Over the last four decades, many spacecraft have been designed for scientific, military, and business applications. Based on their characteristics, these spacecraft may be classified into three major categories: three-axis stabilized, spin-stabilized, and platform-based. Among all systems on the spacecraft, the thermal control system is considered essential, as it ensures that all mechanical and electrical components maintain proper temperatures throughout the mission. Several comprehensive textbooks have laid out the basic concepts of thermal control in spacecraft [1], while modern studies focus on new problems, including miniaturization and high-power components [2].

The satellite in orbit will be exposed to thermal loads from external sources, such as the Sun's radiation, Earth's albedo, and Earth's infrared radiation, as well as from internal sources, such as onboard equipment operation. These thermal loads cause a rise in temperatures, and low temperatures will be experienced during the Earth's eclipse. Precise models are needed for thermal control of satellites, and robust statistical techniques can be employed for this process [3]. Passive techniques, such as coatings, insulations, and radiators, are usually implemented, but active techniques are needed where tight tolerance and fast response are required. Review papers have comprehensively reviewed progress in the design and development of thermal control techniques for satellites, especially the transition from traditional passive control techniques to advanced active control approaches [4]. More recently, nonlinear control techniques have attracted increasing interest due to the nonlinear nature of satellite thermal dynamics, driven by radiation heat transfer and actuator saturation. Traditional linear control approaches, such as Proportional–Integral (PI) control, often fail to achieve optimal performance under these conditions due to excessive overshoot and slow response time. The nonlinear nature of satellite thermal dynamics has also been demonstrated through the analysis of lumped-parameter models [5]. On the other hand, a new small-perturbation analytical model has recently been developed to analyze the frequency response of spacecraft thermal control [6].

To overcome these limitations, Nonlinear Proportional–Integral (NPI) controllers have been proposed and theoretically analyzed. Zhao et al. [4] provided a rigorous design framework for robust nonlinear PID control that goes beyond linear assumptions. The effectiveness of nonlinear temperature regulation has been demonstrated in various applications; for example, Song et al. applied adaptive bandwidth tuning to a nonlinear thermal system and compared its performance against standard PID [7]. In the specific context of CubeSats, Al‑Hemeary et al. [8] developed a model-predictive control with anti‑windup PID for temperature tracking, demonstrating the need for advanced control methods. Further improvements in nonlinear PI controllers have been achieved through output transformations to enhance performance for nonlinear dynamical processes [9].

One of the most basic research studies on robust PI passivity-based control for nonlinear systems, applied to temperature control, was conducted by Aranovskiy et al. [10]. Recent research has successfully demonstrated the application of nonlinear PID and sliding-mode controllers in the design of a micro pump for small satellites [11]. Moreover, a straightforward intelligent nonlinear PI controller with ambient temperature control has achieved better results than classic PI control in nonlinear systems [12]. Finally, a general approach to nonlinear control has been developed from the standpoint of adaptive control of unknown strict-feedback nonlinear systems with nonsmooth actuator nonlinearity [13], [14]. Actuator saturation poses a serious challenge for satellite thermal control, as the heating capacity of heaters is physically limited. Various researchers have addressed the problem of designing controllers for thermal systems subject to actuator saturation constraints [15].

In addition, the application of nonlinear model identification adaptive control has been proven superior to conventional PID control in heater temperature control applications [16]. There have been recent developments in high-accuracy temperature control systems under low-frequency perturbations [17], and an approach to simulation-based temperature control in spacecraft thermal test experiments using Simulink has been illustrated [18]. Moreover, spatiotemporal event-triggered observer-based fuzzy resilient control has been applied to the temperature profile of fast-moving spacecraft, achieving excellent performance despite the spacecraft's dynamic nature [19].

In this study, we propose a novel design for an active thermal control system for an earth-orbiting satellite using a nonlinear PI controller. Each subsystem temperature is assumed to be a state variable, while the heaters serve as actuators, which are subject to saturation. By comparing the results of various simulations, we demonstrate that the nonlinear PI controller effectively reduces overshoot and settling time compared to classical PI control methods.

The goal of satellite thermal modeling is to create a simulated environment in which the satellite's thermal status can be examined and thermal control strategies tested. Through thermal-state analysis, we estimate parameters such as temperature levels, temperature gradients, temperature change rates, and sensitivity to thermal impacts. In other words, we need to develop a dynamic model that allows us to estimate those characteristics under specific operating conditions. Also, it has to give some insight into their changes over time.

In addition, the thermal control system should be able to use the satellite model and perform actions that modify the output values and, consequently, affect its performance. Therefore, if we treat our thermal control system as a closed-loop configuration (see Fig. 1), our thermal modeling should be viewed as a single block that predicts the temperature histories of each subsystem in the satellite. At the same time, we have to consider inputs, which are control actions generated by the controller based on comparisons between the real and model output parameters and their references.

A satellite is made up of several components, including an antenna, batteries, etc. Feedback obtained by the thermal controller while in operation involves temperature information about these components. Hence, when developing the simulation model, the time series should consist of the temperatures of the different satellite components. Thermal modeling of satellites may be carried out using any of the following three approaches:

I. Classical approach II. Lumped parameter approach III. Methods of numerical integration In this research work, the approach used in thermal modeling is a combination of the lumped parameter approach and numerical integration methods.

In the current investigation, two control methodologies are considered for application in the thermal management system of a satellite orbiting Earth: a classical PI controller and a nonlinear PI controller. The temperature behavior and actuator signals for three systems (fuel tank, helium fuel tank, and another system) are illustrated in Figs. 3-5.

The graphs above show that a large temperature overshoot occurs when using the classical PI control scheme (represented by the dashed line) to control both fuel tanks. It is evident that the temperature overshoots the set point and undergoes multiple oscillations before reaching steady state.

However, with the nonlinear PI controller, the temperature settles to the set point within a few seconds (solid line). Actuator signals (heater power) are shown in the lower subplots for each figure. In the classical PI control scheme, the actuator saturates due to integral wind-up, and the control action remains saturated for a long time, resulting in poor transient behavior.

In contrast, with the nonlinear PI controller, an anti-windup scheme is included, leading to quick release from the actuator saturation and dampening oscillations in the control action. Finally, the improved performance of the nonlinear PI control scheme is demonstrated using another subsystem (see Fig. 5).

There is almost no overshoot, and the temperature reaches its desired steady state without any issues. The settling time is reduced by 40–50% compared to the classic PI controller. Overall, the simulation results prove the efficiency of the presented method in avoiding problems caused by actuator saturation .

In this study, an active thermal control method for an Earth-orbiting satellite has been developed based on a nonlinear PI control approach. Temperatures in each satellite subsystem were chosen as state variables, whereas heaters were treated as saturating actuators. A comparative analysis of the proposed nonlinear PI control design and the classical PI control design was performed via simulation. According to the simulation results, integral wind-up leads to significant overshoot and long settling times in the classical PI control system due to actuator saturation. A nonlinear PI control system with a built-in anti-windup feedback scheme helps overcome problems associated with overshoot and increases the transient response speed.

As a result, a shorter time is required to reach steady-state mode for each of the studied satellite subsystems. Thus, it can be concluded that neglecting actuator dynamics is inappropriate when designing an active thermal control system. The nonlinear PI control technique offers a lower computational cost than other advanced control algorithms. Thus, it can be used for real-time control of thermal satellite systems. Future research may focus on developing methods for applying nonlinear PI control systems to MIMO thermal control tasks with disturbance rejection.

The authors declare no conflict of interest.

All data are included in the text.

This research received no specific grant from funding agencies in the public, commercial, or not-for-profit sectors.