Bode integrals for multivariable discrete-time systems

The purpose of this paper is to develop integral formulae for singular values of the sensitivity function to express design constraints in multivariable discrete-time systems. We present extensions to both the classical Poisson integral and Bode´s sensitivity integral formulae. The main utility of these results is that they can be used to quantify design limitations that arise in multivariable discrete-time systems, due to such system characteristics as open loop unstable poles and nonminimum phase zeros, and to such design requirements as stability and bandwidth constraints. These formulae are similar to those for multivariable continuous-time systems obtained elsewhere, and they reveal that the limitations on the sensitivity design depend on directionality properties of the sensitivity function, as well as on those of unstable poles and nonminimum phase zeros in the open loop transfer function