I n d . Eng. Chem. Res. 1987,26, 1523-1536 Patterson, W. A. Anal. Chem. 1954,26, 823. Schuetzle, D.; Carter, R. 0.;Shyu, J.; Dickie, R. A.; Holubka, J. W.; McIntyre, N. S. Appl. Spectrosc. 1986, 40, 641. Shreve, 0. D.; Heether, M. R.; Knight, H. B.; Swern, D. Anal. Chem. 1951, 23, 277. Sojka, S. A.; Moniz, W. B. J . Appl. Polym. Sei. 1976, 20, 1977. Stevens, G . C. J . Appl. Polym. Sei. 1981, 26, 4259.
1523
Strothers, J. B. Carbon-I3Nuclear Magnetic Resonance Spectroscopy; Academic: New York, 1972. Wehrli, F. R.; Wirthlin, T. Interpretation of Carbon-13 NMR Spectra; Heydan and Sons: New York, 1976. Received for review August 19,1986 Accepted April 30, 1987
The Generalized Analytical Predictor Michael C. Wellonst and Thomas F. Edgar* Department of Chemical Engineering, University of Texas, Austin, Texas 78712
The Generalized Analytical Predictor (GAP) is developed for single-input, single-output and multiple-input, multiple-output discrete-time systems, and its relationship t o other dead-time compensators (Smith Predictor, Internal Model Control, and Discrete Analytical Predictor) is examined. The GAP employs a dynamic disturbance prediction scheme to compensate for the effects of process time delay on the regulatory response. IMC uses the steady-state version of this disturbance predictor. The GAP possesses stability properties equivalent to those for IMC and retains the simplicity of IMC controller design and robustness analysis. The disturbance predictor gives GAP an additional degree of freedom compared with IMC, allowing the user to optimize servoresponses and regulatory responses relatively independently. Simulation examples show that significant improvement in regulatory response is possible using GAP compared with IMC for load changes which can be represented as a series of steps. The control of processes containing time delays has recently received considerable interest. The detrimental effects of time delays on control system stability and achievable control quality are well-known; therefore, a number of techniques have been devised to compensate for the effects of time delays; these include the Smith Predictor (SP) (Smith, 1957), Analytical Predictor (AP) (Moore et al., 1970), Discrete Analytical Predictor (DAP) (Doss and Moore, 1982), and Internal Model Control (IMC) (Garcia and Morari, 1982, 1985a,b). Garcia and Morari have shown that a number of conventional control schemes (Smith Predictor, Linear Quadratic Optimal Control, Model Algorithmic Control, and Dynamic Matrix Control) have the same structure as IMC. Several simulation and experimental comparisons of the Smith and Analytical Predictors have been performed (Meyer et al., 1978, 1979), and Wong and Seborg (1986) have shown theoretically that the two compensation techniques are not the same. Additionally, Wong and Seborg generalized the analytical predictive approach to enable the use of any controller rather than the specific form of PI control proposed by Moore et al. The resulting dead-time compensator was labeled the Generalized Analytical Predictor (GAP). Apparently, no theoretical comparison of the analytical predictive approaches (AP,DAP, and GAP) with Internal Model Control has been made. Several researchers (Meyer et al., 1976; Kantor and Andres, 1980; Watanabe et al., 1983) have noted deficiencies in the regulatory capabilities of the Smith Predictor to unmeasured disturbances. Recently, the same weakness has been identified in the Internal Model Control structure (Huang and Stephanopoulos, 1985). The simulation studies performed by Meyer et al. (1978, 1979) showed significantly better regulatory performance for the Analytical Predictor vs. the Smith Predictor. Simulation studies (Wellons, 1985) have shown that improved regulatory response can also be achieved with the Discrete
* To whom all correspondence
should be addressed. Current affiliation: Department of Chemical Engineering, Purdue University, West Lafayette, IN 47907. 0888-5885/87/2626-1523$01.50/0
Analytical Predictor as compared with the Smith Predictor or IMC. In this paper we examine the relationship between Internal Model Control, the Discrete Analytical Predictor, and the Generalized Analytical Predictor. The results of this comparison lead directly to the development of a truly generalized analytical predictive approach, also called the Generalized Analytical Predictor, which contains the Wong and Seborg version and IMC as special cases. Significant improvement in regulatory response is possible using this GAP as compared with IMC.
Model Formulation In this paper we use the z-domain transfer function formulation of the process Y(z)= G,(z)u(z) + d(z) (1) where
B ( z)rN Ab)
G,(z) = B(2)
=
b12-1
+ bzz-2 + ... + b,z-"
A ( z ) = 1 - u~z-'- U~Z-' - ... - u,z-" where Y is the process output, u is the manipulated input, G, is the process transfer function of order n with dead time of N sampling intervals, and d represents the effect of disturbances on the process output. The control action is constant over the sampling interval At.
Dead-Time Compensation Compensation for the effects of time delays necessarily implies the prediction of the effect of the inputs on the process output. In this section we examine the Smith Predictor, Internal Model Control, and Analytical Predictive control structures to compare methods of output prediction. Smith Predictor. The Smith PEedictor, shown in Figure 1, uses a model of theprocess, G,(z), to predict the effect of control actions. If G,(z) = G,(z), then the actual 1987 American Chemical Society
1524 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
Figure 3. Basic Discrete Analytical Predictor structure.
Figure 1. Smith Predictor.
Figure 2. Internal Model Control structure.
output (less any disturbances) will be cancelled by the process model output, and the controller will act u e n the response of the process model without dead time, G,*(z). This results in removal of the time delay from the characteristic equation so that larger controller gains can be used. Internal Model Control. The basic IMC control structure is shown in Figure 2, with controller G,(z), filter F,(z), and set point Y&). Dead-time compensation and output prediction are handled implicitly by IMC. The IMC structure does not explicitly contain the inner feedback of the process model predictor as does the Smith Predictor. Instead, IMC uses the controller G&), which is an approximation of the process model inverse. IMC estimates the effect of disturbances on the process in an inferential manner, and these estimates are handled directly by the controller. The closed-loop response is Y ( z )=
Perfect control can be achieved (with F,(z) = 1) if G,(z) is chosen to be the process model inverse. Generally, perfect control is not possible because the process model inverse is unstable and/or unrealizable. Consequently, a stable and realizable approximation of the process model inverse is used. To obtain such an approximation, the process model is factored so that
GP(z) = G+(z)G_(z)
(3)
where G+(z)contains the dead time and other non-minimum-phase (noninvertible) factors. The controller G,(z) is then chosen as
G,(z) = [G_(z)]-'
(4)
Garcia and Morari (1982) and Zafiriou and Morari (1985) give further details on IMC controller design. The filter F,(z) (usually first-order) is included to improve the robustness of the system; Zdiriou and Morari (1985) proved there exists a filter constant a such that the system is closed-loop stable if and only if the process model gain has the same sign as the actual plant gain. Analytical Predictive Control Structures. The Analytical Predictor (Moore et al., 1970) provides deadtime compensation by including an analytical prediction of the process output in the feedback path. The predictor was derived by continuous-time techniques for a first-order plus dead-time model and includes an additional one-half sampling period correction in the prediction to account for the effects of sampling. To compensate for unmeasured disturbances, a load disturbance is estimated by compar-
Figure 4. Wong and Seborg Generalized Analytical Predictor.
ison of the actual process output and the process model output and is assumed to be constant over the prediction horizon. The load transfer function is assumed to be the same as the process model transfer function; consequently, the load estimates are treated as control actions in the output prediction. Because of the difficulty of extending this approach to higher order models, Doss and Moore (1982) proposed the Discrete Analytical Predictor which relies upon discretetime equations for output prediction. The basic control structure of the DAP is shown in Figure 3, with digital controller C(z). The output predictor is represented conceptually by PN,where N is the process model time delay. The DAP uses the second-order discrete-time equivalent of eq 1, augmented with the most recent load estimate, to recursively predict the future output. As for the Analytical Predictor, the DAP treats the load estimate as an input to the process model. The load estimation scheme contains a tuning parameter KI which is used to adjust the sensitivity of the load estimator to process model error. In their analysis of the Analytical Predictor, Wong and Seborg (1986) assumed the dead time to be an integer multiple of the sampling time and did not include the additional one-half sampling interval correction. Under these conditions it is easy to show that the Analytical Predictor and the first-order analogue of the Discrete Analytical Predictor are the same (Wellons, 1985). Wong and Seborg (1986) showed that the first-order predictor and load estimator can be combined to yield Y * ( z )= Y,*(z) + A * ( z ) [ Y ( z-) Y,(z)] (5) where Y*(z) is the predicted output, Y ( z ) is the actual output, and Ym(z) = G p ( z ) ~ ( z ) Y,*(Z) = G,(z)u(z) A*(z) =
alN
+ (1 - ~
1 - alz-' 1 ~ ) -
1 - a,
In their derivation they substituted the deadbeat value of the load estimation constant. The resulting control structure, which they called the Generalized Analytical Predictor, is shown in Figure 4. The second-order DAP prediction equations can be combined in a similar manner (Wellons, 1985) to yield the output predictor given by eq 5 where A*(z)is now given by A * ( z ) = P*(z) + [l -
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1525 where p ( Z )
= AN
+ U~AN-~Z-'
(8)
FL(z)is a first-order load estimation filter and the constants A N and
AN-1
are given recursively by A0
=1
A1 =
Ai = alAi-1
+ uZA,~
The load estimation filter constant p is related to the DAP load estimation constant by
p = 1 - KI(b, + b,)
(9)
Thus, the Discrete Analytical Predictor can also be represented by the block diagram in Figure 4 by substituting the second-order predictor given by eq 7 and 8 for the first-order predictor given by eq 6. A comparison of the block diagram of the analytical predictive approaches with those for the Smith Predictor and IMC leads to several important observations. (1) The analytical predictive approach provides disturbance prediction. We see from Figure 4 that the pred_iictedoutput is the sum of the process model pre$iction, G,*(z)u(z) and the disturbance prediction A*(z)d(z). In comparison, IMC and the Smith Predictor assume that the disturbance d ( z ) is constant (Le., A*(%)= 1). (2) The process model prediction and disturbance prediction can be separated into distinct procedures. In the original formulation of the output predictor (AP and DAP), the load transfer function is assumed to be the process model transfer function. Obviously the predicted disturbance will be accurate only if the process model and load transfer function are similar. Performing the disturbance prediction separately allows a model of the load transfer function to be used for the disturbance prediction. Additionally, the disturbance predictor should retain the load estimation filter constant ( K , or 8) to provide flexibility to the prediction scheme. We develop such a predictor in the following section. (3) The GAP can be rearranged to the IMC_structure by incorporating the process model predictor Gp*(z) into the controller C(z) to form a new controller block G,(z). This rearrangement allows for direct comparison of the GAP and IMC and aids in controller design and robustness analysis.
Generalized Analytical Predictor Motivated by the observations of the previous section, we develop a more general disturbance prediction scheme for the GAP. A Disturbance Predictor. A current _estimateof the effect of the disturbance on the process, d ( z ) , can be obtained from the difference of the actual output and the process model output: d ( z ) = Y ( z )- Y&) (10) The effect of the disturbance on the future output can be predicted by estimating the entering load and then predicting forward, assuming the load is constant over the prediction horizon. To do this, a model of the load transfer function is required, called the disturbance prediction transfer function (DPTF). For purposes of derivation, a general second-order DPTF is given by BDp(Z) 6 1 Z - l + b22-2 GDp(2) = -(11) 1 - a1z-l - u22-2 -
The numerator and denominator coefficients are overlined
to distinguish them from the process model coefficients. Note that no dead time has been assumed in eq 11. Since the disturbance is unmeasured and determined by output comparison, the actual dead time of the load transfer function is both unknown and irrelevant. Assuming there is no process modeling error and that the load is a step input, it can be easily shown that d ( z ) = HGL(z)L(z) (12) where L ( z ) is the actual load and HGL(z)is the actual load transfer function combined with a zero-order hold (based on the step input in L(z)). Substituting GDp(z) for HGL(z) in eq 12 and writing as a difference equation yields dk
=
dldk-1
+ dpdk-2 + 61Lk-1 + 62Lk-2
Assuming the step load occurs at time k - 2, then Lk-l = and d k = 6ldk-1 a2dk-2 (61 6 2 ) L k - z (13)
Lk-2
+
+
+
Thus, an estimate of the load (L) is given by
or
where a first-order filter has been added for robustness. Note that assuming the step load occurs at time k - 2 is equivalent to placing the pole of the resulting load estimator to zero. This eliminates instability if the DPTF contains inverse response and ringing when 6, and 6, have the same sign. The disturbance prediction is accomplished in the same manner as for the DAP
+ a 2 ; i k - i + (61 + 62)Lk-z d*k+2 7 did*k+l + 8 2 d k + (61 + 6 2 ) L k - z - d * k + N p = dld*k;N,-l + d 2 d * k + ~ ~ + - 2 (61 + b2)Lk-z d*k+l
= didk
(16a) (16b) (16n)
where d*k+i is the predicted disturbance for time k + i made at tipe k and Npis the prediction horizon. The load estimate Lk-2 is used throughout eq 16a-16n because the load is assumed to be a step input. The resulting closedform prediction equation is (Wellons, 1985) d * k + N p = A N p d k + d2A~,-ldk-l+
where A0
=1
A , = a,
Converting to the z domain and substituting for the load estimate (eq 15) gives d*(z)= A*(z)d(z) (19) where d*(z) is the predicted disturbance and
P*(z) = A N p + G ~ A N , - ~ z - ~
(21)
This is the disturbance predictor used in the GAP. The disturbance predictor A * ( z ) of the Analytical Predictor as developed by Wong and Seborg (1986) is a special
1526 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
Figure 6. Alternate representation of the GAP. Figure 5. Wellons and Edgar Generalized Analytical Predictor.
case of the disturbance predictor given by eq 20 and 21. For a first-order system, P ( z ) is given by
P*(z) = dlN Assuming the first-order DPTF to be the process model (al = al) and by use of the deadbeat filter (@= O), eq 20 reduces to
Additionally, if the second-order DPTF is chosen to be the process model (ADp(2) = A ( z ) ) ,the resulting disturbance predictor is equivalent to the DAP predictor given by eq 7. Thus, the prediction scheme derived here contains the DAP and (Wong and Seborg, 1986) GAP disturbance predictors as special cases. Obviously the effects of a load disturbance cannot be predicted a priori. However, once a disturbance is detected, the above predictor will yield perfect prediction of the disturbance when all of the assumptions made in the derivation of the predictor are valid. These _assumptions are (1) there is no process modeling error (GP = Gp), (2) there is perfect load modeling (GDp = GL), and (3) the load disturbance is a step input. Additionally, the second-order predictp assumes the load occurs at time k - 2 , so that Lk-1 = Lk-2. When all of these conditions are satisfied, the recursive prediction given by eq 16a-16n is exact. Since A*(%)is the closed-form representation of this prediction, then d*(z) = A * ( z ) d ( z )= ~ + ~ p d ( z ) (22)
and the predictor can be represented conceptually by z + ~ P . Mathematically A * ( z ) does not reduce to z + ~ Phowever, ; with the above conditions satisfied, d*(z) = A * ( z ) d ( z )is an exact prediction of the effect of the disturbance on the process N sampling intervals into the future. Generahzed Analytical Predictor. Incorporating the disturbance predictor developed in the previous section into the GAP structure results in a truly generalized Analytical Predictor. The block diagram of the new GAP is shown in Figure 5. To facilitate controller design, Figure 5 can be rearranged to an IMC-type control s_tructureby constructing the process model predictor Gp*(z) as a feedback loop around the controller C ( z ) and then combining this inner loop to form a new controller G,(z). As for IMC, an exponential filter F,(z) can be included to improve robustness. The resulting block diagram appears in Figure 6. The closed-loop response for the GAP as shown in Figure 6 is given by Gp(z)Gc(z)Fc(z)[Ys(z)- A*(z)d(z)l Y(z) = d ( 2 ) + (23) 1 + Gc(z)Fc(z)A*(z)(Gp(z)- G,(z)) Comparing the GAP control structure shown in Figure 6 with the IMC structure given by Figure 2, we see the only difference between the two control schemes is the disturbance predictor A * ( z ) in the feedback path of GAP. Due to the similarities in control structure, the GAP
possesses properties equivalent to those of IMC: perfect cpntrol with G,(z) = G,(z)-' and zero offset if Gc(l) = GP(l)-l (Garcia and Morari, 1982). Deadbeat Regulatory Control. To facilitate the comparison of GAP and IMC, we first define the deadbeat regulatory response to an unmeasured disturbance. For servocontrol, the deadbeat response is defined as returning to and maintaining the set-point N + 1sampling intervals after a change in set point. If an unmeasured disturbance occurs at discrete time k, it will begin to affect the output immediately but remain undetected until time k + 1. The control action based on this disturbance estimate cannot affect the output until (after) time k + N + 1. Consequently, deadbeat regulatory response to an unmeasured disturbance is defined as returning to and maintaining the set-point N 2 sampling intervals after the disturbance begins to affect the output. Comparison of IMC and GAP. At steady state (z = l), the disturbance predictor is given by A*(l) = 1. Comparing Figures 2 and 6, we see that the disturbance predictor employed by IMC (d*(z) = 4 2 ) ) is the steady-state version of the predictor used by GAP. If the IMC controller G,(z) = G_(z)-' is employed, the closed-loop GAP response (no model error) is given by
+
Y(z) = G+(z)F,(z)Ys(z) + [1 - G+(z)J',(z)A*(z)ld(z) (24)
Comparing the GAP closed-loop response with that for IMC
Y ( z )= G+(z)J'c(z)Y,(z) + [1 - G+(z)F,(z)ldk) (25) we see that IMC and GAP yield the same servoresponse. When the process contains no dead time, A * ( z ) = 1 and the IMC and GAP regulatory responses are the same as well. However, if the process does contain dead time, the term G+(z) can be further factored so that G+(z) = G , ( z ) ~ - ~ - l
(26)
where G+(z) does not contain any time delays. The additional one unit delay is inherent in all discrete-time systems using a zero-order hold. Upon rearrangement, the IMC regulatory response becomes
Y(z) = [ l - F , ( z ) G + ( z ) z - ~ - l ] d ( z )
(27)
Similarly, the GAP regulatory response is given by
Y(z) = [ l - F,(z)G+(z)z-~-~A*(z)]~(z) (28) Consider the effect of the disturbance predictor A*(z) on the GAP regulatory response. 'If all the assumptions made in the derivation of the predictor are valid, the disturbance prediction is exact, and A * ( z ) can be conceptually represented by z + ~ P If . Np is chosen as N + 1, then once a disturbance is detected and its effect predicted, the GAP regulatory response becomes
Y(z) = (1 - F , ( Z ) G + ( Z ) ) ~ ( Z ) (29) Thus, the disturbance predictor A * ( z ) attempts to compensate for the process time delay in the regulatory
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1527 closed-loop response. If a step load disturbance occurs at time k, ita effect will not be detected until time k + 1and the prediction made for time k 1 ( N 1). Thus, the response given by eq 29 cannot be obtained until N + 2 sampling intervals after the disturbance occurs (and only under the conditions of exact disturbance predietion). Consider now the ability of IMC and GAP to achieve the deadbeat reguiatory response. If G+(z) contains only dead time, then G+(z) = 1. With F,(z) = 1, the IMC regulatory response is (no process model error)
+ + +
Y ( z )= (1 - z-N-')d(z)
(34) For first-order systems P ( z ) = d l N and AD&) = 1- 612-l. Substituting these relations into eq 34 gives the first-order exponential filter (35)
(30)
Clearly IMC can achieve deadbeat regulatory control only for step inputs in d (i.e., step load input and constant load transfer function). If d is not a step, Y ( z )will not return to the set point until d reaches its steady-state value. Under the same conditions, the GAP regulatory response is
Y ( z )= (1- ~ - ~ - l A * ( z ) ) d ( ~ )
Substituting the general expression for A*(z) (eq 20) and rearranging yields
(31) If the conditions permit exact disturbance prediction, then A*(z) can be represented conceptually by z + ~ Pand , eq 31 becomes (with Np = N + 1) Y ( z )= 0 (32) which is valid N + 2 sampling intervals after the disturbance occurs. Thus, GAP can achieve deadbeat regulatory control for step load inputs for the whole class of disturbances that can be modeled by the DPTF given by eq 11 (of which a step input in d is a special case). Typically, the conditions for exact disturbance prediction are not satisfied; however, in most cases the disturbance predictor will provide partial compensation for the effect of the time delay (e.g., disturbance prediction is much closer to the actual disturbance than the steady-state prediction), improving the regulatory response. If G+(z) contains factors other than dead time, deadbeat regulatory control cannot be obtained. However, from the previous discussion it should be apparent that partial or complete compensation for the time delay will yield improved regulatory response when d(z) is not a step. Removal of the time delay in the regulatory response is the goal of the GAP disturbance predictor. It is well-known that the deadbeat response is very sensitive to model errors. Thus, the GAP predictor design consists of two steps: (1) selection of the DPTF and prediction horizon to achieve deadbeat regulatory control (if possible) and (2) selection of the load estimation filter constant /3 to provide robustness to the disturbance prediction scheme. This is the same general procedure used in the design of the IMC controller; therefore, in practice, a and 6 will be detuned to make the system response more robust. IMC Predictor-A Special Case of GAP. The steady-state disturbance predictor employed by IMC arises as a special case of the GAP predictor in two ways: (1)if the disturbance prediction transfer function is chosen as a constant and (2) proper selection of load estimation filter constant in first-order prediction. If the DPTF is chosen as a constant, a step input in load results in g step disturbance in d. Since the IMC predictor (d*(z) = d(z)) assumes that d is constant, selecting the DPTF to be a constant results in the steady-state (IMC) predictor. More importantly, the IMC predictor also occurs as a special case of first-order prediction. Steady-state prediction requires A*(z) = 1 (33)
Thus, selecting the filter constant /3 = d1 where dl is the denominator coefficient of the DPTF results in the IMC disturbance prediction. An alternate expression for the filter constant that has physical significance is T F , the time constant of the filter, where /3 = e - A t f r p . Specifying T F = 0 gives the deadbeat response, while setting the filter time constant equal to the disturbance prediction model time constant yields the steady-state prediction used by IMC. Substituting for P ( z ) and ADp(z)in eq 34 for second-order systems results in a complicated second-order filter that is obviously contrived to obtain the IMC prediction using a second-order DPTF. We see from the above discussion that the GAP contains IMC as a special case of first-order prediction. The GAP essentially contains an additional degree of freedom (the disturbance prediction scheme) compared to IMC. This allows the user to select optimal responses for both the servo and regulatory cases. The IMC controller filter constant represents the trade-off between robustness and quality of response. The same is true for the load estimation filter constant. With /3 = 0 we can achieve deadbeat regulatory control, while detuning /3 results in good, but not optimal, regulatory responses in the presence of load and process modeling error. Example 1. A first-order process with transfer function ,-4s
Gp(s) =
10s + 1
is controlled by using a deadbeat IMC controller ( a = 0). The sampling interval is At = 1. An exact process model is assumed. The load transfer function is 4
GL(s)
=I 7s 1
+
The DPTF used by GAP is the same as the load transfer function. The IMC factorization procedure yields G+(z) = z - ~ therefore, ; the disturbance prediction horizon is selected as Np= 5. The IMC and GAP responses and controller actions for a step load disturbance at time t = 10 are shown in Figure 7. The IMC and GAP disturbance predictions are compared with the actual disturbance in Figure 8. Deadbeat regulatory response requires the output to return to the set point in N + 2 or 6 sampling intervals after the disturbance begins to affect the output. Examining Figure 7 we see that GAP (/3 = 0) yields deadbeat response. In contrast, the IMC response does not return to the set point until the disturbance reaches steady state-almost 40 sampling intervals after the load occurs. However, to achieve the deadbeat response, the GAP controller action is very strong. Filtering the load estimate will lessen the severity of controller output and provide an exponential return to set point. The response for GAP with /3 = 0.6 is also shown in Figure 7. Note that the controller action for GAP ((3 = 0.6) is not nearly as severe as the deadbeat GAP case, while the resulting output response is still far better than IMC. The IMC
1528 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
'"1
1.87
I
I
-0.5
-2
-
!:I
T
!!
U -4
30
28
10
se
qe
'I
! II
,
I 1 I
,
,
,,
-
,
I
4
II
. ,
I
,
I
,
I
.
I
,
., I
,
I
,
I
.
I
,
),
I
-6-
-8
.
l
.
l
,
.
l
l
l
l
.
l
l
.
,
.
.
l
l
(
.
.
l
.
,
10
3e
28
4e
68
SO
TIME
Figure 9. Regulatory response and controller action for example 2 (-) IMC; (- - -) GAP (3 = 0.
i
-1
1 1 10
20
38
40
58
TIME
-1
Figure 8. Disturbance prediction for example 1: (-) IMC, (---) GAP @ = 0; (--) GAP /3 = 0.6.
response can be obtained by GAP with p = e-Il7 = 0.8669. Example 2. A second-order process with transfer function -5(-3s + i)e-48 Gp(s) = (3s 1)(5s + 1)
+
is controlled by using a deadbeat IMC controller ( a = 0). There is no process modeling error. The sampling interval is At = 1. The resulting IMC factorization is z - 1.410 2-5 G + ( z ) = 1 - 1.4102 The load transfer function is
G L ~ =)
1
(2s
+ 1)(7s + 1)
ie
30
20
48
58
68
d; d* for (- - -1
TIYE
Figure 10. Disturbance prediction for example 2: (-) d; d* for (- - -) IMC; (---) GAP /3 = 0.
The DPTF used by GAP is the same as the load transfer function. The prediction horizon is selected as N p = 5 . The IMC and GAP (p = 0 ) responses and controller actions for a step load disturbance of magnitude 1 occurring at time t = 10 are shown in Figure 9. The IMC and GAP disturbance predictions are compared with the actual disturbance in Figure 10. Since the conditions for exact disturbance prediction are satisfied in this example, the GAP regulatory response after time t = 16 ( t = 10 + ( N + 2)At) is given by 1 -- 1.4102
which exhibits sharp inverse response (initial effect of
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1529 control action forces output away from the set point) and then approaches the set point exponentially. The IMC response is given by
1.9
I
z - 1.410 z - 5 ] d ( z ) 1 - 1.4102
The presence of the time delay zd makes the IMC response sluggish in comparison with GAP. Note that the GAP controller actions are quite acceptable in this case. In the previous example, deadbeat regulatory control could be achieved, requiring strong controller action. Due to the inverse response in G+(z), the output cannot be forced back to the set point in one sampling interval. Thus,the output trajectory for this example is not as severe, resulting in more reasonable controller actions. Figure 10 shows that the GAP disturbance prediction is perfect after time t = 17, while the IMC prediction is correct only at steady state. The disturbance predictor requires an additional sampling interval ( N + 3) to achieve exact prediction compared to the previous example. This is because the second-order predictor assumes Lk-2 = Lkl, which is not true for the first load estimate. Robustness of the GAP Control Structure. In this section we examine the robustness of the GAP control structure to disturbance and process modeling errors. The closed-loop response for GAP in the presence of process modeling error is given by
Y ( z ) = d(z) +
Gp(z)Gc(z)Fc(z)[Y,(z) - A*(z)d(z)l 1 + G,(z)FC(z)A*(z)(Gp(z)- Op(z))
(36)
The predictor A*(z) appears in the characteristic equation and therefore will affect the closed-loop stability. Due to the similarity in control structures, the robustness of GAP is similar to IMC. That is, there exists a control filter constant a* (0 Ia* < 1)such that the system is closedloop stable for all a in the range a* Ia < 1 if and only if the gain of the process model and the actual plant have the same sign. The proof of this result is nearly identical with that given by Zafiriou and Morari (1985) and is given in Wellons (1985). Although IMC and GAP have the same requirement for stability, the value of a* for a given plant/model mismatch for GAP is not necessarily the same as for IMC. Since the disturbance predictor A*(z) appears in the characteristic equation of GAP, the disturbance prediction time constant, the prediction horizon and the load estimation filter constant all affect the value of a* necessary for stability. Since the IMC control structure is a special case of the more general GAP structure for constant and fmt-order DPTF's, one would expect the value of a* to vary as the disturbance prediction conditions vary, with the IMC value for a* occurring when the disturbance predictor yields the steady-state prediction. The following example illustrates these points. Example 3. A first-order process with transfer function e-49
G,(s) =
10s + 1
is modeled by e-38
G,(s) = 10s + 1 so there is a 25% error in the dead time. The sampling interval is At = 1. The load transfer function is first order with a time constant of 7. Consequently, the DPTF is selected to be first order with time constant TDP = 7 and prediction horizon Np= 4. A plot of a*,the IMC controller
0 . 0 : . 0.9
.
. I , , . ; . . , I , , . ; . . . 9.2
8.6
9.4
9.8
1.9
BETA
Figure 11. Plot of a* VS. fl for example 3.
filter constant necessary for stability, vs. 0,the load estimation filter constant, appears in Figure 11. The region of instability is below and to the left of the curve, while all points above and to the right are stable. Since the DPTF is first order, the IMC response is obtained when @ = e - A t / 7 ~ por @ = 0.8669. Obviously the presence of the dynamic disturbance predictor in the feedback loop adds instability to the output response compared to IMC when there is process modeling error. The value of a* for IMC stability is approximately 0.33, while for deadbeat GAP it is roughly 0.86. Consequently, when there is significant model error, the disturbance predictor must be detuned toward the steady-state (IMC) prediction to obtain a stable response. This result is not surprising; deadbeat servocontrol is achieved with an IMC filter constant of a = 0. In this example the deadbeat IMC response is unstable and the controller must be detuned. We have shown that GAP can achieve deadbeat regulatory control with = 0 (and a = 0). The presence of modeling error makes deadbeat regulatory control unattainable, and the disturbance predictor must also be detuned. Qualitatively, as model error increases, both the controller and disturbance predictor must be detuned, resulting in a degradation of the achievable set point and regulatory response. As a result, the improvement in regulatory response possible with disturbance prediction decreases with increased model error. The IMC and GAP responses and controller actions for a step load disturbance of magnitude 1at time t = 10 are shown in Figure 12. The IMC controller filter constant is a = 0.75, and the load estimation filter constant is 0 = 0.6 (filter constants selected by using tuning procedure described below). We see that the GAP response is significantly better than IMC. In fact, the GAP response in Figure 12 is quite comparable to the deadbeat IMC response with perfect modeling (see Figure 7). Note from Figure 11 that the GAP response (with a = 0.75) would be unstable for a load estimation filter constant smaller than 0.35. Figure 13 shows the IMC and GAP response (same filter constants) and control actions to a set-point change at time t = 10. The IMC response exhibits smooth exponential rise to the set point with slight overshoot and (relatively) smooth controller actions. The GAP response, although still acceptable, is more oscillatory and not as smooth. The controller actions exhibit similar behavior. An explanation for this behavior is found by comparing the disturbance estimate, d , which for a change in set point reflects the model error, and the predicted disturbance for IMC and GAP, shown in Figure 14. We see that the model error is very similar to an impulse disturbance, returning to zero
1530 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
2-
Y
Y
. 1-
8.8
e
1
-‘-i 4 -e
10
.?e
3 ,
4e
5,
TIME
Figure 12. Regulatory response and controller actions for example 3: (-) (3 = 0.8669 (IMC); ( - - - ) p = 0.6.
at steady state. Modeling errors in the dead time and time constant will exhibit such behavior, while modeling errors in the process gain will yield non-zero steady-state values. Since the GAP disturbance predictor assumes the most recent load estimate is constant over the prediction horizon, GAP overpredicts the “disturbance”, yielding a feedback signal that is more oscillatory than for IMC. Although the controller filter will attenuate much of the signal, the GAP response will be more oscillatory than IMC. GAP is not as sensitive to modeling errors in process gain because the steady-state error is non-zero, much like a disturbance. Therefore, in the presence of process modeling error, improved regulatory response comes at the expense of degraded servocontrol. Since the control filter directly affects the robustness of the closed-loop transfer function, we recommend the following tuning procedure: (1)select a (with p set to the IMC response) to provide a stable response with the desired set-point performance. We have found that the stable IMC set-point response with minimum ISE rating generally exhibits significant overshoot and/or oscillatory behavior with excessive control actions. In addition to ISE, we recommend considering the ITAE rating (more sensitive to tuning) as well as the magnitude of controller actions to fine tune the IMC controller. (2) Adjust the load estimation filter constant for improved regulatory response. Obviously the possible improvement in regulatory response will be a function of the process model error, the type of disturbance, and the relative importance of regulatory control and servocontrol. With the IMC structure, selection of a determines the closed-loop response for both servocontrol and regulatory control. With GAP, the disturbance prediction scheme provides more freedom for the user to select the desired regulatory
et=L----
-2
.?e
10
30
4e
50
TIME
Figure 13. Servoresponse and controller actions for example 3 (-) = 0.8669 (IMC); (- - -) p = 0.6.
, . , .
-1.e
e
-1.e1 I
I
ie
.
,
,
.
,
ze
I
78
,
,
,
., 48
, , , ,
, 50
....I....I....(....(...., lb
3b
.?b
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Figure 14. Disturbance prediction for example 3: (-) for (top) p = 0.8669 (IMC) and (bottom) @ = 0.6.
d; (- - -) d*
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1531 response. Although there is a trade-off between the quality of regulatory response and servoresponse when there is model error, GAP allows the user more flexibility in selecting servoresponses and regulatory responses. Assuming a stable response has been achieved, errors in disturbance modeling (e.g., mismatch between the DPTF and the actual load transfer function) affect only the quality of the regulatory response. The disturbance predictor A*(z) has three modeled parameters. (1) DPTF Steady-State Gain. The generalized disturbance predictor derived in eq 15 contains only the denominator coefficients of the DPTF, which do not depend on the DPTF gain. Consequently, the disturbance prediction is unaffected by errors in the gain, and unity gain is used for all applications. (2) DPTF Time Constant(s). For simplicity of analysis, we will consider first-order systems, although the results are directly applicable to higher order systems. If the DPTF time constant is smaller than the actual LTF time constant, the initial effect of the disturbance on the process is smaller than expected, resulting in underprediction of the disturbance. However, the resulting disturbance prediction will be closer to the actual disturbance than the steady-state (IMC) prediction; thus, the GAP response will always be better than IMC. In fact, the IMC prediction occurs in the limit of such DPTF/LTF mismatch. If the DPTF time constant is larger than the actual LTF time constant, the initial load estimates are too large, resulting in overprediction of the disturbance. This results in controller actions more severe than those required for deadbeat regulatory response and overshoot of the output response. The relative importance of quality of response and the magnitude of controller action is very subjective and will obviously impact the acceptable mismatch between DPTF and LTF time constants. We refer the reader to Wellons and Edgar (1985) for further details. (3) Prediction Horizon. If the process model dead time is in error and Napis larger than N + 1, the disturbance will be overpredicted, resulting in overshoot as the output returns to set point. If Npis smaller than N 1, the disturbance will be underpredicted and the output will not return to the set point as quickly as the deadbeat response, Obviously choosing Np= 0 will result in the IMC response. Since the load estimation filter allows the user to shape the regulatory response, we recommend selecting Np = N 1 (i.e., assuming no process model error) and then detuning the load estimator (which must be done anyway in the presence of process model error) to obtain a satisfactory response.
+
+
Practical Implementation of the GAP There are many processes for which the load transfer function is not well-known or the disturbances are too varied to characterize in a single transfer function. Thus, if the load transfer function is not well-known, using a first-order DPTF with a relatively small time constant would ensure conservative prediction, yet the resulting response would be much better than IMC. Additionally, for those cases where the IMC response is desired, the GAP predictor can be easily detuned to IMC. If the system output is perturbed by more than one disturbance, the DPTF dynamics should be selected to resemble the dynamics of the fastest load transfer function. This would yield optimal regulatory response for one disturbance and conservative (yet better than IMC) regulatory responses to the other disturbances. In the following example, we demonstrate that improved regulatory response can be achieved by using a first-order DPTF with relatively fast dynamics for a variety of load transfer functions.
Table I. Load Transfer Functions for Example 4 Figure load transfer function 15a 1 15b 1/(8s + 1) 15c 1/[(2s + 1)(7s + l)] 15d (-3s + 1 ) / [ ( 2 ~+ 1 ) ( 7 + ~ 1)J
Example 4. A second-order process with transfer function -5e-48 Gp(s) = (3s 1)(5s + 1)
+
is controlled with a deadbeat IMC controller (a = 0). There is no process modeling error. The disturbance prediction transfer function is first order with a time constant of 5.0 and is used for all the simulations in this example. A step load disturbance of magnitude 1 occurs at time t = 10. The simulated responses are shown in parts a-d of Figure 15, with the respective load transfer functions given in Table I. Four responses are plotted for each case, using load estimation filter constants of TF = 5.0, 3.0, 1.0, and 0.0. These curves illustrate the range of available responses from IMC (TF = 5.0) to deadbeat prediction (TF = 0). If the load transfer function is a constant, IMC yields deadbeat regulatory control; therefore, the disturbance predictor should be tuned to the IMC response when the load transfer function is a constant or has very fast dynamics. Figure 15a illustrates this point. For a first-order load transfer function, selecting T D P < T~ results in conservative disturbance prediction but yields responses superior to IMC. This is illustrated in Figure 15b. The same general trend is observed for higher order systems; this is illustrated in Figure 15c, where the first-order DPTF has faster dynamics than the second-order load transfer function. Finally, the results for a load transfer function that has inverse response are shown in Figure 15d. Although the GAP responses have a larger deviation from the set point (due to prediction of the inverse response as a step input), the overall response is still much better than IMC.
Extension to Multivariable Systems The z-domain transfer function matrix description of multiple-input, multiple-output processes is Y(z) = G,(z)u(z) + d ( z )
(37)
where Y E R" is a vector of system outputs, u E R" is a vector of control actions (constant over the sampling interval), and d is a vector describing the effect of disturbances on the process. The process transfer function matrix has the form
where g,,*(z) is a physically realizable rational function of z and the input-output pair ij has a time delay of Ntj sampling intervals. The additional sampling interval delay premultiplying the transfer function matrix is due to the zero-order hold.
Multivariable Internal Model Control Since the disturbance predictor design procedure for the multivariable GAP relies upon the IMC time delay fac-
1532 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987
-0.5
-3
2!a
$5
10
25
3e
10
30
10
4e
50
7
-8.5
1 I 10
ea
3*
4e
56
-0.5
10
a
3e
TIME
Figure 15. Regulatory response for example 4. a (top left), b (bottom left), c (top right), d (bottom right): (-) 7F = 1; (--) 7F = 0.
torization procedure, we present a brief review of the multivariable IMC design procedure (Garcia and Morari, 1985a,b). As for the SISO case, G,(z) mustbe designed as a stable and rgalizable approximation of G,(z)-'. The factorization of G,(z) is represented by
Gp(z) = G+,(z)G+,(z)G-(z)
(39)
and is acsomplished in two steps: (1) G+l(z) is selected to make G,(z)-'G+,(z) realizable, andJ2) G+&) is chosen to eliminate the unstable factors of G,(z)-l such that
Gp(Z)-'G+~ (z )G+,(z) = G-(z1-l is stable. Although no general theory exists for the optimal factorization of G,(z), Garcia and Morari (1985a) present the following factorization procedure for the selection of an "optimal" diagonal factorization matrix G+'(z). Diagonal Factorization Procedure. Assume G,(z) = G,(z) and that G+'(t) is a diagonal of the form
G+l(z) = diag
= max max (0,IVij)
i
j = I, ...,n
TF
5b
= 5; (- - -) TF = 3; (-- -)
mum time delay in each column is on the diagonal. Otherwise a faster response can be achieved by allowing interactions in the closed-loop response. If the process model does not contain any dead time, G+l(z) must be chosen as z-'I to obtain a realizable response. The one sampling interval delay is inherent in all digital systems using the zero-order hold. We refer the reader to Garcia and Morari (1985a,b) and Holt and Morari (1985a) for details on the factorization procedure for G+&). The closed-loop response is
Y ( z )= G,(z)[I + G,(z)(G,(z) G,(z))l-'G,(z)F,(z) Y&) + [I - G,(z)[I + G,(z)(G,(z) - ~p(z))l-'G,(z)F,(z)ld(z) (41) If there is no process modeljng error and the controller is designed as above, the closed-loop response is
Y ( z )= G+(z)F,(z)Y,(z)+ [I - G+(z)F,(z)ld(z)
,...,2-5'-']
[~-'1+-~,~-~2+-l
&et the elements of the process model inverse matrix, G,(z)-', be gij*(z)zm~~+l where g i j * ( z )is semiproper in z. Then T,+
40
TIME
(40)
yields the minimum time delay in the response necessary for realizability of G,(z) = G-(z)-'. It is important to realize that the factorization procedure assumes G+l(z) to be diagonal. It has been shown (Holt and Morari, 198513) that such a factorization is optimal (yigding a closed-loop response with minimum ISE) only if G,(z) can be rearranged by interchanging rows and columns such that the mini-
(42)
where G+(z)= G+l(.z)G+Z(z).Obviously if G+(z)and F,(z) are chosen to be diagonal, the system will be dynamically decoupled. Regardless of dynamic interaction, G+(z) and F,(z) must both be equal to the identity matrix at steady state to guarantee zero offset at steady state. As for the SISO case, the control filter F,(z) is included to improve system robustness.
Multivariable Generalized Analytical Predictor Analogous to the SISO case, the multivariable GAP is achieved by inserting the matrix disturbance predictor A*(z) in the feedback loop of the multivariable IMC structure. The resulting GAP control structure is shown
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1533
Figure 16. Multivariable Generalized Analytical Predictor structure.
IMC occurs as a special case, yielding the IMC response for the ith output. If all the DPTFs are first order, there exists a set of pis so that A*(z) = I, yielding the IMC response for all outputs. Deadbeat Regulatory Control. For multivariable systems, deadbeat regulatory response to an unmeasured disturbance is a function of the process model factorization. If the diagonal elements of G+(z) are of the form gii+(z)= gii+(z)Z-7'+-l
then to achieve deadbeat regulatory response, the ith output must return to its set-point T ~ + 2 sampling intervals after the load disturbance begins to affect the ith output. If G+(z) contains off-diagonal elements, these interactions must be removed as well. Comparison of IMC and GAP, Employing the IMC controller G,(z) = G-(z)-l, the closed-loop GAP response is given by (no model error) Y(z) = G+(z)Fc(z)Y,(z) + [I - G+(z)Fc(z)A*(z)ld(z) (49) Multivariable Disturbance Predictor Design. For the multivariable disturbance predictor, a disturbance Comparing this response with the IMC response given by prediction transfer function 1s assumed for each output eq 42, we see that the multivariable GAP and IMC sery l , so that the disturbance predictor for the ith output is voresponses are the same (with no modeling error). If given by G+(z) = G+,(z)G+,(z) and F,(z) are diagonal, the IMC closed-loop regulatory response can be arranged as ADP,(z) A"(z) = P,*(z) (1- P1*(1))FL,(z) i = 1,..., Y ( z ) = (1 - Fc(z)G+2(z)G+l(z))d(z) (50) ADP,(1) where G+l(z) contains only time delays. Similarly the GAP (45) regulatory response becomes where ADP,(z)is the denominator of the disturbance preY(z) = (1 - Fc(z)G+z(z)G+l(z)A*(z))d(z) (51) diction transfer function for the ith output, FL,(z) is a first-order exponential filter with filter constant p, and the If conditions for exact prediction exist, then by design prediction horizon is NpI= j . For example, A2%) repreG+l(z)A*( z ) = I; the disturbance predictor compensates sents the disturbance predictor for output y 2 with prefor the time delays in the regulatory response. As for SISO diction horizon Npz= 3. p , * ( ~is) given by (for a secondsystems, if F,(z)G+,(z) = I, then IMC can achieve deadbeat order system) regulatory response only for step inputs in d, while GAP can achieve deadbeat response for step load inputs for the Pl*(z)= A, + 62zAl-1z-1 (46) whole class of disturbances modeled by the DPTF of eq where a2,is a denominator coefficient of the ith DPTF and 11. IfG+(z)contains factors other than time delays (G+,(z) A, is a function of the denominator coefficients of the ith # I) or is not diagonal, deadbeat regulatory response DPTF (eq 18). The prediction horizon Np,is selected cannot (in general) be obtained. However, a diagonal based on the discussion below. The matrix predictor A*(z) disturbance predictor can always be employed to comcontains the predictors AbJJ(z) as elements. pensate for the time delays along the diagonal of G+(z). The goal of the disturbance predictor is to compensate As for single-variable systems, the design of the disfor the time delays in the regulatory response. The folturbance predictor A*(z) consists of two steps: (1)selection lowing disturbance predictor design procedure is compliof the DPTFs and prediction horizons to achieve deadbeat mentary to the IMC diagonal factorization procedure. regulatory control (if possible) and (2) selection of the load Disturbance Predictor Design Procedure. Assume estimation filter constants p, to provide robustness to the that G+(z) is a diagonal of the form disturbance prediction scheme. In practice the ais and pis will be detuned from their deadbeat values to reduce the G+( z ) = diag [gll+(z)z-rl+-l,g22+(z)z-rZt-1, ...,gnn+(~)~-7n+-11severity of controller actions and to provide a robust reThen the disturbance predictor A*(z) should be a diagonal sponse. of the form Example: Wood and Berry Model ...,Anflpn(z)] (47) A*(z) = diag [A1flp1(z),A2flp2(z), In this section we compare the multivariable IMC and where the prediction horizon Npcis selected as GAP via simulation of their performance on the Wood and Berry (1973) distillation column model. Wood and Berry i = 1,...,n NpI= T,+ 1 (48) developed an experimentally based model of a pilot-scale Although a diagonal disturbance predictor can always be binary distillation column. Because of the simplicity of implemented, if G+(z) is not diagonal, then A*(z) cannot the model and significant interactions, it has become a (in general) be designed to completely compensate for the standard for comparison of simulation results for singleeffects of the interactions. We refer the reader to Pavloop and multivariable control schemes (e.g., Meyer et al., lechko et al. (1986) for additional examples of the design 1978, 1979; Ogunnaike and Ray, 1979; Garcia and Morari, procedure for A*(z). 1985a,b). The model of the column is given by Since the multivariable disturbance predictor contains single-variable predictors as matrix elements, it possesses properties similar to those of the SISO predictors. If the ith DPTF is first order, there exists a filter constant p, = e-At/7Dpb such that the steady-state predictor employed by in Figure 16, where d*(z) is a vector of predicted disturbances, given by d*(z) = A*(z)d(z) (43) The closed-loop response for the multivariable GAP is Y(z) = G,(z)[I + G,(z)A*(z)(G,(z) - GP(2))]-' X G,(z)F,(z)Y,(z) + [I - Gp(z)[I + G,(z)A*(z)(G,(z) Gp(z))l-lGc(z)F,(z)A*(z)Id(z) (44)
+
~
+
+
1534 Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 Table 11. Variable Definitions for variable description y1 overheads composition y2 bottoms composition u1 reflux flow rate u2 steam flow rate L feed flow rate
Wood and Berry Model steady-state value 96.25% mol of methanol 0.50% mol of methanol 1.95 lb/min 1.71 lb/min 2.45 lb/min
A summary of the variables and their steady-state values is shown in Table 11. Time is measured in minutes. An error-free model of the system is assumed for the simulations. The sampling interval is At = 1 min. In the discrete-time domain, the process model transfer function is of the form
i.ew
Following the time delay factorization procedure outlined earlier, the resulting G+(z)is (53) L
J
Note that the minimum time delays in eq 52 are on the diagonal, so the decoupled factorization is optimal. The resulting controller G&) = G-(z)-l is both stable and realizable. For a complete derivation of G+(z)and the resulting Controller G,(z), we refer the reader to Wellons (1985). Following the design rule for the disturbance predictor, the desired prediction matrix is
"* 1 . m
1
'
20
I
.
I
'
I
-
t
iee
80
68
40
TIME
(54)
The disturbance prediction transfer functions are assumed to be the same as the load transfer functions given in eq 52 (i.e., no load modeling error). Thus, the disturbance predictor for output yl, A1%z), is given by eq 45, using a first-order DPTF with time constant 'Qp, = 14.9 and prediction horizon Npl = 2. Likewise, A2s4(2) uses a first-order DPTF with time constant 7Dp, = 13.2 and prediction horizon Np2= 4. The IMC and GAP responses and controller actions for a step load disturbance of +0.34 Ib/min in the feed flow rate (i.e., L = 2.79 lb/min) occurring a t time t = 10 are shown in Figure 17. The predicted disturbances are compared with the actual disturbance in Figure 18. The deadbeat IMC controller (al = cy2 = 0) was used. For the GAP responses, deadbeat disturbance prediction was also used (PI = P2 = 0). The output responses in Figure 17 show quite clearly the improvement in regulatory response for unmeasured disturbances that can be made by compensating for the time delays in the regulatory response. Since all the conditions for exact disturbance prediction are satisfied, and G + ( z )is diagonal and contains only time delays and F(z) = I, GAP can achieve deadbeat regulatory response. Thus, output y1 returns to its set-point T ~ + + 2 = 3 sampling intervals after the load affects it, and + 2 = 5 sampling output y 2 returns to its set point after T ~ + intervals. In contrast, since IMC cannot compensate for the time delays in G+(z),the IMC regulatory response does not return to set point until the disturbance reaches steady state. Figure 18 clearly illustrates the effects of including the dynamic disturbance predictor in the feedback path. Once the disturbance is detected, the GAP prediction is exact, while the IMC prediction is correct only at steady state. Finally, the IMC response can be obtained by using GAP with = 0.9351 and & = 0.9270.
Figure 17. Regulatory response and controller actions for Wood and Berry model: (-) IMC; (- - -) GAP (pi = 0).
d,
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or
or
4
. ,
I,,'
0.w
e
I
Be
'
I
'
I
68
40
~
Be
I
'
lee
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Figure 18. Disturbance prediction for Wood and Berry model: (-) d; d* for ( - - - ) IMC, (---) GAP (pi= 0).
The deadbeat GAP responses in Figure 17 were obtained with rather strong controller actions. Instead of detuning the IMC controller to reduce the severity of the control actions, which would detune the servoresponse as well, we can detune the GAP load estimators. This results in an exponential return to the set point for the regulatory response rather than the deadbeat response shown in Figure 17. The GAP responses and controller actions with Dl = pz = 0.7 for the same load disturbance as in Figure 17 are compared with the IMC responses in Figure 19. Note that the GAP controller actions are now essentially the same magnitude as IMC; however, the GAP output responses are still clearly superior. If the load transfer functions are not well-known, selecting the DPTF to be first order with relatively fast
I
Ind. Eng. Chem. Res., Vol. 26, No. 8, 1987 1535
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Figure 19. Regulatory response and controller actions for Wood and Berry model: (-) IMC; (---I GAP (8; = 0.7).
Figure 20. Regulatory response and controller actions for Wood and Berry model: (-1 pi = 0.8187 (IMC); (- - -) 8, = 0.6;(- - -) pi = 0.
dynamics will yield safe, conservative disturbance prediction yet improve upon the IMC response. To illustrate this result for the multivariable case, suppose we were unsure of the load transfer function time constants for the Wood and Berry model and selected DPTF time constants of TDP, = rDP2= 5 , quite conservative compared to the process dynamics. The resulting GAP responses and controller actions to a load disturbance of +0.34 lb/min in the feed flow rate are shown in Figure 20. The responses shown are for the IMC response (Bi = 0.8187), & = 0.6, and deadbeat prediction (pi= 0). Obviously the GAP responses (pi< 0.8187) significantly improve upon the IMC result.
the trade-off between robustness and quality of response. The disturbance prediction scheme provides GAP an additional degree of freedom to the design procedure compared with IMC, allowing the user to optimize servoresponses and regulatory responses relatively independently. In the presence of process model error, the disturbance predictor adds instability to the output response and improved regulatory response comes at the expense of degraded servocontrol. The GAP control structure is extended to multivariable systems; the disturbance predictor design procedure is outlined for diagonal time delay factorizations. As for the SISO case, significant improvement in regulatory response is possible using GAP compared with IMC.
Conclusions Comparison of the Smith Predictor and Internal Model Control with the Discrete Analytical Predictor has shown that the analytical predictive control structures contain disturbance prediction. Motivated by this observation, a generalized disturbance predictor was derived which uses a model of the load transfer function to predict the effects of the load disturbance on the process output. Substitution of this generalized disturbance predictor i n t ~ the Analytical Predictor control structure yields the Generalized Analytical Predictor. We have shown that the disturbance predictors used by the Smith Predictor, IMC, and DAP are special cases of the GAP disturbance predictor. The GAP control structure possesses stability properties equivalent to those of IMC and retains the simplicity of the IMC controller design and robustness analysis. A stable closed-loop response in the presence of process model error can always be achieved if the gain of the process model and the actual plant have the same sign. Additionally, the filter constants a and p directly reflect
Acknowledgment Financial support from the National Science Foundation and the E. I. du Pont de Nemours & Co. is gratefully acknowledged. Literature Cited Doss, J. E.; Moore, C. F. "The Discrete Analytical Predictor-A Generalized Dead Time Compensation Technique", ISA Trans. 1982,20, 11. Garcia. C. E.: Morari. M. "Internal Model Control. 1. A Unifvine Review and Some New Results", Ind. Eng. Chem. Process be: Deu. 1982,2I(2), 308-323. Garcia, C. E.; Morari, M. "Internal Model Control. 2. Design Procedure for Multivariable Systems", Ind. Eng. Chem. Process Des. Deu. 1985a,24(2),472-484. Garcia, C. E.; Morari, M. "Internal Model Control. 3. Multivariable Control Law Computation and Tuning Guidelines", Ind. Eng. Chem. Process Des. Deu. 198513,24(2),484-494. Holt, B.R.; Morari, M. "Design of Resilient Processing Plants VI. The Effect of Right-Half-Plane Zeros on Dynamic Resilience",
Ind. Eng. Chem. Res. 1987,26, 1536-1540
1536
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Received for review March 14, 1986 Revised manuscript received May 18, 1987 Accepted May 24, 1987
Kinetics of the Hydrobromination of 10-Undecenoic Acid Suhas P. Bhagwat,? D. H. S. Ramkumar, Raghunathan S. Tirukkoyilur, and Arvind P. Kudchadker* Department of Chemical Engineering, Indian Institute of Technology, Bombay 400 076, India
The kinetics of the reaction between 10-undecenoic acid and hydrogen bromide solution in toluene initiated by benzoyl peroxide was studied in a batch reactor in the temperature range 0-30 O C . The effects of temperature, initiator concentration, and initial mole ratio of HBr to 10-undecenoic acid on the yield of ll-bromoundecanoic acid and reaction rates were experimentally investigated. This reaction was found to exhibit “limiting conversion” behavior in the range of variables studied. A plausible explanation in the form of a mechanistic scheme for the overall reaction is provided. On the basis of the kinetic results, a free-radical chain reaction mechanism for the formation of 11bromoundecanoic acid from hydrogen bromide and 10-undecenoic acid has been postulated. Hydrobromination of 10-undecenoic acid (10-UA) to 11-bromoundecanoicacid (11-BUA)is one of the important steps in the manufacture of nylon-11 from m t o r oil, which is a renewable agricultural resource. Nylon 11 possesses superior mechanical characteristics and also better physical as well as chemical properties than nylon 6 and nylon 66 for specific applications (Aelion, 1956). Hydrobromination of 10-undecenoic acid can be represented as CH2=CH(CH2)&OOH 10-UA
+ HBr
peroxide
Br(CHJloCOOH (1) 11-BUA
In the presence of a peroxide, the terminal addition of hydrogen bromide takes place by anti-Markovnikov’srule (Kharasch and Mayo, 1933). The aim of this work is to present our experimental investigations on the kinetics of this hydrobromination reaction. This reaction was carried out in the past by various investigators (Walker and Lumsden, 1901; Smith, 1935; Urshibara and Takebayashi, 1938; Jones, 1947; Societe Organico, 1957; Adamek and Scheiber, 1959) using different experimental conditions. Presently at Larsen and Toubro Ltd., Bombay 400 072, India.
The common method of carrying out this reaction is to pass dry HBr gas through a solution of 10-undecenoic acid in a suitable solvent. However, in the present investigation, a solution of HBr in toluene was used and the integral method of analysis of batch reactor data was carried out. Toluene was chosen as the solvent, as the solubility of HBr gas is higher in toluene than in other reported solvents (Obrien and Bobaleck, 1940). Benzoyl peroxide was used as initiator in the reaction studied in the temperature range 0-30 “C.
Experimental Section Materials. 10-Undecenoicacid (Jayant Oil Mills, India) was found to be 98.5% pure, as determined by its iodine value, 135. The maximum concentration of hydrogen bromide in toluene (Sontara Organic Industries, India) was 0.4-0.42 kmol/m3. Additional solution of HBr in toluene was prepared by evolving out HBr gas from its solution in acetic acid and dissolving it in AR-grade toluene. Benzoyl peroxide (Robert-Johnson, India) and all other reagents used in the analysis were of analytical reagent grade. Experimental Setup. Experiments were carried out in a stirred batch reactor made of stainless steel (Bhagwat, 1982). Temperatures in the range of 0-30 “C were main-
0888-588518712626-1536$01.50/0 0 1987 American Chemical Society
