OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Continuous-time strong stabilizability
## The System

Consider the continous-time LTI system, $G:L_{2e}\rightarrow L_{2e}$ with state-space realization (A,B,C,0)
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t),\end{aligned}}$
where $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, and it and it is assumed that (A, B) is stabilizable, (A, C) is detectable, and the transfer matrix $G(s)=C(s1-A^{-1})B$ has no poles on the imaginary axis.

## The Data

The matrices $A,B,C$.

## The Optimization Problem

The system G is strongly stabilizable if there exist $P\in \mathbb {S} ^{n}$, $Z\in \mathbb {R} ^{n\times p}$, and $\gamma \in \mathbb {R} _{>0}$, where $P>0$, such that
    ${\begin{aligned}PA+A^{T}+ZC+C^{T}Z^{T}<0\\{\displaystyle {\begin{aligned}{\begin{bmatrix}-P(A+BF)+(A+BF)^{T}P+ZC+C^{T}Z^{T}&&-Z&&-XB\\*&&-\gamma I&&0\\*&&*&&-\gamma I\end{bmatrix}}<0\end{aligned}}}\\\end{aligned}}$

## Conclusion:

where $F=-B^{T}X$ and $X\in S_{n}$ , $X\geq 0$ is the solution to the Lyapunov equation given by
    ${\begin{aligned}XA+A^{T}X-XBB^{T}X=0\end{aligned}}$
Moreover, a controller that strongly stabilizes G is given by the state-space realization
    ${\begin{aligned}{\dot {x}}_{c}=(A+BF+P^{-1}ZC)x(t)-P^{-1}Zu(t)\\y_{C}=-B^{T}Xx(t)\end{aligned}}$

## Implementation

* Example Code

## Related LMIs

* https://en.wikibooks.org/wiki/LMIs_in_Control/Stability_Analysis/Discrete_Time/DiscreteTimeStrongStabilizability - Discrete Time Strong Stabilizability

## External Links

* http://control.asu.edu/MAE598_frame.htm LMI Methods in Optimal and Robust Control- A course on LMIs in Control by Matthew Peet.
* https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory] - A List of LMIs by Ryan Caverly and James Forbes.
* https://web.stanford.edu/~boyd/lmibook/ 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/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/Click here to continue/LMIs in system and stability Theory/DC Gain of a Transfer Matrix

LMIs in Control/Click here to continue/LMIs in system and stability Theory/DC Gain of a Transfer Matrix
The continuous-time DC gain is the transfer function value at the frequency s = 0.

## The System

Consider a square continuous time Linear Time invariant system, with the state space realization $(A,B,C,D)$ and $\gamma \in \mathbb {R} _{>0}$
    ${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y=Cx(t)+Du(t)\end{aligned}}$

## The Data

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

## The LMI: LMI for DC Gain of a Transfer Matrix

The transfer matrix is given by$G(s)=C(sI-A)^{-1}B+D$
The DC Gain of the system is strictly less than $\gamma$ if the following LMIs are satisfied.
    ${\begin{bmatrix}\gamma I&-CA^{-1}B+D\\(-CA^{-1}B+D)'&\gamma I\end{bmatrix}}$${\begin{aligned}>0\end{aligned}}$
OR
    ${\begin{bmatrix}\gamma I&-B^{T}A^{-T}C^{T}+DT\\(-B^{T}A^{-T}C^{T}+DT)'&\gamma I\end{bmatrix}}$${\begin{aligned}>0\end{aligned}}$

## Conclusion

The DC Gain of the continuous-time LTI system, whose state space realization is give by ($A,B,C,D$), is
$K=D-CA^{-1}B$
* Upon implementation we can see that the value of $\gamma$ obtained from the LMI approach and the value of $K$ obtained from the above formula are the same

## Implementation

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

## 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.
* -Mathworks reference to DC Gain
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Continuous-time Stabilizability

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

## The System

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

## The Data

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

## The LMI: Stabilizability LMI

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

## Conclusion:

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

## Implementation

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

## Related LMIs

Hurwitz Stability LMI
Detectability LMI
Controllability Grammian LMI
Observability Grammian LMI

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013, Section 6.1.1 and Table 6.1 pp. 166–170, 192.
* A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Section 2.2.3, pp. 71-73.
# HyperText Markup Language/Links

The web tutorials:
* HTMLCenter, htmlcenter.com, has over 200 HTML tutorials and a forum for questions.
* HTML Tutorials, pickatutorial.com
* HTML Source: HTML Tutorials, yourhtmlsource.com - step by step lessons
* HTML, CSS, and JavaScript Tutorials, References, and Articles, htmldog.com
* Learning Html And Css, c2.com, another list of tutorials
* HTML lessons and tutorials, landofcode.com
* Free tutorials on HTML, CSS and PHP, html.net
* HTML Tutorial, w3schools.com
# HyperText Markup Language/Quotations

There are two kinds of quotations supported by HTML--inline ones and block quotations.

## Inline quotations

An example:
Will be displayed as:
An inline quotation of significant length (25 words, for example) goes here.

## Block quotations

An example:
Example rendering:
   Lorem ipsum dolor sit amet, consectetur adipisicing elit, sed do eiusmod tempor incididunt ut labore et dolore magna aliqua. Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris nisi ut aliquip ex ea commodo consequat. Duis aute irure dolor in reprehenderit in voluptate velit esse cillum dolore eu fugiat nulla pariatur. Excepteur sint occaecat cupidatat non proident, sunt in culpa qui officia deserunt mollit anim id est laborum.
# HyperText Markup Language/Tag List

The following is a list of all elements in HTML 4, in alphabetical order. TODO for the book: update list for HTML5 Click on a element for its description. XHTML 1.0 has the same elements but the attributes differ slightly.
The official list of current standard elements is at Index of the HTML 5 Elements.
You can also view a list of standard attributes
* a
* abbr
* acronym – in most instances use abbr instead. See The Accessibility Hat Trick: Getting Abbreviations Right for some advice on using these elements.
* address
* (applet – deprecated, use object instead.)
* area
* b – use strong or the CSS property font-weight set to the value bold instead.
* base
* (basefont – deprecated, set the CSS property font on the body element instead.)
* bdo
* bgsound Used for inserting background sounds.
* big – the CSS property font-size set to the value larger or a percentage greater than 100% may be more appropriate.
* blink used to make the text blink (Depreciated).
* blockquote
* body Identifies the main content of a Web Page.
* br – use the p element for paragraphs. Use the CSS properties margin and padding to increase or decrease the space between paragraphs. Consider using structured elements such as lists or tables instead.
* button
* caption
* (center – deprecated, use a div element instead and set the CSS property text-align to the value center.)
* cite
* code
* col
* colgroup
* dd
* del
* dfn
* (dir – deprecated, use ul.)
* div
* dl
* dt
* em
* fieldset
* (font – deprecated, use the CSS property font. For finer control use the CSS properties font-style, font-variant, font-weight, font-size, line-height and font-family.)
* form Creates a form.
* frame Specifies information for one frame.
* frameset – avoid using frames if possible.
* head Contains information about a Web Page.
* hr
* html
* h1
* h2
* h3
* h4
* h5
* h6
* i – use em or the CSS property font-style set to the value italic instead.
* iframe
* img
* input
* ins
* (isindex – deprecated, use input.)
* kbd
* label
* legend
* li
* link
* map
* (menu – deprecated, use ul.)
* meta
* nobr is a proprietary element type supported by some web browsers. It is used to prevent automatic wrapping of lines.
* noframes
* noscript Specifies what should be done if there is no javascript found on the browser.
* object
* ol
* optgroup
* option
* p
* param
* pre
* q
* (s – deprecated, use del to indicate deleted text. If the text isn't 'deleted' use the CSS property text-decoration set to the value line-through.)
* samp
* script
* select
* small – the CSS property font-size set to the value smaller or a percentage less than 100% may be more appropriate.
* span
* (strike – deprecated, use del to indicate deleted text. If the text isn't 'deleted' use the CSS property text-decoration set to the value line-through.)
* strong
* style
* sub
* sup
* table
* tbody
* td
* textarea
* tfoot
* th
* thead
* title
* tr
* tt
* (u – deprecated, use the CSS property text-decoration set to the value underline instead.)
* ul
* var

## References

* ↑ http://www.w3.org/TR/html-markup/elements.html
# HyperText Markup Language/Tag List/a
## Use

The anchor element (a) is used to define anchors, or hyperlinks. You can use it to refer to other pages with the href attribute (Hyperlink Reference).

## DTD

Strict, Transitional and Frameset, for both HTML 4 and XHTML 1.0.
a is an inline element.

## Optional attributes
#### charset

This attribute allows you to specify the character encoding of the target location

#### href

With the href attribute you specify the destination of the hyperlink, in other words, the page a visitor will be sent to when he clicks the link.

#### hreflang

This attribute allows you to specify the language of the target page.

#### name

This allows you to specify a name for the hyperlink, for use as a bookmark. Note that in XHTML, you should use the id attribute.

#### rel

Allows you to specify the relation of the current to the target page. In other words, what the target page is, Possible values are alternate, appendix, bookmark, chapter, contents, copyright, designates, glossary, help, index, next, prev, section, start, stylesheet and subsection.

#### rev

Allows you to specify the relation of the target page to the current page. In other words, what the current page is. This should be the same for all hyperlinks in the current page. For possible values, see rel above.

#### target

This attribute can only be used in transitional and frameset XHTML documents. It is used to tell the browser where to open the targeted URL. The user has Four options to choose from:
* _blank: The URL will open in a new window.
* _parent: Will open in the parent frameset.
* _self: Will open in the same window/tag as it was clicked on.
* _top: Will open in the full body window.

#### type

This lets you specify the Multipurpose Internet Mail Extensions type of the targeted URL.

## Standard attributes

* class
* dir
* id
* lang
* style
* title
* accesskey
* id
* xml:lang

## Notes

In Microsoft Internet Explorer, this is the only element on which the CSS pseudo-class selector hover works.

## More info

The <a> tag on W3Schools
# HyperText Markup Language/Lists

In HTML, there are three kinds of lists, each appropriate for a different kind of information. An unordered list made with <ul> and </ul> tags is meant for elements that have no order or the order is unimportant (typically shown with bullets). An ordered list created using the <ol> and </ol> tags is typically shown with numbers and is used for elements whose order matters such as a sequence of steps to perform. Finally, there are definitions lists, created with <dl> and </dl> tags.

## Ordered Lists

Ordered lists provide a list of items, each of which are preceded by an incremental number starting from 1.
Sample HTML:
Example rendering:
* First item
* Second item
* Third item

## Unordered Lists

Unordered lists display a list of items, each of which is prefixed by a bullet.
Sample HTML:
Example rendering:
* First item
* Second item
* Third item

## Definition Lists

Definition lists display a list of bolded terms, followed by the definition on a new line and prefixed by a tab (by default).
Sample HTML:
Example rendering:

### Term 1

    Definition of Term 1

### Term 2

    Definition of Term 2
If two terms share the same definition they can be grouped together like so:
Example rendering:

### CSS

    Definition of Cascading Style Sheets (aka CSS)

### Term 2

    Definition of Term 2
If a term has two alternative definitions use a separate dd element for each definition, e.g.
Example rendering:

### Mouse

    Small mammal
    Input device for a computer
Longer definitions can be broken up into paragraphs by nesting p elements within the dd element.
Example rendering:

### Term 2

    First paragraph of the definition. Second paragraph of the definition.

## Nested Lists

Lists can be nested. An example:
Example rendering:
* Lists
** Numbered Lists
** Unnumbered Lists
When nesting, nested list elements should be within a parent list item element.
An example of incorrect nesting:
A further example of incorrect nesting, with two consecutive UL elements:

## Note on format

The above descriptions of each of the three list types refer to the default rendering of the corresponding HTML code by the web browser. However, by using CSS, you are able to change the formatting of the lists. For example, with CSS you are able to make the lists horizontal as opposed to the vertical.

## Example

An example of using lists to mark up a recipe for pancakes.
# HyperText Markup Language/Music

Users should have control over the background sound or music; therefore, media player controls should always be visible. Don't try to script your own controls since they might not work in all environments.
There are music players for websites that you can find if you search. They should never be set to auto-play, they should only play when the user chooses to play them.
# HyperText Markup Language/Paragraphs and Headings

The bulk of a typical web page often consists of paragraphs structured with the use of headings.

## Paragraphs

The p element is used to split text into paragraphs.

## Headings

There are six levels of headings. The most important heading(s) in a document should be level one. Sub-headings should be level two. Sub-sub-headings should be level three, etc. Do not skip levels. If the default sizes do not suit your document, use CSS to change them. Headings should be used to effectively outline your content. By doing so, users can find information more quickly, and some search engines use headings to help rank page content.

## Example

This example will be used in the next section where we see how to change the appearance of a document.
# HyperText Markup Language/Standard Attributes List

Below is a list of all attributes which are available for most elements in HTML.
You can also view a list of HTML elements.

## Attributes
### class

This attribute allows you to designate an element to be a member of a given class. Multiple elements can be assigned to the same class (eg. <p class="foo"> ... </p> <p class="foo"> ... </p>), as well as a single element belonging to multiple classes (eg. <p class="foo bar"> ... </p>).

### dir

With this attribute you can define which direction the text is written for a given element, either ltr for left-to-right or rtl for right-to-left.

### height

Setting this attribute with a numerical value defines the height of an element in pixels (eg. <div height="150"> ... </div>)

### id

This attribute allows you to define a unique identifier for each element. This would be useful for hyperlinks that link to a specific section of a page or when styling using a style sheet.

### lang

With this attribute you can specify a language that is used for an element.

### style

This attribute allows you to apply specific styling to a given element.

### title

With this attribute you can define what will be displayed when a user hovers the element. It is not available for base, head, html, meta, param, script, style, and title.

### width

Setting this attribute with a numerical value defines the width of an element in pixels (eg. <div width="230"> ... </div>)

## More attributes
### accesskey

The accesskey attribute defines a keyboard shortcut for a hyperlink or form element. The combination of keys need to activate the shortcut varies from browser to browser. In Microsoft Internet Explorer the user must press Alt+accesskey. If the shortcut is for a link the user must then press Enter to follow the link. The choice of Alt+accesskey means that access keys can clash with shortcuts built-in to the browser.
It is quite common to use numbers for the access keys since these don't clash with any major browser's built-in shortcuts, e. g.
* 1 = Home Page
* 0 = List of access keys on this website.
There is no standard way to let users know the access keys that are available on the page. Some suggestions can be found in Accesskeys: Unlocking Hidden Navigation.
# Instructional Technology/Postmodernist Thoughts

Instructional Systems Design's team approach to design is considered postmodern because it rejects the idea of a central authority. Instruction should be designed based on individual needs as individuals are viewed as decentered by postmodernists. Constructivist theory and hypertext both emphasize the individual learner. If we view ourselves as postmodernists, when developing instruction, we should focus on the process rather than the content or information because information changes rapidly.
Although Instructional Design is based on a scientific, systematic and quantitative paradigm, postmodernists perceive it as an art as well as a science and suggest that instructional designers become "Connoisseurs."
Andrew Yeaman (1994) outlined nine ideas for postmodern instructional design:
1. Accept that there a e several workable solutions to every design problem.
2. Expect that students or trainees will interpret instruction in different ways.
3. Examine and learn from instruction that fails as well as instruction that succeeds.
4. Metaphors, symbols and models should be used cautiously.
5. Determine if technological fixes have improved problems or created more.
6. Avoid idealism that all students/trainees will have absolute correspondence in their understanding - there is never a true meeting of the minds.
7. Avoid authoritarian approach - give some control to learners.
8. Look for contradictions in your messages and in other's.
9. Plan instruction based on learners' needs not just technologies.
With UNESCO's Education for Everyone by 2015 initiatives underway, this philosophy is extremely applicable, especially in terms of creating instruction across geographical, linguistic, and cultural boundaries. Culture, language and societal contexts color contextual meaning. Culture not only affects how we learn, but determines how we think, and how we think about learning. As instructional designers, we need to be thinking more holistically about the design of instruction. This is especially true as we move from local design, development and deployment to more global implementations.
Reference:
Yeaman, A.R.J. (1994). Nine ideas for postmodern instructional design. In D.H. Jonassen (Ed.), Handbook of research for educational communications and technology (pp. 285-286). New York: Simon and Schuster Macmillan.