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Moving horizon estimation

Moving horizon estimation (MHE) is an optimization-based method for state estimation in dynamic systems, where the states are inferred by solving a constrained optimization problem over a fixed-length receding horizon of past measurements, incorporating a nonlinear system model, noise statistics, and physical constraints on states and disturbances. Developed in the early 1990s as an extension of receding horizon principles from control to estimation, MHE addresses limitations of traditional filters like the extended Kalman filter (EKF) by directly handling nonlinear dynamics and constraints without linearization. The core formulation of MHE involves minimizing a cost function that penalizes deviations from the system model and measurement predictions, typically comprising a sum of stage costs for process and measurement noise over the horizon plus an arrival cost approximating the influence of prior data.^[1] This arrival cost, often derived from an EKF covariance or a fixed weighting, ensures computational tractability while maintaining estimation stability under conditions like uniform observability and proper horizon length selection. At each time step, the horizon slides forward, discarding the oldest measurement and incorporating the newest one, yielding updated state estimates that are robust to model uncertainties and outliers.^[1] Compared to the EKF, which relies on local linear approximations and Gaussian assumptions, MHE provides superior performance in nonlinear systems by globally optimizing trajectories and enforcing constraints, such as non-negativity in chemical concentrations, though it is computationally more intensive than the EKF; advances in algorithms and hardware have enabled real-time implementations on modern computing platforms as of 2025.^[2] Stability guarantees for MHE, including asymptotic convergence, depend on detectability of the system and tuning of the arrival cost to avoid over- or under-weighting historical information.^[1] MHE finds applications in chemical process control, such as reactor state estimation under disturbances; aerospace systems for trajectory tracking; and robotics for simultaneous localization and mapping with bounded noise. Recent extensions integrate neural networks for faster optimization or handle time-varying parameters in real-time scenarios, enhancing its utility in autonomous systems and networked control.

Introduction

Definition and Motivation

Moving horizon estimation (MHE) is an advanced state estimation technique that formulates the problem as a constrained optimization over a finite receding horizon of past measurements and inputs to determine the current system states.^[3] This approach solves an optimization problem at each time step, using a fixed-length window of recent data to compute estimates that account for system dynamics, measurement noise, and process disturbances.^[4] The primary motivation for MHE arises from the limitations of traditional linear estimators, such as the Kalman filter, which assume linear dynamics and Gaussian noise but often perform poorly in nonlinear systems or when constraints on states and inputs must be respected.^[4] MHE addresses these challenges by explicitly incorporating nonlinear models, inequality constraints, and an arrival cost term that approximates the influence of data prior to the estimation horizon, enabling robust performance in complex, constrained environments like chemical processes or autonomous systems.^[4] In operation, MHE employs an estimation window of length N, incorporating the most recent N measurements and inputs to jointly estimate the current state and past states within the horizon, while discarding the oldest data as the window advances at each sampling instant.^[3] This receding horizon structure ensures computational feasibility by limiting the optimization to a bounded timeframe, balancing accuracy with real-time implementation. For instance, in a vehicle tracking application, MHE can estimate the position using a series of noisy GPS measurements over a short horizon, while enforcing physical constraints such as bounded velocity to mitigate errors from sensor outliers and improve tracking reliability in urban environments.

Core Principles

Moving horizon estimation (MHE) operates through a receding horizon mechanism, wherein at each time step k, an optimization problem is solved over a fixed-length window of the most recent N measurements, spanning from time k-N+1 to k. This formulation allows the estimator to focus computational resources on a finite set of data while discarding older information, thereby maintaining a bounded problem size that does not grow indefinitely with time. Once the optimal state trajectory is obtained for this window, only the current state estimate at time k is retained for use in control or further estimation; the horizon then shifts forward by one step to incorporate the next measurement, repeating the process. This receding approach ensures that the estimator remains responsive to new data and evolving system dynamics.^[5] To account for information prior to the current horizon, MHE incorporates an arrival cost term in the optimization objective, which approximates the influence of historical data through a quadratic penalty on deviations from a prior state estimate at the beginning of the window. Typically formulated as \| \hat{x}_{k-N+1} - \bar{x}_{k-N+1} \|_P^2, where \bar{x}_{k-N+1} is the prior estimate and P is a weighting matrix reflecting uncertainty, this term enforces consistency between the current solution and past estimates, preventing bias accumulation over time. The arrival cost effectively summarizes the conditional probability distribution of the initial state given all previous measurements, often assuming a Gaussian form for tractability. MHE treats process and measurement noise as bounded disturbances rather than purely stochastic, enabling the inclusion of hard constraints on states, inputs, and noise terms within the optimization framework. This bounded noise model, such as |w_k| \leq \bar{w} for process noise w_k and |v_k| \leq \bar{v} for measurement noise v_k, allows the estimator to reject outliers and respect physical limits, such as x \in \mathcal{X} and u \in \mathcal{U}, which are enforced as inequality constraints in the problem. By solving a constrained optimization—often a quadratic program for linear systems—this approach provides robust estimates even in the presence of model mismatches or sensor faults.^[5] For efficient real-time implementation, MHE relies on initialization and warm-starting strategies, where the solution from the previous time step serves as the initial guess for the current optimization. Specifically, the optimal state and input trajectories from time k-1 are shifted forward and augmented with a terminal estimate, providing a feasible starting point that promotes rapid convergence, often within a few iterations. This warm-starting technique is crucial for maintaining computational feasibility in receding-horizon applications, as it leverages the similarity between consecutive problems to reduce solving time.^[6]

Historical Development

Origins in Optimization

Moving horizon estimation (MHE) emerged as the natural estimation counterpart to model predictive control (MPC), both relying on receding-horizon optimization principles to address dynamic systems in real time. MPC, which originated in the chemical engineering industry during the 1970s, employed finite-horizon optimization to predict and control process trajectories while respecting constraints, with early implementations like Dynamic Matrix Control at Shell Oil in 1973 and Model Algorithmic Control by Richalet et al. in 1978. This receding-horizon paradigm, where only the first step of an optimized solution is implemented before shifting the horizon forward, provided a framework for handling constraints and nonlinearities that traditional feedback controllers struggled with, laying the groundwork for similar optimization-based approaches in state estimation.^[7] Early influences on MHE drew from moving horizon control concepts in the 1970s and 1980s, initially developed for trajectory optimization in aerospace and process industries, where finite-time optimization windows were used to approximate infinite-horizon problems while incorporating constraints. These ideas, adapted from optimal control theory, shifted focus toward estimation by leveraging past measurements over a moving window to reconstruct states, particularly in scenarios with model uncertainties or bounded disturbances. By the late 1980s, researchers in chemical engineering began exploring horizon-based optimization specifically for state estimation, motivated by the need to enforce physical constraints on states and disturbances in process models, which classical methods like the Kalman filter could not directly accommodate. The initial formulation of MHE was proposed in the early 1990s as an optimization technique to handle constraints in state estimation for chemical processes, building directly on the least-squares frameworks used in MPC. A seminal contribution came from Robertson and Lee in 1995, who introduced a least-squares state estimation method using a fixed estimation window to balance computational feasibility with accuracy in dynamic systems. This was extended in 1996 by Robertson, Lee, and Rawlings, who formalized the moving horizon estimator as a constrained optimization problem over a receding window of measurements, enabling robust handling of nonlinearities and inequalities in industrial applications like reactor monitoring.^[8]^[9]

Key Advancements

The formulation of moving horizon estimation (MHE) saw major breakthroughs in the 1990s through the work of James B. Rawlings and collaborators, who introduced a general least-squares framework using a fixed-size estimation window to handle constraints on states and disturbances. This approach, detailed in early publications from 1995 to 2000, effectively integrated MHE with model predictive control (MPC) for nonlinear systems, enabling simultaneous state estimation and control in dynamic processes. Key contributions included the development of algorithms that solved optimization problems over a receding horizon, improving accuracy over traditional methods for constrained environments.^[10]^[9] Advancements in constraint handling emerged prominently in this period, with formulations that incorporated hard constraints directly into the optimization objective, ensuring feasible solutions even under physical bounds on system variables. These developments led to robust variants of MHE capable of managing model uncertainties and disturbances, enhancing reliability in real-time applications such as chemical processes. Stability proofs were established for these constrained estimators, demonstrating asymptotic stability under appropriate horizon lengths and weighting schemes. Theoretical guarantees on convergence and consistency were further solidified in early 2000s research, relying on detectability assumptions for the system dynamics to ensure the estimator tracks the true state over time. For instance, sufficient conditions for bounded and asymptotic stability were proven for nonlinear discrete-time systems, confirming that MHE approximations converge to optimal full-information estimates as the horizon extends. These results provided a rigorous foundation for practical deployment.^[11] By the mid-2000s, MHE had achieved standardization in process control software, particularly within advanced MPC frameworks used in industrial settings like gas chromatography and chemical reactors. Open-source implementations also began emerging, such as integrations in platforms like JModelica.org, which facilitated broader experimentation and customization in academic and engineering contexts.^[12]^[13]

Mathematical Formulation

System Model

Moving horizon estimation (MHE) relies on a discrete-time state-space representation of the underlying dynamic system to formulate the estimation problem. The standard model consists of a process equation that describes state evolution and a measurement equation that relates states to observations. Specifically, the state at time k+1 evolves according to

x_{k+1} = f(x_k, u_k, w_k),

where x_k \in \mathbb{R}^n is the state vector, u_k \in \mathbb{R}^m is the known input, w_k \in \mathbb{R}^n is the process noise, and f: \mathbb{R}^n \times \mathbb{R}^m \times \mathbb{R}^n \to \mathbb{R}^n is a nonlinear function. The measurement at time k is given by

y_k = h(x_k) + v_k,

where y_k \in \mathbb{R}^p is the observed output, v_k \in \mathbb{R}^p is the measurement noise, and h: \mathbb{R}^n \to \mathbb{R}^p is a nonlinear function.^[14]^[6] The functions f and h are typically assumed to be continuous, and often Lipschitz continuous with respect to their arguments to ensure well-posedness and facilitate numerical solutions. The process and measurement noises w_k and v_k are modeled as bounded disturbances residing in compact convex sets \mathcal{W} and \mathcal{V} containing the origin in their interior, respectively; this boundedness assumption supports robust estimation under uncertainty. Alternatively, in stochastic formulations, the noises are zero-mean, uncorrelated random variables with known positive definite covariance matrices, such as Gaussian distributions w_k \sim \mathcal{N}(0, Q) and v_k \sim \mathcal{N}(0, R).^[14]^[6] In the full-information model underlying MHE, the estimation incorporates all available past inputs u_{k-N+1}, \dots, u_{k-1} and measurements y_{k-N+1}, \dots, y_k over a fixed-length horizon of N time steps, along with the current measurement y_k, to reconstruct the state trajectory within the window. This contrasts with recursive filters by directly using the horizon data to account for model mismatch and constraints without relying on prior estimates outside the window.^[14]^[6] For systems with unknown parameters, the state-space model can be extended to enable joint state and parameter estimation by augmenting the state vector with the parameters, treating them as constant or slowly varying. If \theta \in \mathbb{R}^q denotes the parameter vector, the augmented dynamics become

\begin{bmatrix} x_{k+1} \\ \theta_{k+1} \end{bmatrix} = \begin{bmatrix} f(x_k, u_k, w_k; \theta_k) \\ \theta_k + w_{\theta,k} \end{bmatrix},

where w_{\theta,k} models parameter uncertainty, often set to zero for constant parameters; the measurement equation is similarly adjusted to depend on \theta_k. This augmentation allows MHE to estimate both states and parameters over the horizon by optimizing over the extended variables.^[15]

Optimization Objective

The optimization objective in moving horizon estimation (MHE) centers on solving a constrained nonlinear program at each sampling instant to reconstruct the most likely state and disturbance trajectory over a receding horizon of length N, given recent measurements and known inputs. This approach explicitly accounts for system constraints and noise statistics, distinguishing it from approximate filters like the Kalman filter. The problem formulation leverages the system model described earlier, where states evolve according to nonlinear dynamics and outputs are corrupted by additive noise. The core objective function minimizes the weighted sum of measurement residuals and process noise penalties over the horizon, augmented by an arrival cost term that encodes information from measurements prior to the horizon. Define the horizon times as t_i = k - N + 1 + i for i = 0, \dots, N-1, so t_0 = k - N + 1 and t_{N-1} = k: \begin{align*} \min_{{x_{t_i|k}}{i=0}^{N-1}, {w{t_i|k}}{i=0}^{N-2}} &\quad |x{t_0|k} - \bar{x}{t_0|k}|{P_k^{-1}}^2 + \sum_{i=0}^{N-1} |y_{t_i} - h(x_{t_i|k})|{R^{-1}}^2 + \sum{i=0}^{N-2} |w_{t_i|k}|_{Q^{-1}}^2, \end{align*} subject to the dynamic constraints x_{t_{i+1}|k} = f(x_{t_i|k}, u_{t_i}, w_{t_i|k}) for i = 0, \dots, N-2, process noise bounds w_{t_i|k} \in \mathcal{W}, and state bounds x_{t_i|k} \in \mathcal{X}. Here, R > 0 and Q > 0 are the measurement and process noise covariance matrices, respectively; y_{t_i} are the observed outputs; and u_{t_i} are the known inputs. In simplified cases assuming identity covariances, the weights reduce to squared Euclidean norms.^[11]^[14]^[6] The arrival cost \|x_{t_0|k} - \bar{x}_{t_0|k}\|_{P_k^{-1}}^2 approximates the cost of all measurements before time k - N + 1, using a quadratic form where \bar{x}_{t_0|k} is the prior state estimate (often from a steady-state or propagated value) and P_k > 0 is the weighting matrix derived from the prior covariance, typically updated via Riccati recursion or filtering approximations to ensure consistency with the noise model. This term prevents discarding historical data and promotes stability by penalizing deviations from plausible initial conditions.^[11]^[3] The horizon length N critically influences performance, presenting a trade-off between estimation accuracy and computational demand: larger N enhances robustness by incorporating more data to resolve ambiguities in noisy or constrained systems, but it expands the optimization problem size, often scaling cubically with N in dense implementations, which can challenge real-time feasibility. Sufficiently large N (typically exceeding the system observability index) is required for convergence, while shorter horizons suffice for mildly nonlinear systems but risk sensitivity to recent disturbances.^[14]^[6]

Numerical Implementation

Solution Algorithms

Moving horizon estimation (MHE) typically involves solving a nonlinear optimization problem at each time step to estimate system states over a receding horizon. For general nonlinear cases, nonlinear programming (NLP) solvers are employed, with interior-point methods and sequential quadratic programming (SQP) being prominent approaches. Interior-point methods, such as those implemented in IPOPT, solve the smoothed Karush-Kuhn-Tucker (KKT) conditions using Newton's method, effectively handling constraints through barrier functions and demonstrating robustness for medium-scale NLPs arising in MHE.^[16] SQP methods, including variants like full-step SQP or those using BFGS Hessian approximations, iteratively linearize the nonlinear constraints and objective around the current iterate to form a quadratic subproblem, offering reliable convergence for constrained MHE formulations.^[16] In mildly nonlinear systems, linearized variants of MHE accelerate computation by approximating the nonlinear dynamics, akin to extended Kalman filter techniques. These approaches, such as successive linearization within SQP iterations or Carleman linearization, transform the problem into a sequence of linear or quadratic programs, reducing the need for full nonlinear solves while preserving constraint handling.^[17] For instance, Carleman linearization embeds the nonlinear model into a higher-dimensional linear system, enabling faster QP-based solutions suitable for real-time estimation in processes with moderate nonlinearity.^[17] MHE solvers operate in either batch or online modes to balance accuracy and computational speed. Batch solving involves full re-optimization of the NLP at each horizon shift, ensuring convergence to a stationary point but potentially exceeding real-time deadlines for large horizons.^[16] In contrast, online solving performs partial updates, such as one iteration of an SQP or interior-point method per time step using warm-starting from prior solutions, which approximates the optimum efficiently for moving horizons.^[16] Software tools facilitate the implementation of these algorithms, with open-source frameworks like CasADi providing symbolic modeling and interfacing with NLP solvers such as IPOPT for MHE problems.^[18] CasADi supports automatic differentiation and structured solvers, enabling efficient handling of the dynamic optimization in MHE, while IPOPT serves as a core engine for interior-point-based solutions in various toolboxes.

Computational Strategies

To enable real-time implementation of moving horizon estimation (MHE), various computational strategies have been developed to mitigate the high demands of solving large-scale optimization problems at each time step. These approaches focus on reducing iteration counts, adapting problem size to system needs, leveraging distributed computing, and analyzing parameter impacts to ensure efficient convergence without sacrificing estimation accuracy. Such techniques are particularly vital for constrained nonlinear systems where standard solvers may falter under tight sampling rates. Warm-starting is a key strategy that initializes the optimization solver with an approximate solution derived from the previous estimation horizon, typically by shifting the prior optimal trajectory forward and appending a terminal guess. This reduces the number of iterations required for convergence, often achieving near-optimal solutions in fewer steps compared to cold starts. For instance, in nonlinear programming-based MHE, sensitivity analysis of the prior solution provides a high-quality warm start, significantly reducing computation time in process control applications.^[16] Horizon length tuning involves adaptively selecting the estimation window size N based on system dynamics, trading off between estimation precision and computational load. Shorter horizons accelerate solving but may degrade accuracy for slowly varying states, while longer ones enhance robustness at the cost of increased complexity. Adaptive schemes dynamically adjust N by monitoring objective function values or estimation residuals, ensuring feasibility in resource-constrained environments like embedded systems.^[19] Parallelization and decomposition exploit the structure of MHE problems for large-scale systems, distributing computations across multiple processors or nodes. Techniques such as operator splitting enable parallel solution of subproblems in distributed sensor networks, while decomposition methods partition the horizon into manageable blocks solved concurrently. This approach has demonstrated speedups in nonlinear state estimation for power systems and robotics, maintaining stability guarantees.^[20] Sensitivity analysis evaluates how tuning parameters, such as weighting matrices in the objective function, influence convergence time and estimation performance. By computing parametric derivatives, analysts identify robust parameter sets that minimize sensitivity to perturbations, often reducing solve times by optimizing trade-offs between arrival cost and stage costs. In practice, this has been applied to refine MHE for multi-rate measurements, ensuring real-time viability without exhaustive retuning. Recent numerical enhancements, as of 2024, include Gaussian process-based formulations for unknown nonlinear systems and physics-informed online learning to improve efficiency in dynamic optimization.^[21]^[22]

Comparisons with Other Methods

Versus Extended Kalman Filter

Moving horizon estimation (MHE) and the extended Kalman filter (EKF) both address state estimation in nonlinear systems, but they differ fundamentally in their approaches to handling nonlinearity. The EKF relies on local linear approximations of the nonlinear dynamics and measurement models at each time step, propagating the state estimate and covariance through these linearizations via recursive updates. In contrast, MHE formulates state estimation as a global nonlinear optimization problem over a receding horizon of past measurements, directly solving for the optimal state trajectory without relying on successive linearizations. This optimization-based strategy allows MHE to capture the full nonlinear behavior more accurately, particularly in systems with strong nonlinearities where EKF linearizations can lead to significant errors or divergence. A key distinction lies in their treatment of constraints. MHE explicitly incorporates state and input constraints into the optimization objective, ensuring that estimates remain physically feasible even under bounded disturbances or model uncertainties. The EKF, however, does not natively handle constraints; attempts to enforce them often require ad hoc modifications like clipping the estimates, which can degrade performance and introduce inconsistencies in the covariance propagation. MHE's batch processing over a horizon of multiple past measurements contrasts with the EKF's recursive, single-step update, providing inherent robustness to measurement outliers by distributing their impact across the optimization rather than amplifying them in sequential corrections.^[23] This multi-step horizon enables MHE to weigh historical data more effectively, reducing sensitivity to isolated erroneous readings that can destabilize the EKF.^[23] In terms of performance, MHE generally outperforms the EKF in constrained nonlinear scenarios, yielding lower estimation errors at the expense of higher computational demand due to repeated optimization solves. For instance, in simulations of a continuous stirred-tank reactor (CSTR) with nonlinear dynamics and state constraints, MHE demonstrated superior convergence and accuracy under disturbances compared to a constrained EKF, avoiding divergence and unphysical states.^[24] While the EKF is computationally efficient for real-time applications with mild nonlinearities, MHE's advantages become pronounced in complex processes where constraints and outliers are prevalent.

Versus Particle Filter

Moving horizon estimation (MHE) and particle filters represent two distinct paradigms for state estimation in nonlinear systems, with MHE relying on deterministic optimization to compute point estimates and particle filters employing stochastic Monte Carlo sampling to approximate the full posterior distribution. In MHE, the estimation problem is formulated as a constrained optimization over a receding horizon of past measurements, yielding a maximum a posteriori (MAP) estimate under Gaussian noise assumptions, which provides a single, optimal point trajectory for the states.^[25] Conversely, particle filters generate a set of weighted samples (particles) that evolve according to the system dynamics and measurement likelihood, offering a nonparametric representation of the state probability density without assuming a specific form like Gaussianity.^[25] This fundamental difference makes MHE more akin to recursive least-squares methods, while particle filters directly address Bayesian inference through sequential importance sampling and resampling.^[25] Particle filters excel in handling highly multimodal posterior distributions, a common challenge in nonlinear systems with non-Gaussian noise or multiple possible state trajectories, as their sampling approach naturally captures multiple modes without requiring convexity.^[25] MHE, however, typically assumes unimodal posteriors and relies on convex optimization for efficiency; in multimodal cases, it may converge to local optima or require heuristic initializations and smoothing techniques to approximate multiple modes, potentially leading to suboptimal estimates if the arrival cost does not adequately summarize prior information.^[25] For instance, in systems with bearing-only tracking or cluttered environments, particle filters maintain diversity in samples to represent ambiguity, whereas MHE's optimization might overlook low-probability modes unless the horizon is extended or constraints are carefully tuned.^[25] Computationally, MHE's complexity scales primarily with the horizon length and the dimensionality of the optimization problem, involving repeated solves of nonlinear programs that can be mitigated by warm-starting but remain demanding for long horizons or high-dimensional states.^[25] Particle filters, in contrast, scale with the number of particles required for accurate approximation, often necessitating thousands or more in high dimensions to avoid degeneracy, though each iteration involves simpler propagation and weighting steps that parallelize well.^[25] In practice, particle filters may demand more particles for comparable accuracy in smooth, low-noise scenarios where MHE's optimization leverages structural assumptions, but MHE incurs higher per-iteration costs due to solver overhead.^[25] Hybrid approaches combining MHE and particle filters have shown promise for leveraging the strengths of both, such as using particle filters to generate diverse initial guesses or arrival costs for MHE optimization, thereby improving robustness in uncertain, multimodal environments while retaining MHE's ability to enforce constraints and compute smoothed estimates.^[25] These integrations, explored in early works, enable particle-based exploration of the state space followed by MHE refinement, reducing the particle count needed and enhancing overall estimation quality in nonlinear, constrained systems.^[25]

Applications

Process Industries

In process industries, moving horizon estimation (MHE) is widely applied to enhance state estimation in complex, constrained systems such as chemical reactors, where it estimates critical variables like reactant concentrations and temperatures while enforcing safety constraints to prevent operational violations.^[26] This approach leverages an optimization framework over a receding horizon to incorporate process models, measurements, and bounds, improving accuracy in the presence of nonlinear dynamics and noise.^[27] A representative case study involves a continuous stirred-tank reactor (CSTR) with exothermic reactions, where MHE estimates concentrations of species A, B, and C (c_A, c_B, c_C) and temperature by solving a nonlinear optimization problem based on mass and energy balances, subject to non-negativity constraints on states to avoid unphysical estimates.^[26] In simulations with measurement noise, MHE achieves swift convergence to true states and reduces estimation variance compared to the extended Kalman filter (EKF), with steady-state errors for c_A of -0.0122 (versus 0.0224 for EKF) and overall lower root-mean-square errors, demonstrating its robustness for safety-critical monitoring.^[26] In polymer production, MHE addresses challenges from nonlinear kinetics and sensor delays in gas-phase polymerization reactors, estimating states such as monomer concentrations, hydrogen mole fraction, and production rates to maintain product quality.^[28] For an industrial Unipol process licensed by Univation Technologies, MHE uses a horizon of 50 minutes (updated every 5 minutes) on a detailed kinetic model with 46,870 variables, adjusting for delays in composition measurements and yielding significant improvements in tracking key variables like hydrogen concentration. This enables offset-free control and disturbance rejection.^[28] For energy systems, MHE facilitates state estimation in distillation columns and power plant components, integrating seamlessly with nonlinear model predictive control (NMPC) to handle large-scale nonlinearities and model-plant mismatches.^[27] In a methanol-propanol distillation column with 40 trays, advanced multi-step MHE estimates mole fractions and liquid holdups with standard errors below 0.1 and online computation times under 0.5 seconds, supporting real-time setpoint tracking in refining processes.^[27] Similarly, in coal-fired power plants, MHE-based soft sensors estimate unmeasurable variables like pulverized coal flow in the fuel preparation system, outperforming EKF with root-mean-square errors of 2.8493 versus 6.0084, while incorporating constraints to mitigate measurement uncertainties.^[29] These applications draw on the optimization objective outlined in the Mathematical Formulation section to ensure constraint satisfaction and predictive accuracy.^[29]

Robotics and Control Systems

In robotics and control systems, moving horizon estimation (MHE) plays a crucial role in enabling precise state estimation for dynamic, high-speed operations, particularly in environments requiring real-time decision-making and constraint handling. For autonomous vehicles, MHE facilitates multi-sensor fusion to estimate position and velocity by solving nonlinear optimization problems over a receding horizon, incorporating kinematic constraints such as limited lateral motion to ensure feasible trajectories. This approach outperforms unscented Kalman filters (UKF) in accuracy, achieving up to 33% lower translation errors (0.059% vs. 0.089%) and 38% better orientation estimates on long-term datasets spanning 45 km, while robustly handling sensor outliers via Mahalanobis distance-based rejection. By integrating data from IMUs, GPS, and LiDAR, MHE supports obstacle-aware estimation, where proximity constraints prevent unrealistic states near detected obstacles, enhancing safety in urban navigation scenarios.^[30] For robotic manipulators, MHE addresses joint state estimation in systems with nonlinear dynamics, such as soft continuum robots actuated by fluidic systems, by simultaneously estimating states and time-varying parameters through constrained optimization. This method incorporates actuator limits as inequality constraints in the cost function, mitigating underactuation and external disturbances without relying on probabilistic noise models, thus providing robust performance in planar and spatial motions. Numerical simulations demonstrate coherent estimation of joint configurations and structural parameters, even under uncertainty, enabling precise control for tasks like object manipulation where traditional filters fail due to model nonlinearities.^[31] In drone localization, particularly for indoor navigation, MHE leverages ultra-wideband (UWB) measurements to achieve high-precision positioning on resource-constrained platforms like small quadrotors. Recent advancements optimize MHE with dynamic step sizes and switching outlier rejection, reducing computational demands to about 25% of available resources on devices such as the Crazyflie 2.1, while maintaining 100% success rates with 5-6 time-of-flight beacons. Compared to extended Kalman filters (EKF), this yields superior accuracy in time-difference-of-arrival setups, with errors below 10-30 cm, supporting agile maneuvers in GPS-denied environments as demonstrated in studies up to 2021.^[32] A representative example of MHE's efficacy in robotics is its integration with simultaneous localization and mapping (SLAM), where it decouples ego-state estimation from landmark updates to improve accuracy over EKF in feature-sparse settings. By optimizing over a horizon with provable robust global exponential stability for states and bounded landmark errors (exponentially decaying with visibility index), MHE achieves lower position errors in simulations, enhancing robot navigation in unknown terrains without the consistency issues plaguing EKF at scale. This formulation briefly leverages efficient computational strategies, such as parallel updates, to meet real-time requirements.^[33]

Recent Advances

Integration with Machine Learning

Neural moving horizon estimation (NMHE) represents a key fusion of machine learning with traditional MHE, where neural networks (NNs) are integrated to enhance state estimation in nonlinear systems by leveraging data-driven learning alongside optimization-based principles. In NMHE, NNs are employed to learn system dynamics models, adapt cost function parameters, or approximate the underlying optimization process, thereby addressing limitations in model accuracy and computational efficiency. A systematic review identifies three primary architectures: NNs improving system model fidelity (Category I), NNs tuning cost functions for adaptability (Category II), and NNs serving as surrogates for the MHE solver to accelerate inference (Category III). Common NN types include multi-layer perceptrons (MLPs), long short-term memory (LSTM) networks, and input convex neural networks (ICNNs), with MLPs being the most prevalent due to their simplicity and effectiveness in capturing nonlinearities.^[34] Physics-informed variants of NMHE incorporate physical constraints directly into NN-based estimators, enabling online learning while respecting domain knowledge such as state bounds and inequality constraints. For instance, in gray-box models, moving horizon estimation is used to jointly optimize states and recurrent neural network (RNN) parameters, embedding constraints like x_{j,\min} \leq \hat{x}_j \leq x_{j,\max} to ensure physically plausible trajectories during adaptation to time-varying dynamics. Another approach employs physics-informed machine learning via Koopman operators to self-tune MHE for nonlinear systems, where NN approximations of the Koopman model are refined online within the horizon optimization, improving robustness to model uncertainties. These methods, developed in the 2020s, facilitate simultaneous state estimation and parameter learning in constrained environments.^[35] Learning-based arrival costs further integrate machine learning into MHE by using data-driven techniques to approximate and tune the prior state covariance, which summarizes information outside the estimation horizon and enhances long-horizon approximations. Neural networks or reinforcement learning policies, such as deterministic policy gradient, parameterize the arrival cost as a quadratic form \| \hat{s}_{k-N} - \tilde{s}_{k-N} \|_{P_{k-N}}^2 with adaptive matrices derived from historical data, compensating for imperfect models in real-time. This approach improves estimation accuracy in scenarios with model mismatch, as validated in simulations of linear systems and chemical reactors.^[36] The integration yields significant benefits, including reduced computational load—such as 98% faster inference compared to standard MHE—and enhanced accuracy with minimal data requirements. In soft robotics applications, NMHE using type-2 fuzzy NNs estimates external forces in bilateral telerobotic manipulators, improving force compliance and human perception by adapting to interaction dynamics with fewer sensors. For parameter estimation, NMHE has been applied to lithium-ion battery state-of-charge tracking, achieving low mean squared error through NN-approximated optimizations.^[34]^[34]

Emerging Applications

In recent years, moving horizon estimation (MHE) has found novel applications in domains requiring robust handling of complex dynamics and uncertainties, extending beyond established industrial and rigid robotic uses. These emerging uses leverage MHE's ability to incorporate constraints and past measurements for accurate state estimation in real-time, challenging environments.^[31] One promising area is soft continuum robotics, where MHE enables simultaneous state and parameter estimation for deformable structures actuated by fluidic systems. In such robots, traditional estimation methods struggle with underactuation, material uncertainties, and external disturbances due to their compliant, infinite-degree-of-freedom nature. A 2024 formulation addresses this by integrating equality constraints into the MHE optimization, akin to tube-model predictive control, to coherently estimate both time-varying states (e.g., configuration and velocity) and structural parameters (e.g., bending stiffness) without relying on probabilistic priors. Numerical simulations of planar motion demonstrate its robustness, achieving accurate trajectory tracking and parameter identification even under noisy measurements and actuation variability, with estimation errors reduced by up to 50% compared to extended Kalman filters in underactuated scenarios. This approach paves the way for safer, more adaptive control of soft manipulators in medical or exploration tasks.^[31] Advancements in multi-sensor fusion have also propelled MHE into enhanced drone localization, particularly integrating ultra-wideband (UWB) ranging with inertial measurement units (IMUs) for indoor navigation. Small unmanned aerial vehicles (UAVs) face computational constraints that limit traditional MHE deployment, but a 2025 improvement introduces dynamic gradient descent step sizes (via linesearch and Newton's method) and switching variables for outlier rejection, enabling real-time on-board processing. This method fuses UWB time-of-flight measurements from multiple anchors with IMU data to estimate 3D position and velocity, outperforming extended Kalman filters in simulations with 100% success rates using 5-6 beacons and reducing root-mean-square errors by 20-30% in dynamic flights. Experimental validation on a Crazyflie 2.1 nano-drone confirms its efficacy, using only 25% of available computational resources while rejecting up to 40% of outlier measurements from multipath effects, thus supporting precise swarming or inspection in GPS-denied spaces.^[37] In biomedical systems, MHE supports real-time physiological modeling through wearable devices by estimating human kinematics and muscle forces from limited sensor data. Wearables equipped with electromyography (EMG) and motion markers benefit from MHE's optimization framework, which resolves ill-posed inverse dynamics problems while enforcing biomechanical constraints. This enables personalized modeling of gait or rehabilitation movements, integrating noisy EMG signals to predict forces with 15-25% improved accuracy over static optimization methods, and holds potential for closed-loop prosthetics or remote health monitoring. Finally, simultaneous MHE and model predictive control (MPC) has emerged for integrated estimation-control in nonlinear systems, addressing bounded disturbances without separate observer loops. A 2023 framework poses a single optimization problem over a shared horizon to jointly estimate states from partial outputs and compute control inputs, ensuring input-to-state stability under persistent noise. Applied to chemical reactors and mechanical oscillators, it reduces mean squared estimation errors by 40% and stabilizes trajectories faster than cascaded MHE-MPC schemes, with recursive feasibility guaranteed via terminal ingredients. This unified strategy minimizes conservatism in safety-critical nonlinear applications, such as autonomous vehicles or energy systems.^[38]

References

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[PDF] CDS 270-2: Lecture 4-2 Moving Horizon Estimation
Apr 19, 2006 · Goals: • To learn how extended Kalman filters (EKF) and moving horizon estimation (MHE) works. • To learn sufficient conditions for ...
[2]
Constrained linear state estimation—a moving horizon approach
Moving horizon estimation (MHE) uses a fixed-size window to bound the problem size in constrained linear state estimation, reformulating it as a quadratic ...Missing: seminal | Show results with:seminal
[3]
Nonlinear Predictive Control and Moving Horizon Estimation
An Introductory Overview. In: Frank, P.M. (eds) Advances in Control. Springer, London. https ...
[4]
https://link.springer.com/chapter/10.1007/978-1-4471-0853-5_19
[5]
[PDF] A Fast Moving Horizon Estimation Algorithm Based on Nonlinear ...
Moving Horizon Estimation (MHE) is an efficient optimization-based strategy for state estima- tion. Despite the attractiveness of this method, its application ...
[6]
[PDF] A survey of industrial model predictive control technology - CEPAC
A brief history of industrial MPC technology is presented first, followed by results of our vendor survey of MPC control and identification technology. A ...
[7]
A least squares formulation for state estimation - ScienceDirect.com
The moving horizon estimator has one more tuning parameter (namely, the horizon size) than many well-known recursive filters. This parameter allows for a ...<|control11|><|separator|>
[8]
A moving horizon‐based approach for least‐squares estimation
A general formulation of the moving horizon estimator is presented. An algorithm with a fixed-size estimation window and constraints on states, disturbances ...
[9]
A least squares formulation for state estimation - ScienceDirect.com
A least squares formulation for state estimation. Author links open overlay ... Lee, J.B. Rawlings. AIChE J. (1994). submitted for publication. 12. J ...
[10]
Constrained state estimation for nonlinear discrete-time systems
One strategy for constrained state estimation is to employ online optimization using a moving horizon approximation. We propose a general theory for constrained ...
[11]
MOVING HORIZON ESTIMATION FOR AN INDUSTRIAL GAS ...
Moving horizon estimation, preceding the nonlinear control optimization, provides an estimate of the current states and unmeasured disturbances. MHE is compared ...Missing: adoption | Show results with:adoption
[12]
[PDF] Moving Horizon Estimation for JModelica.org - CORE
In this thesis a Moving Horizon Estimator (MHE) has been implemented for the. JModelica.org platform. JModelica.org is an open-source software platform for ...
[13]
[PDF] Model predictive control: Theory, computation and design
This chapter gives an introduction into methods for the numerical so- lution of the MPC optimization problem. Numerical optimal control builds on two ®elds: ...
[14]
Nonlinear Moving Horizon State Estimation - SpringerLink
About this chapter. Cite this chapter. Muske, K.R., Rawlings, J.B. (1995). Nonlinear Moving Horizon State Estimation. In: Berber, R. (eds) Methods of Model ...
[15]
A real-time algorithm for moving horizon state and parameter ...
Jan 10, 2011 · A moving horizon estimation (MHE) approach to simultaneously estimate states and parameters is revisited. Two different noise models are considered.
[16]
[PDF] Efficient Numerical Methods for Nonlinear MPC and Moving Horizon ...
This overview paper reviews numerical methods for solution of optimal con- trol problems in real-time, as they arise in nonlinear model predictive control.
[17]
(PDF) Fast Moving Horizon Estimation of nonlinear processes via ...
Aug 7, 2025 · Moving Horizon Estimation (MHE) is an optimization-based technique to estimate the unmeasurable state variables of a nonlinear dynamic system ...<|control11|><|separator|>
[18]
CasADi
yet efficient — implementation of different ...Docs · Gets · Source · Support
[19]
Adaptive horizon size moving horizon estimation with unknown ...
Aug 21, 2024 · In this paper, we propose a novel adaptive horizon size MHE strategy that dynamically adjusts the horizon size based on the value of the objective function.Missing: length | Show results with:length
[20]
Moving-horizon estimation with guaranteed robustness for discrete ...
Robustness is guaranteed with sufficiently large outliers. The effectiveness of the proposed method as compared with the Kalman filter is shown by means of a ...
[21]
https://arxiv.org/abs/2402.04665
[22]
[PDF] A Critical Evaluation of Extended Kalman Filtering and Moving ...
In this paper, we first outline the basics of nonlinear observability, extended Kalman filtering, and moving horizon estimation.
[23]
Advanced-multi-step moving horizon estimation for large-scale ...
In particular, moving horizon estimation (MHE) solves finite-horizon optimization problems to estimate states and parameters [9], [10]. Similar to nonlinear ...
[24]
[PDF] moving horizon estimation for an industrial gas phase ...
Moving horizon estimation, preceding the nonlinear control optimization, provides an estimate of the current states and unmeasured disturbances.
[25]
Nonlinear Modeling and Inferential Multi-Model Predictive Control of ...
This paper proposes an inferential multi-model predictive control scheme based on moving horizon estimation for the fuel preparation system in coal-fired power ...<|separator|>
[26]
https://sites.engineering.ucsb.edu/~jbraw/jbrweb-archives/tech-reports/twmcc-2002-03.pdf
[27]
https://www.sciencedirect.com/science/article/pii/S0959152422001123
[28]
Neural Moving Horizon Estimation: A Systematic Literature Review
Jun 21, 2024 · Title:Neural Moving Horizon Estimation: A Systematic Literature Review ... Abstract:The neural moving horizon estimator (NMHE) is a relatively new ...
[29]
[PDF] Physics-Informed Online Learning by Moving Horizon Estimation
Aug 24, 2024 · A good approximation of the full-information estimation can be achieved by a suitable selection of the arrival cost (Rawlings et al., 2017).
[30]
https://doi.org/10.1177/01423312211011454
[31]
https://asmedigitalcollection.asme.org/computationalnonlinear/article/19/8/081004/1200697/A-Moving-Horizon-Estimation-for-a-Class-of-Soft
[32]
A Moving Horizon Estimation for a Class of Soft Continuum ...
Jun 18, 2024 · The MHE method is a special case of full information estimation that considers not just the finite horizon but all measurements available.
[33]
Improved moving horizon estimation for ultra-wideband localization ...
Moving Horizon Estimation (MHE) offers multiple advantages over Kalman Filters when it comes to the localization of drones.
[34]
Simultaneous moving horizon estimation and control for nonlinear ...
May 28, 2023 · In this work, we address the output–feedback control problem for nonlinear systems under bounded disturbances using a moving horizon ...