Solution of the Spectrum Allocation Problem for a Linear Control System with Closed Feedback

A method for constructing a feedback matrix to solve the spectrum allocation (spectrumcontrol; pole assignment) problem for a linear dynamical system is given. A new proof of thewell-known theorem about the connection between the complete controllability of a dynamicalsystem and the existence of a feedback matrix is formed in the process of constructing the cascadedecomposition method. The entire set of arbitrary elements affecting the nonuniqueness of thematrix is identified. Examples of constructing a feedback matrix in the case of a real spectrumand in the presence of complex conjugate eigenvalues as well as for the case of multiple eigenvaluesare given. The stability of the specified spectrum under small perturbations of system parameterswith a fixed feedback matrix is studied.