OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/pages/Reachable set polytopic
## Reachable sets with unit-energy inputs; Polytopic uncertainty

A Reachable set is a set of system States reached under the condition $u=Kx$. On this page we will look at the problem of finding an controller $K$, that $E$ contains $RS$ - reachable set.

## The System

    ${\begin{aligned}{\dot {x}}&=Ax+B_{w}w+B_{u}u\\u&=Kx\end{aligned}}$
Where:
    ${\begin{aligned}x&\in R^{n}\\w&\in R^{m}\\u&\in R^{k}\\\end{aligned}}$
In case of polytopic uncertainty, we have:
    ${\begin{aligned}A(t)\;\;B_{w}(t)\;\;B_{u}(t)&\in {\textbf {Co}}\{[A_{1}\;\;B_{w,1}\;\;B_{u;1}],...,[A_{L}\;\;B_{w;L}\;\;B_{u;L}]\}\\\end{aligned}}$

## Reachable set

The reachable set can be defined:
    ${\begin{aligned}RS&=\{x(T)|u=Kx;\;\;x(0)=0;\;\;T\geq 0;\;\;\int _{0}^{T}w^{T}wdt<1\}\\\end{aligned}}$
The elipsoid $E=\{\varepsilon \in R^{n}|\varepsilon ^{T}Q\varepsilon \leq 1\}$ contains $RS$

## The Data

The matrices $A,A_{i}\in R^{n\times n};\;B_{w},B_{w;i}\in R^{n\times m};\;B_{u},B_{u;i}\in R^{n\times k};Q\in R^{n\times n}$. And $$

## The Optimization Problem

The following optimization problem should be solved:
    ${\begin{aligned}{\text{Find}}\;&Y>0:\\&QA_{i}^{T}+A_{i}Q+B_{u;i}Y+Y^{T}B_{u;i}^{T}+B_{w;i}B_{w;i}^{T}<0{\text{ for all }}i=1,...n\\&K=YQ^{-1}\end{aligned}}$

## Conclusion:

This LMI allows us to investigate stability for the robust control problem in the case of polytopic uncertainty and gives on the controller for this case

## Implementation:

* - Matlab implementation using the YALMIP framework and Mosek solver

## Related LMIs:

* Reachable sets with unit-energy inputs; Norm-bound uncertainty
* Reachable sets with unit-energy inputs; Diagonal Norm-bound uncertainty
* Quadratic polytopic 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. (3.20.2 page 64)
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

## Return to Main Page:

* Main Page.
# LMIs in Control/pages/Quadratic polytopic stabilization

A Quadratic Polytopic Stabilization Controller Synthesis can be done using this LMI, requiring the information about $A$,$\Delta _{A(t)}$ ,$B$ and $\Delta _{B(t)}$ matrices.

## 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 system consist of uncertainties of the following form
    ${\begin{aligned}\Delta _{A(t)}=A_{1}\delta _{1}(t)+....+A_{k}\delta _{k}(t)\\\Delta _{B(t)}=B_{1}\delta _{1}(t)+....+B_{k}\delta _{k}(t)\\\end{aligned}}$
where $x\in \mathbb {R} ^{m}$,$u\in \mathbb {R} ^{n}$,$A\in \mathbb {R} ^{mxm}$ and $B\in \mathbb {R} ^{mxn}$

## The Data

The matrices necessary for this LMI are $A$,$\Delta _{A(t)}\,ie\,A_{i}$ ,$B$ and $\Delta _{B(t)}\,ie\,B_{i}$

## The Optimization and LMI:LMI for Controller Synthesis using the theorem of Polytopic Quadratic Stability

There exists a K such that
    ${\begin{aligned}{\dot {x}}(t)&=(A+\Delta _{A}+(B+\Delta _{B})K)x(t)\\\end{aligned}}$
is quadratically stable for $(\Delta _{A},\Delta _{B})\in C_{0}((A_{1},B_{2}),...,(A_{k},B_{k}))$ if and only if there exists some P>0 and Z such that
    ${\begin{aligned}(A+A_{i})P+P(A+A_{i})^{T}+(B+B_{i})Z+Z^{T}(B+B_{i})^{T}<0\quad for\quad i=1,...k.\end{aligned}}$

## Conclusion:

The Controller gain matrix is extracted as $K=ZP^{-1}$
Note that here the controller doesn't depend on $\Delta$
* If you want K to depend on $\Delta$ , the problem is harder.
* But this would require sensing $\Delta$ in real-time.

## Implementation

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

## Related LMIs

Quadratic Polytopic $H_{\infty }$ Controller
Quadratic Polytopic $H_{2}$ Controller

## External Links

* 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/Reciprocal Projection Lemma

WIP, Description in progress
Another condition for eliminating a variable in an LMI called reciprocal projection lemma is introduced.

## Reciprocal Projection Lemma

For a given symmetric matricx $\Phi \in \mathbb {S} ^{n}$, there exists a matrix $S\in \mathbb {R} ^{n\times n}$ satisfying
$\Phi +S^{T}+S<0$
if and only if, for an arbitrarily fixed symmetric matrix $P\in \mathbb {S} ^{n}$, there exist a matrix $W\in \mathbb {R} ^{n\times n}$ satisfying
${\begin{bmatrix}\Phi +P-(W^{T}+W)&S^{T}+W^{T}\\S+W&-P\end{bmatrix}}<0$.
WIP, additional references to be added

## External Links

A list of references documenting and validating the LMI.
# LMIs in Control/pages/Reachable set diagonalNB
## Reachable sets with unit-energy inputs; Diagonal Norm-bound uncertainty

A Reachable set is a set of system States reached under the condition $u=Kx$. On this page we will look at the problem of finding an controller $K$, that $E\supseteq RS$ - reachable set.

## The System

    ${\begin{aligned}{\dot {x}}&=Ax+B_{w}w+B_{u}u+B_{q}p\\q&=C_{q}x+D_{qw}w+D_{qu}u+D_{qp}p\\u&=Kx\end{aligned}}$
Where:
    ${\begin{aligned}x&\in R^{n}\\w&\in R^{m}\\u&\in R^{k}\\\end{aligned}}$
In case of Diagonal Norm-bound uncertainty, we have:
    ${\begin{aligned}p_{i}&=\delta _{i}(t)q_{i}\\&|\delta _{i}(t)|\leq 1;\;\;\;\;{\text{for }}i=1,...,N_{q}\end{aligned}}$

## The Data

$N_{p}=N_{q}$
The matrices $A\in R^{n\times n};\;B_{w}\in R^{n\times m};\;B_{u}\in R^{n\times k};B_{p}\in R^{n\times N_{p}};K\in R^{k\times n}$.
$C_{q}\in R^{N_{q}\times n};\;D_{qu}\in R^{N_{q}\times k}D_{qw}\in R^{N_{q}\times m}D_{qp}\in R^{N_{q}\times N_{p}}$.

## Reachable set

The reachable set can be defined:
    ${\begin{aligned}RS&=\{x(T)|u=Kx;\;\;x(0)=0;\;\;T\geq 0;\;\;\int _{0}^{T}w^{T}wdt<1\}\\\end{aligned}}$
The elipsoid $E=\{\varepsilon \in R^{n}|\varepsilon ^{T}Q\varepsilon \leq 1\}\supseteq RS$

## The Optimization Problem

The following optimization problem should be solved:
    ${\begin{aligned}{\text{Find}}\;&M>0:Y\\&{\begin{bmatrix}QA^{T}+AQ+B_{u}Y+Y^{T}B_{u}^{T}+B_{w}B_{w}^{T}+B_{p}MB_{p}^{T}&(C_{q}Q+D_{qu}Y)^{T}\\C_{q}Q+D_{qu}Y&-MI\end{bmatrix}}<0\\&K=YQ^{-1}\end{aligned}}$

## Conclusion:

This LMI allows us to investigate stability for the robust control problem in the case of polytopic uncertainty and gives on the controller for this case

## Implementation:

* - Matlab implementation using the YALMIP framework and Mosek solver

## Related LMIs:

* - Reachable sets with unit-energy inputs; Polytopic uncertainty
* - Reachable sets with unit-energy inputs; norm bound uncertainty
* Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

## 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. (3.20.2 page 64)
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.

## Return to Main Page:

* Main Page.
# LMIs in Control/pages/Robust stabilization of nonlinear systems

LMIs in Control/pages/Robust stabilization of nonlinear systems
Robust Stabilization of Nonlinear Systems

## The Optimization Problem

Consider a non linear system whos dynamics are given by
${\dot {x}}=Ax+Bu+h(t,x)$
where $x\in \mathbb {R} ^{n}$,$A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$ and $h:\mathbb {R} ^{n+1}\rightarrow \mathbb {R} ^{n}$, $A$ is Hurwitz stable and $h(t,x)$ is piecewise continuous in both $t$ and $s$
We assume $(A,B)$ is stabilizable so
$u(x)=Kx$ , $K\in \mathbb {R} ^{m\times n}$
Assume that $h^{T}(t,x)h(t,x)\leq \alpha ^{2}x^{T}H^{T}Hx$
where $\alpha >0$ is the bounding parameter and $H\in \mathbb {R} ^{l\times n}$

## The Data

The matrices required to solve this problem are A, B, H

## The LMI: Nonlinear Systems Robust Stabilization

There exists scalars $\gamma$, $\kappa _{Z}$, and $\kappa _{Y}$, along with the matrices $Y>0$ such that the following optimization problem is feasible.
       ${\begin{aligned}{\text{minimize}}\ &\gamma +\kappa _{Z}+\kappa _{Y}\\{\text{subject to}}\ &Y>0\\&{\begin{bmatrix}AY+YA^{T}+BZ+Z^{T}B^{T}&I&YH^{T}\\I&-I&0\\HY&0&-\gamma I\end{bmatrix}}&<0\\&\gamma -{\frac {1}{\alpha ^{2}}}<0\\&{\begin{bmatrix}-\kappa _{Z}I&Z^{T}\\Z&-I\end{bmatrix}}<0\\&{\begin{bmatrix}-Y&-I\\-I&-\kappa _{Y}I\end{bmatrix}}<0\end{aligned}}$

## Conclusion:

The controller K can be recovered by the relation
$K=ZY^{-1}$

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* Robust stabilization of nonlinear systems: The LMI approach
# Lower Sorbian/Grammar/2nd conjugation

* over-1-syllable-long verbs ending with -aś, except the verbs ending with -ojaś, -ejaś, -šaś, -caś, -žaś, -šćaś
* following verbs ending with -saś and -zaś: měšaś `mix, stir`, pšašaś `ask`, cycaś `suck`, placaś `clap`, pasaś `engird`, smasaś `taste`, tśasaś `kindle`, se wjelicaś `boast`
* the verb ending with -maś - verb se sromaś `be ashamed`

## Present tense

1. sg. źěłam 2. sg. źěłaš 3. pl. źěłaju

## Imperative

2. 3. sg. źěłaj!

## Imperfect

1. sg. źěłach 2. 3. sg. źěłašo

## Other forms

* Past active participle: źěłał
* Infinitive: źěłaś [1]
* Supine: źěłat
* Present active participle: źěłajucy [2]
* Past passive participle: źěłany
* Verbal noun: źěłanje
[1] compounds of 2. conjugation verb form present tense according to the 1. conjugation, while their verbal stem is suffixed with -aj; the present forms formed according to the 2. conjugation have futural meaning, e.g. pytam `I seek`, pśepytajom `I seek through`, pśepytam `I will seek through`
[2] in the present active participle with -ajucy the vowel -u- can be omitted, e.g. spěwajcy = spěwajucy `singing`
# Louisiana French/Greetings
## Basic Vocabulary

Conversation Starters
| Cajun French | English |
| Bonjour | Hello |
| Comment ça va? | How's it going? |
| Comment les affaires? | How are things? |
| Comment c'est? | How is it? |
| Qui ça dit? | What's up? |
| Comment ça se plume? | Literally: How's it plucking? |
| Comment ça se roule? | Literally: How's it rolling? |
Common Responses
| Cajun French | English |
| Doing Well ||
| Ça va très bien | It's going very well |
| C'est magnifique | It's magnificent |
| Joliment bien | Very well |
| Beaucoup bien | Very well |
| Doing OK ||
| Comme ci, comme ça | So-so (ok) |
| Ça peut faire | It'll do |
| Ça va | It's going |
| Pas mal | Not bad |
| C’est pas trop mauvais | Not too bad |
| Doing Bad ||
| Ça va mal | It's going bad |
| Ça va pas bien | It's not going well. |
| Beaucoup mal | Very bad |
| Ça va pas du tout | It's not going at all |

## Conversation

Here is a conversation between Pierre and Dennis
* Pierre: Bonjour Dennis.
* Dennis: Bonjour Pierre. Comment ça va?
* Pierre: Ça peut faire. Et toi?
* Dennis: Ça va bien, merci.
Here Pierre and Dennis first say hello to each other. Then they exchange basic introductory questions.
* Louisiana French
* 01 .
* 02 .
* 03 .
* 04 .
* 05 .
* 06 .
* 07
# Lower Sorbian/Grammar/3rd conjugation

* all over-1-syllable-long verbs ending with -iś, -yś
* most of the verbs ending with -eś
* verbs ending with -ojaś, -ejaś, -šaś, -žaś, -caś, -šćaś

## Present tense

| 1. sg. | pšosym | palim |
| 2. sg. | pšosyš | pališ |
| 3. pl. | pšose | pale |

## Imperative

2. 3. sg. pšos! ø, pal!ø

## Imperfect

1. sg. pšosach, palach 2. 3. sg. pšosašo, palašo

## Other forms

* Past active participle: pšosył; palił
* Infinitive: pšosyś; paliś [3]
* Supine: pšosyt; palit
* Present active participle: pšosecy; palecy
* Past passive participle: pšosony; palony
* Verbal noun: pšosenje, palenje

## Notes

[1] verbs with infinitive ending in -iś, -yś, -aś form imperfect with -ach, -ašo; verbs with -eś form imperfect with -ech, ešo
[2] verbs with infinitive with –iś, -yś, the stems of which end in consonantal group, form an imperative with –i, e.g. modli!
[3] compounds ending in -iś, -yś, -eś form their proper present tense (durative) in 1. conjugation by lengthening the stem with a suffix -ij, -yj, -ej; forms of present tense in 3. conjugation have futural meaning, e.g. se gorjeś `burn`, se zagorjejo `he burns`, se zagorjo `he will burn`

## Verbs not fitting the pattern

rdžaś (džaś) `shiver`: pres.džym, džyš, džy, etc., 3. pl. džyju (džaju); pres. act. part. džujucy (džajucy); imperative: džyj (džaj)
gorjeś se `burn`: in present tense according to 1. conjugation - gorju, gorjoš, gorjo se; pres. act. part. gorjecy; imperf. gorješo se
kśěś `want`
spaś `sleep`: pres. spim, spiš... 3. pl.spě (spiju); imperative spi!; pres. act. part. spicy (spijucy); impf. spach
spomńeś `evoke, remind`: pres.=fut. spomnim, spomniš... spomnjeju or spomnu, spomnjoš... spomnjeju; aor. spomnjech; imperative spomnjej!
wocuśeś `wake`: pres. wocuśeju, wocuśejoš
verbs ending in -ojaś, -ejaś in present tense and aorist omit vowel -a- from the stem ending, e.g. se bojaś - ja se bojm
# Láadan/Lessons/22
## The same as: zhe

"zhe" means "to be as", "to be like", or "to be similar to". You can say that one thing is like another with two sentences put together with zhe, such as "the flower is beautiful - zhe - the mountain is beautiful"
| Láadan | English | Literal | Citation |
| Bíi melirihal babí zhe melirihul mahina wa. | The birds are quite colorful like the flowers are very colorful. | The birds, like the flowers, are very colorful. | Amberwind's lesson 49 |

## Just comparing

You can also compare different values by just giving different degree markers for each.
| Láadan | English | Literal | Citation |
| Bíi mehíthi yu, i mehíthihul mahina, izh mehíthihel mi wa. | The fruit are high, and the flowers are very high, but the leaves are scarcely high. | The flowers are higher than the fruit, but they are both much higher than the leaves. | Amberwind's lesson 49 |

## To surpass: hesho

[(Aux) Verb: "hesho" (Neg) CP–S]
To compare two things, such as in height or in beauty, we state it this way: ObjectA is tall, ObjectB is tall, ObjectA surpasses.
| Láadan | English | Literal | Citation |
| Bíi íthi yu wa. Íthi mi. Hesho mi. | The leaf is higher than the fruit. | The fruit is high. The leaf is high. The leaf surpasses. | Amberwind's lesson 50 |
| Bíi mehíya rul i zhub wa. Hesho zhub. | The insect is smaller than a cat. | A cat and the insect are small. The insect surpasses. | Amberwind's lesson 50 |
| Bíi melaya mi i mahina wi; báa hesho mahina? | Is the flower redder than the leaf? | The leaf and the flower are red (obviously); does the flower surpass? | Amberwind's lesson 50 |
| Bíi melaya mi i mahina wi; hesho bebáa? | Which is redder, the flower or the leaf? | The leaf and the flower are red (obviously); what surpasses? | Amberwind's lesson 50 |

## Notes

* ↑ http://ayadan.moosader.com/archive/LaadanLessons/l-49.html
* ↑ http://ayadan.moosader.com/archive/LaadanLessons/l-49.html
* ↑ http://ayadan.moosader.com/archive/LaadanLessons/l-50.html
* ↑ http://ayadan.moosader.com/archive/LaadanLessons/l-50.html
* ↑ http://ayadan.moosader.com/archive/LaadanLessons/l-50.html
* ↑ http://ayadan.moosader.com/archive/LaadanLessons/l-50.html
# Láadan/Lessons/23

To make an "If (this) then (that)" statement in Láadan, we use the form "bre… ébre".

## Examples

| Láadan | English |
| Ulanin le | I study |
| Bedi le | I learn |
| Aril bre ulanin le ébre bedi le. | If I study, then I will learn. |
# Láadan/Lessons/5
## Pronouns

There are first, second, and third person pronouns in Láadan. These are gender neutral, and change based on the amount of people being talked about.
| Voice | Singular | Plural, 2 to 5 | Plural, 6 or more |
| First Person ("I") | Le | Lezh | Len |
| Second Person ("You") | Ne | Nezh | Nen |
| Third Person ("He", "She") | Be | Bezh | Ben |

### Examples

| Láadan | English | Vocab words |
| Bíi ham le wi! | I am present! | ham = to be present, to attend |
| Bíi meham lezh wi! | We are present! | meham = me- (plural) + ham |

## Objects

Now we'll introduce objects to our sentence. The sentence pattern is [(Aux) Verb (Neg) CP-S CP-O], where CP-O is the Case-Phrase-Object. Now we can see that the sentence structure in Láadan is VSO (verb-subject-object) order.
Objects in Láadan are also marked with the Object Marker, the suffix -th or -eth (depending on whether the last letter of the object is a vowel or consonant).

### Examples

| Láadan | English | Vocab words |
| Bíi yod le wa. | I am eating. | yod = to eat |
| Bíi yod le doyuth wa. | I am eating an apple. | doyu = apple |
| Bíi yod le ódoneth wa. | I am eating cheese. | ódon = cheese |
| Bíi néde le thuzheth wa. | I want a cake. | néde = to want, thuzh = cake |

### Notes

In the case where the subject and object are obvious, the object marker can be left off. For example, if we said "I speak Láadan", it is clear: I can speak Láadan, but Láadan cannot speak me. Therefore, the sentence could be said as "Bíi di le Láadaneth wa." or "Bíi di le Láadan wa." (Suzette Haden Elgin (1988), A First Dictionary and Grammar of Láadan, Second Edition, p. 18)

## Multiple verbs

Sometimes you will want to combine verbs like "to want", "to need", "to must" with another verb - "I want to run", "I need to wait", "I must work". In Láadan, this forms a Verb Complex. The two verbs go together, with the auxiliary going before both, and the negative going afterward, if present. Nothing can go between the two verbs. Both must be marked plural if the subject is plural. (Suzette Haden Elgin (1988), A First Dictionary and Grammar of Láadan, Second Edition, p. 22)

### Examples

| Láadan | English | Vocab words |
| Bíi yod le wa. | I am eating. | yod = to eat |
| Bíi néde yod le wa. | I want to eat. | |
| Bíi menéde meyod lezh wa. | We want to eat. | |

## To try to [VERB]

The prefix "du-" means "to try".

### Examples

| Láadan | English | Vocab words |
| Bíi ulanin le wa. | I am studying. | ulanin = to study |
| Bíi duhulanin le wa. | I am trying to study. | |
# Láadan/Lessons/6
## Introduction

In English, we use prepositions like "in", "at", "on", "to". In Láadan, these concepts are conveyed through suffixes.

## The Goal marker: -di

Pattern: [(Aux) Verb (Neg) CP-S (CP-O) CP-GOAL]
The Goal case phrase is where something is directed. This could be in a direction of travel, or towards some recipient.

### Examples

| Láadan | English | Vocab words |
| Bíi sháad le bethedi wa. | I am going home. | sháad = to come/to go, beth = home |
| Bíi eril ban le doyuth háawithedi wa. | I gave the apple to the child. | eril = past, ban = to give, doyu = apple, háawith = child |

## The Source marker: -de

Pattern: [(Aux) Verb (Neg) CP-S (CP-O) CP-SOURCE]
Sentences using the verb "sháad" (to come/to go) will be very similar to each other, and the only way to tell a "come" from a "go" is based on whether the Goal or the Source marker is used.

### Examples

| Láadan | English | Vocab words |
| Bíi eril sháad le bethedi wa. | I came home. | sháad = to come/to go, beth = home |
| Bíi eril sháad le bethede wa. | I came from home. | sháad = to come/to go, beth = home |

## Alternative to -di

For some people, -de and -di might sound too alike. If you prefer, you can use -dim instead of -di. (Suzette Haden Elgin (1988), A First Dictionary and Grammar of Láadan, Second Edition, p. 30)
# Láadan/Lessons/20
## Introduction

Maybe you want to describe what is happening in a more passive voice; for example, changing "Jane coveted the treasure" (active), "The treasure was coveted by Jane".
The first version, we would order it something like:
Speech-act-morpheme verb subject object evidence-morpheme
Where the subject is Jane and the object is the treasure.

## Passive suffix: -shub marker

In order to change the original active sentence to passive move the subject, which is the Agent doing the action, (Jane) after the verb and the object (treasure) immediately before the verb. The object is now in the focus position.
You would also add the suffix -shub onto the end of the subject (Jane).

### Passive sentence without an Agent

If your sentence does not have an Agent subject (The treasure is coveted), then you will still move your object before the verb, and add the -shub suffix to the verb.

### Examples

| Step | Láadan | English | Vocab words |
| Active | Bíi eril yod le thuzheth wa. | I ate cake. | eril = past, yod = eat, le = I, thuzh = cake |
| Passive, with Agent | Bíi eril thuzheth yod leshub wa. | The cake was eaten by me. | |
| Passive, no Agent | Bíi eril thuzheth yodeshub wa. | The cake was eaten. | |
# Láadan/Lessons/19
## Introduction

Once you're comfortable with basic sentence structures, you may want to include one sentence in another. For example, "Did you know that [grandma died]?", where "Grandma died" is its own sentence, but is included in a larger sentence.

## Embedding statements: -hé marker

The "-hé" marker can be used to embed one statement within another statement. "-hé" will be added to the last word in the internal sentence.

### Examples

| Láadan | English | Vocab words |
| Bíi áya le wa. | I am beautiful. | áya = to be beautiful |
| Bíi lith le [áya lehé] wa. | I think that I am beautiful. | lith = to think |
| Bíi lith be [áya lehé] wa. | She thinks that I am beautiful. | |

## Embedding questions: -hée marker

This marker can be used to embed a question within a sentence. The external sentence can also be a question, or it can be a statement. "-hée" will be added to the last word in the internal sentence.

### Examples

| Láadan | English | Vocab words |
| Bíi áya le wa. | I am beautiful. | |
| Bíi lith be [áya lehée]? | Do you think that I am beautiful? | |
| Bíi lothel ra le [áya lehée] wa. | I don't know whether I am beautiful or not. | lothel = to know |

## Embedding relative clauses: -háa marker

A relative clause modifies a noun. With the statement and question embedding markers above, the internal sentences are relatively standalone ("the rain is cold." "do you think that [the rain is cold]?"). Here, however, the relative clause will modify the sentence as a whole.

### Examples

| Láadan | English |
| Bíi an behid witheth wa. | He knows a woman. |
| Bíi lalom with wa. | The woman sings. |
| Bíi an behid [lalom witheháath] wa. | He knows a woman who sings. |
(Amberwind (2012), Láadan Lessons, p. 295)

## Comparison

Note the difference between using "-hé" and "-háa":
| Láadan | English | Vocab words |
| Bíi dom le [hal withehé] wa. | I remember that [the woman works]. | dom = to remember |
| Bíi dom le [hal witheháa] wa. | I remember the [woman that works]. | |