OpenLLM-France/wikimedia huggingface dataset:
# Navajo/Shopping

| Ké Za' nisin. | I need a pair of shoes |
| Díkwíí yíghíí nínízin? | What size would you like? |
| Díkwíí yíghíí shíí shííghah hóla, haa'í shą' 'ashdla'ágíí 'ádíneesh'ąąh. | I'm not sure, but I would like to try on a size 5. |
| Díkwíí bą́ą́h 'ílį́? | How much does it cost? |
| Díí 'éétsoh díkwíí bą́ą́h 'ílį́? | How much does this jacket cost? |
| Hastą́ą́ béeso dóó bi'ąą náhást'éí dootł'izh dóó bi'ąą łitso bą́ą́h ílį́. | It costs $6.95 |
| T'áá 'ak'aanígíí shee 'ádin. | I don't have change |
| T'aa 'át'éego lá díkwíí bííghah k'ad? | How much is it all together? |
# Neapolitan/501 Neapolitan verbs

* Neapolitan Primer
* edit
* Grammar
* Pronunciation
* Alphabet
* Nouns
* Articles
* Prepositions
* Adjectives
* Pronouns
* Verbs
* Adverbs
* Conjunctions
* Interrogatives
* Doubling
* Syntax
* Usage
* Greetings
* 501 Neapolitan verbs
* Numbers
* Calendar
* Colors
* Weather
* Food
* Parts of the body
* Rooms of the house
* Expressions and figures of speech
* Resources
* Bibliography
* Sites
* abballà
* accattà
* acchiappà
* aggarbà
* aggrannirse
* allicurdà
* alluccà
* arapì
* ascì
* attené
* assapé
* avé
* cagnà: to change (English), cambiare (Italiano)
* cammenà
* canóscere
* campà
* cantà
* carè
* chiagnere
* chiammà
* correre
* dà
* dimannà
* dormì
* èssere
* fà
* fernì
* fràvecà
* fuì
* ghì
* jì appriésso
* jucà
* magnà
* mpazzì
* natà
* rìcere
* rispónnere
* parlà
* penzà
* piglià
* prià
* pruvà
* puté
* sapé
* scetà
* scialà
* sentì
* servì
* sgarrupà
* stà
* suggerì
* sunnà
* tené
* trasì
* truvà
* vedé
* venì
* vèvere
* vulà
* vulé
* zumpà
# Neapolitan/501 Neapolitan verbs/aggarbà
## Presente

* i'/io aggarbo
* tu aggarbe*
* isso/essa aggarba
* nuje aggarbàmmo
* vuje aggarbàte
* lloro/isse aggarbano
# Neapolitan/501 Neapolitan verbs/arapì
## arapì

* (transitive) to open
1. Innecativo
    a. presente
       io arapo tu arape isso/essa arape nuje arapimmo vuje arapite isse/lloro arapeno
Participio passato
       araputo
# Mario Kart 64/Tactics

In addition to knowing what to do in each course, it's also important to know why you're doing it, so you need a good understanding of the game physics. This will also help you pay attention to little details that you might not have thought about.

### Basic Tactics
#### Weight Class

The lightweights, Toad, Peach, and Yoshi, are the fastest, have the best acceleration, and slow down the least on offroad surfaces. The three lightweights are identical to each other. Always use one of the lightweights in time trials.

#### Rocket Boost

Press and hold A just before the light turns blue to get a speed boost at the beginning of a race. To save an extra .01 seconds, start pressing A before Lakitu starts counting down, and just before you would normally press A, let go of A for just long enough that you won't spin out, and then press A again.

#### Reverse

Press B and down to go in reverse. This is never done except to help position the kart to set up for a fastlap attempt on lap 2.

#### Spin Turn

Press A, B, and left or right while standing still to quickly spin around. This is only used in some SC races. You can also press diagonally down and left or right while spin turning to do a reverse spin turn, which is useful for some shortcuts.

#### Turning

Press left or right to turn. Turning normally slows you down, so you should use power slides to go around turns so you can then release a mini-turbo. Turning without power sliding slows you down, so it should be minimized, but sometimes it is needed to make occasional slight direction adjustments when needed. When you do need to turn, you should aways untilt the kart.

#### Hopping

Press R to hop. Hopping repeatedly is done to go around sharp turns. Hop turns are used occasionally to make slight direction adjustments. Hops are also the start of power slides. You can hop over gaps at corners to save distance. Don't make the mistake of hopping repeatedly on straightaways because it slows you down.

#### Power sliding

Press left or right and press and hold R to start a left facing or right facing power slide, and then move the stick back in the opposite direction. Each time you move the control stick back and forth, the smoke will change from grey to yellow, and then to red. Slide around turns and corners as tightly as possible, unless specified otherwise.

#### Mini-Turbos

After changing the smoke of a power slide to red, release the R button to get a speed boost. Mini-turbos should usually be as fast as possible.
# Mario Kart DS

Welcome to the Wikibook on Mario Kart DS. This guide covers the Nintendo DS version of the game, as well as the re-release for the Wii U's Virtual Console, which is nearly identical besides a difference in controller layout.

## Contents

| General Items Emblems Courses Karts Kart Stats Characters | Multiplayer Nintendo WFC Battle Battle Courses |

## Further reading

* Mario Kart DS on the Mario Wiki (Fandom)
* Mario Kart DS on the Super Mario Wiki
# Mario Kart DS/Battle Courses

A list of Courses available in battle mode:
* Block Fort (Mario Kart 64 - Mario Kart 64)
* Nintendo DS (New)
* Palm Shore (New)
* Pipe Plaza (MKDD - Mario Kart Double Dash)
* Tart Top (New)
* Twilight House (New)
# Mario Kart 64/Shortcut

PAL times are converted to the equivalent NTSC time by dividing by 1.2024.
Click on the name of a course to see the stratagies for that course.
Current shortcut world records. :
| Course | 3lap | flap |
| Luigi Raceway | 1'17"53 | 0'21"70 |
| Kalamari Desert | 1'44"09 | 0'31"63 |
| Toad's Turnpike | 1'31"22 | 0'28"94 |
| Frappe Snowland | 0'21"72 | 0'05"04 |
| Choco Mountain | 1'06"71 | 0'17"68 |
| Mario Raceway | 0'51"26 | 0'15"46 |
| Wario Stadium | 0'12"87 | 0'02"13 |
| Royal Raceway | 1'41"24 | 0'31"53 |
| DK's Jungle Parkway | 0'20"92 | 0'03"60 |
| Yoshi Valley | 0'27"93 | 0'05"78 |
| Rainbow Road | 3'18"83 | 1'04"86 |
# Mario Kart 64/Non-shortcut

PAL times are converted to the equivalent NTSC time by dividing by 1.2024.
Click on the name of a course to see the strategies for that course.
Current non-shortcut world records :
| Course | 3lap | flap |
| Luigi Raceway | 1'38"24 | 0'31"35 |
| Moo Moo Farm | 1'12"65 | 0'23"40 |
| Koopa Troopa Beach | 1'19"77 | 0'25"72 |
| Kalamari Desert | 1'44"42 | 0'32"63 |
| Toad's Turnpike | 2'28"95 | 0'49"09 |
| Frappe Snowland | 1'40"65 | 0'32"17 |
| Choco Mountain | 1'36"68 | 0'31"76 |
| Mario Raceway | 1'13"37 | 0'23"11 |
| Wario Stadium | 3'38"87 | 1'12"33 |
| Sherbet Land | 1'36"60 | 0'31"64 |
| Royal Raceway | 2'23"29 | 0'46"42 |
| Bowser's Castle | 1'50"67 | 0'36"05 |
| D.K.'s Jungle Parkway | 1'51"50 | 0'35"47 |
| Yoshi Valley | 1'25"73 | 0'26"18 |
| Banshee Boardwalk | 1'44"41 | 0'34"12 |
| Rainbow Road | 4'54"69 | 1'37"63 |
# Muggles' Guide to Harry Potter/Characters/Mafalda Hopkirk
## Overview

Mafalda Hopkirk works in the Improper Use of Magic department at the Ministry of Magic.

## Role in the Books
### Chamber of Secrets

When Dobby performs a Hover Charm in the Dursley's house, Harry is believed to be the one actually doing magic. The warning letter that arrives is signed by Mafalda Hopkirk.

### Order of the Phoenix

Harry again does magic, this time using the Patronus charm to repel Dementors who are attacking himself and his cousin Dudley. The letter of reprimand that is sent to him by Owl Post is signed by Mafalda Hopkirk, and so is the following letter rescinding the immediate punishment and setting a hearing date.

### Deathly Hallows

When Harry, Ron, and Hermione determine that they must get into the Ministry to search for the locket Horcrux which has been appropriated by Dolores Umbridge, they disguise themselves as Ministry employees, using Polyjuice potion to do so. Hermione ends up disguised as Mafalda Hopkirk; the real Mafalda Hopkirk is left, Stunned, behind a closed door.
Once in the Ministry, Harry and Hermione take the elevator to the top floor, where they understand Umbridge's office is. Umbridge is at the elevator and, recognizing Hopkirk, assumes she was sent by Travers. The two of them return through the Atrium to the courtrooms, where "Mafalda" is to act as recorder for the day's session of the Muggle-Born Registration Commission. Very shortly after Harry re-joins her, they depart the Ministry and Hermione's disguise wears off.

## Questions

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

Magorian, a Centaur, is one of only four Centaurs named in the books. He is a member of the centaur herd that lives in the Forbidden Forest.

## Role in the Books
### Order of the Phoenix

Hagrid asks Harry and Hermione for a favour; he expects that he will be sacked soon, and wants them to do something for him if he gets sent away. As he leads them into the forest, he mentions that Firenze working with Dumbledore has made for problems with the Centaurs; they are not happy, especially not with Hagrid as he stopped them kicking Firenze to death. After they have met Grawp and are on their way back out of the forest, they are met by about half a dozen centaurs, including Magorian and Bane; Magorian warns Hagrid that he is not welcome in the Forest any more. As the centaurs do not harm foals (meaning Harry and Hermione), however, Hagrid will be allowed to pass today.
When Harry is trapped in Dolores Umbridge's office along with Hermione, Hermione manages to convince her that Harry was trying to reach Professor Dumbledore to say that the weapon they had been building for him was complete. Umbridge is lured by the promise of this weapon into the Forbidden Forest, where the three of them are challenged by the Centaur herd. While there are many centaurs present, only three are named: Bane, Ronan, and Magorian. It is Magorian who first speaks to the three of them, and who Umbridge attempts to bind.

### Deathly Hallows

Magorian is apparently one of the centaurs who joins in at the final battle for Hogwarts.

## Questions

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

A Hogwarts student in Harry's year, Mandy Brocklehurst is a member of Ravenclaw house.

## Role in the Books
### Philosopher's Stone

The Sorting Hat places Mandy Brocklehurst in Ravenclaw House.

## Relationships with Other Characters

We are led to believe that Mandy is a friend of Marietta Edgecombe, and so possibly also a friend of Cho Chang. However, she plays no real part in the story.

## Questions

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

Lisa Turpin is a Hogwarts student in Harry's year, in Ravenclaw house.

## Role in the Books
### Philosopher's Stone

The Sorting Hat places Lisa Turpin into Ravenclaw house.

## Questions

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

Malcolm Baddock is a Slytherin who arrived at Hogwarts in Harry's fourth year.

## Role in the Books
### Goblet of Fire

Malcolm Baddock was Sorted into Slytherin House.

### Deathly Hallows
## Strengths
## Questions

Study questions are meant to be left for each student to answer; please don't answer them here.
# LMIs in Control/pages/Delay Independent Time-Delay Stabilization

Stabilization of Time-Delay Systems - Delay Independent Case
Suppose, for instance, there was a system where a time-delay was introduced. In that instance, stabilization would have to be done in a different manner. The following example demonstrates how one can stabilize such a system independent of the delay.

## The System

For this particular problem, suppose that we were given the time-delayed system in the form of:
    ${\begin{aligned}{\begin{cases}{\dot {x}}(t)&=Ax(t)+A_{d}(t-d)+Bu(t),\\x(t)&=\phi (t),t\in [0,d],0<d\leq {\bar {d}},\\\end{cases}}\end{aligned}}$
where
    ${\begin{aligned}&A\,A_{d}\in \mathbb {R} ^{n\times n},B\in \mathbb {R} ^{nxr}{\text{ are the system coefficient matrices,}}\\&\phi (t){\text{ is the initial condition,}}\\&d{\text{ represents the time-delay, and}}\\&{\bar {d}}{\text{ is a known upper-bound of }}d\\\end{aligned}}$
Then the LMI for determining the Time-Delay System for the Delay-Independent case would be obtained as described below.

## The Data

In order to obtain the LMI, we need the following 3 matrices: $A,A_{d},$ and $B$.

## The Optimization Problem

Suppose - for the time-delayed system given above - we were asked to design a memoryless state-feedback control law where $u=Kx$ such that the closed-loop system:
    ${\begin{aligned}{\begin{cases}{\dot {x}}(t)&=(A+BK)x(t)+A_{d}(t-d),\\x(t)&=\phi (t),t\in [0,d],0<d\leq {\bar {d}},\\\end{cases}}\end{aligned}}$
is simultaneously both uniform and asymptotically stable, then the system would be stabilized as follows.

## The LMI: The Delay-Independent Stabilization of Time-Delay Systems

From the given pieces of information, it is clear that the optimization problem only has a solution if there exists a matrix$W\in \mathbb {R} ^{r\times n}$ and 2 symmetric matrices $X>0$ and $Y>0$ that satisfy the following:
       ${\begin{aligned}{\begin{bmatrix}X{A^{T}}+AX+BW+{W^{T}}{B^{T}}+Y&&{A_{d}}X\\X{A_{d}}^{T}&&-Y\end{bmatrix}}&<0\\\end{aligned}}$

## Conclusion:

Given the resulting feedback gain matrix $K=WX^{-1}$, it can be observed that the matrix is asymptotically stable while simultaneously ensuring that the solution is delay-independent from the time-delay system where this gain matrix was derived.

## Implementation

* Example Code - A GitHub link that contains code (titled "DelayIndependentTimeDelay.m") that demonstrates how this LMI can be implemented using MATLAB-YALMIP.

## Related LMIs

      * Delay Dependent Time-Delay Stabilization - Equivalent LMI for delay-dependent time-delay stabilization.

## 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 - A book co-authored by Guang-Ren Duan and Hai-Hua Yu.
# LMIs in Control/pages/Discrete-Time Detectability

Discrete-Time Detectability
A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.
Discrete-Time LTI systems can be made detectable using observer gain L, which can be found using LMI optimization, such that the close loop system is detectable.

## The System

Discrete-Time LTI System with state space realization $(A_{d},B_{d},C_{d},D_{d})$ ${\begin{aligned}&A_{d}\in {\bf {{R^{n*n}},}}&B_{d}\in {\bf {{R^{n*m}},}}&C_{d}\in {\bf {{R^{p*n}},}}&D_{d}\in {\bf {{R^{p*m}}\;}}\\\end{aligned}}$

## The Data

The matrices: System $(A_{d},B_{d},C_{d},D_{d}),P,W$.

## The Optimization Problem

The following feasibility problem should be optimized:
Maximize P while obeying the LMI constraints. Then L is found.

## The LMI:

Discrete-Time Detectability
The LMI formulation
${\begin{aligned}P\in {S^{n}};W\in {R^{m*n}}\;\\&P>0\\{\begin{bmatrix}P&A_{d}^{T}P+C_{d}^{T}W\\*&P\end{bmatrix}}&>0,\\L=P^{-1}W\end{aligned}}$

## Conclusion:

The system is detectabe iff there exits a $P$, such that $P>0$. The matrix $A_{d}+LC_{d}$ is Schur with $L=P^{-1}W$

## Implementation

A link to CodeOcean or other online implementation of the LMI MATLAB Code

## Related LMIs

- Continuous time Detectability

## 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/pages/Discrete-Time H2 Norm

Discrete-Time H2 Norm
A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.
Discrete-Time LTI systems' H2 norm can be found by solving a LMI.

## The System

Discrete-Time LTI System with state space realization $(A_{d},B_{d},C_{d},D_{d})$ ${\begin{aligned}&A_{d}\in {\bf {{R^{n*n}},}}&B_{d}\in {\bf {{R^{n*m}},}}&C_{d}\in {\bf {{R^{p*n}},}}&D_{d}\in {\bf {{R^{p*m}}\;}}\\\end{aligned}}$

## The Data

The matrices: System $(A_{d},B_{d},C_{d},D_{d}),P,Z$.

## The Optimization Problem

The following feasibility problem should be optimized:
$\mu$ is minimized while obeying the LMI constraints.

## The LMI:

Discrete-Time Bounded Real Lemma
The LMI formulation
H2 norm < $\mu$
${\begin{aligned}P\in {S^{n}};Z\in {S^{p}};\mu \in {R_{>0}}\;\\&P>0,&Z>0\\{\begin{bmatrix}P&A_{d}P&B_{d}\\*&P&0\\*&*&I\end{bmatrix}}&>0,\\{\begin{bmatrix}Z&C_{d}P\\*&P\end{bmatrix}}&>0,\\trZ<\mu ^{2}\end{aligned}}$

## Conclusion:

The H2 norm is the minimum value of $\mu \in {R_{>0}}$that satisfies the LMI condition.

## Implementation

A link to CodeOcean or other online implementation of the LMI MATLAB Code

## Related LMIs

- Continuous time H2 norm.

## 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.
* - MATLAB function for the H2 norm.
# LMIs in Control/pages/Detectability LMI
## Detectability LMI

Detectability is a weaker version of observability. A system is detectable if all unstable modes of the system are observable, whereas observability requires all modes to be observable. This implies that if a system is observable it will also be detectable. The LMI condition to determine detectability of the pair $(A,C)$ is shown below.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t),\\x(0)&=x_{0},\end{aligned}}$
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 $C$. There is no restriction on the stability of $A$.

## The LMI: Detectability LMI

$(A,B)$ is detectable if and only if there exists $X>0$ such that
       $AX+XA^{T}-B^{T}B<0$.

## Conclusion:

If we are able to find $X>0$ such that the above LMI holds it means the matrix pair $(A,C)$ is detectable. In words, a system pair $(A,C)$ is detectable if the unobservable states asymptotically approach the origin. This is a weaker condition than observability since observability requires that all initial states must be able to be uniquely determined in a finite time interval given knowledge of the input $u(t)$ and output $y(t)$.

## Implementation

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

## Related LMIs

Stabilizability LMI
Hurwitz Stability 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/pages/Deduced LMI Conditions for H-infinity Index

H-infinity Index Deduced LMI
Although the KYP Lemma, also known as the Bounded Real Lemma, is a basic condition to evaluate an upper bound on the H$_∞$, the verification of the bound on the H$_∞$-gain of the system can be done via the deduced condition.

## The System

A state-space representation of a linear system as given below:
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bw(t)\\y(t)&=Cx(t)+Dw(t)\end{aligned}}$
where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$ and $w(t)\in \mathbb {R} ^{r}$ are the system state, output, and the disturbance vector respectively. The transfer function of such a system can be evaluated as:
    ${\begin{aligned}G(s)=C(sI-A)^{-1}B+D\end{aligned}}$

## The Data

Number of states n, number of outputs m and number of external noise channels r need to be known. Moreover, the system matrices A,B,C,D are also required to be known.

## The Feasibility LMI

For an arbitrary $\gamma$, the transfer function G(s) satisfies
    $\left\|G(s)\right\|_{\infty }<\gamma$
if and only if there exists a symmetric matrix X > 0 and a matrix $\Omega .$ such that:
       ${\begin{aligned}{\text{Find}}\;X,\Omega :&\\\Theta +\Phi ^{\top }\Omega \Psi +\Psi ^{\top }\Omega ^{\top }\Phi <0\end{aligned}}$
where:
       ${\begin{aligned}\Theta &={\begin{bmatrix}0&X&0&0&0\\X&-X&0&0&0\\0&0&-\gamma I_{m}&0&0\\0&0&0&-X&0\\0&0&0&0&\gamma I_{r}\end{bmatrix}}\\\Phi &={\begin{bmatrix}-I_{n}&A^{\top }&C^{\top }&I_{n}&0\\0&B^{\top }&D^{\top }&0&-\gamma I_{r}\end{bmatrix}}\\\Psi &={\begin{bmatrix}I_{n}&0&0&0&0\\0&0&0&0&I_{r}\end{bmatrix}}\end{aligned}}$
The above LMI can be combined with the bisection method to find minimum $\gamma$ to find the minimum upper bound on the H$_∞$ gain of $G(s)$.

## Conclusion:

If there is a feasible solution to the aforementioned LMI, then the $\gamma$ upper bounds the infinity norm of the system G(s).

## 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/Deduced_hinf_example.m

## Related LMIs

Bounded Real Lemma

## External Links

A list of references documenting and validating the LMI.
* LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 5.2.2 pp. 153–156.
# LMIs in Control/pages/Discrete-Time Lyapunov Stability

Discrete-Time Lyapunov Stability

## The System

Discrete-Time System
    ${\begin{aligned}x(t)_{k+1}&=A_{d}x(t)_{k},&A_{d}\in {\bf {{R^{n*n}}\;}}\\\end{aligned}}$

## The Data

The matrices $SystemA_{d},P$.

## The Optimization Problem

The following feasibility problem should be solved:
Find P obeying the LMI constraints.

## The LMI:

Discrete-Time Lyapunov Stability
The LMI formulation
       ${\begin{aligned}P\in {\bf {S^{n}}}\\Find\;&P>0,\\{\begin{bmatrix}A_{d}^{T}PA_{d}-P\end{bmatrix}}&\leq 0\end{aligned}}$

## Conclusion:

If there exists a $P\in {\bf {S^{n}}}$ satisfying the LMI then, $|\lambda _{i}(A_{d})|\leq 1,\forall i=1,2,...,n;$ and the equilibrium point ${\bar {x}}=0$ of the system is Lyapunov stable.

## Implementation

* - MATLAB implementation of the LMI.

## Related LMIs

Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality

## 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.