The Representation of the Solution of the Cauchy Problem for Linear Discrete-Continuous Systems

The article discusses the description of mathematical models of multidimensional multi-loop continuous-discrete control systems. Multi-loop systems are often characterized by the control of several interconnected state elements and the presence of several signal propagation channels and interconnections between individual objects. In the general case, continuous-discrete systems are considered as a MIMO multiloop multi-rate continuous-discrete system, specified as an input-output model. The great importance for the theory of linear continuous-discrete systems (as well as for the theory of any other dynamical systems) are the methods of representing the solution of the Cauchy problem. The problem is to find a way to represent a given solution of an inhomogeneous system from the solutions of a homogeneous system. An important result is the method of constructing conjugate systems for the considered models of systems. The components of the solution of the Cauchy problem for a continuous-discrete system are obtained, which made it possible to obtain a complete solution of the Cauchy problem for the class of continuous-discrete control systems under consideration. The results obtained make it possible to introduce weight matrices of systems of all considered types that are effective for further applications, obtain equations for them with respect to the first argument, clarify their properties, and extend many methods and results of state-space theory to continuous-discrete systems.