Backstepping Synthesis for Feedback Control of First-Order Hyperbolic PDEs with Spatial-Temporal Actuation
Backstepping Synthesis for Feedback Control of First-Order Hyperbolic PDEs with Spatial-Temporal Actuation
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DOI：
10.1155/2014/643640
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发表时间：
2014-08
影响因子：
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通讯作者：
Xin Yu;Chao Xu;Huachen Jiang;Arthi Ganesan;Guojie Zheng
中科院分区：
文献类型：
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作者：
Xin Yu;Chao Xu;Huachen Jiang;Arthi Ganesan;Guojie Zheng
本文研究了具有时空激励的一阶双曲型偏微分方程在全物理域上的镇定问题。我们假设内部执行器可以分解为空间和时间分量的乘积，其中空间分量满足特定的常微分方程(ODE)。利用Volterra积分变换将原始系统转化为简单的目标系统，并采用类反推的方法。与求解偏微分方程组边界控制问题的经典反步法不同，内部激励不能消除导致开环系统不稳定的残余项。因此，引入了一个附加的微分变换来将输入从区域内部转移到边界上。然后，利用经典的反步法设计了一阶双曲型偏微分方程组的反馈控制律，并用半群方法证明了这一点。数值仿真结果表明了该设计的有效性。
This paper deals with the stabilization problem of first-order hyperbolic partial differential equations (PDEs) with spatial-temporal actuation over the full physical domains. We assume that the interior actuator can be decomposed into a product of spatial and temporal components, where the spatial component satisfies a specific ordinary differential equation (ODE). A Volterra integral transformation is used to convert the original system into a simple target system using the backstepping-like procedure. Unlike the classical backstepping techniques for boundary control problems of PDEs, the internal actuation can not eliminate the residual term that causes the instability of the open-loop system. Thus, an additional differential transformation is introduced to transfer the input from the interior of the domain onto the boundary. Then, a feedback control law is designed using the classic backstepping technique which can stabilize the first-order hyperbolic PDE system in a finite time, which can be proved by using the semigroup arguments. The effectiveness of the design is illustrated with some numerical simulations.