Machine Learning-Based Beam Pointing Error Reduction for Satellite–Ground FSO Links

Abstract

Free space optical (FSO) communication, which has the potential to meet the demand for high-data-rate communications between satellites and ground stations, requires accurate alignment between the transmitter and receiver to establish a line-of-sight channel link. In this paper, we propose a machine learning (ML)-based approach to reduce beam pointing errors in FSO satellite-to-ground communications subjected to satellite vibration and weak atmospheric turbulence. ML models are utilized to find the optimal gain, which plays a crucial role in reducing pointing error displacement in a closed-loop FSO system. In designing the FSO environment, we employ several system model parameters, including control and system matrix components of the transmitter and receiver, noise parameters for the optical channel, irradiance, and the scintillation index of the signal. To predict the gain matrix of the closed-loop system, ML methods, such as tree-based algorithms, and a 1D convolutional neural network (Conv1D) are applied. Experimental results show that the Conv1D model outperforms other ML methods in gain value prediction, helping to maintain the beam position centered on the receiver aperture, minimizing beam pointing errors. When constructing a closed-loop system based on the Conv1D model, the error variance of the pointing error displacement was obtained as 0.012 and 0.015 in clear weather and light fog conditions, respectively. In addition, this research analyzes the impact of input features in a closed-loop FSO system, and compares the pointing error performance of the closed-loop setup to the conventional open-loop setup under weak turbulence.

1. Introduction

Since the early 1980s, there has been an increase in the number of satellites launched around the world. Specifically, more than 4700 low Earth orbit (LEO) satellites had been successfully launched by the end of 2021, making up roughly 86% of the total launch count for all types of satellites [1]. During these years, the satellite communications industry transformed from an alternative approach to communications into a mainstream transmission technology [2]. Since its inception, satellite communications have found a plethora of applications, including newsgathering, backhauling, media broadcasts, and other internet-based services. However, due to the ever-increasing demand for services like high-speed internet and video conferencing, bandwidth and capacity requirements have increased [3]. As the demand for data and multimedia grows, the conventional radio frequency spectrum is becoming congested [4]. Hence, to facilitate broadband applications, free space optical (FSO) systems were introduced as promising candidates for the next generation of wireless communications systems [5]. In particular, FSO satellite communications is a versatile technology that can quickly expand traditional communications networks in adverse geographic areas through optical carriers [6]. The German Aerospace Center’s terabit-throughput optical satellite system technology (THRUST) and NASA’s terabyte infrared delivery (TBIRD) program, which demonstrate satellite-to-ground laser communications in low Earth orbit with a transmission rate of 200 Gb/s, are primary standards in the application of FSO satellite-to-ground communications. These missions have already paved the way for future FSO satellite communication missions [7].

In FSO communications systems for LEO satellites, the transceivers are positioned to achieve line-of-sight (LOS) linkage between the satellite and the ground station [5].The optical laser beams utilized in these links need to propagate through the atmosphere over distances of 160 to 2000 km [8].These signals are subjected to atmospheric turbulence, such as absorption, dispersion, and pointing error, depending on the channel environment. The atmospheric turbulence generated by temperature and pressure changes along the propagation path can cause random temporal and spatial irradiance fluctuations in optical beams [9]. Because the optical beam in FSO communications is highly directional with very narrow beam divergence, it is essential to maintain pointing and acquisition throughout communication. Additionally, the atmospherically induced beam-wander phenomenon distorts the beam along its path, resulting in pointing errors [10]. The signal loss from pointing error can be caused by platform jitter, satellite vibration, or any kind of stress in electronic or mechanical devices. In any event, pointing error increases the prospect of link failure, leads to an increase in bit error rates, reduces throughput and drastically reduces the amount of received power, which increases the likelihood of error [11]. Thus, precise pointing is necessary to reduce the loss brought on by the beam-wander effect induced by atmospheric turbulence or due to platform vibrations [12,13].

To achieve LOS connectivity between the transmitter and receiver in FSO systems, and to reduce the impact of pointing error on communication performance, several pointing error reduction techniques have been proposed for optical communications systems. In [14], a detailed analysis was conducted to optimize the transmitted laser beam and the field-of-view of the receiver to minimize the outage probability under different channel conditions. Similarly, in [15], an efficient multiple-input multiple-output (MIMO) configuration was proposed to mitigate the impact of pointing errors on the transmission performance of a bi-directional vertical FSO link. A two-line element (TLE) set is also utilized to determine a satellite’s position and to estimate its point-ahead angle and future location, from which it is possible to minimize the impact of pointing error [16]. A pointing, acquisition, and tracking (PAT) system, which makes use of a satellite’s control system for coarse relative pointing and a fine pointing system (FPS) within the payload to mitigate residual pointing error, has also been employed [17]. The FPS employs a fast-steering mirror to maintain alignment between transmitted and received laser signals. In addition, transmitted beam width and transmitted power are adjusted to optimize the link performance of FSO systems [18,19].

However, these methods failed to consider both the vibration level generated by mechanical components (such as motors, actuators, and gimbals in the FSO system) and the effects of atmospheric turbulence. Because the optical beam in FSO communications is highly directional, the vibration levels of the transmitter and receiver are important, because they affect pointing error and the stability of the communications link. The beam-wander effect induced by atmospheric turbulence also leads to a significant reduction in received power and link quality, leading to communication errors. In [18], the real-time performance of the tracking system that adapts both the beam width and the transmitter power was addressed. During periods of high pointing error, the proposed system increases transmitter power to compensate for the potential signal loss and maintain the quality of the communication link. In [19], the pointing error reduction system utilizes a fine pointing mechanism that includes fast steering mirrors (FSM) or micro-electro-mechanical systems (MEMS) mirrors. To ensure fast and accurate corrections, advanced control algorithms, such as proportional integral derivative (PID) controllers or adaptive control algorithms, were utilized to process the error signal, and determine the required adjustments to the pointing mechanism. Although these mechanisms pursue rapid and precise adjustments to the beam direction, they mainly focus on reducing pointing errors by analyzing and upgrading existing components, such as optical telescopes, position-sensitive photodetectors, and optical filters.

Controller design, in a closed-loop feedback system, may involve the integration of adaptive optics into the system. The control system, which includes a wavefront sensor, measures aberrations in the optical system using a deformable mirror. The measured aberration is then compared to the intended or desired wavefront phase. The difference between the measured and intended wavefront is used to adjust the deformable mirror in real time. The system creates a feedback loop, where corrections are made based on the actual performance. On the other hand, in an open-loop system, the wavefront sensor is placed before the deformable mirror, resulting in the absence of a feedback mechanism. The deformable mirror is adjusted based solely on a pre-calculated model of how it should correct for aberrations [20]. In this manner, a closed-loop control system can effectively mitigate the impact of atmospheric turbulence and vibration levels by utilizing real-time feedback control information. When the transmitter receives feedback signals including the location of the receiving station, it uses this information to adjust the direction of the transmitted optical beam, ensuring the beam is precisely directed towards the receiver. Considering atmospheric turbulence, current pointing error, and transmitter and receiver vibrations, a well-designed controller can provide suitable optical beam direction to improve FSO communications performance.

To overcome the limitations in existing work, we propose a machine learning (ML)-based pointing error reduction technique for FSO satellite- to-ground communications by considering the controller design aspect to lessen the effects of atmospheric turbulence and the levels of vibration, securing a robust communication link. To achieve this, the FSO satellite-to-ground system is designed as a closed-loop feedback system in which the displacement from pointing error is constantly measured and compared to the desired placement. The error signal, which is the difference between the desired output and actual output, is fed back into the system to adjust input parameters to minimize pointing error. Considering this, our ML model is interpreted as finding the optimal gain for the closed-loop FSO system. The gain value plays a crucial role in determining the stability of the closed-loop system by reducing the error signal. Also, it determines the sensitivity of the system’s output, and affects how quickly the error diminishes to the allowable level of pointing error displacement. Optimal gain ensures the stability of the FSO satellite-to-ground closed-loop system, maintaining reduced pointing error, enhancing communication performance between a ground station and an LEO satellite. To develop our proposed ML model for use in predicting the optimal gain in the emulated real-world FSO satellite-to-ground system, we employ several system parameters, including control and system components of the transmitter and receiver, noise parameters in the optical channel, irradiance, and the scintillation index of the signal. Furthermore, we conduct simulations with tree-based ML algorithms (decision tree, random forest, and gradient boosting regressors) and a 1D convolutional neural network (Conv1D) to predict the optimal gain in FSO satellite-to-ground systems. The simulations show that the Conv1D network outperforms other algorithms in precisely predicting the gain value in the FSO system. In addition, we find that irradiance, the scintillation index, and the optical channel state are the most vital elements that go into a controller system for FSO satellite-to-ground communications.

The contributions from this study to pointing error reduction are as follows:

• We propose ML-based pointing error reduction using various parameters of the transmitter, receiver, and optical channel state. These parameters are used to design the FSO satellite-to-ground communications system, and to estimate gain in the closed-loop feedback system, ensuring that pointing error is minimized;
• We prove that the Conv1D model outperforms other ML schemes in precise prediction of gain in the closed-loop feedback system. This model outputs the predicted gain that maintains the optical beam’s center at the receiving aperture, mitigating pointing error displacement;
• We evaluate the important features influencing the controller design of the closed-loop FSO system to identify which input features are the most influential in predicting the gain value;
• We demonstrate a reduction in pointing error with comparisons between an open-loop control system and a closed-loop FSO system, varying the scintillation index values with respect to different atmospheric conditions. Based on this, we evaluate the effectiveness of the feedback mechanism in correcting pointing error in FSO satellite-to-ground communications under various atmospheric conditions.

2. The Proposed Model

The optical channel in the FSO satellite-to-ground link is impacted by the uneven distribution of, or variations in, the temperature and pressure of the atmosphere. That atmospheric turbulence leads to variations in the refractive index of air along the transmission path of the optical signal. These variations result in fluctuations in both the intensity and phase of the received optical signal [21]. Under intense fluctuations, there is a change in the power of the optical signal as it travels through the turbulent atmosphere, which causes the receiver to detect a signal weaker or stronger than expected. Similarly, in the context of phase fluctuation, the relative positions of the peaks and troughs change as the optical signal travels through the atmosphere. This causes the wavefront of the optical signal to distort or shift. These fluctuations are regarded as pointing error on the receiving side, and they limit communication performance. Based on fluctuations in the optical signal, atmospheric turbulence is categorized into three main regimes: weak, moderate, and strong. Weak turbulence occurs most frequently in practical FSO satellite-to-ground communication scenarios, whereas moderate and strong turbulence usually occur under severe atmospheric conditions such as storms and high winds. This paper focuses on weak turbulence, also classified as clean air turbulence [22]. In weak atmospheric turbulence, the variation in the refractive index of the air is small compared to the wavelength of the optical signal, and can be modeled as random fluctuations in the received optical signal. To simulate these variations, the log-normal model is utilized to describe the received optical signal through the FSO channel. The log-normal distribution is mathematically tractable and provides a good fit to simulated data obtained from the measurement of optical signals received under atmospheric turbulence [23].

2.1. The Log-Normal Optical Channel

Weak turbulence conditions are considered in this study of the FSO communication scenario owing to the nature of the link geometry and the atmospheric conditions encountered [24]. This scenario also provides a baseline for performance analysis before considering more severe conditions [25]. The influence of weak turbulence is significant in the performance of FSO links because it causes the scintillation effect that induces random fluctuations at the receiving end of the optical signal [26]. To analyze the weak turbulence effect, log-normal distribution is utilized because it is desirable to have a tractable probability distribution function (PDF) for received optical signal fluctuations that can predict communication performance with acceptable accuracy [27]. The log-normal distribution is suitable for characterizing the irradiance fluctuations in FSO communications under weak turbulence, because the random changes in the optical signal strength follow a log-normal process [28]. To account for the impact of atmospheric turbulence on the optical beam and its relative position on the receiver side, the optical channel state should be modeled. Figure 1 illustrates the relationship between the position of the transmitted optical beam and the PDF of the received signal. By analyzing the PDF assumed for a weak turbulence effect on the received signal at the detector, the position of the optical beam can be determined. The PDF of the received optical signal is affected by variations in the refractive index of the atmosphere. These variations can cause the optical beam to experience random fluctuations, resulting in positional errors. Modeling the data distribution by the log-normal PDF provides insights into the actual fluctuations and their influence on the positioning of the optical beam. The modeling approach is built upon a 1D log-normal distribution along with a corresponding correlation function. This correlation function establishes a mathematical relationship that describes how the received signal strength (depicted by the probability density function of the received optical signal) correlates with the position of the optical beam transmitted within an FSO communication system. This approach effectively characterizes the weak turbulence level of the FSO channel and emulates the variance in atmospheric turbulence. Additionally, the relationship between the received signal strength and the position of the transmitted optical beam is influenced by fluctuations in the refractive index of the atmosphere [29]. Bykhovsky [29] verified the theoretical results of the log-normal distribution process with a discrete time differential equation using the implicit Milstein scheme, which simulates stochastic differential equations involving random fluctuations. In the context of FSO satellite-to-ground communication systems, the simulated optical channel can be considered a moving object because it undergoes random fluctuations due to atmospheric turbulence, wherein the movement is the variation in the received optical signal strength over time, which affects the position of the transmitted optical beam, thus making the position of the optical beam transmitter time-correlated.

To describe the state of the channel and the corresponding position of the optical beam by taking into account the stochastic nature of the signal fluctuations, equations based on the Milstein scheme present the log-normal optical channel state and its relative position as follows [30]:

$$\left\{\begin{array}{c}{x}_{k+1}^p=a^p{x}_k^p+\phi \left(x_k^p\right)+b^p{u}_k^p+r^p{w}_k^p,\\ {\theta }_k=c^p{x}_k^p,\end{array}\right.$$

(1)

where

$$\phi \left(x_k^p\right)=-{\displaystyle \frac{K}{2{\sigma }^2{x}_k^p}}\left[\ln \left(x_k^p/I_0\right)\right], r^p=\sqrt{K\times \Delta t},$$

(2)

in which $K$ is given by

$$K={\displaystyle \frac{2I_0^{2}exp\left({\sigma }^2\right)\left[exp\right({\sigma }^2)-1]}{{\tau }_c}}.$$

(3)

In Equation (2), $k\in {\mathbb{Z}}^{+}$ denotes the discrete time step at which the simulated optical channel state is evaluated every 100 ms based on computational efficiency and system responsiveness, $I_0$ is irradiance, $\mathsf{\Delta }t$ denotes the sample period, and $x_k^p\in \mathbb{R}$ is the state of a simulated optical channel that can be considered a moving object. In Equation (1), ${\theta }_k\in \mathbb{R}$ denotes the relative position of the optical beam transmitter in millimeters, $u_k^p$ designates the bounded control input used to modify the optical channel state and transmission angle, $w_k^p\in \mathbb{R}$ denotes samples of white Gaussian noise, and $a^p\in \mathbb{R}$, $b^p\in \mathbb{R}$, $c^p\in \mathbb{R}$, and $r^p\in \mathbb{R}$ are constant values. Here, $p$ is utilized to distinguish it as a parameter utilized in the transmitter section. In (3), ${\tau }_c$ refers to a predetermined correlation period. The system matrix ($a^p)$ describes how the current state of the simulated optical channel affects the changes in the channel’s state in the future. In addition, $a^p$ plays a role in determining how state variable $x_k^p$ evolves over time in a deterministic manner, i.e., the next state of the system is entirely determined by the current state and parameter $a^p$. The control matrix, represented by $b^p$, relates how the system input impacts the state change, and $b^p$ is associated with the stochastic term $u_k^p$, typically interpreted as an external input. The value of $b^p$ determines how much random input $u_k^p$ affects the evolution of the state variable, $x_{k+1}^p$. Additionally, $c^p$ is the output matrix that connects the position of the optical beam transmitter with the optical beam condition, and $c^p$ can also influence the value of ${\theta }_k,$ which depends on the current state, $x_k^p.$ Larger values of $c^p$ coefficients indicate a stronger dependence on the current state for the position of the optical beam transmitter. Noise matrix $r^p$, which scales random white Gaussian noise, $w_k^p$, symbolizes the uncertainties and disturbances introduced into a closed-loop control system, which can affect the system’s output, performance, and stability. The nonlinearity term $\phi \left(x_k^p\right)$ relates to the full-signal-strength model, which includes optical information such as the scintillation index of the channel ${\sigma }^2$, the state of the optical channel, and irradiance of the signal. This nonlinearity term captures the intricate relationship of factors influencing the signal strength, providing a holistic representation of the optical channel’s dynamic behavior.

Using Equations (1)–(3), both the FSO satellite-to-ground channel and the position of the transmitted optical beam can be described. Here, atmospheric turbulence that affects the transmission of optical light beams can be included. It also explains the vibration level at the transmitter, which is characterized by the simulated optical channel state being considered a moving object. Figure 2a shows the 1-D position of the transmitted optical beam motion versus time, in which ${\sigma }^2=0.0380$, ${\tau }_c=0.1$, $a^p=1$, $b^p=1$, and $c^p=1$. The angular displacement in the y-axis represents the shift in the transmitted optical beam’s position compared to a stationary reference point. As depicted in Figure 1, it also implies that the PDF of the received signal changes accordingly, which has implications in performance degradation that can result in increased pointing error and a reduced signal-to-noise ratio. Notably, angular displacement is calculated by taking the difference between the position of the transmitted optical beam and its position at a previous time step. It also demonstrates that the time-correlated position of the transmitted optical beam depends on control input $u_k^p$ and the constant values $a^p$, $b^p$, $c^p$, and $r^p$.

2.2. Receiver Aperture Model

The uniqueness of the receiving aperture model is found in how it depicts the unpredictable physical vibrations encountered by the receiver aperture of a photodetector in FSO systems. The model effectively portrays the influence of these random vibrations on the receiving aperture’s position, a critical aspect for precisely capturing the optical beam and ensuring the continuity of the LOS connection between transmitter and receiver. Because atmospheric turbulence causes the optical beam to become scattered or distorted during transmission, it reduces the strength of the signal. In addition, the receiver aperture is assumed to have random physical vibrations caused by thermal expansion and voltage jitter [31]. Thermal expansion is the tendency of materials to expand or contract as they heat up or cool down, while voltage jitter relates to irregular fluctuations in voltage that occur in electronic circuits. These vibrations make the receiver aperture move slightly, which can in turn cause the incoming optical beam to be misaligned or distorted. This effect results in a weaker received signal, which is similar to the atmospheric turbulence effect. Moreover, the vibrations caused by thermal expansion and voltage jitter in the receiver aperture of the FSO satellite-to-ground scenario can be thought of as a form of excitation that causes the aperture to move in a random Brownian-like motion. Figure 2b demonstrates the receiving aperture motion that is analogous to Brownian motion, which is the movement exhibited by particles (molecules or atoms suspended in a fluid or gas) due to the collisions between those particles and the surrounding fluid’s molecules or atoms. To represent the motion of these particles through a mathematical model, the following equation is employed [32]:

$$m{\displaystyle \frac{\stackrel{ 2}{d}x(t)}{d{t}^2}}=-\gamma {\displaystyle \frac{dx\left(t\right)}{dt}}-n{\displaystyle \frac{dx\left(t\right)}{dt}}+\sqrt{2k_{B}T\gamma }W(t),$$

(4)

where t denotes time, and $x\left(t\right)$ is the trajectory of the particle with respect to the trap center, where the forces on an optically trapped particle are in equilibrium. The mass of the particle is denoted by $m$, and the friction it experiences from the surrounding medium is $\gamma$; $n$ is the optical trap stiffness, which represents a highly focused beam manipulating the small particles’ inability to move easily, $k_B$ is the Boltzmann constant, and $k_{B}T$ is the thermal energy unit, while $T$ and $W\left(t\right)$ represent absolute temperature and white Gaussian noise, respectively. Utilizing the continuous-time equation in (4), the motion of the receiving aperture is modeled in discrete time by discretizing the continuous-time equation and converting it into a state space model. The following equation represents the receiving aperture model’s discrete-time state space [32]

$$\left\{\begin{array}{c}{x}_{k+1}^l=a^l{x}_k^l+r^l{w}_k^l,\\ {\alpha }_k=c^l{x}_k^l.\end{array}\right.$$

(5)

In Equation (5), $l$ is used to distinguish a parameter in the receiver section. The state space model includes two state variables: the position of the source, $x_k^l$, and the measured position of the receiving aperture motion, ${\alpha }_k$. The position of the source refers to the location of the light source received by the photodetector’s aperture, while the measured position of the received aperture motion is the actual position of the receiver aperture at a given time, which may differ from the expected position due to the random physical vibrations present in the system. The discrete-time system includes standard white Gaussian noise, $w_k^l$, and parameters $a^l,r^l$, and $c^l$, which are derived from the continuous-time equation wherein $\mathsf{\Delta }T$ is the discretization time step, $a^l=\left(1-{\displaystyle \frac{n\Delta T}{\gamma }}\right)$, $r^l=\sqrt{2k_{B}T\gamma }$, and $c^l=1$. Here, $a^l$ represents the system matrix that demonstrates how the current source position impacts the future source position. It scales the current position, $x_k^l,$ to determine the next position, $x_{k+1}^l$. Larger values of $a^l$ imply a stronger dependence on the current position, and $r^l$ signifies the receiver-side noise matrix that describes how noise impacts the source position and the position resulting from the receiving aperture’s motion. It scales the random term $w_k^l$, introducing variability or randomness to the motion. Larger values of $r^l$ amplify the impact of the random term, making the system more susceptible to external noise or uncertainties, and $c^l$ influences how the derived parameter ${\alpha }_k$ depends on the current position. Larger values of $c^l$ indicate a stronger influence from the current position on ${\alpha }_k$.

2.3. The FSO Communications Scenario

To model the FSO satellite-to-ground communications link configuration, the optical transmitter and receiver were both subjected to relative motion in the optical link. The emitted optical beam has a non-uniform Gaussian intensity profile that changes with time [33]. The receiver is assumed to use a photodetector to determine the intensity of the profile of the optical beam. To ensure that the optical beam remains within the receiver’s aperture, the center of the transmitted optical beam should be measured, and information on transmitter position adjustments should be calculated [19]. Then, pointing error reduction is performed via an active pointing mechanism that sends the information from the receiver via feedback link to adjust the position of the transmitter. The receiver can provide information on the center position and intensity of the received beam, which is then used to calculate the necessary adjustments to the transmitter’s position. This ensures that the optical beam remains within the receiver’s aperture, and a stable communication link is established. Figure 3 illustrates an active pointing mechanism for the FSO satellite-to-ground communications channel to minimize pointing error. To model how the relative position or displacement of the transmitted optical beam and receiving aperture affect signal strength, a discrete-time model was developed based on the stochastic state-space model [29]. Here, the displacement of the transmitted optical beam position is denoted by vector ${\theta }_k$, and the location of the receiving aperture is denoted by vector ${\alpha }_k$. Then, we calculate the relative displacement of the transmitted optical beam’s center at the receiving aperture center: $y_k=d\left({\theta }_k-{\alpha }_k\right)$. This deviation is the pointing error in which the displacement is equal to the distance ($d$) between transmitter and receiver times the difference between ${\theta }_k$ and ${\alpha }_k$. Figure 4 shows the relationship between the optical beam in the receiving aperture’s plane and displacement vector $y_k$. The pointing error indicates how far off the transmitted optical beam is from the optimal position, and how much the signal strength is impacted as a result. By modeling the pointing error, we can better understand how to adjust the transmitter’s position to minimize pointing error and improve received signal strength. To model the pointing error $y_k$, which is a linear function of $x_k^p$ and $x_k^l$, using the augmented vectors, the closed-loop system with feedback link can be expressed in the following augmented form [30]:

$$\left\{\begin{array}{c}{x}_{k+1}=\mathbb{A}{x}_k+\phi \left(x_k\right)+\mathbb{B}{u}_k+\mathbb{R}{w}_k,\\ {ϵ}_k=\mathbb{C}{x}_k,\end{array}\right.$$

(6)

where $\mathbb{A}=\left[\begin{array}{cc}{a}^p& 0\\ 0& a^l\end{array}\right],\mathbb{B}=\left[\begin{array}{c}{b}^p\\ 0\end{array}\right],\mathbb{R}=\left[\begin{array}{c}{r}^p\\ r^l\end{array}\right], \mathrm{a}\mathrm{n}\mathrm{d} \mathbb{C}=\left[\begin{array}{cc}{c}^p& -c^l\end{array}\right]$; $x_k=\left[\begin{array}{c}{x}_k^p\\ x_k^l\end{array}\right]\in {\mathbb{R}}^2$ is the augmented state vector, $w_k=\left[\begin{array}{cc}{w}_k^p& w_k^l\end{array}\right]$ is the augmented disturbance vector, $y_k=d{ϵ}_k$ is the pointing error, and ${ϵ}_k={\theta }_k-{\alpha }_k$.

In the closed-loop system shown in Equation (6), two state variables, $x_k^p$ and $x_k^l$, represent the positions of the optical beam and the receiving aperture, respectively. Augmented state vector $x_k$ is a 2 $\times$ 1 vector that combines the original state variables. Augmented vector $\phi \left(x_k\right)$ represents the non-linearities that affect the system’s behavior. Augmented vector $w_k$ includes disturbance variables $w_k^p$ and $w_k^l$, which capture the disturbance in the transmitter and receiver, respectively. In addition, $\mathbb{A}$ is a 2 $\times$ 2 state matrix in augmented system form, where $a^p$ and $a^l$ represent the coefficients for the linear function connecting pointing error $y_k$ with $x_k^p$ and $x_k^l$. $\mathbb{B}$ is the 2 $\times$ 1 input matrix in the augmented system that includes coefficient $b^p$, which corresponds to input variable $u_k$ in the equation. $\mathbb{R}$ is the disturbance matrix in the augmented system, which is a 2 $\times$ 2 matrix. It consists of coefficients $r^p$ and $r^l$ that relate to the disturbance variables $w_k^p$ and $w_k^l$, respectively. $\mathbb{C}$ is the 1 $\times$ 2 output matrix in the augmented system, which contains coefficients $c^p$ and $c^l$ associated with output variable ${ϵ}_k$ in the closed-loop system equation.

In this paper, we consider an ${\mathcal{H}}_{\mathrm{\infty }}$ optimization problem, aiming to ensure the stability of a closed-loop pointing error while maintaining a specified disturbance attenuation level. For this, a pointing error $y_k$ that satisfies disturbance attenuation condition $w_k\ne 0$ is considered, and attenuation factor $\epsilon >0$. The primary objective of this approach is to maintain the FSO communications link, with a focus on keeping the center of the optical beam closely aligned to the center of the receiving aperture within a specified disturbance attenuation level. This can be done by finding a set-point, $u_k=-\mathbb{K}{x}_k$, that stabilizes pointing error $y_k$ with respect to disturbance $w_k$:

$$\Vert y_k\Vert ⩽\epsilon \Vert w_k\Vert ,$$

where $\epsilon$ is a positive real number. This approach reduces the extent of pointing error displacement and ensures controlled disturbance attenuation for the prescribed attenuation level $\epsilon >0.$

In the context of our closed-loop system, the feedback gain matrix, $\mathbb{K}$, plays a crucial role in determining the set-point and in controlling the system’s behavior. It includes information about the current state of the system (represented by augmented state vector $x_k$) and determines the appropriate set-point, $u_k$, that can effectively minimize the pointing error. The set-point is the desired position of the optical beam’s center at the center of the receiving aperture. If the set-point is not maintained, the optical beam’s center will deviate from the center of the receiving aperture, resulting in pointing error. Additionally, feedback gain matrix $\mathbb{K}$ considers the need for attenuation. Disturbances $w_k$ are unwanted external factors that can affect the system’s performance, and it is essential to mitigate their impact. The feedback control strategy, guided by gain matrix $\mathbb{K}$, considers both minimizing the pointing error and ensuring that disturbances are attenuated, leading to a more robust and accurate control system. To find optimal gain matrix $\mathbb{K}$, reducing the pointing error is challenging owing to the fact that the FSO satellite-to-ground system involves relative motion between the satellite-based transmitter and the ground-based receiver. Also, the FSO link is affected by atmospheric turbulence, causing random fluctuations in the received signal’s intensity and phase. Therefore, we introduce ML-based techniques that can predict the gain matrix by feeding in a dataset with information on the transmitter, the receiver, and the FSO satellite-to-ground channel, based on Equations (1)–(6). The dataset synthetically generated is obtained from the proposed control law evaluated using a linear matrix inequality (LMI) condition, as defined in [30]. For this, the YALMIP interface is utilized with the SDPT3 optimization toolbox. YALMIP is a MATLAB-based modeling language for optimization problems and provides a convenient interface for specifying optimization problems in a high-level language. The SDPT3 optimization toolbox is a solver that is used to find a solution that satisfies the LMI condition and ensures the stability of the closed-loop system.

2.4. Dataset Preparation

To train ML models for gain matrix prediction, a dataset that contains input variables (features) and output variables (targets) should be determined.

Table 1. Input parameters and symbols.

Transmitter		Receiver		Optical Channel
Feature	Symbol	Feature	Symbol	Feature	Symbol
System matrix	$a^p$	System matrix	$a^l$	Attenuation	$\epsilon$
Control matrix	$b^p$	Output matrix	$c^l$	Optical channel state	$x_k^p$
Output matrix	$c^p$	Noise matrix	$r^l$	Scintillation index	${\sigma }^2$
Noise matrix	$r^p$	Aperture motion	$x_k^l$
Irradiance	$I_0$	White Gaussian noise	$w_k^l$
White Gaussian noise	$w_k^p$

is the list of input variables employed to generate our dataset, which includes components from the transmitter, receiver, and FSO channel. The prediction of gain matrix $\mathbb{K}$ in the closed-loop system is equivalent to multiple output variables of the ML model. Gain matrix $\mathbb{K}$ plays a crucial role in minimizing pointing error $y_k$, given the augmented state vector $x_k$ that represents the position of the optical beam and the receiving aperture. Using optimal gain $\mathbb{K}$ and augmented state vector $x_k$, set-point $u_k$ is found and applied as feedback in the closed-loop system to keep the centroid of the transmitted optical beam as close as possible to the center of the receiving aperture. The predicted optimal gain in reducing pointing error enables us to proactively adjust the system parameters to minimize the error and maintain the optical communications link. It allows preemptive adjustments based on input variables for anticipated real environments between satellite and ground stations, incorporating factors such as atmospheric turbulence and satellite vibration. By predicting gain values, the system can proactively optimize beam alignment to compensate for disturbances, thereby reducing the likelihood of significant pointing error. Furthermore, predicted gain provides the opportunity for advanced optimization strategies, such as predictive control algorithms, which can continuously adapt the system parameters to maintain optimal beam alignment. This proactive and adaptive approach to gain adjustment can lead to more precise and stable beam control, ultimately reducing pointing error and enhancing communication reliability.

The 14 input variables in Table 1 were used for designing various ML models. The four output variables are elements of a 2 × 2 $\mathbb{K}$ gain matrix. Because multiple outputs should be predicted to construct the predicted gain matrix, we considered multi-output regression models for training with the dataset. Before feeding the data into the ML models, they were subjected to Z-score normalization to standardize the input variables (features), ensuring they are on the same scale. Z-score normalization involves transforming the data so that they have a mean of zero and a standard deviation of 1. This process is crucial because it helps machine learning algorithms converge more efficiently, and ensures that no single feature dominates the learning process. Standardization also eliminates bias caused by features with larger magnitudes, promoting fair learning across all features.

After normalization, the dataset was divided into training and testing subsets to evaluate the performance of the ML models. Typically, 70% of the data are used for training, while the remaining 30% are reserved for testing. This division ensures that the models are trained on a substantial portion of the data while still being validated on a separate, unseen dataset to assess their generalizability and performance in real-world scenarios.

2.5. Multi-Output Regression

With the given dataset, we can train multi-output regression models that can handle the prediction of multiple variables of gain matrix $\mathbb{K}$ in a single model. From the trained model, we can observe the dependencies, patterns, and interactions between input and output, allowing us to identify and represent the unique relationship of each output variable with input features. This information helps the pointing error control system to identify which input is the most important for achieving the desired control objectives, and to design feedback controllers that can effectively regulate the system. To evaluate the performance of the model and determine tuning parameters of ML algorithms, five-fold cross-validation was used. In Section 2.5.1 and Section 2.5.2, we describe the tree-based algorithms and the Conv1D neural network utilized as multi-output regression models to predict the gain matrix of the closed-loop system.

2.5.1. Multi-Output Tree-Based Regressor

We developed a multi-output decision tree regressor that can simultaneously predict multiple output variables based on a given set of input variables by recursively and independently partitioning the data samples based on input features for each output variable. Because a single decision tree regressor is prone to high variance problems, resulting in different tree structures for small changes in training data, a multi-output random forest regressor was also considered. This model uses ensemble learning, combining the predictions of individual trees to improve accuracy and generalization, thus reducing the risk of overfitting, and improving generalization. The randomness introduced during the creation of each tree, such as random feature selection and bootstrap sampling, helps to decorrelate the individual trees and enhance their collective predictive power.

Furthermore, random forest is more robust to noise and variations in the data, because averaging multiple tree predictions helps mitigate the high variance in decision trees, resulting in a more stable and reliable model. Similarly, a multi-output gradient boosting regressor can be utilized to predict multiple target values based on our input feature variables. It is also an ensemble learning technique that combines the prediction of multiple weak learners (typically decision trees) in a sequential manner. In this algorithm, the decision trees are iteratively added to the model to provide gradient boosting, with each tree trained to rectify error from the one before it. This allows it to learn complex relationships and capture non-linear interactions between features and targets effectively.

Table 2. Hyperparameters for multi-output tree-based regressors.

Hyperparameter	Decision Tree Regressor	Random Forest Regressor	Gradient Boosting Regressor
min_samples_split	12	5	2
min_samples_leaf	1	9	4
max_depth	3	5	5
n_estimators		600	100
learning_rate			0.001

lists the optimized hyperparameters for the multi-output decision tree, random forest, and gradient boosting regressors found by five-fold cross validation.

2.5.2. The Conv1D Regression Model

Convolutional neural networks (CNNs) are designed to capture spatial dependencies in the input data, making them well suited to problems involving images, time-series data, or data with a grid-like structure. By utilizing sliding windows, CNNs can capture local patterns and extract meaningful information across different regions of the time-series input. In our study, we utilized a Conv1D model because it excels at learning regional patterns and characteristics in time-series data [34].

Convolutional operations on local sections of the time-series input are crucial to enable the model to capture the connection between subsequent values, allowing it to learn local features with temporal dependencies. Note that the input features in our dataset are temporal data that vary over time, as seen in the scintillation index, Gaussian noise, and level of attenuation. By applying convolutions to these input variables over a sliding window of time, the Conv1D model effectively captures temporal patterns and relationships between features that may be challenging for other models. The implementation of the same set of filters across the entire input sequence enables the model to develop effective parameter sharing, allowing it to distinguish related patterns at various points in the sequence.

Figure 5 illustrates the Conv1D network architecture for multi-output regression, specifically for predicting the gain matrix of the closed-loop system for FSO satellite-to-ground communications. Each Conv1D layer comprises multiple filters that slide over the input using a specified kernel to generate feature maps. In our network architecture, the initial layer employs 64 filters with a kernel size of 7. This choice is motivated by the desire to capture a wide range of local patterns in the input data by employing a larger number of filters. Additionally, the kernel size of 7 achieves a larger receptive field, enabling each filter to gather information from a wider span of neighboring time steps. This characteristic is advantageous for capturing longer-term dependencies and patterns that extend over multiple time steps. To address overfitting and enhance generalization, we incorporate a dropout layer in the Conv1D model with a dropout rate of 0.5. The selected dropout rate deactivates half of the units in the dropout layer during training, striking a balance between retaining sufficient information and effectively regularizing the model. Subsequent Conv1D layers with smaller kernel sizes of 3 and 2 were introduced to capture more detailed and localized patterns from the input data. While the initial Conv1D layer with the larger kernel size (7) captures broader features, the subsequent layers with smaller kernel sizes focus on extracting finer-grained information. This hierarchical approach allows the model to learn features at different levels of abstraction. Moreover, reducing the number of filters in the subsequent Conv1D layers help prevent overfitting and balances the complexity of the model. With a reduced number of filters, the subsequent layers can concentrate on learning more specialized and specific patterns. To down-sample the feature maps and reduce their spatial dimensions, we incorporate a Max Pooling 1D layer. Then, a Flatten layer reshapes the output for a compressed representation of the features extracted from the input data.

As the final layer of the model architecture, a Dense layer is responsible for making predictions based on the learned representations from the preceding layers. By applying an activation function to the weighted sum of the input, the Dense layer introduces non-linearity to the model, enabling it to learn complex and non-linear relationships between the flattened features and the target variables.

Table 3. The network architecture.

Layer	Output Shape	Parameter
Conv1D	(None, 8, 64)	512
Dropout	(None, 8, 64)	0
Conv1D	(None, 6, 32)	6176
Conv1D	(None, 5, 16)	1040
MaxPooling	(None, 2, 16)	0
Flatten	(None, 32)	0
Dense	(None, 32)	1058
Dense	(None, 4)	132

summarizes the network architecture of the Conv1D multi-output regression model.

Table 4. Network parameters.

Parameter	Value
Learning rate	${10}^{-4}$
Epochs	200
Batch size	64
Activation function	ReLU
Optimizer Test size	Adam 30% of the samples

lists the network parameters utilized to predict gain in the FSO-based satellite-to-ground closed-loop feedback system. ReLU is the activation function used in the convolutional layers, introducing non-linearity into the model. The model is compiled using the mean squared error (MSE) loss function and the Adam optimizer.

3. Experiments and Evaluation

In this section, we demonstrate the performance of pointing error reduction by using various ML models. To measure the performance of the regression model, mean absolute error (MAE), MSE, and R-squared score were considered. By predicting the gain matrix of the closed-loop FSO system, we measure the distance between the center of the receiving aperture and the optical beam, which is characterized as pointing error in the FSO satellite-to-ground communications system. Our simulation environment considers FSO communications where the LEO satellite is at a 1000 km altitude [35] using 1550 nm wavelength to send optical data through an atmosphere under weak turbulence. We set input values related to the transmitter, receiver, and FSO channel that are adequate for this scenario and were obtained using Equations (1)–(6). To show the effectiveness of the feedback link, the performances from open-loop and closed-loop pointing error displacement are compared based on changes in atmospheric turbulence factors. The simulation to predict the gain matrix using several ML models was conducted with an RTX 3080 Ti GPU running Windows 10. To implement ML models, Keras 2.6.0 with the TensorFlow 2.6.0 library and Sklearn 1.3.2 in a Python (version 3.9.13) environment were considered. MATLAB R2022a was utilized to plot the simulation for open-loop and closed-loop pointing error displacement.

3.1. Pointing Error Reduction Performance

To examine the impact and severity of atmospheric turbulence on the transmitted optical signals in the FSO satellite-to-ground scenario, we varied the scintillation index of the signals in the simulation. Temperature changes throughout the day causing variations in the refractive index of the air can lead to changes in the scintillation index. Generally, the scintillation index is often greater [36] during the daytime (about noon) at ${\sigma }^2$ = 0.0244, compared to the morning, when ${\sigma }^2$ = 0.0101, and night, when ${\sigma }^2$ = 0.0716. This is primarily due to increased solar radiation and a higher temperature in the daytime, which contribute to severe atmospheric turbulence. In the morning, shortly after sunrise, the atmosphere is typically more stable with lower temperatures. As a result, the scintillation index is relatively lower. Similarly, at night, the absence of solar radiation and cooler temperatures lead to a lower scintillation index compared to daytime. The simulation parameters for the FSO satellite-to-ground link are in

Table 5. Simulation parameters for the FSO link.

Description	Parameter	Value(s)
Transmitter	$\mathrm{Scintillation} \mathrm{index} ({\mathsf{\sigma }}^2$)	0.0101, 00244, 0.0716
	$\mathrm{Sampling} \mathrm{time} (\Delta t$)	0.5 ns
	$\mathrm{Irradiance} (I_0$)	0.8 Watt/m2
	Number of channel samples	${10}^5$
	$\mathrm{Correlation} \mathrm{period} ({\tau }_c$)	0.1
Receiver	$\mathrm{Sampling} \mathrm{time} (\Delta T$)	0.5 ns
	$\mathrm{Trap} \mathrm{stiffness} (n$)	${10}^{-6}$ N/m
	$\mathrm{Friction} \mathrm{coefficient} (\gamma$)	$3.477 \times {10}^{-10}$
	Number of channel samples	${10}^5$
	$\mathrm{Boltzmann} \mathrm{constant} (k_B$)	$1.38 \times {10}^{-2}$ J/K
	$\mathrm{Temperature} (T$)	298.15 K

. By measuring the distance between the center of the receiving aperture and the optical beam, we compared the performance of pointing error reduction from various multi-output regression algorithms.

Table 6. Prediction performance of ML regression algorithms.

Regressors	MAE	MSE	R-Squared
Decision tree	0.019	0.014	0.960
Random forest	0.018	0.013	0.961
Gradient boosting	0.023	0.013	0.961
Conv1D	0.049	0.005	0.968

demonstrates the prediction results of various multi-output regression algorithms in terms of MAE, MSE, and R-squared score. The improvements seen with the Conv1D model as compared to the decision tree regressor are evident in the evaluation metrics. The Conv1D model achieved an R-Squared score of 0.968, higher than other tree-based regressors. This indicates that Conv1D is more effective in predicting the gain matrix, leading to better alignment and reduced pointing errors in the FSO system.

As shown by the MAE, MSE, and R-squared scores in Table 6, the proposed regressors demonstrate robust performance in removing pointing errors. Here, the open-loop system does not consider the feedback path in the satellite-to-ground communications link. Predicted gain matrix $\mathbb{K}$ is then utilized to determine set-point $u_k$ to minimize pointing error. In the scenarios for satellite operation in three different times, Figure 6 demonstrates that the closed-loop FSO system using the proposed ML gain prediction performs better than the open-loop FSO system by effectively reducing pointing error displacement. This is due to the ability of the Conv1D regression model to capture temporal dependencies in the data and effectively distinguish both broad and detailed features, providing a robust solution for predicting the gain matrix, and therefore reducing pointing error. In the figure, we can clearly observe that optimal gain value $\mathbb{K}$ applied to the FSO satellite-to-ground system maintained relatively smaller alignment errors in the closed-loop system than in the open-loop system. For the open-loop system, pointing error displacement ($y_k)$ varied in the ranges 2.00 mm to 3.35 mm, 2.10 mm to 4.51 mm, and 2.35 mm to 2.75 mm in the morning, at noon, and at night, respectively. However, in the closed-loop system, $y_k$ varied in the ranges −0.029 mm to 0.021 mm, −0.058 mm to 0.041 mm, and −0.007 mm to 0.006 mm in the morning, at noon, and at night, respectively. Note that the closed-loop system reduced the ranges of pointing error displacement in the FSO satellite-to-ground scenario by approximately 98%. This shows that the optical beam was stabilized at the center of the receiver, and confirms the superiority of the proposed Conv1D model for predicting the gain matrix in the closed-loop system.

In addition to atmospheric turbulence, an optical beam transmitted from an LEO satellite to a ground station is susceptible to fog. In particular, signal attenuation is frequently caused by fog, because fog particles reduce the visibility of light due to the dispersion of small water droplets. To examine the impact and severity of fog on optical signals transmitted in an FSO satellite-to-ground scenario, we varied attenuation factor ε for light fog and clear weather conditions using Kim’s model [37,38].

Table 7. Attenuation coefficient in foggy and clear weather conditions.

Parameter	Light Fog	Clear Weather
Wavelength	1550 nm	1550 nm
Visibility	0.70 km	18 km
Attenuation	4.540	0.056
Attenuation (dB/km)	19.718	0.245

displays the attenuation coefficient parameters used for light fog and clear weather conditions. Figure 7 shows pointing error displacement under light fog and clear weather conditions for open-loop and closed-loop systems. The figures clearly demonstrate that when optimal gain value $\mathbb{K}$ is applied to the FSO satellite-to-ground system, the closed-loop system exhibits significantly smaller alignment errors than open-loop systems. In open-loop systems, the displacement of pointing error $y_k$ ranges from 0.81 mm to 1.82 mm in foggy conditions, and from 0.55 mm to 1.55 mm in clear weather. However, in the closed-loop system, the range of $y_k$ is much smaller: from 0.01 mm to 0.06 mm in fog, and from −0.01 mm to 0.02 mm in clear weather. Our ML-based approach, especially the Conv1D model, demonstrated exceptional accuracy in predicting gain within closed-loop FSO systems. This capability is critical for ensuring the precise positioning of the optical beam on the receiving aperture, minimizing pointing error. By accounting for diverse input features and varying weather conditions, our method establishes a resilient communications link between LEO satellites and ground stations, ultimately leading to a notable improvement in the closed-loop FSO system’s pointing error displacement. Although more complex models may offer better performance, they require high computational power and high-end hardware specifications. To consider the effect of reduction on a small satellite’s hardware overhead, the Conv1D model can be a viable solution for real-time applications requiring quick adjustments of the optical beam position.

This simulation with various scintillation index and attenuation factors can provide baseline data for system designers who want to manage robust communication links between LEO satellites and ground stations. When the pointing error displacement level and the weather conditions are acquired, the system manager can choose an appropriate LEO satellite to send an optical signal to the designated ground station. In particular, a reduction in fog-induced attenuation allows a stronger and clearer signal, enabling better performance and fewer errors in communication. By choosing an LEO satellite with minimal fog and atmospheric turbulence, it is possible to increase the SNR and achieve higher data rates.

3.2. Feature Importances

In the pointing error reduction problem, it is significant to identify the most relevant features that contribute to predicting gain matrix $\mathbb{K}$ for the closed-loop system. For this, feature importance analysis based on a random forest regression model was conducted. Figure 8 indicates how much each feature contributes to model prediction of the gain matrix. Based on the insights provided by the figure, white Gaussian noise at both the transmitter $w_k^p$ and the receiver $w_k^l$ is the most influential factor in predicting the optimal gain of the closed-loop system. This information can be used to make informed decisions about carefully handling AWGN to improve the model’s performance when predicting optimal gain matrix $\mathbb{K}$. Other features that need to be cautiously controlled are the disturbance matrix for transmitter $r^p$ and receiver $r^l$, which captures the relationship between $w_k^p$ and $w_k^l$ and their impact on state vector $x_k$.

To handle white Gaussian noise in the FSO satellite-to-ground system model, the system designer must utilize a weighting function that emphasizes the frequency range of interest and attenuates noise outside that range [39]. Furthermore, it is necessary to consider the motion of the receiving aperture relative to its center, denoted $x_k^l$. To handle receiving aperture motion, the system designer should implement accurate tracking mechanisms that continuously adjust the position of the receiver’s aperture to keep it aligned with the incoming optical beam. Active tracking systems, such as motorized gimbals or piezoelectric actuators, can be employed to maintain alignment, compensating for any motion, and ensuring optimal reception. Simulated optical channel state $x_k^p$ (the position of the optical beam) should also be considered. This is crucial because it can be utilized to make informed decisions to maintain alignment between the transmitted optical beam and the receiving aperture. This alignment ensures that the optical beam remains within the receiver’s aperture, leading to a stable communications link and maximizing signal strength.

3.3. Statistical Performance Using Gaussian KDE

To analyze the underlying data distribution of the pointing error displacement obtained from a set of observations, histogram data for both open-loop and closed-loop pointing error displacements were collected. The distribution curve was then plotted using Gaussian kernel density estimation (KDE), depicted in Figure 9. During the simulation, weak turbulence in the atmosphere (both light fog and clear weather conditions) was utilized with attenuation rates measured at 19.718 dB/km and 0.0674 dB/km, respectively. In clear weather conditions, the open-loop pointing error ranged from +0.51 mm to +1.18 mm, with output error variance ${\sigma }^2$ = 0.092. Nonetheless, the closed-loop pointing error feedback efficiently reduced the pointing error, stabilizing the beam at the center and narrowing the pointing error displacement range to −0.01 mm to 0.01 mm, with output error variance ${\sigma }^2=0.012.$

For light fog conditions, the results show that the open-loop pointing error varied between +1.5 mm and +3.5 mm, with output error variance ${\sigma }^2=0.110$ leading to inadequate beam stabilization at the center. However, the closed-loop pointing error feedback effectively stabilized the beam at the center, and significantly reduced pointing error displacement, which ranged from −0.02 mm to 0.04 mm, with error output variance ${\sigma }^2=0.015$. The proposed Conv1D model, used to predict optimal gain $\mathbb{K}$, was validated through a numerical simulation, demonstrating that the center of the optical beam aligned closely with the center of the receiving aperture. This verifies the effectiveness and efficiency of our closed-loop feedback system approach to ensuring accurate pointing in FSO satellite-to-ground communications systems. The Gaussian KDE curve provides a smooth and continuous representation of pointing error displacement distribution, aiding in the identification of patterns and trends. In addition, KDE can be used to calculate the probability of a particular pointing error value occurring, and the proportion of the pointing error that falls within a particular interval of a distribution for the given atmospheric condition.

3.4. Communication Performance

To quantify the performance of the FSO satellite-to-ground communication system in fading channels, we considered outage probability in this study. Outage probability is influenced by factors such as atmospheric turbulence, pointing error, and system configuration, all of which play a critical role in determining the overall performance of FSO communications. The outage probability is given as follows [37]:

$$P_o\left(I\right)={\int }_0^{I_0/m}.{\displaystyle \frac{1}{\sqrt{2\pi {\sigma }^2}}}{\displaystyle \frac{1}{I}}exp\left\{-{\displaystyle \frac{{\left(ln\left(I/I_0\right)+{\sigma }^2/2\right)}^2}{2{\sigma }^2}}\right\}dI,$$

(7)

where $\mathit{m}$ is the power margin utilized to address the additional power requirements necessary for mitigating signal fading caused by turbulence. By applying the Chernoff upper bound to Equation (7), one can derive an estimated power margin, expressed as follows [37]:

$$m\approx exp\left(\sqrt{-2ln2P_o{\sigma }^2}+{\sigma }^2∕2\right).$$

(8)

The assessment of the error in outage probability for the FSO satellite-to-ground link under open-loop and closed-loop scenarios is illustrated in Figure 10. To interpret the impacts of these outage probability values on satellite-to-ground communication, a lower outage probability signifies better communication reliability. In practical terms, a lower outage probability implies a reduced likelihood of signal interruption or degradation. In closed-loop conditions, where the output error variance is ${\sigma }^2=0.015,$ the outage probability is reduced by 4.83 dBm, contrasting with the open-loop outage probability error, where the output error variance is ${\sigma }^2=0.110$. For instance, to achieve an outage probability of 10⁻⁶, about 37.56 dBm is required in open-loop conditions. This requirement diminishes to 32.73 dBm in closed-loop conditions as the output error variance decreases to ${\sigma }^2=0.015$. To maintain a given error performance level, less transmission power is required on the closed-loop system compared to the open-loop system, which can reduce the energy consumption designated for FSO satellite-to-ground communications.

4. Conclusions

In this paper, an ML-based pointing error reduction scheme for FSO satellite-to-ground communications links under weak atmospheric turbulence was investigated. ML models were utilized to predict the gain matrix of the closed-loop system, a crucial factor in minimizing pointing error in an FSO communications system. For configuration of the FSO satellite-to-ground environment, we considered components of the control and system matrix regarding both a transmitter and a receiver, the characteristics of noise in the optical channel, irradiance, and the scintillation index of the signal. To predict the optimal gain value, we employed various ML models, including tree-based algorithms and a Conv1D neural network. The simulation results show that the proposed Conv1D model surpassed other ML approaches in accurately predicting gain in closed-loop FSO systems, helping to keep the optical beam’s position centered on the receiving aperture and minimizing pointing error. When building a closed-loop system based on the Conv1D model, the error variance of the pointing error displacement was obtained as 0.012 and 0.015 in clear weather and light fog conditions, respectively. Furthermore, we investigated the significance of the effects various input features on the closed-loop FSO system, and compared performance between closed-loop and open-loop systems under weak turbulence in the atmosphere. Using ML-based regression models, this study provides baseline data for system designers who want a robust communication link between LEO satellites and ground stations by considering the level of pointing error displacement and various weather conditions.