$A_h = -\frac{k_i}{h}V^2$, $A_1 = \frac{k_i}{h}V^2$, $\varpi(\theta) \equiv 1$ and $a_1(\tau) = \cos(4\pi\tau)$, which leads to $\Delta a_i(t) \equiv 0$.

We set $\alpha = 0$ and verify the feasibility of (4.18) and $\delta_{2,x}\eta_\varpi < e^{-\alpha h}$. We fix $\epsilon = h = 0.02$ as in [11] and verify (4.18) and $\delta_{2,x}\eta_\varpi < e^{-\alpha h}$ to maximize $k_i$ which preserves feasibility. The result is $\max k_i = 6.96 \cdot 10^{-4}$, which is 3.03 times larger than $\max k_i = 2.2923 \cdot 10^{-4}$, obtained in [11]. Finally, our approach can handle the case of decoupled $\epsilon$ and $h$. To illustrate this, consider $k_i = 2.2923 \cdot 10^{-4}$, $\epsilon = 0.02$ and set $a_1(t/\epsilon) = \cos(\frac{4\pi t}{\epsilon})$. We verify the conditions of Theorem 4.1 to obtain the largest value of $h$ which preserves feasibility. The result is given by $\max h = 0.0657$.

## 5 Conclusions

We introduced a novel quantitative methodology for deriving ISS-like/stability properties for linear continuous-time systems. The presented methodology relies on a new system presentation, in conjunction with a delay-free system transformation. Compared to the recent time-delay approach to averaging, the new method presents a simpler ISS analysis of the transformed non-delayed system that employs Lyapunov functions and does not need additional solution bounds for times smaller than the time-scale parameter, and significantly improve the results in the numerical examples. However, the time-delay approach is applicable not just to classical averaging as considered in the present paper, but also to Lie-brackets-based averaging [33], [34] where application of the non-delay transformation seems to be questionable. Future work may include applications of the method to control problems that employ averaging.

## References

[1] C. Albea and A. Seuret. Time-triggered and event-triggered control of switched affine systems via a hybrid dynamical approach. *Nonlinear Analysis: Hybrid Systems*, 41:101039, 2021.

[2] N. N. Bogoliubov and I. A. Mitropolskij. Asymptotic methods in the theory of non-linear oscillations, volume 10. CRC Press, 1961.

[3] F. Bullo. Averaging and vibrational control of mechanical systems. *SIAM Journal on Control and Optimization*, 41(2):542–562, 2002.

[4] E. Butcher and B. Mann. Stability analysis and control of linear periodic delayed systems using Chebyshev and temporal finite element methods. In *Delay Differential Equations*, pages 93–129. Springer, 2009.

[5] B. Caiazzo, E. Fridman, and X. Yang. Averaging of systems with fast-varying coefficients and non-small delays with application to stabilization of affine systems via time-dependent switching. *Nonlinear Analysis: Hybrid Systems*, 48:101307, 2023.

[6] X. Cheng, Y. Tan, and I. Mareels. On robustness analysis of linear vibrational control systems. *Automatica*, 87:202–209, 2018.

[7] E. Fridman. Effects of small delays on stability of singularly perturbed systems. *Automatica*, 38(5):897–902, 2002.

[8] E. Fridman. *Introduction to time-delay systems: analysis and control*. Birkhauser, Systems and Control: Foundations and Applications, 2014.

[9] E. Fridman and J. Zhang. Averaging of linear systems with almost periodic coefficients: A time-delay approach. *Automatica*, 122:109287, 2020.

[10] M. A. Gomez, G. Ochoa, and S. Mondié. Necessary exponential stability conditions for linear periodic time-delay systems. *International Journal of Robust and Nonlinear Control*, 26(18):3996–4007, 2016.

[11] R. Griñó, R. Ortega, E. Fridman, J. Zhang, and F. Mazenc. A behavioural dynamic model for constant power loads in single-phase ac systems. *Automatica*, 131:109744, 2021.

[12] J. Hale and S. M. V. Lunel. Strong stabilization of neutral functional differential equations. *IMA J. Math. Contr. Information*, 19:5–23, 2002.

[13] G. Hek. Geometric singular perturbation theory in biological practice. *Journal of mathematical biology*, 60(3):347–386, 2010.

[14] T. Insperger and G. Stépán. *Semi-discretization for time-delay systems: stability and engineering applications*, volume 178. Springer Science & Business Media, 2011.

[15] R. Katz, F. Mazenc, and E. Fridman. Stability by averaging via time-varying lyapunov functions. *22nd IFAC World Congress. To appear.*, 2023.

[16] H. K. Khalil. Nonlinear systems third edition. 2001.

[17] P. V. Kokotovic and H. K. Khalil. *Singular perturbations in systems and control*. IEEE press, 1986.

[18] M. Krstić and H.-H. Wang. Stability of extremum seeking feedback for general nonlinear dynamic systems. *Automatica*, 36(4):595–601, 2000.

[19] B. Lehman and S. P. Weibel. Fundamental theorems of averaging for functional differential equations. *Journal of differential equations*, 152(1):160–190, 1999.

[20] O. N. Letyagina and A. P. Zhabko. Robust stability analysis of linear periodic systems with time delay. *International Journal of Modern Physics A*, 24(05):893–907, 2009.

[21] F. Mazenc and M. Malisoff. Extensions of Razumikhin's theorem and Lyapunov-Krasovskii functional constructions for time-varying systems with delay. *Automatica*, 78:1–13, 2017.

[22] F. Mazenc, M. Malisoff, and M. S. De Queiroz. Further results on strict Lyapunov functions for rapidly time-varying nonlinear systems. *Automatica*, 42(10):1663–1671, 2006.

[23] S. Meerkov. Principle of vibrational control: theory and applications. *IEEE Transactions on Automatic Control*, 25(4):755–762, 1980.
[24] W. Michiels and S.-I. Niculescu. *Stability, control, and computation for time-delay systems: an eigenvalue-based approach*. SIAM, 2014.

[25] L. Moreau and D. Aeyels. Practical stability and stabilization. *IEEE Transactions on Automatic Control*, 45(8):1554–1558, 2000.

[26] E. Mostacciuolo, S. Trenn, and F. Vasca. A smooth model for periodically switched descriptor systems. *Automatica*, 136:110082, 2022.

[27] J. A. Murdock. *Perturbations: theory and methods*. SIAM, 1999.

[28] H. Sandberg and E. Möllerstedt. Periodic modelling of power systems. *IFAC Proceedings Volumes*, 34(12):89–94, 2001.

[29] O. Solomon and E. Fridman. New stability conditions for systems with distributed delays. *Automatica*, 49(11):3467–3475, 2013.

[30] A. R. Teel, L. Moreau, and D. Nesic. A unified framework for input-to-state stability in systems with two time scales. *IEEE Transactions on Automatic Control*, 48(9):1526–1544, 2003.

[31] X. Xie and J. Lam. Guaranteed cost control of periodic piecewise linear time-delay systems. *Automatica*, 94:274–282, 2018.

[32] J. Zhang and E. Fridman. $L_2$-gain analysis via time-delay approach to periodic averaging with stochastic extension. *Automatica*, 137:110126, 2022.

[33] J. Zhang and E. Fridman. Lie-brackets-based averaging of affine systems via a time-delay approach. *Automatica*, 2023.

[34] Y. Zhu and E. Fridman. Extremum seeking via a time-delay approach to averaging. *Automatica*, 135:109965, 2022.
[24] W. Michiels and S.-I. Niculescu. *Stability, control, and computation for time-delay systems: an eigenvalue-based approach*. SIAM, 2014.

[25] L. Moreau and D. Aeyels. Practical stability and stabilization. *IEEE Transactions on Automatic Control*, 45(8):1554–1558, 2000.

[26] E. Mostacciuolo, S. Trenn, and F. Vasca. A smooth model for periodically switched descriptor systems. *Automatica*, 136:110082, 2022.

[27] J. A. Murdock. *Perturbations: theory and methods*. SIAM, 1999.

[28] H. Sandberg and E. Möllerstedt. Periodic modelling of power systems. *IFAC Proceedings Volumes*, 34(12):89–94, 2001.

[29] O. Solomon and E. Fridman. New stability conditions for systems with distributed delays. *Automatica*, 49(11):3467–3475, 2013.

[30] A. R. Teel, L. Moreau, and D. Nesic. A unified framework for input-to-state stability in systems with two time scales. *IEEE Transactions on Automatic Control*, 48(9):1526–1544, 2003.

[31] X. Xie and J. Lam. Guaranteed cost control of periodic piecewise linear time-delay systems. *Automatica*, 94:274–282, 2018.

[32] J. Zhang and E. Fridman. $L_2$-gain analysis via time-delay approach to periodic averaging with stochastic extension. *Automatica*, 137:110126, 2022.

[33] J. Zhang and E. Fridman. Lie-brackets-based averaging of affine systems via a time-delay approach. *Automatica*, 2023.

[34] Y. Zhu and E. Fridman. Extremum seeking via a time-delay approach to averaging. *Automatica*, 135:109965, 2022.