Disturbance Rejection Control

• Disturbance Rejection Control methods are feedback techniques designed to mitigate the impact of external disturbances and model uncertainties on system performance.
• Key principles involve using observers like the Extended State Observer (ESO) or internal models to estimate and compensate for total disturbances in real-time.
• These methods are applied across mechanical, robotic, and process control systems to achieve robust stabilization and high-precision tracking, often showing significant performance gains.

Disturbance rejection control methods are an essential class of feedback control techniques aimed at attenuating or eliminating the impact of exogenous disturbances and model uncertainties on a dynamic system’s performance. These approaches are critical in achieving stability, maintaining high tracking precision, and providing robustness in the presence of unknown or partially known perturbations, both in linear and nonlinear, single- and multi-input multi-output systems.

1. Theoretical Foundations and Core Principles

Disturbance rejection control builds upon the fundamental premise that plant-model mismatches—arising from external signals, unmodeled dynamics, or parametric uncertainties—can be modeled as lumped disturbances affecting the system’s behavior. The genesis of modern strategies can be traced to two key frameworks:

• Disturbance Observer-Based Control (DOBC), which introduces an inner feedback loop to estimate and compensate aggregate disturbances using plant output and a nominal model (2101.02859).
• Active Disturbance Rejection Control (ADRC), where the system's total disturbance (including unknown dynamics and external inputs) is estimated in real-time via an Extended State Observer (ESO) and immediately counteracted in the feedback law (1603.03734).

In both paradigms, key goals are robust stabilization, perfect or exponential disturbance attenuation (when possible), and maintaining nominal closed-loop performance in the presence of bounded uncertainties.

2. Observer Design and Disturbance Estimation

A unifying characteristic of advanced disturbance rejection methods is their reliance on observers to estimate unknown disturbances and model states:

• Extended State Observer (ESO): In ADRC and its extensions, an ESO augments the system state with disturbance estimates, directly feeding them into the control law (1603.03734, 1801.06058). The basic structure is:

$$\dot{\hat{x}} = Ax + Bu + L(y - C\hat{x}) \quad \dot{\hat{d}} = \text{observer update}$$

where $L$ is the observer gain ensuring convergence.

• Internal Model-Based Observer: If the dynamics or structure of the disturbance are known (e.g., sinusoidal or generated by an exosystem), an internal model of the disturbance can be embedded in the observer, enabling perfect estimation and rejection (1603.03734). Key mathematical steps involve solving Sylvester equations for constructing the observer states corresponding to the disturbance model, often resulting in exponential convergence.
• Adaptive and Experience Replay Observers: For unknown or partially known disturbance dynamics, adaptive observers may be employed, including techniques that utilize "experience replay"—the storage and reuse of historical estimation errors—to assure rapid and even finite-time convergence without requiring persistent excitation (2007.14565).

The precision and speed of disturbance estimation directly affect the controller’s rejection capability; observer gains are typically tuned for a trade-off between estimation responsiveness and noise amplification.

3. Control Law Structures and Algorithmic Variants

The control law in disturbance rejection frameworks typically counteracts the estimated disturbance in real-time, often following the canonical template: $u = u_{\text{nominal}} - \hat{d}$ where $\hat{d}$ is the real-time disturbance estimate. Several advanced strategies incorporate further enhancements:

• Internal Model Based ADRC (IADRC): Merges ADRC's model-free robustness with explicit internal modeling when the disturbance’s structure is known, enabling perfect rejection under mild assumptions (e.g., disturbance generated by known exosystem) (1603.03734).
• Nested Observers: To overcome limitations imposed by observer bandwidth (especially in noisy or hardware-constrained environments), methods implementing inner and outer observer loops (N-LESO, N-ADRC) can successively refine the disturbance estimate without exposing the system to high-gain issues (1805.00505).
• Sliding Mode and Nonlinear Extensions: To further enhance robustness—particularly for systems with mismatched or nonlinear disturbances—sliding mode observers and nonlinear error injection structures are used, often smoothing the chattering via boundary layer or hyperbolic tangent approximations (1805.12170).
• Integral and Energy-Shaping Control: For port-Hamiltonian systems or when classical models apply, integral action embedded in the feedback law allows rejection of both matched and unmatched disturbances while preserving physical energy properties, even for underactuated or non-separable Hamiltonians (1710.06070).

These varied control structures ensure that the total disturbance, once estimated, can be rejected promptly, either by direct subtraction or through more sophisticated feedback architectures designed for specific plant and disturbance characteristics.

4. Robustness, Stability, and Design Requirements

The effectiveness and reliability of disturbance rejection control methods rest on several key theoretical and practical considerations:

• Robust Stability: For DOBC and similar inner-loop schemes, robust stability is guaranteed when the plant is minimum-phase and observer/controller gains are chosen to satisfy Hurwitz or Schur conditions over the entire uncertainty set. For Q-filter designs, this entails selecting proper bandwidth and relative degree (2101.02859).
• Nominal Performance Recovery: Both ADRC and DOB structures can achieve performance convergence to the nominal (disturbance-free) closed-loop system, not only at steady state but also for transients, provided that implementation parameters (observer gain, filter bandwidth, controller gains) are set according to derived analytical criteria (1603.03734).
• Practical and Adaptive Tuning: Observer gain (or filter bandwidth) must be balanced between disturbance tracking agility and noise sensitivity. In highly uncertain or signal-dependent environments, adaptive and experience-based adaptation laws can compensate for unknown or time-varying plant and disturbance properties (2007.14565).
• Observer Robustness to Input-Dependent Disturbances: Recent extensions allow consideration of disturbances dependent on control input, explicitly circumventing classical matching conditions by modeling all plant/model mismatch in a lumped disturbance term (1801.06058).

Stability is further validated via Lyapunov-based analyses or algebraic Riccati equations, ensuring that all error signals generally remain bounded and converge, even under complex nonlinear perturbations.

5. Application Domains and Empirical Outcomes

Disturbance rejection methods have been extensively validated across diverse settings:

• Mechanical and Robotic Systems: Handling friction compensation, vibration suppression, and precise trajectory tracking, especially in the presence of periodic or composite disturbances (1603.03734, 1710.06070, 1801.06058, 1805.12170).
• Hydraulic and High-Fidelity Plants: Utilizing nonlinear disturbance estimators combined with model-based control, these strategies have demonstrated substantial improvement in robust tracking and rejection, even in systems with strong actuator nonlinearities and external perturbations (1808.01445).
• Process and Test Stand Control: Multi-input, multi-output coordinated ADRC and observer-based architectures have achieved regulatory precision under strict safety constraints, facilitating practical deployment in intake pressure regulation and networked control (2505.09352).
• Energy-Shaping and Port-Hamiltonian Systems: Integral action controllers, constructed without coordinate transformation and built for general, underactuated settings, have enabled unmatched disturbance rejection for mechanical systems with non-constant mass matrices (1710.06070).

Comparative simulation and experimental results underscore the major performance gains over PID, classical robust control, or model-based approaches—often displaying reductions over 70% in regulation errors and control effort, particularly in high-noise or rapidly-varying disturbance environments.

6. Extensions, Limitations, and Research Directions

Recent research has advanced disturbance rejection beyond its classical boundaries:

• Adaptive and Model-Referenced Advancements: Incorporation of adaptive observers, learning-based policies (including experience replay), and internal model principles allow handling of unknown, time-varying, or partially structured disturbances with rigorous convergence guarantees (2007.14565).
• Event-Triggered and Resource-Constrained Environments: Event-driven implementation of ESOs and ADRC significantly reduces the computational and communication load in networked settings, introducing design principles to exclude Zeno behavior and maintain convergence under (possibly stochastic) sensor and actuator update schedules (2303.04296).
• Data-Driven and Learning-Based Approaches: Recent movement toward direct data-driven controller synthesis for disturbance rejection has enabled optimal parameterization without explicit system models, using one-step prediction error minimization and correlation-based learning (2307.01700).

Ongoing research explores fusion with reinforcement learning, output-only implementation without full state measurement, and practical extension to large-scale multi-agent and distributed systems.

7. Summary Table: Methodological Landscape

Approach	Observer Mechanism	Disturbance Type(s)	Advantage
Classic DOB (2101.02859)	Q-filter + nominal model	Lumped, input-matched	Recovers nominal performance
ADRC/ESO (1603.03734, 1801.06058)	ESO (linear or nonlinear)	Any (model-free)	Model-free, rapid rejection
IADRC (1603.03734)	Internal model + ESO	Modeled + unmodeled	Perfect/exponential rejection
N-ADRC (1805.00505)	Nested ESOs (inner + outer)	General, noisy	Error bound reduction, noise robust
Adaptive/Replay-based (2007.14565)	Adaptive ESO, experience replay	Unknown/time-varying	Finite-time convergence
Integral pH (1710.06070)	Integral state-energy shaping	Matched/unmatched	Physical energy structure preserved
Data-driven (2307.01700)	No model; error correlation	General	Direct controller tuning, flexible

Disturbance rejection control thus constitutes a central pillar of modern feedback design, with methodologies that blend robust estimation, adaptive learning, and physically grounded architectures to ensure system performance in the presence of significant disturbance and uncertainty.