Sensitivity Integral Relations and Design Trade-Offs in Linear Multivariable Feedback Systems

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Original languageEnglish
Pages (from-to)1700-1716
Journal / PublicationIEEE Transactions on Automatic Control
Issue number10
Publication statusPublished - Oct 1995
Externally publishedYes


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. The unique phenomena in multivariable systems are that the trade-offs and limitations on sensitivity design are related to, among other known factors in single-input single-output systems, the directionality properties of the sensitivity function as well as the directions of open-loop unstable poles and nonminimum phase zeros. © 1995 IEEE