Stability Analysis for Linear Systems with a Differentiable TimeVarying Delay via Auxiliary EquationBased Method
Abstract
:1. Introduction
 (1)
 Motivated by the method in [33], the auxiliary equation $\ddot{x}(t)=A\dot{x}(t)+(1\dot{h}(t)){A}_{d}\dot{x}(th(t))$ is utilized to investigate the stability of the systems with a differentiable timevarying 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 $2{\int}_{b}^{a}{\dot{x}}^{T}(s)U\ddot{x}(s)ds={\dot{x}}^{T}(a)U\dot{x}(a){\dot{x}}^{T}(b)U\dot{x}(b)$, 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 delayproducttype augmented LKF is constructed, which can utilize more system information, such as $\ddot{x}(t)$, $\ddot{x}(th(t))$ and $\ddot{x}(th)$. 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/$\mathit{\mu}$  0.1  0.5  0.8  NoDVs 

Theorem 3 [23]  4.8562  3.1831  2.7391  $59.5{n}^{2}+14.5n$ 
Theorem 1 [14]  4.867  3.12  –  $53.5{n}^{2}+8.5n$ 
Theorem 2(C1) [42]  4.940  3.304  2.877  $69{n}^{2}+12n$ 
Theorem 1 [43]  4.945  3.314  2.882  $100.5{n}^{2}+8.5n$ 
Corollary 1(II) [44]  4.966  3.395  2.983  $85{n}^{2}+15n$ 
Theorem 1 [15]  4.996  3.251  2.867  $38{n}^{2}+9n$ 
Theorem 8 (N = 4) [45]  5.01  3.19  2.70  $146.5{n}^{2}+9.5n$ 
Corollary 1  4.8662  3.3349  2.9886  $66{n}^{2}+8n$ 
Theorem 1  5.0213  3.6032  3.2235  $205{n}^{2}+13n$ 
Methods/$\mathit{\mu}$  0.2  0.5  0.8  NoDVs 

Theorem 1 [46]  4.5179  2.4158  1.8384  $142{n}^{2}+18n$ 
Theorem 3 [23]  4.6380  2.5898  2.0060  $59.5{n}^{2}+14.5n$ 
Corollary 1(II) [44]  4.947  2.801  2.137  $85{n}^{2}+15n$ 
Corollary 2 [3]  4.969  2.774  2.117  $235{n}^{2}+34n$ 
Theorem 2 (N = 5) [8]  4.985  2.806  2.148  $103.5{n}^{2}+15.5n$ 
Theorem 2 [17]  4.997  2.814  2.149  $307{n}^{2}+13n$ 
Theorem 1 [2]  5.0035  2.8096  2.1499  $249.5{n}^{2}+15.5n$ 
Corollary 1  4.9481  3.1531  2.7024  $66{n}^{2}+8n$ 
Theorem 1  5.1073  3.3984  2.9053  $205{n}^{2}+13n$ 
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Yin, Z.; Jiang, X.; Zhang, N.; Zhang, W. Stability Analysis for Linear Systems with a Differentiable TimeVarying Delay via Auxiliary EquationBased 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 TimeVarying Delay via Auxiliary EquationBased 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 TimeVarying Delay via Auxiliary EquationBased Method" Electronics 11, no. 21: 3492. https://doi.org/10.3390/electronics11213492