Time maximum disturbance switch curve and isochrones of linear second-order systems with numerator dynamics

It has been established recently that the most stressful bounded (time maximum) disturbance for stabilized second-order systems with numerator dynamics is of bang-bang type. This bang-bang disturbance can be implemented with a switch curve that is constructed by using isochrones as the level sets for the disturbance index in a state-space setting. In this paper, the isochrones of linear second-order systems with numerator dynamics are shown to have a closed analytic form. This closed analytic form requires less computational effort and gives an intuitive feeling of just how the isochrones change with the other parameters of the system. Simulation results show how the isochrones evolve and approach a maximum limit cycle with different damping coefficients and finite zero locations. Furthermore, these isochrones could be utilized directly in synthesizing the time maximum disturbance and therefore, the ideas can be readily extended to higher order systems.