Showing papers on "Lyapunov function published in 1986"
TL;DR: The fundamental idea behind the algorithm presented involves constructing an upper bound for the Lyapunov derivative corresponding to the closed loop system, a quadratic form, which can be found by solving a certain matrix Riccati equation.
825 citations
TL;DR: In this paper, the stability of solitons is reviewed for nonlinear conservative media and the main attention is paid to the description of the methods: perturbation theory, inverse scattering transform, Lyapunov method.
399 citations
01 Dec 1986
TL;DR: A calculus for computing Filippov's differential inclusion which simplifies the analysis of dynamical systems described by differential equations with discontinuous right-hand-side and a variable structure control law for rigid-link robot manipulators is described and its stability is proved.
Abstract: This paper develops a calculus for computing Filippov's differential inclusion which simplifies the analysis of dynamical systems described by differential equations with 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.
381 citations
Book•
01 Jan 1986
TL;DR: In this article, Liapunov's characterization of stable matrices is presented, and an algorithm for computing An is presented for linear systems x' = Ax + f(n).
Abstract: 1. Introduction.- 2. Liapunov's direct method.- 3. Linear systems x' = Ax..- 4. An algorithm for computing An..- 5. A characterization of stable matrices. Computational criteria..- 6. Liapunov's characterization of stable matrices. A Liapunov function for x' = Ax..- 7. Stability by the linear approximation..- 8. The general solution of x' = Ax. The Jordan Canonical Form..- 9. Higher order equations. The general solution of ?(z)y = 0..- 10. Companion matrices. The equivalence of x' = Ax and ?(z)y = 0..- 11. Another algorithm for computing An..- 12. Nonhomogeneous linear systems x' = Ax + f(n). Variation of parameters and undetermined coefficients..- 13. Forced oscillations..- 14. Systems of higher order equations P(z)y = 0. The equivalence of polynomial matrices..- 15. The control of linear systems. Controllability..- 16. Stabilization by linear feedback. Pole assignment..- 17. Minimum energy control. Minimal time-energy feedback control..- 18. Observability. Observers. State estimation. Stabilization by dynamic feedback..- References.
368 citations
Book•
01 Dec 1986
TL;DR: The existence of invariant manifolds for smooth maps with singularities has been established for plane billiards with singularity in this article, where the estimation of entropy from below through Lyapunov characteristic exponents has been shown to be invariant.
Abstract: Existence of invariant manifolds for smooth maps with singularities.- Absolute continuity.- The estimation of entropy from below through Lyapunov characteristic exponents.- The estimation of entorpy from above through Lyapunov characteristic numbers.- Plane billiards as smooth dynamical systems with singularities.
313 citations
TL;DR: The Nose oscillator is a borderline case, not sufficiently chaotic for a fully statistical description, and it is suggested that the behavior of only slightly more complicated systems is considerably simpler and in accord with statistical mechanics.
Abstract: Nos\'e has developed many-body equations of motion designed to reproduce Gibbs's canonical phase-space distribution. These equations of motion have a Hamiltonian basis and are accordingly time reversible and deterministic. They include thermodynamic temperature control through a single deterministic friction coefficient, which can be thought of as a control variable or as a memory function. We apply Nos\'e's ideas to a single classical one-dimensional harmonic oscillator. This relatively simple system exhibits both regular and chaotic dynamical trajectories, depending on the initial conditions. We explore here the nature of these solutions by estimating their fractal dimensionality and Lyapunov instability. The Nos\'e oscillator is a borderline case, not sufficiently chaotic for a fully statistical description. We suggest that the behavior of only slightly more complicated systems is considerably simpler and in accord with statistical mechanics.
268 citations
TL;DR: In this article, a very simple fourth-order electrical circuit was observed, for the first time, from a real physical system: a simple four-order circuit with only one nonlinear element, a three-segment piecewise-linear resistor.
Abstract: Hyperchaos has been observed, for the first time, from a real physical system: a very simple fourth-order electrical circuit. It is autonomous and reciprocal and has only one nonlinear element, a three-segment piecewise-linear resistor. Because of the circuit's simplicity, the laboratory measurements have been confirmed by digital computer simulations. The hyperchaotic nature is confirmed by the two positive Lyapunov exponents associated with the attractor, which is a fractal with a Lyapunov dimension between 3 and 4.
263 citations
TL;DR: The paper proves that the existence of a positive definite solution pair to the 2-D Lyapunov equation is not necessary for stability, disproving a long-standing conjecture.
Abstract: The stability of two-dimensional, linear, discrete systems is examined using the 2-D matrix Lyapunov equation. While the existence of a positive definite solution pair to the 2-D Lyapunov equation is sufficient for stability, the paper proves that such existence is not necessary for stability, disproving a long-standing conjecture.
215 citations
TL;DR: In this paper, the co-moving mutual information flow is introduced, which shows the selective transmission of the information at some speed, and the spatial structures of the vectors are investigated.
212 citations
TL;DR: It is shown that solvability of various output feedback design problems is implied by existence of a solution to a certain constrained Lyapunov problem (CLP), and the CLP can be stated in purely algebraic terms.
Abstract: Given a dynamical system whose description includes time-varying uncertain parameters, it is often desirable to design an output feedback controller leading to asymptotic 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 implied by 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.
154 citations
TL;DR: In this paper, a state transformation is employed to improve the upper bounds on the linear time-varying perturbation of an asymptotically stable linear time invariant system for robust stability.
Abstract: This note addresses the issue of "conservatism" in the time domain stability robustness bounds obtained by the Lyapunov approach. A state transformation is employed to improve the upper bounds on the linear time-varying perturbation of an asymptotically stable linear time-invariant system for robust stability. This improvement is due to the variance of the conservatism of the Lyapunov approach with respect to the basis of the vector space in which the Lyapunov function is constructed. Improved bounds are obtained, using a transformation, on elemental and vector norms of perturbations (i.e., structured perturbations) as well as on a matrix norm of perturbations (i.e., unstructured perturbations). For the case of a diagonal transformation, an algorithm is proposed to find the "optimal" transformation. Several examples are presented to illustrate the proposed analysis.
TL;DR: In this article, the authors consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set Λ of arbitrary shape and show that discretized versions of the system involving one-step numerical methods have nearby attracting sets Λ(h), which are also uniformly asymptotically stable.
Abstract: We consider a dynamical system described by a system of ordinary differential equations which possesses a compact attracting set Λ of arbitrary shape. Under the assumption of uniform asymptotic stability of Λ in the sense of Lyapunov, we show that discretized versions of the dynamical system involving one-step numerical methods have nearby attracting sets Λ(h), which are also uniformly asymptotically stable. Our proof uses the properties of a Lyapunov function which characterizes the stability of Λ.
TL;DR: A graph-theoretic method of extreme simplicity is proposed for decomposition and clustering of complex systems into interconnected subsystems with prescribed interaction strength, suitable for stability analysis via vector Lyapunov functions and decentralized control, and estimation of large-scale systems.
TL;DR: In this article, it was observed that the Lyapunov instability is a ubiquitous feature of interesting dynamical systems and whether or not it affects other properties of such systems in any important way is a question.
Abstract: The basic concepts-mass, force, and acceleration-were first correctly interrelated by Newton. Three hundred years ago he explained the regular motion of the planets about the Sun on the basis of pairwise-additive gravitational forces. The possibility of solving Newton's equations of motion for N-body systems of atoms or molecules had to wait unti11953, when suitable computers had become available at Los Alamos. On a human time scale, the night sky has a relatively stable appearance. It is amusing that, from a mathematical viewpoint, gravitational N-body systems are much less "stable" than the molecular systems studied by molecular dynamics. Mathematical stability can be monitored by observ ing the separation between two neighboring trajectories. In a stable case, the separation grows linearly with time. In the typical unstable case, this distance increases exponentially with time. This "Lyapunov" instability is a ubiquitous feature of interesting dynamical systems. Whether or not it affects other properties of such systems in any important way is a
18 Jun 1986
TL;DR: In this paper, the problem of obtaining stabilizing memoryless state feedback controllers for a class of uncertain systems described by difference equations (discrete-time systems) is considered, and controllers are presented which yield the desired behavior.
Abstract: We consider the problem of obtaining stabilizing memoryless state feedback controllers for a class of uncertain systems described by difference equations (discrete-time systems). Based on Lyapunov functions, controllers are presented which yield the desired behavior.
TL;DR: In this article, a high-dimensional chaotic attractor in an infinite-dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of highdimensional attractors in the phase space.
Abstract: The nature of a very high-dimensional chaotic attractor in an infinite-dimensional phase space is examined for the purpose of studying the relationships between the physical processes occurring in the real space and the characteristics of high-dimensional attractor in the phase space. We introduce two complementary bases from which the attractor is observed, one the Lyapunov basis composed of the Lyapunov vectors and the another the Fourier basis composed of the Fourier modes. We introduce the “exterior” subspaces on the basis of the Lyapunov vectors and observe the chaotic motion projected onto these exteriors. It is shown that a certain statistical property of the projected motion changes markedly as the exterior subspace “goes out” of the attractor. The origin of such a phenomenon is attributed to more fundamental features of our attractor, which become manifest when the attractor is observed from the Lyapunov basis. A counterpart of the phenomenon can be observed also on the Fourier basis because there is a statistical one-to-one correspondence between the Lyapunov vectors and the Fourier modes. In particular, a statistical property of the high-pass filtered time series reflects clearly the difference between the interior and the exterior of the attractor.
TL;DR: In this paper, the eigenvalues and sums and products of the Eigenvalues of the solution of the discrete Riccati and Lyapunov matrix equations and the continuous Lyapinov matrix equation were studied.
Abstract: We present some bounds for the eigenvalues and certain sums and products of the eigenvalues of the solution of the discrete Riccati and Lyapunov matrix equations and the continuous Lyapunov matrix equation. Nearly all of our bounds for the discrete Riccati equation are new. The bounds for the discrete and continuous Lyapunov equations give a completion of some known bounds for the extremal eigenvalues and the determinant and the trace of the solution of the respective equation.
TL;DR: This paper concentrates on the scenario above and in addition, the analysis permits the controller to be nonlinear, which amounts to an extension of the classical separation theorem to the case when the controller is nonlinear.
TL;DR: In this article, the qualitative behavior of the models of cell production systems, in the form of systems of nonlinear delay differential equations, is investigated, and the stability of equations linearized around the equilibria is investigated.
TL;DR: In this paper, the capacity of the graphs of certain fractal functions was analyzed using the Lyapunov exponents of an associated dynamical system, and it was shown that this formula can also be obtained using the exponents in an associated dynamic system.
Abstract: We present a formula for the capacity of the graphs of certain fractal functions. We show that this formula can also be obtained using the Lyapunov exponents of an associated dynamical system.
TL;DR: In this paper, the authors generalize this result to apply directly to the distributed parameter system itself, and show that the classical separation principle for the finite-dimensional reduced-order controller case is dead.
Abstract: One of the major difficulties in designing implementable finite-dimensional controllers for distributed parameter systems is that such systems are inherently infinite dimensional while controller dimension is severely constrained by on-line computing capability. While some approaches to this problem initially seek a correspondingly infinite-dimensional control law whose finite-dimensional approximation may be of impractically high order, the usual engineering approach involves first approximating the distributed parameter system with a high-order discretized model followed by design ofa relatively low-order dynamic controller. Among the numerous approaches suggested for the latter step are model/controller reduction techniques used in conjunction with the standard LQG result. An alternative approach, developed in (36), relies upon the discovery in (31) that the necessary conditions for optimal fixed-order dynamic compensation can be transformed into a set of equations possessing remarkable structural coherence. The present paper generalizes this result to apply directly to the distributed parameter system itself. In contrast to the pair of operator Riccati equations for the "full-order" LQG case, the optimal finite-dimensional fixed-orderdynamic compensator is characterized byfour operator equations (two modified Riccati equations and two modified Lyapunov equations) coupled by an oblique projection whose rank is precisely equal to the order ofthe compensatorand which determines the optimal compensator gains. This "optimal projection" is obtained by a full-rank factorization of the product of the finite-rank nonnegative-definite Hilbert-space operators which satisfy the pair ofmodified Lyapunov equations. The coupling represents a graphic portrayal of the demise of the classical separation principle for the finite-dimensional reduced-order controller case. The results obtained apply to a semigroup formulation in Hilbert space and thus are applicable to control problems involving a broad range of specific partial and functional differential equations.
TL;DR: In this article, the effect of an automatic voltage regulator on power system stability is illustrated by numerical examples, showing some shifts of equipotential curves which are due to changes of voltage regulator gain.
Abstract: The stability of power systems including the effects of automatic voltage regulators is studied via Lyapunov's direct method. The multivariable Popov criterion developed by Moore and Anderson is employed in constructing a Lure-type function for a power system consisting of synchronous machines interconnected by a lossless transmission system. The Lyapunov function constructed for the system with only flux decay is regarded as a special case of the result obtained in this paper. The effect of an automatic voltage regulator on power system stability is illustrated by numerical examples, showing some shifts of equipotential curves which are due to changes of voltage regulator gain.
01 Jan 1986
TL;DR: In this article, an explicit solution to the algebraic Lyapunov matrix equation is obtained in terms of the controllability matrix of the pair of coefficient matrices, which enables us to determine the number of positive eigenvalues of the positive semidefinite solution through the covariance matrix.
Abstract: An explicit solution to the algebraic Lyapunov matrix equation is obtained in terms of the controllability matrix of the pair of coefficient matrices. This enables us to determine the number of positive eigenvalues of the positive semidefinite solution through the controllability matrix. Based on this explicit formula, upper and lower bounds for each eigenvalue of the solution are derived, which always give nontrivial estimates.
TL;DR: In this article, it is shown that the dynamics of the flexible spacecraft can be described by a coupled system of ordinary differential equations and partial differential equations (hybrid system), and simple feedback controls are suggested for stabilization of flexible spacecraft.
Abstract: We consider the problem of rigorous modelling of flexible spacecraft and their stabilization. It is shown that the dynamics of the flexible spacecraft can be described by a coupled system of ordinary differential equations and partial differential equations (hybrid system). Lyapunov's approach is used to prove the stabilizability of the system. Simple feedback controls are suggested for stabilization of flexible spacecraft.
TL;DR: In this article, a Lyapunov-based stability test and linear control system models with time delays are explored. The main result is a generalization of the result that if (A, B ) is a reachable pair of matrices, then square A is a stability matrix if and only if there is a positive-definite matrix K such that AK + KA' = - BB'.
Abstract: Some relationships between a Lyapunov based stability test and linear control system models with time delays are explored. The main result is a generalization of the result that if ( A, B ) is a reachable pair of matrices, then square A is a stability matrix if and only if there is a positive-definite matrix K such that AK + KA' = - BB' .
TL;DR: In this article, the Lyapunov and Riccati differential equations with periodically time-varying coefficients are considered under the assumption of detectability of the underlying periodic system and two inertia theorems are provided linking the inertia of the solution to the so-called monodromy matrix.
Abstract: The Lyapunov and Riccati differential equations with periodically time-varying coefficients are considered. Under the assumption of detectability of the underlying periodic system, two inertia theorems are provided linking the inertia of the solution to the one of the so-called monodromy matrix.
TL;DR: In this article, the Lyapunov dimension of the attractors was calculated for two cases in which the broadening of the power spectra around the dominant peaks occurs, and one of the exponents is positive.
Abstract: Numerical integration of the equations for the evolution of amplitudes of a harmonically forced simply supported elastic beam indicates that for certain values of the frequencies of excitation the response of the beam may be chaotically modulated. In an attempt to confirm the presence of chaos, the Lyapunov exponents have been calculated for two cases in which the broadening of the power spectra around the dominant peaks occurs. In both the cases one of the Lyapunov exponents is positive; this provides further evidence of chaos. The Lyapunov dimension of the attractors in these cases is estimated to be 2.09 and 2.11, respectively.
TL;DR: In this paper, a stability test for large-scale systems with delays by employing both the aggregation technique and solutions of the complex Lyapunov equation is proposed, and an example is given to illustrate the application of the results.
Abstract: A stability test is proposed for large-scale systems with delays by employing both the aggregation technique and solutions of the complex Lyapunov equation. An example is given to illustrate the application of the results
TL;DR: In this paper, the direct stability analysis of a power system in which both active and reactive power flows are considered is considered, and flux decay action is included in the generator model, and the load is modelled with complex load power as a function of both bus voltage magnitude and frequency.