Point In Finite Time Research Articles

Dual sliding mode control of linear maglev synchronous motor based on novel extended state observer

In order to improve the convergence speed and disturbance immunity of the linear maglev synchronous motor (LMLSM) maglev system, this paper proposes a dual sliding mode control strategy based on a novel expanded state observer (NESO). Firstly, a dual terminal sliding mode (DTSM) surface is designed, which exhibits the desired dynamic and steady state performance. The problem of the TSM singularity is solved by the dual sliding mode technique, which turns the control signal into a continuous switching control through the filtering function and effectively reduces the jitter problem. Since the maglev system is susceptible to uncertainties such as nonlinearity, strong coupling and external perturbations, which affect its tracking accuracy, a continuous smooth ESO is designed to estimate and compensate for these uncertainty perturbations, thus solving the problem that the traditional ESO is continuous but not derivable at the switching point. Then, it is proved that the DTSM control strategy based on NESO converges to the equilibrium point in finite time by Lyapunov function. Finally, the effectiveness and superiority of the proposed strategy are verified on the LMLSM maglev system platform.

Non-homogeneous fully nonlinear contracting flows of convex hypersurfaces

Abstract We consider a general class of non-homogeneous contracting flows of convex hypersurfaces in R n + 1 ${\mathbb{R}}^{n+1}$ , and prove the existence and regularity of the flow before extincting to a point in finite time.

A distributed finite‐time optimization algorithm for directed networks of continuous‐time agents

AbstractThis article proposes a new distributed finite‐time optimization algorithm for agents under directed graphs. By employing the nonsmooth technique and graph theory, a distributed discontinuous algorithm for continuous‐time agents subject to strongly convex local cost functions is first designed with a finite‐time distributed estimator, where the gradients of the local cost functions are estimated in finite time. It is shown that for a strongly connected graph and arbitrary initial conditions, the proposed algorithms can achieve consensus, and the systems can converge to the optimal point in finite time. Then, a two‐step approach is proposed to achieve finite‐time optimization of high‐order agents with disturbances under directed graphs. Finally, the validity of the proposed finite‐time optimization algorithm is verified by two numerical examples.

Finite-Time Stability Control of Uncertain Nonlinear Systems With Self-Limiting Control Terms.

In this brief, we define a self-limiting control term, which has the function of guaranteeing the boundedness of variables. Then, we apply it to a finite-time stability control problem. For nonstrict feedback nonlinear systems, a finite-time adaptive control scheme, which contains a piecewise differentiable function, is proposed. This scheme can eliminate the singularity of derivative of a fractional exponential function. By adding a self-limiting term to the controller and the virtual control law of each subsystem, the boundedness of the overall system state is guaranteed. Then the unknown continuous functions are estimated by neural networks (NNs). The output of the closed-loop system tracks the desired trajectory, and the tracking error converges to a small neighborhood of the equilibrium point in finite time. The theoretical results are illustrated by a simulation example.

A novel model-free adaptive terminal sliding mode controller for bridge cranes

To achieve stabilisation control of an underactuated bridge crane system, a new robust control strategy for the sliding mode is proposed in this paper. It can realise finite-time-convergent stabilisation control under the conditions of model uncertainty, parameter perturbation and external interference. In contrast to the existing methods, our method does not need prior information of the dynamic characteristics of the bridge crane system, and can make the system converge to the equilibrium state at the preset time. Specifically, the nonlinear model of the bridge crane system is linearised with partial feedback, and adaptive signals are introduced. Then, according to the form of the transformed system, a fast terminal sliding mode surface is constructed, and an adaptive terminal sliding mode controller is designed. According to strict analysis, the proposed control law ensures that the system converges to the equilibrium point in finite time and provides the convergence time. Finally, the effectiveness and robustness of the proposed control method are verified by comparing the simulation and experimental results with existing methods.

Saturated Nonsingular Fast Sliding Mode Control for the Crane-Form Pipeline System.

The crane-form pipeline (CFP) system is a kind of petrochemical mechanical equipment composed of multiple rotating joints and rigid pipelines. It is often used to transport chemical fluid products in the factory to tank trucks. In order to realize the automatic alignment of the CFP and the tank mouth, the trajectory tracking control problem of the CFP must be solved. Therefore, a saturated nonsingular fast terminal sliding mode (SNFTSM) algorithm is proposed in this paper. The new sliding mode manifold is constructed by the nonsingular fast terminal sliding mode (NFTSM) manifold, saturation functions and signum functions. Further, according to the sliding mode control algorithm and the dynamic model of the CFP system, the SNFTSM controller is designed. Owing to the existence of saturation functions in the controller, the stability analysis using the Lyapunov equation needs to be discussed in different cases. The results show that the system states can converge to the equilibrium point in finite time no matter where they are on the state's phase plane. However, due to the existence of signum functions, the control signal will produce chattering. In order to eliminate the chattering problem, the form of the controller is improved by using the boundary layer function. Finally, the control effect of the algorithm is verified by simulation and compared with the NTSM, NFTSM and SNTSM algorithms. From the comparison results, it is obvious that the controller based on the SNFTSM algorithm can effectively reduce the amplitude of the control torque while guaranteeing the fast convergence of the CFP system state error. Specifically, compared with the NFTSM algorithm, the maximum input torque can even be reduced by more than half.

Finite-Time Tracking Control of Autonomous Underwater Vehicle Without Velocity Measurements

Human-on-the-loop (HOTL) system is regarded as a promising technology to allow autonomous underwater vehicle (AUV) to track the most adequate target point as soon as possible. However, the unique characteristics of the underwater environment make it challenging to perform the tracking task. This article is concerned with a finite-time tracking control issue for AUV, subjected to unavailable velocity signals in the measurement side and uncertain model parameters in physical side. A HOTL system, including operator, buoys, AUV and sensors, is first provided to construct a cooperative tracking network. For such system, operator in surface control center decides the tracking mission based on all available data. Then, a buoy-assisted localization estimator is utilized by AUV to acquire its position, through which a fast terminal sliding mode observer is developed to estimate the velocity of AUV in finite time. With the estimated velocity information, an adaptive-nonsingular fast terminal sliding mode tracking controller is designed to drive AUV to the target point in finite time. For the proposed velocity observer and tracking controller, the signum and differential functions are employed together to improve the convergence speed and reduce the chattering. Besides that, the proposed solution can not only guarantee finite-time velocity observation, but also achieve finite-time tracking control. Finally, simulation and experimental results are both presented to verify the effectiveness.

Optimal spiral-like solutions near a singular extremal in a two-input control problem

<p style='text-indent:20px;'>We study an optimal control problem affine in two-dimensional bounded control, in which there is a singular point of the second order. In the neighborhood of the singular point we find optimal spiral-like solutions that attain the singular point in finite time, wherein the corresponding optimal controls perform an infinite number of rotations along the circle <inline-formula><tex-math id="M1">\begin{document}$ S^{1} $\end{document}</tex-math></inline-formula>. The problem is related to the control of an inverted spherical pendulum in the neighborhood of the upper unstable equilibrium.</p>

Full-Order Terminal Sliding-Mode Control of Brushless Doubly Fed Induction Generator for Ship Microgrids

This paper proposes a full-order terminal sliding-mode (FOTSM) control scheme for brushless doubly fed induction generator (BDFIG)-based islanded microgrids. To deal with mismatched uncertainties in the BDFIG system, virtual control technique-based full-order sliding-mode control is applied to stabilize the amplitude and frequency of terminal voltage. In the current loops, two full-order terminal sliding-mode controllers are designed to make sure that the current tracking errors can reach their equilibrium points in finite time. It is demonstrated by the comprehensive simulations that the proposed method can significantly improve the tracking accuracy, the rapidness, and the robustness to the uncertainties of the BDFIG control system and can enhance the output voltage quality. Furthermore, an experimental study of the proposed control method for BDFIG-based islanded microgrids would be another important future work.

Evolving to non-round Weingarten spheres: integer linear Hopf flows

In the 1950’s Hopf gave examples of non-round convex 2-spheres in Euclidean 3-space with rotational symmetry that satisfy a linear relationship between their principal curvatures. In this paper, we investigate conditions under which evolving a smooth convex rotationally symmetric sphere by a linear combination of its radii of curvature yields a Hopf sphere. When the coefficients of the flow have certain integer values, the fate of an initial sphere is entirely determined by the local geometry of its isolated umbilic points. A variety of behaviours is uncovered: convergence to round spheres and non-round Hopf spheres, as well as divergence to infinity. The critical quantity is the rate of vanishing of the astigmatism—the difference of the radii of curvature—at the isolated umbilic points. It is proven that the size of this quantity versus the coefficient in the flow function determines the fate of the evolution. The geometric setting for the equation is Radius of Curvature space, viewed as a pair of hyperbolic/AdS half-planes joined along their boundary, the umbilic horizon. A rotationally symmetric sphere determines a parameterized curve in this plane with end-points on the umbilic horizon. The slope of the curve at the umbilic horizon is linked by the Codazzi–Mainardi equations to the rate of vanishing of astigmatism, and for generic initial conditions can be used to determine the outcome of the flow. The slope can jump during the flow, and a number of examples are given: instant jumps of the initial slope, as well as umbilic circles that contract to points in finite time and ‘pop’ the slope. Finally, we present soliton-like solutions: curves that evolve under linear flows by mutual hyperbolic/AdS isometries (dilation and translation) of Radius of Curvature space. A forthcoming paper will apply these geometric ideas to non-linear curvature flows.

Disturbance Observer-Based Chattering-Attenuated Terminal Sliding Mode Control for Nonlinear Systems Subject to Matched and Mismatched Disturbances

This work presents a new disturbance observer-based chattering-attenuated terminal sliding mode control for a class of nonlinear systems in the presence of both mismatched and matched disturbances. A nonlinear disturbance observer is typically employed to accurately estimate mismatched disturbances. In this study, a terminal sliding mode control was designed, based on the disturbance estimation results, to counter the effects of disturbances and ultimately stabilize the target system. The utilization of a chattering-attenuated full-order terminal sliding mode structure satisfactorily resolves both chattering and singularity problems in controller design. It was shown by theoretical analyses that both the disturbance estimation error and the system state converge to the equilibrium point in finite time. Two simulation studies, namely a numerical example and an application to an electro hydrostatic actuator system, were conducted to examine the characteristics and to verify the effectiveness of the proposed algorithm.

Robust Stabilization and Synchronization of a Novel Chaotic System with Input Saturation Constraints.

In this paper, the robust stabilization and synchronization of a novel chaotic system are presented. First, a novel chaotic system is presented in which this system is realized by implementing a sigmoidal function to generate the chaotic behavior of this analyzed system. A bifurcation analysis is provided in which by varying three parameters of this chaotic system, the respective bifurcations plots are generated and evinced to analyze and verify when this system is in the stability region or in a chaotic regimen. Then, a robust controller is designed to drive the system variables from the chaotic regimen to stability so that these variables reach the equilibrium point in finite time. The robust controller is obtained by selecting an appropriate robust control Lyapunov function to obtain the resulting control law. For synchronization purposes, the novel chaotic system designed in this study is used as a drive and response system, considering that the error variable is implemented in a robust control Lyapunov function to drive this error variable to zero in finite time. In the control law design for stabilization and synchronization purposes, an extra state is provided to ensure that the saturated input sector condition must be mathematically tractable. A numerical experiment and simulation results are evinced, along with the respective discussion and conclusion.

A novel finite-time q-power recurrent neural network and its application to uncertain portfolio model

This paper presents a novel finite-time q-power recurrent neural network (FT-QPNN) for uncertain portfolio model. An uncertain mean–variance-skewness model under concave transaction costs is discussed. This portfolio model is essentially a nonconvex nonlinear optimization problem with a non-positive definite Hessian matrix of the Lagrange function. The non-positive definite Hessian matrix leads to the failure of many recurrent neural network methods in solving the problem, and many recurrent neural networks cannot converge to the equilibrium point in finite time. To overcome these difficulties, the FT-QPNN is proposed. Combined with finite-time activation function and local convexification method, the FT-QPNN can solve the optimization problem with non-positive definite Hessian matrix and converge to the equilibrium point in finite time. The global finite-time stability and robustness properties of the FT-QPNN are proved theoretically and verified by numerical experiments. Furthermore, the proposed FT-QPNN is applied to solve the uncertain portfolio model. The application simulation results and comparative experiments with other methods respectively illustrate the feasibility and superiority of the FT-QPNN.

Contraction of convex hypersurfaces by nonhomogeneous functions of curvature

A recent article Li and Lv (J. Geom. Anal. 30: 417–447, 2020) considered contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in certain cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article, we extend the result to various other cases that are analogous to those considered in other earlier work, and we show that in all cases, where sufficient pinching conditions are assumed on the initial hypersurface, then under suitable rescaling the final point is asymptotically round and convergence is exponential in the \(C^\infty \)-topology.

Flight quality characteristics and observer-based anti-windup finite-time terminal sliding mode attitude control of aileron-free full-wing configuration UAV

This paper evaluates the flight quality of an aileron-free full-wing configuration UAV and proposes a highly robust attitude controller considering the typical control problems (i.e., manipulation saturation, coupling, susceptibility to the disturbance, nonlinearity, and uncertainty) of the aileron-free full-wing configuration UAV. First, the flight quality characteristics are analyzed through the change of the flight modes at different trim state points in the flight envelope. Then, the back-stepping method is introduced for decoupling and systematic control law design. Based on the Lagrange multiplier approach, the pseudoinverse control allocation is adopted to realize command decoupling and acquire the minimal 2-norm control output. In order to improve the robustness and control accuracy of the controller, a novel continuous finite-time terminal sliding mode control scheme with adaptive law is designed, which has the virtue of suppressing the chattering and avoiding singularity. In addition, an anti-windup compensator (AWC) is developed to ensure the stability of the flight state under manipulation saturation and make the manipulation output exit the saturation region quickly. To further solve the problem that the aileron-free full-wing configuration UAV is susceptible to disturbances, an adaptive super-twisting disturbance observer (ASTDO) is designed to estimate and compensate the uncertain compound disturbance including the uncertain time-varying disturbance and nonlinear dynamic term to improve control performance. The proposed observer can suppress chattering, adjust gain parameters adaptively and have high observation accuracy. Based on the Lyapunov stability theorem, the anti-windup compensation variable in the AWC and the disturbance estimation error in the ASTDO can gradually converge to neighborhoods of the equilibrium points in finite time. Moreover, the practical finite-time stability of the closed-loop control system is proved. According to the flight quality evaluation, the aileron-free full-wing configuration UAV has poor flight quality at the boundary of the flight envelope and exhibits inherent longitudinal high-frequency oscillation characteristics. The simulation results demonstrate that the designed ASTDO-AWFTTSMC guarantees strong robustness, high accuracy, and fast convergence speed for attitude tracking, solves the saturation problem and realizes state decoupling and manipulation decoupling. Moreover, the designed controller overcomes the adverse effects of disturbance, nonlinearity, and uncertainty, greatly suppresses the longitudinal high-frequency oscillation and still exhibits excellent control performance at the flight state points on the flight envelope boundary.

Collapse of n Point Vortices, Formation of the Vortex Sheets and Transport of Passive Markers

In this paper, the motion of the n-vortex system as it collapses to a point in finite time is studied. The motion of vortices is described by the set of ordinary differential equations that we are able to solve analytically. The explicit formula for the solution demands the initial location of collapsing vortices. To find the collapsing locations of vortices, the algebraic, nonlinear system of equations was built. The solution of that algebraic system was obtained using Newton’s procedure. A good initial iterate needs to be provided to succeed in the application of Newton’s procedure. An unconstrained Leverber–Marquart optimization procedure was used to find such a good initial iterate. The numerical studies were conducted, and numerical evidence was presented that if in a collapsing system n=50 point vortices include a few vortices with much greater intensities than the others in the set, the vortices with weaker intensities organize themselves onto the vortex sheet. The collapsing locations depend on the value of the Hamiltonian. By changing the Hamiltonian values in a specific interval, the collapsing curves can be obtained. All points on the collapse curves with the same Hamiltonian value represent one collapsing system of vortices. To show the properties of vortex sheets created by vortices, the passive tracers were used. Advection of tracers by the velocity induced by vortices was calculated by solving the proper differential equations. The vortex sheets are an impermeable barrier to inward and outward fluxes of tracers. Arising vortex structures are able to transport the passive tracers. In this paper, several examples showing the diversity of collapsing structures with the vortex sheet are presented. The collapsing phenomenon of many vortices, their ability to self organize and the transportation of the passive tracers are novelties in the context of point vortex dynamics.

TSM-Based Adaptive Fuzzy Control of Robotic Manipulators with Output Constraints.

This paper proposes an adaptive control scheme based on terminal sliding mode (TSM) for robotic manipulators with output constraints and unknown disturbances. The fuzzy logic system (FLS) is developed to approximate unknown dynamics of robotic manipulators. An error transformation technique is used in the process of controller design to ensure that the output constraints are not violated. The advantage of the error transformation compared to traditional barrier Lyapunov functions (BLFs) is that there is no need to design a virtual controller. Thus, the design complexity of the controller is reduced. Through Lyapunov stability analysis, the system state can be proved to converge to the neighborhood near the balanced point in finite time. Extensive simulation results illustrated the validity of the proposed controller.

Stabilization of a Class of Nonlinear Underactuated Robotic Systems through Nonsingular Fast Terminal Sliding Mode Control

To facilitate the stabilization of nonlinear underactuated robotic systems under perturbation, a novel nonsingular fast terminal sliding mode control method is proposed. Based on the system transformation into an integrator chain, the combination of twisting-like algorithm and a nonsingular fast terminal sliding mode control technique is employed to achieve the stabilization of the studied systems, which can drive the robot states (joint positions and velocities) to the desired region and then maintain the system at the equilibrium point in finite time. The robustness of the proposed method is validated by the Lyapunov direct method. Finally, numerical simulation results further demonstrate that the proposed method has better performance on the convergent speed of the system state (robot joint positions and velocities) than state-of-the-art methods, especially for the underactuated joints.

A continuous strategy for collisionless gathering

Over the past decades, the Gathering problem, which asks to gather a group of robots in finite time given some restrictions, has been intensively studied. In this paper, we are given a group of n autonomous, dimensionless, deterministic, and anonymous robots, with bounded viewing range. Assuming a continuous time model, the goal is to gather these robots into one point in finite time. We introduce a simple convergence criterion that defines a new class of algorithms which perform gathering in O(nd) time, where d is the diameter of the initial robot configuration. We show that some gathering algorithms in the literature belong to this class and propose two new algorithms that belong to this class and have quadratic running time, namely, Go-To-The-Relative-Center algorithm (GTRC) and Safe-Go-To-The-Relative-Center algorithm (S-GTRC). We prove that the latter can perform gathering without collision by using a slightly more complex robot model: non oblivious, chiral, and luminous (i.e. robots have observable external memory, as in [8]). We also consider a variant of the Gathering problem, the Near-Gathering problem, in which robots must get close to each other without colliding. We show that S-GTRC solves the Near-Gathering problem in quadratic time and assumes a weaker robot model than the one assumed in the current state-of-the-art.

Contracting Self-similar Solutions of Nonhomogeneous Curvature Flows

A recent article (Li and Lv, J Geom Anal 30:417–447, 2020) considered fully nonlinear contraction of convex hypersurfaces by certain nonhomogeneous functions of curvature, showing convergence to points in finite time in cases where the speed is a function of a degree-one homogeneous, concave and inverse concave function of the principle curvatures. In this article we consider self-similar solutions to these and related curvature flows that are not homogeneous in the principle curvatures, finding various situations where closed, convex curvature-pinched hypersurfaces contracting self-similarly are necessarily spheres.