论文标题
$ l _ {\ infty} $的计算界限 - 使用同质lyapunov函数的线性时间不变系统的诱导标准
Computing Bounds on $L_{\infty}$-induced Norm for Linear Time-Invariant Systems Using Homogeneous Lyapunov Functions
论文作者
论文摘要
二次lyapunov函数已被广泛用于线性时间不变（LTI）系统的分析，因为它表明存在这种二次lyapunov函数的存在证明了LTI系统的稳定性。在这项工作中，考虑了$ l _ {\ infty} $ - 诱导LTI系统的标准的上限和下限的问题。二次lyapunov函数用于查找$ l _ {\ infty} $诱导的标准的恒星标准，通过界定单位峰值输入可通过不可移动的椭圆机界限。取而代之的是，比使用标准的Quadratic Lyapunov函数获得的更通用的同质Lyapunov函数可在$ L _ {\ infty} $上获得较不保守的上限 - 诱导的标准和更好的保守近似值。 LTI系统的同质Lyapunov函数被认为是通过Kronecker乘积抬起LTI系统获得的高阶系统的二次lyapunov函数。提供了不同的示例，以显示通过使用同质Lyapunov函数获得的边界的显着改进。
Quadratic Lyapunov function has been widely used in the analysis of linear time invariant (LTI) systems ever since it has shown that the existence of such quadratic Lyapunov function certifies the stability of the LTI system. In this work, the problem of finding upper and lower bounds for the $L_{\infty}$-induced norm of the LTI system is considered. Quadratic Lyapunov functions are used to find the star norm, the best upper on the $L_{\infty}$-induced norm, by bounding the unit peak input reachable sets by inescapable ellipsoids. Instead, a more general class of homogeneous Lyapunov functions is used to get less conservative upper bounds on the $L_{\infty}$-induced norm and better conservative approximations for the reachable sets than those obtained using standard quadratic Lyapunov functions. The homogeneous Lyapunov function for the LTI system is considered to be a quadratic Lyapunov function for a higher-order system obtained by Lifting the LTI system via Kronecker product. Different examples are provided to show the significant improvements on the bounds obtained by using Homogeneous Lyapunov functions.
