Let $n,m,p \in \mathbb{N}$. Let $X \subset \mathbb{R}^n$, $U \subset \mathbb{R}^m$, and $Y \subset \mathbb{R}^p$ be open neighbourhoods of $0$. Let $f: X \times U \to \mathbb{R}^n$, $h: X \to Y$, $k: X \to U$, and $q: X \times Y \times U \to \mathbb{R}^n$ be $C^1$ maps satisfying $f(0,0)=0$, $h(0)=0$, and $k(0)=0$.
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Consider the plant, output, observer-based feedback, and observer equations
Assume that the observer is locally exact on plant trajectories: there exists a neighbourhood $X_{\mathrm{ex}} \subset X$ of $0$ such that, for every $x \in X_{\mathrm{ex}}$,
Let $e=\hat{x}-x$. Assume there exist neighbourhoods $X_1 \subset X$ and $E_1 \subset \mathbb{R}^n$ of $0$ such that $x+e \in X$ and $k(x+e)\in U$ for every $(x,e)\in X_1 \times E_1$. Define $F: X_1 \times E_1 \to \mathbb{R}^n \times \mathbb{R}^n$ by
has $x=0$ as a locally exponentially stable equilibrium.
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Assume further that the frozen error subsystem at $x=0$ is locally exponentially stable in the following Lyapunov sense: there exist a neighbourhood $E_0 \subset E_1$ of $0$, constants $c_1,c_2,c_4>0$, and a $C^1$ function $W: E_0 \to \mathbb{R}$ such that, for every $e\in E_0$,
Assume also that there is a neighbourhood $X_0 \subset X_1 \cap X_{\mathrm{ex}}$ of $0$ such that $X_0 \times E_0$ is contained in the domain on which $F$ is defined.
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Then $(x,e)=(0,0)$ is a locally asymptotically stable equilibrium of the closed-loop system