scholarly journals Lyapunov Function for Nonuniform in Time Global Asymptotic Stability in Probability with Application to Feedback Stabilization

2011 ◽  
Vol 116 (1) ◽  
pp. 107-117 ◽  
Author(s):  
Fakhreddin Abedi ◽  
Malik Abu Hassan ◽  
Norihan Md Arifin
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Global Asymptotic Stability ◽  
Feedback Stabilization ◽  
Stability In Probability ◽  
Asymptotic Stability In Probability

On Stability of Solutions of Certain Differential Equations of the Third Order

1967 ◽  
Vol 10 (5) ◽  
pp. 681-688 ◽  
Author(s):  
B.S. Lalli
Keyword(s):  
Differential Equation ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Differential Equations ◽  
Global Asymptotic Stability ◽  
Sufficient Conditions ◽  
Stability Of Solutions ◽  
Third Order ◽  
The Third ◽  
Image Position

The purpose of this paper is to obtain a set of sufficient conditions for “global asymptotic stability” of the trivial solution x = 0 of the differential equation1.1using a Lyapunov function which is substantially different from similar functions used in [2], [3] and [4], for similar differential equations. The functions f1, f2 and f3 are real - valued and are smooth enough to ensure the existence of the solutions of (1.1) on [0, ∞). The dot indicates differentiation with respect to t. We are taking a and b to be some positive parameters.

On the Partial Stochastic Stability of Stochastic Differential Delay Equations with Markovian Switching

2013 ◽  
Vol 765-767 ◽  
pp. 709-712 ◽  
Author(s):  
De Zhi Liu ◽  
Wei Qun Wang
Keyword(s):  
Asymptotic Stability ◽  
Stochastic Stability ◽  
Sufficient Conditions ◽  
Delay Equations ◽  
Markovian Switching ◽  
Differential Delay ◽  
Stability In Probability ◽  
Differential Delay Equations ◽  
Stochastic Differential Delay Equations ◽  
Asymptotic Stability In Probability

In the paper, we are concerned with the partial asymptotic stochastic stability (stability in probability) of stochastic differential delay equations with Markovian switching (SDDEwMSs), the sufficient conditions for partial asymptotic stability in probability have been given and we have generalized some results of Sharov and Ignatyev to cover a class of much more general SDDEwMSs.

Stochastic Response and Stability Analysis of Single Leg Articulated Tower

2003 ◽  
Author(s):  
A. K. Banik ◽  
T. K. Datta
Keyword(s):  
Asymptotic Stability ◽  
Probability Density ◽  
Parametric Excitation ◽  
Stochastic Averaging ◽  
Stochastic Response ◽  
Sea State ◽  
Stationary Response ◽  
Stability In Probability ◽  
Hydrodynamic Damping ◽  
Asymptotic Stability In Probability

The stationary response and asymptotic stability in probability of an articulated tower under random wave excitation are investigated. The articulated tower is modelled as a SDOF system having stiffness nonlinearity, damping nonlinearity and parametric excitation. Using a stochastic averaging procedure and Fokker-Plank-Kolomogorov equation (FPK), the probability density function of the stationary solution is obtained for random sea state represented by a P-M sea spectrum. The method involves a Van-Der-Pol transformation of the nonlinear equation of motion to convert it to the Ito’s stochastic differential equation with averaged drift and diffusion coefficients. The asymptotic stability in probability of the system is investigated by obtaining the averaged Ito’s equation for the Hamiltonian of the system. The asymptotic stability is examined approximately by investigating the asymptotic behaviour of the diffusion process Y(t) at its two boundaries Y = 0 and ∞. As an illustrative example, an articulated tower in a sea depth of 150 m is considered. The tower consists of hollow cylinder of varying diameter along the height, providing the required buoyancy of the system. Wave forces on the structure are calculated using Morrison’s equation. The stochastic response and the stability conditions are obtained for a sea state represented by P-M spectrum with 16m significant wave height. The results of the study indicate that the probability density of the stationary response obtained by the stochastic averaging procedure is in very good agreement with that obtained from digital simulation. Further, the articulated tower is found to be asymptotically stable under the parametric excitation arising due to hydrodynamic damping.

Replacement of Lyapunov Function by Locally Convergent Robust Fixed Point Transformations in Model-Based Control a Brief Summary

2010 ◽  
Vol 14 (2) ◽  
pp. 224-236 ◽  
Author(s):  
József K. Tar ◽  
Keyword(s):  
Fixed Point ◽  
Asymptotic Stability ◽  
Lyapunov Function ◽  
Global Asymptotic Stability ◽  
External Disturbances ◽  
Model Based Control ◽  
Model Based ◽  
Point Transformations ◽  
Tuning Process ◽  
The Cost

Lyapunov’s 2ndMethod is a popular approach in the model-based control of nonlinear systems since normally it can guaranteeglobal asymptotic stability. However, the cost of the application of Lyapunov function frequently can be inefficient and complicated parameter tuning process containing unnecessary number of almost arbitrary control parameters, and vulnerability of the tuning process against not modeled, unknown external disturbances. Improved versions of the original, model-based tuning are especially sensitive to the external perturbations. All these disadvantages can simply be avoided by the application of robust fixed point transformations at the cost of giving up the guarantee of global asymptotic stability. Instead of that simple,stable iterative controlof local basin of attraction can be constructed on the basis of an approximate system model that can well compensate the effects of unknown external disturbances, too. Since the basin of convergence of the method in principle can be left, setting the three adaptive parameters of this controller needs preliminary simulation investigations. These statements are illustrated and substantiated via simulation results obtained for the adaptive control of various physical systems.

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