488
Ind. Eng. Chem. Process Des. Dev. 1983, 22, 488-495
Predictive Control Based on Discrete Convolution Modelst Jaclnto L. Marchettl,: Duncan A. Melllchamp, and Dale E. Seborg’ Department of Chemical and Nuclear Engineering, University of California, Santa Barbara, California 93 106
In several recent studies a discrete convolution representation of a dynamic system has been used to develop new digltal control algorithms based on the predlction of future outputs. This paper presents a fundamental analysis of the concepts involved in designing both predictor and control elements for such algorithms. For single-input, single-output systems, the predicted controller Is equivalent to a dlscrete feedback controller which operates on the current output and past input signals. Simulation and experimental results for several single-input, single-output systems provide a comparison of a predictive control technique with standard PID control. The predictive controller is shown to be quite effective even for systems with inverse response characteristics.
Introduction Advanced control systems are usually designed based on a dynamic model of the process to be controlled. A wide variety of advanced control techniques has been developed based on parametric models such as transfer-function or state-space models (Ray, 1980). An important disadvantage of using a parametric model is that a model structure (e.g., model order) must be assumed. For a process which exhibits unusual dynamic behavior, it can be quite difficult to select an appropriate model structure. In recent years, “predictive control” techniques have been proposed based on a nonparametric model, namely, an impulse response or discrete convolution model. The advantage of using a discrete convolution model is that the model coefficients can be obtained directly from samples of the input and output responses without assuming a model structure. Theoretically, this modeling approach is applicable to any system which can be described by a set of linear, differential equations. In particular, the predictive control techniques are especially useful for processes with unusual dynamics where specifying the model structure for a parametric model would be difficult. Other advantages of predictive control techniques include inherent time delay compensation and the ability to accommodate process constraints. Two predictive control techniques have evoked a great deal of interest recently: Model Algorithmic Control (Richalet et al., 1978; Mehra et al., 1982) and Dynamic Matrix Control (Cutler and Ramaker, 1980; Cutler, 1982). Model Algorithmic Control (MAC) basically involves: (i) an explicit discrete convolution model for system representation and prediction, (ii) a reference trajectory, (iii) an optimality criterion and, (iv) the consideration of state and control constraints. Applications of the method to a number of processes have been reported (Richalet et al., 1978; Mehra et al., 1978, 1982). However, an extensive theoretical analysis of even the simplest case (single-input, single-output system and single-step prediction) has been presented only recently (Rouhani and Mehra, 1982). Dynamic Matrix Control (DMC) exhibits some common features with MAC including both the convolution-based model used to predict the process output and the minimization of a similar performance index. Three DMC papers have appeared so far. Cutler and Ramaker (1980) first presented the technique and illustrated its application
to an industrial furnace-preheater system. Prett and Gillete (1980) described a constrained multivariable version of the controller which was applied to a catalytic cracking unit in a refinery. Cutler (1982) indicated how DMC can be modified for applications to processes which are not self-regulating. For the sake of clarity, the analysis presented in this paper will be limited to single-input, single-output systems without constraints even though the Dynamic Matrix Control and Model Algorithmic Control methods are primarily intended for multiple-input, multiple-output systems with constraints. Since both of these methods are based on a discrete convolution model of the process, the present analysis begins with such a formulation. After development of single-step and multistep predictive relations, the control laws are then developed and analyzed. The relationships between the predictive control laws and existing techniques such as deadbeat control and Dahlin’s algorithm are presented. The effect of the various design parameters on the performance of the predictive control schemes are investigated. Finally, simulation and experimental results are presented to compare the performance of predictive control with conventional PID control. Discrete Convolution Models The derivation of a .single-step, predictive control algorithm is presented below. The algorithm is then compared with two well-known digital controllers based on deadbeat response methods. In the following development, the symbol Qk denotes the predicted value of the output (or controlled) variable at the lzth sampling instant, while Yk denotes the actual output value. Hence, the prediction of the system output at time t = ( K + 1)T8can be written as
where the hi (i = 1, ..., N) are the impulse response coefficients of the process (see Figure l), i.e., eq 1 is simply the convolution summation. A straightforward derivation (Marchetti, 1981) leads to a different, but mathematically equivalent, representation of the predicted output, Qk+l N
where A preliminary version of this paper w a presented ~ a t the 14th Annual AIChE Meeting, New Orleans, Nov 1981. Institute of Technological Development for the Chemical Industry (INTEC), Santa Fe, Argentina.
*
0196-4305/83/1122-0488$01.50/0
hUk
= uk - uk-1
(3)
In order to obtain an accurate estimate of the future system output, the predicted value must be corrected at @ 1983 American Chemical Society
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 489
TS
Figure 2. Block diagram of the feedback control system.
0
T,
3T,
2Ts
4T,
Applying the z-transform technique to eq 12 and rearranging yields the following controller transfer function
5Ts
TIME
Figure 1. Step and impulse response coefficients.
each sampling time. A corrected prediction, Yck+l, can be defined by substituting Y k for j$ in eq 2 or equivalently by (Rouhani and Mehra, 1982) Yck+l
= Qk+l +
(Yk
- 9,)
(4)
In order to calculate uk, the control action to be applied at the present sampling instant, the desired value for the controlled variable at the next sampling instant, ydk+l, needs to be chosen. This can be accomplished by simply selecting the desired set point rk or by using a fdtered value (Richalet et al., 1978)
+ (1- a)rk where a is a filter constant, 0 5 a < 1. ydk+l
= ffyk
(5)
Predictive Control Law from One-Step Prediction If the predicted value is forced to match the desired output of the system after one sampling period, i.e., yck+l = ydk+l, then the resulting control law can be expressed in two equivalent ways (Marchetti, 1981)
or
where Ek = rk - Yk. These two expressions correspond to the two predictive relations in eq 1and 2, differing mainly in computational detail and in the form of the input. Equation 6 is considered to be a special case of Model Algorithmic Control (Mehra et al., 1980; Rouhani and Mehra, 1982),while eq 7, for a = 0, can be seen as a special case of Dynamic Matrix Control (Marchetti, 1981). Suppose that the process model contains an arbitrarily long time delay MT,< time delay I( M + l)TE (8) where M is a positive integer. Then hj = Ofor 1 5 j I M
hj # 0 for j = M
+ 1, M + 2, ...,N
(9) (10)
The effect of the next control action, uk, is not observed in the predicted output until time t = (it + M + 1)TE. Consequently, the requirement that YCk+M+l
= ydk+M+l
yields the control algorithm
(11)
where
HG,(z) = hM+lZ-M-l+ hM+2Z-M-2+ ... + hNYN (14) Thus the predictive control scheme can be represented by the block diagram of Figure 2 which has the following closed-loop transfer function y(z)
-- r(z)
D(z)HG,(z) 1 + D(z)HG,(z)
Substituting eq 13 into eq 15 gives y ( z ) (1- cY)z-M-1 r(z) 1 - az-M-1
--
(15)
(16)
The closed-loop transfer function in eq 16 facilitates comparison of the predictive controller and classical sampled-data aIgorithms such as those obtained from the minimal prototype and Dahlin's design techniques. Comparison with Deadbeat Controllers and Dahlin's Algorithm Minimal prototype or deadbeat control is a type of optimal control developed for sampled-data systems (Luyben, 1973). The basic concept is to specify a desired process response to a specific type of input disturbance, and then to calculate the required controller relationship by substituting the response/input relationship into the closedloop transfer function. Marchetti (1981) has shown that the minimal prototype controller is identical with the predictive controller in eq 13 under certain conditions. In particular, for a step change in the set point, if the output is required to reach the set point in M + 1 sampling intervals, the minimal prototype design method yields the controller in eq 13 when a = 0 (Marchetti, 1981). Martin (1981) has noted that both DMC and MAC behave like deadbeat control under certain circumstances. It is well known that requiring the process output to move from the old set point to the new set point in one sampling interval longer than the dead time is far too stringent a requirement for most real systems (Smith, 1972). Instead of imposing such an unrealistic requirement, Dahlin (1966) suggested that the objective should be to have the closed-loopsystem behave like a continuous first-order system with time delay
Discretizing eq 17 and substituting the pulse transfer function, y ( z ) / r ( z ) ,into eq 15 gives the controller transfer function (1- e-Ts/A)Z-M-l 1 D ( z ) = 1 - e-T./Az-l - (1 - e-T,/A)z-M-l HG,(z) (18)
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Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983
Note that for systems with time delay smaller than T,(Le., M = 01, eq 13 is identical with eq 18 if we set (19) For M # 0, the equivalence of Dahlin’s algorithm and the predictive control method no longer exists. Multistep Predictive Control The single-step predictive method developed above can be extended to include multistep predictions. In this approach, the objective is to minimize the distance between the predicted output trajectory and the desired (or reference) trajectory over the next R time intervals. An R-step predictor can be defined in terms of incremental changes in the manipulated variable (y
= e-Te/X
Ramaker (1980) in the DMC technique. Equation 23 is subtracted from eq 27 to give B = -A‘AU E (28) where A’ and Au are, respectively, the R X R triangular matrix and the R-dimensional vector given in eq 23. The other two vectors are defined by
+
N gk+j
= Qk+j-l
+ &C h=iIA U k + j - i
(20)
for j = 1,2, ...,R where R C N . Feedback information, Y k , allows the prediction to be corrected recursively Yck+j
= gk+j +
(Yck+j-l
- gk+j-l)
(21)
for j = 1, 2, ..., R and y c k = Y k . Finally, substituting eq 20 into eq 21 gives N Yck+j
= Yck+j-l
+ iC=hl i A U k + j - i
(22)
for j = 1, 2, ..., R and y c k = Y k . Equation 22 can be written in a more convenient vector-matrix form by taking the future increments of the manipulated variable out of the summations and rearranging (Marchetti, 1981)
. . . 0 1
0 0
. . . . . .
a1
0 ’
where Ek = rk - y k . Note that both E and E are vectors of predicted errors. The distinction between them concernsJhe future control actions, Auk+j0’ = 0, 1, 2, ...,R - 1). E’is calculated on the basis of past inputs to the system and represents the predicted deviation of the output with respect-to the desired trajectory. However, the calculation of E’ assumes that no disturbances will occur during the time interval [kT,, (K + R)T,] and also, that there will be no control tctions during that same period of time. In other words, E’is a prediction which assumes open-loop behavior dur@ the next R time intervals. The other error vector, E , includes the assumption regarding disturbances, but predicts future errors assuming closed-loop behavior. If a perfect match between the predicted output trajectory of the c l o s e + o p system and the desired trajectory is required, then E = 0 and, from eq 28 0 = -A’Au E (31) The solution to eq 31 is given by
+
where the (ai) are step response coefficients defined by i ai
= Chj j=l
(24)
and i
Pi =
s,
(i = 1, 2,..., R)
E S,
m=l
N
=
hiAUk+,-i
( m = 1, 2, ..., R)
(25) (26)
i=m+l
Desired values for the controlled variable y d k + j 0’ = 1, 2, ...,R ) must be chosen in order to calculate the future control action. The set of values given by Richalet et al. (1978) ydk+j
=
d y k
+ (1- f f j ) r k
(27)
for j = 1, 2, ...,R and 0 5 a C 1 may be taken to be the desired trajectory for the output Y k + j 0’ = 1,2, ..., R). Note that for CY = 0, the trajectory consists of the actual set-point values, a feature that also has been used by Cutler and
Au = (A’)-lE (32) Generally, one would apply only the first control action A u k , observe &+I, correct the predictions, and apply eq 32 again. Thus, at each sampling instant, R future control actions are calculated but only the first one is implemented. The advantage of this procedure is that it keeps the predictions closer to the actual values of the output variable. Moreover, since only Auk (a scalar) needs to be calculated and A’ is a triangular matrix, it is not necessary to solve eq 31 completely but only to solve the first equation corresponding to the prediction Y c k + l . This solution is identical to the control law in eq 7, which was based on a single-step prediction. Unfortunately, the control law in eq 7 tends to be unsatisfactory since it is based on the stringent requirement that the predicted output be driven to the desired value in one sampling period. Two methods have been proposed in the literature to solve this problem. The Model Algorithmic Control technique (Mehra et al., 1982; Richalet et al., 1978) incorporates constraints on the manipulated variable and uses a set of predictions obtained by writing eq 1for R future sampling instants. Since constraints are included, an overdetermined system of equations is obtained. The solution to the problem is found by determining the values of uk+, 0‘ = 0, 1,...,R - 1)which minimize the quadratic performance index
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 R
J [ UI = C bdk+j - Yck+j)2w~ j=l
(33)
The wjare penalty functions used primarily when output constraints are present (Rouhani and Mehra, 1982). In the second approach, Dynamic Matrix Control, an overdetermined system of equations is obtained by arbitrarily reducing the dimension of the A u vector from R to L. Thus only L future control actions are calculated and eq 31 can be written as E=-AAu+E (34) where A, the R X L “Dynamic Matrix”, is defined as the first L columns of A’. The overdetermined system of eq 34 does not have an exact solution. It is possible, however, to obtain the “best solution” in the least-squares sense by minimizing the performance index (Cutler and Ramaker, 1980)
J [ A u ] = @&’
(35)
The solution is AU = (ATA)-~ATE (36) where (ATA)-lATis the pseudoinverse matrix (Penrose, 1956; Greville, 1977). One difficulty with the control law of eq 36 is that it can result in large changes in the manipulated variable (“ringing”)or even unstable responses. These undesirable phenomena occur when the ATA matrix is ill-conditioned. Cutler and Ramaker (1980) overcame this problem by multiplying the diagonal elements of ATA by a number greater than one before performing the matrix inversion. An alternative approach is to modify the performance index by penalizing movements of the manipulated variable
J [ A u ] = @ Q E + AuTRAu
(37)
where Q and R are positive-definite weighting matrices. The resulting control law that minimizes J is
A u = (ATQA + R)-lATQE (38) Note that eq 37 and 38 include weighting matrix Q which allows different penalties to be placed on the predicted errors. Although eq 38 indicates what the next L movements of the manipulated variable should be, normally one would apply only the first control action Auk, observe Yk+l, and then use eq 38 to update the control policy, as discussed earlier. Because only Auk (the first element of A u ) is used, it is convenient to write the control law as uk = uk-1 + h?E
+
e..
Design Parameters In order to implement the prredictive control law in eq 39 a number of design parameters have to be specified including four scalars ( N ,R, L, and sampling interval, T,) and two weighting matrices (Q and R). In the present study, we arbitrarily assume that Q = I which implies that all R predicted errors are weighted equally (cf. eq 37). We also choose R = fl where the scalar quantity, f , serves as a convenient tuning factor for the ’predictive control scheme. Parameter N is the number of Coefficientsthat are used in the discrete convolution model; it should correspond to the settling time of the system expressed as a multiple of the sampling interval. It should also be large enough so that no truncation problems arise in calculating the predicted values using the discrete convolution model. A good rule of thumb is to choose N such that hN is of the order of the error in the measurement of the output variable. Parameter R is the number of predicted values to be used or, equivalently, the dimension of the gain vector I%; L is the number of future control actions that are calculated to reduce the predicted errors. This parameter is also the dimension of the matrix in eq 38 that must be inverted. The selection of R and L depends on the process dynamics, the sampling interval, and the computational facilities available. Small values of L, Le., 2 or 3, allow the controller gain vector K to be calculated by using only a pocket calculator. Note that as L and R become smaller, less computation effort is required to calculate the gains. For instance, if L = 10 and R = 25, then 250 elements of the pseudeinverse matrix must be calculated, but only the 25 elements in the first row are used. However, if L = 2 and R = 5, five of the ten calculated coefficients are required for the gain vector. Moreover, as R is reduced, the computational effort is also reduced since a smaller number of predicted values is required in the calculation of the control action, Auk. The effect of the choice of design parameters on control system performance will be illustrated in the next section. Simulation Results A computer simulation study was performed to compare the performance of the predictive control technique with conventional PID control and to investigate the effects of different design parameters and possible tuning procedures. Three representative process models were selected. system 1: (1 - 9s) (41) Gp(s) = (3s + 1)(15s + 1) system 2:
(39)
where the elements of IP are the elements of the first row of the pseudoinverse matrix. Using a standard z-transform analysis, the controller transfer function, D ( z ) , shown in Figure 2 can be obtained from eq 39 (Marchetti, 1981). D ( z ) = u(z)/E(z) = gO/[(l C1z-l c2z-2 + cN-lz-N+’)(l - z-’)] (40)
+
491
+
where the coefficients go and C, (j = 1, 2, ...,N - 1) are functions of the gains Kj0’ = 1, 2, ...,R ) and the impulse response coefficients. Consequently, although explicit predictions of the process future response are employed, the controller transfer function corresponds to a discrete feedback controller which calculates the controller output from the current error signal and previous controller output siknals. Note that the digital controller in eq 40 contains integral action due to the pole at z = 1.
system 3: (43)
Note that system 1 is a second-order model having inverse response characteristics. Figure 3 compares the performance of the predictive controller and a standard, discrete PID controller for system 1 for a unit step change in the load variable. Ziegler-Nichols settings for the PID controller were used except that the derivative time was reduced to about one-half of the recommended value in order to improve the controller performance. Thus the PID settings for Figures 3 and 4 were K, = 1.04, TI = 16.9, and TD = 2.0.
492
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 18-
..... 0
40
-
....-
L=2 fz0.05
- -0 -L.8,
A
0
f.0.05
Predictive Control Discrete PID Controller
80
120
160
a
TIME
I
I
~
I
~
I
I
L
Figure 3. Comparison of load responses for system 1 ( N = 25, R = 24, L = 10, f = 0, T, = 4).
18
Set-Point Changes, f=O.l
lt 9 -OS
t
Set-Point Changes,
1
-
1
-1.0
0
40
80
120
160
10
TIME
Figure 4. Comparison of setpoint responses for system 1 ( N = 25, R = 24, L = 10, f = 0, T, = 4).
1.0
t-
- t":
0
1
Figure 7. Effect of the number of control actions, L , on control system performance (system 2, T, = 5, N = 25, R = 12, Q = I, R = fr).
1.25
-0.5 0
40
80
120
160
TIME
Figure 5. The effect of time delay on the performance of predictive control for system 2 ( N = 25, R = 24, L = 10, f = 0, T , = 7 ) .
The preference for one response or the other in Figure 3 depends on the criteria used to evaluate such performance. Figure 4,however, shows the clear superiority of the predictive controller method for set-point changes. The simulation results in Figures 3 and 4 were obtained by using filter constant a = 0. The use of LY as a tuning parameter was shown to be of limited utility for system 1 (Marchetti, 1981). Figure 5 shows the load response of the predictive controller for system 2 and four different values of the time delay, t D These results indicate that time delays are easily accommodated using predictive control. The effect of design parameters R and L on control system performance for system 2 is shown in Figures 6 and 7 using the integral absolute error (IAE) as a performance index. Figure 6 indicates that the IAE values for step changes in load are relatively insensitive to the choices of R and L when f = 0.05. However, when f = 0 the IAEvalue decreases as L increases or R decreases. The R dependence is relatively weak for L > 2 and R > 8. Figure 7 provides additional information about the influence of L for both load and set-point changes. When
-0.25
80
40
120
TIME
Figure 8. Effect of the number of control actions, L , on the setpoint response of system 2 (T, = 5, N = 25, R = 12, Q = I, R = 0).
f = 0.1 the IAE values are relatively insensitive to the choice of L. However, for f = 0 (Le., no weighting on the
manipulated variable), the control system performance changes significantly with the choice of L. When f = 0 and L increases, the closed-loop responses tend to become more oscillatory as indicated in Figure 8. The results in Figures 6 and 7 indicate that penalizing the movements of the manipulated variable by choosing f > 0 results in more conservative control action and larger IAE values. The set-point responses in Figure 9 demonstrate that the choice off = 0 provides a better transient response but at the expense of a very large initial move in the manipulated variable. Another serious disadvantage of using f = 0 is that it can result in an ill-conditioned matrix being inverted in the control algorithm of eq 39. When f = 0, then the L X L matrix, ATA needs to be inverted. However, this matrix tends to be ill-conditioned
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983 493
1.5
-1 0
20
40
60 80 TIME
dimension, L
determinant
1 3 5 8 10 15
14.3 0.023 3.0 x 10-5 1.6 x 10-13 3.0 x 1 0 4 9 1.4 x 10-34
1 2 3 4
10 11 12 13
when either large values of L or small values of T,are employed (Marchetti, 1981). Typical results are shown in Table I, where the determinant of ATA is included for several values of L. The ill-conditioning problem can result in severe ringing or even instability. When R I L = 1, A is a square matrix and the solution to the optimization problem is given by eq 32. Then the gain vector I% is given by
... 0 1
hi
1
Table I. nl-Conditioning of Matrix ATA (System 2, t~ = 0,f = 0,R = 20, T, = 5)
i o 0
I
I
I
I
100 120
Figure 9. Effect of weighting the manipulated variable for system 2 (T,= 5,N = 25, R = 12, L = 2).
I%= [
I
(44)
where the dimension of I% is R = L. As was discussed above, the gain vector given by eq 44 leads to a control action which is equivalent to the minimal prototype controller. This theoretical limit for R / L = 1 is shown in Figure 10. The results of the simulation study indicate that using relatively small values of R and L does not significantly degrade control system performance. Furthermore, reducing the size of the R X L “dynamic matrix”, A, greatly reduces the on-line computational requirements since R is the number of predictions that need to be made and L is the number of calculated control actions. Weighting control movements by using f > 0 is recommended since f provides a convenient tuning parameter and the ill-conditioning problems that can occur when f = 0 are eliminated. Experimental Results The results of an experimental application of the predictive control strategy to a stirred-tank heating system are presented below. A digital PID controller was also applied for the sake of comparison. The experimental apparatus is a constant-volume, constant-flow process consisting of two stirred, cylindrical tanks in series. The two tanks are connected by a piping
-0.00244 0.00488 0.0329 0.0646 0.0812 0.0817 0.0769 0.0708 0.0622 0.0529 0.0463 0.0402 0.0305
Table 111. Control Gain Vector Tank Heater (f = 0.1)
K
i
14 15 16 17 18 19 20 21 22 23 24 25
h:
0.0219 0.0195 0.0195 0.0134 0.0109 0.00732 0.00854 0.00365 0.00244 0.00244 0.00488 0.0036
K T for Stirred
= [-0.01747,-0.02086,-0.2509,-0.6660,-1.14701
manifold which is included to produce a time delay. A schematic diagram of this process is shown in Figure 11. Control of water temperature in the second tank is provided by a silicon-controlled rectifier (SCR) connected to an immersion heater located in the first tank. The second tank contains a resistance thermometer connected to a resistance bridge/ampWier network. The output from this amplifier serves as input to an analog-to-digital converter (ADC) which supplies the digital controller (a computer) with the measured value of the controlled variable. More information about the process apparatus and the facilities related to the computer interface is available in the references (Johnson, 1970; Marchetti, 1981). The impulse response coefficients (Table 11) were determined as hj = a, - aj-l, after the step response coefficients aj 0’ = 1, 2, ..., N) were normalized to obtain the response to a unit step change; i.e., the sampled output was divided by the value of the change in the input variable. The dimensions adopted for the A matrix were R = 5 and L = 3. This choice keeps the ratio R / L and the dimension L low enough to test the alternative of working with a reduced matrix, as discussed above. A weighting factor off = 0.1 was selected and resulted in the control gain vector shown in Table 111. Figures 1 2 and 13 show the closed-loop responses obtained for a step change in set point. The predictive control method and the standard PID controller give very similar responses. Figures 14 and 15 show the responses obtained for load changes. In each case, two successive step changes (+200
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Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983
rt+Cl
+
I
I
I
I
I 500
I
I
I
l
I
I
I
I
1
SCR
Controlled Heater
Delay Line
Input
Variable
1.5
-
0
l
1000
1500
2000
TIME (seconds)
Figure 14. Predictive control of the stirred-tank heating system for load changes (T, = 30 8, N = 25, R = 5, L = 3).
3W
Variable output
Figure 11. Bench-scale, stirred-tank heating process.
l.Ol+
0
I
I
500
I
I1
I
I
1000 1500 TIME (seconds)
I1
2000
Figure 15. PID control of the stirred-tank heating system for load changes ( T , = 30 8, K, = 2.5, TI 250 S, TD = 40 s).
1.o 500
1000 1500 TIME (seconds)
2000
Figure 13. PID control of the stirred-tank heating system for setpoint changes (T. = 30 s, K, = 2.5, TI = 250 s, T D = 40 s).
W and -200 W) were introduced manually by use of the heater located in the second tank (cf. Figure 11). Again the two controllers give similar responses. Consequently, in this application the predictive controller does not appear to offer any significant advantage over a well-tuned PID controller. A theoretical analysis of the locations of the closed-loop transfer function poles for the impulse response model in Table I1 revealed the presence of poles outside the unit circle when f = 0 and the other design parameters were kept at their nominal values. For large sampling intervals (T,> 40 s) and large values of R and L, instability did not occur but the matrix ATA was considered to be ill-conditioned since the determinant was close to zero. Hence, the choice off > 0 in this case not only prevented excessive
movement of the control variable but also avoided a potential stability problem. Conclusions and Recommendations This paper provides a detailed analysis of predictive control techniques which are based on discrete convolution models. The advantages of using these nonparametric models are that a model structure does not have to be specified and that the model parameters are easily obtained from input-output data. Although discrete convolution models are “non-minimal”, they do tend to be more robust than minimal (parametric) models. Two popular predictive control techniques, Dynamic Matrix Control (DMC) and Model Algorithmic Control (MAC), have been shown to reduce to deadbeat control and Dahlin’s algorithm, respectively, as limiting cases. The quadratic performance index used in Dynamic Matrix Control can be modified so as to avoid excessive ringing of the manipulated variable and ill-conditioning that can arise during the computation of the controller gain vector. Weighting parameter f provides a convenient means of tuning the predictive controller. The effects of various design parameters on the performance of a predictive control scheme have been examined via a simulation study. A major conclusion is that a significant reduction in the dimensions of the “dynamic matrix” does not significantly degrade control system performance. Small dimensions also tend to reduce the amount of on-line and off-line computations that are required in order to implement predictive control. The simulation study indicated that the predictive control scheme performed better than conventional PID control for three representative processes. However, in an experimental application to a stirred-tank heating system, the predictive control scheme did not provide a significant improvement over a well-tuned PID controller. It should be noted that both the simulated and experimental systems involved single-input,single-output processes without active contraints. By contrast, the predictive control
Ind. Eng. Chem. Process Des. Dev., Vol. 22, No. 3, 1983
techniques are primarily intended for multivariable processes with constraints. Consequently, the simulation and experimental studies did not utilize the full capabilities of predictive control techniques. In conclusion, we believe that predictive control techniques based on discrete convolution models provide a powerful design approach, especially for processes which are difficult to model. Acknowledgmeqt We wish to express our appreciation to CONICET (Consejo Nacional de Investigacionea Cientificas y TBcnicas de la Reptiblica Argentina) for financial support of the first author during the course of this investigation. Nomenclature A = dynamic matrix (R X L) A’ = triangular matrix (R X R) ak = step response coefficient defined in eq 24 Ci = coefficient in eq 40 D = digital controller transfer function E k = error defined as rk - Yk E k = predicted error defined in eq 29 EL = predicted error defined in eq 30 f = tuning parameter G, = process transfer function go = coefficient in eq 40 H = hold device transfer function hk =. impulse response coefficient defined in Figure 1 I = identity matrix J = performance index K = gain vector Kc = proportional gain Kj = element of K L = number of control moves L(s) = load variable in Figure 2 M = integer part of the ratio, tD/T, N = number of terms in the discrete convolution P, = summation defined by eq 25 Q = positive definite R X R weighting matrix R = number of predictions R = positive definite L X L weighting matrix rk = set point S, = summation defined by eq 26 s = Laplace transform variable T = outlet temperature
405
TI = reset time
TD = derivative time Ti = feed temperature T, = sampling interval = time delay u = control variable w = flow rate w j = nonnegative weighting factor y = output variable tD
9 = predicted value of output
y c = corrected prediction of the output Yd = desired value of output z = z-transform variable
Greek Symbols filter constant (0 Ia < 1) A = difference operator 19 = time delay X = time constant a=
Subscript
k = denotes value at the kth sampling instant
Literature Cited Cutler, C. R. ISA Trans. 1982, 21, 1. Cutler, C. R.; Ramaker. B. L. Jolnt Automatlc Control Conference Preprints, Paper WP5-8, San Francisco, 1980. Dahlin, E. B. Instrum. Control. Syst. 1088, 13, 76. Greville. T. N. E. SIAMRev. 1077, 1, 38. Johnson, P. C. MSc. Thesls. University of California, Santa Barbara, 1970. Luyben, W. L. “Process Modelling, Simulation, and Control for Chemical Engineers”; Chapter 15, McQraw-Hill: New York, 1973. Marchetti, J. L. hkSc. Thesis, University of California, Santa Barbara, 1981. Martin, G. D. AIChE J. 1981, 2 7 , 748. Mehra, R. K.; Kessel, W. C.; Rauit, A.; Richalet, J.; Papou, J. I n “Alternatives for Linear Multivariable Control with Turbofan Engine Theme Problem”; Peczkowski, J. L., Ed.; Natlonal Engineering Consortium, Inc.: Chicago, 1978. Mehra, R. K.; Rouhanl, R.; Eterno, J.; Richalet, J.; Rauiet, A. I n “Chemical Process Control 2”; Edgar, T. F.; Seborg, D. E., Ed.; Engr. Found.: New York, 1982. Penrose, P. Roc. Cambridgs Philos. SOC. 1958, 52, 17. Prett, D. M.; Glllette, R. D. Joint Automatic Control Conference Preprints, Paper WP56, San Francisco, 1980. Ray, W. H. “Advanced Control Systems”; Chapter 3, McGraw-Hill: New York, 1980. Richalet, J.; Rault, A.; Testud, J. L.; Papon, J. Automatica 1078, 14, 413. Rouhanl, R.; Mehra, R. K. Automiitlca 1082, 18, 401. Smith, C. L. “Digital Computer Process Control”; Chapter 6, Int. Textbook Co.: Scranton, PA, 1972.
Received for reuiew July 27, 1981 Accepted December 20, 1982
