Lyapunov Stability of Linear Fractional Systems: Part 1 — Definition of Fractional Energy

2013 ◽  
Author(s):  
Jean-Claude Trigeassou ◽  
Nezha Maamri ◽  
Alain Oustaloup
Keyword(s):  
Initial Conditions ◽  
Dissipation Function ◽  
Distributed Energy ◽  
Initial State ◽  
Fractional Systems ◽  
Fractional Integrator ◽  
Derivative System ◽  
The Stability ◽  
Definition Of ◽  
Infinite State

This paper addresses the stability of linear commensurate order fractional systems, Dn (X) = AX 0 < n < 1, using the infinite state approach. First, the energy of a fractional integrator is defined, using the distributed energy of its initial state. Compared to the integer order case, this energy is characterized by a long memory decay, which is the characteristic feature of fractional systems. Then, it is applied to define the energy V(t) of a one derivative system. Numerical simulations exhibit the influence of initial conditions on V(t). Thanks to the definition of a dissipation function, a stability condition is derived. Finally, the general case is investigated and a weighted Lyapunov function is derived, using a positive P matrix, related to the eigenvalues of A matrix.

Lyapunov Stability of Linear Fractional Systems: Part 2 — Derivation of a Stability Condition

2013 ◽  
Author(s):  
Jean-Claude Trigeassou ◽  
Nezha Maamri ◽  
Alain Oustaloup
Keyword(s):  
Stability Condition ◽  
Initial Conditions ◽  
Positive Real ◽  
Fractional Systems ◽  
Direct Derivation ◽  
Energy Part ◽  
Complex Eigenvalues ◽  
The Stability ◽  
Definition Of ◽  
Infinite State

This paper, composed of two parts, addresses the stability of linear commensurate order fractional systems, Dn (X) = A X 0 < n < 1, using the infinite state approach. Whereas Part 1 has been dedicated to the definition of fractional systems energy, Part 2 deals with the derivation of a stability condition. When the eigenvalues of A are real, the modal representation shows that system energy is the sum of independent modal energies, so the derivation of a stability condition is straightforward in this case. On the contrary, when the eigenvalues are complex with positive real parts, unusual energy dynamics depending on initial conditions prevent direct derivation of a stability condition. Thus, an indirect method is proposed to formulate a stability condition in the complex eigenvalues case.

Another Note on Stability of Sliding Mode Dynamics in Suppression of Fractional Chaotic Systems

2015 ◽  
Vol 743 ◽  
pp. 303-306
Author(s):  
J. Yuan ◽  
B. Shi ◽  
Yan Wang
Keyword(s):  
Stability Analysis ◽  
Chaotic Systems ◽  
Sliding Mode ◽  
Fractional Systems ◽  
Infinite Dimensional ◽  
Fractional Integrator ◽  
Fractional Differential ◽  
Continuous Frequency ◽  
The Stability ◽  
Sliding Mode Dynamics

This paper revisits the stability analysis of sliding mode dynamics in suppression of a classof fractional chaotic systems by a different approach. Firstly, we convert fractional differential equationsinto infinite dimensional ordinary differential equations based on the continuous frequency distributedmodel of the fractional integrator. Then we choose a Lyapunov function candidate to proposethe stability analysis. The result applies to both the commensurate fractional systems and the incommensurateones.

Initial Conditions and Initialization of Fractional Systems

2016 ◽  
Vol 11 (4) ◽  
Author(s):  
Massinissa Tari ◽  
Nezha Maamri ◽  
Jean-Claude Trigeassou
Keyword(s):  
Initial Conditions ◽  
Fractional Order Systems ◽  
Fractional Systems ◽  
Infinite Dimensional ◽  
State Variable ◽  
Future Behavior ◽  
Fractional Integrator ◽  
True State ◽  
Fractional System ◽  
Dimensional Variable

In this paper, the initialization of fractional order systems is analyzed. The objective is to prove that the usual pseudostate variable x(t) is unable to predict the future behavior of the system, whereas the infinite dimensional variable z(ω, t) fulfills the requirements of a true state variable. Two fractional systems, a fractional integrator and a one-derivative fractional system, are analyzed with the help of elementary tests and numerical simulations. It is proved that the dynamic behaviors of these two fractional systems differ completely from that of their integer order counterparts. More specifically, initialization of these systems requires knowledge of z(ω,t0) initial condition.

Effects of Initial Conditions and Circuit Parameters on the SBT-Memristor-Based Chaotic Circuit

2019 ◽  
Vol 29 (12) ◽  
pp. 1950171 ◽  
Author(s):  
Gang Dou ◽  
Huaying Duan ◽  
Wenyan Yang ◽  
Hai Yang ◽  
Mei Guo ◽  
...  
Keyword(s):  
Chaotic System ◽  
Initial Conditions ◽  
Chaotic Circuit ◽  
Mathematical Methods ◽  
Initial State ◽  
Dynamical Characteristics ◽  
Dynamical Behaviors ◽  
Initial States ◽  
The Stability ◽  
Stable Points

In the paper, a fourth-order SBT-memristor-based chaotic system described by the flux-controlled model is investigated. The stability of the chaotic system is analyzed, and the effects of initial conditions and circuit parameters on the SBT-memristor-based chaotic circuit are discussed by mathematical methods of Lyapunov exponents spectra, bifurcation diagrams, phase orbits and Poincaré maps. Through simulations, it is observed that the dynamical characteristics vary with initial states and circuit parameters. Complex dynamical behaviors such as stable points, period cycles and chaos can be found in the SBT-memristor-based system. It is also found that the system exhibits multistability, which is closely dependent on the initial state of the SBT memristor. This study provides insightful guidance for the design and analysis of memristor-based circuits towards potential real applications.

Initialization of Riemann-Liouville and Caputo Fractional Derivatives

2011 ◽  
Author(s):  
Jean-Claude Trigeassou ◽  
Nezha Maamri ◽  
Alain Oustaloup
Keyword(s):  
Numerical Simulations ◽  
State Vector ◽  
Fractional Derivatives ◽  
Initial Conditions ◽  
Fractional Integration ◽  
Initial State ◽  
Caputo Fractional Derivatives ◽  
Infinite Dimensional ◽  
Fractional Integrator ◽  
Initial Conditions Problem

Riemann-Liouville and Caputo fractional derivatives are fundamentally related to fractional integration operators. Consequently, the initial conditions of fractional derivatives are the frequency distributed and infinite dimensional state vector of fractional integrators. The paper is dedicated to the estimation of these initial conditions and to the validation of the initialization problem based on this distributed state vector. Numerical simulations applied to Riemann-Liouville and Caputo derivatives demonstrate that the initial conditions problem can be solved thanks to the estimation of the initial state vector of the fractional integrator.

Postmodern More

Moreana ◽  
2003 ◽  
Vol 40 (Number 153- (1-2) ◽  
pp. 219-239
Author(s):  
Anne Lake Prescott
Keyword(s):  
Human Language ◽  
Thomas More ◽  
The Stability ◽  
Definition Of ◽  
Recent Critic

Thomas More is often called a “humanist,” and rightly so if the word has its usual meaning in scholarship on the Renaissance. “Humanist” has by now acquired so many different and contradictory meanings, however, that it needs to be applied carefully to the likes of More. Many postmodernists tend to use the word, pejoratively, to mean someone who believes in an autonomous self, the stability of words, reason, and the possibility of determinable meanings. Without quite arguing that More was a postmodernist avant la lettre, this essay suggests that he was not a “humanist” who stalks the pages of much recent postmodernist theory and that in fact even while remaining a devout Catholic and sensible lawyer he was quite as aware as any recent critic of the slipperiness of human selves and human language. It is time that literary critics tightened up their definition of “humanist,” especially when writing about the Renaissance.

scholarly journals Handling Initial Conditions in Vector Fitting for Real Time Modeling of Power System Dynamics

Energies ◽  
2021 ◽  
Vol 14 (9) ◽  
pp. 2471
Author(s):  
Tommaso Bradde ◽  
Samuel Chevalier ◽  
Marco De Stefano ◽  
Stefano Grivet-Talocia ◽  
Luca Daniel
Keyword(s):  
Dynamical Systems ◽  
Power System ◽  
System Dynamics ◽  
Real Time ◽  
Initial Conditions ◽  
Test System ◽  
Initial State ◽  
Power System Dynamics ◽  
Vector Fitting ◽  
Time Modeling

This paper develops a predictive modeling algorithm, denoted as Real-Time Vector Fitting (RTVF), which is capable of approximating the real-time linearized dynamics of multi-input multi-output (MIMO) dynamical systems via rational transfer function matrices. Based on a generalization of the well-known Time-Domain Vector Fitting (TDVF) algorithm, RTVF is suitable for online modeling of dynamical systems which experience both initial-state decay contributions in the measured output signals and concurrently active input signals. These adaptations were specifically contrived to meet the needs currently present in the electrical power systems community, where real-time modeling of low frequency power system dynamics is becoming an increasingly coveted tool by power system operators. After introducing and validating the RTVF scheme on synthetic test cases, this paper presents a series of numerical tests on high-order closed-loop generator systems in the IEEE 39-bus test system.

scholarly journals Influence of study model, baseline catalytic concentrations and analytical system on the stability of serum alanine aminotransferase

2020 ◽  
Vol 1 (2) ◽  
Author(s):  
Josep Miquel Bauça ◽  
Andrea Caballero ◽  
Carolina Gómez ◽  
Débora Martínez-Espartosa ◽  
Isabel García del Pino ◽  
...  
Keyword(s):  
Alanine Aminotransferase ◽  
Experimental Models ◽  
Individual Variability ◽  
Serum Samples ◽  
Multicentric Study ◽  
Different Temperatures ◽  
The Stability ◽  
Analytical System ◽  
Definition Of ◽  
Analytical Platforms

AbstractObjectivesThe stability of the analytes most commonly used in routine clinical practice has been the subject of intensive research, with varying and even conflicting results. Such is the case of alanine aminotransferase (ALT). The purpose of this study was to determine the stability of serum ALT according to different variables.MethodsA multicentric study was conducted in eight laboratories using serum samples with known initial catalytic concentrations of ALT within four different ranges, namely: <50 U/L (<0.83 μkat/L), 50–200 U/L (0.83–3.33 μkat/L), 200–400 U/L (3.33–6.67 μkat/L) and >400 U/L (>6.67 μkat/L). Samples were stored for seven days at two different temperatures using four experimental models and four laboratory analytical platforms. The respective stability equations were calculated by linear regression. A multivariate model was used to assess the influence of different variables.ResultsCatalytic concentrations of ALT decreased gradually over time. Temperature (−4%/day at room temperature vs. −1%/day under refrigeration) and the analytical platform had a significant impact, with Architect (Abbott) showing the greatest instability. Initial catalytic concentrations of ALT only had a slight impact on stability, whereas the experimental model had no impact at all.ConclusionsThe constant decrease in serum ALT is reduced when refrigerated. Scarcely studied variables were found to have a significant impact on ALT stability. This observation, added to a considerable inter-individual variability, makes larger studies necessary for the definition of stability equations.

scholarly journals Local modal participation analysis of nonlinear systems using Poincaré linearization

2019 ◽  
Vol 99 (1) ◽  
pp. 803-811 ◽  
Author(s):  
Boumediene Hamzi ◽  
Eyad H. Abed
Keyword(s):  
Nonlinear Systems ◽  
Autonomous Systems ◽  
Analytical Formula ◽  
Linear Case ◽  
Local Mode ◽  
Initial Condition ◽  
Initial State ◽  
State Participation ◽  
Symmetry Assumption ◽  
Definition Of

AbstractThe paper studies an extension to nonlinear systems of a recently proposed approach to the definition of modal participation factors. A definition is given for local mode-in-state participation factors for smooth nonlinear autonomous systems. While the definition is general, the resulting measures depend on the assumed uncertainty law governing the system initial condition, as in the linear case. The work follows Hashlamoun et al. (IEEE Trans Autom Control 54(7):1439–1449 2009) in taking a mathematical expectation (or set-theoretic average) of a modal contribution measure over an uncertain set of system initial state. Poincaré linearization is used to replace the nonlinear system with a locally equivalent linear system. It is found that under a symmetry assumption on the distribution of the initial state, the tractable calculation and analytical formula for mode-in-state participation factors found for the linear case persists to the nonlinear setting. This paper is dedicated to the memory of Professor Ali H. Nayfeh.

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