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Crop Growth Modeling-a New Data-driven Approach
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Crop Growth ModelingCrop Growth ModelingCrop Growth Modelinga New Approach CropData-driven Growth Modelinga New Data-driven Approach a New Data-driven Approach a New Data-driven Approach Dirk S¨ offker, Friederike K¨ ogler, Lina Owino.
Dirk S¨ offker, Friederike K¨ ogler, Lina Owino. Dirk S¨ offker, Friederike K¨ ogler, Lina Owino. Dirk S¨ offker, Friederike K¨ oDuisburg, gler, LinaGermany Owino. University of Duisburg-Essen, University of Duisburg-Essen, Duisburg, Germany (e-mail: [email protected]). University of Duisburg-Essen, Duisburg, Germany (e-mail: [email protected]). University of Duisburg-Essen, Duisburg, Germany (e-mail: [email protected]). (e-mail: [email protected]). Abstract: Deficit Deficit irrigation irrigation strategies strategies have have been been employed employed in in mitigation mitigation of of challenges challenges related related Abstract: Abstract: irrigation haveVarious been employed in mitigation of challenges related to efficient efficient Deficit water use use in crop cropstrategies production. Various models have have been developed developed to represent represent to water in production. models been to Abstract: irrigation strategies haveVarious been employed in mitigation challenges related theefficient behaviorDeficit of plants plants under stress This presents aa of state-machine-based to water use in cropwater production. models have been developed to represent the behavior of under water stress conditions. conditions. This work work presents state-machine-based to efficient water use in crop production. Various models have been developed to represent model that defines plant behavior terms of and which are the behavior of plants under water in stress conditions. work presents a state-machine-based model that defines plant behavior in terms of states states This and transitions transitions which are determined determined by by the of plants under waterstatus stress conditions. work presents a state-machine-based model that defines plant behavior in terms of states and which arein bothbehavior current and historical water of the plant.This Thetransitions model is employed employed indetermined prediction by of both current and historical water status of the plant. The model is prediction of model that defines plant behavior in terms of states and transitions which are determined by both currentof historical of the plant. treatments The model during is employed in prediction of the growth growth ofand maize plants water under status different irrigation treatments during the vegetative vegetative stage. the maize plants under different irrigation the stage. both current historical status of the plant. Thegrowth model performance is employed in of The growth new approach provides an accurate estimation of treatments the of prediction maize plants plants the ofand maize plants water under different irrigation during the vegetative stage. The new approach provides an accurate estimation of the growth performance of maize the growth of maize plants under different irrigation treatments during the vegetative stage. during the early vegetative vegetative phase, allowing it to to be beof used used in evaluation evaluation of effects effects of different different The new approach provides an accurate estimation the growth performance of maize plants during the early phase, allowing it in of of The new approach provides accurate estimation the growth performance of maize plants during the early vegetative phase, allowing it to beof used incontrol. evaluation of effects of different irrigation treatments and in in an design of irrigation-based plant control. irrigation treatments and design of irrigation-based plant during thetreatments early vegetative allowing it to be used evaluation of effects of different irrigation and in phase, design of irrigation-based plantincontrol. © 2019, IFAC (International Federation of Automatic Control) Hosting Elsevier Ltd. All rights reserved. irrigation treatments and in design of irrigation-based plantby control. Keywords: Modeling; Modeling; Prediction; Prediction; Agriculture; Keywords: Agriculture; Growth Growth control; control; State State machine. machine. Keywords: Modeling; Prediction; Agriculture; Growth control; State machine. Keywords: Modeling; Prediction; Agriculture; Growth control; State machine. 1. INTRODUCTION INTRODUCTION 1.1 Aims Aims of of the the contribution contribution 1. 1.1 1. INTRODUCTION 1.1 Aims of the contribution 1. INTRODUCTION 1.1 Aims of the contribution In this this contribution previously developed developed state-machinestate-machineIn contribution aa previously based model (K¨ o gler and S¨ o ffker (2018), K¨ gler and and In this contribution a previously developed state-machinebased model (K¨ogler and S¨offker (2018), K¨ oogler In this contribution a previously developed state-machineS¨ o ffker (2019)) is validated within a broader context of based model (K¨ o gler and S¨ o ffker (2018), K¨ o gler and S¨ offker model (2019))(K¨ isogler validated within broaderK¨ context of based and S¨ owith ffker aarespect (2018), ocontrolled gler and irrigation variation especially to S¨ o ffker (2019)) is validated within broader context of variation especiallywithin with arespect to context controlled One of of the the major major challenges challenges for for sustainable sustainable agriculture agriculture and and irrigation S¨ offker (2019)) is validated broader of One irrigation variation especially with stress respect to controlled transitions between different plant plant stress states. Core to to transitions between different states. Core food production occurs in relation to the efficiency of water One of the major challenges for sustainable agriculture and irrigation variation especially with respect to controlled food production occurs in relation to the efficiency of water the control approach is the knowledge about the plant transitions between different plant stress states. Core to One of the majoroccurs challenges for sustainable agriculture and the control approach is the knowledge about the plant food production relation to In thethis efficiency ofdeficit water use with with respect to cropin production. In this context, deficit transitions between different plant stress states. Core to use respect to crop production. context, growth behavior caused by knowledge varying water supply. The control approach is the about the plant food production occurs in relation to thethis efficiency ofdeficit water growth behavior caused by varying water supply. The irrigation methods are of interest. Here control of plants is the use with respect to crop production. In context, the control approach is the knowledge about the plant irrigation methods are of interest. Here control of plants is developed model is is the basis basis for formulating formulating the control control behavior caused by varying water supply. The use withtorespect to crop In this context, deficit developed model the for the related their individually individually optimal water supply state(s). irrigation are ofproduction. interest. Here control of plants is growth growth behavior caused by varying water supply. The related to methods their optimal water supply state(s). developed model is thethis basis for formulating the control control algorithm. The aim of this approach is to directly irrigation methods are of interest. Here control of plants is algorithm. The aim of approach is to directly This control control problem establishes feedback. The plant plant sys- developed model is the basis for formulating the control related to their individually optimal water supply state(s). This problem establishes feedback. The sysgrowth/yield instead of controlling the water availability in algorithm. The aim of this approach is to directly control related to their individually optimal water supply state(s). instead ofthis controlling theiswater availability in This control employs problem state-based establishes feedback. plant system (model) (model) employs state-based irrigation The (in contrast contrast to growth/yield algorithm. The aim ofof approach to directly control tem irrigation (in to availability the soil. The approach provides the missing link to model growth/yield instead controlling the water in This control problem establishes feedback. The plant syssoil. The approach provides thethe missing link to model soil-based fullemploys irrigation). Modernirrigation control approaches approaches are tem (model) state-based (in contrastare to the growth/yield instead controlling water availability in soil-based full irrigation). Modern control plant dynamic growthofbehavior behavior for aamissing controlled deficit irsoil. The approach provides the link deficit to model tem (model) employs state-based irrigation (in are contrast to the plant dynamic growth for controlled irbased on plant models. In this context models used to soil-based full irrigation). Modern control approaches are the soil. The approach provides the missing link to model based on plant models. In this context models are used to plant dynamic growth behavior for a controlled deficit irrigation. Based on experimental results the controllability soil-based full formation irrigation). Modern approaches are Based on experimental results the controllability describe yield (Steduto etcontrol al. models (2008)). Plants dybased onyield plant models. In this context are useddyto rigation. plant dynamic behavior for a controlled deficit irdescribe (Steduto et al. (2008)). Plants of plant plant growth by means of irrigation irrigation sequencing could be be Based growth on means experimental results the controllability based onbehavior plantformation models. In this context models are useddyto rigation. of growth by of sequencing could describe yield formation (Steduto et al. (2008)). Plants namical due to water deficit has been researched rigation. Based on experimental results the controllability namical behavior due to water deficit has been researched demonstrated (K¨ gler andofS¨ S¨ ffker (2019)) (2019)) and the thecould results plant growth(K¨ byoogler means irrigation sequencing be describe yield formation et al. (2008)). Plants dy- of demonstrated and ooffker and results by several several authors introducing linear yield reduction funcnamical behavior due to (Steduto water linear deficit has been researched of plant growth byogler means irrigation sequencing be by authors introducing yield reduction funcdemonstrated (K¨ andofS¨ offker (2019)) and thecould results of specific training sequences (sport) for different control namical behavior due to water deficit has been researched of specific training sequences (sport) for different control tions reflecting the introducing effects of of water water stress onreduction yield (Brisson (Brisson by several authors linear yieldon func- of demonstrated (K¨ o gler and S¨ o ffker (2019)) and the results tions reflecting the effects stress yield specific training sequences (sport) for different control targets shown. Based on the established model in this by several authors introducing linear yield func- targets shown. Based on the established model in this tions reflecting effects ofthe water stress onreduction yield behavior (Brisson et al. al. (2006)) orthe describing related dynamical of specific training sequences et (2006)) or describing related dynamical contribution newBased experimental resultsfor aredifferent reported repretargets shown. on the(sport) established model control inreprethis tions reflecting the effects ofthe water stress onUsing yield abehavior (Brisson contribution new experimental results are reported et al. (2006)) or describing the related dynamical behavior (Kloss et al. (2012), Linker et al. (2016)). related targets shown. Based on the established model inreprethis (Kloss et al. (2012), Linker et al. (2016)). Using a related senting an extended variation of watering sequences. New contribution new experimental results are reported et al. (2006)) or describing the related dynamical behavior an extended variation ofresults watering sequences. New model et theal.plant plant system control loop would provide the senting (Kloss (2012), Linker et al. loop (2016)). Using a related contribution new experimental are reported repremodel the system control would provide the senting an extended variation of watering sequences. New calculations about about related related parameterization parameterization and and threshthresh(Kloss et (2012), Linker et al.water-based (2016)). Using a related opportunity to pursue different optimization model theal.plant system control loop would provide the calculations senting an extended variation of watering sequences. New opportunity to pursue different water-based optimization calculations about related parameterization and thresholds defining are reported, so that the existing approaches model the plant system control loop would provide the olds defining are reported, so that the existing approaches opportunity to pursue different water-based optimization strategies, e.g. e.g. alignment alignment of of growth growth to to predicted predicted precipitaprecipita- calculations about related parameterization and threshstrategies, is validated validated again within aa modified modified and extended context. definingagain are reported, so that the existing approaches opportunity toalignment pursue water-based optimization is within and extended context. strategies, of growth to predicted precipitation events. events.e.g. The water different supply state determination can be be olds olds definingagain are reported, so that the existing approaches tion The water supply state determination can is validated within a modified and extended context. strategies, e.g. alignment of growth to predicted precipitarelated directly to growth factors, with commonlycan used This contribution is structured structured as follows: follows: The state state mation events. The to water supply state with determination be This validated again within a modified and extended context. related directly growth factors, commonly used contribution is as The mation events. The water supply state determination can be is water supply states described as ‘full irrigation’, ‘mild chine model is introduced in section 2. Experimental rerelated directly to growth factors, with commonly used This contribution is structured as follows: The state mawater supply states described as ‘full irrigation’, ‘mild model is introduced in section 2. Experimental rerelated directly to growth withof commonly used chine This contribution isand structured asinfollows: The state mawater states described as ‘full irrigation’, ‘mild stress’,supply and ‘high stress’. Thefactors, distinction different states states sults are presented discussed section 3 followed by chine model is introduced in section 2. Experimental restress’, and ‘high stress’. The distinction of different are presented and discussed in section 3 followed reby water supply states as ‘full irrigation’, ‘mild sults chine model is introduced in section 2. Experimental is mainly mainly defined by described theThe underlying models and related related sults are presented and4. in section 3 followed by a conclusion conclusion in section 4.discussed stress’, anddefined ‘high stress’. distinction of different states is by the underlying models and a in section stress’, anddefined ‘high stress’. distinction of different states sults are presented and discussed in section 3 followed by parameters and thresholds. thresholds. To date, models most stress-related is mainly by theThe underlying and related a conclusion in section 4. parameters and To date, most stress-related is mainly defined by the underlying models and related a conclusion in section 4. thresholds for plant-based output variables described in 2. STATE STATE MACHINE-BASED MODELING parameters and thresholds. To date, most stress-related thresholds for output variables described in 2. MACHINE-BASED MODELING parameters andplant-based thresholds. To date, most stress-related thresholds for plant-based output variables described in 2. STATE MACHINE-BASED MODELING literature refer to stress incipience, i.e. transition of ‘wellPARAMETRIZATION literature refer to stress incipience, i.e. transition of ‘wellPARAMETRIZATION thresholds for to plant-based output described in 2. STATE PARAMETRIZATION MACHINE-BASED MODELING refer incipience, i.e. transition of ‘wellwatered’ state state state of of first stressvariables symptoms (Williams literature watered’ toto aastress state first stress symptoms (Williams literature refer to stress incipience, i.e. transition of ‘wellPARAMETRIZATION In previous previous publications publications (K¨ (K¨ gler and and S¨ S¨ ffker (2018), (2018), and Araujo Araujo (2002), Jones (2004), Jones (2007)). Plant- In watered’ state(2002), to a state of first stressJones symptoms (Williams oogler ooffker and Jones (2004), (2007)). Plantwatered’ state(2002), to for a state of first stress symptoms (Williams K¨ o gler and S¨ o ffker (2019)) first time a state-machine based based thresholds the transition of a ‘mild stress’ state to In previous publications (K¨ o gler and S¨ o ffker (2018), and Araujo Jones (2004), Jones (2007)). Plantogler and S¨offker (2019)) first state-machine based based thresholds for the transition a ‘mild(2007)). stress’ state to K¨ In previous publications (K¨ otime gler aawas and S¨offker (2018), and Araujo Jones (2004),of Plantogler and S¨offker (2019)) time state-machine based based thresholds for the transition ofJones awhether ‘mild stress’ state to K¨ model describing the plantfirst growth established. The a ‘high ‘high stress’(2002), state, or an evaluation whether the detected detected model describing the plant growth was established. The abased stress’ state, or an evaluation the K¨ ogleridea and of S¨offker (2019)) first time awas state-machine based for the transition of awhether ‘mild stress’ state to to model main this model is to combine a set of different symptom still belongs to a ‘mild stress’ state or already describing the plant growth established. The asymptom ‘highthresholds stress’ state, or an evaluation the detected idea of this the model is to combine a set of different still belongs to ‘mild stress’ state or already to main model describing plant growth was The aa ‘high anaadescribed evaluation thealready detected parameterized equations describing plant growth. Each ’high stress’ stress’ state,orare are inwhether few cases as e.g. to in main idea of this model isdescribing to combine aestablished. set of different symptom still state, belongs to ‘mild stress’ state or parameterized equations plant growth. Each asymptom ’high stress’ state, described in few cases as e.g. in main idea of this model is to combine a set of different still belongs to a ‘mild stress’ state or already to Each equation is is related related to different different growthplant statesgrowth. combined by (Ben-Gal et al. al.state, (2010)). equations describing a(Ben-Gal ’high stress’ are described in few cases as e.g. in parameterized to growth states combined by et (2010)). parameterized equations describing Each a ’high stress’ are described in few cases as e.g. in equation equation is related to different growthplant statesgrowth. combined by (Ben-Gal et al.state, (2010)). equation related growth states combined by (Ben-Gal al. (2010)). 2405-8963 ©et2019, IFAC (International Federation of Automatic Control) Hosting by is Elsevier Ltd.toAlldifferent rights reserved.
Copyright © 2019 IFAC 132 Copyright 2019 IFAC 132 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 132 10.1016/j.ifacol.2019.12.510 Copyright © 2019 IFAC 132
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conditioned transitions. Transitions conditions and staterelated parameters are defined by optimization within a training loop compairing measured plant growth and state machine-based estimated ones. This new approach was firstly established in Beganovic and S¨ offker (2017) and applied the machine wear processes. States are defined by the state variables stress, memory, and damage level, distinguishing different states. Here states describe the status of the system (here: plant) within an event-discrete frame. Events are conditioned and described by measurable as well as inner variables in this case also including time. A state machine-based approach lends itself particularly well to modeling of plant growth due to the event discrete nature of plant growth, in this case: full irrigation, mild stress, and high stress for which different growth models are assumed. Additionally, a state machine-based approach allows for the description of structurally variable systems, which is ideal in cases where different behaviors can be assumed, as is the case in plant growth under different irrigation treatments. In Figure 1 the state machine model describing plant growth subjected to irrigation treatments is shown. This model represents the basis for the growth control approach.
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ranges, defined as stress level L=0 (L0) for no stress, stress level L=1 (L1 ) for mild stress, and stress level L=2 (L2 ) for high stress. The state variable ‘stress level’ (L) can therefore take the values 0, 1, or 2. These numbers are used to define irrigation sequences: A sequence of L=0/1/2/0 for instance describes a sequence starting in stress level L0 (above response threshold: full irrigated), followed by substrate desiccation till initially the response threshold is passed (stress level L1) and further till recuperation threshold is passed (stress level L2). Re-irrigation results in a switch back to stress level L0. An overview about the performed irrigation strategies, denoted as stress level trajectories, for the seven experiments is given in Table 1. 2.1 Definition of states The measurement time series of all plants are split into data segments resulting from the labeling procedure and then grouped according to the labels, irrespective of individual plant or time. Each data group is defined as a state. Hence, the plant states S are described by the vector S = [L, M, D]T , (1) with L = Stress level as a function of water demand and water availability, M = Memory as a function of mild stress occurence and interval, and D = Damage as a function of high stress occurence and mild stress duration 2.2 Deficit irrigation state machine model Detailed description for the plant growth model is given in K¨ogler and S¨offker (2019). Details are repeated here briefly for comprehensive introduction into the approach.
Fig. 1. State machine model of plant behavior due to water stress [K¨ ogler and S¨ offker (2019)] In compliance with defined transition conditions (arrows) the plant ‘switches‘ from one state Si to another state Sj (circles). Transition conditions are: L = Stress level with L=0 (L0 ; no stress), L=1 (L1 ; mild stress), L=2 (L2 ; high stress), tM = Maximum memory time, tD = Maximum stress duration time, tL1 = Time duration of stress level L=1 (L1 ), and tL1i −L1j = Time between the successive stress levels L1i to L1j . As described above, response threshold is defined by the onset of growth reduction due to a water deficit. Therefore, response threshold has to be determined at the water content level at which test plants showed initially a reduced growth rate (TLER) after water withdrawal compared to reference plants. Recuperation threshold is defined as point of no return, at which the progressed water deficit leads to irreversible growth because of a damage. The thresholds divide the water content scale into three 133
The starting point for the state machine model is state S1 representing a plant which did not experience any water stress before. State S1 is denoted as (0/0/0) for (‘no stress’, ’memory off’, ‘damage off’). Withdrawal of water resulting in an exceedance of the response threshold (stress level L1: L=1) leads initially to a transition from state S1 to the memory-initiating state S2 denoted as (1/1/0) for (‘mild stress’, ’memory on’, ‘damage off’). Henceforward, the state transitions depend on irrigation schedule: Re-irrigation (return to stress level L0: L=0) initiates the transition to state S1a (0/1/0). Alternatively, further water withdrawal resulting in an exceedance of the recuperation threshold (advance to stress level L2: L=2) initiates the transition to the damage-initiating state S3 (2/1/1); or constant mild stress (continuation in stress level L1: L=1) results in transition to the damageinitiating state S2a (1/1/1). Further transitions along the depicted model states operate correspondingly. End point for the state machine model is not defined, as model target is an intended deficit irrigation till harvest. Within the state-machine-model context the states Si are denoted as • S1 : Vector (0/0/0); reference growth behavior (comparison values), • S2 : Vector (1/1/0); reversibly reduced growth,
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S3 : Vector (2/1/1); irreversibly reduced growth, S1a : Vector (0/1/0); increased growth, S1b : Vector (0/1/1); irreversibly reduced growth, S1c : Vector (0/0/1); irreversibly reduced growth, and S2a : Vector (1/1/1); irreversibly reduced growth.
Here, the states S1 , S2 , and S1a are the desirable states for deficit irrigation purposes, as growth is not irreversibly affected due to water stress. The transition conditions are defined using • Lnj : the current stress level, • Lni : the stress level experienced in the previous instance, • tLni −Lnj : the time between successive stresses, and • tLn : the current stress time duration
as variables used in suitably designed conditions. 2.3 Data driven model parametrization
As illustrated before parameters of growth equations as well as transition conditions are assumed as unknown but constant, and have to be defined. As introduced in Beganovic and S¨ offker (2017) the parameter identification can be realized by multiobjective optimization. This procedure can be denoted as training of an interpretable machine model. Here the unknown parameters are defined by applying NSGA-II optimization. After training the model can be applied for prediction of plant growth. Assuming a known/given topology about the qualitatively distinguishable behavior of plant growth related mathematical equations are applied to describe the state dependent leaf elongation L for the different states depending on the lifetime increment ∆LT. The structural behavior is assumed as similar, the details are assumed as different (different parameter setting). 3. EXPERIMENTAL RESULTS The test rig consists of four growth stations placed in a rectangle and consisting of tables with superstructures for illumination, irrigation, measurements, and plant positioning (Figure 2).
Fig. 2. Test rig with potted maize plants (1). Growth stations (2) equipped with super-structures for illumination (a), irrigation (b), measurements (c), and plant positioning (d). The laboratory is located in a room (ground floor) of approximately 15 m2 2, the windows are blacked and oriented north. Air temperature, air humidity, and ventilation are 134
managed by university building equipment (pre-controlled ambient temperature and ventilation system). Illumination is automated with a clock timer. Seven irrigation experiments with maize plants (zea mays, c Ronaldinio (KWS), vegetative stadium EC11-EC15 / V1-V5) of 15 to 30 days’ duration each were implemented under laboratory conditions (nearly constant temperature, humidity, and ventilation; constant illumination over the day controlled by clock timer) for model parametrization and validation of the control approach. The plants are grown in 200 ml PET pots filled with a granular growth substrate. Evaporation rate of an equally prepared pot without plant showed nearly constant evaporation of +/- 2 g/d. The experimental conditions are summarized in Table 1. Trajectories evaluated for optimization of water use by application of deficit irrigation treatment are shown in bold font. Daily measurements of individual leaf lengths, environmental temperature, and pot mass are manually performed. From the measurements the total leaf length, leaf elongation rate, and water content are calculated. Implementation of the prediction algorithm is based on Monte Carlo simulation, which involves three steps: • Random sampling to obtain training data for the model • Validation of the model also using a random sample of test subjects • Statistical analysis of the results obtained from the validation stage In this case, two groups each consisting of eight randomly selected plants are selected for training and validation. Optimization of the growth model parameters is performed with the aim of minimizing mean ABE, MSE, and RSE across the entire sampled group. A population size of 50 evolving over 600 generations is observed to provide the optimal overall solution, with the performance of the algorithm described by: • Absolute error: 5.42 • Mean squared error: 20.57 • Root squared error: 1.99
The optimization algorithm generates state boundaries which are used to determine the water content corresponding to different stress states. These state boundaries are applied in defining the irrigation treatment employed. An experimental verification of the stress states of the plants has also been performed by considering the leaf elongation rate of test plants as compared to the control group, as well as examining elongation behavior after reirrigation to determine whether damage has occured- which is indicative of high stress. In Figure 3, a sample of optimized outputs from the state machine model for individual plants under different irrigation treatment is shown. For the selected plants under full irrigation and mild stress, the tracking of the growth trajectory is successfully achieved. The use of elongation variables in tracking the growth of the plant shows particular effectiveness in the application of the state machine model, producing reliable estimates of plant growth over time. The incorporation of the distinct stress states and accumulation of growth experienced as
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Table 1. Overview of experimental set-up and environmental conditions. Stress level trajectory describes the sequence of stress levels (L) experienced by the plant
Fig. 3. Example plant growth trajectory estimates. The continuous lines shows the actual total leaf length measured over time with the broken lines showing the estimated trajectory calculated using the state machine model. a result of each individual state over the duration of the plant growth makes it an attractive tool for flexible deficit irrigation scheduling using growth projections based on real-time data input. For plants grown under identical conditions, it is possible to obtain a relatively accurate
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estimate of comparative biomass expected from plants under different irrigation conditions using this model.
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4. CONCLUSION The newly introduced model shows good performance in tracking the growth of plants under different irrigation regimes, with the effects of water stress directly integrated into the design of the states and transitions. Future work will involve application of the state machine model in control of growth trajectories during the vegetative stage and incorporation of plant-based water stress signals in irrigation-related control decisions. ACKNOWLEDGEMENTS The authors acknowledge the work of Ms. Rosmawati Jihin by contributing numerical results applying the NSGAII optimization routines to the developed state machine model. REFERENCES Beganovic, N. and S¨ offker, D. (2017). Remaining lifetime modeling using state-of-health estimation. Mech. Syst. Sig. Process, 92, 107–123. Ben-Gal, A., Kool, D., Agam, N., van Halsema, G.E., Yermiyahu, U., Yafe, A., Presnov, E., Erel, R., Majdop, A., Zipori, I., Segal, E., R¨ uger, S., Zimmermann, U., Cohen, Y., Alchanatis, V., and Dag, A. (2010). Whole-tree water balance and indicators for short-term drought stress in non-bearing ‘barnea’ olives. Agricultural Water Management, 98(1), 124–133. doi: 10.1016/j.agwat.2010.08.008. Brisson, N., Wery, J., and Boote, K. (2006). Fundamental concepts of crop models illustrated by a comparative approach. In Working with Dynamic Crop Models, 257– 280. Elsevier LTD, , Oxford. Jones, H. (2004). Irrigation scheduling: advantages and pitfalls of plant-based methods. J. Exp. Bot, 55, 2427– 2436. Jones, H. (2007). Monitoring plant and soil water status: Established and novel methods revisited and their relevance to studies of drought tolerance. J. Exp. Bot., 58, 119–130. Kloss, S., Pushpalatha, R., Kamoyo, K., and Sch¨ utze, N. (2012). Evaluation of crop models for simulating and optimizing deficit irrigation systems in arid and semiarid countries under climate variability. Water Resour. Manag., 26, 997–1014. K¨ogler, F. and S¨offker, D. (2018). Steuerung des Pflanzenwachstums durch Bew¨ asserung. In 61. Jahrestagung der Gesellschaft f¨ ur Pflanzenbauwissenschaften e.V, Kiel. K¨ ogler, F. and S¨ offker, D. (2019). Sport for plants as a means for growth control: Water-based open-loop control of plant growth. Agricultural Water Management, submitted. Linker, R., Ioslovicha, I., Sylaiosb, G., Plauborgc, F., and Battilanid, A. (2016). Optimal model-based deficit irrigation scheduling using aquacrop: A simulation study with cotton, potato and tomato. Agric. Water Manag., 163, 236–243. Steduto, P., Raes, D., Hsiao, T., Fereres, E., Heng, L., Izzi, G., and Hoogeveen, J. (2008). Aquacrop: a new model for crop prediction under water deficit conditions. In A. L´ opez-Francos (ed.), Drought management: scientific and technological innovations CIHEAM, 136
(Options M´editerran´eennes : S´erie A. S´eminaires M´editerran´eens, volume 80, 285–292. Academic Press, Zaragoza. Williams, L. and Araujo, F. (2002). Correlations among predawn leaf, midday leaf, and midday stem water potential and their correlations with other measures of soil and plant water status in vitis vinifera. J. Am. Soc. Hortic. Sci., 127, 448–454.
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Crop Growth ModelingCrop Growth ModelingCrop Growth Modelinga New Approach CropData-driven Growth Modelinga New Data-driven Approach a New Data-driven Approach a New Data-driven Approach Dirk S¨ offker, Friederike K¨ ogler, Lina Owino.
Dirk S¨ offker, Friederike K¨ ogler, Lina Owino. Dirk S¨ offker, Friederike K¨ ogler, Lina Owino. Dirk S¨ offker, Friederike K¨ oDuisburg, gler, LinaGermany Owino. University of Duisburg-Essen, University of Duisburg-Essen, Duisburg, Germany (e-mail: [email protected]). University of Duisburg-Essen, Duisburg, Germany (e-mail: [email protected]). University of Duisburg-Essen, Duisburg, Germany (e-mail: [email protected]). (e-mail: [email protected]). Abstract: Deficit Deficit irrigation irrigation strategies strategies have have been been employed employed in in mitigation mitigation of of challenges challenges related related Abstract: Abstract: irrigation haveVarious been employed in mitigation of challenges related to efficient efficient Deficit water use use in crop cropstrategies production. Various models have have been developed developed to represent represent to water in production. models been to Abstract: irrigation strategies haveVarious been employed in mitigation challenges related theefficient behaviorDeficit of plants plants under stress This presents aa of state-machine-based to water use in cropwater production. models have been developed to represent the behavior of under water stress conditions. conditions. This work work presents state-machine-based to efficient water use in crop production. Various models have been developed to represent model that defines plant behavior terms of and which are the behavior of plants under water in stress conditions. work presents a state-machine-based model that defines plant behavior in terms of states states This and transitions transitions which are determined determined by by the of plants under waterstatus stress conditions. work presents a state-machine-based model that defines plant behavior in terms of states and which arein bothbehavior current and historical water of the plant.This Thetransitions model is employed employed indetermined prediction by of both current and historical water status of the plant. The model is prediction of model that defines plant behavior in terms of states and transitions which are determined by both currentof historical of the plant. treatments The model during is employed in prediction of the growth growth ofand maize plants water under status different irrigation treatments during the vegetative vegetative stage. the maize plants under different irrigation the stage. both current historical status of the plant. Thegrowth model performance is employed in of The growth new approach provides an accurate estimation of treatments the of prediction maize plants plants the ofand maize plants water under different irrigation during the vegetative stage. The new approach provides an accurate estimation of the growth performance of maize the growth of maize plants under different irrigation treatments during the vegetative stage. during the early vegetative vegetative phase, allowing it to to be beof used used in evaluation evaluation of effects effects of different different The new approach provides an accurate estimation the growth performance of maize plants during the early phase, allowing it in of of The new approach provides accurate estimation the growth performance of maize plants during the early vegetative phase, allowing it to beof used incontrol. evaluation of effects of different irrigation treatments and in in an design of irrigation-based plant control. irrigation treatments and design of irrigation-based plant during thetreatments early vegetative allowing it to be used evaluation of effects of different irrigation and in phase, design of irrigation-based plantincontrol. © 2019, IFAC (International Federation of Automatic Control) Hosting Elsevier Ltd. All rights reserved. irrigation treatments and in design of irrigation-based plantby control. Keywords: Modeling; Modeling; Prediction; Prediction; Agriculture; Keywords: Agriculture; Growth Growth control; control; State State machine. machine. Keywords: Modeling; Prediction; Agriculture; Growth control; State machine. Keywords: Modeling; Prediction; Agriculture; Growth control; State machine. 1. INTRODUCTION INTRODUCTION 1.1 Aims Aims of of the the contribution contribution 1. 1.1 1. INTRODUCTION 1.1 Aims of the contribution 1. INTRODUCTION 1.1 Aims of the contribution In this this contribution previously developed developed state-machinestate-machineIn contribution aa previously based model (K¨ o gler and S¨ o ffker (2018), K¨ gler and and In this contribution a previously developed state-machinebased model (K¨ogler and S¨offker (2018), K¨ oogler In this contribution a previously developed state-machineS¨ o ffker (2019)) is validated within a broader context of based model (K¨ o gler and S¨ o ffker (2018), K¨ o gler and S¨ offker model (2019))(K¨ isogler validated within broaderK¨ context of based and S¨ owith ffker aarespect (2018), ocontrolled gler and irrigation variation especially to S¨ o ffker (2019)) is validated within broader context of variation especiallywithin with arespect to context controlled One of of the the major major challenges challenges for for sustainable sustainable agriculture agriculture and and irrigation S¨ offker (2019)) is validated broader of One irrigation variation especially with stress respect to controlled transitions between different plant plant stress states. Core to to transitions between different states. Core food production occurs in relation to the efficiency of water One of the major challenges for sustainable agriculture and irrigation variation especially with respect to controlled food production occurs in relation to the efficiency of water the control approach is the knowledge about the plant transitions between different plant stress states. Core to One of the majoroccurs challenges for sustainable agriculture and the control approach is the knowledge about the plant food production relation to In thethis efficiency ofdeficit water use with with respect to cropin production. In this context, deficit transitions between different plant stress states. Core to use respect to crop production. context, growth behavior caused by knowledge varying water supply. The control approach is the about the plant food production occurs in relation to thethis efficiency ofdeficit water growth behavior caused by varying water supply. The irrigation methods are of interest. Here control of plants is the use with respect to crop production. In context, the control approach is the knowledge about the plant irrigation methods are of interest. Here control of plants is developed model is is the basis basis for formulating formulating the control control behavior caused by varying water supply. The use withtorespect to crop In this context, deficit developed model the for the related their individually individually optimal water supply state(s). irrigation are ofproduction. interest. Here control of plants is growth growth behavior caused by varying water supply. The related to methods their optimal water supply state(s). developed model is thethis basis for formulating the control control algorithm. The aim of this approach is to directly irrigation methods are of interest. Here control of plants is algorithm. The aim of approach is to directly This control control problem establishes feedback. The plant plant sys- developed model is the basis for formulating the control related to their individually optimal water supply state(s). This problem establishes feedback. The sysgrowth/yield instead of controlling the water availability in algorithm. The aim of this approach is to directly control related to their individually optimal water supply state(s). instead ofthis controlling theiswater availability in This control employs problem state-based establishes feedback. plant system (model) (model) employs state-based irrigation The (in contrast contrast to growth/yield algorithm. The aim ofof approach to directly control tem irrigation (in to availability the soil. The approach provides the missing link to model growth/yield instead controlling the water in This control problem establishes feedback. The plant syssoil. The approach provides thethe missing link to model soil-based fullemploys irrigation). Modernirrigation control approaches approaches are tem (model) state-based (in contrastare to the growth/yield instead controlling water availability in soil-based full irrigation). Modern control plant dynamic growthofbehavior behavior for aamissing controlled deficit irsoil. The approach provides the link deficit to model tem (model) employs state-based irrigation (in are contrast to the plant dynamic growth for controlled irbased on plant models. In this context models used to soil-based full irrigation). Modern control approaches are the soil. The approach provides the missing link to model based on plant models. In this context models are used to plant dynamic growth behavior for a controlled deficit irrigation. Based on experimental results the controllability soil-based full formation irrigation). Modern approaches are Based on experimental results the controllability describe yield (Steduto etcontrol al. models (2008)). Plants dybased onyield plant models. In this context are useddyto rigation. plant dynamic behavior for a controlled deficit irdescribe (Steduto et al. (2008)). Plants of plant plant growth by means of irrigation irrigation sequencing could be be Based growth on means experimental results the controllability based onbehavior plantformation models. In this context models are useddyto rigation. of growth by of sequencing could describe yield formation (Steduto et al. (2008)). Plants namical due to water deficit has been researched rigation. Based on experimental results the controllability namical behavior due to water deficit has been researched demonstrated (K¨ gler andofS¨ S¨ ffker (2019)) (2019)) and the thecould results plant growth(K¨ byoogler means irrigation sequencing be describe yield formation et al. (2008)). Plants dy- of demonstrated and ooffker and results by several several authors introducing linear yield reduction funcnamical behavior due to (Steduto water linear deficit has been researched of plant growth byogler means irrigation sequencing be by authors introducing yield reduction funcdemonstrated (K¨ andofS¨ offker (2019)) and thecould results of specific training sequences (sport) for different control namical behavior due to water deficit has been researched of specific training sequences (sport) for different control tions reflecting the introducing effects of of water water stress onreduction yield (Brisson (Brisson by several authors linear yieldon func- of demonstrated (K¨ o gler and S¨ o ffker (2019)) and the results tions reflecting the effects stress yield specific training sequences (sport) for different control targets shown. Based on the established model in this by several authors introducing linear yield func- targets shown. Based on the established model in this tions reflecting effects ofthe water stress onreduction yield behavior (Brisson et al. al. (2006)) orthe describing related dynamical of specific training sequences et (2006)) or describing related dynamical contribution newBased experimental resultsfor aredifferent reported repretargets shown. on the(sport) established model control inreprethis tions reflecting the effects ofthe water stress onUsing yield abehavior (Brisson contribution new experimental results are reported et al. (2006)) or describing the related dynamical behavior (Kloss et al. (2012), Linker et al. (2016)). related targets shown. Based on the established model inreprethis (Kloss et al. (2012), Linker et al. (2016)). Using a related senting an extended variation of watering sequences. New contribution new experimental results are reported et al. (2006)) or describing the related dynamical behavior an extended variation ofresults watering sequences. New model et theal.plant plant system control loop would provide the senting (Kloss (2012), Linker et al. loop (2016)). Using a related contribution new experimental are reported repremodel the system control would provide the senting an extended variation of watering sequences. New calculations about about related related parameterization parameterization and and threshthresh(Kloss et (2012), Linker et al.water-based (2016)). Using a related opportunity to pursue different optimization model theal.plant system control loop would provide the calculations senting an extended variation of watering sequences. New opportunity to pursue different water-based optimization calculations about related parameterization and thresholds defining are reported, so that the existing approaches model the plant system control loop would provide the olds defining are reported, so that the existing approaches opportunity to pursue different water-based optimization strategies, e.g. e.g. alignment alignment of of growth growth to to predicted predicted precipitaprecipita- calculations about related parameterization and threshstrategies, is validated validated again within aa modified modified and extended context. definingagain are reported, so that the existing approaches opportunity toalignment pursue water-based optimization is within and extended context. strategies, of growth to predicted precipitation events. events.e.g. The water different supply state determination can be be olds olds definingagain are reported, so that the existing approaches tion The water supply state determination can is validated within a modified and extended context. strategies, e.g. alignment of growth to predicted precipitarelated directly to growth factors, with commonlycan used This contribution is structured structured as follows: follows: The state state mation events. The to water supply state with determination be This validated again within a modified and extended context. related directly growth factors, commonly used contribution is as The mation events. The water supply state determination can be is water supply states described as ‘full irrigation’, ‘mild chine model is introduced in section 2. Experimental rerelated directly to growth factors, with commonly used This contribution is structured as follows: The state mawater supply states described as ‘full irrigation’, ‘mild model is introduced in section 2. Experimental rerelated directly to growth withof commonly used chine This contribution isand structured asinfollows: The state mawater states described as ‘full irrigation’, ‘mild stress’,supply and ‘high stress’. Thefactors, distinction different states states sults are presented discussed section 3 followed by chine model is introduced in section 2. Experimental restress’, and ‘high stress’. The distinction of different are presented and discussed in section 3 followed reby water supply states as ‘full irrigation’, ‘mild sults chine model is introduced in section 2. Experimental is mainly mainly defined by described theThe underlying models and related related sults are presented and4. in section 3 followed by a conclusion conclusion in section 4.discussed stress’, anddefined ‘high stress’. distinction of different states is by the underlying models and a in section stress’, anddefined ‘high stress’. distinction of different states sults are presented and discussed in section 3 followed by parameters and thresholds. thresholds. To date, models most stress-related is mainly by theThe underlying and related a conclusion in section 4. parameters and To date, most stress-related is mainly defined by the underlying models and related a conclusion in section 4. thresholds for plant-based output variables described in 2. STATE STATE MACHINE-BASED MODELING parameters and thresholds. To date, most stress-related thresholds for output variables described in 2. MACHINE-BASED MODELING parameters andplant-based thresholds. To date, most stress-related thresholds for plant-based output variables described in 2. STATE MACHINE-BASED MODELING literature refer to stress incipience, i.e. transition of ‘wellPARAMETRIZATION literature refer to stress incipience, i.e. transition of ‘wellPARAMETRIZATION thresholds for to plant-based output described in 2. STATE PARAMETRIZATION MACHINE-BASED MODELING refer incipience, i.e. transition of ‘wellwatered’ state state state of of first stressvariables symptoms (Williams literature watered’ toto aastress state first stress symptoms (Williams literature refer to stress incipience, i.e. transition of ‘wellPARAMETRIZATION In previous previous publications publications (K¨ (K¨ gler and and S¨ S¨ ffker (2018), (2018), and Araujo Araujo (2002), Jones (2004), Jones (2007)). Plant- In watered’ state(2002), to a state of first stressJones symptoms (Williams oogler ooffker and Jones (2004), (2007)). Plantwatered’ state(2002), to for a state of first stress symptoms (Williams K¨ o gler and S¨ o ffker (2019)) first time a state-machine based based thresholds the transition of a ‘mild stress’ state to In previous publications (K¨ o gler and S¨ o ffker (2018), and Araujo Jones (2004), Jones (2007)). Plantogler and S¨offker (2019)) first state-machine based based thresholds for the transition a ‘mild(2007)). stress’ state to K¨ In previous publications (K¨ otime gler aawas and S¨offker (2018), and Araujo Jones (2004),of Plantogler and S¨offker (2019)) time state-machine based based thresholds for the transition ofJones awhether ‘mild stress’ state to K¨ model describing the plantfirst growth established. The a ‘high ‘high stress’(2002), state, or an evaluation whether the detected detected model describing the plant growth was established. The abased stress’ state, or an evaluation the K¨ ogleridea and of S¨offker (2019)) first time awas state-machine based for the transition of awhether ‘mild stress’ state to to model main this model is to combine a set of different symptom still belongs to a ‘mild stress’ state or already describing the plant growth established. The asymptom ‘highthresholds stress’ state, or an evaluation the detected idea of this the model is to combine a set of different still belongs to ‘mild stress’ state or already to main model describing plant growth was The aa ‘high anaadescribed evaluation thealready detected parameterized equations describing plant growth. Each ’high stress’ stress’ state,orare are inwhether few cases as e.g. to in main idea of this model isdescribing to combine aestablished. set of different symptom still state, belongs to ‘mild stress’ state or parameterized equations plant growth. Each asymptom ’high stress’ state, described in few cases as e.g. in main idea of this model is to combine a set of different still belongs to a ‘mild stress’ state or already to Each equation is is related related to different different growthplant statesgrowth. combined by (Ben-Gal et al. al.state, (2010)). equations describing a(Ben-Gal ’high stress’ are described in few cases as e.g. in parameterized to growth states combined by et (2010)). parameterized equations describing Each a ’high stress’ are described in few cases as e.g. in equation equation is related to different growthplant statesgrowth. combined by (Ben-Gal et al.state, (2010)). equation related growth states combined by (Ben-Gal al. (2010)). 2405-8963 ©et2019, IFAC (International Federation of Automatic Control) Hosting by is Elsevier Ltd.toAlldifferent rights reserved.
Copyright © 2019 IFAC 132 Copyright 2019 IFAC 132 Control. Peer review© under responsibility of International Federation of Automatic Copyright © 2019 IFAC 132 10.1016/j.ifacol.2019.12.510 Copyright © 2019 IFAC 132
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conditioned transitions. Transitions conditions and staterelated parameters are defined by optimization within a training loop compairing measured plant growth and state machine-based estimated ones. This new approach was firstly established in Beganovic and S¨ offker (2017) and applied the machine wear processes. States are defined by the state variables stress, memory, and damage level, distinguishing different states. Here states describe the status of the system (here: plant) within an event-discrete frame. Events are conditioned and described by measurable as well as inner variables in this case also including time. A state machine-based approach lends itself particularly well to modeling of plant growth due to the event discrete nature of plant growth, in this case: full irrigation, mild stress, and high stress for which different growth models are assumed. Additionally, a state machine-based approach allows for the description of structurally variable systems, which is ideal in cases where different behaviors can be assumed, as is the case in plant growth under different irrigation treatments. In Figure 1 the state machine model describing plant growth subjected to irrigation treatments is shown. This model represents the basis for the growth control approach.
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ranges, defined as stress level L=0 (L0) for no stress, stress level L=1 (L1 ) for mild stress, and stress level L=2 (L2 ) for high stress. The state variable ‘stress level’ (L) can therefore take the values 0, 1, or 2. These numbers are used to define irrigation sequences: A sequence of L=0/1/2/0 for instance describes a sequence starting in stress level L0 (above response threshold: full irrigated), followed by substrate desiccation till initially the response threshold is passed (stress level L1) and further till recuperation threshold is passed (stress level L2). Re-irrigation results in a switch back to stress level L0. An overview about the performed irrigation strategies, denoted as stress level trajectories, for the seven experiments is given in Table 1. 2.1 Definition of states The measurement time series of all plants are split into data segments resulting from the labeling procedure and then grouped according to the labels, irrespective of individual plant or time. Each data group is defined as a state. Hence, the plant states S are described by the vector S = [L, M, D]T , (1) with L = Stress level as a function of water demand and water availability, M = Memory as a function of mild stress occurence and interval, and D = Damage as a function of high stress occurence and mild stress duration 2.2 Deficit irrigation state machine model Detailed description for the plant growth model is given in K¨ogler and S¨offker (2019). Details are repeated here briefly for comprehensive introduction into the approach.
Fig. 1. State machine model of plant behavior due to water stress [K¨ ogler and S¨ offker (2019)] In compliance with defined transition conditions (arrows) the plant ‘switches‘ from one state Si to another state Sj (circles). Transition conditions are: L = Stress level with L=0 (L0 ; no stress), L=1 (L1 ; mild stress), L=2 (L2 ; high stress), tM = Maximum memory time, tD = Maximum stress duration time, tL1 = Time duration of stress level L=1 (L1 ), and tL1i −L1j = Time between the successive stress levels L1i to L1j . As described above, response threshold is defined by the onset of growth reduction due to a water deficit. Therefore, response threshold has to be determined at the water content level at which test plants showed initially a reduced growth rate (TLER) after water withdrawal compared to reference plants. Recuperation threshold is defined as point of no return, at which the progressed water deficit leads to irreversible growth because of a damage. The thresholds divide the water content scale into three 133
The starting point for the state machine model is state S1 representing a plant which did not experience any water stress before. State S1 is denoted as (0/0/0) for (‘no stress’, ’memory off’, ‘damage off’). Withdrawal of water resulting in an exceedance of the response threshold (stress level L1: L=1) leads initially to a transition from state S1 to the memory-initiating state S2 denoted as (1/1/0) for (‘mild stress’, ’memory on’, ‘damage off’). Henceforward, the state transitions depend on irrigation schedule: Re-irrigation (return to stress level L0: L=0) initiates the transition to state S1a (0/1/0). Alternatively, further water withdrawal resulting in an exceedance of the recuperation threshold (advance to stress level L2: L=2) initiates the transition to the damage-initiating state S3 (2/1/1); or constant mild stress (continuation in stress level L1: L=1) results in transition to the damageinitiating state S2a (1/1/1). Further transitions along the depicted model states operate correspondingly. End point for the state machine model is not defined, as model target is an intended deficit irrigation till harvest. Within the state-machine-model context the states Si are denoted as • S1 : Vector (0/0/0); reference growth behavior (comparison values), • S2 : Vector (1/1/0); reversibly reduced growth,
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S3 : Vector (2/1/1); irreversibly reduced growth, S1a : Vector (0/1/0); increased growth, S1b : Vector (0/1/1); irreversibly reduced growth, S1c : Vector (0/0/1); irreversibly reduced growth, and S2a : Vector (1/1/1); irreversibly reduced growth.
Here, the states S1 , S2 , and S1a are the desirable states for deficit irrigation purposes, as growth is not irreversibly affected due to water stress. The transition conditions are defined using • Lnj : the current stress level, • Lni : the stress level experienced in the previous instance, • tLni −Lnj : the time between successive stresses, and • tLn : the current stress time duration
as variables used in suitably designed conditions. 2.3 Data driven model parametrization
As illustrated before parameters of growth equations as well as transition conditions are assumed as unknown but constant, and have to be defined. As introduced in Beganovic and S¨ offker (2017) the parameter identification can be realized by multiobjective optimization. This procedure can be denoted as training of an interpretable machine model. Here the unknown parameters are defined by applying NSGA-II optimization. After training the model can be applied for prediction of plant growth. Assuming a known/given topology about the qualitatively distinguishable behavior of plant growth related mathematical equations are applied to describe the state dependent leaf elongation L for the different states depending on the lifetime increment ∆LT. The structural behavior is assumed as similar, the details are assumed as different (different parameter setting). 3. EXPERIMENTAL RESULTS The test rig consists of four growth stations placed in a rectangle and consisting of tables with superstructures for illumination, irrigation, measurements, and plant positioning (Figure 2).
Fig. 2. Test rig with potted maize plants (1). Growth stations (2) equipped with super-structures for illumination (a), irrigation (b), measurements (c), and plant positioning (d). The laboratory is located in a room (ground floor) of approximately 15 m2 2, the windows are blacked and oriented north. Air temperature, air humidity, and ventilation are 134
managed by university building equipment (pre-controlled ambient temperature and ventilation system). Illumination is automated with a clock timer. Seven irrigation experiments with maize plants (zea mays, c Ronaldinio (KWS), vegetative stadium EC11-EC15 / V1-V5) of 15 to 30 days’ duration each were implemented under laboratory conditions (nearly constant temperature, humidity, and ventilation; constant illumination over the day controlled by clock timer) for model parametrization and validation of the control approach. The plants are grown in 200 ml PET pots filled with a granular growth substrate. Evaporation rate of an equally prepared pot without plant showed nearly constant evaporation of +/- 2 g/d. The experimental conditions are summarized in Table 1. Trajectories evaluated for optimization of water use by application of deficit irrigation treatment are shown in bold font. Daily measurements of individual leaf lengths, environmental temperature, and pot mass are manually performed. From the measurements the total leaf length, leaf elongation rate, and water content are calculated. Implementation of the prediction algorithm is based on Monte Carlo simulation, which involves three steps: • Random sampling to obtain training data for the model • Validation of the model also using a random sample of test subjects • Statistical analysis of the results obtained from the validation stage In this case, two groups each consisting of eight randomly selected plants are selected for training and validation. Optimization of the growth model parameters is performed with the aim of minimizing mean ABE, MSE, and RSE across the entire sampled group. A population size of 50 evolving over 600 generations is observed to provide the optimal overall solution, with the performance of the algorithm described by: • Absolute error: 5.42 • Mean squared error: 20.57 • Root squared error: 1.99
The optimization algorithm generates state boundaries which are used to determine the water content corresponding to different stress states. These state boundaries are applied in defining the irrigation treatment employed. An experimental verification of the stress states of the plants has also been performed by considering the leaf elongation rate of test plants as compared to the control group, as well as examining elongation behavior after reirrigation to determine whether damage has occured- which is indicative of high stress. In Figure 3, a sample of optimized outputs from the state machine model for individual plants under different irrigation treatment is shown. For the selected plants under full irrigation and mild stress, the tracking of the growth trajectory is successfully achieved. The use of elongation variables in tracking the growth of the plant shows particular effectiveness in the application of the state machine model, producing reliable estimates of plant growth over time. The incorporation of the distinct stress states and accumulation of growth experienced as
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Table 1. Overview of experimental set-up and environmental conditions. Stress level trajectory describes the sequence of stress levels (L) experienced by the plant
Fig. 3. Example plant growth trajectory estimates. The continuous lines shows the actual total leaf length measured over time with the broken lines showing the estimated trajectory calculated using the state machine model. a result of each individual state over the duration of the plant growth makes it an attractive tool for flexible deficit irrigation scheduling using growth projections based on real-time data input. For plants grown under identical conditions, it is possible to obtain a relatively accurate
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estimate of comparative biomass expected from plants under different irrigation conditions using this model.
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4. CONCLUSION The newly introduced model shows good performance in tracking the growth of plants under different irrigation regimes, with the effects of water stress directly integrated into the design of the states and transitions. Future work will involve application of the state machine model in control of growth trajectories during the vegetative stage and incorporation of plant-based water stress signals in irrigation-related control decisions. ACKNOWLEDGEMENTS The authors acknowledge the work of Ms. Rosmawati Jihin by contributing numerical results applying the NSGAII optimization routines to the developed state machine model. REFERENCES Beganovic, N. and S¨ offker, D. (2017). Remaining lifetime modeling using state-of-health estimation. Mech. Syst. Sig. Process, 92, 107–123. Ben-Gal, A., Kool, D., Agam, N., van Halsema, G.E., Yermiyahu, U., Yafe, A., Presnov, E., Erel, R., Majdop, A., Zipori, I., Segal, E., R¨ uger, S., Zimmermann, U., Cohen, Y., Alchanatis, V., and Dag, A. (2010). Whole-tree water balance and indicators for short-term drought stress in non-bearing ‘barnea’ olives. Agricultural Water Management, 98(1), 124–133. doi: 10.1016/j.agwat.2010.08.008. Brisson, N., Wery, J., and Boote, K. (2006). Fundamental concepts of crop models illustrated by a comparative approach. In Working with Dynamic Crop Models, 257– 280. Elsevier LTD, , Oxford. Jones, H. (2004). Irrigation scheduling: advantages and pitfalls of plant-based methods. J. Exp. Bot, 55, 2427– 2436. Jones, H. (2007). Monitoring plant and soil water status: Established and novel methods revisited and their relevance to studies of drought tolerance. J. Exp. Bot., 58, 119–130. Kloss, S., Pushpalatha, R., Kamoyo, K., and Sch¨ utze, N. (2012). Evaluation of crop models for simulating and optimizing deficit irrigation systems in arid and semiarid countries under climate variability. Water Resour. Manag., 26, 997–1014. K¨ogler, F. and S¨offker, D. (2018). Steuerung des Pflanzenwachstums durch Bew¨ asserung. In 61. Jahrestagung der Gesellschaft f¨ ur Pflanzenbauwissenschaften e.V, Kiel. K¨ ogler, F. and S¨ offker, D. (2019). Sport for plants as a means for growth control: Water-based open-loop control of plant growth. Agricultural Water Management, submitted. Linker, R., Ioslovicha, I., Sylaiosb, G., Plauborgc, F., and Battilanid, A. (2016). Optimal model-based deficit irrigation scheduling using aquacrop: A simulation study with cotton, potato and tomato. Agric. Water Manag., 163, 236–243. Steduto, P., Raes, D., Hsiao, T., Fereres, E., Heng, L., Izzi, G., and Hoogeveen, J. (2008). Aquacrop: a new model for crop prediction under water deficit conditions. In A. L´ opez-Francos (ed.), Drought management: scientific and technological innovations CIHEAM, 136
(Options M´editerran´eennes : S´erie A. S´eminaires M´editerran´eens, volume 80, 285–292. Academic Press, Zaragoza. Williams, L. and Araujo, F. (2002). Correlations among predawn leaf, midday leaf, and midday stem water potential and their correlations with other measures of soil and plant water status in vitis vinifera. J. Am. Soc. Hortic. Sci., 127, 448–454.