Figure 9c and 9d shows that the specifications are met for the sensitivity and its complement in the nominal case. In fact, the singular values for both S, and S,., as well as T, and T~ look the same in the nominal case since the plant is inverted and, therefore, the closed-loop transfer functions are well-conditioned (only S~ and T,, are shown). For the plant with 10% error in the input (AL: +10%, AV: -10%), S,, and T, still look unchanged. However, the sensitivity S,, and its complement T,, are completely deteriorated for such an error as shown in Figure 9. This is consistent with Freudenberg's 4 statement: if the controller inverts the plant, the control system may be very sensitive at one (thel°cati°nplant(theinput).plant o u t p u t ) t o uncertainty at the other The inverting property of this weighting scheme is easy to derive heuristically. If the specifications for the control system are tight, all the singular values of the transfer function 7_., from w to z are about equal and constant for all frequencies. Thus, at low frequencies W"Su = U holds, where U is an allpass transfer function.
Robust control of distillation columns: U. Christen et al. a
b
10 2
-
103
101
102
[,oo
1o1
10-1
1
10-2
10-5
~ 100 ~
10-3
10-I
101
~-~
10-1
10-5
Frequency (rad/min) e
.~
~ _ _ ~ - _ //-
e.,
10-3
" / ~
10-1
101
Frequency (rad/min) d 101
10 o
27
~_. .~ 10 0
-nominal plant - - - 10% input error . . . . . . . .
/
~1o-21~
=l
,'
~
I ga?t
'
"
]
10_a . . . . . 10% input error 10-5 10-3 10.1 101 Frequency (tad/rain)
,
X
,
10_2 10 -5
10-3 10.1 Frequency (tad/rain)
\
\
101
Figure 9 Controller designed with the S/KS/T scheme of Figure & (a) Singular values of the weights. (b) singular values and condition number of the controller, (c) singular values of the sensitivity S~ (d) singular values of the complementary sensitivity
In the low-frequency range, the loop gain L,, = K G is much(KG) ~.largerConsequently,than I and hence, S, is approximately
W.S.~ W.G~ K ~ = U
(14)
K = U I W,,G 1
(1 511
v
I
w{ I
~
~
= l l -.
~"~'] ~
--
" z
Figure 10 G S / T weio_htin~ scheme which avoids the reversion of the
Thus, K contains the inverted plant, at least as far as its low-frequency behaviour is concerned, and is as badly conditioned as the plant. A more rigorous derivation of the inversion of the plant in the controller can be found elsewhere. 2° The inversion could be avoided if G was included in the weight W,,, as can be seen by equation (17). However, this would double the order of the controller,
GS/T H:controller design for
ill-conditioned plants Another possibility to include the plant in the weighting of the sensitivity is the G S / T scheme shown in Figure 10. The transfer function from v to y is G S . whence the sensitivity is weighted by the plant and by W,. The resulting controller no longer contains the inverted plant. :~ As before, T:. has constant singular values such that II':GS, ~ U at low frequencies. With the open-loop approximation for the sensitivity, it follows that
K = ('~W~
(17)
plant in
the
controller
Thus, the singular values of K are directly controlled by' W,. Now' a well-conditioned controller with ~:(K) = 1. as requested by Freudenberg, can be obtained by choosing 14:] = %(/co)I. The weight W,7 for the complementary sensitivity T, is chosen to prescribe the same maximum bandwidth as before. Its static gain is smaller than in the previous section in order to demonstrate that modelling of an input uncertainty is not required with this scheme. W~ in the weight for the sensitivity is smaller than W,, before since the sensitivity is additionally weighted by the plant. To allow for S. + T,, = I. W, is chosen such that the weight W,G for the sensitivity and the weight I#',~for the cornplementar) sensitivity T,, intersect below 1 (see Figure 1/el), The plant is again Gv. 40 W - 18000s +1 I 1800s + l W = 0.01 I W/ = 0.02I 2'5s+ 1
(18)
The controller designed with this weighting scheme is much better conditioned. Consequently, the singular
103
Robust control of distillation columns: U. Christen et al.
28
a
b
102
102
100 10-5
10° ~ ¢~ .
10-3 lO-I Frequency (rad/min) e
~
101
10.5 101
/
10 !
-nominal plant i i- - - 10% input error i ' ~
"~ l00
~10-2
10-3 10-I Frequency (rad/min) d
~ 10-1 - - nominal plant - - - 10% input error
10-'4 ,// 10-5 1()-3 1()-l Frequency (rad/min)
10-2i 10-5
101
10-3 10-I Frequency (rad/min)
101
Figure It Controller designed with the GS/T scheme. (a) Singular values of the weights, (b) singular values and condition number of the controller, (c) singular values of the sensitivitySe, (d) singular values of the complementarysensitivity values of the sensitivity and the complementary sensitivity no longer match (Figure lib-d) but they are the same for S, and S~ as well as for T~ and T< (only S,, and T~, are shown). They meet the specifications in the nominal and perturbed case with the input errors of 10%. In fact, the deterioration is nearly indiscernible. The same is true for the nonlinear simulation of two disturbance steps (Figure 12). The performance degradation in the presence of errors in the manipulated variables is insignificant, whereas it would be prohibitive for the inverting controllers,
These simulation results are similar to those obtained with the p-controller (Figure 7). In both cases, the peaks of the temperature differences being smaller than 0.3 K (Figure 12), the limitations given by the resolution of the temperature sensors are nearly exhausted. The similar performance of the two controllers can also be seen in the p-analysis [Figure 13 (left) as compared to Figure 5] for which the same weights as for the p-synthesis are used. At low frequencies, the H=-controller performs slightly better than the p-controller, whereas the peak around I0 -~ rad/min is higher. For the inverting H=-con-
~0.018
0.2
t=0"016 ~ . - ~ - ~ - ~ i - -
'~~'0"1 ' i '
~ 0 . 0 1 4 "~.
~
~ 0.012
o-0.1
""
:~ 0.01 -- - ~ ............. ~'0.008 F = 33 mol/min o 0.O06 120 F = 33 tool/rain
=i00 \
,--0"2 r~ -0.3
360 40 0
10
/ 20 30 Time (h)
",,..f !~: !i F = 33 tool/rain [ . . . . . 10% input error I
-0.4 ~.0.018 ~O 0.016 ~---0--0-~, ~- . . . . . . . . . o ' .... ~ ,
~--~
o
0
40
"~ 0.01 . . . . . . . . . . . ~ 'o.oo8 o V [. . . . F = 20 mol/min J r.) 0.006(~ 10 20 30 40 Time (h)
Figure 12 Nonlinear simulationsof an increase in feed composition from 0.8 to 0.9 mol/mol at t = 0 h and an increase of feed flow rate of 3.6 mol/min at t = 20 h
Robust control of distillation columns: U. Christen et al.
29
robust performance - - - robust stability - -
-~
46 0.5 . . . .
-
010-5
Figure 13
-._-i
.s
- - robust stability 10-3 10-1 Frequency (rad/min)
~41 \ 10 t
=-.0t
10-5
10-3 10-1 Frequency (rad/min)
101
p-analyses for the noninverting (lelk) and the inverting (right) H -controllers
troller designed in the previous section, on the other hand, neither stability nor performance is acceptable (Figure 13 right). There are several possibilities to further improve the Ha- and p-controllers. The two main disturbances, which are easy to model, can be taken into account in t h e augmentation scheme. For the feed flow rate, which is measured, it is even possible to design a feedforward compensator. The disturbance rejection at higher frequencies may be improved by feeding back an additional temperature taken near the feed tray. An H~-controller with all of these improvements has been designed, ~ but space limitations preclude a presentation of the results in this paper,
Conclusions Only MIMO plants can be ill-conditioned and there is no analogue in SISO plants. Consequently, designing a robust controller for an ill-conditioned plant with SISO synthesis methods is not possible. The H=- and p-controllers designed here are not diagonal, even though they have small condition numbers and they both outperform diagonal PI-controllers by far. s, 2-, For the design of the p-controller, a structured uncertainty model has been developed which describes the dynamics of a high-purity distillation column for the entire operating range. All uncertainties in the model can be physically justified, which allows for an easy shaping of the uncertainty bounds. A controller design based on this uncertainty model is distinguished by the high performance for the entire operating range of the column and by low sensitivity to uncertainty in the manipulated variables. However, these advantages are achieved by a large effort for the development of an appropriate uncertainty structure and for the computation of the controllers. For further projects, though, the effort for uncertainty modelling is significantly reduced, In the H~-section, it has been shown that it is vitally important to choose a weighting scheme which guarantees that the plant is not inverted - the GS/T scheme. Then, an H~-controller performing as well as the p-controller can be designed, However, if an S/KS/T scheme is used, even t h e inclusion of an input uncertainty cannot prevent the inversion of the plant in the controller,
Comparisonof the H~- and the /.t-synthesis methods Either method, p- or H~-synthesis, may be used to design a controller for ill-conditioned plants - if appropriate structures for the augmentation of the plant are chosen. However, the two methods differ in the ways in which they arrive at these structures. The p-method requires a profound knowledge of the plant dynamics. Only with this knowledge is modelling disturbances and uncertainties based on first principles possible. The p-method advantageously avoids dealing explicitly with the bad condition of the plant. However, it too must prevent the inversion of the plant in the controller, as indicated by Freudenberg. 4 The inclusion of both an input and an output uncertainty prevents the control system from becoming sensitive to those uncertainties, as may happen with inverting controllers. The incorporation of various operating points guarantees the specified performance for all operating points as long as they vary arbitrarily slowly, t~ For the H~controller, this objective can be assured only by simulation or p-analysis. However, for both methods, nothing can be guaranteed for fast transients between different operating points. Simulations are thus required to prove the controller performance. With the p-approach, the upper bound for the bandwidth of the control system is provided by the uncertainty model, whereas the lower bound is a matter of optimization. For the H~-approach, on the other hand, the control designer must have an idea of the upper and lower bounds for the allowable bandwidth. However, a nominal model of the plant suffices. It can be obtained either by identification or by modelling based on first principles. This minimal knowledge is incorporated in an augmentation scheme which explicitly avoids the inversion of the plant. Once the augmented plant is formed, the p-synthesis requires a lengthy computation. The synthesis procedure is iterative, and its convergence is not guaranteed. The resulting controller is of a much higher order than the plant and subsequently must be reduced. The computations for H~-synthesis are less involved. They take about the same amount of time as the first H~-synthesis step in p-synthesis: less than 1 min on a SPARCstation 2. The order of the resulting controller is only slightly higher than that of the plant because normally only two
Robust control of distillation columns: U. Christen et al.
30
dynamic weightings are needed to augment the plant. Because the computations for H=-controllers are that inexpensive, it is possible to iterate on the weightings to improve the control system, whereas that is nearly impossible for a/.t-controller, necessitating two hours of
6
Skogestad, S. and Morari, M. Chemical Engineering Science
7
1988, 43, 33 Skogestad, S. and Lundstr6m, P. Computers Chem. Engng. 1990, 14, 401
computation time. To conclude,/l-synthesis is ideally suited to deal with complex uncertainty models which take into account such aspects as various operating points. The price o n e has to pay for it is tedious computations. If, however, loop-shaping ideas are used to form the augmented
9
plant, H~-synthesis may be used to advantage. The results are as good as with the /.t-synthesis, but are
obtained with less numerical effort.
aeferellces l
2 3
4 5
Skogestad, S. 'Dynamics and control of distillation columns - A critical survey,' 3rd IFAC Symposium on Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes, College Park, MD, pp. 1-25, 1992 Skogestad, S., Morari, M. and Doyle, J. C. 1988, IEEE Transactions on Automatic Control, 33 1092 Christen, U. and Musch, H. E. 'Robust control of distillation columns: p or H=?' IMRT Report No. 29, Measurement and Control Laboratory, ETH Zurich, 1995 Freudenberg, J. S. Int. J. Control 1990, 51, 365 Musch, H. E. 'Robust control of an industrial high-purity distillation column', PhD thesis, ETH Zurich, 1994
8 Skogestad, S. and Morari, M. 'Control of ill-conditioned plants: high purity distillation' AIChE Annual Meeting, Miami Beach, FL, 1986 Lundstr6m, P., Skogestad, S. and Doyle, J. in Proc. of the 2nd
European Control Conference,Vol. 2, p. 969, 1993 10 McDonald, K. A. in The Shell Process Control Workshop (eds D. M. Prett and M. Morari), Butterworth, Boston, 1987, 279 11 Poolla, K. and Tikku, A. IEEE Transactions on Automatic Control 1995, 40, 1589 12 Doyle, J. C. lEE Proc., 1982, 129, Pt. D., No. 6, 242 13 Balas, G. J., Doyle, J. C., Glover, K , Packard, A. and Smith, R. 'p-Analysis and Synthesis Toolbox, User's Guide' The
MathWorks, Natick, MA, 1993 14 Packard, A. and Doyle, J. C. Automatica, 1993, 29, 71 15 Packard, A., Doyle, J. C. and Balas, G. A S M E Journal of Dynamic Systems, Measurement, and Control 1993, 115, 426 16 Lin, J.-L., Postlethwaite, I. and Gu, D.-W. Automatica, 1993, 29, 219 17 Musch, H. E. and Steiner, M. 1EEE Control Systems Magazine, 1995, 15, 46 18 Musch, H. E., McKay, B., Willis, M. and Barton, G. 'Industrial applications of MATLAB based software for advanced data analysis and control' Preprints Control 95, Melbourne, p. 211, 1995 19 Doyle, J. C., Glover, K., Khargonekar, P. P. and Francis, B. A. 1EEE Transactions on Automatic Control 1989, 34, 831 20 Christen, U. 'Engineering aspects of H~ control'. PhD thesis, ETH Zurich, 1996 21 Raisch J., Lang, L. and Gilles, E-D. Automatisierungstechnik 1993, 41,215 22 Musch, H. E. and Steiner, M. in 'Preprints of the 12th World Congress IFAC', Sydney, Vol. 1, p. 49, 1993
~ ]
~
Cop}right (~ 1996 Elsevier Science Ltd Printed in Great Britain All rights reserved 0939-1524797 $ I 7.0(I + 0.00
ELSEVIER
Pll: S0959--1524(96)00008-X
Robust control of distillation columns: /l- vs H -synthesis Urs Christen, Hans E. Musch and Max Steiner* Measurement and Control Laboratory, Swiss Federal Institute of Technology (ETH) CH-8092 ZSrich, Switzerland Received 5 October 1994; in revised form 11 December 1995
High-purity distillation columns are known to be difficult to control due to their ill-conditioned and nonlinear behaviour. Two approaches for the design of robust controllers yielding high performance are presented. For the first approach, first principles are used to develop an uncertainty model describing the nonlinear dynamics within the entire operating range of an industrial distillation column. This structured uncertainty model is used for/l-synthesis. In a second approach which is based on loop shaping ideas, an H -controller is designed. This controller performs as well as the y-controller. The H~-approach offers the advantage that the burden for uncertainty modelling and computation is greatly reduced. However, the GS/T augmentation scheme, which is developed in this paper, must be used for the design of the H -controller to avoid the inversion of the plant in the controller. The paper concludes with a comparison of the H=- and H-synthesis methods. Copyright %2"1996 Elsevier Science Ltd.
Keywords: distillation column, ill-conditioned plants, g-synthesis, H~-synthesis
Distillation is one of the most important unit operations in the process industry. Almost all chemical plants use distillation columns for the separation of substances according to their relative volatility, and distillation processes are among the largest energy consumers in chemical plants. Tight control of the product compositions is fundamental for an economically and ecologically optimal operation of the entire plant. However, especially in the case of high product purities, distillation columns are known to be difficult to control because their dynamics are ill-conditioned. Furthermore, many distillation columns in the chemical industry operate over a wide range of feed flow rates and compositions, Many robust control design techniques have been successfully applied to this control problem ~ and, at first
trays, a total condenser and a steam-heated reboiler (Figure 1), serves as an example for a typical high-purity distillation process. The feed F is a mixture of mainly two components which is split into the two product streams D and B. The most important operating data including the operating range are summarized in Table l. Due to high investment and maintenance costs of composition analysers, the pressure-compensated ten> peratures on trays 10 and 44 are controlled instead of the product compositions. The manipulated variables chosen to control these two tray temperatures are the reflux k and the boilup V (indirectly controlled by reboiler heat duty). This L V control structure is very common in the chemical industry.
glance, this design task could be considered as solved. Upon closer consideration, however, some drawbacks of the design approaches proposed in the literature become apparent. Since a discussion of these drawbacks necessitates a basic knowledge concerning distillation dynamics, a short summary of the basic principles is given next.
IIl-conditionedplaJlt dynamics As mentioned above, the dynamics of high-purity distillation columns are ill-conditioned, which leads to high sensitivity to uncertainties in the manipulated variables.: Even small errors in these manipulated variables can cause a significant deterioration of the product qualities, 3 a fact which explains why open-loop control of high-purity distillation columns is hardly ever satisfactory. As shown in Figure 2, the column considered here has a condition number
The distillation process An industrial distillation column equipped with 50 sieve
* A u t h o r to w h o m c o r r e s p o n d e n c e s h o u l d be a d d r e s s e d
19
Robust control of distillation columns: U. Christen et al.
20
Vacuum
/~1
•/@I¢' 1 &
Distillate
i~
L[~
Reflux accumulator
102
Condenser
VT
.
.
'" ~ .. .~10 o
.
.
9
Reflux
10 10-5
-Feed - ~
_2.o._
F, x F
Vi / Yi
Figure 2
Yi Gas phase composition
ni Holdup Liquid phase composition
47 -~,8-~t95i3
~
Reboiler B, x B ..._
Bottom product Figure 1 Binary distillation column
Table 1 Steady-state data Column data No. of trays Column diameter (m) Feed tray number Murphree tray efficiency Relative volatility a
50 0.8 20 =0.4
Feed flow rate F (mol/min) Top pressure(mbar)
Nominal operating point
Feed composition (mol/mol)
Feed flow rate (mol/min) Reflux L (mol/min) Boilup V(mol/min)
~¢(G(jo))) = ~ ( G ( j o g ) ) cr(G(jw))
Singular values ( - - ) and condition number ( - -) of the
L i n e a r plant models
1.61
All controller designs within the context of this report are based on linear process models which were calculated by numerical linearization of a rigorous dynamic model. This rigorous model includes the dynamics of the tray compositions and holdups, as well as the
Operating data
Top composition .x',~(tool/tool) Bottom composition xA (moI/mol) Feed composition xF(mol/mol)
10 l
for inputs in the least amplified direction. If the gains of the actuators and therefore the directions of the maximum and minimum gains of the column are uncertain, large amplifications by the controller may be misaligned which causes the controlled outputs to deviate from the intended values. These observations are further clarified by Freudenberg. 4 He has shown that a controller for an ill-conditioned plant must not invert the plant. With plant inversion, the control system becomes very sensitive to uncertainty due to the strong directionality of illconditioned plants. Robustness properties may be good at the output of the plant, but very poor at its input, or vice versa. Freudenberg concludes that the controller should be well-conditioned and consequently, the openloop transfer function of the control system should be as ill-conditioned as the plant itself. This means that there are strong limitations to the achievable closedloop performance.
Ti Temperature
_.., Boilup,, i -" v ~ ~-Ij @
10-3 10-l Frequency (rad/min)
distillation column (nominal model)
xi
t
_
0.99 0.015 0.7-0.9 20~,6 60
0.8
33 65 104
(1)
of several hundred (~(.) and <7(.) denote the maximum and minimum singular values). In other words, the gain for inputs in the direction in which the plant amplifies most can be several hundred times larger than the gain
algebraic equations for the liquid and vapour flow rates, pressure drops and boiling points. Altogether, this model consists of a system of several hundred differential and algebraic equations. 5 In the case of this industrial column, the level of the reflux accumulator is controlled by the top product flow rate D while the reboiler level is controlled by the bottom product flow rate B. Since these levels have tight control, perfect level control and constant holdups in column top and bottom can be assumed. With these assumptions, a linear state-space model of order 102 (52 state variables representing compositions and 50 variables representing holdups) with the following structure can be calculated.
,,~ = A x + Bu + E d y = Cx
(2)
Robust control of distillation columns: U. Christen et al. • &'~D Axi : x = zk,c8 Dn~
[ AL ] u =
AV
V•X"F 1 d =
LAFJ
VATI0 1 y =
LAT,~]
-Ansc (3) The symbol A stands for the deviation from the operating point. The other symbols are as defined in Figure 1. Since the computation time for controller design strongly depends on the model order, an order reduction of these linear models is necessary. Using a balanced truncation, the models of order 102 are easily reduced to an order of 10-15 without significant loss of accuracy. The identification of linear plant models based on recorded step response is an alternative approach to obtain linear models. As pointed out by Skogestad, ~ it is very difficult to identify the dynamics of ill-conditioned plants. Furthermore, the recording of step responses should start at operating points where the product compositions are close to their setpoints and the dynamics are almost at steady-state. While this can be easily achieved in simulations, it is almost impossible in a real high-purity distillation column operated in open-loop (i.e. without composition control)• Therefore, a linearization of the rigorous dynamic model is used in this study,
Control objectives The main objective of the control system is to keep the controlled (pressure compensated) tray temperatures close to their setpoints despite the presence of disturbances. The most significant disturbances are changes in the feed flow rate and in the feed composition, The second design objective is a sufficiently fast setpoint tracking• Direct composition measurements are not installed; rather, as mentioned above, pressure compensated temperatures on trays 10 and 44 are controlled• Therefore, significant changes in the feed composition cause a slight worsening of the product compositions. The plant operator counteracts this effect by changing the setpoints for the temperatures on trays 10 and 44. These two objectives - a best possible disturbance compensation and a reasonable setpoint tracking - must be achieved for the entire operating range of the distillation column•
Design approaches
Basically, two different approaches in robust controller design can be chosen. With a thorough knowledge of the process, the mismatch between the dynamics of the linear model and the real plant dynamics can be modelled from a physical point of view. This includes the operating range of the column, inaccuracies of the sensors and
21
actuators, as well as the frequency content of the disturbances and setpoint signals. Based on a structured uncertainty model which describes the column dynamics within the entire °perating range' a c°ntr°ller is calculated using the ~-synthesis technique. This approach is discussed in the next section. In the second approach, the controller is designed based on loop-shaping ideas. A new augmentation scheme for the design of Ha-controllers is introduced which explicitly avoids the inversion of the plant in the controller by including the plant in the weight of the sensitivity. This ensures that Freudenberg's condition discussed above is satisfied.
N-synthesis Since high-purity distillation columns can be very sensitive to uncertainties in the manipulated variables, it is important for successful implementation that a controller guarantees its performance in the presence of input uncertainties. This particular design task is tYequentty solved by modelling a multiplicative input uncertainty for a nominal plant model and subsequently calculating the controller using p-synthesis. > ') If a distillation column is operated in a rather small range of feed flow rates and feed composition, such a controller can be expected to perform well. However. in the chemical industry the operating conditions of a distillation column often change over a (vide range depending on the actual load of the entire plant. The distillation column described here is a typical example of such a plant, where a controller should perform as well as possible not only, for the nominal operating conditions, but for the whole range of flow rates and compositions fed to the column. One way to handle this task is to design a controller for the nominal plant model (including a multiplicative input uncertainty) and subsequently test its robustness for other operating conditions. 6 Another more straightforward approach is the representation of the varying column dynamics by an extended uncertainty model as proposed by Skogestad et al. 2.* In these papers, the nominal dynamics of a distillation column are approximated by a crude first order model, The nonlinearity of the cotumn is described by time constant and gain uncertainty,. which are represented by divisive uncertainty at plant output and additive uncertainty, respectively• The authors explicitly state that the variations of the time constants and gains are clearly correlated, a fact which is not captured by the uncertainty description they propose. Consequently, this uncertainty, model describes a range of column dynamics which is wider than that of the real column. Therefore, a controller design based on this uncertainty model is always conservative. A similar plant and uncertainty model (using gain and delay uncertainty) has been used, e.g. by Lundstr6m et al2 for a study of a two degree of freedom controller design within the /~-framework. This gain and delay uncertainty, however, fails to represent the correlations in the
Robust control of distillation columns: U. Christen et al.
22
column dynamics, necessitating a more structured model of uncertainty. In the following section, an uncertainty model is developed based on physical considerations which takes the correlation between time constant and gain uncertainty into account and thus avoids this conservatism,
An
uncertainty model for the distillation
As a general guideline, Skogestad et al. 2~ recommend modelling uncertainty where it physically occurs, According to this principle, three types of uncertainty are identified in this report: •
Uncertainty of the manipulated variables (input
•
uncertainty) Uncertainty due to nonlinearity of the process
•
dynamics Unmodelled high-frequency dynamics and uncertainty of the temperature measurements (output certainty),
Input uncertainty As discussed above, ill-conditioned plants can be very sensitive to errors in the manipulated variables. Since measurement errors, parameter variations (e.g. changing heat of evaporation), and unmodelled high frequency dynamics are always present, errors between the setpoints and real flow rates of the reflux and the boilup must be expected? The bounds for the relative errors of the column inputs u are modelled in the frequency domain by a multiplicative uncertainty with two frequency-dependent error bounds w,,. These two bounds are combined in the diagonal matrix IV,, = %1. with (4)
Due to flow dynamics and energy balance, any change e.g. in the reflux may cause a change of the vapour flow rates within the column. Therefore, the errors in the manipulated variables can be interdependent, as represented by the 2 × 2 complex matrix A,(j'to). This matrix is limited in magnitude and only shapes the spatial direction of the error. If all flow measurements are carefully calibrated, an error bound of 10% for the low frequency range is reasonable. Because much higher errors must be assumed in the higher frequency range, the error bounds for input uncertainties are modelled using a first-order lead/lag transfer function with the bound exceeding 100% of the nominal value for frequencies above 0.5 rad/min: 1+ 20s w~(s) = 0.1 - -
1 + 0.02s
For the derivation of an adequate uncertainty model, it must be considered that the column is operated in closed-loop once a controller is designed and implemerited. In the low-frequency range, a reasonable controller will exhibit large gains ('integrating behaviour') in order to keep the controlled variables close to their setpoints. Thus, the composition profile as well as the internal flow rates are maintained very close to their steady-state values. Consequently, the column dynamics
column
~(jto) = [I +A (jto)W,(jto)]u(jto) II A , ( j t o ) ] ] ~ 1
Process nonlinearity
(5)
at low frequencies become a function of feed flow rate and feed composition only. Within the range of these feed conditions, the controller sets the largest internal flow rates for the smallest feed composition and the largest feed flow rate. The smallest internal flow rates, on the other hand, are set for the largest feed composition and smallest feed flow rate. Together, the composition profiles at these two steady-state operating points represent the bounds of the domain of all steady-state profiles.S Therefore, the low-frequency dynamics of a binary high-purity distillation column are bounded by two models representing the maximum and minimum column load. As a basis for further discussions the following three linear models are introduced: Model G~ column at nominal load (F = 33 tool/rain, X F = 0 . 8 tool/tool) Model G~ column at maximum feed flow rate and
minimum feed composition (increased load, F = 46 mol/min, XF = 0.7 tool/tool) Model GR column at minimum feed flow rate and maximum feed composition (reduced load, F = 20 tool/rain, XF = 0.9 mol/mol) The Nyquist plots (Figure 3) for the individual transfer functions between the two manipulated inputs and two controlled outputs demonstrate that a variation in the column load causes a simultaneous increase or decrease of the plant gains. Thus, we can assume that the dynamics must lie somewhere between model G~ and model GR. This can be represented by a linear combination of the two column models G~ (/to) and GR(/'(.o). ~(jto) = Gi(jto) + GR(Jto)+ 6a(jto ) Gx(jto)-GR(jto) 2 2 with II 6a(Jto)I[= < 1,6c ~ C~X~ (6) Since the points at fixed frequencies in the Nyquist plots are not on a straight line, the uncertain parameter 8a is chosen to be complex. This uncertainty model provides a reasonable approximation of the low-frequency dynamics of the controlled column for the whole range of feed conditions. It describes the correlation of gain and time constant uncertainty and avoids circular uncertainty bounds for the nominal model which are known to be very conservative. ~°
Robust control of distillation columns: U. Christen et al.
GTxoL
GTmV
/ .~
,
= .~ ea E
I
.
I
I
x
~+ x
\
/ N
/
I
~
"
= .~ ~ ~ )=,=(
Xx
"/
/
/ \
\
/
/
~.
-10G 6
200
-100~ ()
200 Real axis
GT44L
"--
GT44V ,
t:)
c~
•t:~ =E
,,.
~. . ~
0
I'
m
I ~
~<
,
t:l
I'
/
I
I
\
i
/ "
.=~rj
E
"
x
~
?
\ \
-40( 6
/
/
Real axis 0
23
700
/
-300 6
Real axis
600 Real axis
Figure 3 N y q u i s t p l o t s o f the i n d i v i d u a l t r a n s f e r f u n c t i o n s for m o d e l s r e p r e s e n t i n g d i f f e r e n t c o l u m n l o a d s a n d linear i n t e r p o l a t i o n s -: M o d e l GM . - . : M o d e l G~: - - : M o d e l GR; × : C0= I X I 0 ~ r a d / m i n : +: Co= 1 x 1 0 a r a d / m i n
It is important to note that such an uncertainty model in the frequency domain can only describe uncertain, but time-invariant plants. Therefore, the description is restricted to a description of all fixed or arbitrarily slowly varying ~ operating points within tile operating range of the distillation column. Output uncertainty
Three different sources of uncertainty are lumped together as an output uncertainty: transient effects during disturbance rejection, unmodelled high-frequency dynamics and uncertainty in the temperature measurements. This prevents an overly complex uncertainty model and significantly simplifies the numerical analysis, While disturbances can be compensated very well in the low-frequency range, for higher frequencies the composition on trays 10 and 44 will deviate from their setpoints. Due to the nonlinear vapour/liquid equilibrium, the gains of the individual transfer functions between the two manipulated inputs and controlled outputs may change in opposite directions, e.g. becoming higher for the transfer functions between the inputs and Tl0, and lower for the transfer functions between the inputs and T44.1° This behaviour can be described with independent multiplicative uncertainties for the two outputs of the models and a diagonal weighting matrix W,, = w,.l.
[ [6,~(.jo)) i:,(jco) = I + 0
with
0
]i, ] 4,(.jco) y(jco)
6,:(jco)
I[ 6,,(jco)[I < 1, 6, • C " ' (7)
The uncertainty bound w,(s)=0.1--1+167s 1 + 1.67s
(8)
is chosen to have large gains in the high frequency range to account for transient effects of disturbances and for neglected flow dynamics. At low frequencies, an uncertainty of 10% is assumed for the description of uncertainties in the temperature measurements. PelJbrmance
specification
The uncertainty model developed so far represents the set of all plants for which the performance of a controller should be guaranteed. F i g u r e 4 illustrates the uncertainty model including the performance specification in the frequency domain. The external inputs d and r are shaped by the weights W a = diag [w,.F, wr] and W, = w,l which are chosen based on physical considerations. As mentioned earlier, the most important disturbances entering this distillation column are slow
Robust control of distillation columns: U. Christen et al.
24
T
., I aP'
,I
[
" -
Performance ,specification 4
--..
' 2-21 6 t r------~" ~ ' - - - - "
Figure
____
'
,,
,11
'ILl
I~"_-'~!
[_2
1
,,
II
It ii
l
l
Jl
n
y
/+"
I
8 I j
9 Yl 0]!
ii
N
Input uncertainty ~Nonlinearity
II
jrOutput uncertainty
The structured uncertainty model
changes of the feed composition and step changes of the feed flow rate. The frequency content of these disturbances is shaped by first-order lags. As a worst-case scenario, the following transfer functions are chosen: 0.1 I~'..¢F ( S ) -- - (9) 1 + 180s 6 WF(S ) -- - (10) 1 + 120s It is not necessary for the gains of these transfer functions to correspond to the operating range of the column which is already taken into account by the uncertainty modelling of the nonlinearity. The gains of Wa rather reflect the scaling of the two signals x F and F to each other and with respect to the reference signal r. The changes of the setpoints for the controlled temperatures are small with a magnitude of _+0.2 K. Thus, W, is chosen as W, = 0.2I
(11)
The performance of the control system now is specified as a norm bound on the transfer function from the scaled inputs to the control error e, 2 which is weighted by W,~ = w,,(s)I. The weighting function w~(s) is chosen as a first-order lag with a large static gain of 100 to statically diminish the errors to 1% of the input signals. The selection of the crossover frequency for the first-order lag is a matter of optimization. If the crossover frequency is too high, the performance cannot be achieved for all the plants described by the uncertainty model. If the crossover frequency is too small, the performance of the closed-loop system is not maximized. The best results were obtained using the performance weight 100 I
W~,(s)-
(12)
1 + 27800s
/~-synthesis results The controller design based on the uncertainty model and the performance specification discussed above
require the framework of the structured singular value (SSV or/1). The SSV was introduced some 10 years ago by Doyle? 2 However, only the recent launch of the '/.tSynthesis and Analysis Toolbox '13 has provided a direct and easy access to the analysis and synthesis algorithms. Since a review of the mathematical properties 14 and the design procedures l-~ would go far beyond the scope of this report, any reader not familiar with the theoretical background is asked to consult the references. The numerical solution of the design task K2 HpA(M)fp= < 1
(13)
where the plant M incorporates the process models, the weighting functions and the controller, is difficult. Two iteration algorithms proposed - the DK-iteration ~5 and the more recently developed/,tK-iteration ~6- use an H=minimization alternating with a frequency dependent scaling of the plant. Both algorithms have two considerable drawbacks: firstly, neither algorithm can guarantee convergence. In our case, it was not possible to achieve convergence of the DK-iteration, and the solution of the design task using the/~K-iteration was very difficult. Secondly, both algorithms require a scaling of the plant at each iteration step, thus increasing the order of the plant. Consequently, the computation times become extraordinarily high (typically two hours on a SPARCstation 2 for six iteration steps using yK-iteration). The order of the resulting controller is very high (more than 80 state variables), but by a balanced truncation it is easily reduced to an order of 20. The good performance of this controller is demonstrated by a plot of the structured singular value (Figure 5). Since the SSV is below one for all frequencies except for a small and insignificant peak in the higher frequency range, this controller guarantees the specified performance, as well as stability for the set of all plants described by the uncertainty model. The tracking properties of this controller are adequate, which is illustrated by a plot of the singular values of the transfer function between the reference signals r and the plant output y based on the nominal closed-loop system with the process model G~ (Figure 6a). Up to the mid-frequency
Robust control of distillation columns: U. Christen et al.
_g
I
_ A ~~a,/~/
~ R P
~ = -~'~ 0.5 '~
o0-5
plant such that the H=-norm of the closed-loop transfer function is smaller than a fixed value. Doyle et al. ~ have elegantly solved this latter task for a general augmented plant. The engineering task, however, cannot be solved for the general case, but must be carried out individually for each plant. It is possible, though, to find weighting structures suited for special classes of problems, e.g. for ill-conditioned plants. It has often been suggested (e.g. refs 2,7,~) to use an input uncertainty to account for the effects of the bad condition of the plant. This can be done with the socalled S / K S / T (or mixed sensitivity) scheme weighting the sensitivity S, = (I + KG) -~ and its complement T,, = I - S, at the plant input. However, this leads to a controller which contains the inverted plant. 2° The disastrous consequences for the robustness of such a controller are demonstrated below. The same problem arises with the S / K S / T scheme in which the sensitivity S and its complement T,, at the output of the plant are weighted? That scheme has been used by Raisch et al. 2~ for the control of a binary and ternary distillation column. However, they have not shown the robustness of their controllers against input uncertainty. Moreover, their column is much better conditioned, the steadystate model of the binary column having a condition number of roughly 15. To avoid the inversion of the plant in the controller, an alternative weighting scheme, the G S / T scheme, is introduced. Using the G S / T scheme, the sensitivity is automatically weighted by the plant, and therefore, the controller does not invert it. 2° Hence, this scheme is especially suited for ill-conditioned plants.
/
RS-/ ~ ~
/ ~
10-3 10-~ Frequency (rad/min)
25
10j
Figure 5 Robustperformance (RP) and robust stability (RS) of the p-optimal controller range, the singular values are close to one and the maximum of the upper singular value is sufficiently small, The plots of the singular values of the sensitivity functions S, = [l + GK] ~ demonstrate the good disturbance rejection properties, which are insensitive to input uncertainty (Figure 6b). The two sensitivity functions with the nominal plant and with an input error of +10% in AL and -10% in AV are almost identical, The simulation of step responses using the nonlinear model shows the disturbance rejection capability in the time domain (Figure 7). At minimum as well as at maximum feed flow rates, the maximum control errors are small (i.e. not much higher than both the measurement noise and the resolution of an industrial temperature sensor), and the product compositions are kept close to their setpoints. The steady-state offsets of the product compositions arise from controlling tray temperatures at some distance from the column ends. From an industrial point of view, the low sensitivity of the performance to errors in the manipulated variables is very important. Based on this uncertainty model, multivariable controllers were designed and implemented for two industrial distillation columns with excellent results. 17. ~
H=-eontroller design
S/KS/T
By introducing the auxiliary input signal v in the S / K S / T scheme shown in Figure 8, the sensitivity S,, and the complementary sensitivity 7",, are the transfer functions from v to u and ~, respectively. The sensitivity and its complement are weighted by 14/,, and W,, which are chosen as
H=-synthesis The design of an H=-controller consists of two stages. In the engineering stage, the plant has to be augmented by weightings reflecting the specifications. In the mathematical stage, a controller is derived for the augmented
W, =
100 I 2000s + 1
a
b
10
'r.
r. 0.01 --
//. ,
0.0001
0.01 10_5
10_3
10_1
Frequency (racL/min)
101
10_5
10_3
nominal plant 10% input, error 10_l
Frequency (rad/min)
Figure 6 Singularvalues of the closed-loopsystem. (a) Complementarysensitivity:(b) sensitivityfunctions S,.
10 I
26
R o b u s t c o n t r o l o f distillation c o l u m n s : U. Christen ,_ 0.01S
etal.
~-, 0.018 [
0.016,
"~' .
.
.
.
.
.
.
.
~ .
"\ .._ s:.~ -
"~ O.Ol
.
.
.
.
.
.
.
.
.
.
.
.
.
r..) 0"0060
10
20
I':;~ .
.
.
.
. . . . .
|t .
.
.
.
.
.
.
.
.
.
.
.
................
.1""""
F = 20 mol/min I [ . . . . . 10% input error I
"
~ .
...........
0.008
0'016tt
.
]
30
"~
O.Ol
o~
0.008 [-q 0"0060
~
40
10
1/
L'."" 10% input error 20 30 40
4/;' i.
.0.2 [
e
e 0
10
20
30
-O4
40
70
0
10
20
30
40
160
•
g
50
g
[ F = 20 raol/min
120
I F = 46 tool/rain
, 20
o
lO
20 Time (h)
30
40
150
lO
20 Time (h)
30
40
Figure 7 Simulationresults with the ~t-optimalcontroller for an increase in feed composition (0.8-0.9 mol/mol)at t = 0 h and an increase of feed flow rate (+ 3.6 mol/min) at t = 20 h W~ = 0 . 1 . ! 8 0 S + l l 2.5s + 1 Wj -- 0.0021 (cf. Figure 9a). W, prescribes a maximum gain of 0.01 for the sensitivity S, at low frequencies up to about 10-4 rad/min, thus attenuating disturbances to 1% in that frequency range. The weight W,7 for the complementary sensitivity can be used to specify a multiplicative uncertainty at the input of the plant. In our case, we assign 10% uncertainty in the low-frequency range and more than 100% above 10-I rad/min, thus limiting the bandwidth of the control system. The weight W,/ is only
~~} rV t
, [_ ~
w
Figure 8 S/K,STTscheme weightingat the plant input
z
used to avoid a singular H~-problem and can be small. The plant used for the control design is the nominal plant GN. The resulting controller is as ill-conditioned as the plant (Figure 9b). Figure 9c and 9d shows that the specifications are met for the sensitivity and its complement in the nominal case. In fact, the singular values for both S, and S,., as well as T, and T~ look the same in the nominal case since the plant is inverted and, therefore, the closed-loop transfer functions are well-conditioned (only S~ and T,, are shown). For the plant with 10% error in the input (AL: +10%, AV: -10%), S,, and T, still look unchanged. However, the sensitivity S,, and its complement T,, are completely deteriorated for such an error as shown in Figure 9. This is consistent with Freudenberg's 4 statement: if the controller inverts the plant, the control system may be very sensitive at one (thel°cati°nplant(theinput).plant o u t p u t ) t o uncertainty at the other The inverting property of this weighting scheme is easy to derive heuristically. If the specifications for the control system are tight, all the singular values of the transfer function 7_., from w to z are about equal and constant for all frequencies. Thus, at low frequencies W"Su = U holds, where U is an allpass transfer function.
Robust control of distillation columns: U. Christen et al. a
b
10 2
-
103
101
102
[,oo
1o1
10-1
1
10-2
10-5
~ 100 ~
10-3
10-I
101
~-~
10-1
10-5
Frequency (rad/min) e
.~
~ _ _ ~ - _ //-
e.,
10-3
" / ~
10-1
101
Frequency (rad/min) d 101
10 o
27
~_. .~ 10 0
-nominal plant - - - 10% input error . . . . . . . .
/
~1o-21~
=l
,'
~
I ga?t
'
"
]
10_a . . . . . 10% input error 10-5 10-3 10.1 101 Frequency (tad/rain)
,
X
,
10_2 10 -5
10-3 10.1 Frequency (tad/rain)
\
\
101
Figure 9 Controller designed with the S/KS/T scheme of Figure & (a) Singular values of the weights. (b) singular values and condition number of the controller, (c) singular values of the sensitivity S~ (d) singular values of the complementary sensitivity
In the low-frequency range, the loop gain L,, = K G is much(KG) ~.largerConsequently,than I and hence, S, is approximately
W.S.~ W.G~ K ~ = U
(14)
K = U I W,,G 1
(1 511
v
I
w{ I
~
~
= l l -.
~"~'] ~
--
" z
Figure 10 G S / T weio_htin~ scheme which avoids the reversion of the
Thus, K contains the inverted plant, at least as far as its low-frequency behaviour is concerned, and is as badly conditioned as the plant. A more rigorous derivation of the inversion of the plant in the controller can be found elsewhere. 2° The inversion could be avoided if G was included in the weight W,,, as can be seen by equation (17). However, this would double the order of the controller,
GS/T H:controller design for
ill-conditioned plants Another possibility to include the plant in the weighting of the sensitivity is the G S / T scheme shown in Figure 10. The transfer function from v to y is G S . whence the sensitivity is weighted by the plant and by W,. The resulting controller no longer contains the inverted plant. :~ As before, T:. has constant singular values such that II':GS, ~ U at low frequencies. With the open-loop approximation for the sensitivity, it follows that
K = ('~W~
(17)
plant in
the
controller
Thus, the singular values of K are directly controlled by' W,. Now' a well-conditioned controller with ~:(K) = 1. as requested by Freudenberg, can be obtained by choosing 14:] = %(/co)I. The weight W,7 for the complementary sensitivity T, is chosen to prescribe the same maximum bandwidth as before. Its static gain is smaller than in the previous section in order to demonstrate that modelling of an input uncertainty is not required with this scheme. W~ in the weight for the sensitivity is smaller than W,, before since the sensitivity is additionally weighted by the plant. To allow for S. + T,, = I. W, is chosen such that the weight W,G for the sensitivity and the weight I#',~for the cornplementar) sensitivity T,, intersect below 1 (see Figure 1/el), The plant is again Gv. 40 W - 18000s +1 I 1800s + l W = 0.01 I W/ = 0.02I 2'5s+ 1
(18)
The controller designed with this weighting scheme is much better conditioned. Consequently, the singular
103
Robust control of distillation columns: U. Christen et al.
28
a
b
102
102
100 10-5
10° ~ ¢~ .
10-3 lO-I Frequency (rad/min) e
~
101
10.5 101
/
10 !
-nominal plant i i- - - 10% input error i ' ~
"~ l00
~10-2
10-3 10-I Frequency (rad/min) d
~ 10-1 - - nominal plant - - - 10% input error
10-'4 ,// 10-5 1()-3 1()-l Frequency (rad/min)
10-2i 10-5
101
10-3 10-I Frequency (rad/min)
101
Figure It Controller designed with the GS/T scheme. (a) Singular values of the weights, (b) singular values and condition number of the controller, (c) singular values of the sensitivitySe, (d) singular values of the complementarysensitivity values of the sensitivity and the complementary sensitivity no longer match (Figure lib-d) but they are the same for S, and S~ as well as for T~ and T< (only S,, and T~, are shown). They meet the specifications in the nominal and perturbed case with the input errors of 10%. In fact, the deterioration is nearly indiscernible. The same is true for the nonlinear simulation of two disturbance steps (Figure 12). The performance degradation in the presence of errors in the manipulated variables is insignificant, whereas it would be prohibitive for the inverting controllers,
These simulation results are similar to those obtained with the p-controller (Figure 7). In both cases, the peaks of the temperature differences being smaller than 0.3 K (Figure 12), the limitations given by the resolution of the temperature sensors are nearly exhausted. The similar performance of the two controllers can also be seen in the p-analysis [Figure 13 (left) as compared to Figure 5] for which the same weights as for the p-synthesis are used. At low frequencies, the H=-controller performs slightly better than the p-controller, whereas the peak around I0 -~ rad/min is higher. For the inverting H=-con-
~0.018
0.2
t=0"016 ~ . - ~ - ~ - ~ i - -
'~~'0"1 ' i '
~ 0 . 0 1 4 "~.
~
~ 0.012
o-0.1
""
:~ 0.01 -- - ~ ............. ~'0.008 F = 33 mol/min o 0.O06 120 F = 33 tool/rain
=i00 \
,--0"2 r~ -0.3
360 40 0
10
/ 20 30 Time (h)
",,..f !~: !i F = 33 tool/rain [ . . . . . 10% input error I
-0.4 ~.0.018 ~O 0.016 ~---0--0-~, ~- . . . . . . . . . o ' .... ~ ,
~--~
o
0
40
"~ 0.01 . . . . . . . . . . . ~ 'o.oo8 o V [. . . . F = 20 mol/min J r.) 0.006(~ 10 20 30 40 Time (h)
Figure 12 Nonlinear simulationsof an increase in feed composition from 0.8 to 0.9 mol/mol at t = 0 h and an increase of feed flow rate of 3.6 mol/min at t = 20 h
Robust control of distillation columns: U. Christen et al.
29
robust performance - - - robust stability - -
-~
46 0.5 . . . .
-
010-5
Figure 13
-._-i
.s
- - robust stability 10-3 10-1 Frequency (rad/min)
~41 \ 10 t
=-.0t
10-5
10-3 10-1 Frequency (rad/min)
101
p-analyses for the noninverting (lelk) and the inverting (right) H -controllers
troller designed in the previous section, on the other hand, neither stability nor performance is acceptable (Figure 13 right). There are several possibilities to further improve the Ha- and p-controllers. The two main disturbances, which are easy to model, can be taken into account in t h e augmentation scheme. For the feed flow rate, which is measured, it is even possible to design a feedforward compensator. The disturbance rejection at higher frequencies may be improved by feeding back an additional temperature taken near the feed tray. An H~-controller with all of these improvements has been designed, ~ but space limitations preclude a presentation of the results in this paper,
Conclusions Only MIMO plants can be ill-conditioned and there is no analogue in SISO plants. Consequently, designing a robust controller for an ill-conditioned plant with SISO synthesis methods is not possible. The H=- and p-controllers designed here are not diagonal, even though they have small condition numbers and they both outperform diagonal PI-controllers by far. s, 2-, For the design of the p-controller, a structured uncertainty model has been developed which describes the dynamics of a high-purity distillation column for the entire operating range. All uncertainties in the model can be physically justified, which allows for an easy shaping of the uncertainty bounds. A controller design based on this uncertainty model is distinguished by the high performance for the entire operating range of the column and by low sensitivity to uncertainty in the manipulated variables. However, these advantages are achieved by a large effort for the development of an appropriate uncertainty structure and for the computation of the controllers. For further projects, though, the effort for uncertainty modelling is significantly reduced, In the H~-section, it has been shown that it is vitally important to choose a weighting scheme which guarantees that the plant is not inverted - the GS/T scheme. Then, an H~-controller performing as well as the p-controller can be designed, However, if an S/KS/T scheme is used, even t h e inclusion of an input uncertainty cannot prevent the inversion of the plant in the controller,
Comparisonof the H~- and the /.t-synthesis methods Either method, p- or H~-synthesis, may be used to design a controller for ill-conditioned plants - if appropriate structures for the augmentation of the plant are chosen. However, the two methods differ in the ways in which they arrive at these structures. The p-method requires a profound knowledge of the plant dynamics. Only with this knowledge is modelling disturbances and uncertainties based on first principles possible. The p-method advantageously avoids dealing explicitly with the bad condition of the plant. However, it too must prevent the inversion of the plant in the controller, as indicated by Freudenberg. 4 The inclusion of both an input and an output uncertainty prevents the control system from becoming sensitive to those uncertainties, as may happen with inverting controllers. The incorporation of various operating points guarantees the specified performance for all operating points as long as they vary arbitrarily slowly, t~ For the H~controller, this objective can be assured only by simulation or p-analysis. However, for both methods, nothing can be guaranteed for fast transients between different operating points. Simulations are thus required to prove the controller performance. With the p-approach, the upper bound for the bandwidth of the control system is provided by the uncertainty model, whereas the lower bound is a matter of optimization. For the H~-approach, on the other hand, the control designer must have an idea of the upper and lower bounds for the allowable bandwidth. However, a nominal model of the plant suffices. It can be obtained either by identification or by modelling based on first principles. This minimal knowledge is incorporated in an augmentation scheme which explicitly avoids the inversion of the plant. Once the augmented plant is formed, the p-synthesis requires a lengthy computation. The synthesis procedure is iterative, and its convergence is not guaranteed. The resulting controller is of a much higher order than the plant and subsequently must be reduced. The computations for H~-synthesis are less involved. They take about the same amount of time as the first H~-synthesis step in p-synthesis: less than 1 min on a SPARCstation 2. The order of the resulting controller is only slightly higher than that of the plant because normally only two
Robust control of distillation columns: U. Christen et al.
30
dynamic weightings are needed to augment the plant. Because the computations for H=-controllers are that inexpensive, it is possible to iterate on the weightings to improve the control system, whereas that is nearly impossible for a/.t-controller, necessitating two hours of
6
Skogestad, S. and Morari, M. Chemical Engineering Science
7
1988, 43, 33 Skogestad, S. and Lundstr6m, P. Computers Chem. Engng. 1990, 14, 401
computation time. To conclude,/l-synthesis is ideally suited to deal with complex uncertainty models which take into account such aspects as various operating points. The price o n e has to pay for it is tedious computations. If, however, loop-shaping ideas are used to form the augmented
9
plant, H~-synthesis may be used to advantage. The results are as good as with the /.t-synthesis, but are
obtained with less numerical effort.
aeferellces l
2 3
4 5
Skogestad, S. 'Dynamics and control of distillation columns - A critical survey,' 3rd IFAC Symposium on Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes, College Park, MD, pp. 1-25, 1992 Skogestad, S., Morari, M. and Doyle, J. C. 1988, IEEE Transactions on Automatic Control, 33 1092 Christen, U. and Musch, H. E. 'Robust control of distillation columns: p or H=?' IMRT Report No. 29, Measurement and Control Laboratory, ETH Zurich, 1995 Freudenberg, J. S. Int. J. Control 1990, 51, 365 Musch, H. E. 'Robust control of an industrial high-purity distillation column', PhD thesis, ETH Zurich, 1994
8 Skogestad, S. and Morari, M. 'Control of ill-conditioned plants: high purity distillation' AIChE Annual Meeting, Miami Beach, FL, 1986 Lundstr6m, P., Skogestad, S. and Doyle, J. in Proc. of the 2nd
European Control Conference,Vol. 2, p. 969, 1993 10 McDonald, K. A. in The Shell Process Control Workshop (eds D. M. Prett and M. Morari), Butterworth, Boston, 1987, 279 11 Poolla, K. and Tikku, A. IEEE Transactions on Automatic Control 1995, 40, 1589 12 Doyle, J. C. lEE Proc., 1982, 129, Pt. D., No. 6, 242 13 Balas, G. J., Doyle, J. C., Glover, K , Packard, A. and Smith, R. 'p-Analysis and Synthesis Toolbox, User's Guide' The
MathWorks, Natick, MA, 1993 14 Packard, A. and Doyle, J. C. Automatica, 1993, 29, 71 15 Packard, A., Doyle, J. C. and Balas, G. A S M E Journal of Dynamic Systems, Measurement, and Control 1993, 115, 426 16 Lin, J.-L., Postlethwaite, I. and Gu, D.-W. Automatica, 1993, 29, 219 17 Musch, H. E. and Steiner, M. 1EEE Control Systems Magazine, 1995, 15, 46 18 Musch, H. E., McKay, B., Willis, M. and Barton, G. 'Industrial applications of MATLAB based software for advanced data analysis and control' Preprints Control 95, Melbourne, p. 211, 1995 19 Doyle, J. C., Glover, K., Khargonekar, P. P. and Francis, B. A. 1EEE Transactions on Automatic Control 1989, 34, 831 20 Christen, U. 'Engineering aspects of H~ control'. PhD thesis, ETH Zurich, 1996 21 Raisch J., Lang, L. and Gilles, E-D. Automatisierungstechnik 1993, 41,215 22 Musch, H. E. and Steiner, M. in 'Preprints of the 12th World Congress IFAC', Sydney, Vol. 1, p. 49, 1993