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Feedforward Adaptive Controllers in Chemical Processes
Co pyri g ht © (FAC Adapti ve S, stem s in Control and Signal Processing . Lund . Sweden. (YHti
IMPLEMENTATION OF FEEDBACK! FEEDFORWARD ADAPTIVE CONTROLLERS IN CHEMICAL PROCESSES Ricardo Perez and Lester Kershenbaum Departmellt of Chemical Engineering. Imperial College. Prince Consort Road. London SW7 2BY. L'K
Abstract. Many process systems which are difficult to control with classical controllers show greatly improved performance when adaptive self-tuning controllers are used. Most algorithms previously tested have been purely feedback in structure. In this work, we demonstrate a feedback/feedforward self-tuning controller which can be used in systems with a measurable disturbance, but with unknown (and variable) dynamics. The controller has been studied through simulations and via a series of experiments on a pilot plant characterized by uncertain and variable time delay. The inclusion of the f eedforward made improved control performance and led to more robust behavi out'. INTRODUCTION In recent years, there have been significant advances in adaptive control and, specifically, in the area of self-tuning regulators; the latter have also become widely used in the process industries. Comprehensive surveys of such algorithms and their applications have been presented by Seborg, Shah and Edgar (1983) and Isermann (1985).
(1)
where z-1 is the delay operator, and the polynomials A, B, C are given by: A(z-l) _ 1 + a B(z
fhe majority of these algorithms have been formulated as single-input, single-output feedback controllers, despite the fact that multivariable systems are common in the chemical industry. However, multivariable algorithms require large numbers of parameters to be estimated and also present difficulties in multirate sampled systems.
C(z
-1 -1
+
)
3
b
)
3
Cl +
1
1
• z-l + .... + a • z
n
• z-n
-n+l z
-q
Yk' uk' wk are the output, the control and the measurable disturbance respectively. The operator ~ represents the difference between 2 succes'sive values , viz:
On the other hand, often some of the major disturbances coming into the process can be measured, al though their effect on the outputs may be uncertain.
To simplify the control computations, Eq. can be transformed into a predictive model.
(1)
It has been shown (Fortescue, 1977) that is always possible to find polynomials F and G, such that:
Taking into account the above, it seems natural to consider an adaptive feed forward controller that could be used in multivariable processes, and also in systems with measurable disturbances, avoiding some of the problems of multivariable controllers.
1 - z-T _ F(z-l) • A(z-l) + z-T. G(z-l)
(2)
where T is an integer larger or equal to the time delay in the control, and F, G are polynomials of order T-l and n-l respectively.
Some simple adaptive feedforward controllers have been described by Schumann and Christ (1979), and others more sophisticated, based on LQG self-tuning control, can be seen in Hunt, Grimble and Jones (1986).
Replacing (2) into (I), and conSidering d - 1 and d w - 0 , (their minimum possible vafues) yields:
In
this paper a feedback-feedforward selfregulator is presented. This algorithm retains the robustness and simplicity of the feedback algorithm of Ydstie, Kershenbaum and Sargent (1985).
Y k
~uning
3
Y - + z-T • G(Z-I) • ~Yk + k T
F(Z-I) • B(Z-I) • ~Uk-l + F(Z-I) • C(Z-I) • ~Wk
THE ALGORITHM
(3)
and written in a more compact form:
rhe process can be modelled by the following linear difference equation, in incremental form:
(4)
249
250
R. Perez and Lesler Kershenbaulll It is necessary to use a value of n larger or equal than the control delay. Values of n smaller than du' lead to instability because of identifiability problems, as discussed by Schumann and Christ (1979) and Perez (1986).
where
ii) Delay in the disturbance: Figure 2 shows a perfect control, where the delay in the disturbance is automatically inferred, and the control compensates for variations in 101 at the right time .
.... , 6 q+T )
Through Eq. (4), the parameters of e can be estimated via Recursive Least Squares. The algorithm used also contains a variable forgetting factor to deal with systems with variable dynamics. For more details see Ydstie et aL (1985). Assuming that the parameters given by the estimator are right, the control is computed using a minimum variance extended horizon controller for feedback action (Ydstie et al., 1985), and a minimum variance criterion for the feedforward action. It can be shown (Perez, 1986) that increment in the control is given by: u
1
6U k -
~T·{Y*k+T - i~lai
the
u-1 • 6Y k+ 1 - i -
• 6U k_ i - ~ 6 TH • 6w k+ 1 i=l
-d
i~l~T+i (5)
Where Y*k+T is the set-point at time k+T. In Eq. (5) it was assumed that future controls and disturbances are given by Uk Wk
~
Uk+l =
3
Wk +1 -
(6)
In reality, of course, the control will be re-calculated at the next sampling interval. Expression (6) represents one of the simplest strategies, and good results were obtained with it. The Algorithm must be provided with initial values for n, q and T and a value of a which controls the speed of adaptation for the variable forgetting factor. A similar algorithm can also be designed using state-space variables (Perez, 1986).
RESULTS AND DISCUSSION The performance of the algorithm was studied via both simulation and experiment for systems with several types of time delays. Simulations Figures 1 to 4 show the results obtained with a first-order linear model, with constant parameters, where different delays in the inputs were used. The disturbance W was a measurable square wave. 1) Delay in the control: The control action is delayed 2 sample times (true value of d u - 3). Figure 1 shows a good and stable performance, where variations in the output are due to the delay in the control. Simulations with higher order models show the same behaviour.
In this case, a value of q larger or equal to ~ must be used. i11) Delay in U and W: The control is perfect since both inputs enter the system with the same delay. Figure 3 shows this behaviour. iv) Advanced W: Equation (1) shows that 101 can affect y immediately. Although this is not possible in a continuous model of a physical system, it is clearly possible in a discrete model of a continuous system in which the disturbance enters the process at time, say, k-l/2. Equation (3) takes into account this situation by setting ~ O. 3
Figure 4 shows that an effective control results which surpresses the disturbance as soon as possible. Furthermore, the use of Eq. (3) leads to the estimation of the correct parameters.
In addition, simulations were carried out, in which a value of 1 was used for ~ in Eq. (1). Although the control performance was similar to that of Fig. 4, the parame ters changed considerably each time the value of 101 changed.
Experiments Figure 7 shows a schematic diagram of the pilot plant absortion column used, in which the level of the liquid in the bottom of the packed column was controlled. Wi th normal plant operation the level is controlled by the outlet valve (V1), (configuration /11). However, an artifically difficult arrangement was consturcted, where the control was achieved through the inlet valve (V2), (configuration 82). This abnormal arrangement introduces a large time delay resulting from the trickling flow down the packed column. Furthermore, the time delay was highly dependent on the liquid flow rate itself, the gas flow rate up the column, and the previous state of the column. This delay can vary between 30 and 90 sec. A pro controller is unable to achieve a satisfactory performance under these conditions. a) Configuration Ill: Figure 5 shows the behavIour of the controller when the disturbance has a delay before affecting the output; the result is very similar to the simulations shown in Fig. 2. A perfect control is achieved, despite the large disturbances coming into the systems. b) Configuration 112: In this case the output devIates from the set-point (see Fig. 6) due to the delay in the control action. This behaviour is similar to that of the simulations of Fig. 1.
Implementation of FeedbacklFeedforward Controllers This configuration has been known to cause problems when a feedback-only self-tuner is used (Hiram and Kershenbaum, 1985).
Discussion From figures 1 to 6 it can be seen that the algorithms work perfectly well in those systems affected by time delays, due to an automatic estimation of these delays. Furthermore, based on extensive work in simulations and experiments it has been shown that the proposed regulator is quite robust in terms of the selection of the ini tial values of n, q, T and (J. It is only necessary to be careful in selecting values of nand T larger than the true time delay in the control, and a value of q larger on equal to the delay in the disturbance.
CONCLUSION An adaptive feedback-feedforward regulator was presented and tested with simulations and pilot plant experiments. The regulator has desirable properties of robustness and flexibility as well as excellent performance.
REFERENCES Fortescue, T.R. (1977). "Work on Astrom's Self-tuning Regulator", Department of Chemical Engineering Report, Imperial College, London. Hiram, Y., LoS. Kershenbaum (1985). "Overcoming Difficulties in the Application of Self-Tuning Controllers", Proc. Amer. Control Conf. (Boston), p. 1~ Hunt, K.J., M.J. Grimble and R.W. Jones (1986). "Developments in LQG Selftuning Control", Colloquium on "Advances in Adaptive Control", LE.E. Computing and Control Division, London. Isermann, R. (1985). "Parameter-adaptive Control Systems - A Review on Methods and Applications", 6th IFAC Symnposium on Identification and System Parameter Estimation, York. Perez, J.R. (1986). "Studies on Robust Self-tuning Control", Ph.D. Thesis, to be submitted, University of London. Schumann, R. and T. F. Edgar (1983). "Adaptive Feedforward Controllers for Measurable Disturbances", Joint Automatic Control Conference, Denver. Seborg, D.E., S.L. Sha and T.F. Edgar (1983). "Adaptive Control Strategies for Process Control: A Survey", Presented at AIChE Diamond Jubilee MeetIii&, Washington, D.C. Ydstie, B.E., L.S. Kershenbaum and R.W.H. Sargent (1985). "Theory and Application of an Extended Horizon Controller,"
AIChE Journal, 31, 1771.
251
252
R. Perez and Lester Kershenba ullI
M
II
200 4
100
o -IOlo-J.--""--""'I~OO--1""0--2""0-0-"'2"-50---"30-0-
.0t----:'!"'---::o:---:":""""-........- - - ._ _ 10 15 20 25 30 35 mnls
Fig. 1
Delay in the control
Fig. 5 Configuration 1 M
400 300 200 10
o
.--_..._-.....,...-ooooooy--...--
-IO'()..1._--,_ _
I 0
150
200
20
o
300
8~0-~90~~10-0-..,Ir-10--1~20--13~0--lr40--1~50~
Fig. 2 Delay in the disturbance
mnts
Fig. 6 Configuration 2
-------,
400 300
V2
I I I
MEA
200 100
o
u
JI~
-1 00'--..,....-........--,...--..,....---..--.--_ 150 200 250 300 100
Fi g. 3 Delay in U and Iq
C02 N2 I---'---"""'~-!r - - - - - J
400 V1
300
Fig. 7 Level control in the absortion column
200 100
o -100..1.-_..,...._....,..._--,_ _.__-_-.....,...-300 100
Fig . 4 Advanced W
IMPLEMENTATION OF FEEDBACK! FEEDFORWARD ADAPTIVE CONTROLLERS IN CHEMICAL PROCESSES Ricardo Perez and Lester Kershenbaum Departmellt of Chemical Engineering. Imperial College. Prince Consort Road. London SW7 2BY. L'K
Abstract. Many process systems which are difficult to control with classical controllers show greatly improved performance when adaptive self-tuning controllers are used. Most algorithms previously tested have been purely feedback in structure. In this work, we demonstrate a feedback/feedforward self-tuning controller which can be used in systems with a measurable disturbance, but with unknown (and variable) dynamics. The controller has been studied through simulations and via a series of experiments on a pilot plant characterized by uncertain and variable time delay. The inclusion of the f eedforward made improved control performance and led to more robust behavi out'. INTRODUCTION In recent years, there have been significant advances in adaptive control and, specifically, in the area of self-tuning regulators; the latter have also become widely used in the process industries. Comprehensive surveys of such algorithms and their applications have been presented by Seborg, Shah and Edgar (1983) and Isermann (1985).
(1)
where z-1 is the delay operator, and the polynomials A, B, C are given by: A(z-l) _ 1 + a B(z
fhe majority of these algorithms have been formulated as single-input, single-output feedback controllers, despite the fact that multivariable systems are common in the chemical industry. However, multivariable algorithms require large numbers of parameters to be estimated and also present difficulties in multirate sampled systems.
C(z
-1 -1
+
)
3
b
)
3
Cl +
1
1
• z-l + .... + a • z
n
• z-n
-n+l z
-q
Yk' uk' wk are the output, the control and the measurable disturbance respectively. The operator ~ represents the difference between 2 succes'sive values , viz:
On the other hand, often some of the major disturbances coming into the process can be measured, al though their effect on the outputs may be uncertain.
To simplify the control computations, Eq. can be transformed into a predictive model.
(1)
It has been shown (Fortescue, 1977) that is always possible to find polynomials F and G, such that:
Taking into account the above, it seems natural to consider an adaptive feed forward controller that could be used in multivariable processes, and also in systems with measurable disturbances, avoiding some of the problems of multivariable controllers.
1 - z-T _ F(z-l) • A(z-l) + z-T. G(z-l)
(2)
where T is an integer larger or equal to the time delay in the control, and F, G are polynomials of order T-l and n-l respectively.
Some simple adaptive feedforward controllers have been described by Schumann and Christ (1979), and others more sophisticated, based on LQG self-tuning control, can be seen in Hunt, Grimble and Jones (1986).
Replacing (2) into (I), and conSidering d - 1 and d w - 0 , (their minimum possible vafues) yields:
In
this paper a feedback-feedforward selfregulator is presented. This algorithm retains the robustness and simplicity of the feedback algorithm of Ydstie, Kershenbaum and Sargent (1985).
Y k
~uning
3
Y - + z-T • G(Z-I) • ~Yk + k T
F(Z-I) • B(Z-I) • ~Uk-l + F(Z-I) • C(Z-I) • ~Wk
THE ALGORITHM
(3)
and written in a more compact form:
rhe process can be modelled by the following linear difference equation, in incremental form:
(4)
249
250
R. Perez and Lesler Kershenbaulll It is necessary to use a value of n larger or equal than the control delay. Values of n smaller than du' lead to instability because of identifiability problems, as discussed by Schumann and Christ (1979) and Perez (1986).
where
ii) Delay in the disturbance: Figure 2 shows a perfect control, where the delay in the disturbance is automatically inferred, and the control compensates for variations in 101 at the right time .
.... , 6 q+T )
Through Eq. (4), the parameters of e can be estimated via Recursive Least Squares. The algorithm used also contains a variable forgetting factor to deal with systems with variable dynamics. For more details see Ydstie et aL (1985). Assuming that the parameters given by the estimator are right, the control is computed using a minimum variance extended horizon controller for feedback action (Ydstie et al., 1985), and a minimum variance criterion for the feedforward action. It can be shown (Perez, 1986) that increment in the control is given by: u
1
6U k -
~T·{Y*k+T - i~lai
the
u-1 • 6Y k+ 1 - i -
• 6U k_ i - ~ 6 TH • 6w k+ 1 i=l
-d
i~l~T+i (5)
Where Y*k+T is the set-point at time k+T. In Eq. (5) it was assumed that future controls and disturbances are given by Uk Wk
~
Uk+l =
3
Wk +1 -
(6)
In reality, of course, the control will be re-calculated at the next sampling interval. Expression (6) represents one of the simplest strategies, and good results were obtained with it. The Algorithm must be provided with initial values for n, q and T and a value of a which controls the speed of adaptation for the variable forgetting factor. A similar algorithm can also be designed using state-space variables (Perez, 1986).
RESULTS AND DISCUSSION The performance of the algorithm was studied via both simulation and experiment for systems with several types of time delays. Simulations Figures 1 to 4 show the results obtained with a first-order linear model, with constant parameters, where different delays in the inputs were used. The disturbance W was a measurable square wave. 1) Delay in the control: The control action is delayed 2 sample times (true value of d u - 3). Figure 1 shows a good and stable performance, where variations in the output are due to the delay in the control. Simulations with higher order models show the same behaviour.
In this case, a value of q larger or equal to ~ must be used. i11) Delay in U and W: The control is perfect since both inputs enter the system with the same delay. Figure 3 shows this behaviour. iv) Advanced W: Equation (1) shows that 101 can affect y immediately. Although this is not possible in a continuous model of a physical system, it is clearly possible in a discrete model of a continuous system in which the disturbance enters the process at time, say, k-l/2. Equation (3) takes into account this situation by setting ~ O. 3
Figure 4 shows that an effective control results which surpresses the disturbance as soon as possible. Furthermore, the use of Eq. (3) leads to the estimation of the correct parameters.
In addition, simulations were carried out, in which a value of 1 was used for ~ in Eq. (1). Although the control performance was similar to that of Fig. 4, the parame ters changed considerably each time the value of 101 changed.
Experiments Figure 7 shows a schematic diagram of the pilot plant absortion column used, in which the level of the liquid in the bottom of the packed column was controlled. Wi th normal plant operation the level is controlled by the outlet valve (V1), (configuration /11). However, an artifically difficult arrangement was consturcted, where the control was achieved through the inlet valve (V2), (configuration 82). This abnormal arrangement introduces a large time delay resulting from the trickling flow down the packed column. Furthermore, the time delay was highly dependent on the liquid flow rate itself, the gas flow rate up the column, and the previous state of the column. This delay can vary between 30 and 90 sec. A pro controller is unable to achieve a satisfactory performance under these conditions. a) Configuration Ill: Figure 5 shows the behavIour of the controller when the disturbance has a delay before affecting the output; the result is very similar to the simulations shown in Fig. 2. A perfect control is achieved, despite the large disturbances coming into the systems. b) Configuration 112: In this case the output devIates from the set-point (see Fig. 6) due to the delay in the control action. This behaviour is similar to that of the simulations of Fig. 1.
Implementation of FeedbacklFeedforward Controllers This configuration has been known to cause problems when a feedback-only self-tuner is used (Hiram and Kershenbaum, 1985).
Discussion From figures 1 to 6 it can be seen that the algorithms work perfectly well in those systems affected by time delays, due to an automatic estimation of these delays. Furthermore, based on extensive work in simulations and experiments it has been shown that the proposed regulator is quite robust in terms of the selection of the ini tial values of n, q, T and (J. It is only necessary to be careful in selecting values of nand T larger than the true time delay in the control, and a value of q larger on equal to the delay in the disturbance.
CONCLUSION An adaptive feedback-feedforward regulator was presented and tested with simulations and pilot plant experiments. The regulator has desirable properties of robustness and flexibility as well as excellent performance.
REFERENCES Fortescue, T.R. (1977). "Work on Astrom's Self-tuning Regulator", Department of Chemical Engineering Report, Imperial College, London. Hiram, Y., LoS. Kershenbaum (1985). "Overcoming Difficulties in the Application of Self-Tuning Controllers", Proc. Amer. Control Conf. (Boston), p. 1~ Hunt, K.J., M.J. Grimble and R.W. Jones (1986). "Developments in LQG Selftuning Control", Colloquium on "Advances in Adaptive Control", LE.E. Computing and Control Division, London. Isermann, R. (1985). "Parameter-adaptive Control Systems - A Review on Methods and Applications", 6th IFAC Symnposium on Identification and System Parameter Estimation, York. Perez, J.R. (1986). "Studies on Robust Self-tuning Control", Ph.D. Thesis, to be submitted, University of London. Schumann, R. and T. F. Edgar (1983). "Adaptive Feedforward Controllers for Measurable Disturbances", Joint Automatic Control Conference, Denver. Seborg, D.E., S.L. Sha and T.F. Edgar (1983). "Adaptive Control Strategies for Process Control: A Survey", Presented at AIChE Diamond Jubilee MeetIii&, Washington, D.C. Ydstie, B.E., L.S. Kershenbaum and R.W.H. Sargent (1985). "Theory and Application of an Extended Horizon Controller,"
AIChE Journal, 31, 1771.
251
252
R. Perez and Lester Kershenba ullI
M
II
200 4
100
o -IOlo-J.--""--""'I~OO--1""0--2""0-0-"'2"-50---"30-0-
.0t----:'!"'---::o:---:":""""-........- - - ._ _ 10 15 20 25 30 35 mnls
Fig. 1
Delay in the control
Fig. 5 Configuration 1 M
400 300 200 10
o
.--_..._-.....,...-ooooooy--...--
-IO'()..1._--,_ _
I 0
150
200
20
o
300
8~0-~90~~10-0-..,Ir-10--1~20--13~0--lr40--1~50~
Fig. 2 Delay in the disturbance
mnts
Fig. 6 Configuration 2
-------,
400 300
V2
I I I
MEA
200 100
o
u
JI~
-1 00'--..,....-........--,...--..,....---..--.--_ 150 200 250 300 100
Fi g. 3 Delay in U and Iq
C02 N2 I---'---"""'~-!r - - - - - J
400 V1
300
Fig. 7 Level control in the absortion column
200 100
o -100..1.-_..,...._....,..._--,_ _.__-_-.....,...-300 100
Fig . 4 Advanced W