Extended Dynamics Observer (EDO)

• EDO is a state observer that simultaneously estimates system states and disturbance inputs, integrating partial disturbance models with high-gain techniques.
• The observer architecture augments the plant state with an extended state vector, enabling explicit compensation of modeled disturbances while suppressing unmodeled residuals.
• Performance analyses show that increasing observer bandwidth and internal model dimensions reduces steady-state errors, enhancing robust output feedback control.

An Extended Dynamics Observer (EDO) is a state observer architecture designed to provide simultaneous estimation of both the state and disturbance inputs in dynamical systems. EDOs generalize the classical Extended State Observer (ESO) framework, augmenting it to accommodate partial models of the disturbance alongside high-gain techniques for robust estimation of unmodeled or weakly modeled components. The principal motivation is to unify internal-model-based cancellation strategies with high-gain methodologies, resulting in improved disturbance rejection, rapid convergence, and robustness in the presence of uncertain or partially known disturbance dynamics (Feng et al., 2020).

1. System Model and Disturbance Decomposition

The EDO framework addresses single-input single-output (SISO) finite-dimensional linear systems subject to additive input disturbances, formalized as

$$\dot{x}(t) = A x(t) + B \left[u(t) + d(t)\right], \quad y(t) = C x(t),$$

where $x\in\mathbb{R}^n$, $u,d\in\mathbb{R}$, and $y\in\mathbb{R}$. The disturbance $d(t)$ is modeled as an element of a function space

$$d \in \mathbb{S} = \left\{ s \in L^\infty[0,\infty): \dot{s} \text{ exists weakly, } \dot{s} \in L^\infty[0,\infty) \right\}.$$

In numerous applications, the disturbance can be decomposed as $d = d_1 + d_2$, where $d_1$ admits a partial (often parametric) dynamic model

$$d_1 \in \Omega(G) = \{ Qv : \dot{v} = G v,\, v(0) \in \mathbb{R}^m \}$$

with known matrices $G, Q$, and the residual $x\in\mathbb{R}^n$0 is only known to be bounded but structurally unknown. This decomposition enables explicit use of available a priori information for disturbance rejection while preserving robustness to unmodeled disturbances (Feng et al., 2020).

2. Extended Dynamics Observer Construction

The EDO is constructed by augmenting the plant state with an extended state vector $x\in\mathbb{R}^n$1 whose dynamics combine the known disturbance model and a high-gain injection for unknown residuals: $x\in\mathbb{R}^n$2 where $x\in\mathbb{R}^n$3 denotes the best-approximation operator, $x\in\mathbb{R}^n$4, and $x\in\mathbb{R}^n$5 captures the portion of $x\in\mathbb{R}^n$6 orthogonal to the modeled subspace.

The combined observer system takes the form: $x\in\mathbb{R}^n$7 with gain matrices designed such that $x\in\mathbb{R}^n$8 and $x\in\mathbb{R}^n$9 are Hurwitz for any choice of high observer bandwidth $u,d\in\mathbb{R}$0. The Sylvester equations governing $u,d\in\mathbb{R}$1 ensure correct internal-model matching and high-gain error feedback (Feng et al., 2020). This construction ensures that the modeled disturbance modes are precisely estimated and canceled, while any unmodeled disturbance is suppressed by the high-gain mechanism.

3. Performance Analysis and Error Bounds

Under the standard observability/zero conditions

$u,d\in\mathbb{R}$2

and solvability of the Sylvester equations, the error system is block-triangular and strictly stable for all $u,d\in\mathbb{R}$3. The estimation error satisfies, for suitable constants,

$u,d\in\mathbb{R}$4

so that in steady state,

$u,d\in\mathbb{R}$5

Hence, the residual estimation error is proportional to the size of the unmodeled component and inversely proportional to the observer gain $u,d\in\mathbb{R}$6. Increasing the dimension of the internal model ($u,d\in\mathbb{R}$7) reduces $u,d\in\mathbb{R}$8, and thus the steady-state error (Feng et al., 2020).

4. Special and Limiting Cases: Relation to ESO, High-Gain, and IMP

The EDO subsumes several canonical observer structures as special cases:

• Extended State Observer (ESO): If no disturbance dynamics are modeled ($u,d\in\mathbb{R}$9), the EDO reduces to an ESO, with the disturbance treated as a constant and the observer employing high-gain feedback to estimate and reject it.
• Internal Model Principle (IMP): If all disturbance dynamics are known and exactly matched by $y\in\mathbb{R}$0, the EDO functions as an internal-model-based observer, precisely canceling the disturbance modes with no steady-state error.
• Hybridization: For partially modeled disturbances, the EDO interpolates between the ESO and IMP: precisely modeling certain disturbance harmonics while rejecting the remainder via high gain. This hybrid structure generally outperforms pure ESO (which ignores prior knowledge) and pure IMP (which cannot handle unmodeled residuals) (Feng et al., 2020).

5. Integration with Output Feedback Control

Upon constructing estimates $y\in\mathbb{R}$1, the EDO naturally leads to disturbance-compensating output feedback control: $y\in\mathbb{R}$2 where $y\in\mathbb{R}$3 is a high-gain state feedback designed for the nominal (disturbance-free) plant. The resulting closed-loop system achieves exponential stabilization, and the ultimate boundedness of the state is governed by the unmodeled disturbance and the product of observer and controller gains: $y\in\mathbb{R}$4 Both observation and control bandwidths ($y\in\mathbb{R}$5) provide explicit trade-offs between convergence speed and residual error (Feng et al., 2020).

6. Numerical Validation and Practical Implementation

Empirical studies illustrate the performance of the EDO architecture under various prior knowledge scenarios. For a test system with $y\in\mathbb{R}$6, $y\in\mathbb{R}$7, $y\in\mathbb{R}$8, and $y\in\mathbb{R}$9, three designs were evaluated:

• Pure ESO ($d(t)$0): Residual error of $d(t)$1 persists due to unmodeled harmonics.
• Inexact Internal Model ($d(t)$2): Steady-state error reduced by 50%.
• Exact Internal Model ($d(t)$3): Estimation error driven to zero.

The EDO maintains accuracy and robustness against sensor noise, and the numerical results confirm theoretical predictions regarding error reduction and convergence rate (Feng et al., 2020).

7. Extensions and Related Frameworks

The EDO framework generalizes to nonlinear uncertain systems using high-gain extended observers, enabling estimation of both state and uncertainty even in the presence of time-varying nonlinearities. The design principles—observer gain tuning, robustness-accuracy tradeoff, and integration with variable-structure or sliding mode controllers—remain similar, with high-gain acting as the mechanism for rapid estimation (Wang et al., 2023).

Connections have been established between EDO/ESO methodologies and other disturbance observer architectures, including the General Model-Based ESO (GMB-ESO) and Unknown Input Observer (UIO), particularly in their ability to reconstruct disturbance signals and guarantee finite-step convergence under structural conditions on the plant (Chen et al., 2022). The EDO, ESO, and their discrete or multi-channel variants have found application in robust control of high-performance aerial vehicles, complex manipulators, and nonlinear systems, where rapid disturbance estimation is essential (Abdulmajeed, 2019).

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Key References:

• "Extended Dynamics Observer for Linear Systems with Disturbance" (Feng et al., 2020)
• "Output tracking based on extended observer for nonlinear uncertain systems" (Wang et al., 2023)
• "A General Model-Based Extended State Observer with Built-In Zero Dynamics" (Chen et al., 2022)
• "Autonomous Control of a Quadrotor-Manipulator; Application of Extended State Disturbance Observer" (Abdulmajeed, 2019)