Let $n,m \ge 1$ be integers. Let $D \subset \mathbb{R}^n$ and $U \subset \mathbb{R}^m$ be open sets, and for each $1 \le i \le n$ let $\phi_i:D\times U\to\mathbb{R}$ be continuous. Consider the plant with state map $x:[0,T]\to D$, input map $u:[0,T]\to U$, and output map $y:[0,T]\to\mathbb{R}$ given for $\mathcal{L}^1$-a.e. $t$ by

\begin{align*} \dot{x}_i(t)=x_{i+1}(t)+\phi_i(x(t),u(t)), \qquad 1\le i\le n-1. \end{align*}

\begin{align*} \dot{x}_n(t)=\phi_n(x(t),u(t)). \end{align*}

\begin{align*} y(t)=x_1(t). \end{align*}

Let $K\subset D$ and $U_0\subset U$ be compact sets, and let $N\subset D$ be an open neighbourhood of $K$. Suppose that there exists $L>0$ such that, for every $x,z\in N$ and every $u\in U_0$,

\begin{align*} |\phi_i(z,u)-\phi_i(x,u)|\le L\sum_{j=1}^{i}|z_j-x_j|, \qquad 1\le i\le n. \end{align*}

Let $a_1,\dots,a_n\in\mathbb{R}$ be such that the polynomial $p:\mathbb{C}\to\mathbb{C}$ defined by

\begin{align*} p(s)=s^n+a_1s^{n-1}+\cdots+a_{n-1}s+a_n \end{align*}

is Hurwitz. For $\varepsilon>0$, an absolutely continuous observer state map $\hat{x}:[0,T]\to D$ is said to satisfy the high-gain observer equations if, for $\mathcal{L}^1$-a.e. $t\in[0,T]$,

\begin{align*} \dot{\hat{x}}_i(t)=\hat{x}_{i+1}(t)+\phi_i(\hat{x}(t),u(t))+\frac{a_i}{\varepsilon^i}(y(t)-\hat{x}_1(t)), \qquad 1\le i\le n-1. \end{align*}

\begin{align*} \dot{\hat{x}}_n(t)=\phi_n(\hat{x}(t),u(t))+\frac{a_n}{\varepsilon^n}(y(t)-\hat{x}_1(t)). \end{align*}

Then there exist constants $\varepsilon_0>0$, $C>0$, and $\lambda>0$, depending only on $K$, $N$, $U_0$, $L$, and $a_1,\dots,a_n$, with the following property. For every $0<\varepsilon<\varepsilon_0$, every $T>0$, every measurable input $u:[0,T]\to U_0$, every absolutely continuous plant trajectory $x:[0,T]\to K$ satisfying the plant equations for $\mathcal{L}^1$-a.e. $t\in[0,T]$, and every absolutely continuous observer trajectory $\hat{x}:[0,T]\to N$ satisfying the displayed observer equations for $\mathcal{L}^1$-a.e. $t\in[0,T]$ with the same input $u$ and output $y=x_1$, the error map $e:[0,T]\to\mathbb{R}^n$ defined by $e(t)=\hat{x}(t)-x(t)$ satisfies

\begin{align*} |e(t)|\le C\varepsilon^{-(n-1)}e^{-\lambda t/\varepsilon}|e(0)| \end{align*}

for every $t\in[0,T]$.