A Bode-like integral for discrete-time linear periodic systems

We present a generalization of Bode's integral formula for discrete-time linear periodic systems. It is shown that similar to the classical Bode's integral formula, the sensitivity integral for a discrete-time linear periodic system depends only on the open-loop dynamics; in particular, on the open-loop characteristic multipliers outside the unit circle in the complex plane. The integral is derived by using an asymptotic eigenvalue distribution theorem for block Toeplitz matrices. We do not require the open loop system to be stable. Further, as opposed to the existing results on Bode like integrals for unstable open loop systems, we do not require the disturbance to be Gaussian.