GENERATION OF RANDOM PROCESSES BY DYNAMIC SYSTEMS

Abstract

The article is devoted to the analysis of the behavior of dynamic systems of various nature. Many dynamic systems are represented by one or several differential equations connecting the input signal with the output one. The impulse response h (t, u) is a simple solution when the input signal is the input pulse at time u. All processes in the system are generated by passing white noise through a linear system with time-varying parameters. The article discusses the three main ideas of the description of systems using differential equations. The methods for implementing the system described by an n-th order differential equation with constant coefficients are given. Any non-singular linear transformation of the state vector leads to a new representation of the state of a linear dynamical system. Systems with time-varying parameters and multichannel systems with many inputs and outputs are considered. Using the example of a system that generates two output messages, it was found that the excitation function is a vector function. For the problem of modeling a message, it is assumed that the exciting function is white noise with a matrix covariance function, and the initial conditions can be random variables. The next step was to consider the solution of the equation of state of the system and the properties of the transition state matrix for the case of a system with constant parameters. By performing some transformations, a dispersion equation is obtained that does not contain the received signal, so it can be solved before receiving information of any nature and used to determine the transfer coefficients of a linear dynamic system. In the general case, it is impossible to explicitly obtain an exact analytical solution, but the equation is obtained in a form convenient for integration on a computer. The dispersion equation is the Riccati matrix equation, which is used to identify and predict the state of a dynamic system, namely, to determine the basic matrices of the equations of state and observation.