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.

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.

scholarly journals On the Recursive Sequencexn=A+xn−kp/xn−1r

2009 ◽  
Vol 2009 ◽  
pp. 1-8 ◽  
Author(s):  
Fangkuan Sun ◽  
Xiaofan Yang ◽  
Chunming Zhang
Keyword(s):  
Difference Equation ◽  
Dynamic Behavior ◽  
Positive Solutions ◽  
Initial Conditions ◽  
Real Numbers ◽  
Positive Real ◽  
The Difference ◽  
The Stability

This paper studies the dynamic behavior of the positive solutions to the difference equationxn=A+xn−kp/xn−1r,n=1,2,…, whereA,p, andrare positive real numbers, and the initial conditions are arbitrary positive numbers. We establish some results regarding the stability and oscillation character of this equation forp∈(0,1).

Dynamic Analysis of Three Parallel Beams With Electrostatic Force Interaction

2011 ◽  
Author(s):  
Pezhman A. Hassanpour ◽  
Patricia M. Nieva ◽  
Amir Khajepour
Keyword(s):  
Time Domain ◽  
Stability Condition ◽  
Initial Conditions ◽  
Electrostatic Force ◽  
Equations Of Motion ◽  
Beam Theory ◽  
Stability Margin ◽  
Modal Damping ◽  
The Stability ◽  
Euler Bernoulli Beam

In this paper, the dynamics of a micro-machined structure with three parallel cantilevers is investigated. The cantilevers are electrically charged and apply electrostatic force to each other. The governing equations of motion are derived using Euler-Bernoulli beam theory and considering structural modal damping. The stability condition of the beams for various electric charges is also studied. In addition, the equations of motion are integrated to obtain the response of the beams in time-domain for a range of initial conditions. This response is used to study the behavior of the beams at the stability margin. The end application of the structure under investigation is in the device characterization. The dynamic stability condition and time-domain responses are used to investigate the reliability of the characterization. Once translated back to physical quantities, these results can be used for improving the measurements.

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 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 Computing the Exact Number of Similarity Classes in the Longest Edge Bisection of Tetrahedra

Mathematics ◽  
2021 ◽  
Vol 9 (12) ◽  
pp. 1447
Author(s):  
Jose P. Suárez ◽  
Agustín Trujillo ◽  
Tania Moreno
Keyword(s):  
Data Structure ◽  
Stability Condition ◽  
Open Problem ◽  
Integer Arithmetic ◽  
Exact Number ◽  
Similarity Class ◽  
The Stability ◽  
High Level ◽  
The Cost

Showing whether the longest-edge (LE) bisection of tetrahedra meshes degenerates the stability condition or not is still an open problem. Some reasons, in part, are due to the cost for achieving the computation of similarity classes of millions of tetrahedra. We prove the existence of tetrahedra where the LE bisection introduces, at most, 37 similarity classes. This family of new tetrahedra was roughly pointed out by Adler in 1983. However, as far as we know, there has been no evidence confirming its existence. We also introduce a new data structure and algorithm for computing the number of similarity tetrahedral classes based on integer arithmetic, storing only the square of edges. The algorithm lets us perform compact and efficient high-level similarity class computations with a cost that is only dependent on the number of similarity classes.

scholarly journals Stability Analysis of Distributed Order Fractional Differential Equations

2011 ◽  
Vol 2011 ◽  
pp. 1-12 ◽  
Author(s):  
H. Saberi Najafi ◽  
A. Refahi Sheikhani ◽  
A. Ansari
Keyword(s):  
Stability Analysis ◽  
Characteristic Function ◽  
Differential Equations ◽  
Density Function ◽  
Robust Stability ◽  
Fractional Differential Equations ◽  
Stability Condition ◽  
Fractional Differential ◽  
The Stability ◽  
Distributed Order

We analyze the stability of three classes of distributed order fractional differential equations (DOFDEs) with respect to the nonnegative density function. In this sense, we discover a robust stability condition for these systems based on characteristic function and new inertia concept of a matrix with respect to the density function. Moreover, we check the stability of a distributed order fractional WINDMI system to illustrate the validity of proposed procedure.

scholarly journals On the Solutions of the System of Difference Equationsxn+1=max{A/xn,yn/xn},yn+1=max{A/yn,xn/yn}

2009 ◽  
Vol 2009 ◽  
pp. 1-11 ◽  
Author(s):  
Dağistan Simsek ◽  
Bilal Demir ◽  
Cengiz Cinar
Keyword(s):  
Difference Equations ◽  
Initial Conditions ◽  
Real Numbers ◽  
System Of Difference Equations ◽  
Positive Real

We study the behavior of the solutions of the following system of difference equationsxn+1=max⁡{A/xn,yn/xn},yn+1=max⁡{A/yn,xn/yn}where the constantAand the initial conditions are positive real numbers.

scholarly journals Reliable Learning with PDE-Based CNNs and DenseNets for Detecting COVID-19, Pneumonia, and Tuberculosis from Chest X-Ray Images

Mathematics ◽  
2021 ◽  
Vol 9 (4) ◽  
pp. 434
Author(s):  
Anca Nicoleta Marginean ◽  
Delia Doris Muntean ◽  
George Adrian Muntean ◽  
Adelina Priscu ◽  
Adrian Groza ◽  
...  
Keyword(s):  
Transfer Learning ◽  
Negative Impact ◽  
X Ray ◽  
Domain Specific ◽  
Direct Training ◽  
Public Data ◽  
Normal Chest ◽  
Chest X Ray ◽  
The Stability ◽  
Definition Of

It has recently been shown that the interpretation by partial differential equations (PDEs) of a class of convolutional neural networks (CNNs) supports definition of architectures such as parabolic and hyperbolic networks. These networks have provable properties regarding the stability against the perturbations of the input features. Aiming for robustness, we tackle the problem of detecting changes in chest X-ray images that may be suggestive of COVID-19 with parabolic and hyperbolic CNNs and with domain-specific transfer learning. To this end, we compile public data on patients diagnosed with COVID-19, pneumonia, and tuberculosis, along with normal chest X-ray images. The negative impact of the small number of COVID-19 images is reduced by applying transfer learning in several ways. For the parabolic and hyperbolic networks, we pretrain the networks on normal and pneumonia images and further use the obtained weights as the initializers for the networks to discriminate between COVID-19, pneumonia, tuberculosis, and normal aspects. For DenseNets, we apply transfer learning twice. First, the ImageNet pretrained weights are used to train on the CheXpert dataset, which includes 14 common radiological observations (e.g., lung opacity, cardiomegaly, fracture, support devices). Then, the weights are used to initialize the network which detects COVID-19 and the three other classes. The resulting networks are compared in terms of how well they adapt to the small number of COVID-19 images. According to our quantitative and qualitative analysis, the resulting networks are more reliable compared to those obtained by direct training on the targeted dataset.

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