Continuous-time and discrete-time complementary sensitivity Bode integrals (CSBIs) are investigated via a simplified approach in this note. For continuous-time feedback systems with unbounded frequency domain, the CSBI weighted by 1/ω2 is considered, where this simplified method reveals a more explicit relationship between the value of CSBI and the structure of the open-loop transfer function. With a minor modification of this method, the CSBI of discrete-time system is derived, and illustrative examples are provided. Compared with the existing results on CSBI, neither the Cauchy integral theorem nor the Poisson integral formula is used throughout the analysis, and the analytic constraint on the integrand is removed.

中文翻译：

---

分析连续时间和离散时间系统中互补灵敏度权衡的简化方法


本文通过简化方法研究了连续时间和离散时间互补灵敏度波德积分 (CSBI)。对于无界频域的连续时间反馈系统，考虑以1/ω2加权的CSBI，这种简化的方法揭示了CSBI值与开环传递函数结构之间更明确的关系。通过对该方法的微小修改，导出了离散时间系统的CSBI，并提供了说明性例子。与CSBI现有结果相比，整个分析过程中既没有使用柯西积分定理，也没有使用泊松积分公式，并且取消了对被积函数的解析约束。