In this paper, the theory and effectiveness of two polynomial approaches are compared in the analysis of $L_2\text{-}{L}_{\infty }$ and $H_{\infty }$ performance for a type of periodic piecewise polynomial systems, where the time-varying subsystems can be characterized in Bernstein polynomials. Using the Bernstein polynomial-based lemma and the existing lemma concerning the negativity/positivity of matrix polynomial functions, sufficient conditions are established in tractable forms aimed at the global asymptotic stability and performance analysis. Four cases of optimization constraints are considered based on the proposed conditions. The performance indices obtained via the four cases are compared through a numerical example, and the lower conservatism achieved by the proposed Bernstein polynomial approach is demonstrated.

在本文中，比较了两种多项式方法的理论和有效性。 ${\mathrm{大号}}_{\mathrm{2个}}\text{--}{\mathrm{大号}}_{\infty }$ 和 $H_{\infty }$一类周期性分段多项式系统的性能，时变子系统可以用伯恩斯坦多项式来表征。使用基于伯恩斯坦多项式的引理和关于矩阵多项式函数的负/正性的现有引理，以易于处理的形式建立了充分的条件，旨在进行全局渐近稳定性和性能分析。根据建议的条件，考虑了四种优化约束条件。通过一个数值示例比较了通过这四种情况获得的性能指标，并证明了所提出的Bernstein多项式方法实现的较低保守性。