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Linear time-invariant Systems

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...

Discrete-Time Fourier Series

Discrete-Time Fourier Series

The Discrete-Time Fourier Series (DTFS) is a fundamental concept in signal processing, serving as the discrete-time counterpart to the continuous-time Fourier series. It allows for the representation and analysis of discrete-time periodic signals in terms of their frequency components. Unlike its continuous counterpart, which utilizes integrals, the calculation of DTFS expansion coefficients involves summations due to the discrete nature of the signal.
For a discrete-time periodic signal x[n]...

Sampling Continuous Time Signal

Sampling Continuous Time Signal

In signal processing, a continuous-time signal can be sampled using an impulse-train sampling technique, followed by the zero-order hold method. Impulse-train sampling involves the use of a periodic impulse train, which consists of a series of delta functions spaced at regular intervals determined by the sampling period. When a continuous-time signal is multiplied by this impulse train, it generates impulses with amplitudes corresponding to the signal's values at the sampling points.
In the...

Linear Approximation in Frequency Domain

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....

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,...

Frequency-Domain Interpretation of PD Control

Frequency-Domain Interpretation of PD Control

Proportional-Derivative (PD) controllers are widely used in fan control systems to improve stability and performance. A fan control system can be effectively represented using a Bode plot to illustrate the impact of a PD controller through its transfer function. The Bode plot visually conveys how PD control modifies the fan's response across various frequencies, providing a frequency domain interpretation of the controller's behavior.
The proportional control gain, combined with the...

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Related Experiment Video

Updated: Jul 17, 2026

Optimization of An Air-Based Heat Management System for Dusty Particulate Matter-Covered Lithium-Ion Battery Packs

Optimization of An Air-Based Heat Management System for Dusty Particulate Matter-Covered Lithium-Ion Battery Packs

Published on: November 3, 2023

PID-Optimized Deep Learning for Adaptive Time-Frequency Forecasting in Dynamic Systems: Coal Calorific Value

Hongwei Liu, Ning Liu, Wen Yu

IEEE Transactions on Cybernetics

|March 11, 2026

Summary

This study presents an intelligent monitoring framework using PID-optimized deep learning for accurate forecasting in industrial systems. The novel approach enhances prediction accuracy for nonstationary data, improving efficiency.

Related Experiment Videos

Last Updated: Jul 17, 2026

Optimization of An Air-Based Heat Management System for Dusty Particulate Matter-Covered Lithium-Ion Battery Packs

Optimization of An Air-Based Heat Management System for Dusty Particulate Matter-Covered Lithium-Ion Battery Packs

Published on: November 3, 2023

Area of Science:

Industrial Process Monitoring
Data Science
Machine Learning

Background:

Accurate real-time prediction is essential for optimizing dynamic industrial systems.
Nonstationary industrial data presents significant challenges for traditional forecasting methods.

Purpose of the Study:

To introduce a novel intelligent monitoring framework for adaptive time-frequency forecasting.
To address the challenges of nonstationary industrial data using deep learning and PID optimization.

Main Methods:

Integration of a channel-independent separable dynamic filter (CSDF) for real-time adaptive multivariate data processing.
Application of a closed-loop proportional-integral-derivative (PID) optimization strategy to enhance deep learning model convergence and accuracy.
Utilizing deep learning for adaptive time-frequency forecasting.

Main Results:

The framework demonstrated effectiveness in predicting cleaned coal calorific value.
Achieved a significant 5.36% increase in forecast hit rate (FHR) compared to existing techniques.
Minimized cross-channel interference in multivariate process variable analysis.

Conclusions:

The proposed PID-optimized deep learning framework offers advanced monitoring capabilities for dynamic industrial systems.
The method shows potential for improving optimization and energy efficiency in industrial processes.
The CSDF and PID optimization contribute to enhanced prediction accuracy and model performance.