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Related Concept Videos

Routh-Hurwitz Criterion II01:19

Routh-Hurwitz Criterion II

639
In the application of the Routh-Hurwitz criterion, two specific scenarios can arise that complicate stability analysis.
The first scenario occurs when a singular zero appears in the first column of the Routh table. This situation creates a division by zero issues. To resolve this, a small positive or negative number, denoted as epsilon (∈), is substituted for the zero. The stability analysis proceeds by assuming a sign for ∈. If ∈ is positive, any sign change in the first...
639
Quadratic Models01:23

Quadratic Models

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Quadratic models are mathematical representations used to describe relationships in which the rate of change changes at a constant rate. These models appear in a wide variety of natural and engineered systems, especially those involving motion, forces, and optimization. One common application is analyzing the vertical motion of objects influenced by gravity, such as a ball thrown into the air.In such scenarios, the object's height changes over time in a curved pattern, rising to a maximum point...
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Routh-Hurwitz Criterion I01:15

Routh-Hurwitz Criterion I

416
Consider an electrical power grid, where stability is essential to prevent blackouts. The Routh-Hurwitz criterion is a valuable tool for assessing system stability under varying load conditions or faults. By analyzing the closed-loop transfer function, the Routh-Hurwitz criterion helps determine whether the system remains stable.
To apply the Routh-Hurwitz criterion, a Routh table is constructed. The table's rows are labeled with powers of the complex frequency variable s, starting from the...
416
Quadratic Equations in the Complex Number System01:29

Quadratic Equations in the Complex Number System

118
A quadratic equation in the form ax2+bx+c=0 can have solutions that vary in nature depending on the value of the discriminant, b2−4ac. In this expression, a is the coefficient of the quadratic term x2, b is the coefficient of the linear term x, and c is the constant term. When the discriminant is negative, the equation has no real number solutions. However, by introducing complex numbers through the imaginary unit i, defined by i=-1, these equations can still be solved.The square root of a...
118
Quadratic Equations01:29

Quadratic Equations

99
A quadratic equation is an algebraic expression where a variable is raised to the second power and combined with its first power and a constant; all equated to zero. These equations are frequently used to model relationships involving area, motion, and optimization. The general representation of a quadratic equation iswhere a, b, and c are real values, and a is nonzero to ensure the presence of the squared term.One method for solving a quadratic equation involves rewriting it as a product of...
99
Application of Nonlinear Inequalities01:29

Application of Nonlinear Inequalities

77
A nonlinear inequality describes a comparison involving an expression that curves or behaves more complexly than a straight line. These inequalities often appear in forms that include squares, products, or variables in the denominator.To solve such an inequality, one starts by rewriting it so that zero appears on one side. For example, the inequality:  can be factored as: This form makes it easier to identify the values that cause the expression to equal zero. In this case, the key values...
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Related Experiment Videos

Improved Stability Criteria for Delayed Neural Networks Using a Quadratic Function Negative-Definiteness Approach.

Jun Chen, Xian-Ming Zhang, Ju H Park

    IEEE Transactions on Neural Networks and Learning Systems
    |December 16, 2020
    PubMed
    Summary

    This study enhances neural network stability analysis for time-varying delays. A novel Lyapunov-Krasovskii functional approach offers a less conservative stability criterion for delayed neural networks.

    Related Experiment Videos

    Area of Science:

    • Control Systems Engineering
    • Computational Neuroscience
    • Applied Mathematics

    Background:

    • Neural networks with time-varying delays present significant stability challenges.
    • Existing Lyapunov-Krasovskii (L-K) functional methods can be conservative.
    • Accurate stability analysis is crucial for reliable neural network applications.

    Purpose of the Study:

    • To develop a more general and less conservative stability criterion for neural networks with time-varying delays.
    • To introduce a simplified method for calculating stability-related coefficients.
    • To improve upon existing L-K functional approaches for delayed neural networks.

    Main Methods:

    • Utilizing a quadratic function negative-definiteness approach.
    • Employing a general reciprocally convex combination inequality to augment the L-K functional derivative.
    • Estimating the L-K functional derivative using a novel quadratic function on the time-varying delay.

    Main Results:

    • A novel, less conservative stability criterion for delayed neural networks was derived.
    • A simplified coefficient calculation method was introduced, reducing manual computation.
    • The proposed method demonstrated improved performance over existing techniques in numerical examples.

    Conclusions:

    • The enhanced L-K functional approach provides a more effective method for analyzing the stability of neural networks with time-varying delays.
    • The simplified coefficient calculation enhances the practicality of the method.
    • The derived hierarchical stability criterion offers a significant improvement in reducing conservatism.