## 2 Main Result  
The work presented herein aims to determine the internal and external state of the true physical system given a form of health change. In this context, a health change constitutes any deviation that can be captured within the prescribed decomposition of the true plant, typically manifesting as variations in constitutive constants or internal interactions. Such changes may arise due to environmental influences, aging, operational wear, or shifts in the system's operating point. The initial model and the true system may differ substantially, as long as the true system conforms to the prescribed decomposition, the proposed proof guarantees that the estimation error will converge to a neighborhood around zero. However, there may exist real-world limitations of the theorem, some of which are mentioned in the discussion section of the paper. This study extends our previous findings by considering the case where the true system experiences stochasticities and non-linearities in the dynamics, summarized in Theorem 1. In contrast, earlier results only considered model uncertainties [16]. Here, given an initial model of the true plant, states are estimated using known inputs and the true system's external state. Due to the inherent stochasticities, state error can only be guaranteed to converge in a neighborhood around zero (\(R_{*}\)). In the absence of such stochasticities, our prior results demonstrate that the estimation error asymptotically converges to zero [16]. Although presenting the main theorem prior to introducing the true system and the proposed observer may seem unconventional, this structure is intended to provide a high-level overview before delving into the technical details. The control diagram for Theorem 1 is expressed in Figure 1.  
**Theorem 1**. : _Output Feedback on the Reference Model for Robust Adaptive Plant and State Estimation._  
_Consider the following state error system:_  
\[ \begin{cases}\dot{e}_{x}=(A_{m}-KC)e_{x}+B(\underline{L-L_{*}})y+v\\ \dot{e}_{y}=Ce_{x}\\ \dot{L}=\Delta\dot{L}=-e_{y}y^{\dagger}\gamma_{y}-aL\end{cases} , \tag{1}\]  
_where \(e_{x}\) is the estimated internal state error, \(\dot{e}_{y}\) is the estimated external state error, \(L_{*}\) is the fixed-correction matrix, \(\Delta\dot{L}\) is the variability-uncertainty term, \(K\) is a fixed gain, \(\gamma_{y}\) is an interaction tuning term, '\(a\)' is a positive scalar feedback term, and \(v\) are the unknown stochastic variations present in the system._  
_Given:_  
1. _The triples of_ \((A,B,C)\) _and_ \((A_{m},B,C)\) _are ASD and SPR, respectively._
2. _A model plant_ \((A_{m})\) _must exist._
3. _Input matrix_ \((B)\) _and output matrix_ \((C)\) _are known._
4. _Allow_ \(A\in\operatorname{sp}\{A_{m},BL_{*}C\}\
i A=A_{m}+BL_{*}C\)_._
5. _The set of eigenvalues_ \((\sigma)\) _of the true and reference plants are stable (i.e.,_ \(\operatorname{Re}(\sigma\{A\})<0\)__\(\mathcal{E}\)__\(\operatorname{Re}(\sigma\{A_{m}\})<0\)_)._
6. _Input_ \((u)\) _must be bounded and continuous._
7. _Allow the stochastic variation_ \((v)\) _to be bounded_ \(\
i\exists M_{\mathrm{v}}>0\
i\sup_{t\geq 0}\|v(t)\|\leq M_{\mathrm{v}}<\infty\)_._
8. _Allow 'a' to be a scalar value_ \(\
i a\geq\frac{\lambda_{\min}(Q_{x})}{2\lambda_{\max}(P_{x})}>0\)_, where_ \(\lambda_{\min}(Q_{x})\) _is the minimum eigenvalue of matrix_ \(Q_{x}\) _and_ \(\lambda_{\max}(P_{x})\) _is the maximum eigenvalue of matrix_ \(P_{x}\)_._
9. _Allow_ \(\operatorname{tr}(L_{*}^{\dagger}L_{*})^{\frac{1}{2}}\) _to be bounded_ \(\
i\exists M_{k}\
i M_{k}>0\
i\operatorname{tr}(L_{*}^{\dagger}L_{*})^{\frac{1} {2}}\leq M_{k}\)_._
10. _Allow_ \(\gamma_{y}\) _satisfy_ \(\operatorname{tr}(\gamma_{y}^{-1})\leq\left(\frac{M_{\mathrm{v}}}{M_{k}}\right) ^{2}\)_._If conditions are met, then \(\limsup_{t\to\infty}\|e_{x}\|\leq\frac{1+\sqrt{\lambda_{\max}(P_{x})}}{a\sqrt{ \lambda_{\min}(P_{x})}}M_{v}\equiv R_{*}\) asymptotically, where \(R_{*}\) is the radius confidence around zero.