Ind. Eng. Chem. Res. 1988,27, 956-963
956
V = volume of solution, L Z = adsorbed phase composition, mole fraction Greek Symbols a = constant of two-dimensional van der Waals' equation,
J-m2/mo12 /3 = constant of two-dimensional van der Waals' equation,
m2/mol chemical potential of solute i, J T = spreading pressure, J/m2 II = osmotic pressure, atm
pi =
Superscripts a = adsorbed phase
* = value at very low concentration o = value at single solute
' = value at initial state Subscripts i = solute i T = total
Literature Cited Fritz, W.; Schlunder, E. U. Chem. Eng. Sci. 1974, 29, 1279. Greenbank, M.; Manes, M. J . Phys. Chem. 1981,85, 3050. Hoory, S. E.; Prausnitz, J. M. Chem. Eng. Sci. 1967, 22, 1025. Innes, W. B.; Rowley, H. H. J. Phys. Chem. 1942,46, 548. Langmuir, I. J . Am. Chem. SOC.1916, 38, 2267. Langmuir, I. J . Am. Chem. SOC.1918,40, 1361. Manes, M. Activated Carbon Adsorption of Organics from Aqueous Phase;Ann Arbor Science: Ann Arbor, MI, 1980; Vol. 1, pp 43-64. McCreary, J. J.; Snoeyink, V. L. Am. Water Works Assoc. J . 1977, 69(8), 437-444.
Minka, C.; Myers, A. L. AIChE J . 1973, 19, 453. Myers, A. L.; Prausnitz, J. M. AZChE J. 1965, 11, 121. Radke, C. J.; Prausnitz, J. M. AZChE J. 1972, 18, 761. Rosene, M. R.; Manes, M. J . Phys. Chem. 1976, 80, 953. Rosene, M. R.; Manes, M. J . Phys. Chem. 1977,81, 1646. Ross, S.; Olivier, J. P. On Physical Adsorption; Interscience: New York 1964. Shuval, H. I. Water Renovation and Reuse; Academic: New York, 1977; pp 33-72. Van Ness, H. C. Znd. Eng. Chem. Fundam. 1969,8,464.
Received for review June 9, 1986 Revised manuscript received February 8, 1988 Accepted February 19, 1988
Registry No. C, 7440-44-0; 4-HOC6H4(CHz)zCH3, 645-56-7; 4-HOC6HhCH3, 106-44-5;C6H6, 71-43-2; HO(CH.JiCH3, 71-41-0.
PROCESS ENGINEERING AND DESIGN Predictive Controller Design for Single-Input/Single-Output(SISO) Systems Paul R. Maurath,+ Duncan A. Mellichamp, and Dale E. Seborg* Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, Santa Barbara, California 93106
This gaper presents a fundamental analysis of the stability of single-input/single-output(SISO), closed-loop systems with predictive controllers. T h e analysis can be used t o calculate allowable modeling errors for a given system and controller. Design parameter selection guidelines for predictive controllers in SISO systems have been developed by considering performance, robustness, and ease of tuning. T h e performance and robustness of the resulting controllers are demonstrated in four numerical examples and compared to controllers designed using other parameter choices.
In the late 19705, two new control strategies emerged in the control literature which approached the problem of process control somewhat differently than before. Model Algorithmic Control (MAC) (Richalet et al., 1978) and Dynamic Matrix Control (DMC) (Cutler and Ramaker, 1980) both (i) utilize a discrete convolution-type model to represent the process and (ii) control the process by optimizing the process output(s) over some finite future time interval. These techniques share several characteristics in addition to those mentioned here and constitute a new category of process control referred to as "predictive control". A number of papers have appeared on these and related predictive control algorithms. A complete review is given in Maurath (1985). Predictive controllers have also been examined from the framework of Internal Model Control (Garcia and Morari, +Presentaddress: Procter & Gamble Company, Cincinnati, OH 45201.
1982). Their paper approaches the IMC controller based on impulse response models from a different perspective than DMC, particularly for nonminimum-phase processes. They also made observations on the effects of controller design parameters in the IMC controller which are closely related to the parameters of the DMC-type controller presented here. The parameter effects noted in this paper are consistent with their earlier results. The goal of this paper is to present a series of guidelines which will lead to reasonable starting values for the design parameters in a predictive controller. The present paper is restricted to unconstrained single-inputlsingle-output systems. Maurath et al. (1985) present a more powerful predictive controller design technique for multiinput/ multioutput processes. A brief introduction to predictive control will be presented followed by a discussion of the effects of the controller design parameters on closed-loop performance and robustness. A stability analysis is presented which con-
0 1988 American Chemical Society
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 957 siders plant/model mismatches. Design parameter recommendations along with several illustrative examples of the application of predictive controllers to SISO processes are also included. Development of t h e SISO Predictive Controller The form of the predictive controller which will be used in these studies closely parallels that used in DMC. DMC has been chosen rather than MAC since more detailed information about the control computations in DMC is available in the open literature. The available publications on DMC have presented descriptions of the algorithm but few guidelines on design parameter selection. The effects of the design parameters on the performance of the closed-loop system are not always obvious to the casual observer. The predictive controller is based on the following discrete convolution model of the process: N
Yk = x h i U k - L til
+d
(1)
where Y k is the system output a t the kth sampling time, is the system input at sampling time k-i, and d is the steady-state value o f y when u = 0. The model parameters (hlJare the values of the impulse response for the system in eq 1. They are related to the step response coefficients (a,)by the expression hk = ak - ak-1 (2) In eq 1,N is the number of terms in the sampled impulse response sequence which have been retained. The results of this paper are restricted to linear, self-regulating processes. For such a process, the dynamic model can be represented by a finite sampled impulse response model to any degree of precision required by an appropriate choice for N . The development of the predictive controller formulation is based on the minimization of the predicted error at a finite number (R)of future sampling instants. A more detailed development is available in Marchetti et al. (1983a) and Maurath (1985). The optimization problem can be formulated mathematically by using the following performance index:
J = ETQE where E is a vector of length R and Q is an R
(3)
X R positive-definite weighting matrix. The elements of E are differences between the set point and the predicted output of the process at the next R sampling instants. These predictions can be calculated by using eq 1or an analogous method. The past, current, and future process inputs are required in order to perform these calculations. The changes in the process input a t the next L sampling instants (including the current sampling time) are the independent variables in the minimization of the performance index in eq 3. If eq 1 and 2 are utilized, the effects of past and future inputs on the performance index can be conveniently separated as follows: E = Ef-AAu (4)
where AUk
AU
= [Auk
= uk - uk-1
AUk+l
AUk+2
Ef = ys - yoL
...
(5) A U ~ + L - ~ ]( 6 ~)
(7)
E’ contains the effects of past inputs on the process at the next R sampling instants, y s is a vector of set points for
the next R sampling instants, and yoLis the prediction of the process output using eq 1and 2, assuming that no future control action is taken. The Au vector contains the next L input changes. The AAu term includes the effects of the current and future inputs. A is the R X L “Dynamic Matrix” (Cutler, 1983) and is constructed from the process step response in the following way:
It is important to note that the A matrix contains information about the process dynamics over the next R sampling intervals. Note that complete dynamic information about the process (i.e., al up to a N )is not required in the optimization but is required in the calculation of the error vector E’. The changes in inputs which result from optimal control formulations can often be unacceptably large. In order to prevent excessive changes, weighting of input changes in the performance index has been proposed (Marchetti et al., 1983a). When the above development is utilized, the performance index can be modified as follows:
J = (E’ - A A u ) ~ Q ( E-’ AAu) + fAuTAu (9) where f is a nonnegative scalar called the “move suppression factor”. The resulting optimal control law is Au = (ATQA+ fI)-lATQE’= KE’ (10) A key design decision is the selection of appropriate values for the optimization horizon R, the control horizon L, the move suppression factor f , and the weighting matrix
8.
The control law in eq 10 generates a series of L control movements which minimize the predicted error in the process output at the next R sampling instants. Thus the dimensions of the matrices have physical significance. For the above solution to exist, the A matrix must be of full column rank. This requires that R be greater than L + O/T, rounded up to the nearest integer, where 0 is the process time delay and T, is the sampling period. It should be noted that the choices Q = I and f = 0 result in a one-step deadbeat controller when the minimum value of R is chosen (Marchetti et al., 1983a). If the L control moves computed using eq 10 were implemented without feedback from the process, modeling errors and disturbances would go undetected for that period of time, resulting in a problem for practical applications of the predictive controller. The DMC implementation of predictive control as presented by Cutler and Ramaker (1980) solves this problem by executing only the first move in this calculated sequence. Then a t the next sampling time, a new value of E’ is calculated using the measured process output to correct the predictions and a new L step control sequence is generated. Therefore only the first element of Au , Auk, is ever implemented, and the actual control law can be expressed as auk = KTE‘
(11)
where KT is the first row of the pseudoinverse matrix in eq 10. This method of implementation results in a forward shift in the optimization horizon ( R sampling periods in the future) and the control horizon ( L sampling periods in the future) at each sampling time. The extension of this predictive controller to multivariable systems is straightforward and is presented in Marchetti (1982). It involves partitioning the error, pre-
958 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988
diction, and input vectors and adding additional blocks to the Dynamic Matrix.
Design Parameters for Predictive Controllers Several parameters must be specified to “design” the predictive controller. For a controller of the type described above with move suppression, five parameters must be specified: (1) T,, sampling period; (2) N , model horizon; (3) R, optimization horizon; (4) L, control horizon; (5) f, move suppression parameter. This list assumes that the weighting matrix Q has been specified a priori. For the SISO systems considered in this paper, Q = I will be used. Two groups of investigators have previously considered the effects of design parameters on predictive controllers. Marchetti (1981, 1982) and Marchetti et al. (1983a) have examined the effects of the design parameters of this predictive controller for several example systems. Marchetti et al. (1983a) concluded by noting that small values of both L and R produced good results for SISO systems. Garcia and Morari (1982) have presented the effects of very similar parameters on a version of their Internal Model Controller (IMC). The parameter effects cited in this study are largely consistent with those cited by Garcia and Morari (1982) and Marchetti et al. (1983a). The choices of N and T , art! interrelated and affect the choice of R. The settling time of the system model in eq 1 is NT,. Since it has been assumed that the system is self-regulating, N T , is finite. The model horizon, NT,, must be large enough to reduce the truncation error from any available model to an acceptable level and to adequately represent available experimental data. Use of a settling time which is too small introduces unnecessary modeling error, but N does not directly affect the size or complexity of the optimization calculations, only the process predictions. Small values of N are desirable to reduce necessary computation, but on the other hand, values of T,which are too large result in reduced controller flexibility and impaired disturbance handling. In practical circumstances, N can be quite a large number. If a first-order system is sampled 5 times per time constant and a settling time of four time constants is used, then N = 20. If faster sampling is used, N increases accordingly. The choice of sampling period is an open area of research in the control of sampled systems, and its effect on the performance of predictive controllers has not been a subject of this study. Marchetti (1982) used values of N ranging from 18 to 70 in his simulation studies and experimental applications. The next design parameter to be examined is L , the control horizon. As L is increased, more degrees of freedom are available for controller optimization. This often (but not always) translates into tighter control of the process. Increasing L may result in better control system performance, but a t the expense of larger changes in the manipulated variable and a reduction in the controller’s robustness. The effect of adding one more degree of freedom is greater when L is small; hence, this effect is more pronounced for small values of L. As more degrees of freedom are added to the optimization, more compensating control moves are available, and the controller can force the process harder in the early portion of the response and still compensate with later moves. It is important to recall that, when L < R , the implicit assumption for the controller calculations is that t h e manipulated variable will not change after t h e first L moves. Increasing the optimization horizon R has a generally stabilizing effect on the closed-loop system. Increasing R increases the number of equations which must be satisfied in a least-squares sense by the controller optimization. The
primary reason for using R > L is to avoid the deadbeat result. When R is close to L and both are small, the effects of small changes in R can be dramatic. As R is increased, its impact on the controller decreases. The effect of changing R is even smaller when L is large (greater than about 5). When L is large, enough compensating moves are available that the first control moves in an L-step poliey are most strongly affected by errors early in the optimization integral, while the later moves handle the longer term errors. Since only the first step in the predictive control policy is actually implemented, the closed-loop characteristics of the system are little affected by changes in R when L > 5 and R is large. The move suppression parameter, f, is used in the objective function to weight the changes in the input. Increasing f slows down the closed-loop response and reduces the size of the input changes generated by the controller, thus contributing to increased robustness. It is difficult, if not impossible, to suggest a range of typical values for the move suppression parameter f. A very small value of f (as small as 0.0001 or less) may have a dramatic impact on the process response by restricting the severity of the input changes made by the controller or it may have no effect at all. The selection off can usually be made by trial and error using simulation results to judge whether the controller is too energetic or too sluggish. We suggest initially changing f by factors of 10 in such a screening process. Another technique, principal components analysis (Maurath et al., 19851, can be used to determine appropriate values o f f without repeated simulation. An analogous procedure, “ridge regression”, has been used in least-squares estimation problems over the last 20 years by many researchers (Hoerl and Kennard, 1970). Ridge regression was originally used as a method of “shrinking” parameters in estimated models back to the origin and thus reducing the sensitivity of those values to noise in the data. Significantly, no single method of determining the ”proper” value for the ridge parameter has ever been widely accepted (Dempster et al., 1977),and the use of the ridge parameter has even come under attack as a poor statistical practice (Smith and Campbell, 1980). In the predictive controller optimization procedure, the f parameter has been used in an analogous way to reduce the size of the calculated input changes.
Selection of Control and Optimization Horizons Cutler (1983) presents a method for selecting L, R, and f for the DMC controller. He advocates setting R = L N a n d finding L by increasing it until changes in L have no further effect of the first move of the controller in response to a step change in set point. The main tuning parameter is then the input suppression parameter f which can be chosen as noted above. Computational experience suggests that similar performance can be obtained using many different sets of the three design parameters. Marchetti et al. (1983a) recommend small values of L and R. Do complicated controllers (those using larger values of L and R) enjoy any advantages over simpler (smaller dimension) versions of the predictive controller? The simplest controller results when L = 1. Marchetti (1982) and Otto (1986) have developed predictive controllers using L = 1 and predictions of the future error at a single sampling instant. Garcia and Morari (1982) used the L = 1case for several theorem proofs as a limiting case for small L. The basic implication of their theorems is that small values of L (and large differences between L and R ) contribute to robustness. The gain vector for the L = 1 controller can be written easily in closed form:
+
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 959 1
K T = r [ a l a2 ... aR] i=l
(12)
identical with that in Marchetti (1981) with a change in notation)
ai2
When R is set to ita minimum value (the first step response coefficient that is nonzero), a deadbeat controller results. When R is infinite, the L = 1 controller becomes a steady-state controller; i.e., it makes the single-step input change necessary to bring the process to the new set point. For intermediate values of R, the controller performance can be varied between these two extremes (for processes with inverse responses or underdamped open-loop step responses, the effects are not quite as simple as this). Move suppression is not required to tune the controller since R can serve as a simple tuning parameter. Computer simulations indicate that a reasonable starting value for R in the L = 1controller is one for which aR> 0.5Kpwhere the process gain K , is assumed to be positive. If a controller can be designed for SISO processes using L = 1, which is adjustable from deadbeat control to steady-state control, is there motivation to use L > l ? As L is increased from one to larger values, the controller almost immediately becomes a deadbeat controller, which is very sensitive to process noise or modeling error. Consequently, the move suppression factor, f , must be introduced to make the controller more robust and to slow the response. Thus, the design of the controller becomes more complex when L > 1. One potential benefit would accrue if controllers designed using larger values of L improved the load response of the process. We have not observed in our study that increasing L and using move suppression result in a better controller than using L = 1 and reducing R. But if a particularly fast response is necessary (and a controller with poor robustness can be tolerated), a controller using a larger value of L (2-4) can be more finely tuned using f, a continuous adjustment, than the L = 1controller with the discrete tuning parameter R. Robustness of predictive controllers is an important topic and has been cited by Cutler (1983) as a major advantage. Does a controller with L > 1which uses move suppression have better robustness properties than a comparable L = 1controller? Before we can address that question, a stability analysis is required.
A Stability Analysis for Predictive Control Since predictive control was originally developed by industrial groups, their primary concern was to obtain good system performance. Therefore, theoretical analyses of these techniques have been slow to appear. Marchetti (1981) derived a controller transfer function for a predictive controller. For the special case where R = N - 1, Marchetti et al. (1983b) developed a closed-loop characteristic equation for use in pole assignment. Their analysis assumed an exact process model. In this section, the closed-loop transfer function and characteristic equation will be developed for any predictive controller of the form shown in eq 12, regardless of how the gain vector KT has been calculated. Different convolution models can be used for the process and for the model used to design the process controller, thus allowing the analysis to include the effects of modeling errors. The subsequent derivation assumes that the process is openloop stable and can therefore be represented by a finite sampled impulse response. The details of the derivation are given by Maurath (1985). By use of a classical feedback structure, the controller transfer function can be written as (this equation is
where R
B=
CK, r=l
and e = y - y , z is the z-transform variable, Kr is the rth element of the gain vector KT in eq 11, and h, is the vth sampled impulse response coefficient of the process model used to design the controller and used by the controller for process output predictions. This model will be referred to as the controller model. The process transfer function is (Marchetti, 1981)
where hj is the jth element of the process sampled impulse response. The closed-loop characteristic equation of the system is 0 = 1 + D(z)HG,(z) (15) or R+l
0=1 N-1
s+R
CZ-'[ 9=2
R
+ z-'[ u=2 t=u-1 Kth, + Bhl - 11 + o=s+1
K,,h,
+ B(h, - As)] + z-N[B(hN- h ~ ) (16) ]
This equation contains two convolution models: a "true" p_rocessmodel {hi]and a model used to design the controller (hi). The order of this equation is N. For the case of no modeling error, the order reduces to N - 1. The number of terms in the model, N , in most cases will be fairly large, between 20 and 50. Since the elements of the controller gain vector are complicated functions of the controller design parameters and the process model, it is not possible to directly calculate stability limits as functions of these controller design parameters. However, this development does allow direct determination of whether a particular closed-loop system is stable and what degree of plant/ model mismatch can be tolerated by the controller. Since the roots of eq 16 can be very sensitive to small changes in the polynomial coefficients, the determinations of controller stability reported in this study have been verified via closed-loop simulations as well as the solution of eq 16. Cutler (1983) stated that a stability analysis is not required for DMC because it is so robust. However, the stability analysis presented above is a valuable design tool for determining the degree of robustness of the controller and the trade-off between robustness and closed-loop system performance in the design of the controller. These considerations affect the choice of design parameters for predictive controllers. The robustness characteristics of simple and more complicated predictive controllers can now be compared objectively for several example systems.
Simulation Results Several simulation examples will now be presented to compare the performance and robustness of L = 1 predictive controllers to more complicated predictive controllers. Three numerical examples are presented in Table I. Note that system 3 has a right-half-plane zero which
960 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 e.
Table I. Simulation Examples system 1: G(s) =
e+ 10s + 1
T,= 2
e+ (5s + 115 (1 - 9s) system 3: G(s) = (15s + 1)(3s + 1) system 2: G(s) =
-
T, = 2 L.
T , = 1.5 > 4
ii 0
1 ' 2 . W
1 00
Y
3
2 0
m 0)
u
U
o .so o -
+ 0 W
0 .75
N U
-I
I:
.25
L-1 R-7 L-1 R-5 1-4 R-30 f-0.1
L
a
0
0.
zb
u
L-1 R-7 L - 1 R-5 'L-4 R=30 t - 0
1
4.01
H P
1.
I
0.
3.0
0
1
2
I 3
NORMALIZED PROCESS GAIN
Figure 2. Stability regions for system 1 when the time constant is known.
0 01 0
6
12
!E
24
I 30
TIME
Figure 1. System 1 set-point responses.
produces an inverse response to a step change in input. Figure 1shows closed-loop step responses for system 1 and three predictive controllers. The L = 4 controller response lies between the responses for the two L = 1 controllers. The responses for the two L = 1 controllers demonstrate that increasing R results in a more conservative controller. All three controllers would be acceptable in many situations. This example shows how the L = 1 controller can be easily tuned to provide the desired response by adjusting R. The design of the L = 4 controller required selection of the L, R, and f parameters. Many different combinations of these parameters result in controllers with similar performance characteristics. Also, the design calculations for the L = 4 controller require the inversion of a 4 X 4 matrix while the L = 1 controller can be written in the closed form of eq 12. Figures 2 and 3 illustrate the robustness characteristics of the three controllers for system 1. These and similar figures presented later in this section show the allowable variations in the model used to design the controller (Le., the "controller model") relative to the fxed process model. The location of the process model in the parameter space is designated with a Figure 2 shows the stability limits for the three controllers in the gainjtime delay parameter space for the controller model when the process time constant is known exactly. The slower controller ( L = 1, R = 7) is somewhat more robust than the faster controllers in this parameter space. Figure 3 shows the stability regions in the time constantjtime delay parameter space for the models used to design the controller. Figure 4 shows the closed-loop step responses for system 2 using three predictive controllers, including an L = 1
"+".
> W J
O W CI
+ W 0 M N
-I
I 0
0
1
2
3
NORMALIZE0 T I M E CONSTANT
Figure 3. Stability regions for system 1 when the process gain is known (see Figure 2 for legend).
controller with R = 12. (The design criterion applied to this system would yield R = 15 as a starting value of R.) The responses of these controllers and the size of the input variable changes are quite similar. Note that increasing f results in a more conservative controller, as would be expected. The L = 1 controller exhibits less oscillation in its approach to set point. The oscillation would increase if R were reduced. Figure 5 shows the responses of the
Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 961 1.25r
3.0
1
I
I
I
I
I
1.00U
3 0
/
L-10 R-50 f-0.01 L-10 R-50 f-0.1
-
.~'--'-'-'-
u 3 n H
VI
VI
m
0 0 L
a
20
0
so
40
L-1 R-12 L-10 R-50 t-0.01 L-10 R-50 t-0.1
--.
I
I
..-.-
80
TIME
I
Figure 4. System 2 set-point responses.
I
1
0
I
2
3
NORMALIZED PROCESS GAIN
Figure 6. Stability regions for system 2 when the time constant is known. U
3
a
0
U
J
0
J
m
U
,,-='-s=-
a 3 0
U
0
aL
i
U
V I o
O
VI
In
m
U
L-10 R-50 t - 0 . 1
0
I
-0.25
L
a
- --
iff
L-4 R-30 t - 0 . 5
.-
0.51
-*.* 0
20
40
60
80
I
I
100
TIME
Figure 5. System 2 load response.
0.0 0
closed-loop systems to a step change in load (the load and process transfer functions were assumed to be equal). Figure 6 shows the stability regions for these three controllers in the gain/time delay plane. The stability regions are comparable in size and shape for the L = 1 controller and the more complicated controllers. The shapes of the regions are somewhat similar to those for the first-order system shown in Figure 2. Figure 7 presents closed-loop step responses for system 3 and three predictive controllers. The recommended initial value of R is 15 for the L = 1 controller. Because system 3 is a nonminimum-phase system, the speed of response of the L = 1 controller goes through a maximum as R is increased. For small values of R, the L = 1 con-
14
29
44
59
74
TIME
Figure 7. System 3 set-point responses.
troller is unstable. The lower limit for stability is the minimum value of R that satisfies R
Cui > 0 i=l
For nonminimum-phase systems, the influence of R on system response is different than for minimum-phase systems. Instead of a smooth change in'performance, with performance being improved (at the expense of robustness) as R is reduced, the speed of response of the closed-loop system passes through a maximum as R is reduced from
962 Ind. Eng. Chem. Res., Vol. 27, No. 6, 1988 1.25
1
L-1 R - 1 0
L-3 R-5 1 - 0 . 1
a
L-1 A - 1 0 L-3 R - 5 t-0.1
L -0.25
a
-0.50
-0.25
u
LI
3
0.
P U C
P H u)
v)
Lo
Lo
u
U O
-0.
m
al 0 L
-1.
a
".V--
0
:0
20 TIME
30
40
Figure 8. Set-point responses for simulated stirred-tank heating system.
a very large number. As R is further reduced, the response becomes slower until the system becomes unstable as R becomes less than the stability limit in eq 17. The stability limit for this system and sampling period occur at R = 12. Controller 2 (with R = 20) shows the fastest response for any L = 1 controller. The third controller was designed using L = 4,R = 30, and f = 0.5. For smaller values off, the controller would exhibit a larger inverse response and a marginally faster rise to the new set point. Such a large inverse response (more than 50% of the set-point change) would not be acceptable in many practical situations. In an earlier paper, Marchetti et al. (1983a) presented an experimental application of a predictive controller to a stirred-tank heating system. They designed a controller using L = 3, R = 5,and f = 0.1, with the goal being to keep the ratio LIR small. The simulation results in Figures 8 and 9 compare their predictive controller with an L = 1 controller designed using R = 10. (These closed-loop responses were generated from the convolution model that Marchetti et al. (1983a) used to design their controller.) The L = 1 controller results in a superior set-point response with a slightly slower load response. This example illustrates the effectiveness of the L = 1 predictive controller. It also provides an indirect comparison with a well-tuned PID controller since the controller of Marchetti et al. (1983a) was roughly comparable to a well-tuned PID controller.
Conclusions The design of a predictive controller has been reviewed for any unconstrained stable single-inputlsingle-output linear process. A stability analysis has been developed that allows the computation of closed-loop system stability limits in the presence of modeling errors. The effects of controller design parameter choices on performance and robustness have been examined for several representative systems. These studies have led to recommendations for the controller design parameters that are based on the process dynamics. The use of a control horizon of one
..-0
10
20
30
40
TIME
Figure 9. Load responses for simulated heating system.
sampling period (L= 1)produces an effective controller that can be easily tuned using the optimization horizon R as the principal tuning parameter. Control systems designed using these recommendations were evaluated for four process models and were found to be comparable to considerably more complicated predictive controllers with regard to both performance and robustness.
Acknowledgment Financial support from the National Science Foundation in the form of a Graduate Fellowship for the first author is gratefully acknowledged.
Nomenclature ak = step response coefficient A = Dynamic Matrix B = defined in eq 13 d = steady-state bias D = controller transfer function e=yS- Y E = vector of predicted errors E' = predicted errors based on past inputs f = move suppression parameter G = transfer function hk = impulse response coefficient hk = impulse response coefficient for model HG, = transfer function for process and zero-order hold I = identity matrix J = performance index K = controller gain matrix K T = first row of K K , = rth element of KT L = control horizon N = number of terms in the process model Q = diagonal weighting matrix R = optimization horizon s = Laplace transform variable T , = sampling period uk = input at time k y k = output at time k y s = vector of set points yoL= predicted outputs based on past inputs
Ind. Eng. Chem. Res. 1988,27,963-969 z = z-transform variable
Greek Symbol AU = uk - uk-1
Literature Cited Cutler, C. R. Ph.D. Thesis, University of Houston, Houston, 1983. Cutler, C. R.; b a k e r , B. L. Proc. Am. Control Conf. San Francisco 1980,WP5-B (also presented at 83rd National AIChE Meeting, Houston, 1979). Demuster. A. P.: Schatzoff. M.: Wermuth. N. J. Am. Statist. Assoc. 1i77,72,77. ' Garcia, C. E.; Morari, M. Znd. Eng. Chem. Process Des. Dev. 1982, 2, 308-323. Hoerl, A. E.;Kennard, R. W. Technometrics 1970,12, 55. Marchetti, J. L. M.S. Thesis, University of California, Santa Barbara, 1981. I
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963
Marchetti, J. L. Ph.D. Thesis, University of California, Santa Barbara, 1982. Marchetti, J. L.; Mellichamp, D. A,; Seborg, D. E. Znd. Eng. Chem. Process Des. Dev. 1983a,22, 488. Marchetti, J. L.; Mellichamp, D. A.; Seborg, D. E. Proc. Joint Autom. Con,trol Conf., San Francisco, 1983b, 193. Maurath, P. R. Ph.D. Thesis, University of California, Santa Barbara, 1985. Maurath, P. R.; Seborg, D. E.; Mellichamp, D. A. Proc. Am. Control Conf., Boston, 1985,1059. Otto, R. E. Paper presented at the 1986 AIChE Annual Meeting, Miami Beach, 1986. Richalet, J.; Rault, A,; Testud, J. L.; Papon, J. Automatica 1978,14, 413. Smith, G.; Campbell, F. J . Am. Statist. Assoc. 1980, 75,74. Received for review April 23, 1987 Accepted October 13, 1987
Estimated Effects of Process Variables on Jet Temperature in a U-Gas Reactor J. M. Beeckmans* Faculty of Engineering Science, The University of Western Ontario, London, Ontario, Canada N6A 5B9
J. F. Large Centre d'Etudes et Recherches des Charbonnages de France, BP No. 2, Verneuil-en-Halatte, 60550 France
A steady-state model was developed of the jet in a fluidized bed containing coke and ash particles. T h e combustion of the particles of coke is included in the model, which was used t o predict longitudinal gas and solids temperature profiles for different operating conditions and scales of operation. T h e model was developed t o assist in scale-up design in going from pilot plant to demonstration or commercial plant size and also in extrapolating effects of increases in pressure. The temperature profile in the jet is important because of its effects on the kinetics of agglomeration of ash particles. It was concluded t h a t a scale-up in size and/or pressure in going from a pilot plant t o a full scale plant will result in a significantly higher solids temperature in the jet. The principle of the U-gas process (developed a t the Institute of Gas Technology in Chicago) has been described in numerous publications (e.g., Patel (1980), Palat et al. (1983), and Schora et al. (1985)). A steam/oxygen or steam/air jet is injected a t the base of a fluidized bed having a conical bottom. The bed is composed of a mixture of ash and char particles originating from the devolatilization and gasification of particles of coal in the body of the bed. The char ignites and burns in the jet, providing sensible heat required to balance losses and to drive the endothermic gasification reactions occurring in the bed. The jet temperature is sufficiently high to cause fusion and agglomeration of particles of ash, but the bed itself is below the ash fusion temperature (Mason and Patel, 1980). This feature permits controlled agglomeration of the ash and prevents the formation of large agglomerates and defluidized masses of clinker in the bed, which would prevent it from functioning as intended. Instead, agglomerates which reach a certain size fall into an ash pit through a venturi constriction at the jet inlet and are thus prevented from growing beyond a certain size. At steady state the rate at which inert matter enters with the coal must be in approximate balance with the rate of discharge of ash agglomerates, although some inert material is also recovered in the cyclones. This implies that the rate of agglomeration of ash must be controlled, since too rapid a rate will cause the inventory of fluidizable ash particles in the bed to diminish to the point that the system becomes inoperable. In addition, the rate of ag0888-5885/88/2627-0963$01.50/0
glomeration must be controlled in order to discharge agglomerates having a low carbon content and therefore achieve a high carbon conversion in the gasifier (Mason and Patel, 1980). It is difficult to study precisely the effects of the operating variables, especially the temperature of the jet, on the rate of agglomeration in a gasifier. However, it has been shown that the temperature range between incipient deformability and fusibility in coal ash is of the order of 125 O C (Mason and Patel, 19801, which suggests that the temperature range in the jet of the gasifier for maintaining controlled agglomeration is relatively small. Studies on model systems have shown that the rate of agglomeration of a fusible particulate material in a fluidized bed with a central jet is very sensitive to jet temperature, bed temperature, and cone angle (Arastoopur et al., 1986). For instance, it was found that the rate of agglomeration of polyolefin particles increased 5-fold for an increase in jet temperature of only 3 "C. To date, stable, extended operation of the U-gas process has been demonstrated with various coals a t IGT's pilot plant, which has a capacity of approximately 25 ton of coal per day at a maximum pressure of 5 bar, and in a smaller Process Development Unit which is operable at 20 bar. All commercial reactor would, however, involve considerable scale-up in the size of the reactor and in the severity of operating conditions. In view of the demonstrated sensitivity to temperature of agglomeration kinetics in a fluidized bed, it appeared 0 1988 American Chemical Society
