On global trajectory tracking control of robot manipulators with a delayed feedback

In this paper, the trajectory tracking control problem of a robot manipulator with cylindrical joints is considered by means of a nonlinear PD controller taking into account the delayed feedback structure. The conclusion about stability of a closed-loop system is obtained on the basis of the development of the direct Lyapunov method in the study of the stability property for a non-autonomous functional differential equation by constructing a Lyapunov functional with a semi-definite time derivative.

Analysis of the most common methods for determining the stability of energy systems

Problem. There are many methods for determining the stability of the energy system. In normal operating condition (normal rated mode), the power system must reliably ensure the consumption of electricity of normalized quality. However, in addition to the normal state, there are emergency and transient states caused by various transients. This is due to the fact that the energy system is constantly changing its parameters. Such changes are determined by variations in the amount of power produced and consumed, as well as the changes in system configuration. Goal. The goal is studying the possibilities of various methods of determining the power systems stability and drawing up the general algorithm of actions for maintenance of their stability. Methodology. When determining the stability of energy systems by the Lyapunov method, two methods can be used: the direct method and the first approximation method. Lyapunov direct method refers to differential methods. To conclude about the stability of the system we do not find a general or particular solution of differential equations, but with their help we find a mathematical function, the complete derivative of which over time allows to obtain a conclusion about the stability of the system. Results. Many methods can be used to determine whether a sustainable energy system is stable or not. The most common are the Lyapunov methods and the Moiseev method. It is determined that the direct Lyapunov method refers to differential methods. The application of the direct Lyapunov method for energy problems is limited. Currently, it can be used only for some individual cases. The method of the first approximation (Lyapunov first method) has received wider application in the solution of power problems. When applying this method, which belongs to the group of methods of full integration, the right-hand sides of the equations are decomposed into power series. Originality. It is determined that one of the perspective directions of increasing the efficiency of the mathematical device work is using the methods of the second order in modeling and optimization of operating modes of electric power systems. This allows you to increase the speed and reliability of the convergence of iterative processes. Practical value. Based on the analysis of various existing methods for solving the problems of stability of energy systems, an algorithm of actions is proposed and developed, which will help to solve the problem of stability in practice.

MATHEMATICAL MODELS OF ANGULAR MOTION OF SPACE VEHICLES AND THEIR USE IN ORIENTATION CONTROL PROBLEMS

A general structure of the kinematic equations for attitude evolution of a spacecraft (SC) (coordinate system associated with a spacecraft (SCS)) relative to the reference coordinate system (RCS) is proposed. It is assumed that the origins of the coordinate systems coincide and are located at an arbitrary point of the spacecraft. Each of the coordinate systems rotates at an arbitrary absolute angular velocity (relative to the inertial space) specified by the projections on their axes. Attitude parameters can be the Euler–Krylov angles, Rodrigues–Hamilton parameters, and modified Rodrigues parameters. It is shown that the well-known representations of the attitude evolution equations of the SCS relative to the RCS using the Rodrigues-Hamilton parameters (components of normalized quaternions) can be simply obtained from the solution of the Erugin problem of finding the entire set of differential equations with a given integral of motion. The advantages and disadvantages of use for each of the specified attitude parameters are considered. A method of attitude control synthesis is proposed which is common for all these equations and based on the decomposition of the original problem into kinematic and dynamic ones and the use of well-known generalizations of the direct Lyapunov method for their solution. The property of structural roughness according to Andronov–Pontryagin [27–29] of the obtained algorithm is illustrated with the help of computer simulation. Particularly, a specific example illustrates the possibility for even a structurally simplified algorithm of stabilizing a specified constant spacecraft attitude to track the program of its change with sufficient accuracy. The tracking task is typical for the control of spacecraft docking, spacecraft de-orbiting, and performing route surveys of the Earth's surface.

A Type-2 Fuzzy Approach to Driver-Automation Shared Driving Lane Keeping Control of Semi-autonomous Vehicles under Imprecise Premise Variable

The driver-automation shared driving is a transition to fully-autonomous driving, in which human driver and vehicular controller cooperatively share the control authority. This paper investigates the shared steering control of semiautonomous vehicles with uncertainty from imprecise parameter. By considering driver’s lane-keeping behaviour on the vehicle system, a driver-automation shared driving model is introduced for control purpose. Based on the interval type-2 (IT2) fuzzy theory, moreover, the driver-automation shared driving model with uncertainty from imprecise parameter is described using an IT2 fuzzy model. After that, the corresponding IT2 fuzzy controller is designed and a direct Lyapunov method is applied to analyze the system stability. In this work, sufficient design conditions in terms of linear matrix inequalities are derived, to guarantee the closed-loop stability of the driver-automation shared control system. Meanwhile, an H ∞ performance is studied to ensure the robustness of the control system. Finally, simulationbased results are provided to demonstrate the performance of the proposed control method. Furthermore, an existing type-1 fuzzy controller is introduced as comparison, to verify the superiority of the proposed IT2 fuzzy controller.

High Velocity Lane Keeping Control Method Based on the Non-Smooth Finite-Time Control for Electric Vehicle Driven by Four Wheels Independently

In order to improve the output response and robustness of the lane keeping controller for the electric vehicle driven by four wheels independently (EV-DFWI), the article proposes a lane keeping controller based on the non-smooth finite-time (NoS-FT) control method. Firstly, a lane keeping control (LKC) model was built for the EV-DFWI. Secondly, a tracking method and error weight superposition method to track error computing for the lane keeping control based on the LKC model are proposed according to the lane line information. Thirdly, a NoS-FT controller was constructed for lane keeping. It is proved that the NoS-FT controller can stabilize the system by the direct Lyapunov method. Finally, the simulations were carried out to verify that the NoS-FT controller can keep the vehicle running in the desired lane with the straight road, constant curvature road, varied curvature road, and S-bend road. The simulation results show that the NoS-FT controller has better effectiveness than the PID controller. The contributions of this article are that two kinds of tracking error computing methods of lane keeping control are proposed to deal with different conditions, and a Non-FT lane keeping controller is designed to keep the EV-DFWI running in the desired lane suffering external disturbances.

Asymptotic synchronization of fractional order non-identical complex dynamical networks with Parameter Uncertainties

This article specically deals with the asymptotic synchronization of non-identical complex dynamic fractional order networks with uncertainty. Initially, by using the Riemann-Liouville derivative, we developed a model for the general non-identical complex network, and based on the properties of fractional order calculus and the direct Lyapunov method in fractional order, we proposed that drive and response system if nonidentical complex networks ensuring asymp-totic synchronization by using neoteric control. Second, taking into account the uncertainties of non-identical complex networks in state matrices and evaluating theirrequirements forasymptotic synchronization. In addition, to explain the eectiveness of the proposed approach, two numerical simulations are given.

Exponential stability of axially moving Kirchhoff-beam systems with nonlinear boundary damping and disturbance

<p style='text-indent:20px;'>This paper examines the stabilization problem of the axially moving Kirchhoff beam. Under the nonlinear damping criterion established by the slope-restricted condition, the existence and uniqueness of solutions of the closed-loop system equipped with nonlinear time-delay disturbance at the boundary is investigated via the Faedo-Galerkin approximation method. Furthermore, the solution is continuously dependent on initial conditions. Then the exponential stability of the closed-loop system is established by the direct Lyapunov method, where a novel energy function is constructed.</p>

Improved Model-Free Sliding Mode Control Algorithm for Control Non-Linear Systems

When testing the performance of the model-free sliding mode control algorithm, it was found that it could not maintain good performance when the system was exposed to noise. this research suggests designing a noise-resistant model-free sliding mode control algorithm. The importance of this algorithm is that it takes into account the effect of the noise, where the noise value is implemented in the model-free algorithm. The sliding surface of the controller is designed based on the improved relationship and to ensure the stability of the system in the closed-loop the control signal was derived based on the direct Lyapunov method. To minimize the effects of chattering in the control signal, the control law was reconfigured using a boundary layer. The improved algorithm was implemented to a second-order non-linear system and the simulation results showed the system's ability to track the desired signal in spite of the presence of the noise as well as its ability to maintain the stability of the controlled system

On the stability of solutions of fractional non conformable differential equations

In this note we obtain sufficient conditions under which we can guarantee the stability of solutions of a fractional differential equations of non conformable type and we obtain some fractional analogous theorems of the direct Lyapunov method for a given class of equations of motion.

Conditions for the local and global asymptotic stability of the time–fractional Degn–Harrison system

Fractional calculus has been shown to improve the dynamics of differential system models and provide a better understanding of their dynamics. This paper considers the time–fractional version of the Degn–Harrison reaction–diffusion model. Sufficient conditions are established for the local and global asymptotic stability of the model by means of invariant rectangles, the fundamental stability theory of fractional systems, the linearization method, and the direct Lyapunov method. Numerical simulation results are used to illustrate the theoretical results.