⚡ electrical engineering

Robust stability of event-triggered nonlinear moving horizon estimation

This paper proposes a robustly stable event-triggered moving horizon estimation scheme for remote state estimation of general nonlinear systems, featuring a novel triggering rule that ensures global exponential stability and a variable horizon length to achieve tighter error bounds.

Original authors: Isabelle Krauss, Victor G. Lopez, Matthias A. Müller

Published 2026-08-21

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Isabelle Krauss, Victor G. Lopez, Matthias A. Müller

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the world of modern engineering, machines and systems are rarely isolated; they are part of vast, interconnected networks where sensors constantly monitor conditions and controllers make split-second decisions. To keep these systems running safely and efficiently, a computer must know exactly what the system is doing at any given moment. This is the job of state estimation: the process of figuring out the hidden internal conditions of a machine, like the pressure inside a chemical tank or the angle of a robot's arm, based on noisy and imperfect measurements. For complex machines that do not follow simple, straight-line rules, engineers often use a powerful technique called moving horizon estimation. This method works like a sliding window, looking back at a short history of past data to calculate the most likely current state, constantly updating its guess as new information arrives. However, in real-world applications, these systems often run on battery power or share limited communication bandwidth. Sending a constant stream of data to a central computer is expensive and wasteful, especially when the system is behaving predictably and no new information is needed. The challenge, then, is to design a system that only speaks up when it truly has something important to say, without losing the ability to track the machine's state accurately.

Researchers at Leibniz University Hannover have developed a new method to solve this problem for complex, non-linear systems. They created a system that combines the powerful tracking abilities of moving horizon estimation with a smart "event-triggering" rule. In their approach, the sensor does not send data continuously. Instead, it constantly checks a specific condition to decide if a transmission is necessary. If the system is behaving as expected, the sensor stays silent, and the remote computer simply predicts the next state based on the last known information. Only when the system deviates enough to trigger an "event" does the sensor send a single, fresh measurement. Upon receiving this single data point, the remote computer solves a complex optimization problem to update its entire view of the system's state. This is a significant shift from previous methods, which often required sending a long sequence of past measurements every time an event occurred. By sending just one measurement, the new method drastically reduces the amount of data that needs to be transmitted, saving energy and bandwidth while avoiding the risk of large data packets failing to arrive.

The team proved mathematically that this approach is robustly stable, meaning the estimation error remains small and predictable even when the system is disturbed by noise or external factors. They demonstrated that as long as the system has certain inherent properties that make it observable, the error will shrink over time, converging toward zero once the disturbances fade. To further improve performance, they also explored a variation where the "window" of time the computer looks back at is not fixed but can change length depending on how many events have recently occurred. This flexible approach allowed them to achieve even tighter bounds on the estimation error, though it required slightly more computational effort. The researchers tested their ideas on two distinct examples: a chemical batch reactor, a standard benchmark in the field where simpler methods often fail, and a two-link robot arm moving in a plane. In these simulations, the new method successfully tracked the true state of the systems with high accuracy.

The results showed a clear trade-off controlled by a design parameter that determines how sensitive the system is to changes. When the system was set to be very sensitive, it triggered events frequently, sending many measurements and achieving very low error. When set to be less sensitive, it sent far fewer messages, conserving resources while still maintaining a reliable estimate. Crucially, when compared to older methods that sent sequences of data, the new approach transmitted significantly fewer total measurements. In one test with the chemical reactor, the older method sent sequences of up to 32 measurements per event, whereas the new method sent exactly one measurement per event, even though it triggered slightly more events overall. This highlights a key finding: for applications where the total volume of data is the limiting factor, sending single, timely updates is far more efficient than sending large batches of data less frequently. The researchers also found that the extra mathematical constraints they added to ensure stability had almost no negative impact on the actual performance, confirming that the theoretical safeguards were practical and non-restrictive.

While the method with a fixed window size is simpler to implement and computationally lighter, the version with the varying window length offered a measurable improvement in accuracy, reducing the average estimation error by about three percent in the robot arm simulation. This suggests that for applications where every fraction of a percent of accuracy matters, the extra computational cost of adjusting the window size is a worthwhile investment. The study concludes that this event-triggered framework provides a reliable, efficient way to monitor complex systems remotely. It balances the need for accurate state information with the practical constraints of limited energy and communication bandwidth, offering a solution that is both theoretically sound and practically effective for the next generation of networked control systems.

Technical Summary: Robust Stability of Event-Triggered Nonlinear Moving Horizon Estimation

Problem Statement
The paper addresses the challenge of remote state estimation for general nonlinear systems subject to process disturbances and measurement noise. In networked control systems, resources such as communication bandwidth, energy, and computational power are often limited. While Moving Horizon Estimation (MHE) offers robust stability guarantees for nonlinear systems under mild detectability conditions (specifically incremental input/output-to-state stability, or i-IOSS), standard implementations typically require frequent data transmission and high computational loads. Previous event-triggered MHE (ET-MHE) schemes for nonlinear systems either required transmitting sequences of measurements upon an event (increasing packet size and failure risk) or focused on distributed systems rather than remote plant-to-estimator communication. The specific problem addressed is designing an ET-MHE scheme that minimizes both the frequency of channel access and the volume of data transmitted (sending only the most recent measurement) while maintaining robust global exponential stability (RGES).

Methodology
The authors propose a novel ET-MHE scheme where the optimization problem is solved only when a specific event is triggered. The core components of the methodology include:

Event-Triggering Mechanism (ETM): A binary variable $\gamma_t$ determines transmission. If $\gamma_t = 1$ , a single measurement $y_{t-1}$ is sent to the remote estimator. If $\gamma_t = 0$ , no data is transmitted, and the estimator updates the state using an open-loop prediction based on system dynamics.
Optimization Problem: The MHE formulation minimizes a cost function over a horizon window. The cost function includes a prior weighting term and a stage cost penalizing estimated disturbances and output mismatches. Crucially, the cost function is parameterized consistent with the exponential i-IOSS Lyapunov function to facilitate stability proofs.
Handling Missing Data: When no event occurs, the estimator relies on open-loop predictions. To ensure stability, the authors introduce an additional constraint (Equation 6) in the optimization problem. This constraint forces the estimated output sequence during periods without measurements to remain close to a previously obtained optimal sequence, weighted by the system's decay rate.
Stability Analysis: The authors prove that under the assumption of exponential i-IOSS, the proposed scheme achieves Robust Global Exponential Stability (RGES). The proof demonstrates that the estimation error converges exponentially to zero in the absence of disturbances and remains bounded by a weighted sum of past disturbances.
Varying Horizon Extension: A modified scheme is proposed where the horizon length $M_t$ varies dynamically based on the history of events. This extension aims to provide tighter error bounds and a smaller minimum required horizon length, albeit with increased computational complexity.

Key Contributions

Single-Measurement Transmission: Unlike previous works (e.g., [16]) that transmit sequences of measurements upon triggering, this scheme transmits only the most recent measurement. This reduces packet size, mitigating risks associated with limited bandwidth and transmission failures.
Computational Efficiency: The optimization problem is solved exclusively when an event occurs. Between events, the state estimate is updated via open-loop prediction, significantly reducing computational load.
Robust Stability Proof: The paper establishes RGES for the proposed ET-MHE scheme under a suitable detectability condition (exponential i-IOSS). The analysis accounts for the specific structure of the event-triggered updates and the additional constraint used to handle missing data.
Varying Horizon Scheme: The introduction of a varying horizon length allows for tighter theoretical error bounds and a reduced minimum horizon requirement, offering a trade-off between performance and computational demand.

Results
The effectiveness of the proposed method is validated through two numerical examples: a batch reactor system and a two-link robot arm.

Batch Reactor: Simulations show that the ET-MHE scheme with $\alpha=5$ (a design parameter controlling sensitivity) achieves estimation performance nearly identical to a standard MHE ( $\alpha=0$ ) but with significantly fewer transmissions. Comparisons with the scheme from [16] reveal that while the proposed method triggers more events (14 vs. 6.5 on average), it transmits only single measurements per event, whereas the [16] scheme transmits sequences averaging 23 measurements (max 32).
Robot Arm: Results confirm that increasing the parameter $\alpha$ reduces the number of events but slightly increases the estimation error, consistent with theoretical bounds. The scheme demonstrates convergence to a region near zero despite disturbances.
Constraint Impact: Numerical experiments indicate that the additional constraint (6) required for stability proofs rarely affects the optimization solution (active in <15% of cases) and has a negligible impact on estimation error (average reduction of 0.25% when active).
Varying Horizon Performance: The varying horizon scheme showed marginal average improvements (1% for the reactor, 3% for the robot arm) but demonstrated significant advantages (up to 21% improvement) in specific scenarios where consecutive events occurred, allowing the estimator to utilize more historical information.
Computation: The varying horizon scheme incurs higher computational costs due to larger and structurally changing optimization problems. The additional constraint (6) also increases computation time, though the authors note this is acceptable given the theoretical guarantees.

Significance
The paper claims to present the first ET-MHE scheme for general nonlinear systems that requires transmitting only a single measurement per event while guaranteeing robust global exponential stability. The significance lies in addressing the dual constraints of communication bandwidth (by minimizing packet size) and computational resources (by solving the optimization only when necessary). The work provides a theoretical framework for balancing estimation performance with resource constraints in nonlinear remote state estimation, offering a practical alternative to schemes that require large data packets or frequent computations. The authors modestly note that while the varying horizon extension offers tighter bounds, the fixed horizon scheme remains simpler to implement and less computationally demanding.

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