On the stability of periodic motions

Айзерман, М.А.; Gantmakher, F.R.

doi:10.1016/0021-8928(58)90033-9

Cited by 48 publications

(43 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Unclassified

Order By: Relevance

“…Hiskens et al used this model to find the required jump map. Very briefly, just before the switching, the two conditions that must be satisfied are (14) where and are the state and algebraic variables immediately before the switching. Furthermore, by using the chain rule, the derivative of the state vector just before the switching is given by (15) where is the instant just before the switching.…”

Section: B Nonsmooth Systemsmentioning

confidence: 99%

“…If there is no discontinuity (i.e., ), by using the linear approximation given in (7) for both sides of the switching, the map reduces to a simple form similar to the saltation matrix [14] and part of the Jacobian of the Poincaré map [27], [28].…”

Section: B Nonsmooth Systemsmentioning

confidence: 99%

“…In an unconnected line of development occurring in the erstwhile Soviet Union, scientists like Aizerman, Gantmakher, and Filippov [14], [15] developed a mathematical formalism applicable to systems with a discontinuous right-hand side. The method was subsequently applied to mechanical systems undergoing stick-slip vibrations or impacting motion [16], [17], with very fruitful results.…”

mentioning

confidence: 99%

See 2 more Smart Citations

Stability Analysis of the Continuous-Conduction-Mode Buck Converter Via Filippov's Method

Giaouris

Banerjee

Zahawi

et al. 2008

IEEE Trans. Circuits Syst. I

204

148

View full text Add to dashboard Cite

Abstract-To study the stability of a nominal cyclic steady state in power electronic converters, it is necessary to obtain a linearization around the periodic orbit. In many past studies, this was achieved by explicitly deriving the Poincaré map that describes the evolution of the state from one clock instant to the next and then locally linearizing the map at the fixed point. However, in many converters, the map cannot be derived in closed form, and therefore this approach cannot directly be applied. Alternatively, the orbital stability can be worked out by studying the evolution of perturbations about a nominal periodic orbit, and some studies along this line have also been reported. In this paper, we show that Filippov's method-which has commonly been applied to mechanical switching systems-can be used fruitfully in power electronic circuits to achieve the same end by describing the behavior of the system during the switchings. By combining this and the Floquet theory, it is possible to describe the stability of power electronic converters. We demonstrate the method using the example of a voltage-mode-controlled buck converter operating in continuous conduction mode. We find that the stability of a converter is strongly dependent upon the so-called saltation matrix-the state transition matrix relating the state just after the switching to that just before. We show that the Filippov approach, especially the structure of the saltation matrix, offers some additional insights on issues related to the stability of the orbit, like the recent observation that coupling with spurious signals coming from the environment causes intermittent subharmonic windows. Based on this approach, we also propose a new controller that can significantly extend the parameter range for nominal period-1 operation.

show abstract

Section: B Nonsmooth Systemsmentioning

confidence: 99%

Section: B Nonsmooth Systemsmentioning

confidence: 99%

mentioning

confidence: 99%

See 1 more Smart Citation

Stability Analysis of the Continuous-Conduction-Mode Buck Converter Via Filippov's Method

Giaouris

Banerjee

Zahawi

et al. 2008

IEEE Trans. Circuits Syst. I

204

148

View full text Add to dashboard Cite

show abstract

“…The eigenvalues of this matrix are called Floquet multipliers. For piecewise linear systems, as is the case for the system considered in this study, the monodromy matrix can be constructed from the product of the state transition matrices corresponding to each sub-cycle and the corresponding saltation matrix [Leine and Nijemeijer, 2004;Aizerman and Gantmakher, 1958;Fillipov, 1988;Giaouris et al, 2008]. Suppose a trajectory x(t) starts at time instant t i and is passing from the Configuration C i described by the vector field A i x+B i (t) := f i (x, t), intersects the switching boundary described by the equation σ i (x, t) = 0 at t i , and goes to Configuration C i+1 given by the vector field A i+1 x + B i+1 (t) := f i+1 (x, t).…”

Section: Floquet Theory and Fillipov Methodsmentioning

confidence: 99%

“…The first approach will be based on a Poincaré map function [El Aroudi et al, 2007]. The second approach is the Floquet theory ( [Aizerman and Gantmakher, 1958]) together with Fillipov methods for systems with discontinuous vector field [Fillipov, 1988], [Leine and Nijemeijer, 2004], [Giaouris et al, 2008]. The third approach is based on the FDM together with Floquet theory [Nayfeh et al, 2009], [Nayfeh and Balachandran, 1995].…”

Section: Closed-form Solutions Corresponding To the Different Linear mentioning

confidence: 99%

Analysis of Bifurcation Behavior of a Piecewise Linear Vibrator with Electromagnetic Coupling for Energy Harvesting Applications

Aroudi

Ouakad

Benadero

et al. 2014

Int. J. Bifurcation Chaos

View full text Add to dashboard Cite

Recently, nonlinearities have been shown to play an important role in increasing the extracted energy of vibration-based energy harvesting systems. In this paper, we study the dynamical behavior of a piecewise linear (PWL) spring-mass-damper system for vibration-based energy harvesting applications. First, we present a continuous time single degree of freedom PWL dynamical model of the system. Different configurations of the PWL model and their corresponding state-space regions are derived. Then, from this PWL model, extensive numerical simulations are carried out by computing time-domain waveforms, state-space trajectories and frequency responses under a deterministic harmonic excitation for different sets of system parameter values. Stability analysis is performed using Floquet theory combined with Fillipov method, Poincaré map modeling and finite difference method (FDM). The Floquet multipliers are calculated using these three approaches and a good concordance is obtained among them. The performance of the system in terms of the harvested energy is studied by considering both purely harmonic excitation and a noisy vibrational source. A frequency-domain analysis shows that the harvested energy could be larger at low frequencies as compared to an equivalent linear system, in particular for relatively low excitation intensities. This could be an advantage for potential use of this system in low frequency ambient vibrational-based energy harvesting applications.

show abstract

Foldings and grazings of tori in current controlled interleaved boost converters

Giaouris

Banerjee

Stergiopoulos

et al. 2013

Circuit Theory & Apps

View full text Add to dashboard Cite

SUMMARYInterleaved boost converters (IBCs) are used when energy conversion is required at high current levels. Such converter systems may undergo various nonlinear phenomena which can affect their performance adversely. In this paper, we study an IBC and demonstrate the first instability through a Neimark-Sacker bifurcation, resulting in a torus. An analysis based on the calculation of the monodromy matrix reveals that the torus has a rather strange form as the complex Floquet multipliers that became unstable have a real value close to À1. We show that further variation in a parameter can result in novel nonlinear phenomena where the torus itself folds and grazes a switching manifold, resulting in a 'wobbling' of the closed loop that represents the torus in discrete time. Numerical and analytical results validate our work.

show abstract

On the stability of periodic motions

Cited by 48 publications

References 0 publications

Stability Analysis of the Continuous-Conduction-Mode Buck Converter Via Filippov's Method

Stability Analysis of the Continuous-Conduction-Mode Buck Converter Via Filippov's Method

Analysis of Bifurcation Behavior of a Piecewise Linear Vibrator with Electromagnetic Coupling for Energy Harvesting Applications

Foldings and grazings of tori in current controlled interleaved boost converters

Contact Info

Product

Resources

About