Active Disturbance Rejection Control (ADRC)

• Active Disturbance Rejection Control (ADRC) is a robust, model-agnostic control strategy that uses an Extended State Observer to estimate and cancel total disturbances in real time, improving stability in uncertain and nonlinear systems.
• It employs bandwidth-based tuning and advanced observer designs, such as nested and fractional-order ESOs, to reduce noise amplification and enhance disturbance rejection.
• ADRC has been effectively applied in industrial, motion, and process control, demonstrably reducing tracking errors, oscillations, and improving overall system performance compared to traditional methods.

Active Disturbance Rejection Control (ADRC) is a robust control framework distinguished by its model-agnostic approach to handling uncertainties and disturbances. Rather than relying on precise modeling, ADRC actively estimates and cancels internal and external disturbances in real time using an Extended State Observer (ESO) and a disturbance-compensating feedback law. This methodology underpins a range of advanced control solutions for uncertain, nonlinear, and noise-prone plants in industrial, motion, and process control (Lakomy et al., 2021, Herbst, 2019, Herbst et al., 2020, Louyue et al., 14 May 2025).

1. Fundamental Structure and Methodology

The core ADRC paradigm consists of two principal modules: the Extended State Observer (ESO) and the state-feedback control law. The plant of interest, assumed to be SISO and of known relative order $n$, is recast in a form where all unknown dynamics and disturbances—including unmodeled internal structure, external perturbations, and input gain mismatch—are lumped as a generalized or "total disturbance" $f(t)$. This yields the system representation: $y^{(n)}(t) = b_0 u(t) + f(t)$ with $b_0$ a nominal input gain (Herbst, 2019, Lakomy et al., 2021).

The $n+1$-dimensional ESO reconstructs both the system states $x_1,\,...,\,x_n$ and the total disturbance $x_{n+1}$ using high-gain or bandwidth-parametrized observer techniques. The canonical ESO ODE is

$$\dot{\hat{x}} = A\hat{x} + b u + l(y - c^\top \hat{x})$$

where $A$ is an $n+1 \times n+1$ integrator-chain matrix, $b$ is the input vector, $l$ is the observer gain vector (typically via bandwidth parameterization), and $c^\top$ extracts the measured output (Lakomy et al., 2021, Herbst et al., 2020).

The ADRC control law performs active disturbance compensation: $$u(t) = (1/b_0)\left[k_1 r(t) - (k_1,...,k_n) \hat{x}_{1:n}(t) - \hat{x}_{n+1}(t)\right]$$ where $k_i$ are pole-placement or bandwidth-parametrized gains for the nominal $n$th-order integrator with unity gain.

This two-stage structure allows ADRC to achieve closed-loop stabilization and disturbance rejection independent of precise plant knowledge, provided the relative order and approximate gain $b_0$ are available (Lakomy et al., 2021, Herbst, 2019, Herbst, 2020).

2. Observer and Gain Tuning Principles

2.1 Bandwidth Parameterization

A hallmark of ADRC is bandwidth-based tuning. Both the controller and observer gains are set such that all observer poles are placed at $-\omega_o$, and all closed-loop controller poles at $-\omega_c$. Explicitly: $$k_i = \frac{n!}{(n-i+1)!(i-1)!}\,\omega_c^{n-i+1},\quad i=1..n$$

$$l_i = \frac{(n+1)!}{(n + 1 - i)! i!}\,\omega_o^i, \quad i=1..n+1$$

with $\omega_o$ typically chosen much larger than $\omega_c$ (common practice: $\omega_o \approx 5$ to $10 \,\omega_c$) (Lakomy et al., 2021, Herbst et al., 2020, Herbst, 2019).

2.2 Half-Gain and α-Controller Variants

Recent work connects bandwidth-parameterized ADRC with optimal control via an algebraic Riccati equation (ARE) approach. The resulting "half-gain" tuning rule, derived analytically for integrator-chain plants, prescribes gains precisely one-half those resulting from traditional bandwidth tuning: $$k_{\alpha} = 0.5\,k_{BW}(\omega_{CL}),\quad l_{\alpha} = 0.5\,l_{BW}(k_{ESO}\,\omega_{CL})$$ This configuration reduces high-frequency noise sensitivity with minimal loss in closed-loop performance, especially when applied to the observer block (Herbst et al., 2020).

3. Nonlinear and Distributed Extensions

ADRC generalizes to nonlinear and distributed systems via nonlinear ESOs (NESO), fractional-order observers, and distributed/decentralized controller arrangements.

• Nonlinear ESO (NESO): Incorporates nonlinear injection terms (e.g., "fal" functions) to enhance robustness against measurement noise and aggressively reject nonlinear disturbances (Saag et al., 26 Jun 2025, Leblebicioglu et al., 2021).
• Distributed/Coordinated ADRC: Multi-agent and multi-channel systems (e.g., coordinated multi-valve pressure control) extend ADRC by uniting local ESOs/controllers with global coordination terms, such as gradient-based penalty approaches for enforcing safety or optimization constraints while retaining robust disturbance rejection (Louyue et al., 14 May 2025).

4. Variants and Implementation Strategies

4.1 Discrete-Time and Embedded Forms

ADRC readily adapts to digital control environments. The discrete-time current observer (also known as ZOH with current injection) maintains exact pole placement, and minimal-footprint implementations (using transposed direct form II) drastically reduce RAM and computational demands without loss of rigor, making ADRC suitable for low-cost or high-speed embedded systems (Herbst, 2021, Herbst, 2019).

4.2 Incremental and Rate-Limited ADRC

Incremental ADRC formulations, where the control computation is carried out through recursive increments, enable seamless incorporation of control magnitude and rate limitation, vital for power electronics and actuator saturation. Feeding the limited signal back into the observer prevents windup without auxiliary logic. Proven "bumpless transfer" strategies allow for smooth controller activation, observer enabling, and on-the-fly parameter changes (Herbst, 2019).

5. Advanced Observers and Robustness Enhancements

5.1 Nested and Cascade Observers

To address the notorious noise amplification in high-gain observer designs, nested and cascade ESOs are employed. A low-bandwidth inner observer restricts noise inflow, while a higher-bandwidth outer observer cancels the residual disturbance, achieving a 69.87% reduction in integral time absolute error under noise versus single-level LESO approaches (Abdul-Adheem et al., 2018, Łakomy et al., 2020).

5.2 Fractional-Order and Internal Model ESOs

Fractional-order ESOs (e.g., IFO-ESO) further reduce the number of required observer states while achieving higher estimation bandwidth and improved robustness to parameter variations, yielding faster settling times, less overshoot, and superior disturbance rejection compared to both integer-order ADRC and classical FO-ADRC (Li et al., 2022).

Internal model–based ADRC methods exploit prior knowledge of disturbance exosystem dynamics to realize perfect disturbance estimation and cancellation in the presence of known periodic or deterministic disturbances. With appropriate observer and filter structure, the IADRC exactly reconstructs exosystem-driven disturbances even under parameter uncertainty, provided mild persistence of excitation holds (Pan et al., 2016).

6. Applications and Practical Performance

ADRC is validated across a diverse array of applications, including converter-fed motor drives (Madonski et al., 2020), unmanned surface vehicles subjected to stochastic wave and current disturbances (Saag et al., 26 Jun 2025), multi-valve pressure systems (Louyue et al., 14 May 2025), gimbal stabilization with neural network–enhanced ADRC (Leblebicioglu et al., 2021), and unmanned tracked vehicles under real-world slippage and measurement noise (Amokrane et al., 2023).

Quantitatively, ADRC achieves:

• RMS cross-track error reductions of 30–40% for USV trajectories under real and simulated disturbances, with improved battery usage profile under strong currents (Saag et al., 26 Jun 2025).
• 77.1% lower maximum pressure error and 81.5% reduction in valve oscillation in coordinated pressure control compared to PID control (Louyue et al., 14 May 2025).
• Up to 85.4% tracking error reductions in neural network–assisted ADRC for gimbal systems (Leblebicioglu et al., 2021).
• Significant suppression of noise amplification and chattering with cascade observers or advanced differentiators (SOND, SMESO), alongside faster disturbance rejection and improved output regulation (Łakomy et al., 2020, Ibraheem et al., 2018, Ibraheem et al., 2018).

In all cases, the practical superiority of ADRC is most pronounced under substantial modeling uncertainty, unmodeled disturbances, or measurement noise—domains traditionally problematic for conventional PID and fixed-model controllers.

7. Connections to Classical Control and Design Integration

A formal equivalence has been established between error-domain ADRC (eADRC) and classical PI/PID controllers: any first- or second-order eADRC can be expressed as a PI or PID in series with a strictly proper low-order filter, and vice versa (Stankovic et al., 2023). This insight enables "backward compatibility" with industrial control standards, facilitating adoption of ADRC's disturbance rejection properties in PI/PID-dominated environments without requiring a paradigm shift in implementation infrastructure.

Control design workflows now admit direct recipes for tuning, conversion, and integration of ADRC and PI/PID controllers, preserving robust disturbance rejection (due to the embedded ESO) while allowing for 1-DOF and 2-DOF constructions with strict reference pre-filtering and classical feedback structures.

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References:

• (Herbst et al., 2020) Half-Gain Tuning for Active Disturbance Rejection Control
• (Louyue et al., 14 May 2025) Coordinated Multi-Valve Disturbance-Rejection Pressure Control for High-Altitude Test Stands via Exterior Penalty Functions
• (Lakomy et al., 2021) Active Disturbance Rejection Control (ADRC) Toolbox for MATLAB/Simulink
• (Herbst, 2019) A Simulative Study on Active Disturbance Rejection Control (ADRC) as a Control Tool for Practitioners
• (Herbst, 2021) A Minimum-Footprint Implementation of Discrete-Time ADRC
• (Herbst, 2020) Transfer Function Analysis and Implementation of Active Disturbance Rejection Control
• (Abdul-Adheem et al., 2018) Novel Active Disturbance Rejection Control Based on Nested Linear Extended State Observers
• (Li et al., 2022) An Improved Active Disturbance Rejection Control for Bode's Ideal Transfer Function
• (Łakomy et al., 2020) Active Disturbance Rejection Control Design with Suppression of Sensor Noise Effects in Application to DC-DC Buck Power Converter
• (Amokrane et al., 2023) Active disturbance rejection control for unmanned tracked vehicles in leader-follower scenarios: discrete-time implementation and field test validation
• (Saag et al., 26 Jun 2025) Active Disturbance Rejection Control for Trajectory Tracking of a Seagoing USV: Design, Simulation, and Field Experiments
• (Stankovic et al., 2023) From PID to ADRC and back: expressing error-based active disturbance rejection control schemes as standard industrial 1DOF and 2DOF controllers
• (Pan et al., 2016) Internal Model Based Active Disturbance Rejection Control
• (Ibraheem et al., 2018) An Improved Active Disturbance Rejection Control for a Differential Drive Mobile Robot with Mismatched Disturbances and Uncertainties
• (Ibraheem et al., 2018) A Novel Second-Order Nonlinear Differentiator With Application to Active Disturbance Rejection Control
• (Leblebicioglu et al., 2021) NN Based Active Disturbance Rejection Controller for a Multi-Axis Gimbal System