Introduction. Controlled-environment agriculture (CEA) is viewed as a key platform for implementing Agriculture 4.0 concept, in which digitalization, Internet of Things, cyber-physical systems, and data analytics are integrated into a unified, highly automated production environment [1]. Measures aimed at providing food security of the Russian Federation are also driving the industry toward active digitalization and increased production efficiency [2]. CEA technology entails the widespread implementation of wireless sensor networks [3], intelligent climate and nutrient control systems, computer vision, and digital twins of greenhouses and vertical farms, enabling real-time adaptation of growing conditions to the requirements of specific crops and target product quality metrics [4]. In such Agriculture 4.0 architecture, the light control subsystem evolves from a standard automation component into a key cyber-physical module, closely integrated with cloud analytics, decision support systems, and artificial intelligence algorithms [5].

LED supplementary lighting has become one of the most energy-intensive and controllable components of modern greenhouse complexes: it can account for 60–85% of a protected-cultivation facility energy consumption, and 20–40% of production costs [6]. At the same time, the shift from local manual control to programmable luminaires, distributed controllers, wireless sensor networks, and dynamic control algorithms increases the number of digital signal conversion stages between the light regime setpoint and the actual photon flux at the leaf level. Therefore, the accuracy of the PPFD setting becomes not only an electrical characteristic of dimming but also a quality parameter of the control action applied to the plant.

With PWM control, bit depth of the controller determines the minimum step size for the duty cycle: ΔD = 1/(2N – 1). Given a linear relationship between the average light flux and the duty cycle, this corresponds to a PPFD adjustment step ΔPPFD = PPFDmax/(2N – 1). For example, within a range of 0–1000 µmol·m⁻² s⁻¹, a single step in 8-bit control corresponds to approximately 3.9 µmol m⁻² s⁻¹, whereas in 12-bit control, it corresponds to about 0.24 µmol m⁻² s⁻¹. In multichannel systems where the light flux is adjusted based on sensor data and varies throughout the photoperiod, such discreteness can become significant within PPFD ranges where the photosynthetic response is highly sensitive [7]. Consequently, the relevance of this issue stems not from the emergence of a new physical phenomenon, but from the expanding use of digital, networked, and adaptive control for LED supplementary lighting, where the precision of setting the light output becomes integral to the efficient operation of the entire system.

From a biological perspective, the phenomenon of intermittent lighting was extensively studied as early as the 1950s and 1960s. Pulsed lighting at frequencies of tens to hundreds of Hz, with specific duty cycles, yields a photochemical efficiency comparable to or exceeding that of continuous lighting, while achieving energy savings of up to 67 % [8]. Paper [9] demonstrates that the net photosynthetic rate (Pn) of lettuce under pulsed irradiation depends on the interplay between photosynthetic photon flux density, frequency, and the duty cycle of the light pulses. Comparable average illuminance does not guarantee an identical photosynthetic response when the temporal structure of the light signal differs. However, this study considers deliberately imposed pulsed irradiation regimes and does not analyze parameters such as the discreteness of the digital duty-cycle setting, quantization error, or the accuracy with which the set average PPFD is reproduced.

Other studies, notably [10], also demonstrate the variable nature of plant responses to pulsed lighting. In lettuce, photosynthetic activity may decrease at low frequencies and low duty cycles compared to continuous light, whereas at high frequencies and appropriate duty cycles, differences may be minimal or accompanied by improvements in specific growth and power consumption parameters. Paper [11] describes experiments involving various light pulse shapes at an average PPFD of 100 µmol m⁻² s⁻¹. The results show that the net photosynthetic rate of lettuce under a 50 Hz full-wave rectified signal is higher than that under 100 Hz rectangular pulses with a 50% duty cycle and does not differ statistically from continuous lighting. This indicates that not only frequency and average PPFD but also the waveform itself can be biologically significant.

Engineering research on PWM drivers for horticultural lighting has focused primarily on dimming range, efficiency, transient response, current ripple, thermal performance, electromagnetic compatibility, and emission color characteristics. For example, in [12], the accuracy of current regulation and the stability of color coordinates are assessed on PWM drivers for LED. With a current ripple coefficient of up to 30%, the photometric characteristics of LED can remain close to constant. At the same time, spectral effects are generally not considered a photobiological factor, since the engineering objective of such studies is to provide the electrical and photometric characteristics of the light source rather than to evaluate plant response. Consequently, the results of such studies cannot be directly used to determine the permissible PWM bit depth required to maintain stable photosynthesis.

Thus, it can be stated that the aforementioned publications constitute two complementary yet loosely connected lines of research: the photobiological and the engineering approaches. The photobiological approach examines plant responses to specifically designed pulsed light regimes. However, the quantization error associated with the digital setting of the average PPFD is typically not singled out in these studies. Engineering studies analyze the characteristics of PWM drivers and digital dimming, but the error in setting the light flux is not translated into photosynthetic response indicators. It follows that no quantitative model links the bit depth of PWM control to the error in setting the average PPFD and, further, to the relative variation of the net photosynthetic rate at a given operating point on the light response curve.

The objective of this study is to develop and validate — using a model tomato light response curve — a calculation criterion for selecting the bit depth of a digital PWM control system for LED supplementary lighting. This criterion establishes a quantitative relationship between the duty cycle quantization step, the error in reproducing the average PPFD level, and the relative deviation of the net photosynthetic rate Pn. This criterion is intended to determine the minimum bit depth of a digital controller at which the error in the discrete PPFD setpoint — at a specific operating point on the light response curve — does not cause the relative variation Pn to exceed a defined threshold. The study is limited to steady-state supplementary lighting conditions with constant temperature, CO2 concentration, and humidity, as well as constant plant water and mineral status. Current ripple within the PWM period, jitter, wireless communication errors, LED spectral shifts, and spatial non-uniformity of the light field are not accounted for in the calculation model.

Materials and Methods. Analysis of literature sources. The study is based on a review of publications regarding the research topics in IEEE Xplore, ScienceDirect, MDPI, Frontiers, and SpringerLink. The following query combinations were used: PWM dimming horticulture LED, pulsed light photosynthesis, LED driver quantization noise, flashing light effect microalgae, duty cycle PPFD lettuce, PWM AlGaInP spectral shift, and conducted EMI LED PWM dimming. Publications from 2015 to 2026 were analyzed.

At the first stage, more than 300 publications on the topic were identified. After removing duplicates, 258 remained. Based on titles and abstracts, 208 publications were excluded as they did not pertain to artificial plant lighting, LED systems, or photosynthesis response. The full texts of 50 research papers were evaluated against the selection criteria. Consequently, 22 sources were included in the analytical review. These sources covered the following parameters:

No direct publications quantitatively evaluating the sequence “PWM bit depth → mean PPFD reproduction error → net photosynthesis rate variability” were identified in the selected sample. Therefore, the model rationale was based on a comparison of the results regarding these three components.

Methodology for assessing photosynthetic activity. To evaluate the impact of quantization noise on plant photosynthesis, it is crucial to determine the target parameters for light systems and photosynthetic metrics. The current standard GOST R 58461-20191 introduces specialized units of measurement for plants. An analysis of the literature, particularly [13], has shown that the primary units for measuring the light flux affecting greenhouse plants are photon-based units — specifically PPFD and Daily Light Integral (DLI) — since these are directly linked to the quantum nature of photosynthesis. Radiometric quantities are required for analyzing energy and thermal conditions, whereas photometric quantities play an auxiliary role and should not serve as a basis for standardizing controlled environment light regimes without converting to photon-based metrics.

The DLI parameter, defined as the PPFD integral over the course of a day [14], characterizes the daily dose of light absorbed by plants and is used to describe both the natural climate regime and the total effect of natural and artificial lighting. Taking into account the need to estimate instantaneous values, this parameter is not considered in this work.

Empirical relationship of PPFD and Pn [15] was used to conduct the study:

(1)

where Pnmax — maximum photosynthesis rate (user-defined); PPFDsat — light saturation point (PPFD value at which the photosynthesis rate reaches ≈ 90 % of Pnmax, if d = 1); d — shading coefficient (ranging from 0.1 to 0.3), which affects the slope of the initial section of the curve: the smaller d, the slower Pn increases with rising light intensity.

Pn, in turn, affects plant growth, as reflected by a simplified model of the daily increase in plant dry mass [16]:

where ΔW — dry mass gain over time T; SL — total photosynthesis leaf area; R — carbon consumption for respiration and other losses (in carbon units).

In this work, tomato is used as a basic parameterization object, being one of the most studied greenhouse crops, for which quantitative dependences Pn(PPFD) under CEA conditions are presented in the literature. It should be noted that the shape of the light curve, the position of the light saturation region, and the value of local sensitivity dPn/dPPFD depend on the biological characteristics of the crop, variety, phase of ontogenesis, and the current physiological state of plants. Consequently, the quantitative estimates obtained below should be interpreted as results for the model object “tomato”, whereas the proposed approach remains applicable to other crops as well, provided the parameters of function Pn(PPFD) are replaced with species- and phase-specific ones.

Mathematical modeling of the effect of PWM control quantization for LED supplementary lighting on plant photosynthetic response. To establish realistic boundary conditions and parameter ranges for the simulation, an analysis was conducted of the technical specifications of the LED light system at the “Solnechny Dar” greenhouse complex in the Stavropol Territory, Russian Federation (45.295220, 41.468056). Covering an area of approximately 20 hectares, the complex employs a LED supplementary light system based on YAR-2-1040W-78R9G5B+8FR (GOLDEN SCORPION (AH) CO., LTD, China) and CS-1000-900100 (Nanolux, China) LED luminaires. The first type of luminaire generates the primary photosynthetically active flux (~3200 µmol·s⁻¹) dominated by red and far-red radiation, whereas the second type is used for spectral correction and for tuning the “light recipe” to the various stages of tomato ontogenesis. The luminaire groups are controlled via a wireless link (Fig. 1), and the light flux is regulated using a PWM signal from a control microprocessor. However, no experimental studies were conducted directly on this specific facility as part of the work. Instead, its parameters served solely as a typical example of an industrial CEA system and a source of realistic ranges for PPFD, bit depth, and PWM frequencies used to verify and tune the proposed model.

The established relationship between the net photosynthetic rate (Pn) and the PPFD allows the light regime to be viewed as a key control input in greenhouse systems utilizing artificial lighting. However, in practical digital LED supplementary lighting systems, the target PPFD level is generated not continuously but discretely, governed by PWM signals, the controller finite bit depth, and driver clocking. As a result, the actual light flux may contain small amplitude and time deviations from the specified value due to quantization noise. In this regard, based on the mathematical expression of the photosynthesis light curve Pn = f(PPFD), it is considered how the quantization parameters in the control channel are converted into PPFD fluctuations.

To quantitatively assess the impact of quantization noise in the LED supplementary lighting control channel on the photosynthetic response, a two-stage model was employed: the conversion of quantization parameters into PPFD fluctuations, and the relationship between PPFD fluctuations and variations in the net photosynthetic rate Pn. A rectangular hyperbola was adopted as the baseline light response function [17]:

(2)

where Pmax — photosynthetic rate at light saturation; α — quantum yield at low PPFD values.

In the scenario under consideration, other environmental factors — such as leaf and air temperatures, CO2 concentration, air humidity, water supply, and mineral nutrition — were assumed to be constant. This allowed for an isolated assessment of the contribution specifically of PPFD fluctuations induced by digital control parameters. While this assumption is justified for a theoretical analysis of system sensitivity, it does not capture the full complexity of actual processes in CEA systems, where the photosynthetic response emerges from the combined effect of light and non-light factors.

If the ideal PPFD value (set by the controller) in the leaf zone is designated as PPFD0, then the actual irradiance — accounting for quantization and the discreteness of PWM control — is described by expression:

(3)

where δPPFDq(t) simulates fluctuations in the light flux caused by amplitude and temporal quantization noise. It is assumed that δPPFDq(t) is a stationary random process with zero mean and variance , whose value is determined by the quantization step, PWM bit depth, and driver switching frequency.

Substituting expression (3) into (2) and linearizing the relationship Pn = f(PPFD) with respect to small δPPFDq(t) yields the approximation:

(4)

From (4), it follows that when

,

the average value of photosynthesis remains equal to Pn(PPFD0) and the variance of the photosynthetic response is determined by the expression:

(5)

Thus, the sensitivity of the system to quantization noise is given by the product of the local slope of the light response curve and variance , induced by the quantization parameters in the LED supplementary lighting control channel.

For practical application, a criterion for the permissible level of instability of photosynthesis is introduced in the form of a limit on the relative standard deviation:

(6)

where ε — acceptable relative variation (e.g., 0.01–0.05). If we substitute (5) into (6), we will obtain the requirement for σPPFD and, consequently, for the PWM/DAC bit depth and light signal filtering parameters:

(7)

Relation (7) will be used in the future to calculate the minimum permissible bit depth, modulation frequency, and PPFD pulsation depth, at which quantization noise does not lead to biologically significant instability Pn when growing plants in greenhouses with artificial lighting.

In a more general case, the dependence used in the work can be expanded to a multifactorial form, in which the net rate of photosynthesis is considered as a function of not only PPFD, but also microclimate parameters and plant resource supply: , where T — temperature; — carbon dioxide concentration; VPD — water vapor pressure deficit; W — water status index; N — mineral nutrient supply.

In this formulation, the sensitivity of the photosynthetic response to quantization noise is determined not only by the position of the operating point on the light curve, but also by the current physiological and microclimatic context.

Research Results. Figure 2 shows a typical light response curve: a rapid rise in Pn at PPFD up to ~500–600 µmol·m⁻²·s⁻¹, followed by a gradual approach to a plateau at 1500–2000 µmol·m⁻² s⁻¹ (for all three regimes: W300/W500/W700). In terms of the model, this means:

Figure 3 shows a processed version of the graph in Figure 2, displaying the dependence of the net photosynthetic rate (Pn) on the PPFD for three lighting regimes (W300, W500, W700), simulating different levels of total illuminance under greenhouse conditions.

Figure 4 reflects the sensitivity of photosynthesis to PPFD fluctuations with an illustration of the numerically calculated derivative dPn/dPPFD for the fitted light curve in Figure 2. Maximum values dPn/dPPFD occur in the intermediate PPFD range (200–600 µmol m⁻² s⁻¹), while in the light saturation region (PPFD ≥ 150 µmol·m⁻² s⁻¹), the derivative tends to zero. This fact indicates a sharp decrease in the influence of light flux fluctuations on the photosynthetic response. The presented dependence is used to quantify the effect of quantization noise through the model .

Figure 5 shows the relative standard deviation of PPFD (σPPFD/PPFD0) as a function of PWM control bit depth for several modulation frequencies (1, 5, and 20 kHz). The data in the plot provides a link between the digital control system parameters and the variability of the light regime.

Figure 6 shows the results obtained based on the proposed mathematical model (7). The dependence reflects the relative standard deviation of the net photosynthesis rate from the given PPFD0 level for four PWM bit depth options (8, 10, 12, and 16 bits). It is shown that with 8-bit quantization in the moderate PPFD range (300–600 µmol·m⁻² s⁻¹) — corresponding to the steep section of the light response curve — the relative variation Pn reaches several percent, exceeding the conventional threshold ε = 2 %. Conversely, with 12–16-bit resolution, the variation remains below this threshold across the entire operating range. In the light saturation region, the impact of quantization noise on Pn is minimal regardless of bit depth, highlighting the combined importance of selecting the PPFD operating range and quantization parameters.

Figure 7 a shows the time-domain waveforms of PPFD(t) for 8-bit and 12-bit PWM control at the same average value PPFD0. The plots illustrate larger and more irregular light intensity fluctuations at lower bit depths. Figure 7 b displays the corresponding time series of the photosynthetic response Pn(t), obtained by substituting PPFD(t) into the light response curve model.

Based on the modeling results and established lighting quality requirements for greenhouse complexes, engineering recommendations were formulated regarding digital control parameters for LED supplementary lighting across various PPFD operating ranges (Table 1). These parameters link the target PPFD0 range — specific to particular crops and cultivation regimes — with the minimum required PWM bit depth and frequency. These characteristics keep the relative standard deviation of the net photosynthetic rate caused by quantization noise at or below 1–3%. When designing specific systems, these values can be refined to account for the spectral characteristics of the luminaires, driver-specific features, and agronomic requirements.

Discussion. The PPFD range of 200–600 µmol·m⁻² s–1 represents a critical zone of sensitivity to quantization noise, beyond which the impact of PPFD fluctuations on Pn diminishes sharply. Based on data from [18], it can be concluded that this parameter ranges from 200 to 850 µmol·m⁻² s⁻¹ across various vegetative stages. Consequently, PPFD fluctuations exert a significant influence on photosynthetic efficiency and, by extension, crop yield [13][19].

Figure 6 shows that increasing the bit depth leads to an exponential reduction in the relative level of PPFD noise fluctuations, while increasing the PWM frequency further diminishes the effective impact of quantization through shifting a portion of the noise spectrum into the high-frequency range, which is less significant for the photosynthetic apparatus of plants. The obtained light curves Pn(PPFD) shown in Figure 6, along with their approximation by hyperbolic models, are consistent with general concepts regarding photosynthetic light saturation [15]: when PPFD increases, the photosynthetic rate rises quasi-linearly, whereas upon the onset of saturation and further increases in irradiance, the rate reaches a plateau. This pattern of dependence underlies approaches to quantitative assessment of plant productivity based on integral resources (water, CO2, light) and has been confirmed for a wide range of crops. In this context, the use of the derivative dPn/dPPFD as a measure of the sensitivity of photosynthesis to PPFD fluctuations is a natural development of approaches to the analysis of light curves accepted in plant physiology [20].

At the same time, it should be noted that the tomato parameterization used in this study does not capture the full biological variability of crops grown in protected cultivation systems. Light saturation points, quantum yields, the slopes of the initial section of the light-response curve, and the rates of transition to the plateau phase can differ significantly among plant species and cultivars. This results in varying sensitivities of the photosynthetic apparatus to PPFD fluctuations of the same amplitude. Furthermore, even within a single crop, sensitivity to light fluctuations varies across ontogenetic phases — from vegetative growth to flowering and fruiting — and depends on the plant physiological state, including stress caused by water, temperature, or food compounds. Consequently, when implementing the proposed approach, it is advisable to refine the engineering specifications for PWM resolution and frequency based on Pn(PPFD) characteristics determined for the specific crop, cultivar, and developmental stage.

Based on the data obtained in the study (Fig. 7 a, b), we can conclude that the nonlinear pattern of dependence Pn = f(PPFD) partially smoothes out high-frequency oscillations. However, with 8-bit quantization, noticeable pulsations Pn remain, while at the 12-bit level, they practically disappear. These results confirm that increasing the PWM bit depth and decreasing PPFD fluctuations lead to a decrease in the temporal instability of photosynthesis.

On the other hand, studies on plant phenotyping and the analysis of resources–growth relationships [21] emphasize that even with identical total resource inputs (e.g., the same cumulative light dose), differences in supply dynamics can lead to variations in biomass accumulation and functional traits. In the moderate PPFD range, where dPn/dPPFD is at its maximum, high-frequency PPFD fluctuations caused by quantization noise in digital control systems can induce additional Pn variation on the order of a few percent. Criterion proposed in this study efficiently supplements the recommendations for target PPFD and DLI used in CEA and greenhouse practice through introducing a formal constraint on the permissible level of light pulsation associated with quantization parameters [20].

A significant limitation of the present model is its single-factor description of the external environment using only PPFD. In actual CEA systems, the net photosynthetic rate is determined by the combined parameters of light, temperature, CO2 concentration, humidity, and water and mineral nutrient supply. Moreover, the factors listed can alter both the absolute level of Pn, and the local derivative dPn/dPPFD, which underlies the proposed sensitivity criterion. Therefore, for the same average PPFD, identical quantum fluctuation depths can produce biological effects of different magnitudes, depending on the microclimate and the plant physiological state. From this perspective, it is advisable to link the further development of the model to a transition from the single-factor relationship Pn = f(PPFD) to multifactor response functions that account for key environmental parameters and their cross impact.

Research on digital control of LED lighting and drivers [12] typically focuses on conversion efficiency, thermal stability, and electromagnetic compatibility, with high PWM bit depth and modulation frequencies employed primarily to enable dimming and eliminate visible flicker. Such studies rarely address the impact of PWM bit depth and frequency on the biological characteristics of the crops being cultivated.

The results of this modeling show that, at PPFD of approximately 300–500 µmol·m⁻² s⁻¹, switching from 8-bit to 12–16-bit PWM at frequencies ≥ 2–4 kHz can reduce the modeled variability of Pn, caused by quantization noise from several percent to values <1–2%, that is, to a level comparable to microclimate and spatial light distribution inhomogeneities. This fact allows PWM parameters to be viewed not only as a means of enhancing the visual quality of lighting but also as a factor directly affecting the stability of the photosynthetic process [22].

The selection of 8-, 12-, and 16-bit resolution levels for the study is based on the fact that this range reflects the most common implementations of digital dimming in modern LED drivers. 8-bit depth represents the lower end of the spectrum, typical of simplified or budget-friendly control systems, whereas 12- to 16-bit depths are characteristic of drivers requiring smoother regulation and high precision in setting the light flux. Similarly, the selection of a frequency range in the order of 2–4 kHz is driven by engineering feasibility for industrial supplementary light systems, which must simultaneously provide the absence of visible flicker, acceptable electromagnetic interference levels, and permissible switching losses. Thus, the investigated PWM parameter combinations represent not merely abstract simulation modes, but a practically significant range of characteristics for LED supplementary lighting drivers actually in use.

The study results align with the systems approach to productivity assessment, wherein growth and gas exchange are viewed as the outcome of the temporal integration of responses to dynamically changing environmental conditions [23]. Unlike purely agronomic studies, such as [8], where pulsed light regimes are analyzed primarily in terms of energy savings and average yield comparisons — the model proposed here explicitly describes the end-to-end impact of digital constraints within the “quantization → PPFD noise → Pn variation” chain.

The research results lay the foundation for the further integration of light control models into broader Agriculture 4.0 and CEA digital platforms, where requirements regarding bit depth, frequency, and control algorithms can be derived not only from electrical and economic criteria but also from photobiological ones.

Conclusion. The authors have developed a computational model analyzing the impact of quantization in digital PWM control of LED supplementary lighting on the net photosynthetic rate of plants. The model captures the chain of relationships “PWM bit depth → error in maintaining average PPFD → relative variability of Pn”, using a constraint on the relative variability of the photosynthetic response as the criterion for determining sufficient bit depth.

A model light-response curve for tomato shows that the sensitivity of Pn errors in PPFD reproduction is highest in the 200–600 µmol·m⁻² s–1 range, where the slope of the light-response curve is steepest. In the light-saturation region, the impact of a PPFD maintenance error of a given magnitude on Pn is substantially reduced. Based on the adopted simulation conditions, recommended bit-depth ranges for digital PWM control were calculated: in the moderate PPFD range, the use of 12-bit or higher resolution provided compliance with the criterion of limiting the relative variability of Pn to no more than 1–2%.

The practical significance of this research is that the derived relations allow designers of greenhouse artificial light systems to move from empirical selection of bit depth and PWM frequency to their quantitative justification based on a photobiological criterion of acceptable instability of Pn. The proposed parameter ranges can serve as guidelines for the development of LED supplementary lighting drivers and controllers within the Agriculture 4.0 framework, where the light subsystem is viewed as a cyber-physical module subject to specific requirements regarding light signal quality.

The results obtained refer to stationary light regimes and parameterization of the photosynthetic light curve of tomato. The effects of current ripple, jitter, spectral effects, wireless transmission errors, and light field inhomogeneity were not taken into account. It is advisable to direct further research towards experimental verification of the model on real crops in greenhouses and on vertical farms, as well as its expansion taking into account spectral effects, nonlinear phenomena such as intermittent lighting and the combined effect of lighting noise and other environmental factors (temperature, CO2, water supply).

1. GOST R 58461-2019 Plants Illumination in Greenhouses. Terms and Definitions. (In Russ.). URL: https://files.stroyinf.ru/Data/715/71513.pdf (accessed: 01.07.2026).