The practical use of variation principles in the determination of the stability of non linear systems

1961 ◽  
Vol 2 (2) ◽  
pp. 153-188 ◽  
Author(s):  
J. N. Lyness ◽  
J. M. Blatt
Keyword(s):  
Linear Systems ◽  
Initial Conditions ◽  
Mathieu Equation ◽  
Iteration Procedure ◽  
Variation Principle ◽  
Non Linear ◽  
Non Linear Systems ◽  
The Stability ◽  
Special Case

AbstractWe are interested in the motion of non linear systems. In this paper we use a variation principle and an iteration procedure in order to treat the stability of free oscillations against small disturbances of the initial conditions. It is found that approximations to the low lying stability lines can be obtained using the Rayleigh-Ritz variation principle and that these approximations can be consistently improved using an iteration procedure. These approximations are compared with the tabulated values in the special case of the Mathieu Equation. The results are in general a considerable improvement on those obtained using the usual Perturbation Theory, and have a much wider range of validity.

scholarly journals The practical use of variation principles in non-linear mechanics

1962 ◽  
Vol 2 (3) ◽  
pp. 357-368 ◽  
Author(s):  
J. M. Blatt ◽  
J. N. Lyness
Keyword(s):  
Initial Conditions ◽  
Variation Principle ◽  
Linear Vibration ◽  
Average Value ◽  
True Motion ◽  
Non Linear ◽  
Vibration Problems ◽  
The Stability ◽  
Non Linear Mechanics ◽  
Small Disturbances

AbstractContrary to the general impression that variation principles are of purely theoretical interest, we show by means of examples that they can lead to considerable practical advantages in the solution of non-linear vibration problems. In this paper, we develop a variation principle for the period of a free oscillation, as a function of the average value of the Lagrangian over one period. Even extremely simple-minded approximations to the true motion lead to excellent values for the period. The stability of such free oscillations against small disturbances of the initial conditions is treated in a previous paper.

Algorithm for the Determination of the Resonances of Anharmonic Damped Oscillators

1987 ◽  
Vol 42 (8) ◽  
pp. 803-807 ◽  
Author(s):  
G. Reiser ◽  
A. Hübler ◽  
E. Lüscher
Keyword(s):  
Linear Systems ◽  
New Method ◽  
Resonance Spectroscopy ◽  
Driving Mechanism ◽  
Non Linear ◽  
Non Linear Systems ◽  
Damped Oscillators ◽  
The Way

Non-linear damped oscillators can be resonantly driven by an aperiodic force. An algorithm is described which shows how such a driving mechanism can be calculated. This new method paves the way for resonance spectroscopy in non-linear systems.

Non-linear disturbance observer-based adaptive composite anti-disturbance control for non-linear systems with dynamic non-harmonic multisource disturbances

2017 ◽  
Vol 40 (12) ◽  
pp. 3458-3465 ◽  
Author(s):  
Zheng Wang ◽  
Jianping Yuan
Keyword(s):  
Linear Systems ◽  
Disturbance Observer ◽  
Control Structure ◽  
Non Linear ◽  
Linear Damping ◽  
Non Linear Systems ◽  
Complex Features ◽  
The Stability ◽  
Linear Disturbance ◽  
Adaptive Composite

In this paper, an adaptive composite anti-disturbance control structure is constructed for a class of non-linear systems with dynamic non-harmonic multisource disturbances. The key point of this paper is that a kind of non-harmonic disturbance, which has non-linear internal dynamics and complex features, is involved. A non-linear exogenous system is employed to describe the dynamic non-harmonic disturbances and several useful assumptions are introduced. By introducing a non-linear damping term, a novel adaptive non-linear disturbance observer is constructed. Based on the disturbance/uncertainty estimation and attenuation (DUEA) schemes, a composite anti-disturbance control structure is synthesized. Meanwhile, a new sufficient condition is derived and the stability of the closed-loop system is proved. Several illustrative examples are employed to demonstrate the effectiveness of the proposed method.

Inversion control of non-linear systems with an inverse NARX model identified using genetic algorithms

2000 ◽  
Vol 214 (4) ◽  
pp. 259-271 ◽  
Author(s):  
G-C Luh ◽  
C-Y Wu
Keyword(s):  
Genetic Algorithms ◽  
Linear Systems ◽  
Inverse Dynamics ◽  
Feedback Controller ◽  
Input Estimation ◽  
Non Linear ◽  
Non Linear Systems ◽  
The Stability ◽  
Identification Model ◽  
Narx Model

The inverse dynamics approach has been widely utilized in the control problem of various practical non-linear systems in recent years. This paper demonstrates a feedforward-feedback controller scheme of a non-linear plant whose dynamics are unknown and uncertain. The feedforward controller, an inverse NARX model (non-linear autoregressive model with exogenous inputs), provides only coarse control, whereas the feedback controller is used to handle unmodelled dynamics and disturbance. The inverse NARX model is derived by inverting the forward NARX model identified using genetic algorithms. A parallel-type NARX model whose outputs of the identification model are fed back into the identification model is adopted in the identification procedure to include the stability examination numerically. Both experimental and simulation results demonstrate that the proposed controller provides very good performance in the problems of input estimation and output tracking.

On the stability of a class of general non-linear systems

2001 ◽  
Vol 291 (1) ◽  
pp. 11-16 ◽  
Author(s):  
I.M. Gléria ◽  
A. Figueiredo ◽  
T.M. Rocha Filho
Keyword(s):  
Linear Systems ◽  
Non Linear ◽  
Non Linear Systems ◽  
The Stability
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