Closed-form boundary state feedbacks for a class of 1-D partial integro-differential equations

被引:440
作者
Smyshlyaev, A [1 ]
Krstic, M [1 ]
机构
[1] Univ Calif San Diego, Dept Mech & Aerosp Engn, La Jolla, CA 92093 USA
基金
美国国家科学基金会;
关键词
backstepping; boundary control; distributed parameter systems; Lyapunov function; partial differential equations (PDEs); stabilization;
D O I
10.1109/TAC.2004.838495
中图分类号
TP [自动化技术、计算机技术];
学科分类号
0812 ;
摘要
In this paper, a problem of boundary stabilization of a class of linear parabolic partial integro-differential equations (P(I)DEs) in one dimension is considered using the method of backstepping, avoiding spatial discretization required in previous efforts. The problem is formulated as a design of an integral operator whose kernel is required to satisfy a hyperbolic P(I)DE. The kernel P(I)DE is then converted into an equivalent integral equation and by applying the method of successive approximations, the equation's well posedness and the kernel's smoothness are established. It is shown how to extend this approach to design optimally stabilizing controllers. An adaptation mechanism is developed to reduce the conservativeness of the inverse optimal controller, and the performance bounds are derived. For a broad range of physically motivated special cases feedback laws are constructed explicitly and the closed-loop solutions are found in closed form. A numerical scheme for the kernel P(I)DE is proposed; its numerical effort compares favorably with that associated with operator Riccati equations.
引用
收藏
页码:2185 / 2202
页数:18
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