Stability Analysis for Linear Systems with a Differentiable Time-Varying Delay via Auxiliary Equation-Based Method
Abstract
:1. Introduction
- (1)
- Motivated by the method in [33], the auxiliary equation is utilized to investigate the stability of the systems with a differentiable time-varying delay, and thus the information of delay derivative can be captured well and be used to derive a less conservative stability condition.
- (2)
- Inspired by the fact that , two state augmented zero equalities are introduced, which can help reduce the conservatism of the obtained stability condition.
- (3)
- On the basis of the system equation and the auxiliary equation, a new delay-product-type augmented LKF is constructed, which can utilize more system information, such as , and . Then, based on the LKF and by employing some vital lemmas, adding zero terms, and the convex analysis method, a relaxed stability condition is proposed. Finally, to illustrate the merit of the obtained stability condition, two typical numerical examples are given.
2. Problem Statement and Preliminaries
3. Stability Conditions
Algorithm 1: Obtaining the optimal value of h based on Theorem 1 or Corollary 1. |
|
4. Numerical Examples
5. Conclusions
Author Contributions
Funding
Institutional Review Board Statement
Informed Consent Statement
Data Availability Statement
Conflicts of Interest
References
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Methods/ | 0.1 | 0.5 | 0.8 | NoDVs |
---|---|---|---|---|
Theorem 3 [23] | 4.8562 | 3.1831 | 2.7391 | |
Theorem 1 [14] | 4.867 | 3.12 | – | |
Theorem 2(C1) [42] | 4.940 | 3.304 | 2.877 | |
Theorem 1 [43] | 4.945 | 3.314 | 2.882 | |
Corollary 1(II) [44] | 4.966 | 3.395 | 2.983 | |
Theorem 1 [15] | 4.996 | 3.251 | 2.867 | |
Theorem 8 (N = 4) [45] | 5.01 | 3.19 | 2.70 | |
Corollary 1 | 4.8662 | 3.3349 | 2.9886 | |
Theorem 1 | 5.0213 | 3.6032 | 3.2235 |
Methods/ | 0.2 | 0.5 | 0.8 | NoDVs |
---|---|---|---|---|
Theorem 1 [46] | 4.5179 | 2.4158 | 1.8384 | |
Theorem 3 [23] | 4.6380 | 2.5898 | 2.0060 | |
Corollary 1(II) [44] | 4.947 | 2.801 | 2.137 | |
Corollary 2 [3] | 4.969 | 2.774 | 2.117 | |
Theorem 2 (N = 5) [8] | 4.985 | 2.806 | 2.148 | |
Theorem 2 [17] | 4.997 | 2.814 | 2.149 | |
Theorem 1 [2] | 5.0035 | 2.8096 | 2.1499 | |
Corollary 1 | 4.9481 | 3.1531 | 2.7024 | |
Theorem 1 | 5.1073 | 3.3984 | 2.9053 |
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Yin, Z.; Jiang, X.; Zhang, N.; Zhang, W. Stability Analysis for Linear Systems with a Differentiable Time-Varying Delay via Auxiliary Equation-Based Method. Electronics 2022, 11, 3492. https://doi.org/10.3390/electronics11213492
Yin Z, Jiang X, Zhang N, Zhang W. Stability Analysis for Linear Systems with a Differentiable Time-Varying Delay via Auxiliary Equation-Based Method. Electronics. 2022; 11(21):3492. https://doi.org/10.3390/electronics11213492
Chicago/Turabian StyleYin, Zongming, Xiefu Jiang, Ning Zhang, and Weihua Zhang. 2022. "Stability Analysis for Linear Systems with a Differentiable Time-Varying Delay via Auxiliary Equation-Based Method" Electronics 11, no. 21: 3492. https://doi.org/10.3390/electronics11213492