## 1 Introduction  
In control theory, the problem of regulation is commonly addressed due to recurring issues in robotics, electrical, and mechatronic systems, among others. Its origins date back to 1976 when linear time-invariant systems were studied [1, 2]. Later, in 1990, regulation theory was extended to a nonlinear framework [3]. Regulation theory establishes the conditions for the controller, ensuring that the output of the closed-loop system tracks reference signals and rejects disturbances. For this purpose, a known dynamical exogenous system (also called exosystem) must generate the reference/disturbance signals [4]. Nevertheless, in some cases, such dynamics are unavailable, for example, if an aerial vehicle follows an unknown terrestrial vehicle. Hence, a need arises to develop control algorithms to track unknown but measurable signals. Some works that have studied the HGO can be found in [5, 6, 7]. Specifically, in [5], the author surveyed different results of implementing HGO in nonlinear feedback control, while in [6], the authors proposed a cascade HGO to address the peaking phenomenon. In [7], the authors have proved that the HGO can be employed in the hybrid linear output regulation theory for minimum phase linear systems with a relative degree greater than one. In addition to the above, a methodology similar to HGO that has gained some interest among researchers is the Active Disturbance Rejection Control approach, developed by Professor [PERSON] [8]. This approach consists of designing an extended state observer that allows for estimating system disturbances (both external and internal). The estimated disturbance is then added to the control strategy to cancel these disturbances in the closed-loop system. This technique has experienced significant growth in recent years due to its ease of working with linear and nonlinear systems, as well as the numerous applications found in various systems [9; 10; 11]. Other control approaches have been developed to address the problem involving non-smooth and non-periodic exogenous signals. For instance, in [12], an adaptive observer is designed under a relaxed Uniform Persistence of Excitation (UPE) condition to estimate both the state \(x\) and the initial value \(\omega_{0}\) from the output \(e\). This approach interprets the exogenous excitation (EE) as equivalent to a non-resonant condition. In [13], a nonlinear Luenberger observer combined with adaptive online gain tuning is used to select the best parameter based on an optimization criterion. A specific least-squares identifier is also proposed for general operating conditions. On the other hand, fault diagnosis allows the avoidance of physical disrepair or malfunction in the system [14]; such faults may occur in sensors, actuators, or the process [15; 16]. In [17], a UI observer is designed to estimate the state of the system and detect faults in actuators and sensors using the linear approximation at the origin; the design conditions are in terms of LMIs [18], due to its broad applicability and flexibility in achieving such performances. Once the faults are estimated, these signals can be used by the controller to form a fault-tolerant control scheme [19]. Similarly, in [20], the output regulation problem is addressed for cooperative heterogeneous linear multi-agent systems under communication and actuator faults. An adaptive dual event-triggered compensator is introduced to estimate the exosystem's state under such faults, while also reducing chattering and avoiding Zeno behavior. In output regulation theory, the avoidance of such perturbations is considered. However, such perturbations must be modeled, or in the case of regulation of unmodeled signals [21; 22; 23], at least be measurable. The problem arises when incipient faults appear in actuators and, unfortunately, are not measurably available for estimation within the scope of unmodeled signal regulation.