A bode sensitivity integral for linear time-periodic systems

For linear time-invariant systems Bode's sensitivity integral is a well-known formula that quantifies some of the limitations in feedback control. In this paper we show that a very similar formula holds for linear time-periodic systems. We use the infinite-dimensional frequency-response operator called the harmonic transfer function to prove the result. It is shown that the harmonic transfer function is an analytic operator and a trace class operator under the assumption that the periodic system has roll-off 2. A periodic system has roll-off 2 if the first time-varying Markov parameter is equal to zero.