A Simplified Approach to Analyze Complementary Sensitivity Trade-offs in Continuous-Time and Discrete-Time Systems

A simplified approach is proposed to investigate the continuous-time and discrete-time complementary sensitivity Bode integrals (CSBIs) in this note. For continuous-time feedback systems with unbounded frequency domain, the CSBI weighted by $1\mathrm{/}{\omega }^2$ 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 Cauchy integral theorem nor Poisson integral formula are used throughout the analysis, and the analytic constraint on the integrand is removed.