球对称n维反应扩散方程的边界观测与输出反馈控制

Boundary Observer and Output-feedback Control for a Class of n-dimensional Symmetric Advection-diffusion Equations

许多实际系统可用n维超球坐标系来描述,并且系统有球对称的性质,因而可通过研究半径方向的状态变化,得到系统的全局动态过程.通过将高维的对称系统转化为等价的径向一维方程,本文采用边界Backstepping方法设计了球对称反应扩散方程的输出反馈控制器.使用容易测量的边界状态值,设计了状态观测器来估计系统在空间域的所有状态,从而实现输出反馈控制.本文扩展了连续Backstepping方法,提出了n维球坐标的Volterra积分映射,从而求出了显式表达的控制器和状态观测器.论文用Lyapunov函数法证明了输出反馈系统在H1范数下指数稳定,表明状态对初值的连续依赖,确保控制系统具有较好的性质,不会在空间某点发散.最后进行了数值仿真,仿真结果表明系统在输出反馈控制律的作用下收敛到稳态值.

Many systems in application are modelled in n- sphere coordinates and also have symmetric properties. For symmetric system, we can obtain the dynamics of n-D states by investigating 1-D states along the radial direction. This pa- per uses the boundary backstepping method to design an output feedback control law for a symmetrical n-D reaction-diffusion equation by studying an equivalent 1-D radial system. We de- sign an observer by boundary sensor to estimate the whole states in space which are needed in the feedback control. We extend the continuous backstepping method by proposing an n-D sphere Volterra integration mapping, which makes it possible to get ex- plicit controller and observer. We have proved the exponential stability in the H1 norm by using Lyapunov function, which means that the states are continuous in terms of initial condi- tions and that the system has good properties in that it will not diverge in some special point. Finally, we do some simulations to illustrate that the system can converge to the equilibrium actuated by the output feedback control.