[ Article ]

The Journal of Korean Institute of Information Technology - Vol. 23, No. 3, pp.227-230

ISSN: 1598-8619 (Print) 2093-7571 (Online)

Print publication date 31 Mar 2025

Received 13 Feb 2025 Revised 12 Mar 2025 Accepted 15 Mar 2025

DOI: https://doi.org/10.14801/jkiit.2025.23.3.227

A Refined Stability Criterion for Time-Delay Systems Via a New Lyapunov-Krasovskii Functional

Bum Yong Park^*

; JaeWook Shin^**

; Won Il Lee^**

*Department of IT Convergence Engineering, Kumoh National Institute of Technology
**School of Electronic Engineering, Kumoh National Institute of Technology

Correspondence to: Won Il Lee School of Electronic Engineering, Kumoh National Institute of Technology Tel.: +82-54-478-7429, Email: wilee@kumoh.ac.kr

Abstract

This paper concerns the problem of stability analysis for time-delay systems. A Lyapunov-Krasovskii Functional(LKF) is newly designed through the introduction of double integral terms that contain the information about the time-varying delay over the integral range. Based on an appropriate integral inequality, an improved stability criterion is derived. A numerical example is presented to exhibit the enhancements in the proposed stability condition.

초록

본 논문에서는 시간지연 시스템의 안정성 해석 문제를 다룬다. 시간지연의 정보를 적분 구간에 충분히 반영하는 이중 적분항을 도입한 새로운 LKF(Lyapunov-Krasovskii functional)을 제안하고, 이를 위한 적절한 적분 부등식을 활용하여 보다 개선된 시간지연 시스템의 안정성 조건을 도출한다. 제안하는 안정성 조건의 우수성을 보이기 위해 수치 예제를 제시한다.

Keywords:

time-delay systems, stability analysis, lyapunov-krasovkii functionals, time-varying delays

Ⅰ. Introduction

Stability analysis of time-varying delay systems has been investigated in two main directions[1]. The first is to derive less conservative stability criteria that can show stability under larger time delays[2]-[6], and the second is to derive less complex stability criteria with fewer decision variables[7][8].

In this paper, we further investigate methods to establish a less complex stability result for delayed systems. The contributions of the paper can be described in two ways: (1) an effective LKF is newly designed based on new double integral terms. (2) an enhanced stability criterion is derived via a choice of suitable integral inequality for the new LKF.

A well-known example is presented to illustrate the enhancements of the proposed approach. Notations. Throughout the paper, $R n$ and $R m × n$ denote the n-dimensional Euclidean space and the set of real matrices with m×n dimensions, respectively. X>0 means that X is a real symmetric positive definite matrix. The notation Sym{A} stands for A+A^T. I_n and 0_m×n represent n×n identity matrix and m×n zero matrix, respectively.

Ⅱ. Main Results

Consider the following time-varying delay system:

x ˙ t = A x t + A d x t - τ t, x t = ψ t, t ∈ - τ ¯, 0

(1)

where $x t ∈ R n$ represents the state vector and ψ(t) is the initial condition, and τ(t) is the time-varying delay such that $0 ≤ τ t ≤ τ ¯, μ 1 ≤ τ ˙ t ≤ μ 2 < 1, τ ~ t = τ ¯ - τ t$ , and $τ^t = 1 - τ ˙ t$ .

Before proceeding our main results, the notations used in this section are denoted as follows:

η 1 t = x T t x T t - τ t x T t - τ ¯ ∫ t - τ t t x T s d s ∫ t - τ ¯ t - τ t x T s d s 1 τ t ∫ - τ t 0 ∫ t + θ t x T s d s d θ 1 τ ¯ - τ t ∫ - τ ¯ - τ t ∫ t + θ t - τ t x T s d s d θ T, η 2 t, s = x ˙ T s x T s x T t - τ t ∫ s t x ˙ T θ d θ ∫ t - τ ¯ s x ˙ T θ d θ ∫ t - τ t s x ˙ T θ d θ T, ξ t = x T t x T t - τ t x T t - τ ¯ x ˙ T t - τ t x ˙ T t - τ ¯ 1 τ t ∫ t - τ t t x T s d s 1 τ ¯ - τ t ∫ t - τ ¯ t - τ t x T s d s 1 τ t 2 ∫ - τ t 0 ∫ t + θ t x T s d s d θ 1 τ ¯ - τ t 2 ∫ - τ ¯ τ t ∫ t + θ t - τ t x T s d s d θ T, e i = 0 n × i - 1 n I n 0 n × 9 - i n, i = 1,2, ⋯, 9

Theorem 1. For specified scalars $τ ¯ > 0$ and $μ 1 < μ 2 < 1$ , asymptotic stability of the system (1) is guaranteed if symmetric matrices $P ∈ R 7 n × 7 n > 0, Q 1 ∈ R 6 n × 6 n > 0, Q 2 ∈ R 6 n × 6 n > 0, Z 1 ∈ R n × n > 0, Z 2 ∈ R n × n > 0$ and matrices $N 1 ∈ R 9 n × 3 n$ and $N 2 ∈ R 9 n × 3 n$ exist, satisfying LMIs (2) and (3) for i=1, 2:

Ω 0, μ i + S y m N 1 M 1 + N 2 M 2 τ ¯ N 2 τ ¯ N 2 T - Z 2 ― < 0

(2)

Ω τ ¯, μ i + S y m N 1 M 1 + N 2 M 2 τ N 1 τ ¯ N 1 T - Z 1 ― μ i < 0

(3)

where

Ω τ t, τ ˙ t = S y m Ξ 0 T τ t P Ξ 1 τ ˙ t + Π 1 T Q 1 Π 1 - Π 3 T Q 2 Π 3 + τ^t Π 2 T Q 2 - Q 1 Π 2 + e 0 T τ t Z 1 + τ ~ t Z 2 e 0 + S y m Π 4 T Q 1 Π 5 + + Π 6 T Q 2 Π 5

with

Ξ 0 τ t = e 1 T e 2 T e 3 T τ t e 6 T τ ~ t e 7 T τ t e 8 T τ ~ t e 9 T T, Ξ 1 τ ˙ t = e 0 T τ^t e 4 T e 5 T e 1 T - τ^t e 2 T τ^t e 2 T - e 3 T e 1 T - τ^t e 6 T - τ ˙ t e 8 T τ^t e 2 T - e 7 T + τ ˙ t e 9 T T, Π 1 = e 0 T e 1 T e 2 T 0 9 n × n e 1 T - e 3 T e 1 T - e 2 T T, Π 2 = e 4 T e 2 T e 2 T e 1 T - e 2 T e 2 T - e 3 T 0 9 n × n T, Π 3 = e 5 T e 3 T e 2 T e 1 T - e 3 T 0 0 n × n e 3 T - e 2 T T, Π 4 = e 1 T - e 2 T τ t e 6 T τ t e 2 T τ t e 1 T - e 6 T τ t e 6 T - e 3 T τ t e 6 T - e 2 T T, Π 5 = 0 9 n × n 0 9 n × n τ^t e 4 T e 0 T - e 5 T - τ^t e 4 T T, Π 6 = e 2 T - e 3 T τ ~ t e 7 T τ ~ t e 2 T τ ~ t e 1 T - e 7 T τ ~ t e 7 T - e 3 T τ ~ t e 7 T - e 2 T T, M 1 = e 1 - e 2 e 1 + e 2 - 2 e 6 e 1 - e 2 + 6 e 6 - 12 e 8, M 2 = e 2 - e 3 e 2 + e 3 - 2 e 7 e 2 - e 3 + 6 e 7 - 12 e 9, Z ¯ 1 τ ˙ t = d i a g Z^1 τ ˙ t, 3 Z^1 τ ˙ t, 5 Z^1 τ ˙ t, Z 2 ― = d i a g Z 2, 3 Z 2, 5 Z 2

and $Z^1 τ ˙ t = τ^t Z 1 + τ ˙ t Z 2, e 0 = A e 1 + A d e 2$

Proof. Consider a Lyapunov-Krasovskii functional as

V t = η 1 T t P η 1 t + ∫ t - τ t t η 2 T t, s Q 1 η 2 t, s d s + ∫ t - τ ¯ t - τ t η 2 T t, s Q 2 η 2 t, s d s + ∫ - τ t 0 ∫ t t + θ t x ˙ T s Z 1 x ˙ s d s d θ + ∫ - τ ¯ - τ t ∫ t + θ t x ˙ T s Z 2 x ˙ s d s d θ

(4)

The time derivative of (4) can be computed as follows:

V ˙ t = ξ T t Ω τ t, τ ˙ t ξ t - ∫ t - τ t t x ˙ T s Z^1 τ ˙ t x ˙ s d s - ∫ t - τ ¯ t - τ t x ˙ T s Z 2 x ˙ s d s

(5)

Applying the generalized free-matrix-based integral inequality(GFMBII) [4] to the resulting two integral terms in (5) gives

- ∫ t - τ t t x ˙ T s Z^1 τ ˙ t x ˙ s d s ≤ ξ T t S y m N 1 M 1 - τ t N 1 Z ¯ 1 τ ˙ t - 1 N 1 T ξ t

(6)

- ∫ t - τ ¯ t - τ t x ˙ T s Z 2 x ˙ s d s ≤ ξ T t S y m N 2 M 2 - τ ~ t N 2 Z ¯ 2 - 1 N 2 T ξ t

(7)

Combining (6)-(7) into (5) yields

V ˙ t ≤ ξ T t Φ τ t, τ ˙ t ξ t

(8)

where

Φ τ t, τ ˙ t = Ω τ t, τ ˙ t + S y m N 1 M 1 + N 2 M 2 - τ t N 1 Z 1 ― τ ˙ t - 1 N 1 T - τ ~ t N 2 Z ¯ 2 - 1 N 2 T

By applying the Schur complement, we can get $Φ τ t, τ ˙ t < 0$ if LMIs (2-3) are satisfied. Since $V ˙ t < 0$ is ensured, also asymptotic stability of the system (1) is guaranteed by the Lyapunov stability theorem, which ends the proof.

Remark 1. Recently, in [5], new double integral terms have been proposed in the LKF which contain the augmented state vector and have the information on the time-varying delay on their integral range. Compared to the LKF in [5], the integral ranges are identical but the double integral terms in (4) only utilize a single state derivative term, whereas those in [5] employ the augmented vectors. The augmented LKF can diminish the conservatism of the developed stability criterion but also increases the number of variables. Therefore, the proposed LKF in (4) can lead to a less complex stability condition. Nonetheless the performance behavior remains the same with the choice of a suitable integral inequality, which will be shown in the next section.

Remark 2. In [5], the generalized integral inequality based on free matrices(GIIBFM) has been developed and employed to effectively deal with the augmented LKF. Meanwhile, in (5), the GFMBII [4] is adopted since it provides identical bounds to the GIIBFM with fewer decision variables when no augmented vectors are used in the integral terms.

Ⅲ. Numerical Example

Example 1. Consider the system (1) with

A = - 2.0 0.0 0.0 - 0.9, A d = - 1.0 0.0 - 1.0 - 1.0

where μ₂=-μ₁. For various μ₂, the maximum admissible upper bounds of the delay( $τ ¯$ ) and the number of decision variables(N_v) are summarized in Table 1 alongside the results from previous studies.

Table 1.

Admissible upper bounds for different μ2

As shown in Table 1, Theorem 1 outperforms the stability conditions proposed in [3]-[5]. In terms of the performance, Theorem 1 can yield larger delay upper bounds than those derived in [3][4]. Furthermore, Theorem 1 can yield same delay upper bounds as those in [5] with much smaller number of decision variables, which clearly shows that Theorem 1 is more refined than the stability condition in [5] from a complexity perspective.

Ⅳ. Conclusion

This paper has investigated stability criterion for time-delay systems via the novel LKF. The refined stability condition has been derived based on the new LKF and the use of an appropriate integral inequality. A numerical example has verified the superiority of the proposed approach.

Acknowledgments

This research was supported by Kumoh National Institute of Technology(2022)

References

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Table 1.

Admissible upper bounds for different μ2

μ₂(=-μ₁)	0.1	0.2	0.8	N_v
[3]	4.83	4.14	2.71	142n² + 18n
[4] (N = 2)	4.92	4.21	2.79	115n² + 10n
[5] (N = 2)	4.93	4.23	2.80	154.5n² + 11.5n
Theorem 1 (proposed)	4.93	4.23	2.80	115.5n² + 10.5n