Learning-based model identification for greenhouse climate control

1.1 Problem statement and contribution

The objective of this work is to maintain optimal temperature and humidity in a greenhouse using natural ventilation. This task is challenging due to the complex, nonlinear interactions between indoor climate variables and external weather conditions, which vary by region and depend on greenhouse-specific factors such as orientation with respect to the wind and sun and structure of the greenhouse. In addition, simultaneous control of temperature and humidity often results in conflicting control actions, requiring weighted cost functions. Safety mechanisms are essential to prevent unsafe conditions, such as open vents during strong winds.

In contrast to existing data-driven approaches that use simplified linear models [23], [24] or artificially generated data [27], this work proposes a fully data-driven nonlinear MPC that learns directly from real greenhouse data. Compared to prior neural-network-based MPC methods [26], which focus solely on temperature, this approach jointly optimizes temperature and humidity while accounting for their coupled dynamics and constraints.

The main contribution of this work is the integration of all these factors into a single method, addressing gaps left by previous approaches and advancing state-of-the-art greenhouse climate control. This is achieved through a learning-based MPC strategy to regulate greenhouse temperature and humidity using natural ventilation. In this context, the term “learning-based” refers to approaches in which the system dynamics is represented by machine learning models, such as NNs. Unlike traditional models, which require frequent recalibration, and prior data-driven models that rely on pre-generated datasets, the approach presented in this work enables continuous adaptation to the changing greenhouse conditions, such as changes in dynamics due to seasonal changes or wear. NNs were chosen over other learning-based approaches for this application due to the need for scalability, transfer learning, and online learning capabilities in greenhouse environments to allow for automatic adjustments to the current region and structure [31], [32]. The NPC strategy is tested in simulation using weather data from a traditional greenhouse in Southeastern Spain.

1.2 Paper outline

The remainder of this paper is structured as follows: Section 2 describes the greenhouse setup and the experimental data used for model learning. The background of learning-based MPC is explained, including NN modeling approaches. The proposed NPC strategy for greenhouse climate control is then presented in Section 3, covering the identification of the environmental model and the design of the control framework. Section 4 presents and discusses the results. Finally, Section 5 summarizes the main findings.

1.3 Notation

Vectors are written as bold lowercase letters, matrices are denoted as bold uppercase letters, and sets are denoted with calligraphic font, e.g., x , A , and X , respectively. The variable x _k is the state at time k, and x ̂ i | k is the prediction i steps ahead from k. Predicted sequences are denoted as uppercase vectors, i.e., X _k , U _k . The weighted norm is x Q 2 = x ⊤ Q x . The function diag : R n × n → R n maps diagonal elements of a matrix into a column vector. The maximum operator max : R n × R n × ⋯ → R n acts on each argument element-wise. Percentage points are denoted as %pt.

2.1 Greenhouse data

The data used in this work is from the greenhouse shown in Figure 1, located at “Las Palmerillas” Experimental Station of the Cajamar Foundation, in El Ejido, Almería, Spain. It is a traditional Almería-type greenhouse with an area of 877 m² (37.80 m × 32.20 m) and a polyethylene cover. The greenhouse is equipped with several actuation systems that control the climate. In particular, the natural ventilation system consists of five roof vents (8.36 m × 0.73 m) and two lateral vents (32.75 m × 1.90 m) on the north and south sidewalls. As shown in Figure 1, the roof vents have an angled opening and the lateral vents are opened by rolling up a plastic film. The vents are motor-controlled and can be opened from 0 % to 100 % of their ventilation area.

Figure 1:

Traditional Almería-type greenhouse.

The main climatic variables were recorded every 30 s using two sensors (each measuring both temperature and humidity). One is placed outside and one inside central in the greenhouse. The following variables were measured for this work: solar global radiation R in W m 2 (pyranometers model LP02, Hukseflux, Delft, The Netherlands), inside and outside air temperature in °C, ϑ _in and ϑ _out, inside and outside relative humidity in %, φ _in and φ _out (probes model HC2S3, Campbell Scientific Ltd., Shepshed, UK), and outside wind speed v _wind in m s (anemometer model A100L2, Vector Instruments, Rhyl, UK).

The opening ratio of each vent in %, u _ven, was recorded every 30 s, as the electric motors are connected to a supervisory and control data acquisition (SCADA) system in which manual and automatic actions are performed to regulate the air temperature inside the greenhouse by natural ventilation.

Formally, for a given time k, the input u _k = u _ven,k is the opening percentage, meaning that at 100 % all windows are fully open and at 0 % they are fully closed. All windows are controlled identically, so u _ven,k is a scalar value applied to each window. For a traditional low-tech greenhouse, there are no other actuators that heat, cool, humidify, or dehumidify the greenhouse. The state of the greenhouse is described by x k = ϑ in , k , φ in , k ⊤ . The weather data that act as measurable and predictable disturbances on the system is p k = φ out , k , R k , ϑ out , k , v wind , k ⊤ at each time step k. Using this data, the objective is to develop a model that can predict greenhouse temperature and humidity in the next time step based on current temperature, humidity levels, current weather data, and vent positions. This single-step predictive model allows to predict the sequences up to the prediction horizon using the state predictions from the previous time steps and the weather forecast as input for subsequent predictions.

The next section introduces NPC which is a standard learning-based MPC.

2.2 Learning-based model predictive control

For prediction of the state sequence for a specified input sequence, the system dynamics are modeled using a nonlinear discrete-time state-space representation

where x ̂ i | k is the predicted state, u ̂ i | k is the scalar input producing the prediction, and p ̂ i | k is the predicted weather data. The index [⋅] _i|k denotes a prediction variable where k is the initial time and i is the prediction steps ahead. The nonlinear function f _p describes the predicted update of the state Δ x ̂ i | k based on the current state, input, and system parameter. The standard MPC paradigm, explained in Section 2.2.1, uses a nonlinear function f _p derived from first principles, typically based on physics-based modeling or system identification techniques where parameters are estimated through regression or optimization on experimental data. In contrast, the learning-based MPC paradigm replaces the first principles-derived model with a machine learning model, specifically an NN in this work. In contrast to traditional system identification, which assumes a predefined functional structure (e.g., linear state-space models) and estimates the parameters accordingly, the NN learns a more flexible representation of f _p directly from the data. This modeling approach will be explained in Section 2.2.2.

3.1 Greenhouse model identification

For the training of the NN model, it is essential to use a dataset containing the most informative data and with sufficient excitation for the relevant features. Note that in a greenhouse, the indoor climate is a consequence not only of the effect of the ventilation, but also of the external climate that acts as a disturbance. The excitation required to capture the dynamics of the indoor variables is therefore mainly imposed by the inherent variations of the external climate. As for the natural ventilation system, it has a strongly nonlinear effect on the indoor climate. Therefore, as shown in Figure 3, data with opening and closing of the greenhouse vents have been used, covering the whole actuator operation range. The vents remain open or closed for different intervals of time each day, and under different external climate conditions, especially wind speed, which significantly affects the ventilation effect.

Figure 3:

Dataset for neural networks. Green background: data used for learning and validation in a k-fold cross-validation scheme; blue background: data solely used for testing.

Following an exploratory data analysis, relevant features were selected from the Mediterranean greenhouse data (see Section 2.1) to ensure that the model is trained on the most meaningful information, also ensuring coverage of diverse weather conditions. To that end, the dataset presented in Figure 3, comprising 81 days between October and December, was divided into three subsets: training data, validation data, and test data. The majority of the data (73 days, representing 90 % of the total data) was allocated for training and validation, while eight days from different weather scenarios were designated for testing. For training and hyperparameter tuning, a k-fold cross validation approach [34] was used with k = 5, meaning that in each ensemble, approximately 15 days out of the 73 were used for validation. Since this approach dynamically assigns training and validation data, it is not explicitly represented in Figure 3.

The following considerations were taken into account when using the data to synthesize the NN models. Although the greenhouse has roof and lateral vents, a single input is used to indicate the opening percentage of all the vents (i.e., all the vents are opened and closed at the same time). Similarly, internal radiation data is omitted as it follows external radiation, differing only by a factor influenced by the transmission coefficient of the cover. The inputs and outputs are then restructured to match the state-space representation, as shown in Figure 4. The network receives the data for one time step d k = x k ⊤ , u k , p k ⊤ ⊤ and predicts the changes in the inside temperature and humidity for the next timestep y _k = Δ x _k . Therefore, the prediction and the simulation model f _p( d _k ) and f _s( d _k ) are represented as f ( d _k , W , b ) with the learned parameterization W , b based on different sampling times and network sizes. During the training of the respective NN, this output is compared to the actual values in the training dataset.

Figure 4:

Setup for neural network training. The network inputs are states, inputs, and parameters from the training. Its output predicts state changes for the next time step.

In contrast to the standard NN approaches, the proposed models of the greenhouse climate are in the form of a state space representation, and only the update of the state is modeled by the NN, as introduced in (1). The training was conducted in an open-loop setup, where the NN was trained using data from the dataset at each time step. The input of the NN is data from a single timestep, and the output is the state of the next time step, e.g., inside temperature and humidity. The chosen NNs for the predictive and simulation models are both feedforward networks composed of fully connected layers. The choice of activation functions α _j for each layer j should reflect the expected nonlinear dynamics. The predictive network is used in the gradient-based optimization of the MPC approach and must therefore remain differentiable, making the commonly used ReLU activation α(z) = max(0, z) unsuitable. Instead, the smooth approximation softplus activation α x = log 1 + exp x is used [35].

The presented identification method is applicable to all kinds of greenhouses if sufficient data is available, as it purely relies on the given data for a specific greenhouse.

3.2 Greenhouse climate control

Relying solely on the opening and closing of windows to track the desired indoor temperature and humidity reference simultaneously is a challenging task. This difficulty arises in particular because the single decision variable, the ratio at which the vents are open, has a significant impact on the humidity and the temperature, and the required control action for successful tracking of temperature and humidity is often in conflict. In order to obtain the best possible results for this trade-off in greenhouse climate control, optimal control is used, i.e., MPC, in combination with the NN models, as described in (1). Therefore, the proposed NPC problem is a specific adaptation of the general MPC framework, as described in Section 2.2.1, tailored to the specific needs of greenhouse climate control and is defined as follows

The optimization is based on the current measurement of the state x _k defined in (3e). Based on this measurement, the state trajectory is predicted with the learned predictive model (1), and the input trajectory results from minimizing the cost function J. The cost function J in (6) will be explained in Section 3.2.2. It contains a smoothing part, therefore it also requires the previous input, defined in the problem in (3f). The weather data p ̂ i | k are chosen from the weather forecast. The constraints on the inputs, introduced in (3c) and (3d), will be further specified in Section 3.2.1.

4.1 Data and modeling results

The predictive model f _p is a fully connected NN with L _p = 2 layers with r _p = 64 neurons and a sampling time T _p = 10 min. The p in the subscript, indicates the hyperparameter of the predictive model. The deeper and more accurate simulation network f _s consists of L _s = 4 layers with r _s = 208 neurons each and the real-time sampling time of T _s = 30 s, where the s in the subscript denotes parameter of the simulation model. This finer sampling interval is chosen as it aligns with the original 30 s sampling of the data, providing a more precise system model than that of the predictive network needed for the controller. In order to prevent overfitting, several regularization techniques were applied, including k-fold cross-validation (k = 5), dropout [37], learning rate reduction on plateaus, and weight decay. Training and validation losses were continuously monitored to detect early signs of overfitting, but no such issues were observed in the selected models. Further increasing the network size did not yield significant performance improvements. Both networks were trained using the mean squared error (MSE) loss with the Adam [38] optimizer. All hyperparameters, including the number of layers, neurons per layer, learning rate, weight decay, and dropout rate, were optimized using Optuna [39], which efficiently identified the best-performing configurations. This automated approach provides a more systematic and scalable optimization process compared to manual tuning. The selected hyperparameters include a weight decay of 6.71 × 10⁻⁶ for the predictive network and 1.95 × 10⁻⁶ for the simulation network, along with dropout rates of 0.1 and 0.125, respectively. The learning rates were set to 2.2 × 10⁻⁴ for the predictive network and 1.4 × 10⁻³ for the simulation network. The predictive network was trained for 850 epochs, while the simulation network underwent 750 epochs of training.

The networks were trained on the dataset presented in Figure 3, as explained in Section 3.1. The metrics for the test of both NNs are given in Table 1, which summarizes the mean, maximum, and variance of the absolute prediction errors for temperature and humidity. These metrics are computed solely for a one-step prediction, as the networks were trained to predict the state in the next time-step based on the current state and the current input. Both networks demonstrate strong performance, with the prediction network achieving low mean absolute errors of 0.080 K and 4.002 %pt for temperature and humidity, respectively, where %pt denotes percentage points. The simulation network also performs well, with minimal errors in both temperature (0.048 K) and humidity (4.442 %pt), indicating effective generalization to the test data.

Figure 5 presents results for test data. In the temperature and humidity plots, the gray line represents measured data, while the black and orange lines correspond to predictions from the NNs. Given the correct data from prior time steps, both networks closely match the actual dynamics for the next step. These results confirm that the model has effectively learned the training data, ensuring a highly accurate one-step prediction, which is an essential foundation for reliable multi-step forecasts and control applications. Since, in practice, predictions cannot rely on real measurements of the states for future time steps, the performance of the network is also evaluated over longer horizons without measurement updates. This is especially relevant for the prediction network that will be used in the NPC controller. However, in this work, the control method will be tested in a closed-loop simulation with the simulation network, making the accuracy of the simulation network relevant as well.

Figure 5:

Model evaluation on test data. The test includes the performance of the model on days 3–6, with corresponding weather conditions and the vent inputs used during this period.

Figure 6 shows this evaluation for both networks. Both networks effectively capture the dynamics over the extended prediction horizon, although they both show slight deviations from the real data. On average, the simulation network remains closer to the actual temperature values than the predictive network, making it more suitable choice for control evaluation purposes. For humidity, both networks perform similarly well. The general dynamics of temperature and humidity are predicted correctly; however, absolute errors can be relatively large at certain moments, exceeding 5 K for temperature and 20 %pt for humidity. Unlike typical cases where prediction errors accumulate over time, Figure 6 shows that the errors do not propagate indefinitely. This stability can be attributed to external weather conditions, which act as stabilizing factors for temperature and humidity, preventing uncontrolled error growth. This qualitative evaluation highlights that the networks do not merely memorize the training data but generalize well over longer prediction horizons.

Figure 6:

Model prediction performance on test data. In the comparison, the networks predict states over days 3–6 without updates from real measurements, relying solely on their own state predictions at each time step.

4.2 Control results

The NPC controller is based on a learned PyTorch model [40], which is used in the optimal control problem described in (3). The problem is solved with CasADi [41] with the IPOPT solver. The PyTorch model is translated to CasADi by using the L4CasADi toolbox [22], [42].

A prediction horizon of N = 30 samples was selected, and with the controller sampling time T _p = 10 min, this resulted in a prediction time of 5 h. Note that this prediction horizon was selected according to the dynamics of the effect of natural ventilation on the indoor temperature. In that sense, natural ventilation is effective to control temperature during the day, particularly around midday, so the time gap for this control is usually 5–8 h depending on the season (i.e., depending on the hours with solar radiation). In addition, the natural ventilation effect is strongly influenced by the external weather conditions, especially by wind speed and the difference of temperature between indoor and outdoor, so it is difficult to know in advance and with certainty (i.e., even with greater prediction horizons) the values of these variables. With a 5-h forecast time, it is possible to react to changes in the weather forecast in sufficient time while the computing load is still manageable with a time step of 10 min. The opening speed of the vents is also taken into account by the time specification, as it takes approximately 5 min to open all vents from fully closed to fully open.

The state reference x _ref contains the reference of the inside temperature and humidity. The indoor temperature reference is a sinusoidal function that is 28 °C at noon and 18 °C at midnight, while the humidity reference is constant at 70 %. The inside temperature bounds are at 5 K below and above the temperature reference, and the preferred humidity is in the interval [50 %, 90 %]. The weights in the cost function (6) are determined empirically, such that the results reflect the desired behavior. The weights of the state reference tracking are Q k = diag 1 ⋅ 1 0 4 , 1 at day and Q k = diag 1 ⋅ 1 0 4 , 10 at night. During the day, the primary focus is on maintaining precise temperature tracking. At night, while the temperature remains essential, there is an additional emphasis on humidity control to ensure the humidity stays below a critical value. The penalties of the soft constraints are weighted with S = diag 1 ⋅ 1 0 8 , 100 . The change of the input is penalized with R = 0.5. The wind hard constraint ensures that the vents must be closed when the wind speed exceeds 8 m/s. In both scenarios, the calculations to solve the optimization problem with the NPC approach take around 0.5 s for each iteration, which is adequate for working within the 10 min that are used for the controller sampling time but is also adequate for working within the 30-s threshold required for real-time system operation. Note that the optimization was done on a standard laptop. The weather forecast in the simulation is based on the data from the real greenhouse described in Section 2.1, where zero-mean noise with a standard deviation of 1 % of the value range is added to the dataset. This noise reflects the uncertainty in real-world weather forecasts, ensuring a realistic evaluation of the NPC.

The control method was tested for two different weather scenarios. The first scenario, shown in Figures 7 and 9, covers days 3–7 of the dataset and represents a sunny period. The second scenario, shown in Figure 8, covers days 10–14 and represents a period of cloudy and windy days.

Figure 7:

Control results for four sunny days (day 3–7 in the dataset). Periods, where the wind speed exceeds 8 m/s and windows must be closed, are marked with gray vertical lines.

Figure 8:

Control results for four days with weather fluctuations (days 10–14 in the dataset). Periods, where the wind speed exceeds 8 m/s and windows must be closed, are marked with gray vertical lines.

In the sunny weather scenario in Figure 7, a clear control pattern can be observed: during the day, the vents open just enough to maintain the reference temperature, while at night, they remain closed to slow the cooling of the greenhouse. Also, around the 38-h time step, the vents are closed due to high wind speeds. The controller successfully maintains the temperature within the acceptable range for almost the entire period, mainly compensating the external weather disturbances by changing the vents opening. At daytime, when no saturation of the control signal occurs, the temperature is less than 3 K and on average 1.1 K away from the reference. While the nighttime humidity successfully remains below the maximum limit, it occasionally falls below the desired minimum for short intervals during the day. In those short daytime intervals and according to the observed vents opening, the controller is giving priority to regulating the greenhouse temperature since it is more important for crop growth. In 81.4 % of the time, the humidity is within the bounds.

In Figure 8, it is clear that the controller adapts its behavior based on changing weather conditions. As before, the vents close when wind speeds exceed 8 m s . During the first night, the outside temperature remains relatively high, causing the controller to keep the vents open for an extended period of time, allowing for effective humidity and temperature tracking. On the third day, however, minimal sunlight limits the daytime heating of the greenhouse, which means, it does not even reach the reference temperature when the vents are mostly closed. On the fourth day, as the weather improves, the greenhouse is maintained in more optimal conditions. In addition, the effect of natural ventilation to compensate for the weather disturbances and to follow the temperature reference is shown on the second and fourth day. In particular, on the second day (between the 33-h and 43-h time steps), the controller significantly modifies the vents opening when the outside temperature changes, so that the temperature inside the greenhouse does not deviate from the reference.

A different setting of NPC is presented in Figure 9. The temperature reference is kept constant at 25 °C for both day and night, and the cost for humidity deviation is set to zero. That is, only the humidity soft constraints are used, without a specific humidity reference. The resulting temperature trajectory is controlled at the set point when feasible. On the second day, however, the ventilation openings are temporarily closed due to the wind protection, which temporarily leads to a higher temperature. At night, the temperature setpoint cannot be maintained, leading to a drop in temperature. Therefore, changing references such as in Figure 7 are beneficial, as they are better to reach and incorporate the knowledge of realizable temperature trajectories, taking into account the experience of farmers.

Figure 9:

Control results for four sunny days (day 3–7 in the dataset) with different temperature reference and zone control of humidity. Periods, where the wind speed exceeds 8 m/s and windows must be closed, are marked with gray vertical lines.