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A simplified approach is proposed to investigate the continuous-time anddiscrete-time complementary sensitivity Bode integrals (CSBIs) in this note.For continuous-time feedback systems with unbounded frequency domain, the CSBIweighted by $1/\omega^2$ is considered, where this simplified method reveals amore explicit relationship between the value of CSBI and the structure of theopen-loop transfer function. With a minor modification of this method, the CSBIof discrete-time system is derived, and illustrative examples are provided.Compared with the existing results on CSBI, neither Cauchy integral theorem norPoisson integral formula are used throughout the analysis, and the analyticconstraint on the integrand is removed.

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