L2-Gain Analysis of Periodic Event-Triggered Control and Self-Triggered Control Using Lifting

Nard Strijbosch; Geir E. Dullerud; Andrew R. Teel; W.P.M.H. Heemels

doi:10.1109/TAC.2020.3025304

L₂-Gain Analysis of Periodic Event-Triggered Control and Self-Triggered Control Using Lifting

Nard Strijbosch (Corresponding author)
, Geir E. Dullerud
, Andrew R. Teel
, W.P.M.H. Heemels

Research output: Contribution to journal › Article › Academic › peer-review

23 Citations (Scopus)

Abstract

We analyze the stability, and L2-gain properties of a class of hybrid systems that exhibit time-varying linear flow dynamics, periodic time-triggered jumps, and arbitrary nonlinear jump maps. This class of hybrid systems encompasses periodic event-triggered control, self-triggered control, and networked control systems including time-varying communication delays. New notions on the stability, and contractivity (L2-gain strictly smaller than 1) from the beginning of the flow, and from the end of the flow are introduced, and formal relationships are derived between these notions, revealing that some are stronger than others. Inspired by ideas from lifting, it is shown that the internal stability, and contractivity in L2-sense of a continuous-time hybrid system in the framework is equivalent to the stability, and contractivity in ℓ2-sense (meaning the ℓ2-gain is smaller than 1) of an appropriate time-varying discrete-time nonlinear system. These results recover existing works in the literature as special cases, and indicate that analysing different discrete-time nonlinear systems (of the same level of complexity) than in existing works yield stronger conclusions on the L2-gain.

Original language	English
Article number	9201403
Pages (from-to)	3749-3756
Number of pages	8
Journal	IEEE Transactions on Automatic Control
Volume	66
Issue number	8
DOIs	https://doi.org/10.1109/TAC.2020.3025304
Publication status	Published - Aug 2021

Funding

Manuscript received January 14, 2020; revised July 14, 2020; accepted September 10, 2020. Date of publication September 21, 2020; date of current version July 28, 2021. The work of Geir E. Dullerud was supported in part by NSF Grant 1932735. The work of Andrew R. Teel was supported in part by the Air Force Office of Scientific Research under Grant FA9550-18-1-0246. Recommended by Associate Editor S. N. Dashkovskiy. (Corresponding author: Nard Strijbosch.) Nard Strijbosch and W.P.M.H. (Maurice) Heemels are with the Department of Mechanical Engineering, Eindhoven University of Technology, 5632AZ Eindhoven, The Netherlands (e-mail: [email protected]; [email protected]).

Keywords

Networked control systems
periodic event-triggered control (PETC)
piecewise affine (PWA) systems
self-triggered control (STC)

Access to Document

10.1109/TAC.2020.3025304

Cite this

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title = "L2-Gain Analysis of Periodic Event-Triggered Control and Self-Triggered Control Using Lifting",

abstract = "We analyze the stability, and L2-gain properties of a class of hybrid systems that exhibit time-varying linear flow dynamics, periodic time-triggered jumps, and arbitrary nonlinear jump maps. This class of hybrid systems encompasses periodic event-triggered control, self-triggered control, and networked control systems including time-varying communication delays. New notions on the stability, and contractivity (L2-gain strictly smaller than 1) from the beginning of the flow, and from the end of the flow are introduced, and formal relationships are derived between these notions, revealing that some are stronger than others. Inspired by ideas from lifting, it is shown that the internal stability, and contractivity in L2-sense of a continuous-time hybrid system in the framework is equivalent to the stability, and contractivity in ℓ2-sense (meaning the ℓ2-gain is smaller than 1) of an appropriate time-varying discrete-time nonlinear system. These results recover existing works in the literature as special cases, and indicate that analysing different discrete-time nonlinear systems (of the same level of complexity) than in existing works yield stronger conclusions on the L2-gain. ",

keywords = "Networked control systems, periodic event-triggered control (PETC), piecewise affine (PWA) systems, self-triggered control (STC)",

author = "Nard Strijbosch and Dullerud, \{Geir E.\} and Teel, \{Andrew R.\} and W.P.M.H. Heemels",

note = "Funding Information: Manuscript received January 14, 2020; revised July 14, 2020; accepted September 10, 2020. Date of publication September 21, 2020; date of current version July 28, 2021. The work of Geir E. Dullerud was supported in part by NSF Grant 1932735. The work of Andrew R. Teel was supported in part by the Air Force Office of Scientific Research under Grant FA9550-18-1-0246. Recommended by Associate Editor S. N. Dashkovskiy. (Corresponding author: Nard Strijbosch.) Nard Strijbosch and W.P.M.H. (Maurice) Heemels are with the Department of Mechanical Engineering, Eindhoven University of Technology, 5632AZ Eindhoven, The Netherlands (e-mail: [email protected]; [email protected]). Publisher Copyright: {\textcopyright} 1963-2012 IEEE.",

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AU - Heemels, W.P.M.H.

N1 - Funding Information: Manuscript received January 14, 2020; revised July 14, 2020; accepted September 10, 2020. Date of publication September 21, 2020; date of current version July 28, 2021. The work of Geir E. Dullerud was supported in part by NSF Grant 1932735. The work of Andrew R. Teel was supported in part by the Air Force Office of Scientific Research under Grant FA9550-18-1-0246. Recommended by Associate Editor S. N. Dashkovskiy. (Corresponding author: Nard Strijbosch.) Nard Strijbosch and W.P.M.H. (Maurice) Heemels are with the Department of Mechanical Engineering, Eindhoven University of Technology, 5632AZ Eindhoven, The Netherlands (e-mail: [email protected]; [email protected]). Publisher Copyright: © 1963-2012 IEEE.

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N2 - We analyze the stability, and L2-gain properties of a class of hybrid systems that exhibit time-varying linear flow dynamics, periodic time-triggered jumps, and arbitrary nonlinear jump maps. This class of hybrid systems encompasses periodic event-triggered control, self-triggered control, and networked control systems including time-varying communication delays. New notions on the stability, and contractivity (L2-gain strictly smaller than 1) from the beginning of the flow, and from the end of the flow are introduced, and formal relationships are derived between these notions, revealing that some are stronger than others. Inspired by ideas from lifting, it is shown that the internal stability, and contractivity in L2-sense of a continuous-time hybrid system in the framework is equivalent to the stability, and contractivity in ℓ2-sense (meaning the ℓ2-gain is smaller than 1) of an appropriate time-varying discrete-time nonlinear system. These results recover existing works in the literature as special cases, and indicate that analysing different discrete-time nonlinear systems (of the same level of complexity) than in existing works yield stronger conclusions on the L2-gain.

AB - We analyze the stability, and L2-gain properties of a class of hybrid systems that exhibit time-varying linear flow dynamics, periodic time-triggered jumps, and arbitrary nonlinear jump maps. This class of hybrid systems encompasses periodic event-triggered control, self-triggered control, and networked control systems including time-varying communication delays. New notions on the stability, and contractivity (L2-gain strictly smaller than 1) from the beginning of the flow, and from the end of the flow are introduced, and formal relationships are derived between these notions, revealing that some are stronger than others. Inspired by ideas from lifting, it is shown that the internal stability, and contractivity in L2-sense of a continuous-time hybrid system in the framework is equivalent to the stability, and contractivity in ℓ2-sense (meaning the ℓ2-gain is smaller than 1) of an appropriate time-varying discrete-time nonlinear system. These results recover existing works in the literature as special cases, and indicate that analysing different discrete-time nonlinear systems (of the same level of complexity) than in existing works yield stronger conclusions on the L2-gain.

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