Multivariable gain-phase and sensitivity integrals

In this paper we present several extensions of the classical Bode integral relations to multivariable systems. Our main contribution includes an extended gain-phase integral formula and an extended sensitivity integral relation. These results are useful for analyzing certain fundamental design limitations that arise in linear multivariable feedback systems. In particular, they provide important insights into feedback properties which are unique to multivariable systems and which cannot be inferred from the scalar integral relations. Both results show that design limitations in multivariable systems depend strongly upon certain directionality properties.<>