Ind. Eng. Chem. Process Des. Dev. 1984,
K , = physical gas-liquid equilibrium constant for species i, or the distribution coefficient K , = y J x , k, = rate constant for reaction i ko = rate constant at temperature To L = characteristic length of catalyst pellet, volume/area n, = total molar flow rate of species i, combined number of moles in gas and liquid phases, mol/time np = initial molar feed rate of species i, mol/time ni = molar flow rate of species i in the liquid phase, mol/time n b = total molar flow rate of the liquid phase, mol/time n$ = total molar flow rate of the gas phase, mol/time P = pressure P, = partial pressure of sulfur compound PH = partial pressure of hydrogen R = gas constant r = reaction rate r, = reaction rate for reaction i T = temperature To= average temperature for calculatingthe Arrhenius constant W = weight of catalyst x = conversion of desulfurization reaction x , = mole fraction of species i in the liquid phase x , = conversion of the water gas shift reaction 2, = catalyst bed depth Greek Letters a, = fraction of total no. of moles species i in liquid phase, = nt/n, 6 = constant = ( Y B T / 6 ( 1 - 6) 6 = total liquid fraction = nh/nT
t
23,93-101
93
= porosity of catalyst pellet
4p = Thiele modulus
pp = pellet density \k = fraction of total number of moles in the gas phase =
n&/nT Registry No. Benzothiophene, 95-15-8. L i t e r a t u r e Cited Bischoff, K. B.; Froment, G. F. "Chemical Reactor Analysis and Design"; Wiiey: New York, 1979; pp 710-722. Cecil, R. R.; Mayer, F. X.; Cart, E. N. Paper 12a presented at the 61st AIChE meeting, Los Angeles, CA, 1968. Daly, F. P. J. Catal. 1978, 5 1 , 221. Frye, C. G.; Mosby, J. F. Chem. Eng. Prog. 1987, 6 3 , 66. Furimsky, E.: Amberg, C. H. Can. J. Chem. 1976, 5 4 , 1507. Gates, 8. C.; Schult, G. C. A. AIChE J. 1973, 19, 417. Givens, E. N.; Venuto, P. B. Am. Chem. SOC.Div. Pet. Chem. Prepr. 1970, 75(4), A183. Goto, S.; Smith, J. M. AIChE J. 1975, 2 1 , 706. Guin, J. A.; Lee, J. M.: Fan, C. W.; Curtis, C. W.; Lioyed, J. C.; Tarrer, A. R. Ind. Eng. Chem. ProcessDes. Dev. 1980, 19, 440. Hornbeck, R. W. "Numerical Methods"; Quantum Publishers Inc.: New York, 1975. van Klinken, J.; van Dongen, R. H. Chem. f n g . Sci. 1980, 3 5 , 59. Kumar, M. M. S. Thesis, Texas ABM University, 1982. Lee, H. H.: Smlth, J. M. Chem. Eng. Sci. 1982, 3 7 , 223. Metcalfe, T. B. Chim. Ind. Gen. Chim. 1969, 102, 1300. Morooka, S.; Hamrin, C. E., Jr. Chem. f n g . Sci. 1977, 3 2 , 125. Newsome, D. S. Catel. Rev. Sci. Eng. 1980, 27(2), 275. Satterfield, C. N. AIChE J. 1875, 2 7 , 209. Sedriks, W.; Kenney, C. N. Chem. Eng. Sci. 1973, 2 8 , 559.
Received for review October 4 , 1982 Revised manuscript received March 17, 1983 Accepted May 9, 1983
Robustness Analysis of Process Control Systems. A Case Study of Decoupling Control in Distillation Yaman Arkun, Billy Manouslouthakls, and Ahmet Palazdlu Department of Chemical Engineering and Environmental Engineering, Rensselaer Polytechnic Institute, Troy, New York 72 78 7
A general analysis procedure is developed to assess the robustness properties of process control systems. Various mathematical tools are given to quantify the degree of robustness with respect to sensitivity and stability. As a case study, decoupling control of dlstillation columns is analyzed to illustrate the use of the methodology. The study shows that the results available on decoupling control of distillation columns can now be explained and unified in a rigorous framework within which dynamics and more general types of modeling errors can be easily included.
Introduction
In chemical process control many simplifications are made in obtaining dynamic models. Theoretical models used for controller design are most often simplified by neglecting nonlinearities, time-varying parameters, and higher order dynamics. Experimental models derived from methods such as reaction curve or frequency response data always exhibit uncertainties at high frequencies, and they are valid only for a limited range of operating conditions. Thus the practicing engineer is always confronted with the problem of mismatch between the approximate model and the real plant which he has to address carefully during controller design. The ability of a given system to maintain its important properties, such as stability and sensitivity, despite the modeling uncertainties will henceforth be referred to as the robustness of the system with respect to these properties. Design of robust controllers is a pressing problem
in process industry and it is now receiving more and more recognition in academia (Garcia and Morari, 1982; Brosilow, 1979; Arkun and Ramakrishnan, 1982). The general design philosophy is depicted by Figure 1. The crucial point is that the controller is derived from a model but implemented on the real plant. If the controller design does not take into account the modeling errors, the closed-loop performance of the plant deteriorates from that predicted by the model and often the onset of instability prevails. For the single input-single output systems, classical analysis and design techniques which use gain and phase margins (Bode, 1945; Horowitz, 1963) to cope with uncertainties are reliably implemented in process industry. However, the extension of these classical design procedures to multivariable feedback control is by no means trivial and fully developed. Only recently new concepts for the analysis and design of robust multivariable systems have emerged in the control literature (MacFarlane and Kou0 1983 American Chemical Society
94
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
'
Design Object1 v e s
I-' I
-A-11
Models
+ Y
b Design Phase
Figure 2. Multivariable feedback control.
L j L .
Disturbances
U
True Plant
useful mathematical preliminaries will be introduced. Singular Value Decomposition (SVD) SVD is a fundamental tool for numerical analysis (Klema and Laub, 1980) and is now being used extensively in the control field (MacFarlane and Kouvaritakis, 1977; Zames, 1981; Doyle and Stein, 1981; Cruz et al., 1981) to calculate the singular values of a matrix. The singular values ui of a matrix A are given by outputs
Figure 1. General approach toward design of controllers.
varitakis, 1977; Zames, 1981; Doyle and Stein, 1981). Meanwhile, most of the attempts in chemical industry have focused on trial-and-error approaches which are not rigorous and systematic. One common approach, which has been exploited also in academia (Luyben, 1970; Weischedel and McAvoy, 1980; Hammarstrom et al., 1981) is to test the control strategies derived from linear models on more detailed nonlinear models through dynamic simulation. The method has serious shortcomings. First of all, it cannot directly analyze and quantify the robustness properties of a given design without resorting to a detailed dynamic simulation. Secondly, the modeling errors are preconditioned by the selected nonlinear model which may not fully account for all the uncertainties present. In this paper we present mathematical analysis tools which can be systematically used to study the robustness properties of process control systems. Within the mathematical framework developed, the degree of robustness can be quantified and a comparative assessment of different designs can be tackled. Designs that are potentially inferior can be ruled out right from the start and extensive simulation studies need be carried out only for the most promising candidates at the end. Here we should emphasize that this paper will focus only on analysis. Furthermore, adaptive schemes using real-time plant measurements to update the controller and/or the model (see Figure 1) will not be discussed in this paper. Thus our major objective here is to focus on the robustness analysis which will be performed during early design stages. In order to demonstrate the utility of our methods, we will dwell on a case study of distillation columns with decoupling control. The industrial importance of distillation control, wide acceptance of decoupling in industry, and the abundance of research in this area make decoupling in distillation the best candidate for such a study. However, the methodology is general and can be applied to any system. Four distillation columns are studied in this paper. These are the low and high purity columns of Weischedel and McAvoy (1980),the column by Toijala and Fagervik (1972), and the experimental column by Wood and Berry (1973). The analysis procedure to be presented unifies the distillation control results obtained by the previous researchers. All the problems associated with decoupling control of distillation columns can now be predicted by a rigorous technique. The method of analysis is not restricted to steady-state conditions and specific uncertainties as in Jafarey and McAvoy (1978) and McAvoy (1979), but it rather treats dynamics and uncertainties in a general setting. Before discussing the robustness analysis, some
ui = Xi'/'[AHA]
(1)
where H denotes the conjugate transpose and hi are the eigenvalues. The maximum and minimum singular values, u*[A] and o,[A] respectively, serve as measures of the magnitude of a matrix and are called the spectral norms (Stewart, 1973)
g*[A] = llAl12
max IIAnIl = Xmax'/' [AHA]
(2)
iixli=l
where l.z is the Euclidean norm. The next sections describe how the singular values are used for robustness analysis. Robustness Analysis I. Mathetmatical Tools. Robustness analysis will be introduced to quantify the effect of model/plant mismatch on the closed-loop performance of the plant shown in Figure 2. One method of representing uncertainties for the plant transfer function G,(s) is by parametrizing it. Let a be the vector of uncertain parameters with a€ LA where LA is a finite dimensional vector space. It is assumed that a lie in a "ball" of uncertainty %CA centered around the known nominal parameters cyo and containing the true parameters. Thus the nominal model is given by one of the elements in the set of possible models A, = {G,(s,a); a € % ) . Robustness for the feedback system of Figure 2 can now be formally defined as follows. Definition. Let the controller K ( s ) be fixed. The corresponding closed-loop system for the plant G, is said to be robust with respect to a property x, if the closed-loop system for every element in the set of models A, = ((2,(s,a);a € % ) possesses the property x. Prerequisite properties for a feedback system are (a) stability and (b) reduction of sensitivity to uncertainties. The following sections discuss robustness with respect to each of these properties. Robustness with Respect to Stability The following theorem gives the conditions for a closed-loop plant to maintain its stability in the face of uncertainties. Theorem 1. The closed-loop system of Figure 2 is robust with respect to stability if and only if G,(s,a)EA, contains the same number of unstable poles for all a € % and u.[T(jw,a)] > 0 for every w E R + and a € % (4) where w is the frequency and R+ is the set of positive real numbers and T(jw,a) = I + G,(jw,a)K(jw) is the return
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yop' = Gp[I f KG,,,]-lKysP Figure 3. Open-loop feedforward control.
difference operator. The proof will not be given here; for details the reader should refer to Doyle and Stein (1981). Equation 4 proves to be useful for analysis if the set A, contains finite number of models. This would be the case if parameters a€% take a finite number of deterministic values. An important industrial example is when plants operate at different optimal steady-state set-points (Arkun and Stephanopoulos, 1980). Linear models G,(s,ai) for each regime i = 1, ..., N are known, and the designer is interested in the robustness of the controller K(s) for all these operating regimes. Thus (4) is tested for known values of ai, i = 1, ...,N. However (4) is not convenient to use when a is completely uncertain, Le., when it can take any value inside the uncertainty ball 9. An alternative test to parametrization is needed for such cases. This is accomplished by utilizing a different representation of uncertainty given by the multiplicative form Gp(jw) = [I + L(jw)lGm0(jw)
(5)
u*[L(jw)] < l,(w) for all w E R +
(6)
with where I,(w) is a positive scalar function and Gm0 is the nominal model. The true plant model lies somewhere in the neighborhood of Gm0which contains the set of perturbed nominal models generated by (5) and (6) for different L's. We denote this set of perturbed models by Ab Theorem 2. The closed-loop system of Figure 2 is robust with respect to stability if and only if all the perturbed models Gp(s) EALcontain the same number of unstable poles and the same imaginary poles as the nominal model Gmo(s) and if (a) the nominal closed-loop system is stable, and (b) u l [ l + (Gmo(jw)K(jw))-'] > Z,(w) for every WEER'. For the proof, see Doyle and Stein (1981). ul[I + (G,OK)-'] is the magnitude of the maximum uncertainty that can be tolerated before instability occurs. u, can be easily calculated and interpreted as the degree of stability robustness of the plant. Thus it provides a measure of "gain stability margin" for multi-input-multi-output control systems. In later sections, ut[I + (G,OK)-'] will be used to discriminate between designs that possess different degrees of stability robustness. Robustness with Respect to Sensitivity When the plant model is perfectly known, either closed-loop feedback control (Figure 2) or an equivalent open-loop feedforward control (Figure 3) can be used to achieve the same overall transfer function between y and the set-points ysP. Thus both systems give the same performance. However, when the plant model has uncertainties, feedback controller can be designed to have useful properties, e.g., reduction of sensitivity to uncertainties, which feedforward cannot offer. When there is no uncertainty, G, G, and the closedand open-loop outputs are y, = [ I + GmK]-'GmKySp (7) Yop
= GmGFYsp
(8)
Both control systems have the same overall transfer function if GF = [I + KGJIK. However, when the model/plant mismatch occurs, G, # G, and the outputs are no longer the same but rather are given by y,' = [ I + G&]-'G&ySP (9)
We define the induced errors in the outputs due to uncertainty by e, = Y, - Y,, eop = Yop - Yop' and use the integral quadratic error
J ( e ) = l m0 e T ( t ) e ( tdt )
(13)
to compare the sensitivities of the open- and closed-loop systems. The feedback system is said to have reduced output comparison sensitivity to uncertainties when J(eJ < J(e,), i.e., when the closed-loop feedback is made less sensitive to modeling errors than the equivalent open-loop feedforward system. The feedback system of Figure 2 will be called robust with respect to sensitivity when for every possible representation of the plant belonging to the set A,: (Gp(s,a); a € 91,the closed-loop has reduced sensitivity compared to the equivalent open-loop. Sufficient conditions for robustness are given by the following theorem. Theorem 3. The feedback system of Figure 2 is robust with respect to sensitivity if the system is stable for all a€% and ul[T(jw,a)] > 1 for every WErCR+ and for every a€ 8,where I' is the frequency band of interest for the system under consideration. For proof see Cruz et al. (1981). These are, however, sufficient conditions for reduced sensitivity; i.e., when they are violated, the analysis is inconclusive and does not mean that the feedback system has increased sensitivity. The concept of increased sensitivity is introduced in this paper and a complementary theorem will be given to define the conditions for increased sensitivity. A feedback system is said to exhibit increased comparison sensitivity when J(eJ > J(eop). Theorem 4. The feedback system of Figure 2 is not robust with respect to sensitivity when there exists a model G p ( s , a ) € A a for some a€% for which the closed-loop system has increased sensitivity compared to the open-loop system. Sufficient conditions for this to occur are: (a) the system is stable for all a€% and if (b) there exists a WE rCR+ and an a E 8 for which u*[T(jw,a)]< 1. Proof. See Appendix. When the above conditions hold, the feedback makes the open-loop system more sensitive to modeling errors; consequently, the design under consideration is not acceptable. In theorems 3 and 4 the frequency range of interest r is a low-frequency band less than a specified frequency wb and is classically referred to as the bandwidth of the control system (Rosenbrock,1974; Kwakernaak and Sivan, 1972). For single-input-single-outputsystems wb is defied to be the smallest frequency at which the closed-loop amplitude ratio falls below -3 dB (decibels). For multivariable control systems, we define wb and the bandwidth r in a similar fashion. If we consider u*([I + (GmK)-l]-l) as the gain of the closed-loop system y = (I + GmK)-'G,Kysp = [I + (GmK)-1]-1y8P (14)
wb is defined by u* ([I+ (GmK)-ll-l)lu=ub =
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Start
6-i
Reject D e s i g n
1
L
@-+
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b
-
-
M
T
+
Gyp ( 5 )
! Figure 5. Decoupling control.
i A c c e p t Design
(a)
Procedure P
1
Inconclusive
(b)
Procedure B
Figure 4. (a) Procedure A; (b) procedure B.
Thus for a giuen controller design, wb and r are easily determined from the gain plot U,[I+(G,K)-~]vs. frequency w. The sensitivity robustness of the given design is then evaluated for this frequency band .'I We must finally note that the bandwidth I' over which we would like to achieve reduced sensitivity is usually limited by stability robustness requirements. Thus if we were to design a control system, theorem 3 would have to be satisfied up to a frequency wb above which modeling uncertainty becomes significant enough to affect stability. For the remaining high-frequencyband above wb,stability robustness conditions of theorem 2 would then have to be satisfied. In this paper no attempt has been made to design a controller to achieve stability and sensitivity robustness which would automatically determine the best achievable bandwidth for a process with given modeling uncertainties. Instead for given control system designs robustness with respect to sensitivity and stability are analyzed using the calculated bandwidths as described above.
XI. Analysis Procedure Utilizing the above theorems, two robustness analysis procedures can be followed during design. Each procedure will make use of the following tests. Test 1. For all wER+ and a E B , the following should hold for robustness with respect to stability: u*[T(jw,a)] > 0. Test 2. If there exists a w E R + and an aEB for which a , [ T ~ w , a )=] 0, then the plant has lack of robustness with respect to stability. Test 3. For all wErCR+ and a E B , the following should hold for reduced sensitivity: at[T(jw,a)]> 1. Test 4. If there exists a wEI'CR+ and an a € % for which u*[T(jw,a)]< 1,then the plant has increased sensitivity. If all the models that could possibly represent the uncertain plant are available, procedure A given in Figure 4a is followed. Note that the procedure is conclusive where designs are either accepted or rejected. However, such extensive information is not often available to the designer. Usually, only a nominal model is all the designer has. In this case, procedure B given in Figure 4b is preferred. Only the nominal model is tested by this procedure. Consequently, the procedure has two disadvantages: (a) It might be inconclusive. (b) It does not quantify the degree of the failure of a design. It is, however, possible to improve the procedure by coupling it with theorem 2 whereby the magnitude of uncertainty which a given design can accomodate is easily judged by evaluating 1, = u r [ l + (GmoK)-l] for each design. Thus a comparative assessment of the
degree of robustness for different designs can be accomplished. In the following section, the procedure B is going to be applied to distillation columns to illustrate each analysis step in detail. Finally, we finish the theoretical developments by mentioning that the theorems and procedures given so far are completely general. Either procedure A or B can be used depending on the problem and the amount of information available about the plant. A Case Study of Decoupling Control in Distillation Being the major energy consumers in a chemical plant (Shinskey, 1977a), distillation columns offer most challenging design and control problems. In order to save energy, dual composition control has been proposed and its merits have been extensively studied in the literature (Shinskey, 1977b;Luyben, 1975). However, control of both top and bottoms compositions usually results in undesirable interactions among the control loops. To be able to cope with such interactions, a lot of research effort has been devoted to decoupling control (Luyben, 1970; Weischedel and McAvoy, 1980; Jafarey and McAvoy, 1978; McAvoy, 1979; Fagervik et al., 1981; Ryskamp, 1981; Schwanke et al., 1977; Shinskey, 1977b;Wood and Berry, 1973; Waller, 1974). Because of the simplicity and transparency of the design procedure decoupling is now the most popular control strategy in distillation. However, the need for accurate models to achieve decoupling limits the success of the method. Luyben (1970) reports that ideal decoupling for high-purity columns has very small tolerance to modeling errors and can exhibit stability problems. The same observation is supported by Weischedel and McAvoy (1980) for their moderate and high-purity columns. Weischedel and McAvoy (1980) further report that perfect decoupling is not feasible for high-purity columns. Their results are based on a steady-state sensitivity analysis which investigates the sensitivity of the Relative Gain Array (RGA) (Bristol, 1966) to decoupler gain errors. So far this method together with Shinskey's similar approach (Shinskey, 1977b) are the only attempts to investigate the sensitivity issues in decoupling control. However, these methods have their limitations. First of all they are only steady-state measures; thus they have to be justified by dynamic simulations as done in Weischedel and McAvoy (1980). Secondly, they treat only the uncertainties in decoupler gains and not directly the more realistic uncertainties in the process model itself. In contrast, our robustness analysis discussed above is a direct method which rigorously includes dynamics in a more general model/plant mismatch framework. Before we discuss our results, we will give a short overview of the various decoupling control schemes we will study. Decoupling Control Schemes. The general decoupling control system is shown in Figure 5. The decoupler is denoted by D(s) which acts as a precompensator to decouple and isolate the individual single-input-singleoutput control loops for which the classical PID controllers can be designed one at a time. K ( s ) denotes these controllers. The general design equation for the decoupler is G,(s)D(s) = W s ) (17)
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 97
d 0 0
3
Y
a
: 3
a
u +
Y
a
0
b
3
M
E
;j Y
a
a
3
I
b
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9
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n
u
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-
n
cs:
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o
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uN
9 0, both columns are nominally stable. However, because both c*[Z‘l and u * [ q lie below unity for some wEr (i.e., test 3 is not satisfied and test 4 is satisfied), the simplified decoupling control of low- and high-purity columns is not robust with respect to sensitivity. The frequency range r was found to be within the bandwidth of the composition control loops indicating that these loops cannot handle modeling errors effectively. In the presence of uncertainty, there is not only a deterioration from perfect decoupling but more importantly the sensitivity of the closed-loop system to model/plant mismatch is increased. Figure 6 also shows that the high-purity (HP) column is more sensitive to modeling errors than the low-purity (LP) column. This result is shown more pronounced in Figure 7 when ideal decoupling is used. For the HP column, the singular values stray away from unity more and approach closer to zero. Thus, decoupling control of the HP column is not only more sensitive to uncertainties but its stability margin is also smaller than that of the LP column. These results are consistent with the steady-state predictions and dynamic simulations of Weischedel and McAvoy (1980) which conclude that (a) the most sensitive columns are H P columns and (b) simplified decoupling works better than ideal decoupling. Our robustness
Figure 10. Uncertainty bounds for the low purity column with (a) ideal decoupling; (b) simplified decoupling; (c) one-way decoupling; and (d) no decoupling.
analysis further concludes that, whether simplified or ideal decoupling, designs as given in the literature are all prone to sensitivity problems. The same conclusion holds for WOBE and TOFA columns as illustrated by Figures 8 and 9. The peaks of the singular values at relatively low frequencies suggest that the control loops have increased sensitivity to modeling errors. B. Results on Stability. We make use of theorem 2 to determine the maximum magnitude of uncertainty 1, that can be tolerated by a given design. Thus the degree of stability robustness for different decoupling schemes and purity columns can be compared. This is easily accomplished by plotting a r [ l + (Gm0DK)-’] = l,(w) in Figures 10 and 11. Comparing the values.of lm’s, the following conclusions are in order. (a) For both LP and HP columns simplified decoupling can tolerate more uncertainty than any other decoupling scheme (i.e., min, I,(w) takes its largest value in simplified decoupling). Luyben (1970), Weischedel and McAvoy (1980) also arrive at the same conclusion. (b) One-way decoupling is comparable to simplified decoupling in both columns. Similar results are given by Fagervik et al. (1981), Jafarey and McAvoy (1978), and Weischedel and McAvoy (1980). Weischedel and McAvoy (1980) report that the dynamic simulation results for one-way and simplified decoupling are essentially the same, which we confirm by our singular value analysis.
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984 99
where A = 1 - (G12G21/G11G22). We see that the difference between the robustness of ideal and simplified decoupling should be attributed to A(s). Therefore, we will give some general results connected to this factor. We say that simplified decoupling is more robust with respect to stability than ideal decoupling when
Figure 11. Uncertainty bounds for high purity column with (a) ideal decoupling;(b) simplified decoupling;(c) one-way decoupling;(d) no decoupling.
L
-I
As it is shown in the Appendix, (22) is assured if (IA-'(jw)l - l)u,[F(jw)]
1x102t
I
I
I 1x 1 ~ - 3
I
I
I j 1
5
I1 1x 1 ~ - 2
,
II
5
1xio-l
..I
I 1
1
I
TP 5Tp
W
Figure 12. Gain plot of the factor (1 - (G21G12/G22G11))-1 for highand low-purity columns.
(c) Ideal decoupling is the least robust decoupling scheme for LP and HP columns. The same result has been obtained by many authors (Luyben, 1975; Weischedel and McAvoy, 1980; Waller, 1974). (d) For the LP column, all the decoupling schemes improve the robustness of the closed-loop system. However, for the HP column, ideal decoupling makes the closed-loop system less robust. This can be deduced from Figure 11 where min, Z(w) for ideal decoupling is smaller than min, I(w) for no-decoupling (i.e., ideal decoupling can tolerate less uncertainty; its stability margin is smaller). Using detailed dynamic simulation tests Weischedel and McAvoy (1980) report that ideal decoupling for the LP column works but for the HP column it produces a very oscillatory response. Again our robustness analysis via singular values predicts and explains these results. (e) The H P column is less robust than the LP column. This is seen by comparing Figures 10 and 11 and it explains the results of Weischedel and McAvoy (1980). In order to get more insight into these results, we will analyze the matrix N = [I (Gm0DK)-l] for which we have plotted the minimum singular values a*[NI. For ideal decoupling
+
and for simplified decoupling
(23)
where1 . 1 is the absolute value. Similarly, it can be proved that ideal decoupling is better than simplified if (1- IA-'(jw)l)u,[F(jw)]
1 x 10-1
> 2 for all w E r
> 2 for all w E F
(24)
Thus (23) and (24) give the conditions under which simplified or ideal decoupling should be used. When IA-'(jw)l >> 1for w E r ,then simplified decoupling must be preferred over ideal. Figure 1 2 gives plots of lA-l(jw)l for w within the bandwidth of interest for LP and HP columns. These plots show why simplified decoupling is more robust for both columns as also previously predicted by the singular value plots of Figures 10 and 11. According to Figure 12 IA-'(jw)l for the H P column is much greater than unity which explains why ideal decoupling performs worse in HP columns than in LP columns. Finally note that lA-llu,o is the steady-state relative gain array element which has been used in the literature (McAvoy, 1979; Shinskey, 1977b) for steady-state sensitivity analysis. Feeling at this point the reader has realized the capabilities of the method of analysis, we reveal some more of its uses by explicitly calculating magnitudes of tolerable uncertainty limits. It has been already predicted by procedure B that WOBE and TOFA columns are sensitive and may also exhibit stability problems. In Figure 13, Z,(w) curves are plotted. The critical magnitudes of tolerable uncertainty can be computed as 1,(4 rad/min) N 0.22 for TOFA decoupled 1,(3 rad/min) 1,(0.7 rad/min) 1,(0.4 rad/min)
0.12 for TOFA undecoupled N N
0.19 for WOBE decoupled
0.17 for WOBE undecoupled
In view of eq 5 and 6, one way of interpreting these numbers would be as percent changes in the gains. For example, feedback control of TOFA without the decouplers can tolerate 12% change in the gain of the plant before it becomes unstable. Considering the fact that TOFA column is difficult to control and stability problems can easily arise as reported by Fagervik et al. (1981), it is not unreasonableto assume that such a magnitude of modeling error can in fact occur and result in stability problems. From the above values, we can also conclude that decoupling improves stability robustness of both columns.
Ind. Eng. Chem. Process Des. Dev., Vol. 23,No. 1, 1984
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P
Figure 15. Individual closed-loop gains for high-purity column with simplified and ideal decoupling schemes: (a) ideal decoupling top loop; (b) ideal decoupling bottom loop; (c) simplified decoupling top loop; (d) simplified decoupling bottom loop.
”
f
/
+
Figure 14. Individual closed-loop gains for simplified and ideal decoupling schemes for low-purity columns: (a) ideal decoupling top loop; (b) ideal decoupling bottom loop; (c) simplified decoupling top loop; (d) simplified decoupling bottom loop.
C. Individual Loop Analysis. Referring back to eq 19 and 20, we recognize that the singular values of NGw) are the gains of its diagonal elements, i.e., INLiO’w)lfor i = 1, 2. Figures 14 and 15 give these gain-plots for the top and bottom composition loops when simplified and ideal decoupling are used. Since I,(w) plots are directly obtained from the minimum values of these gain plots (e.g., compare Figures 10 and 11 with Figures 14 and 15), one can study the sensitivity and stability robustness of the individual composition loops by analyzing the gain-plots. For the LP column, the top loop is less robust than the bottom loop at low frequencies (w < 0.6) and for both ideal and simplified decoupling. Also it is the crucial loop for the robustness of the overall system since I,(w) takes its minimum value on that loop. For the HP column, the bottom loop is less robust at lower frequencies and it is more robust at higher frequencies. The top loop is the more crucial one for the overall stability since l,(w) takes its minimum value on that loop. It is anticipated that this type of information will be very useful for design of the controllers where the designer is
guided towards the right direction in which he can improve the robustness. Conclusion An analysis procedure which studies the robustness of process control systems with respect to stability and sensitivity has been presented. The analysis tools developed are very general and easy to implement. As the powerful results on distillation columns indicate, the technique can be effectively used in making a comparative assessment of different designs in terms of how well they can operate in the presence of modeling errors which exist in a real industrial environment. Acknowledgment Financial support from the National Science Foundation through Grant CPE80-09435is gratefully acknowledged. Proof of Theorem 4. The performance of the closedloop system is worse than the open-loop system in the integral quadratic error sense if and only if
Assume that the inverse return difference operator is stable. Then following Cruz and Perkins (1964), a sufficient condition for (A.1) to be satisfied is developed
[THO’w,.)]-’[TO’w,.)-’] - z > 0 for some a€% and some w E r (A.2) From the positive definiteness condition (A.2) and using the identity X(Z A) = 1 + h[A], (A.2) is satisfied if and only if Xm,,[(TH)-lT1] > 1 Q X,[T(jw,a)THO’w,a)] < 1
+
which is satisfied if and only if
~max[TO’w,a)l< 1 The Derivation of Eq 23 from 22. Following inequalities of singular values are used in the derivation ut[A] - 1 Iat[Z + A] Iat[A] + 1 (A.3) u*[A] a*[B] Ia,[AB] (A.4) Derivation u.[I + A-lFI > u,[Z + F ] (22)
Ind. Eng. Chem. Process Des. Dev., Vol. 23, No. 1, 1984
(A.5) (A.6) (A*7) (23) where1 . 1
is the absolute value.
Nomenclature A = finite dimensional vector space fB = subspace of A D = the decoupling compensator e = parameter-induced error on output G, = plant transfer function G, 7 model of the plant I = identity matrix j = complex variable J = performance index K = controller transfer function I , = boued on uncertainty L = uncertainty matrix R+ = positive real numbers s = Laplace variable T = return difference matrix w = frequency, rad/min y = plant output Greek Letters a = parameter vector
r = frequency range of interest
A = eigenvalue of a matrix u = singular value of a matrix
Subscripts
c = closed-loop i = indexing variable 0 = nominal variable Op = open-loop sp = set-point * = minimum Superscripts H = conjugate transpose HP = high purity column
101
LP = low purity column TF = TOFA column WB = WOBE column * = maximum Literature Cited Arkun, Y.; Ramakrishnan, S. "Structural Sensitivity Analysis in Controller Synthesis", American Control Conference Proceedings, 1982; p 1109. Arkun, Y.; Stephanopoulos, G. AIChE J. 1980, 2 6 , 975. Bode, H. W. "Network Analysls and Feedback Amplifier Design"; Van Nostrand: Princeton, NJ 1945; p 453. Brosiiow, C. B. "The Structure and Design of Smith Predictors from the Viewpoint of Inferential Control"; Joint Automatic Control Conference Proceedings, 1979. Bristoi, E. H. I€€€ Trans. Autom. Control 1966, AC- 1 1 , 1966. Cruz, J. B.; Perkins, W. R. I€€€ Trans. Aufom. Control 1964, AC-9, 216. Cruz, J. B.; Freudenberg, D. P.; Looze, D. P. I€€€ Trans. Aufom. Control 1981, AC-26, 66. Dongarra, J. J. et ai. LINPACK User's Guide, Philadelphia, PA: SIAM, 1979. Doyle, J. C.; Stein, G. IEEE Trans. Autom. Control 1981, AC-26, 4. Fagervik. K. C.; Wailer. K. V.; .Ha"arstrom, L. G. "One-way and Two-way Decoupling in Distillation", AB0 AKADEMI Report 81-3, 1981. Garcia, C. E.; Morari, M. Ind. Eng. Chem. Process Des. D e v . 1982, 21, 308. Hammarstrom. L. G.; Waiier, K. V.; Fagervlk, K. C. "Model Mismatching in Multivariable Distillation Control". ABO AKADEMI Report 81-4, 1981. Horowitz, I . M. "Synthesis of Feedback Systems"; Academic Press: New York, 1969. Jafarey, A.; McAvoy, T. J. Ind. Eng. Chem. Process Des. D e v . 1978, 17. 485. Kiema, V. C.; Laub, A. J. IEEE Trans. Autom. Control 1980, AC-25, 164. Kwakernaak, H.; Sivan, R. "Llnear Optimal Control Systems"; Wiiey-Interscience: New York, 1972, p 145. Luyben. W. L. AIChE J. 1970, 16, 198. Luyben. W. L. Ind. Eng. Chem. Fundam. 1975, 14, 321. MacFarlane, A. G. J.; Kouvaritakis. B. Inf. J. Control 1977, 2 5 . 837. McAvoy, T. J. Ind. Eng. Chem. Fundam. 1979, 16, 269. Rosenbrock, H. H. "Computer Aided Control System Design"; Academic Press: New York; 1974; p 69. Ryskamp, C. J. "Expiiclt versus Implicit Decoupiing in Distillation Control", Eng. Foundations Conference, GA, 1981. Schwanke, C. D.; Edgar, T. F.; Hougen, J. 0. ISA Trans. 1977, 16, 69. Shinskey. F. G. "Distillation Control"; McGraw-Hill: New York, 1977a. Shinskey. F. G. "The Stability of Interacting Control Loops with and without Decoupling"; Roc. IFAC Multhrarlable Technotogical Systems Conf., Univ. of New Brunswick, 1977b; p 21. Stewart, G. W. "Introduction to Matrix Computations", Academic Press: New York, 1973. Toijaia (Waller), K. V. T.; Fagervik, K. C. Kem. Teolllsuus 1972, 2 9 , 5 . Waiier (Toijala), K. V. T. AIChEJ. 1974, 2 0 , 592. Weischedei, K.; McAvoy, T. J. Ind. Eng. Chem. Fundam. 1980, 19, 379. Wood, R. K.; Berry, M. W. Chem. Eng. Sci. 1973, 26, 1707. Zames, G. IEEE Trans. Aufom. Control 1981, AC-26, 301.
Received for review October 12, 1982 Accepted April 28, 1983
