Adaptive Disturbance Observer (ADO)

• Adaptive Disturbance Observers (ADO) are estimators that dynamically reconstruct external disturbances using adaptive laws and time-varying gains, ensuring effective real-time control under uncertainty.
• ADOs integrate methods like filtered regressors and Lyapunov-based guarantees to achieve finite-time, exponential, or prescribed-time convergence in disturbance estimation.
• Embedding ADOs in feedback control loops enhances tracking performance and safety by compensating for both constant and time-varying disturbances in nonlinear systems.

An Adaptive Disturbance Observer (ADO) is an estimator designed to reconstruct external disturbances acting on a dynamical system, typically in real time and without precise a priori knowledge of disturbance structure, magnitude, or time variation. ADOs leverage parameter and state adaptation mechanisms to maintain estimation accuracy under model uncertainty, time-varying disturbance statistics, and limited measurement information. Their integration is central to robust, high-performance control in the presence of unknown or changing exogenous inputs.

1. Mathematical Formulation and Key Structures

The core structure of an ADO consists of a state estimator dynamically updated based on input/output data, augmented with adaptive laws that tune observer gains, estimate disturbance model parameters, and/or learn the real-time disturbance profile. For a nonlinear system with lumped disturbance $d(t)$, a prototypical ADO (as in (Wang et al., 2021)) takes the form: $$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= f(\bar{x}, t) + d(\bar{x}, t) \ \hat{d} &= L_{1d}(t) |s_d|^{\frac{m-1}{m}}\mathrm{sgn}(s_d) + L_{2d}(t) s_d + \varphi_d \ \dot{\varphi}_d &= L_{3d}(t) |s_d|^{\frac{m-2}{m}}\mathrm{sgn}(s_d) + L_{4d}(t) s_d \end{aligned}$$ where $s_d=x_2-\hat{x}_2$ is the disturbance estimation error, and all $L_{id}(t)$ are adaptive, time-varying gains. The adaptation law for $L_d(t)$ is "switched": it increases when $|s_d|$ exceeds a small threshold $\varepsilon_d$ and holds otherwise: $$\dot{L}_d = \begin{cases} \kappa & \text{if } |s_d| \ge \varepsilon_d \ 0 & \text{otherwise} \end{cases}$$ with positive constants $\kappa, \varepsilon_d$.

These structures generalize across ADO designs, from smooth fractional power incrementations (Wang et al., 2021), filtered-regressor frameworks (Li et al., 2020), to observer-canonical and overparameterized forms in the linear regime (Glushchenko et al., 2023, Bui et al., 2023). The essential feature is the automatic adjustment of observer dynamics based on online tracking of estimation errors or data-driven indicators of unmodeled behavior.

2. Disturbance Models and Adaptation Mechanisms

Disturbance models in ADO frameworks are often weakly parameterized, adopting only boundedness and slow variation assumptions. For the adaptive smooth disturbance observer (ASDO) (Wang et al., 2021), the only requirement is an unknown but bounded disturbance derivative: $$| \dot{d}_1(\bar{x}, t) | \leq \delta_1, \quad \delta_1 \geq 0$$ The adaptive update of observer gains $$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= f(\bar{x}, t) + d(\bar{x}, t) \ \hat{d} &= L_{1d}(t) |s_d|^{\frac{m-1}{m}}\mathrm{sgn}(s_d) + L_{2d}(t) s_d + \varphi_d \ \dot{\varphi}_d &= L_{3d}(t) |s_d|^{\frac{m-2}{m}}\mathrm{sgn}(s_d) + L_{4d}(t) s_d \end{aligned}$$0 ensures that the ADO remains effective for both constant and slowly time-varying disturbances, without knowledge of $$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= f(\bar{x}, t) + d(\bar{x}, t) \ \hat{d} &= L_{1d}(t) |s_d|^{\frac{m-1}{m}}\mathrm{sgn}(s_d) + L_{2d}(t) s_d + \varphi_d \ \dot{\varphi}_d &= L_{3d}(t) |s_d|^{\frac{m-2}{m}}\mathrm{sgn}(s_d) + L_{4d}(t) s_d \end{aligned}$$1.

Other approaches (e.g., (Li et al., 2020, Glushchenko et al., 2023, Bui et al., 2023)) characterize the disturbance as the output of an unknown autonomous linear generator or exosystem: $$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= f(\bar{x}, t) + d(\bar{x}, t) \ \hat{d} &= L_{1d}(t) |s_d|^{\frac{m-1}{m}}\mathrm{sgn}(s_d) + L_{2d}(t) s_d + \varphi_d \ \dot{\varphi}_d &= L_{3d}(t) |s_d|^{\frac{m-2}{m}}\mathrm{sgn}(s_d) + L_{4d}(t) s_d \end{aligned}$$2 with $$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= f(\bar{x}, t) + d(\bar{x}, t) \ \hat{d} &= L_{1d}(t) |s_d|^{\frac{m-1}{m}}\mathrm{sgn}(s_d) + L_{2d}(t) s_d + \varphi_d \ \dot{\varphi}_d &= L_{3d}(t) |s_d|^{\frac{m-2}{m}}\mathrm{sgn}(s_d) + L_{4d}(t) s_d \end{aligned}$$3 unknown. The ADO then contains adaptation laws to identify these generator parameters online—possibly using experience replay or overparameterized regression—for guaranteed convergence when the regressor signals provide sufficient excitation.

3. Stability, Convergence, and Performance Guarantees

ADO architectures are analyzed through Lyapunov-based methods or their discrete-time analogs, yielding finite-time, exponential, or prescribed-time convergence depending on the structure. For example, the ASDO in (Wang et al., 2021) achieves:

• Finite-time convergence of the disturbance estimation error to the origin (for constant disturbances) or to an arbitrarily small region (for bounded-rate-varying disturbances), under a quadratic Lyapunov function with fractional power dissipation terms.
• Uniform ultimate boundedness of errors, controlled via an explicit gain adaptation and auxiliary $$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= f(\bar{x}, t) + d(\bar{x}, t) \ \hat{d} &= L_{1d}(t) |s_d|^{\frac{m-1}{m}}\mathrm{sgn}(s_d) + L_{2d}(t) s_d + \varphi_d \ \dot{\varphi}_d &= L_{3d}(t) |s_d|^{\frac{m-2}{m}}\mathrm{sgn}(s_d) + L_{4d}(t) s_d \end{aligned}$$4-modification law for residual approximation error.

In (Li et al., 2020), exponential convergence of parameter and disturbance estimates is ensured by an experience-replay augmentation, provided a finite-rank condition on the past regressor stack is satisfied: $$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= f(\bar{x}, t) + d(\bar{x}, t) \ \hat{d} &= L_{1d}(t) |s_d|^{\frac{m-1}{m}}\mathrm{sgn}(s_d) + L_{2d}(t) s_d + \varphi_d \ \dot{\varphi}_d &= L_{3d}(t) |s_d|^{\frac{m-2}{m}}\mathrm{sgn}(s_d) + L_{4d}(t) s_d \end{aligned}$$5 For prescribed-time ADOs (Vahidi-Moghaddam et al., 2020), observer and system errors are forced to zero within a fixed, user-specified time using sliding mode and nonlinear gain terms proportional to $$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= f(\bar{x}, t) + d(\bar{x}, t) \ \hat{d} &= L_{1d}(t) |s_d|^{\frac{m-1}{m}}\mathrm{sgn}(s_d) + L_{2d}(t) s_d + \varphi_d \ \dot{\varphi}_d &= L_{3d}(t) |s_d|^{\frac{m-2}{m}}\mathrm{sgn}(s_d) + L_{4d}(t) s_d \end{aligned}$$6.

4. Integration with Control Architectures

ADOs are most effective when embedded directly in feedback control loops for real-time disturbance compensation. In backstepping designs, the disturbance estimate $$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= f(\bar{x}, t) + d(\bar{x}, t) \ \hat{d} &= L_{1d}(t) |s_d|^{\frac{m-1}{m}}\mathrm{sgn}(s_d) + L_{2d}(t) s_d + \varphi_d \ \dot{\varphi}_d &= L_{3d}(t) |s_d|^{\frac{m-2}{m}}\mathrm{sgn}(s_d) + L_{4d}(t) s_d \end{aligned}$$7 enters explicitly into the virtual and physical control laws, providing direct cancellation. For example, the attitude tracking control of a small unmanned helicopter in (Wang et al., 2021) employs: $$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= f(\bar{x}, t) + d(\bar{x}, t) \ \hat{d} &= L_{1d}(t) |s_d|^{\frac{m-1}{m}}\mathrm{sgn}(s_d) + L_{2d}(t) s_d + \varphi_d \ \dot{\varphi}_d &= L_{3d}(t) |s_d|^{\frac{m-2}{m}}\mathrm{sgn}(s_d) + L_{4d}(t) s_d \end{aligned}$$8 suppressing both direct and residual disturbance errors.

In robust output regulation and tracking (Bui et al., 2023), ADOs combine full-order unknown-input observers for state reconstruction, with adaptive disturbance observers for $$\begin{aligned} \dot{x}_1 &= x_2 \ \dot{x}_2 &= f(\bar{x}, t) + d(\bar{x}, t) \ \hat{d} &= L_{1d}(t) |s_d|^{\frac{m-1}{m}}\mathrm{sgn}(s_d) + L_{2d}(t) s_d + \varphi_d \ \dot{\varphi}_d &= L_{3d}(t) |s_d|^{\frac{m-2}{m}}\mathrm{sgn}(s_d) + L_{4d}(t) s_d \end{aligned}$$9, and synthesis of disturbance-compensating control via Sylvester/Francis regulator equations. In nonlinear safety-critical control (Yang et al., 2024), the disturbance estimate parametrizes control barrier function (CBF) constraints, adaptively shrinking or swelling safety bounds in real time as disturbance estimates evolve.

5. Practical Implementation and Tuning Considerations

Effective ADO implementation relies on:

• Adaptive gain selection: Gains are typically set as nonlinear or time-varying functions of error integrals or system excitation, and freeze when error contracts below threshold, ensuring robustness and noise suppression (Wang et al., 2021).
• Use of filters and data stacks: Filtered regressors (Li et al., 2020), history stacks for experience replay, and first-order filters for state and regressor estimation reduce reliance on noisy derivatives.
• Online parameter estimation: Recursive least squares and memory-augmented regression are adopted in linear plant ADOs (Sariyildiz et al., 2019, Bui et al., 2023), enabling both plant and environment parameter identification to inform adaptive observer/force controller tuning.
• Computational complexity control: The structure often avoids differentiation altogether; implementation is simplified especially when only low-order auxiliary dynamics and normalized regressions are required (Muramatsu et al., 2020, Vahidi-Moghaddam et al., 2020).

Table 1 summarizes selected ADO instantiations:

Reference	System Type	Adaptation Law	Convergence Type
(Wang et al., 2021)	Nonlinear, 3-DOF	Fractional-power, error-driven gain	Fast finite-time
(Li et al., 2020)	Nonlinear, exosystem	Filtered regressor, exp. replay	Exponential (finite-time)
(Glushchenko et al., 2023)	Linear, overparam.	Scalar regressor, Lyap. projection	Exponential
(Vahidi-Moghaddam et al., 2020)	SISO, nonlinear	Prescribed-time, SMC/TSMC	Fixed-time
(Bui et al., 2023)	Linear, MIMO	Gradient, memory regressor	Exponential or $s_d=x_2-\hat{x}_2$0

6. Quantitative Performance and Comparative Analysis

Extensive simulation and experimental results demonstrate the superiority of ADOs over fixed-gain or classical disturbance observer designs:

• Helicopter attitude tracking (Wang et al., 2021): ASDO-disturbance estimation error reaches zero in $s_d=x_2-\hat{x}_2$1 s for constant, and contracts to $s_d=x_2-\hat{x}_2$2 rad/s$s_d=x_2-\hat{x}_2$3 for time-varying disturbances (vs. larger, oscillatory errors in ASOSMO). The resulting control action is smoother, and on trajectory tracking, settling times are halved relative to observer-based command-filtering (CFB) methods.
• Adaptive safety bounds (Yang et al., 2024): In adaptive CBF with disturbance-aware bounds, the required inter-vehicle safe distance in cruise control adapts with road grade, reducing conservatism by 10–20%, and improving input smoothness by up to 42% across scenarios, compared to worst-case-fixed methods.
• General nonlinear plants (Li et al., 2020): The experience-replay ADO achieves finite-time disturbance rejection without state derivative measurement, which is not possible for classical disturbance observers relying on numerical differentiation.

7. Extensions and Broader Context

ADO ideas extend to multifunctional observers (e.g., reaction force observers (Sariyildiz et al., 2019)), periodic and frequency-adaptive observers (Muramatsu et al., 2020), and systems with significant modeling uncertainty. Overparameterized observer forms (Glushchenko et al., 2023) enable state and disturbance estimation without transformation to canonical coordinates, and with weaker excitation conditions than those required in classical persistent excitation arguments. Advanced ADOs interoperate with integral terminal sliding mode (Li et al., 2020, Vahidi-Moghaddam et al., 2020), reinforcement memory (Li et al., 2020), and real-time optimization strategies (e.g., PSO and EKF for parameter/state estimation (Vahidi-Moghaddam et al., 2020)), reflecting their adaptability to highly nonlinear, multivariable, and uncertain environments.

In summary, Adaptive Disturbance Observers are now established as rigorously analyzable, practically effective schemes for system disturbance estimation and rejection, capable of supporting advanced real-time control, robust output regulation, and safety-critical operation under challenging and uncertain disturbance regimes (Wang et al., 2021, Li et al., 2020, Yang et al., 2024, Bui et al., 2023, Glushchenko et al., 2023).