- Email: [email protected]
An Adaptive Optimizer for Process Control
Copyright <0 IFAC Control Applications and Ergonomics in Agriculture, Athens, Greece, 1998
AN ADAPTIVE OPTIMIZER FOR PROCESS CONTROL
N. Rerras, A.Anastasiou, K. Feredinos, N.Sigrimis Department of Agricultural Engineering Agricultural University of Athens 75 Iera Odes St. Athens 11855, Greece e-mail: [email protected]
Abstract: An optimization based methodology for control of irrigation and nutrient supply is developed using common measurements of greenhouse climate. The method allows onsite on-line identification of plant water needs and as an added benefit provides information for the creation of crop transpiration models. Copyright © 1998 IFAC Keywords: models, intelligent control.
1. INTRODUCTION
drain measurement and model estimate the coefficients of the model are iteratively adapted. The adaptation process is continuous so that the model accounts for seasonal variation of crop growth.
Contemporary greenhouse operations require precise control of irrigation and nutrient supply in order to optimize crop growth and minimize cost and to effluents. Moreover in pollution due Mediterranean countries there is a need to minimize water waste due to seasonal shortages. In modern greenhouses nutrient supply is computer controlled and based on measuring salinity and compensating deficiencies by a mix of clean water and two stock nutrient solutions. The process of applying this solution to the crop presents several control problems such as time delays and seasonal variations due to plant growth. Several proposals to the nutrient supply problem have been proposed such as direct feedback control of drain water flow in both closed and open growing systems using flow measurement of the drain water (Gieling, 1997). On the other hand progress has been achieved in model prediction of crop irrigation needs. Usually a transpiration model (Stanghellini 1987) is used that predicts plant transpiration based on ambient conditions (temperature, solar radiation, CO2 concentration and vapor saturation deficit). This approach leads to open loop control of water and nutrient supply based on model estimation of plant needs (Hamer 1997).
2. MATERIALS AND METHODS
In common greenhouse growing practice the irrigation water is supplied in specified time cycles. In each cycle the amount of water supplied, (designated V o) , is determined by the grower manually in order to satisfy crop requirements. The approach taken is to ensure that the growing medium saturates by the presence of an amount of drain water. The desired drain is typically around 20% of the irrigation amount (Vo) , required to secure desired nutrients concentration in the root environment. If we define the fraction of drain water as f, and the true crop requirement, which is the amount of real crop evapotranspiration, as E·, then the current growing practice is expressed by E·
=(1-f) V
(1)
o
This practice can be depicted in the leftmost part of Fig.I. Assume that a model exists relating crop transpiration to climate conditions, of the form E=aS+~(1-RH)+yT
This work proposes an intermediate approach where a crop transpiration model of convenient form is used to predict the necessary supply of water. At the same time drain water flow from the crop is measured using an appropriate device. Using the error between
(2) 2
where crop transpiration E (kglm /s) is related to solar radiation S(yVIrn2), humidity deficit (I -RH) and ambient air temperature T (0C), through suitable
189
irrigat.dose=Vo
e=Em -E· Vo :
T*
o
Figure 1: The concept of model prediction of irrigation needs and its error appearance in measured drain. coefficients (a,p,y), then crop transpiration E* could be calculated by integrating this model for the duration of the irrigation cycle. However in the general case the model of Eq.2 has unknown parameters dependent on factors such as crop type, growth stage (LA!) etc.
The alternative proposed here is to utilize this measurement in order to determine the unknown parameters of Eq.2. Then with well tuned parameters the irrigation process could be controlled in a feedforward loop. Moreover if the parameters of Eq.2 are continuously estimated then the system can account for process drift. Parameter tuning can be accomplished using a minimization algorithm if an appropriate performance function is devised. In this case the performance criterion to minimize is simple to create. During the irrigation cycle the integral of Eq.2 is monitored so that the estimated evapotranspiration (Em) is available
However if a measurement of the drain water is available then the true crop transpiration can be determined by integrating drain water rate and using the equation
E· =Vo - Vdm
(3)
Where: Vdm is the integrated drain water. In fact this measurement is used in feedback control of irrigation in closed systems (Gieling 1997). This type of control requires careful design due to time delays present in the irrigation process. Also due to system drifts the controller design should incorporate robustness. This is shown in fig.2 where the drain water is shown accumulated at the end of the irrigation cycle. This time delay can be many minutes, and depends on the length of the irrigated line and the characteristics of the substrate.
As depicted in fig. 1 if for example eq2 underestimates the transpiration this will lead to a longer irrigation cycle, that is delayed application of irrigation dose. The real transpiration shall be higher than estimated and this results to more water retained in the substrate and less drain will be measured. This measured error in the expected drain is actually the error of transpiration estimate:
(5)
190
The optimizer module is an iterative search algorithm that can adjust the model parameters in order to minimize the specified error metric. The algorithm is an enhanced variation of the Hooke-Jeeves (Shoup, 1979) minimization algorithm, and is embedded for general use in the MACQU greenhouse control software (Anastasiou et aI, 1998). It does not need any process model or gradient information to conduct parameter searching. Instead it explores the parameter space modifying one parameter at a time searching for a gradient. After all parameters have been explored the composite direction of steepest descent is formed. The algorithm then enters a phase of "pattern" moves where this direction is used until progress is exhausted. The algorithm cycles between exploration and pattern searches until a local minimum is reached. Earlier experience with application of this algorithm has shown it to be effective in optimizing the parameters for a model based mist control system (Sigrimis and Rerras, 1996). In this case the search for better parameters never stops in order for the algorithm to detect drifts of the system characteristics (i.e. plant growth etc.). In order to achieve better fitting of the search parameters and shorter search times the parameter space is split into regions of similar environmental conditions as determined by solar radiation and temperature. A separate set of parameters is created for each of these regions resulting in the formation of a parameter dictionary. Each time the environmental conditions shift to a different region the optimizer switches to the appropriate parameter set from the dictionary and continues with a new experiment.
In a form more suitable for the optimization algorithm the performance criterion J is the percent relative error squared
(6) All the pieces have come together for the presentation of the algorithm that accomplishes the task of tuning the parameters and controlling irrigation.
Irrigation Optimization algorithm l.
2. 3. 4.
5. 6. 7. 8.
Define Vo, f Select initial parameters a,~,y Apply Vo Do Loop Calculate Em (from Eq. 4) Measure drain and integrate Vdm While Em < (1-f) Vo Compute E* (from Eq.3) Compute J (from Eq.6) Call OPTIMIZER to update [a,~,y] using J GoTo step 3
The proposed method is based on common measurements of ambient dry-bulb temperature, relative humidity and crop level solar radiation which are used as a means for controlling the greenhouse environment. The method additionally requires the existence of a flow sensor (such as a tipping bucket) on the drainage of the irrigation channel. The estimation of the model parameters proceeds with each irrigation cycle. The estimation is iterative and is based on the availability of irrigation drain flow measurement. If drain water doesn't appear at all then there is no estimate of error, and the method must take special steps to recover. A full irrigation cycle is shown in Fig.2.
3. RESULTS Preliminary results were produced using a reduced size (3 x 3) optimization dictionary with the following ranges: for temperature T(°C) = {(16 .. 18), (18 .. 20), (20 .. 22) } and for solar radiation SR(W/m2) = { (25 ..75), (75 .. 125), (125 ..200) }. The initial values for the model parameters in the dictionary were assigned without specific knowledge of true plant transpiration. Even so the initial algorithm errors were not excessive and in any case they are reduced quickly by the process of optimization.
I I I
:
Em
~
I I I
Results from running with this dictionary can be summarized in the charts of Fig. 3. In each chart the surface represents the performance map after a certain number of experiments represented by the number of successive pattern searches. Each pattern search is preceded by a maximum of six explorations. Each exploration demands a performance evaluation (one irrigation cycle that lasts approximately one hour). Progress is relatively fast especially at high solar radiation conditions that occur more frequently during the daytime.
I I I
:I
Vdm~:I
I
I
I
I
t Figure.2
Depiction of irrigation dose (Vo), momentary plant transpiration (Em), and measured drain water flow (Vdm), in a typical irrigation cycle.
191
Initial Performance
Pattern #3 6000
6000
J
5000
.... ,... .... ~....
5000
4000 3000
...,........;....
4000
.... ,.... .
2000
...... :.....
1000
.............;.... J 3000 ....~ ... .... ;... .::: ::'
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: .'
::: :: '.
21 7' 19
2000
.. ,
,,':.
.... . ......
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(Oe)
21 19 7'(0
C)
Pattern #
6000 5000 4000
J
3000 2000
attern #9
... -':". ...... ..:' .... ..... .... ... ..;.. ..... : ...... ..; .. . ... ~ .. ...... ...~ .........~ . . . ..1. .. .~ ........l... . ~ : : ~ ~ ~
6000
~
liE·!;;
5000 4000
J 3000 2000 1000
Pattern #15
1000
J
.. '
Pattern #20
800
· ·····;··· · ·· ·· ·· ~··· ·· · ···r · · ·
...... :.......;.. ······1····
50~~--~~~~~--"
600
· · ····'······ · ···~······ ·· ·T···
····T·····'········! .. ··
40
400 200
.... . ~ .... .......~ ..... .
... ...
30
J 20
... ....:.... .
Figure 3. Global performance map at different stages of optimization. Performance is defined as percent relative error squared ( J= [ 100 ( E - E* )/ E*J 2)
192
The progress in a single set of parameters is shown in FigA. It can be observed that after an initial set of pattern searches the error settles at less than 20% in fewer than 20 evaluations. In this chart all explorations are shown (regardless of success). It can be observed that the algorithm oscillates within a small range of errors. This oscillation is explained because experiments are not conducted at a specified point, but within a specified range of conditions. Consequently the error cannot be constantly reduced because the same "best" parameters produce somewhat different errors under varying conditions.
REFERENCES Anastasiou, A, N. Sigrimis, N. Rerras, 1998 Experiments with a process control optimizer. 3d IFAC workshop on Intelligent Control Systems in Agriculture, Makuhari, Japan, April 26-27, 1998. Gieling T. , 1997, Monitoring and control of the supply of water and nutrients. MACQU/AIR Program 11 Final Report Hamer P., 1997, Optimisation of irrigation and nutrition. MACQUIAIR Program 11 Final Report Monteith J.L. and Unsworth M., 1990. Principles of environmental physics. Edward Amold, London,
.......... :...... ...... ~ ............ ~ ............ ~ ......... -.. ~ ........... ,; ....... -.... ~ ..
0.4
:
"-
Shoup T.E. 1979. A practical guide to computer methods for engineers. Prentice Hall, Inc., Englewood Cliffs, NJ.
:
~---i---t-~--- t
0.2
e
::
UK.
0.0
Sigrimis N.A. and N.Rerras 1995. Prefield tests of an electronic leaf sensor. Acta Horticulturae, No 406, pp471-481.
"-
UJ
_
c
-0.2 1--+ "-
Q)
........ :........... .: ............ :........ ....:............ : ............ ;............; .. .. ..
~ -0 .4
Q)
a..
: T:=(16 .18) :R=(~5 ..7$) ~ ·······1············i ···· ······ .. i············[············i .. ··········i············;··
t
-0 .8 -1.0
Stanghellini C , 1987. Transpiration of greenhouse crops, an aid to climate management. Ph.D. Dissertation, Agricultural University, Wageningen, 150pp.
~ . . t ...P.: ~m~e~ ........ nt E·r.r.Or. ..;~ .. .... .....;~ . ..Expenme.o
-0.6
U-L...L...l....I....L..I-'-I...I....L..J....I....l..J...J...J...J...L-W....L...l....L..W-'-I...w..'-'-L..I...l...Jj
o
20
40
60
80
100
120
140
experiment Figure 4. Estimation error vs experiment number for a single set of parameters (T=(16 .. 18) SR=(25 .. 75). All experiment steps are displayed. A wellcome side effect of this method is that when tuned it can provide data for the evaluation of plant transpiration of a specific crop. By gathering data sets from the experiment operating records the development of crop transpiration models can be accomplished without further experimentation. Discussion
The preliminary tests of the optimization algorithm show that it is a viable alternative to the techniques used for irrigation and nutrient supply control. Compared with robust control designs it requires virtually no effort for its application on a specific site. Compated with model based control techniques it offers the ability of on-site, on-line tuning which removes the need for exact knowledge of plant transpiration model for a specific crop. The next step in development will be the application to a production site to evaluate growers feedback and operation data from different crops.
193
AN ADAPTIVE OPTIMIZER FOR PROCESS CONTROL
N. Rerras, A.Anastasiou, K. Feredinos, N.Sigrimis Department of Agricultural Engineering Agricultural University of Athens 75 Iera Odes St. Athens 11855, Greece e-mail: [email protected]
Abstract: An optimization based methodology for control of irrigation and nutrient supply is developed using common measurements of greenhouse climate. The method allows onsite on-line identification of plant water needs and as an added benefit provides information for the creation of crop transpiration models. Copyright © 1998 IFAC Keywords: models, intelligent control.
1. INTRODUCTION
drain measurement and model estimate the coefficients of the model are iteratively adapted. The adaptation process is continuous so that the model accounts for seasonal variation of crop growth.
Contemporary greenhouse operations require precise control of irrigation and nutrient supply in order to optimize crop growth and minimize cost and to effluents. Moreover in pollution due Mediterranean countries there is a need to minimize water waste due to seasonal shortages. In modern greenhouses nutrient supply is computer controlled and based on measuring salinity and compensating deficiencies by a mix of clean water and two stock nutrient solutions. The process of applying this solution to the crop presents several control problems such as time delays and seasonal variations due to plant growth. Several proposals to the nutrient supply problem have been proposed such as direct feedback control of drain water flow in both closed and open growing systems using flow measurement of the drain water (Gieling, 1997). On the other hand progress has been achieved in model prediction of crop irrigation needs. Usually a transpiration model (Stanghellini 1987) is used that predicts plant transpiration based on ambient conditions (temperature, solar radiation, CO2 concentration and vapor saturation deficit). This approach leads to open loop control of water and nutrient supply based on model estimation of plant needs (Hamer 1997).
2. MATERIALS AND METHODS
In common greenhouse growing practice the irrigation water is supplied in specified time cycles. In each cycle the amount of water supplied, (designated V o) , is determined by the grower manually in order to satisfy crop requirements. The approach taken is to ensure that the growing medium saturates by the presence of an amount of drain water. The desired drain is typically around 20% of the irrigation amount (Vo) , required to secure desired nutrients concentration in the root environment. If we define the fraction of drain water as f, and the true crop requirement, which is the amount of real crop evapotranspiration, as E·, then the current growing practice is expressed by E·
=(1-f) V
(1)
o
This practice can be depicted in the leftmost part of Fig.I. Assume that a model exists relating crop transpiration to climate conditions, of the form E=aS+~(1-RH)+yT
This work proposes an intermediate approach where a crop transpiration model of convenient form is used to predict the necessary supply of water. At the same time drain water flow from the crop is measured using an appropriate device. Using the error between
(2) 2
where crop transpiration E (kglm /s) is related to solar radiation S(yVIrn2), humidity deficit (I -RH) and ambient air temperature T (0C), through suitable
189
irrigat.dose=Vo
e=Em -E· Vo :
T*
o
Figure 1: The concept of model prediction of irrigation needs and its error appearance in measured drain. coefficients (a,p,y), then crop transpiration E* could be calculated by integrating this model for the duration of the irrigation cycle. However in the general case the model of Eq.2 has unknown parameters dependent on factors such as crop type, growth stage (LA!) etc.
The alternative proposed here is to utilize this measurement in order to determine the unknown parameters of Eq.2. Then with well tuned parameters the irrigation process could be controlled in a feedforward loop. Moreover if the parameters of Eq.2 are continuously estimated then the system can account for process drift. Parameter tuning can be accomplished using a minimization algorithm if an appropriate performance function is devised. In this case the performance criterion to minimize is simple to create. During the irrigation cycle the integral of Eq.2 is monitored so that the estimated evapotranspiration (Em) is available
However if a measurement of the drain water is available then the true crop transpiration can be determined by integrating drain water rate and using the equation
E· =Vo - Vdm
(3)
Where: Vdm is the integrated drain water. In fact this measurement is used in feedback control of irrigation in closed systems (Gieling 1997). This type of control requires careful design due to time delays present in the irrigation process. Also due to system drifts the controller design should incorporate robustness. This is shown in fig.2 where the drain water is shown accumulated at the end of the irrigation cycle. This time delay can be many minutes, and depends on the length of the irrigated line and the characteristics of the substrate.
As depicted in fig. 1 if for example eq2 underestimates the transpiration this will lead to a longer irrigation cycle, that is delayed application of irrigation dose. The real transpiration shall be higher than estimated and this results to more water retained in the substrate and less drain will be measured. This measured error in the expected drain is actually the error of transpiration estimate:
(5)
190
The optimizer module is an iterative search algorithm that can adjust the model parameters in order to minimize the specified error metric. The algorithm is an enhanced variation of the Hooke-Jeeves (Shoup, 1979) minimization algorithm, and is embedded for general use in the MACQU greenhouse control software (Anastasiou et aI, 1998). It does not need any process model or gradient information to conduct parameter searching. Instead it explores the parameter space modifying one parameter at a time searching for a gradient. After all parameters have been explored the composite direction of steepest descent is formed. The algorithm then enters a phase of "pattern" moves where this direction is used until progress is exhausted. The algorithm cycles between exploration and pattern searches until a local minimum is reached. Earlier experience with application of this algorithm has shown it to be effective in optimizing the parameters for a model based mist control system (Sigrimis and Rerras, 1996). In this case the search for better parameters never stops in order for the algorithm to detect drifts of the system characteristics (i.e. plant growth etc.). In order to achieve better fitting of the search parameters and shorter search times the parameter space is split into regions of similar environmental conditions as determined by solar radiation and temperature. A separate set of parameters is created for each of these regions resulting in the formation of a parameter dictionary. Each time the environmental conditions shift to a different region the optimizer switches to the appropriate parameter set from the dictionary and continues with a new experiment.
In a form more suitable for the optimization algorithm the performance criterion J is the percent relative error squared
(6) All the pieces have come together for the presentation of the algorithm that accomplishes the task of tuning the parameters and controlling irrigation.
Irrigation Optimization algorithm l.
2. 3. 4.
5. 6. 7. 8.
Define Vo, f Select initial parameters a,~,y Apply Vo Do Loop Calculate Em (from Eq. 4) Measure drain and integrate Vdm While Em < (1-f) Vo Compute E* (from Eq.3) Compute J (from Eq.6) Call OPTIMIZER to update [a,~,y] using J GoTo step 3
The proposed method is based on common measurements of ambient dry-bulb temperature, relative humidity and crop level solar radiation which are used as a means for controlling the greenhouse environment. The method additionally requires the existence of a flow sensor (such as a tipping bucket) on the drainage of the irrigation channel. The estimation of the model parameters proceeds with each irrigation cycle. The estimation is iterative and is based on the availability of irrigation drain flow measurement. If drain water doesn't appear at all then there is no estimate of error, and the method must take special steps to recover. A full irrigation cycle is shown in Fig.2.
3. RESULTS Preliminary results were produced using a reduced size (3 x 3) optimization dictionary with the following ranges: for temperature T(°C) = {(16 .. 18), (18 .. 20), (20 .. 22) } and for solar radiation SR(W/m2) = { (25 ..75), (75 .. 125), (125 ..200) }. The initial values for the model parameters in the dictionary were assigned without specific knowledge of true plant transpiration. Even so the initial algorithm errors were not excessive and in any case they are reduced quickly by the process of optimization.
I I I
:
Em
~
I I I
Results from running with this dictionary can be summarized in the charts of Fig. 3. In each chart the surface represents the performance map after a certain number of experiments represented by the number of successive pattern searches. Each pattern search is preceded by a maximum of six explorations. Each exploration demands a performance evaluation (one irrigation cycle that lasts approximately one hour). Progress is relatively fast especially at high solar radiation conditions that occur more frequently during the daytime.
I I I
:I
Vdm~:I
I
I
I
I
t Figure.2
Depiction of irrigation dose (Vo), momentary plant transpiration (Em), and measured drain water flow (Vdm), in a typical irrigation cycle.
191
Initial Performance
Pattern #3 6000
6000
J
5000
.... ,... .... ~....
5000
4000 3000
...,........;....
4000
.... ,.... .
2000
...... :.....
1000
.............;.... J 3000 ....~ ... .... ;... .::: ::'
"
: .'
::: :: '.
21 7' 19
2000
.. ,
,,':.
.... . ......
1000
(Oe)
21 19 7'(0
C)
Pattern #
6000 5000 4000
J
3000 2000
attern #9
... -':". ...... ..:' .... ..... .... ... ..;.. ..... : ...... ..; .. . ... ~ .. ...... ...~ .........~ . . . ..1. .. .~ ........l... . ~ : : ~ ~ ~
6000
~
liE·!;;
5000 4000
J 3000 2000 1000
Pattern #15
1000
J
.. '
Pattern #20
800
· ·····;··· · ·· ·· ·· ~··· ·· · ···r · · ·
...... :.......;.. ······1····
50~~--~~~~~--"
600
· · ····'······ · ···~······ ·· ·T···
····T·····'········! .. ··
40
400 200
.... . ~ .... .......~ ..... .
... ...
30
J 20
... ....:.... .
Figure 3. Global performance map at different stages of optimization. Performance is defined as percent relative error squared ( J= [ 100 ( E - E* )/ E*J 2)
192
The progress in a single set of parameters is shown in FigA. It can be observed that after an initial set of pattern searches the error settles at less than 20% in fewer than 20 evaluations. In this chart all explorations are shown (regardless of success). It can be observed that the algorithm oscillates within a small range of errors. This oscillation is explained because experiments are not conducted at a specified point, but within a specified range of conditions. Consequently the error cannot be constantly reduced because the same "best" parameters produce somewhat different errors under varying conditions.
REFERENCES Anastasiou, A, N. Sigrimis, N. Rerras, 1998 Experiments with a process control optimizer. 3d IFAC workshop on Intelligent Control Systems in Agriculture, Makuhari, Japan, April 26-27, 1998. Gieling T. , 1997, Monitoring and control of the supply of water and nutrients. MACQU/AIR Program 11 Final Report Hamer P., 1997, Optimisation of irrigation and nutrition. MACQUIAIR Program 11 Final Report Monteith J.L. and Unsworth M., 1990. Principles of environmental physics. Edward Amold, London,
.......... :...... ...... ~ ............ ~ ............ ~ ......... -.. ~ ........... ,; ....... -.... ~ ..
0.4
:
"-
Shoup T.E. 1979. A practical guide to computer methods for engineers. Prentice Hall, Inc., Englewood Cliffs, NJ.
:
~---i---t-~--- t
0.2
e
::
UK.
0.0
Sigrimis N.A. and N.Rerras 1995. Prefield tests of an electronic leaf sensor. Acta Horticulturae, No 406, pp471-481.
"-
UJ
_
c
-0.2 1--+ "-
Q)
........ :........... .: ............ :........ ....:............ : ............ ;............; .. .. ..
~ -0 .4
Q)
a..
: T:=(16 .18) :R=(~5 ..7$) ~ ·······1············i ···· ······ .. i············[············i .. ··········i············;··
t
-0 .8 -1.0
Stanghellini C , 1987. Transpiration of greenhouse crops, an aid to climate management. Ph.D. Dissertation, Agricultural University, Wageningen, 150pp.
~ . . t ...P.: ~m~e~ ........ nt E·r.r.Or. ..;~ .. .... .....;~ . ..Expenme.o
-0.6
U-L...L...l....I....L..I-'-I...I....L..J....I....l..J...J...J...J...L-W....L...l....L..W-'-I...w..'-'-L..I...l...Jj
o
20
40
60
80
100
120
140
experiment Figure 4. Estimation error vs experiment number for a single set of parameters (T=(16 .. 18) SR=(25 .. 75). All experiment steps are displayed. A wellcome side effect of this method is that when tuned it can provide data for the evaluation of plant transpiration of a specific crop. By gathering data sets from the experiment operating records the development of crop transpiration models can be accomplished without further experimentation. Discussion
The preliminary tests of the optimization algorithm show that it is a viable alternative to the techniques used for irrigation and nutrient supply control. Compared with robust control designs it requires virtually no effort for its application on a specific site. Compated with model based control techniques it offers the ability of on-site, on-line tuning which removes the need for exact knowledge of plant transpiration model for a specific crop. The next step in development will be the application to a production site to evaluate growers feedback and operation data from different crops.
193