论文标题
混合$ l_ \ infty \ times \ ell_ \ infty $ - 绩效分析和线性时间变化的冲动和切换的正系统
Hybrid $L_\infty\times\ell_\infty$-Performance Analysis and Control of Linear Time-Varying Impulsive and Switched Positive Systems
论文作者
论文摘要
最近的作品表明，$ L_1 $和$ L_ \ infty $ - 是线性正面系统的自然性能标准，因为它们可以使用线性程序进行表征。通过混合$ l_1 \ times \ ell_1 $ -gain的概念，这些性能措施也已扩展到线性正冲动和切换系统。对于LTI阳性系统，已知$ L_ \ infty $ gain与$ l_1 $ - 转移系统的$ L_1 $相吻合，因此，人们可以使用线性共振lyapunov函数来表征$ l_ \ l_ \ infty unfty $ inti-lti阳性系统。不幸的是，这在随时间变化的设置中不存在，也无法表征混合$ l_ \ iftty \ times \ ell_ \ eld_ \ infty $ - 在杂种$ l_1 \ times \ times \ ell_1 $中，是线性阳性冲动系统的gain。提出了一种基于线性共同最大分离lyapunov函数的替代方法。我们首先证明了表征指数稳定性的必要条件和足够的条件，以及$ l_ \ infty \ times \ ell_ \ infty $ - 和$ l_1 \ times \ times \ ell_1 $ - 使用线性max-separable共阳性和线性共态和可行的共阳性共阳性lyapunov lyapunov lyapunov函数。表征稳定性和混合$ L_ \ infty \ ell_ \ indty $ - 在任意，恒定，最小值和范围停留时间约束下的线性正脉冲系统的$ l_ \ infty \ ell_ \ infty $ gain的结果。然后从先前获得的一般结果中得出。然后利用这些条件通过状态反馈产生建设性凸的稳定条件。通过将线性正向开关系统重新定义为具有多个跳跃图的冲动系统，也可以为线性正开关系统获得稳定性和稳定条件。众所周知，所获得的条件概括了现有文献。
Recent works have shown that the $L_1$ and $L_\infty$-gains are natural performance criteria for linear positive systems as they can be characterized using linear programs. Those performance measures have also been extended to linear positive impulsive and switched systems through the concept of hybrid $L_1\times\ell_1$-gain. For LTI positive systems, the $L_\infty$-gain is known to coincide with the $L_1$-gain of the transposed system and, as a consequence, one can use linear copositive Lyapunov functions for characterizing the $L_\infty$-gain of LTI positive systems. Unfortunately, this does not hold in the time-varying setting and one cannot characterize the hybrid $L_\infty\times\ell_\infty$-gain of a linear positive impulsive system in terms of the hybrid $L_1\times\ell_1$-gain of the transposed system. An alternative approach based on the use of linear copositive max-separable Lyapunov functions is proposed. We first prove very general necessary and sufficient conditions characterizing the exponential stability and the $L_\infty\times\ell_\infty$- and $L_1\times\ell_1$-gains using linear max-separable copositive and linear sum-separable copositive Lyapunov functions. Results characterizing the stability and the hybrid $L_\infty\times\ell_\infty$-gain of linear positive impulsive systems under arbitrary, constant, minimum, and range dwell-time constraints are then derived from the previously obtained general results. These conditions are then exploited to yield constructive convex stabilization conditions via state-feedback. By reformulating linear positive switched systems as impulsive systems with multiple jump maps, stability and stabilization conditions are also obtained for linear positive switched systems. It is notably proven that the obtained conditions generalize existing ones of the literature.
