between the biological and physical subsystems and also to the strong coupling of
the two main control variables: temperature and humidity (Gurban and Andreescu
2012). However, PID controllers have mainly been applied to control air temperature, (Zhou et al. 2013) humidity, irrigation, and nutrients supply (Jaimes-Ponce
et al. 2013). In several research works, PI/PID controllers are used as references
(Cheng et al. 2013) because they are well known and also because researchers are
looking for improvement in PID controllers performance by combining them with
modern control approaches such as neural networks (Qu et al. 2011; Zuo et al.
2012) and genetic algorithms (Bounaama and Draoui 2011).
14.2 Optimal Control
14.2.1 Problem Statement
A greenhouse system is a complex biosystem with biological and climatic components that make it a mathematical challenge for its modeling and control. The
dynamic mathematical models that describe the behavior of the greenhouse crop
and microclimate generally are nonlinear and non-convex (Chalabi and Zhou
1996). In addition they are stiff because of the existence of several different
timescales in the system. The time constants of the climatic variables and plant
physiological processes such as photosynthesis and transpiration are too small in
comparison to the crop growth (van Straten and van Henten 2010; van Straten
et al. 2011). Research carried out during the last two decades (van Straten et al.
2011) has proved that time constants in the greenhouse vary from 1 to 2 months in
case of the crop, 1–2 days for the soil and greenhouse climate, to 10–20 min in
case of light (van Henten 1994; Tap 2000). Furthermore, there is a high uncertainty
related to the input variables of the models such as external weather, but also to the
initial conditions of the states and model parameters. The optimal control of the
greenhouse climate implies the manipulation of the control variables in such a way
that the profit of the grower is maximized. According to the optimal control theory
(Kirk 1998) in order to state an optimal control problem it is required to define a
dynamic mathematical model of the system, a set of physical constraints, and also
a performance measure. Furthermore, due to the model complexity and nonlinearities, numerical algorithms are needed to solve the optimal control problem. In
the greenhouse the main state variables are air temperature (T g ), humidity (V i ), and
carbon dioxide concentration (C i ), but also for instance structural (W s ) and nonstructural (W n ) biomass of the crop. The crop behavior in time inside the greenhouse is influenced by the external weather and control actions. On one hand, the
most relevant disturbances are solar radiation (G), temperature (T o ), wind velocity
(w), CO 2 concentration (C o ), and humidity (R o ) outside the greenhouse. On the
other hand, most important control variables are manipulated variables that allow
modifying the greenhouse environment such as windows opening (r w ), heating
402
I. L. López-Cruz et al.
the two main control variables: temperature and humidity (Gurban and Andreescu
2012). However, PID controllers have mainly been applied to control air temperature, (Zhou et al. 2013) humidity, irrigation, and nutrients supply (Jaimes-Ponce
et al. 2013). In several research works, PI/PID controllers are used as references
(Cheng et al. 2013) because they are well known and also because researchers are
looking for improvement in PID controllers performance by combining them with
modern control approaches such as neural networks (Qu et al. 2011; Zuo et al.
2012) and genetic algorithms (Bounaama and Draoui 2011).
14.2 Optimal Control
14.2.1 Problem Statement
A greenhouse system is a complex biosystem with biological and climatic components that make it a mathematical challenge for its modeling and control. The
dynamic mathematical models that describe the behavior of the greenhouse crop
and microclimate generally are nonlinear and non-convex (Chalabi and Zhou
1996). In addition they are stiff because of the existence of several different
timescales in the system. The time constants of the climatic variables and plant
physiological processes such as photosynthesis and transpiration are too small in
comparison to the crop growth (van Straten and van Henten 2010; van Straten
et al. 2011). Research carried out during the last two decades (van Straten et al.
2011) has proved that time constants in the greenhouse vary from 1 to 2 months in
case of the crop, 1–2 days for the soil and greenhouse climate, to 10–20 min in
case of light (van Henten 1994; Tap 2000). Furthermore, there is a high uncertainty
related to the input variables of the models such as external weather, but also to the
initial conditions of the states and model parameters. The optimal control of the
greenhouse climate implies the manipulation of the control variables in such a way
that the profit of the grower is maximized. According to the optimal control theory
(Kirk 1998) in order to state an optimal control problem it is required to define a
dynamic mathematical model of the system, a set of physical constraints, and also
a performance measure. Furthermore, due to the model complexity and nonlinearities, numerical algorithms are needed to solve the optimal control problem. In
the greenhouse the main state variables are air temperature (T g ), humidity (V i ), and
carbon dioxide concentration (C i ), but also for instance structural (W s ) and nonstructural (W n ) biomass of the crop. The crop behavior in time inside the greenhouse is influenced by the external weather and control actions. On one hand, the
most relevant disturbances are solar radiation (G), temperature (T o ), wind velocity
(w), CO 2 concentration (C o ), and humidity (R o ) outside the greenhouse. On the
other hand, most important control variables are manipulated variables that allow
modifying the greenhouse environment such as windows opening (r w ), heating
402
I. L. López-Cruz et al.