An Insight into Stability Conditions of Discrete-Time Systems With Time-Varying Delay

Abstract

This note studies an interesting phenomenon for stability conditions of discrete-time systems with time-varying delay. The underlying reason behind this phenomenon is revealed, and thereafter some conclusions are drawn: (i) Stability conditions of discrete-time systems with time-varying delay are generally divided into two types: those obtained by summation inequalities with free-matrix variables and those obtained by the combination of summation inequalities without free-matrix variables and the reciprocally convex lemma; (ii) The conservatism between the two types of stability conditions can not be theoretically compared. To clearly demonstrate this interesting phenomenon and meanwhile, to further verify these conclusions, several bounded real lemmas are obtained via different bounding-inequality methods and applied to a numerical example.INDEX TERMS Discrete-time system, Lyapunov functional, stability, summation inequality, time-varying delay.JUN CHEN received the Ph.D. degree in control theory and control engineering from the

A survey of inequality techniques for stability analysis of time‐delay systems

“…Remark The discrete‐time case for different stability conditions obtained by different inequality techniques was studied in Reference 118, in which similar conclusions were made.…”

“…To estimate the single-summation terms arising in (9), a great number of summation inequalities have been developed, for example, Jensen summation inequality, 88,111 Wirtinger-based summation inequality, 107,108,110 auxiliary-function-based summation inequalities, 43,[112][113][114] Bessel summation inequalities, 115 FMB summation inequalities, 116,117 and generalized FMB inequalities. 118 It is found that there usually exists a counter-part summation inequality corresponding to an integral one. So just like integral inequalities, these summation inequalities can also be classified into two types: ones with free matrices involved and ones without free matrices involved.…”

Control for Multiplicative Noise Systems With Packet Losses and Measurement Delays

“…In order to use the induction method, let any l with θ ≤ l ≤ N , and for j > l + 1, we assume that H j in 10is strictly positive definite, the optimal feedback controller is presented by (14), A j , B j satisfy the coupled Riccati difference equations (12), (13), and the relationship between λ j−1 and state x j is assumed to satisfy (17).…”

“…To be discussing clearly, A k and B k in (12) and (13) are relabeled as A k (N ), B k (N ), respectively. In the meanwhile, we denote the terminal values as A N +1 (N ) = 0 and B N +1 (N ) = 0 to analyse the infinite horizon case.…”

“…However, contention based networks may exhibit bandwidth limitations, date packet disordering, limitations arising from data quantization and time-varying sampling period(see [9], [10], [11]). Time delays are normally caused by signal transmission and traffic congestion, and packet losses are by virtue of nodes failure or long time delays(see [12], [13], [14], [15]). For nonlinear NCSs with packet losses, there are some developments in virtue of applications of practical control systems.…”