A class of reset linear systems: the reset-delayed linear systems and their structural and stability properties

This work introduces a new class of reset linear systems, called reset-delayed linear systems, which consists of multivariable dynamical systems featuring a continuous-time state evolution interrupted by discontinuities of some, or even all the state variables at isolated time instants. In particular, these discontinuities abide by an algebraic equation imposing that the state variables involved take the same values they respectively had a certain amount of time before the time instant when the discontinuity is triggered. In this way, the evolutions of the state variables involved turn out to be delayed by the amount of time considered in the reset operation. In the presence of suitably chosen forcing actions, reset-delayed linear systems can effectively model repetitive behaviors which imply a discontinuity at the junction between one cycle and the subsequent one. Herein, some structural properties of this class of multivariable linear dynamical systems are studied and the geometric notions of invariance and controlled invariance are formalized and applied to the solution of the disturbance decoupling problem.