Vector Lyapunov functions for stability and stabilization of differential repetitive processes

Speech Spectral Transfer Function

“…The connection between the degree of operator's fatigue and parameters of his speech is described in [4]. This relation is based on the A. M. Lyapunov's theory of stability [5,6]. The influence of the acoustic noises on the speech recognition performance score is analysed in papers [7,8].…”

Finite-region boundedness and stabilization for 2D continuous-discrete systems in Roesser model

“…As such, the research on two-dimensional continuous-discrete systems has been a hot area in control field. Especially, the stability analysis for two-dimensional continuous-discrete systems has attracted much attention of some researchers over the last few decades, and several interesting findings in linear and non-linear frameworks have been obtained (see Xiao, 2001;Benton et al, 2002;Knorn & Middleton, 2013a;Chesi & Middleton, 2014;Knorn & Middleton, 2016;Galkowski et al, 2016;Pakshin et al, 2016;Wang et al, 2017 and references therein). For example, in the linear setting, Xiao (2001) considered three models of two-dimensional continuous-discrete systems and gave sufficient and necessary conditions for their Lyapunov asymptotic stability (LAS)-based two-dimensional characteristic polynomial.…”

“…For non-linear systems, Knorn & Middleton (2013b) modeled the homogeneous, unidirectional non-linear vehicle strings as a general two-dimensional continuous-discrete non-linear system and presented a sufficient condition for stability of this system. In Galkowski et al (2016), the exponential stability conditions for non-linear differential repetitive processes were established by using a vector Lyapunov function-based approach. However, most of these results related to stability were focused on LAS or exponential stability.…”

“…Until recently, the authors in Zhang & Wang (2016a,b) studied the problems of finite-region stability (FRS) and finite-region boundedness (FRB) for two-dimensional discrete Roesser models and Fornasini-Marchesini second models. For two-dimensional continuousdiscrete systems, it is worth mentioning that there are some results on the problems of exponential stability, weak stability and asymptotic stability over bounded region for repetitive processes (see Rogers et al, 2007;Galkowski et al, 2016;Pakshin et al, 2016 and references therein), where the variable x h (t, k) changes on finite interval [0, T]. Clearly, these problems are completely different from the finiteregion control problem considered in Zhang & Wang (2016a,b) in that their methods can exhibit the transient performance of two-dimensional discrete systems over a given finite region by setting the state variables of the system less than a particular threshold.…”