OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/pages/MatrixNormMinimization

LMI for Matrix Norm Minimization
This problem is a slight generalization of the eigenvalue minimization problem for a matrix. Calculating norm of a matrix is necessary in designing an $H_{2}$ or an $H_{\infty }$ optimal controller for linear time-invariant systems. In those cases, we need to compute the norm of the matrix of the closed-loop system. Moreover, we desire to design the controller so as to minimize the closed-loop matrix norm.

## The System

Assume that we have a matrix function of variables $x$:
${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}$
where ${\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}$ are symmetric matrices.

## The Data

The symmetric matrices $A_{i}$ (${\begin{aligned}A_{0},A_{1},...,A_{n}\end{aligned}}$) are given.

## The Optimization Problem

The optimization problem is to find the variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ in order to minimize the following cost function:
${\begin{aligned}J(x)=||A(x)||_{2}\end{aligned}}$
where $J(x)$ is the cost function and $||.||_{2}$ indicates the norm of the matrix function $A$.
According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent:
${\begin{aligned}A^{T}A-t^{2}I\leq 0\iff {\begin{bmatrix}-tI&A\\A^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

## The LMI: LMI for matrix norm minimization

This optimization problem can be converted to an LMI problem.
The mathematical description of the LMI formulation can be written as follows:
${\begin{aligned}&{\text{min}}\quad t&\\&{\text{s.t.}}\quad {\begin{bmatrix}-tI&A(x)\\A(x)^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

## Conclusion:

As a result, the variables $x_{i},\quad i=1,2,...,n$ after solving this LMI problem and we obtain $t$ that is the norm of matrix function $A(x)$.

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization

## Related LMIs

LMI for Matrix Norm Minimization
LMI for Generalized Eigenvalue Problem
LMI for Maximum Singular Value of a Complex Matrix
LMI for Matrix Positivity

## External Links

A list of references documenting and validating the LMI.
* - LMI in Control Systems Analysis, Design and Applications

## Return to Main Page

LMIs in Control/Tools
# LMIs in Control/pages/Minimum Gain Lemma

LMIs in Control/pages/Minimum Gain Lemma
Minimum Gain Lemma

## The System

Suppose there is a continuous-time LTI system where one must develop a controller for such system that is unstable. The minimum gain of a system would be obtained by taking the infimum of ratio between norms of the outputs and inputs over all nonzero inputs. With the Large Gain theorem expressing that if such an unstable system contains a finite value for the minimum gain of the system which is also non-zero, then any controller is capable of stabilizing the closed-loop feedback system, so long as the the controller also has a large minimum gain.
With this theorem led to the development of the Minimum Gain Lemma where applicable analysis can determine whether a closed loop system achieves a non-zero minimum gain value despite being inherently unstable in the open-loop case. By having an LTI system where the system $G$ corresponds to the following matrices $A\in \mathbb {R} ^{m\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, and $D\in \mathbb {R} ^{p\times m}$.

## The Data

A system of the plant $G$ will be required with the following matrices: $A\in \mathbb {R} ^{m\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, and $D\in \mathbb {R} ^{p\times m}$.

## The LMI: Minimum Gain Lemma

The following two LMI's are equivalent, with the same variable:
Suppose there exists $P\in \mathbb {S} ^{n}$, and $\upsilon \in \mathbb {R} _{\geq 0}$, Where $P\geq 0$ and:
       ${\begin{aligned}{\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}D\\(PB-C^{T}D)^{T}&\upsilon ^{2}I-D^{T}D\end{bmatrix}}&\leq 0\end{aligned}}$
Or (Obtained using Schur compliment):
       ${\begin{aligned}{\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}D&0\\(PB-C^{T}D)^{T}&-D^{T}D&\upsilon I\\0&\upsilon I&-I\end{bmatrix}}&\leq 0\end{aligned}}$

## Conclusion:

Solving this LMI will give the minimum gain from the LTI system. This minimum gain can then be used if the system has been made stable based on the value of $\upsilon$ obtained from the optimization.

## Implementation

* Minimum Gain Lemma Example

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* The minimum gain lemma - A paper about Minimum gain lemma by Bridgeman and Forbes
# LMIs in Control/pages/Minimum Singular Value of a Complex Matrix

LMIs in Control/pages/Minimum Singular Value of a Complex Matrix
Minimum Singular Value of a Complex Matrix

## The System

Consider $A\in \mathbb {C} ^{n\times m}$ as well as $\gamma$. A minimum singular value of a matrix $A$ is greater than $\gamma$ if and only if $AA^{H}>\gamma ^{2}I$ or $A^{H}A>\gamma ^{2}I$, where $A^{H}$ is the conjugate transpose or Hermitian transpose of the matrix $A$. the inequality used depends on the size of matrix $A$.

## The Data

The matrix $A$ is the only data required.

## The LMI: Minimum Singular Value of a Complex Matrix

The following LMIs can be constructed depending on the size of $A$:
if $A\in \mathbb {C} ^{n\times m}$, where $n\leq m$, then:
       ${\begin{aligned}{\bar {\sigma }}(A)>\gamma \\AA^{H}>\gamma ^{2}I\end{aligned}}$
Else if $n\geq m$, then:
       ${\begin{aligned}{\bar {\sigma }}(A)>\gamma \\A^{H}A>\gamma ^{2}I\end{aligned}}$

## Conclusion:

The results from this LMI will give the maximum complex value of the matrix $A$:
       ${\begin{aligned}{\bar {\sigma }}(A)>\gamma \end{aligned}}$
This answer can also be proven using the following solution. Note that this solution only works if the matrix $A$ is a square, invertible matrix: $\sigma _{min}=1/||A^{-1}||_{2}$.

## Implementation

* Minimum Singular Value Example

## Related LMIs

* Maximum singular value of a complex matrix

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/pages/Maximum Singular Value of a Complex Matrix

LMIs in Control/pages/Maximum Singular Value of a Complex Matrix
Maximum Singular Value of a Complex Matrix

## The System

Consider $A\in \mathbb {R} ^{n\times m}$ as well as $\gamma$. A maximum singular value of a matrix $A$ is less than $\gamma$ if and only if $AA^{H}<\gamma ^{2}I$, where $A^{H}$ is the conjugate transpose or Hermitian transpose of the matrix $A$.

## The Data

The matrix $A$ is the only data required.

## The LMI: Maximum Singular Value of a Complex Matrix

Using the Shur complement procedure, the following LMIs can be constructed:
       ${\begin{aligned}{\bar {\sigma }}(A)<\gamma \\{\begin{bmatrix}\gamma I&A\\A^{H}&\gamma I\end{bmatrix}}&>0\\\end{aligned}}$
The following LMI is also equivalent:
       ${\begin{aligned}{\bar {\sigma }}(A)<\gamma \\{\begin{bmatrix}\gamma I&A^{H}\\A&\gamma I\end{bmatrix}}&>0\\\end{aligned}}$

## Conclusion:

The results from this LMI will give the maximum complex value of the matrix $A$:
       ${\begin{aligned}{\bar {\sigma }}(A)<\gamma \end{aligned}}$

## Implementation

* Maximum Singular Value Example

## Related LMIs

* Minimum singular value of a complex matrix

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/pages/LMIforMinimizingMatrixNorm

LMIs in Control/pages/LMIforMinimizingMatrixNorm

## The System

    ${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}$
Note that ${\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}$ are symmetric matrices.

## The Data

    ${\begin{aligned}A_{0},A_{1},...,A_{n}\quad {\text{are given matrices.}}\end{aligned}}$

## The Optimization Problem

Find
${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$
to minimize,
${\begin{aligned}J(x)=||A(x)||_{2}\end{aligned}}$
According to Lemma 1.2 in [1] page 11, the following statements are equivalent
${\begin{aligned}A^{T}A-t^{2}I\leq 0\iff {\begin{bmatrix}-tI&A\\A^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

## The LMI: LMI for matrix norm minimization

Mathematical description of the LMI formulation:
       ${\begin{aligned}{\text{min}}\;\quad t:&\\{\text{s.t.}}\quad {\begin{bmatrix}-tI&A(x)\\A(x)^{T}&-tI\end{bmatrix}}\leq 0\\\end{aligned}}$

## Conclusion:

This problem is a slight generalization of the eigenvalue minimization problem for a matrix.
${\begin{aligned}x_{i},\quad i=1,2,...,n\quad {\text{and}}\quad t>0\end{aligned}}$ are parameters to be optimized

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization

## Related LMIs

LMI for Eigenvalue Minimization

## External Links

A list of references documenting and validating the LMI.
* - LMI in Control Systems Analysis, Design and Applications
# LMIs in Control/pages/MatrixEigenValueMinimization

LMI for Minimizing Eigenvalue of a Matrix
Synthesizing the eigenvalues of a matrix plays an important role in designing controllers for linear systems. The eigenvalues of the state matrix of a linear time-invariant system determine if the system is stable or not. The system is stable if all the eigenvalues of the state matrix are located in the left half of the complex plane. Thus, we may desire to minimize the maximal eigenvalue of the state matrix such that the minimized eigenvalue is placed in the left half-plane, which guarantees that the system is stable.

## The System

Assume that we have a matrix function of variables $x$:
${\begin{aligned}A(x)=A_{0}+A_{1}x_{1}+...+A_{n}x_{n}\end{aligned}}$
where ${\begin{aligned}A_{i},\quad i=1,2,...,n\end{aligned}}$ are symmetric matrices.

## The Data

The symmetric matrices $A_{i}$ (${\begin{aligned}A_{0},A_{1},...,A_{n}\end{aligned}}$) are given.

## The Optimization Problem

The optimization problem is to find the variables ${\begin{aligned}x=[x_{1}\quad x_{2}...x_{n}]\end{aligned}}$ to minimize the following cost function:
${\begin{aligned}J(x)=\lambda _{\text{max}}(A(x))\end{aligned}}$
where $J(x)$ is the cost function and $\lambda _{\text{max}}(.)$ indicates the maximim eigenvalue of a matrix.
According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent
${\begin{aligned}\lambda _{max}(A(x))\leq t\iff A(x)-tI\leq 0\end{aligned}}$
where $t$ is defined as the maximim eigenvalue of the matrix $A$.

## The LMI: LMI for eigenvalue minimization

This optimization problem can be converted to an LMI problem.
The mathematical description of the LMI formulation can be written as follows:
${\begin{aligned}&{\text{min}}\quad t\\&{\text{s.t.}}\quad A(x)-tI\leq 0\end{aligned}}$

## Conclusion:

As a result, the variables $x_{i},\quad i=1,2,...,n$ after solving this LMI problem.
Moreover, we obtain the maximum eigenvalue, $t$, of the matrix $A(x)$.

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Minimizing-the-Maximum-Eigenvalue-of-Matrix

## Related LMIs

LMI for Generalized Eigenvalue Problem
LMI for Matrix Norm Minimization
LMI for Maximum Singular Value of a Complex Matrix
LMI for Matrix Positivity

## External Links

* - LMI in Control Systems Analysis, Design and Applications
* State-space Representation of a System
* Eigenvalues and Eigenvectors of a Matrix

## Return to Main Page

LMIs in Control/Tools
# How To Search/AND-plus

AND - "+"
This is a list of searches that use "+" for the AND boolean operator. For example typing the example above in a search box would find all items searched which have both of the words "skunk" and "tomato" in it.
# How To Search/Directory of Open Access Journals

Directory of Open Access Journals (DOAJ) - Journal Search
* Purpose: "Increase the visibility and ease of use of open access scientific and scholarly journals thereby promoting their increased usage and impact."
* Audience: Researchers.
* Scope: "The Directory aims to be comprehensive and cover all open access scientific and scholarly journals that use a quality control system to guarantee the content."
* Refresh Rate:
* Further Information: While DOAJ started as a journal directory, it has expanded to catalog at the article level also. This page is for the journal level searching although, they seem to use the same operators at the moment. There are no written instructions for searching so the below information was found by trial and error.

## Boolean Operators

| Boolean Operator | Syntax |
| AND | {Template only} |
| OR | {Template only} |
| NOT | {Not possible} |

## Other Features

The DOAJ searches, both journal and article level, seem to have primitive searching mechanisms at the moment. Using this site as a directory for finding open access journals seems to be its forte at this time. 07/29/04
# How To Search/AND-default

AND - {default}
This is a list of searches that use the AND boolean operator by default. For example typing the above example in a search box would find all items searched which have both of the words "skunk" and "tomato" in it.

## Search Engines/Software/Interfaces

* BioMed Central
* Google
* PubMed
* Yahoo
# How To Search/AND-AND

AND - AND
This is a list of searches that use "AND" or "and" for the AND boolean operator. For example typing the example above in a search box would find all items searched which have both of the words "skunk" and "tomato" in it.
Items below marked with "lower case also" may use "AND" or "and" as a boolean operator in a search. Other searches use only "AND".

## Search Engines/Software/Interfaces

* BioMed Central
* Library of Congress
* Millennium-III "lower case also"
* MOBIUS Consortium
* OhioLINK
* Stanford Encyclopedia of Philosophy
# How To Search/Citation index list

The following list contains citation indexes that are described in this book.
# How To Search

Purpose:
* Provide a quick reference guide for electronic information searching.
* Teach information seekers how to form accurate searches in electronic environments.
Scope: Any electronic search environment. Examples can be found in the "Searches A-Z" section of the book.
Audience: All information seekers in electronic environments.

## Reference Guide

This reference guide quickly answers how to use various electronic searches. This may include an Internet search engine such as Google or Yahoo, a library web catalog such as Library of Congress, a personal computer file searcher such as grep, or any other electronic search mechanism. Some knowledge of Boolean logic is helpful and if you don't know how to use the AND, OR, NOT operators, please read the Teaching How to Search section of this Wikibook.
Each search page should list Boolean operators, other operators, and available search fields as well as the purpose, audience, and scope of that search.

### How to search...

* Google
* Amazon
* BioMed Central
* Directory of Open Access Journals
* Google
* Library of Congress
* Millennium-III
* MOBIUS Consortium
* MS Windows file search
* OhioLINK
* PubMed
* Stanford Encyclopedia of Philosophy
* Wikia
* Yahoo
* The United States Patent and Trademark office ( http://uspto.gov/ )
* Feedster
* Other search type list

### Search Filters

* Against Obscene Content
* For Certain Address Endings

## Teaching How to Search: Theory and Instruction

The following book sections give general theory and instruction about searching in an electronic environment:
* Types of resources
* Boolean Logic
* The information search process

## Related books

* Searching for files and within files on Windows
* Grep
# How To Search/BioMed Central

BioMed Central
* Purpose: "BioMed Central is an independent publishing house committed to providing immediate free access to peer-reviewed biomedical research."
* Audience: People researching topics in the field of biomedicine.
* Scope: Biomedical research.
* Refresh Rate: ?
* Further Information: In order to perform advanced searches on BioMed Central, you are required to create a login name and password. This is free but somewhat inconvenient.

## Boolean Operators

| Boolean Operator | Syntax | Example |
| AND | AND | skunk AND tomato |
| OR | OR | skunk OR tomato |
| NOT | NOT | skunk NOT tomato |

## Other Operators

| Syntax | Example | Description |
| "" | "skunk smell removal" | phrase search |

## Search Fields

| Syntax | Example | Description |
| [PG] | 125 [PG] | article number or first page number |
| [AU] | rogers [AU] | author name |
| [AD] | hopkins [AD] | institutional affiliation or author addresses |
| [TIAB] | brain [TIAB] AND cancer [TIAB] AND jones [TIAB] | search only the citation and abstract |
| [DISC] | 'inaccurate results" [DISC] | search the discussion/conclusion section of texts |
| [DOI] | | search the Digital Object Identifier (DOI) |
| [FIG] | "rate of change" [FIG] | search figure legends |
| [INTR] | doctoral [INTR] | search the introduction section |
| [IP] | | search by issue |
| [TA] | neuroscience [TA] | search the journal title, abbreviation, or ISSN |
| [METH] | "statistical sample" [METH] | search the methods section of an article |
| [RES] | innaccurate [RES] | search the results section of an article |
| [REF] | barnes [REF] AND metcalf[REF] | search reference sections |
| [BDY] | "cranial splint" [BDY] | search text bodies (excludes citation and abstract) |
| [TI] | "stomach cancer" [TI] | search titles |
| [TW] | "stomach cancer" [TW] | search title, abstract, and text (excludes abstract) |
| [VI] | 5 [VI] | search volume number |
| [DP] | 1999 [DP] | search year of publication |

## Other Features

BioMed Central also allows a number of searches which can only be done using templates on their web pages. These include content sets (BioMed Central Journals, The Scientist news), date ranges, and "added in last" (last 1 day, 7 days, 30 days).
Other features not directly impacting search results include number of results shown per page, order in which results are shown, and whether to show the abstract or not.

## More Example Searches

The Following example will find all articles written by someone named either "jones" or "smith" and that contains the word "assessment" and the word "pain" in the body of the text.
# How To Search/Boolean Logic

Boolean logic symbolically represents relationships between entities. There are three Boolean operators:
AND Use the AND operator to retrieve a set in which each citation contains all the search terms. This operator places no condition on where the terms are found in relation to one another; the terms simply have to appear somewhere in the same citation.
OR Use the OR operator to retrieve documents that contain at least one of the specified search terms.
Use OR when you want to pull together articles on similar subjects.
NOT Use the NOT operator to exclude the retrieval of terms from your search.

## Sources

Originally copied from PubMed Tutorial.

## See also

Wikipedia Articles:
* http://en.wikipedia.org/wiki/Boolean_algebra
* http://en.wikipedia.org/wiki/Logical_operator
* http://en.wikipedia.org/wiki/Venn_diagram
# How To Search/Feedster

Feedster
* Purpose:
* Audience:
* Scope:
* Refresh Rate:

## Boolean Operators

| Boolean Operator | Syntax | Example |
| AND | AND | skunk AND tomato |
| OR | OR | skunk OR tomato |
| NOT | NOT | skunk NOT tomato |

## Other Operators

| Syntax | Example | Description |

## Search Fields

| Syntax | Example | Description |
# LaTeX/Creating 3D graphics

* LaTeX
* Getting Started
* Introduction
* Installation
* Installing Extra Packages
* Basics
* How to get help
* Common Elements
* Document Structure
* Text Formatting
* Paragraph Formatting
* Colors
* Fonts
* List Structures
* Special Characters
* Internationalization
* Rotations
* Tables
* Title creation
* Page Layout
* Customizing Page Headers and Footers‎
* Importing Graphics
* Floats, Figures and Captions
* Footnotes and Margin Notes
* Hyperlinks
* Labels and Cross-referencing
* Initials
* Mechanics
* Errors and Warnings
* Lengths
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* Technical Text
* Mathematics
* Advanced Mathematics
* Theorems
* Chemical Graphics
* Algorithms
* Source Code Listings
* Linguistics
* Special Pages
* Indexing
* Glossary
* Bibliography Management
* More Bibliographies
* Special Documents
* Scientific Reports (Bachelor Report, Master Thesis, Dissertation)
* Letters
* Presentations
* Teacher's Corner
* Curriculum Vitae
* Academic Journals (MLA, APA, etc.)
* Creating Graphics
* Introducing Procedural Graphics
* MetaPost
* Picture
* PGF/TikZ
* PSTricks
* Xy-pic
* Creating 3D graphics
* Programming
* Macros
* Plain TeX
* Creating Packages
* Creating Package Documentation
* Themes
* Miscellaneous
* Modular Documents
* Collaborative Writing of LaTeX Documents
* Export To Other Formats
* Help and Recommendations
* FAQ
* Tips and Tricks
* Appendices
* Authors
* Links
* Package Reference
* Sample LaTeX documents
* Index
* Command Glossary
For creating three-dimensional graphics, there is basic functionality in the PGF/TikZ package, although drawing 3D graphics with PGF/TikZ is very non-flexible, mainly because it lacks functionality for identifying the surfaces that are covered by other surfaces and should be excluded from the rendered image.
A package that can handle this correctly is the pst-solides3d package.
Another way to create 3D graphics is to use Asymptote.
Yet another way to create 3D graphics is to use Sketch.
# Opticks Developer Guide/Introduction

Opticks is an open source, remote sensing application. Using Opticks you can create products from satellite and airborne imagery, video (motion imagery), Synthetic Aperture Radar (SAR), multispectral, and hyperspectral data. Opticks is also a developer framework, empowering software developers to extend its functionality. This guide helps C++ developers build capability for Opticks.
This developer guide is divided into five sections. The first section gives a general overview of features of the Opticks remote sensing development framework. The second section helps developers get started. This section describes how to download Opticks and get the necessary components to start developing with Opticks. The third section give an overview of Opticks development concepts and terminology. The fourth section describes how to setup your build environment, and the last section describes how to get started building the sample plug-ins.