Greenhouse climate management : an optimal control approach

In this thesis a methodology is developed for the construction and analysis of an optimal greenhouse climate control system.In chapter 1, the results of a literature survey are presented and the research objectives are defined. In the literature, optimal greenhouse climate management systems have been commonly presented and analysed as hierarchical systems. The main reasons were the inherent complexity of the system and existing differences in dynamic response times of the process variables involved. In general terms, the hierarchical control schemes contained two layers. An upper layer emphasizes control of the crop growth dynamics and a lower layer is concerned with control of the greenhouse climate. With respect to optimal greenhouse climate management it has not become completely clear from the literature how the hierarchical structure should be derived, how the relations between the different layers should be defined nor what kind of performance criterion should be used at each level. It is one of the objectives of this thesis to investigate the hierarchical decomposition of greenhouse climate management.Until now, in research on optimal greenhouse climate management, the main emphasis was on control of the carbon dioxide concentration and temperature in the greenhouse. Since in horticultural practice as well as in research, the humidity level in the greenhouse is considered to be an important climate variable determining the quality and quantity of crop production, a second objective of this research is to explicitly include humidity control in the control system design.For application of optimal control in horticultural practice, it is necessary to have a quantative model which describes the dynamic response of the greenhouse crop production process to the control and exogenous inputs with some accuracy. A third objective of this thesis is to analyse and validate dynamic models of the greenhouse crop production process.In optimal greenhouse climate control, predictions of the auction price and the weather are important. A final objective of this thesis is to investigate briefly the predictability of auction prices and to develop a methodology for greenhouse climate control which deals with errors in the model and the predictions of both auction price and weather, while maintaining near optimal performance.In this thesis the production of a lettuce crop is used as a vehicle for the illustration of the methodology developed.In chapter 2 the optimal greenhouse climate control problem is defined. The objective of optimal greenhouse climate control during the production of a lettuce crop is defined as to control the greenhouse climate such that the net return is maximized. The net return is defined as the difference between the revenues of the harvested lettuce crop and the climate conditioning costs. In the control problem physical limitations of the control inputs are included. Also simple bound constraints are imposed on the state of the greenhouse climate. These constraints represent unmodelled effects of the greenhouse climate on the quality and quantity of crop production.In chapter 3, dynamic models of crop growth and greenhouse climate are presented and analysed. With data collected in two lettuce growth experiments in a greenhouse, the models used in this research are calibrated and validated. It is shown that the models quite accurately describe the dynamics of the process variables considered in this research. Also it is illustrated that in the crop production process differences in response times do exist. The greenhouse climate responds rapidly to changes in control and external inputs, whereas crop growth responds comparatively slow to changes in the environmental conditions.A methodology for a first-order sensitivity analysis of dynamic models is presented and one of the lettuce growth models is analysed. It is shown that in this model of lettuce growth, only a few parameters mainly determine crop growth. The solar radiation level and carbon dioxide concentration are important environmental conditions. The temperature is relatively less important in this model.At the end of the chapter the models of lettuce growth and greenhouse climate are integrated to yield a model of the crop production process. This model is also validated.In chapter 4, the performance criterion used in optimal greenhouse climate control is defined in more detail. It is shown that a linear relation between the harvest fresh weight of the crop and the auction price of lettuce exists. The time-varying parameters of this linear relation show distinct diurnal trends which offer an opportunity for predicting the auction price.In chapter 5, the methodology for the solution and analysis of optimal control problems is presented. Necessary conditions for the existence of an optimal control trajectory are derived for a control problem with state and control constraints.It is shown that, in optimal control, the necessary conditions for optimality have a meaningful economic interpretation which can be used in the analysis of optimal greenhouse climate management.Considerable attention is paid to the solution of control problems in which both slow and fast dynamics interact. Using the framework of singular perturbed systems, the control problem in which both slow and fast dynamics are included, is decomposed into two subproblems. One sub-problem emphasizes efficient control of the slow dynamics, the other sub-problem emphasizes efficient control of the fast dynamics. It is shown that the performance criteria used in both sub-problems have a clear relationship with the main objective of the original control problem.The methodology of a first-order sensitivity analysis of openloop optimal control problems is derived.An algorithm is presented which is used to solve the optimal control problem iteratively on a digital computer is described.Finally, a feedback, feedforward control scheme based on the framework of optimal control is derived which is expected to yield near optimal performance in the presence of perturbations in the model and in the external inputs.In chapter 6 the methodology derived in chapter 5 is applied to the optimal greenhouse climate control problem using the model defined in chapter 3 and the performance criterion defined in chapter 4. For the full control problem, in which both greenhouse climate dynamics and crop growth dynamics are included, the necessary conditions are derived. Also the equations of the slow sub-problem, aiming at economic optimal control of the crop growth dynamics, and of the fast sub-problem, emphasizing efficient control of the greenhouse climate dynamics, are explicitly stated. The equations are analysed to gain insight into the operation of optimal greenhouse climate control.In chapter 7, optimal greenhouse climate control is investigated with simulations. Using the measured data obtained during one of the lettuce growth experiments in the greenhouse, optimal greenhouse climate control is compared with climate control supervised by the grower. The simulations show that using optimal control, greenhouse heating, carbon dioxide supply and ventilation are more efficiently used. Also, it is shown that the humidity level in the greenhouse largely determines the ventilation rate and strongly affects the performance of optimal greenhouse climate control.The feedback, feedforward control scheme is evaluated in simulations. It is shown that using a short-term weather prediction, the control scheme is able to achieve near optimal performance in the presence of large perturbations of the state and the weather from precalculated nominal trajectories. These results suggest that accurate long term weather predictions no longer limit the application of optimal greenhouse climate control in practice.In simulations, the decomposition of the greenhouse climate control problem is successfully evaluated. The performance of both sub-problems closely approximates the performance of the full control problem in which both slow and fast dynamics are included. Also, the control trajectories resulting from the decomposition closely approximate the control trajectories calculated for the full problem. Another simulation shows the importance of using an explicit economic criterion for control of the greenhouse climate dynamics. The decomposition supplies valuable information on the form of the objective function appropriate for use in control of the greenhouse climate dynamics.Finally, a sensitivity analysis supplies insight into the performance sensitivity of open-loop optimal greenhouse climate for perturbations in the model parameters and initial conditions.Chapter 8 contains a synthesis of the results obtained in the previous chapters. Two concepts for the implementation of optimal greenhouse climate management in horticultural practice are presented. The concepts are compared and their operation is discussed. It is indicated that both control systems are able to deal with uncertainty in the weather and the auction prices. Also, practical aspects like the amount of CPU- time needed, the flexibility of the hierarchical structure, the role of the grower in greenhouse climate management as well as the contribution of optimal control to the improvement of the efficiency of greenhouse crop production are discussed. Finally, it is indicated how the methodology developed in this thesis can be applied to multiple harvest crops like tomatoes and cucumbers.Chapter 9 contains concluding remarks as well as suggestions for further research.