Methods for numerical optimization of regulators has a number of advantages as compared with the analytical methods. The use of modern software, together with well-known and newest methods of forming of the optimization criteria in the form of cost functions provides a set of methods, successfully tested on a number of tasks. These methods, in contrast to analytical ones, allow effectively calculation of robust regulators for objects even in the case of the simultaneous presence in these of non-linear elements, units of lag, discrete elements (such as sampling-and-hold analog storage) and other complicating singularities.

This paper proposes a new structure and substantiates the choice of a technique for controller design using fractional integration and fractional differentiation. The method of fractional integration is based on extrapolation using a series of integrating and differentiating links, the time constant of which changes symmetrically from one step to another. The new controller structure does not contain the three traditional links (proportional, integrating, and differentiating) but contains six links, three of which are incomplete integration, incomplete differentiation, and redundant (for example, one and a half) integration. The proposed technique consists of modeling a controller with the indicated six paths and using the numerical optimization method to calculate the coefficients of each such path. Alternative controllers are known PID controllers and PI^λD^µ controllers. Both of these species contain only three tracts. The proposed structure has never been used before, and publications with examples of calculation of such structures were not found in the literature. The effectiveness of the proposed method is confirmed by modeling and examples.

In this paper, a model of a car impact against a fixed roadside objects/safety barrier is presented and discussed to solve a mixed task: to determine the pre-impact velocity of the vehicle center of mass taking into consideration the recorded post-impact final rest position. The principle of conservation of mechanical energy was applied, whereas the energy of the rigid barrier deformation was estimated by the finite element method/FEM for an elasto-plastic body. FEM analysis was performed by integrating SolidWorks - SW Simulation software, the output data being the deformation of the safety barrier in the three dimensional space subjected to limited boundary conditions. The post-impact car motion was modelled by a multiscale spatial model, well established in modern accident reconstruction. The findings of the study point out the precision and accuracy of the crash pattern revealed in the accident reconstruction and essential in clarifying the situation with appropriate real facts. Two cases involving real-world head-on accident collisions against a fixed flexible barrier and a tank body are considered, which exemplify the adopted method.

Various methodologies and tools applied to identification of vehicle and collision impact seek to present more and more accurate solutions to reproduce, restore, recreate and investigate the casualty. Modern computer technology and software provide the tools to solve specific problems developing mathematical modelling of complex mechanical systems involving vehicles and other objects in a road accident. Scientists generally utilize the Standard Test Method for Impact Testing calculating the energy of deformation of both vehicles, however, one of its limitations is the evaluation of the kinetic energy of the vehicles in post-collision taking into consideration vehicle rotation and linear displacement. To improve the analysis, dynamic traffic simulation is used, taking into account the variations in the coefficient of friction, suspension elasticity and damping. The proposed method is based on a system of two equations derived from two principles: the Principle of Conservation of Mechanical Energy and the Principle of Conservation of Momentum in the impact phase. The new approach is conducted on mathematical modelling and computer simulation of vehicle motion after the impact, wherefrom the linear and angular velocities are analysed. This is achieved by the numerical solution of the differential equations of motion of the cars after the impact, and the given initial conditions that satisfy the solution are used to solve the system of equations. The main findings of the study can be grouped as follows: 1) The positions of the vehicles prior to the moment of first impact and the post-impact orientation of velocity vectors are more precise. 2) The variability of the tire-road friction coefficient is taken into consideration. 3) The value of coefficient of restitution according to Newton’s theory of impact is unnecessarily determined.