Showing papers on "Lyapunov equation published in 1992"

Adaptive nonlinear regulation: estimation from the Lyapunov equation

Abstract: A stabilizing adaptive controller for a nonlinear system depending affinely on some unknown parameters is presented. It is assumed that this system is feedback stabilizable. A key feature of the method is the use of the Lyapunov equation to design the adaptive law. A result on local stability, two different conditions for global stability, and a local result where the initial conditions of the state of the system only are restricted are given. >

Controller design with regional pole constraints

Abstract: A design procedure is developed that combines linear-quadratic optimal control with regional pole placement. Specifically, a static and dynamic output-feedback control problem is addressed in which the poles of the closed-loop system are constrained to lie in specified regions of the complex plane. These regional pole constraints are embedded within the optimization process by replacing the covariance Lyapunov equation by a modified Lyapunov equation whose solution, in certain cases, leads to an upper bound on the quadratic cost functional. The results include necessary and sufficient conditions for characterizing static output-feedback controllers with bounded performance and regional pole constraints. Sufficient conditions are also presented for the fixed-order (i.e. full- and reduced-order) dynamic output-feedback problem with regional pole constraints. Circular, elliptical, vertical strip, parabolic, and section regions are considered. >

Local Lyapunov exponents computed from observed data

Abstract: We develop methods for determining local Lyapunov exponents from observations of a scalar data set. Using average mutual information and the method of false neighbors, we reconstruct a multivariate time series, and then use local polynomial neighborhood-to-neighborhood maps to determine the phase space partial derivatives required to compute Lyapunov exponents. In several examples we demonstrate that the methods allow one to accurately reproduce results determined when the dynamics is known beforehand. We present a new recursive QR decomposition method for finding the eigenvalues of products of matrices when that product is severely ill conditioned, and we give an argument to show that local Lyapunov exponents are ambiguous up to order 1/L in the number of steps due to the choice of coordinate system. Local Lyapunov exponents are the critical element in determining the practical predictability of a chaotic system, so the results here will be of some general use.

Vector norms as Lyapunov functions for linear systems

Abstract: A unified theory of quadratic and piecewise-linear Lyapunov functions for continuous and discrete-time linear systems is presented. The key to this work is the description of these Lyapunov functions by vector norms. The main results are sufficient and necessary conditions for a vector norm to be a Lyapunov function as well as a method (based on these conditions) of constructing such Lyapunov functions. >

Identification of True and Spurious Lyapunov Exponents from Time Series

Abstract: A new method for the identification of true and spurious Lyapunov exponents computed from time series is presented. It is based on the observation that the true Lyapunov exponents change their signs upon time reversal whereas the spurious exponents do not. Furthermore by comparison of the spectra of the original data and the reversed time series suitable values for the free parameters of the algorithm used for computing the Lyapunov exponents (e.g., the number of nearest neighbors) are determined. As an example for this general approach an algorithm using local nonlinear approximations of the flow map in embedding space by radial basis functions is presented. For noisy data a regularization method is applied in order to get smooth approximating functions. Numerical examples based on data from the Henon map, a four-dimensional analog of the Henon map, a quasiperiodic time series, the Lorenz model, and Duffing’s equation are given.

Learning and convergence analysis of neural-type structured networks

Abstract: A class of feedforward neural networks, structured networks, has recently been introduced as a method for solving matrix algebra problems in an inherently parallel formulation. A convergence analysis for the training of structured networks is presented. Since the learning techniques used in structured networks are also employed in the training of neural networks, the issue of convergence is discussed not only from a numerical algebra perspective but also as a means of deriving insight into connectionist learning. Bounds on the learning rate are developed under which exponential convergence of the weights to their correct values is proved for a class of matrix algebra problems that includes linear equation solving, matrix inversion, and Lyapunov equation solving. For a special class of problems, the orthogonalized back-propagation algorithm, an optimal recursive update law for minimizing a least-squares cost functional, is introduced. It guarantees exact convergence in one epoch. Several learning issues are investigated. >

Lyapunov exponents for one-dimensional cellular automata

Abstract: In the paper we give a mathematical definition of the left and right Lyapunov exponents for a one-dimensional cellular automaton (CA). We establish an inequality between the Lyapunov exponents and entropies (spatial and temporal).

Lyapunov exponents as a nonparametric diagnostic for stability analysis

Abstract: SUMMARY The common observation made in the empirical nonlinear dynamics literature is the constraints imposed by the availability of a limited number of observations in the implementation of the existing algorithms of Lyapunov exponents. The algorithm discussed here can estimate all n Lyapunov exponents of an unknown n-dimensional dynamical system accurately with limited number of observations. This makes the algorithm attractive for applications to economic as well as financial time-series data. The implementation of the algorithm is carried out by multilayer feedforward networks which are capable of approximating any function and its derivatives to any degree of accuracy.

Domains of analytic continuation for the top Lyapunov exponent

Abstract: For a product of random positive matrices with Markovian dependence, we show the top Lyapunov exponent depends realanalytically on the transition probabilities (under an ergodicity assumption) and determine explicit domains of analytic continuation.

Two-dimensional digital filters without overflow oscillations and instability due to finite word length

Abstract: New criteria for the absence of finite word-length effects, such as overflow oscillations and instability, in two-dimensional digital filters are presented. The criteria are formulated using the state-space representations and are based on results concerning the 2D Lyapunov equation. Several examples illustrate the theoretical results. >

Adaptive regulation: Lyapunov design with a growth condition

Abstract: We propose a new Lyapunov design of an adaptive regulator under some restriction on the dependence of a Lyapunov function on the parameters. This restriction has been introduced by Praly et al. Its interest is to involve only a Lyapunov function and not explicitly the system non-linearities. We show it is satisfied by strict pure feedback systems with polynomial growth non-linearities and some other non-feedback linearizable systems. Our new Lyapunov design leads to an adaptive regulator where the adapted parameter vector is transformed before being used in the control law; namely, the so-called certainty equivalence principle is not applied. Unfortunately, the implementation of this regulator needs the explicit solution of a fixed point problem, so in a second stage we propose a more practical solution obtained by replacing the fixed point static equation by a dynamical system with this fixed point as equilibrium.

Lower summation bounds for the discrete Riccati and Lyapunov equations

Abstract: Lower eigenvalue summation (including trace) bounds for the solution of the discrete algebraic Riccati and Lyapunov matrix equations are presented. These are tighter than, or supplement, existing results. >

Solution to the general mixed H 2 /H ∞ control problem - necessary conditions for optimality

Abstract: A synthesis method is developed to design an output feedback controller to minimize the two-norm of one transfer function while ensuring the infinity-norm of another is held below a chosen level. This is known as the general mixed H2/H∞ optimization problem. The solution minimizes the actual two-norm, rather than an upper bound to it, and therefore is not conservative. Seven coupled nonlinear matrix equations are derived which represent the necessary conditions such a solution must satisfy. By analyzing the case where the desired controller has the same order as the plant, it is found that the solution lies on the boundary of the infinity-norm constraint whenever the two objectives are competing. In this case, the mixed controller requires the neutrally stabilizing solution to a Riccati equation and solution of a Lyapunov equation which has no unique solution.

Entropy and the Hausdorff Dimension for Infinite-Dimensional Dynamical Systems

Abstract: We define a sequence of uniform Lyapunov exponents in the setting of Banach spaces and prove that the Hausdorff dimension of global attractors is bounded from above by the Lyapunov dimension of the tangent map. This result generalizes the papers by Douady and Oesterle (1980) and Ledrappier (1981) in finite dimension and Constantinet al. (1985) for Hilbert spaces.

Control design for robust eigenstructure assignment in linear uncertain systems

Abstract: A generalized eigenstructure assignment procedure for designing a controller that has the best eigenstructure achievable while simultaneously maintaining stability robustness to time-varying parametric variations is presented. The approach taken is the constrained minimization of the difference between the actual and desired eigenstructure. The minimization is made subject to the constraints of the eigenstructure equation and the closed-loop Lyapunov equation. The capability of the formulation is illustrated in the design of a mode-decoupling roll-yaw autopilot for a generic, non-axisymmetrical airframe. >

Upper summation and product bounds for solution eigenvalues of the Lyapunov matrix equation

Abstract: Upper bounds for summations including the trace, and for products including the determinant, of the eigenvalues of the solution of the continuous algebraic Lyapunov matrix equation are presented. The majority of the bounds are tighter than those in the literature, and some are new. >

On the solution of the continuous-time Lyapunov matrix equation in two canonical forms

Abstract: A novel algorithm is presented in this paper for the solution to the continuous-time Lyapunov matrix equation in controllable and/or observable canonical forms by using the dimension-reduced method. The solution evaluation by this method is more accurate and considerably faster than any previously published methods. Numerical examples are given with systems in stable minimal and in unstable non-minimal to demonstrate the computational procedure.

On the analysis of eigenvalue assignment robustness

Abstract: The robust eigenvalue assignment of systems subject to parameter perturbations is addressed. Based on some essential properties of induced norms and matrix measures, the authors derive some sufficient conditions which ensure the assignment of the system's eigenvalues in the specified region irrespective of the system perturbations. The robustness bounds for eigenvalue assignment are obtained without the need to solve the Lyapunov equation. An example is given to illustrate the effectiveness and ease of the proposed analysis methods. >

Robust stabilization of time-varying discrete interval systems

Abstract: A sufficient condition given by P.H. Bauer and K. Premaratne (1990) for the robust stability of time-varying discrete interval systems is rederived using a diagonal Lyapunov function. This allows extension of the stability condition to a class of nonlinear time-varying systems. When the result is applied to the robust stabilization of time-varying discrete interval systems, it permits the relaxation of a nonnegativity constraint required by the time-invariant result reported by B. Shafai and C.V. Hollot (1991); greater flexibility in the choice of feedback gains is therefore allowed. A natural analog of D-stability called Hadamard stability is introduced and characterized. >

Lyapunov Exponent of SU(3) Gauge Theory

Abstract: The classical SU(3) gauge theory is shown to be deterministic chaotic. Its largest Lyapunov exponent is dertermined, from which a short time scale of thermalization of a pure gluon system is estimated. The connection to gluon damping rate is discussed.

Zeta function for the Lyapunov exponent of a product of random matrices.

Abstract: A cycle expansion for the Lyapunov exponent of a product of random matrices is derived. The formula is nonperturbative and numerically effective, which allows the Lyapunov exponent to be computed to high accuracy. In particular, the free energy and heat capacity are computed for the one-dimensional Ising model with quenched disorder. The formula is derived by using a Bernoulli dynamical system to mimic the randomness.