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Related Concept Videos

Linear Approximation in Time Domain01:21

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, the...
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.
Classification of Systems-I01:26

Classification of Systems-I

Linearity is a system property characterized by a direct input-output relationship, combining homogeneity and additivity.
Homogeneity dictates that if an input x(t) is multiplied by a constant c, the output y(t) is multiplied by the same constant. Mathematically, this is expressed as:
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...
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...
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...

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Related Experiment Video

Updated: Jul 7, 2026

Assembly and Characterization of Biomolecular Memristors Consisting of Ion Channel-doped Lipid Membranes
08:07

Assembly and Characterization of Biomolecular Memristors Consisting of Ion Channel-doped Lipid Membranes

Published on: March 9, 2019

Complete memory structures for approximating nonlinear discrete-time mappings.

B W Stiles1, I W Sandberg, J Ghosh

  • 1Dept. of Electr. and Comput. Eng., Texas Univ., Austin, TX.

IEEE Transactions on Neural Networks
|January 1, 1997
PubMed
Summary

This study presents a novel two-stage structure for approximating nonlinear discrete-time systems. A "complete memory" concept is introduced, enabling accurate modeling of complex system dynamics for artificial neural network design.

Related Experiment Videos

Last Updated: Jul 7, 2026

Assembly and Characterization of Biomolecular Memristors Consisting of Ion Channel-doped Lipid Membranes
08:07

Assembly and Characterization of Biomolecular Memristors Consisting of Ion Channel-doped Lipid Membranes

Published on: March 9, 2019

Area of Science:

  • Control Systems Engineering
  • Nonlinear Dynamics
  • Computational Neuroscience

Background:

  • Nonlinear discrete-time systems are prevalent in various scientific and engineering fields.
  • Accurate modeling of these systems is crucial for analysis and control.
  • Existing modeling structures may have limitations in capturing complex dynamics.

Purpose of the Study:

  • To introduce a general, two-stage structure for approximating nonlinear discrete-time systems.
  • To define and utilize the concept of "complete memory" for enhanced modeling capabilities.
  • To provide a template for designing artificial neural networks for spatiotemporal processing.

Main Methods:

  • A two-stage structure comprising a dynamical stage and a memoryless nonlinear stage.
  • Development of a theorem establishing necessary and sufficient conditions for modeling capability.
  • Introduction and application of the "complete memory" concept.

Main Results:

  • A theorem proves that specific structures with a "complete memory" dynamical stage can approximate a wide class of nonlinear discrete-time systems.
  • Demonstration that bounded-input bounded-output, time-invariant, causal memory structures approximate system dynamics if and only if they possess "complete memory".
  • Presentation of linear and nonlinear examples of "complete memory" structures.

Conclusions:

  • The proposed "complete memory" structure offers a powerful and general approach to modeling nonlinear discrete-time systems.
  • This structure serves as a foundational template for developing advanced artificial neural networks tailored for nonlinear spatiotemporal data processing.
  • The findings advance the theoretical understanding of system approximation and provide practical design guidelines.