Extended State Observer (ESO) Overview

• Extended State Observer (ESO) is a state estimation approach that augments the system state with additional variables to capture model uncertainties and external disturbances.
• It employs coordinate transformations and eigenvalue assignment to enable fast, robust feedback control and accurate disturbance estimation in both linear and nonlinear systems.
• ESO architectures facilitate safe control by balancing fast convergence with measurement noise amplification through careful observer gain tuning.

An extended state observer (ESO) is an observer design paradigm that augments the system state with additional "extended" state variables representing lumped model uncertainties and external disturbances. The ESO dynamically estimates both the plant state and these total disturbances—either for robust feedback control, disturbance rejection, or state reconstruction under uncertainty. In contemporary nonlinear and linear control theory, the ESO framework underpins the architecture of Active Disturbance Rejection Control (ADRC), robust model-based control with bounded uncertainty, and robust safe control under uncertainty with limited measurements.

1. Mathematical Structure and Formulation

The canonical ESO formulation is defined for a system of the form

$$\dot{x} = f(x) + g(x) u + d(x, t), \qquad y = h(x),$$

where $d(x, t)$ lumps all model uncertainties and exogenous disturbances affecting the system, and $y$ is the measured output.

Under coordinate transformation (normal form), a single-input, single-output (SISO) system of relative degree $r$ is mapped to a chain of integrators with a "lumped" disturbance: $$\begin{aligned} z_1 &= y, \ \dot{z}_1 &= z_2, \ &~\,\,\, \vdots \ \dot{z}_r &= b(z)\!+\!a(z)u + d(t), \ y &= z_1. \end{aligned}$$ The extended state $f(t) \equiv d(t)$ is introduced as a dynamic variable, typically modeled as a slowly-varying or bounded function.

The ESO augments the system by including $f$ as a new state. For order-$r$ SISO systems in companion (Brunovsky) form: $$\begin{aligned} \dot{x} &= A_0 x + B_0 (b(x)+a(x)u + d(t)), \ y &= C_0 x, \end{aligned}$$ where $A_0$, $B_0$, $C_0$ are canonical shift, column, and row selection matrices. The augmented system becomes: $$\begin{bmatrix}\dot{x} \ \dot{f} \end{bmatrix} = \begin{bmatrix}A_0 & B_0 \ 0 & 0\end{bmatrix} \begin{bmatrix} x \ f \end{bmatrix} + \begin{bmatrix} B_0 \ 0 \end{bmatrix} (b(\hat{x}) + a(\hat{x})u) + D \dot{f},$$ with output $y = C_0 x$ (Chen et al., 2023).

A standard continuous-time linear ESO for this structure is: $$\begin{bmatrix} \dot{\hat{x}} \ \dot{\hat{f}} \end{bmatrix} = A\begin{bmatrix}\hat{x} \ \hat{f}\end{bmatrix} + B (b(\hat{x}) + a(\hat{x}) u) + L\left(y - C [\hat{x}; \hat{f}] \right),$$ where $L$ is the observer gain vector that places the eigenvalues of $A - L C$ at a prescribed location, typically $-\omega_o$ with multiplicity $r+1$.

For multi-input, multi-output (MIMO) and discrete-time extensions, the ESO is formulated on the augmented state with per-channel disturbance estimates and eigenvalue allocation for multivariable systems, subject to existence conditions such as no invariant zeros between disturbance channels and plant outputs (Chen et al., 1 Oct 2025, Chen et al., 2022).

2. Existence, Design, and Error Characterization

Necessary and sufficient conditions for the existence of ESOs (model-based or ADRC-style) in linear time-invariant systems require:

• Plant observability: $[C_0; C_0 A_0; \ldots; C_0 A_0^{n-1}]$ full rank,
• Absence of invariant zeros blocking the disturbance-output path, i.e., for all $z$,

$\rank\left[\begin{matrix} A_0 - zI & E_0 \ C_0 & 0 \end{matrix}\right] = n+1,$

where $E_0$ couples disturbance to the plant state (Chen et al., 2022, Chen et al., 1 Oct 2025).

For nonlinear or time-varying plants, the chain-of-integrators structure must be attainable via coordinate transformation, with smooth $f, g, h$ and their Lie derivatives (Chen et al., 2023).

Under boundedness of the disturbance and its rate ($|f(t)| \leq b_f$, $|\dot{f}(t)| \leq l_f$) (Chen et al., 2023), explicit closed-form $L_\infty$ error bounds can be computed for both state and disturbance estimates: $|f(t) - \hat{f}(t)| \leq \gamma(\omega_o, T),$ where $\gamma$ is a function of the observer bandwidth and sampling period, constructed from the convolution of the disturbance derivative bound and an impulse-response kernel depending on the observer's eigenvalues (Chen et al., 2023, Chen et al., 1 Oct 2025, Chen et al., 2022).

Further, for multivariable/discrete systems, the disturbance estimation error is given as a convolution: $$e_{w_i}[k] = w_i[k] - \hat{w}_i[k] = \sum_{j=0}^{k} h_i[j] \Delta w_i[k-j],$$ with $h_i$ tied to the observer pole assignment and relative degree (Chen et al., 1 Oct 2025).

If all observer poles are placed at the origin in discrete time, the ESO achieves dead-beat estimation of the disturbance after a delay equal to the plant's relative degree plus one (coinciding with unknown-input observer results (Chen et al., 1 Oct 2025, Chen et al., 2022)).

3. Output-Only State and Disturbance Reconstruction

A defining property of the classical ESO is that it reconstructs both the state vector and total disturbance using output-only injection; i.e., direct access to the state is not required. The measurement injection term $L(y - C[\hat{x}; \hat{f}])$ ensures coupling between the measured output and all augmented states, achieving uniform exponential convergence to a bounded region determined by the disturbance's smoothness (Chen et al., 2023, Łakomy et al., 2020).

Further, the ESO architecture can be extended to account for:

• MIMO systems by block-diagonalizing the observer gains or constructing per-channel variants (Chen et al., 1 Oct 2025).
• Fractional-order plants by employing observer equations with Caputo or Grünwald-Letnikov derivatives and establishing BIBO stability in the fractional domain (Li et al., 2021).
• Time-delay augmentation for non-observable pairs by Taylor-approximation of disturbance derivatives (Nguyen, 5 May 2024).

4. Observer Gain Tuning and Practical Implementation

Observer gain or bandwidth selection is critical for the performance/sensitivity balance. High bandwidths ($\omega_o$ large) yield fast convergence and better disturbance tracking at the cost of increased measurement noise amplification, often scaling as a higher power of $\omega_o$, causing peaking and elevated noise floors (Łakomy et al., 2020, Łakomy et al., 2020). Closed-form expressions for the noise and disturbance attenuation trade-off have been derived: $$\|\text{error}\| \leq O(1/\omega_o) (\text{disturbance derivative bound}) + O(\omega_o^n) (\text{measurement noise}),$$ where $n$ is the observer's order (Łakomy et al., 2020, Łakomy et al., 2020).

Recent advances include:

• Neural network-based automated tuning for multi-objective performance (e.g., state error, control effort, estimation error) directly from closed-loop tests, using architectures that regress control indices from observer pole choices and plant signatures (Kicki et al., 2021).
• Switchable ensembles of ESOs (parallel or cascaded) with distinct orders/bandwidths, where a supervisor chooses the estimator yielding the minimal predicted tracking error, combining robustness tailored to both disturbance characteristics and sensor noise (Tang et al., 2023).
• Cascade-ESO architectures that sequentially estimate disturbance components, substantially mitigating noise amplification while retaining fast adaptation (Łakomy et al., 2020).

5. Theoretical Extensions and Integration with Robust/Adaptive Control

The ESO philosophy extends naturally to:

• Nonlinear output regulation and adaptive semiglobal stabilization by merging high-gain ESOs with internal models and adaptive parameter identifiers, ensuring robustness to nonlinearly parameterized exogenous signals under minimal phase/well-posedness assumptions (Wang et al., 2019).
• Safety-critical control, where explicit error bounds from the ESO quantify the uncertainty in control barrier function (CBF)-based robust safety filters, securing formal guarantees of constraint satisfaction even with output-only measurements (Chen et al., 2023).
• Geometric and finite-time ESOs on Lie groups (e.g., $SE(3)$) via Hölder-continuous differentiator structures for disturbance/torque estimation in underactuated vehicle dynamics, achieving (practical) finite-time convergence under bounded/noisy conditions (Wang et al., 2023).
• Integral-chain, event-triggered, and nonlinear/homogeneous observer formulations, with triggering rates and Lyapunov exponents optimized for stochastic, partially observed, or networked multi-agent systems (Wu et al., 2023, Liu et al., 2022, Łakomy et al., 2020).

6. Limitations, Modifications, and Application Scope

The ESO approach relies on specific system properties:

• Observability of the augmented pair and the absence of invariant zeros are necessary for reconstructable disturbance estimation in model-based ESOs (Chen et al., 2022, Chen et al., 1 Oct 2025).
• High-gain in static ESOs can cause peaking; saturation-like nonlinear injection, finite-time/homogeneous structures, or cascaded observers have been proposed to address these limitations (Ibraheem et al., 2018, Wang et al., 2023, Łakomy et al., 2020).
• In systems with partial model knowledge (e.g., subspaces of known disturbance dynamics), the framework can be generalized via internal-model augmentations (e.g., EDO, GMB-ESO) to maximize disturbance cancellation efficiency (Feng et al., 2020, Chen et al., 2022).
• For plants with zero dynamics or fractional-order dynamics, model-conscious modifications such as built-in zero-dynamics (GMB-ESO) or fractional-order ESOs have been rigorously justified (Chen et al., 2022, Li et al., 2021).

A representative table summarizes selected ESO variants and design features:

ESO Variant	Key Features	Main References
Linear Output-Injection ESO	Standard ADRC core; SISO/MIMO; output only	(Chen et al., 2023, Łakomy et al., 2020)
Model-Based / GMB-ESO	State-space; explicit MIMO design	(Chen et al., 1 Oct 2025, Chen et al., 2022)
Cascade-ESO	Sequential noise attenuation, SISO/MIMO	(Łakomy et al., 2020)
Fractional-Order ESO (FESO)	Fractional dynamics, frequency analysis	(Li et al., 2021)
Finite-Time/Homogeneous ESO	Nonlinear, event-triggered, fast response	(Ibraheem et al., 2018, Wu et al., 2023)
Geometric/SE(3) ESO	Lie group, fast finite-time stability	(Wang et al., 2023)

ESO-based architectures are widely adopted in high-precision motion, robust vehicle guidance, distributed multi-agent systems under uncertainty, and adaptive or safe output regulation scenarios. The design methodology enables explicit trade-offs between estimation error, convergence speed, and robustness to measurement noise and model uncertainty, forming a foundational technology for high-performance robust and adaptive control (Chen et al., 2023, Tang et al., 2023, Chen et al., 1 Oct 2025, Feng et al., 2020).