Transfer Function Analysis and Implementation of Active Disturbance Rejection Control

TL;DR

This work reframes linear ADRC in a realizable transfer-function framework, enabling direct comparison with classical controllers and facilitating practical implementation. It presents explicit continuous-time transfer-function representations for the feedback, prefilter, and feedforward paths, and provides a thorough frequency-domain analysis of how tuning parameters such as $k_ ext{ESO}$ and $b_0$ shape disturbance rejection and noise sensitivity. An exact discrete-time transfer-function realization is derived for first- and second-order ADRC, including accumulator-windup considerations and discretized ESO, yielding low-footprint implementations suitable for embedded hardware. Collectively, the approach lowers barriers to industrial adoption by enabling intuitive, predictable tuning and robust, resource-efficient ADRC deployments.

Abstract

To support the adoption of active disturbance rejection control (ADRC) in industrial practice, this article aims at improving both understanding and implementation of ADRC using traditional means, in particular via transfer functions and a frequency-domain view. Firstly, to enable an immediate comparability with existing classical control solutions, a realizable transfer function implementation of continous-time linear ADRC is introduced. Secondly, a frequency-domain analysis of ADRC components, performance, parameter sensitivity, and tuning method is performed. Finally, an exact implementation of discrete-time ADRC using transfer functions is introduced for the first time, with special emphasis on practical aspects such as computational efficiency, low parameter footprint, and windup protection.

Figures (9)

• Figure 1: Continuous-time state-space implementation of linear ADRC. The parameters for controller and extended state observer (ESO) are given in (\ref{['eqn:ADRC_A_B_C']}) and (\ref{['eqn:ADRC_K_L']}).
• Figure 2: Control loop with transfer function based ADRC implementation consisting of feedback controller $C_\mathrm{FB}(s)$, reference signal prefilter $C_\mathrm{PF}(s)$, and reference signal feedforward $C_\mathrm{FF}(s)$. Disturbances (at plant input, $d$, and plant output, $n$) are present in order to derive the gang-of-six transfer functions in Sect. \ref{['sec:FrequencyDomain_ClosedLoop']}.
• Figure 3: Frequency-domain illustration of the critical gain parameter $b_0$ for first- (left-hand side) and second-order (right-hand side diagram) plant variations with the same value of $b_0$ (in these examples: $b_0 = 1$). The critical gain parameter can be obtained from magnitude plots of the plant transfer function by extending the straight-line approximation (red dashed line) of the $-20$ dB/decade (first-order case) or $-40$ dB/decade segment (second-order case) in order to find the $0$ dB crossover frequency (encircled in the diagrams). The crossover angular frequency amounts to $b_0$ in the first-order case and $\sqrt{b_0}$ in the second-order case.
• Figure 4: Bode plots of the feedback controller $C_\mathrm{FB}$ of first- and second-order ADRC with variation of $k_\mathrm{ESO}$ using bandwidth parameterization. The exponentially increasing values of $k_\mathrm{ESO}$ are ranging from $k_\mathrm{ESO} = 1$ (purple) through $k_\mathrm{ESO} = 5$ (black) to $k_\mathrm{ESO} = 25$ (orange). The remaining tuning parameters are fixed at $b_0 = 1$ and $\omega_\mathrm{CL} = 2\piup$ (i. e. $1\,\mathrm{Hz}$).
• Figure 5: Sensitivity of the gang-of-six transfer functions to variations of $k_\mathrm{ESO}$ for second-order ADRC. Normalized plant in this example: $P(s) = \frac{1}{1 + 2s + s^2}$. Fixed tuning parameters are $b_0 = 1$ and $\omega_\mathrm{CL} = 0.4\piup$ (i. e. $0.2\,\mathrm{Hz}$). The exponentially increasing values of $k_\mathrm{ESO}$ are ranging from $k_\mathrm{ESO} = 1$ (purple) through $k_\mathrm{ESO} = 5$ (black) to $k_\mathrm{ESO} = 25$ (orange).

Theorems & Definitions (1)

• Proof 1