Partial equiasymptotic stability of nonlinear dynamic systems

The stability of nonlinear dynamical system of relative rotation is studied. Firstly, the dynamics equation of relative rotation autonomous nonlinear dynamical system with commonly damped force and forced excitation is deduced. Secondly, the stability of relative rotation nonlinear dynamical system is studied. For the nonlinear dynamical system, it is proved that the closed orbit bifurcation can occur under some conditions. Finally, The approximate solution of the equation under forced excitation is obtained by the method of multiple scales.

Exponential and global stability of nonlinear dynamical systems relative to initial time difference

In this paper, a new approach to stability analysis of nonlinear dynamics of an underactuated autonomous underwater vehicle (AUV) is presented. AUV is a highly nonlinear robotic system whose dynamic model includes coupled terms due to the hydrodynamic damping factors. It is difficult to analyze the stability of a nonlinear dynamical system through Routh's stability approach because it contains nonlinear dynamic parameters owing to hydrodynamic damping coefficients. It is also difficult to analyze the stability of AUVs using Lyapunov's criterion and LaSalle's invariance principle. In this paper, we proposed the extended-Routh's stability approach to verify the stability of such nonlinear dynamic systems. This extended-Routh's stability approach is much easier as compared to the other existing methods. Numerical simulations are presented to demonstrate the efficacy of the proposed stability verification of the nonlinear dynamic systems, e.g., an AUV system dynamics.

The paper considers the issues of control and stabilization of nonlinear dynamic systems described by a set of differential and difference equations, the latter of which contain a control vector. The states of these systems have both continuous and discrete components, so such systems are called continuous-discrete or hybrid. Necessary and sufficient features of controllability of nonlinear hybrid systems with a constant discretization step are established, which imply a transition from these systems to equivalent, in the natural sense, nonlinear discrete dynamic systems. A transformation is presented that allows reducing a linear discrete system to the canonical Brunovsky form and constructing a stabilizing control on its basis for the corresponding continuous-discrete system with scalar control. An algorithm for reducing a first approximation system of a nonlinear discrete system with scalar control to the canonical Brunovsky form and an algorithm for constructing a stabilizing control for nonlinear hybrid systems with scalar control are developed and illustrated with examples. Sufficient signs of stabilization of nonlinear hybrid systems are presented both without and with the feedback controller.

This paper is concerned with tele‐presence and tele‐operation of autonomous vehicles over the Additive White Gaussian Noise (AWGN) channel. This wireless communication channel is subject to transmission noise and transmission power constraint. An encoder, decoder and controller are proposed by implementing a novel linearization method for linearizing the nonlinear dynamic systems at operating points. For the linearized systems, the encoder, decoder and controller are implemented that were previously proposed for output tracking as well as stability of linear dynamic systems. Two novel linearization methods are proposed and their satisfactory performances are proved for output tracking as well as stability of nonlinear dynamic system. The first method is based on the fixed linearization rate and the second one is based on the variable (optimal) linearization rate. The decoder that is based on the second method is in fact the extended Kalman filter with the optimal linearization rate. The performances of these two methods are compared with each other and the other proposed methods by implementing them to the problems of the tele‐presence and tele‐operation of autonomous vehicles over the AWGN channel. Their satisfactory performances are illustrated. It is illustrated that the second method has higher computational complexity with slightly better performance and is applicable to a larger class of reference trajectories.

Two direct methods, the method of tangent hyperplanes and the method of tangent hypersurfaces, are applied to an elementary nonlinear dynamic system for transient stability assessment. The former method is based on the approximation of the asymptotic stability boundary by hyperplanes at a certain class of unstable singular points in the state-space, and the latter replaces hyperplanes by hypersurfaces. The applicability and accuracy of both methods are evaluated through a comparison of results.

In this paper, we discuss the quantitative characterization problems of the exponential stability and trajectory convergence property of nonlinear discrete dynamical systems on a bounded set of the state space. Through introducing two new concepts-the Lip constant of a nonlinear operator and strongly equivalent metrics of a norm-we show that the nonlinear discrete dynamical systems are exponentially stable on the bounded set if and only if the corresponding nonlinear operator is contractive under some strongly equivalent metrics, or if and only if the Lip constant of the nonlinear operator is less than one. In the latter case, we further show that the infimum of the exponential bounds of trajectories of the system equals exactly to the Lip constant. Based on the obtained results, we clearly explain how trajectory convergence properties of the systems are determined quantitatively by the Lip constant and strongly equivalent metrics. The obtained results not only are of importance in understanding the essence of exponential stability and trajectory convergence properties of nonlinear discrete dynamical systems on bounded sets of the state space but also provide some new, useful criteria of testing exponential stability and estimating convergence speed of trajectories of the systems.

Two-dimensional nonlinear dynamic system model of interspecific interaction and numerical simulation research on it

It is can identify three main classes of problems broadly characterizing stability of a dynamical systems, viz., (1) stability with respect to a part of the variables of the zero equilibrium position (Lyapunov-Rumyantsev stability problem), (2) stability of the partialzero equilibrium position, and (3) stability with respect to a part of the variables of the partialzero equilibrium position. In the problem of stability with respect to a part of the variables of the zero equilibrium position of systems of ordinary differential equations with continuous right-side assumes the domain of initial perturbations to be a sufficiently small neighborhood of the zero equilibrium position. Along with this statement, the case then initial perturbations can be large with respect to one part of non-controlled variables and arbitrary with respect to their other part is also considered. On the other hand, for stability problem of partial zero equilibrium positions of systems of ordinary differential equations also naturally assume that initial perturbations of variables that do not define the given equilibrium position can be large with respect to one part of the variables and arbitrary with respect to their other part. Contrary the assumptions that initial perturbations of this variables are either only arbitrary or only large the combined assumption made it possible an admissible trade-off between the meaning sense for notion of stability and the respective requirements on the Lyapunov functions. The article studies the problem of stability for nonlinear discrete-time systems: stability with respect to a part of the variables of partial equilibrium position. Initial perturbations of variables that do not define the given equilibrium position can be large (belonging to an arbitrary compact set) with respect to one part of the variables and arbitrary with respect to their other part. A conditions of stability of this type are obtained in the context of a discrete analog of the Lyapunov functions method, which generalize a number of existing results. Example is given. The problem of unification of process of studying stability problems of stationary and non-stationary nonlinear discrete-time systems is also discussed

This volume brings together internationally recognized authorities in both geometric and nonsmooth analysis methods in optimal control and its applications. The topics covered include geometric and nonsmooth analysis techniques in various problems of optimal control, and stabilization for nonlinear dynamical systems governed by ordinary and partial differential equations and differential inclusions. The powerful mathematical techniques that have been developed in deterministic optimal control theory make it possible to derive more detailed information about the structure of solutions then could have been obtained in the past. This volume addresses this topic as well as supports new algorithmic approaches to the calculation of solutions. This volume is suitable for mathematicians and students.

The 2011 issue of Nonlinear Dynamics presents articles on stability of non-linear dynamic structures and systems. Alijani and co-researchers investigate geometrically nonlinear vibrations of rectangular plates with Functionally Graded Materials (FGM) in thermal environments via a multi-modal energy approach. Demeio and co-researchers investigate the primary non-linear resonance response of a one-dimensional continuous system, which can be regarded as a model for semiinfinite cables resting on an elastic substrate reacting in compression only, and subjected, to a constant distributed load and to a small harmonic displacement applied to the finite boundary. Stoykov and Ribeiro investigate the geometrically non-linear periodic vibrations of beams with rectangular cross section under harmonic forces using a p-version finite element method. Krylov and co-researchers investigate the feasibility of two-directional switching of an initially curved or prebuckled electrostatically actuated micro-beams using a single electrode fabricated from the same structural layer.

This paper presents a method for obtaining optimal stable control for general nonlinear nonautonomous dynamical systems. The approach is inspired by recent developments in analytical dynamics and the observation that the Lyapunov criterion for stability of dynamical systems can be recast as a constraint to be imposed on the system. A closed-form expression for control is obtained that minimizes a user-defined control cost at each instant of time and enforces the Lyapunov constraint simultaneously. The derivation of this expression closely mirrors the development of the fundamental equation of motion used in the study of constrained motion. For this control method to work, the positive definite functions used in the Lyapunov constraint should satisfy a consistency condition. A class of positive definite functions has been provided for mechanical systems that meet this criterion. To illustrate the broad scope of the method, for linear systems it is shown that a proper choice of these positive definite functions results in conventional LQR control. Control of the Lorenz system and a multi-degree of freedom nonlinear mechanical system are considered. Numerical examples demonstrating the efficacy and simplicity of the method are provided.

Partially stable systems involve dynamical systems whose motion lie in a subspace of the state space resulting in system stability with respect to part of the system's states. In this paper, we develop partial stability theorems for nonlinear continuous-time and discrete-time dissipative feedback systems. Specifically, by invoking additional structural constraints on the forward loop and feedback loop system storage functions, we develop feedback interconnection partial stability results for dissipative nonlinear dynamical systems. Our results provide extensions of the positivity and small gain theorems for guaranteeing partial stability of feedback interconnected systems.

Direct control method for improving stability and reliability of nonlinear stochastic dynamical systems

Conditions were obtained under which the uniform stability (uniform asymptotic stability) in one part of the variables of the zero equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position relative to another, larger part of variables. Conditions were also obtained under which the uniform stability (uniform asymptotic stability) in one part of variables of the “partial” (zero) equilibrium position of the nonlinear nonstationary system of ordinary differential equations implies the uniform stability (uniform asymptotic stability) of this equilibrium position. These conditions complement a number of the well-known results of the theory of partial stability and partial detectability of the nonlinear dynamic systems. Application of the results obtained to the problems of partial stabilization of the nonlinear control systems was considered.