Showing papers on "Lyapunov function published in 1983"

Absolute stability of global pattern formation and parallel memory storage by competitive neural networks

Abstract: Systems that are competitive and possess symmetric interactions admit a global Lyapunov function. However, a global Lyapunov function whose equilibrium set can be effectively analyzed has not yet been discovered. It remains an open question whether the Lyapunov function approach, which requires a study of equilibrium points, or an alternative global approach, such as the Lyapunov functional approach, which sidesteps a direct study of equilibrium points will ultimately handle all of the physically important cases.

Stabilization with relaxed controls

Abstract: cannot in general be stabilized using a continuous closed loop control U(X), even if each state separately can be driven asymptotically to the origin. (An example is analyzed in Section 2.) In this paper we examine the possibility of stabilizing such systems with a continuous closed loop relaxed control. We find, indeed, that the family of systems stabilizable with relaxed controls is larger than the family of those stabilizable with ordinary controls. An even larger class is obtained if the continuity of the closed loop at the origin is not required. The latter class includes all one dimensional systems for which states can be driven asymptotically to the origin. This result does not hold in two dimensional systems and we provide a counter-example. It should be pointed that relaxed control-type stabilization is used both in theory and in practice; the method is known as dither. We shall comment on the similarities. Lyapunov functions for the system (1) help us in the construction of the continuous closed loop stabilizers. In fact, we find that the existence of a smooth Lyapunov function is equivalent to the existence of a stabilizing closed loop which is continuous except possibly at the origin; an additional condition on the Lyapunov function implies the continuity at the origin as well. We present these results in Section 4, after a brief introduction of closed loop relaxed controls, notations and terminology in Section 3. Prior to that, in Section 2, we discuss an example illustrating the power of relaxed controls. In the particular case of systems linear in the controls, relaxed controls can be replaced by ordinary controls, this is discussed in Section 5. The role of Lyapunov functions in the stability and stabilization theories is of course well known. Examples of systems with Lyapunov functions are available in the literature. We display some in Section 6, along with general comments on the construction, applications and counterexamples, including one which cannot be continuously stabilized, yet possesses a nonsmooth Lyapunov function. Closed loop stabilization with ordinary controls is analyzed extensively in the literature, see Sontag [8], Sussmann [ll] and references therein. Lyapunov functions techniques in stabil-

A Lyapunov-Like Characterization of Asymptotic Controllability

Abstract: It is shown that a control system in ${\bf R}^n $ is asymptotically controllable to the origin if and only if there exists a positive definite continuous functional of the states whose derivative can be made negative by appropriate choices of controls.

Attractors of partial differential evolution equations and estimates of their dimension

Abstract: CONTENTSIntroduction § 1. Maximal attractors of semigroups generated by evolution equations § 2. Examples of parabolic equations and systems having a maximal attractor § 3. The Hausdorff dimension of invariant sets § 4. Estimate of the change in volume under the action of shift operators generated by linear evolution equations § 5. An upper bound for the Hausdorff dimension of attractors of semigroups corresponding to evolution equations § 6. A lower bound for the dimension of an attractor § 7. Differentiability of shift operators § 8. Estimates of the Hausdorff dimension of an attractor of a two-dimensional Navier-Stokes system § 9. Upper and lower bounds for the Hausdorff dimension of attractors of parabolic equations and parabolic systems § 10. Attractors of semigroups having a global Lyapunov function § 11. Regular attractors of semigroups having a Lyapunov functionReferences

Qualitative Theory of Stochastic Systems

Abstract: Publisher Summary This chapter presents the problems and results of the qualitative theory of stochastic dynamical systems. Qualitative theory studies the general nature of a solution in the entire time interval. Attention has been attracted mainly by the white noise case, in particular, by nondegenerate diffusion processes. The methods used were mainly Lyapunov function techniques and the interrelation between Markov processes and deterministic partial differential equations. White noise is a generalized Gaussian stationary process with constant spectral density on the whole real line. Qualitative theory studies the general nature of a solution on the entire time interval (asymptotic or long-term behavior) without solving the equation.

A discrete analogue of the inequality of Lyapunov

Abstract: and the inequality is sharp. In view of the obvious similarity between the equations (1) and (2), we expect to fifind discrete analogoues of (4) which are necessary for the existence of a non-trivial solution of (1) satisfying certain boundary conditions. In the next section, we shall assume that p(k) is a non-negative function defined on the set \\{1, 2, \\cdots, N\\} and derive a condition which is necessary for (1) to have a non-trivial solution x(k) satisfying x(0)=0 and x(N+1)=0. Under the same assumption on p(k) , we then derive in the third section a more general condition which is necessary for the same equation to have a nontrivial solution x(k) satisfying x(0)+\\sigma x(1)=0 and x(N+1)+\\lambda x(N)=0 where \\sigma and \\lambda are non-negative real numbers. We could have omitted the next section entirely but include it here for constrasting the principles and computations involved. In the final section, we use a comparison theorem to deal with the case when p(k) can take on nonpositive values. 2. In the sequel, the smallest integer which is larger than or equal to the real number t will be denoted by t^{+} . Let A_{n}=(a(i,j)) be the n by n tridiagonal matrix defined by

Power system stability: Analysis by the direct method of Lyapunov

Abstract: gineering and Science: Algebra and Analysis. Engelewood Cliffs, NJ. Prentice Hall, 1981. F. Brauer and I . A. Nohel. Qualitative Theory of Ordinary Different ial Equations. New York: Benjamin, 1%9. E. A. Coddington and N. Levinson, Theory of Ordinary Differential Equations. New York: McCraw-Hill, 1955. M. W. Hirsch and 5. Smale, Differential Equations, Dynamical Systems a n d Linear Algebra. New York: Academic Press, 1974. R. K . Miller and A. N. Michel, Ordinary Differential Equations.

Realization algorithms and approximation methods of bilinear systems

Abstract: High-dimensional mathematical models of bilinear control systems are often not amenable due to the difficulty in implementation. In this paper, we address the problem of order-reduction for both discrete and continuous time bilinear systems. Two model-reduction algorithms are presented; one is based on the singular value decomposition of the generalized Hankel matrix (the Hankel Approach) and the other is based on the eigenvalue / eigenvector decomposition of the product of reachability and observability Gramians (the Gramian Approach). Equivalence between these two algorithms is established. The main result of this paper is a systematic approach for obtaining reduced-order bilinear models. Furthermore, the bilinear reachabiiity and observability Gramians are shown to be obtainable from the solutions of generalized Lyapunov equations. Computer simulations of a neutron-kinetic system are presented to illustrate the effectiveness of the proposed model-reduction algorithms.

General Instability Results for Interconnected Systems

Abstract: General results giving conditions for an interconnected system to be input-output and/or Lyapunov unstable are considered. These results are derived in terms of the theory of dissipative systems. This enables a very simple formulation of the requirements for instability. In particular, the restrictions of linearity and unstable subsystems, that appear in previous results, are seen to be unnecessary. Consequently, the relationship between instability and stability conditions is made clearer. A wide variety of useful instability criteria can be easily obtained as special cases.

Explicit optimality conditions for fixed-order dynamic compensation

Abstract: We consider steady-state, linear-quadratic fixed-order dynamic compensation in the presence of disturbance and observation noise. First-order necessary conditions for the optimization problem are derived in a new and highly simplified form. These necessary conditions constitute a system of two modified Riccati equations and two modified Lyapunov equations coupled by a projection which plays an essential role in defining the geometric structure of the compensator. When the order of the compensator is equal to the dimension of the plant, the classical linear-quadratic-Gaussian results are immediately obtained.

Hamiltonian-type Lyapunov functions

Abstract: A flexible and systematic way for the construction of Lyapunov functions which lends itself to the use of nonlinear programming techniques is presented. The class of Lyapunov functions obtained is shown to be energy functions (Hamiltonians) for an equivalent system.

Simultaneous attainability of central lyapunov and bohl exponents for ode linear systems

Abstract: Millionkikov's Accessibility Theorem for the central Lyapunov exponent of a linear ODE system is extended to simultaneous attainability of both central Lyapunov and Bohl exponents. 1. Let

Lyapunov functions or adaptive systems

Abstract: A method for construction of Lyapunov functions for a class of adaptive systems is given. Stability in the sense of Lyapunov and exponential convergence are shown.

Prolongations and Lyapunov functions in control systems

Abstract: In contrast to the case of a single dynamical system, the asymptotic stability of orbits of control systems cannot be characterized in terms of suitably defined Lyapunov functions. It is shown that the existence of Lyapunov functions corresponds to a stronger type of asymptotic stability, which is defined by introducing higher prolongations.

Reduction of large-scale systems via generalized Gramians

Abstract: System analysis and control design of large-scale dynamic systems are important in many applications. However, due to high dimensionality of the system, practical implementation of the theoretic results in large-scale systems is a difficult and expensive task, even with the aid of modern computers. In this paper, a different approach is undertaken to overcome the computational difficulty which is often associated with the nearly singular large-scale mathematical models. Instead of using the standard dynamic equations to represent a large-scale system, mathematical models in the descriptor form (generalized dynamic equations) are used. It is shown that via numerically reliable algorithms, the generalized dynamic model can be balanced in the sense that the balanced model is equally controllable and observable. Moreover, the balancing transformation can be obtained by solving generalized Lyapunov equations for observability and controllability Gramians, Based upon the balanced model, reduced-order models which closely match the input-output behavior of the system can be derived. Computer simulations of a power machine system will be presented to illustrate the effectiveness of the model reduction algorithm.

Multivariable Adaptive Control

Abstract: This thesis treats the problem of direct adaptive control of linear multivariable systems. The parametrization problem of adaptive control is discussed extensively. A pole-placement problem and a model-matching problem are formulated and interpreted in terms of model reference control. The problem is solved via a discussion on system invariants of multivariable systems as presented by Pernebo. The attention is then directed towards problems of identification and two different estimation schemes are formulated. Parameter convergence is guaranteed provided some conditions on /a priori/ information are satisfied. The requested prior knowledge is formulated in terms of the non-invertible system for the suggested prediction error identification algorithm. The second parameter adjustment law is shown to converge when a certain approximant of the left structure matrix, /i.e./, the system invariant is known. This result relaxes a result by Elliott /et al./ where the interactor is required to be known.The important question of stability of adaptive systems is also treated. The major result is a method for construction of Lyapunov functions for a class of single-input systems. Stability in the sense of Lyapunov and exponential convergence are shown. (Less)

Computer generated Lyapunov functions for interconnected systems: Improved results with applications to power systems

Abstract: We establish new results to analyze the stability of interconnected systems and to obtain estimates for the domain of attraction of asymptotically stable equilibrium points in interconnected systems by making use of the constructive algorithm due to Brayton and Tong. These results are applied in the stability analysis of interconnected power systems. In doing so, we introduce a new Lyapunov function in the analysis of power systems.

Identification of stochastic time-varying systems

Abstract: The parameter identification of a class of time-varying stochastic systems is considered in this paper. Online schemes which track the time-varying parameters in real time are proposed. Conditions which guarantee either almost sure convergence of the estimation error to the null or bounded mean-square error are obtained. The analysis is based on stochastic Lyapunov functions. This allows the convergence conditions to be weakened, and makes investigation of the stability problem of the proposed identification schemes possible.

Real-time parameter identification in a class of distributed systems using Lyapunov design method Parti. Theory

Abstract: A method is presented for the real-time identification of parameters in distributed parameter systems governed by parabolic and hyperbolic partial differential equations. The method uses a finite clement technique to reduce the system's governing equation into a set of ordinary differential equations. A suitable performance index is then formed as a function of identification error. The Lyapunov design technique is used to develop proportional and integral type identification algorithms. The method can be used either on-line or off-line for identification purposes.

On the stability of a class of kinetic equations

Abstract: Stability properties of kinetic model equations (including discrete versions of the master equation and Boltzmann's equation) are derived by means of Lyapunov's direct method. The construction of suitable Lyapunov functions leads to results about the structural stability of the dynamical systems and makes it possible to compose more complicated systems from the given ones, preserving automatically some form of stability.

Some algorithms for parameter estimation in water resources systems

Abstract: This paper is concerned with the development of algorithms for the estimation of the unknown model parameters of some water resources systems. The proposed algorithms are of recursive form and therefore are suitable for real-time estimation purposes. Proportional and proportional plus integral type algorithms are developed by using Lyapunov's stability theorem. This way, asymptotic convergence of the algorithms is guaranteed in the space of unknown parameters. The finite-element method is used to discretize the spatial domain of the distributed system. The method developed does not require prior knowledge of the unknown parameters and the system initial conditions. The numerical examples presented include the convection-diffusion equation and Burgers' equation.

Flows of stochastic dynamical systems : ergodic theory of stochastic flows

Abstract: In this thesis we present results and examples concerning the asymptotic (large time) behaviour of the flow of a nondegenerate smooth stochastic dynamical system on a smooth compact manifold. In Chapter 2 we prove a stochastic version of the Oseledec (Multiplicative Ergodic) Theorem for flows (theorem 2.1), in which we define the Lyapunov spectrum for the stochastic flow. Then we obtain stochastic analogies (Theorems 2.2.1, 2.2.2) of the Stable Manifold Theorems of Ruelle [16]. These theorems are proved by adapting Ruelle'S techniques to our situation. Also we discuss the implications of ‘Lyapunov stability', which we define to be the situation when the Lyapunov spectrum is strictly negative. In this situation the trajectories of the flow cluster in a certain way. (Proposition 2.3.3) In Chapter 3 we give some examples of systems for which we can calculate the Lyapunov spectrum. We can choose our parameters such that these systems are Lyapunov stable, and in this case we can calculate the flows and their asymptotic behaviour completely. In Chapter 4 we give a formula for the Lyapunov numbers which is analogous to that of Khas’ninskii [9] for a linear system. Then we use this formula to prove a theorem on the preservation of Lyapunov stability under a stochastic perturbation.