Second-order Nonlinear Systems Research Articles

Optimal adaptive neural PI full-order sliding mode control for robust fault tolerant control of uncertain nonlinear system

This paper proposes a robust fault tolerant control scheme for a class of second-order uncertain nonlinear systems. First, a novel PI full-order sliding mode (PI-FOSM) control, which integrates a new PI-FOSM sliding surface and a continuous control law, is developed. The crucial parameters of the controller are optimally selected by Bat algorithm so that the nearly optimal performance of the controller can be achieved. In addition, the unknown system dynamics is approximated by using a radial basic function neural network (RBFNN) so that the proposed controller does not require an exact model of the system. Compared with other existing sliding mode controllers for fault tolerant control system, the proposed method provides very strong robustness, low oscillation, fast convergence and high precision. The superior performance of the proposed robust fault tolerant controller is proved through simulation results for attitude control of a spacecraft.

A Novel Approach to Fixed-Time Stabilization for a Class of Uncertain Second-Order Nonlinear Systems

This paper is concerned with the problem of fixed-time stabilization for a class of uncertain second-order nonlinear systems. By delicately introducing extra manipulations in the feedback domination and revamping the technique of adding a power integrator, a new approach is developed, by which a state feedback controller, together with a suitable Lyapunov function, which is critical for verifying fixed-time convergence, can be explicitly organized to render the closed-loop system fixed-time stable. The major novelty of this paper is attributed to a subtle strategy that offers a distinct perspective in controller design as well as stability analysis in the problem of fixed-time stabilization for nonlinear systems. Finally, the proposed approach is applied to the attitude stabilization of a spacecraft to demonstrate its merits and effectiveness.

Adaptive Neural Network Leader-Follower Formation Control for a Class of Second-Order Nonlinear Multi-Agent Systems With Unknown Dynamics

In this article, an adaptive leader-follower formation control on the basis of neural network (NN) is developed for a class of second-order nonlinear multi-agent systems with unknown dynamics. Unlike the first-order formation control that only needs to govern the position states, the second-order formation control needs to govern both the position and velocity variables. Hence the second-order formation is more challenging and interesting than the first-order case. In the control design, the adaptive NN approximator is employed to compensate the nonlinear uncertainties, so that the control design difficulty coming from the unknown dynamics is effectively overcome. Through Lyapunov stability analysis, it is demonstrated that the proposed control method can complete the control tasks. To further demonstrate the effectiveness, the formation method is implemented to a numerical simulation, and it shows the desired results.

A Non-Singular Fast Terminal Sliding Mode Control Based on Third-Order Sliding Mode Observer for a Class of Second-Order Uncertain Nonlinear Systems and its Application to Robot Manipulators

This paper proposes a controller-observer strategy for a class of second-order uncertain nonlinear systems with only available position measurement. The third-order sliding mode observer is first introduced to estimate both velocities and the lumped uncertain terms of system with high accuracy, less chattering, and finite time convergency of estimation errors. Then, the proposed controller-observer strategy is designed based on non-singular fast terminal sliding mode sliding control and proposed observer. Thanks to this combination, the proposed strategy has some superior properties such as high tracking accuracy, chattering phenomenon reduction, robustness against the effects of the lumped uncertain terms, velocity measurement elimination, finite time convergence, and faster reaching sliding motion. Especially, two period times, before and after the convergence of the velocity estimation takes place, are considered. The finite time stability of proposed controller-observer method is proved by using the Lyapunov stability theory. Final, the proposed strategy is applied to robot manipulator system and its effectiveness is verified by simulation results, in which a PUMA560 robot manipulator is employed.

Observer-Based Fixed-Time Consensus Control for Nonlinear Multi-Agent Systems Subjected to Measurement Noises

This article investigates the fixed-time consensus control problem for nonlinear second-order multi-agent systems subjected to measurement noises and in absence of the second-order state measurements of the leader and the followers under directed communication topology graphs. For each follower, an innovative augmented fixed-time disagreement error observer (AFTDEO) is first designed to reconstruct the disagreement error within fixed time. On one hand, the proposed AFTDEO can not only operate well only depending on the noisy measurements of the first-order states of the leader and the followers, but also suppress the measurement noises of the leader and the followers. On the other hand, the proposed AFTDEO can guarantee that the observation errors are convergent to the origin in fixed time independent of the initial observation errors. Based on the proposed AFTDEO, a novel fixed-time consensus control scheme under a directed graph having a spanning tree is constructed to address the consensus problem without the second-order measurements of the leader and the followers by utilizing fixed-time nonsmooth backstepping technique. The fixed-time convergence of the tracking error is guaranteed through Lyapunov stability analysis. Finally, simulation results demonstrate the effectiveness of the proposed consensus control scheme.

Fixed-Time Leader-Following Flocking for Nonlinear Second-Order Multi-Agent Systems

This work focuses on the fixed-time leader-following flocking for multi-agent systems. Different from the previous continuous inherent dynamic f(x <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ,v <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</sub> ,t) for agent i, a new nonlinear discontinuous one is firstly proposed in this paper. Employing non-smooth techniques, graph theory and fixed-time stability theory, the fixed-time leader-following flocking is achieved. Moreover, an upper bound of the settling time is independent of initial states. In addition, two illustrative examples are given to verify the effectiveness of the theoretical results.

Adaptive Fast Finite-Time Consensus for Second-Order Uncertain Nonlinear Multi-Agent Systems With Unknown Dead-Zone

This paper presents an adaptive fast finite-time consensus (FTC) for second-order uncertain nonlinear (UN) multi-agent systems (MASs) with unknown nonsymmetric dead-zone and external disturbances. In the process of control protocols design, based on the estimation of the dead-zone width that is obtained by using adaptive method, a fuzzy logic dead-zone compensator is adopted to deal with the nonsymmetric dead-zone input phenomenon. Combining finite-time control technique and Lyapunov's relevant theory, a new fast FTC protocol is developed. Based on the basis of radial basis function neural networks (RBFNNs) theories, the unknown nonlinear functions are approximated. Under the presented consensus protocols and adaptive laws, it can be proved that the position errors of arbitrary two agents can reach a small region of zero in finite time as well as the velocity errors. Ultimately, the effectiveness of the designed method is tested via two numerical examples.

Mixed H∞ and passive consensus sampled-data control for nonlinear systems

This paper studies the consensus problem of a second-order nonlinear multi-agent system with directed topologies. A distributed control protocol is proposed for each agent using the relative states among neighboring agents. A mixed H∞ and passivity-based control is maneuvered to deal the bounded disturbances enduring in the system. Based on the theory of the sampled-data control technique and Lyapunov stability theory, some novel conditions are given to realize the consensus of a class of second-order multi-agent nonlinear systems. A new set of delay dependent sufficient conditions is derived in terms of linear matrix inequalities, which guarantees that all agents asymptotically converge to the convex hull with the prescribed H∞ and passive performance. Finally, an example with simulation results is given to verify the theoretical results.

Leadership Hierarchy-based Formation Control via Adaptive Chaotic Pigeon-inspired Optimization

Abstract Formation control of multi-agent systems (MASs) is a significant research subject in the field of cooperative control. In this paper, we propose a novel consensus-based formation control approach with minimal resource cost and excellent adaptability for second-order nonlinear multi-agent systems. Specifically, an improved constrained adaptive chaotic pigeon-inspired optimization algorithm (ACPIO) is proposed for tuning parameters, which promotes the automation of controller design and alleviates the workload of conventional designer. Moreover, a variant of pinning control method integrating with hierarchical leadership model of pigeon flocks is introduced, which achieves excellent adaptability and reduces computational complexity simultaneously. Additionally, sufficient conditions are derived for achieving the desired formation pattern based on Lyapunov stability theory and matrix theory. Numerical simulation results demonstrate the feasibility and effectiveness of the proposed method for formation control of second-order nonlinear MASs.

Novel Fixed-Time Nonsingular Fast Terminal Sliding Mode Control for Second-Order Uncertain Systems Based on Adaptive Disturbance Observer

In this paper, a novel fixed-time nonsingular fast terminal sliding mode control incorporating with an adaptive disturbance observer is proposed for second-order uncertain nonlinear systems to achieve fast stabilization and robust control. Based on the theory of fixed-time convergence, a novel fixed-time stable system is first investigated. Using this system, a novel fixed-time nonsingular fast terminal sliding mode controller is developed, which can achieve system stabilization within bounded convergence time regardless of initial states and provide nonsingularity and fast convergence. Moreover, an adaptive disturbance observer is designed to improve the control performance and compensate for uncertain disturbances. The fixed-time stability of the sliding surface and the system states under the proposed composite control scheme are demonstrated by the Lyapunov stability theory. Both theoretical analysis and simulation results are presented to verify the feasibility and superiority of the proposed method.

Robust Tracking Control of Aerial Robots Via a Simple Learning Strategy-Based Feedback Linearization

To facilitate accurate tracking in unknown/uncertain environments, this paper proposes a simple learning (SL) strategy for feedback linearization control (FLC) of aerial robots subject to uncertainties. The SL strategy minimizes a cost function defined based on the closed-loop error dynamics of the nominal system via the gradient descent technique to find the adaptation rules for feedback controller gains and disturbance estimate in the feedback control law. In addition to the derivation of the SL adaptation rules, the closed-loop stability for a second-order uncertain nonlinear system is proven in this paper. Moreover, it is shown that the SL strategy can find the global optimum point, while the controller gains and disturbance estimate converge to a finite value which implies a bounded control action in the steady-state. Furthermore, utilizing a simulation study, it is shown that the simple learning-based FLC (SL-FLC) framework can ensure desired closed-loop error dynamics in the presence of disturbances and modeling uncertainties. Finally, to validate the SL-FLC framework in real-time, the trajectory tracking problem of a tilt-rotor tricopter unmanned aerial vehicle under uncertain conditions is studied via three case scenarios, wherein the disturbances in the form of mass variation, ground effect, and wind gust, are induced. The real-time results illustrate that the SL-FLC framework facilitates a better tracking performance than that of the traditional FLC method while maintaining the nominal control performance in the absence of modeling uncertainties and external disturbances, and exhibiting robust control performance in the presence of modeling uncertainties and external disturbances.

Network-based deployment of the second-order multi agents: a PDE approach

Deployment of a second-order nonlinear multi agent system over a desired open smooth curve in 2D or 3D space is considered. We assume that the agents have access to their velocities and to the local information of the desired curve and their displacements with respect to their closest neighbors, whereas in addition a leader is able to measure his absolute position. We assume that a small number of leaders transmit their measurements to other agents through a communication network. We take into account the following network imperfections: the variable sampling, transmission delay and quantization. We propose a static output-feedback controller and model the resulting closed-loop system as a disturbed (due to quantization) nonlinear damped wave equation with delayed point state measurements, where the state is the relative position of the agents with respect to the desired curve. To manage with the open curve we consider Neumann boundary conditions. We derive linear matrix inequalities (LMIs) that guarantee the input-to-state stability (ISS) of the system. The advantage of our approach is in the simplicity of the control law and the conditions. Numerical example illustrates the efficiency of the method.

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.

Adaptive Consensus of Two Coupled Heterogeneous Networked Systems With Bidirectional Actions

The outer consensus between two coupled heterogeneous networked systems with bidirectional actions and nonidentical topologies is addressed in this paper. Based on the adaptive theory, a control protocol is designed for two coupled heterogeneous networked systems, where each system consists of double-integrator and single-integrator nodes with different nonlinear dynamics. In the light of Lyapunov method and LaSalle’s invariance principle, we prove that the coupled heterogeneous networked systems can reach outer consensus by using the reported controller. Meanwhile, the adaptive controller is also given for the second-order nonlinear systems under identical and nonidentical topologies. The correctness of the obtained theory is demonstrated by two simulation examples.

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.

Distributed fixed-time consensus for nonlinear heterogeneous multi-agent systems

The distributed fixed-time consensus problem for a class of heterogeneous nonlinear leader–follower multi-agent systems is investigated in this paper. The considered multi-agent systems comprise several first-order nonlinear systems and second-order nonlinear systems of which the inherent nonlinearities are allowed to be lower-order and higher-order. By designing distributed fixed-time observers and fixed-time tracking controllers, it is theoretically shown that distributed consensus for the heterogeneous nonlinear multi-agent systems can be achieved in a fixed time, i.e., the achievement time for consensus is independent of any initial states. A simulation example is provided to show the effectiveness of the designed consensus controllers.

A new time-varying feedback RISE control for second-order nonlinear MIMO systems: theory and experiments

This study focuses on the development of a new class of the Robust Integral of the Sign of the Error (RISE) control law adequate for second-order nonlinear multi-input–multi-output systems. A revisit for the original RISE is done by altering some static feedback gains into time-varying nonlinear ones depending on the system states. The proposed controller takes advantage of both RISE control robustness towards uncertainties and the special behaviour of nonlinear feedback gains towards time-varied parameters. A Lyaponuv-based stability analysis to prove the semiglobal asymptotic tracking of the proposed new controller is included. In order to validate the relevance of the proposed controller, real-time experimental results are presented and discussed. Experiments have been conducted on a Delta parallel manipulator, in different operating conditions including payload and speed variations.

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.

Event-triggered fixed-time cooperative tracking control for uncertain nonlinear second-order multi-agent systems under directed network topology

In this work, we concern with the fixed-time event-triggered cooperative tracking control problem for uncertain nonlinear second-order multiagent systems (MASs) under directed communication topology. Firstly, to obtain fixed-time tracking control, a fractional-order-based nonlinear algorithm is designed. Secondly, based on our proposed fixed-time consensus tracking controller, an event-triggered mechanism is further developed to reduce the updating frequency of the controllers. Different from the finite-time control algorithm, the fixed-time strategy can achieve consensus tracking within a settling time regardless of the initial values. In addition, a strictly positive lower bound of the inter-event intervals is provided to demonstrate that the Zeno-behavior is avoided before the consensus tracking is achieved. Finally, several simulations are performed to illustrate the validity of the proposed algorithms.

Leader-following bipartite consensus of second-order time-delay nonlinear multi-agent systems with event-triggered pinning control under signed digraph

This paper investigates the leader-following bipartite consensus problem for second-order multi-agent systems with nonlinear dynamics, node-delay and signed digraph topology. By using pinning control strategies, an event-triggered controller is designed to achieve bipartite consensus in multi-agent systems, where a novel event-triggered function is proposed. Two cases are discussed in this paper according to whether the node-delay is differentiable or not. When the node-delay is differentiable, an LMI (Linear Matrix Inequality) condition is derived for reaching bipartite consensus in multi-agent systems by using M-matrix theory. When the node-delay is non-differentiable, some bipartite consensus conditions are developed based on Jensen inequality and the reciprocally convex approach. Moreover, it is shown that Zeno-behavior can be avoided for the designed event-triggered controller. Finally, some numerical examples are provided to illustrate the effectiveness and correctness of the theoretical analysis.