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Related Concept Videos

State Space Representation01:27

State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
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 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.
Multi-input and Multi-variable systems01:22

Multi-input and Multi-variable systems

Cruise control systems in cars are designed as multi-input systems to maintain a driver's desired speed while compensating for external disturbances such as changes in terrain. The block diagram for a cruise control system typically includes two main inputs: the desired speed set by the driver and any external disturbances, such as the incline of the road. By adjusting the engine throttle, the system maintains the vehicle's speed as close to the desired value as possible.
In the absence of...
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...

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Related Experiment Video

Updated: Jul 9, 2026

Quasi-light Storage for Optical Data Packets
07:45

Quasi-light Storage for Optical Data Packets

Published on: February 6, 2014

Space bandwidth-efficient realizations of linear systems.

M A Kutay, M F Erden, H M Ozaktas

    Optics Letters
    |December 19, 2007
    PubMed
    Summary

    Multistage and multichannel fractional Fourier-domain filters offer flexible trade-offs between accuracy and efficiency for linear systems. This approach systematically exploits matrix structure for efficient computation in various applications.

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    Gain-compensation Methodology for a Sinusoidal Scan of a Galvanometer Mirror in Proportional-Integral-Differential Control Using Pre-emphasis Techniques

    Published on: April 4, 2017

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    Last Updated: Jul 9, 2026

    Quasi-light Storage for Optical Data Packets
    07:45

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    Published on: February 6, 2014

    Gain-compensation Methodology for a Sinusoidal Scan of a Galvanometer Mirror in Proportional-Integral-Differential Control Using Pre-emphasis Techniques
    09:01

    Gain-compensation Methodology for a Sinusoidal Scan of a Galvanometer Mirror in Proportional-Integral-Differential Control Using Pre-emphasis Techniques

    Published on: April 4, 2017

    Area of Science:

    • Signal Processing
    • Applied Mathematics
    • Computational Science

    Background:

    • Linear systems and matrix-vector products are fundamental in numerous scientific and engineering applications.
    • Efficiently realizing these computations is crucial for performance and resource optimization.
    • Existing methods may not fully exploit the inherent structure of matrices.

    Purpose of the Study:

    • To introduce a novel scheme for realizing linear systems and matrix-vector products.
    • To enable flexible trade-offs between computational accuracy and efficiency.
    • To systematically exploit matrix regularity for improved implementation.

    Main Methods:

    • Implementation using multistage or multichannel fractional Fourier-domain filters.
    • Systematic exploitation of matrix or linear system structure.
    • Flexible adjustment of filter number and configuration.

    Main Results:

    • Achieved space-bandwidth-efficient systems.
    • Obtained useful approximations with acceptable accuracy decreases.
    • Demonstrated a systematic approach to exploiting hidden matrix structures.

    Conclusions:

    • Fractional Fourier-domain filters provide a powerful tool for efficient linear system realization.
    • The proposed method allows for adaptable accuracy-efficiency balances.
    • This scheme offers a systematic way to leverage matrix regularity in computational tasks.