Causal impact modeling of state dependent impulsive affine systems using Non-standard Analysis

A causal modeling for an impulsive system with a state dependent switching surface is defined and analyzed on a new extended real space, denoted as Krylov hyperreals, which is based on the nonstandard analysis (NSA). The recent work of the authors contains the detailed construction of the extended space, and the generalized function on that space. In the first part of the paper, important concepts of NSA, and the suggested function space are reviewed. Next, a new generalization of a continuous but not differentiable function will be defined on the Krylov hyperreals in order to properly define a composition between singular and non differentiable function. By using an analogy to a spring and damper model, the authors suggest an equivalent causal model of the state dependent impulsive system by introducing the powers of singular control in the original continuous dynamics. A motivational example of a bouncing ball moving on a horizontal surface is analyzed to show the effectiveness.