$2 \times 2$非线性双曲型系统的前馈边界控制及其在Saint-Venant方程中的应用
Feedforward boundary control of $2 \times 2$ nonlinear hyperbolic systems with application to Saint-Venant equations
摘要：
因为它们代表具有传播延迟的物理系统，所以双曲系统非常适合于前馈控制。当扰动和输出之间的延迟大于控制延迟时，尤其如此。在本文中，我们解决的前馈控制器的设计一般类的$2 \times 2$双曲型系统具有一个单一的干扰输入位于一个边界和一个单一的控制驱动在另一个边界。目标是设计一种前馈控制，使系统输出对测量的干扰输入不敏感。我们证明了，对于这类系统，存在一个有效的理想前馈控制器是因果和稳定的。首先在频域中对一个简单的线性系提出并研究了这个问题。然后，我们的主要贡献是展示如何理论可以扩展，在时域，一般的非线性双曲系统。该方法被示出与应用程序的控制由圣维南方程表示的明渠，其目标是使输出水位不敏感的输入流量的变化。最后，我们解决了一个更复杂的应用程序的级联池的完美前馈控制的盲目应用可能会导致有害的振荡。提出了一种改进控制律的实用方法，并通过仿真实验进行了验证。
英文摘要：
Because they represent physical systems with propagation delays, hyperbolic systems are well suited for feedforward control. This is especially true when the delay between a disturbance and the output is larger than the control delay. In this paper, we address the design of feedforward controllers for a general class of $2 \times 2$ hyperbolic systems with a single disturbance input located at one boundary and a single control actuation at the other boundary. The goal is to design a feedforward control that makes the system output insensitive to the measured disturbance input. We show that, for this class of systems, there exists an efficient ideal feedforward controller which is causal and stable. The problem is first stated and studied in the frequency domain for a simple linear system. Then, our main contribution is to show how the theory can be extended, in the time domain, to general nonlinear hyperbolic systems. The method is illustrated with an application to the control of an open channel represented by Saint- Venant equations where the objective is to make the output water level insensitive to the variations of the input flow rate. Finally, we address a more complex application to a cascade of pools where a blind application of perfect feedforward control can lead to detrimental oscillations. A pragmatic way of modifying the control law to solve this problem is proposed and validated with a simulation experiment.
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