Using a single bit to stabilise an n-dimensional quantised non-linear feedforward system

Abstract

This study investigates the global asmptotic stabilisability of an n-dimensional quantised feedforward non-linear system. The system’s dynamics are assumed to be locally Lipschitz. So the ranges of the state variables are tightly related to the parameters of the quantisation procedure, that is, there exists coupling between quantisation and control. To save network transmission bandwidth, less quantisation bits (per sample) are preferred. In De Persis’ paper, n bits (per sample) are shown to be enough to stabilise the n-dimensional system by taking a time-invariant quantiser, which assigns 1 bit to each state variable (dimension). We design a time-varying quantiser and can globally stabilise the system with only 1 bit, instead of n bits. That single bit is assigned to the most ‘important’ state variable at each sampling instant, which yields the time-varying coupling among the quantisation errors of state variables besides the coupling between quantisation and control. Owing to the well-designed structure of our quantiser, we can place exponentially converging upper bounds on the quantisation errors of state variables and well handle both types of couplings, so that the global stability of the system is guaranteed with 1 bit. An example is used to demonstrate the effectiveness of the proposed quantiser.