OpenLLM-France/wikimedia huggingface dataset:
# High School Calculus/Infinite Limits

The limit as $x$ approaches gives you the horizontal asymptote.
    ${\begin{aligned}&\lim _{x\to -\infty }x\\&\lim _{x\to \infty }x=L\end{aligned}}$
You can use L'Hopital rule to help you with these limits
    ${\begin{aligned}&f(x)=\lim _{x\to \infty }{\frac {2x}{2x^{2}}}\\&f(x)={\frac {\infty }{\infty }}\end{aligned}}$
This is an indeterminate form so you can use L'Hopital's rule.
To do this you need to take ${\frac {f'(x)}{g'(x)}}$ . From this you get
    $\lim _{x\to \infty }{\frac {2}{4x}}$
This gives you
    ${\frac {2}{\infty }}$
This is no longer in indeterminate form so, we get
    $L=0$
# High School Calculus/Integration by Substitution
### Integration by Substitution

There is a theorem that will help you with substitution for integration. It is called Change of Variables for Definite Integrals.
what the theorem looks like is this
$\int _{a}^{b}f(x)\operatorname {d} x=\int _{\alpha }^{\beta }f(g(u))g\prime (u)\operatorname {d} u$
In order to get $\alpha$ you must plug a into the function g and to get $\beta$ you must plug b into the function g.
The tricky part is trying to identify what you want to make your u to be. Some times substitution will not be enough and you will have to use the rules for integration by parts. That will be covered in a different section
Ex. 1
$\int _{0}^{2}x(x^{2}+1)^{2}\operatorname {d} x$
Instead of making this a big polynomial we will just use the substitution method.
Step 1
Identify your u
Let $u=x^{2}+1$
Step 2
Identify $\operatorname {d} u$
$\operatorname {d} u=2x\operatorname {d} x$
Step 3
Now we plug in our limits of integration to our u to find our new limits of integration
When $x=0,u=0^{2}+1=1$
and when $x=2,u=2^{2}+1=5$
Now our integration problem looks something like this
${\frac {1}{2}}\int _{0}^{5}(x^{2}+1)^{2}(2x)\operatorname {d} x$
Step 4
write your new integration problem
When we plug in our u it looks like
${\frac {1}{2}}\int _{0}^{5}(u)^{2}\operatorname {d} u$
Step 5
Evaluate the Integral
${\frac {1}{2}}\left[{\frac {1}{3}}u^{3}\right]_{0}^{5}$
${\frac {1}{2}}\left[\left({\frac {1}{3}}*5^{3}\right)-\left({\frac {1}{3}}*0^{3}\right)\right]$
${\frac {1}{2}}\left[{\frac {1}{3}}*125\right]$
${\frac {1}{2}}\left[{\frac {125}{3}}\right]$
${\frac {125}{6}}$
As you can see this all simplified fairly nice. Using substitution will be hard, for most people, at first. Once you get the hang of doing this it should come to you faster and faster each time.
I'll give you some other problems to work on as well.
Ex. 2
$\int _{0}^{\frac {\pi }{2}}\sin(x)\cos(x)\operatorname {d} x$
Ex. 3
$\int _{-1}^{2}{\sqrt {x^{2}+4}}(2x)\operatorname {d} x$

### Solutions

Ex. 2
Let $u=\sin(x)$
Then $\operatorname {d} u=\cos(x)\operatorname {d} x$
When x = 0 $\sin(0)=0$ and when $x={\frac {\pi }{2}}$ $\sin \left({\frac {\pi }{2}}\right)=1$
Therefore,
$\int _{0}^{1}u\operatorname {d} u$
$\left[{\frac {1}{2}}u^{2}\right]_{0}^{1}$
$\left[{\frac {1}{2}}\sin ^{2}(1)\right]-\left[{\frac {1}{2}}\sin ^{2}(0)\right]$
$={\frac {1}{2}}\sin ^{2}(1)$
Ex. 3
Let $u=x^{2}+4$
Then $\operatorname {d} u=2x$
plug in our limits to get new limits
When x = -1 $(-1)^{2}+4=5$ and when x = 2 $2^{2}+4=8$
Our new integration problem is
$\int _{5}^{8}(u)^{\frac {1}{2}}\operatorname {d} u$
Giving us
$\left[{\frac {2}{3}}u^{\frac {3}{2}}\right]_{5}^{8}$
$=\left[{\frac {2}{3}}*(x^{2}+4)\right]_{5}^{8}$
$=\left[{\frac {2}{3}}*(8^{2}+4)\right]-\left[{\frac {2}{3}}*(5^{2}+4)\right]$
$=\left[{\frac {136}{3}}\right]-\left[{\frac {58}{3}}\right]$
$=26$
# High School Calculus/L'Hôpital's Rule
### L'Hôpital's Rule

This is a really fun rule to deal with. As explained in a different section the way you have learned to deal with fractional limits is by factoring. L'Hôpital's Rule is a much easier way to deal with these fractional limit functions.
When you use this rule you must first prove that you get an indeterminate form when evaluating the limit.
Indeterminate form can come in a variety of different ways.
Indeterminate forms: ${\frac {0}{0}}$, ${\frac {\infty }{\infty }}$, $0*\infty$, ${\frac {-\infty }{-\infty }}$, $\infty -\infty$, ${\frac {-\infty }{\infty }}$, ${\frac {\infty }{-\infty }}$, $0^{0}$, $1^{\infty }$, $\infty ^{0}$
There are a lot of them, but the more you work with them and become familiar with these indeterminate forms they will be committed to memory.
Ex: 1
Show that the limit is in an indeterminate form
$\lim _{x\to 0}{\frac {x^{2}}{x}}$
The next step to L'Hôpital's Rule is to look at the fraction as ${\frac {f(x)}{g(x)}}$
As long as $g(x)\neq 0$ you can use L'Hôpital's Rule.
The formula for L'Hôpital's Rule is $\lim _{x\to c}{\frac {f(x)}{g(x)}}$ as long as f and g are differentiable on the open interval (a,b)
Ex: 2
Determine if you can use L'Hôpital's Rule. If you can then evaluate the limit using L'Hôpital's Rule.
$\lim _{x\to 0}{\frac {3x^{3}-x}{x-5}}$
You will notice when trying to evaluate some limits using L'Hôpital's Rule that the function just switches back and forth. When this happens you will probably have multiply a variable into the function, as long as it equals one.
Ex: 3
Determine if you can use L'Hôpital's Rule. If you can then evaluate the limit using L'Hôpital's Rule
$\lim _{x\to \infty }{\frac {\sqrt {x}}{\sqrt {x+1}}}$
Solution to Ex: 3
$\lim _{x\to \infty }{\frac {\sqrt {x}}{\sqrt {x+1}}}={\frac {\infty }{\infty }}$
$\lim _{x\to \infty }{\frac {\frac {1}{2*{\sqrt {x}}}}{\frac {1}{2*{\sqrt {x+1}}}}}=\lim _{x\to \infty }{\frac {\sqrt {x+1}}{\sqrt {x}}}$
$\lim _{x\to \infty }{\frac {\frac {1}{2*{\sqrt {x+1}}}}{\frac {1}{2*{\sqrt {x}}}}}=\lim _{x\to \infty }{\frac {\sqrt {x}}{\sqrt {x+1}}}$
As you can see we are getting no where fast. Something else needs to be done
$\lim _{x\to \infty }{\frac {\sqrt {x}}{\sqrt {x+1}}}*{\frac {\frac {1}{x^{2}}}{\frac {1}{x^{2}}}}$
$\lim _{x\to \infty }{\frac {\sqrt {\frac {x}{x}}}{\sqrt {{\frac {x}{x}}+{\frac {1}{x}}}}}$
$\lim _{x\to \infty }{\frac {1}{\sqrt {1+{\frac {1}{x}}}}}$
${\frac {1}{\sqrt {1+{\frac {1}{\infty }}}}}$
${\frac {1}{\sqrt {1+0}}}=1$
$L=1$
# High School Calculus/Infinite Series

* Sequences
* Infinite Series Explored
* Series Without Negative Terms
* Series With Odd or Even Negative Terms
* Ratio Test, Root Test, and Summary of Tests
* Power Series
* Taylor and Maclaurin Series
* Calculations with Series
# High School Calculus/Integration

* Indefinite Integrals Calculus
* The Initial Value Problem
* The Area Problem
* Definite Integrals
* Evaluating Definite Integrals
* The Fundamental Theorems of Calculus
* Integration by Substitution
* Numerical Integration
# High School Calculus/Integration by Partial Fractions
### Integration by Partial Fractions

There are a couple different methods of integrating partial fractions. The one I will be showing you is the one I am most comfortable with.
The important thing is to realize how to split up the denominator into two or more fractions. The numerator will have place holders like A, B, C, and so forth. You may have a Ax + B or some variation thereof depending on how the denominator needs to be split.
Because I think this will be better explained in an example, I'll give you one.
Ex. 1
$\int {\frac {1}{x^{2}-4}}\operatorname {d} x$
By doing some factoring we can see that $x^{2}-4$ also equals $(x+2)(x-2)$
I'm going to leave off the integral sign right now so you don't get confused and so you can see what I'm doing with the fraction
${\frac {A}{x+2}}+{\frac {B}{x-2}}$
In order to get the numerator we need to multiply by the LCD
$1=A(x-2)+B(x+2)$
we set it equal to 1 because 1 is the number in the original numerator
Now we can choose to solve for A or B first. In order to do this we must find when one of them equals 0
when x = 2 $1=A(0)+B(4)$ $B={\frac {1}{4}}$
Now let's find A
When x = -2 $1=A(-4)+B(0)$ $A=-{\frac {1}{4}}$
This means that ${\frac {\left(-{\frac {1}{4}}\right)}{x-2}}+{\frac {\left({\frac {1}{4}}\right)}{x+2}}$
Now that we know what the partial fractions are we can now integrate
$\int {\frac {1}{4(x+2)}}\operatorname {d} x-\int {\frac {1}{4(x-2)}}\operatorname {d} x$
$={\frac {\ln(\left\vert {x+2}\right\vert )}{4}}-{\frac {\ln(\left\vert x-2\right\vert )}{4}}$
$={\frac {1}{4}}\ln \left({\frac {\left\vert x+2\right\vert }{\left\vert x-2\right\vert }}\right)$
Other problem to work on for practice
${\frac {x^{2}+3x+2}{(x-2)^{2}(x+1)}}$
# High School Calculus/Integration Techniques

* Integration by Substitution
* Integration by Parts
* Integration by Partial Fractions
* Trigonometric Integrals
* Trigonometric Substitutions
* Improper Integrals
* Ordinary Differential Equations
# High School Calculus/Integration by Parts

i would like to make life easy for the junior mathematicians in integration by parts.There is a bit trouble on the choice of the variable "U" in integration by parts,because some students tend to vice verse their U and V when using integration by parts,how ever i will like to tell a certain story which reads as follows;
there was a certain discussion which was made between the zimbabwean maths pupils and the egyptian maths pupils some time long back.When this discussion went on and on conflict arose between the two groups and the zimbabwean pupils analysed the conflict from bottom to top very well ,and they realised that but it seems as if the "EGYPTIAN PUPILS ARE LAZY TO THINK",
THE POINT IS: WHEN YOU ARE A GIVEN A FUNCTION TO INTEGRATE BY PARTS (PRODUCT OF CAUSE) WE WIL USE ;
L-----AZY are P-----UPILS E-----GYIPTIAN;
WE READ THIS FROM BOTTOM TO TOP AS EGYPIAN PUPILS are LAZY
WHERE E----- EXPONENTIAL; L-----LOGARITHM; P-----POLYNOMIAL; . SO WHEN EVER WE ARE GIVEN A FUNCTION TO INTEGRATE BY PARTS WE FIRST CHECK WHETHER THER IS A LOGARITHM EG (lnx),IF IT IS THERE IT WILL BE OUR "U",IF NOT WE MOVE ON TO CHECK FOR THE POLYNOMIAL FUNCTION (X,2X,X^2,X^3 ETC), IF THERE THEN THAT WILL BE OUR U.IF NO WE CHOSE EXPONENTIAL (e^x) ,AS OUR U.IN THIS ORDER. IT CAN BE SEEN THAT HERE TRIG FUNCTION WILL NEVER BE U, THIS CAN BE TRIED BY EXAMPLES AND DEFINITELY WE ARE NOW GOING TO WIN OUR INTEGRATION BY PARTS ,JUST AS EASY AS IT IS NOW

### Other Problems to Work On

Ex. 2
$\int \cos(x)x^{2}\operatorname {d} x$
Ex. 3
$\int \operatorname {arcsec}(x)\left({\frac {1}{x}}\right)\operatorname {d} x$
# History of wireless telegraphy and broadcasting in Australia/Topical/Publications/Australasian Radio World/Issues/1947 06
## P.01 - Front Cover

The Australasian Radio World
Registered at the G.P.O., Sydney, for transmission by post as a periodical.
1/-
Vol. 12 - No. 1; JUNE 15, 1947

## P.03 - Contents Banner

THE AUSTRALASIAN RADIO WORLD
Devoted entirely to Technical Radio
and incorporating
ALL-WAVE ALL-WORLD DX NEWS
VOL. 12 - JUNE, 1947. - No. 1

## P.03 - Publication Notes

Editor, Publisher, Proprietor — A. G. HULL, Balcombe St., Mornington, Vic.
Short-wave Editor — L. J. KEAST, 6 Fitzgerald Road, Ermington, N.S.W. 'Phone: WL1101
Ham Notes By — D. B. KNOCK (VK2NO), 43 Yanko Av., Waverley, N.S.W.
Advertising Representative for Vic. — W. J. LEWIS, 20 Queen St., Melbourne, 'Phone MU5154
Advertising Representative for N.S.W. - Amalgamated Publications Pty. Ltd., 83 Pitt St., Sydney, Phone: B1077
Subscription Rates: 6 issues - 5/3; 12 issues - 10/6; 24 issues - £1; Post free to any address in the world.
Address for all correspondence - Australasian Radio World, Balcombe St., Mornington, Victoria
Printed by Bridge Printery Pty. Ltd., 117 Reservoir Street, Sydney, N.S.W., for the proprietor of the "Australasian Radio World," Balcombe St., Mornington, Vic. (Footnote P.44)

## P.03 - Contents

CONTENTS
TECHNICAL —
The "Six-Ten" . . . . 5
Unusual Condensers . . . . 13
Can You Answer These? . . . . 15
Car Radio . . . . 17
The "Voyager" . . . . 19
Teleconda 3 . . . . 27
Pepping up No. 4 Set . . . . 31
SHORTWAVE REVIEW —
Notes From My Diary . . . . 38
THE SERVICE PAGES —
Answers . . . . 42

## P.03 - Editorial

EDITORIAL
In last month's issue there was an article on the problems of trade discounts. It is evident that the article has been misunderstood by a number of people. After glancing over my mail I have decided not to publish the second article of the series until I have been able to expand one or two points so that there can be no chance that they will be misunderstood. What I did not say, but what quite a few accuse me of having said, is that I am in favour of the public being able to buy at trade prices. I am all in favour of the recognised proper trading methods, but I do claim that many radio components are of such a nature that they need no retail price or a long discount. No ordinary man-in-the-street is likely to buy a signal generator, but neither is he likely to buy intermediate frequency transformers. Both require specialised knowledge for their proper application. Another point where several of my correspondents have gone off the track is in regard to so-called "trade" publications. Just because "Australasian Radio World" is sold over counters of bookshops as well as by direct subscription, it does not mean that it is read by those in the trade. Conversely it does not follow that the trade prices openly advertised in "trade" magazines cannot be read by "back-yarders". The wide-awake back-yarder subscribes to trade papers. So why should the trade association bring pressure to bear to stop wholesalers from advertising in the technical press? A. G. HULL.
# ID Board Review/Bacterial Diseases/Respiratory Infections

The ABIM ID blueprint does not elaborate on which specific Bacterial Respiratory Infections will be tested. As such, please understand that this page is filled with topics that the Wikibook author(s) believe are likely to be high-yield based on their knowledge, and not strictly based on the ABIM blueprint.

## Respiratory Infections

* Empyema
* Lower Respiratory Tract Infection (LRTI)
** Bacterial Pneumonia
** Bronchitis (mostly viral)
** Bronchiectasis
* Lung Abscess
* Upper Respiratory Tract Infection (URTI)
# ID Board Review/Bacterial Diseases/Rickettsia
## Rickettsia Species

* Rickettsia conorii
* Rickettsia akari
* Rickettsia rickettsii
* Rickettsia prowazekii
* Rickettsia typhi
* Rickettsia parkeri
* Rickettsia africae
* Coxiella burnetii
* Orientia tutsugamushi
# ID Board Review/Bacterial Diseases/Neurologic Infections

The ABIM ID blueprint does not elaborate on which specific Bacterial Neurologic Infections will be tested. As such, please understand that this page is filled with topics that the Wikibook author(s) believe are likely to be high-yield based on their knowledge, and not strictly based on the ABIM blueprint.

## Neurologic Infections

* Brain Abscess
* Meningitis
# ID Board Review/Bacterial Diseases/Ophthalmologic Infections

The ABIM ID blueprint does not elaborate on which specific Bacterial Ophthalmologic Infections will be tested. As such, please understand that this page is filled with topics that the Wikibook author(s) believe are likely to be high-yield based on their knowledge, and not strictly based on the ABIM blueprint.

## Ophthalmologic Infections

* Keratitis
* Red Eye
# LMIs in Control/Click here to continue/Optimal control systems/Discrete-time Hinf-Optimal Full State Feedback Control

Discrete-Time H∞-Optimal Full-State Feedback Control
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.
A full state feedback controller design for a Discrete Time system, to minimize the H∞ norm of the closed loop system with exogenous input $w_{k}$ and performance output $z_{k}$.

## The System

Discrete-Time LTI System with state space realization $(A_{d},B_{d},C_{d},D_{d})$ ${\begin{aligned}x_{k+1}&=A_{d}x_{k}+B_{d1}w_{k}+B_{d2}u_{k}\\z_{k}&=C_{d1}x_{k}+D_{d12}u_{k}\\y_{k}&=x_{k}\\\end{aligned}}$

## The Data

The matrices: System $(A_{d},B_{d1},B_{d2},C_{d1},D_{d12}),P,F_{d}$.

## The Optimization Problem

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

## The LMI:

Discrete-Time H∞-Optimal Full-State Feedback Control
The LMI formulation
H∞ norm < $\gamma$
${\begin{aligned}P\in {S^{n_{x}}};&F_{d}\in {R^{n_{u}*n_{x}}};\gamma \in {R_{>0}}\;\\&P>0\\{\begin{bmatrix}P&A_{d}P-B_{d2}F_{d}&B_{d1}&0\\*&P&0&PC_{d1}^{T}-F_{d}^{T}D_{d12}^{T}\\*&*&-\gamma I&D_{d11}^{T}\\*&*&*&-\gamma I\end{bmatrix}}&>0,\\\end{aligned}}$

## Conclusion:

The H∞-optimal full-state feedback gain is recovered by $K_{d}=F_{d}P^{-1}$

## Implementation

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

## Related LMIs

- Continuous Time H∞ Optimal Full State Feedback Control

## 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/Optimal control systems/Discrete time H2 optimal filter
## Introduction

The goal of optimal filtering is to design a filter that acts on the output z of the generalized plant and optimizes the transfer matrix from w to the filtered output. An H2-optimal filter is designed to minimize the $H_{2}$ norm of ${\overline {P}}(s)$(will be defined below).

## System Dynamics

Consider the discrete-time generalized LTI plant with minimal states space realization
${\begin{aligned}&\quad \quad {\begin{cases}X_{k+1}=A_{d}X_{k}+B_{d1}w_{k},\\Z_{k}=C_{d1}X_{k}+D_{d11}w_{k},\\Y_{k}=C_{d2}X_{k}+D_{d21}w_{k},\\\end{cases}}\end{aligned}}$
where it is assumed that A_{d} is Schur. A discrete-time dynamics LTI filter with state-space realization
${\begin{aligned}&\quad \quad {\begin{cases}X_{f,k+1}=A_{f}X_{f,k}+B_{f}Y_{k},\\{\hat {Z}}_{k}=C_{f}X_{f,k}+D_{f}y_{k},\\\end{cases}}\end{aligned}}$
is to be designed to optimize the transfer function from w{k} to ${\overline {z}}_{k}=z_{k}-{\hat {z}}_{k}$, given by ${\overline {P}}(z)={\overline {C}}(d1)(zI-{\overline {A}}_{d})^{-1}{\overline {B}}_{d1}+{\overline {D}}_{d11}$,
where
${\overline {A}}_{d}={\begin{bmatrix}A_{d}&0\\B_{f}C_{d2}&A_{f}\end{bmatrix}},{\overline {B}}_{d1}={\begin{bmatrix}B_{d1}\\B_{f}D_{d21}\end{bmatrix}},{\overline {C}}_{d1}={\begin{bmatrix}C_{d1}-D_{f}C_{d2}-C_{f}\\\end{bmatrix}},{\overline {d}}_{d11}=D_{d11}-D_{f}D_{d21}.$

## The Optimization Problem

Solve for $A_{n}\in \mathbb {R} ^{n_{x}Xn_{x}},B_{n}\in \mathbb {R} ^{n_{x}Xn_{y}},C_{f}\in \mathbb {R} ^{n_{z}Xn_{x}},D_{f}\in \mathbb {R} ^{n_{z}Xn_{y}},X,Y\in \mathbb {S} ^{n_{z}},and$ $v\in \mathbb {R} >0$ that minimize $J(v)=v$ subject to $X>0,Y>0,Z>0.$

## LMI

${\begin{bmatrix}YA+A^{T}Y+B_{n}C_{2}+C_{2}^{T}B_{n}^{T}&A_{n}+C_{2}^{T}B_{n}^{T}+A^{T}X&YB_{1}+B_{n}D_{21}\\*&A_{n}+A_{n}^{T}&XB_{1}+B_{n}D_{21}\\*&*&-I\end{bmatrix}}$ < 0 ,
${\begin{bmatrix}-Z&C_{1}-D_{f}C_{2}&-C_{f}\\*&-Y&-X\\*&*&-X\end{bmatrix}}$ < 0 ,
$D_{11}-D_{f}D_{21}=0,$
$Y-X>0,$
$tr(Z)<v.$

## Conclusion

The filter is recovered by the state-space matrices $A_{f}=X^{-1}A_{n},Bf=X^{-1}B_{n},C_{f},andD_{f}.$
# LMIs in Control/Click here to continue/Optimal control systems/H2-Optimal Observer

State observer is a system that provides estimates of internal states of a given real system, from measurements of the inputs and outputs of the real system.The goal of $H_{2}$ -optimal state estimation is to design an observer that minimizes the $H_{2}$ norm of the closed-loop transfer matrix from w to z. Kalman filter is a form of Optimal Observer.

## The System

Consider the continuous-time generalized plant $P$ with state-space realization
    ${\begin{aligned}{\dot {x}}&=Ax+B_{1}w(t),\\y&=C_{2}x+D_{21}w\\\end{aligned}}$

## The Data

The matrices needed as input are $A,B,C,D$.

## The Optimization Problem

The task is to design an observer of the following form:
    ${\begin{aligned}{\dot {\hat {x}}}=A{\hat {x}}+L(y-{\hat {y}}),\\{\hat {y}}=C_{2}{\hat {x}}\\\end{aligned}}$

## The LMI: $H_{2}$ Optimal Observer

LMIs in the variables $P,G,Z,\nu$ are given by:
       ${\begin{aligned}{\begin{bmatrix}PA+A^{T}P-GC_{2}-{C_{2}}^{T}G^{T}&&PB_{1}-GD_{21}\\\star &&-1\end{bmatrix}}<0\\trZ<\nu \end{aligned}}$

## Conclusion:

The $H_{2}$ -optimal observer gain is recovered by $L=P^{-1}G$ and the $H_{2}$ norm of T(s) is $\mu ={\sqrt {\nu }}$

## Implementation

https://github.com/Ricky-10/coding107/blob/master/H2%20Optimal%20Observer

# 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.
* https://onlinelibrary.wiley.com/doi/abs/10.1002/rnc.1310
# LMIs in Control/Click here to continue/Optimal control systems/H2-optimal Control with Regional Pole Assignment via State Feedback
## The System:

Consider a linear time-invariant continuous system in the form of
# LMIs in Control/Click here to continue/Optimal control systems/H2-optimal Full-state Feedback control
## Full State Feedback Optimal $H_{2}$ Control

Full State Feedback in general has the goal of positioning a system's closed loop poles in a desired location. This allows us to specify the performance of the system such as requiring stability or bounding the overshoot of the output. By minimizing the $H_{2}$ norm of this system we are minimizing the effect noise has on the system as part of the performance specifications, particularly when there is information about the distribution of the noise.

## The System

The system is represented using the 9-matrix notation shown below.
       ${\begin{bmatrix}{\dot {x}}\\z\\y\end{bmatrix}}={\begin{bmatrix}A&B_{1}&B_{2}\\C_{1}&D_{11}&D_{12}\\C_{2}&D_{21}&D_{22}\end{bmatrix}}{\begin{bmatrix}x\\w\\u\end{bmatrix}}$
where $x(t)\in \mathbb {R} ^{n}$ is the state, $z(t)\in \mathbb {R} ^{p}$ is the regulated output, $y(t)\in \mathbb {R} ^{q}$ is the sensed output, $w(t)\in \mathbb {R} ^{r}$ is the exogenous input, and $u(t)\in \mathbb {R} ^{m}$ is the actuator input, at any $t\in \mathbb {R}$.

## The Data

$A$, $B_{1}$, $B_{2}$, $C_{1}$, $C_{2}$, $D_{11}$, $D_{12}$, $D_{21}$, $D_{22}$ are known.

## The LMI: Optimal Output Feedback $H_{\infty }$ Control LMI

The following are equivalent.
1) There exists a $K$ such that $||S(K,P)||_{H_{2}}<\gamma$
2) There exists $X>0$, $Z$ and $W$ such that
       ${\begin{bmatrix}A&B_{2}\end{bmatrix}}{\begin{bmatrix}X\\Z\end{bmatrix}}+{\begin{bmatrix}X&Z^{T}\end{bmatrix}}{\begin{bmatrix}A^{T}\\B_{2}^{T}\end{bmatrix}}+B_{1}B_{1}^{T}<0$
       ${\begin{bmatrix}X&*^{T}\\C_{1}X+D_{12}Z&W\end{bmatrix}}>0$
       $trace(W)<\gamma ^{2}$
where $K=ZX^{-1}$

## Conclusion:

This LMI solves the $H_{2}$ optimal full state feedback problem and finds the upper bound of the $H_{2}$ norm of the system, $\gamma$. In addition to this the controller $K$ is also found in the process.

## Implementation

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/FSF_H2.m

## Related LMIs

Full State Feedback Optimal $H_{\infty }$ LMI

## 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 Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013.
* A Course in Robust Control Theory: a Convex Approach, - by Geir E. Dullerud and Fernando G. Paganini, Springer, 2011, Chapter 6.