Cascade EHGO: Extended High-Gain Observer

• Cascade EHGO is an observer architecture that augments system states and organizes estimation in cascaded stages for robust disturbance reconstruction.
• It is applied in nonlinear systems, UAV tracking, and active disturbance rejection control to enhance convergence and mitigate noise.
• Practical implementations balance high-gain trade-offs with distributed bandwidth and multi-stage designs to handle uncertainties and measurement noise.

Cascade Extended High-Gain Observer (EHGO) denotes a family of observer architectures in which unknown dynamics, disturbances, or higher-order signal components are promoted to additional states and estimated through high-gain output-error injection, while the estimation task is organized in a cascade of stages, blocks, or hierarchical subsystems rather than by a single monolithic observer. In the cited literature, this designation covers several closely related constructions: augmented-state high-gain observers for uncertain nonlinear systems, cascade extended state observers (ESOs) for active disturbance rejection control (ADRC) under measurement noise, cascaded first-order disturbance estimators for perching multirotors, multi-block EHGOs coupled with actuator dynamics for UAV tracking, and cascaded high-gain observers for continuous triangular systems whose nonlinearities do not satisfy the regularity conditions required by a classical high-gain observer (Wang et al., 2023, Łakomy et al., 2020, Li et al., 16 Sep 2025, Boss et al., 2021, Bernard et al., 2017).

1. Terminology and conceptual scope

The adjective extended refers to state augmentation. In representative formulations, the observer state includes not only the plant state but also an auxiliary variable representing a lumped disturbance, model mismatch, reference-signal derivative, or disturbance dynamics. Examples include the augmented uncertainty state $x_{n+1}=f(x_1,\ldots,x_n,t)$ in nonlinear output tracking, the disturbance state $v$ in the Extended Dynamics Observer (EDO), the total disturbance $z_3=F^*$ in ADRC, and the translational, rotational, or reference-related disturbance channels $\sigma_\rho$, $\varsigma_\xi$, and $\sigma_{xc}$ in multirotor tracking (Wang et al., 2023, Feng et al., 2020, Łakomy et al., 2020, Boss et al., 2021).

The adjective high-gain refers to output-error injection scaled by a small singular-perturbation parameter or by an observer bandwidth. In the nonlinear strict-feedback setting, observer poles are placed at $-\lambda_i/\varepsilon$, so shrinking $\varepsilon$ makes the observer arbitrarily fast. In ADRC and related linear extended-state observers, gain vectors are parameterized by powers of an observer bandwidth $\omega_o$. Across these designs, the same technical trade-off appears: smaller $\varepsilon$ or larger $v$0 improves convergence speed and disturbance reconstruction, but increases sensitivity to measurement noise and peaking (Wang et al., 2023, Łakomy et al., 2020, Łakomy et al., 2020).

The adjective cascade is used in several distinct but related senses. Some papers use it for a serial observer chain in which later stages estimate residual errors left by earlier stages. Some use it for a two-stage first-order observer decomposition of derivative and disturbance estimation. Others use it for a block-triangular or hierarchical estimation architecture in which one observer provides the input required by a second observer, or in which the observer error system becomes triangular after a coordinate transformation. This suggests that “cascade EHGO” is best understood as an architectural descriptor rather than a single canonical set of equations (Łakomy et al., 2020, Li et al., 16 Sep 2025, Feng et al., 2020, Mateus et al., 2022).

2. Extended-state formulation and disturbance augmentation

A canonical nonlinear formulation appears in output tracking for uncertain single-input single-output systems of relative degree $v$1. The plant is rewritten in chain-of-integrators form by defining

$v$2

which yields

$v$3

The unknown disturbance or uncertainty is therefore treated as an extra state $v$4, and its derivative is assumed to exist and be bounded together with the $v$5-st derivative of the reference signal. This is the extended-state formulation in its most explicit form (Wang et al., 2023).

Tracking-error coordinates are then defined by

$v$6

with the boundedness condition

$v$7

The corresponding extended high-gain observer estimates the full augmented error state: $v$8

$v$9

where the coefficients are chosen so that

$z_3=F^*$0

has roots at the desired high-gain locations. In compact form, the estimation error satisfies

$z_3=F^*$1

so the unknown disturbance enters only through the last augmented channel and acts as a bounded input to the observer error system (Wang et al., 2023).

A linear generalization is given by the EDO for disturbed SISO systems. There, disturbance estimation is not limited to a single lumped uncertainty state. Instead, the disturbance is decomposed into a modeled part with known dynamics and a residual part, so that known disturbance dynamics are canceled exactly and unknown disturbance dynamics are absorbed by high-gain. When no disturbance dynamics are known, the EDO reduces to an extension of the well-known extended state observer or high-gain observer. This formulation broadens the meaning of “extended” from simple state augmentation to the explicit use of prior disturbance models whenever available (Feng et al., 2020).

3. Principal cascade realizations

One major realization is the residual cascade ESO used in ADRC under measurement noise. The first stage is a standard ESO driven directly by the noisy output,

$z_3=F^*$2

while later stages estimate residual observation error: $z_3=F^*$3 The key design idea is that the first observer stage uses moderate bandwidth and is directly exposed to noisy output, whereas later stages refine the estimate using internal observer signals rather than the raw measurement. In both the general ADRC setting and the buck-converter implementation, this distributed-bandwidth structure is presented as a way to improve estimation performance in the presence of noise without relying on one extremely high-gain front-end observer (Łakomy et al., 2020, Łakomy et al., 2020).

A second realization is the cascade of first-order observers used for the perching drone inspired by the Venus flytrap. For the position loop, two first-order observer stages are cascaded: $z_3=F^*$4 with

$z_3=F^*$5

An analogous construction is used for the attitude loop. The first stage estimates the derivative-like state from the measured output; the second stage estimates the disturbance or uncertainty using the first-stage estimate and the input term. The paper explicitly states that this new EHGO can be viewed as a cascade of two first-order high-gain observers and contrasts it with a standard third-order EHGO (Li et al., 16 Sep 2025).

A third realization is the multi-block EHGO used in multirotor tracking. The observer state is partitioned into translational, rotational, and reference-system blocks,

$z_3=F^*$6

and the full observer is written in block-diagonal form with separate high-gain matrices for each subsystem. The actuator dynamics are included in the same fast estimation layer, producing a generalized cascade between the EHGO and actuator error subsystems (Boss et al., 2021, Boss et al., 2020).

A fourth realization appears in the theory of continuous non-locally Lipschitz triangular systems. When the Hölder-type conditions needed for a classical high-gain observer fail, a cascaded high-gain observer is built from blocks of increasing dimension. Block $z_3=F^*$7 estimates a subsystem of dimension $z_3=F^*$8, and later blocks use estimates from earlier blocks inside the nonlinear terms. The price is increased observer dimension, up to $z_3=F^*$9, and more elaborate gain selection (Bernard et al., 2017).

A related but distinct construction is the block-triangular EDO, where a Sylvester-type coordinate transformation makes the observer error effectively triangular. Another related construction is the observer cascade for camera-velocity recovery and multiple Manhattan-world line estimation, in which a first observer recovers the scale of the camera linear velocity and a second observer uses that estimate for line mapping. The latter is explicitly described as an observer cascade, but not as an EHGO in the classical naming sense (Feng et al., 2020, Mateus et al., 2022).

4. Convergence theory and stability mechanisms

For the augmented nonlinear output-tracking observer, two central convergence statements are established. First,

$\sigma_\rho$0

Second, for fixed $\sigma_\rho$1,

$\sigma_\rho$2

The same framework yields the closed-loop tracking bound

$\sigma_\rho$3

The proof uses the explicit variation-of-constants formula for the observer error, the Hurwitz property of the high-gain matrix, and a composite Lyapunov function for the coupled tracking and observer errors. The theoretical meaning is that the observer estimates the derivatives up to order $\sigma_\rho$4 and the unknown lumped disturbance, while the closed-loop tracking error can be made arbitrarily small by shrinking $\sigma_\rho$5 (Wang et al., 2023).

For the perching drone’s cascaded EHGO, the position-loop theorem states that for bounded initial state and any $\sigma_\rho$6, there exists $\sigma_\rho$7 such that for all $\sigma_\rho$8 and $\sigma_\rho$9,

$\varsigma_\xi$0

and

$\varsigma_\xi$1

The proof combines a Lyapunov equation $\varsigma_\xi$2, compact invariant sets, contradiction arguments for boundedness, and singular perturbation-like estimates for the observer errors (Li et al., 16 Sep 2025).

For cascade ESO under bounded disturbance and measurement noise, the analysis is recursive and ISS-based. Each stage is locally ISS with respect to disturbance derivative and measurement noise, and later stages remain bounded because they are driven by already bounded earlier-stage errors. The transformed errors satisfy Lyapunov inequalities of the form

$\varsigma_\xi$3

outside a residual-dependent ball. The paper states asymptotic convergence to zero when $\varsigma_\xi$4 and $\varsigma_\xi$5, and practical convergence to a small neighborhood when bounded noise and disturbance are present (Łakomy et al., 2020).

For non-Lipschitz triangular systems, the convergence taxonomy is more nuanced. The classical high-gain observer gives asymptotic convergence in the Lipschitz case, but under weaker Hölder conditions it guarantees only arbitrary small error. The cascaded high-gain observer is introduced precisely when those Hölder restrictions fail; under continuity and boundedness assumptions it still yields arbitrary-small-error convergence, and if the measurement disturbance and disturbance-approximation errors vanish, the block estimates converge exactly after some finite time (Bernard et al., 2017).

For multirotor trajectory estimation and tracking, the EHGO and actuator subsystem form a cascade analyzed by composite Lyapunov functions and singular perturbation arguments. The main result states that for sufficiently small $\varsigma_\xi$6, the estimation error converges exponentially to an $\varsigma_\xi$7 neighborhood of the origin, where $\varsigma_\xi$8 bounds the disturbance term. This practical, rather than exact, convergence is typical of perturbed high-gain observers (Boss et al., 2021).

5. Control integration and application domains

Cascade EHGO structures are almost always embedded in output-feedback controllers that cancel estimated disturbances or use them in dynamic compensation. In nonlinear output tracking, the observer estimates are inserted into a sliding-mode-like law based on

$\varsigma_\xi$9

and the controller combines estimated dynamics cancellation with a discontinuous robust term $\sigma_{xc}$0. Once the sliding surface is reached, the remaining tracking-error subsystem is a stable linear system driven by the observer error, so small observer error implies small ultimate tracking error (Wang et al., 2023).

In ADRC for the DC-DC buck converter, the cascade ESO replaces the conventional single high-gain ESO to suppress sensor-noise over-amplification. Experimental indices reported for the observer-only comparison show $\sigma_{xc}$1 decreasing from $\sigma_{xc}$2 for the standard ESO ($\sigma_{xc}$3) to $\sigma_{xc}$4 for CESO ($\sigma_{xc}$5), while $\sigma_{xc}$6 decreases from $\sigma_{xc}$7 to $\sigma_{xc}$8. The same study reports that the combination of CESO with a low-pass filter yields a noise oscillation amplitude of approximately $\sigma_{xc}$9, compared with $-\lambda_i/\varepsilon$0 for ESO+LPF and $-\lambda_i/\varepsilon$1 for CESO alone in the compared case (Łakomy et al., 2020).

In self-driving vehicle path following, two third-order EHGOs estimate the lateral and yaw subsystem states and the lumped perturbations $-\lambda_i/\varepsilon$2 and $-\lambda_i/\varepsilon$3. The controller was implemented experimentally on flat, inclined, and banked roads, with estimation errors converging quickly, typically in about $-\lambda_i/\varepsilon$4–$-\lambda_i/\varepsilon$5 s, and with comparable tracking performance on inclined and banked roads relative to the flat road under a range of longitudinal velocities (Al-Nadawi et al., 2020).

In multirotor UAV tracking, EHGO estimates enable output-feedback feedback linearization when only position and orientation of the UAV and position of the reference system are measured. The observer estimates translational and rotational states, disturbance terms, reference-system higher derivatives, and actuator effects, and this estimation layer supports trajectory tracking and landing on a moving ground vehicle with unknown trajectory and dynamics (Boss et al., 2021, Boss et al., 2020).

In aerial robotics under impact and wind disturbance, the cascaded EHGO is used as the disturbance-estimation backbone of a perching drone. During perching, compared with standard EHGO and PID, the cascaded EHGO achieves 25.1% and 67.6% reduction in average error and 79.3% and 83.6% decrease in standard error. Under wind disturbance, relative to standard EHGO and PID, it gives 17.7% and 21.4% lower average error and 22.31% and 33.3% lower standard error (Li et al., 16 Sep 2025).

More broadly, observer-cascade ideas closely related to cascade EHGO also appear outside disturbance-rejection control. In multiple line estimation for structured environments, a first observer retrieves the scale of the camera linear velocity and a second observer uses that estimated velocity in a Manhattan-world line observer; the structured model reduces the state dimension from at least $-\lambda_i/\varepsilon$6 to $-\lambda_i/\varepsilon$7, and the full cascade is proved asymptotically stable (Mateus et al., 2022).

6. Design trade-offs, limitations, and recurring misconceptions

The dominant design issue is the classical high-gain trade-off. Higher observer bandwidth improves convergence speed and disturbance estimation, but also amplifies measurement noise and produces peaking. In ADRC, this is the central reason for replacing a single ESO with a cascade ESO. The cited papers repeatedly emphasize that the cascade does not eliminate the trade-off completely; rather, it provides an extra degree of freedom by distributing bandwidth across stages. The first stage remains the most noise-sensitive design element, and if its bandwidth is made too small the observer becomes too slow, whereas if it is made too large the high-gain noise problem reappears (Łakomy et al., 2020, Łakomy et al., 2020).

A second limitation is architectural complexity. Cascade structures introduce more observer states, more tuning parameters, and sometimes substantially higher dimension. In continuous triangular systems, the observer dimension may grow to $-\lambda_i/\varepsilon$8. In practice, saturation is often added to prevent peaking from contaminating the plant or the control channel, as in the perching drone, multirotor tracking, and autonomous-vehicle path-following implementations (Bernard et al., 2017, Li et al., 16 Sep 2025, Boss et al., 2021, Al-Nadawi et al., 2020).

A third issue is model dependence. Some designs assume bounded derivatives of the disturbance channels, persistent excitation, or smoothness of reference trajectories. The EDO partly addresses this by using prior disturbance dynamics whenever available, and the perching-drone paper explicitly presents the cascaded EHGO as less sensitive to model mismatch because it only requires the control gain symbols to be consistent, without the need for precise estimation of the gain magnitude (Feng et al., 2020, Li et al., 16 Sep 2025).

A common misconception is that every observer cascade is automatically a cascade EHGO in the strict nonlinear-observer sense. The literature does not support that simplification. The Manhattan-world line-estimation paper is an observer cascade, but not an extended high-gain observer in the classical naming sense. The nitrification-process paper includes sensor dynamics as additional observer states and uses a standard high-gain observer with LMI-based gain design, yet explicitly does not present a classical cascade extended high-gain observer. This suggests that cascade EHGO should be distinguished from the broader category of hierarchical observer cascades and from high-gain observers that are merely “extended” by plant augmentation (Mateus et al., 2022, Schmidt et al., 2022).

A second misconception is that “cascade” always means a serial chain of identical ESOs. In the cited works, cascade can mean residual-estimation stages, a two-stage first-order decomposition, a block-triangular transformed error system, or a generalized fast observer–actuator interconnection. It can even coexist with parallelization, as in the multiple-model bank of EHGOs for hexrotor actuator-failure recovery, where one EHGO is run per candidate failure model and disturbance estimates are used for model selection rather than for serial stage-to-stage propagation (Łakomy et al., 2020, Li et al., 16 Sep 2025, Feng et al., 2020, Boss et al., 2021).