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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...

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Individual molecules in a gas move in random directions, but a gas containing numerous molecules has a predictable distribution of molecular speeds, which is known as the Maxwell-Boltzmann distribution, f(v).
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Multi-input and Multi-variable systems

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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.
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Transfer Function to State Space

Transfer Function to State Space

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.
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Sequence Networks of Rotating Machines

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Related Experiment Videos

Learning to represent spatial transformations with factored higher-order Boltzmann machines.

Roland Memisevic¹, Geoffrey E Hinton

¹Department of Computer Science, University of Toronto, Toronto M5S 3G4, Canada. roland@cs.toronto.edu

Neural Computation

|February 10, 2010

Summary

This summary is machine-generated.

This study introduces a low-rank approximation for modeling image transformations using restricted Boltzmann machines. The method efficiently learns optimal filter pairs for representing transformations and performing visual analogy tasks.

Related Experiment Videos

Area of Science:

Artificial Intelligence
Machine Learning
Computer Vision

Background:

Restricted Boltzmann machines (RBMs) traditionally model pairwise interactions.
Modeling transformations between images requires complex, high-order interactions.
Previous methods, like Memisevic and Hinton's (2007), used three-way interactions leading to cubic parameter growth.

Purpose of the Study:

To develop a computationally efficient method for modeling image transformations using RBMs.
To introduce a low-rank approximation for the three-dimensional interaction tensor in RBMs.
To demonstrate the model's ability to learn optimal filter pairs for image transformation representation.

Main Methods:

A low-rank approximation of the three-way interaction tensor is proposed, represented as a sum of factorized outer products.
Each factor is interpreted as a pair of image filters.
The model is trained on synthetic and real image sequences to learn these filter pairs.

Main Results:

The low-rank approximation enables efficient learning of transformations between larger image patches.
The model successfully learns optimal filter pairs for representing image transformations.
The unsupervised network trained on transformations demonstrates human-like perception of transparent dot pattern motions.

Conclusions:

The proposed low-rank approximation significantly improves the efficiency of modeling image transformations in RBMs.
Learned filter pairs effectively capture essential transformation characteristics.
The model shows potential for unsupervised learning of visual perception and analogy tasks.