Generalizable Disturbance Estimation Framework

• The paper introduces a modular architecture that decomposes heterogeneous disturbances into additive, multiplicative, and recessive channels.
• It employs a two-stage design using a disturbance observer and composite filtering to achieve robust, adaptive state estimation with LMI-based performance guarantees.
• The framework’s modular design allows for easy adaptation and extension to complex, high-dimensional environments common in modern cyber-physical systems.

A generalizable disturbance estimation framework provides a principled, modular architecture for state and disturbance estimation in dynamical systems subject to multi-source, heterogeneous, and isomeric (additive, multiplicative, recessive) disturbances. Unlike traditional filtering schemes that lump all disturbances into a single “equivalent noise” input, generalizable frameworks explicitly decompose and model each distinct disturbance channel, enabling tailored rejection, attenuation, and adaptive filtering. This approach is essential in high-dimensional, complex, data-rich environments where disturbances are physically multi-source, mathematically heterogeneous, and deeply coupled with system dynamics (Guo et al., 2023).

1. Multi-Channel System Model and Disturbance Classes

The foundation of a generalizable disturbance estimation framework is an explicit multi-channel system model that recognizes the diversity of disturbance types. For a continuous-time (or discrete-time) nonlinear plant, the canonical model is

$$\dot x(t)= f(x(t),u(t)) + B_a d_a(t) + B_m(x(t)) d_m(t) + B_r(x(t),u(t),t) d_r(t)$$

$$y(t) = h(x(t)) + D_a d_a(t) + D_m(x(t)) d_m(t) + D_r(x(t),u(t),t) d_r(t)$$

where

• $d_a(t)$: unknown additive disturbances (e.g., bias, drift, exogenous signals),
• $d_m(t)$: multiplicative disturbances (e.g., actuator efficiency uncertainty, loss of effectiveness) through state-dependent $B_m$,
• $d_r(t)$: recessive disturbances (e.g., unknown parameters, unmodeled statistics),
• $f$, $h$: known, possibly uncertain system functions,
• $B_a$, $B_m$, $$y(t) = h(x(t)) + D_a d_a(t) + D_m(x(t)) d_m(t) + D_r(x(t),u(t),t) d_r(t)$$0, $$y(t) = h(x(t)) + D_a d_a(t) + D_m(x(t)) d_m(t) + D_r(x(t),u(t),t) d_r(t)$$1, $$y(t) = h(x(t)) + D_a d_a(t) + D_m(x(t)) d_m(t) + D_r(x(t),u(t),t) d_r(t)$$2, $$y(t) = h(x(t)) + D_a d_a(t) + D_m(x(t)) d_m(t) + D_r(x(t),u(t),t) d_r(t)$$3: input and output disturbance channels.

Each disturbance class has distinct mathematical and physical properties, dictating the appropriate observer structure and filter design (Guo et al., 2023). Additive disturbances may be dynamic (obeying their own evolution), multiplicative effects modulate input/output gains, and recessive uncertainties cover constant drifts or unknown noise statistics.

2. Composite Disturbance Vector Construction

Rather than treating all disturbances as a single aggregate, the framework builds a composite disturbance vector reflecting the separability of disturbance types:

$$y(t) = h(x(t)) + D_a d_a(t) + D_m(x(t)) d_m(t) + D_r(x(t),u(t),t) d_r(t)$$4

Here,

• $$y(t) = h(x(t)) + D_a d_a(t) + D_m(x(t)) d_m(t) + D_r(x(t),u(t),t) d_r(t)$$5: Unknown-Dynamic Signals, often low-order and potentially observable via an unknown-input observer (UIO)/DO,
• $$y(t) = h(x(t)) + D_a d_a(t) + D_m(x(t)) d_m(t) + D_r(x(t),u(t),t) d_r(t)$$6: norm-bounded disturbances, e.g., bounded parameter drift,
• $$y(t) = h(x(t)) + D_a d_a(t) + D_m(x(t)) d_m(t) + D_r(x(t),u(t),t) d_r(t)$$7: stochastic signals, typically noise with possibly uncertain statistics.

Channels and measurement matrices are stacked accordingly, preserving the mapping from each $$y(t) = h(x(t)) + D_a d_a(t) + D_m(x(t)) d_m(t) + D_r(x(t),u(t),t) d_r(t)$$8 to system dynamics and measurements. This stacking maintains design modularity—each disturbance class is addressed using its own filter mechanisms (DO for $$y(t) = h(x(t)) + D_a d_a(t) + D_m(x(t)) d_m(t) + D_r(x(t),u(t),t) d_r(t)$$9, $d_a(t)$0 for $d_a(t)$1, stochastic filtering for $d_a(t)$2), yet the composite framework handles all sources simultaneously (Guo et al., 2023).

3. Two-Stage Filter Design: Disturbance Observer and Composite Filtering

The core architecture is a two-stage structure (“X–DO plus Y–Filter”):

(a) Disturbance Observer (DO)/Unknown-Input Observer (UIO)

• For $d_a(t)$3 with known or structurally specified dynamics (e.g., $d_a(t)$4), an observer is designed for $d_a(t)$5:

  $d_a(t)$6

• Observer gain $d_a(t)$7 is computed (via LMI, Riccati, or pole assignment) to stabilize the error dynamics and guarantee specified disturbance rejection margins.

(b) Composite Filtering/Attenuation

• With $d_a(t)$8 estimated, measurements and state evolution are compensated:

  $d_a(t)$9

• The resulting system is filtered to attenuate $d_m(t)$0 ($d_m(t)$1 filter, norm-bounded) and $d_m(t)$2 (stochastic filter, e.g., Kalman, particle, SDF).
• For non-Gaussian $d_m(t)$3, particle or variational Bayes filters are employed, using $d_m(t)$4 in proposal/prediction steps to prevent model-mismatch degeneracy.

For linearized models, the composite LMI simultaneously ensures error convergence and attenuation levels: $d_m(t)$5 with $d_m(t)$6, $d_m(t)$7 bounding $d_m(t)$8 and $d_m(t)$9 responses (Guo et al., 2023).

4. Stability Certification and Performance Guarantees

Stability and robustness analysis is conducted via Lyapunov functions for the augmented observer-filter system:

• Exponential error decay is proved for the DO in the absence of $B_m$0, $B_m$1.
• Disturbance attenuation properties are certified through $B_m$2 and $B_m$3 bounds, computable via computed LMI solutions.
• Stochastic properties (unbiasedness, bounded error covariance, Monte-Carlo error) are checked in particle or stochastic filter-based stages.
• The modular structure allows for easy adaptation to new classes of disturbances by augmenting the composite disturbance vector.

5. Practical Implementation and System Mapping

Generalization to new systems and scenarios requires careful mapping of disturbances to observer/filter structures:

• Disturbances with low-order, known or estimable dynamics are classified under $B_m$4.
• Unknown, norm-bounded effects become $B_m$5.
• Known or partially known statistical noise forms $B_m$6.

Observer selection (classic UIO, high-gain DO, extended/unscented/sliding/adaptive DO) is tailored to the underlying system nonlinearity and $B_m$7 dynamics. The filtering stage (Kalman, $B_m$8, particle, SDF) is matched to statistics of $B_m$9 and $d_r(t)$0, with real-time or adaptive selection for changing environments.

Tuning of attenuation/rejection thresholds $d_r(t)$1, $d_r(t)$2, and observer-filter bandwidth is managed via LMI/Riccati synthesis, balancing rejection capability and noise sensitivity. System discretization, observer scheduling, and real-time data-paths are adapted for low-latency, high-throughput requirements (Guo et al., 2023).

6. Modularity and Future Extensions

The generalizable framework is inherently modular:

• Additive, multiplicative, and recessive disturbances are treated in parallel but interoperating observer/filter pathways.
• Certificates of performance (attenuation, stability) are provided by convex LMI or ARE solutions.
• The architecture supports enhancements, such as adaptive or switching observers, event-triggered or hybrid update schemes, and seamless integration with emerging sensor modalities or fault-detection routines.

Adaptation to new systems is achieved by updating channel mappings, redesigning observer-filter gains, and re-parameterizing composite disturbance vectors without refactoring the core estimation infrastructure. Extensions to hybrid systems, strong stochasticity, and high-dimensional settings are directly compatible with the underlying composite disturbance logic.

Summary Table: Key Elements of the Generalizable Disturbance Estimation Framework

Component	Mathematical Formulation	Design Principle
Disturbance Decomp.	$d_r(t)$3	Explicit, separable channels
Observer Stage	$d_r(t)$4	UIO/DO for dynamic unknowns
Filter Stage	$d_r(t)$5, $d_r(t)$6, particle, SDF	Tailored for norm-bounded, stochastic
Stability Cert.	Lyapunov, LMI/ARE	Provable rejection, attenuation
Modularity	Composite stacking, adaptive gain	Extensible, retunable

7. Significance and Applications

The composite disturbance filtering paradigm enables real-time, high-confidence estimation in applications characterized by nontrivial, high-dimensional, heterogeneous disturbances: navigation, multilayer localization, inertial alignment, and data fusion in the presence of coupled environmental and model uncertainties. The generalizable disturbance estimation framework marks a departure from traditional single-disturbance filtering, providing a scalable and certifiable approach for the signal processing and control of modern cyber-physical systems (Guo et al., 2023).