This paper addresses the problem of determining the response peak of dynamical systems, a classical problem in control systems. The paper starts by considering the impulse response of linear time-invariant (LTI) systems. A linear matrix inequality (LMI) condition for establishing upper bounds of the sought peak is proposed by embedding the trajectory onto the level set of a polynomial and by introducing a projection technique for evaluating the extension of this set. This condition is sufficient for any degree of the polynomial, which has to be chosen a priori, and is also necessary whenever this degree is large enough. Hence, the proposed methodology is extended to address the impulse response of polytopic linear time-varying (LTV) systems, in particular, linear systems affected linearly by structured time-varying uncertainty constrained into a polytope. Lastly, generalizations to various responses and specializations to some structures are also presented. As shown by several examples, which include randomly generated systems and physical systems, the proposed conditions may provide significantly less conservative results than the existing LMI methods.

本文解决了确定动态系统响应峰值的问题，这是控制系统中的经典问题。本文首先考虑了线性时不变（LTI）系统的脉冲响应。通过将轨迹嵌入多项式的水平集并引入用于评估该峰集扩展的投影技术，提出了用于建立所需峰上限的线性矩阵不等式（LMI）条件。该条件对于必须先验选择的多项式的阶数就足够了，并且只要该阶数足够大，该条件也是必要的。因此，所提出的方法已扩展为解决多主题线性时变（LTV）系统的脉冲响应，特别是，受结构化时变不确定性线性影响的线性系统，该线性系统被约束为多面体。最后，还介绍了对各种响应的概括和对某些结构的专门化。如几个示例所示，其中包括随机生成的系统和物理系统，与现有的LMI方法相比，所提出的条件提供的保守性结果可能要低得多。