The modal sensitivity, robustness and roughness of dynamic systems (review article)

The review focuses on the development of theories of sensitivity and modal sensitivity, robustness and roughness of dynamic systems. In modern theory of dynamic systems and automatic control systems, there is a need to study the properties of sensitivity, robustness and roughness of systems in their interconnection. An import application of sensitivity theory is associated with the design and creation of high-precision and low-sensitivity systems. The most significant results were achieved in the development of differential methods for the analysis and synthesis of lowsensitivity systems. One approach to the analysis and synthesis of linear systems of low parametric sensitivity in the space of states was developed using the functions of modal sensitivity or, in other words, the method of modal sensitivity. A review of robustness theory considers methods of studying and ensuring the robust stability of interval dynamical systems, paying attention to both algebraic and frequency directions of robust stability. The work presents the main results of the original algebraic method of robust stability of interval dynamical systems for continuous and discrete time. The section “Basic development stages of the theory of roughness of systems” describes the main principles of the theory and method of topological roughness of dynamical systems, based on the concept of roughness according to Andronov–Pontryagin. Applications of the topological roughness method to synergetic systems and chaos have been used to investigate many systems, such as the Lorenz and Rössler attractors, the Belousov–Zhabotinsky, Chua, predatorprey, Hopf bifurcation, Schumpeter and Caldor economic systems, etc. The review proposes applying the approach of analogies of theoretical-multiple topology and the abstract method for studying weakly formalized and informal systems. Further research suggests the development of theories of sensitivity, robustness, roughness and bifurcation for complex nonlinear dynamic systems.