Robustness in the Graph Topology of a Common Adaptive Controller
Robustness in the Graph Topology of a Common Adaptive Controller
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通用自适应控制器图拓扑的鲁棒性
DOI：
10.1137/0506336371
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
E. Ryan
中科院分区：
文献类型：
--
作者：
M. French;A. Ilchmann;E. Ryan
对于任意m输入，m输出的有限维线性最小相位对象P，其第一个马尔可夫参数的谱在开右半复平面上，已知自适应输出反馈控制C，由u=-ky，k = y ^2$给出，产生一个状态收敛到零的闭环系统[P，C]，信号k收敛到一个有限的极限，所有其他信号都属于L^2类。首先证明了在存在L^2 $-输入和L^2 $-输出扰动的情况下，这些性质仍然成立。在鲁棒稳定性的非线性间隙度量方法的概念框架内，通过建立一个适当的闭环算子的增益函数稳定性，证明了当被控对象P在图拓扑中的一个足够小的邻域内被一个可镇定和可检测的线性被控对象P1所取代时，这些性质仍然存在，只要电厂的初始数据和扰动的大小足够小。Georgiou和Smith的实施例9 [IEEE Trans.Automat. Control，42（1997），pp. 1200-1221.第1200-1221章重新来过大的初始条件和/或大的L^2 $扰动的不稳定行为，表明从L^2 $理论得到的边界是定性紧：这与Georgiou和Smith的L^2\infty$-鲁棒性分析，这是不够紧，预测小的初始条件和零扰动的稳定行为。
For any $m$-input, $m$-output, finite- dimensional, linear, minimum-phase plant $P$ with first Markov parameter having spectrum in the open right half complex plane, it is well known that the adaptive output feedback control $C$, given by $u=-ky,\ \dot k= \|y\|^2$, yields a closed- loop system $[P,C]$ for which the state converges to zero, the signal $k$ converges to a finite limit, and all other signals are of class $L^2$. It is first shown that these properties continue to hold in the presence of $L^2$-input and $L^2$-output disturbances. Working within the conceptual framework of the nonlinear gap metric approach to robust stability, and by establishing gain function stability of an appropriate closed-loop operator, it is proved that these properties also persist when the plant $P$ is replaced with a stabilizable and detectable linear plant $P_1$ within a sufficiently small neighborhood of $P$ in the graph topology, provided that the plant initial data and the $L^2$ magnitude of the disturbances are sufficiently small. Example 9 of Georgiou and Smith [IEEE Trans. Automat. Control, 42 (1997), pp. 1200-1221] is revisited. Unstable behavior for large initial conditions and/or large $L^2$ disturbances is shown, demonstrating that the bounds obtained from the $L^2$ theory are qualitatively tight: this contrasts with the $L^\infty$-robustness analysis of Georgiou and Smith, which is insufficiently tight, to predict the stable behavior for small initial conditions and zero disturbances.