Closed analytic form of time maximum disturbance for stable linear systems

It has been recently established that the time maximum disturbance for some stable linear systems is of bang-bang type. We present a closed analytic form for the disturbance design. For a stable linear system, we describe how to construct the disturbance switch curve. Using the switch curve, we show the existence of the maximum limit cycle. The severity index is maximized as all states oscillate along the limit cycle on and on. Switch curve and maximum limit cycle refer to the second order system in state space. When the system is third or higher order it is replaced by switch surface and maximum limit body. Simulation results show how the switch curves can be used to realize the disturbance in state feedback form.