#### Journal Article

## A Bode sensitivity integral for linear time-periodic systems

### Henrik Sandberg, Bo Bernhardsson

##### Abstract

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.

##### Keywords

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

##### Links

http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=1428859

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