Sensitivity integral relations and design tradeoffs 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 integral relations is that they can be used to quantify the fundamental limitations which arise due to system characteristics such as open loop unstable poles and nonminimum phase zeros, and those due to such fundamental design requirements as stability and bandwidth constraints. We present extensions to both the classical Bode sensitivity integral relation and its extension, and the classical Poisson integral formula. These extended integral relations exhibit important insights toward the tradeoffs that must be performed between the sensitivity reduction and increase due to the aforementioned system characteristics and design constraints. Most importantly, these integral relations display new phenomena concerning the design limitations in multivariable systems which have no analog in single-input single-output systems. The unique phenomena in multivariable systems are that the tradeoffs and limitations on the sensitivity design are related to the directionality properties of the sensitivity function, and to those of open loop unstable poles and nonminimum phase zeros, in addition to factors present in single-input single-output systems.