HVAC SYSTEMS FOR GREENHOUSE ENVIRONMENTS USING NONLINEAR MODEL PREDICTIVE CONTROL

УДК 681.511.4

ABSTRACT

The purpose of this work is to develop and evaluate a Nonlinear Model Predictive Control (NMPC) framework for greenhouse HVAC systems that overcomes the limitations of linear MPC and decoupled PID approaches by integrating a physics-based nonlinear model with an AI-augmented transpiration estimator for coordinated multi-actuator climate control. This article presents such a framework that integrates a seven-state nonlinear dynamic model derived from first-principles energy and mass balances with an artificial-intelligence-augmented crop transpiration estimator. The controller simultaneously coordinates five actuator subsystems – heating, ventilation, fog-based humidification, CO2 injection, and shading – through a single constrained multi-input multi-output (MIMO) optimization problem. The methodology employs a physics-based model capturing coupled thermal, hygric, and chemical dynamics of the greenhouse microclimate, augmented by a neural network surrogate trained on Penman-Monteith transpiration outputs for computational acceleration. The NMPC optimization uses Sequential Quadratic Programming with a one-hour prediction horizon and 10-minute sampling time. A 24-hour simulation of a 1000 m² polyethylene greenhouse in a representative continental climate location under summer conditions demonstrates a 73.5% reduction in temperature tracking error and a doubling of comfort-zone occupancy time (from 13.9% to 31.2%) compared to uncontrolled operation, with mean relative humidity reduced from 93.2% to 75.6% and CO2 concentration maintained at 600-800 ppm for enhanced photosynthesis. The article contextualizes these results within a cross-paradigm review of five HVAC control methodologies – MPC, reinforcement learning, fuzzy logic, classical robust control, and digital twins – identifying the structural advantages and open challenges of the NMPC approach for protected cultivation, including the simulation-to-reality gap and the generalization challenge across greenhouse facilities.

АННОТАЦИЯ

Целью данной работы является разработка и оценка системы нелинейного прогнозного управления (NMPC) климатом теплицы, преодолевающей ограничения линейного MPC и децентрализованных ПИД-регуляторов за счёт интеграции физико-основанной нелинейной модели с нейросетевым оценщиком транспирации для координированного многоактуаторного управления. В статье представлена такая система, интегрирующая семимерную нелинейную динамическую модель, построенную на основе первых принципов энерго- и массообмена, с нейросетевым оценщиком транспирации растений. Регулятор одновременно координирует пять исполнительных подсистем – отопление, вентиляцию, туманообразование, подачу CO₂ и затенение – посредством MIMO-оптимизации с ограничениями. Методология включает физико-основанную модель связанных тепловых, влажностных и химических процессов, дополненную нейросетевым суррогатом транспирации по модели Пенмана–Монтейта. Оптимизация NMPC выполняется методом последовательного квадратичного программирования (SQP) с горизонтом прогноза 1 час и тактом дискретизации 10 мин. 24-часовое моделирование теплицы площадью 1000 м² в регионе показало снижение ошибки регулирования температуры на 73,5% и удвоение времени пребывания в зоне комфорта (с 13,9% до 31,2%), со снижением средней относительной влажности с 93,2% до 75,6% и поддержанием концентрации CO₂ на уровне 600–800 ppm. Результаты контекстуализированы в рамках межпарадигмального обзора пяти методологий управления ОВКВ.

Keywords: greenhouse HVAC, nonlinear model predictive control, NMPC, microclimate, crop transpiration, MIMO optimization, AI-augmented control.

Ключевые слова: ОВКВ теплиц, нелинейное прогнозное управление, NMPC, микроклимат, транспирация, MIMO-оптимизация, ИИ-управление.

Introduction

Buildings account for 30% of global final energy consumption and 26% of energy-related emissions [1]. HVAC systems represent the largest single end-use within this sector, consuming roughly 38% of building energy [2]. For greenhouse operations, where the controlled environment is the fundamental production tool rather than a comfort amenity, the energy stakes are even higher. With global cooling demand projected to rise 45% by 2050 [3], the development of robust and energy-efficient climate control strategies for protected cultivation has become a critical research priority.

The core difficulty in greenhouse climate control lies in the interaction between a nonlinear, multi-input multi-output (MIMO) thermodynamic system and a stochastic environment. Outdoor weather, solar radiation, wind, and crop physiological processes create a continuously shifting operating landscape that demands adaptive, predictive control strategies [4, p. 3], [5]. Unlike conventional commercial buildings where the thermal envelope is relatively static, greenhouses feature a dynamic biological load, the growing crop itself, that simultaneously absorbs solar radiation, transpires moisture, consumes CO₂ through photosynthesis, and modifies the thermal and mass transfer characteristics of the enclosure [6].

Traditional control approaches, on/off controllers, PID regulators, and rule-based systems, rely on fixed parameters and cannot adapt to real-time variations [7; 8]. These methods treat each controlled variable independently, leading to cross-coupling conflicts where, for example, increasing ventilation for temperature reduction simultaneously dilutes enriched CO₂ and reduces humidity below optimal levels [9]. Model Predictive Control (MPC) addresses these limitations by optimizing control actions over a receding horizon using a mathematical model [8], with energy savings of 17-54% documented [5; 10]. However, standard linear MPC formulations are fundamentally limited for greenhouse systems because the underlying physics are governed by nonlinear phenomena: fourth-power radiation exchange, saturating photosynthetic kinetics, humidity-dependent transpiration, and bilinear coupling between ventilation rate and temperature difference [11; 12].

The aim of this study is to develop and evaluate a Nonlinear Model Predictive Control (NMPC) framework for greenhouse HVAC systems, integrating a physics-based seven-state nonlinear dynamic model with an AI-augmented crop transpiration estimator. The tasks include: (1) formulating a coupled nonlinear dynamic model of the greenhouse microclimate; (2) developing a neural network surrogate for the crop transpiration estimator; (3) implementing a constrained MIMO NMPC optimization; and (4) evaluating the controller through simulation under realistic summer conditions in a region.

Materials and methods

HVAC control paradigms. A cross-paradigm review of 23 studies published between 2020 and 2026 identifies five principal control methodologies [13]. MPC uses a mathematical model to predict thermal behavior and solve constrained optimization over a receding horizon [4; 5; 8]. Robustness strategies include polytopic uncertainty with linear matrix inequalities (LMIs), achieving 24% improvement over nominal MPC [14]; reduced-order multiple-model banks with gap metric switching [15]; and ontology-based semantic models within digital twins [16]. Reinforcement learning (RL) agents learn policies through trial-and-error interaction, eliminating the model development bottleneck [7; 17], with energy savings of 10–26% reported [18]. Fuzzy logic controllers achieve up to 37% savings but face exponential rule-base growth [19]. Classical robust methods such as sliding mode control provide formal Lyapunov stability guarantees but are conservative [14; 15]. Digital twin approaches report up to 50% savings but a persistent simulation-to-reality gap remains [20; 21]. Table 1 summarizes the comparison.

Table 1.

Comparative analysis of five HVAC control paradigms

Nonlinear greenhouse microclimate model. The greenhouse microclimate is described by a system of seven coupled nonlinear ODEs derived from first-principles energy and mass balance laws, following the framework of Vanthoor et al. [6] and Rodríguez et al. [22]. The state vector x ∈ ℝ⁷ comprises indoor air temperature, cover temperature, soil temperature, canopy temperature, air humidity ratio, CO₂ concentration, and leaf area index (LAI). The control input vector u ∈ ℝ⁵ includes heating power, ventilation fraction, fogging rate, CO₂ injection rate, and shading fraction. The disturbance vector d ∈ ℝ⁴ captures outdoor temperature, outdoor humidity, solar radiation, and wind speed.

The air temperature dynamics are governed by an energy balance accounting for heating input, convective exchange with the canopy, soil, and cover surfaces, ventilation heat transfer, and evaporative cooling. The ventilation rate introduces a bilinear nonlinearity through the product of the control input and the temperature difference [11]. The cover temperature balance includes a Stefan-Boltzmann radiation term with fourth-power nonlinearity; the external convective coefficient is modulated by wind speed following Papadakis et al. [23]. The soil temperature follows Beer’s law of radiation extinction through the canopy [24]. The canopy temperature is governed by absorbed solar radiation, sensible heat exchange, and latent heat loss through transpiration, establishing the critical coupling between temperature and humidity dynamics [25]. The humidity mass balance accounts for crop transpiration, fog injection, ventilation exchange, and condensation [26]. The CO₂ mass balance follows a simplified Farquhar model [27]. The leaf area index follows logistic growth driven by net photosynthesis [28].

AI-augmented crop transpiration estimator. A neural network surrogate is trained on Penman-Monteith transpiration outputs [29] following Salzmann et al. [30]. The network takes six inputs, canopy temperature, air temperature, humidity ratio, solar radiation, LAI, and stomatal resistance, and produces the transpiration rate. The architecture consists of two hidden layers (12 and 8 neurons) with hyperbolic tangent and ReLU activations. Training uses 2000 samples spanning the full operating range. The surrogate achieves evaluation in under 0.1 ms, compared to almost 5 ms for the full Penman-Monteith computation [31], enabling real-time use within the NMPC optimization loop and establishing a framework for data-driven refinement [32].

NMPC problem formulation. The NMPC controller solves a finite-horizon constrained optimization at each sampling instant following the receding-horizon principle [33]. The cost function comprises output tracking terms for temperature, relative humidity, and CO₂; control increment penalties; and absolute control penalties for energy efficiency. Predicted outputs are obtained by integrating the nonlinear ODEs using a fourth-order Runge-Kutta solver [12]. The optimization uses Sequential Quadratic Programming (SQP) [34]. The prediction horizon is 6 steps (1 hour), control horizon 2 steps (20 minutes), sampling time 10 minutes [35]. All five actuators are optimized simultaneously within a single cost function, accounting for cross-coupling through coupled ODEs [36].

Simulation scenario. The controller was evaluated through a 24-hour simulation of a 1000 m² polyethylene greenhouse during a representative summer day. Weather: outdoor temperature 18-35°C, peak solar radiation 800 W/m². Crop: mature tomato canopy, initial LAI 2.5 m²/m². Setpoints per tomato cultivation recommendations [37]: daytime 22-25°C, nighttime 18°C, RH 65%, CO₂ 800 ppm.

Results and discussion

The NMPC maintains air temperature within 18-28°C throughout most of the day, preventing 35°C overheating during afternoon hours. The controller proactively activates shading and ventilation as solar radiation builds, demonstrating the predictive capability of the one-hour horizon. CO₂ concentration is maintained at 600-800 ppm through coordinated injection and ventilation, versus depletion below 400 ppm in the uncontrolled case. Mean RH is reduced from 93.2% to 75.6% [38].

Actuator coordination reveals the MIMO optimization logic. Heating operates at 40–50 W/m² during nighttime, reducing as solar gain increases. Ventilation ramps from 5% to 50% between 8:00 and 16:00, modulated by the cooling-versus-CO₂-dilution tradeoff. Shading activates at ≈10:00, reaching 40% at peak radiation. CO₂ injection concentrates in the 6:00-10:00 window before ventilation rates become too high.

Table 2 summarizes the quantitative performance metrics.

Table 2.

Performance comparison: NMPC+AI vs. uncontrolled greenhouse

The NMPC+AI controller achieves a 73.5% reduction in integral squared error for temperature tracking. Comfort zone time (simultaneous satisfaction of temperature 18-28°C, humidity 50-80%, CO₂ 350-1200 ppm) more than doubles from 13.9% to 31.2%. These results are consistent with Wang et al. [38]. Energy consumption of 819.9 kWh/day falls within the 0.5-1.2 kWh/(m²·day) range for continental operations [39].

The framework demonstrates three structural advantages. First, the physics-based nonlinear model captures the transpiration-mediated temperature-humidity coupling that linear MPC approximates poorly. Second, MIMO coordination eliminates actuator conflicts inherent in decoupled PID architectures. Third, receding-horizon optimization exploits the predictability of solar radiation for proactive control.

Several limitations warrant acknowledgment. Results are based on a 24-hour simulation with idealized sensors; real deployments face sensor noise, actuator delays, and model-plant mismatch. The broader HVAC literature shows that simulation results consistently overestimate real-world performance [13]. The comfort zone occupancy of 31.2%, while double the baseline, reflects physical constraints of a greenhouse without active dehumidification. The central unresolved question is whether any advanced controller can be deployed across facilities without per-site engineering [13].

Conclusion

This article has presented a comprehensive NMPC framework for greenhouse HVAC systems integrating a seven-state nonlinear model with an AI-augmented transpiration estimator to coordinate five actuator subsystems through constrained MIMO optimization. The 24-hour simulation demonstrates 73.5% reduction in temperature tracking error, doubling of comfort zone occupancy, and maintenance of elevated CO2 at an energy cost consistent with industry benchmarks [39].

Comparison of the obtained results with known studies confirms the competitiveness of the proposed approach. Wang et al. [38] reported temperature tracking improvements of comparable magnitude using predictive control for greenhouse environments, and the energy consumption of 819.9 kWh/day (0.82 kWh/m²·day) falls within the 0.5–1.2 kWh/(m²·day) range established by Lin et al. [39] for continental greenhouse operations. The 73.5% ISE reduction exceeds the 17–54% energy savings typically reported for linear MPC formulations in the broader HVAC literature [5; 10], suggesting that explicit nonlinear modeling provides a meaningful advantage for greenhouse applications where bilinear and radiation-driven nonlinearities are pronounced. Relative to RL-based approaches reporting 10–26% energy savings [18], the NMPC framework achieves superior tracking performance while providing constraint satisfaction guarantees that model-free methods cannot offer. However, unlike the Nagpal et al. [14] robust MPC approach with formal uncertainty certificates via LMIs, the present NMPC formulation does not explicitly account for parametric uncertainty, which remains an area for improvement. The comfort zone occupancy of 31.2%, while substantially higher than the 13.9% baseline, reflects the inherent physical constraints of a naturally ventilated greenhouse without active dehumidification, consistent with observations in the broader controlled-environment agriculture literature [9; 13].

The NMPC approach accepts modeling cost in exchange for constraint-aware, interaction-managing, anticipatory control suited to greenhouses where physics are well-understood [4; 5; 13]. Four priorities for future work emerge: experimental validation with real crops; integration of weather forecast uncertainty through stochastic NMPC [36]; extension to multi-week phenological development; and comparative field trials against RL controllers [17; 18].

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