Showing papers on "Lyapunov function published in 1982"
TL;DR: The continuous-time Lyapunov equation has a unique Hermitian solution, X, if and only if Xt + X~j ^ 0 for all i and j (Barnett, 1975) as discussed by the authors.
Abstract: is called the continuous-time Lyapunov equation and is of interest in a number of areas of control theory such as optimal control and stability (Barnett, 1975; Barnett & Storey, 1968). The equation has a unique Hermitian solution, X, if and only if Xt + X~j ^ 0 for all i and j (Barnett, 1975). In particular if every Xt has a negative real part, so that A is stable, and if C is non-negative definite then X is also non-negative definite (Snyders & Zakai, 1970; Givens, 1961; Gantmacher, 1959). In this case, since X is nonnegative definite, it can be factorized as
406 citations
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01 Jan 1982
TL;DR: The Lyapunov's second method was used in this paper to prove stability and asymptotic equivalence of systems with impulsive behavior.Preliminaries.
Abstract: Preliminaries.- Existence and uniqueness.- Stability and asymptotic equivalence.- Impulsive systems.- Lyapunov's second method.
166 citations
TL;DR: In this paper, a concept called small signal L p -stability is defined, and its relationship to Lyapunov stability is discussed, and the relationship between the two concepts is discussed.
Abstract: In this note we define a concept called "small signal L p -stability," and show its relationship to Lyapunov stability.
87 citations
TL;DR: Two new algorithms to estimate the domain of attraction of the equilibriumx=0 of a nonlinear systemx=f(x) are presented and an interpretation of the comparison principle in terms of stability preserving maps is given.
Abstract: We present two new algorithms to estimate the domain of attraction of the equilibriumx=0 of a nonlinear systemx=f(x) One of these algorithms utilizes quadratic Lyapunov functions while the second algorithm makes use of norm Lyapunov functions Both of these procedures yield estimates for the domain of attraction which are comparable to those obtained by existing methods; however, the present algorithms appear to be significantly more efficient than existing algorithms
78 citations
TL;DR: A complete stability analysis of a new power system model is presented that facilitates a dynamic representation directly in terms of the network structure and the multivariable Popov criterion is used to obtain general Lure-Postnikov type Lyapunov functions which rigorously allow for the presence of real loads.
Abstract: This paper presents a more complete stability analysis of a new power system model which was presented in [1]. The essential feature of the model is the assumption of frequency dependent loads. This facilitates a dynamic representation directly in terms of the network structure. Consequently, concepts and results from circuit theory can play a strong role in the stability analysis of the model. The multivariable Popov criterion is used to obtain general Lure-Postnikov type Lyapunov functions which rigorously allow for the presence of (frequency dependent) real power loads. This has not been possible with the previously used model. Results are given for both local (dynamic) stability and for determination of regions of asymptotic (transient) stability.
64 citations
01 Jan 1982
TL;DR: In this article, a complete stability analysis of a new power system model is presented, where the essential feature of the model is the assumption of frequency dependent loads, which facilitates a dynamic representation directly in terms of the network structure.
Abstract: A complete stability analysis of a new power system model is presented. The essential feature of the model is the assumption of frequency dependent loads. This facilitates a dynamic representation directly in terms of the network structure. Consequently, concepts and results from circuit theory can play a strong role in the stability analysis of the model. The multivariable Popov criterion is used to obtain general Lure-Postnikov type Lyapunov functions which rigorously allow for the presence of real loads. This has not been possible with the previously used model.
56 citations
TL;DR: In this article, the minimax approach is used to study numerical selection of a Lyapunov function from a multiparameter class of functions and a numerical method is suggested for plotting an absolute stability region in the space of system gains for a control system with several nonstationary nonlinearities.
51 citations
TL;DR: In this paper, a Lyapunov-e direct method is used to find stabilizing state feedback laws for systems with uncertain state or control matrices, where the uncertainties are modelled as sector bounded non-linearities, and feedback laws ensuring global asymptotic closed loop stability are obtained using LyAPunov'e direct methods.
Abstract: Existence criteria and computational methods are presented for finding stabilizing state feedback laws for systems with uncertain state or control matrices. The uncertainties are modelled as sector bounded non-linearities, and feedback laws ensuring global asymptotic closed loop stability are obtained using Lyapunov'e direct method. The algorithms essentially amount to repeatedly solving a parameter dependent Riccati equation until the maximal solution becomes positive definite.
50 citations
TL;DR: In this paper, the analysis of complex dynamical systems (which may be of high dimension) is accomplished in terms of the qualitative properties of the free subsystems and the interconnections of such systems.
Abstract: In two recent papers, Brayton and Tong developed some significant results which are the basis of a constructive approach in the stability analysis of dynamical systems. Although their algorithm is powerful, it taxes the capabilities of most modern computers when the dimension of a system is high or even when the dimension is moderate in size. In this paper we remove these difficulties to a certain extent by generalizing the work of Brayton and Tong to interconnected dynamical systems. In our approach, the analysis of complex dynamical systems (which may be of high dimension) is accomplished in terms of the qualitative properties of the free subsystems and the interconnections of such systems.
39 citations
TL;DR: In this paper, the majorization result of Wimmer relating the eigenvalues of the matrices involved in the Lyapunov equation is extended to the algebraic Riccati equation.
Abstract: The majorization result of Wimmer [2] relating the eigenvalues of the matrices involved in the Lyapunov equation is extended to the algebraic Riccati equation. This result, coupled with certain results on the eigenvalue bounds for sum and product of matrices, yields several lower and upper bounds for the eigenvalues of the solution matrix of the algebraic Riccati and Lyapunov equations.
39 citations
01 Sep 1982
TL;DR: In this paper, a necessary and sufficient condition for a tridiagonal complex matrix A to be stable was given, which involves a positive semi-definite image under a Lyapunov map and real and imaginary parts of A. This condition is then used to characterize the real tridagonal matrices which are D-stable, and those which are totally D -stable.
Abstract: A new necessary and sufficient condition is given for an $n \times n$ complex matrix A to be stable. It involves a positive semi-definite image under a Lyapunov map and the real and imaginary parts of A. This condition is then used to characterize the real tridiagonal matrices which are D-stable, and those which are totally D-stable.
TL;DR: A stability test is proposed for large scale systems with delays by employing both the aggregation technique based on a Lyapunov function and the strictly quasi-diagonal dominance.
Abstract: A stability test is proposed for large scale systems with delays by employing both the aggregation technique based on a Lyapunov function and the strictly quasi-diagonal dominance.
TL;DR: In this paper, a general form for describing nonisothermal reactions in closed chemical systems in terms of the Marcelin-de-Donder kinetics and explicit forms of the Lyapunov functions for the systems treated under various conditions are suggested.
Abstract: A general form for the description of nonisothermal reactions in closed chemical systems in terms of the Marcelin-de-Donder kinetics and explicit forms of the Lyapunov functions for the systems treated under various conditions are suggested.
01 May 1982
TL;DR: In this paper, the Euler-Newton equations for a system of connected rigid bodies, written in a special state space form, provide a systematic method of arriving at the differential equations of the system, amenable to programming and symbolic algebraic manipulation.
Abstract: Euler-Newton's equations for a system of connected rigid bodies, written in a special state space form, provide a systematic method of arriving at the differential equations of the system This method is amenable to programming and symbolic algebraic manipulation The elimination of some or all forces of constraint is by projection, implementing the principle of virtual work, and is done by inner products The computation of these forces requires symbolic inversion of a matrix for which an iterative scheme is proposed here A method for construction of Lyapunov functions for stability of such systems in the vicinity of an arbitrary operating point is proposed This construction may be achieved by symbolic manipulations and supplements applications of the Euler-Newton method
TL;DR: A two-level hierarchical computational structure is developed to determine the decentralized gains for linear, interconnected dynamical systems and results show that this structure needs significantly less computational effort than previous approaches.
TL;DR: The Lyapunov equation appears in several areas of control theory such as stability theory, optimal control, stochastic control and in the solution of the algebraic Riccati equation using Newton's method as discussed by the authors.
TL;DR: In this article, the authors investigate the solutions of those autonomous systems with quadratic nonlinearities in a N-dimensional vector space together with the first variational equation systems by means of the Carleman embedding.
Abstract: We investigate the solutions of those autonomous systems with quadratic nonlinearities in a N‐dimensional vector space together with the solutions of their first variational equation systems by means of the Carleman embedding. An iterative procedure based on this result is developed to evaluate the Lyapunov exponents of the considered systems. We test the method by giving some results for the Lyapunov exponents of the Lorenz model.
14 Jun 1982
TL;DR: An overview of the theory of transformations from nonlinear systems to linear systems is presented in this paper, including necessary and sufficient conditions for transformations to exist, a method of constructing transformations, robustness in design based on transformation theory and Lyapunov functions.
Abstract: This paper is presenting an overview of the theory of transformations from nonlinear systems to linear systems Topics covered include (1) necessary and sufficient conditions for transformations to exist, (2) a method of constructing transformations, (3) robustness in design (based on transformation theory) and Lyapunov functions, (4) estimation theory, and (5) the relationship between transformation theory and "nonlinear zeroes" Application of these results to automatic flight control is presented in another paper at this session
TL;DR: In this paper, the main results obtained by the author in this direction in 1965 [6], [7] and the extensions and applications obtained partially with other authors since then and published in different journals [8]-[10] are surveyed.
Abstract: In a publication by Y. P. Harn and C. T. Chen [1] a proof of a discrete stability test by using the Lyapunov method was given. In this note the main results obtained by the author in this direction in 1965 [6], [7] and the extensions and applications obtained partially with other authors since then and published in different journals [8]-[10] are surveyed. The similarity to the results in [1] is obvious.
TL;DR: In this article, it was shown that the Lyapunov indices of a morphism belong to the second Baire class, but do not belong to any prior class. But the problem of unimprovability of this theorem was not solved.
Abstract: Then their properties were studied as functions on B. In particular, Theorem i, which shows that the functions ~(-),k = i ..... n belong to the second Baire class (see [4, 5]), was proved. Since, by definition, each Baire class is contained in the next one, there naturally arises the problem of unimprovability of this theorem, i.e., about the strict belongingness of the Lyapunov indices to the second Baire class (see [5]). For abstractly defined morphism (I), this problem cannot have a unique solution. Thus, e.g., if the base consists of a single point, then each function on it is of zero class. We can give examples of the morphisms (i) with discontinuous Lyapunov indices. However, the present problem will be solved in this case if we indicate objects for which the functions ~(.), k= I,..., n belong to the second, but do not belong to any preceeding class.
TL;DR: The proof of a previously published conjecture on the stability of linear discrete systems is obtained using Lyapunov's second stability theorem in this paper, which is based on the first stability theorem.
Abstract: The proof of a previously published conjecture on the stability of linear discrete systems is obtained using Lyapunov's second stability theorem.
09 Aug 1982
TL;DR: In this paper, a method based on the maximization of dissipation energy due to control of flexible structures is presented. But this method is based on a Lyapunov flow.
Abstract: A method is presented, w hich allows an integrated determination of actuator/sensor positions and feedback gains for control of flexible structures. T his method is based on the maximization of dissipation energy due to control a ction. The optimality criterion is determined via a Lyapunov e quation, and
TL;DR: In this paper, the Lyapunov number for a noisy 2 n -period doubling bifurcation cycle is derived from the derivatives of a deterministic one-dimensional map, and the scaling factor is refined to 6.6190.
Abstract: A stochastic one-dimensional map which produces a sequence of period doubling bifurcations is theoretically studied. We obtain analytic expressions, to a second-order approximation, of the local distribution function of fluctuating orbital points and the Lyapunov number for a noisy 2
n
cycle. The expressions satisfy scaling laws and well agree with the results of numerical experiments when the external noise is weak. A scaling factor for the noise level is formulated in terms of the derivatives of a deterministic map. From it, the scaling factor is refined to be 6.6190 .... The Lyapunov number shows that, when the external noise is weaker than some extent, the noisy orbit is more stable rather than the deterministic one.
TL;DR: In this paper, an approach based on Lyapunov's stability theory is proposed to study the electric arc near current zero by means of mathematical models, which allows a qualitative analysis of the nonlinear differential equations describing the phenomenon.
01 Dec 1982
TL;DR: In this paper, the authors introduced feedforward of the reference input to provide another degree of freedom in the overall system optimization of multivariable robust servomechanisms, and derived explicit formulas for the feedforward matrices in terms of Lyapunov functions.
Abstract: Feedforward of the reference input is introduced to provide another degree of freedom in the overall system optimization of multivariable robust servomechanisms. This provides the designer with substantial control over the shape of the transient response. Introduction of the feedforward matrices has no effect on the pole locations or robustness properties of the system and creates new transmission zeros, the location of which can be influenced by proper selection of feedforward matrices. Loss functions such as the integral square error can be used to determine the feedforward matrices. Explicit formulas are derived for the feedforward matrices in terms of Lyapunov functions.
TL;DR: A theorem due to Arnold about the nonlinear stability of steady plane curvilinear flows of an inviscid fluid is applied to a new class of loop and bend flows recently obtained by Shercliff as discussed by the authors.
Abstract: A theorem due to Arnold about the nonlinear stability of steady plane curvilinear flows of an inviscid fluid is applied to a new class of loop and bend flows recently obtained by Shercliff. The stability (in the Lyapunov sense) of some of these flows is thus proved.
14 Jun 1982
TL;DR: In this article, two parametrical optimization problems are defined in order to improve the stability conditions given by the scalar approach and the vectorial approach of the Lyapunov method for interconnected systems.
Abstract: Two parametrical optimization problems are defined in order to try to improve the stability conditions given by the scalar approach and the vectorial approach of the Lyapunov method for interconnected systems. An approximate solution is provided in the vectorial case, and discussed with respect to previous works. The paper ends on a critical discussion of the Lyapunov approach in the decentralized control problem.
01 Oct 1982
TL;DR: In this paper, a stability inequality in frequency domain which incorporates the slope information of the nonlinearity is derived for a class of nonlinear feedback systems using Lyapunov's direct method.
Abstract: Using Lyapunov's direct method, a stability inequality in frequency domain which incorporates the slope information of the nonlinearity is derived for a class of nonlinear feedback systems. The present Lyapunov function contains an integral involving the slope of the nonlinearity, i.e., unlike the conventional Lure-type Lyapunov function.
TL;DR: In this paper, the problem of stabilisation of interconnected linear systems is considered by means of scalar Lyapunov functions and aggregation, and a sufficient condition is given for an interconnected system to be stabilised using only local state feedback.
Abstract: The problem of stabilisation of interconnected linear systems is considered by means of scalar Lyapunov functions and aggregation. A sufficient condition is given for an interconnected system to be stabilised using only local state feedback. The aim is to provide the least restrictive condition possible, which is termed ‘optimal’, by properly choosing the state-variable weighting matrices for each subsystem.