Journal Article

A Bode sensitivity integral for linear time-periodic systems

Henrik Sandberg, Bo Bernhardsson


For linear time-invariant systems Bode's sensitivity integral is a well-known formula that quantifies some of thelimitations in feedback control. In this paper we show that a very similar formula holds for linear time-periodicsystems. 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 isan 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.


trace class operator, linear systems, time-varying systems, transfer functions, infinite-dimensional frequency-response operator, harmonic transfer function, Bode sensitivity integral, analytic operator, time-varying Markov parameter, linear time-periodic


IEEE Transactions on Automatic Control, 50:12, pp. 2034–2039, December 2004.