تجزیه و تحلیل حساسیت از مساله کنترل بهینه در مدیریت آب و هوای گلخانه ای

Optimal control systems are based on a performance measure to be optimised and a model description of the dynamic process to be controlled. When on-line implementation is considered, the performance of optimally controlled processes will depend on the accuracy of the model description used. Sensitivity analysis offers insight into the impact of uncertainty in the model parameters on the performance of the optimally controlled process. Additionally, sensitivity analysis may reveal the mechanisms underlying optimal process operation. This paper describes the methodology and results of a sensitivity analysis of an optimal control problem in greenhouse climate management. The methodology used, is based on variational arguments and requires a single solution of the optimal control problem, resulting in a computationally efficient technique. The example considered deals with economic optimal greenhouse climate management during the cultivation of a lettuce crop. The sensitivity analysis produced valuable insight into the performance sensitivity and operation of the controlled process. Both the model description of crop growth and production as well as the outside climate conditions have a strong impact on the performance. Humidity control plays a dominant role in economic optimal greenhouse climate management, emphasising the need for an accurate description of humidity effects on crop growth and production, either in terms of quantitative models or time-varying constraints on the humidity level in the greenhouse. Finally, the study revealed that the dynamic response times in the greenhouse climate are not limiting factors for economic optimal greenhouse climate control.

The optimal control methodology is a powerful technique to facilitate the design and analysis of optimally controlled systems. Optimal control systems are based on a model description of the dynamic process to be controlled and are designed in such a way that a performance criterion is optimised with respect to the control action applied to the system (e.g. Pontryagin et al., 1962). In practice, the structure as well as the parameter values of the model rarely coincide exactly with the real process. Since the control system is designed to be optimal with particular regard to the nominal structure and parameter values of the model used, it can be expected that the control system is sensitive to modelling errors which may reduce the performance of an optimal control system in practice. Therefore, sensitivity considerations are among the fundamental aspects of the synthesis and analysis of optimal control systems. One way to assess performance sensitivity is to substitute one by one the original values of the model parameters by slightly perturbed values and to compute the new optimal control and corresponding value of the performance criterion. This, however, is a rather time consuming procedure. In this research, a first-order approach to the sensitivity analysis of open-loop optimal control problems was used as derived by Courtin and Rootenberg (1971) and Evers (1979, 1980). Using variational arguments, the methodology requires a single calculation of the open-loop optimal control and corresponding state and costate trajectories. These are then used to calculate a first-order approximation of the performance sensitivity, thus saving a considerable amount of computation time. The optimal control problem considered in this paper, deals with economic optimal operation of the climate conditioning equipment in a greenhouse. To improve the economic performance of greenhouse crop production, in this approach, greenhouse climate control is based on an explicit trade-off between costs of operating the climate conditioning equipment and the economic return of the crop production process. This optimal control approach has received considerable attention in the agricultural engineering society (e.g. Chalabi, 1992; Hwang, 1993; Van Henten, 1994; Seginer & Ioslovich, 1998; Tap, 2000). However, parameter sensitivity issues have hardly been investigated in this field of research. Chalabi and Bailey (1991) as well as Van Henten and Van Straten (1994) performed sensitivity analyses of dynamic models of the greenhouse climate and crop growth of lettuce, respectively. The results, though being of interest from a modelling point of view, only allow for qualitative conclusions about the impact of model uncertainty on the performance of an optimal control system based on such a model. It is the objective of this paper to directly address the performance sensitivity of an economic optimal greenhouse climate control problem with respect to small perturbations in the model parameters.

With respect to the methodology used in this research the following conclusions are drawn. First of all, it was found that the first-order sensitivity analysis is a simple and straightforward way to obtain deeper insight into the operation of the optimal control problem and the relative importance of the model parameters, the initial conditions of the state variables and the external inputs, without having to go through extensive recalculations of the optimal control strategies. Secondly, the present study confirmed that a sensitivity analysis of the model to parameter variations and a sensitivity analysis of an optimal control problem including the same model might yield different results. Though the process model has an undisputed role in optimal control, it is the balancing of various objectives such as costs, revenues and penalties that determine the optimal control strategies. Finally, the intermediate variables in the solution of the optimal control problem such as the Hamiltonian and the costate trajectories, were found to be instrumental for a better understanding of the role of the model and model parameters in the determination of the optimal control strategies. Application of the sensitivity analysis to an optimal control problem in greenhouse crop production, led to the following conclusions. (1) The constraint on the humidity strongly influences the performance of optimal greenhouse climate management. Since these constraints were included as a first step to deal with adverse effects of high relative humidity on the quality of the crop, future research on greenhouse climate management should focus on a proper assessment of these effects in terms of quantitative models or modified climate constraints. (2) In optimal greenhouse climate management, the dynamics of crop growth play a dominant role and require accurate models of crop growth and development. (3) The relatively small performance sensitivity to changes in the heat and mass capacities of the greenhouse air indicate that the response time of the greenhouse climate is not a limiting factor for economic optimal control. (4) The outside climate conditions such as solar radiation, carbon dioxide concentration and humidity, and to a lesser extent the temperature, are important in greenhouse climate management. Consequently their accurate measurement and prediction is required.