带有非线性边界和扰动输入的波动方程的输出反馈控制

Output feedback control for wave equations with nonlinear boundary condition and disturbance inputs

无穷维系统的输出反馈控制是控制理论中重要的研究课题,相对于线性边界输入而言,非线性边界条件更多应用于实际的数学模型中,容易引起各种不同的动力学行为,如混沌声振动、倍周期分岔、方波等.本文研究了左端具有非线性位移边界条件,右端带有总扰动输入的一维波动方程的输出反馈镇定问题.首先,利用算子半群理论证明了开环系统的适定性;其次,由于内部非线性项和外部扰动的存在,通过构造无穷维扰动估计器,证明了该估计器能够实时在线估计总扰动;紧接着,借助于原系统的量测输出信号设计状态观测器,构造输出反馈控制器并得到了闭环系统;最后,证明了该闭环系统的适定性和渐近稳定性.

The output feedback control of infinite dimensional systems is an important research topic in control theory.Compared with linear boundary input,nonlinear boundary conditions are more applied to practical mathematical models,which can cause various dynamic behaviors,such as chaotic acoustic vibration,period-doubling bifurcation,square wave,and so on.In this paper,the output feedback stability problem of one dimensional wave equation with nonlinear displace-ment boundary condition at left end and total disturbance input at right end is studied.Firstly,the well-posedness of open loop systems is proved by using operator semigroup theory.Secondly,due to the existence of internal nonlinear terms and external disturbances,we prove that the estimator can estimate total disturbances by constructing an infinite-dimensional disturbance estimator.Then,the state observer is designed by means of the measured output signal of the original system,and the stability controller is constructed and the closed-loop system is obtained.Finally,the well-posedness and asymptotic stability of the closed-loop system are proved.