Liapunov's Direct Method Research Articles

The stability of a sleeping top with damping torque

The sufficient condition of the stability of a sleeping top with damping torque is obtained by Liapunov direct method. The condition obtained is the same as the necessary and sufficient condition of the stability of sleeping top without damping torque.

Dynamic stability of viscoelastic shells under time-dependent membrane loads

The stability of the undeflected middle surface of a uniform Voigt-Kelvin cylindrical circular shell is studied. The shell is being subjected to a time-varying axial compression as well as a time-varying uniformly distributed radial loading. Using the direct Liapunov method sufficient criteria for the asymptotic stability and the almost sure asymptotic stability are obtained. Asymptotic stability regions as functions of a reduced retardation time, geometric and material parameters of shell and characteristics of loading are calculated. It was shown that the stability regions do not change qualitatively in going from a Gaussian force to a harmonic one. The change of membrane loading direction from the axial direction to the circumferential one on the stability regions is also discussed.

Stability Analysis of the Motion of a Tuned Gyroscope

In this paper, we use the Liapunov direct method to study the stability of motion of the rotor of a single-gimbal, elastically supported tuned gyroscope with the input rate ø̇1 = ø̇2 = 0 in three cases: (1) rotor-gimbal damping coefficients D1 = D2 = 0, rotor winding coefficient Dw = 0; (2) D1 ≠ 0, D2 ≠ 0, Dw = 0; (3) D1 ≠ 0, D2 ≠0, Dw ≠ 0.

Asymptotic stability criteria for delay-differential equations

SynopsisThis paper is concerned with the problem of showing uniform stability and equiasymptotic stability of thezero solution of functional differential equations with either finite or infinite delay. The investigations are based on Liapunov's direct method and attention is focused on those equations whose right-hand sides are unbounded for bounded state variables.

Feedback stabilization of the nonlinear pendulum under uncertainty: A robustness issue

We examine the control of a (prototype) nonlinear pendulum under uncertainty in modelling. The uncertainty is sufficiently small and is a function of both state and time. We apply Liapunov's direct method to prove the stability of an equilibrium point. Then, we apply Melnikov's method to introduce a criterion for the control to avoid deforming the actual region of stability of the equilibrium point. We demonstrate the cooperative relationship between the Liapunov and Melnikov methods in assessing stability. Consequently, we propose preserving the size and shape of the region of stability, and preventing the creation of complicated dynamics on its boundary, as indicative of robust control. In essence, robustness of controllers for nonlinear systems must be treated from (nonlocal) dynamical systems viewpoint. A simulation result is given which quantitatively supports the analysis.

Microprocessor-Based Decoupled Control of Manipulator Using Modified Model Following Method with Sliding Mode

A methodology of discontinuous feedback and continuous feedforward control is developed to achieve accurate decoupled tracking in a class of nonlinear time varying systems in the presence of disturbances, parameter variations and nonlinear dynamic interactions. The method is based on an improved variable structure control with a sliding mode. The convergence of the error state to the origin is proved by using the "direct method of Liapunov".Then the developed method is practically applied to decoupled tracking control for a two-degrees-of-freedom manipulator. The joint position trajectories are observed to be smooth and to track the desired trajectories accurately.

Improved Sliding Mode Method Applied to Digital Decoupled Water Level Control for Two Tanks Connected with Pipes

A methodology of discontinuous feedback and continuous feedforward control is developed to achieve accurate decoupled tracking in a class of nonlinear systems in the presence of disturbances and parameter variations. The method is based on an improved variable structure control with a sliding mode. First, a desired output time response (forced model) is generated in time series with a microprocessor. Then discontinuous feedback and continuous feedforward controls are applied in order to force the output to track its desired time response. By introducing this modified model following control, the error between the output of the plant and model is made very small. Thus this mitigates the extent of undesirable chattering due to switching delays, neglected small time constants, etc. The convergence of the error state to the origin is proved by using “direct method of Liapunov.” With the rapid progress in microelectronics and powerelectronics high-performance actuators, sensors are now available. Therefore the proposed control strategy can be easily implemented resulting in revolutionary high-quality control in processes.The methodology is practically applied to decoupled water-level controls for two tanks connected with pipes, in which water-levels are interacting on one another. All the control strategies are implemented by a high performance microprocessor which is now available at reasonable cost. It is found that this novel controller obtained performs extremely satisfactorily and is superior to the “conventional” controller for the process. In addition, the controller can be simply designed, since it does not require accurate modeling. but it is sufficient to know only the bounds of model parameter values and the magnitudes of the disturbances.

A new parameter estimate in singular perturbations

A new upper bound is obtained for the singular perturbation parameter of an asymptotically stable singularly perturbed system. General time-invariant systems with a single small parameter are considered. The paper employs a Riccati equation whose solution is known to facilitate the exact decoupling of fast and slow dynamics. An application of the Brouwer fixed point theorem to the Riccati equation and of Liapunov's direct method to the fast and slow subsystems results in the desired upper bound. Computation of the estimate requires only the solution of two Liapunov matrix equations.

V-functions in the control of motion

The geometry and application of K-functions is studied which provide a generalization of the Liapunov direct method in stability theory to allow the control of motion. A sufficiency theorem is proved that provides a means for quantitative design and investigation of dynamical systems.

On the stability of general dynamic systems using a Liapunov's direct method approach

A technique is presented for studying the stability of equilibrium of linear lumped-parameter systems involving general types of forces such as dissipative, non-conservative, gyroscopic, and circulatory. A modified approach to solving the Liapunov equation is used to provide a function V which is exploited to present different stability criteria for the equilibrium of such systems. It was shown that this approach is advantagous to the solution of the Liapunov equation, since it is more computationally attractive. Additionally, this may be used to determine the effects of different parameters on the systems stability or to design a controller for an actively controlled system. Several previously developed related works are studied and compared with this work. Furthermore, examples are used to illustrate the presented approach and some of its applications.

Dynamic Stability of a Nonlinear Cylindrical Shell

The stability of the undeflected middle surface of a uniform elastic cylindrical shell governed by Ka´rma´n’s equations is studied. The shell is being subjected to a time-varying axial compression as well as a uniformly distributed time-varying radial loading. Using the direct Liapunov method sufficient conditions for deterministic asymptotic as well as stochastic stability are obtained. A relation between stability conditions of a linearized problem and that of Ka´rma´n’s equations is found. Contrary to the stability theory of nonlinear plates it is established that the linearized problem should be modified to ensure the stability of the nonlinear shell. The case when the shell is governed by the Itoˆ stochastic nonlinear equations is also discussed.

On the application of Liapunov's direct method to discrete dynamic systems with stochastic parameters

A Liapunov function specially suitable for the study of the almost sure asymptotic stability of a class of linear discrete systems, described by a set of second order differential equations with stochastic parameters, is presented. A theorem and related corollaries, applicable to systems involving general types of forces, are obtained. The proposed technique is shown to be useful in minimizing the computational efforts associated with relatively large dynamic systems. Several examples, including systems involving follower forces, are included to demonstrate the effectiveness of the method.

Stability Boundary of Slip Power Recovery Drive by Liapunov's Direct Method

This paper presents an integrated approach to the study of dynamic stability of adjustable speed slipring induction motor controlled by solid state switches under different conditions of operation. It attempts to bring a global picture covering stability boundary of 3 distinct zones: (A) Sub-synchronous Operation: Sub-synchronous speed(B) Super-synchronous Operation: Sub-synchronous speed(C) Super-synchronous Operation: Super-synchronous speedBy assuming a simple quadratic form of Liapunov's function some light has been thrown on this stability boundary of the above zones of operations considering the inherent non-linearity. Scope for further study has been suggested.

Global Stability of a System of Two Interacting Species with Convective and Dispersive Migration

Using Liapunov's direct method, effects of convective and dispersive migration on the global stability of the equilibrium state of a system of two interacting species are investigated. It is shown that the stable equilibrium state without dispersal remains so with dispersal. Further, it is pointed out that stability or instability of the equilibrium state of the system is not affected by convective migration. These results are justified in cases of a system of mutualistic interactions of species and a prey-predator system with functional response.

On-Line Transient Stability Evaluation by System Decomposition-Aggregation and High Order Derivatives

An on-line transient stability evaluation method is presented in this paper. Under some classical assumptions, the change of generator rotor angles during the transient period is described by a set of second order differential equations. This set of equations is solved by single-step high order derivative explicit integration. A new system decomposition-aggregation technique is proposed to decompose the power system into subsystems and then to aggregate the subsystems by interconnection boundary conditions. Based on the aggregation model, the calculation of high-order derivatives is quite efficient to reduce the requirements of both computer memory and computational time for a large scale power system. The relation of memory and execution time requirements to the system decomposition is discussed. Two systems, the IEEE 118 bus test system with 34 generators and a 293 bus system with 63 generators, are tested by simulation on a CDC-6600 computer. The results demonstrate that the proposed method is effective for on-line application and may be faster than Liapunov's direct method.

On the stability of linear conservative gyroscopic systems

A simple criterion concerning the stability of linear gyroscopic conservative systems is developed. First, a sufficient condition for stability is established. The proof is based on a transformation converting the original autonomous system into a nonautonomous system, and applying Liapunov's direct method to the latter. Then a theorem is given which provides a necessary and sufficient condition for a restricted class of systems. Illustrative examples are provided.

On-Line Control of Nonlinear Flexible Structures

A simple yet efficient method is presented for the on-line control of nonlinear, multidegree-of-freedom systems responding to arbitrary dynamic environments. The control procedure uses pulse generators located at selected positions throughout a given system. The degree of system oscillation near each controller determines the controller’s activation time and pulse amplitude. The direct method of Liapunov is used to establish that the response of the controlled nonlinear system is Lagrange stable. Simulation studies of three example systems (conducted with digital and analog computers) demonstrate the feasibility, reliability, and robustness of the proposed active-control method. These systems, which include one with a hysteretic nonlinearity, are structures representative of modern tall buildings; they are subjected to nonstationary random excitation representative of earthquake ground motions.

Stability relationships between gyrostats with free, constant-speed,and speed-controlled rotors

The stability properties of a class of gyrostat systems with rotor spin rates regulated by feedback control are discussed. The concepts of proportional and rate feedback are generalized to the multiple rotor case, with the spin rate feedback gain matrix assumed to be positive definite, and that of position feedback positive semidefinite. In the past, gyrostat stability analyses have emphasized the idealized limiting case of inifinite feedback gains corresponding to a constant-speed rotor, or, in a few cases, studies of the idealized frictionless free-rotor case. The aim of this work is to study the relationship between stability results using these idealized models and the stability results obtained using the more realistic feedback controlled rotor model for the full range of feedback gains. The state space is divided into two subspaces, one of which is free of damping due to feedback (perhaps zero-dimensional), and this subspace exhibits certain appealing mathematical properties. It is shown that for the controlled systems to be asymptotically stable, a necessary and sufficient condition is that the corresponding restrained system be in the Lagrange region (stable according to Liapunov's direct method) and that the damping-free subspace be zero-dimensional (pervasive damping). Two sufficient conditions for instability are also proved. As a special case, the stability of gravity gradient gyrostat satellites with one symmetric rotor spinning around an axis fixed in the main rigid body are analyzed in detail, and numerical examples are given.

Effects of dispersal on the stability of a gonorrhea endemic model

Using Liapunov's direct method, effects of dispersal on the linear and nonlinear stability of the endemic equilibrium state of the system governing the spread of gonorrhea are investigated. It is noted that the equilibrium state, which is nonlinearly asymptotically stable in the feasible region of the phase plane in the absence of dispersal, remains so with self-dispersal also (cross-dispersal being absent). However, in the presence of both self- and cross-dispersal, the equilibrium state can still remain nonlinearly asymptotically stable in the entire feasible region provided a certain condition involving self- and cross-dispersal coefficients is satisfied. It is also seen in this case that, for the linearly stable equilibrium state, there exists a subregion of the feasible region where it is nonlinearly asymptotically stable.

Effects of dispersal on the stability of a prey-predator system with functional response

Using Liapunov's direct method, effects of dispersal on the linear and nonlinear stability of the equilibrium state for a prey-predator system with functional response are investigated. It is noted that the functional response has a destabilizing effect. It is shown that an otherwise linearly or nonlinearly stable equilibrium state of the system remains so with dispersal as well, even with functional response. It is further established that if the equilibrium state is linearly stable a subregion of the positive quadrant can be found in the phase plane where it is nonlinearly stable with or without dispersal.