A. Yu. Aleksandrov

Saint Petersburg State University | SPBU · Faculty of Applied Mathematics and Control Processes

A linear mechanical system with constant matrix of dissipative forces and continuous and bounded matrices of positional forces is studied. It is assumed that there are a large positive multiplier at the vector of dissipative forces and a constant delay in positional forces. The case is considered where the associated delay-free averaged system is a...

In this paper, Lur’e indirect control systems with sector-type nonlinearities and a constant delay in the feedback law are studied. With the aid of an original construction of complete-type Lyapunov–Krasovskii functional, new conditions of the delay-independent asymptotic stability for the zero solutions of the considered systems are obtained. Furt...

A system of nonlinear differential equations is considered that describes the interaction of two coupled subsystems, one of these subsystems is linear, and the other is nonlinear and homogeneous with an order of homogeneity greater than one. It is assumed that this system is affected by nonstationary perturbations with zero mean values. Using the a...

For a canonical form of mechanical systems defined through gradients of potential energy and dissipative terms, the conditions of finite-time and fixed-time (integral) input-to-state stability are derived by finding suitable Lyapunov functions. The proposed stability conditions are constructive, which is demonstrated in several applications.

For two canonical models of mechanical systems with disturbances presented by Rayleigh‐ and Liénard‐type equations, several designs of Lyapunov functions are investigated. Under given restrictions, the existence of these functions implies global or local input‐to‐state stability property for the systems. The proposed stability conditions are constr...

The paper addresses the problem of attitude stabilization of an artificial Earth satellite with the aid of an electrodynamic control system. Our objective is to stabilize the satellite in a special coordinate system, whose axes are directed along the Lorentz force vector and the geomagnetic induction vector. Thus, natural magneto-velocity coordinat...

A mechanical system under strongly nonlinear potential and dissipative forces, with nonlinear nonstationary perturbations having zero mean values, is studied. Proposing a special construction of Lyapunov function, the conditions are found, under which the perturbations do not influence the asymptotic stability of the trivial equilibrium position of...

The problem of triaxial attitude stabilization of a satellite in the orbital frame is considered. The problem is raised about the possibility of implementing such a system of electrodynamic attitude control by the type of a PID controller in which the restoring component of the control torque contains a distributed delay. A constructive analytical...

The problem of delay-independent stability is studied for a class of mechanical systems under dissipative, non-conservative and potential forces, which are described by homogeneous terms. The obtained conditions are extended to the case of switched force model under arbitrary and restricted commutation laws. The proof is based on analysis of a comp...

The problem of attitude stabilization of a rigid body exposed to a nonstationary perturbing torque is investigated. The control torque consists of a restoring component and a dissipative one. Linear and nonlinear variants of restoring and perturbing torques are analyzed. Conditions of the asymptotic stability of the programmed orientation of the bo...

On the basis of the comparison method, sufficient conditions of polystability are derived for a class of nonlinear complex systems. The developed approach is applied for the stability analysis of mechanical systems composed of interacted subsystems with dissipative, gyroscopic and potential forces. Conditions of polystability of the trivial equilib...

The problem of diagonal Riccati stability is studied for a class of complex systems describing the interaction of the second-order subsystems. It is assumed that the connection graph has a special structure and there is a constant delay in connections between the subsystems. Conditions under which the problem of diagonal Riccati stability for an or...

This letter addresses to the problem of mobile agent deployment on a line segment. It is assumed that each agent receives information from some of its right and some of its left neighbors. A linear control protocol is proposed ensuring the agent convergence to the equidistant distribution on the segment. In addition, the problem of nonlinearly-unif...

This article explores a discrete-time multiagent system on a line. This requires the design of a control protocol providing equidistant agent deployment on a given segment of the line under the constraint that each agent receives information about distances to its neighbors via an auxiliary agent. An approach to the solution of the stated problem i...

The problem of the equidistant deployment of mobile agents on a line segment is studied. It is assumed that agent dynamics are modeled by linear second-order models. Two types of control protocols are studied: with and without relative velocity measurements. For any positive damping coefficient of agents we propose a control protocol that ensures t...

The problem of angular stabilization of a rigid body with an arbitrary triaxial ellipsoid of inertia is solved. The control strategy is based on the type of PID controller, in which, instead of the classical integral term, a more flexible control option is used, which assumes the presence of a distributed delay (integral term) in the control torque...

Some classes of continuous and discrete generalized Volterra models of population dynamics are considered. It is supposed that there are relationships of the type "symbiosis", "compensationism" or "neutralism" between any two species in a biological community. The objective of the work is to obtain conditions under which the investigated models pos...

The conditions of (integral) input-to-state stability and input–output-to-state stability are established for a class of generalized Persidskii systems. The proposed conditions are formulated using linear matrix inequalities. Based on these results the conditions of convergence are derived for discretizations of this class of models obtained by the...

In this work, stability analysis for a class of switched nonlinear time-delay systems is performed by applying Lyapunov–Krasovskii and Lyapunov–Razumikhin approaches. It is assumed that each subsystem in the family is homogeneous (of positive or negative degree) and asymptotically stable in the delay-free setting. The cases of existence of a common...

An artificial Earth satellite (AES) with three different principal central moments of inertia is under consideration. The AES moves along a Keplerian circular equatorial near-Earth orbit. The AES is equipped with electrodynamic attitude control system that simultaneously generates Lorentz and magnetic control torques. The possibility of using an el...

The problem of the equidistant deployment of a group of mobile agents on a line segment is studied. Both cases where agent dynamics is modeled by the first-order integrators and double integrators are considered. It is assumed that each agent receives information from its neighbors not directly, but via an auxiliary agent. The impact of communicati...

In this article, the stability of switched time‐delay Persidskii systems is studied. Both cases of synchronous and asynchronous commutation are considered. A comparison of stability conditions obtained with the aid of Lyapunov–Razumikhin and Lyapunov–Krasovskii approaches is provided. The developed theory is applied to the stability analysis of mec...

For a class of generalized Persidskii systems, whose dynamics are described by superposition of a linear part with multiple sector nonlinearities and exogenous perturbations, the conditions of practical stability, instability and oscillatory behavior in the sense of Yakubovich are established. For this purpose the conditions of local instability at...

The problem of attitude stabilization of a rigid body is under consideration. The issue of realizing such a control system of the PID-controller type with distributed delay (integral term) is addressed. A theorem on the asymptotic stability of the stabilized position of the body, which justifies the possibility of constructing the desired control s...

An Earth-pointing dynamically symmetric satellite with Lorentz attitude control system is under consideration. The problem of the satellite attitude stabilization in the orbital reference frame in the presence of gravitational disturbing torque is studied. The averaging technique is developed and applied in the problem. A rigorous mathematical just...

This paper is devoted to the stability analysis of a class of switched nonlinear positive systems with nonlinearities of a sector type. A special construction of common Lyapunov function is proposed for the family of subsystems associated with a switched system and conditions of the existence of such a function are derived. The obtained results are...

The article analyzes a linear mechanical system with a large parameter at the vector of velocity forces and a distributed delay in positional forces. With the aid of the decomposition method, conditions are obtained under which the problem of stability analysis of the original system of the second-order differential equations can be reduced to stud...

A dynamically symmetric satellite in a circular orbit of small inclination is considered. The problem of the Lorentzian stabilization of the satellite in the orbital coordinate system in the indirect equilibrium position is solved under the conditions of the perturbing effect of the gravitational torque. To solve this problem, which is characterize...

The convergence conditions for a class of generalized Persidskii systems and their discretized dynamics are introduced, which can be checked through linear (matrix) inequalities. The case of almost periodic convergence for this class of systems with almost periodic input is also studied. The proposed results are applied to a Lotka–Volterra model an...

This article is devoted to stability analysis of homogeneous time‐delay systems applying the Lyapunov‐Krasovskii theory, and a generic structure of the functional is given that suits for any homogeneous system of nonzero degree (and can also be used for any dynamics admitting a homogeneous approximation). The obtained stability conditions are utili...

Considering a retarded nonlinear system, this note proposes several modifications of the Lyapunov–Razumikhin approach guaranteeing the existence of an upper estimate on convergence rate of the system solutions. The cases of exponential, finite-time and fixed-time (with respect to a ball) convergences are studied. The proposed approach is illustrate...

An Earth-pointing satellite with Lorentz attitude control system is under consideration. The problem of the satellite attitude stabilization in the orbital reference frame in the presence of disturbances and complicated by underactuation is studied. The averaging technique is developed and applied in the problem. A rigorous mathematical justificati...

Nonlinear differential systems with nonlinearities satisfying sector constraints and with constant delays are studied. Such systems belong to well-known class of Persidskii-type systems, and they are widely used for modeling automatic control systems and neural networks. With the aid of the Lyapunov direct method and original constructions of Lyapu...

The conditions of existence of oscillations in the sense of Yakubovich are considered for a class of generalized nonlinear Persidskii systems. To this end, the conditions of local instability at the origin and global boundedness of solutions are presented in the form of linear matrix inequalities. The proposed theory is applied for robustness analy...

Numeric approximations to the solutions of asymptotically stable homogeneous systems by Euler method, with a step of discretization scaled by the state norm, are investigated (for the explicit and implicit integration schemes). It is proven that for a sufficiently small discretization step the convergence of the approximating solutions to zero can...

This paper provides stability analysis results for a linear mechanical system with a large parameter at the vector of gyroscopic forces and with delay in positional forces. Both cases of discrete and distributed delay are studied. Using the decomposition method and Lyapunov–Krasovskii functionals, conditions are found under which delay does not dis...

The deployment of the agents over the line segment under protocols with communication delays and switching is studied. It is assumed that no information about the value of the communication delay and no information about switching policy in the communication graph is available. It is shown that neither delay values nor switching policy in a natural...

This paper deals with the problem of uniaxial stabilization of the angular position of a rigid body exposed to a nonstationary perturbing torque. The perturbing torque is represented as a linear combination of homogeneous functions with variable coefficients. It is assumed that the order of homogeneity of perturbations does not exceed the order of...

A complex system describing interaction of subsystems of the second order with delay in connections between them is studied. Necessary and sufficient conditions of the existence of a diagonal Lyapunov–Krasovskii functional for the considered system are derived. The obtained results are applied for the stability a nalysis of a mechanical system and...

We consider linear positive systems with delay and switchings of operation modes. We establish conditions under which it is possible to construct a common Lyapunov–Krasovskii diagonal functional for the family of subsystems corresponding to the system with switchings in consideration. These conditions are formulated in terms of the feasibility of a...

A criterion is established for the diagonal stability of positive linear difference-differential systems with discrete and distributed delay. The problem is studied of existence of a common diagonal Lyapunov–Krasovskii functional for a family of these systems. The approach developed is used to obtain conditions of the absolute stability and estimat...

The problem of attitude stabilization of a rigid body with the use of restoring and dissipative torques is studied. The possibility of implementing a control system in which the restoring torque tends to zero as time increases, and the only remaining control torque is a linear time-invariant dissipative one, is investigated. Both cases of linear an...

The problem of monoaxial attitude control of a rigid body subjected to nonstationary perturbations is investigated. The control torque consists of a dissipative component and a restoring one. The cases of linear and nonlinear restoring and perturbing torques are analyzed. Two theorems on asymptotic stability of the programmed orientation are proved...

The problem of monoaxial stabilization of a rigid body is studied. It is assumed that a linear time-invariant dissipative torque and a time-varying restoring torque vanishing as time increases act on the body. Both the case of linear restoring torque and that of essentially nonlinear one are considered. With the aid of the decomposition method, con...

A mechanical system describing by the second order linear differential equations with a positive parameter at the velocity forces and with time delay in the positional forces is studied. Using the decomposition method and Lyapunov–Krasovskii functionals, conditions are obtained under which from the asymptotic stability of two auxiliary first order...

The paper deals with a dynamically symmetric satellite in a circular near-Earth orbit. The satellite is equipped with an electrodynamic attitude control system based on Lorentz and magnetic torque properties. The programmed satellite attitude motion is such that the satellite slowly rotates around the axis of its dynamical symmetry. Unlike previous...

The paper addresses the asymptotic stability problem for a class of difference systems with nonlinearities of a sector type and time-delay. A new approach to Lyapunov–Krasovskii functionals constructing for considered systems is proposed. On the basis of the approach, delay-independent asymptotic stability conditions and estimates of the convergenc...

The paper deals with the problem of monoaxial attitude stabilization of a rigid body. The possibility of implementing such a control system in which the restoring torque tends to zero as time increases is studied. With the aid of the Lyapunov direct method and the differential inequalities theory, conditions under which an equilibrium position of t...

The paper deals with the problem of diagonal stability of nonlinear difference-differential systems. Certain classes of complex systems with delay and nonlinearities of a sector type are studied. It is assumed that these systems describe the interaction of two-dimensional blockswith a delay in connections between the blocks. Two kinds of structure...

The paper is devoted to the problem of delay-independent stability for a class of nonlinear mechanical systems. Mechanical systems with linear velocity forces and essentially nonlinear positional ones are studied. It is assumed that there is a delay in the positional forces. With the aid of the decomposition method and original constructions of Lya...

The paper presents the problem of triaxial stabilization of the angular position of a rigid body. The possibility of implementing a control system in which dissipative torque tends to zero over time and the restoring torque is the only remaining control torque is considered. The case of vanishing damping considered in this study is known as the mos...

A class of discrete-time nonlinear positive time-delay switched systems with sector-type nonlinearities is studied. Sufficient conditions for the existence of common and switched diagonal Lyapunov--Krasovskii functionals for this system class are derived; these are expressed as feasibility conditions for systems of linear algebraic inequalities. Co...

We consider a hybrid dynamical system composed of a family of subsystems of nonlinear differential equations and a switching law which determines the order of their operation. It is assumed that subsystems are homogeneous with homogeneity degrees less than one, and zero solutions of all subsystems are asymptotically stable. Using the Lyapunov direc...

A satellite in a circular near-Earth orbit is under consideration. The three-axis stabilization of the satellite in the orbital coordinate system with the use of electrodynamic attitude control system is studied. No constraints are imposed on the Earth’s magnetic field approximation. The gravity gradient disturbing torque acting on the satellite at...

Systems of differential equations with nonlinearities of a sector type and almost periodic perturbations are studied. With the aid of the Lyapunov direct method, conditions are found under which the considered systems admit globally asymptotically stable almost periodic solutions. Moreover, it is shown that the proposed approach permits us to deriv...

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 l...

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 nonsta...

In this paper, a class of nonlinear switched systems with separable nonlinearities is studied. With the aid of multiple Lyapunov functions method, conditions on switching law are derived under which the zero solutions of the considered systems are globally asymptotically stable. Some examples are presented to illustrate the obtained results.

We present an extension of a recent characterisation of diagonal Riccati stability and, using this, extend a result of Kraaijevanger on diagonal Lyapunov stability to Riccati stability of time-delay systems. We also describe a class of transformations that preserve the property of being diagonally Riccati stable and apply these two results to provi...

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 fo...

We deal with the state consensus problem of a general heterogeneous linear multi-agent system under a time-invariant and directed communication topology. First we adopt a general linear consensus protocol consisting of two parts. One is a state feedback of the agent for independently regulating its dynamics, and the other is a cooperative term in a...

The stability of the trivial solution for a class of difference systems with switching and sector-type nonlinearities is studied. Different approaches to common Lyapunov function design for the family of subsystems corresponding to the considered switched system are proposed. Sufficient conditions making the trivial solution asymptotically stable f...

The problem of preservation of stability under discretization is studied. A class of nonlinear switched difference systems is considered. Systems of the class appear as computational schemes for continuous-time switched systems with homogeneous right-hand sides. By using the Lyapunov direct method, some sufficient conditions of the asymptotic stabi...

Stability of the trivial equilibrium position for a class of hybrid mechanical systems with nonswitched linear velocity forces and switched nonlinear nonhomogeneous positional forces is studied. Sufficient conditions in terms of linear matrix inequalities are obtained to guarantee the existence of a common Lyapunov function for the family of subsys...

An artificial Earth satellite in a circular equatorial orbit is considered. We analyze the possibility of a satellite’s three-axis stabilization in the König coordinate system using an electrodynamic control system exploiting Lorentz and magnetic control torques. Conditions allowing the electromagnetic control to solve the problem for a gravitation...

This paper addresses the stability problem for a set of switched nonlinear difference equations with parametric uncertainties. For the corresponding family of subsystems, a regularization procedure is suggested, and a multiple Lyapunov function is constructed. With the aid of the Lyapunov function, classes of switching signals are determined for wh...

We deal with the state consensus problem of a general Linear Interconnected Multi-Agent System (LIMAS) under a time-invariant and directed communication topology. Firstly, we propose a linear consensus protocol in a general form, which consists of state feedback of the agent itself and feedback form of the relative states between the agent and its...

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 o...

A switched system generated by the family of homogeneous subsystems with homogeneity orders less than one is studied. It is assumed that the zero solution of each subsystem is asymptotically stable. On the basis of the dwell-time approach, conditions on switching law are determined under which a given spherical neighborhood of the origin is contain...

A nonlinear time-delay system with nonlinearities of a sector type and nonstationary perturbations is studied. It is assumed that the zero solution of the corresponding unperturbed delay-free system is asymptotically stable. With the aid of the Razumikhin theorem, conditions are obtained under which the perturbations do not destroy the asymptotic s...