The significance of crop co-states for receding horizon optimal control of greenhouse climate

While a tomato crop grows on the time-scale of weeks, the greenhouse climate changes on a time-scale of minutes. The economic optimal control problem of producing good quality crops against minimum input of resources is tackled by a two time-scale decomposition. First, the sub-problem associated to the slow crop evolution is solved off-line, leading to a seasonal pattern for the co-states of the amount of assimilates produced by photosynthesis, and the fruit and leaf weights. These co-states can be interpreted as the marginal prices of a unit of assimilate, leaf and fruit. Next, they are used in the goal function of an on-line receding horizon control (RHOC) of the greenhouse climate, thus balancing costs of heating and CO 2-dosage against predicted benefits from harvesting, while profiting as much as possible from the available solar radiation. Simulations using the time-varying co-states are compared to experimental results obtained with fixed co-states. It appears that the on-line control is sensitive to the time evolution of the co-states, suggesting that it is advantageous to repeat the seasonal optimisation from time to time to adjust the co-states to the past weather and realised crop state.