ISS and integral-ISS disturbance attenuation with bounded controls

We consider the problem of achieving disturbance attenuation in the ES and integral-ES sense for nonlinear systems with bounded controls. For the ISS case we derive a “universal” formula which ext,ends an earlier result of Lin and Sontag to systems with disturbances. For the integral-ISS case we give two constructions, one resulting in a smooth control law and the other in a switching control law. We also briefly discuss some issues related to input-to-state stability of switched and hybrid systems. is more complicated (except when the values of the disturbances can be directly measured and used for cont.rol): the construction involved “patching” together several control laws defined on appropriate regions of the state space. An interesting source of motivation for the integral-input-to-state stabilization problem is discussed in [7].