Closed-form boundary State feedbacks for a class of 1-D partial integro-differential equations
Closed-form boundary State feedbacks for a class of 1-D partial integro-differential equations
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DOI：
10.1109/tac.2004.838495
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发表时间：
2004-12
影响因子：
6.8
通讯作者：
A. Smyshlyaev;M. Krstić
中科院分区：
文献类型：
--
作者：
A. Smyshlyaev;M. Krstić
本文研究了一类一维线性抛物型偏积分-微分方程组(P(I)DES)的边界镇定问题，避免了以往工作所要求的空间离散化。该问题被描述为一个积分算子的设计，该积分算子的核需要满足双曲P(I)DE。然后将核P(I)DE化为等价的积分方程，并利用逐次逼近的方法建立了该方程的适定性和核的光滑性。给出了如何将该方法推广到设计最优镇定控制器的方法。为了降低逆最优控制器的保守性，提出了一种自适应机制，并给出了性能上界。对于广泛的物理激励的特殊情况，显式构造了反馈律，并以闭合形式求出了闭环解。给出了核P(I)DE的一种数值格式，其数值结果优于与Riccati算子方程有关的数值结果。
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.