A Bode-Like Integral for Discrete Linear Time-Periodic Systems

We present a generalization of the Bode integral formula for discrete-time linear periodic systems. It is shown that similar to the classical Bode integral formula, the sensitivity integral for a discrete-time linear periodic system depends only on the open-loop dynamics of the system; 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, which does not require the open loop system to be stable or the disturbance to be Gaussian. The result is demonstrated through application to a class of multi-rate sampled data systems with commensurate rates.