Showing papers on "Lyapunov function published in 1985"
TL;DR: In this article, the authors present the first algorithms that allow the estimation of non-negative Lyapunov exponents from an experimental time series, which provide a qualitative and quantitative characterization of dynamical behavior.
8,128 citations
[...]
TL;DR: In this paper, a detailed analysis of the geometric structure of a chaotic attractor observed from an extremely simple electrical circuit is given. And the chaotic nature of the attractor is further confirmed by calculating its associated Lyapunov exponents and Lyapeunov dimension.
Abstract: A detailed analysis is given of the geometric structure of a chaotic attractor observed from an extremely simple autonomous electrical circuit. It is third order, reciprocal, and has only one nonlinear element: a 3-segment piecewise-linear resistor. Extensive laboratory measurements from this circuit and a detailed geometrical analysis and computer simulation reveal the following rather intricate "anatomy" of the associated strange attractor. In addition to a microscopically infinite sheet-like composition the attractor has a macroscopic "double-scroll" structure, i.e., two sheetlike objects are curled up together into spiral forms with infinitely many rotations. (See frontispiece.) The chaotic nature of this circuit is further confirmed by calculating its associated Lyapunov exponents and Lyapunov dimension. The double-scroll attractor has one positive, one zero and one negative Lyapunov exponent. The Lyapunov dimension turns out to be a fractal between 2 and 3 which agrees with the observed structures. The power spectra of the three associated state variables obtained by both measurement and computer simulation show a continuous broad spectrum typical of chaotic systems.
602 citations
TL;DR: This paper presents various theoretical and computational methods for estimating the domain of attraction of an autonomous nonlinear system based on the concept of a maximal Lyapunov function, which is introduced in this paper.
317 citations
TL;DR: In this paper, the first-order necessary conditions for quadratically optimal reduced-order modeling of linear time-invariant systems are derived in the form of a pair of modified Lyapunov equations coupled by an oblique projection which determines the optimal reduced order model.
Abstract: First-order necessary conditions for quadratically optimal reduced-order modeling of linear time-invariant systems are derived in the form of a pair of modified Lyapunov equations coupled by an oblique projection which determines the optimal reduced-order model. This form of the necessary conditions considerably simplifies previous results of Wilson [1] and clearly demonstrates the quadratic extremality and nonoptimality of the balancing method of Moore [2]. The possible existence of multiple solutions of the optimal projection equations is demonstrated and a relaxation-type algorithm is proposed for computing these local extrema. A component-cost analysis of the model-error criterion similar to the approach of Skelton [3] is utilized at each iteration to direct the algorithm to the global minimum.
260 citations
TL;DR: In this article, it was shown that within the setting of a difference equation, it is possible to link ergodicity with stability via the physical notion of energy in the form of a Lyapunov function.
Abstract: We have shown that within the setting of a difference equation it is possible to link ergodicity with stability via the physical notion of energy in the form of a Lyapunov function.
228 citations
TL;DR: In this article, it was shown that preservation of cones leads to non-vanishing of Lyapunov exponents of billiards, and that geodesic flows on manifolds of non-positive sectional curvature can be treated from this point of view.
Abstract: We show that in several cases preservation of cones leads to non-vanishing of (some) Lyapunov exponents It gives simple and effective criteria for nonvanishing of the exponents, which is demonstrated on the example of the billiards studied by Bunimovich It is also shown that geodesic flows on manifolds of non-positive sectional curvature can be treated from this point of view
174 citations
TL;DR: This paper attempts to give a unified overview of how direct methods solve the transient stability problem of large-scale power systems by focusing on the derivation of stability indices, intended for on-line monitoring, contingency evaluation and security control.
173 citations
TL;DR: Fisher's Fundamental Theorem of Natural Selection is extended to the selection mutation model with mutation rates ɛij=ɛii.e.
Abstract: Fisher's Fundamental Theorem of Natural Selection is extended to the selection mutation model with mutation rates epsilon ij = epsilon i, i.e. depending only on the target gene, by constructing a simple Lyapunov function. For other mutation rates stable limit cycles are possible.
150 citations
TL;DR: In this paper, the first-order necessary conditions for optimal, steady-state, reduced-order state estimation for a linear, time-invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form.
Abstract: First-order necessary conditions for optimal, steady-state, reduced-order state estimation for a linear, time-invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form. In contrast to the lone matrix Riccati equation arising in the full-order (Kalman filter) case, the optimal steady-state reduced-order estimator is characterized by three matrix equations (one modified Riccati equation and two modified Lyapunov equations) coupled by a projection whose rank is precisely equal to the order of the estimator and which determines the optimal estimator gains. This coupling is a graphic reminder of the suboptimality of proposed approaches involving either model reduction followed by "full-order" estimator design or full-order estimator design followed by estimator-reduction techniques. The results given here complement recently obtained results which characterize the optimal reduced-order model by means of a pair of coupled modified Lyapunov equations [7] and the optimal fixed-order dynamic compensator by means of a coupled system of two modified Riceati equations and two modified Lyapunov equations [6].
98 citations
TL;DR: In this paper, the applicability of the constructive stability algorithm of Brayton and Tong in the stability analysis of fixed-point digital filters is demonstrated. But the authors only consider direct and coupled digital filters and do not consider lattice filters.
Abstract: We demonstrate the applicability of the constructive stability algorithm of Brayton and Tong in the stability analysis of fixed-point digital filters. In the present paper, we consider direct form and coupled form filters while in a companion paper we treat wave digital filters and lattice filters. We compare our results with existing ones which deal with either the global asymptotic stability of digital filters or with existence (resp., nonexistence) of limit cycles in digital filters. Several of the present results are new while some of the present results constitute improvements over existing results. In a few cases, the present results are more conservative than existing results. It is emphasized that whereas the existing results are obtained by several diverse methods, the present results are determined by one unified approach.
98 citations
TL;DR: In this paper, necessary and sufficient conditions are given for stability analysis of two-dimensional (2D) systems based on a Lyapunov approach using the Roesser state-space model.
Abstract: Some necessary and sufficient conditions are given for stability analysis of two-dimensional (2-D) systems based on a Lyapunov approach. The study was carried out using the Roesser state-space model, which when combined with the Lyapunov theory provides the new checkable tests for stability. Also, the results lead to techniques for selecting stabilizing state feedback gain matrices for the 2-D systems.
TL;DR: In this paper, four characterizations of stabilizability and detectability of linear periodic systems are considered, two of them look as natural extensions of the classical definitions given for time-invariant systems and the remaining two are modal characterizations which turn out to be useful in the analysis of the periodic Lyapunov and Riccati equations.
TL;DR: In this article, it is shown that if a system is to have this stronger property, the matching condition must necessarily be satisfied, which is not a necessary condition for stabilizability.
Abstract: This paper investigates one aspect of the problem of stabilizing an uncertain linear system. That is, the systems under consideration contain uncertain parameters which are unknown but bounded. The question arises as to whether such a system can be stabilized via feedback control. In some of the previous papers in this area, the system is assumed to satisfy a so-called “matching-condition;” this type of assumption is used to assure that the uncertain system can be stabilized. It is known, however, that the matching condition is not a necessary condition for stabilizability. This paper introduces a strengthened notion of stabilizability referred to as structural stabilizability via a nominally determined quadratic Lyapunov function. It is then shown that if a system is to have this stronger property, the matching condition must necessarily be satisfied.
TL;DR: By relaxing the observability assumption to detectability, an extended version of the lemma is obtained, where the system stability is linked with the existence of a periodic positive semidefinite solution of the Lyapunov equation.
TL;DR: With an important class of automata networks, a Lyapunov function is associated and by doing so its dynamic behaviour is characterized (transient and cycle lengths), which helps to study threshold and majority networks.
TL;DR: In this paper, a class of discrete-time Lyapunov min-max controllers were obtained for linear, scalar input, phase variable systems with bounded parameter and input uncertainties.
TL;DR: In this paper, a formula for the Lyapunov exponents of the flow of a nonlinear stochastic system is presented, which characterises the asymptotic behaviour of the derivative flow, and negative exponents are associated with clustering.
Abstract: We present a formula for the Lyapunov exponents of the flow of a nonlinear stochastic system (These exponents characterise the asymptotic behaviour of the derivative flow, and negative exponents are associated with clustering of the flow) This formula is analogous to that of Khas'minskii, who deals with a linear system We use this fojoruila to show that if we have an ordinary dynamical system which is Lyapunov stable (ie all the exponents are negative) then so are certain stochastic perturbations of it
19 Jun 1985
TL;DR: In this paper, the first-order necessary conditions for optimal, steady-state, reduced-order state estimation for a linear, time-invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form.
Abstract: First-order necessary conditions for optimal, steady-state, reduced-order state estimation for a linear, time-invariant plant in the presence of correlated disturbance and nonsingular measurement noise are derived in a new and highly simplified form. In contrast to the lone matrix Riccati equation arising in the full-order (Kalman filter) case, the optimal steady-state reduced-order estimator is characterized by three matrix equations (one modified Riccati equation and two modified Lyapunov equations) coupled by a projection whose rank is precisely equal to the order of the estimator and which determines the optimal estimator gains. This coupling is a graphic reminder of the suboptimality of proposed approaches involving either model reduction followed by "full-order" estimator design or full-order estimator design followed by estimator-reduction techniques, The results given here complement recently obtained results which characterize the optimal reduced-order model by means of a pair of coupled modified Lyapunov equations ([7]) and the optimal fixed-order dynamic compensator by means of a coupled system of two modified Riecati equations and two modified Lyapunov equations ([6]).
TL;DR: In this article, the authors give sufficient conditions for the nonlinear stability of possibly nonsmooth stationary solutions of the two-dimensional Euler equation in symmetric bounded domains and investigate the stability of smooth solutions under perturbations of the boundary.
Abstract: We give sufficient conditions for the nonlinear stability of possibly nonsmooth stationary solutions of the two-dimensional Euler equation in symmetric bounded domains. We use, as Lyapunov functions, first integrals due to the symmetry of the problem. Moreover, we investigate the stability of smooth solutions under perturbations of the boundary. The last result is based on a generalization of the well known Arnold approach.
TL;DR: In this paper, a concept of strong stability of an equilibrium point of an electric power system is introduced and a complete local analysis of the stability of power system equilibria in the presence of transfer conductances is given.
Abstract: In this paper, a concept of strong stability of an equilibrium point of an electric power system is introduced. It is shown that almost all stable equilibria of the standard transient stability model are strongly stable and that strong stability is a necessary and sufficient condition for the existence of a local energy-like Lyapunov function for all small perturbations of the nominal system. Such a Lyapunov function is explicitly constructed. A complete local analysis of the stability of power system equilibria in the presence of transfer conductances is given.
TL;DR: In this article, a generalized kinematic similarity transformation is used to diagonalize linear differential systems, and the relation to Lyapunov's classical stability theory is explored, and asymptotic estimates of fundamental solutions are given.
Abstract: A new theory is presented, in which a generalized kinematic similarity transformation is used to diagonalize linear differential systems. No matrices of Jordan form are needed. The relation to Lyapunov’s classical stability theory is explored, and asymptotic estimates of fundamental solutions are given. Finally, some possible numerical applications of the presented theory are suggested.
TL;DR: In this article, the authors characterized Lyapunov diagonally stable real H-matrices with respect to principal submatrix rank property. And they applied their results to the numerical abscissas of real matrices.
TL;DR: In this paper, the authors incorporated rescaling directly into the equations of motion, preventing Lyapunov instability by using an effective constraint force, and used the constraint force to prevent the instability of neighboring phase-space trajectories.
TL;DR: In this paper, the integrated density of statesH(ω2) of a chain of harmonic oscillators with a binary random distribution of the masses was studied, and it was shown that there is a dense set of values of the squared frequency for which the difference H(ω 2+α)-H(α) has a singularity of the type ¦β¦2α, multiplied by a periodic function of ln ¦ ǫ 2α, where the exponent α and the period depend continuously on
Abstract: We study the integrated density of statesH(ω2) of a chain of harmonic oscillators with a binary random distribution of the masses We show in particular that there is a dense set of values of the squared frequency for which the differenceH(ω2+ɛ)-H(ω2) has a singularity of the type ¦ɛ¦2α, multiplied by a periodic function of ln ¦ɛ¦, where the exponent α and the period depend continuously onω2 In the region where α < 1/2,H is not differentiate on a dense set of points The same type of singularities is also present in the Lyapunov coefficient
TL;DR: In this article, the robustness problem for a controlled discrete-time system subject to a Lur'e type feedback nonlinearity together with random multiplicative noises effective on state, input and non-linearity is discussed.
Abstract: The general robustness problem is discussed for a controlled discrete-time system subject to a Lur'e type feedback non-linearity together with random multiplicative noises effective on state, input and the non-linearity. A sufficient condition is given utilizing the discrete stochastic version of the second method of Lyapunov to guarantee that any stabilizing controller designed for the deterministic linear part of the system renders the closed-loop system with perturbations, asymptotically stable with probability one. Then, specializations to systems given in controllability and controller canonical forms are made. Independent noise signals are assumed to be effective on individual parameters of these canonical forms whose inputs are designed to assign the closed-loop poles to zero. In these cases, stronger results are obtained in terms of system parameters and known characteristics of perturbations.
TL;DR: In this paper, the authors used the constructive stability algorithm of Brayton and Tong in the stability analysis of fixed-point digital filters which are in the direct form and in the coupled form.
Abstract: In a companion paper [4], we utilize the constructive stability algorithm of Brayton and Tong in the stability analysis of fixed-point digital filters which are in the direct form and in the coupled form. We continue this work in the present paper by considering wave digital filters and lattice digital filters. We believe that the results of the present paper and its companion paper demonstrate that the constructive algorithm constitutes an effective and general approach in the qualitative analysis of fixed-pointed digital filters.
TL;DR: In this paper, the dynamic approach for stability analysis is used to obtain an approximate closed-form expression for the "viscoelastic critical load" of perfect columns made of a linear three-element model material.
TL;DR: In this paper, the output roundoff noise variance and the scaling of the internal registers are associated with two impulse response sequences, and two Lyapunov equations are established, corresponding to those which have been already used for the noise analysis and optimal structure synthesis in the 1-D case.
Abstract: The output roundoff noise variance and the scaling of the internal registers are associated with two impulse response sequences. Furthermore, two Lyapunov equations are established, corresponding to those which have been already used for the noise analysis and optimal structure synthesis in the 1-D case. An analytic formula is given for the solution of the Lyapunov equations. Thus the known techniques from the 1-D filters can be directly extended.
TL;DR: In this article, a method for calculating an approximation to the reachable set from the origin of a class of linear control systems is presented, which involves the construction of a Lyapunov-like function, and ultimately requires the numerical solution of a simple optimization problem.
Abstract: A method is presented for calculating an approximation to the reachable set from the origin of a class of linear control systems. The method involves the construction of a Lyapunov-like function, and ultimately requires the numerical solution of a simple optimization problem. Advantages of the method are that it is applicable to n-dimensional systems and does not require calculation of orbits or a choice of boundary conditions. Further, the system need not be completely controllable for the method to be applicable. However, while the method ensures that the reachable set lies within a specified set, a closeness-of-fit criterion is not available.
19 Jun 1985
TL;DR: In this paper, the authors show that solvability of various output feedback design problems is equivalent to existence of a solution to a Constrained Lyapunov Problem (CLP).
Abstract: Given a dynamical system whose description includes time-varying uncertain parameters, it is often desirable to design an output feedback controller leading to uniform stability of a given equilibrium point. When designing such a controller, one may consider static (i.e., memoryless) or dynamic compensation. In this paper, we show that solvability of various output feedback design problems is equivalent to existence of a solution to a certain Constrained Lyapunov Problem (CLP). The CLP can be stated in purely algebraic terms. Once the CLP is described, we provide necessary and sufficient conditions for its solution to exist. Subsequently, we consider application of the CLP to a number of robust stabilization problems involving static output feedback and observer-based feedback.