A Linearization-Based Approach to Vibrational Control of Second-Order Systems

2013 ◽  
Author(s):  
I. P. M. Wickramasinghe ◽  
Jordan M. Berg
Keyword(s):  
Inverted Pendulum ◽  
Second Order ◽  
Periodic Forcing ◽  
Averaging Methods ◽  
Vibrational Control ◽  
Electrostatic Mems ◽  
Mathieu’S Equation ◽  
Comb Drives ◽  
The Stability ◽  
Second Order Systems

We present an alternative to averaging methods for vibrational control design of second-order systems. This method is based on direct application of the stability map of the linearization of the system at the desired operating point. The paper focuses on harmonic forcing, for which the linearization is Mathieu’s equation, but somewhat more general periodic forcing functions may be handled. When it is applicable, this method achieves significantly greater functionality than previously reported approaches. This is demonstrated on two sample systems. One is the vertically driven inverted pendulum, and the other is an input-coupled bifurcation control problem arising from electrostatic MEMS comb drives.

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

1995 ◽  
Vol 117 (B) ◽  
pp. 145-153 ◽  
Author(s):  
D. S. Bernstein ◽  
S. P. Bhat
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Viscous Damping ◽  
Stability And Instability ◽  
The Stability ◽  
Second Order Systems ◽  
Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

Sufficient conditions for the stability of a class of second order systems

2015 ◽  
Vol 84 ◽  
pp. 1-6 ◽  
Author(s):  
Marco M. Nicotra ◽  
Roberto Naldi ◽  
Emanuele Garone
Keyword(s):  
Sufficient Conditions ◽  
Second Order ◽  
The Stability ◽  
Second Order Systems

Application of stability region centroids in robust PI stabilization of a class of second-order systems

2011 ◽  
Vol 34 (4) ◽  
pp. 487-498 ◽  
Author(s):  
Mohammad Amin Rahimian ◽  
Mohammad Saleh Tavazoei
Keyword(s):  
Parameter Space ◽  
Stability Region ◽  
Linear Time ◽  
Second Order ◽  
Stable Point ◽  
Convex Shape ◽  
Centroid Method ◽  
Controller Parameter ◽  
The Stability ◽  
Second Order Systems

In this paper we offer a tuning method for the design of stabilizing PI controllers that utilizes the stability region centroid in the controller parameter space. To this end, analytical formulas are derived to describe the stability boundaries of a class of relative-degree-one linear time invariant second-order systems, the stability region of which has a closed convex shape. The so-called centroid stable point is then calculated analytically and the resultant set of algebraic formulas are utilized to tune the controller parameters. The freedom to choose the surface density function in the calculation of centroid stable point provides the designer with the possibility to incorporate optimal or robustness requirements in the controller design process. The proposed method uses the stability regions in the controller parameter space to ensure closed-loop stability, and, while offering robust stability properties, it does not rely on predetermined information with regard to the nature or range of parameter variations and coefficient uncertainty bounds. Being situated away from the boundaries of the stability region in the controller parameter space, controllers designed based on the centroid method are both robust and non-fragile.

On the Stability of Second Order Time Varying Linear Systems

2005 ◽  
Vol 128 (3) ◽  
pp. 408-410 ◽  
Author(s):  
M. Tadi
Keyword(s):  
Asymptotic Stability ◽  
Linear Systems ◽  
Stiffness Matrix ◽  
Linear Time ◽  
Sufficient Conditions ◽  
Second Order ◽  
Time Varying ◽  
The Stability ◽  
Second Order Systems

This note considers the stability of linear time varying second order systems. It studies the case where the stiffness matrix is a function of time. It provides sufficient conditions for stability and asymptotic stability of the system provided that certain conditions on the stiffness matrix are satisfied.

Lyapunov Stability, Semistability, and Asymptotic Stability of Matrix Second-Order Systems

1995 ◽  
Vol 117 (B) ◽  
pp. 145-153 ◽  
Author(s):  
D. S. Bernstein ◽  
S. P. Bhat
Keyword(s):  
Asymptotic Stability ◽  
Lyapunov Stability ◽  
Sufficient Conditions ◽  
Second Order ◽  
Necessary And Sufficient Conditions ◽  
Viscous Damping ◽  
Stability And Instability ◽  
The Stability ◽  
Second Order Systems ◽  
Necessary And Sufficient

Necessary and sufficient conditions for Lyapunov stability, semistability and asymptotic stability of matrix second-order systems are given in terms of the coefficient matrices. Necessary and sufficient conditions for Lyapunov stability and instability in the absence of viscous damping are also given. These are used to derive several known stability and instability criteria as well as a few new ones. In addition, examples are given to illustrate the stability conditions.

scholarly journals Averaging Methods for Second-Order Differential Equations and Their Application for Impact Systems

Mathematics ◽  
2020 ◽  
Vol 8 (6) ◽  
pp. 916
Author(s):  
Michal Fečkan ◽  
Július Pačuta
Keyword(s):  
Differential Equations ◽  
Inverted Pendulum ◽  
Averaging Method ◽  
Mechanical Systems ◽  
Second Order ◽  
Periodic Systems ◽  
Weakly Nonlinear ◽  
Averaging Methods ◽  
Second Order Differential Equations ◽  
Averaging Theorem

In this paper, we discuss the averaging method for periodic systems of second order and the behavior of solutions that intersect a hyperplane. We prove an averaging theorem for impact systems. This allows us to investigate the approximate dynamics of mechanical systems, such as the weakly nonlinear and weakly periodically forced Duffing’s equation of a hard spring with an impact wall, or a weakly nonlinear and weakly periodically forced inverted pendulum with double impacts.

scholarly journals On the problem of stability of a motion of essentially nonlinear systems

2021 ◽  
pp. 3-12
Author(s):  
A.A. Martynyuk ◽  
V.O. Chernienko
Keyword(s):  
Nonlinear System ◽  
Nonlinear Systems ◽  
Lyapunov Functions ◽  
Sufficient Conditions ◽  
Second Order ◽  
Systems Of Equations ◽  
Affine Systems ◽  
The Stability ◽  
Second Order Systems

This article discusses essentially nonlinear systems. Following the approach of applying the pseudolinear inequalitiesdeveloped in a number of works, new estimates for the variation of Lyapunov functions along solutionsof the considered systems of equations are obtained. Based on these estimates, we obtain sufficient conditionsfor the equiboundedness of solutions of second-order systems and sufficient conditions for the stability of anessentially nonlinear system under large initial perturbations. Conditions for the stability of affine systems arealso obtained.

Stability of radical anions derived from substituted 5-nitrofurans investigated by ESR spectroscopy

1985 ◽  
Vol 50 (7) ◽  
pp. 1594-1601 ◽  
Author(s):  
Jiří Klíma ◽  
Larisa Baumane ◽  
Janis Stradinš ◽  
Jiří Volke ◽  
Romualds Gavars
Keyword(s):  
Radical Anion ◽  
High Stability ◽  
Esr Spectroscopy ◽  
Reaction Path ◽  
Order Reaction ◽  
Second Order ◽  
Radical Anions ◽  
Second Order Reaction ◽  
Unpaired Electron Density ◽  
The Stability

It has been found that the decay in dimethylformamide and dimethylformamide-water mixtures of radical anions in five of the investigated 5-nitrofurans is governed by a second-order reaction. Only the decay of the radical anion generated from 5-nitro-2-furfural III may be described by an equation including parallel first- and second-order reactions; this behaviour is evidently caused by the relatively high stability of the corresponding dianion, this being an intermediate in the reaction path. The presence of a larger conjugated system in the substituent in position 2 results in a decrease of the unpaired electron density in the nitro group and, consequently, an increase in the stability of the corresponding radical anions.

scholarly journals ℋ∞ Control Tunning to Guarantee the Output Performance of LTI Second-Order Systems

2020 ◽  
Vol 53 (2) ◽  
pp. 4611-4616
Author(s):  
Ramón I. Verdés ◽  
Luis T. Aguilar ◽  
Yury Orlov
Keyword(s):  
Second Order ◽  
Output Performance ◽  
Second Order Systems
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