Sensitivity reduction in linear systems

A new measure of sensitivity with respect to initial conditions in linear multi-input time-varying systems is defined. Remarkable amongst its properties are its non-dependence on open-loop control and its invariance under a regular linear time-invariant transformation of the states. A system is shown to be less sensitive according to the new measure when driven by a feedback control, optimal with respect to a quadratic performance index, than when driven by any open-loop control. A relation between the control cost and the sensitivity measure is established, which aids in examining trade-off solutions in design procedures. An example illustrates the definition and the results.