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Related Concept Videos

Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

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...
Application of Linearization and Approximation01:29

Application of Linearization and Approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...
Linear time-invariant Systems01:23

Linear time-invariant Systems

A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be calculated...
Linear Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.

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Related Experiment Video

Updated: Jun 19, 2026

WheelCon: A Wheel Control-Based Gaming Platform for Studying Human Sensorimotor Control
08:18

WheelCon: A Wheel Control-Based Gaming Platform for Studying Human Sensorimotor Control

Published on: August 15, 2020

Optimal linear-consensus algorithms: an LQR perspective.

Yongcan Cao1, Wei Ren

  • 1Electrical and Computer Engineering Department, Utah State University, Logan, UT 84322-4120, USA. yongcan.cao@aggiemail.usu.edu

IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics : a Publication of the IEEE Systems, Man, and Cybernetics Society
|November 4, 2009
PubMed
Summary
This summary is machine-generated.

This study optimizes linear-consensus algorithms for multi-vehicle systems using Laplacian matrices. Researchers derived optimal nonsymmetric Laplacian matrices for interaction-free systems and optimal scaling factors for interaction-related systems.

Related Experiment Videos

Last Updated: Jun 19, 2026

WheelCon: A Wheel Control-Based Gaming Platform for Studying Human Sensorimotor Control
08:18

WheelCon: A Wheel Control-Based Gaming Platform for Studying Human Sensorimotor Control

Published on: August 15, 2020

Area of Science:

  • Control Theory
  • Robotics
  • Graph Theory

Background:

  • Laplacian matrices are fundamental to linear-consensus algorithms.
  • Multi-vehicle systems with single-integrator dynamics require efficient consensus protocols.

Purpose of the Study:

  • To derive optimal linear-consensus algorithms for multi-vehicle systems.
  • To investigate both continuous-time and discrete-time settings.
  • To propose and analyze novel cost functions.

Main Methods:

  • Utilized a linear-quadratic-regulator-based method.
  • Derived optimal nonsymmetric Laplacian matrices for interaction-free cost functions.
  • Determined optimal scaling factors for interaction-related cost functions.

Main Results:

  • The optimal nonsymmetric Laplacian matrix corresponds to a complete directed graph.
  • Any symmetric Laplacian matrix can be inverse optimal under specific cost functions.
  • Optimal scaling factors were derived for prespecified symmetric Laplacian matrices.

Conclusions:

  • The study provides a framework for designing optimal linear-consensus algorithms.
  • Findings are applicable to both continuous-time and discrete-time multi-vehicle systems.
  • The derived optimal matrices and scaling factors enhance system coordination and performance.