Transient Response in Matrix Discrete-Time Linear Systems

• Published in: Automation and remote control Vol. 80; no. 9; pp. 1645 - 1652
• Main Authors: Polyak, Boris, Smirnov, Georgi
• Format: Journal Article
• Language: English
• Published: Moscow Maik Nauka/interperiodica 01-09-2019; Pleiades Publishing; Springer Nature B.V
• Subjects: CAE) and Design; Calculus of Variations and Optimal Control; Optimization; Ciências Naturais; Computer-Aided Engineering (CAD; Control; Discrete time systems; Infinite degree of stability; Initial conditions; Large deviations; Linear systems; Matemáticas; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix methods; Mechanical Engineering; Mechatronics; Multidimensional systems; Optimization; Robotics; Science & Technology; Stability; Systems Theory; Topical Issue; Trajectories; Transient response; large deviations; discrete-time systems; transient response; infinite degree of stability; Heavy-ball method; multidimensional systems; stability
• ISSN: 0005-1179; 1608-3032
• DOI: 10.1134/S0005117919090066

The behavior of trajectories of multidimensional linear discrete-time systems with nonzero initial conditions is considered in two cases as follows. The first case is the systems with infinite degree of stability (the processes of a finite duration); the second case is the stable systems with a spectral radius close to 1. It is demonstrated that in both cases, large deviations of the trajectories from the equilibrium may occur. These results are applied to accelerated unconstrained optimization methods (such as the Heavy-ball method) for explaining the nonmonotonic behavior of the methods, which is observed in practice.
Russian Science Foundation, project no. 16-11-10015