Browsing by Author "Sadovoi, Aleksandr"

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    Lyapunov Function in the Hyper-Complex Phase Space
    (Springer, Singapore, 2022) Voliansky, Roman; Volianska, Nina; Kuznetsov, Valeriy; Sadovoi, Aleksandr; Kuznetsov, Vitaliy V.; Kuznetsova, Yevheniia; Ostapchuk, Oleksandr
    ENG: The paper deals with the development of background for defining Lyapunov functions for a wide range of linear dynamical objects. This background is based on assuming that the Lyapunov function is redundant energy in the considered object, and this energy is dissipated only during controlled motion. We assume the full derivative of the Lyapunov function for an autonomous motion of the control objects equals zero, and we use its summands to define linear algebraic equations. The solution of these equations allows us to find unknown terms of the Lyapunov function. The use of these terms, while the Lyapunov equation is being written down, shows that the left-hand expression in the Lyapunov equation is equal to the zero matrix. Thus, we avoid subjective assuming of quadratic form terms in the right-hand of the Lyapunov equation. We extend the proposed approach to the class dynamical system with uncertainty. This extension is performed by using interval methods, which allow defining object motions for minimal and maximal values of parameters. We show that for the control object, which parameters are not exactly known, one should consider two equations of object motions, which correspond to its trajectories on the boundaries of the intervals. Lyapunov functions are defined for these boundary trajectories. Since such an approach increases the number of the considered equations, we offer to decrease them by using hyper-complex numbers while object equations are written down.