ANote onOvershootEstimation inPolePlacements

In this note we show that for a given controllable pair (A,B) and any λ > 0, a gain matrix K can be chosen so that the transition matrix e of the system ẋ = (A + BK)x decays at the exponential rate e and the overshoot of the transition matrix can be bounded by Mλ for some constants M and L that are independent of λ. As a consequence, for any h > 0, a gain matrix K can be chosen so that the magnitude of the transition matrix e can be reduced by 1 2 (or by any given portion) over [0, h]. An interesting application of the result is in the stabilization of switched linear systems with any given switching rate (see [1]).