OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/Matrix and LMI Properties and Tools/Congruence Transformation

LMIs in Control/Matrix and LMI Properties and Tools/Congruence Transformation
This methods uses change of variable and some matrix properties to transform Bilinear Matrix Inequalities to Linear Matrix Inequalities. This method preserves the definiteness of the matrices that undergo the transformation.

## Theorem

Consider $Q\in \mathbb {S} ^{n},W\in \mathbb {R} ^{n\times n}$, where $\operatorname {rank} (W)=n$. The matrix inequality $Q<0$ is satisfied if and only if $WQW^{T}<0$ or equivalently, $W^{T}QW<0$.

## Example

Consider $A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},K\in \mathbb {R} ^{m\times p},C^{T}\in \mathbb {R} ^{n\times p},P\in \mathbb {S} ^{n}$ and $V\in \mathbb {S} ^{p}$, where $P>0$ and $V>0$. The matrix inequality given by
${\begin{aligned}\ Q&={\begin{bmatrix}\ A^{T}P+PA&-PBK+C^{T}V\\\ *&-2V\\\end{bmatrix}}\ <0,\end{aligned}}$
is linear in variable $V$ and bilinear in the variable pair $(P.K)$. Choose the matrix $\operatorname {diag} (P^{-1},V^{-1})$ to obtain the equivalent BMI given by
${\begin{aligned}\ WQW^{T}&={\begin{bmatrix}\ P^{-1}A^{T}+AP^{-1}&-BKV^{-1}+P^{-1}C^{T}\\\ *&-2V^{-1}\\\end{bmatrix}}\ <0,\end{aligned}}$
Using a change of variable $X=P^{-1},U=V^{-1}$ and $F=KV^{-1}$, the above equation becomes
${\begin{aligned}\ WQW^{T}&={\begin{bmatrix}\ XA^{T}+AX&-BF+XC^{T}\\\ *&-2U\\\end{bmatrix}}\ <0,\end{aligned}}$
which is an LMI of variables $X,U$ and $F$. The original variable $K$ is recovered by doing a reverse change of variable $K=FU^{-1}$.

## Conclusion

A congruence transformation preserves the definiteness of a matrix by ensuring that $Q<0$ and $W^{T}QW<0$ are equivalent. A congruence transformation is related, but not equivalent to a similarity transformation $TQT^{-1}$, which preserves not only the definiteness, but also the eigenvalues of a matrix. A congruence transformation is equivalent to a similarity transformation in the special case when $W^{T}=W^{-1}$.

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.
# LMIs in Control/Local Minimum of Polynomial via SOS Method

[UNDER CONSTRUCTION] - CME
The local minimum of a certain polynomial functions f(x) can be found using Sum-of-Squares (SOS) methods. Here we assume the local set has a P-Compact representation, and that this representation is used.

## Required Data

A polynomial function f(x) whose minimum is desired.
A P-Compact representation of the set.

## First Steps

Touch up on the Putinar Positivstellensatz and what it means for us.

## Why does it work

Based in S-Procedure, with the Positivstellensatz all about the right representation.

## Example

Code example in SOStools probably worthwhile!

## Reference

SOS tools manual is a good one

## Related LMIs

SOS Basics
Global Minimum
# LMIs in Control/Lyapunov Inequality

LMIs in Control/Lyapunov Inequality
The System:
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)\\x(0)&=x_{0}\end{aligned}}$
The Optimization Problem:
The LMI: The Lyapunov Inequality
       ${\begin{aligned}{\text{Find}}\;X:&\\{\begin{bmatrix}X\end{bmatrix}}&>0\\{\begin{bmatrix}A^{T}X+XA\end{bmatrix}}&<0\end{aligned}}$
Conclusion:
If feasible, System is internally exponentially Stable for any $x_{0}\in \mathbb {R} ^{n}$.
Related LMIs:
# LMIs in Control/Matrix and LMI Properties and Tools/Concatenation Of Matrices
## Introduction

Matrix concatenation is the process of joining one or more matrices to make a new matrix. This process is similar for concatenating the LMIs as well.

## Concatenation of LMIs

A useful property of LMIs is that multiple LMIs can be concatenated together to form a single LMI. For example, satisfying the LMIs
is equivalent to satisfying the concatenated LMI:
${\begin{bmatrix}A&&0\\0&&B\end{bmatrix}}$ ${\begin{aligned}<0\end{aligned}}$,
More generally, satisfying the LMIs $Ai<0$, where $i=1,...,n$ is equivalent to satisfying the concatenated LMI:
${\begin{bmatrix}A&&0&&0&&.&&.&&0\\0&&B&&0&&.&&.&&0\\0&&0&&C&&.&&.&&0\\.&&.&&.&&.&&.&&0\\.&&.&&.&&.&&.&&0\\0&&0&&0&&0&&0&&A_{n}\\\end{bmatrix}}$ ${\begin{aligned}<0\end{aligned}}$,

## External Links

A list of references documenting and validating the LMI.
# LMIs in Control/Matrix and LMI Properties and Tools/Continuous Time

      * Schur Complement
      * LMI for Eigenvalue Minimization
      * LMI for Matrix Norm Minimization
      * LMI for Generalized Eigenvalue Problem
      * LMI for Linear Programming
      * LMI for Feasibility Problem
      * Structured Singular Value
      * Eigenvalue Problem
      * LMI for Minimizing Condition Number of Positive Definite Matrix
      * Continuous Quadratic Stability
      * Exterior Conic Sector Lemma
      * Modified Exterior Conic Sector Lemma
      * DC Gain of a Transfer Matrix
# LMIs in Control/Matrix and LMI Properties and Tools

      * Basic Matrix Theory
      * Notion of Matrix Positivity
      * Matrix Inequalities and LMIs
      * KYP Lemmas
      * Continuous Time
      * Discrete Time
      * Deduced LMI Conditions for H-infinity Index
      * Deduced LMI Conditions for H$_2$ Index
      * Dissipativity of Systems
      * D-Stabilization
      * H-Stabilization
      * H-2 Norm of the System
      * Algebraic Riccati Equation
      * System Zeros without feedthrough
      * System zeros with feedthrough
      * Negative Imaginary Lemma
      * Small Gain Theorem
      * Tangential Nevanlinna-Pick Interpolation
      * Nevanlinna-Pick Interpolation with Scaling
      * Generalized $H_{2}$ Norm
      * Passivity and Positive Realness
      * Non-expansivity and Bounded Realness
      * Change of Subject
      * Congruence Transformation
      * Finsler's Lemma
      * Young’s Relation (Completion of the Squares)
      * Young’s Relation-Based Properties
      * Schur Stabilizability
      * Variable Reduction Lemma
      * Ellipsoidal inequality
      * Dilation
      * Quadratic Schur Stability
      * D-Stability Rise Time Poles
      * D-Stability Settling Time Poles
      * D-Stability Max Percent Overshoot Poles
      * Iterative Convex Overbounding
      * Strict Projection Lemma
      * Non Strict Projection Lemma
      * Concatenation Of Matrices
      * Dicrete-time Algebraic Riccati Equation
      * Frobenius Norm
      * Dualization Lemma
      * Negative Imaginary System DC Constraint
# LMIs in Control/Matrix and LMI Properties and Tools/Change of Subject

LMIs in Control/Matrix and LMI Properties and Tools/Change of Subject
A Bilinear Matrix Inequality (BMI) can sometimes be converted into a Linear Matrix Inequality (LMI) using a change of variables. This is a basic mathematical technique of changing the position of variables with respect to equal signs and the inequality operators. The change of subject will be demonstrated by the example below.

## Example

Consider $A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},K\in \mathbb {R} ^{m\times n}$, and $Q\in \mathbb {S} ^{n}$, where $Q>0$.
The matrix inequality given by:
$QA^{T}+AQ-QK^{T}B{T}-BKQ<0$ is bilinear in the variables $Q$ and $K$.
Defining a change of variable as $F=KQ$ to obtain
$QA^{T}+AQ+-F^{T}B^{T}-BF<0$,
which is an LMI in the variables $Q$ and $F$.
Once this LMI is solved, the original variable can be recovered by $K=FQ^{-1}$.

## Conclusion

It is important that a change of variables is chosen to be a one-to-one mapping in order for the new matrix inequality to be equivalent to the original matrix inequality. The change of variable $F=KQ$ from the above example is a one-to-one mapping since $Q^{-1}$ is invertible, which gives a unique solution for the reverse change of variable $K=FQ^{-1}$.

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.
# Guitar/Tremolo Picking
## What is Tremolo Picking?

Tremolo means a modulation in volume; in the context of stringed instruments, usually refers to repeatedly striking or bowing a single string in a steady rhythm, especially the fastest rhythm the player can maintain. (This technique is particularly common on the acoustic mandolin.) In guitar literature, this is called tremolo picking, and one of the few places the term "tremolo" is consistently used "correctly" in guitar literature (whose convention usually reverses tremolo and vibrato). This technique has nothing to do with a "tremolo bar" (really a vibrato bar) or a "tremolo" effects box.

## How to hold the Pick

Tremolo picking, though appearing hard at first, is actually quite easy. It is merely alternate picking at a faster speed. To start off, a pick makes tremolo picking much easier and is highly recommended when attempting it, but even though most people find tremolo picking much easier with a pick, it is possible without a pick. The best way to hold your pick is between your thumb and the side of the first knuckle of your pointing finger, but if you feel more comfortable holding it another way, such as with your thumb and middle finger then go ahead.

## How to Pick

The movement should come mostly from the wrist. A little bit of arm movement is okay, but shouldn't be done intentionally. It is possible to tremolo with the elbow, but the wrist is actually easier and faster for most people with practice.
The motion done with the wrist should be like drawing quick zig zags, or Vs. Picking should feel just like writing. Imagine drawing as many connected V's as possible.
Do not play with your hand parallel to the strings. Pick like you write, with your wrist at an angle.

## Grip

An important aspect of tremolo picking that many beginners fail to realise is that you must have a relaxed grip on the pick, as when you try to pick when holding the pick tensely, you will find that the pick hits the string harder therefore making it harder to pass through the string, causing it to sound sloppy. Maintaining a relaxed grip becomes harder when playing faster, but you will get used to it.

## Things to Remember

When tremolo picking make sure you use just your wrist, as this will make it much easier to pass through the string. Also, when you pick the string, make sure your hand doesn't go to far away from it, as this will slow you down. The impact from hitting the string usually forces your hand to leave the string, but after practice, avoiding this will become easier.
It's also important to remember that many beginners start to use the thin side of the pick to tremolo, since the thin side has a smaller surface area and passes through the strings easily. This is incorrect, as the pick start to not only damage the strings, but also causes damage to the wrists, and may further start to ring other strings. Use the flat side of the pick to tremolo, not the thin border.
# Guitar/Tuning Your Ear

Almost as important as learning how to tune your guitar is learning how to tune your ear. Most people do not have "perfect pitch", which is an umbrella term for anyone who is really accurate at determining or singing notes. Although you can naturally be more or less inclined to hear correct pitches, this is a skill that can be learned, and there are several ways of learning it. However, the truth of the matter is that no one is absolutely "perfect" when it comes to pitch, and indeed, many songs you listen to could be deliberately composed slightly sharp or flat.
Some guitar players prefer their instruments not to be exactly tuned, because it gives some extra texture. Tuning the thinner strings slightly sharp can make chords sound brighter, while tuning thicker strings down slightly can give chords some "bite". Another important reason to learn to tune your ear is that one day you will sit down to jam with someone, and have no tuner available, and you will be forced to tune to one another. This is especially true if you have a band, and you are playing live shows; being able to quickly adjust your instrument can be the difference between a great performance and a terrible one.

## Using a Human Voice

If you are in a band, and you have a singer that can never quite hit those notes perfectly, sometimes it is easier to retune your guitar rather than try to retune the singer. Learning to tune your strings to a note someone is singing is a valuable skill, and is surprisingly similar to other regular tuning methods, especially if you can tune to a piano.
First the singer should sing a note, preferably an A or an E, and in the most comfortable range of their singing voice. If they can sing high notes, or different notes on command, it is sometimes easiest to tune each string to the singer, and then quickly do some fine adjustments to make sure the guitar strings are consistent.

## Using Dissonance

Players that have been playing for a long time begin to learn what a chord or interval is "supposed" to sound like. Sometimes this is because a favourite song starts with this chord, or because they always play a particular chord type (like a power chord, major barre chord, etc.).
With this method of tuning, you simply play the appropriate chord, and then adjust your strings until it sounds "right". For example, switching into Drop D tuning is often easy for many people, because the open power chord has a certain tone when it is properly tuned. Power chords are especially easy to tell when they are out of tune, because they contain only two notes.

### Chords for Tuning

These are some common chords that players use to tell if their guitar is properly tuned.
The power chord is a useful tool in tuning as well as it mimics the natural overtone series of the guitar. Playing a note and a fifth above it, on a piano for example, will allow you to more precisely tune your instrument.
# Guitar/Understanding the Strings

To properly understand the strings, the guitar must be in tune. The strings should be tuned to EADGBE. That is, the thickest string should be a low E, the next an A, and so on. This is called standard tuning. Other tunings exist, but we should not concern ourselves with them yet.

## The fifth (standard A) and fourth (standard D) strings

The fifth string behaves similarly to the others: each fret is a single semitone up from the previous fret. The fifth fret on the fifth string is a D, which is the same note as the open fourth string. Likewise, the fifth fret on the fourth string is a G, the same as the open third string. This is the fret used to tune the string above it.

## The third string

This string is slightly different from every other string on the guitar. The reason is the fourth fret on this string, not the fifth, has the same note as the second string.

## The second string

The fifth fret is an E and the same as the open first string.
# Guitar/Vibrato Bar Techniques

The tremolo bar was originally only found on Fender guitars, but now they are on many types of electric guitar. Unfortunately, it has an inappropriate name, because "tremolo" means a fast succession of two different tones. A more accurate but less common name is the vibrato bar, and they are also known as whammy bars.
There are several different types of tremolo bars, details of which can be found in the Anatomy of the Guitar section, but certain types can only perform certain techniques. Thus, you should make sure the tremolo bar you have can do what you want it to do, before you buy it.
In general, it is good to learn to hold the tremolo bar between your third and fourth fingers, so you can use the bar and hold a pick at the same time.
This section will provide a description of how to accomplish various techniques, but it will be up to the guitarist to discover how to perform them.

## Dive Bomb

A Dive Bomb may be achieved by striking a natural harmonic then lowering the tone. An "explosion" may added by keeping the bar pressed down and flicking the low E string repeatedly.
Also, there is an alternative way to do a dive bomb, by flicking a string, dipping the bar down, tapping a harmonic, then manipulating the resulting note however you want. This technique is also known as a "Squeal", or "Dime Squeal" named after Pantera guitarist, Dimebag Darrel.

## Dipping

Dipping is a technique that allows you to make note changes a little more interesting. Before you change to a higher note, use the bar to quickly lower and then raise the pitch.

## Cat Purr

With this technique, a pitch is held for a beat, and then raised up a tone. The lever is moved slowly, and once you reach the upper or lower tone, you immediately hold, and then reverse direction. This results in a sound that can sounds remarkably like a cat.

## Ruler Sound

If you press the tremolo bar down, and then suddenly release it upwards and quickly alternate between high ups and down, it makes a snap-away sounds, like a ruler vibrating off the edge of a table. The principle behind this is similar to the cat purr.

## Windmill

The "windmill" develops if you just keep turning the tremolo bar in a circle. Naturally, the tone moves up and down at a regular pace. However, this can sound very "outer space" and can easily be over done, and you should use this sparingly.

## String Choke

If the strings are really slack, you can quickly whip the tremolo bar back up until it clicks, making a string choke. Sometimes overtones will remain, and you can get some interesting sounds and harmonies. However, these tend to disappear quickly as it is drowned out by the harmonics of the new string pitch.
# Guitar/The four quarters strumming patterns/Song Examples
### D major chord and 4/4 strumming pattern

* Brother Jakob
# Georgia Water/Laws/Contributors

The structure of this guide and the initial summaries were created by two students at the University of Georgia School of Law, Christian Perrin ('06) and Wilson Barmeyer ('06). The work was done under the direction of Professor Kathy Hatcher (School of Ecology) and Professor Laurie Fowler (School of Ecology and School of Law).
# Georgia Water/Laws/Endangered Species Act

U.S. Endangered Species Act (1973)

## General Description

Congress enacted the ESA in 1973 to conserve both endangered and threatened species and also to protect the ecosystems upon which those species depend. The ESA requires the Secretary of the Interior to take action to avoid jeopardizing the continued existence of a species. The Secretary also has a duty to conserve threatened and endangered species until they are no longer threatened or endangered. The ESA requires the secretary to give the highest priority to protecting the endangered species.

## Listing a Species

The Secretary of the Interior has been given the authority under the ESA to determine whether any species is endangered or threatened.

## Limits on Federal Agency Action

ESA prohibits an agency from engaging in any action that is likely to jeopardize the continued existence of endangered or threatened species or that destroys or adversely affects the critical habitat of such species.

## Limits on Private Development

The ESA prohibits any individual/entity to "take" an endangered or threatened species. The definition of "take" includes harass, harm, hunt, kill, shoot, capture, collect, etc.
# Georgia Water/Laws/Environmental Policy Act

Environmental Policy Act, O.C.G.A. § 12-16-1

## Summary of the Act

This Act requires the State to prepare an environmental effects report for any state funded construction project that may have a significant adverse effect on the environment. This requirement applies to “land disturbing activities,” which includes scraping, clearing, dredging, grading, excavating, or building a dam. It applies to government projects, projects funded by government grants, or municipal projects receiving more than $250,000 or more than 50% government funding, but it does not apply to agriculture or private developments.

## Specific provisions of the Act
### Environmental Effects Reports

    If the project will have a significant effect on the environment, an environmental effects report must be prepared. The report should explain any environmental impacts, including effect on water quality and quantity, and also environmental effects that cannot be avoided, possible alternatives, and proposed mitigation.

### Openness to the Public

    The government agency must make the environmental effects report available to the public, and notify the public about the project. If at least 100 people request a public hearing, then the state must hold a hearing a take public comments.

## References

* O.C.G.A. § 12-16-1
# Georgia Water/Laws/Conservation Use Act of 1991

Conservation Use Act, O.C.G.A. § 48-5-7.4

## General Description

The Conservation Use Act allows property owners to designate all or portions of their property for “bona fide conservation use” which will entitle the land owner to tax breaks on undeveloped land.

## Qualifying Uses

Qualifying uses may include wetlands, storm-water wetlands and runoff areas, groundwater recharge areas, river and stream corridors and buffers, habitat for endangered species, wildlife habitat in its natural state, high elevation areas such as ridge tops and crests, and farmland.

## Requirements

To qualify, the landowner must agree to have the land continuously devoted to a qualifying use for ten years. There is a 2,000 acre maximum. To maintain conservation use status, the property owner must submit an annual inspection report from a licensed engineer certifying that the property is being maintained in the state for which it was intended. Property subject to this current use assessment shall be separately classified from other property on the tax digest.

## References

* O.C.G.A. § 48-5-7.4
# Georgia Water/Laws/Convention on International Trade in Endangered Species

U.S. Convention on International Trade in Endangered Species (1975)

## General Description

With over 100 parties, the Convention sets up a complex system of import and export permits and regulations to safeguard endangered species from over-exploitation.

## Structure

A Convention-appointed Secretariat prepares scientific and technical studies as well as coordinates recordkeeping and reporting as required by the Convention. The Convention also establishes a Conference of State Parties which meets every two years to adapt the Convention to evolving conditions.

## Functioning of the Permit System

The permits under the system are nationally administered and keyed to categories of endangered species. Exemptions from permits system exist, and states are permitted to take stricter measures if they wish.
# Georgia Water/Laws/Endangered Wildlife Act of 1973

Georgia Endangered Wildlife Act of 1973, O.C.G.A. §§ 27-3-130

## General Description

This law, similar to (but somewhat different from) the Federal Endangered Species Act, prohibits the capture, killing, or selling of protected species and protects the habitat of these species on public lands.

## Limited to public lands

Georgia law specifically states that rules and regulations related to the protection of state protected species shall not affect rights in private property.

## Primacy requirements not met

The Georgia Endangered Wildlife Act is much narrower in scope than the Federal Endangered Species Act. Because this State law is not as stringent that the Federal Act, Georgia does not receive primacy (Georgia does not have the responsibility to administer federal laws within the state).

## Currently protected species

A list of the currently protected species in Georgia is available at:

## References

* O.C.G.A. §§ 27-3-130