By computing Lyapunov functions of a certain, convenient structure, Lyapunov-based methods guarantee stability properties of the system or, when performing synthesis, of the relevant closed-loop or error dynamics. In doing so, they provide conclusive affirmative answers to many analysis and design questions in systems and control. When these methods fail to produce a feasible solution, however, they often remain inconclusive due to (a) the method being conservative or (b) the fact that there may be multiple causes for infeasibility, such as ill-conditioning, solver tolerances or true infeasibility. To overcome this, we develop linear-matrix-inequality-based theorems of alternatives based upon which we can guarantee, by computing a so-called certificate of nonexistence, that no poly-quadratic Lyapunov function exists for a given linear parameter-varying system. We extend these ideas to also certify the nonexistence of controllers and observers for which the corresponding closed-loop/error dynamics admit a poly-quadratic Lyapunov function. Finally, we illustrate our results in some numerical case studies.

中文翻译：

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用于LPV系统稳定性、稳定性和可检测性分析的不存在证书


通过计算某种方便结构的李雅普诺夫函数，基于李雅普诺夫的方法保证了系统的稳定性，或者在执行综合时保证了相关闭环或误差动力学的稳定性。在此过程中，他们为系统和控制中的许多分析和设计问题提供了结论性的肯定答案。然而，当这些方法无法产生可行的解决方案时，由于（a）方法保守或（b）可能有多种原因导致不可行，例如病态、求解器容差或真实情况，它们通常仍然没有结论。不可行。为了克服这个问题，我们开发了基于线性矩阵不等式的替代定理，基于这些定理，我们可以通过计算所谓的不存在证明来保证，对于给定的线性参数变化系统，不存在多二次李亚普诺夫函数。我们扩展这些想法来证明控制器和观察器不存在，相应的闭环/误差动力学承认多二次李亚普诺夫函数。最后，我们在一些数值案例研究中说明了我们的结果。