Causal Impact Modeling of State Dependent Impulsive Affine Systems using Non-Standard Analysis Nak-seung Patrick Hyun 1 and Erik I. Verriest 2 Abstract— 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. I. INTRODUCTION A model of mechanical systems which interacts with the environment by an impacting behavior involves an instanta- neous change in their motion, usually in the velocity, at the time of the impact. Classically, such a system is modeled by an impulsive system, which contains a single continuous dynamical part, and a discrete dynamical part that models the resetting events of the state. The classical theory of impulsive systems may be referred to as an effect system since the discrete jump dynamics shows only the result of the impact not the cause of this effect. Fundamental theory on the impulsive systems can be found in [1]. In this paper, we are interested in utilizing singular controls in the original continuous dynamics to find the equivalent causal representation which generate the same jump dynamics as in the effect system. The objective is to better obey the physics in the model rather than approximate by the discrete jumps in the effect system. An application of controlling the impact can be found in [2], which analyzes the impact force in a relatively short duration to generate desired gait for a quadrupedal robot. Nevertheless, applying a singular control, as defined by Schwartz distribution, encounters many difficulties for non- linear systems since the multiplication between singular functions and a discontinuous test functions are not well defined, and powers of singular function cannot be well defined in the distribution: Schwartz impossibility theorem [3]. Therefore, most of the work on modeling the singular control in nonlinear system theory has been done by first The authors are with the School of Electrical and Computer Engi- neering, Georgia Institute of Technology, Atlanta, GA, USA. Email: 1. nhyun3@gatech.edu, 2. erik.verriest@ece.gatech.edu This work was supported by NSF grant CPS-1544857. regularizing the singular control system, and then sequen- tially approximating the solution to generalize the solution. Selected analysis of singular control in nonlinear systems can be found in [4]–[10] where the measure driven model is used in [4], [5]. A sequentially defined infinitesimal model was proposed in [6]–[9]. Although most of the paper provides a generalized solution of the impulse driven system, the limiting process itself lacks physical meaning. In this paper, we propose a new generalized solution which can avoid using the limiting process but rather define the solution point-wise at each infinitesimal moments of impact in Non-standard Analysis (NSA) framework, [11]. The key idea of NSA is to extend the real space so that the infinitesimals and infinity can be formally defined as a proper element in the extended space: hyperreal space. The infinitesimal elements and infinitely large elements are not unique, and there are uncountably many. Since NSA is less known to the control system society, up to our knowledge, we briefly recap important definitions in Section III-A. Detailed reference for NSA can be found in [12]. The usage of NSA in the generalized function theory can be found in [13] which translate a Colombeau algebra, a quotient algebra on a set of generalized function, [14], into the NSA framework. These works shows that the multiplication between nonsmooth and singular functions are well-defined either on hyperreals or in a Colombeau algebra. An application to the impulsive equations with an initial condition at the singular moment is considered in [15]. Recently in [16], we proposed a simple and intuitive way to extend the reals with an algebraically structured countably infinite dimensional subspace of original hyperreal space. In this space, we consider only countably many infinitesimals which are generated by geometric sequences with a different ratio. Therefore, by using the successive operation, similar to the Krylov method in numerical linear algebra but with the scaling and translation operators, we generalized the function space. Two important features of using the new generalized function is that the space is closed under multiplication, and the discontinuous function can be continuized in the infinitesimal time with a certain shape function. In this paper, we extend our previous results to analyze the impulsive system which the switching time depends on the state values. The first step is to extend the space of shape functions in order to properly define a composition between δ function and a continuous but non differentiable function at a point of the impact. This composition is necessary when the switching surface is state dependent. A motivational example of a bouncing ball moving on a