## 1 Introduction  
Dynamic systems can often experience unwanted actuation caused by an exogenous input. For aero-space systems, the input can be a gust of wind, changes in atmospheric pressure, and variation in temperature, among other factors [1, 2, 3]. These outside conditions can affect the dynamics and overall performance of the system. Two main methods that deal with exogenous inputs are disturbance accommodating control (DAC) and robustness techniques. Several variations in DAC exist; one notable technique involves identifying the disturbance's basis and utilizing adaptive control to determine the appropriate linear combination of the basis to remove the unknown input [4, 5]. Notably, the adaptive control method of DAC is derived from a fixed gain method, which requires solving a set of matching conditions--also known as solvability conditions--before implementing DAC [6]. The fixed gain method requires significant knowledge of the system. Conversely, the DAC can be implemented for the adaptive case if system stability conditions are satisfied. An alternative viewpoint is treating disturbances as unknown inputs to the system, and an observer-estimator can be formulated to estimate the disturbance [7, 8]. In many instances, the exogenous-unknown input behaves stochastically, and the basis set of the disturbance remains unidentified. In 2002, [PERSON], [PERSON] and [PERSON] explored the case of Robust Adaptive DAC, given that a set of known and unknown input basis functions exist [9]. The emergence of robustness techniques stemmed from the limitations of the Luenberger-like estimators, the inability to account for stochastic noise adequately [10; 11]. This limitation ultimately led to the development of traditional Kalman-like filters, which operate under the assumption that the stochastic variation in the unknown input follows a Gaussian distribution centered around zero [12; 13]. For Luenberger and Kalman-like estimators to work, there must be minimal uncertainty about the true system. Luenberger and Kalman-like estimator's sensitivity to minimal uncertainty about the true system led to the creation of plant-robust techniques [14; 15]. The proposed control technique expands our original finding by accounting for unknown input stochastic variation, where the input basis and plant model uncertainty are unknown [16]. The control technique can accommodate significant changes to the true system's plant, as defined in the derivation, which may arise from environmental factors, aging, or usage effects. Changes to the plant can involve modifications to internal interactions and constitutive constants. In 2022, [PERSON] and [PERSON] developed an open-loop coupled approach for robust adaptive plant and input estimation. Plant estimation depends on adaptive control, and input estimation uses a fixed gain approach [17]. The proposed control technique is a closed-loop approach, enabling the decoupling of plant and input estimation. The closed-loop method allows access to the true system's input and output to update the reference model adaptively. The proposed control scheme is designed for a generic system and can, in theory, be implemented in any system that falls within the assumptions and constraints of the proof. Nonetheless, practical and numerical limitations may arise, some of which will be mentioned later in the text. While the proof is presented for globally linear time-invariant (LTI) systems, the proposed observer-estimator can also be applied to local LTI systems, provided that the assumptions and constraints of both the local LTI system and the derived theorem are satisfied. Given that it is common engineering practice to linearize nonlinear systems around an operating point, the proposed scheme is well suited to aid state estimation in such cases [18; 19]. Moreover, the proposed method can either complement or replace gain estimation scheduling techniques, which are often employed to estimate the state of a system under varying operational conditions or non-linear dynamics [20].