I n d . Eng. Chem. Res. 1990, 29, 1875-1889 function, part 1 and 2. Numer. Math. 1981,38, 83-127. Schittkowski, K. On the convergence of a sequential quadratic programming method with an augmented Lagrangian line search function. Math. Operationsforsch Statist., Ser. Opt. 1983, 24, 197-216.
Schittkowski, K. NLPQL a Fortran subroutine solving constrained nonlinear programming problems. Ann. Op. Res. 1985, 5, 485-500.
Seinfeld, J. H.; Lapidus, L. Mathematical Methods in Chemical Engineering, uol. 3, Process Modelling, Estimation and Identi-
1875
fication; Prentice-Hall: Englewood Cliffs, NJ, 1974. Sincovec, R. F.; Erisman, A. M.; Yip, E. L.; Epton, M. A. Analysis of descriptor systems using numerical algorithms. IEEE Trans. Automat. Control 1981,26, 139-147. StaniSkis, J.; Levigauskas, D. An adaptive algorithm for fed-batch culture. Biotechnol. Bioeng. 1984,26, 419-425. Received f o r review August 23, 1989 Revised manuscript received February 19, 1990 Accepted April 24, 1990
Dynamics and Control of a Distillation Train without Recycle Streams Beatriz Brignole, Carlos Ruiz,? and Jose A. Romagnoli* Planta Piloto de Ingenieria Quimica, UNS-CONICET, 12 de Octubre 1842, 8000 Bahia Blanca, Argentina
A detailed study on the dynamics and control of a complex configuration of two distillation columns without recycle has been carried out. The train separates a mixture of benzene, toluene, and o-xylene (BTX) and consists of a prefractionator and a side-stream column. The dynamic aspects were explored with the help of time and frequency domain plots. The addition of controllers a t the prefractionator minimizes the disturbances affecting column 2 most severely. After the variables were properly paired, the open-loop plots were identified and the controllers were designed. A comparison of the results yielded by different multivariable control design methods for some input-output configurations is presented. The resulting schemes had good spectral properties, and they were also tested via linear and nonlinear simulations with satisfactory results.
Introduction In the chemical and petrochemical industries, we usually face processes that aim at separating multicomponent mixtures through distillation. This is a typical energyconsuming operation. It can be responsible for about 40% of the plant energy consumption according to Shinskey (1977). Much effort has therefore been devoted to the development and evaluation of cheaper distillation configurations. Though it surely means more difficult design and control problems, it may be worthwhile to choose a complex set provided it yields significant savings. This paper explores different design alternatives for a given sample problem, keeping in mind that the resulting schemes should also be easily operable and controllable. We analyze the dynamics and control of a train of two columns arranged in series, which separates a ternary mixture of benzene, toluene, and o-xylene (BTX) (relative volatilities of 6.7, 2.4, and 1). Figure 1 shows the configuration and the control schemes adopted in this work. Doukas and Luyben (1978) explored four different ways of dealing with the mixture mentioned above. The most economic configuration according to their research has been adopted in this paper without design changes. Further design improvements that might lead to a more balanced operation or higher yields are an interesting subject for future work. Table I lists its main features. The first column (19 trays) is a prefractionator that makes a rough preliminary split and feeds its overhead and bottom products to the second column (40 trays) at trays 4 and 24, respectively. Column 2 yields the lightest component as the top product and the heaviest as the bottom one. The intermediate one is removed from tray 20 as a liquid stream. This location corresponds to the toluene molar fraction peak. Previous work on control of distillation trains makes use of classical design approaches only, either for thermally decoupled systems (Doukas and Luyben, 1981; Elaahi and Luyben, 1985) or for thermally
* To whom
all correspondence should be addressed. Present address: PASA (Petroquimica Argentina Sociedad Anonima) San Lorenzo, Rosario, Argentina.
Table I. Configuration Details" no. of components: 3 name of components: benzene (B),toluene (T),xylene (X) feed flow rate: 272.16 kmol/hr feed condition: saturated liquid feed liquid molar fraction: B, 0.4; T, 0.4; X, 0.2 feed enthalpy: -1893.6 kcal/kmol trays* kind sieve trays weir length: 3.5 m weir height: 0.06 m downcomer* kind segmental cross-sectional area: 2.92 m2 efficiency* Murphee tray efficiency: 1.00 configuration (see Figure 1) column 1: prefractionator colum 2: side-stream column item column 1 column 2 no. of trays 19 40 fresh feed tray 9 4/24 feed trays (connections) side-stream drawoff tray 20 column diameter, m 1.9 2.0 top pressure, kPa 101.325 101.325 bottom pressure, kPa 101.325 101.325 1.1 reflux ratio 0.75 distillate rate, kmol/h 114.59 146.48 112.00 side-stream drawoff rate, kmol/h 45.57 bottom rate, kmol/h 125.48 0.949 0.741 top composition (B) 5.1 X top composition (T) 0.258 side-stream composition (T) 0.920 7.9 x 10-2 side-stream composition (X) 1.3 X bottom composition (T) 0.565 bottom composition (X) 0.999 0.433 4.4 x 108 3.7 x 108 reboiler duty, kcal/h 22.8 19.7 holdup in reflux drum, kmol 19.3 23.3 holdup in column base, kmol "The symbol * means that the information holds for all trays of both columns.
coupled columns (Tyreus and Luyben, 1976; Frey et al., 1984). Controllers are generally tuned based on steady-
08S8-58S5/90/2629-lS15~02.50/0 0 1990 American Chemical Society
1876 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
equations and a set of procedures. The latter, which provides the algebraic equations, is used to calculate physical and hydraulic properties, controller responses, and the like. The set of equations is Y = fb,P,t)
(1)
where the procedures P can be written as
p = gb,z,t)
state information or are tuned empirically. Doukas and Luyben (1981) proposed two alternative control schemes for the configuration described here, where they sense and manipulate variables of the second column only. In both cases, top and bottom compositions are held constant by manipulating the reflux rate and the reboiler heat duty. One scheme manipulates the side-stream drawoff flow rate and location in order to control the side-stream composition. The second scheme uses the overhead distillate product rate from the prefractionator to control the side-stream composition. Both of these schemes leave the first column uncontrolled. This paper aims at learning whether or not it is convenient to control the prefractionator carefully. The goal is to reduce all changes as much as possible before they pass on to the second column. Another essential feature of this paper is the use of advanced methods both for analysis and design in order to achieve better control. The huge theoretical background embodied by the modern frequency domain approach for control system design is used here to study the main characteristics of the columns and to design controllers. Several control schemes are thoroughly investigated by means of mathematical and design tools (Hung and MacFarlane, 1983; Agamennoni et al., 1988). These tools can be systematically employed to determine the properties of each scheme. Within this framework, the degrees of interaction, sensitivity, and robustness can be quantified and controllers of different degrees of sophistication can be designed. The work has been divided into five stages: (1)analysis of the open-loop dynamics for both columns; (2) process identification without controllers; (3) variable pairing, sensitivity, and interaction analysis for different input-output schemes; (4) design of multivariable multiple input-multiple output, MIMO) controllers; and (5) test of the control schemes via dynamic simulation and modern frequency domain techniques.
Dynamic Simulation The performance curves resulted from the nonlinear simulation by means of the DYNAM package. Its dynamic model, built by Gani et al. (1986b), has yielded satisfactory results for continuous nonstationary operations, for the start-up of a single column (Rub, 1984,1986)and for trains of multiple columns with and without recycles (Gani et al., 1985, 1986a). It reproduces phase equilibria and tray hydraulics, and then it integrates numerically the resulting system. The model consists of a set of ordinary differential
(2)
The differential variables y are the component and energy holdups for the stages of the column. The algebraic variables z include tray pressures, pressure drops, physical properties, and concentrations of components. In this work, we make use of this model in its simplified version (Gani et al., 1986b),which takes into account composition and hold-up changes but leaves out most energy balances by regarding enthalpy changes as instantaneous. The equilibrium properties are calculated with the ChaoSeader package, and the hydraulic properties are continually checked so that neither weeping nor flooding occurs. As the model is stiff, the integration package adopted is the BDFSH dense option (Cameron et al., 1986). Either stiff or nonstiff discretization methods are used for each step in the calculation, according to the eigenvalues of the corresponding Jacobian. Thus, the package can solve effectively many design and operability problems like design and verification of control systems, start-up procedures, design and verification of tray hydraulics, and interactions between columns. One of the major problems of the model is how to handle the large number of differential equations involved. The numerical methods used must not only be A stable and implicit but they must also possess a mechanism for the control of the integration step in order to cope effectively with the inherent stiffness of the system. For this reason, it is indispensable either to generate sparse Jacobians of great dimensions or, even worse, to find their inverse. The plots obtained for this paper have been generated with the rigorous model described in the preceding paragraphs. The curves agree with those obtained by Crespo et al. (1987), who used a reduced model based on the method of orthogonal collocation. This model drastically reduces the magnitude of the system. Discrete variables like compositions and flow rates are regarded as if they were continuous. The only balances involved are the ones that correspond to the collocation points. The choice of convenient locations for those points varies for each problem, but, in general terms, key spots like feed or drawoff trays must be included.
Analysis of the Open-Loop Dynamics Firstly, the problem was initialized by setting the derivatives equal to zero and solving the resulting nonlinear system of equations for each column separately. The Naphtali-Sandholm method modified by Christiansen et al. (1979) was used for this stage. It employs the Newton-Ftaphson technique to get the solution for the linear system. Then the sequence was simulated with the DYNAM package in order to get a steady state for reference. Afterwards, the plant was tested by introducing step disturbances, one by one, around the normal operating point. The open-loop transient responses were obtained until they reached a new steady state, and lastly, the dynamic responses to those changes were analyzed with regard to operability and control. The step tests performed consist of disturbances to some commonplace manipulative variables; the reflux rate, the distillate rate, and the reboiler heat duty of any of the two
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1877 0.24
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Figure 2. Some parametric plob for the prefractionator. (a, top) Temperature at different locations versus benzene composition at the top, (b, bottom) temperature at different locations versus xylene composition a t the bottom.
columns and the side-stream rate for column 2. Here, we present responses for the prefractionator, and we also analyze their effect on the second column. Step sizes of rt3% in the reflux rate revealed fairly symmetric responses for the prefractionator, whereas the same disturbances yielded highly nonlinear responses for column 2. The downstream column revealed a severe lack of symmetry for step sizes higher than 1% in any variable. As it is more sensitive and nonlinear than the first, it is advisable to generate a control sequence that disturbs column 2 as slightly as possible. Figure 2 shows some parametric plots where time is the parameter. Curves like these sum up the information from several temporal plots in only one graph. Whenever a curve becomes perpendicular to a coordinate axis, the variable on that axis has reached its steady state. The curves for holdup versus composition showed that the hydraulic variables settle more quickly than the thermodynamic variables, which is also true for the start-up behavior (Ruiz, 1986; Ruiz et al., 1988). When indirect measurements are used for control purposes, it is useful to verify that both the measured and the controlled variables move together because this is not always the case if compositions on a certain tray are monitored via the temperature on another tray. So the trajectories become useful for the choice of sensor location. In the prefractionator's case, greater precision in temperature measurements is required to monitor the compositions appropriately when the sensors are located away from the ends, because considerable variations in composition take place during the first 5 min with relatively small temperature changes. Afterwards, the curves become parallel, and this indicates that, at this stage, moving the probes will not make any difference in this respect. The column with the side-stream drawoff is greatly affected in a nonlinear way by disturbances on the prefractionator. This behavior is understandable if we take into account two aspects of this process. Firstly, column 2 receives products from the prefractionator at two dif-
ferent locations (tray 4 and 24). Secondly, the transient disturbances reach the side-stream column with a certain time delay. Both feeds to the second column are introduced at a tray whose composition resembles that of the incoming stream. The temperature gap is not wide; 10 K for the top feed and 1-4 K for the bottom feed. Figure 3 shows how the bottom changes in column 1 affect tray 24 of column 2, which is the bottom feed location. This variable, as well as other downstream currents, suffers a reversion in the direction of the response when the disturbance in the first column actually reaches the second. The single step disturbance introduced upstream reaches column 2 as several composition and flow changes put together. The result of adding several disturbances is a nonlinear composition response in the second column feed trays. Apart from the changes described above, feed rate variations of &lo% in column 1 were also simulated in order to get two new operating conditions for the train (each with a different load) and find the dynamic openloop paths for the feed disturbances that should be controlled. The dynamic behavior of the first column when it undergoes these changes in the feed flow rate is highly asymmetric (Figure 4). This feature, which shows the nonlinearity of the process, is easily noticeable for toluene molar fractions. Though shapes are similar, the response of the top region for negative disturbances is more sluggish than the one corresponding to positive variations. At the bottom, the former exhibits steeper deviations than the latter. The temperature plot agrees with the composition ones, revealing that the bottom behaves more asymmetrically and possesses higher time constants than the top. As for the changes in the purity of the products, negative disturbances look more harmful. In the second column, they ruin the key molar fractions for both the distillate and the side-stream drawoffs though they do not affect the bottom product at all. The +lo% disturbance ruins the purity of xylene at the bottom but only after half an hour has elapsed, thus offering enough time for taking appropriate action. It augments the benzene fraction at the top and generates an oscillatory response for toluene at tray 20. For the prefractionator drawoffs, 10% variation exhibits lower gains and shorter settling times. Thus, from this viewpoint, both columns seem to absorb positive disturbances better. The lack of symmetry becomes more evident for xylene at the bottom of column 2, because of the higher purity involved. The stream remains practically unaffected by those disturbances that cause purification of the bottom product, whereas gains are higher for those that ruin the xylene yield. Nevertheless, this only happens after significant delays. The configuration of this separation process accounts for its high degree of interaction. The step change intro-
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1879 Table 11. Open-Loop Transfer Functions. Change of Characteristic Parameters According to the Location of the Temperature Sensors0 +3% reflux rate +3% reboiler heat duty tray 1 2 3 4 5 6 12 14 15 16 17 18 19
Kb -0.0529 -0.0360 -0.0338 -0.0317 -0.0274 -0.0222 -0.0135 -0.0244 -0.0394 -0.0615 -0.0846 -0.0943 -0.0807
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'Time constants and time delays in minutes. bSide-stream gain. cTime constant. dTime delay. Table 111. Open-Loop Transfer Functions for Column 2a reflux rate distillate rate side-stream drawoff (a) Some Models for Changes in the Manipulated Variables XM 0.99182/ -0.M208/ [-0.095126(0.5076s+ (0.47962s+ 1) (0.08491s+ 1) l)]/(O.oS412s~+ 0.331s+ 1) XT, 1.16447/ [-0.00507(0.4379~ + -1.11991/ (0.24124s+ 1) 1)]/(0.073&*+ (0.27453s+ 1) 0.4807s+ 1) upper feed lower feed (b) Matrix of Disturbances XM 0.91338/(0.54697~ + 1) [0.091521(0.65356~+ 1)]/ (0.10337~~ + 0.4s + 1) XT, 1.36158/(0.24424~ + 1) 0.78518/(0.3827& + 1) xM: benzene composition at distillate. side stream.
x T ~ :toluene
composition at
Two different approaches were considered from the point of view of controller design. Each column could be treated as a separate unit by designing two MIMO independent controllers or the train could be regarded as a single unit and, in this case, only one controller matrix of larger order should be designed. The dynamic analysis can help a t this point. The curves that represent the interaction between the columns had inverse responses, which means more difficult control problems. A 4 X 4 joint controller design with such entries taken into account at the design step would lead to polynomials of high degree that are difficult to handle. Moreover, the values of the steady-state gains for each column differ by 2 orders of magnitude. Thus, it is not advisable to treat the design problem as a whole without proper scaling. The drawbacks mentioned above, i.e., inverse responses and unbalanced steady-state gains, suggested that treating each column separately would be a simpler approach provided that the resulting controllers were robust enough to counteract the interaction between columns effectively.
Control Strategy The control problem treated here is how to get tight closed-loop composition control for the three product streams of the train. Two different alternatives are available. The first approach, already treated by Doukas and Luyben (1981), is to operate with the first column uncontrolled. The second alternative is to add a couple of controllers for the top and bottom sections of the prefractionator in order to reduce disturbances to the sidestream column. The studies on the dynamics of the train revealed that the second column behaves in a more non-
linear way. The range of symmetric behavior extends to *3% for the manipulated variables of the prefractionator and only to f l % for the ones of the side-stream column. As the behavior of the controllers is guaranteed on the linear region, the control over the prefractionator would allow the system to absorb a higher range of disturbances. Therefore, it would be advisable to design controllers for both columns. But, then, the design alternatives mentioned in the last paragraph of the preceding section naturally come out. The open-loop analysis indicated it would be easier to handle the columns one by one, and this is the approach that we will follow. It was assumed that the inventory loops were faster than the quality loops. Therefore, a simplified model that regards pressure, bottom, and condenser levels as constant was used for the runs. In both columns, the pressure was controlled by manipulating the condenser duty; the level at the condenser, by means of the distillate flow rate; and the bottom level, using the bottom flow rate. This paper deals basically with the design of the quality loops. Those loops that had time constants of similar magnitude were designed simultaneously. For the prefractionator, the molar fractions of benzene at the top and xylene at the bottom were controlled by the manipulation of the reflux flow rate and the reboiler heat duty. For the second column, the molar fractions of benzene at the top and toluene at the side stream were controlled by means of the reflux flow rate and the side-stream flow rate. The dynamic curves showed that the bottom stream from column 2 remained insensitive for about half an hour before any disturbance affected it. The existence of 16 trays between the lowest feed and the bottom stream produces a pinch region, which lessens the effect of all disturbances. As a result of this long delay and the lack of interaction (proved via frequency domain responses), the corresponding controller was designed separately with single-variable concepts. Dual-temperature control was applied to the prefractionator. Where possible, composition measurements were chosen for the second column because the requirements on quality are stricter there. The modem design methods employed for the prefractionator were normalization at low and high frequencies and design via the process inverse. Though the controllers for the prefractionator succeeded in minimizing the disturbances in composition affecting the second column, they also caused flow rate oscillations that ruin the purity of the final products. There are two ways of counteracting this effect. A couple of drums whose
1880 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
capacities stabilize the output flows can be added in between or the controller for column 2 should be capable of counteracting this effect. The last alternative was tried here. An effective 2 X 2 PID feedback structure was designed via the process inverse, and the behavior with and without control of the prefractionator was compared. Choice of Measured and Manipulated Variables. Determining the structure of the control system is a topic of utmost importance because proper pairing counters upsets efficiently. The location of both sensors and manipulated variables should be studied carefully. Temperature sensors were chosen for the first column because they are inexpensive, reliable, and responsive enough. Composition devices were placed for the top and sidestream drawoff of second column, just at the final product streams, in order to meet the more stringent requirements through a direct measurement of product quality. For the bottom product from column 2, the composition measurements were not sensitive enough. Therefore, a temperature sensor was located at tray 27 to monitor the xylene yield. It is difficult to decide on the location of the probes for the configuration under study. End trays minimize the lags in the control loop. For instance, control for the top product on any tray other than the condenser results in a steady-state error because the component maintained at zero deviation through the reset action is not the one that should remain constant according to the control purpose. Nevertheless, there are several reasons against controlling end temperatures. Impurities become more concentrated at these points, pressure fluctuations may have more influence on temperature than composition ones, and sensors further down the column detect any feed change faster than those located at the extremes. Other aspects should also be taken into account. First, the sensors should be located in a sensitive region, and then, the pairing should minimize interaction and oscillations from the corresponding controllers. Thus, we face a compromise situation for the choice of regions where monitoring and change of variables is bound to be more effective from the standpoint of control action. Brignole et al. (1985) define the enrichment factor as a sensitivity index stating that the sensors should be located where the composition differences are the largest, Le., where the enrichment factor is high. As compositons of temperatures go together, this quantity also gives the optimum location for temperature sensors. Process gains were compared to enrichment factors, and the direct link mentioned by Brignole et al. (1985) was ratified. Although the former are static quantities and the latter are dynamic ones, the zones of maximum process gains do correspond to the regions of maximum enrichment factors. The highest enrichment factors correspond to trays 1-5, 9,10, and 14-19 for the prefractionator and to trays 1-12 and 16-26 for the side-stream column. The feed tray zones are unsuitable regions because they do not mirror effectively the influence of changes on the column as a whole. The entrance of feed streams altered the profiles, causing discontinuities that increased the enrichment factors without affecting the process gains. The lowest factors for the second column correspond to a pinch zone ranging from tray 30 to tray 40. A t those places, the system experiences almost no changes, and variations that should be controlled are not detected because most disturbances do not reach this region soon enough. If sensors were placed there, the modeling, measurement, and computational errors would be amplified. Therefore, if a temperature probe were placed to monitor the bottom, it
should be located several trays from the end of the column, above the pinch region, where purity is lower and gains are larger. For the first column, several pairings combining trays 1-5 and 14-19 were left. The final choice was made via frequency domain analysis of open-loop responses. Frequency response analysis is a method for investigating the characteristics of a process. Given a transfer function, it provides easy-to-use information on stability margins, interaction, and robustness. Thus, it brings sound grounds for the acceptance or rejection of structures. The sensors were finally placed at trays 1-19 because the sensitivity indexes were higher and the degree of misalignment was lower (see Appendix A). This indicates that the tray 1-19 transfer function matrix is closer to normality, thus having a well-conditioned associated eigenvalue problem that results in less sensitivity to disturbances. Closed-loop responses that are much faster than the corresponding open-loop behavior cannot be generated for they surely have a detrimental effect on stability. As to the choice of manipulated variables, it is therefore useful to have a look at the time constants for different structures. The shorter a controlled variable takes to get stabilized the better. For this train, temperatures and compositions at the top become stable sooner when subjected to changes in reflux flow rate than when they suffer from changes in distillate rate. The structures were also examined by using frequency domain analysis. The plots for the prefractionator revealed similar behavior for LO-QB and DR-QB combinations, and the most conventional structure LO-QB, whose top region got stabilized more quickly, was picked out. For column 2, the product stream a t the bottom exhibited no interaction due to the existence of a pinch region, and therefore, the corresponding loop was designed as a single input-single output (SISO) loop. The DR-SD and the Lo-SD schemes were tried out for dual-composition control of the remaining product streams. Both combinations were interactive, but the Lo-SD structure was finally chosen because it exhibited an adequate alignment.
Open-Loop Analysis via Frequency Domain Techniques The open-loop transfer function matrices belonging to the same column were arranged in 2 X 2 matrices and were examined by means of frequency domain plots (see Appendix A). The low frequencies can be associated to the final steady state, whereas the intermediate and high frequencies represent the unsteady-state behavior. All poles of the open-loop functions lie on the left half plane, and therefore, the plants are open-loop stable. The principal gains are low and different (Figures 5a and 6a), and the controllers should make both singular values almost equal for a wide range of frequencies. This is a desirable behavior because it means a low condition number. From a physical viewpoint, the condition number represents the ratio of the maximum and minimum open-loop, decoupled gains of the system. Therefore, a large condition number shows that the relative sensitivity of the system in one direction is very weak. If several strategies for the same system are to be compared, the numbers reveal, on a relative basis, the difficulty of the control problem (Moore, 1986). For both columns, the open-loop condition numbers do not differ greatly. In terms of overall controllability, the magnitude of the largest singular value is important because if it is small the system is not sensitive enough to control. On the other hand, when it is too large, the problem is poorly scaled again because the controller must have very low gains. For
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1881 50
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column 1, the largest singular value is never higher than 0.32 (-10 db or so) and it reaches 0.003 (about -50 db) at 10 rad/min. Therefore, the controller should also increase this quantity at all frequencies. In Figure 5a, frequencies higher than 10 rad/min have not been plotted because the singular values for the prefractionator exhibit oscillations due to the phase rotation introduced by the delay terms. As to sensitivity, the misalignment plots are given in Figures 5b and 6b. We must bear in mind that the horizontal line at 1.41 represents the upper bound this quantity can possibly reach for a 2 X 2 matrix G,thus meaning the greatest degree of misalignment. For our problem, the curves approach this worst case in a great portion of the frequency range. However, these results depend on the sensor locations, since moving the probes
far from the ends of the column worsened the results. The most convenient location, in terms of spectral sensitivity, was found for the pair 1-19, which shows better normality measures (the index being around 0.5) at low frequencies. The controller should reduce this degree of misalignment as much as possible in order to get a near-aligned, almost normal matrix. However, normality only gives a sufficient condition for low sensitivity. We can resort to another parameter that quantifies spectral sensitivity. Postlethwaite (1982) proposes to use the sensitivity indexes that are equal to 1 for normal matrices. For column 1 (Figure 5c), the indices are better again for the 1-19 pairing because they are closer to l , with values of 0.9 at low frequencies. Then the indexes worsen and become oscillatory, thus showing that the sensitivity of the eigenvalues has
1882 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 Table IV. Closed-Loop Transfer Functionso temp on tray 19 B comp a t top temp on tray 1 (a) PI Normalizer Controller for the Prefractionator Lo (0.082s- 2229.3)/~ (0.053s+ 833.76)/~ QB (-0.026s- 684.84)/~ (0.03s+ 404.88)/~
Lo QB
Lo
QB
LO
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T comp at ss
(b) PI Controller Based on the Process Inverse for the Prefractionator (-440.1s- 7641)/~ (391.9s+ 1709.1)/~ (-226.2s- 2178.5)/~ (208.1s+ 906.6)/~ (-6s' - 113s - 422)/ [s(0.89738s+ I)] (-s' - 34.7s - 150)/ [s(0.90742s+ l)]
(c) PID Controller Based on the Process Inverse for the Prefractionator (0.8s' + 106s + 922)/
[s(0.91121s+ l)] (2.4s' + 86s + 398)/ [s(0.89456s+ l)]
(d) PID Controller Based on the Process Inverse for Column 2 (99s' + 166s + 8.97)/[~(18.202~ + 111 (-0.2~~ - 0.5s - 0.4)/[~(0.26086~ + l)] (107s' + 169s + 9.3)/[~(17.802~ + l)] (-31.6s' - 81.4s - 4)/[~(20.307~l)]
+
"Lo= reflux flow rate in terms of deviations from the initial steady state. QB = reboiler heat duty in terms of deviations from the initial steady state. SD = side-stream flow rate in terms of deviations from the initial steady state. Time in hours.
increased. For column 2 (Figure 6c), they remain close to 0.85 at frequencies lower than 1 and the problem is well-conditioned. With respect to interaction, the global angles of interaction are plotted in Figures 5d and 6d. The angles tend to 45 deg as the frequency increases, and they are never below 15 deg, thus revealing that the systems are tightly coupled. The total angle of interaction (not plotted) is slightly more conservative at low frequencies, the difference being negligible a t all frequencies. As the existence of interaction cannot be neglected, multivariable schemes will be adopted to account for this problem. In short, all the open-loop plants exhibit a high degree of interaction and the singular value plots reveal an unbalanced behavior. For the prefractionator, the 1-19 pairing is the most promising alternative, showing better spectral sensitivity measures that are known to have important consequences on the stability of the nominal system.
Design of the Controllers Control of the Prefractionator. Classical synthesis methods, which work quite effectively for SISO problems, can sometimes be adapted to handle multivariable compensator design. One of them is Rosenbrock's method (1976), which is simple to use and may be employed for diagonally dominant systems. The well-known Gershgorin bands are a graphical test for dominance. This test came out positive for frequencies above 1rad/min. In cases like this, Rosenbrock proposes to achieve dominance by means of pre- and postcompensators. However, in our case it was impossible to design any matrix that modified these plants in the desired way, thus making it necessary to resort to more complex techniques like those based on the quasiclassical approach (Hung and MacFarlane, 1983) and on the process inverse (Agamennoni et al., 1988) described in Appendix B. Quasi-Classical Approach. As discussed in Appendix B, this approach tries to design a controller that normalizes the closed-loop transfer function matrix a t particular frequencies. In the case of the prefractionator, the singular values of the open-loop transfer function matrix are the following: zero frequency, u1 = 0.3036474 and a2 = 0.014 790 84; infinite frequency, ul = 0.045 9997 and u2 = 0.000 957 96. The method of normalization at specific points consists of proposing singular values for the controller so that the resulting closed-loop matrix has a condition number equal to 1 a t frequencies of 0 and m. The lowest frequency
within range (0.01 rad/min) was chosen so that it behaved just as zero frequency. Therefore, normalization a t 0.01 rad/min is effective for all choices. Extremely high frequencies lack importance for systems with time delays because of the oscillation of all parameters. Several multipliers are equally possible. As this trial-and-error procedure imposes no restriction on intermediate frequencies, the best normalizer controller is the one that also exhibits the most acceptable behavior within the full range of interest. Sometimes, satisfactory results cannot be achieved. Such being the case, normalization a t some intermediate frequencies becomes a must. Table IVa shows the transfer functions for the controllers designed by this method. Though the behavior at low and high frequencies was good, it was impossible to avoid a noticeable deviation from normality at medium frequencies. This indicates that the closed-loop system can be sensitive to modeling errors, and consequently, we should be careful when studying the stability through the nominal system. The roblem of the appearance of a higher misalignment index can be solved by designing a precompensator that normalizes the plant at the critical frequencies. Nevertheless, it was unnecessary to do so because the behavior at the simulation stage was good enough (Figure 7). Controller Design Based on the Process Inverse. The controllers that were obtained via the process inverse are shown in Table IV,b and c. This methodology changes soundly the behavior of the plant at all frequencies. The first step is to determine the aifunctions so that the resulting controller becomes proper. The difference between the degrees of the polynomials can be either one or zero. Therefore, a single time constant must be chosen for each control loop and the choice depends on the characteristics of the plant. The desired time constant is naturally smaller or equal to the corresponding open loop one. Suitable controllers that fulfill these requirements are found readily for systems without delays. In this case, the rigorous model stated the following open-loop time constants: ri = 0.3759 and r2 0.092 59. As delays are important, it was necessary to loosen the requirements and choose higher time constants for the closed-loop system. Picking the same ai functions for each loop, the best results were obtained for ai= 2 for both plants. Control of the Side-Stream Column. The controller design was performed by means of the method based on the process inverse. Molar fractions of benzene at the top andtoluene a t the side stream were controlled by manipulating the reflux and side-stream flow rates. Table IVd
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1883 factorily (Figure 7) with no offsets, good decay ratios (about 1/3), and acceptable settling times (15-20 min). The controller designed via the process inverse yielded better results than the normalizer controller for overshoots ( 0 ) Open loop are lower and settling times are shorter. The effect on the ( 1 I P I Normalizer Controller second column, when it remained uncontrolled, was de( 2 1 PI Controller b a p d ibe trimental to the quality of the final products. The feed f 0765 pro:ess inverse flow rates to the second column change considerably when the valves are manipulated. As a result, purity changes quickly in the distillate and side-stream drawoff and the 0.735 I I I reaction is very sluggish in the bottom, where the dis0. 0.3 a6 0.9 1.2 1.5 turbance ruins this stream only after 20 min has elapsed time I hrsl with higher overshoots and longer settling times for control under negative disturbances in feed flow rate. The control -____ 0.44 under those negative load changes was also effective, A 7 though asymmetricdue to the nonlinearities of the process. 0.4296 Set point and load changes were also simulated for other pairings like 3-16 in order to compare the results. The 101 O p n loop 04192 amount of‘overshoot and the stabilization times increased U (11 P I brmnlizer controller as the sensors were shifted away from the ends of the ( 2 ) P I Controller h s e d on !he e 0.4088 process i i w r s e column. This ratified the results obtained via spectral A s analysis that recommended locating the probes at trays 0384 1 and 19. (01 The best controllers were found by means of the process inverse, though normalizers proved to be more robust. The PI and PID control schemes developed were tested on the time lhrs) process. They provided effective control with good siFigure 7. Comparison between open-loop and closed-loop behavior multaneous top and bottom regulation under a variety of using different controllers, Responses to a +lo% disturbance in feed load and set point changes. The closed-loop frequency flow rate. (a, top) Benzene molar fraction in the distillate of the responses revealed that the controllers managed to reduce prefractionator; (b, bottom) xylene molar fractionin the bottoms of interaction and sensitivity to uncertainties over a wide the prefractionator. range of frequencies, as will be shown in the next section. shows the controller (PID structure) with closed-loop time Positive and negative changes in feed flow rate and feed constants of 0.17 and 0.25 h for the top and side-stream composition were explored in order to test the regulatory loops, respectively. It managed to keep the product comaction of the controllers. A variation in feed condition from positions under control both under set point and load saturated liquid feed to a mixture containing 70% vapor changes. The system was subjected to variations in feed in the stream was also tried out with good results. The flow rate, feed composition, temperature set points, and goal was to maintain process variables at their set points composition set points. The matrix of disturbances shown in the face of disturbances. Apart from disturbance rein Table IIIb was used to model the effects of feed disjection, *3% changs in the manipulated variables together turbances for the linear simulations, and Figure 8 shows with f0.55% set point changes in the controlled variables its behavior. For mild disturbances, there is good agreewere also studied. ment between the linear and nonlinear responses, which Both columns make more overheads than bottoms. means that the linear approximations are adequate repTherefore, heavy material is likely to be in the overheads. resentations of the system’s behavior. This unbalanced operation yields maximum purity at the bottom, and any change in the prefractionator affects this Evaluation of the Closed-Loop Performances zone much more slightly. When bottom purity increases Closed-Loop Time Responses. The original DYNAM as a result of a change, it does so very slightly because it package (Gani et al., 1986b) introduced additional difis already very pure. This was the case for a 0.55% deferential equations to simulate SISO, PI, an PID concrease in top temperature without control of column 2. At trollers but could manage neither MIMO nor higher order the same time, the distillate and side-stream drawoff did structures. Consequently, some changes were introduced vary noticeably. in the package to widen the scope of closed-loop problems As the magnitude of the step disturbances in the feed that could be solved. The main facility incorporated is the flow rate increased, the system moved toward nonlinearity, possibility of giving the controllers in terms of their which meant a more difficult control problem. The 2 X transfer function matrices. In this case, the SIMIMO 2 scheme for column 2 without control of the prefractinator routines (Agamennoni et al., 1988) are employed to genworked well for disturbances 1to 10% in feed flow rate, erate the corresponding difference equations. The program giving stable, quicker responses. The 4 X 4 scheme hanuses a periodic sampler and calculates one control action dled greater disturbances better (Figure 8c). When the for each sample, the action remaining constant until the controllers for the prefractionator and column 2 worked beginning of the following sample interval. For a given together, they counteracted the effect of a wider range of simulation run, it is possible to define some controllers disturbances because the controller at the first column (SISO, PI, or PID) as a part of the model, i.e., as additional minimized the changes in composition for the feeds to differential equations, and others by means of the concolumn 2. troller matrix. The simulations performed for this work have been made in this way. Closed-Loop Frequency Responses. Figures 5 and For column 1,controllers were designed for various lo6 show comparative graphs for the evolution of characcations of the temperature sensors. The PI controller for teristic parameters versus frequency, before and after the the 1-19 pairing managed to handle disturbances satisaddition of different controllers. The pursued goals are I):)
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1884 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 0 95
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Figure 8. Controllers for column 11. (a, top left) Linear and nonlinear behavior under PID control for a +1%change in feed flow rate; (b, bottom left) both columns under PID control for a +5% disturbance in feed flow rate; (c, right) comparison of nonlinear plots with and without control of the prefractionator.
to reduce interaction, to normalize the plants, to minimize their invertibility indexes, and to lower their sensitivity to disturbances. For the profractionator, the controller designed via the process inverse (Table 1%) yields better spectral results than those designed via normalization (Table IVa). This statement agrees with the conclusion drawn by simulation (Figure 7), though in that case other objectives regarding purity had been tested. Controllers based on the process inverse satisfy the requirements from both viewpoints most satisfactorily. As to the frequency responses for column 2 (Figure 6), the closed-loop behavior revealed an improvement with regard to sensitivity and interaction. A t this point, it would be advisable to go into some detail. All poles of the open-loop function lie on the left half plane. Then, the plants are open-loop stable. The plots of the characteristic gain loci for all frequencies show that there are no encirclements around the point (-1,O). As there are no unstable poles, it can be concluded from the Nyquist generalized criterion that the systems are closed-loop stable. Unluckily, the eigenvalue decomposition does not describe adequately the system gains before all possible inputs, the exception being the particular case of normal matrices. It generally gives accurate information about the nominal model only, so a more detailed study is indispensable to get a more complete view of the process. Figure 5a shows plots of principal gains that take into account rotations and scaling. A system has large loop gains if the minimum singular value CT, of the open-loop
transfer function is much greater than 1. Conversely, it has small gains if the maximum singular value ul is much lower than 1. For frequencies where gains are large, the output of the closed-loop system is almost insensitive to disturbances and model uncertainties. The ratio ul/um is called the invertibility index. Its minimization is a usual control objective. For both plants, the open-loop principal gains are very different. Controllers succeed in making them closer to each other. They are similar up to 0.02 and 0.07 rad/min for normalizer controllers and for those based on the process inverse, respectively. Then, the performance of the controller designed via the process inverse is better. The magnitude of u1 is also important because loop gains must be high enough in order to generate a model that is properly scaled. The controllers heighten them from -10 to 20 db and 40 db. Bandwidth, defined as the lowest frequency w, at which the minimum singular value of the open-loop transfer function becomes unity, must also augment. It is related to velocity: the higher the bandwidth, the quicker the response. Normalizer controllers and controllers designed via the process inverse make w, shift to the right up to 0.07 and 0.95 rad/min, respectively. For column 2 (Figure 6a), the controllers managed to make both singular values adopt similar values for all frequencies and to increase their values significantly. The first characteristic makes the system more balanced in all directions, and the second one reveals good scaling. Some considerations regarding closed-loop spectral sensitivity will be made in the paragraphs that follow. The
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1885 degree of misalignment is plotted in Figures 5b and 6b. The horizontal line at 1.41 represents the upper bound this parameter can reach for 2 X 2 matrices. This quantity corresponds to the highest degree of misalignment. The system is aligned when the index tends to zero. The controllers drastically reduce its value. A matrix that is nearly aligned is almost normal, and normal matrices mean well-conditioned eigenvalue problems. The plots reveal that the controllers designed via the process inverse yield lower values, which means better conditioning and thus less sensitivity to modeling errors. Column 2 under control exhibits the best performance (Figure 6b). The closed-loop graphs for column 1 (Figure 5b) show a peak of higher degree of misalignment located at about 0.7 or 0.8 rad/min and a zone of 0.6 deg a t the interval ranging from 0.1 rad/min to 0.9 rad/min for normalizers. Controllers based on the process inverse only left two peaks at medium and high frequencies. As normality is a sufficient condition for spectral sensitivity, nothing can be concluded with regard to sensitivity a t those places. Another tool that might give insight into the topic of spectral sensitivity is the use of sensitivity indexes (Postlethwaite, 1982). Very low indexes indicate that the characteristic gain loci are very sensitive to small disturbances either before or after the plant. In such cases, little changes can upset the system so much that it goes unstable although the nominal plant shows acceptable stability margins. Column 2 shows an index of about 0.85 for most frequencies within range, which again means good spectral sensitivity characteristics. In brief, it was not possible to design a controller that normalizes the closed-loop system for all frequencies. However, better results were obtained through the controllers designed via the process inverse. Insofar as interaction is concerned, column 1 does not improve sensibly after adding the normalizer controller (Figure 5d). Angles remain higher than 15 deg at all frequencies. A t frequencies lower than 0.1 rad/min, decoupling is almost total for controllers designed via the process inverse, which exhibit better results again. For column 2, Figure 6d confirms the small degree of interaction that the system shows under control. For the closed-loop systems, the values for the global and total interaction angles differ greatly: the global angle diminished whereas the total angle experienced little variations. The latter offers a more conservative index. The gap between them becomes evident as the plant reaches normality. Finally the measures of robustness are considered. For column 1,the upper bounds for the norm of both additive and multiplicative errors are amin = -10 db (normalizer controller) and dmin = -15 db (controller based on the process inverse). The requirements of good sensitivity and robust stability are compatible for frequencies from 0.01 to 0.1 rad/min. For normalizers, the peaks of lowest 6 coincide with the regions of high degree of misalignment. For controllers designed via the process inverse, the peak of highest misalignment occurs at 0.7 rad/min and the peak of lowest 6 is situated at 0.95 rad/min. These are zones of robust stability with uncertain sensitivity. In short, controllers designed via normalization admit a wider range of disturbances. The results of the closed-loop simulations (Figure 7) revealed that the normalizer controller was slower than the one designed via the process inverse. Therefore, it is natural to expect the former to be more robust. Then, the choice of only one structure is a compromise situation. As slow responses should be avoided, we decided for the controller designed via the process
inverse that admitted a reasonably great amount of uncertainties and has better speed of response. For column 2, the lower bounds for the singular values of the return difference operators are never lower than 0 db for both kinds of errors, thus meaning more robust controllers than those designed for the prefractionator. In terms of these bounds, the 4 X 4 structure was found to even more robust than the 2 X 2 submatrix for column 2, thus assessing an advantage of controlling both columns.
Conclusions It was possible to design effective controllers for the train under study by means of the methodology described here. The frequency domain techniques and the linear simulations were quick, efficient tools for a preliminary analysis. The use of a controller at the prefractionator would make the system more robust and proves to be particularly useful when the train is expected to suffer from severe changes. As the main disturbances affecting column 2 were the variations in its two feed flow rates, it might be worthwhile to try a feedback/feedforward scheme that employed those rates as measured variables for the anticipatory action. Another possibility is to work on a 4 X 4 joint MIMO design that would also take into account the block that represents interaction between the columns at the design step. These are both interesting suggestions for future research. Nomenclature DR = distillate rate E = matrix space F = return difference operator (RDO) for the nominal matrix Fd = RDO for the perturbed matrix G = open-loop nominal transfer function matrix H = closed-loop transfer function matrix k = spectral condition number of G ki = ith element of a diagonal matrix K = controller matrix Lo = reflux rate, kmol/h Lo = dimensionless reflux rate m = degree of misalignment P, = characteristic equations for the poles of ai P = vector of procedures Q = open-loop transfer function matrix that includes the controller QB = reboiler heat duty, kcal/h Q B = dimensionless reboiler heat duty ri = orders of the infinite zeros of G ( s ) s = particular frequency s, = ith sensitivity index SD = side-stream drawoff t = time T, = temperature at the ith tray Ti= dimensionless temperature at the ith tray U(s) = unitary matrix of left singular vectors V(s) = unitary matrix of right singular vectors W i= ith rotation matrix x , = characteristic equation for the poles of the ith closed loop y = vector of differential variables y i = ith normalized left eigenvector of G z = vector of algebraic variables zi = ith normalized right eigenvector of G Greek Symbols a, = characteristic parameter for Fd p = eigenvalue of G tB 6 = upper bound for the norm of errors A = uncertainty 4 = global angle of interaction y i = numerator of the ith element of a diagonal matrix for
+
s-m
1886 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
6 = total angle of interaction A, = ith eigenvalue = additive perturbation for the RDO ui = ith singular value Z = diagonal matrix of singular values ri = ith time constant .$ = magnitude of the perturbations Symbols ll~= l ~ Frobenius norm 1 ~ - I l 2 = spectral norm Subscripts d = perturbed PI = proportional-integral normalizer controller - = minimum (under greek symbols) - = decoupled part of a matrix (otherwise) (Y = particular frequency Superscripts SS = steady state T = transpose - = maximum (under greek symbols) - = coupled part of a matrix (otherwise) * = conjugate transpose
+
Appendix A: System Analysis through Modern Frequency Domain Techniques The evolution of certain parameters in the frequency domain provides insight into the quality of a multivariable system. The performance of controllers of different degrees of sophistication can be tested via quantification of the degrees of interaction, sensitivity, and robustness. One of the main features of this approach is that the overall aspects of the system under study are expressed in terms of gain and phase parameters, associated with graphic displays. The design, control, and simulation package DACSIM,developed at PLAPIQUI (Rotstein et al., 1986), is used for this purpose. A brief description of each parameter will be given next. Principal Gains. The singular value decomposition (SVD) of a matrix G(mXm) results in three component matrices as follows: G(s) = U(s)Z(s)V*(s) (Al) where U(s)is the unitary matrix of left singular vectors, V(s) is the unitary matrix of right singular vectors, and Z(s) is the diagonal matrix of scalars called the singular values or principal gains Z(s) = diag lal(s), ~ ( s )..., , a,(s)l (A21 with Ul(S)
> az(sd) > ... > a&) > 0
The SVD yields insight into the major and minor control directions. V provides a prerotation of the inputs, Z represents a scaling of the transferred variables, and U may be viewed as a postrotation for the outputs. In this way, the singular values provide the ideal decoupled gains of the open-loop process. If the multivariable interactions were removed by an ideal decoupler, the singular values would be the open-loop gains of the noninteracting loops. The ratio of the largest singular value to the smallest, ul/u,, is the condition number, and it is a measure of the difficulty of the control problem. The condition number is greater than or equal to 1. Strictly speaking, this quantity is an invertibility index (Stewart, 1973). Its minimization is a usual control objective (Doyle and Stein, 1981). Misalignment and Sensitivity Indexes. A matrix is normal if and only if it commutes with its conjugate
transpose. If the eigenvalues of G are X and those of G + [B and
H-'GH = diag [A)
(A31
then
IA - PI
I EK(H)IIBII~
(444)
k(H) = llH-111211HIIz (A5) As B can be interpreted as a small change in G, the theorem shows that the overall sensitivity of the eigenvalues of G depends on It, the spectral condition number of G. For normal matrices It = 1; therefore, normal matrices mean well-conditioned eigenvalue problems because their eigenvalues are relatively insensitive to disturbances. Approximately normal matrices are also well-conditioned. The concept of misalignment (associated to the SVD theory) can be interpreted as a measure of the departure from normality because if G is aligned and then G is normal. Moreover, if G has distinct singular values, then the converse also holds. Two unitary matrices U and V are aligned if there exists a matrix Y = diag {exp(jri)] (A6) such that
u*v = Y If U and V are the matrices of singular vectors of G, the singular vector frame misalignment of G(s) is defined as m(G) = sup min IIU*V - diag (exp(jyi)lllz (A81
UVY In the general case, 0I m(G) I 2
and for 2
X
(A91
2 matrices 0 Im(G) 5 2lI2
6410) The closer to zero m(G) becomes, the more aligned G is. Although normality is an adequate structural measure for the conditioning of the eigenvalue problem, it is only a sufficient condition for small spectral sensitivity. Spectral sensitivity may also be measured by means of the sensitivity indexes (Postlethwaite, 1982). Let ziand yi be the right and left normalized ith eigenvectors associated with the ith eigenvalue Xi. Then the sensitivity indices are si = lziTyii
i = 1, ..., m
(All)
with (-412) For normal matrices, si = 1, i = 1, ..., m, and the eigenvalues are least sensitive. The inverse does not necessarily hold. If sitends to zero, the sensitivity of Xi increases, and then, the eigenvalue problem is worse conditioned. Angles of Interaction. In general terms, interaction means the influence of the ith input on the ith outputs for i # i. Naturally, a designer may aim at reducing the interaction. In other words, he may wish to make the ith output respond to the ith input alone, without disturbing the other outputs significantly. If y(s) is the vector of outputs and u(s) is the vector of inputs and Y(S) = G(s)u(s) (A13) then to require low interaction is equivalent to saying that the off-diagonal terms of G(s) should be smaller than the OISl51
Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990 1887 diagonal ones. This comparison of elements is just one way of measuring the degree of interaction. Another tool for the same purpose is the use of the angles that will be defined below. Let G belong to-the space E. Let us divide E in two subspaces E a n d E. G belongs to L is and only if G is decoupled. Otherwise, G belongs to E. Let us call c t o the projection of matrix G on the decoupled subspace The interaction can naturally be interpreted as the angle between the matrix and its projection on a noninteractive subspace. From well-known properties of orthogonal projections, it is given by cos 4 = IlGll/llGII
(A141
This angle, which is called the global interaction angle is an appropriate assessment of interaction. Lau et al. (1985) show how to relate this angle to the SVD. The SVD of a matrix G can be written as n
Wi
G(s) = i=l
where Wi is the rotation matrix. The concept of rotation matrices Wi leads to the following definition of the total angle of interaction
c
ai2
i=l
where
Wi is the ideally decoupled part of Wi. Angles close to zero mean a noninteractive system. As they tend to 45 deg, the system is more and more interactive. It can be proved that 4 S 8; thus, the total interaction angle gives a more conservative estimation of the degree of interaction. Robustness. A feedback control system designed to have desirable properties when the plant behaves according to the nominal model should retain these properties in spite of uncertainties. Assessing the stability is crucial for the design. The robustness tests used are based on a generalization of the Nyquist stability criterion (Postlethwaite and Foo, 1985). Two sets of disturbed plant models have been studied: additive and multiplicative errors. Though they are not unique, they are adequate representations of a wide number of possible plant uncertainties. Let Q = GK be the open-loop transfer function matrix and let us define H = (I + Q)-'Q as the nominal closedloop system. G represents the nominal open-loop model and K is the controller. If H is stable and the disturbed open-loop operator for additive errors is Qd=Q+A
(A18)
+
Q, and it is, at the same time, the upper bound for the maximum singular value of the perturbation A. al(A) is proportional to the magnitude of the disturbance. Thus, if a,(I + Q) is calculated, then 6 can be conservatively taken as equal to this value. Then, 6 can be interpreted as an index of the highest admissible perturbation in terms of singular values according to inequality (A18). The higher 6 is the better. For multiplicative errors, Qd
with Q~(A) 6
Hd will remain stable for all A if and only if u,(I + Q-') > 6 The quantity plotted is a,(I + Q-l), which is taken as equal to 6. b is related to the maximum admissible multiplicative perturbation in the way explained above. The orders of G and K make Q increase as s 0. Therefore, plots for additive errors show high values for low frequencies. For s m, (GK) 0 and a,(I + Q) a,(I) = 1 (0 db). Q-' behaves in the opposite way; therefore, a,(I + Q-l) tends to 0 db for low frequencies and augments for high frequencies. No robustness problems are encountered in these regions. Attention must be focused on the intervals of intermediate frequencies where the peaks of minimum 6 reveal the restrictions imposed on the norm of the perturbation. In short, this computer-aided analysis approach helps the designer to learn about the inherent qualitities of the system under study with regard to performance and stability. Thus, he can obtain a valuable interpretation that aids him to pose his needs in mathematical terms. Alignment, robustness, and low interaction are possible design specifications. Bearing in mind the desired targets, open-loop and closed-loop responses can be compared in order to evaluate effectively the behavior of controllers.
-
al(A) 5 6
-
-
Appendix B: Multivariable Control Methodologies Quasi-Classical Approach. Normalization occurs at particular frequencies (Hung and MacFarlane, 1983). Normal and approximately normal matrices have nice spectral properties. Therefore, normalizing a system at chosen frequencies may be useful for design purposes. At some particular frequency s, E C, G(s,) has a singular value decomposition given by G(s,) = U,Z,V*, (B1) The skewness (lack of normality) can be annihilated by introducing a constant precompensator
K = V, diag (.)U*,
(B2)
to get
G(s,)[V, diag (.)U*,] = UJZ, diag (-)lU*,(B3)
then the disturbed closed-loop system
G(0) = UoZoVoT (A19)
will remain stable for all A if and only if
(B4)
in which the frames Uo and Vo are orthogonal. The constant precompensator is defined as follows:
+ Q) > 6
(A20) 6 is the lower bound for the minimum singular value of I am(I
-
Thus, the system can be normalized, or standardized, by orthogonalizing its eigenframe. At s = 0, G(0) is real. Then it has a real singular value decomposition
where
Hd = (1 + Qd)-'Qd
= (1 + A)Q
For Is1
-
Ko(kl,kz,...,k,) = Vo diag (ki)UoT G(s) becomes
a,
(B5)
1888 Ind. Eng. Chem. Res., Vol. 29, No. 9, 1990
G(s) = U(s) diag (yi/sri)V*(s)
(B6)
where the y's are real, the ri(s are orders of infinite zeros of G(s), and U(s) and V(s) tend to the orthogonal matrices U, and V,, respectively. The compensator is given by
KJk1,k2,.,.,k,,,)= V, diag (IZJ UWT
(B7)
H(s) = (I
+ GKd)-'GKd = (I + $)-'$
=
diag { a i / ( l+ ai)l (B16)
where it is clear that the zeros of 1 + aidefine the time constants of each control loop. The set of rational functions (ai(s)lcan be defined as follows:
By means of this technique, simple controllers such as (B8) Kp~b= ) Km+ (Ko/s) can be designed. At frequencies 0 and m, G has a decomposition in which U and V are real. At other intermediate frequencies, these matrices are usually complex. The precompensated system G(s,)K, has an orthogonal eigenframe at s = CY for a = 0 and CY 63. If K, is orthogonal, the singular values of G(s) and G(s)K, coincide. For s = 0 in particular, the eigenvalues of G(s)& are given by Zo diag ( k i ) . Methodology Based on the Process Inverse (Agamennoni et al., 1988). The method for design of controllers based on this theory makes use of the relationship between the singular values of the return difference operator and the time constants of the system. The first step is to design an ideal controller that is normal and perfectly decoupled. These characteristics guarantee minimal sensitivity to disturbances, no interaction, and an acceptable degree of robustness and stability. A diagonal matrix of elements ai must be created for this purpose. This method gives the designer freedom to choose the closed-loop time constants T~ because they are the inverse of the coefficients cyi. It is convenient to pick values of ai of the same order of magnitude to ensure the good conditioning of the system. The ideal controller is expressed in terms of frequency response through time. Its order can be reduced by minimizing the sum of the square errors at each frequency so as to get simpler controllers, such as PI or PID. This reduction naturally implies an error that acts in detriment of the quality of the resulting controller. Therefore, the designer faces a compromise situation between simplicity and accuracy. The RDO for a multiple input-multiple output system is F=I+GK (B9)
where xi(s)and pi(s)are polynomials in s. As the diagonal elements of H(s) become
-
When the additive perturbation AK appears, the new controller is Kd = K + AK The RDO for the perturbed controller can be expressed as Fd = F(I $) This expression can be arranged to give
AK = (G-l
+ K)$
0312)
The new controller Kd modifies the singular values and the determinant of the RDO according to U i ( F d ) = (1+ ail~i(F) det (Fd)= det (F)
n(l + ai)
0313)
1
Further simplications can be made if K is a null matrix, which means there is no control. In this case, Ui(Fd) = 1 1 + ffil (B14) det (Fd) = n(1+
CY,)
1
Thus, the closed-loop matrix becomes
(B15)
the roots of xi(s) are the poles of the ith closed loop and the roots of pi(s) are the poles of ai. pi(s) should have a root in zero because an integrator guarantees null steady-state error. The synthesis is developed by solving an order reduction problem to adjust a low-order controller. Literature Cited Agamennoni, 0.; Desages, A.; Romagnoli, J. A. Robust Controller Design Methodology for Multivariable Chemical Processes. Chem. Eng. Sci. 1988,43 (ll),2937-2950. Albuja, C.; Ruiz, C. IDENTIFY: Manual del usuario. Internal Report; Planta Piloto de Ingenieria Quimica: Bahia Blanca, Argentina, 1985. Brignole, E.; Gani, R.; Romagnoli, J. A. A Simple Algorithm for Sensitivity and Operability Analysis of Separation Processes. Ind. Eng. Chem. Process Des. Deu. 1985, 24, 42-48. Cameron, I.; Ruiz, C.; Gani, R. A Generalized Dynamic Model for Distillation Columns. Part 11: Numerical and Computational Aspects. Comp. Chem. Eng. 1986,lO (3), 199-211. Christiansen, L.; Michelsen, M.; Fredenslund, A. Naphtali Sandholm Distillation for NGL Mixtures near the Critical Region. Comp. Chem. Eng. 1979,3, 535-542. Crespo, N.; Ruiz, C.; Romagnoli, J. A. Modelo D i n h i c o Reducido para Columnas de Destilacibn. Actas X I V J.AADICIQA, Santa Fe, 1987,2, 149-155. Doukas, N.; Luyben, W. Economics of Alternative Distillation Configurations for Separation of Ternary Mixtures. Ind. Eng. Chem. Process Des. Dev. 1978,17, 272-281. Doukas, N.; Luyben, W. Control of an Energy-Conserving Prefractionator/sidestream Distillation System. Ind. Eng. Chem. Process Des. Dev. 1981,20, 147-153 Doyle, J.; Stein, G. Multivariable Feedback Design: Concepts for Classical/Modern Synthesis. IEEE Trans. Autom. Control 1981, AC-26, 4. Elaahi, A.; Luyben, W. Control of an Energy Conservative Complex Configuration of Distillation Columns for Four-Component Separations. Ind. Eng. Chem. Process Des. Deu. 1985,24, 368-376. Frey, R.; Doherty, M.; Douglas, J.; Malone, M. Controlling Thermally Linked Distillation Columns. Ind. Eng. Chem. Process Des. Deu. 1984,23, 483-490. Gani, R.; Ruiz, C.; Cameron, I. Studies in the Dynamics of Distillation Trains. Inst. Chem. Eng., Symp. Ser. 1985, 92, 353-364. Gani, R.; Albuja, C.; Ruiz, C.; Karim, N. Analysis of the Dynamic Behavior of Distillation Trains. Proceedings of DYCORD'86 IFAC, Bournemouth, England, 1986a. Gani, R.; Ruiz, C.; Cameron, I. A Generalized Model for Distillation Columns. Part I: Model Description and Applications. Comp. Chem. Eng. 1986b, 10 (3), 181-198. Hung, Y. S.; MacFarlane, A. G. J. Multivariable Feedback: A Quasi-Classical Approach; Springer-Verlag: New York, 1983. Lau, H.; Alvarez, J.; Jensen, K. Synthesis of Control Structures by Singular Value Analysis: Dynamic Measure of Sensitivity and Interaction. AIChE J. 1985, 31, 427-453. Moore, C. Application of Singular Value Decomposition to the Design, Analysis and Control of Industrial Processes. Presented at the American Control Conference (ACC), Seattle, WA, 1986. Postlethwaite, I. Sensitivity of the Characteristic Gain Loci. Autom a t i c ~1982, 18, 6.
Znd. Eng. Chem. Res. 1990,29, 1889-1893 Postlethwaite, I.; Foo, Y. Robustness with Simultaneous Pole and Zero Movement Across the jw-axis. Automatica 1985, 21 (4), 433-443. Rosenbrock, H. H. Computer-Aided Control System Design; Academic Press: New York, 1976. Rotstein, H.; Desages, A.; Romagnoli, J. A.; Karim, N. Analysis and Control of Distillation Columns: A Quasi-Classical Approach. Proceedings of DYCORD86, IFAC, Bournemouth, England, 1986; pp 215-220. Ruiz, C. Estudio D i n h i c o de las Operaciones de Destilacibn. Magister in Chem. Eng., Thesis, Universidad Nac. del Sur, Bahia Blanca, Argentina, 1984. Ruiz, C. Desarrollo de una Politica de Control para Operaciones de Puesta en Marcha de Columnas de Destilacibn. Ph.D. Thesis, Universidad Nac. del Sur, Bahia Blanca, Argentina, 1986.
1889
Ruiz, C.; Cameron, I.; Gani, R. A Generalized Dynamic Model for Distillation Columns. Part 111: Study of Start up Operations. Comp. Chem. Eng. 1988,12 (l), 1-14. Shinskey, F. G. Distillation Control for Productivity and Energy Conservation; McGraw-Hill: New York, 1977; Chapters 2,8, and 10. Stewart, G. W. Error and Perturbation Bounds for SubsDaces Associated with Certain Eigenvalue Problems. SZAM R e i 1973,15 (4). 727-752. Tyreus, B.; Luyben, W. L. Controlling Heat Integrated Distillation Columns. Chem. Eng. Prog. 1976,59-66.
Receioed for review June 17, 1988 Revised manuscript received May 7, 1990 Accepted May 15, 1990
SEPARATIONS Selective Adsorption of Cationic Surfactants on Cross-Linked Poly(p -hydroxystyrene) Nariyoshi Kawabata,* Koichi Sumiyoshi, and Minoru Tanaka Department of Chemistry and Materials Technology, Faculty of Engineering and Design, Kyoto Institute of Technology, Matsugasaki, Sakyo-ku, Kyoto 606, Japan
Cross-linked poly@-hydroxystyrene) (PHSresin) was found to be an excellent and selective adsorbent for cationic surfactants in aqueous solution. Its adsorption capacity was remarkably higher than those observed with cation-exchange resins and porous resins not possessing an ion-exchange functional group. In addition, ita adsorption capacity was not reduced by the presence of hydrochloric acid, sodium hydroxide, or sodium chloride and other inorganic salts. Elution of the adsorbed surfactants from PHS resin was easily accomplished by treatment with methanol, and the resin was effectively regenerated, in sharp contrast to adsorption on conventional cation-exchange resins. Cationic surfactants were concentrated into about 10 wt % solution in methanol. A significance for the acid-base interaction between the cationic surfactant and the phenolic hydroxyl group of PHS resin was suggested, and the action of an ion-exchange mechanism was excluded as a possibility.
Introduction Previous reports from this laboratory have demonstrated excellent and highly selective adsorption of phenol (Kawabata and Ohira, 1979) and carboxylic acids (Kawabata et al., 1981b) in aqueous solution on cross-linked poly(4vinylpyridine) (PVP resin). The resin showed excellent adsorption capacities for these acidic organic solutes. This phenomenon was explained in terms of an acid-base interaction between the acidic solutes and the pyridyl group of the resin and on the basis of hydrophobic interaction between the solutes and the resin surface (Kawabata et al., 1981a). In addition, since the pK, of pyridine is reported to be 5.17 at 25 "C (Brown and Mihm, 1955), pyridine is a much weaker base than primary, secondary, and tertiary aliphatic amines. Therefore, the adsorption of these acidic organic solutes on PVP resin did not proceed through an ion-exchange mechanism, and the presence of hydrochloric acid, sodium hydroxide, or sodium chloride and other inorganic salts did not reduce the adsorption capacity. As a result, PVP resin showed a highly preferential adsorption of acidic organic solutes over in-
* Corresponding author. 0888-5885/90/2629-1889$02.50/0
organic ions, in sharp contrast to the adsorption of these solutes on conventional anion-exchange resins, for which the capacity of adsorption was conspicuously reduced in the presence of inorganic salts (Kawabata and Ohira, 1979; Kawabata et al., 1981b). On the basis of these observations, basic resins can be classified into three categories (Kawabata, 1989): (A) strongly basic anion-exchange resins containing a quaternary ammonium group, which are effective ion exchangers over a wide pH range; (B) weakly basic anionexchange resins containing aliphatic amino groups, which are effective ion exchangers only at an acidic pH range; and (C) resins containing a pyridyl group, which do not exhibit ion-exchangefunction with inorganic ions and are selective adsorbents for acidic organic materials. According to this concept, acidic resins can be classified into three categories: (D)strongly acidic cation-exchange resins containing a sulfonic acid group, which are effective ion exchangers in a wide pH range; (E) weakly acidic cation-exchangeresins containing a carboxyl group, which are effective ion exchangers only at a basic pH range; and (F)resins containing a weakly acidic functional group, such as a phenolic hydroxyl group, which are expected to exhibit no ion-exchange function with inorganic ions and to behave 0 1990 American Chemical Society
