DETERMINING ROBUST PARAMETERS IN STABILIZING SET OF BACKSTEPPING BASED NONLINEAR CONTROLLER FOR SHIP COURSE KEEPING

In order to solve the problem that backstepping method cannot effectively guarantee the robust performance of the closed-loop system, a novel method of determining parameter is developed in this note. Based on the ship manoeuvring empirical knowledge and the closed-loop shaping theory, the derived parameters belong to a reduced robust group in the original stabilizing set. The uniformly asymptotic stability is achieved theoretically. The training vessel “Yulong” and the tanker “Daqing232” are selected as the plants in the simulation experiment. And the simulation results are presented to demonstrate the effectiveness of the proposed algorithm.

DETERMINING ROBUST PARAMETERS IN STABILIZING SET OF BACKSTEPPING BASED NONLINEAR CONTROLLER FOR SHIP COURSE KEEPING

The authors have addressed an important topic that is needed for backstepping algorithm to guarantee the robust performance of the closed-loop system. A novel method of determining parameters was presented based on ship maneuvering empirical knowledge and closed-loop shaping theory, and theoretical proof had shown the uniformly asymptotic stability of an established nonlinear Nomoto model. However, the following important points are suggested for the improvement of this paper.

Delay-Dependent Stability, Integrability and Boundedeness Criteria for Delay Differential Systems

This paper deals with non-perturbed and perturbed systems of nonlinear differential systems of first order with multiple time-varying delays. Here, for the considered systems, easily verifiable and applicable uniformly asymptotic stability, integrability, and boundedness criteria are obtained via defining an appropriate Lyapunov–Krasovskiĭ functional (LKF) and using the Lyapunov– Krasovskiĭ method (LKM). Comparisons with a former result that can be found in the literature illustrate the novelty of the stability theorem and show new contributions to the qualitative theory of solutions. A discussion of two illustrative examples and the obtained results are presented.

Uniformly asymptotic stability of second-order linear time-varying systems

This paper is concerned with the stability of second-order linear time-varying systems. By utilizing the Lyapunov approach, a generally uniformly asymptotic stability criterion is obtained by adding the system matrices into the quadratic Lyapunov candidate function. In the case of the derivative of the Lyapunov candidate function is semi-positive definite, the stability criterion is also efficient. Based on the proposed results, the systems with uncertain disturbances such as structured independent and structured dependent perturbations are considered. Using the matrix measure and the singular value theory, the bounds of the uncertainties are obtained that guarantee the system uniformly asymptotically stable, while the bounds of state feedback control input are also derived to stabilize the second-order linear time-varying systems. Finally, several numerical examples are given to prove the validity and correctness of the proposed criteria with existing ones.

A New Set of Stability Criteria Extending Lyapunov’s Direct Method

A dynamical system is a mathematical model described by a high dimensional ordinary differential equation for a wide variety of real world phenomena, which can be as simple as a clock pendulum or as complex as a chaotic Lorenz system. Stability is an important topic in the studies of the dynamical system. A major challenge is that the analytical solution of a time-varying nonlinear dynamical system is in general not known. Lyapunov's direct method is a classical approach used for many decades to study stability without explicitly solving the dynamical system, and has been successfully employed in numerous applications ranging from aerospace guidance systems, chaos theory, to traffic assignment. Roughly speaking, an equilibrium is stable if an energy function monotonically decreases along the trajectory of the dynamical system. This paper extends Lyapunov's direct method by allowing the energy function to follow a rich set of dynamics. More precisely, the paper proves two theorems, one on globally uniformly asymptotic stability and the other on stability in the sense of Lyapunov, where stability is guaranteed provided that the evolution of the energy function satisfies an inequality of a non-negative Hurwitz polynomial differential operator, which uses not only the first-order but also high-order time derivatives of the energy function. The classical Lyapunov theorems are special cases of the extended theorems. the paper provides an example in which the new theorem successfully determines stability while the classical Lyapunov's direct method fails.

A New Set of Stability Criteria Extending Lyapunov’s Direct Method

A dynamical system is a mathematical model described by a high dimensional ordinary differential equation for a wide variety of real world phenomena, which can be as simple as a clock pendulum or as complex as a chaotic Lorenz system. Stability is an important topic in the studies of the dynamical system. A major challenge is that the analytical solution of a time-varying nonlinear dynamical system is in general not known. Lyapunov's direct method is a classical approach used for many decades to study stability without explicitly solving the dynamical system, and has been successfully employed in numerous applications ranging from aerospace guidance systems, chaos theory, to traffic assignment. Roughly speaking, an equilibrium is stable if an energy function monotonically decreases along the trajectory of the dynamical system. This paper extends Lyapunov's direct method by allowing the energy function to follow a rich set of dynamics. More precisely, the paper proves two theorems, one on globally uniformly asymptotic stability and the other on stability in the sense of Lyapunov, where stability is guaranteed provided that the evolution of the energy function satisfies an inequality of a non-negative Hurwitz polynomial differential operator, which uses not only the first-order but also high-order time derivatives of the energy function. The classical Lyapunov theorems are special cases of the extended theorems. the paper provides an example in which the new theorem successfully determines stability while the classical Lyapunov's direct method fails.

Almost periodic solutions of a commensalism system with Michaelis-Menten type harvesting on time scales

In this paper, we consider an almost periodic commensal symbiosis model with nonlinear harvesting on time scales. We establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. Our results show that the continuous system and discrete system can be unify well. Examples and their numerical simulations are carried out to illustrate the feasibility of our main results.

Almost periodic solution of a discrete competitive system with delays and feedback controls

A discrete nonlinear almost periodic multispecies competitive system with delays and feedback controls is proposed and investigated. We obtain sufficient conditions to ensure the permanence of the system. Also, we establish a criterion for the existence and uniformly asymptotic stability of unique positive almost periodic solution of the system. In additional, an example together with its numerical simulation are presented to illustrate the feasibility of the main result.

Permanence and Almost Periodic Solutions for N-Species Nonautonomous Lotka-Volterra Competitive Systems with Delays and Impulsive Perturbations on Time Scales

We investigate a class of nonautonomous N-species Lotka-Volterra-type competitive systems with time delays and impulsive perturbations on time scales. By using comparison theorems of impulsive dynamic equations on time scales, we obtain sufficient conditions to guarantee the permanence of the system. Then based on the Massera-type theorem for impulsive dynamic equations on time scales, we establish existence and uniformly asymptotic stability of the unique positive almost periodic solution of the system. Finally, an example is employed to illustrate our main results.

Adaptive control for a rewinding process of a roll-to-roll system

In this paper, transverse vibration and transport velocity controls of a moving web in a rewinding section of a roll to roll system are investigated. The moving web is modeled as an axially moving beam. Two independent adaptive control schemes are proposed. The first control scheme using a control force exerted from a hydraulic actuator is to suppress transverse vibrations of the moving web of unknown mass per unit length under a spatially varying tension and a time-varying transport velocity. The second control scheme using a control torque applied to the rewind roller is to maintain the transport velocity levels of the moving web in spite of disturbances such as the variations of rotating elements and unknown bearing friction. From the decentralized control viewpoint, the uniformly exponential stability for suppressing the transverse vibrations and the uniformly asymptotic stability for maintaining the transport velocity are achieved. However, as a whole, the uniformly asymptotic stability is concluded. Simulations for demonstrating the effectiveness of the proposed control schemes are presented.