Linear ADRC is equivalent to PID with set-point weighting and measurement filter

TL;DR

This paper addresses whether linear ADRC tuned with the bandwidth method can be implemented as a standard PI(D) controller with set-point weighting and measurement filtering. By deriving transfer-function and state-space mappings, it demonstrates exact equivalence in the measurement response for both first- and second-order plants, with a minor approximation in the reference response for first-order systems. It provides explicit expressions that map ADRC tuning parameters to PI gains and filters, and shows through numerical comparisons that an equivalent PI(D) controller matches ADRC performance while often offering similar robustness under plant variations. The practical impact is that linear ADRC can be implemented with ubiquitous PID-based controllers, simplifying adoption and tuning in engineering practice.

Abstract

We show that linear Active Disturbance-Rejection Control (ADRC) tuned using the "bandwidth method" is equivalent to PI(D) control with set-point weighting and a lowpass filter on the measurement signal. We also provide simple expressions that make it possible to implement linear ADRC for first and second-order systems using commonplace two degree-of-freedom PID implementations. The expressions are equivalent to ADRC in the response from measurements, and a slight approximation in the response from references.

Figures (8)

• Figure 1: Step response from $r$ to $y$ of the closed loop with different values of the plant gain $K$.
• Figure 2: Step response from $r$ to $y$ of the closed loop with different values of the plant time constant $T$.
• Figure 3: Bode plot of the controllers.
• Figure 4: Gang-of-seven plot for the first-order system. The ADRC controller is shown in blue, the PI controller suggested in herbst2013simulative is shown in orange, and the equivalent PI controller proposed in this paper is shown in green. The ADRC and the equivalent PI controller are identical in most transfer functions (all solid lines, including only the loop-transfer function) and are hard to distinguish in the plots. The equivalent PI controller does differ slightly from the ADRC controller in the response from references (dashed lines). The plots show the sensitivity function $S = 1/(1+PC)$ (top left), the input disturbance rejection $PS = P/(1+PC)$ (top right), the measurement noise amplification in the control signal $CS = C/(1+PC)$ (bottom left), and the complementary sensitivity function $T = PC/(1+PC)$ (bottom right). Where relevant, the transfer function post multiplied by $F = -C_r / C_y$ is shown as well, indicating the response from references.
• Figure 5: Step response from $r$ to $y$ of the closed loop with different values of the plant gain $K$. Second-order plant.