Bode integrals for multivariable discrete-time systems

Abstract

The purpose of this paper is to develop integral formulae for singular values of the sensitivity function to express design constraints in multivariable discretetime 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 i u niultivariable discretetime systems, due to such system characterist,ics as open loop unstable poles and nonininiinum 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 nonn~inin~uni phase zeros i n the open loop transfer function.