OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/pages/TDSDC
## The System

The problem is to check the stability of the following linear time-delay system on a delay dependent condition
    ${\begin{aligned}{\begin{cases}{\dot {x}}(t)&=Ax(t)+A_{d}x(t-d)\\x(t)&=\phi (t),t\in [-d,0],0<d\leq {\bar {d}},\\\end{cases}}\end{aligned}}$
where
    ${\begin{aligned}{A,A_{d}}\in \mathbb {R} ^{n\times n},A\in \mathbb {R} ^{n\times r}{\text{ are the system coefficient matrices,}}\\\end{aligned}}$
$\phi (t)$ is the initial condition
$d$ represents the time-delay ${\bar {d}}$ is a known upper-bound of $d$
For the purpose of the delay dependent system we rewrite the system as
    ${\begin{cases}{\dot {x}}(t)&=Ax(t)+A_{d}x(t-d)\\{\dot {x}}(t)&=(A+A_{d})x(t)-A_{d}(x(t)-x(t-d)\end{cases}}$

## The Data

The matrices $A,A_{d}$ are known

## The LMI: The Time-Delay systems (Delay Dependent Condition)

From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a symmetric positive definite matrix $X$ and a scalar $0<\beta <1$ such that
    ${\begin{bmatrix}\Phi (X)&{\bar {d}}XA^{T}&{\bar {d}}XA_{d}^{T}\\{\bar {d}}AX&-d\beta I&0\\{\bar {d}}A_{d}X&0&-{\bar {d}}(1-\beta )I\end{bmatrix}}$${\begin{aligned}<0\end{aligned}}$
Here $\Phi (X)=X)A+A_{d})^{T}+(A+A_{d})X+{\bar {d}}A_{d}A_{d}^{T}<0$
This LMI has been derived from the Lyapunov function for the system. It follows that the system is asymptotically stable if
$P(A+A_{d})+(A+A_{d})^{T}P+{\bar {d}}PA_{d}A_{d}^{T}P+{\frac {\bar {d}}{\beta }}A^{T}A+{\frac {\bar {d}}{1-\beta }}A_{d}^{T}A_{d}<0$
This is obtained by replacing $X$ with $P^{-1}$

## Conclusion:

We can now implement these LMIs to do stability analysis for a Time delay system on the delay dependent condition

## Implementation

The implementation of the above LMI can be seen here
https://github.com/yashgvd/LMI_wikibooks

## Related LMIs

Time Delay systems (Delay Independent Condition)

## 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.
* D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.
# LMIs in Control/pages/TDSIC
## The System

The problem is to check the stability of the following linear time-delay system
    ${\begin{aligned}{\begin{cases}{\dot {x}}(t)&=Ax(t)+A_{d}x(t-d)\\x(t)&=\phi (t),t\in [-d,0],0<d\leq {\bar {d}},\\\end{cases}}\end{aligned}}$
where
    ${\begin{aligned}{A,A_{d}}\in \mathbb {R} ^{n\times n}{\text{, }}{A}\in \mathbb {R} ^{n\times r}{\text{ are the system coefficient matrices,}}\\\end{aligned}}$
$\phi (t)$ is the initial condition
$d$ represents the time-delay ${\bar {d}}$ is a known upper-bound of $d$

## The Data

The matrices $A,A_{d}$ are known

## The LMI: The Time-Delay systems (Delay Independent Condition)

From the given pieces of information, it is clear that the optimization problem only has a solution if there exists two symmetric matrices $P,S\in \mathbb {S} ^{n}$ such that
$P>0$
    ${\begin{bmatrix}A^{T}P+PA+S&PA_{d}\\A_{d}^{T}P&-S\end{bmatrix}}$${\begin{aligned}<0\end{aligned}}$
This LMI has been derived from the Lyapunov function for the system. By Schur Complement we can see that the above matrix inequality is equivalent to the Riccati inequality
$A^{T}P+PA+PA_{d}S^{-1}A_{d}^{T}P+S<0$

## Conclusion:

We can now implement these LMIs to do stability analysis for a Time delay system on the delay independent condition

## Implementation

The implementation of the above LMI can be seen here
https://github.com/yashgvd/LMI_wikibooks

## Related LMIs

Time Delay systems (Delay Dependent Condition)

## 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.
* D. d. S. Madeira and J. Adamy, "Static output feedback: An LMI condition for stabilizability based on passivity indices," 2016 IEEE Conference on Control Applications (CCA), Buenos Aires, 2016, pp. 960-965.
# LMIs in Control/pages/Switched systems H2 Optimization

LMIs in Control/pages/Switched systems H2 Optimization
Switched Systems $H_{2}$ Optimization

## The Optimization Problem

This Optimization problem involves the use of the State-feedback plant design, with the difference of optimizing the system with a system matrix which "switches" in properties during optimization. This is similar to considering the system with variable uncertainty; an example of this would be polytopic uncertainty in the matrix $A$. which ever matrix is switching states, there must be an optimization for both cases using the same variables for both.
This is first done by defining the 9-matrix plant as such: $A\in \mathbb {R} ^{m\times m}$, $B_{1}\in \mathbb {R} ^{m\times n}$, $B_{2}\in \mathbb {R} ^{m\times p}$, $C_{1}\in \mathbb {R} ^{n\times m}$, $D_{11}\in \mathbb {R} ^{n\times n}$, $D_{12}\in \mathbb {R} ^{n\times p}$, $C_{2}=I$, $D_{21}=0$, and $D_{22}=0$. Using this type of optimization allows for stacking different LMI matrix states in order to achieve the controller synthesis for $H_{\infty }$.

## The Data

The data is dependent on the type the state-space representation of the 9-matrix plant; therefore the following must be known for this LMI to be calculated: $A\in \mathbb {R} ^{m\times m}$, $B_{1}\in \mathbb {R} ^{m\times n}$, $B_{2}\in \mathbb {R} ^{m\times p}$, $C_{1}\in \mathbb {R} ^{n\times m}$, $D_{11}\in \mathbb {R} ^{n\times n}$, $D_{12}\in \mathbb {R} ^{n\times p}$, $C_{2}=I$, $D_{21}=0$, and $D_{22}=0$. What must also be considered is which matrix or matrices will be "switched" during optimization. This can be denoted as $A(\delta )$.

## The LMI: Switched Systems $H_{2}$ Optimization

There exists a scalar $\gamma$, along with the matrices $X>0$, $W$, and $Z$ where:
       ${\begin{aligned}||S(P,K(0,0,0,F))||_{H_{2}}^{2}\leq \gamma \\\\traceW<\gamma \\\\AX+XA^{T}+B_{2}Z+Z'B_{2}'+B_{1}B_{1}^{T}<0\\\\{\begin{bmatrix}X&(C_{1}X+D_{12}Z)^{T}\\C_{1}X+D_{12}Z&W\\\end{bmatrix}}&>0\\\end{aligned}}$
Where $F=ZX^{-1}$ is the controller matrix. This also assumes that the only switching matrix is $A$; however, other matrices can be switched in states in order for more robustness in the controller.

## Conclusion:

The results from this LMI gives a controller gain that is an optimization of $H_{2}$ for a switched system optimization.

## Implementation

* H2 switched state-feedback optimization example

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
# LMIs in Control/pages/Unsat Inp Stabilization

LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control
The LMI in this page gives the feasibility conditions which, if satisfied, imply that the correstponding system can be stabilized.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+BSat(u(t)),\\x(0)&=x_{0},\\Sat(u)_{i}=,\end{aligned}}$
where $x\in \|R^{n}$ is the state, $u\in \|R^{m}$ is the control input.
For the system given as above, its symmetrical saturated control form can be derived by following the procedure in the original article. The new system will have the form:
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+{\tilde {B}}sat(z(t))+Ew\end{aligned}}$
where $w_{i}=u_{i}-{\frac {\alpha _{i}-\beta _{i}}{2}},z_{i}=w_{i}{\frac {2}{\alpha _{i}+\beta _{i}}}$

## The Data

The system matrices $(A,{\tilde {B}},E)$, the saturation bounds $(\alpha _{i},\beta _{i})$ of the control inputs. Positive scalars $\rho ,\eta$.

## The LMI: The Stabilization Feasibility Condition

       ${\begin{aligned}&{\text{Find}}\;X,Y,Z:\\&{\text{subj. to: }}X>0,\\&\quad {\begin{bmatrix}AX+{\tilde {B}}(D_{s}Y+{\hat {D}}_{s}^{-}Z)\end{bmatrix}}+{\begin{bmatrix}AX+{\tilde {B}}(D_{s}Y+{\hat {D}}_{s}^{-}Z)\end{bmatrix}}^{\top }+{\frac {\eta }{\rho }}X+{\frac {1}{\eta }}EE^{\top }<0,\\{\begin{bmatrix}\mu &Z_{i}\\*X\end{bmatrix}}>0,i=1,...,{\bar {m}}\end{aligned}}$
Here $D_{s}$ is a diagonal matrix with a component either 0 or 1, and $D_{s}+D_{s}^{-}={\frac {\Lambda +\mathrm {T} }{2}}$ and ${\hat {D}}_{s}^{-}=e_{f_{m}}\times D_{s}^{-}$

## Conclusion:

The feasibility of the given LMI implies that the system is stabilizable with control gains $K=YX^{-1},H=ZX^{-1}$.

## Implementation

A link to CodeOcean or other online implementation of the LMI

## External Links

* Stabilization of Systems with Unsymmetrical Saturated Control: An LMI Approach Link to the original article.
# Muggles' Guide to Harry Potter/Magic/Confringo
## Overview

Confringo is a spell or curse used to make things explode.

## Extended Description

As Harry is making his escape from the Dursleys' during the fourth chapter (The Seven Potters) of the Harry Potter and the Deathly Hallows, the side car that has been bolted on to Sirius' flying motorbike, in which Harry is riding, breaks away due to the strain of flight. Harry is rescued by Hagrid, but then uses this spell to destroy the abandoned side car, disabling one of the attacking Death Eaters with the fragments.
This spell is used again in the battle with Nagini; here, Hermione casts it, and it ricochets around the room before hitting something it can blow up. Hermione later identifies it as the Blasting Curse.

## Analysis

This is similar in effect to a differently-named spell used in some fan fiction; one may wonder whether the name was chosen specifically to avoid similarity to the name in the fan fiction. It also appears similar in effect to the Expulso and Deprimo spells.

## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.
# Muggles' Guide to Harry Potter/Magic/Conjunctivitus
## Overview

The Conjunctivitus curse apparently causes temporary blindness.

## Extended Description

In Harry Potter and the Goblet of Fire, Harry is required to get past a Dragon in the First Task of the Triwizard Tournament. Dragonhide is nearly impervious to most spells – it takes several wizards working in concert to Stun a dragon, for instance – so in order to disable a dragon single-handed, you must attack its only vulnerable area - its eyes. The Conjunctivitus curse is the one you use to do that.
We are told about this spell, but we do not see it actually executed, so we do not know the incantation.

## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.
# Muggles' Guide to Harry Potter/Magic/Crumple Horned Snorkack
## Overview

The Crumple-Horned Snorkack is mentioned in passing, but never described in detail.

## Extended Description

Luna Lovegood mentions the Crumple-Horned Snorkack a number of times in Harry Potter and the Order of the Phoenix; apparently, her father, the publisher of The Quibbler, believes in the existence of this and other magical creatures that nobody else has ever seen. This is a sore point for Hermione, who will not believe in their existence without seeing some proof.
At the end of the book, Luna says that some of the money her father has received for the right to publish Harry's interview about the return of Voldemort will be spent on an expedition to find the Snorkack, an expedition that we later find out is fruitless.
The Snorkack is mentioned again in Harry Potter and the Deathly Hallows, when Harry visits Luna's father, Xenophilius. Apparently, Xeno has purchased what he thinks is a Snorkack horn, to give to Luna for Christmas; in fact, it is an Erumpent horn, which explodes, nearly destroying Xeno's house.
Later in the book, when Luna and Dean Thomas, among others, have been rescued from Malfoy Manor, we overhear Luna describing the Snorkack to Dean. The only description we hear, though, is that it has tiny, purple ears.

## Analysis

The major reason for introducing the Crumple-Horned Snorkack, one must suppose, is to show Luna Lovegood's mild strangeness; her insistence that this creature exists is a harmless form of insanity and is therefore seen as laughable rather than dangerous. Her steadfast holding to this belief is another part of the limning of her character; Luna, though believing things that just aren't so (according to accepted wizard lore) holds steadfastly to her beliefs, to her father's beliefs, and similarly to her friends.

## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.
* Presumably, the Crumple Horned Snorkack has at least one crumpled horn. What do you think it looks like? What characteristics would it have?
* Luna's father says that they will be hunting for it in Sweden; can you think of characteristics that would make it more likely to be there than, say, Florida?
# Muggles' Guide to Harry Potter/Magic/Decoy Detonator
## Overview

Decoy Detonators are products of Weasleys' Wizard Wheezes. They are small devices which scurry off into shadows and then make a noise, thus decoying people away.

## Extended Description

We first see Decoy Detonators when Harry, Ron, Ginny, and Hermione, along with Mr. and Mrs. Weasley, visit the Twins' new joke shop in Diagon Alley. Fred, giving Harry the tour of the shop, takes him into the back room where the more serious magical items are kept, and there shows him the Decoy Detonators, saying they can't keep them on the shelves.
When the Trio are planning to break into the Ministry in Harry Potter and the Deathly Hallows, Decoy Detonators are one of the things on Hermione's list. Harry then uses one of them to distract a room full of witches and wizards so that he can enter Umbridge's office undetected. The Detonator scuttles silently to the far side of the room, then blows itself up, making a sharp noise and a cloud of smoke. The witches and wizards examine it for a long time, finally deciding that it is an escapee from the department of Experimental Charms.

## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.
# Muggles' Guide to Harry Potter/Magic/Crup
## Extended Description

Mentioned only in passing in Harry Potter and the Order of the Phoenix: Professor Grubbly-Plank, in response to a question from Professor Umbridge, says that she will be training the class to recognize Knarls and Crups.

## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.
# Muggles' Guide to Harry Potter/Magic/Confundo
## Overview

The Confundus charm makes the subject confused. It is fairly advanced magic and is cast by means of the incantation "Confundo." The Confundus charm can apparently also be cast silently.

## Extended Description

In Harry Potter and the Prisoner of Azkaban, Severus Snape knows that his story of what happened in the Shrieking Shack will differ radically from Harry's, so he claims that Harry was obviously suffering under the effects of a Confundus charm.
Professor Moody, in Harry Potter and the Goblet of Fire, suggests that the Goblet may have been Confunded to believe that there were four schools. While it is never explicitly stated that the Confundus charm was used, no other spell has been mentioned that would have similar effects. This is an indication that the Confundus charm may also work on inanimate objects that have been magically granted some level of reasoning ability, such as the Goblet and the Sorting Hat.
In Harry Potter and the Half-Blood Prince, Hermione casts this charm on Cormac McLaggen, a potential goal-keeper, to prevent him looking better than Ron at the team try-outs.
It is mentioned in Harry Potter and the Deathly Hallows that the Auror Dawlish, despite his other qualities, is very easily Confunded.
In that same book, Harry uses the Confundus charm to avoid being interrogated by human guards at Gringotts with Probity Probes.
We later see that Snape may have put Mundungus Fletcher under the Confundus charm to secretly give him the idea of having multiple Harrys flying to multiple safe-houses.
Finally, in the Epilogue, Ron admits to having Confunded his driving examiner into overlooking an error (checking a wing mirror) that Ron considers trivial.

## Analysis

Cormac McLaggen's response to this charm would indicate that there are physical as well as mental effects; McLaggen, some time after he is Confunded, has trouble navigating stairs.
As seems to be the case with many other spells, there would appear to be a substantial mental component to this charm. The effects on McLaggen seem to be diverse and diffuse; however, Snape's discussion with Fudge, and Harry's using the charm to achieve a specific goal (convincing the guard that he has already been probed), suggests that the Confundus charm can, with the appropriate mental imagery imposed, cause the subject to believe something specific, other than what has happened.
Some characters, apparently, are more susceptible to this charm than others. We are told that Dawlish seems to be easily Confunded. We suspect that this probably relies on the same sort of mental control mechanism that underlies both the attraction of the Veela, and the Imperius curse. We would expect, in that case, that it would be harder to Confund Harry, as he has shown ability to throw off the Imperius curse completely. Ron, however, with his demonstrated susceptibility to both Veela and the Imperius curse, likely would be very easy to Confund.

## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.
# Meitei Culture/Agriculture

Meitei agriculture has a strong relationship with the geography of the Meitei realms. According to ancient sources as well as modern day geological studies, the land of Kangleipak (present day Manipur) was a hilly and mountainous place, once having an oval shaped huge lake (which will later become the present day oval shaped lacustrine valley). During those times, different groups of the Meitei people and other ethnic groups used to live in different parts of the hills and mountains of Kangleipak. In those time, they practised shifting cultivation as their mode of agriculture.
However, in due course of time, the water level of the very lake decreased. Due to the formation of passage ways (possibly because of tectonic activities, soil erosion and massive land slides) in some parts of the hills of Kangleipak, the waters of the central lake flowed towards the plains of Myanmar (Burma), passing through the hills and mountains of Kangleipak. This eventually lowered the water level of the lake, thereby forming a lacustrine valley in its former position, though numerous smaller lakes were still present there.
This led to the migration of the Meiteis and other ethnic groups (which will all get Meiteised in later centuries) from the highlands to the newly formed lowlands of Kangleipak. Thus, their mode of agriculture got transformed into arable farming from that of shifting cultivation.
The fertile Imphal valley (also known as Manipur valley) is drained by the Imphal River and her tributaries (the Iril R., the Kongba R., the Nambul R., etc.) which will eventually become the Manipur R. to unite with the Chindwin Irrawaddy River System of Myanmar (Burma) to drain ultimately into the Bay of Bengal in the Indian Ocean.
With the flourishing of agriculture in the fertile river banks of the Imphal R. and her tributaries, the civilization of the Meiteis developed.
# Meitei Culture/Literature

Meitei literature has both written as well as oral forms.
All the Meitei language literary works, produced before the 18th century CE, were written in the traditional Meetei Mayek writing system. During and after the 18th century CE, Bengali script replaced the Meitei script for writing the Meitei language. However, in the later parts of the 20th century CE, Meitei script rose once again to replace the Bengali script for writing Meitei language.
With the changes of writing system, the usage of words in Meitei literature, also changes, simultaneously, due to acculturation. Meitei script enthusiasts usually love to write linguistically pure Meitei language, free from any foreign or borrowed or corrupted words. On the other hand, Bengali script enthusiasts usually love to mix Meitei language writings, with the words borrowed or derived from Bengali language and Bengali's ancestor language, Sanskrit. Thus, in case of Meitei culture, the writing systems matter to the language of literary works.
Meitei people believes that "language" is their mother and "script" is their father. And if Bengali script or Latin script is used to write their language, they believe that it is equivalent to their mother being under the control of another man, who is not their father. This notion leads the masses to revive the traditional Meetei Mayek writing system.
Meitei oral literature (orature) is mainly composed of folktales and folk songs. As oral retellings may become varied from one narrator to another, folktales and folk songs usually exist in multiple variations. On the other hand, the possibility of variations is very limited in case of written literature.
# Meitei Culture/Law & Governance

The traditional law and governance in Meitei civilisation was predominantly monarchical in nature. In 429 CE, during the reign of King Naophangba, a proto-Constitution was established in Kangleipak. The law went on expanded, through several amendments, until it was finalised in 1110 CE in the form of a written Constitution of the Meitei kingdom. Since it was enacted by King Loiyumba (1074-1122 CE), it was known as the "Loiyumpa Silyel" in the later centuries.
# Meitei Culture

* Law & Governance
* Early Society
* Military
* Domestic Lifestyle (Textiles, Jewelry, Family, Cuisine)
* Agriculture (Farming, Livestock, Domestication)
* Economy (Occupations, Trade, Coinage, Poverty, Wealth, Mines)
* Education
* Entertainment
* Religion (Priesthood, Rituals, Calendar, Cults, Funerals)
* Philosophy (Philosophers, Maxims, Texts)
* Mythology (Mortal Heroes, Mortal Heroines, Gods, Goddesses, Mythical creatures, Dragons)
** Images of Kanglasha
** Images of Pakhangba
** Images of Panthoibi
* Language & Literature (Poetry, Plays, Prose)
* Architecture (Temples, Gates)
* Arts (Dance, Paintings, Pottery, Puppetry, Martial Arts, Music, Theatre)
* Technology (Medicine, City-planning, Metallurgy, Writing systems, Musical Instruments)
* Sex & Gender (Women; Mother; LGBT)
* Slavery
* Non-Meiteis (Historical Alliances, Historical Enemies)
# Meitei Culture/Architecture

The traditional Meitei art of designing and constructing buildings, specifically in the realms of Kangleipak, is known as "Meitei architecture", or "Meetei architecture", "Manipuri architecture", "Kanglei architecture". Most of the buildings are semi-concrete or wooden in medium but heavily intricate in designs as well as in details.
Best known examples are standing in the forms of temples, gates and public houses, other than royal palaces and royal court buildings.
# Meitei Culture/Mythology

A collection of myths, belonging to the religious and cultural traditions of the Meiteis, are often termed as Meitei mythology, or Meetei mythology, or Manipuri mythology, or Kanglei mythology. It includes a vast range of myths, legends, folklore, folktales, about different Mortal Heroes, Mortal Heroines, Gods, Goddesses, Mythical Creatures (including Dragons), among many others.
# Meitei Culture/Mythology/Goddesses

In Meitei mythology, there are numerous goddesses. A few of them are listed below:
* Helloi
* Ichum Lairembi
* Imoinu
* Irai Leima
* Khamnung Kikoi Louonbi
* Khunu Leima
* Konthoujam Tampha Lairembi
* Koucheng Leima
* Kounu
* Laikhurembi
* Lainaotabi
* Leimalen
* Loktak Ima
* Maheigi Lairembi
* Ngaleima
* Nganu Leima
* Nongthang Leima
* Panthoibi
* Phouoibi
* Shapi Leima
* Thongak Leima
* Thoudu Nungthen Leima
* Thumleima
* Yumchao Leima