# Discrete-Time Asymptotic Tracking Control System for a Satellite with a Solar Panel
## 1 Introduction
### Related Works  
Usual synthesis methods for attitude control systems present limitations that can be circumvented through the use of modern advanced control design strategies. In recentyears, this has become an active area of research for both control theorists and aerospace engineers. For an engineering point of view on the attitude control system for an observation satellite, we refer the reader to [1; 2]. For a good review of applications of advanced control algorithms from an engineering perspective, see [3] and the references cited therein. The paper [4] addressed the challenges posed by flexible appendages by proposing a damper to mitigate vibrations, ensuring high-accuracy attitude control despite structural flexibilities. Many authors have employed PDE-based mathematical models of satellites with flexibility to model vibrations and developed sophisticated suppression algorithms for attitude control problems; see, for example, [5; 6; 7; 8; 9]. Appendages increase the surface area exposed to aerodynamic drag, solar radiation pressure, and gravity gradient torques. These external disturbances increase the burden on the attitude control system, and handling these disturbances requires efficient design and control strategies. As traditional PID controllers are insufficient for this purpose, there is a growing need for model-based control, adaptive control, and robust AI/ML-based control methods. In a study [10], the authors introduced robust observer-based methods to isolate and identify actuator faults in microsatellite attitude control systems, accounting for various space disturbance torques and gyro drifts. The proposed methodologies enhance the reliability and safety of on-orbit microsatellites. In turn, [11] considered a PDE-ODE model of a flexible satellite that is composed of two identical flexible solar panels and a central rigid body. For this model, the authors constructed two internal model-based controllers--a passive controller and an observer-based controller--such that the linear and angular velocities of the central rigid body converge to the given sinusoidal signals. The work [12] investigated the use of deep reinforced learning to control satellites with under-actuated systems, where trained neural network-based policies were shown to enable satellites to perform large-angle maneuvers even in the case of actuator failures. The present study falls into the category of advanced control theory, as we develop a discrete-time control algorithm that allows the attitude angle of a satellite with an attached solar panel to track a prescribed periodically changing reference signal with zero asymptotic error. We work within the framework of the general regulation theory for the state space setup (see, e.g., [13; 14]) and combine it with the Cayley-Tustin transformation (see, e.g., [15; 16]). As an example, the Cayley-Tustin transformation has been used in the context of the satellite attitude control problem in order to discretize continuous-time transfer functions, thus enabling digital implementation, as well as in the context of satellite signal processing (see, e.g., [17; 18]). However, the combination of the Cayley-Tustin transformation and general regulators for the purpose of satellite attitude control seems to be novel. The main reason for using the general regulation theory is that it is tailored to the design of controllers for error tracking control systems with time-varying reference signals and zero asymptotic tracking error. At the same time, the resulting controllers are robust with respect to parametric uncertainties in the associated model and can handle disturbances. This approach has been successfully applied to continuous-time control systems for satellite attitude asymptotic tracking; see [19; 20]. This study takes the above-mentioned results into consideration, which cover continuous-time controllers, in order to develop a discrete-time error feedback controller. The obtained discrete-time controller is then applied in a digital control system with zero-order hold.