A. A. Kosov

On the asymptotic stability of switched homogeneous systems

Abstract The stability of switched systems generated by the family of autonomous subsystems with homogeneous right-hand sides is investigated. It is assumed that for each subsystem the proper homogeneous Lyapunov function is constructed. The sufficient conditions of the existence of the common Lyapunov function providing global asymptotic stability of the zero solution for any admissible switching law are obtained. In the case where we can not guarantee the existence of a common Lyapunov function, the classes of switching signals are determined under which the zero solution is locally or globally asymptotically stable. It is proved that, for any given neighborhood of the origin, one can choose a number L > 0 (dwell time) such that if intervals between consecutive switching times are not smaller than L then any solution of the considered system enters this neighborhood in finite time and remains within it thereafter.

Analysis of hybrid systems’ dynamics using the common Lyapunov functions and multiple homomorphisms

Consideration was given to the hybrid systems obeying the nonlinear common differential equations with switched right-hand sides and state jumps (pulses). Conditions for availability of dynamic characteristics like stability, attraction, invariance, and boundedness were formulated in terms of the common Lyapunov functions or multiple homomorphisms.

Stability and stabilization of mechanical systems with switching

Hybrid mechanical systems with switched force fields, whose motions are described by differential second-order equations are considered. We propose two approaches to solving problems of analysis of stability and stabilization of an equilibrium position of the named systems. The first approach is based on the decomposition of an original system of differential equations into two systems of the same dimension but of the first order. The second approach is in direct specifying a construction of a general Lyapunov function for a mechanical system with switching.

Analysis of stability and stabilization of nonlinear systems via decomposition

We establish necessary and sufficient conditions for the solvability of a Lyapunov-type system of PDEs in the class of homogeneous functions. Using these, we propose an approach to studying the stability of an equilibrium of an essentially nonlinear system of ODEs in the critical case of n zero roots and n pure imaginary roots. The approach bases on decomposition of the system in question into two separate subsystems of half dimension.

On the control problem of a switched mechanical system

The problem of stabilization of the equilibrium position for the mechanical systems described by the Lagrange equations of the second kind with possible switchings of both a kinetic energy, and the forces field is investigated. The simple power law of a feed-back guaranteeing the transfer to equilibrium position for finite time from the small neighborhood with the subsequent retention for any admissible switching law is offered.

Stability analysis of hybrid mechanical systems with switched nonlinear nonhomogeneous positional forces

Mechanical systems with linear gyroscopic forces, nonlinear homogeneous dissipative forces and switched nonlinear nonhomogeneous positional forces are studied. Using the decomposition method, conditions are found providing the asymptotic stability of equilibrium positions of the considered systems for an arbitrary admissible switching law. On the basis of the obtained results, new approaches to stabilization of nonlinear switched mechanical systems are proposed.

Stabilization of the Equilibrium Position of a Magnetic Control System with Delay

A model of magnetic suspension control system of a gyro rotor is studied. A delay in the feedback control scheme and dissipative forces occurring due to energy losses at the interaction of the magnetic field with currents in the control loops are taken into account. Two approaches to the synthesis of controls stabilizing the equilibrium position of the considered system are proposed. The results of a computer simulation are presented to demonstrate effectiveness of the approaches. INTRODUCTION Nonlinear oscillatory systems are widely used for the modeling charge particles motions in cyclotrons in neighborhoods of equilibrium orbits [1–3]. They are also applied for the analysis and synthesis of magnetic control devices [4, 5]. In particular, magnetic systems of retention of power gyro rotors are used in modern control systems of spacecraft orientation with long periods of autonomous operation. An actual problem for such systems is stabilization of their operating modes. It is worth mentioning that realistic models of numerous control systems should incorporate delay in feedback law [6]. It is well-known that delay may seriously affect on system’s dynamics. Therefore, it is important to obtain restrictions for delay values under which stability can be guaranteed. In this paper, analytical and numerical investigations of stability of the equilibrium position for a nonlinear oscillatory system are presented. The system can be treated as a mathematical model of magnetic suspension control system of a gyro rotor [5]. A delay in the feedback control scheme and dissipative forces occurring due to energy losses at the interaction of the magnetic field with currents in the control loops are taken into account. Two approaches to the synthesis of stabilizing controls are proposed. With the aid of a computer simulation of dynamics of closed-loop systems, a comparison of these approaches is fulfilled, and their features and conditions of applicability are determined. The research was partially supported by the Saint Petersburg State University (project No. 9.37.157.2014), and by the Russian Foundation for Basic Research (grant Nos. 15-08-06680-a and 16-01-00587-a). † a.u.aleksandrov@spbu.ru STATEMENT OF THE PROBLEM Consider the control system