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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.
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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...

State Space to Transfer Function

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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:

Free Energy Changes for Nonstandard States

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Transformers with Off-Nominal Turns Ratios

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In scenarios involving parallel transformers with disparate ratings, developing per-unit models requires accommodating off-nominal turns ratios. This situation arises when the selected base voltages are not proportional to the transformer’s voltage ratings. Consider a transformer where the rated voltages are related by the term a. If the chosen voltage bases satisfy a relationship involving term b, term c is defined as the ratio of these bases. This ratio is then substituted into the rated...

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The Region of Convergence (ROC) is a fundamental concept in signal processing and system analysis, particularly associated with the Laplace transform. The ROC represents an area in the complex plane where the Laplace transform of a given signal converges, determining the transform's applicability and utility.
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Related Experiment Videos

Plaquette renormalization scheme for tensor network states.

Ling Wang¹, Ying-Jer Kao, Anders W Sandvik

¹Department of Physics, Boston University, 590 Commonwealth Avenue, Boston, Massachusetts 02215, USA.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics

|July 7, 2011

Summary

This summary is machine-generated.

We developed a variational method for tensor network contraction, optimizing wave functions. This approach significantly improves accuracy for quantum phase transitions compared to mean-field methods.

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Area of Science:

Condensed Matter Physics
Quantum Many-Body Systems
Computational Physics

Background:

Tensor network methods are crucial for simulating quantum many-body systems.
Efficient contraction of large tensor networks remains a computational challenge.
Variational approaches offer a path to optimize approximations in these simulations.

Purpose of the Study:

To introduce a novel variational method for contracting square-lattice tensor networks in two dimensions.
To enable efficient and accurate approximation of quantum wave functions.
To improve the study of quantum phase transitions.

Main Methods:

The method employs auxiliary tensors for successive truncations (renormalization) of eight-index tensors to four-index tensors.
Approximations are performed directly on the wave function, allowing for tensor network interpretation.
Tensors are optimized variationally by minimizing the system's energy.

Main Results:

The developed method successfully contracts tensor networks by reducing tensor complexity.
Variational optimization of tensors leads to improved accuracy.
Tests on the transverse-field Ising model show superior results compared to mean-field states, even with minimal tensor sizes.

Conclusions:

The proposed variational tensor network contraction method is effective for simulating quantum systems.
It offers a significant improvement over simpler approximation techniques like mean-field theory.
This method provides a powerful tool for investigating quantum phase transitions and other many-body phenomena.