OpenLLM-France/wikimedia huggingface dataset:
# ans Most recent answer

Julia's ans is functionally basically identical, though note that it is available only at the REPL.
# Aerospace Toolbox

Maps functionality from the MATLAB Aerospace Toolbox to equivalent(s) in Julia.

## Related Julia packages

* ReferenceFrameRotations.jl
# Desktop Environment
## Startup and Shutdown

See Getting Started in the Julia manual.

#### startup Start up function for user-defined options

Equivalently, the file ~/.julia/config/startup.jl is executed whenever Julia is run.
# Graphics

See Introducing Julia/Plotting for an overview of the plotting ecosystem in Julia.
# Kapampangan
## Table of contents

* Disclaimer
* Introduction
* How to use this book
* Lesson 1: Pronouncing Kapampangan
* Lesson 2: Greeting
* Lesson 3: Introducing Yourself
* Lesson 4: Introducing Yourself, Part Two
* Lesson 5: Where Are You From?
* Lesson 6: Inclusive And Exclusive We
* Lesson 7 Nouns
* Lesson 7.1Other Pronouns
* Lesson 9: Numbers To Twenty
* Lesson 8: Numbers To One Hundred
* Lesson 10: Numbers Above One Hundred
* Lesson 11: A Little Bit More Vocabulary
* Lesson 12: Verbs At The Future Tense
* Lesson 13: Verbs At The Present Tense
* Lesson 14: Verbs At The Past Tense
* Lesson 15: Ergativity
* Lesson 16: Days Of The Week
* Lesson 17: Months
* Kapampangan To English Lexicon
* English To Kapampangan Lexicon
* Solutions to all exercises
* Alphabetical Index
* References

## TODO list

* plural of y is di, plural of ing is deng
* verbs: insert um for the future, in for the past, duplicate the syllable for the present
* mag verbs: mág for present, mig for past
* ning means of sing., plural is dari
* ku -> ku, ka -> mu, ya -> na
* at means and
* a from mayap a bengi can become ng if final letter of adjective is vowel
* adjective-noun construction
* cang yang lang and cung meaning
* clean country list
* numbers
* add examples, exercices and solutions for the days of the week
* add examples, exercices and solutions for the months
* explain da for the introcing yourself lesson
# Kannada/Talking about oneself

| Kannada | English Transliteration | English Translation |
| ನನ್ನ ಹೆಸರು ಮಾಯ. | Nanna hesaru Maya. | My name is Maya. |
| ನಿಮ್ಮ ಹೆಸರು ಏನು? | Nimma hesaru yenu? | What is your name? (formal/plural) |
| ನಿನ್ನ ಹೆಸರು ಏನು? | Ninna hesaru yenu? | What is your name? (informal/singular) |
| ನನ್ನ ಊರು ಬೆಂಗಳೂರು. | Nanna Ooru Bengaluru. | My native place is Bangalore. (More commonly worded as "I am from Bangalore.") |
# Kannada/Questions

| Kannada | Transliteration | English |
| ನಿಮ್ಮ ಹೆಸರೇನು? | Nimma Hesarenu? | What's your name? |
| ಹೇಗಿದ್ದೀರಾ? | Hegiddira? | How are you? |
| ಏನು ಮಾಡುತಿದ್ದೀರಾ? | Yenu Maaduthiddira? | What are you doing? |
# Kannada/Verb

| English | Transliteration in Kannada | Kannada |
| eat | thinnu / unnu | ತಿನ್ನು / ಉಣ್ಣು |
| come | baa | ಬಾ |
| go | hogu | ಹೋಗು |
| drink | kudi | ಕುಡಿ |
| sleep | Malagu | ಮಲಗು |
| read | odhu | ಓದು |
| write | bari | ಬರಿ |
| sit | Kulituko / kutko / kooru / kundru | ಕುಳಿತುಕೋ / ಕೂತ್ಕೋ / ಕೂರು / ಕುಂಡ್ರು |
| stand | ninduko / nill(u) | ನಿಂತ್ಕೋ / ನಿಲ್ಲು |
| ask | kelu | ಕೇಳು |
| call | kare / kari | ಕರೆ / ಕರಿ |
| do | madu | ಮಾಡು |
| gulp | nungu | ನುಂಗು |
| walk | nade / nadi | ನಡೆ / ನಡಿ |
| run | odu | ಓಡು |
| give | kodu | ಕೊಡು |
| take | tagedko / tago | ತೆಗೆದುಕೊ / ತಗೊ |
| shout | Kirachu / odaru | ಕಿರಚು / ಒದರು |
| talk | matadu | ಮಾತಾಡು |
| tell | helu | ಹೇಳು |
| jump | egaru / jigi / kudi | ಎಗರು / ಜಿಗಿ / ಕುದಿ |
| bend | baggu / baagu | ಬಗ್ಗು / ಬಾಗು |
| push | Tallu / doodu / nooku / dabbu | ತಳ್ಳು / ದೂಡು / ನೂಕು / ದಬ್ಬು |
| snatch | kitthuko / keelu | ಕಿತ್ತುಕೊ / ಕೀಳು |
| look | nodu / kaanu | ನೋಡು / ಕಾಣು |
| listen | kelisiko / aalisu | ಕೇಳಿಸಿಕೊ / ಆಲಿಸು |
# Kannada/Plants
## Leaves names

| English | Transliteration | Kannada names of trees |
| Tree(s) | mara(gaLu) | ಮರ(ಗಳು) |
| Sacred fig (Bodhi) | arali | ಅರಳಿ |
| Banyan | aala | ಆಲ |
| Bamboo | bidiru | ಬಿದಿರು |
| Banana | baaLe | ಬಾಳೆ |
| Cashew | godambi | ಗೋಡಂಬಿ |
| Jack Fruit | halasu | ಹಲಸು |
| Rosewood | beete | ಬೀಟೆ |
| Teak | thega | ತೇಗ |
| Mango | mavina mara | ಮಾವಿನ ಮರ |
| coconut tree | Thengina mara | ತೆಂಗಿನ ಮರ |
| Cedar | Matthi mara | ಮಥಿ ಮರದ |
| Teak | Saguvani | ಸಾಗುವಾನಿ |
| Drumstick Tree | Nuggemara | ನುಗ್ಗೆಮರ |
| Sandalwood Tree | Sriganadamara | ಶ್ರೀಗಂಧದ ಮರ |
| Neem Tree | Bevinamara | ಬೇವಿನ ಮರ |
| Tamarind Tree | Hunasemara | ಹುಣಸೆ ಮರ |
| Golden Champa Tree | sampige mara | ಸಂಪಿಗೆ |
| Curry Tree | karibevu mara | ಕರಿಬೇವಿನ ಮರ |
| Arecanut tree | adaki mara | ಅಡಿಕೆ ಮರ |
# Kannada/Vegetables

| English | Kannada Transliteration | Kannada |
| Vegetable | Tarakari | ತರಕಾರಿ |
| Beans /Green Beans | Hurali kaayi | ಹುರಳೀಕಾಯಿ |
| Bitter gourd | Haagala kaayi | ಹಾಗಲಕಾಯಿ |
| Bottle gourd | Sore kaaye | ಸೋರೆಕಾಯಿ |
| Brinjal | Badane kaayi | ಬದನೇಕಾಯಿ |
| Cabbage | Yele kosu | ಎಲೆ ಕೋಸು |
| Capsicum | Donne Menasina kaayi / Dappa Menasina kaayi | ದೊಣ್ಣೆ ಮೆಣಸಿನಕಾಯಿ / ದಪ್ಪ ಮೆಣಸಿನಕಾಯಿ |
| Carrot | Gajjari | ಗಜ್ಜರಿ |
| Cauliflower | Hoo kosu | ಹೂ ಕೋಸು |
| Chow-chow | Seemebadne | ಸೀಮೆಬದನೆ ಕಾಯಿ |
| Coriander leaves | Kotthambari Soppu | ಕೊತ್ತಂಬರಿ ಸೊಪ್ಪು |
| Cucumber | Southe kaayi | ಸೌತೇಕಾಯಿ |
| Curry leaves | Karibevu | ಕರಿಬೇವು |
| Flat Beans | Avarekāyi | ಅವರೆಕಾಯಿ |
| Garlic | Belulli | ಬೆಳ್ಳುಳ್ಳಿ |
| Knol khol | GeDDee kosu | ಗೆಡ್ಡೆ ಕೋಸು/ನವಿಲು ಕೋಸು |
| Ginger | Shunti | ಶುಂಠಿ |
| Green chilli | Hasiru menasina kaayi | ಹಸಿರು ಮೆಣಸಿನಕಾಯಿ |
| Greens | Soppu | ಸೊಪ್ಪು |
| Lady's finger / Okra | Bende kaayi | ಬೆಂಡೇಕಾಯಿ |
| Lemon | Nimbe hannu | ನಿಂಬೆ ಹಣ್ಣು |
| Onion | Eerulli, Ullagadde | ಈರುಳ್ಳಿ, ಉಳ್ಳಾಗೆಡ್ಡೆ |
| Peas | Batani | ಬಟಾಣಿ |
| Potato | Aaloo gadde | ಆಲೂ ಗಡ್ಡೆ |
| Radish | Moolangi Mulli | ಮೂಲಂಗಿ |
| Pumpkin | Kumbal kaayi | ಕುಂಬಳಕಾಯಿ |
| Raw plantain | Baale kaayi | ಬಾಳೇಕಾಯಿ |
| Plantain Stem | Bale Dindu | ಬಾಳೆ ದಿಂಡು |
| Ridge Gourd/Chinese Okra | Hīrekāyi | ಹೀರೆಕಾಯಿ |
| Snake gourd | Padavala kaayi | ಪಡವಲಕಾಯಿ |
| Sweet Potato | Genasu | ಗೆಣಸು |
| Tomato | Tamtekayi, Tometo, Tomyato/Gore hannu | ಟಮಟೆಕಾಯಿ,ಟೊಮೆಟೊ/ಟೊಮ್ಯಾಟೊ/ಗೊರೆಹಣ್ಣು |
| Coconut | Thengina kayi | ತೆಂಗಿನಕಾಯಿ |
| Beetroot | Kanduberu | ಕಂದು ಬೇರು(ಕೆಂಪು ಮೂಲಂಗಿ) |
| Cocina/Ivy Gourd | Tondekayi | ತೊಂಡೆಕಾಯಿ |
| Elephant Yam | suvarna Gedde | ಸುವರ್ಣ ಗೆಡ್ದೆ |
| Mushroom | Nayi Kode/Anabe | ನಾಯಿ ಕೊಡೆ/ಅಣಬೆ |
| Mint leaves | Pudina soppu | ಪುದಿನ ಸೊಪ್ಪು |
| Fenugreek leaves | Menthya soppu | ಮೆಂತ್ಯ ಸೊಪ್ಪು |
| Spinach | palak soppu | ಪಾಲಕ್ ಸೊಪ್ಪು |
| cluster beans | Chavalikai/Gorikai | ಚವಳಿಕಾಯಿ |
| Malabar spinach | Basale soppu | ಬಸಳೆ ಸೊಪ್ಪು |
| Amaranth | rajgira | ರಾಜಗಿರಿ ಸೊಪ್ಪು |
| Bread fruit | Beru halasinakayi | ಬೇರು ಹಲಸಿನಕಾಯಿ |
| Taro root | kesu gedde | ಕೆಸುವಿನ ಗೆಡ್ಡೆ |
| Groundnut | sheng/kadle kayi | ಶೇಂಗಾಕಾಳು/ಕಡ್ಲೆಕಾಯಿ |
taciona/Cassava maragenasu ಮರಗೆಣಸು
# Kannada/Zodiac Signs

| Zodiac/ರಾಶಿಗಳು | Transl. | Zodiac Signs | Gloss/ಅರ್ಥ | Dates/ ದಿನಾಂಕ | Planet/ಗ್ರಹ (Symbol/ಚಿನ್ಹೆ) | Element/ಅಂಶ (Symbol/ಚಿನ್ಹೆ) | Polarity/ ಧ್ರುವೀಯತೆ | Modality/ಕ್ರಮಬದ್ಧತೆ (Symbol/ಚಿನ್ಹೆ) |
| ಮೇಷ | Mesha | Aries | The Ram/ ಟಗರು | Mar 21 - Apr 19 | Mars/ಮಂಗಳ | Fire/ಅಗ್ನಿ | Positive(+) | Cardinal/ಕ್ರಿಯಾತ್ಮಕ |
| ವೃಷಭ | Vrushabha | Taurus | The Bull/ ಗೂಳಿ | Apr 20 - May 20 | Venus/ಶುಕ್ರ | Earth/ಭೂಮಿ | Negative (-) | Fixed/ಸ್ಥಿರ ( ) |
| ಮಿಥುನ | Mithuna | Gemini | The Twins/ ಗಂಡು ಹೆಣ್ಣು (ಅವಳಿ) | May 21 - Jun 20 | Mercury/ಬುಧ | Air/ವಾಯು | Positive (+) | Mutable/ ಬದಲಾಗುವಂಥದ್ದು |
| ಕರ್ಕಾಟಕ | Karkataka | Cancer | The Crab/ ಏಡಿ | Jun 21 - Jul 22 | Moon/ಚಂದ್ರ | Water/ಜಲ | Negative (-) | Cardinal/ಕ್ರಿಯಾತ್ಮಕ |
| ಸಿಂಹ | Simha | Leo | The Lion/ ಸಿಂಹ | Jul 23 - Aug 22 | Sun/ಸೂರ್ಯ | Fire/ಅಗ್ನಿ | Positive (+) | Fixed/ಸ್ಥಿರ ( ) |
| ಕನ್ಯಾ | Kanya | Virgo | The Maiden/ ಕನ್ಯೆ | Aug 23 - Sep 22 | Mercury/ಬುಧ | Earth/ಭೂಮಿ | Negative (-) | Mutable/ ಬದಲಾಗುವಂಥದ್ದು |
| ತುಲಾ | Thula | Libra | The Scales/ ತಕ್ಕಡಿ | Sep 23 - Oct 22 | Venus/ಶುಕ್ರ | Air/ವಾಯು | Positive (+) | Cardinal/ಕ್ರಿಯಾತ್ಮಕ |
| ವೃಶ್ಚಿಕ | Vrishchika | Scorpio | The Scorpion/ ಚೇಳು | Oct 23 - Nov 21 | Mars/ಮಂಗಳ | Water/ಜಲ | Negative (-) | Fixed/ಸ್ಥಿರ ( ) |
| $^{***}$ಧನು | Dhanu | Sagittarius | The Archer/ ಬಿಲ್ಲುಗಾರ | Nov 22 - Dec 21 | Jupiter/ಗುರು | Fire/ಅಗ್ನಿ | Positve (+) | Mutable/ ಬದಲಾಗುವಂಥದ್ದು |
| ಮಕರ | Makara | Capricorn | The Crocodile/ ಮೊಸಳೆ | Dec 22 - Jan 21 | Saturn/ಶನಿ | Earth/ಭೂಮಿ | Negative (-) | Cardinal/ಕ್ರಿಯಾತ್ಮಕ |
| ಕುಂಭ | Kumbha | Aquarius | The Water -bearer/ ನೀರು ಹೂಜಿ | Jan 22 - Feb 18 | Saturn/ಶನಿ | Air/ವಾಯು | Positve (+) | Fixed/ಸ್ಥಿರ ( ) |
| ಮೀನ | Meena | Pisces | The Fish/ ಮೀನು | Feb 19 - Mar 20 | Jupiter/ಗುರು | Water/ಜಲ | Negative (-) | Mutable/ ಬದಲಾಗುವಂಥದ್ದು |
$^{***}$ Ophiuchus (Kannada:ಒಫಿಯುಕಸ್) (November 29 to December 18) has sometimes been used in sidereal astrology as a thirteenth sign in addition to the twelve signs of the tropical Zodiac.

## References

* ↑ "About the Controversial 13th Zodiac Sign". cosmopolitan.com. 20 September 2016. Retrieved 20 September 2016.
* ↑ "Born under the sign of Ophiuchus?". earthsky.org. 16 August 2017. Retrieved 16 August 2017.
# Kannada/Objects

| English | Kannada |
| Chair | ಕುರ್ಚಿ |
| Bed | ಹಾಸಿಗೆ / ಗಾಧಿ |
| Cot | ಮಂಚ |
| Table | ಮೇಜು |
# OpenSSL/d2i ECPKParameters

The ECPKParameters encode and decode routines encode and parse the public parameters for an EC_GROUP structure, which represents a curve.
d2i_ECPKParameters attempts to decode len bytes at *in. If successful a pointer to the EC_GROUP structure is returned. If an error occurred then NULL is returned. If px is not NULL then the returned structure is written to *px. If *px is not NULL then it is assumed that *px contains a valid EC_GROUP structure and an attempt is made to reuse it. If the call is successful *in is incremented to the byte following the parsed data.
i2d_ECPKParameters encodes the structure pointed to by x into DER format. If out is not NULL is writes the DER encoded data to the buffer at *out, and increments it to point after the data just written. If the return value is negative an error occurred, otherwise it returns the length of the encoded data.
If *out is NULL memory will be allocated for a buffer and the encoded data written to it. In this case *out is not incremented and it points to the start of the data just written.
d2i_ECPKParameters_bio is similar to d2i_ECPKParameters except it attempts to parse data from BIO bp.
d2i_ECPKParameters_fp is similar to d2i_ECPKParameters except it attempts to parse data from FILE pointer fp.
i2d_ECPKParameters_bio is similar to i2d_ECPKParameters except it writes the encoding of the structure x to BIO bp and it returns 1 for success and 0 for failure.
i2d_ECPKParameters_fp is similar to i2d_ECPKParameters except it writes the encoding of the structure x to BIO bp and it returns 1 for success and 0 for failure.
These functions are very similar to the X509 functions described on the OpenSSL d2i_X509 manual page, where further notes and examples are available.
The ECPKParameters_print and ECPKParameters_print_fp functions print a human-readable output of the public parameters of the EC_GROUP to bp or fp. The output lines are indented by off spaces.

## Return Values

d2i_ECPKParameters, d2i_ECPKParameters_bio and d2i_ECPKParameters_fp return a valid EC_GROUP structure or NULL if an error occurs.
i2d_ECPKParameters returns the number of bytes successfully encoded or a negative value if an error occurs.
i2d_ECPKParameters_bio, i2d_ECPKParameters_fp, ECPKParameters_print and ECPKParameters_print_fp return 1 for success and 0 if an error occurs.
# OpenVOGEL/User's guide

The following tutorial pages are intended for end users. Hence, here you will not find information about how the program has been built, but about how it works and how to use it in practical aircraft design.

## Graphic interface
### Part 1: simulating a simple wing model

In this guide we show you how to:
* Create and edit a lifting surface.
* Load polar curves to simulate skin drag using XFoil or other data sources.
* Start a steady simulation.

### Part 2: creating a fuselage

In this guide we show you how to:
* Create and edit a native OpenVOGEL fuselage.
* Import fuselages from OpenVSP or other sources.

### Part 3: propellers

In this guide we show you how to:
* Create and edit a propeller.

## Calculation console
### Part 1: introduction to the Console

In this guide we show you how to:
* Introduce commands and work with the cosole.
* Run the simulation remotely from Tucan.
* Setup Intel MKL for improved calculation performance.

### Part 2: running the simulation from the console

In this guide we show you how to:
* Run a steady analysis in the Console.
* Run different standard batch simulations and interpret the results.
# OpenVOGEL
## Table of Contents

* Introduction
* User's guide
* Source code
* Aerodynamics
* Aeroelasticity
* Free flight
* Validation
# Pinyin/Abortion

See also
# Pinyin/Abortion Law

See also
# Pinyin/Above

See also
# Pinyin/Abstraction

See also
# Pinyin/About

See also
# Pinyin/Absorb

See also
# Pinyin/Accent

See also
An accent is a mark on a letter to show that it should be pronounced in a particular way.
biahnyin-furhaoh shih xiaanshih tehdihng fayin fangfaa d zihmuu shahng d jihhaoh.
# Pinyin/Absolute truth

See also
English
Absolute truth is an unchangeable and permanent fact. In the development of the Universe, the correct understanding of the specific process of various development stages, is the approximate and incomplete reaction of the objective world. Relative truth and absolute truth are in a dialectical unity. Absolute truth exists in the relative truth. Relative truth contains the ingredients of absolute truth. The sum of innumerable relative truths is the absolute truth.
Pinyin
juerduih zhenlii shih yoongherng bubiahn d shihshir. zaih Yuuzhouh d fazhaan guohcherng zhong, geh'g fazhaan jieduahn d juhtii guohcherng d zhehngqueh rehnshir, shih duih kehguan shihjieh jihnsih d, buwarnquarn d faanyihng. xiangduih zhenlii her juerduih zhenlii shih biahnzhehng toongyi d. juerduih zhenlii curnzaih yur xiangduih zhenlii zhizhong. xiangduih zhenlii harnyoou juerduih zhenlii d cherngfehn. wurshuh xiangduih zhenlii d zoongher jiuhshih juerduih zhenlii.
# Pinyin/Accident

See also
An accident is an unexpected problem. In physical accidents: such as collision, injuries and falls. In non-physical accidents: such as when secrets are leaked carelessly, when some important things are forgotten, or when important computer files are deleted.
Yìwài shì yùcè-bùdào de wèntí. Wùlǐ de yìwài rú pèngzhuàng, shòushāng hé diēdǎo. Fēi-wùlǐ de yìwài rú bù xiǎoxīn xièlù mìmì, wàngjì yīxiē zhòngyào de shìqíng, huò shānchú zhòngyào de diànnǎo wénjiàn.
# Pinyin/Combination

A combination is two or more things joined or mixed together to form a single unit.
jierhertii shih 1'g huoh duo'g shihwuh zuucherng d danyitii.
# Pinyin/College

See also
# Pinyin/Cold war

Search Pinyin war
# Pinyin/Collaborative

See also
# Pinyin/Collapse

Search Pinyin collapse
# Pinyin/Its

Search its
# Pinyin/Jade

Search Pinyin Jade
# Pinyin/January

January is the first month of a year.
Yiyueh shih meeiniarn d dih-1'g yueh.
# Pinyin/It

See also
# Pinyin/Italian

* Italy d
* Italy rern
* Italy-wern
# Pinyin/Ivory Coast

Ivory Coast is a country in the west of Africa.
Ivory Coast shì Fēizhōu xībù de yīgè guójiā.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Negative Imaginary Lemma

LMIs in Control/Click here to continue/LMIs in system and stability Theory/Negative Imaginary Lemma Positive real systems are often related to systems involving energy dissipation. But the standard Positive real theory will not be helpful in establishing closed-loop stability. However transfer functions of systems with a degree more than one can be satisfied with the negative imaginary conditions for all frequency values and such systems are called "systems with negative imaginary frequency response" or "negative imaginary systems".

## The System

Consider a square continuous time Linear Time invariant system, with the state space realization $(A,B,C,D)$
    ${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y=Cx(t)+Du(t)\end{aligned}}$

## The Data

$A\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{n\times m},C\in \mathbb {R} ^{m\times n},D\in \mathbb {S} ^{m}$

## The LMI: LMI for Negative Imaginary Lemma

According to the Lemma, The aforementioned system is negative imaginary under either of the following equivalent necessary and sufficient conditions
* There exists a $P$ ∈ ${\begin{aligned}\mathbb {S} \end{aligned}}$$^n$,where $P\geq 0$, such that,
    ${\begin{bmatrix}A^{T}P+PA&PB-A^{T}C^{T}\\B^{T}P-CA&-(CB+B^{T}C^{T})\end{bmatrix}}$${\begin{aligned}\leq 0\end{aligned}}$
* There exists a $Q$ ∈ ${\begin{aligned}\mathbb {S} \end{aligned}}$$^n$,where $Q\geq 0$, such that,
    ${\begin{bmatrix}QA^{T}+AQ&B-QA^{T}C^{T}\\B^{T}-CAQ&-(CB+B^{T}C^{T})\end{bmatrix}}$${\begin{aligned}\leq 0\end{aligned}}$

## Implementation

A link to the Matlab code for a simple implementation of this problem in the Github repository:
https://github.com/yashgvd/ygovada

## Related LMIs

Positive Real Lemma

## External Links

* - LMI in Control Systems Analysis, Design and Applications
* 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.
* Xiong, Junlin & Petersen, Ian & Lanzon, Alexander. (2010). A Negative Imaginary Lemma and the Stability of Interconnections of Linear Negative Imaginary Systems.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Peak-to-Peak Norm

LMIs in Control/Click here to continue/LMIs in system and stability Theory/Peak-to-Peak Norm
Peak-to-peak norm performance of a system

## The System

Considering the following system:
       ${\begin{aligned}{\dot {x}}=Ax+Bu\\z=Cx+Du\\\end{aligned}}$
Where $x(t)\in \mathbb {R} ^{n}$ is the state signal, $u(t)\in \mathbb {R} ^{m}$ is the input signal, and $z(t)\in \mathbb {R} ^{p}$ is the output. When given an initial condition $x(0)=0$, the system can be defined to map the output and input signals for the peak-to-peak performance.
       ${\begin{aligned}||T||_{\infty ,\infty }:=\sup \limits _{0<||u||_{\infty }<\infty }{\frac {||z||_{\infty }}{||u||_{\infty }}}\\\end{aligned}}$

## The Data

The matrices $A$, $B$, $C$, and $D$ are the only data sets required for this optimization problem.

## The Optimization Problem

Consider a continuous-time LTI system, $G:L_{2e}\to L_{2e}$, given that: $A$, $B$, $C$, and $D$ $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, and $D\in \mathbb {R} ^{p\times m}$. Given that the matrix $A$ is Hurwitz,The peak-to-peak norm of $G$ is given as:
       ${\begin{aligned}||T||_{\infty ,\infty }:=\sup \limits _{0<||w||_{\infty }<\infty }{\frac {||z||_{\infty }}{||u||_{\infty }}}\\\end{aligned}}$

## The LMI: Peak-to-Peak norm

There exists a matrix $P\in \mathbb {S} ^{n}$ and $\gamma$, $\epsilon$, $\mu \in \mathbb {R} _{>0}$, where the following constraints are used: $\min \mu$
       ${\begin{aligned}P<0\\{\begin{bmatrix}A^{T}P+PA+\gamma P&PB\\B'P&\epsilon I\end{bmatrix}}&<0\\{\begin{bmatrix}\gamma P&0&C^{T}\\0&(\mu -\epsilon )I&D^{T}\\C&D&\mu I\end{bmatrix}}&>0\\\end{aligned}}$
Since this optimization has $\gamma P$ in the constraints, this does make this optimization bi-linear. attempting to solve this LMI is not feasible unless some type of substitute is implemented to the variables $\gamma P$.

## Conclusion:

The results from this LMI will give the peak to peak norm of the system:
       ${\begin{aligned}\sup \limits _{0<||u||_{\infty }<\infty }{\frac {||z||_{\infty }}{||u||_{\infty }}}<\mu \end{aligned}}$

## Implementation

* Peak-to-Peak Norm Example

## External Links

* 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.
* Linear Matrix Inequalities in Control - A downloadable book on LMIs by Carsten Scherer and Siep Weiland
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Positive Orthant Stabilizability

Positive Orthant Stabilizability
The positive orthant stability of a linear system refers to the property of the system states being real and positive for all $t\geq 0$ and decaying down to zero over time. In this section, the feasibility problem for systems to be positive orthant stable, and the stabilizability conditions to make the system positive orthant stable will be covered.

## The System

Consider a linear state-space representation of a system as:
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\end{aligned}}$
where $x(t)\in \mathbb {R} ^{n}$ and $u(t)\in \mathbb {R} ^{r}$ are the system state and the input vector respectively. A and B are system coefficient matrices of appropriate dimensions.

## The Data

Number of states n and number of control inputs r need to be known. Moreover, the system matrices A,B are also required to be known.

## The Feasibility LMI

An LTI system is positive orthant stable if $x(0)\in \mathbb {R} _{+}^{n}$ implies that $\forall t\geq 0,x(t)\in \mathbb {R} _{+}^{n}$. Moreover, as $t\rightarrow \infty$, $x(t)\rightarrow 0$. This is possible if and only if the following conditions hold:
       ${\begin{aligned}A_{ij}&\geq 0,\forall i\neq j,\\\exists P&>0{\text{ s.t. }}PA^{\top }+AP<0,\end{aligned}}$
The above LMI feasibility is the positive orthant stability criteria. To convert it into a positive orthant stabilizability check, the problem can be modified so that we check if ${\dot {x}}=(A+BK)x$ is positive orthant stable. As $K$ is also a design variable here, the second inequality in the above LMI will result in bilinearity. A simple change of variables can overcome that to result in the following LMI feasibility problem for checking positive orthant stabilizability of the LTI system:
       ${\begin{aligned}{\text{Find }}Q,Y&{\text{ subj. to:}}\\&Q>0,(AQ+BY)_{ij}\geq 0,\forall i\neq j,\\&{\text{ s.t. }}QA^{\top }+AQ+BY+Y^{\top }B^{\top }<0,\end{aligned}}$
If the above LMI is feasible, the LTI system is stabilizable with controller $K=YQ^{-1}$.

## Conclusion:

The feasibility of the above LMIs guarantees that the system is positive orthant stable if the first LMI is feasible or stabilizable with a controller if the second LMI holds.

## Implementation

To solve the feasibility LMI, YALMIP toolbox is required for setting up the feasibility problem, and SeDuMi is required to solve the problem. The following link showcases an example of the feasibility problem:
https://github.com/smhassaan/LMI-Examples/blob/master/Positive_Orthant_LMI.m

## External Links

A list of references documenting and validating the LMI.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Quadratic Hurwitz Stabilization for Polytopic Systems

This section studies the Quadratic Hurwitz stabilization for polytopic systems.

## The System

Given a state-space representation of a linear system
    ${\begin{aligned}\ {\dot {x}}(t)=A(\delta (t)x(t)+B\delta (t)u(t)\\A(\delta (t))=A_{0}+\delta _{1}(t)A_{1}+\delta _{2}(t)A_{2}\\B(\delta (t))=B_{0}+\delta _{1}(t)B_{1}+\delta _{2}(t)B_{2}\end{aligned}}$         (1)

## LMI Condition

With $\Delta =\Delta _{p}$ , the quadratic Hurwitz Stabilization problem has a solution if and only if there exists a symmetric positive definite matrix $P$ and a matrix $W$ satisfying the below LMI :
    ${\begin{aligned}(A_{0}+A_{i})P+P(A_{0}+A_{i})^{T}+(B_{0}+B_{i})W+W(B_{0}+B_{i})^{T}<0,i&=1,2,....k\end{aligned}}$         (2)
In this case, a solution to the problem is given by
    ${\begin{aligned}K=WP^{-1}\\\end{aligned}}$         (3)

## Conclusion

Stability of a system does not guarantee quadratic stability. Since quadratic stability can represent infinite LMI constraints, it is not tractable. Therefore, to make it feasible and tractable, polytopic sets are helpful.

## External Links

* 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.
* LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013