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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...

Classification of Systems-II

Classification of Systems-II

Continuous-time systems have continuous input and output signals, with time measured continuously. These systems are generally defined by differential or algebraic equations. For instance, in an RC circuit, the relationship between input and output voltage is expressed through a differential equation derived from Ohm's law and the capacitor relation,

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, 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 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 calculated...

BIBO stability of continuous and discrete -time systems

BIBO stability of continuous and discrete -time systems

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.
To determine the BIBO stability, the convolution integral is utilized when a bounded continuous-time input is applied to a Linear Time-Invariant (LTI) system.

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Related Experiment Video

Updated: Jul 6, 2026

Inherent Dynamics Visualizer, an Interactive Application for Evaluating and Visualizing Outputs from a Gene Regulatory Network Inference Pipeline

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Self-organizing radial basis function network for real-time approximation of continuous-time dynamical systems.

Jianming Lian¹, Yonggon Lee, Scott D Sudhoff

¹School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907, USA. jlian@purdue.edu

IEEE Transactions on Neural Networks

|March 13, 2008

Summary

This study introduces a self-organizing radial basis function (RBF) network for real-time approximation of complex dynamical systems. The adaptable network structure ensures accuracy and computational efficiency, outperforming traditional methods.

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Area of Science:

Control Systems Engineering
Artificial Intelligence
Computational Mathematics

Background:

Dynamical systems with numerous inputs pose significant approximation challenges.
Existing real-time approximation methods often lack adaptability and efficiency.
Radial Basis Function (RBF) networks offer a framework for function approximation.

Purpose of the Study:

To develop a novel self-organizing RBF network for real-time approximation of continuous-time dynamical systems.
To dynamically adjust network structure for maintaining approximation accuracy and computational efficiency.
To analyze the performance of Gaussian RBF (GRBF) and raised-cosine RBF (RCRBF) within the proposed framework.

Main Methods:

Implementation of a self-organizing RBF network with online addition/removal of RBFs.
Dynamic structural adaptation to meet prescribed approximation accuracy.
Comparative analysis of GRBF and RCRBF for network training and evaluation.
Application to both linear and nonlinear dynamical systems, including higher-order systems.
Lyapunov stability analysis to prove uniform ultimate boundedness of approximation error.

Main Results:

The variable structure RBF network effectively approximates complex dynamical systems in real-time.
Raised-cosine RBF (RCRBF) demonstrated advantages in training speed and output evaluation compared to Gaussian RBF (GRBF).
The proposed method showed particular effectiveness for higher-order dynamical systems.
Theoretical proof of uniform ultimate boundedness for the approximation error was established.

Conclusions:

The self-organizing RBF network provides an efficient and accurate real-time approximation scheme for continuous-time dynamical systems.
The dynamic adaptability of the network ensures computational efficiency and maintains approximation quality.
The RCRBF variant offers practical advantages for real-time implementation.