High-Gain Observer: Theory & Design

• High-gain observers are state estimation structures that use large observer gains to achieve fast exponential convergence in nonlinear and linear systems.
• Limited-gain observer designs partition the state into cascaded 2D blocks, reducing numerical sensitivity and noise amplification for enhanced robustness.
• The approach balances trade-offs between convergence speed and robustness through tailored gain selection and eigenvalue assignment, with practical benefits in disturbance rejection.

A high-gain observer is a state estimation structure for control systems—generally nonlinear or linear in canonical observability form—characterized by the use of large observer gains to achieve arbitrarily fast convergence of estimation error. The standard paradigm increases the observer gain as a function of a tunable high-gain parameter (often denoted ℓ or 1/ε), with the aim of making the error dynamics contract rapidly. While this approach enables rapid state reconstruction, it introduces trade-offs between speed, robustness, numerical sensitivity, and noise amplification. High-gain observer theory and design methods are fundamental in nonlinear control, disturbance rejection, and robust stabilization, and continue to be refined for better practical properties and broadened applicability.

1. Fundamental Structure and Canonical Forms

Let $(A_n, B_n, C_n)$ denote the prime form triple associated with an $n$-dimensional system in canonical observability coordinates. A prototypical nonlinear system considered in high-gain observer design is

$$\dot{x} = A_n x + B_n \varphi(x) + d(x,t), \qquad y = C_n x + \nu(t)$$

where $\varphi(\cdot)$ is locally Lipschitz and $d(\cdot, t)$, $\nu(t)$ represent process and noise disturbances.

The traditional high-gain observer for such a system is constructed with observer state $\hat{x} \in \mathbb{R}^n$ and gain matrix

$$D_n(\ell) = \operatorname{diag}(\ell, \ell^2, \ldots, \ell^n)$$

where $\ell \gg 1$ is the high-gain parameter. The correction term in the observer scales with powers of $\ell$ up to $\ell^n$: $$\dot{\hat{x}} = A_n \hat{x} + B_n \varphi(\hat{x}) + D_n(\ell) K (y - C_n \hat{x})$$ with $K$ a gain vector ensuring $(A_n - D_n(\ell) K C_n)$ is Hurwitz. This structure yields (theoretically) exponential convergence: $$\| \hat{x}(t) - x(t) \| \leq c_1 \ell^{n-1} e^{-c_2 \ell t} \| \hat{x}(0) - x(0) \| + \dots$$ However, the rapidly increasing entries of $D_n(\ell)$ for large $n$ or large $\ell$ lead to significant peaking and severe numerical and robustness limitations (Astolfi et al., 2015).

2. Block-Structured Limited-Gain High-Gain Observers

To mitigate the peaking and sensitivity issues of classical designs, the block-structured observer introduced in (Astolfi et al., 2015) raises a critical alternative. Here, the observer state $\xi \in \mathbb{R}^{2n-2}$ is organized as a cascade of $(n-1)$ coupled 2D subsystems, each with block-diagonal gain scaling

$D_2(\ell) = \operatorname{diag}(\ell, \ell^2)$

Thus, the maximal gain power applied in any coordinate is $2$, independent of $n$.

The observer equations for each block are: $$\begin{aligned} & \text{For } i = 1, \ldots, n-2:\quad \dot{\xi}_i = A \xi_i + N \xi_{i+1} + D_2(\ell) K_i e_i \ & \text{For } i = n-1:\quad \dot{\xi}_{n-1} = A \xi_{n-1} + B \varphi_s(\cdot) + D_2(\ell) K_{n-1} e_{n-1} \end{aligned}$$ where $A$, $B$, $C$ define a 2D prime-form system, $N$ is a fixed $2 \times 2$ matrix (e.g., $$N = \begin{bmatrix} 0 & 0\ 0 & 1\end{bmatrix}$$), $K_i$ are to be chosen via a block-tridiagonal eigenvalue assignment, and the error signals $e_i$ are recursively defined: $$\begin{aligned} & e_1 = y - C \xi_1 \ & e_i = B^\top \xi_{i-1} - C \xi_i, \quad i = 2, \ldots, n-1 \end{aligned}$$ This structure achieves similar asymptotic convergence guarantees to the standard high-gain observer, but the effective amplification by $\ell$ is capped at $\ell^2$ in each block, yielding improved numerical robustness and noise attenuation (Astolfi et al., 2015).

3. Error Dynamics and Performance Bounds

The scaled error system, after coordinate and gain scaling, leads to error dynamics of the form

$$\dot{\varepsilon} = \ell M \varepsilon + \ell^{-(n-1)} \left( B_{2n-2} \Delta\varphi_\ell(\varepsilon,x) + \upsilon_\ell(t) + n_\ell(t) \right)$$

where $M$ is a block-tridiagonal matrix with assignable eigenvalues (via $K_i$), and the disturbance/measurement effects appear in the last term at a lower gain-dependent scaling. The Lyapunov argument yields the explicit bound

$$\| \hat{x}(t) - x(t) \| \leq \max \left\{ c_1 \ell^{n-1} e^{-c_2 \ell t} \| \hat{x}(0) - x(0) \|,\;\; c_3 \| (1/\ell) \Gamma(\ell) d(\cdot) \|_\infty,\;\; c_4 \ell^{n-1} \|\nu(\cdot)\|_\infty \right\}$$

with $$\Gamma(\ell) = \operatorname{diag}(\ell^{n-1}, \ldots, \ell, 1)$$ and problem-dependent constants $c_i > 0$.

This form demonstrates that, although the observer state dimension is doubled, the noise amplification is polynomial rather than exponential in the noise frequency, and the deleterious peaking phenomenon is significantly reduced for most components (Astolfi et al., 2015).

4. Trade-offs: Dimensionality, Numerical Properties, and Design Complexity

A comparison of the standard and limited-gain observer designs is summarized below:

Feature	Standard HGO (dim = n)	Limited-Gain HGO (dim = 2n–2)
Max gain power	ℓⁿ	ℓ² in each 2D block
Numerical peaking	Severe for large n or ℓ	Substantially reduced
Sensitivity to noise	Amplified by ℓⁿ⁻¹	Polynomial decay with frequency
Observer dimension	n	2n–2
Eigenvalue assignment	Simple, companion form	Block-tridiagonal, requires procedure
Noise attenuation (linear)	Error $\sim$ ω_N⁻¹	Error $\sim$ ω_N^{-(r_i'-1)}, $r_i' > 1$

The design thus offers a trade-off: increased observer order for improved numerical performance and noise robustness, at the cost of more intricate gain selection and higher computational complexity per integration step (Astolfi et al., 2015).

5. Extensions and Practical Implications

Simulation with canonical nonlinear examples such as the Van der Pol oscillator demonstrates that the limited-gain observer achieves substantially lower noise amplification than a standard high-gain observer under equivalent tuning and measurement noise (Astolfi et al., 2015). In linear systems, higher-state estimates in the chain are particularly sensitive to noise in the standard design; this effect is mitigated in the limited-gain approach through polynomial attenuation.

For high-order systems, the limited-gain observer allows the designer to avoid excessively large gains (and attendant sensitivity), which is especially crucial when high-frequency noise is present or when implementation is subject to finite-precision arithmetic.

6. Relation to Other High-Gain Observer Developments

The block-structured observer belongs to a broader set of innovations in high-gain observer theory. Alternative frameworks, such as homogeneous observers (Bernard et al., 2017), adaptive-gain sliding-mode observers (Bahrami et al., 2020), and multi-observer frameworks for optimal trade-offs in robustness vs. speed (Petri et al., 2022), further expand the domain of high-gain observer applicability, each introducing new trade-offs among convergence speed, noise amplification, and computational complexity.

Within this landscape, limiting the maximal observer gain power without sacrificing convergence rate is a principled response to the core bottleneck of high-gain observer technology. The main results of (Astolfi et al., 2015) establish that, for a broad class of nonlinear systems in canonical observability form, this strategy yields convergence and noise attenuation properties competitive with classical designs, but with advantages regarding numerical implementation and robustness.

7. Summary of Key Formulas

• Standard high-gain observer gain:

$$D_n(\ell) = \operatorname{diag}(\ell, \ell^2, \ldots, \ell^n)$$

• Limited-gain observer block:

$D_2(\ell) = \operatorname{diag}(\ell, \ell^2)$

• Cascaded observer dynamics:

$$\begin{aligned} \dot{\xi}_i & = A \xi_i + N \xi_{i+1} + D_2(\ell) K_i e_i, \quad (i = 1, \ldots, n-2) \ \dot{\xi}_{n-1} & = A \xi_{n-1} + B \varphi_s(\cdot) + D_2(\ell) K_{n-1} e_{n-1} \end{aligned}$$

• Error bound:

$$\| \hat{x}(t) - x(t) \| \leq \max \left\{ c_1 \ell^{n-1} e^{-c_2 \ell t} \| \hat{x}(0) - x(0) \|,\;\;\dots,\;\; c_4 \ell^{n-1} \|\nu(\cdot)\|_\infty \right\}$$

This design paradigm provides a rigorous and practically appealing alternative for observer-based state estimation in nonlinear systems where classical high-gain approaches are limited by noise sensitivity and numerical artifacts (Astolfi et al., 2015).