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Stochastic Event-triggered Sensor Schedule for Remote State Estimation

Duo Han, Yilin Mo, Junfeng Wu, Sean Weerakkody, Bruno Sinopoli, Ling Shi

arXiv:1402.0599v1cs.IT

TL;DR

Remote state estimation must schedule transmissions despite limited bandwidth and sensor resources, while deterministic event triggers can destroy innovation Gaussianity and force approximate nonlinear filtering. The paper proposes open-loop and closed-loop stochastic event-triggered schedules that preserve Gaussian structure and yield closed-form MMSE estimators. Simulations show both event-based schedules outperform random and periodic offline schedulers at the same considered communication setting.

  • Problem

    Deterministic event-triggered mechanisms can destroy innovation Gaussianity, creating a difficult nonlinear filtering problem that requires approximation techniques.

  • Method

    The paper proposes stochastic event-triggered schedules for open-loop and closed-loop remote estimation and derives exact recursive MMSE estimators.

  • Results

    Both open-loop and closed-loop event-based schedulers outperform the random and periodic offline schedulers in the reported scalar stable-system comparison.

  • Takeaways & Limitations

    The stochastic schedules preserve Gaussian innovation properties, reducing estimation to a simple linear filtering problem without the approximations required by prior deterministic event-based approaches.

Abstract

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We propose an open-loop and a closed-loop stochastic event-triggered sensor schedule for remote state estimation. Both schedules overcome the essential difficulties of existing schedules in recent literature works where, through introducing a deterministic event-triggering mechanism, the Gaussian property of the innovation process is destroyed which produces a challenging nonlinear filtering problem that cannot be solved unless approximation techniques are adopted. The proposed stochastic event-triggered sensor schedules eliminate such approximations. Under these two schedules, the MMSE estimator and its estimation error covariance matrix at the remote estimator are given in a closed-form. Simulation studies demonstrate that the proposed schedules have better performance than periodic ones with the same sensor-to-estimator communication rate.

I. INTRODUCTION

The paper addresses remote state estimation under unequal measurement importance, sensor energy limits, and finite shared bandwidth by proposing stochastic event-triggered schedules for open- and closed-loop systems. It develops exact estimators and analyzes communication, estimation performance, stability, and event-parameter design.

  • Motivation: Unequal signal fluctuations motivate allocating more sampling and scheduling effort to periods with more variable measurements.
  • Motivation: Wireless sensors require energy-aware scheduling because they are often battery-powered and difficult to replace.
  • Motivation: Shared channel bandwidth can limit simultaneous sensor-to-estimator communication, motivating scheduling under communication constraints.
  • Contributions: The paper proposes a general stochastic decision rule with practical open-loop and closed-loop event-triggered schedules.
  • Contributions: The proposed framework yields exact MMSE estimators in simple recursive form, avoiding the intractable nonlinear estimation problem associated with prior event-based approaches.
  • Contributions: The analysis covers estimator stability, communication rates, prediction-error covariance bounds, and optimization of the event parameter for communication–quality tradeoffs.

III. MMSE ESTIMATOR DESIGN

The paper derives the MMSE estimator for the open-loop stochastic event-triggered schedule by analyzing measurement updates according to whether the sensor transmission is received. The resulting estimator preserves Gaussian conditional distributions and modifies the no-transmission update through an enlarged measurement-noise covariance.

  • Open-loop MMSE estimator: The open-loop MMSE estimator is stated as a recursive theorem for the conditional Gaussian state estimates before and after measurement updates.
  • Measurement updates: The proof separates measurement updates into the cases where the estimator does not receive or does receive the current measurement.
  • Measurement updates: When no measurement arrives, conditional independence and Gaussian joint structure preserve a Gaussian state distribution for the estimator update.
  • Measurement updates: When the measurement arrives, conditional joint Gaussianity permits the standard Kalman filtering update.
  • Open-loop MMSE estimator: For a missing measurement, the covariance recursion uses the Kalman-gain form with enlarged measurement-noise covariance R + Y^-1, while the posterior estimate is a scaled prior estimate.

B. Closed-Loop Stochastic Event-Triggered Scheduling

The closed-loop schedule feeds the estimator’s prior measurement estimate back to the sensor and preserves Gaussian conditioning for MMSE estimation. Its covariance recursion retains the Kalman-filter form with a modified gain when no packet arrives.

  • B. Closed-Loop Stochastic Event-Triggered Scheduling: The closed-loop scheduler feeds the prior measurement estimate back to the sensor and yields an MMSE estimator through Theorem 2.The theorem states Gaussian conditional distributions for the state before and after incorporating the event-triggered information.
  • B. Closed-Loop Stochastic Event-Triggered Scheduling: When γk = 0, the error-covariance recursion keeps the standard Kalman-filter form but uses a modified Kalman gain.Because the event uses the zero-mean innovation zk, the posterior estimate is the prior estimate itself rather than a scaled prior estimate.

IV. PERFORMANCE ANALYSIS

The performance analysis studies communication rate and estimation covariance for stochastic schedules, using matrix functions and fixed-point properties to characterize their behavior. The stochastic transmission sequence makes the MMSE covariance iteration random, so the analysis focuses on statistical properties.

  • IV. PERFORMANCE ANALYSIS: The analysis evaluates average sensor-to-estimator communication rate together with estimation performance under the proposed schedules.The scheduler’s stated goal is to reduce communication frequency while retaining estimation performance.
  • IV. PERFORMANCE ANALYSIS: The stochastic decision rule makes γk random, so the MMSE estimator iteration is stochastic and only statistical properties of P−k can be obtained.The analysis therefore studies mean stability and asymptotic bounds on E[P−k].
  • IV. PERFORMANCE ANALYSIS: The matrix functions gW(X) and ΓW(X) are introduced, with gW(X) and ΓW(X) increasing in X and a unique positive-definite fixed point under Proposition 1.These properties support the subsequent performance analysis.

A. Open-Loop Schedule

For the open-loop schedule, the communication rate and covariance behavior are characterized under a stable system. Packet-arrival events recur almost surely, covariance remains bounded, and performance oscillates between matrix bounds.

  • A. Open-Loop Schedule: For ρ(A) < 1, Theorem 3 gives the open-loop communication rate γ for scheduler (11)–(12).The derivation uses the zero-mean Gaussian distribution of yk.
  • A. Open-Loop Schedule: The events of l sequential packet arrivals and l sequential packet drops recur almost surely, and the time-average communication rate equals the expected rate γ.The equality holds for almost every sample path.
  • A. Open-Loop Schedule: Under stability, the open-loop prior covariance P−k is uniformly bounded and eventually satisfies lower and upper bounds around X0 and Xol.The bounds hold for all sufficiently large k within any specified ε margin, and corresponding almost-sure inequalities hold infinitely often.
  • A. Open-Loop Schedule: The proposed open-loop estimator remains stable regardless of the packet-arrival process and permits Y to be adjusted for arbitrarily small communication rates.This contrasts with the cited deterministic event-triggered scheduler, which requires a critical minimum transmission rate for mean stability.
  • A. Open-Loop Schedule: X0 and Xol represent the best- and worst-case performance of OLSET-KF because P−k oscillates between them.The expected covariance is also asymptotically bounded using Xol.

B. Closed-Loop Schedule

The closed-loop analysis addresses communication rate and covariance without assuming stability of A, but packet-dependent innovations make the analysis more difficult. At equal communication rates, feedback improves performance relative to open-loop scheduling.

  • B. Closed-Loop Schedule: Unlike the open-loop case, closed-loop analysis imposes no assumption on A, but packet-dependent innovations make ζk’s distribution more complicated.This dependence makes the closed-loop analysis more difficult.
  • B. Closed-Loop Schedule: The closed-loop covariance bounds involve Xcl and X0, obtained by setting every γk to 0 and 1, respectively, in the covariance recursion.Theorems 7 and 8 characterize communication-rate bounds and covariance behavior using these solutions.
  • B. Closed-Loop Schedule: At the same communication rate, the closed-loop schedule achieves better performance than the open-loop schedule because zk has smaller covariance than yk.The open-loop schedule is easier to implement because it does not require estimator feedback.

V. DESIGN OF EVENT PARAMETER

The design framework selects event parameters to balance communication rate and estimation quality, using analytical bounds and convex relaxations for open-loop scheduling. For vector measurements, the relaxed optimization is solved through a convex formulation, while closed-loop design uses an upper bound on the communication rate.

  • Design objectives can minimize communication rate under an estimation-performance requirement, or optimize estimation performance under a communication-rate constraint.
  • For vector-state systems, a covariance constraint can correspond to multiple event parameters and communication rates, motivating optimization over metrics such as worst-case covariance.
  • For vector measurements, minimizing communication rate directly is troublesome because the objective is log-concave in Y, so a convex upper bound is used.
  • The communication-rate relaxation replaces minimizing γ with minimizing tr(ΠY), using the bounds 1 − (1 + tr(ΠY))^-1/2 < γ < 1 − exp(−tr(ΠY)/2).
  • The relaxed optimization problem has an optimal Y* obtainable by solving the convex problem stated in Theorem 11.
  • The resulting suboptimal solution is close to the true optimum, as indicated by the plotted upper bound on the optimality gap κ.

A. Performance of OLSET-KF and CLSET-KF

Simulations examine OLSET-KF and CLSET-KF across stable and unstable systems, including comparisons with offline schedulers. The bounds tighten as communication rate γ increases, and event-based schedulers outperform the offline baselines in the reported scalar stable case.

  • The simulations consider stable and unstable systems for OLSET-KF and CLSET-KF, respectively.
  • The bounds for both cases become tighter as the communication rate γ increases.
  • In a scalar stable system with A = 0.8, C = 1, Q = 1, and R = 1, both event-based schedulers outperform random and periodic offline schedulers.
  • The closed-loop event-based scheduler performs better than the open-loop event-based scheduler in the reported comparison.
  • Problem 9 addresses the tradeoff between communication rate and estimation quality when designing an OLSET-KF.
  • The suboptimal solution obtained by varying ϖ is close to the true optimal solution, while the optimality gap κ is also bounded.

C. Comparison between CLSET-KF and DET-KF

The target-tracking experiments compare CLSET-KF with DET-KF at communication rates 0.65 and 0.25. The stochastic schedule preserves Gaussian filtering properties, whereas DET-KF’s theoretical covariance can diverge from empirical covariance.

  • Target-tracking setup: The target state comprises position, speed, and acceleration in the tracking model.The sensor periodically measures all three components.
  • Target-tracking setup: At communication rate 0.65, Fig. 6 compares target-position error variance for CLSET-KF and DET-KF.CLSET-KF is shown on the left and DET-KF on the right.
  • Target-tracking setup: At communication rate 0.25, Fig. 7 compares target-position error variance for CLSET-KF and DET-KF.The two filters are again displayed separately for the same tracking task.
  • Comparison between schedulers: DET-KF’s theoretical error covariance does not match its empirical covariance, indicating that its approximate MMSE estimator is invalid in this setting.The corresponding measurement update therefore needs to be re-examined.
  • Comparison between schedulers: The stochastic scheduler preserves Gaussian measurement-update properties, avoiding the nonlinear filtering approximation required by deterministic event-based schedules.This preservation yields a simpler linear filtering problem.

APPENDIX

The appendix develops technical results supporting the stochastic scheduling analysis. It establishes determinant inequalities, ergodic properties, convergence, and recurrence results used in the proofs.

  • Ergodic arguments: The Markov process formed by the state sequence has a stationary and unique probability measure when the system matrix A is stable.The associated process is ergodic under the shift operator.
  • Ergodic arguments: Birkhoff’s Ergodic Theorem converts time averages of indicator functions into almost-sure probabilities for the event analysis.This supports long-run frequency arguments for event occurrences and packet-drop sequences.
  • Theorem 5: The proof of Theorem 5 bounds the covariance sequence by induction and uses convergence to Xol to establish asymptotic inequalities.The argument also shows that relevant events occur infinitely often.
  • Theorem 5: Theorem 5’s final step selects a sufficiently large l so that event occurrence implies the covariance is within ε of Xol infinitely often.The opposite inequality is obtained analogously.
  • Technical lemmas: The appendix uses Cholesky decomposition and Sylvester’s determinant theorem to rewrite determinant expressions.It then bounds the determinant using trace and exponential terms.
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