Dynamic Model-Based Estimation

• Dynamic model-based estimation is a method that employs explicit dynamic models, such as state-space or differential equations, to infer system states, parameters, and structure from data.
• It integrates domain knowledge via recursive, Bayesian, and neural-augmented algorithms, enabling real-time tracking and improved robustness in noisy, nonstationary environments.
• Key challenges include managing partial observability, computational complexity, and adapting to abrupt system changes while ensuring reliable performance.

Dynamic model-based estimation refers to a family of methodologies that exploit explicit dynamic models—typically in state-space or differential equation form—to infer system states, parameters, or structure from observed data. In contrast to purely data-driven or static estimation paradigms, dynamic model-based estimation systematically integrates domain-knowledge regarding system evolution, stochasticity, and algebraic constraints, often yielding improved robustness, physical interpretability, and reliable performance under nonstationary and noisy operating conditions. Key developments span linear, nonlinear, and hybrid dynamical systems; algorithms range from recursive estimators (e.g., Kalman filters, moving-horizon estimators) to regression-based, Bayesian, and neural-augmented approaches. Applications are ubiquitous: power system stability, UAV navigation, microgrid state estimation, reinforcement learning, and real-time tracking in robotics.

1. Mathematical Foundations and Problem Formulations

Core dynamic model-based estimation relies on explicit dynamical system equations:

• Linear time-invariant (LTI) or stochastic differential equations, e.g.,

$\dot{x}(t) = A x(t) + B \xi(t)$

for power system small-signal models (Sheng et al., 2019), or

• Parameterized ODE models,

$$\dot{y} = G(y, x) \quad \text{with unknown parameters}~x$$

for synthesis modeling (Lukšič et al., 2019).

The estimation targets may include:

• The state trajectory $x(t)$ given partial/noisy measurements,
• Unknown model parameters $\theta$ embedded in $f(x,\theta)$, or
• System structural properties (e.g., Jacobians for stability/structure analysis).

Many methods recast the estimation objective into regression, Bayesian inference, or stochastic filtering frameworks. Notably, in power system dynamic state Jacobian estimation, the stationary covariance and lagged correlation encode $A$ via Ornstein-Uhlenbeck process regression theorems (Sheng et al., 2019). In general nonlinear models, techniques such as recursive nonlinear least squares, variational methods, MCMC, Gaussian process regression with model constraints, or surrogate optimization are deployed, depending on tractability and computational constraints (Bhaumik et al., 2014, Linden et al., 2022, Zhou et al., 18 Sep 2024).

2. Key Estimation Algorithms and Methodological Variants

Dynamic model-based estimation embraces a rich algorithmic ecosystem:

Measurement-based/Regression Techniques:

• For ambient power systems, the dynamic Jacobian is extracted via sample covariance and cross-correlation, leveraging theoretical results for multivariate Ornstein-Uhlenbeck processes (Sheng et al., 2019).
• The regression estimator exploits the analytic relationship

$$A = \frac{1}{\Delta t} \log[\hat{G}(\Delta t) \hat{C}^{-1}]$$

for the state matrix from time-lagged statistics.

Recursive/Online Algorithms:

• Real-time suitability is ensured by recursive exponential smoothing of mean, covariance, and correlation, with efficient Sherman–Morrison updates for covariance inversion and adaptive smoothing during topology change events (Sheng et al., 2019).

State–Parameter Joint Estimation:

• Extended Kalman filters (EKF) embed nonlinear, parameterized process models, with parameters estimated online via auxiliary identification methods (e.g., MRFT–DNN for UAVs (Wahbah et al., 2021), or via ODE-Net neural parameterizations for microgrids (Feng et al., 2022)).
• Observer-based techniques (e.g., DREM-based for power system flux-decay models) provide global boundedness in the presence of measurement disturbances (Lorenz-Meyer et al., 2022).

Bayesian and Two-Step Approaches:

• Bayesian parameter inference (via MCMC, Laplace, or variational methods) is standard for nonlinear and high-dimensional ODE models, with full uncertainty propagation (Linden et al., 2022).
• Two-step procedures fit a nonparametric model (e.g., B-spline regression) to observed trajectories, then estimate parameters by matching the estimated derivatives to those required by the ODE, achieving $\sqrt{n}$-consistency for $\theta$ (Bhaumik et al., 2014).

Hybrid, Surrogate, and Meta-model Frameworks:

• Surrogate-based frameworks employ learned regression models (e.g., RF, TREE, SVM) as cheap proxies for expensive simulator-based objective evaluations. Meta-models dynamically switch between surrogate and true objectives based on relevancy predictors, considerably reducing expensive ODE simulation calls (Lukšič et al., 2019).
• Model-embedded Gaussian process regression (ME-GPR) integrates dynamic constraints into the GPR framework, enabling joint inference of solutions and parameters, with piecewise linearization for nonlinear systems (Zhou et al., 18 Sep 2024).

Filter-based and Koopman Methods:

• Interacting Multiple Model (IMM) filters blend multiple model-based Kalman filters to address mode-switching and nonstationary behavior efficiently (Dingler, 2022).
• Koopman operator and dynamic mode decomposition approaches lift nonlinear dynamics into linear predictors for use with convex moving-horizon estimators, improving performance in highly dynamic legged locomotion (Khorshidi et al., 20 Mar 2024).

Neural Augmentation:

• Recent approaches utilize neural ODEs (ODE-Nets) to parameterize unknown physics, either for direct dynamic state estimation or combined with trainable Kalman/gain structures (e.g., KalmanNet) to address modeling uncertainties and nonlinearity (Feng et al., 2022).

3. Algorithmic Properties, Implementation, and Robustness

Dynamic model-based estimation methods are characterized by their adaptability to real-time operation, noise robustness, computational tractability, and ability to handle partial information.

• Computational Complexity: Many routines are $O(n^2)$ or $O(n^3)$ per update, with explicit parallelization potential (Sheng et al., 2019, Wang et al., 2015).
• Real-time Performance: Recursive and online variants (e.g., exponentially-weighted covariance updates or MRFT–DNN for UAVs) support sub-millisecond updates at sensor frequencies (25–60 Hz for PMUs, 1 kHz for robotic MHE) (Sheng et al., 2019, Wahbah et al., 2021, Khorshidi et al., 20 Mar 2024).
• Robustness to Noise and Incomplete Sensing: PMU-based Jacobian estimation is resilient to missing sensors, moderate measurement noise, and can localize faults via principal submatrix discrepancy (Sheng et al., 2019).
• Adaptation to Structural Change: Sudden network topology changes, actuator/sensor faults, or unmodeled dynamics can be detected and promptly incorporated, exploiting adaptive smoothing, change detection, or online neural parameter re-identification (Sheng et al., 2019, Feng et al., 2022).
• Guaranteed Estimation Boundedness: Observer-based methods for nonlinear generators furnish explicit error bounds under bounded measurement disturbances (ultimate boundedness property) (Lorenz-Meyer et al., 2022).

4. Applications and Empirical Performance

Dynamic model-based estimation has demonstrated efficacy across numerous domains:

• Power System Monitoring:
  • Online topology error detection/localization by comparing measurement-based and model-based Jacobians, with immediate response to faults (Sheng et al., 2019, Wang et al., 2017).
  • Real-time stability metrics and modal participation computation for oscillation source identification (Wang et al., 2015).
• Robotics and Navigation:
  • Koopman/MHE frameworks for centroidal state recovery under highly dynamic locomotion surpass EKF in handling hybrid nonlinearities, contact events, and sensor noise (Khorshidi et al., 20 Mar 2024).
  • UAV state estimation with online, data-driven dynamics parameterization enables aggressive control under sparse position sensing (Wahbah et al., 2021).
• Microgrid and Energy Systems:
  • Neural-dynamic DSE and variants (Neuro-DSE, Neuro-KalmanNet) provide rapid convergence (3–5 s) and robust tracking even with substantial topology/model mismatch (Feng et al., 2022).
• Systems Biology:
  • Bayesian ODE parameter estimation quantifies uncertainty, detects nonidentifiability, and integrates prior biochemical knowledge (Linden et al., 2022).
  • Surrogate and meta-model frameworks (TREE surrogate/RF relevator) have achieved up to 77% reduction in true objective calls for biological network parameter fitting (Lukšič et al., 2019).
• Model-based RL:
  • Dynamic-horizon model-based value expansion leverages model confidence for state-adaptive horizon selection, improving sample efficiency and return in pixel-based RL domains (Wang et al., 2020).

<table> <thead> <tr> <th>Domain</th> <th>Representative Method(s)</th> <th>Empirical Outcome</th> </tr> </thead> <tbody> <tr> <td>Power system monitoring</td> <td>OU-regression estimator, hybrid covariance-based Jacobian</td> <td><5% error in A-matrix; real-time topology change localization</td> </tr> <tr> <td>UAV estimation</td> <td>DNN–MRFT parameterization + decoupled EKFs</td> <td>Position RMSE <3mm (Hi-rate), <10cm (lo-rate)</td> </tr> <tr> <td>Legged robots</td> <td>Koopman-MHE with DMDc/DLK</td> <td\>1–2 order of magnitude RMSE reduction vs EKF</td> </tr> </tbody> </table>

5. Limitations, Extensions, and Open Challenges

Despite the advances, several limitations and frontiers remain:

• Model Assumptions: Many methods rely on accurate model order and structure (e.g., classical swing equations or assumed observability). Extensions to accommodate higher-order dynamics or incomplete models have been tested but remain areas for further development (Sheng et al., 2019, Wang et al., 2017, Feng et al., 2022).
• Partial Observability: Recovery of complete state matrices or parameters with limited sensors (PMUs, process variables) generally requires structural assumptions or augmentation (principal submatrix estimation, state augmentation) (Sheng et al., 2019, Wahbah et al., 2021).
• Computational Bottlenecks: High-dimensional, nonlinear, or stiff dynamics may preclude real-time solutions or intractably large optimization problems. Kernel methods (ME-GPR) and surrogate/meta-models address some scenarios, but scaling to very large systems remains open (Zhou et al., 18 Sep 2024, Lukšič et al., 2019).
• Uncertainty Quantification and Identifiability: Bayesian approaches provide full posterior distributions and explicit identifiability diagnostics, but their computational burden can be prohibitive for real-time or high-dimensional applications (Linden et al., 2022, Bhaumik et al., 2014).
• Adaptation and Learning: While neural-augmented estimators can significantly improve tracking and noise robustness, theoretical guarantees (stability, convergence) under online adaptation and changing regimes are an active research area (Feng et al., 2022, Khorshidi et al., 20 Mar 2024).
• Event Detection and Cybersecurity: Automated and robust event detection (e.g., undetected topology change, data injection attacks) is an open problem, with event-localization via dynamic Jacobian discrepancies a promising but not definitive method (Sheng et al., 2019, Lorenz-Meyer et al., 2022).

6. Directions for Future Research and Broader Impact

Areas highlighted for further exploration include:

• Distributed and scalable implementations for very large-scale dynamic networks (e.g., sparse ODE-Nets, networked observers) (Feng et al., 2022).
• Formal stability guarantees and contraction analysis for neural-enhanced filters and learning-based MHE frameworks (Feng et al., 2022, Khorshidi et al., 20 Mar 2024).
• Multi-objective, constrained, and discrete/continuous parameter estimation, extending surrogate/meta-model approaches (Lukšič et al., 2019).
• Formal integration of Bayesian and model-embedded GPR methods with streaming data and online adaptation (Zhou et al., 18 Sep 2024).
• Automated detection and localization of structural/modeling faults for critical infrastructure security and resilience (Sheng et al., 2019, Lorenz-Meyer et al., 2022).
• Domain-specific hybridizations, exploiting physics knowledge, neural architectures, and uncertainty quantification for robust dynamic state and parameter estimation in novel and rapidly evolving domains.

Dynamic model-based estimation continues to evolve rapidly, synthesizing advances in stochastic process theory, computational optimization, system identification, and machine learning to meet the challenges posed by increasingly complex, data-rich, and safety-critical dynamical systems.