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Second Order systems II

Second Order systems II

In an underdamped second-order system, where the damping ratio ζ is between 0 and 1, a unit-step input results in a transfer function that, when transformed using the inverse Laplace method, reveals the output response. The output exhibits a damped sinusoidal oscillation, and the difference between the input and output is termed the error signal. This error signal also demonstrates damped oscillatory behavior. Eventually, as the system reaches a steady state, the error diminishes to zero.

Types of Damping

Types of Damping

If the amount of damping in a system is gradually increased, the period and frequency start to become affected because damping opposes, and hence slows, the back and forth motion (the net force is smaller in both directions). If there is a very large amount of damping, the system does not even oscillate; instead, it slowly moves toward equilibrium. In brief, an overdamped system moves slowly towards equilibrium, whereas an underdamped system moves quickly to equilibrium but will oscillate about...

Linear Approximation in Time Domain

Linear Approximation in Time Domain

Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...

Damped Oscillations

Damped Oscillations

In the real world, oscillations seldom follow true simple harmonic motion. A system that continues its motion indefinitely without losing its amplitude is termed undamped. However, friction of some sort usually dampens the motion, so it fades away or needs more force to continue. For example, a guitar string stops oscillating a few seconds after being plucked. Similarly, one must continually push a swing to keep a child swinging on a playground.
Although friction and other non-conservative...

Second Order systems I

Second Order systems I

A servo system exemplifies a second-order system, featuring a proportional controller and load elements that ensure the output position aligns with the input position. The relationship between these components is described by a second-order differential equation. Applying the Laplace transform under zero initial conditions yields the transfer function, showing how inputs are converted to outputs in the system.
By reinterpreting the system, one can derive the closed-loop transfer function, which...

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...

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Related Experiment Video

Updated: Dec 24, 2025

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

Published on: October 28, 2022

A second-order dynamical approach with variable damping to nonconvex smooth minimization.

Radu Ioan Boţ¹, Ernö Robert Csetnek¹, Szilárd Csaba László²

¹Faculty of Mathematics, University of Vienna, Vienna, Austria.

Applicable Analysis

|April 8, 2020

Summary

This summary is machine-generated.

This study introduces a second-order dynamical system for minimizing nonconvex functions. The system

Keywords:

65K10 90C26 90C30 Boris Mordukhovich Kurdyka–Łojasiewicz inequality Second-order dynamical system convergence rate nonconvex optimization

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Last Updated: Dec 24, 2025

Design and Application of a Fault Detection Method Based on Adaptive Filters and Rotational Speed Estimation for an Electro-Hydrostatic Actuator

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Published on: October 28, 2022

Studying Large Amplitude Oscillatory Shear Response of Soft Materials

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Published on: April 25, 2019

Area of Science:

Optimization Theory
Dynamical Systems
Non-convex Analysis

Background:

Minimizing non-convex functions is challenging due to local minima.
Second-order dynamical systems offer potential for efficient optimization.
Nesterov's accelerated gradient method provides a benchmark for convex optimization.

Purpose of the Study:

To investigate a second-order dynamical system with variable damping for non-convex function minimization.
To analyze the convergence properties of this dynamical system.
To establish convergence rates based on the Kurdyka-Lojasiewicz property.

Main Methods:

Formulating a second-order dynamical system inspired by Nesterov's accelerated gradient method.
Applying regularization to the objective function.
Analyzing the system's trajectory convergence under the Kurdyka-Lojasiewicz property.

Main Results:

The generated trajectory converges to a critical point under specific regularization conditions.
Convergence rates are provided, dependent on the Lojasiewicz exponent.
The variable damping mechanism is shown to be effective.

Conclusions:

The proposed second-order dynamical system is a viable approach for non-convex function minimization.
The Kurdyka-Lojasiewicz property is crucial for guaranteeing convergence.
The study offers theoretical insights into the convergence behavior of accelerated methods for non-convex problems.