Sensitivity integral relations and design trade-offs in linear multivariable feedback systems

The purpose of this paper is to develop integral relations regarding the singular values of the sensitivity function in linear multivariable feedback systems. The main utility of these integrals is that they can be used to quantify the fundamental limitations in feedback design which arise due to system characteristics such as open-loop unstable poles and nonminimum phase zeros and to such fundamental design requirements as stability and bandwidth constraints. We present extensions to both the classical Bode sensitivity integral relation and Poisson integral formula. These extended integral relations exhibit important insights toward trade-offs that must be performed between sensitivity reduction and sensitivity increase due to the aforementioned system characteristics and design constraints. Most importantly, these results display new phenomena concerning design limitations in multivariable systems which have no analog in single-input single-output systems