Showing papers on "Lyapunov function published in 1979"

Mathematical Aspects of Reacting and Diffusing Systems

Abstract: Preface and General Introduction.- 1. Modeling Considerations.- 1.1 Basic hypotheses.- 1.2 Redistribution processes.- 1.3 Boundaries and interfaces.- 1.4 Reactions with migration.- 1.5 The reaction mechanism.- 1.6 Positivity of the density.- 1.7 Homogeneous systems.- 1.8 Modeling the rate functions.- 1.9 Colony models.- 1.10 Simplifying the model by means of asymptotics.- 2. Fisher's Nonlinear Diffusion Equation and Selection-Migration Models.- 2.1 Historical overview.- 2.2 Assumptions for the present model.- 2.3 Reduction to a simpler model.- 2.4 Comments on the comparison of models.- 2.5 The question of formal approximation.- 2.6 The case of a discontinuous carrying capacity.- 2.7 Discussion.- 3. Formulation of Mathematical Problems.- 3.1 The standard problems.- 3.2 Asymptotic states.- 3.3 Existence questions.- 4. The Scalar Case.- 4.1 Comparison methods.- 4.2 Derivative estimates.- 4.3 Stability and instability of stationary solutions.- 4.4 Traveling waves.- 4.5 Global stability of traveling waves.- 4.6 More on Lyapunov methods.- 4.7 Further results in the bistable case.- 4.8 Stationary solutions for x-dependent source function.- 5. Systems: Comparison Techniques.- 5.1 Basic comparison theorems.- 5.2 An example from ecology.- 6. Systems: Linear Stability Techniques.- 6.1 Stability considerations for nonconstant stationary solutions and traveling waves.- 6.2 Pattern stability for a class of model systems.- 7. Systems: Bifurcation Techniques.- 7.1 Small amplitude stationary solutions.- 7.2 Small amplitude wave trains.- 7.3 Bibliographical discussion.- 8. Systems: Singular Perturbation and Scaling Techniques.- 8.1 Fast wave trains.- 8.2 Sharp fronts (review).- 8.3 Slowly varying waves (review).- 8.4 Partitioning (review).- 8.5 Transient asymptotics.- 9. References to Other Topics.- 9.1 Reaction-diffusion systems modeling nerve signal propagation.- 9.2 Miscellaneous.- References.

Uncertain dynamical systems--A Lyapunov min-max approach

Abstract: A class of dynamical systems in the presence of uncertainty is formulated by contingent differential equations. Asymptotic stability (in the sense of Lyapunov) is then developed via generalized dynamical systems (GDS's). The uncertainty is deterministic; the only assumption is that its value belongs to a known compact set. Application to variable structure and model reference control systems are discussed.

Global stability of predator-prey interactions

Abstract: A Lyapunov function is given that extends functions used by Volterra, Goh, and Hsu to a wide class of predator-prey models, including Leslie type models, and a biological interpretation of this function is given. It yields a simple stability criterion, which is used to examine the effect on stability of intraspecific competition among both prey and predators, of a refuge for the prey, and of Holling type II and type III functional responses. Although local stability analysis of these specific models has been done previously, the Lyapunov function facilitates study of global stability and domains of attraction and provides a unified theory which depends on the general nature of the interactions and not on the specific functions used to model them.

Kolmogorov entropy of a dynamical system with an increasing number of degrees of freedom

Abstract: Lyapunov characteristic numbers are used to estimate numerically the Kolmogorov entropy of an isolated one-dimensional self-gravitating system consisting of $N$ plane parallel sheets with uniform density It appears that the Kolmogorov entropy increases linearly when the number of degrees of freedom is greater than or equal to 2

Lyapunov Redesign of Model Reference Adaptive Control Systems (Part I)

Abstract: A new approach for geaeraticn of Lyapunov function, due to Nagaraja aid Chalam (1974) has been applied to the redesign of first-order and second-order model-reference adaptive control systems considered earlier by Parks (1966) and Phillipson (1967). They chose the V function rather arbitrarily, whereas a systematic method is presented here for choosing an appropriate V function. In the present work, V includes all state variables, whereas in the previous work of Parks (1966), V included only error e for first-order system and e for second-order system. Inclusion of all state variables in V ensures that if the gain-parameter misalignment tends to zero, the error will also tend to zero. The same approach was applied to modified first-order and second-order systems (Phillipson 1967).

An Approximate Method for Probabilistic Assessment of Transient Stability

Abstract: Power-system transient-stability evaluation deals with the performance of the system when subjected to aperiodic disturbances such as faults, sudden loss of load or generation, and line switching. Under these conditions stability is lost when one or more generators fall out of synchronism with the rest of the system. Numerous disturbances can lead to system instability and the probabilities of their occurrences are quite different. Most system-design criteria require system survival under specific fault conditions in the normal system state. Systems are therefore designed with a set of severe criteria which may be extremely unlikely. In some cases the initial criteria become too expensive to maintain and they are modified or relaxed. The probabilities of occurrences of the disturbances should be included in the assessment to develop a realistic appraisal of system adequacy. Probabilistic considerations in the stability of simple single and multi-machine systems involves simulating the system dynamics during the disturbance. In large-scale systems, such a simulation can be computationally expensive. Approximate methods of stability evaluation are quite adequate in initial system planning to identify the critical areas. This paper deals with an approximate method using Lyapunov functions to obtain a transient stability risk index which is useful in initial system planning and design studies. The procedure for considering the probabilities associated with the type, location, and clearance of faults is illustrated for a simple multimachine system. The effect of system load on the risk index is also demonstrated.

An Approximate Incorporation of Field Flux Decay into Transient Stability Analyses of Multimachine Power Systems by the Second Method of Lyapunov

Abstract: This paper presents a new approach to incorporate in an approximate manner field flux decay into transient stability analysis of multimachine power systems by the second method of Lyapunov; this has not been successful so far in other investigations. In this approach, changes in field flux linkages are not dealt with as other state variables, but treated as parameter variations, because of their small changes, in the conventional stability analysis under the assumption of constant field flux linkages. Therefore, the approach can utilize any Lyapunov function which has been proved to be effective for multimachine power systems, and hence never requires the laborious and very difficult process of constructing an effective Lyapunov function with inclusion of field flux decay. Simulation results have demonstrated the proposed method's practicability and its clear advantage over the exact treatment of the field flux decay attempted by other authors for a simple one machine connected to an infinite bus system.

Existence of limit cycle and stabilization of induction motor via new nonlinear state observer

Abstract: In this paper the boundedness of the solutious of the bilinear and nonlinear differential equations, which describe the dynamic behavior of an ideal three-phase squirrel cage induction motor, is shown using a Lyapunov function. It is then proved by sampling combined with a digital simulation that an unstable machine has limit cycle. Utilizing these results a new bilinear and nonlinear reduced-order state observer, which is globally asymptotically stable, is constructed to estimate the unmeasurable state variables. By using this observer a new two-step procedure for stabilizing an unstable machine, which has a limit cycle, is proposed. This scheme can be easily implemented resulting in an asymptotically stable overall system. These results are numerically verified by simulation.

Conditions for Global Stability Concerning a Prey-Predator Model with Delay Effects

Abstract: An integrodifferential system concerning prey-predator interaction is studied. By means of appropriately constructed Lyapunov functions, sufficient conditions concerning the lag effects are derived for global stability of the equilibrium. The effects of various forms of continuously distributed delays are examined.

Load frequency control using Lyapunov's second method: Bang-bang control of speed changer position

Abstract: This letter reports a bang-bang load frequency control policy based on the second method of Lyapunov. The proposed method is simple and practically feasible for implementation. A numerical illustration on a two-area load frequency control system is presented in order to verify the practicality of the proposed method.

A new lower bound on the cost of optimal regulators

Abstract: A new lower bound on the quadratic cost of linear regulators is derived using some recently established results on norm bounds for the algebraic matrix Riccati and Lyapunov equations. The bound thus obtained is very attractive computationally and is independent of the initial conditions.

Stability of Charged Solitons

Abstract: The stability of charged solitons described by the relativistic complex scalar field is investigated by the direct Lyapunov method. It is shown that the stability of pulson-type solitons can only be conditional. Some necessary and sufficient conditions for the stability of stationary solitons with fixed charge are established. Several examples are considered.

Stability of two-level control schemes subjected to structural perturbations

Abstract: The paper deals with the problem of stability of large-scale interconnected systems with a two-level control subjected to structural perturbations. By means of Lyapunov vectors function, sufficient degrees of stability for each isolated subsystem are determined in order that the overall structure remains stable when perturbations occur between the two levels of control. Numerical features are discussed by means of two examples.

Time delays in large-scale systems

Abstract: The paper examines classes of large scale systems problems which have been tackled by methods associated with M-matrices and vector Lyapunov functions, and shows that the insertion of time delays into the interconnection terms will preserve exponential stability, assuming that such stability is present when no time delays exist. The key is to use new types of Lyapunov-like functions for analysing the behaviour of the delay-differential equations which are encountered. These functions allow the derivation of results for time-varying systems and systems including delay functions other than a pure delay.

Stability Bounds for the Control of Large Space Structures

Abstract: Balas (1977) has discussed the stability problem of reduced-order regulators and estimators in terms of control and observation 'spillover'. The term 'control spillover' was used to define that part of the feedback control which excites the uncontrolled (or residual) modes, and 'observation spillover' was used to define that part of the measurement which is contaminated by residual modes. In this paper, two sufficient conditions are derived via Lyapunov methods for asymptotic stability of large space structures using a class of reduced-order controllers. These conditions give allowable bounds on the spectral norms of control and observation spillover terms. The sufficient condition given by a specified inequality equation appears to be less conservative, and should be useful as a design tool for the control of large space structures.

Desensitizing observers for LQG feedback control

Abstract: This paper considers the problem of designing an observer-estimator for use as part of the compensation scheme in the LQG control problem such that the usual quadratic performance index is least sensitive to small errors in the values of the plant parameters. A composite performance index consisting of a linear combination of the usual quadratic performance index and the squared norms of the sensitivity matrices of that performance index with respect to the plant parameters is used to design the observer-estimator. The design equations for the optimal observer-estimator parameter matrices are determined by requiring that the first variations of the composite index are each zero for all variations of the observer-estimator parameter matrices. These non-linear equations are solved iteratively using Picard's method. The calculations involved in each iteration are linear and are concerned mainly with the solution of linear matrix Lyapunov equations.

On the Construction of Nonautonomous Stable Systems with Applications to Adaptive Identification and Control

Abstract: In this paper, the uniform asymptotic stability of nonautonomous ordinary differential equations with Lyapunov functions whose derivatives are negative semidefinite is studied. A general framework for constructing and analyzing such systems is established, and applications to adaptive schemes for identification and control are described. Specific rates of convergence and robustness estimates are also given.

Stability from the bifurcation function

Abstract: Publisher Summary Recently, de Oliveira and Hale presented a relationship between the bifurcation function obtained by the method of Liapunov–Schmidt and the flow on the center manifold for periodic n-dimensional systems for which the linear approximation has one characteristic multiplier equal to one and the remaining ones inside the unit circle. This chapter presents a more elementary proof of this result and presents an application to stability for autonomous systems in the critical case of one zero root. This allows the extension and simplification of the classical results of Liapunov to the case of C-vector fields. The approach used by Liapunov cannot be extended to this case as he employs the theory of transformations to approximate the vector fields by simpler vector fields.

The concept of repulsivity in dynamical systems as motivated by persistence problems in population biology

Abstract: An alternative to stability analysis in population biology is proposed for cases in which persistence problems such as coexistence of species and protectedness of genetic polymorphisms are of primary interest. In order to arrive at a general mathematical description, the concept of repulsivity of certain sets with respect to dynamical systems (continuous-time as well as discrete-time) defined on metric spaces is introduced. A first basic result linking this concept to the existence of Lyapunov functions is derived in analogy to the respective results from stability theory.

Iterations of continuous mappings on metric spaces Asymptotic stability and Lyapunov functions

Abstract: The stability of difference equations, which represent discrete-time motions, is studied on general metric spaces. An analogous theorem to the equivalence between the asymptotic stability of invariant sets and the existence of Lyapunov functions for continuous-time motions is proved. One consequence of the results is the reduction of the asymptotic stability to an invariance condition.

A Simplified Stability Criterion for Nonconservative Systems

Abstract: Dynamic stability of a general nonconservative system of n degrees of freedom is investigated. A sufficient and necessary condition for the stability of such a system is developed. It represents a simplified criterion based on the famous Lyapunov’s theorem which is proved afresh using λ-matrix methods only. When this criterion is adopted, it reduces the number of equations in Lyapunov’s method to less than half. A systematic procedure is then suggested for the stability investigation and its use is illustrated through a numerical example at the end of the paper.

Asymptotic stability of systems: Results involving the system topology

Abstract: In this paper we answer the following question for a large class of (linear and nonlinear) dynamical systems. Given is a system with dissipation and given is the associated conservative system. Suppose the associated conservative system is stable. What properties of the system topology (system configuration) will ensure that the overall system with dissipation is asymptotically stable? Both linear and nonlinear (Hamiltonian) systems are treated. For the linear case, necessary and sufficient conditions for asymptotic stability are established, while for the nonlinear case, sufficient conditions and also some necessary and sufficient conditions for asymptotic stability are obtained. It is emphasized that the application of the present results to specific problems will usually not require a search for appropriate Lyapunov functions. Indeed, a stability analysis by the present method involves the following two steps: (a) given a system with dissipation, the stability of its trivial solution (equilibrium) is ascertained by determining the stability of the associated conservative system, i.e., by determining whether the potential energy is a minimum at the equilibrium; and (b) attractivity of the equilibrium of the entire system (with dissipation) is determined from the system topology (system configuration). This approach to stability analysis appears to be new. Furthermore, since the present method involves concepts from control theory (namely, the notion of observability), these results provide further insight into the mechanisms of stability (and stabilization). To provide motivation and to demonstrate the applicability of the results, some specific examples are considered.