论文标题
退化梯度流的最差指数衰减率受到持续激发
Worst Exponential Decay Rate for Degenerate Gradient flows subject to persistent excitation
论文作者
论文摘要
在本文中，我们估计了根据自适应控制理论发出的退化梯度流量$ \ dot x = -s x $的最坏衰减率。在持续的激发假设上，我们在正半明确矩阵$ s $上，我们为这种衰减速率提供了上限，符合先前已知的下限和类似稳定性的结果，以持续激发信号的更一般类别。证明策略包括将最坏的衰减率与最佳控制问题联系起来并详细研究其解决方案。 作为我们分析的副产品，我们还获得了对时变线性控制系统的最差$ L_2 $的估计值$ \ dot x = -cc^\ top x+u $，其中信号$ c $持续兴奋，从而解决了A. Rantzer在1999年提出的开放问题。
In this paper we estimate the worst rate of exponential decay of degenerate gradient flows $\dot x = -S x$, issued from adaptive control theory. Under persistent excitation assumptions on the positive semi-definite matrix $S$, we provide upper bounds for this rate of decay consistent with previously known lower bounds and analogous stability results for more general classes of persistently excited signals. The strategy of proof consists in relating the worst decay rate to optimal control questions and studying in details their solutions. As a byproduct of our analysis, we also obtain estimates for the worst $L_2$-gain of the time-varying linear control systems $\dot x=-cc^\top x+u$, where the signal $c$ is persistently excited, thus solving an open problem posed by A. Rantzer in 1999.
