Parallel Disturbance Observer

• Parallel disturbance observer is a control architecture that integrates a disturbance estimator running parallel to a baseline controller for robust rejection of uncertainties.
• It employs a Q-filter for real-time disturbance estimation, ensuring high-bandwidth performance while preserving nominal controller behavior.
• Practical implementations include linear, adaptive, and learning-based variants, making it essential for high-precision applications in motion control and power electronics.

A parallel disturbance observer (parallel DOB) is a feedback-control architecture in which a disturbance estimation module acts in parallel with a primary baseline controller to achieve robust rejection of disturbances and uncertainties, as well as recovery of nominal closed-loop performance. This scheme encompasses a broad class of designs, including classic linear disturbance observers, extended state observers, model-based and data-driven observers, adaptive/learning observers, and their nonlinear or switching extensions. The parallel configuration contrasts with cascade-based observer–compensator structures, enabling concurrent estimation and rejection of lumped disturbances with minimal coupling to nominal controller performance. Parallel DOBs are central to modern robust control engineering, supporting high performance even under severe plant-model uncertainty and dynamic, time-varying disturbances.

1. Theoretical Foundations and Core Architecture

The canonical parallel DOB architecture consists of the true plant $P(s)$, a baseline controller $C(s)$ designed for the nominal model $P_n(s)$, and a disturbance estimation branch operating on plant input-output signals. Let $u(s)$ denote the total input and $y(s)$ the measured output. The observer computes a disturbance estimate $\hat d(s)$ by filtering a residual error signal: $e_d(s) = y(s) - P_n(s)u(s)$ through a low-pass $Q$-filter: $\hat d(s) = Q(s)\,e_d(s)$ The compensated plant input is then

$$u(s) = u_1(s) - \hat d(s), \quad u_1(s) = C(s)[r(s)-y(s)]$$

This realizes a closed-loop structure where the DOB acts as an inner loop in parallel with the primary controller, targeting both nominal performance and robust disturbance rejection (Shim, 2021).

The parallel implementation extends seamlessly to state-space and nonlinear regimes, for example with sliding mode controllers (SMC) using disturbance estimates in the sliding manifold, and to multi-observer architectures combining multiple reduced- or high-order ESOs in parallel (Tang et al., 2023).

2. Design Methodologies and Variants

Linear Parallel DOB Designs

The classic linear DOB is formulated in the Laplace domain with plant model uncertainty, disturbances at plant input, and measurement noise. The Q-filter $Q(s)$ is constructed as a strictly proper, stable, unity-gain DC low-pass filter, whose order covers the relative degree of $P_n(s)$. The principal design requirements are:

• High Q-filter bandwidth for accurate disturbance estimation in the relevant frequency range.
• Maintenance of small-gain stability: $\|\Delta\|_\infty\|T_d\|_\infty < 1$, where $T_d(s)$ is the transfer function from the model uncertainty $\Delta(s)$ into the loop (Shim, 2021).

The optimal bandwidth for $Q(s)$ is chosen based on the disturbance frequency content, the robustness margin to model uncertainty, and noise sensitivity.

Parallel Multi-Observer Architectures

Recent extensions leverage parallel observer banks, each with distinct order or bandwidth (e.g., third-order and fourth-order ESOs), running simultaneously. A real-time selection mechanism or switching law selects the observer whose current estimate leads to the least computed tracking error. Formally, for $M$ parallel ESOs, the chosen disturbance estimate at time $t$ is indexed by

$q(t) = \arg\min_j |z^{(j)}(t)|$

where $z^{(j)}(t)$ is a computable surrogate for the output tracking error associated with ESO $j$ (Tang et al., 2023). This switching approach yields an adaptive disturbance estimation process, combining speed and precision (high-order observers) with noise immunity (low-order observers).

3. Nonlinear and Learning-Based Parallel DOBs

For nonlinear plants and time-varying, mismatched uncertainties, parallel DOBs integrate model-based observers with adaptive, data-driven structures. A representative architecture is the Self-Learning Disturbance Observer (SLDO), where a Basic Nonlinear Disturbance Observer (BNDO) and an adaptive Neuro-Fuzzy Structure (NFS) operate in parallel (Kayacan, 2021, Kayacan et al., 2021). The functional structure is:

• BNDO provides a model-based disturbance estimate $\hat d_{BN}$ with fast transient response under bounded $\dot{d}$.
• The neuro-fuzzy branch receives derivatives of $\hat d_{BN}$ as input, uses a fuzzy inference system to refine adaptive rules, and outputs a learned correction $\tau_n$.
• The total SLDO update is: $$\dot{\hat d}_{SL} = \dot{\hat d}_{BN} + \tau_c - \tau_n$$ where $\tau_c$ is a conventional estimation law, and $\tau_n$ the NFS output.

Feedback-error learning ensures that as learning progresses, $\tau_c$ vanishes, yielding full adaptation to time-varying disturbances. Lyapunov stability analysis guarantees finite-time convergence of the error signals under appropriate learning rates and bounded disturbance acceleration (Kayacan, 2021).

Table 1 summarizes parallel observer structures:

Approach	Parallel Elements	Main Application Domain
Linear DOB	Model-based DOB + baseline	SISO/MIMO linear systems, robust stabilization
Multi-ESO ADRC	Multiple ESO (varied order)	Precision motion, high-bandwidth rejection
SMC-SLDO	BNDO + NFS in parallel	Nonlinear systems, mismatched/time-varying dist.

4. Robustness, Stability, and Performance Guarantees

Parallel DOB schemes inherit robust stability from their design: the core requirement is that all individual subsystems (e.g., each branch of the observer and nominal controller) maintain stability and passivity-type properties. The overall closed loop enjoys robust stability to model uncertainty as long as the small-gain criteria are satisfied (for linear DOBs) or the Lyapunov conditions hold (for nonlinear/learning DOBs).

Performance metrics demonstrated in the literature include:

• Bounded tracking error: With sufficiently fast observer dynamics (high-gain), the residual error induced by unmodeled disturbance derivatives is suppressed to a neighborhood scaling inversely with observer bandwidth (Feng et al., 2020, Shim, 2021).
• Nominal performance recovery: With high observer bandwidth and accurate disturbance estimation, the reference-to-output transfer function approaches that of the nominal plant-plus-controller system (Shim, 2021).
• Chattering mitigation: In SMC-SLDO, the controller gain required for stability is linked to the estimation error bound (e.g., $k > \lambda e_d^*$) rather than the full disturbance upper bound, allowing substantial reduction in chattering amplitude without sacrificing robustness (Kayacan, 2021).
• Enhanced tracking and transient suppression is demonstrated by mean absolute error (e.g., SMC-SLDO error 0.049 vs. SMC 1.88), and faster settling times due to real-time adaptation (Kayacan, 2021, Tang et al., 2023).

5. Practical Implementations and Industrial Applications

Parallel disturbance observers have been realized in diverse applications including high-speed motion control, power electronics, and RF field regulation.

• In high-speed precision motion stages, parallel multi-ESO ADRC achieves lower integral absolute error (IAE) and smaller control signal jumps than traditional single-ESO architectures. For example, IAE reductions from 105.95 (3rd-order ESO), 86.15 (4th-order ESO) to 82.67 (parallel ESO) are reported (Tang et al., 2023).
• In particle accelerator RF regulation, FPGA-based realization of a parallel DOB suppresses SSPA phase drift by an order of magnitude, maintaining field phase within ±1° over 40-min warm-up, and containing beam-pulse amplitude error within ±0.2% under fast loading transients. The DOB path never compromises the gain/phase margin of the main feedback controller (Kwon et al., 2022).
• Nonlinear, learning-based SLDOs running in real time (computation per step ≈0.18 ms on commodity hardware) demonstrate noise robustness and negligible steady-state error, even under strong nonstationary disturbances (Kayacan et al., 2021).

6. Extensions, Limitations, and Research Directions

Key extensions to the parallel DOB paradigm include:

• High-order state and disturbance estimation: The Extended Dynamics Observer (EDO) generalizes to cases where prior disturbance dynamics are partially known, interpolating between classic high-gain observers and internal-model principles. Multiple EDO channels can be run in parallel, tuned for structured and unstructured disturbance spectra (Feng et al., 2020).
• Adaptivity and switching: Parallel observer banks with online switching laws, or adaptive observer-scheduler schemes, can further optimize performance in the presence of changing disturbance statistics.
• MIMO and higher-order generalizations: Designing parallel observers for MIMO plants with both matched and unmatched disturbance channels remains an active area.
• Limitation: For very rapidly changing disturbances, observer gain or learning rates may need to be set high, inducing noise sensitivity or computational burden. In most designs, performance guarantees are ultimately restricted by the highest bounded derivative of the lumped disturbance and the accuracy of the nominal model (Kayacan, 2021, Tang et al., 2023, Feng et al., 2020).

A plausible implication is that further integration of model-based and data-driven estimation (e.g., combination of deep learning with structure-exploiting observers) may overcome some of the residual limitations in noise robustness and adaptation speed.

7. Summary and Significance

Parallel disturbance observers fundamentally extend classical disturbance-rejection paradigms by providing modular, inner-loop architectures capable of high-bandwidth disturbance estimation and rejection with minimal compromise of nominal system characteristics. Theoretical underpinnings (small gain, Lyapunov analysis), implementational diversity (linear, nonlinear, adaptive, switching), and strong performance in experimental applications position parallel DOBs as essential tools in precision control systems across industries (Shim, 2021, Kwon et al., 2022, Tang et al., 2023, Kayacan, 2021, Kayacan et al., 2021, Feng et al., 2020).