- Email: [email protected]
Nitrate Control of Leafy Vegetables - a Classical Dynamic Optimization Approach
Copyright @ IF AC Modelling and Control in Agriculture, Horticulture and Post-Harvest Processing, Wageningen, The Netherlands, 2000
NITRATE CONTROL OF LEAFY VEGETABLES - A CLASSICAL DYNAMIC OPTIMIZATION APPROACH J.D. Stigter· G. van Straten·
• Wageningen University System and Control Group Bomenweg -4 6703 llLJ Wageningen The Netherlands e-mail: [email protected]
Abstract: Control of nitrate content and other growth characteristics of leafy vegetables such as greenhouse lettuce is an increasingly topical subject in a European context. This preliminary study focuses on the control of characteristics in greenhouse lettuce at harvest time on the basis of dynamic optimization techniques in which a novel lettuce growth model of the plant is embedded. In order to gain insight in the growth model itself first various constant input responses are presented, followed by some results of a more challenging problem in which dynamic input trajectories are assumed. The 'step-by-step' approach starts with a concise model definition and progresses via the constant input response and the definition of an appropriate cost (defining optimality) towards a dynamic optimal solution. The method leads to some interesting design considerations and suggestions for further research, which are made in a concluding section. Copyright @2000 IFAC Keywords: nitrate content of leafy vegetables, dynamic optimization, Pontryagin's minimum principle, greenhouse lettuce
1. INTRODUCTION
ing that only the mass fluxes in the dynamical balance are presented. Second, the characteristics of the model are further investigated under constant input conditions which immediately sets the stage for a detailed definition of the cost function defining (although implicitly) optimal growth behaviour. Third, the time invariant input profiles will then be refined and put in a dynamic setting so that time dependent optimal trajectories can be iteratively calculated. After having discussed some optimal dynamic input trajectories, the paper concludes with some observations and suggestions for a further refinement of the methodology used.
1.1 Brief Summary Current standards commissioned by the European Union have set bounds on the nitrate content of leafy vegetables (such as greenhouse lettuce) at harvest time. This study attempts to utilize a mathematical model of the growth process in a systems and control setting, i.e. we will apply dynamic optimization methods to a growth model of greenhouse lettuce in a controlled environment, meaning that temperature, sunlight, and artificial radiation, are determined in some optimal way through a dynamic optimization algorithm. The paper develops as follows: First, the mathematical model is discussed on an introductory level, mean-
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be observed that at low light the temperature is not greatly affecting the head-weight at harvest time. Figure 6, however, shows an optimum temperature value for each (fixed) carbon dioxide concentration meaning that temperature dependence is observable. From the head-weight plots it is observable that light intensity and carbon dioxide concentration are both distinctively correlated with temperature for the higher temperature regimes.
3. OPTIMAL GROWTH BEHAVIOUR: DYNAMIC INPUTS
°oL---~,o~--~~----~~~--~e~--~ ~----~~
time (days)
Having gained some understanding of the carbon source-sink model in the previous section using constant input profiles a natural step to make at this stage is to find some dynamical input profiles which lead to a desired behaviour. Given the complex and nonlinear carbon source-sink model this is not an easy task and, also, some characteristics of optimal growth behaviour have to be defined before this question can be addressed. Since for the current preliminary study our first interest lies in minimization of energy cost (notably artificial light) under some terminal (desired) values for head-weight and nitrate content, the following optimization problem was defined:
Fig. 8. Optimal input profiles for light and temperature calculated with a gradient search algorithm. The results for the more natural situation of an open final time optimization problem will indeed be considered at a later stage of the project. The initial input profiles for the temperature and light Hux Io(t) and To(t) were chosen as truly constant values for all t. These constant values were determined on the basis of figures 4 and 7 in such a way that the terminal constraints Yhw (t,) = 400 g and YNOs (t ,) = 3500 mg were (almost) satisfied. The optimization algorithm was then directed to minimize the light integral J(I) while staying as close as possible to the terminal constraints for nitrate content and head weight. This was achieved by changing the segments ([ui(tA:), Ui(tHd, k = 0, ... ,N - 1} where N is the number of discrete intervals that constitute the mesh, i is an integer value indicating the iteration number, and {ui(t), tA: < t < tHl} is assumed constant and equal to u(tA:)' In the steepest descent algorithm the optimal value is simply achieved by adjusting the input values on each interval on basis of the gradient ~! with H the Hamiltonian associated with the optimization problem (Kirk 1970, A. E. Bryson (Jr.) Y. Ho 1975, A. E. Bryson (Jr.) 1999).
Given the lettuce source-sink model (1) - (2), denoted here as f(Scv,Scs,J,T,C0 2 ,Q.), and a fixed harvest time t,: Which temperature and light input 1 profiles determine a desired headweight Yhw and nitrate content YNOs at harvest time t, under minimal cost of the light Hux J(t),O < t < t,? Or, more specifically, let the cost J (J) be defined as
! t/
J(J) =
J(r)dr
(3)
o Then the optimization problem is to find an optimal input sequence {(J(t) T(t»,t = O. . . t,} which minimizes J(J) under the dynamical model constraints (1) - (2) and the terminal constraints
Yhw(t,) =Yhw(t,)
(4)
YNOs(t,) =YN03 (t,)
(5)
The results for this exercise are summarized in figures 8-9. Figure 9 shows clearly that the terminal constraints are satisfied meaning that the algorithm has performed satisfactorily although, of course, no global minimum for this specific case study can be guaranteed. From this figure we can observe that it is obviously cheaper to reach the final nitrate content level of 3500 mg by applying an impulse of light at the very end of the simulation interval while preserving a low light intensity during the first part of the simulation - just sufficient to ensure growth towards a lettuce with 400 g final head weight. The final light impulse increases the sugar content in the vacuoles thereby reducing its nitrate content as desired. Numerical experiments with the dynamic optimization algorithm demon-
The above optimization problem was solved using a gradient search for the associated two point boundary value problem, assuming a fixed harvest time t, = 60 days (A. E. Bryson (Jr.) 1999). The reason for choosing a fixed harvest time of 60 days is merely for computational convenience. I For simplicity it is assumed that the carbon dioxide concentration in the air is held constant at 0.02 mol/m 3
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NITRATE CONTROL OF LEAFY VEGETABLES - A CLASSICAL DYNAMIC OPTIMIZATION APPROACH J.D. Stigter· G. van Straten·
• Wageningen University System and Control Group Bomenweg -4 6703 llLJ Wageningen The Netherlands e-mail: [email protected]
Abstract: Control of nitrate content and other growth characteristics of leafy vegetables such as greenhouse lettuce is an increasingly topical subject in a European context. This preliminary study focuses on the control of characteristics in greenhouse lettuce at harvest time on the basis of dynamic optimization techniques in which a novel lettuce growth model of the plant is embedded. In order to gain insight in the growth model itself first various constant input responses are presented, followed by some results of a more challenging problem in which dynamic input trajectories are assumed. The 'step-by-step' approach starts with a concise model definition and progresses via the constant input response and the definition of an appropriate cost (defining optimality) towards a dynamic optimal solution. The method leads to some interesting design considerations and suggestions for further research, which are made in a concluding section. Copyright @2000 IFAC Keywords: nitrate content of leafy vegetables, dynamic optimization, Pontryagin's minimum principle, greenhouse lettuce
1. INTRODUCTION
ing that only the mass fluxes in the dynamical balance are presented. Second, the characteristics of the model are further investigated under constant input conditions which immediately sets the stage for a detailed definition of the cost function defining (although implicitly) optimal growth behaviour. Third, the time invariant input profiles will then be refined and put in a dynamic setting so that time dependent optimal trajectories can be iteratively calculated. After having discussed some optimal dynamic input trajectories, the paper concludes with some observations and suggestions for a further refinement of the methodology used.
1.1 Brief Summary Current standards commissioned by the European Union have set bounds on the nitrate content of leafy vegetables (such as greenhouse lettuce) at harvest time. This study attempts to utilize a mathematical model of the growth process in a systems and control setting, i.e. we will apply dynamic optimization methods to a growth model of greenhouse lettuce in a controlled environment, meaning that temperature, sunlight, and artificial radiation, are determined in some optimal way through a dynamic optimization algorithm. The paper develops as follows: First, the mathematical model is discussed on an introductory level, mean-
95
be observed that at low light the temperature is not greatly affecting the head-weight at harvest time. Figure 6, however, shows an optimum temperature value for each (fixed) carbon dioxide concentration meaning that temperature dependence is observable. From the head-weight plots it is observable that light intensity and carbon dioxide concentration are both distinctively correlated with temperature for the higher temperature regimes.
3. OPTIMAL GROWTH BEHAVIOUR: DYNAMIC INPUTS
°oL---~,o~--~~----~~~--~e~--~ ~----~~
time (days)
Having gained some understanding of the carbon source-sink model in the previous section using constant input profiles a natural step to make at this stage is to find some dynamical input profiles which lead to a desired behaviour. Given the complex and nonlinear carbon source-sink model this is not an easy task and, also, some characteristics of optimal growth behaviour have to be defined before this question can be addressed. Since for the current preliminary study our first interest lies in minimization of energy cost (notably artificial light) under some terminal (desired) values for head-weight and nitrate content, the following optimization problem was defined:
Fig. 8. Optimal input profiles for light and temperature calculated with a gradient search algorithm. The results for the more natural situation of an open final time optimization problem will indeed be considered at a later stage of the project. The initial input profiles for the temperature and light Hux Io(t) and To(t) were chosen as truly constant values for all t. These constant values were determined on the basis of figures 4 and 7 in such a way that the terminal constraints Yhw (t,) = 400 g and YNOs (t ,) = 3500 mg were (almost) satisfied. The optimization algorithm was then directed to minimize the light integral J(I) while staying as close as possible to the terminal constraints for nitrate content and head weight. This was achieved by changing the segments ([ui(tA:), Ui(tHd, k = 0, ... ,N - 1} where N is the number of discrete intervals that constitute the mesh, i is an integer value indicating the iteration number, and {ui(t), tA: < t < tHl} is assumed constant and equal to u(tA:)' In the steepest descent algorithm the optimal value is simply achieved by adjusting the input values on each interval on basis of the gradient ~! with H the Hamiltonian associated with the optimization problem (Kirk 1970, A. E. Bryson (Jr.) Y. Ho 1975, A. E. Bryson (Jr.) 1999).
Given the lettuce source-sink model (1) - (2), denoted here as f(Scv,Scs,J,T,C0 2 ,Q.), and a fixed harvest time t,: Which temperature and light input 1 profiles determine a desired headweight Yhw and nitrate content YNOs at harvest time t, under minimal cost of the light Hux J(t),O < t < t,? Or, more specifically, let the cost J (J) be defined as
! t/
J(J) =
J(r)dr
(3)
o Then the optimization problem is to find an optimal input sequence {(J(t) T(t»,t = O. . . t,} which minimizes J(J) under the dynamical model constraints (1) - (2) and the terminal constraints
Yhw(t,) =Yhw(t,)
(4)
YNOs(t,) =YN03 (t,)
(5)
The results for this exercise are summarized in figures 8-9. Figure 9 shows clearly that the terminal constraints are satisfied meaning that the algorithm has performed satisfactorily although, of course, no global minimum for this specific case study can be guaranteed. From this figure we can observe that it is obviously cheaper to reach the final nitrate content level of 3500 mg by applying an impulse of light at the very end of the simulation interval while preserving a low light intensity during the first part of the simulation - just sufficient to ensure growth towards a lettuce with 400 g final head weight. The final light impulse increases the sugar content in the vacuoles thereby reducing its nitrate content as desired. Numerical experiments with the dynamic optimization algorithm demon-
The above optimization problem was solved using a gradient search for the associated two point boundary value problem, assuming a fixed harvest time t, = 60 days (A. E. Bryson (Jr.) 1999). The reason for choosing a fixed harvest time of 60 days is merely for computational convenience. I For simplicity it is assumed that the carbon dioxide concentration in the air is held constant at 0.02 mol/m 3
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