The main provisions of the method of research the roughness of dynamic systems, based on the concept of coarseness according to Andronov-Pontryagin and called the "topological roughness method," are considered. Definitions of concepts of maximum roughness and minimum non-roughness of dynamic systems are given. The corresponding theorems are formulated on the necessary and sufficient conditions for the reachability of maximum roughness and minimum non-roughness, as well as the emergence of bifurcations of topological structures of the phase space of dynamic systems, which were proved in the author's works. The method allows you to control the roughness of control systems based on the theorem formulated using the Sylvester matrix equation. The method can be used for studies of roughness and bifurcations of dynamic systems, as well as synergetic systems and chaos of various physical nature. In the author's works, the method has been tested for studying the synergetic systems of Lorenz, Rössler, Belousov-Zhabotinsky, "predator-prey", Chua, Rikitake dynamo, Henon maps, Hopf bifurcations, and models of economic systems of the Schumpeter and Kaldor types.

The problem of determining the conditions of experiment planning, when the value of the quality estimates of processes in a linear multivariable system coincides exactly with the ellipsoidal quality estimates obtained using the apparatus of ellipsoidal estimation in the form of majorants and minorants, is considered. The apparatus allows to solve the problem of transferring the basic quality estimates developed in the theory of SISO-systems to the case of MIMO-systems, providing their geometric interpretability. The properties of the components of singular value decomposition (SVD) of matrices are used to solve the problem.

A problem of studying the sensitivity of ellipsoidal frequency estimates of quality of multivariable dynamical systems to parameter variations is considered. The apparatus of sensitivity functions of extreme elements of singular values of real-valued transfer matrices is used for solving of this problem. Combined use of the frequency sensitivity apparatus and the state space method makes it possible to construct sensitivity models which can be used to define ellipsoidal estimates of the frequency sensitivity functions for the state, output and error of linear multivariable continuous systems in the form of a majorant and a minorant of these functions. The calculations use a singular value decomposition of matrices composed of frequency parametric sensitivity functions. The resulting ellipsoidal estimates have the property of minimum sufficiency due to the meaningfulness of the singular values decomposition of matrix. The method allows to select in the state, output and error spaces the subspaces characterized at each frequency value by the largest and the smallest normal variation, using the elements of the left singular basis corresponding to the extreme singular values. The use of the right singular basis permits to identify in the parameter space the subspaces that produce the largest and the smallest normal variation of the amplitude frequency response. This approach allows to solve the problem of “optimal nominal”, that is, the problem of choosing the nominal value of the vector of plant parameters, which delivers the lowest value of ellipsoidal estimates of frequency sensitivity functions to multivariable controlled processes and to compare the course of multivariable controlled processes by ellipsoidal estimates of parameter frequency sensitivity.

The algebraic method of researches of robust stability of continuous and discrete interval dynamical systems is considered, as well as the method of ellipsoidal quality estimates of multidimensional control systems. This paper presents the original results obtained for the research of the stability of continuous and discrete linear interval dynamical systems, called the Algebraic method of robust stability. Ellipsoidal quality estimates based on majorant and minorant estimates calculated using of singular values of corresponding criterion matrices of these properties are proposed for investigation of qualitative properties of multivariate control systems. The computational apparatus for determining ellipsoidal estimates is based on singular value decomposition matrix using standard SVD-procedures. Basic concepts of ellipsoidal estimation method are given.