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    Alexander Andreev

    Leo Tolstoi str., 42, Ulyanovsk, 432017, Russia
    Ulyanovsk State University


    Andreev A. S., Peregudova O.
    In this paper, the stability and stabilization problems for nonlinear Volterra integrodifferential equations with unlimited delay are considered. The development of the direct Lyapunov method in the study of the limiting properties of the solutions of these equations is carried out by using Lyapunov functionals with a semidefinite time derivative. The topological dynamics of these equations has been constructed revealing the limiting properties of their solutions. The assumption of the existence of a Lyapunov functional with a semidefinite time derivative gives a more complete solution to the positive limit set localization problem. On this basis new theorems on sufficient conditions for the asymptotic stability and instability of the zero solution of nonlinear Volterra integro-differential equations are proved. These theorems are applied to the problem of the equilibrium position stability of the hereditary mechanical systems as well as the regulation problem of the controlled mechanical systems using a proportional-integro-differential controller. As an example, the regulation problem of a mobile robot with three omnidirectional wheels and a displaced mass center is solved using the nonlinear integral controllers without velocity measurements.
    Keywords: Volterra integro-differential equation, stability, Lyapunov functional, limiting equation, regulation problem
    Citation: Andreev A. S., Peregudova O.,  On the Stability and Stabilization Problems of Volterra Integro-Differential Equations, Rus. J. Nonlin. Dyn., 2018, Vol. 14, no. 3, pp.  387-407
    Andreev A. S., Peregudova O.
    In the paper the problem on stabilization of program motion for two-link manipulator with elastic joints is solved. Absolutely rigid manipulator links are connected by elastic cylindrical joint and via the same one the first link is fixed to the base. Thus, the manipulator can perform motion in a vertical plane. Motions of the manipulator are described by the system of Lagrange equations of the second kind. The problem on synthesis of motion control of such a system consists in the construction of the laws of change of control moments that allow the manipulator to carry out a given program motion in real conditions of external and internal disturbances, inaccuracy of the model itself. In this paper the mathematical model of controlled motion of the manipulator is constructed for the case of the control actions in the form of continuous functions. Using vector Lyapunov functions and comparison systems on the base of the cascade decomposition of the system we justified the application of these control laws in the problem of stabilization of the program motion of the manipulator. The novelty of the results is to solve the problem of stabilization of nonstationary and nonlinear formulation, without going to the linearized model. The graphs for the coordinates and velocities of the manipulator links confirm the theoretical results.
    Keywords: multi-link manipulator, elastic joint, stabilization, program motion, comparison system, Lyapunov vector-function
    Citation: Andreev A. S., Peregudova O.,  On control for double-link manipulator with elastic joints, Rus. J. Nonlin. Dyn., 2015, Vol. 11, No. 2, pp.  267-277

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