Ind. Eng, Chem. Rea. 1991,30, 482-490
482
with i = A (H2 or CO), B (H20 or COP),and KI, j = [k1,](1 + 1/KI, j)ttl-'
(A.13)
(A.15)
(A.18) Registry No. Fe, 7439-89-6; hematite, 1317-60-8.
Literature Cited Brown, K. M. Computer Oriented Algorithms for Solving Systems of Simultaneous Nonlinear Algebraic Equations. In Numerical Solution of Systems of Nonlinear Algebraic Equations; Byme, G. D., Hall, Ch. A., Eds.; Academic Press Inc.: New York, 1973;pp 281-348. Hara, Y.; Sakawa, M.; Kondo, S. Mathematical Model of the Shaft Furnace for Reduction of Iron-Ore Pellet. Tetsu to Hagane 1976, 62,315-323. Hughes, R.;Kam,E. K.T. Direct Reduction of Iron Ore in a Moving-Bed Reactor: Analysed by Using the Water Cas Shift Reaction. In Chemical Reaction Engineering-Boston; Wei, J., Georgakii, Ch., Eds.;ACS Symposium Series; American Chemical Society: Washington, DC, 1982;pp 29-38. Kam, E.K. T.; Hughes, R. A Model for the Direct Reduction of Iron Ore by Mixtures of Hydrogen and Carbon Monoxide in a Moving Bed. Trans. Inst. Chem. Eng. 1981,59,196-206. Kaneko, D.; Takenaka, Y.;Kimura, Y.; Narita, K. Production of Reduced Iron by Model Plant of Shaft Furnace. Trans. ISIJ 1982,22,88-97.
Negri, E. D. Doctoral Thesis, Universidad Nacional del Litoral, Santa Fe, Argentina, 1987. Negri, E. D.; Alfano, 0. M.; Chiovetta, M. G. Heat and Mass Transfer in the Modeling of Noncatalytic Moving Bed Reactors and its Application to Direct Radudion of Iron Oxides: A Review. Lat. Am. J. Heat Mass Transfer 1985,9,86129. Negri, E.D.;Alfano, 0. M.; Chiovetta, M. G. Direct Reduction of Hematite in a Moving Bed. Comparison between One- and Three-Interface Pellet Model. Chem. Eng. Sci. 1987, 42, 2472-2475. Negri, E. D.; Alfano, 0. M.; Chiovetta, M. G. Optimal Operating Conditions of the Direct Reduction of Hematite in a Shaft Furnace. Lat, Am, Appl, Res. 1988,18, 93-104. Perry, R,H., Chilton, C. H., Eds.Chemical Engineer's Handbook, 5th ed.; McGraw-Hill: Tokyo, 1973. Reid, R, C,; Prautnitz, J. M,; Shemood, T. K,The Properties of Gases and Liquids, 3rd ed.; McGraw-Hill: New York, 1977. Sen Gupta, A.; Thodos, G. Mass and Heat Transfer in the Flow of Fluids Through Fixed and Fluidized Beds of Spherical Particles. AIChE J. 1962,8,608-610. Takenaka, Y.; Kimura, Y.; Narita, K.; Kaneko, D. Mathematical Model of Direct Reduction Shaft Furnace and Its Application to Actual Operations of a Model Plant. Comp. Chem. Eng. 1986,10, 67-75. Tsay, Q. T.; Ray, W. H.; Szekely, J. The Modeling of Hematite Reduction with Hydrogen Plus Carbon Monoxide Mirtures. Part I-The Behavior of Single Pellets. AZChE J. 1976a, 22, 1064-1072, Tsay, Q. T.; Ray, W. H.; Szekely, J. The Modeling of Hematite Reduction with Hydrogen Plus Carbon Monoxide Mixtures. Part 11-The Direct Reduction Process in a Shaft Furnace Arrangement. AIChE J , 1976b,22, 1072-1079. von Bogdandy, L.; Engell, H. J. The Reduction of Iron Ores; Springer Verlag: Berlin, 1971. Wagner, C. Adsorbed Atomic Species as Intermediates in Heterogeneous Catalysis. Adu. Catal. 1970,21,323-381. Weast, R. C., Ed. Handbook of Chemistry and Physics, 57th ed.; CRC Press: Cleveland, 1976. Yagi, J.; Szekely, J. The Effect of Cas and Solids Maldistribution on the Performance of Moving-bed Reactors: The Reduction of Iron Oxide Pellets with Hydrogen. MChE J. 1979,26,800-810. Yanagiya, T.; Yagi, J.; Omori, Y.Reduction of Iron Oxide Pellets in Moving Bed. Ironmaking Steelmaking 1979,3,93-100. Yu, K. 0.; Gillis, P. P. Mathematical Simulation of Direct Reduction. Met. Trans. B 1981,12B,111-120.
Received f o r review November 21, 1989 Reuised manuscript receiued July 23, 1990 Accepted August 1,1990
Development of a Multivariable Forward Modeling Controller Kelvin T.Erickson* Department of Electrical Engineering, University of Missouri-Rolla, Rolla, Missouri 65401
Robert E. Otto Monsanto Company, St. Louis, Missouri 63198
The Forward Modeling Controller (FMC), a recently developed model-based predictive digital controller for single-input, single-output processes, is extended to multiinput, multioutput processes. The multivariable FMC is a promising approach to the control of complex industrial processes with many inputs and outputs. The theory presented in this paper includes stability analysis plus other features necessary for robustness in industrial control. The controller has only two types of adjustments: a robustness/performance setting for each controller variable and the controller sample interval. The performance of the multivariable FMC is demonstrated on a distillation column simulation and is compared with the performance of the Dynamic Matrix Controller. Introduction complex multiinput, multioutput ( ~ 1 ~chemical 0 ) industrial processes such distillation we *Author to whom correspondence should be addressed.
difficult to control. Manv of these Drocesses have large dead times and unusual djllamics and are often affeccd by persistent disturbances. Automatic control of these Drocesses is usually troublesome due to the interaction inherent in the process, requiring highly skilled operators to maintain acceptable product quality.
0888-6886/91/2630-0482$02.60/0Q 1991 American Chemical Society
Ind. Eng. Chem. Res., Vol. 30,No. 3,1991 483 Modern control theory, such as optimal control, while used frequently in aerospace and aircraft applications, has been used in only a small number of chemical process applications. The lack of applications appears to be due to the inability of modern control theory to deal with the typically imprecise knowledge of the process and disturbance characteristics. In addition, most process control engineers are unfamiliar with modern control theory and therefore tend to adapt traditional control techniques to solve their control problems. The limited successful application of modern control theory to chemical process control motivated the development of digital model-based predictive control algorithms such as Dynamic Matrix Control (DMC) (Cutler and Ramaker, 1980), Model Algorithmic Control (MAC) (Richalet et al., 1978; Rouhani and Mehra, 19821, and Internal Model Control (IMC) (Garcia and Morari, l982,1985a,b). These multivariable digital control algorithms use an impulse reponse or step response model of the process to predict the trend of the process outputs and to compute the required change in the process inputs to bring the outputs to their desired values. These model-based predictive control schemes were formulated to deal with the dead time and unusual behavior of complex MIMO chemical industrial processes. These control algorithms have also been extended to nonlinear systems (Economou et al., 1986) and to constrained systems (Garcia and Monhedi, 1984, Ricker, 1985). Consequently, these control algorithms have been applied to many chemical processes with favorable results (e.g., Wassick and Camp, 1988; Ricker et al., 1989; Van Hoof et al., 1989). Recently, the forward modeling controller (FMC), which circumvents the difficulties of the other model-based predictive controllers, was developed for use on single-input, single-output (SISO) proceases by Otto (1986). In this paper, the basic FMC algorithm is extended for use on multivariable processes. Erickson and Otto (1989) is a condensed version of this paper. A single-point modelbased controller was independently developed by Marchetti and co-workers (1983) and Marchetti (1982). The multivariable FMC algorithm is conceptually similar to other model-based predictive digital controllers but is a computationally simpler algorithm. Unlike the DMC, the MAC, or the IMC, the FMC calculates the controller action to minimize the error at only one future sample (the output horizon) of the predicted outputs. Other model-based predictive controllers minimize the error at multiple samples of the predicted outputs. Therefore, the matrix that must be inverted in the FMC is much smaller than the matrix used in the DMC (Cutler and Ramaker, 1980). In contrast to the MAC, the FMC does not require the on-line solution of an optimization problem or the solution of an off-line Ricatti difference equation for processes with nonminimum phase (Rouhani and Mehra, 1982). The multivariable FMC also does not require the derivation of an approximate process inverse as in the IMC (Garcia and Morari, 1985a). The response models are used directly in the calculation of the controller outputs. In contrast to the recently developed simplified model predictive controller (Arulalan and Deshpande, 1987), the multivariable FMC does not use an off-line optimization procedure to determine the tuning parameters. The multivariable FMC is also similar to the single-step controller of Brosilow and Zhao (1988) but does not factor out the process dead time and then add filtering to get less vigorous control action. The aim of this paper is to present the multivariable FMC for unconstrained processes. The basic FMC can be extended to handle hard constraints
using the approach of Ricker (1985). The feature of the multivariable FMC that makes it unique among other model-based predictive controllers is the use of the output horizon as the only major tuning parameter. Each controlled variable may have a different output horizon, depending on how tightly each is controlled. No external filter factors or move suppression factors are used. A comparison of the operation of the FMC and the DMC in this paper shows equivalent performance and reduced computational burden for the FMC. The remainder of the paper is organized as follows. The transfer function form of the controller is developed in the next section and used to examine the system for stability. Robustness issues, feedforward control, and controller tuning are also discussed in this section. In the section Algorithm Implementation, the algorithm is reformulated to use a prediction of each process output. This is the form of the algorithm that is actually implemented. In the section FMC Performance, the performance of the multivariable FMC is evaluated on a simulation of the Wood and Berry (1973) methanol/water distillation column. Regulatory and servo behavior are evaluated for various values of controlled variable closed-loop settling times and under model-plant mismatch. The performance of the FMC is also compared with the performance of the DMC. The last section of the paper summarizes the advantages of the multivariable FMC. Algorithm Development In the multivariable FMC, the process has q outputs and r inputs and is modeled by a discrete impulse response model with N samples: yi = Hlmi-, + H2mj-2+ ... + H N ~ ~+- di N (1) where yi = [yl y2 ... yJT, a q X 1vector of process output measurements; mi = [mlm2 m,lT, an r x 1 vector of manipulated variables; Hk is a q X r matrix of the kth impulse response coefficient; (hij)kis the kth impulse response coefficient of the response between the j t h manipulated variable and the ith process output; and di = [d, d2 dJT, a q X 1 vector of the current discrepancies between the model and the measurement. An impulse response model of the process is also employed by others (Cutler and Ramaker, 1980; Marchetti et al., 1983; Richalet et al., 1978; Rouhani and Mehra, 1982), and has the following significant advantages: (1) No a priori assumptions about model order, time delay, etc. are necessary. (2) Unusual process dynamics are handled naturally and do not require the specification of model structure. (3) The coefficients of the model can be obtained from simple step response data. (4) The step response model, obtained by integrating the impulse response model, has high intuitive appeal to process operators. ( 5 ) With prediction error identification methods, multivariable moving average (impulse response) models will, in the limit, converge to the true parameter values (Stoica and Soderstrom, 1982). The chief disadvantages of an impulse response model are as follows: (1) Nonminimality of the representation, Le., the relatively larger number of parameters used in the model, compared with the smaller number of parameters associated with a loworder transfer function model. (2) An a priori assumption of the time to steady state; that is, N must be chosen sufficiently large. However, we feel the advantages of an impulse response model far outweigh the disadvantages. It is important to note that the value of di is determined by two separate factors: unmeasured process disturbances and modeling errors. It is impossible to separate the two without making further assumptions.
...
...
484 Ind. Eng. Chem. Res., Vol. 30, No. 3, 1991
The multivariable FMC computes a prediction of the process output, Ti,into the future with the following assumptions: 1. The manipulated variable is held constant into the future: mi = mi+l = mi+2- ...
Figure 1. Conventional control system configuration.
2. The future discrepancies remain at their current value: di = di+l = di+2 = ...
The process output,predictions are given by the following equations (where TJjis the vector of process outputs at the j t h sample in the future, given the information of the current, ith, sample):
Tli = Hlmi + H2mi-, + ... + HNmi-N+l+ di T2i = Hlmi + H2mi+ H3mi-, + ... + HNmi++, + di Tii = (H,
+ H2 + ... + Hj)mi + Hj+lmi-l+ ... +
HNmi-N+j+ di
= Aimi
+ Hj+lmi-l+ ... + HNmi-N+j+ di
TNi= ANmi+ di
,
(2)
where Ai = &HL is the j t h "matrix coefficient" of the process step response. Note that
+
Tji = Tj+li-l Aj(mi - mi-l) + di - di-l = Tjtli-l
+ AjAmi + Adi
(3)
where Ami = mi - mi-l and Adi = di - di-l. Expressed in words, the prediction of the j t h sample in the future is updated from the previous prediction of the j+lst sample by the change in the unmeasured disturbances and the changes in the manipulated variables. We will design a controller that yields at sample i a controller output, mi, which, if it were kept constant from here on, would minimize V,the Euclidean norm of the error between the prediction and the set point, at P sample intervals in the future:
V = [spi- TPilT[sPi- TPi]= [gilTePi (4) where spiand giare the set-point vector and error vector, respectively, P sample intervals in the future. The details of the actual computation of mi are given in the subsequent section. Here, we assume that the controller output is calculated to drive the prediction after P sample intervals in the future to the set point, spi,minus an error vector, epi:
T P i = sPj - e p ,
+ ... + HNmj-N+P+ di
= Apmi + Hp+lmi-l
(5) For systems where the number of manipulated variables is greater than or equal to the number of controller variables (r 1 q), it is theoretically possible to calculate a controller output that makes the error, gr,zero. However, for systems where r < q, or for constrained systems where r 1 q, it may not be possible to achieve the desired set point. For convenience, we define si
sPi
I
- ePi
(6)
The value of si thus represents the set point that can be
I
I
Figure 2. Internal model control configuration.
achieved, given the limitations of the system. Substituting for di from (1)into ( 5 ) , and using (6) si- yi = Apmi + Hp+,mi_,+ ... + HNmi-N+P H1mi-, - ... - HNmi-N(7) Converting to z transforms S ( Z ) - Y(Z) = [(Ap + Hp+lZ-l + ... + HNgN+9(Hlz-' + ... + HNfN)]m(z)(8) In conventional controller terms (Figure 11, the controller transfer function C ( z ) is m(z) = C ( z ) ( s ( z )- Y(z)) Therefore, in conventional controller terms the transfer function of the FMC is C ( Z )= [(Ap + Hp+lZ-' + ... + HN@+') (H1z-l + ... + HNz-~)](-~) (9) where A(-1)indicates the left pseudoinverse of a nonsquare matrix. Of course, if A is square, then the left pseudoinverse is the usual matrix inverse. The output of the closed-loop system in Figure 1 is expressed as y(z) = [I + G(z)C(z)](-')G(z)C(z)s(z) (10) where I is the identity matrix. One way to examine stability of the closed-loop system is to examine the system poles. Obviously, this task is not trivial for (10). However, if we convert the controller to the internal model configuration (Figure 2), stability analysis is simpler. Garcia and Morari (1985a) show that the output of the system in Figure 2 is given by Y(Z) G(z)[I + G,(t)(G(z) - Gm(z))](-l) x G,(z)(s(z) + d(z) (11) Now, if exact process modeling is assumed, then G, is set to G and (11) becomes Y(Z) = G(z)G,(z)(s(z) - d(z)) + d(z) (12) Therefore, the closed-loopsystem is stable if G,(z) and G(z) are stable. We will assume that the process G(z) is stable. Therefore, stability of the controller, G,(z), is sufficient for closed-loop stability. Consequently, for system stability, the controller poles must lie within the unit circle. For the multivariable FMC, G,(z) is developed as follows. From Figure 2 ab) Y(Z) - G,(z)m(z) Therefore 4 2 ) - Y(Z) = S ( Z ) - &z) - G,(z)m(z) (13) Now, if exact process modeling is assumed, then G, is set to G and (13) becomes
Ind. Eng. Chem. Res., Vol. 30, No. 3, 1991 485 S(2) -
y(2) =
S(2)
- a(2) -
(H1Z-l -k ... -k HNz-N)m(z) (14)
Equating (14) and (8) s ( z ) - d(z) = [AP
+ Hp+l~-l+ ... + HNz-N+Sm(z)
(15)
Now
m(z) = G , ( z ) [ s ( z -) &)I
sign on the model gains) by filtering heavily enough. With filtering, the calculation of the manipulated variable (16) becomes m(z) = G,(z)F(z)[s(z) - d(d1 (19) In Garcia and Morari (1985a),the filter, F(z), is diagonal and of the exponential type
(16)
Therefore G,(z) = [Ap + H p + l ~ - + l ... + H N Z - ~ + ~ ] ((17) -~) For system stability, the controller poles (given by the roots of det (G,(t))= 0 in (17) when G&) is square) must lie within the unit circle. Unfortunately, the task of finding the minimum P for stability of the controller and, hence, stability of the closed loop system is not trivial even for a system with the same number of manipulated variables and system outputs. Note that if P is picked equal to N (steady-state control), stability is guaranteed if the modeling is exact and AN(-1)exists. We have not yet been able to formulate stability theorems as for the SISO FMC (Otto, 1986). However, a stability analysis procedure for square and nonsquare systems is presented in the paper by Raney and Erickson (1989). Other than the stability theorems proven by Garcia and Morari (1985a),there are no known stability results for other MIMO model predictive controllers. However, results are reported for a SISO DMC by Balhoff and Lau (1985) and for a SISO Simplified Model Predictive Controller by Arulalan and Deshpande (1987). The preceding development assumed that P, the output horizon, was the same for each controlled variable. However, each controlled variable may have a different value of P, depending on how tightly each one is to be controlled. To accomodate different values of P for each process output, we redefine TP as TP = [Tpl TP2 ... TPqIT where Pi is the output horizon for the ith process output. Similarly, AP is redefined as
L
where ajkPiis the step-response model coefficient at output horizon Pi of the model between the kth process input and the jth process output. The expression for the controller transfer function (17) becomes similarly complex and is not shown here. The minimum value of each Pi can be calculated with the use of stability analysis (Raney and Erickson, 1989). In general, increasing the output horizon results in smaller control moves. Filtering for Robustness. If modeling errors exist, stability cannot be guaranteed even for all Pi= N . Furthermore, even if one had exact modeling, a policy of setting Pi = N may produce a controller that moves the manipulated variables too vigorously when large amounts of noise are present in the measurements. We need to modify the controller to produce robustness (tolerance to modeling errors) and noise rejection. Garcia and Morari (1985a) have shown that if a filter, F(z), is added to the controller input, the closed-loop system can be made stable for arbitrarily large modeling errors (other than the wrong
F(z) = diag
1-,
05
ai
1, i = 1, ..., 4
Similar to Otto (1986) we assume the filter, F(z), is diagonal and of the form
F(z) = diag = [I
where as
1
+ fi - fiZ-1 ,
+ @ - @z-’]-’
fi
> 0, i = 1,
..a,
4
(20)
= diag Ifl f2 ...f q / . As in Otto (1986), fi is picked I
When a particular Pi is larger than the length of the step response model, the ith controlled variable is filtered and provides robustness to modeling error. Therefore, adjustment of the output horizon, P, can move the controller from high-performance control to noise-rejecting,sluggish, robust control. Feedforward Control. Feedforward control is easily accommodated by the MIMO FMC. Given a model of the effect of a vector of measured disturbances, x, on the process outputs of the same form as the discrete impulse model used for the process yi = H’IXi-1 + H’2~i-2+ ... + H’Nxi-N (22) where the matrices, H’j, are the appropriate dimensions. Under the additional assumption that measured disturbances remain at their present value, the prediction vector at the j t h sample in the future is Tji = Ajmi + Hj+lmi-l+ ... + HNmi-N+j+ di + A ’ j ~ i+ H’j+i~i-i + ... +H’Nxi-N+j where A’j = Ci=lH’kis the jth “matrix coefficient” of the measured disturbances step response. With this change, (3) becomes Tii = T’+li-l+ AjAmi + A’jAxi + Adi (23) Therefore, the only change to the control algorithm is to update the process variable predictions with the expected effect of the measured disturbances. No other change to the control algorithm is necessary. Controller Tuning. The multivariable FMC has only two types of adjustments. The above development focused on the output horizons, Pi, which smoothly take the control action from (1)extremely aggressive control by setting all Pi to their minimum values, to (2) steady-state control where the controller moves the manipulated variables only to statically compensate the process, to (3) extremely sluggish, noise-rejecting, robust control which should be stable in all practical situations. There is only one output horizon for each controlled variable. In addition to the output horizons, the controller has only one other tuning parameter, the control interval, At. The number of points in the model, N , is usually fixed.
486 Ind. Eng. Chem. Res., Vol. 30, No. 3, 1991 TP+=
+I -
TP Figure 3. Forward modeling controller configuration.
For ease of communication with the engineer commissioning the loop and with operating personnel, the two types of parameters can be renamed: (1)the closed-loop settling time of the controlled variable, PiAt, and (2) the open-loop settling time of the process, N A t . In practice, the open-loop settling time of the process is rarely changed. The closed-loop settling time is approximate (except for all Pi = N with exact modeling) but is useful for explaining the tradeoff between loop performance and robustness. Decreasing the closed-loop settling time tends to make ,the system less tolerant to modeling errors. This feature has strong intuitive appeal and should be well received by operating personnel. Algorithm Implementation If the controller were implemented in the form of (171, the controger output would be based on the current error signal, s - d, and the past values of m. By themselves, the past values of m are not useful to process operators. An indication of where the process is going is far more useful to process operators. Therefore, the controller is implemented to recursively process a prediction of the future trends of the process variables. No past values of m are needed. When a predictor is used within the control loop, the structure of the controller changes to the configuration shown in Figure 3. The new configuration is called a forward modeling controller. The FMC algorithm implements the following steps at the beginning of a new control cycle: A. Predictor: Update the process variable predictions with disturbance and feedforward corrections. B. Selector: Select the prediction vector, TP+. C. Controller: Compute the changes in the manipulated variables to minimize the error between the set point and TP,. D. Predictor correction: Correct the predictions for the effects of the changes in the manipulated variables. Each of the algorithm steps is explained as follows. Predictor. The process variable predictions, one prediction for each controlled variable, are updated by using all available current information and assuming that the manipulated variables, measured disturbances, and unmeasured disturbances remain at their current values. The change in the value of the disturbance at sample i is calculated as the difference between the actual output and the predicted output Adi = yi - T'i-1 (24) An intermediate set of predictions is calculated as Tj+ = Tj+li-l + Adi + A'jAxi for j = 1, 2, .,,, N
TN+1+= TN, (25) At this point in the algorithm, the prediction contains all known effects on the process variables, except that caused by moving the manipulated variables. Selector. Since each controlled variable may have a different value of P depending on how tightly each one is to be controlled, the vector Tp+is formed from the predictions as
Tpl+ ... TPq+IT
[TP1+
(26)
where Pi is the output horizon for the ith process output. Controller. Ignoring filtering, the change in the manipulated variable vector, Ami, is calculated so that, if it were held constant, the norm of the error between the prediction and set point at P sample intervals in the future is minimized. The Euclidean norm of the error, V, is defined as V = [sPi- TPJT[sPi- TPi]= [ePilTePi (27)
si,
Using (3), (24), and (25), the error vector, can also be expressed as epi = spi - Tpi = spi - (Tp++ ApAmi) (28) where AP is defined by (18). So, the calculation of Ami can be expressed as the minimization problem: min V = min [sPi- TP+- ApAmilT[sPi- Tp+- ApAmi] Ami Ami (29) The unconstrained solution to this problem is found by solving
-av aAmi
a2v > 0 - 0 and dam?
which gives a change in the manipulated variable Ami = (ApTAp)-lApT(sPi - Tp+) = Ap(-l)(sPi - TP,)
(30)
If AP is square, then Ami = AP-'(sPi- TP+) For robustness and noise rejection, we add filtering to the controller. With filtering, the change in the manipulated variable is Ami = AP(-l)[I+ @]-l(spi- Tp+) (31) where
[I + @]-l= diag
:1.1
11
and f i is chosen according to (21). Note that the matrix inversion in (31) only needs to be performed when an output horizon, Pi, is changed. Increasing the output horizons generally results in smaller control moves. The same result may be obtained by adding a penalty term to the objective function (29) min V = min {[sPi- Tp+ - APAmilT[spi- Tp+Ami Ami ApAmi] + AmiTRAmi] where R is a weighting matrix. However, the variation in the weighting matrix needed to produce a change in performance or robustness for one process output is not obvious to operating personnel. On this basis, we selected the output horizons as the tuning parameters instead of a weighting matrix. Predictor Corrector, Having applied the above control action to the process, the prediction is corrected for the effects of the changes in the manipulated variables: Tji = Tj+ + AjAmi for j = 0, 2, ..., N In this step is very important to use the actual Ami applied to the process. If for some reason valves stick or hit limits, the actual valve movement should be used for the correction if it is possible to measure it.
Ind. Eng. Chem. Res., Vol. 30,No. 3, 1991 487 Cooling Water
m1 Feed d
+ &
I----
’ ----@____,
Figure 4. Diagram of Wood and Berry column, 5
0
5
97.51
f
I
‘0
10
20
30
40
Time (min)
’W! Time (min)
1.51
Figure 6. Column response to a set-point change in bottoms composition: (-) Pl = 2 and Pz= 4; (---) Pl = 4 and Pz= 6 (time in minutes). 10
0
20
30
40
Time (min)
the comparison of multivariable control algorithms (Arulalan and Despande, 1987; Garcia and Morari, 1985a,b; Marchetti, 1982). The model for the column is
c
5
lo
10
20
30
40
1
I
Time (min)
n
“1.” 2
!i
‘
.
5
n.&”
....._
1.5 ’0
10
20
30
,
40
Time (min)
Figure 5. Column response to a set-point change in overhead composition: (-) P1 = 2 and Pz= 4; (---) PI = 4 and Pz = 6 (time in minutes).
The above algorithm leaves the predictions with a “best estimate” of the future process variable trends assuming the controller is put in manual operation. The only dynamic data stored between control cycles are the process variable predictions, T,and the last controller outputs, mi. FMC Performance The features and performance of the multivariable FMC are evaluated on a simulation of the Wood and Berry (1973) methanol/water distillation column (Figure 4). This process model is used frequently in the literature for
L
L
where time is measured in minutes. The physical meaning of the variables and nominal operating conditions for the column are given in Table I. As is usual in the literature, an error-free system model is assumed for the simulations. The multivariable FMC was implemented with a sampling period of 1 min. The transfer function model (33) was converted to matrices of impulse response models of 60 samples each. Using stability analysis (Jury, 1964) on the poles of G,(z) in (17), the minimum output horizons are PImin = 2 and P2min = 4. Note that this result is also obtained by Garcia and Morari (1985a) but does not require the selection of a diagonal factorization matrix. The manipulated variable moves were calculated with the algorithm in the previous section. Servo Behavior. The system response to changes in the overhead composition set point to 97.0 for two sets of output horizons is shown in Figure 5. Note that, for both output horizons at their minimum values, the system ex-
488 Ind. Eng. Chem. Res., Vol. 30, No. 3, 1991 Table I. Wood and Berry Column Variable Summary variable descrirttion operating steady state y1 overhead composition 96.25 mol 70methanol yz bottoms composition 0.50 mol '70 methanol m, reflux flow rate 1.95 Ib/min steam flow rate 1.71 lb/min feed flow rate 2.45 lb/min
h
I
dml
Time (min) I
8
96.5
E io A
8
20 30 Time (min)
40 0
10
20
30
40
Time- (mi")
I
1.57
lt/------l ...... ......
a....
.5
0
10
.................
20
30
40
a E u
Time (min)
2Ez.13
10
10
20
30
40
Time (min) h
5 3 I h
3
v
3
"
1
0
10
20
30
40
Time (min) A
Figure 8. Column response to a disturbance change of +0.34 lb/ min, feedforward control: (-1 PI = 2 and P2 = 4; (- - -) P1 = 4 and P2 = 6 (time in minutes).
'E - j ; ; ; ; ; ; ; 1 '0
10
20
Time (min)
30
40
Figure 7. Column response to a disturbance change of +0.34 lb/ min, no feedforward control: (--) PI = 2 and P2= 4; (- - -) P, = 4 and P2= 6 (time in minutes).
hibits deadbeat response. However, the deadbeat response comes at the expense of strong input actions. The input actions are moderated by increasing both output horizons, but at the expense of some interaction at the bottoms composition output. Although not shown, the input actions can also be alleviated by increasing the output horizon for just the bottoms composition, presuming some degradation in control can be tolerated. However, the resulting interaction at the bottoms composition output is more severe than the response shown. As noted in Figure 6, the system responds in a similar manner to changes in the bottoms composition set point. The servo behavior of the system with the FMC is similar to the response of the system with the IMC (Garcia and Morari, 1985a). In fact, the responses for both controllers are identical when each is set for the most aggressive control action. The servo behavior of the system with the FMC is better than the performance of the system with the simplified model predictive controller of Arulalan and Deshpande (1987). For the example used by Arulalan and Despande (1987),the integrated absolute error (IAE) due to a 1% change in the overhead set point is 7.4 for the simplified model predictive controller. Under the same conditions, the IAE is 1.7 for the FMC with the output horizons at their minimums.
Regulatory Behavior. Responses to a disturbance change of 0.34 lb/min feed flow rate are shown in Figure 7. As with the servo response, increasing the output horizon reduces the input actions but causes more deviation at the system outputs. The regulatory behavior of the system with the FMC is similar to the response of the system with the IMC (Garcia and Morari, 1985a). When models of the disturbance effects are added to the multivariable FMC to produce feedforward control, the disturbance effects are completely canceled for the most aggressive control (Figure 8). Increasing the output horizons causes some degradation in the system response. However, the system response is still much better than the most aggressive control without feedforward compensation. Comparison with the DMC. Responses of the system to a change in overhead composition set point with the FMC and DMC are shown in Figure 9. Here, the sampling period is 2.5 min. The DMC tuning parameters are number of predictions included for each output, R = 10; number of future control moves considered, L = 5; filter factor, f = 0.1. The DMC tuning parameters R and L, determined by Marchetti (1982), represent a tradeoff between the integrated absolute error for load and set-point changes. For the FMC, the closed-loop settling time was set to 5 min (two samples) for each process output. There are 40 samples in the step response models for both the FMC and DMC. As can be seen from Figure 9, there is little difference in the response of the FMC and DMC. The IAE is 2.9 for the FMC, and the IAE is 3.0 for the DMC. Similar results were also obtained by Marchetti (1982). The FMC and DMC performance is compared under model-plant mismatch in Figure 10. In this comparison,
Ind. Eng. Chem. Res., Vol. 30, No. 3, 1991 489
a0 4
“
E= -
1
’E
s
-
0o
ta
Time (min)
0
10
‘0
10
20 30 Time (min)
40
30
40
20
za
30
40
30
40
lime (min)
L
$I! l 1*0” r ” -101
20
Time (min)
Time (min)
Figure 9. Column response to a set-point change in overhead composition: (-) FMC, PI = Pz= 5; (- - -) DMC, R = 10, L = 5, f = 0.1 (time in minutes).
Figure 10. Column response to a set-point change in overhead composition under model-plant mismatch (-) FMC, PI = Pz = 5; (---) DMC, R = 10,L = 5, f = 0.1 (time in minutes).
the model in the controller corresponds to (331, but the process is changed so that the gains in the first column of the process matrix in (33) are doubled. This scenario corresponds to a change in the dynamic gain of the final control element for ml.As shown in Figure 10, the response of the system to an overhead composition set-point change with FMC is similar to the response with the DMC. The IAE is 6.3 for the FMC, and the IAE is 7.2 for the DMC. Although not shown, similar results are obtained for bottoms composition set-point changes and disturbances under the same type of model-plant mismatch. Similar results are also obtained under different modelplant mismatch scenarios, for example, mismatch in the gains in one row of the process matrix (33). Although not one of the major advantages, the FMC does have a smaller computational burden than the DMC. T o calculate controller outputs, the DMC must invert a 20 X 20 matrix. The FMC only inverts a 2 X 2 matrix, a considerable savings in computational time. In reality, though, the matrix inversion only needs to be done when tuning parameters or process models change. Apart from the matrix inversion, at every sample period, the DMC must multiply a 2 X 20 and a 20 X 1 matrix to obtain the controller action. The FMC multiplies a 2 X 2 and a 2 X 1 matrix to calculate the control action. All other tasks in the DMC and FMC (e.g., error or prediction update) are comparable, so the overall reduction in computational burden for a nonadaptive system is not significant.
1. Each controlled variable has only one tuning parameter, the output horizon, also called the closed-loop settling time. The adjustment of these tuning parameters can move the controller from high-performance control to noise-rejecting, sluggish, robust control. 2. The minimum value of each output horizon is calculated with the use of stability analysis, providing the operator with an indication of the maximum performance available. 3. Feedforward compensation is trivial. 4. The multivariable FMC is computationally simpler than other model-based predictive digital controllers. 5. Graphical display of the process models and process variable predictions have intuitive appeal to process operators. A comr>arisonof the oDeration of the FMC and DMC under identical conditiois shows equivalent performance for the FMC.
Summary The major advantages of the multivariable FMC include the following:
Acknowledgment Financial support from the Weldon Spring Endowment Fund is gratefully acknowledged.
Literature Cited Arulalan, G. R.; Deshpande, P. B. Simplifed Model Predictive Control. Ind. Eng. Chem. Res. 1987,26, 347-356. Balhoff, R. A.; Lau, H. K. A Transfer Function Form of the Dynamic Matrix Control and its Relationship with Some Classical Controllers. Presented at the American Control Conference, Boston, MA; 1986; paper TP7-230. Broeilow, C.; Zhao, G. Q. A Linear Programming Approach to Constrained Multivariable Process Control. In Control and Dynamic
Znd. Eng. C h e m fies. 1991,30,490-501
490
Systems; Leondes, C. T., Ed.; Academic: New York, 1988, Vol. 27, pp 141-181. Cutler, C. R.;Ramalcer, B. L. Dynamic Matrix Control-A Computer Control Algorithm. Proceedings of the 1980 JACC, San Francisco, CA; American Automatic Control Council: Green Valley, AZ, 1980; paper WP5-B. Economou, L. G.; Morari, M.; Paleson, B. 0.Internal Model Control. 5. Extensions to Nonlinear Svstems. Ind. Eng. Chem. Process Des. Dev. 1986,25, 403-411. Erickson, K. T.; Otto, R. E. Development of A Multivariable Forward Modeling Controller. In Model Based Process Control; McAvoy, T. J., Arkun, Y., Zafiriou, Eds.; Pergamon Press: Oxford, 1989.
Garcia, C. E.; Morari, M. Internal Model Control. 1. A Unifying Review and Some New Results. Ind. Eng. Chem. Process Des. Deu. 1982,21, 308-323. Garcia, C. E.; Morari, M. Inteknal Model Control 2. Design Procedures for Multivariable Svstems. Ind. Ena. - Chem. Process Des. Dev. 1986a, 24,472-484. Garcia, C. E.; Morari, M. Internal Model Control. 3. Multivariable Control Law Computation and Tuning Guidelines. Znd. Eng. Chem. Process Des. Dev. 198513, 24, 484-494. Garcia, C. E.; Morshedi, A. M. Quadratic Programming Solution of Dynamic Matrix Control (QDMC). Presented at the America1 Control Conference, San Diego, CA; 1984, paper TA4-1000. Jury, E. I. Theory and Application of the Z-Transform Method; Huntington: New York, 1964. M. rchetti, J. L. Predictive Compute :Control of a Distillation Column. Ph.D. Dissertation, University of California, Sank Barbara, 1982.
Marchetti, J. L.; Mellichamp, D. A.; Seborg, D. E. Predictive Control Based on Discrete Convolution Models. Ind. Eng. Chem. Process Des. Deu. 1983,22,48&495. Otto, R.E. Forward Modeling Controllers: A Comprehensive SISO Controller. Presented at the AIChE Annual Meeting, November, 1986.
h e y , A. D.; Erickson, K. T. Stability Analysis of a Multivariable Model-based Predictive Controller. In Proceedings of the 27th Allerton Conference on Commdcation, Control and Computing, Monticello, IL; University of Illinois: Urbana-Champaign, 1989; pp 243-252. Richalet, J.; Rault, A.; Testud, J. L.; Papon, J. Model Predictive Heuristic Control: Applications to Industrial Processes. Automatica 1978,14,413-428. Ricker, N. L. Use of Quadratic Programming for Constrained Internal Model Control. Ind. Eng. Chem. Process Des. Deu. 1986, 24,925-936.
Ricker, N. L.; Subrahmanian, T.; Sim, T. Case Studies of Model Predictive Control in Pulp and Paper Production. In Model Based Process Control; McAvoy, T. J., Arkun, Y., Zafkiou, Eds.; Pergamon Press: Oxford, 1989. Rouhani, R.; Mehra, R. K. Model Algorithmic Control (MAC);Basic Theoretical Properties. Automatica 1982, 18, 401-414. Stoica, P.; SGderstr6m,T. Uniqueness of Prediction Error Estimates of Multivariable Moving Average Models. Automatica 1982,18, 617-620.
Van Hoof,A. H.; Cutler, C. R.;Finlayson, S. G. Application of a Constrained Multi-variable Controller to a Hydrogen Plant. In Proceedings of the 1989 American Control Conference, Pittsburgh, PA; American Automatic Control Council: Green Valley, AZ, 1989; pp 1555-1560. Wassick, J. M.; Camp, D. T. Internal Model Control of an Industrial Extruder. In Proceedings of the 1988 American Control Conference, Atlanta, GA; American Automatic Control Council: Green Valley, AZ, 1988; pp 2347-2352. Wood, R, K.; Berry, M. W. Terminal Composition Control of a Binary Distillation Column. Chem. Eng. Sci. 1973,28, 1707-1717.
Received for review February 1, 1990 Reuised manuscript received September 11, 1990 Accepted September 24, 1990
SEPARATIONS Extractive Crystallization of Salts from Concentrated Aqueous Solution David A. Weingaertner, Scott Lynn,*and Donald N. Hanson Department of Chemical Engineering, University of California, Berkeley, California 94720
&tractive crystallization is considered as an alternative to evaporation in processes for the purifkation of raw salts. Purified concentrated salt solution is mixed with a selected solvent, and salt is forced to crystallize by the mutual solubility of water and solvent. After removal of the solid salt, the mother liquor of solvent, water, and residual salt is re-formed into a regenerated relatively dry solvent phase and a diluted aqueous phase by change in temperature and contact with additional concentrated solution. Both phases are directly recycled within the process, allowing creation of a continuous, industrially useful process. Solvents can be employed that are either miscible or immiscible with water. Process energy and capital requirements are shown to be substantially below those of conventional multieffect evaporation processes. Detailed examination is made for processes to produce NaCl and Na2C03.
Introduction Many inorganic salts are produced industrially from aqueous solutions. The final production step involves the crystallization of the salt from solution. If the solubility of the salt being produd,changes little with temperature, the crystallization must be effected by removing water. Simple boiling is prohibitively expensive, since typically 2-3 lb of water muat be evaporated to recover 1lb of salt. Energy costs are lowered by the use of multieffect evap-
oration or vapor recompression; however, these processes are still energy intensive. Because large amounts of heat must be exchanged, they are also capital intensive. If the separation of a salt from water could be done without vaporizing water, substantial energy savings would be possible. It is well documented (Seidell, 1940; Washburn, 1928) that the addition of an organic solvent to an aqueous solution can reduce the solubility of some inorganic solutes.
0888-5885/91/2630-0490$02.50/0Q 1991 American Chemical Society
