Stability In Sense Of Lyapunov Research Articles

Secure Communication in Complex Dynamical Networks via Time-Delayed Feedback Control

In this article, a synchronization method of complex dynamical networks (CDNs) with time-varying delay feedback control is proposed to secure communication between the command system and each node of CDNs. To advance the security of the communication between the command system and each node of CDNs, the original information signal transmitted from the command system is encrypted with the techniques of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathcal {N}$ </tex-math></inline-formula> -shift cipher and public key. Based on the Lyapunov stability sense and linear matrix inequality (LMI) framework, a new delay-dependent synchronization criterion is established to restore the original information signal on each node of the CDN as well as to ensure stable synchronization for secure communication of all nodes of the command system and CDN. With the support of numerical simulation, the validity of the proposed method is verified.

Fuzzy Adaptive Protective Control for High-Speed Trains: An Outstretched Error Feedback Approach

This paper presents a fuzzy adaptive protective control method for autonomous high-speed trains (HSTs) automatic operation using a new outstretched error feedback design approach. In order to stabilizing the error dynamics with respect to target position and speed profiles of controlled HSTs, nonlinear transformation in prescribed performance control methodology is used to convert running states subject to protective (constrained) information, imposed by automatic train protection (ATP) subsystem, to new coordinates in unconstrained form. By blending a new outstretched error feedback and fuzzy approximation, it is guaranteed above-mentioned errors are kept within regions characterized by error boundary (or prescribed performance) functions and control parameters simultaneously, which can be adjusted to arbitrarily small even without error decreasing boundary functions. Fuzzy approximation is used in compensating unknown running resistances. It is rigorously proved that the resulting closed-loop system is stable in sense of Lyapunov stability in the presence of unknown resistance, containing basis and aerodynamic resistances with uncertain parameters and piecewise continuously slope resistance over varying gradient profile. The proposed control is demonstrated to be effective by comparative simulations of train G1 running on Beijing-Shanghai railway line.

Distributed event-driven adaptive three-dimensional formation tracking of networked autonomous underwater vehicles with unknown nonlinearities

This paper presents a distributed event-driven adaptive formation control strategy for networked uncertain nonlinear autonomous underwater vehicles (AUVs) in three-dimensional space. It is assumed that the leader information is only transmitted to a subset of AUV followers under a directed graph and the nonlinearities of the AUV dynamics are unknown. A distributed error transformation method is presented to address the distributed formation tracking problem of AUV followers in three-dimensional space. Then, a distributed event-driven adaptive control method using neural networks and stabilizing auxiliary signals is developed to ensure the stability of the event-driven closed-loop system and accomplish three-dimensional formation tracking in the Lyapunov stability sense. Additionally, it is shown that Zeno behavior does not occur in the resulting event-triggering strategy. A simulation demonstrates the effectiveness of the proposed theoretical methodology.

Dynamic Preservation for a Class of Semi-Discrete Recurrent Neural Networks

The dynamic preservation in discrete simulations of the recurrent neural networks (RNNs) with discrete and infinite distributive asynchronous time delays is addressed. Firstly, we formulate the corresponding discrete-time model by semi-discrete technique. Secondly, we derive several mild algebra conditions to guarantee that the discussed discrete-time system is the global exponential stability in Lyapunov sense. It is shown analytically that the discrete-time technique is able to maintain the uniqueness of equilibrium point and its dynamic behavior of the continuous-time model under the same conditions. Meanwhile, we also show that there exists some degree of deviation in the spatial position among the equilibrium points of the continuous-time model and its discrete-time analogue due to the rounding error. Finally, we verify the validity of the main obtained results by comparing one continuous-time numerical example with its discrete-time counterpart.

Adaptive finite-time quantized synchronization of complex dynamical networks with quantized time-varying delayed couplings

This paper investigates the finite-time synchronization problem of the complex dynamical networks (CDNs) with quantized time-varying delayed couplings and adaptive coupling strengths. A novel finite-time adaptive quantized control algorithm and the corresponding updated laws are developed to address the aforementioned problem. Then based on the convex combination technique and nonsmooth analysis, the sufficient conditions are derived to guarantee the finite-time synchronization in the sense of Lyapunov stability. Finally, numerical simulations are performed to verify the effectiveness and feasibility of the proposed algorithm.

Rigid-body attitude control guaranteeing finite-time convergence

This study proposes an effective solution to the problem of attitude control for a rigid body satisfying angular velocity constraint as well as providing fault-tolerant capability. More specifically, a finite-time sliding surface containing attitude quaternion and angular velocity is first defined. Then, a novel tan-type prescribed performance control (PPC) with simple structure is presented to confine the sliding surface within a predefined performance boundary. Not only the attitude quaternion and angular velocity are indirectly constrained, but also it is thoroughly proved that the rotation velocity constraint is met even when severe actuators faults occur. The closed-loop attitude system is confirmed to be finite-time stable in the sense of Lyapunov stability. Numerical simulations clearly illustrate the effectiveness and usefulness of the suggested finite-time PPC despite actuator faults and environmental disturbances.

Stability analysis for uncertain differential equation by Lyapunov’s second method

Uncertain differential equation is a type of differential equation driven by Liu process that is the counterpart of Wiener process in the framework of uncertainty theory. The stability theory is of particular interest among the properties of the solutions to uncertain differential equations. In this paper, we introduce the Lyapunov’s second method to study stability in measure and asymptotic stability of uncertain differential equation. Different from the existing results, we present two sufficient conditions in sense of Lyapunov stability, where the strong Lipschitz condition of the drift is no longer indispensable. Finally, illustrative examples are examined to certify the effectiveness of our theoretical findings.

Asymptotical stability for complex dynamical networks via link dynamics

From the perspective of large‐scale system, a complex dynamical network can be regarded as the interconnected system with the node subsystem and link subsystem, which implies that the node subsystem and the link subsystem are the two main bodies of dynamical behaviors of network. Therefore, the whole stability of network is influenced by not only the dynamics of node subsystem but also the dynamics of link subsystem. According to the above view, the wholly asymptotical stability (WAS) is defined in this paper for the complex dynamical network with the model of differential equations. The WAS is used to describe the node subsystem achieves asymptotical stability when link subsystem achieves also the asymptotic stability in Lyapunov sense. For the WAS of the complex dynamical network, the corresponding criteria are derived by checking whether certain matrices are Hurwitz. The analysis results show that even if the isolated nodes are not asymptotically stable in Lyapunov sense, employing the dynamics of link can also force the node subsystem to achieve the asymptotical stability. Finally, the simulation examples show the validity of methods in this paper.

A cooperative fault-tolerant control method for the coupled system based on interaction effect utilization

Abstract An active fault-tolerant control scheme is proposed in this paper for the strongly coupled MIMO systems which subject to total actuator failures. A control-oriented interaction indicator is defined in the Lyapunov stability sense and then is utilized to design the cooperative fault-tolerant control law. The proposed scheme can achieve robust tracking performance with globally uniformly ultimate boundlessness and is capable of improving transient performance (fault-tolerant ability) by wisely using the interactions. Simulation results obtained on a flight attitude control system illustrates the benefit of the proposed techniques.

Fixed time terminal sliding mode trajectory tracking design for a class of nonlinear dynamical model of air cushion vehicle

This manuscript proposes a robust fixed time terminal sliding mode prototype for trajectory tracking of the nonlinear dynamics of an under-actuated air cushion vehicle. Nonlinearity, external disturbances, internal uncertainties and unmodeled dynamics are the main difficulties that an amphibious vehicle is faced with in its maneuver. The main contribution of the proposed methodology is to overcome these problems based on both the guaranteed stability in sense of Lyapunov and the fixed time tracking error even if the initial values are changed. Robustness against uncertainties and disturbances, fixed time convergence of tracking error to zero are other merits of the proposed approach. The simulation results demonstrate the effectiveness and superiority of suggested scheme.

Maximum integrator gain of PID on performing position control of DC motor in the presence of Stribeck friction: Kalman conjecture approach

The application of PID to perform position control in the presence of Stribeck friction is believed to provoke the system trapped in limit cycle oscillation. This research aims to investigates the formulation of PID that will stabilize the position control of DC motor in the presence of Stribeck friction. In order to diminish the limit cycle oscillation, the PID gain is set to meet global asymptotic stability in Lyapunov sense as Kalman conjecture is fully applied. The analysis shows that the stability of the system depends on the friction properties near Stribeck velocity. Furthermore, the proposed PID formulation is found to be easy to be applied as the maximum integrator gain applied to the system is directly proportional to the given proportional gain.

On stabilizing hydraulic motion experiencing Stribeck friction via PI controller: Circle criterion approach

A Friction is inevitable in a hydraulic actuator due to the presence of seal. Unfortunately, the existence of friction makes the system prone to limit cycle oscillation as PI controller is applied. This research investigates the maximum integrator constant of PI controller that can be introduced to the system while maintaining global asymptotic stability in Lyapunov sense. This condition is achieved as the system complies with circle criterion. In this research, a simplified hydraulic system which behaves linearly is adopted. Thus, the only source of nonlinearity is the Stribeck friction. The main contribution of this research is a formulation of stable PI strategy that is free from the effect of inertia. The PI strategy is suitable for a hydraulic testing machine of which the mass of device under the test is varying.

Adaptive neural tracking control for automotive engine idle speed regulation using extreme learning machine

The automotive engine idle speed control problem is a compromise among low engine speed for fuel saving, minimum emissions and disturbance rejection ability to prevent engine stall. However, idle speed regulation is very challenging due to the presence of high nonlinearity and aging-caused uncertainties in the engine dynamics. Therefore, the engine idle speed system is a typical uncertain nonlinear system. To address the problems of inherent nonlinearity and uncertainties in idle speed regulation, an extreme learning machine (ELM)-based adaptive neural control algorithm is proposed for tracking the target idle speed adaptively. The purpose of ELM is to rapidly deal with the uncertain nonlinear engine system. Since the original ELM is not designed for adaptive control, a new adaptation law is designed to update the weights of ELM in the sense of Lyapunov stability. Experiment is conducted to validate the performance of the proposed control method. Experimental result indicates that the ELM-based adaptive neural control outperforms the classical proportional–integral–derivative (PID), fuzzy-PID and back-propagation-neural-network-based controllers in terms of tracking performance under the variation of engine load.

Barrier-Certified Adaptive Reinforcement Learning With Applications to Brushbot Navigation

This paper presents a safe learning framework that employs an adaptive model learning algorithm together with barrier certificates for systems with possibly nonstationary agent dynamics. To extract the dynamic structure of the model, we use a sparse optimization technique. We use the learned model in combination with control barrier certificates which constrain policies (feedback controllers) in order to maintain safety, which refers to avoiding particular undesirable regions of the state space. Under certain conditions, recovery of safety in the sense of Lyapunov stability after violations of safety due to the nonstationarity is guaranteed. In addition, we reformulate an action-value function approximation to make any kernel-based nonlinear function estimation method applicable to our adaptive learning framework. Lastly, solutions to the barrier-certified policy optimization are guaranteed to be globally optimal, ensuring the greedy policy improvement under mild conditions. The resulting framework is validated via simulations of a quadrotor, which has previously been used under stationarity assumptions in the safe learnings literature, and is then tested on a real robot, the brushbot, whose dynamics is unknown, highly complex and nonstationary.

Stabilisation of a relative equilibrium of an underactuated AUV on SE(3)

ABSTRACTThis paper presents stabilisation of a relative equilibrium of an underactuated autonomous underwater vehicle (AUV) with three independent control inputs on Lie group SE(3), since SE(3) can describe the evolution of a rigid body without singularities or ambiguities. First, by using the interconnection and damping assignment (IDA) method through kinetic energy shaping, a controller is proposed to stabilise the relative equilibrium in the sense of Lyapunov stability in cotangent bundle T*SE(3) modulo an isotropy subgroup when a marine vehicle immerses in an ideal fluid. Then, for an AUV submersed in a viscous fluid, an asymptotical static control law consisting of three terms is designed to stabilise the relative equilibrium via the IDA method, which these three control terms are derived from kinetic energy shaping, potential energy shaping and damping reassignment, respectively. The control term derived from kinetic energy shaping preserves stability of this motion modulo translations along the shortest semi-axis in any direction and rotations about the same axis. Moreover, the static controller is developed to a dynamic one via adaptive neural network approach, which is robust for uncertainties in the system's damping and mass. Simulation results complete our work.

A New Motion Planning Framework based on the Quantized LQR Method for Autonomous Robots

This study addresses an argument on the disconnection between the computational side of the robot navigation problem with the control problem including concerns on stability. We aim to constitute a framework that includes a novel approach of using quantizers for occupancy grids and vehicle control systems concurrently. This representation allows stability concerned with the navigation structure through input and output quantizers in the framework. We have given the theoretical proofs of qLQR in the sense of Lyapunov stability alongside with the implementation details. The experimental results demonstrate the effectiveness of the qLQR controller and quantizers in the framework with realtime data and offline simulations.

PREDICTION OF MULTI-DIMENSIONAL MILLING BEHAVIOR

Milling process is a multi-dimensional cutting process accompanied by chatter vibrations because of material removal discontinuity. Also, chatter phenomenon has a great impact on final milled products quality. In this paper, an adequate criterion derived from mathematical literature is used to predict stability of multi-dimensional chatter milling. For that, firstly, the chatter milling system is represented by a system of three Degrees Of Freedom (DOF) and its dynamics is modeled by a system of Retarded Differential Equations (RDEs). Afterward, stability in Lyapunov sense is computed on the basis of the quasi-polynomial characteristic function in frequency domain. Finally, this method is used to predict stability of chatter milling process under different machining conditions and the output related to each case is verified by resolving the system of RDEs and visualizing the displacements in time domain using Matlab software.

미지의 제어 방향성과 비어파인 비선형성을 고려한 신경망 기반 외란 관측기와 추종기 설계

A disturbance-observer-based adaptive neural tracker design problem is investigated for a class of perturbed uncertain non-affine nonlinear systems with unknown control direction. A nonlinear disturbance observer (NDO) design methodology using neural networks is presented to construct a tracking control scheme with the attenuation effect of an external disturbance. Compared with previous control results using NDO for nonlinear systems in non-affine form, the major contribution of this paper is to design a NDO-based adaptive tracker without the sign information of the control coefficient. The stability of the closed-loop system is analyzed in the sense of Lyapunov stability.

Leader–follower consensus for multi-agent systems with three-layer network framework and dynamic interaction jointly connected topology

This paper addresses the leader–follower consensus for linear and nonlinear multi-agent systems with three-layer network framework and dynamic interaction jointly connected topology. By introducing a novel concept of mirroring nodes for different roles of evolution agents, we remove the general assumptions that the topology among followers is specific and continuously contains a spanning tree, and propose a simple criterion to determine if a network is of three-layer framework. This criterion divides the agents of systems into leader-layer, middle-layer and follower-layer nodes according to their functions in cooperative behaviors. Moreover, by combining the state information of agents in different layers, two classes of linear and nonlinear leader–follower consensus protocols are designed with bounded consensus speed. In the sense of Lyapunov stability, it is proved that the leader–follower consensus for the closed-loop linear and nonlinear multi-agent systems can be achieved by employing novel error forms that decouple the leader–follower consensus errors of the three-layer network framework. Two simulation examples are presented to verify the proposed approach and demonstrate its effectiveness.

Stability of Intelligent Transportation Network Dynamics: A Daily Path Flow Adjustment considering Travel Time Differentiation

A theoretic formulation on how traffic time information distributed by ITS operations influences the trajectory of network flows is presented in this paper. The interactions between users and ITS operator are decomposed into three parts: (i) travel time induced path flow dynamics (PFDTT); (ii) demand induced path flow dynamics (PFDD); and (iii) predicted travel time dynamics for an origin-destination (OD) pair (PTTDOD). PFDTT describes the collective results of user’s daily route selection by pairwise comparison of path travel time provided by ITS services. The other two components, PTTDOD and PFDD, are concentrated on the evolutions of system variables which are predicted and observed, respectively, by ITS operators to act as a benchmark in guiding the target system towards an expected status faster. In addition to the delivered modelings, the stability theorem of the equilibrium solution in the sense of Lyapunov stability is also provided. A Lyapunov function is developed and employed to the proof of stability theorem to show the asymptotic behavior of the aimed system. The information of network flow dynamics plays a key role in traffic control policy-making. The evaluation of ITS-based strategies will not be reasonable without a well-established modeling of network flow evolutions.