Showing papers on "Lyapunov equation published in 1991"
Book•
31 Mar 1991
TL;DR: In this article, several Lyapunov functions are discussed, and extensions are proposed for applications in the context of applications with a focus on energy efficiency and energy efficiency, respectively.
Abstract: 1. Why several Lyapunov functions?.- 2. Refinements.- 3. Extensions.- 4. Applications.- References.
294 citations
TL;DR: In this article, the authors introduce the idea of local Lyapunov exponents which govern the way small perturbations to the orbit of a dynamical system grow or contract after an infinite number of steps along the orbit.
Abstract: We introduce the idea of local Lyapunov exponents which govern the way small perturbations to the orbit of a dynamical system grow or contract after afinite number of steps,L, along the orbit. The distributions of these exponents over the attractor is an invariant of the dynamical system; namely, they are independent of the orbit or initial conditions. They tell us the variation of predictability over the attractor. They allow the estimation of extreme excursions of perturbations to an orbit once we know the mean and moments about the mean of these distributions. We show that the variations about the mean of the Lyapunov exponents approach zero asL → ∞ and argue from our numerical work on several chaotic systems that this approach is asL
−v. In our examplesv ≈ 0.5–1.0. The exponents themselves approach the familiar Lyapunov spectrum in this same fashion.
180 citations
TL;DR: In this paper, a new procedure is presented for solving the Lyapunov matrix equation, where the system is reduced to tridiagonal form with Gaussian similarity transformations and then the resulting system is solved with Alternating-Direction-Implicit (ADI) iteration.
Abstract: In this report, a new procedure is presented for solving the Lyapunov matrix equation. First, the system is reduced to tridiagonal form with Gaussian similarity transformations. Then the resulting system is solved with Alternating-Direction-Implicit (ADI) iteration. A matrix commutation property essential for “model problem” convergence of ADI iteration applied to elliptic difference equations is not needed for this application. All stable Lyapunov matrix equations are model ADI problems.
138 citations
TL;DR: In this paper, the design of a state feedback law for an affine nonlinear system to render a (as small as possible) compact neighborhood of the equilibrium of interest globally attractive is discussed.
Abstract: The design of a state feedback law for an affine nonlinear system to render a (as small as possible) compact neighborhood of the equilibrium of interest globally attractive is discussed. Following Z. Artstein's theorem (1983), the problem can be solved by designing a so-called control Lyapunov function. For systems which are in a cascade form, a Lyapunov function meeting Artstein's conditions is designed, assuming the knowledge of a control law stabilizing the equilibrium of the head nonlinear subsystem. In particular, for planar systems, this gives sufficient and necessary conditions for a compact neighborhood of the equilibrium to be stabilized. >
87 citations
11 Dec 1991
TL;DR: In this paper, a framework for parameter-dependent Lyapunov functions, a less conservative refinement of 'fixed' LyAPunov function, is developed. And an immediate application of this framework is a reinterpretation of the classical Popov criterion as a parameter dependent Lyapinov function. And this observation is exploited to obtain conditions for robust controller synthesis with full-state, full-order, and reduced-order controllers in H/sub 2/ and H/ sub 2/
Abstract: A framework for parameter-dependent Lyapunov functions, a less conservative refinement of 'fixed' Lyapunov functions, is developed. An immediate application of this framework is a reinterpretation of the classical Popov criterion as a parameter-dependent Lyapunov function. This observation is exploited to obtain conditions for robust controller synthesis with full-state, full-order, and reduced-order controllers in H/sub 2/ and H/sub 2//H/sub infinity / settings. >
77 citations
TL;DR: An analysis of some important qualitative properties of such symmetrically interconnected systems focussing on the spectrum characterization, controllability and observability, and the solutions of the algebraic Riccati equation and the matrix Lyapunov equation is conducted.
76 citations
01 Jan 1991
TL;DR: The main theme of as discussed by the authors is the connections between various Riccati equations and the closed loop stability of control schemes based on Linear Quadratic (LQ) optimal methods for control and estimation.
Abstract: The main theme of this Chapter will be the connections between various Riccati equations and the closed loop stability of control schemes based on Linear Quadratic (LQ) optimal methods for control and estimation. Our presentation will encompass methods applicable both for discrete time and continuous time, and so we discuss concurrently the difference equations (discrete time) and the differential equations (continuous time) — the intellectual machinery necessary for the one suffices for the other and so it makes sense to dispense with both cases in one fell swoop.
62 citations
TL;DR: In this article, the spectral radii of local and global Lyapunov exponents are related by redefining them as the spectral radius of some positive operators on a space of continuous functions and utilizing the theory developed by Choquet and Foias.
Abstract: Various properties of Local and Global Lyapunov exponents are related by redefining them as the spectral radii of some positive operators on a space of continuous functions and utilizing the theory developed by Choquet and Foias These results are then applied to the problem of estimating the Hausdorff dimension of the global attractor and the existence of a critical trajectory, along which the Lyapunov dimension is majorized, is established Using this new estimate, the existing dimension estimate for the global attractor of the Lorenz system is improved Along the way a simple relation between topological entropy and the fractal dimension is obtained
61 citations
TL;DR: This paper treats the three-dimensional aspects of tethered satellite deployment and retrieval and shows that the controller designed by the second method of Lyapunov is superior to that of the former, primarily from the out-of-plane thrust usage point of view.
Abstract: This paper treats the three-dimensional aspects of tethered satellite deployment and retrieval. Feedback control laws with guaranteed closed-loop stability are obtained using the second method of Lyapunov. Tether mass and aerodynamic effects are not included in the design of the control laws. First, a coordinate transformation is presented that partially uncouples the in-plane and out-of-plane dynamics. A combination of tension control as well as out-of-plane thrusting is shown to be adequate for fast retrieval. Next, a unified control design method based on an integral of motion (for the coupled system) is presented. It is shown that the controller designed by the latter, method is superior to that of the former, primarily from the out-of-plane thrust usage point of view. A detailed analysis of stability of the closed-loop system is presented and the existence of limit cycles is ruled out if out-of-plane thrusting is used in conjunction with tension control. Finally, a tether rate control law is also developed using the integral of motion mentioned previously. The control laws developed in the paper can also be used for stationkeeping.
TL;DR: In this article, the authors use a singular system setting to provide a geometric theory for dynamical systems under derivative feedback and define the relevant subspace and provide computational design techniques in terms of a generalized Sylvester or Lyapunov equation for which efficient solution techniques are well known.
Abstract: The authors use a singular system setting to provide a geometric theory for dynamical systems under derivative feedback. They define the relevant subspace and provide computational design techniques in terms of a generalized Sylvester or Lyapunov equation for which efficient solution techniques are well-known. The authors provide both geometric and algebraic characterizations of the effects of derivative feedback, drawing connections with previous work in state-variable systems as well as extending that work to singular systems. >
TL;DR: Two three-dimensional structured networks are developed for solving linear equations and the Lyapunov equation and the training algorithms for the two networks are proved to converge exponentially fast to the correct solutions.
Abstract: Two three-dimensional structured networks are developed for solving linear equations and the Lyapunov equation. The basic idea of the structured network approaches is to first represent a given equation-solving problem by a 3-D structured network so that if the network matches a desired pattern array, the weights of the linear neurons give the solution to the problem: then, train the 3-D structured network to match the desired pattern array using some training algorithms; and finally, obtain the solution to the specific problem from the converged weights of the network. The training algorithms for the two 3-D structured networks are proved to converge exponentially fast to the correct solutions. Simulations were performed to show the detailed convergence behaviors of the 3-D structured networks. >
TL;DR: In this article, several new characterizations of Lyapunov diagonal stability are presented, and one of the characterizations is used to derive a simple necessary and sufficient condition for the stability of a 3 × 3 matrix.
TL;DR: An algorithm for the numerical solution of the real discrete-time non-negative definite Lyapunov equation is discussed in this paper, where an improved rank 2 updating formula is proposed for the Cholesky factor that arises during numerical solution.
TL;DR: In this paper, small random perturbations of matrix cocycles over Lipschitz homeomorphisms of compact metric spaces are considered, and the results are applicable to dynamical systems, particularly to volume-preserving diffeomorphisms.
Abstract: We consider small random perturbations of matrix cocycles over Lipschitz homeomorphisms of compact metric spaces. Lyapunov exponents are shown to be stable provided that our perturbations satisfy certain regularity conditions. These results are applicable to dynamical systems, particularly to volume-preserving diffeomorphisms.
11 Dec 1991
TL;DR: In this article, Grobner bases are used when choosing parameters in Lyapunov functions for nonlinear dynamic systems in an optimal way, and the method requires the nonlinearities of the system and the LyAPunov function to be of a polynomial type.
Abstract: The author shows how Grobner bases can be used when choosing parameters in Lyapunov functions for nonlinear dynamic systems in an optimal way. The method requires the nonlinearities of the system and the Lyapunov function to be of a polynomial type. Some concrete examples of how to apply the method are provided. >
TL;DR: The almost-sure asymptotic stability of a class of two-degree-of-freedom linear systems subjected to wide-band random parametric excitation of small intensity is investigated in this paper.
07 Apr 1991
TL;DR: In this paper, it is shown that B.C. Moore's balancing model (1981) can be extended to unstable systems by utilizing two weighted Grammian matrices and the J omega axis can be shifted to handle unstable systems.
Abstract: It is shown that B.C. Moore's balancing model (1981) can be extended to unstable systems by utilizing two weighted Grammian matrices. It is proposed that the J omega axis can be shifted to handle unstable systems. The balanced approach involves the solution of two modified Lyapunov equations. Two numerical examples are given to illustrate the results. >
TL;DR: The concept of Lyapunov stability instead or the M-matrix or quasi-diagonal dominance techniques are used to treat the robust stability problem for large-scale time-delay systems and the allowable bounds of structured and unstructured uncertainties are derived.
Abstract: We use the concept of Lyapunov stability instead or the M-matrix or quasi-diagonal dominance techniques to treat the robust stability problem for large-scale time-delay systems. The allowable bounds of structured and unstructured uncertainties which maintain the stability of the systems are derived. For the unstructured uncertain system, the robustness bounds are obtained directly from the system matrices even without solving the Lyapunov equation. Two main results are also applicable to the stability test and stabilization design.
01 Mar 1991
TL;DR: The use of Walsh functions to analyse singular bilinear systems is investigated in this paper, where it is shown that the nonlinear implicit differential system equation may be converted to a set of linear algebraic Lyapunov equations to be solved iteratively for the coefficients of the semistate x(t) in terms of the Walsh basis functions.
Abstract: The use of Walsh functions to analyse singular bilinear systems is investigated. It is shown that the nonlinear implicit differential system equation may be converted to a set of linear algebraic Lyapunov equations to be solved iteratively for the coefficients of the semistate x(t) in terms of the Walsh basis functions. Solution of the iterative algorithm is uniformly convergent to the exact solution of the algebraic generalised Lyapunov equation of the singular bilinear system. The present method is slightly more complicated than a similar one arising from the analysis of linear singular systems. In fact, it is a hybrid between the analyses of usual linear singular and bilinear regular systems.
TL;DR: This work first characterize the existence of such state evaluation functions through the properties on their difference functions, then applies the results for networks of McCulloch-Pitts type model neurons to see when there can be Lyapunov functions.
TL;DR: In this article, a method of Lyapunov functions on product spaces was developed for studying stability properties of equations with delay, where, except conceptually, the knowledge of solutions is not demanded.
TL;DR: In this article, lower bounds for the eigenvalues and certain sums and products of the Eigenvalues of the solution of the discrete Lyapunov matrix equation are presented, which are stronger than the majority of the relevant bounds shown in the literature.
Abstract: Some lower bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete Lyapunov matrix equation are presented These bounds are stronger than the majority of the relevant bounds shown in the literature They complete some known bounds such as the extremal eigenvalues, the determinant and the trace of the solution of the above equation
01 Nov 1991
TL;DR: In this paper, a simple method for solving the matrix Lyapunov equation for linear continuous systems with the system matrix in companion form is proposed, which does not involve solution of a linear system of equations or matrix inversion and hence is computationally efficient compared to other techniques.
Abstract: A simple method for solving the matrix Lyapunov equation for linear continuous systems with the system matrix in companion form is proposed. If the system is asymptotically stable, the solution of the Lyapunov equation can be obtained directly from the entries of a Routh table. The proposed method does not involve solution of a linear system of equations or matrix inversion and hence is computationally efficient compared to other techniques. It is illustrated by a numerical example.
TL;DR: In this paper, a reduced-order dynamic compensator design with stability robustness for linear discrete systems is presented, which is based on an unstructured perturbation stability bound for time varying perturbations.
Abstract: A reduced-order dynamic compensator design is presented with stability robustness for linear discrete systems, by including a stability robustness component in addition to the standard quadratic state and control terms in the performance criterion. The robustness component is based on an unstructured perturbation stability bound for time varying perturbations. The controller design is developed by the parameter optimization technique and involves the solution of five algebraic matrix equations, four of which are discrete-time Lyapunov matrix equations. >
11 Dec 1991
TL;DR: The author addresses the problem of computing a low rank estimate Y of the solution X of the Lyapunov equation without computing the matrix X itself, which has applications in both the reduced-order modeling and control of large dimensional systems.
Abstract: The author addresses the problem of computing a low rank estimate Y of the solution X of the Lyapunov equation without computing the matrix X itself. This problem has applications in both the reduced-order modeling and control of large dimensional systems as well as in a hybrid algorithm for the rapid numerical solution of the Lyapunov equation via the alternative direction implicit method. >
TL;DR: In this article, a Lyapunov characterization of scattering Schur, reactance Schur and immittance Schur properties of multidimensional polynomials is given.
Abstract: Lyapunov characterization of scattering Schur, reactance Schur, and immittance Schur properties of multidimensional polynomials are given. Since these polynomials arise in the description of passive systems, energy-like functions can be conveniently defined for such systems, thus making Lyapunov-type characterization readily available. It is shown that the notion of modal observability and its further variants for multidimensional systems are necessary ingredients of such considerations. Two approaches are followed. One is essentially passive synthesis based, whereas the other is that of viewing multidimensional systems as a parametric family of one-dimensional systems. A nontrivial example is included to illustrate the former method. >
Book•
01 Jan 1991
TL;DR: In this article, Lyapunov exponents and invariant measures for nonlinear stochastic differential equations have been derived for random dynamical systems on grassmannians, where Lyapounov functions have been shown to have almost sure exponential stability.
Abstract: Random dynamical systems.- Lyapunov exponents and asymptotic behaviour of the product of random matrices.- Lyapunov exponents of random dynamical systems on grassmannians.- Eigenvalue representation for the Lyapunov exponents of certain Markov processes.- Analytic dependence of Lyapunov exponents on transition probabilities.- A second order extension of Oseledets theorem.- The upper Lyapunov exponent of Sl(2,R) cocycles: Discontinuity and the problem of positivity.- Linear skew-product flows and semigroups of weighted composition operators.- Filtre de Kalman Bucy et exposants de Lyapounov.- Invariant measures for nonlinear stochastic differential equations.- How to construct stochastic center manifolds on the level of vector fields.- Additive noise turns a hyperbolic fixed point into a stationary solution.- Lyapunov functions and almost sure exponential stability.- Large deviations for random expanding maps.- Multiplicative ergodic theorems in infinite dimensions.- Stochastic flow and lyapunov exponents for abstract stochastic PDEs of parabolic type.- The Lyapunov exponent for products of infinite-dimensional random matrices.- Lyapunov exponents and complexity for interval maps.- An inequality for the Ljapunov exponent of an ergodic invariant measure for a piecewise monotonic map of the interval.- Generalisation du theoreme de Pesin pour l'?-entropie.- Systems of classical interacting particles with nonvanishing Lyapunov exponents.- Lyapunov exponents from time series.- Lyapunov exponents in stochastic structural dynamics.- Stochastic approach to small disturbance stability in power systems.- Lyapunov exponents and invariant measures of equilibria and limit cycles.- Sample stability of multi-degree-of-freedom systems.- Lyapunov exponents of control flows.
01 Jan 1991