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A Simulative Study on Active Disturbance Rejection Control (ADRC) as a Control Tool for Practitioners

Gernot Herbst

arXiv:1908.04596v2eess.SY

TL;DR

ADRC addresses practical control design by reducing reliance on precise process models while retaining a simple tuning approach. The paper introduces and simulates linear ADRC for generic first- and second-order plants, then develops a low-latency discrete-time formulation. It reports robustness to substantial process-parameter variation and an implementation requiring only one addition and one multiplication after sensor input becomes available.

  • Problem

    Practical control design needs an alternative that combines simple applicability with robustness while avoiding the explicit process models required by model-based approaches.

  • Method

    The paper presents linear ADRC, evaluates it with simulations of generic first- and second-order plants, and develops an optimized discrete-time formulation using precomputation to reduce latency.

  • Results

    ADRC adapts to heavily varying process parameters without maintaining an explicit process model, while its optimized discrete-time form computes the output with only one addition and one multiplication after sensor input becomes available.

  • Takeaways & Limitations

    ADRC is presented as an appealing control-tool alternative for practitioners because it needs little process knowledge and offers easy parameterization in continuous- and discrete-time cases.

  • Takeaways & Limitations

    ADRC’s potential depends on balancing process dynamics, observer dynamics, sampling time, and measurement noise, with practical constraints including limited observer dynamics and actuator saturation.

Abstract

from arXiv · show

As an alternative to both classical PID-type and modern model-based approaches to solving control problems, active disturbance rejection control (ADRC) has gained significant traction in recent years. With its simple tuning method and robustness against process parameter variations, it puts itself forward as a valuable addition to the toolbox of control engineering practitioners. This article aims at providing a single-source introduction and reference to linear ADRC with this audience in mind. A simulative study is carried out using generic first- and second-order plants to enable a quick visual assessment of the abilities of ADRC. Finally, a modified form of the discrete-time case is introduced to speed up real-time implementations as necessary in applications with high dynamic requirements.

1 Introduction

ADRC is presented as a practitioner-oriented alternative that combines simple applicability with model-based control capabilities while requiring only a coarse process model. The article introduces linear ADRC, evaluates it through simulations, and proposes a low-delay discrete-time implementation.

  • ADRC combines the easy applicability of PID-type methods with the power of modern model-based approaches.
  • An observer jointly treats disturbances and modeling uncertainties, so ADRC requires only a coarse process model for controller design.
  • ADRC leaves modeling errors to be handled as disturbances, potentially sacrificing performance relative to controllers built around precise process or reference models.
  • The article introduces linear ADRC step by step, then uses simulations to examine varying process parameters, structural uncertainties, and tuning-parameter effects.
  • For discrete-time ADRC, the article presents an optimized formulation intended to achieve very low input-output delay.

2 Linear Active Disturbance Rejection Control

Linear ADRC uses an extended state observer to estimate generalized disturbances, allowing first- and second-order plants to be controlled with coarse canonical models. Its tuning specifies closed-loop and observer poles, while the paper relates the approach to state-space control with disturbance estimation and compensation.

  • 2 Linear Active Disturbance Rejection Control: ADRC combines accessible tuning with model-based control capabilities by estimating disturbances and modeling uncertainties through an observer.Only a coarse process model is required, while modeling errors and parameter variations are handled as generalized disturbances.
  • 2.1 First-Order ADRC: When estimation is accurate, the first-order loop behaves as a normalized integrator under proportional control, with closed-loop pole s_CL = −K_P.The controller design can therefore be selected from a desired first-order settling time, while observer dynamics are tuned faster than the closed loop.
  • 2.1 First-Order ADRC: The first-order design requires modeling, a proportional disturbance-rejection structure with an extended state observer, closed-loop tuning, and observer-pole placement.For first-order behavior, the observer poles are placed left of the closed-loop pole, commonly using bandwidth parameterization.
  • 2.2 Second-Order ADRC: For second-order plants, generalized-disturbance estimation reduces the remaining process model to a double integrator controlled by a modified PD law.The proportional and derivative gains can be selected for adjustable second-order dynamics, including critically damped behavior with a desired 2%-settling time.
  • 2.2 Second-Order ADRC: Observer poles in the second-order design follow the same faster-than-closed-loop rule, with s_ESO ≈ (3 . . . 10) · s_CL and s_CL ≈ −6/T_settle.Observer gains are then computed from the characteristic polynomial of the observer error dynamics.
  • 2.3 Relation to Linear State Space Control with Disturbance Estimation and Compensation: Linear ADRC has an observer-and-control structure equivalent to state-space control with disturbance estimation and compensation, but assumes an integrator model rather than an accurate plant model.The comparison identifies the same parameter values while retaining ADRC’s deliberate treatment of modeling errors through disturbance estimation.

3 Simulative Experiments

The simulations examine continuous-time ADRC under varying process conditions, observer limitations, actuator constraints, noise, and sampling-time considerations. They show that practical performance involves compromises beyond the ideal disturbance-rejection case.

  • ADRC can theoretically suppress nearly all effects of disturbances and parameter variations with noise-free measurements and unlimited ideal actuators.Practical designs must compromise when observer dynamics are limited or controller outputs saturate.
  • The experiments confront one fixed controller design with a heavily varying process to visualize ADRC abilities and limitations.The study uses Matlab/Simulink-based simulations and examines ESO pole placement and actuator limitations.
  • Further discrete-time simulations address the effects of measurement noise and sampling time.

3.1 First-Order ADRC with a First-Order Process

Continuous-time first-order ADRC simulations test fixed tuning against parameter variation, ESO pole placement, actuator saturation, dead time, and higher-order dynamics. ADRC generally preserves desired behavior under parameter changes, but actuator limits, dead time, and structural uncertainty introduce specific degradations.

  • Parameter variation: Fixed ADRC parameters were tested on first-order processes whose DC gain and time constant varied from 10% to 1000% of nominal values.The controller was designed with noise-free variables, ideal measurements, b0 = K/T = 1, Tsettle = 1, KP = 4, and observer poles at -40 unless otherwise noted.
  • Parameter variation: The closed-loop step response remained similar or nearly identical to the desired one-second settling behavior for most parameter settings.Larger overshoots appeared mainly when the time constant increased five- or ten-fold.
  • ESO pole placement: Faster ESO poles better preserve desired dynamics under modeling errors, whereas slower observers produce larger deviations and overshoots for stronger low-pass process variations.The fastest setting tested was sESO = 100 · sCL; the baseline was sESO = 10 · sCL, and a slower setting was sESO = 5 · sCL.
  • Actuator saturation: Feeding the limited actuator signal ulim(t) to the observer prevents controller-output windup, so no further anti-windup measures are necessary for the demonstrated ADRC structure.The controller output converges to a steady-state value when saturation takes effect, and the controller recovers well afterward.
  • Actuator saturation: Actuator saturation at |ulim| ≤ 5 prevents the reference from being reached for reduced process gains K ≤ 0.2.For slower dynamics T = 5 and T = 10, settling time increases considerably while overshoot remains almost absent.
  • Dead time: Unknown dead time up to 10% of the process time constant causes oscillations, especially in the controller output.Delaying the observer’s controller-input signal by an approximately known dead time reduces these oscillations, even when the delay estimate does not match exactly.
  • Structural uncertainties: Adding an unknown second pole with T2 ≤ 0.1 produces transient oscillations as the higher-order pole approaches the dominant pole.The results remain acceptable, but ADRC’s advantage over standard PI controllers is smaller than in the parameter-robustness case.
  • Comparison to PI control: ADRC keeps closed-loop dynamics similar under major parameter variations, while a comparably tuned PI controller’s dynamics vary heavily.For an input disturbance applied from t = 2 until t = 4, ADRC compensates the disturbance impact much faster than PI control.

3.2 Second-Order ADRC with a Second-Order Process

The second-order ADRC study evaluates robustness to process-parameter changes and structural uncertainties, including observer placement, saturation, dead time, and higher-order dynamics. Simulations also compare ADRC with PI and PID control for parameter sensitivity and disturbance rejection.

  • Setup: The nominal second-order process uses K=1, D=1, and T=1, with b0=1 and desired closed-loop settling time T_settle=5.The controller design assumes perfect process knowledge and noise-free control variables and measurements.
  • Sensitivity to Process Parameter Variations: Large changes in K and D barely affect closed-loop behavior, while very small K slows the response and very large D introduces overshoot.The study varies K from 0.1 to 5 and D from 0.1 to 10.
  • Observer Pole Locations: Larger T values increase overshoot and oscillations more than K or D changes, motivating experiments with different observer pole locations.Observer poles are tested at 5, 10, and 100 times the closed-loop pole locations.
  • Observer Pole Locations: Placing observer poles farther left can produce nearly ideal behavior, but requires faster actuators and larger controller outputs and increases sensitivity to noise and sampling restrictions.The paper describes this trade-off as a consequence of faster observer dynamics.
  • Actuator Saturation: Actuator saturation prevents the desired output when K is reduced, but the controller output does not wind up.Increased damping also lengthens settling time under saturation without producing additional post-saturation oscillations.
  • Dead Time: With unknown dead time T_dead≤0.3, the second-order ADRC output is hardly affected, and delaying the observer input reduces controller oscillations even when the assumed delay mismatches reality.The second-order case is reported as an improvement over the corresponding first-order case.
  • Comparison to PI and PID Control: Across parameter-variation simulations, ADRC surpasses PI and PID by a large margin in sensitivity, while ADRC also compensates an input disturbance faster with a smaller control-variable effect.The comparison uses Figure 16 for parameter variations and Figure 17 for a disturbance d=0.5 applied for ten seconds.

4 Discrete Time ADRC

The discrete-time ADRC section develops observer discretization and pole placement, then simulates sampling time, measurement noise, and observer-speed effects. The results expose practical limits and the trade-off between tracking speed and noise rejection.

  • Discrete-Time Formulation: A discrete-time ADRC can be formed by discretizing the extended state observer while retaining proportional state feedback, provided sampling is sufficiently fast.The section evaluates discretization and measurement-noise effects through simulations.
  • Discretisation of the State Observer: The prediction observer uses discretized process matrices and current measurement y(k) to correct the estimate for the subsequent time step.Its observer error dynamics are governed by A_d−L_c·C_d·A_d, whose eigenvalues must match the desired observer poles.
  • Observer Pole Placement: Observer poles are mapped from the s-plane to the z-plane by z_ESO=e^(s_ESO·T_sample), with first- and second-order designs using two or three observer states.The implementation procedure distinguishes first-order and second-order process models.
  • Effect of Sample Time: Larger sample times from T_sample=0.01 to 0.20 increasingly produce controller-output oscillations and cause closed-loop dynamics to depart from the desired first-order behavior.Sampling restrictions can prevent the desired process behavior from being achieved.
  • Effect of Measurement Noise: Higher measurement-noise variance produces controller-output oscillations, which can be mitigated by designing an observer with slower dynamics.The simulations add normally distributed output noise with variance up to σ²_noise=0.001.
  • Effect of Observer Pole Locations: Fast observers support desired first-order tracking but amplify noise sensitivity, whereas slower observers suppress noise effects while weakening dynamics under parameter changes.This is the reported trade-off controlled by the observer-pole factor k_ESO.

5 Optimized Discrete-Time Implementation

The optimized implementation reduces discrete-time ADRC latency by transforming observer states, precomputing time-invariant terms, and separating immediate output computation from later observer updates. The resulting schedule makes the new controller output available early in each sampling period.

  • Motivation: Discrete-time ADRC has a larger computational footprint than classical PID, motivating an implementation focused on minimizing input-output lag.The optimization targets computations needed to deliver the controller output within a sampling period.
  • Modified ADRC Structure: A coordinate transformation rescales estimated states so multiplications by b0, K_P, and K_D can be omitted from the modified ADRC structure.The transformed observer matrices are obtained through similarity transformations using T.
  • Modified ADRC Structure: The transformed observer updates its states using the current measurement and the previous input and state estimate, while the controller uses the resulting variables for feedback.The optimized formulation retains the observer-based control structure.
  • Minimizing Latency by Precomputation: Precomputing terms from time k−1 allows u(k) to be updated using the new measurement, and known or slowly changing references permit further precomputation.The controller output equation reduces the measurement-time calculation to one multiplication and one addition.
  • Minimizing Latency by Precomputation: The latency-optimized schedule outputs u(k) after its second step, then updates observer states and precomputes the controller output for k+1.The actual observer states need not be explicitly updated in the output-critical portion.

6 Conclusions

Simulations with generic first- and second-order plants demonstrate ADRC's potential as a practical control tool, including adaptation to strongly varying process parameters without an explicit process model. An optimized discrete-time formulation supports low-delay implementation, while performance depends on balancing observer dynamics, sampling time, and measurement noise.

  • ADRC was demonstrated as a powerful control tool through simulations with generic first- and second-order plants.
  • ADRC adapts to heavily varying process parameters without maintaining an explicit process model.
  • One addition and one multiplication suffice to compute the discrete-time controller output after sensor input becomes available.
  • ADRC's potential depends on the relation among process dynamics, observer dynamics, sampling time, and measurement noise.

Errata

The updated preprint corrects Equation (25) and its derivation, including the observer gain l2 result that was incorrect in the original publication.

  • The updated preprint corrects Equation (25) and its derivation.
  • The corrected result concerns the observer gain l2.
  • The incorrect l2 equation appeared previously in the original publication.
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