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Practical Active Disturbance Rejection Control: Bumpless Transfer, Rate Limitation and Incremental Algorithm

Gernot Herbst

arXiv:1908.04610v1eess.SYcs.SY

TL;DR

Practical ADRC must accommodate control-signal limits and bumpless switching, beyond control performance. The article gives discrete-time implementation guidelines, including incremental ADRC, and demonstrates them in simulations. Those simulations report bumpless behavior under stated conditions and noise attenuation with rate limitation.

  • Problem

    Practical ADRC implementations need to handle magnitude or rate limits and bumpless controller switching, aspects often overlooked in theory-focused studies.

    Source: §I · p. 1

  • Method

    The article gives implementation guidelines for non-incremental and incremental discrete-time ADRC with signal limits and bumpless controller changes.

    Source: §I · p. 1

  • Results

    Simulations show no jumps in state estimates or control signal during parameter changes when r = y holds, and rate limitation attenuates measurement noise without affecting loop stability.

    Source: A. Rate Limitation · p. 6, D. Bumpless Parameter Changes · p. 7

  • Takeaways & Limitations

    Rate limitation can provide guaranteed control-signal rise times after reference or disturbance steps, and the guidelines cover practical ADRC variants with or without signal limitations.

    Source: §VII · p. 8

    Avoiding integral windup under saturation or rate limitation requires feeding the limited control signal into the observer.

    Source: §IV · p. 4

Abstract

from arXiv · show

Practical applications of controllers often impose further requirements on the implementation beyond the actual control performance, such as the ability to switch between manual and automatic control or between different control laws or controller parameter settings, known as bumpless transfer. Another common requirement is to limit the control signal in magnitude and/or rate. This article examines and extends several discrete-time variants of active disturbance rejection control (ADRC), which is increasingly being applied especially in the field of power electronics and drives, in this regard. Detailed guidelines for practical ADRC implementations with these abilities are presented, and all features are being demonstrated with the help of simulation examples.

I. INTRODUCTION

The article develops practical discrete-time ADRC guidance, including incremental formulations, signal limitations, and bumpless controller changes. It also reviews linear ADRC’s disturbance-estimation approach and controller design.

A. Brief Summary of Continuous-Time ADRC

  • ADRC is presented as reducing the need for detailed plant models and making controller tuning straightforward.Source: §I · p. 1

I. INTRODUCTION

  • The article develops non-incremental and incremental discrete-time ADRC guidance for magnitude or rate limitation and bumpless controller changes.It addresses practical implementation aspects often overlooked in theory-focused studies.Source: §I · p. 1

A. Brief Summary of Continuous-Time ADRC

  • ADRC models plant uncertainty and disturbances as a generalized disturbance estimated by an extended-state observer.The model uses an approximate process input gain, while the observer includes a constant input disturbance.Source: A. Brief Summary of Continuous-Time ADRC · pp. 1–2
  • The estimated states feed a disturbance-rejecting state-space controller whose feedback gains are tuned to the dominant plant dynamics.For first- and second-order cases, the controllers resemble P and PD controllers, respectively.Source: A. Brief Summary of Continuous-Time ADRC · p. 2
  • A common bandwidth-parameterized choice places all closed-loop poles at sCL = −ωCL and all extended-state-observer poles at approximately 3–10 times sCL, farther left in the complex plane.Source: A. Brief Summary of Continuous-Time ADRC · p. 2

B. Discretisation

The discrete-time implementation discretizes the process model and uses a current observer to incorporate the latest measurement; the control law retains its continuous-time form.

B. Discretisation

  • The process-model matrices are converted to discrete time using zero-order-hold discretization.Source: B. Discretisation · p. 2
  • The current observer uses the latest measurement y(k) to estimate the state at sample k.The paper contrasts this with a predictive-observer approach.Source: B. Discretisation · p. 2
  • The discrete-time control law retains the same feedback gain vector as the continuous-time formulation.Source: B. Discretisation · p. 2

C. Parameterisation

Controller parameters are selected from process knowledge and desired closed-loop dynamics, while observer pole placement balances disturbance rejection and noise sensitivity.

C. Parameterisation

  • The approximate process gain b0 represents the process knowledge used for design, while KP and, for second-order ADRC, KD can be selected from desired settling time.Source: C. Parameterisation · p. 2
  • Observer poles are commonly placed about 3–10 times farther left than closed-loop poles, with a compromise between disturbance rejection and noise sensitivity.Source: C. Parameterisation · p. 3

D. Lag-Reduced Form and Implementation

A transformed state vector simplifies the feedback computation in the lag-reduced ADRC form, and the equations can be further arranged for rapid control-signal updates.

D. Lag-Reduced Form and Implementation

III. INCREMENTAL ADRC

The section introduces an incremental ADRC formulation that computes an incremental control signal. It retains the control performance of non-incremental ADRC while simplifying rate limitation and switching among incremental controllers.

III. INCREMENTAL ADRC

  • The new incremental ADRC formulation produces a control law for the incremental control signal Δu(k).It is derived by substituting the incremental observer equations into the existing observer and control equations.Source: §III · p. 3
  • The incremental observer retains the same equation structure in lag-reduced form, with transformed matrices, state, and feedback gain substituted.The transformed feedback gain from (12) is used in (18).Source: §III · p. 3
  • Incremental and non-incremental ADRC have identical functionality and control performance.The incremental form makes rate limitation more straightforward and facilitates switching among incremental controllers sharing an integrator or using actuators with integral behavior.Source: §III · p. 3

IV. MAGNITUDE AND RATE LIMITATION

The section addresses saturation and rate limits in discrete-time ADRC, including windup prevention and the additional measures required for incremental implementations. A carry-over term preserves the desired control signal when limited increments are accumulated.

IV. MAGNITUDE AND RATE LIMITATION

  • Feeding the saturated control signal to the observer avoids integral windup in ADRC, and the same holds for rate-limited signals.The complete loop uses the limited control value u_lim(k−1), rather than u(k−1), in the observer.Source: §IV · p. 4
  • The limiter constrains control-signal magnitude, while the modified incremental law supports magnitude and/or rate limitation.Fig. 10 shows the complete discrete-time incremental ADRC loop with both types of limitation.Source: §IV · p. 4
  • Incremental ADRC needs additional measures to maintain performance with magnitude- or rate-limited control signals.A limiting integrator accumulates limited increments, which may differ from the unlimited increments computed by the controller.Source: §IV · p. 4, Figure 8 · p. 4
  • The incremental control law carries forward the difference between the desired and actual limited increment from the previous step.This keeps the desired control signal from being forgotten; Fig. 9 depicts the modified law with incremental output and limited-signal feedback.Source: §IV · p. 4, Figure 9 · p. 4

V. BUMPLESS TRANSFER

The section presents bumpless-transfer measures for enabling controllers, switching controllers, and changing parameters. Its guidelines assume stationary operation with r = y and adjust the estimated disturbance to keep the control signal constant; enabling the ESO also requires suitable initialization.

V. BUMPLESS TRANSFER

  • Bumpless transfer avoids control-signal jumps when enabling a controller, switching controllers, or changing parameters while enabled.The guidelines cover incremental and non-incremental ADRC in both lag-reduced and standard forms.Source: §V · pp. 4–5
  • The guidelines assume stationary operation with r = y, and adjust the estimated disturbance to keep the control signal constant.The authors state that the measures can also be extended to changes during transients.Source: §V · p. 5

A. Enabling the ESO

  • The ESO is enabled by its first update; preferably, enable it before the controller first computes a control signal.For faster startup or simultaneous enabling, initialize the state vector using the process output and control signal; apply the transformation if using lag-reduced state variables.Source: A. Enabling the ESO · p. 5

B. Enabling the Controller

Incremental ADRC can be enabled by initializing its next control increment from the current reference, observer state, and actual limited control action. Bumpless parameter changes require updates appropriate to the changed plant model, observer, or control law.

B. Enabling the Controller

  • Incremental ADRC requires the observer to be enabled before the controller, with its initial control increment computed from the reference, observer state, and preceding limited control action.The initialisation is assumed to occur immediately before the first control action is computed.Source: B. Enabling the Controller · p. 5

C. Parameter Changes

  • Parameter changes are grouped into plant-model, observer-pole, and control-law changes, with any required computations performed before the observer and controller equations at that time step.Source: C. Parameter Changes · p. 5

1) Moving the closed-loop poles:

  • When moving closed-loop poles, transformed-state variants must update the transformation and retransform observer matrices and states; incremental variants must also recompute the control increment.Source: 1) Moving the closed-loop poles · p. 5

2) Moving the observer poles:

Bumpless parameter updates require changes to observer quantities when observer poles move and coordinated rescaling when the plant-model gain changes.

2) Moving the observer poles:

  • Moving observer poles requires updating the observer gain and matrices, then transforming those updated quantities for variants that use transformed state variables.Source: 2) Moving the observer poles · p. 6

3) Changing the plant model:

  • Changing b0 scales the observer matrices; in transformed-state variants its influence cancels from two matrices, so the observer gain and state vector are rescaled, including opposite scaling of the disturbance state.Incremental variants also require adjusting the control increment after the plant-model change.Source: 3) Changing the plant model · p. 6

VI. EXAMPLES

The examples use a simulated, discrete-time ADRC implementation on a buck converter with ADRC in the outer voltage loop. The plant is treated as first-order, and the model information needed for ADRC is reduced to b0.

VI. EXAMPLES

  • The examples demonstrate practical ADRC features using a simulated plant and a discrete-time controller, with a buck converter in peak current-mode control as the power-electronics example.Source: §VI · p. 6
  • The rate-limitation example uses a 1 A/ms limit, and lower-voltage transients are slower because the converter output capacitor cannot be actively discharged.Source: Figure 11 · p. 6
  • The converter is dominated by first-order low-pass behavior, motivating first-order ADRC; b0 is obtained from the DC gain and dominant time constant.The article describes this as a major simplification compared with the modelling effort traditionally needed for controller design.Source: §VI · p. 6

A. Rate Limitation

The examples demonstrate rate limitation and bumpless transfers among manual control, ADRC, incremental PI, and parameter settings. The conclusion presents incremental ADRC, signal limits, and bumpless controller changes as practical implementation features.

A. Rate Limitation

  • A 1 A/ms rate limit attenuates measurement-noise effects on the control signal without impacting control-loop stability.The experiment compares reference tracking with rate limitation disabled and enabled.Source: A. Rate Limitation · p. 6

B. Bumpless Transfer to ADRC

  • Manual-to-ADRC transfers produce no bumps in the control signal or process output when r = y; enabling the observer first is recommended for transfers outside stationary operation.Source: B. Bumpless Transfer to ADRC · p. 7

C. Bumpless Switch Between Incremental Control Laws

D. Bumpless Parameter Changes

  • Reducing observer bandwidth by 50% yields less noisy control signals and state estimates but slightly degrades controller performance; parameter changes cause no visible jumps when r = y.The experiment also increases closed-loop settling time by 400% before restoring the original parameters.Source: D. Bumpless Parameter Changes · p. 7, Figure 14 · p. 8

VII. CONCLUSION

  • The conclusion presents incremental ADRC, magnitude and rate limits, and bumpless controller or parameter changes as practical implementation features.It notes that rate limitation can provide guaranteed control-signal rise times after reference or disturbance steps.Source: §VII · p. 8
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