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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...

Modeling with Differential Equations

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Population dynamics can be described mathematically by considering the population size P(t) as a function of time. The rate of change of the population is then represented by the derivative of P(t). A simple assumption is that the rate of growth is proportional to the size of the population itself. This leads to an exponential growth model, where the population increases rapidly without bound. While this is a useful first approximation, it does not reflect realistic long-term...

Diffusion

Diffusion

Diffusion is a type of passive transport. In passive transport, a substance tends to move from an area of high concentration to an area of low concentration until the concentration is equal across the space. For example, take the diffusion of substances through the air. When someone opens a perfume bottle in a room filled with people, the perfume is at its highest concentration in the bottle and is at its lowest at the edges of the room. The perfume vapor will diffuse, or spread away, from the...

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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.
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State Space to Transfer Function

State Space to Transfer Function

The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
The transformation process begins with the state-space representation, characterized by the state equation and the output equation. These equations are typically represented as:

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Related Experiment Video

Updated: Jun 16, 2026

Single-Molecule Tracking Microscopy - A Tool for Determining the Diffusive States of Cytosolic Molecules

Single-Molecule Tracking Microscopy - A Tool for Determining the Diffusive States of Cytosolic Molecules

Published on: September 5, 2019

Structured populations with diffusion in state space.

Karl Peter Hadeler¹

¹School of Mathematical and Statistical Sciences, Arizona State University, Tempe, AZ 85287, United States. hadeler@uni-tuebingen.de

Mathematical Biosciences and Engineering : MBE

|January 29, 2010

Summary

This summary is machine-generated.

Classical population models face realism issues. Introducing a diffusion term in partial differential equations resolves these, enabling more accurate modeling of populations, diseases, and metapopulations.

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Last Updated: Jun 16, 2026

Single-Molecule Tracking Microscopy - A Tool for Determining the Diffusive States of Cytosolic Molecules

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Published on: September 26, 2016

Area of Science:

Population Dynamics
Mathematical Biology
Epidemiology

Background:

Classical population models exhibit limitations in biological realism.
These models assume structure variables always increase.
Individuals within cohorts remain identical throughout their lifespan.

Purpose of the Study:

To address limitations in classical population modeling.
To introduce a novel mathematical approach for enhanced realism.
To apply the new method across diverse biological systems.

Main Methods:

Incorporation of a diffusion term into partial differential equations.
Mathematical modeling equivalent to adding viscosity.
Identification of appropriate boundary (recruitment) conditions.

Main Results:

The diffusion term resolves issues of increasing structure variables.
The model allows for variation among initially identical individuals.
Successful application to size-structured populations, metapopulations, and disease dynamics.

Conclusions:

The diffusion approach enhances the biological realism of population models.
This method provides a flexible framework for diverse ecological and epidemiological studies.
Accurate recruitment conditions are crucial for the efficacy of this modeling technique.