Stabilization of Hybrid Systems by Event-Triggered Control Based on Discrete-Time State Observations

Given an unstable hybrid stochastic differential equation (SDE) ⁢⁡()=(⁡(),⁡(), )⁢+⁡(⁡(),⁡(),)⁢⁢⁡(), we can design a feedback control ⁡(⁡(⌊/⌋⁢),⁡(),) so that the controlled SDE ⁢⁡()=(⁡(⁡(),⁡(),)+⁡(⁡(⌊/⌋⁢),⁡(),))⁢⁢+(⁡(),⁡(), )⁢⁡() becomes stable, where >0 and ⌊/⌋ is the integer part of /. Noting that the control ⁡(⁡(⌊/⌋⁢),⁡(),) is based on the state observations at discrete times 0,,2⁢,⋯, we see that this stabilization problem is significantly different from the classical one where the feedback control ⁡(⁡(),⁡(),) is based on continuous-time state observations. This stabilization problem was initiated by Mao in 2013 and has been intensively studied by many authors for the past ten years. In this paper, we will further develop an event-triggered control based on a trajectory of the state or output observations at discrete times. This is significantly different from most of the existing papers where an event-triggered mechanism for SDEs is designed based on a criterion in terms of the mathematical expectation of the system state. The novel approach in this paper is to relate the stability of the event-triggered controlled SDE to the stability of the corresponding uncertain SDE.