研究一类不确定二阶双曲型PDE-ODE级联系统的自适应镇定控制问题. 本质不同于现有相关文献中的结果, 文章所研究系统边界条件上容许存在未知参数, 这使得已有的控制设计方法难以适用. 为此, 利用基于投影算子的自适应技术构造动态补偿器克服参数未知性, 并在此基础上借助反推技术成功地设计出自适应边界控制器, 确保系统状态一致有界, 且收敛到原点的一个小邻域内. 需指出的是, 动态补偿器的引入导致所得闭环系统具有较强的非线性, 因此利用非线性发展方程理论和能量函数构造法, 提出了适用性更广的系统适定性和稳定性分析方法. 最后, 以带有柔性缆绳的桥式起重机为例, 通过仿真验证了理论结果的有效性.

Adaptive stabilizing control is investigated for a class of uncertain second-order hyperbolic PDE-ODE cascade systems. Essentially different from the results in the existing related literature, an unknown parameter is allowed to appear in the boundary condition of the considered system, which leads to that the existing control design methods are difficult to be applied to solve the problem in this paper. For this, by using the adaptive technique that based on the projection operator, a proper dynamic compensator is introduced to deal with the unknowns of the parameter. Based on this and by using the backstepping technique, a desired adaptive boundary controller is successfully constructed for the considered system, which guarantees that all the states of the closed-loop system are uniformly bounded and ultimately converge to a small neighborhood of the origin. It is necessary to point out that the introduction of dynamic compensator makes the resulting closed-loop system have strong nonlinearity. Therefore, by the theory of nonlinear evolution equations and constructing an appropriate energy function, a more general method of analyzing the well-posedness and stability of a system is presented. At last, by taking an overhead crane with flexible cable as an example, numerical results are given to illustrate the effectiveness of the theoretical results.