I n d . Eng. Chem. Res. 1989, 28, 557-563
557
Adaptive Inferential Control for Chemical Processes with Intermittent Measurements Gwo-Chyau Shent and Won-Kyoo Lee* Department of Chemical Engineering, The Ohio S t a t e University, Columbus, Ohio 43210
An adaptive inferential control algorithm is proposed t o improve the control performance of the inferential control system which has been developed to address the problem of intermittent measurements in the presence of unmeasured disturbances. A stable approximation to the inverse of a primary process model forms the basis of the inferential control law. While frequent measurements of the secondary process output are used t o update the secondary process model and to estimate the effect of load disturbance, the inferential controller is adapted by using intermittent measurements of the primary process output. T h e use of the adaptive inferential controller based on a discrete convolution model is examined and compared with one based on a discrete transfer function model. Simulation results show that the proposed adaptive inferential control system performs well and provides improvements over conventional PID and inferential control in the face of unmeasured load changes and process variations. 1. Introduction Quite often, chemical processes are inadequately controlled because frequent measurements of the primary process outputs are not feasible and processes are often subject to unmeasurable disturbances that vary slowly or change abruptly. To cope with the problems of measurement limitation and unmeasurable disturbances, a control scheme called inferential control was developed by Brosilow and his co-workers (Joseph and Brosilow, 1978a,b; Brosilow and Tong, 1978). This inferential controller uses measurements of secondary process outputs to infer and then counteract the effect of unmeasured disturbances on the primary outputs through the use of the inverse of a primary process model as the controller. The performance of the inferential control algorithm depends heavily on the goodness of the process model due to the nature of its control structure. Unfortunately, most chemical processes are nonlinear and often time-varying in nature, resulting in changes in their dynamic characteristics during operation. These changes can degrade the performance of an inferential control system designed with a particular process model. This implies a need for redesigning the inferential control system to accommodate changing process characteristics. Adaptive inferential control systems were proposed (Wright et al., 1977; D’Hulster and Van Cauwenberghe, 1981) by using a direct estimation of the primary process outputs from secondary process outputs and manipulated variables. The estimator parameters were updated on-line using infrequent measurements of the primary process output and then used in their adaptive feedback controllers. However, it did not work satisfactorily (Wright et al., 1977), or it required a priori knowledge of the secondary process model (D’Hulster and van Cauwenberghe, 1981). It is also worth noting that these two approaches to an adaptive inferential control system are feedback in nature, requiring the closed-loop stability to be taken into consideration for the design of the control system. In this paper, the adaptive inferential control algorithm (Shen and Lee, 1985,1988) is extended for chemical processes with intermittent measurements. This adaptive inferential control algorithm makes use of the advantages *To whom all correspondence concerning this paper should be addressed. Current address: Inland Steel Company, Research Laboratories, East Chicago, IN 46312.
of the adaptive and the inferential control schemes by combining these two approaches in an appropriate way. More specifically, the main structure of the inferential control scheme is used to estimate the effect of unmeasured disturbances on the primary controlled variables, while the process models and the controller are being updated by using a recursive parameter estimation method. Since a stable approximation to the inverse of the primary process model is used as the controller in the inferential control structure, the use of adaptive inverse modeling based on a discrete convolution model and intermittent measurements of the primary process output (Shen and Lee, 1985, 1988) is examined. Moreover, the controller structure of Vogel-Edgar’s adaptive dead-time compensator (Vogel and Edgar, 1980) is investigated for its possible use as an alternative design for the inverse controller. A numerical example is employed to illustrate and compare the performance of the proposed adaptive inferential control system with those of the conventional PID controller and the inferential control.
2. Inferential Control In the inferential control system developed by Brosilow and his co-workers (Joseph and Brosilow, 1978a,b;Brosilow and Tong, 1978), available secondary measurements are selected to minimize the steady-state error of a linear estimator, and then the selected measurements are used to infer and counteract the effect of unmeasured disturbances on primary process outputs, as shown in Figure 1. The closed-loop response of this control system, excluding GL, G,, and G,, is as follows:
where
Basically, the appropriate choice for the controller G,(s) is an stable inverse of the process transfer function C(s). The general design methods for the con_troller,GI(s), the estimator, a ( s ) ,and the process model, P(s),are discusFed in their work. It has been pointed out that, even if d ( s ) # d ( s ) and GI(s) # C-l(s), this control system will be stable if the controller, GI(s), is stable and P(s) is a sufficiently good approximation to P(s). Furthermore, the
0888-5885/89/2628-0557$01.50/0 0 1989 American Chemical Society
558
Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989
CONTROLLER
Figure 1. Inferential control structure for processes with intermittent measurements.
problem of matching P(s)to P(s) is crucial to the stability of the inferential control system when the gain of either P(s) or a ( s ) is high. Thus, when P(s) is varies, the inferential control system with fixed model parameters may not work satisfactorily and may even become unstable when a certain type of modeling error arises. Under these circumstances, it is suggested that other control algorithms be considered. The inferential control approach is similar to a feedforward contro! with measurable disturbances. More specifically, if P(s) = P ( s ) , the dynamic behavior of the inferential control system with imperfect estimation resembles the feedforward control system with imperfect compensation. Since the pioneering work of Brosilow and his co-workers, comparative studies of the inferential control system with the conventional feedback control system based on direct secondary measurement have been performed by using experiments or simulations (Patke and Deshpande, 1980, 1982; Lee and Kim, 1984). All results show that the inferential control system is superior to a conventional feedback control scheme in rejecting unmeasured load disturbances. However, steady-state offset was found to be a potential problem with the inferential control system (Patke and Deshpande, 1982). In the inferential control system, a zero steady-state offset in the primary controlled variable can be achieyed only if the following three conditions are satisfied: (i) P(0) = P(O),(ii) C(O)G,(O) = I , and (iii) R(0) = 0. It is clear, therefore, that addition of a slow feedback control with integral action is always desirable to eliminate steady-state offset. Generally, a PI controller with a low gain and a small amount of integral action can serve this purpose, e.g., the G,(s) in Figure 1. Moreover, it has been shown that an appropriate choice for GL(s) in Figure 1 is
G L ~ =) G,(s)C(s)G~(sl
(3)
where G,(s) is the measurement lag (Brosilow and Tong, 1978). With GL(s), the response of the inferential control system can match that of the feedback control system. 3. Adaptive Inferential Control System Design The block diagram of the adaptive inferential control system is shown in Figure 2. For adaptation of the inferential control system, the recursive least-squares (RLS) method with variable forgetting factor (Fortescue et al., 1981) and random walk (Holst and Poulsen, 1984; Wellstead and Zanker, 1982) is used to estimate the model parameters for both processes and to adjust the controller. A process model between the input and the secondary output, P(z-l),is updated at a regular sampling time using frequent measurements of the secondary output. On the
Figure 2. Adaptive inferential control structure for processes with intermittent measurements.
other hand, the primary process output-input model, d(z-l), is updated a t a smaller sampling frequency due to the intermittent measurements of the primary output. This model is then used for adaptation of the controller, G1(z-l). Two types of controllers are investigated for use in the adaptive inferential control systems: the adaptive deconvolution model (Shen and Lee, 1985,1988) and the controller of Vogel-Edgar’s adaptive dead-time compensator (Vogel and Edgar, 1980). Both of these provide a stable approximate inverse to the primary process transfer function even if it may be a nonminimum-phase system. Thus, a factorization of the process model in real-time is not required in the design of the adaptive inferential control system. However, additional modifications to the adaptive deconvolution model as well as the primary process model are necessary due to the multirate sampling problem. Design details of the proposed adaptive inferential control system are described below. 3.1. Process Model. Process models are required to estimate the effect of unmeasured disturbances on the process outputs and to design the inferential controller. Two types of model structures are used in this study, and their effects on the control performance were examined: transfer function and convolution models. A general discrete transfer function model can be expressed by using a difference equation n
y ( k ) = X a y ( k - i) t=1
n
+ X b , u ( k - m - i) + d
(4)
1=1
where y is the process output, u is the process input, n is the model order, m is the dead time expressed as a multiple of sampling period, and d is a bias term. Without including the bias term, d, the model parameters would be erroneously estimated when unmeasured load disturbances are present. However, special care needs to be taken to obtain correct model parameters when eq 4 is used with RLS, as will be mentioned later. Due to the intermittent measurements of the primary process output, use of the ordinary transfer function model, eq 4, for the primary process was found to be unsatisfactory in this study. This can be explained as follows: Suppose the primary process output is available every Nth sampling time; a first-order model for the primary process would be y ( k ) = a,y(k - N ) + b1u(k - N ) + d 15) Because the secondary process model is updated every sampling time based on the frequently available secondary process output, control action is adjusted equally fast to accommodate the process changes and consequently result
Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989 559 in an efficient control. As a result, any change in the control action occurring between k and k - N sampling instants will not be accounted for when eq 5 is updated. Thus, a sudden change in the controller output due to set point or load change will lead to a failure in adaptation of the primary process model, which in turn results in oscillation of the updated controller parameters and, therefore, a very unsatisfactory control. A remedy for this is to incorporate all the inputs between k and k - N sampling instants in the process model. Thus, eq 5 is modified as
It should be noted that the bi parameters in eq 6 obtained through RLS no longer accurately represent the influence of each individual input term on the process output since this model is updated every Nth sampling time. However, this will not lead to any problem in the construction of the inferential controller because these b parameters are not used individually in adaptation of the controller parameters, as will be seen later in the section on controller design. The second type of process model used is a discrete convolution model represented by P
y(k) = Ch,u(k - m - i) i=l
+d
(7)
where the hi’s are impulse response coefficients of the process. Again, for the primary process model, the hi coefficients obtained through performing RLS every Nth sampling time are no longer good representatives of the individual influences of the input terms on the process output. Since adaptation of a deconvolution controller depends only on the model input and output but not the convolution coefficients, such a misrepresentation is really of little importance. One advantage of the discrete convolution approach is that the model coefficients can be determined easily from simple experimental tests, such as from step response data. The discrete convolution models have become more and more popular recently to represent process dynamics in the development of digital predictive control algorithms. In particular, discrete convolution models do not require a priori knowledge of the order of a process, but an approximate knowledge of the settling time is useful in determining the number of coefficients to be used in the model. The disadvantage of using the discrete convolution model with adaptive control is the significant amount of computation work needed if the ratio of the process settling time to the sampling time is large, even when dead time is not present. 3.2. Deconvolution Controller. As stated previously, a stable approximation to the inverse of a process model is used as the controller in the inferential control system. As a result, a factorization is sometimes required in conjunction with a filter with adjustable parameters. To resolve this problem, the controller is designed based on a discrete inverse convolution model (deconvolution model) which approximates the inverse of the process model. Readers are referred to Shen and Lee (1985, 1988) for details of the deconvolution model. To update the deconvolution controller, the process model output with the bias term, d , being excluded is used instead of the process output for the following reason: In the presence of unmeasured load disturbances, the use of the process output yields an erroneous inverse which additionally accounts for a change in the process output occurring as a result of the load disturbances.
Furthermore, additional modification to the deconvolution controller needs to be considered due to the multirate sampling problem. Adaptation of the deconvolution controller is performed every Nth sampling time at the same time when the primary process model is updated. Discrepancy in the time structure arises when this deconvolution controller is used every sampling time to update control action. As a result, such a control performance would be very sluggish, depending on the difference between the two sampling rates. One way to resolve this problem is to reduce the equivalent time constant of the deconvolution controller to make it more responsive. This can be achieved by using a filtered model output for adaptation of the deconvolution coefficients. More specifically, the primary process model output from either eq 6 or eq 7 is passed through a first-order filter before it is used in eq 6 to update the deconvolution gi coefficients (8) where y is the model output and yf is the filtered model output. Since a deconvolution model approximates the inverse of its corresponding process model, such a damped process model output will make the deconvolution model more responsive. This modification has been tested in our simulation study, and the result was found to be satisfactory. The constant in eq 8 must be chosen according to an approximate time constant of the primary process and the difference between the two sampling rates. Generally, large primary model time constants and large differences between the two sampling rates require a small value of 0. An off-line test is usually required for a satisfactory performance. To further ensure that the steady-state gain of the deconvolution model is a reciprocal of that of the process model, the controller is modified by the following equation: P
(9) i=l
where C(1) is the steady-state gain of the primary process model and GI(z-’) is the deconvolution controller. It has been mentioned previously that the stability problem for the inferential co_ntrolsystem is trivial when the secondary process model, P(s),is a good approximation to the secondary process, P(s). However, stability problem may arise when a certain type of modeling error occurs. The stability problem becomes even more serious when IP(0)l or Ia(0)l is large. This problem has been resolved to some extent by updating the secondary process model to accommodate process changes. To further improve the stability, a stabilizing first-order filter is employed in the controller (Garcia and Morari, 1982) based on the structural equivalence between the inferential control and the Internal Model Control (IMC) (Brosilow, 1979): where 7 is an adjustable constant, u is the controller output, and u, is the filtered controller output which is actually used as a control input to the process. In addition to accounting for expected modeling errors, this filter also serves as a detuner to prevent excessive control action. 3.3. Vogel-Edgar’s Controller. Vogel and Edgar’s (1980) adaptive dead-time compensator has been analyzed and found to have a very similar structure to the internal model control system (Shen and Lee, 1985,1988). When reconstructed from the viewpoint of Internal Model Control, the Vogel-Edgar’s dead-time compensator has an
560 Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989
equivalent inverse controller of the following torm: n
1 GI(2-l)
=
+ i=l Caiz-' (1 - exp(-T/X))z-'
(2bi)z-'
1 - exp(-T/X)z-'
(11)
i=l
where Tis the sampling period and X is the tuning constant in Dahlin's algorithm. This controller can be viewed as a combination of a first-order filter and a discrete convolution model approximating the stable inverse of the process model. An obvious advantage of this type of inverse controller is that the controller parameters can be obtained directly from updated process parameters without any complicated calculations. However, due to the multirate sampling problem, the a and b parameters of the process model need to be adjusted before they are used in the controller of eq 11. This can be illustrated using a simple first-order process model: If the process is ~ / ( T s+ 11, the corresponding discrete equation based on N sampling periods is expressed as eq 6 with a, = exp(-NT/i)
(12)
Since the controller is used every sampling time to update the control action, the desired al should be exp(-T/r). Thus, a conversion from exp(-NT/r) to exp(-T/r) can be easily done by taking the appropriate exponential of the estimated a , where is used for the controller and is the estimated parameter. The sum of the b parameters is then converted based on the fact that the steady-state gain is independent of the sampling period. Therefore, we have
A similar but more complicated conversion procedure is required when a higher order process model is employed for the primary process. 3.4. Estimator. An estimator is used to infer the effect of unmeasured disturbances on the primary process output from frequent measurements of the secondary output. An optimal estimator via the Kalman filter method can be used for this purpose, but it results in a complex optimal estimator which is generally less attractive to implement (Joseph and Brosilow, 1978a,b). Furthermore, it has been reported that there is little difference in the performance between optimal estimator and suboptimal estimator of a lead-lag network type (Joseph and Brosilow, 197813). On the basis of these findings, the estimator of the lead-lag filter type is used in the adaptive inferential control system. Since the stability problem has been reduced by updating the secondary process model, as well as use of a stabilizing filter with the controller, the use of a possible high gain and the presence of modeling errors in the estimator may not ,pose serious problems. Therefore, a discrete version of the lead-lag network with constant parameters is used in this study. Nevertheless, steadystate offset will result when the estimator has an incorrect steady-state gain. This problem can be resolved by adding a slow PI feedback controller to the adaptive inferential control system, as suggested by Brosilow and Tong (1978). 3.5. Parameter Estimation Algorithm. The recursive least-squares (RLS) method is chosen here for parameter estimation because of its simplicity and reliable conver-
gence. However, use of a standard RLS in the presence of load disturbances was found very unsatisfactory even if a bias term had been included in the process model. Any change in load disturbances will immediately lead to incorrect estimates of model parameters and may last for a relatively long period before accurate parameter estimation can be resumed. A similar phenomenon was also observed by others (Vogel and Edgar, 1982; Lee and Hang, 1983, 1985). Method for solving the problem of abrupt process changes have been suggested (Vogel and Edgar, 1982; Hagglund, 1984; Holst and Poulsen, 1984). The main feature of the adaptive inferential control is its ability to maintain good control performance in the presence of process parameter changes and sudden load changes. Since a sudden load change mostly degrades the parameter estimation and the performance of the control system, the emphasis has been placed on adapting the process model so that the influence of sudden load changes can be minimized. This is done by modifying the standard RLS with the variable forgetting factor (Fortescue et al., 1981) and a very cautious use of random walk (Holst and Poulsen, 1984; Wellstead and Zanker, 1982) as well as several other modifications. It should be noticed that great care is required when choosing the tuning parameters for either the variable forgetting factor or the random walk. Inappropriate use of them may cause numerical problems. In addition, proper excitement of the process is needed to provide sufficient dynamic information for parameter estimation, particularly when the variable forgetting factor and random walk are used. A weighted M sequence with appropriate magnitude (Koshiwagi, 1974) has been found very useful in this regard. 4. Simulation Results and Discussion
In this section, simulation results of one numerical example are presented and discussed. In order to compare the performances of several different controllers, step changes in set point and load disturbances were introduced as follows: time 0.0 300.0 400.0
set point 0.0 0.5 0.5
650.0
0.2
load 0.0 0.0 30.0 30.0
Upon load change, all the steady-state gains and time constants of the four process transfer functions are changed accordingly, as shown below: C(s) =
~
0.02 16s 1
+
-
-0.05 P(s) = 4s
B(s) =
+1
-0.01 14s 1
___
+
0.03 6s 1
-
A(s) = -
+
0.018 14.4s 1
+
___
-0.04 3.2s 1
+
(16)
-0.008 12.6s 1
(17)
+
0.027 4.8s + 1
~
In this simulation study, it is assumed that the analysis of the primary process output takes 12 time units. More precisely, the primary output is available every 12 time units with a time delay of 12 time units, while the secondary process output is sampled every 2 time units without measurement lag. Figure 3 shows the performance of a PID feedback control system using the intermittent measurements of the primary output. Owing to the long dead time associated with the sampling of the primary output, the control re-
Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989 561 8
8 I C U I T H I N I T I A L L Y PERFECT MODEL
si si
d.lk
d.28
d.57
d.42
T
d.71
d.85
Figure 3. PID feedback control using infrequently measured primary process output.
s
51
I C U I T H I N I T I A L L Y PERFECT MODEL
0.85
1.00 I
o
m
I C WITH I N I T I A L L Y PERFECT MODEL
"1 "1
si .,I
I-
00.00
0.71
Figure 5. Estimate of load disturbance effect on secondary process output using inferential control system with initially perfect model. o Y )
1'sl 1'sl
$ j!
d.q.1030.57
i.00
a103
, 0.14
, 0.28
,
, 0.12
T
0.57
, 0.71
, 0.85
, 1.00
d.14
.io3
Figure 4. Inferential control with initially perfect model.
sponse to load change is relatively poor, as illustrated in Figure 3. An even worse result is highly possible with a longer sampling period. Nevertheless, the PID controller was able to bring the primary output back to its set point after a long enough time given. An inferential control system with initially correct model parameters is then applied to the process. The controller used is a discrete equivalent of the following lead-lag element: GI(s) = 50.0-
d.28
d.k2
T .I@
0.57
d.71
d.85
i.00
Figure 6. Estimate of load disturbance effect on primary process output using inferential control system with initially perfect model. 0
0
AICIDC
o/ 8/
16s + 1 6s + 1
The control performance is presented in Figure 4. Compared to Figure 3, this inferential controller provides a good disturbance rejection. However, due to changing process dynamics, this inferential control system with fixed model and controller parameters fails to bring the controlled variable back to the set point. Figures 5 and 6 show estimates of the effect of load disturbance on the .secondary and primary outputs, respectively. As can be seen in these two figures, the estimator has a problem obtaining good estimates of the effect of load disturbance due to modeling errors. The adaptive inferential control system is then investigated using the same process. First, an adaptive inferential control using the convolution model and deconvolution controller is applied to the process. The estimator has correct parameters initially. In addition, six coefficients with no time delay were employed in the deconvolution controller. Since all the convolution and deconvolution coefficients were assumed zero initially, a weighted
Figure 7. Adaptive inferential control with deconvolution controller based on convolution model.
M sequence with a maximum magnitude of 5.0 was used for the first 120 time units to obtain initial estimates of all model and controller parameters. This maximum magnitude was then reduced to 1.0 to continually excite the process. The control performance is presented in Figures 7-9. With p = 0.1 and 7 = 0.2 in eq 8 and 10, the adaptive inferential controller does a reasonably good job in disturbance rejection. However, as can be seen in Figure 7, steady-state offset still seems to be a problem because of modeling errors in the estimator. This is reflected in Figures 8 and 9. Figure 8 shows the improved estimate
Ind. Eng. Chem. Res., Vol. 28, No. 5 , 1989
562
“1
1 ,k------”.
8
8
0
0
0.00
0.14
0.28
0.42
T
0
51 RJ 0.57
6.71
d.05
700
‘ 0 0 . 0 0
Figure 8. Estimate of load disturbance effect on secondary process output using adaptive inferential control system based on convolution model.
d.20
d.42
T
g RIC*OC
d.57
d.71
d.05
/,OO
a103
Figure 1 1. Adaptive inferential control with Vogel-Edgar’s inverse controller based on transfer function model. o
E
d.14
a103
AICWE
* PI
-1
0 0
: d.14
d.28
d.42
T
E
000
d.57
d.71
0.85
.io3
Figure 9. Estimate of load disturbance effect on primary process output using adaptive inferential control system based on convolution model. 4
AICtDC
014
PI
-1
020
057
042
T
7.00
071
085
IO0
mi03
Figure 12. Adaptive inferential control with Vogel-Edgar’s inverse controller plus slow PI controller.
applied, and the control performance is presented in Figure 11. The filter constant of 0.2 is used, which is the same as the previous case. Again, offset exists due to modeling error in the estimator. With the addition of a slow PI feedback controller, the control performance is improved, as seen in Figure 12. 5. Conclusions
“1
81 “pd.00
0
9 d.l+
d.2E
d.42
T .I@
d.57
6.71
d.E
i.00
Figure 10. Adaptive inferential control with convolution model plus slow PI controller.
of load effect on the secondary output, while estimation error in load effect on the primary output still exists, as seen in Figure 9. To eliminate the steady-state offset, a slow PI controller with a very small controller gain and a small integral action is added to the adaptive inferential control system. As can be seen, in Figure 10, the offset problem is reduced and the control performance is improved to some extent. Finally, the adaptive inferential control system using the transfer function model and Vogel-Edgar’s controller is
In the adaptive inferential control system for processes with intermittent measurements in the presence of unmeasurable disturbances, the adaptive deconvolution controller and Vogel-Edgar’s inverse controller are used in conjunction with on-line parameter estimation of the process models to improve the robustness of the inferential control system. The stability problem of the inferential control system is reduced to some degree by updating the secondary process model to minimize modeling errors. In addition, a controller filter is used to further ensure the stability of the control system, with the price of a little sluggish control response paid. Both the deconvolution controller and the Vogel-Edgar’s inverse controller provide a current stable approximation to the inverse of the primary process model based on the intermittent measurements of the primary output. Furthermore, either the minimum-phase or nonminimum-phase process can be handled effectively without a factorization often required in the design of an inferential controller. Simulation results demonstrate that the proposed adaptive inferential control provides an improved regulatory response in the presence of process variations and
Ind. Eng. Chem. Res., Vol. 28, No. 5, 1989 563 intermittent measurements over conventional PID and the inferential control with fixed parameters. Compared with the Vogel-Edgar’s inverse controller based on a transfer function model, the deconvolution controller coupled with a convolution model exhibits the comparable results even though its adaptation is computationally intensive. However, steady-state offset seems to be a potential problem with the proposed adaptive inferential control system when the estimator is not properly designed. Addition of a slow PI feedback is required to alleviate this problem. Although direct adaptation of the estimator is not considered here, a recursive parameter estimation could be used as a self-tuning aid to provide useful information for design of the estimator. Acknowledgment Partial financial support from the Office of Research and Graduate Studies of The Ohio State University under the University Seed Grant program is gratefully acknowledged. Nomenclature A = secondary disturbance process B = primary disturbance process a, b = coefficients for model transfer function ai,,, bi,c = model coefficient used for controller ai,e,bi,e.= estimated model coefficient C = primary process $ = effect of load disturbance on primary process output d = estimate of d F = defined by eq 2a G, = PI controller GL = filter used in PI control feedback loop GI = inferential controller G, = measurment lag g = discrete deconvolution model coefficient h = impulse response coefficient L = unmeasured load disturbance N = sampling time for primary process output n = order of discrete model transfer function = secondary process P = secondary process model p = order of discrete deconvolution model q = order of discrete convolution model R = defined by eq 2b r = order of overparameterized model u = process input u, = filtered process input y = primary process output yf = filtered process output Greek L e t t e r s a = estimator p = filter constant for estimated model parameters y = delay used in discrete deconvolution model 9 = filter constant for inferential controller 0 = secondary process output X = tuning constant in Dahlin’s controller T
= process model time constant
L i t e r a t u r e Cited Brosilow, C. B. The Structure and Design of Smith Predictors from the Viewpoint of Inferential Control. Proceedings of the Joint
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