OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Observer D-stability

LMIs in Control/Click here to continue/LMIs in system and stability Theory/Observer D-stability
This is a WIP Page.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)\\x(0)&=x_{0}\end{aligned}}$

## The Data

In order to properly define the acceptable region of the poles in the complex plane, we need the following three pieces of data: rise time ($t_{r}$), settling time ($t_{s}$), and percent overshoot ($M_{p}$). From this, we have to then define the acceptable region in the complex plane that the poles can lie on using the following inequality constraints:
Rise Time: $\omega _{n}{\leq }{1.8 \over t_{r}}$
Settling Time: $\sigma {\leq }{-4.6 \over t_{s}}$
Percent Overshoot: $\sigma {\leq }{-ln({M_{p}}) \over {\pi }}|{\omega _{d}}|$
Assume that $z$ is the complex pole location, then:
       ${\begin{aligned}{\omega _{n}}^{2}=\|z\|^{2}&=z^{*}z\\{\omega _{d}}=Im{z}&={(z-z^{*}) \over 2}\\{\sigma }=Re{z}&={(z+z^{*}) \over 2}\end{aligned}}$
This then allows us to modify our inequality constraints as:
Rise Time: $z^{*}z-{1.8^{2} \over {t_{r}}^{2}}{\leq }0$
Settling Time: ${(z+z^{*}) \over 2}+{4.6 \over t_{s}}{\leq }0$
Percent Overshoot: $z-z^{*}+{{\pi } \over ln({M_{p}})}|z+z^{*}|{\leq }0$

## The Optimization Problem

A description of the Problem to be solved, if appropriate.

## The LMI: The Observer D-Stability

Title and mathematical description of the LMI formulation.
       ${\begin{aligned}{\text{Find}}\;X:&\\{\begin{bmatrix}X\end{bmatrix}}&>0\\{\begin{bmatrix}A^{T}X+XA\end{bmatrix}}&<0\end{aligned}}$

## Conclusion:

Interpretation of the results of the LMI

## Implementation

A link to CodeOcean or other online implementation of the LMI

## Related LMIs

      * Controller D-Stability
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Output Energy Bound for Lure's Systems
## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q(t)&=C_{q}x(t),\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}$

## The Data

The matrices $A,B_{p},B_{w},C_{q},C_{z},x(0)$.

## The Optimization Problem:

The following optimization problem should be to find the tightest upper bound for the output energy of the above Lur'e system.
       ${\begin{aligned}&\min _{P\succ 0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0}x^{\top }(0)(P+C_{q}^{\top }\Lambda C_{q})x(0)\\&\quad \quad \quad \quad \quad \quad \quad {\begin{bmatrix}A^{\top }P+PA&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T\end{bmatrix}}\preceq 0\\\end{aligned}}$

## Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/blob/master/LMIs%20for%20Output%20Energy%20Bounds%20of%20Lure's%20Systems

## Conclusion

The value function returns the the lowest bound for the energy function of the Lure's systems, i.e., $J=\int _{0}^{\infty }z^{\top }z\ dt$ with initial conditions $x(0)$.

## Remark

The key step in the proof is to satisfy ${\frac {d}{dt}}V(x)+z^{\top }z\leq 0$, where $V(.)$ is Lyapunov function in a special form.

## 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/Click here to continue/LMIs in system and stability Theory/Polytopic Quadratic Stability

An important result to determine the stability of the system with uncertainties

## The System:

Consider the system with Affine Time-Varying uncertainty (No input)
    ${\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta A(t))x(t)\\\end{aligned}}$
where
    ${\begin{aligned}\Delta A(t)=A_{1}\delta _{1}(t)+....+A_{k}\delta _{k}(t)\\\end{aligned}}$
where $\delta _{i}(t)$ lies in either the intervals
    ${\begin{aligned}\delta _{i}\in [\delta _{i}^{-},\delta _{i}^{+}]\\\end{aligned}}$
or the simplex
    ${\begin{aligned}\delta (t)\in {\delta :\Sigma \alpha _{i}=1,\alpha \geq 0}\end{aligned}}$
where $x\in \mathbb {R} ^{m}$ and $A\in \mathbb {R} ^{mxm}$

## The Data

The matrix A and $\Delta _{A(t)}$ are known

## The Optimization

The Definitions: Quadratic Stability for Dynamic Uncertainty
The system
    ${\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta A(t))x(t)\\\end{aligned}}$
is Quadraticallly Stable over $\Delta$ if there exists a P > 0
Theorem $(A+\Delta ,{\boldsymbol {\Delta }})$ is quadratically stable over ${\boldsymbol {\Delta }}:=Co(A_{1},...,A_{k})$ if and only if there exists a P > 0 such that
    ${\begin{aligned}(A+A_{i})^{T}P+P(A+A_{i})<0\quad for\quad all\quad i=1,....,k\end{aligned}}$
The theorem says the LMI only needs to hold at the EXTREMAL POINTS or VERTICES of the polytope.
* Quadratic Stability MUST be expressed as an LMI

## The LMI

    ${\begin{aligned}(A+\Delta )^{T}P+P(A+\Delta )<0\quad for\quad all\quad \Delta \in {\boldsymbol {\Delta }}\end{aligned}}$

## Conclusion:

Quadratic Stability Implies Stability of trajectories for any $\Delta$ with $\Delta \in {\boldsymbol {\Delta }}$ for all $t\geq 0$
Quadratic Stability is CONSERVATIVE.
There are Stable System which are not Quadratically stable. Quadratic Stability is sometimes referred to as an "infinite-dimensional LMI"
* Meaning it represents an infinite number of LMI constraints.
* One for each possible value $\Delta$ with $\Delta \in {\boldsymbol {\Delta }}$
* Also called a parameterized LMI
* Such LMIs are not tractable.
* For polytopic sets, the LMI can be made finite.

## Implementation

A link to implementation of the LMI
https://github.com/JalpeshBhadra/LMI/blob/master/polytopicstability.m

## Related LMIs

* Parametric Norm Bounded Uncertain System Quadratic Stability

## 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/Click here to continue/LMIs in system and stability Theory/Output Energy Bound for Discrete-Time Autonomous LTI Systems

Output Energy Bound for Autonomous LTI Systems
Autonomous systems are initialized under a given set of initial conditions, and then run without any additional inputs. It is useful to know ahead of time the bounds such a system will operate within. This analysis can be used to determine the upper bound on the output of a given autonomous LTI system.

## The System

Consider the continuous-time, LTI autonomous system with state space representation
${\mathbf {\dot {x}}}={\mathbf {Ax}}$,
${\mathbf {y}}={\mathbf {Cx}}$,
where ${\mathbf {A}}\in \mathbb {R} ^{n\times n},{\mathbf {C}}\in \mathbb {R} ^{p\times n}$ and ${\mathbf {x}}(0)={\mathbf {x}}_{0}.$

## Determining an Upper Bound

The output of this system will satisfy
${\sqrt {\int _{0}^{T}{\mathbf {y^{T}y}}dt}}=\left\vert \left\vert {\mathbf {y}}\right\vert \right\vert _{2T}\leq \gamma \left\vert \left\vert {\mathbf {x}}_{0}\right\vert \right\vert _{2},\forall T\in \mathbb {R} _{\geq 0}$
if there exists some matrix ${\mathbf {P}}\in \S ^{p}$and scalar $\gamma \in \mathbb {R} _{>0}$ such that
${\mathbf {P}}>0$,
${\mathbf {P}}-\gamma {\mathbf {I}}\leq 0$,
${\begin{bmatrix}{\mathbf {PA}}+{\mathbf {A^{T}P}}&{\mathbf {C^{T}}}\\*&-\gamma {\mathbf {I}}\end{bmatrix}}\leq 0$.

## Conclusion

Given an autonomous system with an initial operating condition, the parameter $\gamma$ can be used to determine the largest feasible bound on the output of that system.

## 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/Click here to continue/LMIs in system and stability Theory/Parametric, Norm-Bounded Uncertain System Quadratic Stability

LMIs in Control/Click here to continue/LMIs in system and stability Theory/Parametric, Norm-Bounded Uncertain System Quadratic Stability
Given a system with matrices A,M,N,Q the quadratic stability of the system with parametric, norm-bounded uncertainty can be determined by the following LMI. The feasibility of the LMI tells if the system is quadratically stable or not.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Mp(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+Qp(t),&&\Delta \in \mathbf {\Delta } \;:=\{\Delta \in \mathbb {R} ^{n\times n}:\|\Delta \|\leq 1\}\\\end{aligned}}$

## The Data

The matrices $A,M,N,Q$.

## The LMI:

       ${\begin{aligned}{\text{Find}}\;&P>0,\mu \geq 0:\\{\begin{bmatrix}AP+PA^{T}&PN^{T}\\NP&0\end{bmatrix}}+\mu {\begin{bmatrix}MM^{T}&MQ^{T}\\QM^{T}&QQ^{T}-I\end{bmatrix}}<0\\\end{aligned}}$

## Conclusion:

The system above is quadratically stable if and only if there exists some mu >= 0 and P > 0 such that the LMI is feasible.

## Implementation

https://github.com/mcavorsi/LMI

## Related LMIs

Stability of Structured, Norm-Bounded Uncertainty
Stability under Arbitrary Switching
Quadratic Stability Margins

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
# Pinyin/Preparing

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# Pinyin/Prescription

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# Pinyin/Preterm Birth

Preterm birth, occurring before 37 completed weeks of pregnancy, is the number one cause of newborn deaths and the second leading cause of deaths in children under five. Preterm babies are at increased risk of illness, disability and death. However, some of these early births and a majority of the resulting deaths can be prevented with proven, low cost interventions.
Zǎochǎn de dìngyì shì zài wánchéng 37 zhōu rènshēn zhīqián chūshēng de yīng'ér. Zǎochǎn shì xīnshēng'ér sǐwáng de shǒuyào yuányīn, bìngqiě shì jì fèiyán zhīhòu 5 suì yǐxià értóng sǐwáng de dì-2 dà yuányīn. Xǔduō cúnhuó xiàlái de értóng yě yào miànlín zhōngshēng cánjí, bāokuò xuéxí zhàng'ài hé shìlì, tīnglì wèntí. Rán'ér, jíbiàn méiyǒu zhòngzhèng jiānhù shèshī, jùyǒu chéngběn xiàoyì de xiànyǒu gānyù cuòshī yě néng wǎnjiù 3/4 de zǎochǎn'ér sǐwáng.
# Pinyin/Prevent

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# Pinyin/Prevent and treat

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# Pinyin/Present tense

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# Pinyin/President

Search president
# Pinyin/Pressure

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# Mathematics for Chemistry/Further reading
## Further reading
### Books

* Paul Monk, Maths for Chemistry: A Chemist's Toolkit of Calculations, (Oxford, 2006), ISBN 978-0199277414.
* Bostock and Chandler, Core Maths for A-level, Third Edition, (Nelson Thornes, 2000), ISBN 978-0748755097.
* M. C. R. Cockett and G. Doggett, Maths for Chemists: Numbers, Functions and Calculus v. 1, (Royal Society of Chemistry, London, 2003), ISBN 978-0854046775
* M. C. R. Cockett and G. Doggett, Maths for Chemists: Power Series Complex Numbers and Linear Algebra v. 2, (Royal Society of Chemistry, London, 2003), ISBN 978-0854044955
* G. Currell and T. Dowman,Mathematics and Statistics for Science, (Wiley, 2005), ISBN 978-0470022290.
* Stephen K. Scott, Beginning Maths for Chemistry, (Oxford, 1995), ISBN 978-0198559306
* Peter Tebbutt, Basic Mathematics for Chemists, 2nd edition, (Wiley, 1994), ISBN 978-0471972839

### Online resources

There is much useful free material relevant to this book, including downloadable DVDs, funded by the HEFCE Fund for the Development of Teaching & Learning and the Gatsby Technical Education Project in association with the Higher Education Academy at Math Tutor.
Discover Maths for Chemists from the Royal Society of Chemistry is a a one-stop site designed by chemists for chemists. This new free-to-use site brings together all the best resources to help you combine maths and chemistry.
Maths for Chemistry is an online resource providing interactive context-based resources which explain how various aspects of maths can be applied to chemistry. There are quizzes and downloadable files to check understanding.
# Mathematics Worksheet/Social Arithmetic/Loss and Profit

Do the following story problems carefully and round to the nearest cent if applicable. (6 questions)
# Mathematics for Chemistry

This book was initially derived from a set of notes used in a university chemistry course. It is hoped it will evolve into something useful and develop a set of open access problems as well as pedagogical material.
For many universities the days when admission to a Chemistry, Chemical Engineering, Materials Science or even Physics course could require the equivalent of A-levels in Chemistry, Physics and Mathematics are probably over for ever. The broadening out of school curricula has had several effects, including student entry with a more diverse educational background and has also resulted in the subject areas Chemistry, Physics and Mathematics becoming disjoint so that there is no co-requisite material between them. This means that, for instance, physics cannot have any advanced, or even any very significant mathematics in it. This is to allow the subject to be studied without any of the maths which might be first studied by the A-level maths group at the ages of 17 and 18. Thus physics at school has become considerably more descriptive and visual than it was 20 years ago. The same applies to a lesser extent to chemistry.
This means there must be an essentially remedial component of university chemistry to teach just the Mathematics and Physics which is needed and not too much, if any more, as it is time consuming and perhaps not what the student of Chemistry is most focused on. There is therefore also a need for a book Physics for Chemistry.

## Quantitative methods in chemistry

There are several reasons why numerical (quantitative) methods are useful in chemistry:
* Chemists need numerical information concerning reactions, such as how much of a substance is consumed, how long does this take, how likely is the reaction to take place.
* Chemists work with a variety of different units, with wildly different ranges, which one must be able to use and convert with ease.
* Measurements taken during experiments are not perfect, so evaluation and combination of errors is required.
* Predictions are often desired, and relationships represented as equations are manipulated and evaluated in order to obtain information.

## Table of contents

* Introduction (Dec 15, 2008)

### Numbers

* Number theory (Dec 15, 2008)
* Functions (Dec 15, 2008)
* Units and dimensions (Dec 15, 2008)
* Statistics (Dec 15, 2008)
* Plotting graphs (Dec 15, 2008)
* Complex Numbers (Dec 15, 2008)

### Algebra, trigonometry

* Trigonometry (Dec 15, 2008)
* Vectors (Dec 15, 2008)
* Matrices and Determinants (Dec 15, 2008)

### Calculus

* Differentiation (Dec 15, 2008)
* Integration (Dec 15, 2008)
* Some Useful Aspects of Calculus (Dec 15, 2008)
* Enzyme kinetics (Dec 15, 2008)

### Tests and examples

* Some Mathematical Examples applied to Chemistry (Dec 15, 2008)
* Tests and Exams (Dec 15, 2008)

### Further reading

* Further reading (Dec 15, 2008)
# Mathematics for Chemistry/Functions
## Functions as tools in chemistry
### The quadratic formula

In order to find the solutions to the general form of a quadratic equation,
$ax^{2}+bx+c=0$
there is a formula
$x={\frac {-b\pm {\sqrt {(b^{2}-4ac)}}}{2a}}$
(Notice the line over the square root has the same priority as a bracket. Of course we all know by now that ${\sqrt {a+b}}$ is not equal to ${\sqrt {a}}+{\sqrt {b}}$ but errors of priority are among the most common algebra errors in practice).
There is a formula for a cubic equation but it is rather complicated and unlikely to be required for undergraduate-level study of chemistry. Cubic and higher equations occur often in chemistry, but if they do not factorise they are usually solved by computer.
Solve:
$2x^{2}-14x+9$
$1.56(x^{2}+3.67x+0.014)$
Notice the scope or range of the bracket.
$2x^{2}-4x+2$
$-45.1(1.2[A]^{2}-57.9[A]+4.193)$
Notice here that the variable is a concentration, not the ubiquitous $x$.
# Mathematics Worksheet/Social Arithmetic

This list contains a material related to the Social Arithmatics module.
* Discount (Jun 28, 2009)
* Loss and Profit (Jun 28, 2009)
* Tax (Jun 28, 2009)
* Bank interests (Jun 28, 2009)
# Mathematics Worksheet/Venn diagram

Draw a Venn diagram showing:
# Pinyin/Cairo

Search Cairo
# Pinyin/Calm

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# Pinyin/Calorie

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# Pinyin/Camp

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# Pinyin/Call

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# Pinyin/Called

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# Pinyin/Calcium

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# Pinyin/Camera

A camera is a piece of equipment used for making television programs, movies, or videos.
shehxiahngji shih zhihzuoh diahnshih jiermuh, diahnyiing, huoh shihpirn d shehbeih.
# Object Oriented Programming/Superclasses

Superclasses are the parents of subclasses. In relationships between two objects, a superclass is the name given to the class that passes down it's 'traits'(language entities.) An example of a superclass would be an object like 'car' which would hold information like, make, model, mileage etc. An example of a Superclasses's format can be found below.

## Virtual methods

From the wikipedia page on inheritence: "If the superclass method is a virtual method, then invocations of the superclass method will be dynamically dispatched. Some languages require that methods be specifically declared as virtual (e.g. C++), and in others, all methods are virtual (e.g. Java). An invocation of a non-virtual method will always be statically dispatched (i.e. the address of the function call is determined at compile-time). Static dispatch is faster than dynamic dispatch and allows optimisations such as inline expansion."

## References

* ↑ https://en.wikipedia.org/wiki/Inheritance_(object-oriented_programming)
* ↑ https://www.thoughtco.com/what-is-inheritance-2034264
* ↑ https://en.wikipedia.org/wiki/Inheritance_(object-oriented_programming)
# Object Oriented Programming/Subclasses

Subclasses, also referred to as derived classes, heir classes, or child classes, are classes that inherit one or more language entities from another class/classes. While their may be some minor differences in how class inheritance works between different languages, generally, the subclass automatically inherits the instance variables and member functions of it's superclasses or parent classes. From the wikipedia page on inheritance, you can see how a subclass is generally defined.
The colon at the end of 'Subclass' means that it inherits from the "SuperClass" at the end.

## Non-subclassable classes

Some languages have non-subclassable classes. These are classes that have no subclasses. Non-subclassable classes can be created by adding certain class modifiers to the class declaration. After being created the non-subclassable classes will restrict reusability of code, especially when source code is not accessiable. A few examples of class modifiers that creat Non-subclassable classes include Java's 'final' keyword or C#'s 'sealed' keyword. These modifiers are added to the class declaration before the class keyword and identifier declaration.
Since a non-subclassable class has no subclasses, references or pointers to objects of that class can be easily deduced to be referencing instances of that class and not instances of subclasses or superclasses since a non-subclassable class has no subclasses, and an instance to a superclass would be violating the type system. Early binding(static dispatch) can also be used instead of late binding(dynamic dispatch) due to the exact type of the object referenced being known before execution.

## References

* ↑ https://en.wikipedia.org/wiki/Inheritance_(object-oriented_programming)
* ↑ Herbert Schildt (2003). The complete reference C++. Tata McGraw Hill Education Private Limited. p. 417. ISBN 978-0-07-053246-5.
# Object Oriented Programming/State is evil
## "State" is Evil!

Something that we don't see reiterated enough: State (as opposed to change) is evil! Or, (perhaps better said) maintaining unnecessary state is the root of all (er... many) bugs. Let's look at an example, now in Ruby:
* We're displaying something over time, so we're dealing
* with both pixels and seconds. Based on the zoom level,
* there is a correlation between them: pixels per second,
* or PPS.
Class Timeline
} In this example, we're maintaining gratuitous state — which is to say, we're holding the position in both seconds AND pixels. This is convenient, and important for users of this class, but we've messed up the encapsulation here. Whenever you set the value in one unit, you've destroyed the correctness for the other unit. Okay, you say, here's a simple fix:
* We're displaying something over time, so we're dealing
* with both pixels and seconds. Based on the zoom level,
* there is a correlation between them: pixels per second,
* or PPS.
Class Timeline {
public:
} This fixes the obvious error with the previous example, but you've created a lot of work for yourself. Now you have to keep every calculation in every method in the Timeline class duplicated to maintain consistency. There could be dozens more where this came from (trust me). And what about when you add in other units, say inches? You have to duplicate all of that again. Presumably, you see where this is heading: calculated properties, logical properties, virtual properties, whatever you like to call them. Let's see this example again:
Class Timeline {
public:
} Now, we're presuming that the conversion factors of PPS and IPS are set correctly for simplicity, but otherwise we've vastly simplified our problem. In twenty other functions we now write, we only have to worry about seconds, there's no consistency problems to worry about. Additionally, if we need to change our base units from seconds to pixels, the Timeline users need never know.
# Static vs Dynamic

In Object-Oriented-Programming languages, exists static and dynamic methods as well as properties. Here are the key differences.

## Static

* Cannot access dynamic methods of its own or other class's
* Cannot access dynamic properties of its own or of another class's
* Are only instantiated once since they are static and separate instances will always be identical

## Dynamic

* Can access dynamic methods of its own or other class's
* Can access dynamic properties of its own or of another class's
* Can be instantiated multiple times, each time with its own unique instance working with potentially unique data

## Example

In the code above, you can see that the method squared takes self as a parameter. This allows it to access other properties and methods of the class. As you can see, it takes an initial passed in value, does some modification to it, and then sets the class's value property to the result of the calculation. Since it can access other class methods/properties and modify the class's state, it is considered a dynamic method.
On the other hand, you can see that multiply_value does NOT have self as a parameter. All that this method does is multiply the two passed in values together and return the result. It never has or requires access to its class's properties or methods, it just does the calculation. Since it doesn't need to access unique data of its class, it is only loaded once as any other instance would be identical and a waste of memory, making this a static method.

## See also

* Wikipedia:Method(computer programming)
* Python Programming/Classes