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Event-triggered Learning with Application to Wireless Sensor Networks

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Communication load is a limiting factor in many real-time systems. Event-triggered state estimation and event-triggered learning methods reduce network communication by sending information only when it cannot be adequately predicted based on previously transmitted data. This research aims at developing event-triggered learning approaches for nonlinear discrete-time systems with cyclic excitation. The methods automatically recognize cyclic patterns in data – even when they change repeatedly – and reduce communication load whenever the current data can be accurately predicted from previous cycles. Nonetheless, a bounded error between measurements and the receivers' output is guaranteed. The employed model for predictions is updated hierarchically, and a nonparametric statistical test enforces that model updates happen only if the model changed with high probability. The effectiveness of the proposed methods is demonstrated using the application example of wireless real-time pitch angle measurements of a human foot. They reduce the communication load by 70 % while the root-mean-square error between the sender angle and the receiver angle is less than 1°.

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Many applications require real-time transmission of signals over communication channels with bandwidth limitations. A typical example is given by wireless sensor networks in feedback-controlled systems. The number of agents (i.e., network nodes) and their communication rate is limited by the amount of information the wireless network can transmit in real-time. It is, therefore, desirable to reduce the communication load without compromising the accuracy of the transmitted signals.

Well known approaches are event-based sampling (EBS) and event-triggered state estimation (ETSE, sometimes referred to as model-based event-based sampling): At each sampling instant, the receiving agent independently predicts the state, which is measured by the sender, based on previous estimations and a model. The sender performs the identical prediction and communicates the measured state if the error between prediction and measurement exceeds a predefined threshold. Otherwise, there is no communication and the receiving agent uses the model-based prediction as estimation (cf. Fig. 1).

Fig. 1: Event-triggered learning in a two-agents network. If the measured signal (e.g., a human foot angle) can be described by a cyclically excited system model, the model is shared with the receiver and the communication is reduced to those samples that cannot be predicted using the model and previously transmitted data. Inaccuracy of the model (e.g., due to change in walking pattern) is detected and a new model is identified and shared.

Since the prediction accuracy heavily depends on the quality of the utilized model, it was recently proposed to learn and update models in an event-triggered fashion as well. Occurrence of communication is treated as a random variable, and the model is updated when empirical data does not fit the expected probability distribution inferred based on the given model.

We build on this idea and develop event-triggered learning (ETL) methods for cyclically excited systems. The main contributions are:

  • Extension of the concept of ETL to specifically target systems with a locally cyclic excitation. Locally means that cycles that are close in time are almost identical; however, cycles that are far apart are not necessarily similar. This class of systems is useful for describing both biological and technical processes (e.g., human motion, breath, heartbeat, and production cycles).
  • Utilization of a learning trigger tailored to the problem at hand (one-sided Kolmogorov-Smirnov test), which fires with high probability in case of a model change and with low probability otherwise.
  • Introduction of the novel idea of a hierarchical model learning strategy, which updates and communicates only a reduced number of model parameters whenever that is sufficient.
  • Demonstration of significant communication savings (70 %) using experimental data from cyclic human motion collected with a wearable inertial sensor network. This is the first application of ETL on real-world network data.

We consider a discrete-time system with a state, which is measured by the sending agent. The system is assumed to be influenced by a cyclic input (excitation) and by noise. The recursive state update law of the system is characterized by dynamics, which is assumed to be known. The excitation and its cyclicity are unknown and may change with time. In the following, whenever the term model<\> is used, it refers to an approximation of the cyclic excitation.

We consider architectures with one sending and one receiving agent. However, the methods can be directly applied to multi-agent systems and yield the same advantages therein. The sending and receiving agents have the following capabilities:

  • the sender can transmit measured data samples to the receiver, i.e., perform <i>state updates<\i>;
  • sender and receiver can estimate current data samples from previously transmitted data and a model, i.e., perform <i>predictions<\i>;
  • the sender can estimate excitation trajectories from measured data, i.e., perform <i>model identification<\i>;
  • the sender can send model parameters to the receiver, i.e., perform <i>model updates<\i>.

The main objective is to find a joint strategy for the sender and the receiver such that the amount of communication (state and model updates) is reduced while the error between the actual measurement signal and the signal estimate on the receiver side remains small in the sense of a suitable metric.

Fig. 2: One sending and one receiving agent with the typical event-triggered state estimation architecture in white and event-triggered learning in gray. The process provides the measured state at every sampling instant. At the same time, the state is estimated by the prediction blocks of the sender and the receiver using the previous estimate and a trajectory model of the excitation. If the prediction differs significantly from the measured state, then a state update is triggered, and the internal states of both prediction blocks are set to the measured state. Too frequent state updates indicate poor model quality and, therefore, trigger model learning. This can either lead to only an adjustment of certain parameters of the current model trajectory or to a completely new excitation trajectory. In any case, the new model information is shared between sender and receiver.

The proposed event-triggered learning approach for cyclically excited systems is illustrated in Fig. 2. It can be described by the following building blocks:

  • Two identical predictors that estimate the measured state based on an internal state and an estimated excitation trajectory of one cycle.
  • A binary state-update trigger that determines when to update the internal state of the predictors with the measurement to ensure a bounded error of the estimation.
  • A model learning block, which estimates the excitation trajectory that is used by the predictors or updates certain parameters (e.g., cycle length or amplitude) of the current trajectory.
  • A binary learning trigger that determines when to update the internal excitation trajectory model of the predictors. Ideally, learning shall be triggered if and only if the rate of state updates increases due to a false or inaccurate model.

Details on these four steps can be found in our publications on ETL. Additionally, some MATLAB code is provided in the section below.

To demonstrate the effectiveness of the proposed algorithms, consider a network of wearable sensor units comprising at least an inertial measurement unit (IMU) chip, a wireless communication module, and a microcontroller, which is attached to a human body during gait. Such measurement systems are used for real-time biofeedback and control of robotic systems and neuroprostheses. The rate at which the network can communicate reliably in real time is limited by the number of sensors. If the communication load between each sensor and the receiver can be reduced, higher sampling rates or a larger number of sensors can be used. We apply the proposed ETL methods to this application and consider the specific example of real-time measurement of the foot pitch angle in a feedback-controlled neuroprostheses (cf. Fig. 1). The angle dynamics is modeled by cyclic increments and zero-mean additive Gaussian noise.

Fig. 3: Number of transferred values with respect to full communication (left) and resulting RMSE (right). We compare full communication (full), decimation with factor 2 (decim), event-triggered state-estimation (ETSE) and two parametrizations of event-triggered learning (ETL) for about 0.5 h of highly variable gait; The results show that ETL reduces the communication significantly without changing the accurancy of the transferred signal in comparison to ETSE.

The proposed ETL method is compared to alternative strategies by determining the number of transmitted values and the RMSE between the measured angle and the output of the receiver. Results are visualized in Fig. 3. The trivial strategy of sending every second sample yields a RMSE of almost 2° at 50 % communication rate. The same reduction of communication is achieved by pure ETSE (i.e., the scheme in Fig. 2, white without ETL); however, at a RMSE of less than 1°. ETL achieves a reduction of traffic to even less than 30 %, while the RMSE is still below 1°$. Finally, optimizing the heuristically chosen ETL parameters with nested cross-validation leads to small further improvements (27 %).

Fig. 4 shows how a small model update is triggered due to a change of the cycle length of the measured process. The figure also provides evidence that the absolute estimation error is bounded by 2°.

Fig. 4: Small model update triggered at 98.9 s due to a change of the cycle length (gait velocity). (A) measured and estimated state; (B) estimation error and state-update trigger threshold; (C) probability of the KS-test and significance level. Model learning is triggered because too many state updates occur and the estimated probability falls below the significance level for the minimum holding time.

In contrast to previous approaches, the current methods exploit explicitly the periodicity of the dynamics and account for time-variant behavior. The proposed methods are shown to yield significant resource savings in a wireless body sensor network for orientation tracking of a human foot. The experimental results demonstrate that a large reduction of communication load (by 70 %) and a small bounded estimation error (by 2°) can be achieved at the same time. This means that up to three times as many sensors could be used without jeopardizing latency of the real-time communication. Moreover, the examined sampling rate of 50 Hz is relatively low; up to 1 kHz is common in IMU networks (200 Hz in wireless ones), which gives even more potential for resource savings with the proposed methods.


Matlab script and documentation for cycle length estimation

Related Publications

J. Beuchert, F. Solowjow, J. Raisch, S. Trimpe, T. Seel. Hierarchical Event-triggered Learning for Cyclically Excited Systems with Application to Wireless Sensor Networks. IEEE Control Systems Letters, 4 (1):103–108, 2019.
T. Zhang, D. Laidig, T. Seel. Stop Repeating Yourself: Exploitation of Repetitive Signal Patterns to Reduce Communication Load in Sensor Network. In accepted for European Control Conference (ECC), Naples, Italy, 2019.

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