Showing papers on "Lyapunov function published in 1987"
TL;DR: A calculus for computing Filippov's differential inclusion is developed which simplifies the analysis of dynamical systems described by differential equations with a discontinuous right-hand side and the rigorous stability analysis of variable structure systems is routine.
Abstract: This paper develops a calculus for computing Filippov's differential inclusion which simplifies the analysis of dynamical systems described by differential equations with a discontinuous right-hand side. In particular, when a slightly generalized Lyapunov theory is used, the rigorous stability analysis of variable structure systems is routine. As an example, a variable structure control law for rigid-link robot manipulators is described and its stability is proved.
648 citations
01 Aug 1987
TL;DR: In this article, the authors present an in-depth introduction to chaos in dynamical systems, and present several practical techniques for recognizing and classifying chaotic behavior, such as poincare map, Lyapunov exponents, capacity, information dimension, correlation dimension, and the reconstruction of attractors from a single time series.
Abstract: This tutorial presents an in-depth introduction to chaos in dynamical systems, and presents several practical techniques for recognizing and classifying chaotic behavior. These techniques include the poincare map, Lyapunov exponents, capacity, information dimension, correlation dimension, Lyapunov dimension, and the reconstruction of attractors from a single time series.
418 citations
TL;DR: In this article, the Lyapunov analysis is used to study the high-dimensional chaotic states of a class of delay-differential equations with time delayed feedback such as nonlinear optical resonators.
355 citations
TL;DR: In this article, the authors derived a set of differential equations for the eigenvalues and eigenvectors of the stability matrix of a dynamical system, as well as for the Lyapunov exponents and the corresponding eigenvector.
276 citations
TL;DR: In this article, the authors present a numerically reliable algorithm to compute the balanced realization of a stable state-space system that may be arbitrarily close to being unobservable and/or uncontrollable.
Abstract: In this paper we present a numerically reliable algorithm to compute the balanced realization of a stable state-space system that may be arbitrarily close to being unobservable and/or uncontrollable. The resulting realization, which is known to be a good approximation of the original system, must be minimal and therefore may contain a reduced number of states. Depending on the choice of partitioning of the Hankel singular values, this algorithm can be used either as a form of minimal realization or of model reduction. This illustrates that in finite precision arithmetic these two procedures are closely related. In addition to real matrix multiplication, the algorithm only requires the solution of two Lyapunov equations and one singular value decomposition of an upper-triangular matrix.
223 citations
TL;DR: In this paper, a model equation of fully developed three-dimensional turbulence is proposed which exhibits the Kolmogorov's similarity law in its chaotic state, and the structure of the chaotic attractor is investigated by examining the Lyapunov spectrum for several values of viscosity.
Abstract: A model equation of fully developed three-dimensional turbulence is proposed which exhibits the Kolmogorov's similarity law in its chaotic state The structure of the chaotic attractor is investigated by examining the Lyapunov spectrum for several values of viscosity The Lyapunov spectrum has a scaling property in the interior of the attractor It appears that the distribution function of the Lyapunov exponents has a singularity at a null Lyapunov exponent in the inviscid limit
157 citations
TL;DR: In this paper, a simple sliding mode control scheme for robot manipulators that does not rely upon the construction of individually stable discontinuity surfaces was presented, thus greatly reducing the complexity of design.
Abstract: We present a simple sliding mode control scheme for robot manipulators that does not rely upon the construction of individually stable discontinuity surfaces, thus greatly reducing the complexity of design. We utilize the structure of the manipulator dynamics and Lyapunov's second method in order to establish a sliding surface on the intersection of the switching surfaces in a direct manner. A simple numerical example accompanies the theoretical development.
141 citations
TL;DR: In this article, the sufficiency tests are applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems, and the modified Riccati equation approach of Petersen and Hollot is generalized in the static case and extended to dynamic compensation.
Abstract: Three parallel gaps in robust feedback control theory are examined: sufficiency versus necessity, deterministic versus stochastic uncertainty modeling, and stability versus performance. Deterministic and stochastic output-feedback control problems are considered with both static and dynamic controllers. The static and dynamic robust stabilization problems involve deterministically modeled bounded but unknown measurable time-varying parameter variations, while the static and dynamic stochastic optimal control problems feature state-, control-, and measurement-dependent white noise. General sufficiency conditions for the deterministic problems are obtained using Lyapunov's direct method, while necessary conditions for the stochastic problems are derived as a consequence of minimizing a quadratic performance criterion. The sufficiency tests are then applied to the necessary conditions to determine when solutions of the stochastic optimization problems also solve the deterministic robust stability problems. As an additional application of the deterministic result, the modified Riccati equation approach of Petersen and Hollot is generalized in the static case and extended to dynamic compensation.
130 citations
TL;DR: In this article, the authors proposed a security measure to indicate vulnerability to voltage collapse in electric power systems based on expected exit time from the region of attraction for the stable operating point.
Abstract: This paper proposes a security measure to indicate vulnerability to voltage collapse in electric power systems. The voltage-collapse phenomenon is modeled as a large deviation in the nonlinear dynamics of the system, driven by small magnitude, broad spectrum disturbances in load. This model relates the collapse to shrinking of the region of attraction for the operating point. The security measure indicating vulnerability to voltage collapse is based on expected exit time from the region of attraction for the stable operating point. It is shown that expected exit time can be related to a natural Lyapunov function for the power system, and that a security measure based on this expected exit time can be calculated simply by evaluating an exponential function of the Lyapunov "energy" at a neighboring load flow solution.
109 citations
TL;DR: In this paper, the authors extract information from a time displacement matrix through its singular values, and the associated orthogonal matrices, and show that asymptotic, after a long time, they depend only on the location in phase space, and not on the orbit.
97 citations
Book•
18 Mar 1987
TL;DR: In this paper, a unified theory of stability for systems of rigid bodies with finite degrees of freedom and for continuous elastic bodies with infinite degree of freedom is described, which involves extending the direct method of Lyapunov so that it can be used to evaluate the stability of continuous elastic body.
Abstract: This book describes a unified theory of stability for systems of rigid bodies with finite degrees of freedom and for continuous elastic bodies with infinite degrees of freedom. This involves extending the direct method of Lyapunov so that it can be used to evaluate the stability of continuous elastic bodies. Special attention has also been paid to a dynamical treatment of the stability of these bodies, and the existence of nonconservative follower forces has been taken into account. For this second edition the author has fully revised and updated his material; in particular, the section on elastomechanics has been completely rewritten to include the major developments which have taken place in this area. Exercises for the reader have been added at the end of each chapter.
01 Dec 1987
TL;DR: In this paper, the problem of robustification of a given stabilizing controller to make the closed loop system remain stable for prescribed ranges of variations of a set of physical parameters in the plant is treated in the state space and transfer function domains.
Abstract: This paper considers the problem of robustification of a given stabilizing controller to make the closed loop system remain stable for prescribed ranges of variations of a set of physical parameters in the plant. The problem is treated in the state space and transfer function domains. In the state space domain a stability hypersphere is determined in the parameter space using Lyapunov theory. The radius of this hypersphere is iteratively increased by adjusting the controller parameters until the prescribed perturbation ranges are contained in the stability hypersphere. In the transfer function domain a corresponding stability margin is defined and optimized based on the recently introduced concept of the largest stability hypersphere in the space of coefficients of the closed loop characteristic polynomial. The design algorithms are illustrated by examples.
TL;DR: In this article, the problem of choosing the Lyapunov function from the class of quadratic forms for differential inclusions is reduced to that of searching saddle points of a suitable function.
TL;DR: A special Lyapunov function technique is used to show that the method is convergent with probability 1 to stationary points and recovers the subgradient and Lagrange multipliers that appear in necessary conditions of optimality.
Abstract: For a stochastic programming problem with a nonsmooth objective a recursive stochastic subgradient method is proposed in which successive directions are obtained from quadratic programming subproblems The subproblems result from linearization of original constrains and from approximation of the objective by a quadratic function involving stochastic ϵ-subgradient estimates constructed in the course of computation A special Lyapunov function technique is used to show that the method is convergent with probability 1 to stationary points and recovers the subgradient and Lagrange multipliers that appear in necessary conditions of optimality
TL;DR: In the weak noise limit of Fokker-Planck models, the stationary probability density can be expressed in the form P∼exp(-1/ϵϕ(χ) where ϵ measures the strength of the noise as discussed by the authors.
Abstract: In the weak noise limit of Fokker-Planck models, the stationary probability density can be expressed in the form P∼exp(-1/ϵϕ(χ)), where ϵ measures the strength of the noise. We show firstly that, although in general ϕ is not differentiable, it is regular enough to allow its use as a Lyapunov function for the underlying deterministic dynamical system. This justifies the interpretation of ϕ as a generalized thermodynamic potential for systems away from equilibrium.
Secondly, we show that the nondifferentiability of ϕ does not presuppose any kind of irregular behavior in the underlying deterministic system, nor the presence of several stationary points. We construct a simple model with one single attractor and trivial deterministic dynamics, which has a nondifferentiable quasipotential ϕ. We relate the nondifferentiability to the existence of caustics and shock lines in the associated variational problem.
TL;DR: The Lyapunov dimension of chaotic attractors is found to be almost equal to the delay time divided by the correlation time of the feedback driving force for three dynamical systems: Mackey-Glass model for white-cell production, optical bistable hybrid system, and nonlinear ring cavity.
Abstract: The Lyapunov dimension of chaotic attractors is found to be almost equal to the delay time divided by the correlation time of the feedback driving force for three dynamical systems: Mackey-Glass model for white-cell production, optical bistable hybrid system, and nonlinear ring cavity. This discovery will enable experimentalists to estimate the complexity of a high-dimension system much more easily than by time-series methods, as illustrated by a hybrid experiment.
TL;DR: In this paper, a Lyapunov stability analysis for ideal Hall magnetohydrodynamics (HMHD) in two and three dimensions is presented, which is based on the remarkable array of nonlinear conservation laws for HMHD that follow from its Hamiltonian structure.
Abstract: Hall electric fields produce circulating mass flow in confined ideal‐fluid plasmas The conservation laws, Hamiltonian structure, equilibrium state relations, and Lyapunov stability conditions are presented here for ideal Hall magnetohydrodynamics (HMHD) in two and three dimensions The approach here is to use the remarkable array of nonlinear conservation laws for HMHD that follow from its Hamiltonian structure in order to construct explicit Lyapunov functionals for the HMHD equilibrium states In this way, the Lyapunov stability analysis provides classes of HMHD equilibria that are stable and whose linearized initial‐value problems are well posed (in the sense of possessing continuous dependence on initial conditions) Several examples are discussed in both two and three dimensions
TL;DR: In this paper, if an underlying Markovian state-space system for the controlled process is available, and if this process possesses stationary transition probabilities, then the powerful ergodic theory of Markov processes may be applied.
Abstract: The principal techniques used up to now for the analysis of stochastic adaptive control systems have been 1) super-martingale (often called stochastic Lyapunov) methods and 2) methods relying upon the strong consistency of some parameter estimation scheme. Optimal stochastic control and filtering methods have also been employed. Although there have been some successes, the extension of these techniques to a broader class of adaptive control problems, including the case of time-varying parameters, has been difficult. In this paper a new approach is adopted: if an underlying Markovian state-space system for the controlled process is available, and if this process possesses stationary transition probabilities, then the powerful ergodic theory of Markov processes may be applied. Subject to technical conditions, such as stability, one may deduce 1) the existence of an invariant measure for the process and 2) the convergence almost surely of the sample averages of a function of the state process (and of its expectation) to its conditional expectation. The technique is illustrated by an application to a previously unsolved problem involving a linear system with unbounded random time-varying parameters.
TL;DR: A new order parameter is available which reflects the marked change in the stochasticity properties of the motion of the system as the temperature is varied in the vicinity of the critical temperature.
Abstract: We present a molecular-dynamics simulation of a two-dimensional system of coupled rotators [O(2) planar Heisenberg model]. We observe that if the maximal characteristic Lyapunov exponent for the system is plotted versus the temperature, a clear-cut ``knee'' appears precisely in correspondence with the Kosterlitz-Thouless transition temperature. A new order parameter is therefore available which reflects the marked change in the stochasticity properties of the motion of the system as the temperature is varied in the vicinity of the critical temperature.
10 Jun 1987
TL;DR: In this article, the stability of nonlinear multiparameter singularly perturbed systems is analyzed and sufficient conditions for existence of a Lyapunov function and uniform asymptotic stability are derived.
Abstract: Asymptotic stability of nonlinear multiparameter singularly perturbed systems is analyzed. Sufficient conditions for existence of a Lyapunov function and uniform asymptotic stability are derived. The new feature of these conditions over earlier results is that there is no restriction on the relative magnitudes of the small singular perturbation parameters. Moreover, the class of systems under consideration can be nonlinear in both the slow and fast variables, while earlier results were limited to systems linear in the fast variables.
01 Dec 1987
TL;DR: Using a Lyapunov analysis, this paper shows that, for linear systems and a class of nonlinear systems with discretized feedback, some stability properties can be preserved if the sampling frequency is properly chosen.
Abstract: When a continuous time control law is implemented using a digital computer, the closed loop system may not have the same stability properties as the system with a true continuous controller due to delay and digitization errors. Using a Lyapunov analysis, this paper shows that, for linear systems and a class of nonlinear systems with discretized feedback, some stability properties can be preserved if the sampling frequency is properly chosen. In particular, we propose a variable sampling interval scheme for linear systems. This scheme is desirable when (1) computer resources are tightly shared by many tasks or (2) power consumption is critical. The effect of truncation error is also studied.
TL;DR: The exact Lyapunov dimension of the universal attractor of the complex Ginzburg-Landau partial differential equation agrees with an estimate of the number of degrees of freedom based on a simple linear stability analysis and mode-counting argument.
Abstract: We present an exact analytic computation of the Lyapunov dimension of the universal attractor of the complex Ginzburg-Landau partial differential equation for a finite range of its parameter values. We obtain upper bounds on the attractor's dimension when the parameters do not permit an exact evaluation by our methods. The exact Lyapunov dimension agrees with an estimate of the number of degrees of freedom based on a simple linear stability analysis and mode-counting argument.
10 Jun 1987
TL;DR: In this article, a robust stability and performance analysis technique for harmonic oscillators with uncertain but constant natural frequency was developed. And the authors proved that the majorant Lyapunov equation predicts unconditional stability.
Abstract: A new robust stability and performance analysis technique is developed. The approach involves replacing the state covariance by its block-norm matrix, i.e., the nonnegative matrix whose elements are the norms of subblocks of the covariance matrix partitioned according to subsystem dynamics. A bound (i.e., majorant) for the block-norm matrix is given by the majorant Lyapunov equation, a Lyapunov-type nonnegative matrix equation. Existence, uniqueness, and computational tractability of solutions to the majorant Lyapunov equation are shown to be completely characterized in terms of M matrices. Two examples are considered. For a damped simple harmonic oscillator with uncertain but constant natural frequency, the majorant Lyapunov equation predicts unconditional stability. And, for a pair of nominally uncoupled oscillators with uncertain coupling, the majorant Lyapunov equation shows that the range of nondestabilizing couplings is proportional to the frequency separation between the oscillators, a result not predictable from quadratic or vector Lyapunov functions.
TL;DR: In this paper, the amplitude of the feedback controller is used to stabilize an uncertain nonlinear system, where the amplitude is defined by a state equation which contains uncertain parameters which are unknown but bounded.
TL;DR: In this paper, the Lyapunov spectra were measured for a 3D many-body dense fluid at equilibrium and in the presence of an isoenergetic nonequilibrium field generating a pair of equal and opposite currents.
01 Jan 1987
TL;DR: It is found that the degree of temporal disorder is minimal in a “resting awake” state and increases in sleep as well as in fluroxene induced general anesthesia, and the statistical errors appear to decrease, which can be interpreted as a transition to a more uniform dynamical state.
Abstract: We apply several different methods from nonlinear dynamical systems to the analysis of the degree of temporal disorder in data from human EEGs. Among these are methods of geometrical reconstruction, dimensional complexity, mutual information content, and two different approaches for estimating Lyapunov characteristic exponents. We show how the naive interpretation of numerical results can lead to a considerable underestimation of the dimensional complexity. This is true even when the errors from least squares fits are small. We present more realistic error estimates and show that they seem to contain additional, important information. By applying independent methods of analysis to the same data sets for a given lead, we find that the degree of temporal disorder is minimal in a “resting awake” state and increases in sleep as well as in fluroxene induced general anesthesia. At the same time the statistical errors appear to decrease, which can be interpreted as a transition to a more uniform dynamical state.
TL;DR: In this article, the robustness of nonlinear discrete-time systems is analyzed based on the existence of a stationary solution of the dynamic programming equation (DPE), which provides directly a Lyapunov function associated to the closed-loop system.
Abstract: In this paper the robustness of nonlinear discrete-time systems is analyzed. The nominal plant is supposed to be controlled by means of a feedback control law which is optimal with respect to some given criterion. The robustness of the closed-loop system is studied for two different classes of perturbations in the control law, which are called gain and additive nonlinear perturbations. The results are entirely based on the existence of a stationary solution of the dynamic programming equation (DPE), which provides directly a Lyapunov function associated to the closed-loop system. The convexity of that solution and the use of the Taylor formula appear to be the key to establish the robustness properties of the nominal plant. Two examples are solved in order to show an interesting fact: the existence of a compromise between the robustness of the system subjected to the two different classes of perturbations.
01 Aug 1987
TL;DR: An integrated software toolkit for the analysis of nonlinear dynamical systems is introduced, which includes software that calculates and displays trajectories, bifurcation diagrams, and two-dimensional phase portraits.
Abstract: An integrated software toolkit for the analysis of nonlinear dynamical systems is introduced. This user-friendly, graphically oriented collection of interactive programs includes software that calculates and displays trajectories, bifurcation diagrams, and two-dimensional phase portraits. Also included are programs that locate periodic solutions, calculate and display invariant manifolds of two-dimensional Poincare maps, as well as compute Lyapunov exponents, Lyapunov dimension, fractal dimension, information dimension, and correlation dimension. The toolkit runs under both the UNIX and PC-DOS operating systems.
TL;DR: In this paper, an N-link planar robot holding an unknown load and driving its end-effector along a prespecified trajectory is studied, and the effect of the unknown load is discussed.
Abstract: An N-link planar robot holding an unknown load and driving its end-effector along a prespecified trajectory is studied. The effect of the unknown load is discussed, and the nonlinear dynamic equation with the unknown load is derived. A non linear model-reference adaptive control algorithm is devel oped for tracking of a trajectory that is a piecewise continuous function of time. The tracking error dynamics represent a linear stable system with persistent disturbances, and the Germaidze-Krasovskii theorem in the Lyapunov second method is applied to show the error stability. Digital com puter simulations show the effectiveness of the adaptive con trol algorithm for the unknown load with mass 2 kg, 5 kg, and 10 kg. The effect of the computation time delay on the system dynamics is also addressed.
TL;DR: In this article, the trace of the solution of the Lyapunov matrix differential equation is shown to be obtained as a solution to simple scalar differential equations, and upper and lower bounds for the trace are derived.
Abstract: Upper and lower bounds for the trace of the solution of the Lyapunov matrix differential equation are derived. It is shown that they are obtained as solutions to simple scalar differential equations. As a special case, the bounds for the stationary solution give ones for the solution to the Lyapunov algebraic equation.