论文标题
一种有效的算法，用于计算有限地平线的规范，线性时间变化系统
An Efficient Algorithm to Compute Norms for Finite Horizon, Linear Time-Varying Systems
论文作者
论文摘要
我们提出了一种有效的算法来计算有限马线性时变（LTV）系统的诱导规范。该公式包括$ \ Mathcal {l} _2 $和Euclidean Norm惩罚。现有的计算方法包括RICCATI微分方程（RDE）的功率迭代和一分解。电源迭代的计算时间较低，但总体收敛可能很慢。相反，RDE条件在诱导的增益上提供了保证的界限，但单个RDE集成可能很慢。将这两种算法的互补特征组合在一起，以开发一种新算法，该算法既快速又在所需公差内提供了可证明的上限和下限。该算法还提供了最坏的案例干扰输入，该输入在标准上实现了下限。我们还提供了一个新的证明，该证明表明，此问题的功率迭代单调收敛。最后，我们显示了一种基于Gramian的可控性简单计算方法，用于诱发$ \ Mathcal {l} _2 $ -TO-EUCLIDEAN NORM。这可以用来在地平线上随时计算可及的集合。提供了数值示例以证明所提出的算法。
We present an efficient algorithm to compute the induced norms of finite-horizon Linear Time-Varying (LTV) systems. The formulation includes both induced $\mathcal{L}_2$ and terminal Euclidean norm penalties. Existing computational approaches include the power iteration and bisection of a Riccati Differential Equation (RDE). The power iteration has low computation time per iteration but overall convergence can be slow. In contrast, the RDE condition provides guaranteed bounds on the induced gain but single RDE integration can be slow. The complementary features of these two algorithms are combined to develop a new algorithm that is both fast and provides provable upper and lower bounds on the induced norm within the desired tolerance. The algorithm also provides a worst-case disturbance input that achieves the lower bound on the norm. We also present a new proof which shows that the power iteration for this problem converges monotonically. Finally, we show a controllability Gramian based simpler computational method for induced $\mathcal{L}_2$-to-Euclidean norm. This can be used to compute the reachable set at any time on the horizon. Numerical examples are provided to demonstrate the proposed algorithm.
