## 4 Discussion  
The theorem and illustrative example presented in this paper pertain to global LTI systems. However, since many real-world problems are inherently non-linear, the proposed theorem can also be applied to any local linear approximation of a non-linear system, provided that the assumptions and constraints of both the theorem and the linear approximation are met. While there are theoretical hard constraints, such as that the decomposition of the true plant must follow \(A=A_{m}+BL_{s}C\), these constraints are essential for the proof. Future research could explore ways to relax or circumvent these constraints. Moreover, the de  
Figure 8: AdaptiveGain \(L(t)\) approaching a steady state with the use of a fixed gain and \(aL\
eq 0\). Figure 9: Sub-figures (**a,b**) illustrate the internal state error (\(\varepsilon_{x}\)) and external state error (\(\hat{\varepsilon}_{y}\)), respectively, in the absence of reference model updating to account for changes in the true system health. Both error states exhibit significantly larger estimation errors compared to their counterparts obtained using the adaptive scheme. composition of the true plant \((A)\) may appear unrealistic for many applications; however, if additional actuators and sensors can be incorporated, thereby rendering the true system over-controllable and over-observable, the feasibility of satisfying the decomposition improves. If adding extra sensors is not an option, employing sensor blending or fusion techniques can aid in meeting the decomposition constraints. Sensor blending or fusion techniques are routinely used in signal processing and state estimation applications [22; 23]. Although the necessary decomposition of the true system may initially seem restrictive, given an initial model, the theory enables the error states to converge to a neighborhood around zero, regardless of how large or small the values inside \(L_{*}\) are. In practice, numerical limitations may exist in the theorem due to the integration of large errors. Knowledge of the input and output matrices \(\{B,C\}\) is essential for the theoretical development and stability proof. This assumption is common in control design as the input matrix \((B)\) denotes how the input \((u)\) influences the system dynamics, and the output matrix \((C)\) specifies the linear combination of the internal states that are measured-observed. Practical issues may arise if these matrices degrade over time or with use, but those types of degradation are outside the scope of this work. The current design of the proposed estimator is non-invasive, meaning that none of the estimated states are fed back into the true system. As a result, the estimation scheme can be implemented without any risk of harming the true system. One potential application of the estimator is to operate in conjunction with governing inputs that regulate the true system response. This parallel implementation can assist in monitoring or refining the equations of motion that govern the true system dynamics. A significant change in a specific constitutive constant or internal interaction could alert an operator to inspect a particular area for potential failures or indicate that a component within the system needs to be replaced. The theorem only guarantees an asymptotic convergence of the error states, which may limit the potential real-time, online use. In principle, the theorem can be applied to both fast and slow dynamics systems. However, several factors may limit the practical implementation of the theorem. These include the accuracy of the initial system model, the operational range of the true system, and whether the estimated states are used in control design. For instance, consider a scenario involving a mechatronic system with a poorly identified initial model, where the goal is trajectory tracking using the estimated states. In such cases, the implementation of the proposed theorem may not be feasible for online use. However, the scenario can still be evaluated in an offline setting to assess its performance.