Generalized Extended State Observer

• GESO is an advanced observer design paradigm that estimates both the intrinsic system state and aggregated uncertainties such as disturbances and nonlinear dynamics.
• It extends classical observer frameworks by augmenting the state with generalized quantities, enabling accurate reconstruction in systems with parameter variations and unmodeled dynamics.
• Integrated into robust control architectures, GESO facilitates rapid convergence, stability under uncertainty, and effective performance in adaptive, distributed, and data-driven control applications.

A Generalized Extended State Observer (GESO) is an observer design paradigm that aims to provide real-time estimates of both the intrinsic state and aggregated uncertainties (such as external disturbances, unmodeled dynamics, or unknown parameters) of a dynamical system. By broadening the standard extended state observer (ESO) framework to accommodate more complex classes of systems and uncertainties, GESOs play a central role in modern robust control, estimation, and learning-based control architectures.

1. Conceptual Foundations

GESO extends the classical ESO principle—originally formulated for systems such as those targeted by Active Disturbance Rejection Control—by systematically augmenting the system state with "generalized" quantities representing disturbances, uncertainties, or nonlinear dynamics. The objective is to simultaneously reconstruct the system's state and the generalized "extended" state using only available measurements and known inputs.

The state augmentation typically follows:

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

with an extended observer of the form:

$$\begin{aligned} \dot{\hat{x}} &= F(\hat{x}, y, u) + \hat{d} \ \dot{\hat{d}} &= G(\hat{x}, y, u, \hat{d}) \end{aligned}$$

where $\hat{d}$ seeks to capture all mismatches between the actual plant and the nominal system model, including external disturbances, parameter variations, and modeling inaccuracies.

GESO generalizes the choice of the extended state, the observer dynamics, and the disturbance model, moving beyond low-order integrators and accommodating nonlinearities, nonparametric terms, and higher-order or group-valued systems.

2. Observer Architectures and Synthesis Strategies

GESO design methodologies span a broad spectrum, including but not limited to:

• Parameter-Dependent Multi-Observer Schemes: Multiple state observers—each tuned to a nominal parameter vector—run in parallel, with a supervisory law that selects or blends the observers based on real-time monitoring of an output prediction signal. This selection or blended estimate can track both state and parameters to arbitrary precision as the observer bank is refined, given persistent excitation and robust error dynamics (Chong et al., 2014).
• Finite-Time and Nonlinear GESO: Nonlinear observer gains using saturation-like or homogeneous correction functions (e.g., "fal" or fast finite-time stable differentiators) mitigate peaking and achieve finite-time convergence, critical for applications under sharp uncertainties or discontinuous disturbances (Ibraheem et al., 2018, Wang et al., 2023).
• Adaptive and Identification-Integrated GESO: Immersion of the state estimation task into an online parameter estimation problem via Generalized Parameter Estimation-Based Observers (GPEBO), transforming nonlinear dynamics into a regression form and leveraging identification routines (LS, DREM, wavelet- or neural-based) for both state and parameter estimation. Algebraic procedures for immersing general nonlinear systems into state-affine forms augment the classical PDE-based transformation, greatly increasing practitioner scope (Ortega et al., 2020, Kozachek et al., 2023, Ortega et al., 17 Nov 2024).
• Model-Based and Group-Valued GESO: Immersing dynamics on Lie groups (e.g., SE(3), two-frame navigation systems) into high-dimensional linear time-varying models allows for the design of Kalman-like observers coupled with geometric or optimization-based reconstructions, yielding global or semi-global stability even in non-Euclidean settings (Liu et al., 24 Jul 2025, Wang et al., 2023).
• Geso with Enhanced Disturbance Modeling: Explicit inclusion of additional disturbance dynamics using artificial delays and higher-order Taylor expansions improves accuracy when disturbances are not well-approximated by low-order integrators, especially in poorly observable systems (Nguyen, 5 May 2024).

A tabular summary of representative GESO synthesis approaches:

Class	Core Mechanism	Applicability
Multi-Observer	Observer bank with supervisory selection	Parametric uncertainty, PE
Nonlinear/FTS GESO	Nonlinear correction (e.g., saturating gain)	Fast/robust convergence
GPEBO/Immersion	State-affine immersion + online parameter ID	General nonlinear, nonparametric
Model-based GESO	State-space augmentation from full model	Linear/quasi-linear systems
Group-valued GESO	LTV immersion + geometric reconstruction	Lie group/state manifold systems

3. Convergence and Robustness Properties

GESOs are constructed to guarantee robust convergence of the state and disturbance (or uncertainty) estimates under a variety of system classes and uncertainty profiles:

• Input-to-State Stability (ISS): Robustness is analyzed by constructing Lyapunov functions certifying that estimation errors are bounded by deterministic functions of the disturbance magnitude and measurement noise (Łakomy et al., 2020, Bin et al., 2020).
• Finite-Time and Practical Stability: Use of homogeneity or fast finite-time stable (FFTS) correction terms ensures rapid suppression of estimation errors, with provable finite-time convergence under constant or slow disturbance conditions; for time-varying disturbances, practical finite-time stability (PFTS) bounds the estimation error within a neighborhood determined by the disturbance rate and noise characteristics (Ibraheem et al., 2018, Wang et al., 2023).
• Excitation Conditions: Modern GESO formulations, particularly those based on parameter estimation, shift away from persistent excitation (PE) requirements to weaker "interval excitation" or similar conditions, allowing for convergence even when regressors decay (as in certain chemical/biological reactors), broadening applicability (Ortega et al., 2020, Ortega et al., 17 Nov 2024).
• Structured Adaptation to Prior Information: Model-based GESO variants explicitly integrate prior system or disturbance dynamics knowledge, harnessing known zero dynamics or disturbance patterns to reduce observer burden and improve estimation accuracy (Chen et al., 2022).

4. Applications in Advanced Control Systems

GESOs are integral to robust, adaptive, and learning-based control architectures across a diversity of domains:

• Active Disturbance Rejection Control (ADRC): GESOs are embedded to estimate and compensate for the cumulative effect of unmodeled dynamics and disturbances, enabling robust tracking in complex, highly uncertain environments. Applications include quadrotor-manipulators, mobile robots, and industrial machines (Abdulmajeed, 2019, Wang et al., 2023, Wang et al., 2019).
• Distributed Optimization and Nash Equilibrium Seeking: In multi-agent networked systems, extended state observers (including both smooth PI-based and RISE-based GESO) enable distributed agents to robustly seek Nash equilibria and consensus in the presence of time-varying disturbances and unmodeled dynamics (Ye, 2019).
• Data-Driven and Hybrid Model/Observer Integration: Linear GESO modules are combined with learning-based architectures (e.g., Koopman operator formulations) to provide robustness against real-world disturbances while retaining computationally efficient, data-driven predictive models. In robot manipulator experiments, the use of GESO-augmented Koopman-MPC achieves an order of magnitude improvement in tracking RMSE under external loads relative to alternatives (Singh et al., 21 Sep 2025).
• Adaptive Output Regulation and Nonlinear Internal Models: GESO frameworks augmented with internal models accommodate uncertain, nonlinearly parameterized exosystems and address general nonequilibrium output regulation, relevant for process control and uncertain oscillatory systems (Wang et al., 2019).

5. Implementation, Tuning, and Practical Considerations

GESO implementation strategies address various trade-offs in performance, complexity, and computational cost:

• Observer Gain Tuning: Observer bandwidth or gain selection involves balancing convergence speed against noise amplification. Cascade and multi-stage GESO structures mitigate noise sensitivity by decoupling high-gain correction from direct measurement injection (Łakomy et al., 2020).
• Data-Driven Tuning: Neural network-based performance assessment, using basic closed-loop experimental data, allows multi-objective tuning of observer parameters (error, control effort, smoothness, disturbance tracking), extending easily to multi-degree-of-freedom GESO structures (Kicki et al., 2021).
• Computational Complexity: Dynamic (“zoom-in”) sampling policies in multi-observer GESO architectures reduce computational burden while preserving estimation accuracy, by adaptively focusing observer resources on the most promising parameter regions (Chong et al., 2014).
• Generalization to High-Dimensional and Non-Euclidean Systems: Algebraic immersion, Lie group symmetry exploitation, and geometric optimization underpin GESO applicability to high-dimensional and group-valued systems, including navigation and robot pose estimation (Ortega et al., 17 Nov 2024, Liu et al., 24 Jul 2025).

6. Theoretical Advances and Distinctive Features

Key theoretical distinctions set GESO apart from classical ESO and traditional observer methods:

• Redundancy-Free Augmentation: GESO leverages minimal required augmentation, sometimes by extracting nonlinearly parameterized system components through algebraic matching rather than brute-force augmentation.
• Explicit Connection to Parameter Estimation: The GESO framework merges observer design with parameter estimation theory, inheriting convergence proofs, rate control, and identifiability criteria from the latter.
• Relaxed Excitation/Structural Requirements: GESO removes, whenever possible, restrictive canonical form or strict observability requirements, encapsulating system uncertainty structure in regression or group-immersion frameworks.
• Integration with Modern Learning-Based and Physical-Modeling Paradigms: GESO is central to next-generation robust learning-control architectures, where explicit observer modules are required to mitigate the sensitivity of data-driven models to exogenous perturbations (Singh et al., 21 Sep 2025).

7. Comparative Analysis, Limitations, and Future Directions

A GESO may offer clear structural and robustness advantages over classical single-observer, high-gain, or output-injection observers, but may introduce additional design and computational complexity, especially when observer banks, high-order augmentation, or system immersion is nontrivial. In some applications (e.g., rapidly time-varying disturbances, nonminimal phase or singularly perturbed systems), careful tailoring of the GESO structure (e.g., built-in zero dynamics (Chen et al., 2022), delay-based approximation (Nguyen, 5 May 2024), or group-based immersion (Liu et al., 24 Jul 2025)) is essential for practical success.

Future research directions include:

• Systematic methods for selecting optimal observer augmentation (to balance minimality and robustness).
• Integration of GESO modules with deep reinforcement learning or black-box model-predictive control.
• Extension of GESO principles to the estimation of distributed parameters and large-scale interconnected systems.
• Tuning algorithms that balance between adaptation, noise sensitivity, and real-time computational constraints across general nonlinear and group-valued systems.

GESO thus provides a rigorous, flexible, and practically validated foundation for robust estimation and control in uncertain, complex, and high-performance systems.