Moving Horizon Estimators (MHE)

• Moving Horizon Estimators (MHE) are optimization-based techniques that reconstruct state trajectories over a receding horizon using process models, input/output data, and prior information.
• MHE explicitly manages estimation uncertainties and system constraints, making it effective for nonlinear, high-dimensional, and partially unknown systems.
• Recent advances integrate data-driven, Koopman-lifting, and learning-based methods, such as neural networks and Gaussian processes, to improve accuracy and computational efficiency.

Moving Horizon Estimators (MHE) are optimization-based state estimation schemes that reconstruct the state trajectory of a dynamical system over a receding time window on the basis of process models, input/output data, and prior information. MHE frameworks are widely employed in control, filtering, and system identification, offering rigorous quantification of estimation uncertainties and explicit handling of system constraints. The recent development of data-driven, learning-enhanced, and robust moving horizon estimation methods has further extended the domain of applicability to nonlinear, high-dimensional, and partially unknown systems.

1. Core Principles and Mathematical Formulation

Moving horizon estimation addresses the state estimation problem for a discrete- or continuous-time dynamical system of the form: $$x_{k+1} = f(x_k, u_k, w_k),\quad y_k = h(x_k, u_k, v_k)$$ where $x_k$ denotes the system state, $u_k$ the control input, $w_k$, $v_k$ are process and measurement noise, and $y_k$ the measured output.

Over a chosen horizon $N$, at each time $t$, an MHE solves an optimization problem to estimate the sequence $\{x_{t-N}, ..., x_t\}$, typically of the form: $$\min_{x_{t-N},...,x_t,\{w,\nu\}} \left[ \|x_{t-N} - \hat x_{t-N}\|_P^2 + \sum_{k=t-N}^{t-1} \|x_{k+1} - f(x_k, u_k, w_k)\|_Q^2 + \sum_{k=t-N}^t \|y_k - h(x_k, u_k, \nu_k)\|_R^2 \right]$$ subject to model constraints (possibly including physical state and input constraints). The first term is the arrival cost, summarizing prior information; the remaining terms penalize deviations from the process model and measurement fit.

MHE is a generalization of the Kalman filter for nonlinear/constrained systems and allows for the explicit enforcement of constraints on both state and noise sequences. The arrival cost is essential for recursive feasibility and convergence, and its design (e.g. via Riccati recursion in the linear case) is a key technical component (Al-Matouq et al., 2014).

2. Data-Driven, Koopman and Learning-Based MHE

Recent work has focused on MHE for systems where the process dynamics $x_k$0 and output map $x_k$1 are partially or wholly unknown. Two principal approaches have emerged:

• Trajectory- and Hankel-based Data-Driven MHE: Leveraging Willems' fundamental lemma, it is possible to encode the system behavior via Hankel matrices constructed from persistently exciting input-output (and possibly state) data, without identifying explicit state-space models (Wolff et al., 2022, Wolff et al., 2021). For a horizon length $x_k$2, the key constraint is:

$x_k$3

where $x_k$4 denotes the depth-$x_k$5 Hankel matrix, and $x_k$6 are trajectory coefficients. The optimization simultaneously estimates the state and slack variables, ensuring robust performance even with bounded measurement or data noise.

• Koopman-Lifting and Learning-based MHE: For nonlinear systems, a powerful approach is to learn a high-dimensional "lifted" representation in which the nonlinear process admits an (approximately) linear, parameter-varying structure. Neural networks are used to learn lifting maps $x_k$7 and scheduling maps $x_k$8, producing a convex MHE in the lifted space. State reconstruction is performed by a learned linear map $x_k$9 (Xiaojie et al., 15 Feb 2026). The online MHE then solves a real-time quadratic program over lifted variables, subject to data-driven Hankel constraints and implicit LPV updates.
• Learning Models with Gaussian Processes (GP): GP-based moving horizon estimation replaces the process/measurement models with GP posterior means, and incorporates GP predictive variance into the (possibly time- and trajectory-dependent) cost weights, thus accounting for epistemic uncertainty directly within the estimation scheme (Wolff et al., 2023, Wolff et al., 2024, Choo et al., 2023). The GP-MHE online problem uniquely adjusts the trust in the learned model locally in the state space, ensuring robust stability and improved estimation accuracy.

3. Robustness, Stability, and Theoretical Guarantees

A significant advantage of MHE frameworks is the provision of explicit estimation error guarantees under suitable detectability and persistency-of-excitation (PE) conditions.

• Incremental Input/Output-to-State Stability (i-IOSS): Many recent analyses assert stability via a Lyapunov-type (δ-IOSS) condition. This property, which typically requires detectability of the pair $u_k$0 or its nonlinear counterpart, ensures that the estimation error shrinks exponentially with time, up to a disturbance- and model-mismatch-dependent bound (Wolff et al., 2022, Xiaojie et al., 15 Feb 2026, Wolff et al., 2024).
• Practical Robust Exponential Stability (pRES): pRES is established for data-driven MHE methods, including cases with bounded noise in both the offline training data and online measurements, and for GP-driven MHE with function-approximation error. Analytical bounds are derived that express estimation error as a combination of decaying initial error, filtered process/measurement noise, and a bounded model-mismatch bias (Wolff et al., 2022, Wolff et al., 2023, Xiaojie et al., 15 Feb 2026).
• Viability under Non-Persistent Excitation and Parametric Uncertainty: In the absence of sufficient excitation, standard augmented-state MHE can exhibit parameter drift and loss of stability. Regularization-based formulations, which penalize deviation from a fixed prior on the unknown parameters, guarantee robust stability (δ-IOpS) without requiring PE, albeit with a bias that vanishes with increasing horizon length (Muntwiler et al., 2023).
• Extension to Nonlinear, High-Dimensional, and Uncertain Systems: IQC-based frameworks extend stability analysis to MHE for general constrained systems with parametric or static nonlinear uncertainties, providing robust input-to-state stability guarantees for closed-loop systems under general nonparametric uncertainty (Guo et al., 5 Nov 2025).

4. Algorithmic and Computational Advances

Solver efficiency and scalable implementation are recurring challenges in practical MHE deployment, especially for large-scale or nonlinear systems.

• Distributed and Partition-Based MHE: For large-scale or networked systems, partitioning techniques (PMHE1/2/3, multiple window) reduce computational and communication burden by assigning local MHE problems to subsystems, treating interconnections as frozen disturbances or via iterative coupling updates (Farina et al., 2024, Al-Matouq et al., 2014). Stability is retained under mild coupling and observability conditions, and convergence rates closely approach centralized MHE.
• Time-Splitting and ALADIN/SQP Methods: By decomposing the full horizon into sub-windows and enforcing local consistency by ALADIN or inexact Newton/SQP steps, nonlinear MHE problems can be distributed across parallel subproblems. Closed-form updates and sensitivity-assisted refinements further improve scalability (Wu et al., 30 Mar 2025).
• Iterative Preconditioning and Fast First-Order Methods: Preconditioned gradient-descent techniques exploit the block-tridiagonal structure of the MHE cost, significantly lowering per-iteration complexity. Local linear convergence is established under convexity, with computational advantages over standard Newton-type solvers for large horizons or state dimensions (Liu et al., 2023).
• Differentiable Convex Optimization Layers: For systems with parametric uncertainty, the integration of differentiable convex MHE layers enables joint parameter and state estimation via stochastic gradient descent, leveraging automatic differentiation through KKT conditions (Muntwiler et al., 2021).

5. Extensions to Irregular Sampling, PDEs, and Special Sensors

MHE methodologies have also been extended to accommodate challenging real-world constraints:

• Irregular/Infrequent Sampling: The development of sample-based MHE addresses cases where measurements arrive at non-uniform or infrequent time points. Detection and RGES guarantees follow from sample-based i-IOSS properties, established by analyzing the observability of the underlying sampling pattern (Krauss et al., 28 Oct 2025).
• PDE Systems: For infinite-dimensional systems (e.g., hyperbolic/parabolic PDEs), explicit MHE realizations can be derived via backstepping techniques, leading to analytic moving-horizon estimators requiring only past measurements and inputs. This bypasses online PDE simulation yet inherits the stability properties of infinite-dimensional observers (Bhan et al., 2024).
• Binary and Limited Sensors: With only quantized or thresholded sensor information, piecewise quadratic and switching-instants-driven MHE formulations achieve bounded error and stability, provided suitable observability-from-switchings is obtained (Battistelli et al., 2016).

6. Practical Applications and Empirical Results

MHE has demonstrated effectiveness in a breadth of applications including:

• Bioprocess State Estimation: A Koopman-learner--based data-enabled MHE provided real-time accurate state estimation on a 65-state membrane bioreactor, achieving sub-1% RMSE and >35% error reduction over linear data-driven MHE (Xiaojie et al., 15 Feb 2026).
• SLAM and Robotics: Robust MHE with rigorous estimation error bounds for SLAM problems supports scalable ego-state and landmark estimation, with parallelizable update structure and strong noise robustness (Trisovic et al., 2024).
• High-Speed Quadrotor Flight: GP-augmented GP-MHE significantly improved angular and velocity RMSE under aerodynamic modeling uncertainty and heavy sensor noise (Choo et al., 2023).
• Target Tracking & Networked Systems: Variational Bayes MHE delivered substantial accuracy gains even under unknown time-varying covariances (Dong et al., 2021), while partition-based estimators enabled efficient estimation across large-scale interconnected networks (Farina et al., 2024).

Empirical studies consistently report that advanced (data-driven, learning-based, or robustified) MHE matches or surpasses model-based alternatives in estimation fidelity, stability, and computational tractability, even in regimes with incomplete models or partial observability. Parallel and distributed methods enable real-time deployment for problems far beyond the reach of classical MHE architectures.

7. Summary Table: Representative MHE Variants

Approach	Model Requirement	Problem Structure	Stability Guarantee	Reference
Standard model-based MHE	Explicit first-principles model	Convex/Nonlinear QP	RGES under detectability	(Al-Matouq et al., 2014)
Data-driven/Hankel-based MHE	Persistently exciting historic data	Convex QP (via Hankel)	RGES/pRES	(Wolff et al., 2022)
Koopman-lifted learning-based MHE	(Partial) state and input trajectories	Convex QP in lifted space	i-UEIOSS/pRES	(Xiaojie et al., 15 Feb 2026)
GP-based MHE	Dense historic data, GP models	Nonlinear NLP, time-varying	pRES, δ-IOSS	(Wolff et al., 2024)
Partition/Distributed MHE	Modular process model	Multiple local QPs	Local RGES under coupling	(Farina et al., 2024)
Regularized MHE (param uncertainty)	Mild Lyapunov/IOpS existence	Nonlinear NLP, augmented	δ-IOpS, no PE needed	(Muntwiler et al., 2023)
PDE/backstepping explicit MHE	Linear/PIDE model, measured boundaries	Analytic windowed operator	Exponential contraction	(Bhan et al., 2024)

These developments collectively place moving horizon estimation at the forefront of nonlinear, robust, and data-enabled state estimation methodologies, with proven capability in large-scale, high-dimensional, and model-uncertain environments.