Proof Of Finite-time Stability Research Articles

A Finite-Time Control Design for the Discrete-Time Chaotic Logistic Equations

Finite-time control theory has been widely used as a mathematical tool to design robust controllers. By manipulating the finite-time convergence proof of this theory, we developed a new control design appropriately tuned for the finite-time control of the chaotic logistics system. In this experimental setup, the logistic equation is programmed into a PIC microcontroller, and a part of the controller was conceived using analog electronics. Because the system to be controlled is in the discrete-time domain, and the finite-time stability proof is stated in the continuous-time representation, our finite-time control approach is a good example for designing control algorithms in both time domain schemes. Hence, our experimental results support our main contribution. Pulse width modulation (PWM) is the format used to translate digital signals into the continuous-time field. Therefore, the main contribution of this article is the theoretical foundations for creating a recent controller that satisfies the convergence criterion in finite time and its construction using an 8-bit microcontroller. All this contributes to the chaotic logistical map.

A finite-time path-tracking control algorithm for nonholonomic mobile robots with unknown dynamics and subject to wheel slippage/skid disturbances

Path planning and tracking control are two performance-critical tasks for wheeled mobile robots, particularly when nonholonomic constraints are imposed on robots in dynamically uncertain conditions. Accomplishing certain performance and safety considerations related to path-tracking, such as global stability, transient performance, and smooth finite-time convergence, becomes more difficult for nonholonomic robots. This paper is concerned with proposing a new adaptive robust finite-time tracking control approach for a large class of differential drive autonomous nonholonomic wheeled mobile robots (NWMRs) that are subject to structured uncertainties and extraneous disturbances with fully unknown dynamics. For this purpose, nonlinear kinodynamics of a type of rear-wheel drive NWMRs are developed by incorporating the skid/slippage constituents of the wheel motion. Then, a path-tracking controller is proposed using a continuous finite-time adaptive integral sliding mode control coupled with an integral backstepping approach (FTAISM-IBC). For the adaptive controller design, the entire nonlinear dynamics of the robot, including nonlinear vector functions and control gain functions, together with extraneous disturbances, are estimated by leveraging the universal approximation capabilities of radial basis neural networks (RBFNNs). The finite-time stability proof is presented by utilizing the Lyapunov stability theorem. Furthermore, the adaptive gains are derived to ensure the finite-time stability of the system subject to unknown functions, parametric variations, and unknown but bounded disturbances. Finally, the effectiveness of the proposed controller is evaluated through simulations in terms of several key performance indicators against several reported studies.

Fast finite-time backstepping for helicopters under input constraints and perturbations

This work addresses the issue of trajectory tracking of helicopters under disturbances and input constraints via the proposed fast finite-time backstepping framework. Backstepping with the property of fast finite-time convergence exhibits fast transient convergence both at a distance from and at a close range of the equilibrium. Moreover, finite-time disturbance observers based on multivariable super-twisting are employed to counteract the effect of perturbations. Aiming at the adverse effect of input saturation, a novel auxiliary system is developed to avoid the singularity by transforming auxiliary variables for three times. In addition, the framework is also applied into helicopter control problem, in which thrust magnitude and thrust direction are used as intermediate control variables, connecting external translational dynamics and internal angular dynamics. A rigorous proof of finite-time stability of the closed-loop system is derived from Lyapunov theory. Finally, the effectiveness and superiority of the proposed framework are verified by comparative simulation.

Continuous Finite-Time Torque Control for Flexible Assistance Exoskeleton with Delay Variation Input

SUMMARYAccurate torque control is a critical issue in the compliant human–robot interaction scenario, which is, however, challenging due to the ever-changing human intentions, input delay, and various disturbances. Even worse, the performances of existing control strategies are limited on account of the compromise between precision and stability. To this end, this paper presents a novel high-performance torque control scheme without compromise. In this scheme, a new nonlinear disturbance observer incorporated with equivalent control concept is proposed, where the faster convergence and stronger anti-noise capability can be obtained simultaneously. Meanwhile, a continuous fractional power control law is designed with an iteration method to address the matched/unmatched disturbance rejection and global finite-time convergence. Moreover, the finite-time stability proof and prescribed control performance are guaranteed using constructed Lyapunov function with adding power integrator technique. Both the simulation and experiments demonstrate enhanced control accuracy, faster convergence rate, perfect disturbance rejection capability, and stronger robustness of the proposed control scheme. Furthermore, the evaluated assistance effects present improved gait patterns and reduced muscle efforts during walking and upstair activity.

Adaptive terminal sliding mode control of high-order nonlinear systems

In recent decades, controller design has been attracted a great deal of interest of many researchers in control community which can make the job of many other researchers in different area of research easier. The present study aims to design a finite time adaptive control input for a high-order nonlinear system in presence of a variety of mismatched uncertainties and external disturbances. Adaptive terminal sliding mode control (ATSMC) method is used to design robust controller in a finite time. Also, adaptive concept is employed in ATSMC to estimate the upper bound of mismatched uncertainties and external disturbances and their estimations are used in control input. The finite time stability proof is performed by defining a proper candidate Lyapunov function. Numerical simulation results are carried out in Simulink/MATLAB to reveal the correctness of proposed design in this research. Finally, the performance criterion, integral of the square value (ISV), is defined to provide a numerical comparison between the proposed adaptive controller and non-adaptive controller.

Adaptive terminal sliding mode control of high-order nonlinear systems

In recent decades, controller design has been attracted a great deal of interest of many researchers in control community which can make the job of many other researchers in different area of research easier. The present study aims to design a finite time adaptive control input for a high-order nonlinear system in presence of a variety of mismatched uncertainties and external disturbances. Adaptive terminal sliding mode control (ATSMC) method is used to design robust controller in a finite time. Also, adaptive concept is employed in ATSMC to estimate the upper bound of mismatched uncertainties and external disturbances and their estimations are used in control input. The finite time stability proof is performed by defining a proper candidate Lyapunov function. Numerical simulation results are carried out in Simulink/MATLAB to reveal the correctness of proposed design in this research. Finally, the performance criterion, integral of the square value (ISV), is defined to provide a numerical comparison between the proposed adaptive controller and non-adaptive controller.

Two Novel Approaches of NTSMC and ANTSMC Synchronization for Smart Grid Chaotic Systems

The presence of uncertainties and external disturbances is one of the unavoidable problems with various practical systems which might be unavailable in real-time. Sliding Mode Control (SMC) is one of the effective robust control methods to deal with these uncertainties and external disturbances. In this paper, two novel controllers are designed by using Nonsingular Terminal SMC (NTSMC) and Adaptive Nonsingular Terminal SMC (ANTSMC) methods for synchronization of dual smart grid chaotic systems with various uncertainties and external disturbances. Indeed, both adaptive and non-adaptive controllers based on NTSMC are proposed to provide two alternatives which can adjust by changing operating conditions and dynamics. The concept of SMC method guarantees controller robustness against various uncertainties and external disturbances. Elimination of the undesirable chattering phenomenon is addressed in this study which is one of the common deficiencies with conventional SMC method. Additionally, finite time concept is used to speed up the convergence rate. Finite time stability proof is performed by using Lyapunov stability theory. The numerical simulation is carried out in Simulink/MATLAB to reveal the validity of the proposed controllers for the smart grid chaotic system. A comprehensive comparison is made by performing simulation for the Fractional Order Adaptive Sliding Mode Control (FOASMC) controller and defining three performance criteria, among the proposed controllers in this study and FOASMC controller.