Improving the performance of linear systems by adding a hybrid loop

Considering a linear system, we construct hybrid loops consisting in a (possibly non stabilizing) given linear dynamic continuous-time controller with a jump rule when some conditions of the state occur. This allows us to study the performance improvement problem. More precisely, we firstly show the interest of an hybrid loop to maximize the decay rate when the gain of the controller is limited. Secondly, we show how adding a hybrid loop can be useful to reduce the overshoot of an output. The proposed approaches are illustrated on two examples.