Stability Analysis Using Monodromy Matrix for Impacting Systems

2018 ◽  
Vol E101.A (6) ◽  
pp. 904-914 ◽  
Author(s):  
Hiroyuki ASAHARA ◽  
Takuji KOUSAKA
Keyword(s):  
Stability Analysis ◽  
Monodromy Matrix ◽  
Impacting Systems

scholarly journals Nonlinear Dynamics and Stability Analysis of a Three-Cell Flying Capacitor DC-DC Converter

2021 ◽  
Vol 11 (4) ◽  
pp. 1395
Author(s):  
Abdelali El Aroudi ◽  
Natalia Cañas-Estrada ◽  
Mohamed Debbat ◽  
Mohamed Al-Numay
Keyword(s):  
Steady State ◽  
Stability Analysis ◽  
Monodromy Matrix ◽  
Dynamical Behavior ◽  
Period Doubling ◽  
Simple Expression ◽  
Buck Converter ◽  
State Variables ◽  
Bifurcation Phenomena ◽  
The Stability

This paper presents a study of the nonlinear dynamic behavior a flying capacitor four-level three-cell DC-DC buck converter. Its stability analysis is performed and its stability boundaries is determined in the multi-dimensional paramertic space. First, the switched model of the converter is presented. Then, a discrete-time controller for the converter is proposed. The controller is is responsible for both balancing the flying capacitor voltages from one hand and for output current regulation. Simulation results from the switched model of the converter under the proposed controller are presented. The results show that the system may undergo bifurcation phenomena and period doubling route to chaos when some system parameters are varied. One-dimensional bifurcation diagrams are computed and used to explore the possible dynamical behavior of the system. By using Floquet theory and Filippov method to derive the monodromy matrix, the bifurcation behavior observed in the converter is accurately predicted. Based on justified and realistic approximations of the system state variables waveforms, simple and accurate expressions for these steady-state values and the monodromy matrix are derived and validated. The simple expression of the steady-state operation and the monodromy matrix allow to analytically predict the onset of instability in the system and the stability region in the parametric space is determined. Numerical simulations from the exact switched model validate the theoretical predictions.

Simulation and stability analysis of impacting systems with complete chattering

2009 ◽  
Vol 58 (1-2) ◽  
pp. 85-106 ◽  
Author(s):  
Arne B. Nordmark ◽  
Petri T. Piiroinen
Keyword(s):  
Stability Analysis ◽  
Impacting Systems

scholarly journals Stability Analysis in Positive Output and Negative Output DC to DC Converters Used in Renewable Energy Applications

2019 ◽  
Author(s):  
Maheswari Ellappan ◽  
Kavitha Anbukumar
Keyword(s):  
Renewable Energy ◽  
Stability Analysis ◽  
Output Voltage ◽  
Monodromy Matrix ◽  
Period Doubling ◽  
Current Mode ◽  
Power Converter ◽  
Control Technique ◽  
Step Down ◽  
The Stability

The renewable energy source plays a major role in the grid side power production. The stability analysis is very essential in the renewable energy converters. In this paper the bifurcation is analyzed in ZETA converter and Continuous input and output(CIO) power Buck Boost converter. The ZETA converter gives positive step down and step up output voltage and the CIO power converter gives the negative step up and step down output voltage. These converters are used in the DC micro grid with renewable energy as the source. The current mode control technique is applied to analyze the bifurcation behavior and the reference current is taken as the bifurcation parameter. When the reference current is varied, both the converters loses its stability and it enters into chaotic region through period doubling bifurcation. The simulation results are presented to study the performance behavior of both the converters. The stability region of both the converters are determined by deriving the Monodromy matrix approach.

Linear stability analysis of the figure-eight orbit in the three-body problem

2007 ◽  
Vol 27 (6) ◽  
pp. 1947-1963 ◽  
Author(s):  
GARETH E. ROBERTS
Keyword(s):  
Stability Analysis ◽  
Hamiltonian Systems ◽  
Linear Stability Analysis ◽  
Body Problem ◽  
Monodromy Matrix ◽  
Three Body Problem ◽  
Wide Range ◽  
Linear Differential ◽  
Three Body ◽  
Characteristic Multipliers

AbstractWe show that the well-known figure-eight orbit of the three-body problem is linearly stable. Building on the strong amount of symmetry present, the monodromy matrix for the figure-eight is factored so that its stability can be determined from the first twelfth of the orbit. Using a clever change of coordinates, the problem is then reduced to a 2×2 matrix whose entries depend on solutions of the associated linear differential system. These entries are estimated rigorously using only a few steps of a Runge–Kutta–Fehlberg algorithm. From this, we conclude that the characteristic multipliers are distinct and lie on the unit circle. The methods and results presented are applicable to a wide range of Hamiltonian systems containing symmetric periodic solutions.

Response and Stability Analysis of Periodic Delayed Systems With Discontinuous Distributed Delay

2012 ◽  
Vol 7 (3) ◽  
Author(s):  
Oleg A. Bobrenkov ◽  
Morad Nazari ◽  
Eric A. Butcher
Keyword(s):  
Stability Analysis ◽  
Continuous Time ◽  
Delay Differential Equations ◽  
Monodromy Matrix ◽  
Distributed Delay ◽  
Mathieu Equation ◽  
Adaptive Meshing ◽  
Delayed Systems ◽  
Time Approximation ◽  
Delay Differential

In this paper, the analysis of delay differential equations with periodic coefficients and discontinuous distributed delay is carried out through discretization by the Chebyshev spectral continuous time approximation (ChSCTA). These features are introduced in the delayed Mathieu equation with discontinuous distributed delay which is used as an illustrative example. The efficiency of stability analysis is improved by using shifted Chebyshev polynomials for computing the monodromy matrix, as well as the adaptive meshing of the parameter plane. An idea for a method for numerical integration of periodic DDEs with discontinuous distributed delay based on existing MATLAB functions is proposed.

Approximate Stability Analysis and Computation of Solutions of Nonlinear Delay Differential Algebraic Equations with Time Periodic Coefficients

2010 ◽  
Vol 16 (7-8) ◽  
pp. 1235-1260 ◽  
Author(s):  
Venkatesh Deshmukh
Keyword(s):  
Stability Analysis ◽  
Monodromy Matrix ◽  
Original System ◽  
Differential Algebraic Equations ◽  
Algebraic Equations ◽  
Periodic Coefficients ◽  
Dimensional Approximation ◽  
Delay Differential ◽  
Time Periodic ◽  
Nonlinear Delay

Approximate stability analysis of nonlinear delay differential algebraic equations (DDAEs) with periodic coefficients is proposed with a geometric interpretation of evolution of the linearized system. Firstly, a numerical algorithm based on direct integration by expansion in terms of Chebyshev polynomials is derived for linear analysis. The proposed algorithm is shown to have deeper connections with and be computationally less cumbersome than the solution of the underlying semi-explicit system via a similarity transformation. The stability of time periodic DDAE systems is characterized by the spectral radius of a “monodromy matrix”, which is a finite-dimensional approximation of a compact infinite-dimensional operator. The monodromy matrix is essentially a map of the Chebyshev coefficients (or collocation vector) of the state from the delay interval to the next adjacent interval of time. The computations are entirely performed with the original system to avoid cumbersome transformations associated with the semi-explicit form of the system. Next, two computational algorithms, the first based on perturbation series and the second based on Chebyshev spectral collocation, are detailed to obtain solutions of nonlinear DDAEs with periodic coefficients for consistent initial functions.

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