# Using Autoregressive with Exogenous Input Models to Study Pulsatile Flows
## 1 Introduction  
One of the main characteristics of dynamical systems is their dependency on past information to forecast future information [1]. System Identification (SI) methods characterize dynamic systems by employing mathematical-based models using the measured data from these systems. One of the most common techniques in this area is the AutoRegressive with eXogenous (ARX) input model. Its reliability to replicate a broad spectrum of scenarios and its parametrization make this model an ideal approach to work with single-input single-output systems. The ARX model works as a black-box containing a set of parameters (in this case, coefficients) needed to construct a transfer function, which describes the behavior of the dynamic system. Regarding its applications, we can find examples for different fields such as gas conditioning for cement plants [2], thermal load prediction for buildings [3], and stock market forecasting [4]. Some other mechanical and electronic applications include the estimation of driving behavior [5; 6], gearbox failure detection [7], vibration analysis for manipulators during grinding operations [8], controlling the position of a steel ball in a magnetic levitation system [9], an inverted pendulum [10], and the estimation of the state ofcharge for lithium-ion battery cells [11]. Besides, this method is able to handle applications involving biomedical signals such as interstitial glucose during physical activity [12], aortic blood pressure [13], middle cerebral arterial blood velocity [14], or in a recent work, an extension to the ARX including a moving average term to analyze the behavior of a simple leaflet structure within a fluid domain [15]. This last work is a simple representation of an experimental setup that simulates the behavior of artificial cardiac valves under a pulsatile flow. Taking into consideration this experimental setup, this paper continues the experimentation on the behavior of simple leaflet structures moving as a consequence of a pulsatile flow. For this purpose, Fluid-Structure Interaction (FSI) numerical simulations are one of most significant alternatives to understand the behavior of a solid material under certain flow conditions. This method consists of a combination of two analyses: the dynamics of a solid body using a time-dependent Finite Element Method (FEM) and a Computational Fluid Dynamics (CFD) analysis for fluid motion. These approaches provide the facility of measuring information such as velocity fields, vorticity, pressure differences, stress fields, displacements, and opening area (in the case of leaflets), among others. The setup of an FSI simulation often follows a mesh-based configuration. Specifically, the main approaches are the Arbitrary Lagrange Eulerian (ALE) approach [16; 17] and the Immersed Boundary Method (IBM) [18; 19; 20]. However, despite the vast number of studies, the most significant limitations of using mesh-based models consist of the necessity for continuous re-meshing and, as a consequence, a substantial computational cost, numerical noise, and the difficulty to simulate contact between solid materials [20; 21]. Extensions to the IBM have been proposed as well [19; 20]. Other alternative methods such as the Fictitious Domain Method (FDM) [22; 23; 24] and the Immersogeometric Analysis (IGA) [25; 26] had been employed in the corresponding bibliography: A novel technique proposed to overcome these disadvantages is the use of mesh-free methods. For instance, the Smoothed Particle Hydrodynamics (SPH) approach [21; 27; 28] is based on a Lagrangian particle method where the fluid is discretized as a set of particles distributed along the solution domain without the necessity for a spatial mesh [21]. The main limitations of SPH are observed on the boundary layers, and this is because they are not fully constrained with a non-slip boundary condition.