Liapunov's Direct Method Research Articles

Identification of bilinear systems

A new technique for the identification of general bilinear systems is presented. The model reference adaptive systems (MARS) technique together with the direct method of Liapunov are used to construct the identifier for the system parameters. The plant is assumed to be multivariable, continuous and all of the states are accessible for measurements. The proposed technique can be applied on a certain class of nonlinear systems in which the nonlinearities could be approximated by quadratic terms without a significant loss in the overall accuracy. The technique is used to identify the parameters of a binary distillation column.

A stability theorem for dissipative magnetohydrodynamic equilibria

The global stability of stationary equilibria of dissipative magnetohydrodynamics is studied using the direct Liapunov method. Sufficient conditions for stability of the linearized Euler–Lagrangian system with the full dissipative operators are given for the first time. The case of the two-fluid isentropic flow is discussed.

Multispecies food webs with diffusion

Using Liapunov's direct method, in this paper, it has been shown that the general Lotka-Volterra food web is stable without and with diffusion under each case of homogeneous reservoir and flux boundary conditions. However, for a three species food web with Holling's functional response the above general result regarding stability is not necessarily true. In such a case, conditions and regions for non-linear stability, without and with diffusion, have been derived. It is shown that such an otherwise unstable system may become stable with diffusion at least in a subregion of the positive octant of the state space.

A criterion of asymptotic stability of the trivial solution of an autonomous set of ordinary differential equations

This paper presents a new criterion of asymptotic stability for the trivial solution of an autonomous set of ordinary differential equations. Using the concepts of Liapunov's direct method the criterion is based on analysis of the sign of the integral from the derivative of the Liapunov function along the solutions of the sets the integral being taken for fixed, finite interval of time. The conditions of the criterion are weaker than those of Liapunov's theorem.

On the deterministic and stochastic stability of dissipative magnetohydrodynamic equilibria

The deterministic and stochastic stability of stationary equilibria of dissipative magnetohydrodynamics is discussed using the direct Liapunov method. The influence of random fluctuations on the stability is studied using three stochastic models.

On the stochastic stability of systems with discrete parameters and arbitrary circulatory forces

The stochastic stability of linear dynamic systems with finite dimensions and arbitrary non-conservative circulatory forces is studied by the direct Liapunov method. Sufficient conditions for stability in probability are given for three stochastic models.

On stability under constantly acting perturbations

A general formulation of the Volterra theorem on the parametric stability of steady motion of a gyrostat (/1/ page 261) ∗∗ ∗∗ V. V. Rumiantsev had repeatedly pointed out Volterra's foresight in anticipating in /1/ a number of later developments in the theories of the gyrostat and of stability. His theorem and its general character were brought to the writer's attention by V. N. Rubanovskii. is presented. The fundamental aspects of its proof together with the concept of Liapunov's direct method are extended to more general theorems on parametric stability. Volterra's geometric constructions are generalized so that his proofs can be applied to certain theorems on stability under constantly acting perturbations, and that these and theorems on parametric stability can be considered from a single point of view. Some applications of Volterra's theorem are examined, and a few examples presented.

Astability thorem for systems with discrete parameters and arbitrary circulatory forces

The stability of linear, time-independent dynamic systems with finite dimensions is studied by the direct Liapunov method. Sufficient and necessary conditions for stability in the sense of Liapunov are given for systems with arbitrary non-conservative circulatory forces.

A Few Liapunov Functions for a Synchronous Generator Infinite Bus Power System Problem

The transient stability of a single machine infinite bus power system problem was examined through the direct method of Liapunov. It was observed, through the direct method, that the system can be operated at a larger exciter gain if an additional feedback signal proportional to shaft speed deviation is used.

A New Approach to the Determination of Transient Stability of Large Power Systems

The determination of transient stability of multimachine power systems is important from the point of view of power system planning and Control. Numerical integration algorithms for the multimachine case give accurate results but are time consuming. The direct method of Liapunov has been used for the determination of critical clearing time. This method gives quicker but conservative answers. The conservativeness of the Liapunov method has prevented its wide acceptance by the practising engineers. Even though the Liapunov méthod gives conservative answers for the critical clearing time based on the values of V-functions along the trajectory of the faulted system, the method gives other useful information. For example, the V-Functions can be evaluated at the unstable equilibrium points of the postfault system and the critical unstable equilibrium point determined corresponding to the minimum of the V-function values at. various u.e.p.s. The critical u.e.p. corresponds to one particular machine losing synchronism or a particular group of machines losing synchronism. Once the particular critical machine or group is spotted out by the use of V-functions, the system may be reduced into two or three sets of machines (depending upon the critical u.e.p.) each of which may be combined to yield an equivalent machine (or zone). The resulting two or three machine system may be solved rigorously to get all information pertaining to transient stability. This approach combines the simplicity of the Liapunov method and the accuracy of the numerical integration techniques. The results obtained by this method are compared with those obtained by other methods for the case of a single machine going out of step.

Stochastic stability of MHD equilibria

The stochastic stability of stationary magnetohydrodynamic or plasma equilibria under the influence of stationary random fluctuations is studied using the direct Liapunov method. The destabilizing effect of fluctuations is demonstrated.

On the extent of asymptotic stability for output variables by Liapunov's direct method

For the determination of the extent of asymptotic stability by Liapunov's direct method, La Salle's theorem is extensively applied. This article presents required extensions to La Salle's result to determine regions of attraction for output variables of dynamic systems, even if the attracting set in the state space is unbounded.

A stability theory for perturbed differential equations

The problem of determining the behavior of the solutions of a perturbed differential equation with respect to the solutions of the original unperturbed differential equation is studied. The general differential equation considered is urn:x-wiley:01611712:media:ijmm589604:ijmm589604-math-0001 and the associated perturbed differential equation is urn:x-wiley:01611712:media:ijmm589604:ijmm589604-math-0002 .The approach used is to examine the difference between the respective solutions F(t, t0, x0) and G(t, t0, y0) of these two differential equations. Definitions paralleling the usual concepts of stability, asymptotic stability, eventual stability, exponential stability and instability are introduced for the difference G(t, t0, y0) − F(t, t0, x0) in the case where the initial values y0 and x0 are sufficiently close. The principal mathematical technique employed is a new modification of Liapunov′s Direct Method which is applied to the difference of the two solutions. Each of the various stabillty‐type properties considered is then shown to be guaranteed by the existence of a Liapunov‐type function with appropriate properties.

Calculation of Separatrices in Multistable Chemical Reaction Systems

In multistable chemical reaction systems the space of variables is partitioned by separatrices into different cells which are the domains of attraction of the respective critical points. By numerically solving an initial value problem, which is based on the particular stability properties of separatrix manifolds, we achieve a direct approximation of the stability boundaries in the phase space. The method is principally applicable to nonlinear systems of any number of variables and, other than the direct method of Liapunov, also in regions of the phase space which are far from the critical points.

Stability of a Simply Supported or Clamped Silo Wall Subjected to Frictional Tangential Follower Loads

In this paper, the stability of a silo wall has been investigated without considering the edge effect caused by adjacent walls. The silo wall is assumed to be subjected to tangential, compressive, exponentially varying following loads. Janssen, in his publications, justified using such loads. Hence, since the turn of the century loading of such a type has been widely used for silo design. The direct method of Liapunov is used for the stability investigation. Following this method, one determines the stability conditions without solving explicitly the boundary value problem of the system’s motion. Since no specific form for the solution is assumed when working with Liapunov’s approach, one does not know whether the structure will fail by buckling or by flutter. However, one does obtain the critical load beyond which the safety of the structure is endangered. The results are obtained in the form of three stability conditions. Two of these conditions may be used to determine the dimensions of the silo. The third condition then prescribes the required material damping. The paper may be considered as a first attempt into the investigation of silo stability with Liapunov’s direct method. A complete investigation has to include edge effects and may be left for further research.

On the stiffness property of motion

It is well known that the axis of a rapidly rotating gyroscope is very little responsive to large perturbing forces, i.e., possesses stiffness [1]. It was noticed that under specific conditions the stiffness property is inherent in systems with gyroscopes [2]. Many mechanical systems possess a property skin in some sense to gyroscopic stiffness. In this paper stiffness is interpreted as a distinctive stability. Theorems establishing tests for stiffness, analogous to the theorems of Liapunov's direct method, are formulated. The stiffness property is described by using a separation of variables as is done in problems on stability with respect to a part of the variables (see [3, 4], etc.). Individual questions on the stiffness of motion were examined in [5, 6].

Closed loop control, stability and transient behaviour of certain dynamic economic models

Dynamic optimizing models in economics have usually been analysed using the Maximum Principle, which yields an open-loop solution. The Hamilton Jaccbi Method which yields closed-loop solutions has been rarely applied in economic models. The present paper uses the Hamilton-Jacobi approach and the associated Riccati equation for analyzing dynamic linear economic systems with quadratic performance criterion. The systematic outpfit method due to Kalman is utilized to study the stability properties of the models. The closed loop solution in conjunction with the direct method of Liapunov is applied to study the settling time of the dynamic systems. The models selected for illustrative purposes are (i) Friedman's model of optimal money supply (ii) Phillips' model of stabilization policy (iii) a standard inventory-control model.

Multivariable model reference adaptive control of a pilot scale packed bed tubular reactor

A pilot scale nonadiabatic packed bed tubular reactor carrying out the hydrogenolysis of butane over a nickel on silica gel catalyst is used to evaluate the performance of a multivariable model reference adaptive control algorithm. The multiple reactions are highly exothermic causing steep radial gradients within the reactor. Measurements of temperature along the center of the reactor bed and of exit concentrations are used. The model reference algorithm is based upon Liapunov's direct method. The reference model is based upon a low order state space model derived from a three dimensional partial differential equation description of the process. Model parameters were fitted to reactor data to form the basic model. The model reference algorithm is implemented on a minicomputer which is interfaced to the reactor. The performance of the control algorithm is compared with results from low order conventional controllers and with an empirical low order controller. Problems related to derivation for, implementation on, and performance of the model reference technique on the highly nonlinear reactor system are discussed.

Liapunov Type Design of Lumped System in Conflict with Environment

After introducing a few propositions on a two-player differential game-against-nature, we discuss synthesis (design) of a fairly general, nonlinear lumped system with storage-accumulative and dissipative characteristics under random-but-bounded environmental input. The discussion is made in terms of the above game via the Liapunov Direct Method. Sufficient criteria for qualitative and optimal Payabilities are proposed. A storage-oriented strategy for specific local behaviour of the system in the state space is then derived from the above criteria.

Liapunov direct method in approaching bifurcation problems

A concept of total stability for continuous or discrete dynamical systems and a generalized definition of bifurcation are given: it is possible to show the link between an abrupt change of the asymptotic behaviour of a family of flows and the arising of new invariant sets, with determined asymptotic properties. The theoretical results are a contribution to the study of the behaviour of flows near an invariant compact set. They are obtained by means of an extension of Liapunov's direct method.