The article considers a class of automatic control systems that is described by a multidimensional system of ordinary differential equations. The right hand-side of the system additively contains a linear part and the product of a control matrix by a vector that is the sum of a control vector and an external perturbation vector. The control vector is defined by a nonlinear function dependent on the product of a feedback matrix by a vector of current coordinates. The authors solve the problem of constructing a matrix of a nonsingular transformation, which leads the matrix of the linear part of the system to the Jordan normal form or the first natural normal form. The variables included in this transformation allow us to vary the system settings, which are the parameters of both the control matrix and the feedback matrix, as well as to convert the system to an integrable form. Integrable form is understood as a form in which the system can be integrated in a final form or reduced to a set of subsystems of lower orders. Furthermore, the sum of the subsystem orders is equal to the order of the original system. In the article, particular attention is paid to cases when the matrix of the linear part has complex conjugate eigenvalues, including multiple ones.