耦合线性常微分方程和双曲型偏微分方程的H$_1$-ISS分析与边界控制设计
H$_1$-ISS Analysis and Boundary Control Design for Coupled Linear ODEs and Hyperbolic PDEs
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中文总结 AI 辅助
研究耦合线性ODEs和双曲型PDEs系统的边界控制设计，将基于李雅普诺夫的ISS条件扩展到H$_1$设置，推导稳定性分析和控制设计条件，通过数值例子验证所提方法的有效性和潜力。
中文摘要 AI 辅助
本文考虑分布式参数系统边界控制器设计的一般问题。控制目标是闭环系统的渐近稳定性以及影响测量和边界条件的外部干扰的干扰衰减。具体研究由耦合线性常微分方程（ODEs）和齐次线性双曲型偏微分方程（PDEs）控制的系统的输入到状态稳定性（ISS）和镇定，其中激励、传感和干扰输入位于系统边界。首先，在对干扰输入的温和假设下，将先前为类似系统建立的基于李雅普诺夫的ISS条件扩展到H$_1$设置。该扩展不仅确保由非零干扰和兼容初始条件驱动的系统状态的H$_1$范数有界，还提供了系统H$_1$范数的上界。随后，应用这些理论结果推导出以线性矩阵不等式表示的稳定性分析和控制设计条件。提供了数值例子来说明所提方法的有效性和潜力。
英文摘要
This paper considers the general problem of the design of boundary controllers for distributed parameter systems. The control objectives are the asymptotic stability of the closed-loop system, as well as the disturbance attenuation of exogenous disturbances affecting the measurements and boundary conditions. More specifically, this work studies input-to-state stability (ISS) and stabilization for systems governed by coupled linear ordinary differential equations (ODEs) and homogeneous linear hyperbolic partial differential equations (PDEs), where actuation, sensing, and disturbance inputs are located at the system boundaries. First, we extend a previously established Lyapunov-based ISS condition for a similar class of systems to the H$_1$ setting under mild assumptions on the disturbance inputs. This extension not only ensures that the H$_1$-norm of the system states driven by non-vanishing disturbances and compatible initial conditions are bounded, but also provides an upper bound on the system H$_1$-norm. Subsequently, these theoretical results are applied to derive stability analysis and control design conditions expressed in terms of linear matrix inequalities. Numerical examples are provided to illustrate the effectiveness and potential of the proposed approach.
