We solve a model reference adaptive control problem for a class of linear 2 × 2 hyperbolic partial differential equations (PDEs) with uncertain system parameters subject to harmonic disturbances, from a single boundary measurement anticollocated with the actuation. This is done by transforming the system into a canonical form, from which filters are designed so that the states can be expressed as linear combinations of the filters and uncertain parameters, a representation facilitating for the design of adaptive laws. A stabilizing controller is then combined with the adaptive laws to make the measured signal asymptotically track the output of a reference model. The reference model is taken as a simple transport PDE. Moreover, pointwise boundedness of all variables in the closed loop is proved, provided the reference signal is bounded. The theory is demonstrated in a simulation.

中文翻译：

---

$ 2 \times 2$ 耦合线性双曲偏微分方程的模型参考自适应控制


我们通过与驱动反配置的单个边界测量，解决了一类线性 2 × 2 双曲偏微分方程 (PDE) 的模型参考自适应控制问题，该方程具有受谐波扰动影响的不确定系统参数。这是通过将系统转换为规范形式来完成的，从中设计滤波器，以便状态可以表示为滤波器和不确定参数的线性组合，这种表示形式有利于自适应律的设计。然后将稳定控制器与自适应定律相结合，使测量信号渐近跟踪参考模型的输出。参考模型被视为简单的传输偏微分方程。此外，假设参考信号有界，则证明了闭环中所有变量的点有界性。该理论在模拟中得到了证明。