OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/Stability Analysis/Stabilizability LMI

Stabilizability LMI
Stabilizability is a essentially a weaker version of the controllability condition. A system is stabilizable if all unstable modes of the system are controllable. This implies that if the system is controllable, it will also be stabilizable. The LMI condition for stabilizability of pair $(A,B)$ is shown below.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\x(0)&=x_{0},\end{aligned}}$
where $x(t)\in \mathbb {R} ^{n}$, $u(t)\in \mathbb {R} ^{m}$, at any $t\in \mathbb {R}$.

## The Data

The matrices necessary for this LMI are $A$ and $B$. There is no restriction on the stability of A.

## The LMI: Stabilizability LMI

$(A,B)$ is stabilizable if and only if there exists $X>0$ such that
       $AX+XA^{T}-BB^{T}<0$,
where the stabilizing controller is given by
       $u(t)=-{\frac {1}{2}}B^{T}X^{-1}x(t)$.

## Conclusion:

If we are able to find $X>0$ such that the above LMI holds it means the matrix pair $(A,B)$ is stabilizable. In words, a system pair $(A,B)$ is stabilizable if for any initial state $x(0)=x_{0}$ an appropriate input $u(t)$ can be found so that the state $x(t)$ asymptotically approaches the origin. Stabilizability is a weaker condition than controllability in that we only need to approach $x(t)=0$ as $t\rightarrow \infty$ whereas controllability requires that the state must reach the origin in a finite time.

## Implementation

This implementation requires Yalmip and Sedumi.
https://github.com/eoskowro/LMI/blob/master/Stabilizability_LMI.m

## Related LMIs

WIP: Will be linked once they have been created.
Controllability LMI
Hurwitz Stability LMI
Schur Stability LMI
Observability LMI
Detectability LMI
Controllability Grammian LMI
Observability Grammian LMI

## 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.
* 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 Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
* A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.
# LMIs in Control/Time-Delay Systems/Continuous Time

      * LMI for Time-Delay system on delay Independent Condition
      * LMI for Time-Delay system on delay Dependent Condition
      * LMI for Stability of Retarded Differential Equation with Slowly-Varying Delay
      * LMI for Robust Stability of Retarded Differential Equation with Norm-Bounded Uncertainty
      * Bounded Real Lemma under Slowly-Varying Delay
      * LMI for L2-Optimal State-Feedback Control under Time-Varying Input Delay
# LMIs in Control/Tools

      * Basic Matrix Theory
      * Notion of Matrix Positivity
      * Schur Complement
      * Elimination of Variables
      * Maximum Singular Value of a Complex Matrix
      * Minimum Singular Value of a Complex Matrix
      * Structured Singular Value
      * Eigenvalue Problem
# Irish/Vocabulary
## Home

House • Kitchen • Living room • Bedroom • Bathroom • Garden

## Body

Body parts • Clothes

## Food

Fruit • Meals • Vegetables

## Travel

Nationalities • Countries

## Nature

Animals

## Work

Professions

## Society

School subjects • Insults

## Parts of speech

Verbs
# Irish/Vocabulary/Bedroom

| English | Irish (singular) |
| bedroom | seomra leapa |
| bed | leaba |
| lamp | lampa |
| shelf | seilf |
| wardrobe | vardrús |
| alarm clock | clog aláraim |
| blanket | blaincéad, pluid |
| sheet | braillín |
| quilt | clúdach leapa |
| pillow | piliúr |
| pillowcase | clúdach piliúir |
| headboard | clár cinn |
| bedstead | fráma leapa, stoc leapa |
| radiator | radaitheoir |
| heater | téitheoir leictreach |
| nightstand | bord cois leapa |
| chest of drawers | cófra tarraiceán |
| clothes basket | ciseán éadaigh |
| bookshelf | seilf leabhar |
| toy | bréagán |
| toy box | bosca bréagán |
| game | cluiche |
| doll | bábóg |
| ball | liathróid |
| doll's house | teach bábóige |
| games console | consól cluichí |
| video game | físchluiche |
| contoller | rialtán |
| cot | leaba linbh (f), cliabhán |
| mobile | móibíl (f) |
| dehumidifier | díthaisire |
# Irish/Vocabulary/Animals

Lessons
Unit One: 1 2 3 4
Unit Two: 1 2 3 4 5 6
Unit Three: Lesson Index
Language Reference
Spelling and Pronunciation - Grammar
Wikiversity
More Irish language resources can be found at Wikiversity's Department of Irish Studies
This lesson is part of Unit 3 of WikiBooks Irish, a comprehensive open-source course in the Irish language. Lessons in Unit 3 are intended for the advanced Irish learner and contain specialized vocabulary and language notes for specific subject areas.

## Talking About Animals

Vocabulary :
| animal | ainmhí | pronunciation |
When discussing ainmhí (animals) as gaeilge, bear in mind the following:

### Pronouns

Because Irish has no neutral pronouns, animals take the pronoun appropriate to the grammatical gender of the word in question. Most general names for animal species are masculine.

### Eating Animals

As in English, in some cases different words are used for the animal's meat when used as food than when discussing the living animal. These words will be considered separately in the Unit 3 lesson on food.

## Peataí

Vocabulary :
| peata | pet | pronunciation |
| cat | cat |
| dog | madra |
| kitten | puisín |
| puppy | coileán |

## Ainmhithe na hÉireann

| bird | éan |
| butterfly | féileacán |
| deer | fia |
| duck | lacha |
| mouse | luch |
| owl | ulchabhán |
| spider | damhán alla |
| rabbit | coinín |
| rat | francach |
| seal | rón |
| turtle | turtar |

## Ainmhithe na feirme

Vocabulary :
| feirme | farm | pronunciation |
| cow | bó |
| calf | lao, gamhain |
| foal | searrach |
| goat | gabhar |
| hen | cearc |
| horse | capall |
| pig | muc |
| lamb | uan |
| sheep | caora |

## Ainmhithe an zú

Vocabulary :
| zú | zoo | pronunciation |
| giraffe | sioráf |
| crocodile | crogall |
| bear | béar |
| elephant | eilifint |
| hippopotamus | dobhareach |
| kangaroo | cangarú |
| wolf | mac tíre |
| zebra | séabra |
| koala | cóála |
| lion | leon |
| monkey | moncaí |
| panda | panda |
| penguin | piongain |
| tiger | tíogar |

## Ainmhithe uisceach

| cod | trosc |
| crab | portán |
| dolphin | deilf (f) |
| fish | iasc |
| lobster | gliomach |
| salmon | bradán |
| shark | siorc |
| whale | míol mór |

## See also

Other Unit 3 lessons relevant to animals may include:
* Food (culinary names for animals' meat)
* Sport (terms related to horse and dog racing)
# Irish/Unit 3

Lessons
Unit One: 1 2 3 4
Unit Two: 1 2 3 4 5 6
Unit Three: Lesson Index
Language Reference
Spelling and Pronunciation - Grammar
Wikiversity
More Irish language resources can be found at Wikiversity's Department of Irish Studies
Unit 3 of WikiBooks Irish consists of a series of detailed lessons on specialized subjects, intended for the advanced learner. These lessons will contain detailed and comprehensive discussions of the vocabulary, terminology, and, where appropriate, grammar and syntax related to a specific subject.
Unlike the numbered lessons of Unit 1 and Unit 2, these lessons are purely subject-based and may be explored in any order. They are listed below by subject matter.
These lessons address real world subjects, like clothing, cooking, plumbing, or sport, not general language topics like pronouns or plurals. Comprehensive overviews of language can be found instead in the Irish reference pages.
* Editor's Note: This page is in the process of being converted from a group of vocabulary lists to a set of more learner-friendly lessons.

## General

Verbs

## Home

House • Kitchen • Living room • Bedroom • Bathroom • Garden

## Body

Body parts • Clothes

## Food

Fruit • Meals • Vegetables

## Geography

* Tíortha an Domhain (Countries of the World)

## Nature

Animals

## Work

Professions

## Society

School subjects • Insults
# Irish/Unit 2/Lesson 6

Lessons
Unit One: 1 2 3 4
Unit Two: 1 2 3 4 5 6
Unit Three: Lesson Index
Language Reference
Spelling and Pronunciation - Grammar
Wikiversity
More Irish language resources can be found at Wikiversity's Department of Irish Studies

## Love

Tá grá agam duit or Tá mé i ngrá leat = I love you
Le gach dea-ghui = With best wishes

## Discussing the Weather

Aimsir = Weather
Conas atá an aimsir inniu? = How is the weather today?
Tá an grian ag taitneamh = The sun is shining
Tá sé ag cur/stealladh báistí = It is raining/"pouring"
Feicim scammail sa spéir = I see clouds in the sky
Beidh sé ag cur sneachta amárach = It will be snowing tomorrow
Chonaic/ní fhaca mé réamháisnéis na haimsire = I saw/didn't see the weather forecast
# Irish/Vocabulary/Bathroom

| English | Irish (singular) |
| bathroom | seomra folctha |
| bathtub | folcadán |
| sink | doirteal |
| plug | stopallán |
| toilet | leithreas |
| tap | sconna |
| mirror | scáthán |
| shower | cithfholcadán |
| shower head | sconna cithfholcadáin |
| soap | gallúnach |
| razor | rásúr |
| shaving foam | sobal bearrtha |
| bath mat | mata folcadáin |
| toothbrush | scuab fiacla |
| toothpaste | taos fiacla |
| shampoo | seampú |
| shower gel | glóthach chithfholctha (f) |
| medicine cabinet | cófra cógas |
| first aid kit | fearas garchabhrach |
| clippers | clipéar |
| towel | tuáille |
| flannel | tuáille éadain |
| deodorant | díbholaíoch |
| shower curtain | cúirtín cithfholctha |
| comb | cíor (f) |
| hair brush | scuab ghruaige |
# Learning anatomy/Oral cavity
## Tooth
### Anatomy

The teeth (singular: tooth) are made up of parts shown below.
* Crown
** Enamel
** Dentin
* Neck
** Gingiva
** Pulp cavity
** Pulp
* Root
** Cementum
** Root canal
** Periodontal ligament
** Nerves and blood vessels
** Alveolar bone
* Jaw
** Maxillae
** Mandible

### Function

The teeth function to chew food. The types of teeth are the incisors, canines, premolars, and molars. Incisors cut the food. The canines grip the food. The premolars and molars (from Latin: mola = grindstone) grind the food. A baby has 20 teeth, but they are not visible until they erupt through the gingiva (gums) and an adult has 32 teeth. Maxillary (upper) teeth and their associated periodontal ligament are innervated by the superior alveolar nerves, branches of the maxillary division, termed the posterior superior alveolar nerve, anterior superior alveolar nerve, and the variably present middle superior alveolar nerve. The teeth help us speak.

## Salivary gland
### Anatomy

The parotid, sublingual, and submandibular glands are major salivary glands. The other ones are minor salivary glands.

### Function

The salivary glands make saliva. Saliva travels through ducts and then into the mouth.

#### Saliva

Saliva has alpha-amylase which breaks down starch and lingual lipase which breaks lipids. Saliva also cleans the mouth.

## Lingua
### Anatomy

The lingua (tongue) is made up of the base (root), body, and apex.

### Function

The lingua helps in speech. Saliva helps taste the food. The chemicals from the food travel from the lingual papillae to the taste buds. The lingua and saliva from the food bolus. It gets sent down the pharynx.

## Air

The mouth is also a passage for air.
# Learning anatomy/Stomach, Gaster, or Gastric Region

Esophagus - Small bowel

## Anatomy

The stomach is a J-shaped ”bag” that connects the esophagus to the small bowel. The parts of the stomach are the cardiac region, gastric fundus, gastric body, and pyloric region. The pyloric region is made up of the pyloric antrum, and the pyloric canal.

## Function

The first sphincter of the stomach is the cardiac sphincter or lower esophageal sphincter which guards the entrance to the stomach. If the sphincter weakens, stomach contents can damage the lining of the esophagus and cause acid reflux so, please take care of your lower esophageal sphincter.
The stomach has folds called rugae which look wrinkled when no food is in the stomach. The stomach stores food. If the food is poisonous, it triggers the vomiting reflex. In this case, the body protects itself from the acid. But if it’s not poisonous, the food continues to go. The stomach will make acid, digestive enzymes, and mucin. Hydrochloric acid breaks down all foods into chyme. Amylase breaks down starch. Pepsinogen is the inactive form of pepsin. When pepsinogen reaches the acidic ph of the stomach, it turns into pepsin which digests proteins. Gastric lipase breaks down lipids. Hydrochloric acid in the stomach kills germs. Gastrin and somatostatin are hormones. Gastrin inhibits acid secretion. Somatostatin does a lot of things.
# Learning anatomy/Nephron
## Anatomy

The parts of the nephron are the bowman’s capsule, proximal tubule, a loop of Henle, distal tubule, and collecting tube.

## Function

The urine starts in the bowman's capsule and then into the proximal tubule and then into the loop of Henle and then into the distal tubule and then into the collecting tubes which takes the urine to the ureters.
# Learning anatomy/Nose
## Anatomy

The parts of the nose are:
* Nasion
* Root
* Bridge
* Ala
* Apex
* Nostril
* Columella
* Nasal cavity
** Vestibule
** Olfactory region
** Respiratory region

## Function

The air goes inside the nasal cavity. The nasal cavity warms the air and traps dust particles with tiny hairs and mucous. The smell of things in the air is caused by the olfactory tract which sends signals to the brain about the smell.
# Learning anatomy/Pancreas exocrine

Accessory organs
* Liver
* Biliary tract
* Pancreas exocrine
* Pancreas endocrine
* Kidney
_______________________________________________________________________
Biliary tract - Pancreas endocrine

## Anatomy

Acinar cells in the pancreas.

## Function

Acinar cells make digestive enzymes and bicarbonate. Bicarbonate neutralizes the hydrochloric acid from the stomach. Amylase breaks down starch. Lipase breaks down lipids. Trypsin breaks down proteins. Chymotrypsinogen is activated by trypsin to make chymotrypsin. Chymotrypsin breaks proteins in the chyme.
# Learning anatomy/Spleen

Pancreas endocrine - Heart

## Anatomy

The red and white pulps.

## Function

The spleen makes bilirubin which is sent to the liver to make bile. The spleen also fights off sickness.
# Learning anatomy/Pancreas endocrine

Accessory organs
* Liver
* Biliary tract
* Pancreas exocrine
* Pancreas endocrine
* Kidney
_______________________________________________________________________
Pancreas exocrine - Spleen

## Anatomy

Endocrine cells in the pancreas.

## Function

Beta cells make insulin. Alpha cells make glucagon. Insulin is like a key for opening up the cells so that glucagon can get in.
# Learning anatomy/Small bowel

Stomach - Large bowel Template:Toc

## Anatomy

The small bowel or small intestine connects the stomach to the large intestine. The parts of the small intestine are the duodenum, jejunum, and ileum.

## Function

The small intestine absorbs most of the nutrients from the chyme. The pyloric sphincter lets only a tiny bit of chyme go into the small intestine at a time. The small finger-like projections are called villi. Villi help in the absorption. The jejunum absorbs sugars, amino acids, and fatty acids. The ileum absorbs bile acids, fluid, and vitamin B12.
# Italian/Contents
## General Pages

* Introduction ~ Introduzione
* About the Book ~ Del libro
* Layout~ Disposizione
# Italian/Grammar
## Grammatica dell'italiano standard

* Pronunciation
* Articles
* Nouns
* Pronouns
* Adjectives
* Introduction to Verbs
* Verbs Tenses and Moods
* Verb Conjugation Reference
* Some Important Irregular Verbs
* Adverbs
* Prepositions
* Conjunctions
* Interrogatives
* Comparatives and superlatives
* If sentences (Periodi ipotetici)
* Modifiers (Suffixes)
* Sentence Structure
* Direct and Indirect Speech
# Italian/Grammar/Conjunctions
## Conjunctions / Congiunzioni

The Italian language offers a variety of conjunctions that are used to connect words, phrases, and clauses. Here are some commonly used conjunctions and their meanings:
* affinché (so that): used to express purpose or intention.
* perché (because): used to give a reason or cause.
* benché, per quanto, sebbene (although): used to introduce a contrast or contradiction.
* a meno che (unless): used to express a condition that must be met in order for something to happen.
* prima che (before): used to indicate that an action or event takes place before another.
* dopo che (after): used to indicate that an action or event takes place after another.
* purché (provided that): used to express a condition or requirement.
* senza che (without): used to indicate the absence of something.
* ma (but): used to introduce a contrast or contradiction.
* e (and): used to connect two or more elements.
* o, oppure (or): used to present alternative options.
* invece (instead): used to indicate a contrast or alternative.
* dunque, perciò, quindi (so, therefore): used to indicate a consequence or result.
* Nonostante (despite): Nonostante is used to express a contrast or opposition between two elements. It is used to introduce a subordinate clause that contradicts or goes against the main clause.
* Finché (as long as): Finché is used to express a condition or duration of time. It is equivalent to the English phrase "as long as" and introduces a subordinate clause that explains a condition that needs to be met.
* Se (if): Se is used to introduce a conditional clause. It is equivalent to the English word "if" and shows a possibility or requirement for something to happen.
* Poiché (since): Poiché is used to indicate a cause or reason for something. It is similar to the English word "since" and introduces a subordinate clause that explains the cause behind the main clause.
* Anche se (even though): Anche se is used to express a contrast or concession. It is translated as "even though" in English and introduces a subordinate clause that contradicts or goes against the main clause.
* Tuttavia (however): Tuttavia is used to introduce an alternative or a contrast to what has been stated previously. It is equivalent to the English word "however" and shows a different perspective or opinion.
* Quindi (so): Quindi is used to indicate a consequence or a conclusion. It is translated as "so" in English and shows the result of the actions or circumstances described in the main clause.
By using these conjunctions correctly, you can enhance your Italian writing and better express relationships between different elements in a sentence.
# Italian/Grammar/Adjectives
## Adjectives / Aggettivi

Unlike in English, adjectives may be placed both before or after the noun.
Example:
As a general trend, adjectives tend to follow nouns rather than precede them.
Adjectives must agree with the noun they modify both in gender (masculine or feminine) and in number (singular or plural).
Examples:
Endings for adjectives:
Adjectives that end in -e do not need to modify their endings for gender. Their endings change only depending on whether the noun is singular or plural.
Example:
See also
* Italian/Vocabulary/Adjectives
* Italian/Grammar/Comparatives and superlatives
# Italian/Grammar/Adverbs

An adverb is a word or phrase that modifies or describes a verb, adjective, or other adverb. In Italian, adverbs play an important role in adding detail to a sentence and expressing various aspects such as time, place, manner, degree, and frequency. This chapter will provide an overview of Italian adverbs, including their forms, usage, and common examples.

## Forms of Adverbs

Italian adverbs are generally derived from adjectives by adding the suffix -mente to the feminine singular form of the adjective. For example, the adjective veloce (fast) becomes the adverb velocemente (quickly).
However, not all adverbs are formed by this pattern. Some adverbs have irregular forms, while others are identical to their corresponding adjectives. For instance:
* Adjective: bello (beautiful)
* Adverb: bello (beautifully)
* Adjective: forte (strong)
* Adverb: forte (strongly)
In addition to these regular and irregular forms, there is a group of adverbs that have unique forms and do not derive from adjectives. Examples of this group include bene (well), male (badly), presto (quickly), tardi (late), and mai (never).

## Usage

Italian adverbs are versatile and can be used in various ways to modify different parts of speech. Here are some common usage patterns:

### Modifying Verbs

Adverbs in Italian often modify verbs, indicating aspects such as manner, frequency, time, or degree. They usually come after the verb they modify. For example:
* Cammina velocemente. (He walks quickly.)
* Parlo spesso con mia madre. (I often speak with my mother.)
* Mangio sempre pasta. (I always eat pasta.)
* Studia molto. (He studies a lot.)

### Modifying Adjectives

Adverbs can also modify adjectives to express the degree or intensity of the adjective. In this case, they usually come before the adjective. For example:
* È molto bello. (It's very beautiful.)
* Sono abbastanza stanco. (I am quite tired.)
* La casa è estremamente grande. (The house is extremely big.)

### Modifying Other Adverbs

Similar to their usage with adjectives, adverbs can modify other adverbs to show the degree or intensity. Again, they typically come before the adverb they modify. For example:
* Parla molto chiaramente. (He speaks very clearly.)
* Lavora abbastanza velocemente. (He works quite quickly.)
* Ho sempre vissuto qui. (I have always lived here.)

## Common Adverbs

Italian has a wide range of adverbs, each serving a specific purpose. Here are some commonly used adverbs in Italian:
* bene - well
* male - badly
* spesso - often
* sempre - always
* mai - never
* troppo - too much/too many
* poco - little/few
* abbastanza - enough/quite
* molto - very/much
* così - so/this way
* presto - soon/quickly
* tardi - late
* qui - here
* lì - there
# Italian/About the book
## Aim

This book aims to teach people, with no or little previous knowledge, how to read, write and maybe speak some Italian.

## About the book

The book is still in an early development stage.

## Lessons Information

* Lesson Planning
* Concept Lessons

## Authors

* Fitch
* Angelo Romano
* Sinnefa
* Laurusnobilis
NOTE: Feel free to add your (real or user) name to this list if you have written important / major contributions to this book.
# Italian/Grammar/Articles
## Determinative article (the) / L'articolo determinativo

The English article the has several translations into Italian, according to:
* gender: masculine or feminine
* number: singular or plural
* initial of the following word
Basic articles:
Articles must have the same gender and number of the referred noun.
Examples:
When the following word begins with z, s+consonant, gn, pn, ps or x, the masculine article is changed as follows:
Masculine articles before impure consonant sounds:
Examples:
When the following word begins with a vowel, articles are changed as follows:
Articles before a vowel:
The form l' is not colloquial, it is the normal form even in very polite and refined speech.
Examples:
Note that when the vowel is an i pronounced as a semi-vowel, lo and la are used, eg lo iodio, la iarda.

## Indeterminate and Partitive Articles (a/an) / Gli articoli partitivi e indeterminativi (un/uno/una)

Basic indeterminate articles:
These articles do not have any plural but instead the Partitive Article can be used. These articles all have the meaning of "some" and they are similar to some prepositions but they are not prepositions, they work like articles.
Partitive articles or plural indeterminative: