Class Of Second-order Nonlinear Systems Research Articles

Practical fixed-time non-singular sliding mode control of second order nonlinear dynamic systems with chattering and overshooting avoidance

This paper introduces a Practical Fixed-Time Stabilization technique (PFxT) designed for a specific class of nonlinear second-order systems. The closed-loop systems, when appropriately parameterized, exhibit convergence within a predetermined time frame to a confined region near the origin. This outcome is achieved by amalgamating a nonlinear sliding mode approach with a practical fixed-time tracking virtual trajectory. The control algorithmś design incorporates considerations for overshoot reduction and chattering cancellation. Furthermore, the Lyapunov method is employed to establish the practical fixed-time stability of the proposed solution, providing insights into the limits within which the algorithm can be effectively deployed. To complete the algorithm specification, a parameter selection approach is introduced, enabling customization of the desired settling time. Comprehensive simulations are conducted to validate the effectiveness and viability of the proposed PFxT technique.

A new fixed-time terminal sliding mode control for second-order nonlinear systems

In this paper, the fixed-time control for a class of second-order nonlinear systems with matched disturbance is addressed by using terminal sliding mode control technique. To improve the control performance of traditional fixed-time control method, a new fixed-time stability theorem is constructed by introducing a simple linear term, and theoretical analysis shows that the proposed control method provides faster convergence speed and more accurate estimate of the upper bound of settling time. Moreover, the practical fixed-time stability is also discussed in details. Based on the proposed fixed-time stability theorem, a fixed-time terminal sliding mode controller for a class of second-order nonlinear system is designed to obtain a bounded settling time independently of the initial conditions of the system. Simulations are carried out to verify the feasibility of the proposed control method.

An Observer-Based Extremum Seeking Controller Design for a Class of Second-Order Nonlinear Systems

An Observer-Based Extremum Seeking Controller Design for a Class of Second-Order Nonlinear Systems

Adaptive critic learning for approximate optimal event-triggered tracking control of nonlinear systems with prescribed performances

This paper studies the approximate optimal event-triggered tracking control problem for a class of second-order nonlinear systems with prescribed performances. By employing a prescribed performances function, restrictions on tracking errors are removed for prescribed performances including convergence rate, maximum overshoot, and steady-state error. On this basis, a complex optimal tracking control problem can be transformed into an optimal stabilising control problem by constructing an augmented system. Subsequently, an event-triggered mechanism is introduced to reduce communication resources. Then, an adaptive critic learning algorithm is used to solve the Hamilton-Jacobi-Bellman equation, where the weights in the critic networks are tuned through the gradient descent approach and experience replay technology. Through the developed learning method, the persistence of excitation condition is released, and the data efficiency is improved. The Lyapunov stability theory is introduced to verify the stability of the close-loop system and the tracking errors are uniformly ultimately bounded. Finally, simulation results are presented to show the effectiveness of the proposed approach.

Data‐based learning control for optimization of nonlinear systems

Data‐based learning control for optimization of nonlinear systems

A fixed-time distributed extended state observer for uncertain second-order nonlinear system

This paper investigates the fixed-time distributed estimation problem for a class of second-order nonlinear systems with uncertain input, unknown nonlinearity and matched perturbation. A fixed-time distributed extended state observer (FxTDESO) consisting of a group of local observer nodes under directed communication topology is proposed, and each node can reconstruct both the full state and unknown dynamics of the system. To achieve fixed-time stability, a Lyapunov function is elaborated, and based on this, sufficient conditions for the existence of the FxTDESO are established. Under time-invariant and time-varying disturbance, the observation errors can converge to the origin and a small region of the origin within a fixed time, respectively, where the upper bound of the settling time (UBST) is irrelevant to the initial conditions. Compared to the existing fixed-time distributed observers, the proposed observer can reconstruct both the unknown states and uncertain dynamics, and only the output of the leader and 1-dimensional output estimates from the neighboring nodes are needed in the observer design which effectively reduces the communication load. The paper also extends previous finite-time distributed extended state observer to the case of time-variant disturbance and eliminates the complex linear matrix equation assumption that guarantees the finite-time stability. Furthermore, the FxTDESO design for a class of high-order nonlinear systems is also discussed. Finally, simulation examples are conducted to demonstrate the effectiveness of the proposed observer.

Prescribed Performance Adaptive Tracking Control With Small Overshoot for a Class of Uncertain Second-Order Nonlinear Systems

This brief investigates the problem of prescribed performance adaptive tracking control for a class of second-order nonlinear systems with uncertainties. First, a new performance function is defined to guarantee that the tracking error has small overshoot and converges to a predetermined region for a user-defined time. Then, to effectively diminish the control input energy consumption caused by the introduction of constraints, a novel coordinate transformation function is designed. Next, an improved fixed-time backstepping control framework is established to achieve the tracking control objective. According to the theoretical analysis, all signals of the closed-loop system are bounded, and the tracking error converges to a preset domain with small overshoot for a user-designed time. Finally, numerical simulations confirm the effectiveness of the presented control strategy.

Antisaturation Adaptive Fixed-Time Sliding Mode Controller Design to Achieve Faster Convergence Rate and Its Application

This brief proposes an antisaturation adaptive fixed-time sliding mode controller for a class of second-order nonlinear systems with the saturation constraint. Based on the developed fixed-time stable system, a novel nonsingular fast fixed-time sliding mode surface is designed to circumvent the singularity and acquire a faster convergence rate. To avoid choosing inappropriate control gain, an adaptive mechanism is designed. Furthermore, an auxiliary compensation system is developed to solve the saturation constraint. It is rigorously proved that the system is practical fixed-time stable and the tracking error will fast converge to bounded regions within a fixed time. Finally, the obtained results are employed to realize the quadrotor attitude stabilization, and superior control performance can be illustrated via simulations and experiments.

Reinforcement learning-based optimised control for a class of second-order nonlinear dynamic systems

This paper presents an optimised tracking control scheme based on reinforcement learning (RL) for a class of second-order nonlinear systems with unknown dynamics. Different from the first-order dynamic system control, the second-order case is required to synchronously steer two variables of position and velocity, hence it makes this optimised control more challenging to accomplish. To achieve the optimised control, first, neural network (NN) is employed to approximate the solution of Hamilton–Jacobi–Bellman (HJB), and then an RL is performed by constructing both critic and actor based on the NN approximation. Since the RL training laws are derived from the negative gradient of a simple positive function generated in accordance with the partial derivative of HJB equation, it can make the control algorithm significantly simple to compare with the existing optimal control methods. Meanwhile, it can also release the condition of persistence excitation and compensate for the nonlinear uncertainty. Finally, the proposed adaptive control method can guarantee the desired results that are demonstrated by theorem, proof and simulation.

A Comparative Study of Nonsingular Terminal Sliding Mode and Backstepping Schemes for the Coupled Two-Tank System

This paper presents an implementation of two radically different control schemes for a state-coupled two-tank liquid-level system. This is due to the purpose of transferring theoretical studies to industrial systems. The proposed schemes to be introduced and compared are the nonsingular terminal sliding mode control (NTSMC) and the backstepping control (BC). The performances of the developed methods are experimentally tested on a particular class of second-order nonlinear systems. The main purpose of the considered control schemes is to achieve a tracking trajectory for a coupled-tank system. It is proved that the designed robust controllers guarantee the stability of the corresponding closed loop systems. The obtained results are verified with the same setup test to ensure a suitable basis for their comparison. During the experiments, we resorted to adding an integrator to the backstepping control so that we improve the results, leading to the appearance of the integrator backstepping control (IBC). To focus on the adequacy and applicability of the suggested control layout, theoretical comparisons as well as experimental results are afforded and debated.

A Novel Tracking Control Algorithm With Finite-Time Disturbance Observer for a Class of Second-Order Nonlinear Systems and its Applications

This study aims to build a novel tracking control algorithm using a finite-time disturbance observer which obtains fast convergence within a predetermined amount of time and strong stability for a class of second-order nonlinear systems. Firstly, a nonlinear sliding mode manifold with fast finite-time convergence is introduced. Then, according to the designed manifold for the guarantee of finite-time convergence and robustness stabilization, a nonlinear control algorithm based on theory of finite-time control is developed. Specifically, the information of the lumped uncertainty was achieved by a new finite-time Disturbance Observer (DO). Thanks to the synthetic advantages of the above techniques, the designed controller marked with powerful features including a practical design, fast convergence rate, high precision, a convergence of the control errors in finite-time, along with impressive small chattering in the control actions. Furthermore, the control proposal also eliminates the necessity of the upper boundary of the uncertainties affecting the system and its finite settling time can be estimated in advance by designating the appropriate design parameters. The finite-time stability of the proposed DO, sliding surface, and control algorithm has been fully confirmed by Lyapunov principle. Trajectory tracking simulation for a 3-DOF manipulator and trajectory tracking experiment for a Maglev System (MLS) has been performed under different operating conditions using MATLAB/SIMULINK to testify the effectiveness and feasibility of the suggested strategy.

Adaptive proxy-based sliding mode control for a class of second-order nonlinear systems and its application to pneumatic muscle actuators

The human-centered robotic systems demand safe and robust controllers in many applications. This paper proposes an adaptive proxy-based sliding mode control approach for a class of typical second-order nonlinear systems. A new PID-type virtual coupling is designed between a virtual proxy and the physical object. Considering the unknown bound of lumped disturbances, an adaptation law is applied to online adjust the gain of a sign function which ensures the proxy to track the reference accurately. By using the Lyapunov theorem, the closed-loop system stability is proved. Both simulations and experiments are conducted to verify the proposed method based on a real-world pneumatic muscle actuator control platform. The results show that the proposed adaptive proxy-based sliding mode control approach presents better tracking accuracy, safety, and robustness than the conventional PID control and sliding mode control.

Attitude Control of UAVs Based on Event-Triggered Supertwisting Algorithm

An event-triggered attitude control algorithm is developed for quadrotor unmanned aerial vehicles (UAVs) subject to external disturbances. In this article, first an event-triggered supertwisting stabilizing control strategy for a class of second-order nonlinear systems is proposed. Then, a Lyapunov-based stability analysis is provided for the closed-loop system, and the Zeno-free execution of triggering sequence is guaranteed via rigorous analysis. Furthermore, the proposed control strategy is applied on attitude control of UAVs to reduce the computing cost without degrading the performance of the system. Finally, the efficiency of the developed method is validated by numerical simulation.

A Fast Nonsingular Terminal Sliding Mode Control Method for Nonlinear Systems With Fixed-Time Stability Guarantees

This paper proposes a nonsingular terminal sliding mode control scheme with fast fixed-time convergence for a class of second-order nonlinear systems in the presence of matched uncertainties and perturbations. First, based on fixed-time stability theory, a novel stable system is proposed. Then, using the fixed-time stable system, a fast fixed-time nonsingular terminal sliding surface is derived. The settling time is independent of the initial system state and can be set in advance with the design parameters; the upper-bound of convergence time is derived from the Lyapunov theory. Moreover, the proposed control scheme has an advantage in convergence rate over existing results and achieves better control performance with low control energy cost. The simulation results for a tracking system with a single inverted pendulum are presented to validate the effectiveness and superiority of the proposed control method.

Novel Dynamic-Sliding-Mode-Manifold-Based Continuous Fractional-Order Nonsingular Terminal Sliding Mode Control for a Class of Second-Order Nonlinear Systems

A novel dynamic-sliding-mode-manifold-based continuous fractional-order nonsingular terminal sliding mode control is proposed for a class of second-order nonlinear systems. By designing the parameter in the continuous fractional-order nonsingular terminal sliding mode manifold as an exponential function of the tracking error, a dynamic sliding mode manifold can be obtained by adjusting the parameter online. Even if reference signals change, the parameter does not need repetitive offline optimization. By combining the fast-terminal-sliding-mode-type reaching law, the system states are attracted to the manifold quickly, enhancing the controller’s robustness. When a large initial error exists, the control system can still accelerate response and reduce overshoot simultaneously owing to the dynamic changing characteristic of the manifold. The stability and finite-time convergence of the closed-loop system are proven by the Lyapunov stability theory. Simulation results on SISO and MIMO nonlinear systems show that for different reference signals, the proposed method has a better tracking performance than the general fractional-order nonsingular terminal sliding mode control.

Robust tracking control for mechanical systems using only position measurements

A discontinuous control law is presented to solve the tracking problem for a class of second-order nonlinear systems and fully actuated n degrees of freedom (nDOF) mechanical systems. Moreover, the control algorithm can compensate for viscous friction, Coulomb friction, and bounded external perturbations. Only position measurements are available for feedback. In this way, this proposal does not require to measure or estimate another signal but the position of the mechanical system to achieve the tracking control objective, this is, the controller does not need velocity measurements as feedback; this constitutes the main contribution of the present approach. In the stability analysis, a strict Lyapunov function and its conditions are studied to prove asymptotic stability for second-order systems and Lagrangian systems. In a mass–spring–damper system is tested the closed-loop performance through numerical simulations. Also, simulations and real-time experiments are carried out in a (2DOF) Scara robot. As a result, the proposed discontinuous algorithm is proved to be suitable to solve the tracking problem, and also that the equilibrium points of the closed-loop system are globally stable.

Design of Robust Sliding Mode Control With Adaptive Reaching Law

In this paper, a novel robust sliding mode control with adaptive reaching law (RSMC-ARL) is developed for a class of second-order nonlinear systems. First, a novel adaptive reaching law (ARL), utilizing a time-varying sliding surface, is designed to achieve fast convergence as well as maintain high accuracy for RSMC-ARL. Second, an adaptive tuning approach is developed to adjust the control law without frequency switching and avoid undesired chattering phenomenon. Then, the global robustness can be obtained against the uncertainties and external disturbances for nonlinear systems. Third, by constructing the Lyapunov function, the reachability and stability of RSMC-ARL can be guaranteed. Finally, three different second-order nonlinear systems are used to test the effectiveness of RSMC-ARL. The results have demonstrated that the proposed method can achieve fast convergence and high accuracy in comparison with some existing methods.

An Extended Proxy-Based Sliding Mode Control of Pneumatic Muscle Actuators

To solve the problem of controlling an intrinsically compliant actuator, pneumatic muscle actuator (PMA), this paper presents an extended proxy-based sliding mode control (EPSMC) strategy. It is well known that the chattering phenomenon of conventional sliding mode control (SMC) can be effectively solved by introducing a proxy between the physical object and desired position, which results in the so-called proxy-based sliding mode control (PSMC). To facilitate the theoretical analysis of PSMC and obtain a more general form of controller, a new virtual coupling and a SMC are used in our proposed EPSMC. For a class of second-order nonlinear system, the sufficient conditions ensuring the stability and passivity are obtained by using the Lyapunov functional method. Experiments on a real-time PMA control platform validate the effectiveness of the proposed method, and comparison studies also show the superiority of EPSMC over the conventional SMC, PSMC, and PID controllers.

Prescribed performance fixed-time tracking control for a class of second-order nonlinear systems with disturbances and actuator saturation

This paper investigates the prescribed performance fixed-time tracking control problem for a class of second-order nonlinear systems with bounded disturbance and actuator saturation limit. In order to facilitate the controller development, we introduce the performance function and output error transformation technique to transform the tracking error dynamics with inequality performance constraints to an equivalent unconstrained error one. Different from the existing work on the guaranteeing prescribed performance control, we incorporate the fixed-time sliding mode surface into the control design procedure to cope with the disturbances and actuator saturation limit. Specifically, a robust adaptive fixed-time tracking controller, which requires no upper bound of the disturbance, is designed to solve the prescribed performance fixed-time tracking control problem. The rigorous mathematical stability proof is given. Finally, numerical simulations for the attitude tracking control are presented to demonstrate the effectiveness and robustness of the proposed controller.

Closed‐loop parameter identification of second‐order non‐linear systems: a distributional approach using delayed reference signals

This work presents a closed-loop parameter identification algorithm for a class of second-order non-linear systems affected by constant disturbances, quantisation, and state estimation errors. The proposed scheme permits obtaining a linear parameterisation of the non-linear system by developing a simplified procedure that allows using the distributional framework approach straightforwardly. The parametrisation stage requires signals with known delays. These delays are introduced to the system through the reference signal. Then, the linear parametrisation is used by a least-squares (LS) algorithm and a state estimator to generate the estimated values of the system parameters and the constant disturbance. The proposed algorithm is compared to a standard off-line LS algorithm in numerical simulations. Besides, the effectiveness and robustness of the proposed methodology are verified using a Monte Carlo simulation by considering that the system's output is corrupted by white noise. The results indicate that the proposed parameter identification scheme outperforms the LS algorithm, but without requiring any pre-processing stage.