Abstract We develop a hybrid boundary feedback law for a class of scalar, linear, first-order hyperbolic PDEs, for which the state measurements or the control input are subject to quantization. The quantizers considered are Lipschitz functions, which can approximate arbitrarily closely typical piecewise constant, taking finitely many values, quantizers. The control design procedure relies on the combination of two ingredients—A nominal backstepping controller, for stabilization of the PDE system in the absence of quantization, and a switching strategy, which updates the parameters of the quantizer, for compensation of the quantization effect. Global asymptotic stability of the closed-loop system is established through utilization of Lyapunov-like arguments and derivation of solutions’ estimates, providing explicit estimates for the supremum norm of the PDE state, capitalizing on the relation of the resulting, nonlinear PDE system (in closed loop) to a certain, integral delay equation. A numerical example is also provided to illustrate, in simulation, the effectiveness of the developed design.

中文翻译：

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尽管几乎量化的测量和控制输入，线性一阶双曲偏微分方程的混合边界稳定

摘要 我们为一类标量、线性、一阶双曲偏微分方程开发了一种混合边界反馈律，对于这些偏微分方程，状态测量或控制输入要进行量化。考虑的量化器是 Lipschitz 函数，它可以任意近似地近似典型的分段常数，取有限多个值，量化器。控制设计过程依赖于两个成分的组合——一个名义反步控制器，用于在没有量化的情况下稳定 PDE 系统，以及一个切换策略，它更新量化器的参数，用于补偿量化效果。闭环系统的全局渐近稳定性是通过利用类似李雅普诺夫的论证和解估计的推导来建立的，为 PDE 状态的最高范数提供明确的估计，利用由此产生的非线性 PDE 系统（在闭环中）与某个积分延迟方程的关系。还提供了一个数值示例，以在模拟中说明所开发设计的有效性。