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State Space Representation

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 Approximation in Time Domain

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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.
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System stability is a fundamental concept in signal processing, often assessed using convolution. For a system to be considered bounded-input bounded-output (BIBO) stable, any bounded input signal must produce a bounded output signal. A bounded input signal is one where the modulus does not exceed a certain constant at any point in time.
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Linear time-invariant Systems

Linear time-invariant Systems

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Transfer Function to State Space

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State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC...

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Related Experiment Video

Updated: Jul 1, 2026

Microstate and Omega Complexity Analyses of the Resting-state Electroencephalography

Microstate and Omega Complexity Analyses of the Resting-state Electroencephalography

Published on: June 15, 2018

Synchronization and state estimation for discrete-time complex networks with distributed delays.

Yurong Liu¹, Zidong Wang, Jinling Liang

¹Department of Mathematics, Yangzhou University,Yangzhou 225002, China. liuyurong@gmail.com

IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics : a Publication of the IEEE Systems, Man, and Cybernetics Society

|September 12, 2008

Summary

This study addresses synchronization and state estimation for complex discrete-time networks with time delays. Novel methods using Lyapunov-Krasovskii functionals and linear matrix inequalities (LMIs) ensure global asymptotic stability for synchronization and estimation.

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Area of Science:

Control Theory
Network Science
Nonlinear Systems

Background:

Complex networks, including neural and social networks, are ubiquitous.
Synchronization and state estimation are critical for network function.
Time delays, both discrete and distributed, pose significant challenges in network analysis.

Purpose of the Study:

To investigate the synchronization problem for coupled complex discrete-time networks with discrete and distributed time delays.
To develop a state estimation strategy for these networks to ensure global asymptotic stability of the estimation error.
To provide a unified framework for analyzing synchronization and state estimation under general nonlinearities and delays.

Main Methods:

Definition of distributed infinite time delays in the discrete-time domain.
Development of a novel Lyapunov-Krasovskii functional.
Application of the Kronecker product and linear matrix inequalities (LMIs) for analysis.
Design of a state estimator based on available output measurements.

Main Results:

Synchronization conditions derived based on the feasibility of LMIs.
State estimation guarantees global asymptotic stability of the estimation error for all admissible delays.
The proposed methods are effective even for networks with unstable nominal subsystems.
Simulation examples validate the proposed synchronization and state estimation conditions.

Conclusions:

The developed LMI-based approaches provide effective solutions for synchronization and state estimation in complex discrete-time networks with time delays.
The methodology is robust, handling general nonlinearities and delays, and applicable to unstable network components.
This work contributes to the theoretical understanding and practical application of control strategies for complex dynamical systems.