Showing papers on "Lyapunov function published in 1984"

The Stability of a Dynamic Model of Traffic Assignment—An Application of a Method of Lyapunov

Abstract: This paper considers a dynamic model of traffic assignment in which drivers change their route choices to take advantage of cheaper routes. Using a method due to Lyapunov, we show that if the cost-flow function is monotone and there are no explicit capacity restrictions then any solution trajectory of our dynamical system converges to the set of Wardrop equilibria as time passes.

The optimal projection equations for fixed-order dynamic compensation

Abstract: First-order necessary conditions for quadratically optimal, steady-state,fixed-order dynamic compensation of a linear, time-invariant plant in the presence of disturbance and observation noise are derived in a new and highly simplified form. In contrast to the pair of matrix Riccati equations for the full-order LQG case, the optimal steady-state fixed-order dynamic compensator is characterized by four matrix equations (two modified Riccati equations and two modified Lyapunov equations) coupled by a projection whose rank is precisely equal to the order of the compensator and which determines the optimal compensator gains. The coupling represents a graphic portrayal of the demise of the classical separation principle for the reduced-order controller case.

A theorem on average Liapunov functions

Abstract: A theorem on average Liapunov functions for dynamical systems is generalized. As an illustration the result is used to establish a rather strong coexistence criterion for an ecological system.

The Lyapunov dimension of a nowhere differentiable attracting torus

Abstract: The fractal dimension of an attracting torus Tk in lR x T" is shown to be almost always equal to the Lyapunov dimension as predicted by a previous conjec­ ture. The cases studied here can have several Lyapunov nulJ!bers greater than I and several less than I

Intrinsic stochasticity with many degrees of freedom

Abstract: We study a simple model equation describing a system with an infinity of degrees of freedom which displays an intrinsically chaotic behavior. Some concepts of fully developed turbulence are discussed in relation to this model. We also develop an approach based on Lyapunov exponent measurements. Numerical results on the distribution of Lyapunov numbers and the power spectrum of the associated Lyapunov vectors are presented and briefly discussed.

Qualitative behaviour of stochastic delay equations with a bounded memory

Abstract: A dichotomy is proved concerning recurrence properties of the solution of certain stochastic delay equations. If the solution process is recurrent, there exists an invariant measure π on the state space C which is unique (up to a multiplicative constant) and the tail-field is trivial. If π happens to be a probability measure, then for every initial condition, the distribution of the process converges to it as t→∞. We will formulate a sufficient condition for the existence of an invariant probability measure (ipm) in icrnia of Lyapunov junctionals and give two examples, one Heing the stochastic-delay version of the famous logistic equation of population growth. Finally we study approximations of delay equations by Markov chains.

The Lyapunov Characteristic Exponents — Applications to Celestial Mechanics

Abstract: After a presentation of Lyapunov characteristic exponents (LCE) we recall their basic properties and numerical methods of computation. We review some numerical computations which are concerned with LCEs and mainly related to celestial mechanics problems.

On the existence of energy function for power systems with transmission losses

Abstract: In this paper we show that any energy (Lyapunov) function to analyze the stability of equilibrium points of systems with transmission losses would, in general, not be a continuous (with respect to transmission losses) deformation of the standard energy function used for a lossless system. This result implies that stability analysis using Lyapunov functions for systems with losses would require Lyapunov functions that are substantially different from the energy function used for lossless systems.

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 [1]. 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.

Transient Stability Analysis of Large-Scale Power System by Lyapunov's Direct Method

Abstract: In this paper, a method for reducing computation memory required by Lyapunov' s direct method has been presented. This requirement gets critical when the direct method is applied to transient stability studies of large-scale power systems containing more than 100 generators. Firstly, the system equations, the Lyapunov function, and the linear equations associated with the calculation of the stable equilibrium state, have been formulated in such way that the extreme sparcity of the network admittance matrix could be taken advantage of. Secondly, the programming techniques such as optimally ordered triangular factorization and efficient storage scheme of sparse matrices have been introduced. Lastly, the effectiveness of the method has been investigated with the 107-generator and 363-bus power system represented by the detail generator model including automatic voltage regulators ard power system stabilizers.

Application of singular perturbation techniques to power system transient stability analysis

Abstract: This work extends Lyapunov techniques for power system transient stability analysis to system models incorporating voltage magnitude variation, real and reactive power loads, and flux decay effects. A method of estimating the domain of attraction of an asymptotically stable equilibrium in the resulting system of differential equations with algebraic constraints is examined.

Method for calculating a Lyapunov exponent

Abstract: A new method is presented for calculating the leading Lyapunov exponent directly from experimental data for systems having a strange attractor with dimensionality near 2. The method is exact for one-dimensional maps and gives good results for systems that have approximate one-dimensional maps associated with them even in the presence of some noise. Numerical examples are given.

Discrete systems, analysis, control, and optimization

Abstract: 1 Discrete Models in Systems Engineering.- 1.1 Introduction.- 1.2 Some Illustrative Examples.- 1.2.1 Direct Digital Control of a Thermal Process.- 1.2.2 An Inventory Holding Problem.- 1.2.3 Measurement and Control of Liquid Level.- 1.2.4 An Aggregate National Econometric Model.- 1.3 Objectives and Outline of the Book.- 1.4 References.- 2 Representation of Discrete Control Systems.- 2.1 Introduction.- 2.2 Transfer Functions.- 2.2.1 Review of Z-Transforms.- 2.2.2 Effect of Pole Locations.- 2.2.3 Stability Analysis.- 2.2.4 Simplification by Continued-Fraction Expansions.- 2.2.5 Examples.- 2.3 Difference Equations.- 2.3.1 The Nature of Solutions.- 2.3.2 The Free Response.- 2.3.3 The Forced Response.- 2.3.4 Examples.- 2.3.5 Relationship to Transfer Functions.- 2.4. Discrete State Equations.- 2.4.1 Introduction.- 2.4.2 Obtaining the State Equations.- A. From Difference Equations.- B. From Transfer Functions.- 2.4.3 Solution Procedure.- 2.4.4 Examples.- 2.5 Modal Decomposition.- 2.5.1 Eigen-Structure.- 2.5.2 System Modes.- 2.5.3 Some Important Properties.- 2.5.4 Examples.- 2.6 Concluding Remarks.- 2.7 Problems.- 2.8 References.- 3 Structural Properties.- 3.1 Introduction.- 3.2 Controllability.- 3.2.1 Basic Definitions.- 3.2.2 Mode-Controllability Structure.- 3.2.3 Modal Analysis of State-Reachability.- 3.2.4 Some Geometrical Aspects.- 3.2.5 Examples.- 3.3 Observability.- 3.3.1 Basic Definitions.- 3.3.2 Principle of Duality.- 3.3.3 Mode-Observability Structure.- 3.3.4 Concept of Detectability.- 3.3.5 Examples.- 3.4. Stability.- 3.4.1 Introduction.- 3.4.2 Definitions of Stability.- 3.4.3 Linear System Stability.- 3.4.4 Lyapunov Analysis.- 3.4.5 Solution and Properties of the Lyapunov Equation.- 3.4.6 Examples.- 3.5 Remarks.- 3.6 Problems.- 3.7 References.- 4 Design of Feedback Systems.- 4.1 Introduction.- 4.2 The Concept of Linear Feedback.- 4.2.1 State Feedback.- 4.2.2 Output Feedback.- 4.2.3 Computational Algorithms.- 4.2.4 Eigen-Structure Assignment.- 4.2.5 Remarks.- 4.2.6 Example.- 4.3 Deadbeat Controllers.- 4.3.1 Preliminaries.- 4.3.2 The Multi-Input Deadbeat Controller.- 4.3.3 Basic Properties.- 4.3.4 Other Approaches.- 4.3.5 Examples.- 4.4 Development of Reduced-Order Models.- 4.4.1 Analysis.- 4.4.2 Two Simplification Schemes.- 4.4.3 Output Modelling Approach.- 4.4.4 Control Design.- 4.4.5 Examples.- 4.5 Control Systems with Slow and Fast Modes.- 4.5.1 Time-Separation Property.- 4.5.2 Fast and Slow Subsystems.- 4.5.3 A Frequency Domain Interpretation.- 4.5.4 Two-Stage Control Design.- 4.5.5 Examples.- 4.6 Concluding Remarks.- 4.7 Problems.- 4.8 References.- 5 Control of Systems with Inaccessible States.- 5.1 Introduction.- 5.2 State Reconstruction Schemes.- 5.2.1 Full-Order State Reconstructors.- 5.2.2 Reduced-Order State Reconstructors.- 5.2.3 Discussion.- 5.2.4 Deadbeat State Reconstructors.- 5.2.5 Examples.- 5.3 Observer-Based Controllers.- 5.3.1 Structure of Closed-Loop Systems.- 5.3.2 The Separation Principle.- 5.3.3 Deadbeat Type Controllers.- 5.3.4 Example.- 5.4 Two-Level Observation Structures.- 5.4.1 Full-Order Local State Reconstructors.- 5.4.2 Modifications to Ensure Overall Asymptotic Reconstruction.- 5.4.3 Examples.- 5.5 Discrete Two-Time-Scale Systems.- 5.5.1 Introduction.- 5.5.2 Two-Stage Observer Design.- 5.5.3 Dynamic State Feedback Control.- 5.5.4 Example.- 5.6 Concluding Remarks.- 5.7 Problems.- 5.8 References.- 6 State and Parameter Estimation.- 6.1 Introduction.- 6.2 Random Variables and Gauss-Markov Processes.- 6.2.1 Basic Concepts of Probability Theory.- 6.2.2 Mathematical Properties of Random Variables.- A. Distribution Functions.- B. Mathematical Expectation.- C. Two Random Variables.- 6.2.3 Stochastic Processes.- A. Definitions and Properties.- B. Gauss and Markov Processes.- 6.3 Linear Discrete Models with Random Inputs.- 6.3.1 Model Description.- 6.3.2 Some Useful Properties.- 6.3.3 Propagation of Means and Covariances.- 6.3.4 Examples.- 6.4 The Kalman Filter.- 6.4.1 The Estimation Problem.- A. The Filtering Problem.- B. The Smoothing Problem.- C. The Prediction Problem.- 6.4.2 Principal Methods of Obtaining Estimates.- A. Minimum Variance Estimate.- B. Maximum Likelihood Estimate.- C. Maximum A Posteriori Estimate.- 6.4.3 Development of the Kalman Filter Equations.- A. The Optimal Filtering Problem.- B. Solution Procedure.- C. Some Important Properties.- 6.4.4 Examples.- 6.5 Decentralised Computation of the Kalman Fikter.- 6.5.1 Linear Interconnected Dynamical Systems.- 6.5.2 The Basis of the Decentralised Filter Structure.- 6.5.3 The Recursive Equations of the Filter.- 6.5.4 A Computation Comparison.- 6.5.5 Example.- 6.6 Parameter Estimation.- 6.6.1 Least Squares Estimation.- A. Linear Static Models.- B. Standard Least Squares Method and Properties.- C. Application to Parameter Estimation of Dynamic Models.- D. Recursive Least Squares.- E. The Generalised Least Squares Method.- 6.6.2 Two-Level Computational Algorithms.- A. Linear Static Models.- B. A Two-Level Multiple Projection Algorithm.- C. The Recursive Version.- D. Linear Dynamical Models.- E. The Maximum A Posteriori Approach.- F. A Two-Level Structure.- 6.6.3 Examples.- 6.7 Problems.- 6.8 References.- 7 Adaptive Control Systems.- 7.1 Introduction.- 7.2 Basic Concepts of Model Reference Adaptive Systems.- 7.2.1 The Reference Model.- 7.2.2 The Adaptation Mechanism.- 7.2.3 Notations and Some Definitions.- 7.2.4 Design Considerations.- 7.3 Design Techniques.- 7.3.1 Techniques Based on Lyapunov Analysis.- 7.3.2 Techniques Based on Hyperstability and Positivity Concepts.- A. Popov Inequality and Related Results.- B. Systematic Procedure.- C. Parametric Adaptation Schemes.- D. Adaptive Model-Following Schemes.- 7.3.3 Examples.- 7.4 Self-Tuning Regulators.- 7.4.1 Introduction.- 7.4.2 Description of the System.- 7.4.3 Parameter Estimators.- A. The Least Squares Method.- B. The Extended Least Squares Method.- 7.4.4 Control Strategies.- A. Controllers Based on Linear Quadratic Theory.- B. Controllers Based on Minimum Variance Criteria.- 7.4.5 Other Approaches.- A. Pole/Zero Placement Approach.- B. Implicit Identification Approach.- C. State Space Approach.- D. Multivariable Approach.- 7.4.6 Discussion.- 7.4.7 Examples.- 7.5 Concluding Remarks.- 7.6 Problems.- 7.7 References.- 8 Dynamic Optimisation.- 8.1 Introduction.- 8.2 The Dynamic Optimisation Problem.- 8.2.1 Formulation of the Problem.- 8.2.2 Conditions of Optimality.- 8.2.3 The Optimal Return Function.- 8.3 Linear-Quadratic Discrete Regulators.- 8.3.1 Derivation of the Optimal Sequences.- 8.3.2 Steady-State Solution.- 8.3.3 Asymptotic Properties of Optimal Control.- 8.4 Numerical Algorithms for the Discrete Riccati Equation.- 8.4.1 Successive Approximation Methods.- 8.4.2 Hamiltonian Methods.- 8.4.3 Discussion.- 8.4.4 Examples.- 8.5 Hierarchical Optimization Methodology.- 8.5.1 Problem Decomposition.- 8.5.2 Open-Loop Computation Structures.- A. The Goal Coordination Method.- B. The Method of Tamura.- C. The Interaction Prediction Method.- 8.5.3 Closed-Loop Control Structures.- 8.5.4 Examples.- 8.6 Decomposition-Decentralisation Approach.- 8.6.1 Statement of the Problem.- 8.6.2 The Decoupled Subsystems.- 8.6.3 Multi-Controller Structure.- 8.6.4 Examples.- 8.7 Concluding Remarks.- 8.8 Problems.- 8.9 References.

A stability inequality for nonlinear feedback systems with slope-restricted nonlinearity

Abstract: Using Lyapunov's direct method, a stability inequality in frequency domain is derived for nonlinear feedback systems with slope-restricted nonlinearity. In the present approach, a transformed system that involves the slope of the nonlinearity is considered, thus leading to a stability inequality that incorporates the slope information (but not the sector information) of the nonlinearity. The approach is essentially an extension of a recent idea of the author.

Stabilizing optimal control of bilinear systems with a generalized cost

Abstract: Stabilizing and optimizing feedback control policies are derived for the important class of bilinear systems with generalized quadratric cost functions. These policies as well as the resulting optimal costs are quadratic in the state. The optimal cost function is shown to be a Lyapunov function for the bilinear system at hand. The resulting optimal and stabilized closed-loop system is of third order with respect to the state. One illustrative example is included.

Robustness of Luenberger Observers: Linear Systems Stabilized via Nonlinear Control

Abstract: Given a dynamical system whose state equations include time-varying uncertain parameters, it is often desirable to design a state feedback controller leading to uniform asymptotic stability of a given equilibrium point If, however, the controller operates on some estimate of the state, instead of the true state itself, it is of interest to know whether the desired stability will be preserved; eg, suppose that the measured output is processed by a Luenberger observer This paper concentrates on the scenario above and in addition, our analysis permits the controller to be nonlinear As a first step, inequalities are developed which have implications on the system's robustness; that is, when the uncertain parameters satisfy these inequalities, it becomes possible to separately design controller and observer This amounts to an extension of the classical separation theorem to the case when the controller is nonlinear It is also of interest to note that the approach given here enables us to guarantee stability for some nonzero range of admissible parameter variations This is achieved by introducing a certain "tuning parameter" into the Lyapunov function which is used to assure the stability of the cambined plant-observer-controller system

Lyapunov functional approach to raiative metrics

Abstract: Lyapunov's second method is applied to the spherical radiative Robinson-Trautman vacuum space-times to prove that they asymptotically settle down to Schwarzschild space-time. This class of Robinson-Trautman metrics is characterized by the surfaceS being topologically a two-sphere, whereS is invariantly defined by the intersection of the hypersurfacesu=const andr=const. It is shown that ∝ S K 2 dσ is a Lyapunov functional, whereK is the Gaussian curvature anddσ is the invariant measure onS. The critical point occurs atK=0 or, equivalently, at ð2 K=0, which condition is shown to characterize Schwarzschild space-time.

Scaling Hamiltonian Systems

Abstract: This paper presents a detailed discussion of scaling techniques for Hamiltonian systems of equations. These scaling techniques are used to introduce small parameters into various systems of equations in order to simplify the proofs of the existence of periodic solutions. The discussion proceeds through a series of increasingly more complex examples taken from celestial mechanics. In particular, simple proofs are given for Lyapunov’s center theorem, the continuation theorem of Hadjidemetriou, and several theorems on periodic solutions by the author.

Stability analysis of PWM feedback control systems with PID regulators

Abstract: Stability analysis is of great significance in those feedback control systems in which the power amplifier is operated as a pulse-modulator device, since under these circumstances the whole control system is highly non-linear. Stability in PWM feedback control systems with a proportional type regulator has been amply described in the literature. Only recently, however, have such studies been extended to include systems with a proportional-plus-integral regulator. In this paper the problem is considered for the case involving a PWM control system, where the regulator is a proportional-plus-integral-plus-derivative, the PWM modulator is of a very general type and the controlled process is of arbitrary order. The stability of the system is analysed by means of a discrete version of the second Lyapunov method ; this method in turn leads to an investigation of the positivity region of a quadratic form defined in the parameter space of the regulator. To improve the stability region obtained, a procedur...

A note on the discrete Lyapunov and Riccati matrix equations

Abstract: Inequalities which are satisfied by the determinants of the positive definite solutions of the discrete algebraic Riccati and Lyapunov matrix equations are presented. The results give lower bounds for the product of the eigenvalues of the matrix solutions. Also for a discrete Lyapunov equation, an algorithm is presented to determine under what conditions a positive diagonal solution will exist. If all the conditions are satisfied, the algorithm also provides such a diagonal solution.

Synthesis of systems with periodic solutions satisfying Upsilon (X) = 0

Abstract: A number of papers in the last decade dealt with synthesizing a set of n coupled differential equations \dot{x} = f(x) which have particular globally stable desired solutions, usually periodic. The methods for deriving these differential equations and verifying the properties of the solution have been, at best, ad hoc. This paper investigates the generic and stable synthesis of such systems which have the common property that the desired particular solutions satisfy m constraint equations \Upsilon (x) = 0 . The stability and generic properties are inherent and easily derived from basic properties of the function \Upsilon . First, Lyapunov techniques are used to guarantee that solutions satisfy the constraints. Next, well-known properties of manifolds are used to show that satisfying the constraint equations is a natural way to guarantee that solutions have particular useful properties. Further, these properties are generic in that almost all such possible \Upsilon have them. The synthesis properties are reapplied to the problems of the earlier papers. The resulting systems \dot{x} = f(x) are more general and/or simpler to implement than those originally devised.

Stability of large systems.

Abstract: We use the May-Wigner Stability Theorem (Geman (1984) preprint, Brown University; Hastings (1984) preprint, Hofstra University), to study the Lyapunov and structural stability of “real” large systems. Here are our new main results. For large systems which satisfy certain natural scaling relations (Harrison, Am. Natur., 113 (1979) 659; May (1979) Blackwell Scientific, Oxford), Lyapunov stability tends to increase with increasing complexity. However, at least one aspect of structural stability decreases: both competitive and cooperative effects can rapidly destabilize such a system. Finally, we observe that random matrices which satisfy the hypotheses and stability criterion of the May-Wigner theorem are asymptotically of the form ‘rotation followed by multiplication by γ,γ