Stability Of Nonlinear Dynamical Systems Research Articles

Stabilization of Nonlinear Dynamical Systems by Delayed Feedback Control or by Delayed Connections

Stabilization of Nonlinear Dynamical Systems by Delayed Feedback Control or by Delayed Connections

Polynomial optimization with applications to stability analysis and control - Alternatives to sum of squares

In this paper, we explore the merits of various algorithms for solving polynomial optimization and optimization of polynomials, focusing on alternatives to sum of squares programming. While we refer to advantages and disadvantages of Quantifier Elimination, Reformulation Linear Techniques, Blossoming and Groebner basis methods, our main focus is on algorithms defined by Polya's theorem, Bernstein's theorem and Handelman's theorem. We first formulate polynomial optimization problems as verifying the feasibility of semi-algebraic sets. Then, we discuss how Polya's algorithm, Bernstein's algorithm and Handelman's algorithm reduce the intractable problem of feasibility of semi-algebraic sets to linear and/or semi-definite programming. We apply these algorithms to different problems in robust stability analysis and stability of nonlinear dynamical systems. As one contribution of this paper, we apply Polya's algorithm to the problem of $H_\infty$ control of systems with parametric uncertainty. Numerical examples are provided to compare the accuracy of these algorithms with other polynomial optimization algorithms in the literature.

Study on Stability and Reactive Power Optimization of Active Distribution Network Based on Bifurcation Theory

as the basic method used to analyze the stability of nonlinear dynamic system, it is able to more deeply discuss the stability problems of power system in the vicinity of a critical point in comparison with traditional analysis method. Active power distribution network is a complicated nonlinear dynamic system. The change of voltage stability is the process to convert from stable status to bifurcation in essence. Taking 10-node active distribution system as an example, the author studies the influence of reactive power optimization on voltage stability based on bifurcation theory. The author explains the reactive optimization algorithm based on numerical simulation, namely, it is the decision - making algorithm in order to realize multiple purposes of having minimum power network loss, largest reactive power margin with stable operation of power distribution network and minimum voltage floating in system fluctuation, which is proved to be effective in elevating power distribution system stability, reducing power loss and improving voltage quality.

Robustness analysis of global exponential stability of non‐linear systems with time delays and neutral terms

The global stability of non-linear dynamical systems has been investigated extensively in recent decades. It is well known that time delay and neutral term could derail the stability of non-linear systems. This study presents new results on the robustness of the global exponential stability of non-linear systems with respect to time delay and neutral term. Given globally exponentially stable non-linear systems, the problems to be addressed herein are how much time delay and neutral term contraction coefficient are allowed so that the non-linear systems can remain to be globally exponentially stable, in the presence of time delay and neutral term. Upper bounds of allowable time delay and neutral term contraction coefficient will be derived for non-linear systems to sustain their global exponential stability. A numerical example is provided to illustrate the results.

Summarization on Stability of Wind Power Systems Based on Bifurcation Theory

With wind farm capacity continuously increasing, remarkable characteristics of wind power make nonlinear dynamic characteristics of wind power system more apparent than conventional power system. Domestic and international experience shows that the voltage instability is main forms of wind power system instability. The bifurcation theory is one of the most powerful tools in analysis of structure stability of nonlinear dynamic system. Therefore application of bifurcation theory not only reveals the voltage stability mechanism, but also has important theoretic value and practical value for the security analysis of the whole power system, especially for voltage stability of wind power system.

On stability analysis via Lyapunov exponents calculated based on radial basis function networks

The concept of Lyapunov exponents is a powerful tool for analysing the stability of nonlinear dynamic systems, especially when the mathematical models of the systems are available. For real world systems, such models are often unknown. Estimating Lyapunov exponents using a time series has the advantage in that no mathematical model is required. However the time-series-based methods are believed to be reliable only for estimating positive exponents. Furthermore, when nonlinear mapping is applied for deriving the neighbourhood-to-neighbourhood matrices, the loads of mathematical deduction and programming increase significantly, which makes it unfeasible to nonlinear systems with high dimensions. In contrary, the model-based methods are constructive and reliable for calculating both positive and non-positive exponents. The use of the system Jacobians is the key to the advantage of the model-based methods. In this article, a novel approach is proposed, where the system Jacobians are derived based on system approximation using the radial basis function network. The proposed method inherits the advantage of the model-based methods, yet no mathematical model is required. Two case studies are presented to demonstrate the efficacy of the proposed method. We believe that the work can contribute to the stability analysis of nonlinear systems of which the dynamics are either difficult to model due to complexities or unknown.

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

The exponential and global stability of nonlinear differential dynamical systems with different initial times are investigated. Several criteria for the stability of nonlinear dynamical systems relative to initial time difference are obtained by means of vector Lyapunov functions. The obtained criteria have been applied to a proposed differential dynamic system. The numerical simulation validates our conclusions.

Reply to “Comment on ‘Adaptive steady-state stabilization for nonlinear dynamical systems’ ”

In a recent article [D. J. Braun, Phys. Rev. E 78, 016213 (2008)], the author has proposed a controller to stabilize a steady-state of a nonlinear dynamical system without requiring explicit knowledge of the system dynamics. In the related Comment [W. Lin, Phys. Rev. E 81, 038201 (2010)], Lin has criticized the analytical verification of the controller, and provided an alternative proof to verify its feasibility. While this proof is acknowledged here, we will show that the example used by Lin to demonstrate failure of the controller is a counterexample on the Comment itself.

Comment on “Adaptive steady-state stabilization for nonlinear dynamical systems”

Instead of the problematic reasoning presented in the work of D. J. Braun [Phys. Rev. E 78, 016213 (2008)], a rigorous argument is provided to show the validity of the adaptive controller proposed by Braun under some particular assumptions. Without these assumptions, this controller may be failed to stabilize the unsteady state, which is numerically shown by specific examples. Also, the choice of parameters to guarantee the validity of these assumptions is illustrated.

Adaptive steady-state stabilization for nonlinear dynamical systems

By means of LaSalle's invariance principle, we propose an adaptive controller with the aim of stabilizing an unstable steady state for a wide class of nonlinear dynamical systems. The control technique does not require analytical knowledge of the system dynamics and operates without any explicit knowledge of the desired steady-state position. The control input is achieved using only system states with no computer analysis of the dynamics. The proposed strategy is tested on Lorentz, van der Pol, and pendulum equations.

Finite-time stabilization of nonlinear dynamical systems via control vector Lyapunov functions

Finite-time stability involves dynamical systems whose trajectories converge to an equilibrium state in finite time. Since finite-time convergence implies nonuniqueness of system solutions in reverse time, such systems possess non-Lipschitzian dynamics. Sufficient conditions for finite-time stability have been developed in the literature using Hölder continuous Lyapunov functions . In this paper, we develop a general framework for finite-time stability analysis based on vector Lyapunov functions. Specifically, we construct a vector comparison system whose solution is finite-time stable and relate this finite-time stability property to the stability properties of a nonlinear dynamical system using a vector comparison principle. Furthermore, we design a universal decentralized finite-time stabilizer for large-scale dynamical systems that is robust against full modeling uncertainty. Finally, we present two numerical examples for finite-time stabilization involving a large-scale dynamical system and a combustion control system.

Strict practical stability of nonlinear impulsive systems by employing two Lyapunov-like functions

This paper develops the concepts of strict practical stability of ordinary differential equations to impulsive differential system. Strict practical stability, known as stability in tube-like domain, can be made to estimate upper bound and lower bound of the solutions of impulsive differential equations. This note provides several stability criteria for strict practical stability of nonlinear dynamical systems with impulse effects by employing two Lyapunov-like functions under general restrictions. It may provide a greater prospect to solve problems which exhibit impulsive effects.

Stability of nonlinear dynamical system of relative rotation and approximate solution under forced excitation

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.

On a method for studying stability of nonlinear dynamic systems

A method for studying the stability of the equilibrium points of linearized, nonlinear dynamic systems of arbitrary order is considered. The method is based on the fact that, due to the nature of the mutual arrangement of the trajectories of the corresponding linearized system, and the boundaries of some simply-connected, bounded neighborhood of its equilibrium point, one can judge the asymptotic stability and instability of both this point and the equilibrium point of the nonlinear system. Necessary and sufficient conditions of asymptotic stability and sufficient conditions of instability of equilibrium points of linear systems are given. Together with the theorems of the first Lyapunov method, these conditions determine the sufficient conditions of asymptotic stability and instability of equilibrium points of nonlinear systems. In some cases, the proposed conditions may turn out to be preferable to the known ones.

An approximate stability analysis of nonlinear systems described by Universal Learning Networks

Stability is one of the most important subjects in control systems. As for the stability of nonlinear dynamical systems, Lyapunov's direct method and linearized stability analysis method have been widely used. But, it is generally recognized that finding an appropriate Lyapunov function is fairly difficult especially for the nonlinear dynamical systems, and also it is not so easy for the linearized stability analysis to find the locally asymptotically stable region. Therefore, it is crucial and highly motivated to develop a new stability analysis method, which is easy to use and can easily study the locally asymptotically stable region at least approximately, if not exactly. On the other hand, as for the calculation of the higher order derivative, Universal Learning Networks (ULNs) are equipped with a systematic mechanism that calculates their first and second order derivatives exactly. So, in this paper, an approximate stability analysis method based on @h approximation is proposed in order to overcome the above problems and its application to a nonlinear dynamical control system is discussed. The proposed method studies the stability of the original trajectory by investigating whether the perturbed trajectory can approach the original trajectory or not. The above investigation is carried out approximately by using the higher order derivatives of ULNs. In summarizing the proposed method, firstly, the absolute values of the first order derivatives of any nodes of the trajectory with respect to any initial disturbances are calculated by using ULNs. If they approach zero at time infinity, then the trajectory is locally asymptotically stable. This is an alternative linearized stability analysis method for nonlinear trajectories without calculating Jacobians directly. In the method, the stability analysis of time-varying systems with multi-branches having any sample delays is possible, because the systems are modeled by ULNs. Secondly, the locally asymptotically stable region, where asymptotical stability is secured approximately, is obtained by finding the area where the first order terms of Taylor expansion are dominant compared to the second order terms with @h approximation assuming that the higher order terms more than the third order are negligibly small in the area. Simulations of an inverted pendulum balancing system are carried out. From the results of the simulations, it is clarified that the stability of the inverted pendulum control system is easily analyzed by the proposed method in terms of studying the locally asymptotically stable region.

Stabilization of Nonlinear Dynamic Systems Using the System State Estimates Made by the Asymptotic Observer

Consideration was given to asymptotic stabilization of the equilibria of nonlinear dynamic systems using the dynamic output feedbacks, that is, the feedbacks in the estimate of system state made by the asymptotic observer. Presented were the basic methods of constructing the asymptotic observers for the nonlinear dynamic systems with control and the approaches to system stabilization using the system state estimate made by the observer.

Partial stability and control: The state-of-the-art and development prospects

Problems in bordering fields associated with partial stability and stabilization of nonlinear dynamic systems, including partial stability and stabilization by part of coordinates of the phase vector, are reviewed. They are classified, and their relation with other stability and stabilization problems, including recent ones, are considered. Main development trends in theory and research methods are surveyed and certain results and applications are described. The partial control notion in border disciplines is examined. A long list of references is given.

Partial equiasymptotic stability of nonlinear dynamic systems

For nonlinear nonautonomous dynamic systems, partial equiasymptotic stability of the zero equilibrium position, equiasymptotic stability of the "partial" equilibrium position, and partial equiasymptotic stability of the "partial" equilibrium position are studied by the Lyapunov functions method. Examples are given.

Attractors and partial stability of nonlinear dynamical systems

This paper represents a study of the key concepts of attracting sets (submanifolds) and partially stable systems aimed at establishing relations between the local properties and the unification of the methodologies of their analysis. The use of recent results of stability theory and techniques of geometric control enables one to generalize Lyapunov-like sufficient conditions and propose simplified solutions for the problems of set attractivity and partial stability.

Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions. Pure and Applied Mathematics, Vol 246

1R11. Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions. Pure and Applied Mathematics, Vol 246. - AA Martynyuk (Stability of Processes Dept, Inst of Mech, Natl Acad of Sci, Kiev, Ukraine). Marcel Dekker, New York. 2002. 301 pp. ISBN 0-8247-0735-4. $150.00. Reviewed by RA Ibrahim (Dept of Mech Eng, Wayne State Univ, 5050 Anthony Wayne Dr, Rm 2119 Engineering Bldg, Detroit MI 48202).This addition to the series of pure and applied mathematics monographs deals with the modern theory of dynamics of continuous, discrete-time, and impulsive nonlinear systems using Liapunov matrix-valued functions. It is known that this theory is originally rooted in the developments of Poincare´’s and Liapunov’s ideas for treating nonlinear systems of differential equations. The book is devoted to introduce mathematical theorems for analyzing Liapunov matrix-valued functions in five chapters. The first chapter introduces the mathematical statements of qualitative methods of the general equations of continuous nonlinear systems. The definitions of various types of stability are introduced for nonlinear non-autonomous systems. Scalar, vector, and matrix-valued Liapunov functions, and the comparison principle were introduced to allow the estimation of the distance from every point of the system integral curve to the origin when the time changes from the fixed value. Other stability theorems, based on the work of the author and others, are stated with their proofs. Some methods for analyzing continuous nonlinear systems of hierarchical structure are presented in Chapter 2. These methods are supported by an example of third-order systems. Some stability theorems of systems with regular hierarchy subsystems, large systems, and their extension to overlapping decomposition are discussed. The problem of poly-stability of nonlinear systems with separable motion is analyzed as an application of the matrix-valued function. Chapter 2 includes the concepts of integral and Lipschitz stability based on the use of the principle of comparison with a matrix-valued Liapunov function. Chapter 3 presents the qualitative analysis of discrete-time systems that model mechanical systems with impulse control, digital computing devices, population dynamics, chaotic dynamics of economical systems, and many others. These systems are usually described in terms of difference equations whose stability conditions are defined in terms of the matrix-valued functions method. Chapter 4 introduces the stability of nonlinear dynamical systems subjected to impulsive perturbations. The impulsive system of differential equations are stated for general class of dynamical systems. The stability definitions presented in Chapter 2 for ordinary differential equations are adapted for the impulsive systems. Conditions and definitions of uniqueness, continuity, boundedness, and stability of solutions of impulsive systems are presented. Chapter 5 culminates the theorems and general results presented in the first four chapters by introducing some applications. They include numerical algorithms of constructing a point network supported by illustrative examples. The oscillations and stability of coupled mechanical systems are demonstrated for three pendulums through elastic springs and coupled two non-autonomous parametric oscillators. Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions is recommended to researchers who are studying the mathematical stability theory of dynamical systems. The author is commended for introducing illustrative examples from different applications to support the idea of Liapunov’s matrix functions.