Alexander Aleksandrov

Delay-Independent Stability Conditions for Some Classes of Nonlinear Systems

Some classes of nonlinear time-delay systems are studied. It is assumed that the zero solution of a system is asymptotically stable when delay is equal to zero. By the Lyapunov direct method, and the Razumikhin approach, it is shown that in the case when the system is essentially nonlinear, i.e., the right-hand side of the system does not contain linear terms, the asymptotic stability of the trivial solution is preserved for an arbitrary positive value of the delay. Based on homogeneous approximation of a time-delay system some stability conditions are found. We treat large-scale systems with nonlinear subsystems. New stability conditions in certain cases, critical in the Lyapunov sense, are obtained. Three examples are given to demonstrate effectiveness of the presented results.

Stability analysis and uniform ultimate boundedness control synthesis for a class of nonlinear switched difference systems

In this paper, we deal with stability analysis of a class of nonlinear switched discrete-time systems. Systems of the class appear in numerical simulation of continuous-time switched systems. Some linear matrix inequality type stability conditions, based on the common Lyapunov function approach, are obtained. It is shown that under these conditions the system remains stable for any switching law. The obtained results are applied to the analysis of dynamics of a discrete-time switched population model. Finally, a continuous state feedback control is proposed that guarantees the uniform ultimate boundedness of switched systems with uncertain nonlinearity and parameters.

Asymptotic Stability Conditions and Estimates of Solutions for Nonlinear Multiconnected Time-Delay Systems

This paper examines certain classes of multiconnected (complex) systems with time-varying delay. Delay-independent stability conditions and estimates of the convergence rate of solutions to the origin for those systems are derived. It is shown that the exponents in the obtained estimates depend on the parameters of Lyapunov functions constructed for the corresponding isolated subsystems. The problem of computing parameter values that provide the most precise estimates is investigated. Some examples are presented to demonstrate the effectiveness of the proposed approaches.

Ultimate boundedness conditions for a hybrid model of population dynamics

This paper addresses the ultimate boundedness and permanence analysis for a Lotka-Volterra type system with switching of parameter values. Two new approaches for the constructing of common Lyapunov function for the family of subsystems corresponding to the switched system are suggested. Sufficient conditions in terms of linear inequalities are obtained to guarantee that the solutions of the considered system are ultimately bounded or permanent for an arbitrary switching signal. An example is presented to demonstrate the effectiveness of the proposed approaches.

Stability analysis and design of uniform ultimate boundedness control for a class of nonlinear switched systems

The problems of stability analysis and synthesis for a class of nonlinear switched systems are addressed in the paper. For arbitrary switching laws and any admissible nonlinear dynamical equations, sufficient linear inequalities conditions are investigated to make such systems are absolutely stable. Next, based on the constructive approach of the common Lyapunov function, a uniform ultimate boundedness controller is developed to guarantee the practical stability of the switched systems with the nonlinear and time-varying uncertain disturbance. A numerical example is given to demonstrate the efficiency of the proposed approach.

Stability analysis of some classes of nonlinear switched systems with time delay

ABSTRACT Stability of certain classes of nonlinear time-delay switched systems is studied. For the corresponding families of subsystems, conditions of the existence of common Lyapunov–Razumikhin functions are found. The fulfilment of these conditions provides the asymptotic stability of the zero solutions of the considered hybrid systems for any switching law and for any nonnegative delay. Some examples are presented to demonstrate the effectiveness of the obtained results.

Stability analysis of nonlinear mechanical systems with switched force fields

The stability of the trivial equilibrium position of a nonlinear mechanical system with switched dissipative and potential forces is studied. It is assumed that dissipative forces are linear, while potential forces are nonlinear and homogeneous. By the use of multiple Lyapunov functions approach and dwell time approach, the conditions on switched law guaranteeing asymptotic stability and practical stability of the equilibrium position are obtained. An example is presented to demonstrate the effectiveness of the proposed approaches.

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.

Partial stability analysis of some classes of nonlinear systems

Abstract A nonlinear differential equation system with nonlinearities of a sector type is studied. Using the Lyapunov direct method and the comparison method, conditions are derived under which the zero solution of the system is stable with respect to all variables and asymptotically stable with respect to a part of variables. Moreover, the impact of nonstationary perturbations with zero mean values on the stability of the zero solution is investigated. In addition, the corresponding time-delay system is considered for which delay-independent partial asymptotic stability conditions are found. Three examples are presented to demonstrate effectiveness of the obtained results.

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