I have a linear time invariant dynamic system which can be described via following state space model
$$ \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{x} = \mathbf{A}\cdot\mathbf{x} + \mathbf{B}\cdot\mathbf{u} $$
I have designed a Luenberger observer for that system in following form
$$ \frac{\mathrm{d}}{\mathrm{d}t}\hat{\mathbf{x}} = \mathbf{A}_n\cdot\hat{\mathbf{x}} + \mathbf{B}_n\cdot\mathbf{u} + \mathbf{L}\cdot\left(\mathbf{y} - \hat{\mathbf{y}}\right) = \mathbf{A}_n\cdot\hat{\mathbf{x}} + \mathbf{B}_n\cdot\mathbf{u} + \mathbf{L}\cdot\left(\mathbf{C}\cdot\mathbf{x} - \mathbf{C}_n\cdot\hat{\mathbf{x}}\right) $$
My goal is to analyze error in the state estimate provided by the observer in a situation where there are unscertainties in the system parameters i.e. in a situation when the system model used in the observer differs from the actual system (a situation, which can be expected in practice). I have attempted to model those unscertainties via $$ \begin{eqnarray} \mathbf{A} &=& \mathbf{A}_n - \Delta\mathbf{A}\\ \mathbf{B} &=& \mathbf{B}_n - \Delta\mathbf{B}\\ \mathbf{C} &=& \mathbf{C}_n - \Delta\mathbf{C} \\ \end{eqnarray} $$ The dynamic of the error in estimate can be discribed via
$$ \frac{\mathrm{d}}{\mathrm{d}t}\tilde{\mathbf{x}} = \frac{\mathrm{d}}{\mathrm{d}t}\hat{\mathbf{x}} - \frac{\mathrm{d}}{\mathrm{d}t}\mathbf{x} = \left(\mathbf{A}_n - \mathbf{L}\cdot\mathbf{C}_n\right)\cdot\tilde{\mathbf{x}} + \left(\Delta\mathbf{A} - \mathbf{L}\cdot\Delta\mathbf{C}\right)\cdot\mathbf{x} + \Delta\mathbf{B}\cdot\mathbf{u} $$
My expectation is that in case the
$$ \mathbf{A}_n - \mathbf{L}\cdot\mathbf{C}_n$$
has stable eigenvalues the $\lim_{t \to \infty}\tilde{x} = 0$ in case there are no unscertainties i.e. $\Delta\mathbf{A} = \mathbf{0}$, $\Delta\mathbf{C} = \mathbf{0}$, $\Delta\mathbf{B} = \mathbf{0}$ and $\lim_{t \to \infty}\tilde{\mathbf{x}} = -\left(\mathbf{A}_n - \mathbf{L}\cdot\mathbf{C}_n\right)^{-1}\cdot\left[\left(\Delta\mathbf{A}- \mathbf{L}\cdot\Delta\mathbf{C}\right)\cdot\mathbf{x} + \Delta\mathbf{B}\cdot\mathbf{u}\right]$
in case there are unscertainties i.e. $\Delta\mathbf{A}$, $\Delta\mathbf{B}$, $\Delta\mathbf{C}$ are nonezero matrices. Is my expectation correct?