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Decentralized Event-Triggering for Control of Nonlinear Systems

Pavankumar Tallapragada, Nikhil Chopra

arXiv:1302.4019v3eess.SYcs.SYmath.OC

TL;DR

Decentralized sensing requires event-triggers that operate with local information rather than full-state data. This paper designs asynchronous triggers for nonlinear systems and establishes stability and positive sensor inter-transmission-time lower bounds, with global stability and scale invariance in the LTI case.

  • Problem

    In decentralized sensing, each sensor must decide whether to transmit using only locally available information, rather than full-state information.

    Source: §I · p. 1

  • Method

    The paper designs decentralized asynchronous event-triggers using locally available information to schedule sensor transmissions to a central controller.

    Source: §I · p. 1

  • Results

    The design guarantees asymptotic stability and positive lower bounds on sensor inter-transmission times, while the LTI case is globally asymptotically stable and scale invariant.

    Source: A. Linear System Example · p. 9, §VI · p. 12

  • Takeaways & Limitations

    Under scale invariance, the time and number of transmissions needed to reduce the state norm to a fixed fraction of its initial value are independent of the initial norm’s magnitude.

    Source: A. Linear System Example · p. 9

    Controller-to-sensor communications in the modified nonlinear design are assumed synchronous; allowing them to be asynchronous is left for future work.

    Source: §VI · p. 12

Abstract

from arXiv · show

This paper considers nonlinear systems with full state feedback, a central controller and distributed sensors not co-located with the central controller. We present a methodology for designing decentralized asynchronous event-triggers, which utilize only locally available information, for determining the time instants of transmission from the sensors to the central controller. The proposed design guarantees a positive lower bound for the inter-transmission times of each sensor, while ensuring asymptotic stability of the origin of the system with an arbitrary, but priorly fixed, compact region of attraction. In the special case of Linear Time Invariant (LTI) systems, global asymptotic stability is guaranteed and scale invariance of inter-transmission times is preserved. A modified design method is also proposed for nonlinear systems, with the addition of event-triggered communication from the controller to the sensors, that promises to significantly increase the average sensor inter-transmission times compared to the case where the controller does not transmit data to the sensors. The proposed designs are illustrated through simulations of a linear and a nonlinear example.

I. INTRODUCTION

The paper addresses decentralized sensing, where each sensor must make transmission decisions using only local information, a setting where full-state triggering information is unavailable. It proposes asynchronous decentralized event-triggers for nonlinear systems, with stability and inter-transmission-time guarantees under stated conditions.

I. INTRODUCTION

  • Decentralized sensing requires each sensor to decide whether to transmit using only locally available information, unlike triggering methods that use full-state information.The paper notes that dynamic output feedback is another setting where full-state information is unavailable.Source: §I · p. 1
  • The design guarantees asymptotic stability for a fixed compact region of attraction and positive lower bounds on each sensor’s inter-transmission times.For LTI systems, it guarantees global asymptotic stability and preserves scale invariance; the controller-to-sensor variant’s average-time improvement is not mathematically proved.Source: §I · p. 1
  • For nonlinear systems, the approach assumes a global ISS Lyapunov function for ease of exposition, although a semi-global one suffices for its semi-global stability results.The authors describe the semi-global requirement as less severe and note that such a function may be derived from Lyapunov functions guaranteeing asymptotic stability.Source: §I · p. 1
  • The scheme uses timers at sensors and supports one-way sensor-to-controller communication while guaranteeing semi-global asymptotic stability, globally for LTI systems.The authors highlight that sensors need no receivers, which they describe as useful in low-energy applications.Source: §I · p. 2

II. PROBLEM SETUP

The problem is to stabilize a nonlinear system using a sampled-data controller fed by distributed sensors that transmit asynchronously. Each sensor’s trigger must use local information and ensure a positive minimum time between transmissions.

II. PROBLEM SETUP

  • The measurement error is defined as the difference induced by sampling, excluding sensor noise and quantization as sources of error.The paper treats the sampled-data vector and measurement-error vector as distinct quantities.Source: §II · p. 2
  • Each state component is sensed at a different location and transmitted intermittently to a central controller, which uses sampled sensor data.Sensors sense their respective components continuously, while the resulting sampled signals are piecewise constant.Source: §II · p. 2
  • The triggers must use only local information to define non-identical transmission sequences that stabilize the origin and keep each inter-transmission time positively lower bounded.Without such a bound, event-triggered intervals may approach zero or exhibit Zeno behavior.Source: §II · p. 2

III. DECENTRALIZED ASYNCHRONOUS EVENT-TRIGGERING

The nonlinear design relies on an ISS Lyapunov condition with componentwise measurement-error bounds and Lipschitz regularity on compact sets. Its objective is asymptotic stability over a chosen compact region of attraction while retaining positive minimum inter-transmission times.

III. DECENTRALIZED ASYNCHRONOUS EVENT-TRIGGERING

  • The closed-loop system is assumed to be ISS with respect to measurement error, with componentwise bounds relating each sensor’s error to the state.The paper also assumes the relevant functions are Lipschitz on compact sets.Source: §III · p. 3
  • The design aims to stabilize the origin within a desired compact sublevel-set region while guaranteeing positive minimum inter-sample times.The region may be taken as a compact sublevel set of the Lyapunov function.Source: §III · p. 3

A. Centralized Asynchronous Event-Triggering

The centralized asynchronous design first identifies error limits that preserve stability over a chosen region, then selects triggers that enforce those limits. The resulting trigger condition guarantees asymptotic stability and a positive lower bound on each sensor’s inter-transmission time.

A. Centralized Asynchronous Event-Triggering

  • The design first bounds the functions γi on sets Ei so the measurement-error constraints needed for stability hold within the chosen region.The centralized lemma characterizes a family of triggers that keep |xi,e| ≤ Mi(c)|x|; the decentralized design cannot directly use |x| at each sensor.Source: A. Centralized Asynchronous… · pp. 3–4
  • If the componentwise error bounds |xi,e| ≤ Mi(c)|x| hold for all time, the origin is asymptotically stable with S(c) included in its region of attraction.The result follows from the assumed ISS Lyapunov function and the resulting decrease of the Lyapunov function on S(c).Source: A. Centralized Asynchronous… · p. 3
  • Triggering sensor i when |xi,e| ≥ wi|x| guarantees asymptotic stability with S(c) in the region of attraction and a positive lower bound Ti on its inter-transmission times.The thresholds satisfy 0 < wi ≤ Mi(c), with Mi(c) determined by the chosen region.Source: A. Centralized Asynchronous… · p. 4

B. Decentralized Asynchronous Event-Triggering

The design uses locally available sensor information and asynchronous triggers with enforced dwell times. With suitable trigger parameters, it guarantees asymptotic stability over a priorly chosen region and a positive lower bound on each sensor’s inter-transmission times.

B. Decentralized Asynchronous Event-Triggering

  • After an explicit minimum dwell time, each sensor checks a state-based condition using locally sensed data, enabling decentralized asynchronous triggering.The dwell time is given by T_i; the trigger condition uses the sensor’s local state measurement.Source: B. Decentralized Asynchronous… · pp. 4–5
  • With T_i = 0, the related triggers in, do not generally guarantee a positive lower bound on inter-transmission times, even under weak coupling.The paper attributes this to the set where x_i = 0 not generally being an equilibrium set.Source: B. Decentralized Asynchronous… · p. 5
  • Theorem 2 guarantees asymptotic stability with S(c) included in the region of attraction and a positive lower threshold for each sensor’s inter-transmission times.The guarantee holds when the trigger parameters satisfy the theorem’s conditions.Source: B. Decentralized Asynchronous… · p. 5
  • The desired region S(c) must be chosen at installation, since w_i cannot be selected decentrally unless M_i(c), and thus c, is fixed beforehand.For t > 0, sensors need only transmit data to the controller; they do not need to receive further communication.Source: B. Decentralized Asynchronous… · p. 5
  • Fixed Lipschitz constants may make transmissions more conservative as the state contracts within S(c), motivating controller updates of w_i and T_i.The authors note that the cost of increased transmissions may outweigh the energy saved by sensors not listening.Source: B. Decentralized Asynchronous… · p. 5

C. Decentralized Asynchronous Event-Triggering with Intermittent Communication from the Central Controller

With intermittent controller feedback, the controller uses asynchronously sampled data to bound the Lyapunov function and update sensor-trigger parameters. The design preserves asymptotic stability and positive inter-transmission-time bounds, though average transmission-time improvement is not mathematically established.

C. Decentralized Asynchronous Event-Triggering with Intermittent Communication from the Central Controller

  • The controller updates w_i, T_i, and c when its sampled data indicate the state lies in a sufficiently small positively invariant subset of S(c).Because it sees only asynchronously sampled data, it can determine only an upper bound on the state Lyapunov function.Source: C. Decentralized Asynchronous… · p. 5
  • Theorem 3 guarantees asymptotic stability with S(c) in the region of attraction under the stated parameter and triggering conditions.The controller's sampled Lyapunov upper bound is non-increasing, keeping the state in S(Vs) as parameters are updated.Source: C. Decentralized Asynchronous… · p. 6
  • With Assumption (A3), each sensor's inter-transmission times have a uniform positive lower bound.Without further information, positive bounds at individual updates alone do not rule out their convergence to zero.Source: C. Decentralized Asynchronous… · p. 6
  • There exist choices of w_i that make T_i non-decreasing, while choosing the largest possible T_i is a multi-objective resource-allocation problem.The paper also notes that a systematic characterization of the state-space regions where Corollary 2's conditions hold is left for future work.Source: C. Decentralized Asynchronous… · p. 7
  • The paper does not mathematically establish whether this design improves average inter-transmission times.It identifies the lack of tools to compare average inter-transmission times across different aperiodic sampled-data systems as outside the paper's scope.Source: C. Decentralized Asynchronous… · p. 7

IV. LINEAR TIME INVARIANT SYSTEMS

For LTI systems, the proposed asynchronous decentralized triggers guarantee global asymptotic stability and a positive minimum inter-transmission time. They also preserve scale invariance: transmission timing does not depend on the magnitude of the initial condition.

IV. LINEAR TIME INVARIANT SYSTEMS

  • The LTI design guarantees global asymptotic stability and explicitly enforces a positive minimum inter-transmission time.The stated setup assumes the closed-loop matrix A + BK is Hurwitz.Source: §IV · pp. 7–8
  • The paper attributes the LTI global guarantee to globally Lipschitz dynamics and functions, which yield trigger constants that hold globally.The asynchronous triggers are described as under-approximations of centralized event-triggering, while preserving its scale invariance.Source: §IV · p. 8
  • Scaling the initial conditions by a scalar scales the state trajectories by the same factor without changing inter-transmission times.The paper describes this scale invariance as keeping average network usage the same over large portions of the state space.Source: §IV · p. 8

V. SIMULATION RESULTS

The proposed decentralized asynchronous sensing mechanism is illustrated in one linear and one nonlinear example.

V. SIMULATION RESULTS

  • The simulation section illustrates the proposed sensing mechanism with a linear example and a nonlinear example.The supplied passage identifies the examples but reports no simulation outcomes.Source: §V · p. 8

A. Linear System Example

The batch-reactor simulation illustrates the LTI design’s inter-transmission behavior and cumulative frequency distribution. The reported intervals are compared with prior methods, and the distribution depends on the initial condition’s phase rather than magnitude.

A. Linear System Example

  • The simulation presents the Lyapunov function and its derivative along the closed-loop trajectory, sensor inter-transmission times, and a frequency distribution of those times over a specified interval.Source: A. Linear System Example · p. 9
  • The cumulative frequency distribution depends on the initial condition’s phase, not its magnitude, and a slower rise indicates greater justification for event-triggering over time-triggering.A sharp rise to 100% indicates that the trigger is not much better than a time-trigger.Source: A. Linear System Example · p. 9
  • Average inter-transmission times were [24.9, 27.7, 34.5, 34.2]ms, about an order of magnitude lower than those reported in,.The authors note that the averages are of the same order as in [12].Source: A. Linear System Example · p. 9
  • Minimum inter-transmission times were [11, 15.4, 12.6, 19.9]ms, matching the example’s computed minimum thresholds.The authors report these thresholds as a few orders of magnitude above the guaranteed minima and an order above the observed minima in,.Source: A. Linear System Example · p. 9

B. Nonlinear System Example

The nonlinear examples evaluate decentralized sensor triggering through Lyapunov behavior and sensor inter-transmission times. A simulation with controller-to-sensor updates reports longer average intervals for Sensor 1, while Sensor 2’s average changes only marginally.

B. Nonlinear System Example

  • The example uses a sampled-data controller based on measurement error, with a nonlinear term −(x1 + x1,e)^3 in its control input.Here, xe denotes measurement error resulting from sensor-data sampling.Source: B. Nonlinear System Example · p. 9
  • For chosen parameters and initial conditions, sensors 1 and 2 had average inter-transmission times of 4.2 and 26.2 ms, with minimum intervals of 4 and 3.4 ms, respectively.The simulation reported 2366 and 382 transmissions, respectively; sensor 1’s average interval was only marginally above its minimum.Source: B. Nonlinear System Example · p. 10
  • Over the 10-second simulation, Sensor 1’s cumulative distribution rises sharply, indicating event-triggered transmission is nearly equivalent to time-triggered transmission, while Sensor 2’s slow rise demonstrates event-triggering’s usefulness.Source: B. Nonlinear System Example · p. 10, Figure 2 · p. 11
  • With controller updates to trigger parameters, average inter-transmission times were [50.5, 31.1]ms, and transmissions numbered 198 for Sensor 1 and 322 for Sensor 2.Minimum inter-transmission times were 4.2ms and 9ms, respectively; trigger parameters were updated 16 times.Source: B. Nonlinear System Example · p. 11
  • With controller updates, Sensor 1’s trigger parameters w1 and T1 rise over time, while w2 stays constant and T2 does not rise monotonically.The figure also plots sensor intervals and their cumulative frequency distributions.Source: B. Nonlinear System Example · p. 11

VI. CONCLUSIONS

The paper develops decentralized event-triggers for nonlinear systems with distributed sensors and a central controller. It guarantees positive lower bounds on sensor inter-transmission times and asymptotic stability within a fixed region; the LTI case is globally asymptotically stable, with stated scope limits.

VI. CONCLUSIONS

  • The architecture uses full-state feedback, a central controller, and distributed sensors; each sensor’s trigger uses local information for asynchronous transmissions.The design aims to eliminate or drastically reduce sensor listening to other sensors or the controller.Source: §VI · p. 12
  • The design guarantees a positive lower bound on each sensor’s inter-transmission times and asymptotic stability within an arbitrary, priorly fixed region of attraction.In the special LTI case, the region of attraction is global, with no need for sensors to listen.Source: §VI · p. 12
  • The plant’s control input is updated whenever the controller receives sensor information, so the design does not guarantee a positive lower bound on controller update intervals.Receptions from different sensors can be arbitrarily close; controller-side triggering or an explicit minimum update threshold could be added, as in.Source: §VI · p. 12
  • Controller-to-sensor communication is assumed synchronous, and the paper does not mathematically demonstrate improved inter-transmission times for the modified scheme.The authors identify asynchronous communication and analytical characterization of the frequency distribution as future work.Source: §VI · p. 12
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