We treat input-to-state stability-like (ISS-like) estimates for perturbed linear continuous-time systems with multiple time-scales, under the assumption that the averaged, unperturbed, system is exponentially stable. Such systems contain rapidly-varying, piecewise continuous and almost periodic coefficients with small parameters (time-scales). Our method relies on a novel delay-free system transformation in conjunction with a new system presentation, where the rapidly-varying coefficients are scalars that have zero average. We employ time-varying Lyapunov functions for ISS-like analysis. The analysis yields LMI conditions, leading to explicit bounds on the small parameters, decay rate and ISS-like gains. The novel system presentation plays a crucial role in the ISS-like analysis by allowing to derive essentially less conservative upper bounds on terms containing the small parameters. The obtained LMIs are accompanied by suitable feasibility guarantees. We further extend our approach to rapidly-varying systems subject to either discrete (constant/fast-varying) or distributed delays, where our approach decouples the effects of the delay and small parameters on the stability of the system, and leads to LMI conditions for stability of systems with non-small delays. Extensive numerical examples show that, compared to the existing results, our approach essentially enlarges the small parameter and delay bounds for which the ISS-like/stability property of the original system is preserved.

中文翻译：

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通过无延迟变换实现基于平均的稳定性的构造方法

我们假设平均的、未扰动的系统是指数稳定的，对具有多个时间尺度的扰动线性连续时间系统进行类输入状态稳定性（ISS 类）估计。此类系统包含快速变化、分段连续且几乎周期性的系数以及小参数（时间尺度）。我们的方法依赖于一种新颖的无延迟系统变换以及一种新的系统表示，其中快速变化的系数是平均值为零的标量。我们采用时变李雅普诺夫函数进行类似 ISS 的分析。分析得出 LMI 条件，从而对小参数、衰减率和类似国际空间站的增益产生明确的界限。新颖的系统表示在类似国际空间站的分析中发挥着至关重要的作用，因为它允许在包含小参数的项上导出本质上不太保守的上限。获得的 LMI 附有适当的可行性保证。我们进一步将我们的方法扩展到受离散（恒定/快速变化）或分布式延迟影响的快速变化系统，其中我们的方法解耦了延迟和小参数对系统稳定性的影响，并导致 LMI 条件具有非小延迟的系统的稳定性。大量的数值例子表明，与现有的结果相比，我们的方法本质上扩大了小参数和延迟界限，从而保留了原始系统的类国际空间站/稳定性特性。