In the paper, the control system synthesis by means of the full state vector is considered when using fractional derivatives in the description of this system. To conduct research in the synthesized system with fractional derivatives in the Caputo--Fabrizio representation, a fundamental matrix of the system is formed, which also allows us to analyze the influence of initial conditions on the processes within the system. In particular, the finding of the fundamental matrix of the system in the case of multiple roots of a characteristic polynomial, which are obtained by transforming the syn
Based on an analysis of the influence of the roots of a characteristic polynomial on the increment of the argument of the frequency characteristic of the system, the frequency criterion of stability of a system with fractional-order derivatives has been suggested. The boundaries of the zone of location of the roots of the characteristic polynomial of a stable system have been determined in a complex plane when the index $\alpha$ of the basis of the characteristic polynomial changes.