Barbashin Krasovskii Theorem Research Articles

Sliding Mode-Based Distributed Trajectory Tracking Control of Four-body Train Systems

This paper considers the speed tracking of a four-body train system modelled mathematically based on Newton’s second law, which is described by a large-scale interconnected system with four subsystems. Uncertainties are included in the systems to represent the potential impacts on system performance caused by mechanical wear and external environmental changes. An adaptive sliding mode technique is employed to design a distributed control scheme to guarantee tracking accuracy. Coordinate transformations are introduced to transfer the model of train systems to a system in regular form to facilitate the design of the hyperplane and controllers. The Barbashin-Krasovskii theorem is employed to show the reachability of the hyperplane. In simulations, the Gaussian function is chosen as the desired signal, representing time-varying characteristics relevant to real-world situations, and the result demonstrates the feasibility of the proposed control strategy.

Chaos control and chaos synchronization of a multi-wing chaotic system and its application in multi-frequency weak signal detection

In this paper, first, a nonlinear feedback controller for achieving chaos control of a novel multi-wing chaotic system is presented. The nonlinear feedback controller has two parts. The first part is used to compensate an equilibrium point for the multi-wing chaotic system. The second part is a linear state feedback controller. The nonlinear feedback controller can globally asymptotically stabilize the multi-wing chaotic system to the equilibrium point. Stability conditions are given by using the Barbashin–Krasovskii theorem. Then, a linear state feedback controller for achieving chaos synchronization of the multi-wing chaotic system is presented. The linear state feedback controller can asymptotically stabilize the chaos synchronization error system to the origin. Stability conditions are given by using the passivity-based theory. Finally, a multi-frequency weak signal detection method is presented based on chaos control of the multi-wing chaotic system. The detection method can detect the frequencies of the weak signal and does not need to determine the critical point.

A speed regulator for a force-driven cart-pole system

In this paper, we present a speed regulator for a force-driven cart-pole system. The proposed controller allows bringing the pole towards its upright position, while the cart moves asymptotically at desired constant speed, recovering the position regulation when we assign a desired constant position of the cart. The main motivation for addressing speed regulation is to broaden the scope of control of underactuated mechanical systems, because so far, the control of this class of mechanical systems has been focused primarily on position regulation; and therefore, there is few information and results available on broader control objectives, such as speed regulation. In particular, we design a speed regulator for the cart-pole system, as this system is one of the more representative test benches of underactuated mechanical system found in many automatic control research laboratories. Another novelty is the introduction of an alternative energy shaping approach for the velocity control design of a class of underactuated mechanical systems. A complete asymptotic stability analysis based on the Lyapunov theory and the Barbashin–Krasovskii theorem is presented. Local asymptotic stability is concluded. Simulation results upon a force-driven cart-pole model illustrate the performance of the proposed controller.

A Speed Regulator for A Torque–Driven Inertia Wheel Pendulum

In this paper, we present a speed regulator for a torque-driven inertia wheel pendulum. The proposed controller allows bringing the nonactuated pendulum towards its upright position, while the wheel moves asymptotically at desired constant speed, recovering the popular position regulation control objective for both pendulum and wheel when the desired wheel speed is zero. Also, a complete stability analysis based on the Lyapunov theory and the Barbashin-Krasovskii theorem is presented. Simulation results upon a torque-driven inertia wheel pendulum model illustrate the performance of the proposed controller.

Asymptotical stability of the motion of mechanical systems with partial energy dissipation

We consider a linear mechanical system under the action of potential, gyroscopic and dissipative (partial) forces. The classical Kelvin–Chetaev theorems are not applicable here, and another approach, which is based on Barbashin–Krasovskii theorem, is suggested. This approach is based on decomposition of the whole system and is convenient for systems of high dimension or with uncertain parameters. Some advantages of the proposed method are demonstrated by examples.

Stability and synchronization of oscillators: New Lyapunov functions

The analysis of asymptotic stability of nonlinear oscillator is one of the classical problems in the theory of oscillations. Usually, it is solved by exploiting Lyapunov functions having the meaning of the full energy of the system. However, the Barbashin–Krasovskii theorem has to be used along this way, and no estimates can be found for the rate of convergence of the trajectories to equilibria points. In this paper we propose a different Lyapunov function which lacks transparent physical meaning. With this function, both the rate of convergence and the domain of attraction of equilibria points can be estimated. This result also enables an efficient analysis of another problem, synchronization of oscillations of two oscillators. We formulate conditions that guarantee frequency synchronization and, on top of that, phase synchronization. Generalizations to the case of arbitrary number of oscillators are also discussed; solution of this problem is crucial in the analysis of power systems.

Analysis of stability boundaries of satellite’s equilibrium attitude in a circular orbit

An asymmetric satellite equipped with control momentum gyroscopes (CMGs) with the center of mass of the system moving uniformly in a circular orbit was considered. The stability of a relative equilibrium attitude of the satellite was analyzed using Lyapunov’s direct method. The Lyapunov function V is a positive definite integral of the total energy of the perturbed motion of the system. The asymptotic stability analysis of the stationary motion of the conservative system was based on the Barbashin–Krasovskii theorem on the nonexistence of integer trajectories of the set \(\dot V\), which was obtained using the differential equations of motion of the satellite with CMGs. By analyzing the sign definiteness of the quadratic part of V, it was found earlier by V.V. Sazonov that the stability region is described by four strict inequalities. The asymptotic stability at the stability boundary was analyzed by sequentially turning these inequalities into equalities with terms of orders higher than the second taken into account in V. The sign definiteness analysis of the inhomogeneous function V at the stability boundary involved a huge amount of computations related to the multiplication, expansion, substitution, and factorization of symbolic expressions. The computations were performed by applying a computer algebra system on a personal computer.

Takagi-Sugeno fuzzy logic controller for Liu-Chen four-scroll chaotic system

Chaos modelling, chaos control and chaos synchronisation are active areas in chaos theory, while research in these areas has potential applications in many fields like nonlinear optics, oscillations, chemical reactions, biology, neurology, secure data encryption, secure communications, robotics, neural networks, fractals and nonlinear circuits design. Especially, chaos control is an important research area, and it has a wide range of applications like control of turbulence, dynamics of lasers, plasma, chemical reactions such as Belousov-Zhabotinsky reaction, ecological systems and cardiac arrhythmias. Until now, many methods for chaos control such as open-loop control based on periodic system excitation, closed-loop control such as OGY method and time-delayed feedback control such as Pyragas method have been used. In this paper, a fuzzy logic control using fuzzy sets and fuzzy logic for the control of chaotic systems has been proposed. In more detail, this work presents a fuzzy logic controller design of a chaotic system using Takagi-Sugeno fuzzy model. The stability analysis of the fuzzy logic controller system is carried out using Barbashin-Krasovskii theorem. For the proposed fuzzy logic control, the Liu-Chen four-scroll chaotic system was used. Finally, the simulation results of the proposed control method present the effectiveness of this method.

Global stability and stabilization of more general stochastic nonlinear systems

This paper considers the global stability and stabilization of more general stochastic nonlinear systems. Due to the absence of the conventional assumptions (e.g., Lipschitz condition), the stochastic nonlinear systems under investigation may have more than one weak solution. However, the most associated results are only applicable to the stochastic systems having a unique strong solution, and therefore, it is meaningful to refine and extend the relevant concepts and methods to the more general case. In this paper, the concepts of stochastic stability in the more general sense are first introduced to cover the stochastic nonlinear systems having more than one weak solution. Then, the generalized stochastic Barbashin–Krasovskii theorem and LaSalle theorem are established, which present the criterions of stochastic stability for more general stochastic nonlinear systems. As one of the main contributions in this paper, we rigorously prove the generalized stochastic Barbashin–Krasovskii theorem. Moreover, based on the generalized theorems, the output-feedback and state-feedback stabilization are accomplished respectively for two classes of high-order stochastic nonlinear systems under rather weaker assumptions comparing to the existing literature.

Barbashin-Krasovskii theorem for stochastic differential equations

A system of stochastic differential equations which has a zero solution is considered. It is assumed that there exists a positive definite function such that the corresponding operator is nonpositive. It is proved that if the set does not include entire semitrajectories of the system almost surely, then the zero solution is asymptotically stable in probability.

Analysis of the stability of the equilibria of a satellite carrying a two-degree-of-freedom powered gyroscope with dissipation in the axis of the frame

A method that enables a conclusion to be drawn regarding the nature of the Lyapunov stability of the equilibria positions of autonomous systems with partial dissipation from the nature of the secular stability and truncated equations of the linear approximation is developed based on the Barbashin–Krasovskii theorem. The method is used to investigate the stability of the equilibria of a satellite carrying a two-degree-of-freedom powered gyroscope with dissipation in the axis of the frame in a circular orbit. For the case when the axis of the gyroscope frame is oriented parallel to one of the principal axes of inertia of the satellite, the nature of the Lyapunov stability of all the equilibria, with the exception of a few branching points, is determined. It is established that gyroscopic stabilization, which is possible when there is no dissipation in the axis of the frame for certain equilibria that are unstable in the secular sense, breaks down when dissipation occurs.

The equilibria and stability of a dynamically symmetrical satellite with a two-degree-of-freedom powered gyroscope

Abstract For a dynamically symmetrical satellite carrying a two-degree-of-freedom powered gyroscope, all the relative equilibria in a circular orbit are found as a function of the angular momentum of the rotor and the angle between the precession axis of the gyroscope and the plane of dynamical symmetry. The case with no spring on the axis of the gyroscope frame and the case with a spring whose stiffness satisfies definite conditions are considered. The secular stability of the equilibria is investigated. For a system with dissipation in the axis of the gyroscope frame, the Barbashin–Krasovskii theorem is used to perform a detailed analysis, which enables the character of the Lyapunov stability of all the equilibria to be determined, with the exception of a few points. The results of a numerical solution of the problem of the optimal values of the system parameters, for which asymptotically stable equilibria are obtained with maximum speed, are presented.

The construction of Lyapunov functions with sign-definite derivative for systems satisfying the Barbashin–Krasovskii theorem

The asymptotic stability of the zero solution of autonomous systems of differential equations is considered. For systems satisfying the Barbashin–Krasovskii theorem positive-definite functions are constructed having a negative-definite derivative. The investigation is based on the method of invariant relations.

Lorenz System Stabilization Using Fuzzy Controllers

The paper suggests a Takagi Sugeno (TS) fuzzy logic controller (FLC) designed to stabilize the Lorentz chaotic systems. The stability analysis of the fuzzy control system is performed using Barbashin-Krasovskii theorem. This paper proves that if the derivative of Lyapunov function is negative semi-definite for each fuzzy rule then the controlled Lorentz system is asymptotically stable in the sense of Lyapunov. The stability theorem suggested here offers sufficient conditions for the stability of the Lorenz system controlled by TS FLCs. An illustrative example describes the application of the new stability analysis method.

Stability analysis of steady rotations of a rigid body bearing two-degree-of-freedom control moment gyros with dissipation in gimbal suspension axes

To study the stability of steady rotations of a control moment gyro system with internal dissipation, we use the Barbashin-Krasovskii theorem and the relation, established in [1], between the Lyapunov function and steady motions. Taking into account the special properties of the original problem, we reduce it to a lower-dimensional problem.

Partial asymptotic stability of abstract differential equations

We consider the problem of partial asymptotic stability with respect to a continuous functional for a class of abstract dynamical processes with multivalued solutions on a metric space. This class of processes includes finite-and infinite-dimensional dynamical systems, differential inclusions, and delay equations. We prove a generalization of the Barbashin-Krasovskii theorem and the LaSalle invariance principle under the conditions of the existence of a continuous Lyapunov functional. In the case of the existence of a differentiable Lyapunov functional, we obtain sufficient conditions for the partial stability of continuous semigroups in a Banach space.

The global stability of two-dimensional systems for controlling angular orientation

A two-dimensional dynamic system with an angular coordinate, describing the operation of a two-position automatic steering device, the plane rotations of a spacecraft controlled by an angular orientation system, and phase-lock control frequency system with a proportional-plus-integral filter is considered. An analogue of the Barbashin-Krasovskii theorem is obtained for differential inclusions with angular coordinates. This is used to prove the global stability of two-dimensional systems of angular orientation when there is no external force moment. It is shown that, when there is a constant external force moment in phase space, a region of initial conditions exists that corresponds to circular motions of a spacecraft.

Stabilization of the position of a Lagrangian system with elastic elements and bounded control, with and without measurement of velocities

It is required to position a Lagrangian system whose free and controllable degrees of freedom are elastically linked. The equations of motion of such systems describe, in particular, the dynamics of a robot manipulator with elastic joints. The proposed control laws enable restrictions on the value of the control impulse to be taken into account. In particular, attention is given to the situation in which the velocities are not accessible to measurement. The analysis of the proposed control laws is based on Lyapunov's direct method or, more specifically, on the Barbashin-Krasovskii theorem on asymptotic stability in the large. The proof uses an original method to verify that an auxiliary non-linear function, analogous to the total mechanical energy of a system, closed by a control law, is positive-definite.