Global stability and robustness analysis of oscillators with application to biology and robotics

在本研究项目中，我们建议创建两种新的全局分析方法，它们可以被视为获得振子一般系统理论的第一组元素和工具的一部分。然后，这些方法将用于实现两种算法，旨在数值证明由复杂（高维）振子模型产生的极限环振荡的存在性和全局渐近稳定性，并研究它们的鲁棒性和参数敏感性。我们建议在本项目中深入探索的新方法将依赖于PI和RA的最新研究成果。第一种方法将推广由PI开发的分段线性系统数值分析方法。另一方面，第二种方法将探索用积分二次约束重新表述RA提出的无源振荡器定理中出现的所有耗散条件的可能性，这些约束已经存在有效的数值验证算法。

In this research project, we propose the creation of two new global analysis methodologies which can be seen as part of a first set of elements and tools for obtaining a general system theory for oscillators. These methodologies will then be used to implement two algorithms aimed at numerically proving existence and global asymptotic stability of limit cycle oscillations generated by complex (high dimensional) models of oscillators, and at studying their robustness and parameter sensitivity. The new methodologies that we propose to explore in depth in this project will rely on the recent research results of the PI and the RA. The first methodology will generalise the piecewise linear systems numerical analysis methodology developed by the PI. On the other hand, the second methodology will explore the possibility of reformulating all dissipativity conditions appearing in the passive oscillator theorems proposed by the RA in terms of Integral Quadratic Constraints for which efficient numerical verification algorithms already exist.