Elsevier

Automatica

Volume 47, Issue 10, October 2011, Pages 2319-2322
Automatica

Technical communique
On event design in event-triggered feedback systems

https://doi.org/10.1016/j.automatica.2011.05.027Get rights and content

Abstract

This paper studies event design in event-triggered feedback systems. A novel event-triggering scheme is presented to ensure exponential stability of the resulting sampled-data system. The scheme postpones the triggering of events over previously proposed methods and therefore enlarges the intersampling period. The resulting intersampling periods and deadlines are bounded strictly away from zero when the continuous time system is input-to-state stable with respect to measurement errors.

Introduction

This paper studies event-triggered feedback systems, which update the control with new state information when some error exceeds a specified threshold. Prior work (Tabuada, 2007) designs state-dependent thresholds that ensure that a suitably chosen storage function decreases in a monotone manner over time. However, to guarantee the chosen stability concept, such monotonically decreasing behavior is not necessary. For switched systems, one may tolerate small increases in the storage function, provided that the values that the storage function takes after each sampling instant are monotonically decreasing (Wang & Lemmon, 2008). With this idea, this paper presents an event-triggering scheme for exponential stability in which the state is sampled when the storage function intersects suitably chosen exponentially decreasing functions.

Section snippets

Problem formulation

Notation: We denote by Rn the n-dimensional real vector space and by R+ the real positive numbers. Let R0+=R+{0}. denotes 2-norm of a vector or the induced matrix norm. We use to denote the logical operator OR and to denote the logical operator AND. For a function V(t), we denote the limit of V at t from the right by V(t+)=limst+V(s).

Consider a sampled-data system. Let rk denote the time when the kth control task is released for execution on the computer and fk denote the time when

Event-triggered feedback systems

This section introduces the event-triggered feedback scheme to ensure exponential stability of the sampled-data system. The main idea is to enforce that V(t) is bounded by an exponentially decreasing function h:R0+×RnR0+. It is shown in Fig. 1, where the horizontal axis is time, the vertical axis is the energy V, the solid curve is the trajectory of V(x(t|x0)) (for simplicity, we sometimes denote it by V(t) if it is clear in the context), the dashed curves are the thresholds, and the dotted

Conclusions

This paper proposed a new event-triggering scheme that ensures exponential stability of the system. We show that the intersampling periods and deadlines generated by our scheme are bounded strictly away from zero. Simulation examples can be found in Wang and Lemmon (2008).

Proofs

Proof of Lemma 3.1

We prove the satisfaction of (19) by contradiction. Suppose that it does not hold. Then, since V(rk+1)=hkeδāηk+1<hkeδˆāηk+1, there must exist t[rk+1,fk+1) and a positive constant ϵ1 such that t+ϵ1fk+1,V(t)hkeδˆā(ηk+1+trk+1),t[rk+1,t],V(t)>hkeδˆā(ηk+1+trk+1),t(t,t+ϵ1). By (2), (7), we have x(t)LV(t). Eq. (23), therefore, implies x(t)hkLeδˆāηk+1 for any t[rk+1,t]. Consequently, for any t[rk+1,t], ek(t)hkLeδˆāηk+1+x(rk). Applying this inequality into (3),

Xiaofeng Wang was born in Jiangsu, China. He received B.S. and M.S. in mathematics from East China Normal University in 2000 and 2003, respectively. After working for one year at IBM, China, he went to the University of Notre Dame and received a Ph.D. in electrical engineering in 2009. He is currently a postdoctoral research associate in the Department of Mechanical Science and Engineering at the University of Illinois at Urbana-Champaign. Dr. Wang’s research interests include networked control

References (2)

  • P. Tabuada

    Event-triggered real-time scheduling of stabilizing control tasks

    IEEE Transactions on Automatic Control

    (2007)
  • Wang, X., & Lemmon, M. (2008). Event design in event-triggered feedback control systems. In Proc. IEEE conference on...

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Xiaofeng Wang was born in Jiangsu, China. He received B.S. and M.S. in mathematics from East China Normal University in 2000 and 2003, respectively. After working for one year at IBM, China, he went to the University of Notre Dame and received a Ph.D. in electrical engineering in 2009. He is currently a postdoctoral research associate in the Department of Mechanical Science and Engineering at the University of Illinois at Urbana-Champaign. Dr. Wang’s research interests include networked control systems, adaptive control systems, real-time systems, distributed systems, and optimization.

Michael D. Lemmon is a professor of electrical engineering at the University of Notre Dame. He received a B.S. in electrical engineering from Stanford University (1979) and a Ph.D. in electrical and computer engineering from Carnegie-Mellon University (1990). Dr. Lemmon has served as an associate editor for the IEEE Transactions on Neural Networks and the IEEE Transactions on Control Systems Technology. He chaired the first IEEE working group on hybrid dynamical systems and was the program chair for a hybrid systems workshop in 1997. Most recently he helped forge a consortium of academic, private and public sector partners to build one of the first metropolitan scale sensor–actuator networks (CSOnet) used in monitoring and controlling combined-sewer overflow events. Dr. Lemmon’s research deals with real-time networked control systems with an emphasis on understanding the impact that reduced feedback information has on overall system performance. This work has been funded by a variety of state and federal agencies that include the National Science Foundation, Army Research Office, Defense Advanced Research Project Agency, and Indiana’s 21st Century Technology Fund.

The authors gratefully acknowledge the partial financial support of the National Science Foundation (NSF-ECCS-0925229). The material in this paper was not presented in any conference. This paper was recommended for publication in revised form by Associate Editor Maurice Heemels under the direction of Editor André L. Tits.

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