OpenLLM-France/wikimedia huggingface dataset:
# SAP ERP/Business user

Business users of SAP® ERP use the system to perform daily operations, such as posting financial documents, creating vendors or customers, or displaying reports. They are not expected and not authorized to change the way how the system behaves. If they notice an error or request a new functionality, then they should contact an IT specialist within their organization through an appropriate channel.
Although the "standard" SAP ERP is the same everywhere, there is a great degree of flexibility in setting it up for the needs of each enterprise. An organization may choose to use or not to use certain functionality; screen fields can be made required, optional or hidden from view; in some cases, the business need may require that the normal functionality is modified through enhancements or custom-built transactions. Also, each company has different internal rules on how fields should be filled, what naming standard to follow or when certain transactions can or should be executed. Therefore, this guidebook can only give you a general overview, how SAP is structured and how it handles basic business scenarios. It does not replace any internal training that you got - or will get - in your job. However, you may use it as a basis for your discussion with an IT specialist, what functionality may be activated or changed.
As a business user, you will perform certain business scenarios with the help of SAP transactions. You may encounter informational, error or warning messages. You can customize your screens by setting up own defaults.
From a business user perspective, there are two types of data:
* Master data - relatively stable, entered once and used repeatedly for many transactions. Their quality is of high importance because an error is easily transferred on other data. Example: general ledger accounts, vendors, customers, materials, projects, rental units.
* Transactional data - relating to single business transactions, such as: accounting documents, goods receipts, purchase orders, vendor and customer invoices.
In relation to various master and transaction data, you will (or will not) be authorized to:
* Create - enter new vendors, customers, financial documents, etc. Creation is usually irreversible - to assure transparency, once an object is created, it usually cannot be physically deleted but only "blocked" or "marked for deletion" (there are technical steps to physically delete data, though, through "archiving" after exceeding a certain holding period).
* Change - modify existing vendors, customers, documents etc. Rules set up by standard SAP and by IT specialists in your company determine, which fields in which objects can and which cannot be changed. Most of the time, each change creates a log that shows previous value, new value, date, time and user who made it.
* Display - view the data without the ability to modify it
* Display change - view log of previous changes
* Execute reports - getting an overview of data from many objects (e.g., list of open vendor invoices, catalog of materials, usage of budget on investment projects etc.)
# SAP Event Management

SAP's standard, on-premise, solution for process and object track and trace functionality. Businesses today can operate in a highly distributed, outsourced and networked environment. Globalization is upon us which means that we have to be more intelligent with the way that we operate. We have to collaborate efficiently and effectively. We need to ensure that regulations are enforced during the entire process. How do we do that?
* We need to track and trace our products and processes across the supply chain measuring performance and proactively reacting to deviations to the plan in real time.
* We need complete visibility of our end-to-end processes across all the geographical and partner boundaries.
* We need the ability to quickly adapt to changing conditions in our process.
* We need to head off issues before they become an issue...

## Overview

* SAP Event Management (SAP EM) is an application within the component SAP SCM that belongs to the mySAP Supply Chain Management solution. SAP EM can be connected to other mySAP Business Suite components such as SAP R/3, SAP CRM, and SAP BW. It can also be connected to non SAP systems using BAPI or XML / XI calls. IDocs are also a popular way to get data to your SAP EM system.
* With the SAP Event Management application your company can monitor and manage events across your distributed processes involving partners, inventories and assets. It captures events from your system and your partners systems, analyzes them against it's predefined plan and alerts or workflows a response to the required people to react of deviations are found. It comes integrated with SAP NetWeaver BI and SAP Auto-Id Infrastructure (RFID enabled scenarios).

## Key Points

* Provides monitoring capability (visibility) to any business process
** Partners - See how your partners are performing and react to deviations to planned partners performing certain events in your process
** Locations - If your process is intended to have Event A end up at Location A but it arrives at location B then raise exception!!
** Measurements - You're transporting frozen goods from point A to point B and the temperature must not go above 10 deg. If it arrives at point B at 11 deg then raise exception!!!
* Tracks status changes to YOUR business processes
* Matches the planned process with the actual process (TO UNCOVER EXCEPTIONS)
* Real time notification of events
** Alert framework
** Workflow engine
* 2 Types of events:
** Those that you expect to happen (the plan)
** Those that actually happen (the actual)
*** Regular event => Events that occur within the planned period
*** Early / Late event => Events occurring outside the planned period
*** Unexpected event => Events that you never planned for
*** Unreported event => Event that you expected to occur but never did
* Manage the sequencing of your events
* Provides both operational and analytical reporting capabilities
** Operational => Web GUI application - Where are my issues now?
** Analytical => SAP NetWeaver BI integration - How well did I do and where are my improvement points?
# SAS

SAS
* SAS Basics
** Introduction
** Basics
** Settings
** Documentation
** Math Functions
** String Functions
** Data Management
** Graphics
** Programming
** IML
** Publication Quality Output
** Descriptive Statistics
* Methods
** Maximum Likelihood
** Method of Moments
* Topics
** Linear Models
** Binomial Outcome Models
** Multinomial Models
** Duration Analysis
** Factor Analysis
This book is intended to be a guide to SAS programming. SAS is a commercial statistical software. SAS help is known to be messy and it is often difficult to find relevant documentation on a specific issue in SAS programming. Note that the WPS software is another interpreter of the SAS language. There are lots of alternative to SAS. R is a free software and powerful statistical package. SPSS is easy to use. Stata is widely used in economics. SAS is especially adapted if you deal with very large datasets such as census data because it works directly on the data files without loading them in memory. Your contributions to this book are welcome.
There is also a Wikiversity course, Data Analysis using the SAS Language, which is organized by SAS functions and not by tasks.
# SAP ERP/Transactions

Transactions in SAP® ERP enable a user or an IT specialist to access a certain functionality. There are about 120 thousand of them. A transaction can be called by filling its code in a box on the top of the screen, or by double-clicking on its node in a menu.
When you log-on to SAP, then you normally enter the main menu (main screen). In a standard SAP, it corresponds to transaction S000. Each user may change the start-up transaction by entering it into his user defaults (trans. SU3), but it is usually practical only in certain industry solution (for example, SAP Retail proposes to start with W10T which is better suited to the needs of retail companies).
On the main screen, you can usually choose between showing your "user menu" (it only contains a limited range of transactions which you are authorized to) or "SAP menu" (showing all available SAP transactions organized into hierarchies) by clicking on an appropriate icon.
To display transaction codes on the main screen, go to Extras - Settings (on the horizontal menu) and then tick "Display technical names".
Due to the enormous quantity of SAP transactions, only a limited selection will be described in this handbook. If you know the code (see above how to obtain it), then you may search for it in an alphabetically sorted list under Category:Book:SAP ERP (note: the list contains all chapters of this book). Else, look for the appropriate Business scenario, IMG activity (for IT specialists) or Generally useful transactions.
# SAP ERP/IT specialist

IT specialists on SAP® ERP implementation projects or support assignments work with business users and with other IT specialists to build or maintain a solution that satisfies requirements and supports predefined business scenarios.
SAP specialists usually focus on certain areas - modules or their parts. These are:
* Logistics - contains Material Management (MM), Sales and Distribution (SD), Global Trade Management, Compensation Management and Agency Business
* Financials (Accounting) - contains Financial Accounting (FI), Controlling (CO), Enterprise Controlling (EC), Strategic Enterprise Management (SEM), Project System and Investment Management (PS-IM), Flexible Real Estate Management and Banking Applications
* Human Resources
In addition, each project and support group employs "cross-application" experts, e.g. in the area of:
* Authorizations
* ABAP™ programming
* ALE (automatic link enablement, design of interfaces)
* Basis (database administration)
* Project management
Due to the integrated nature of the software, it is an important task for any IT specialist to understand how his work impacts on that of others.
# LMIs in Control/Tools/Schur Complement

WIP, Description in progress
An important tool for proving many LMI theorems is the Schur Compliment. It is frequently used as a method of LMI linearization.

### The Schur Compliment

Consider the matricies $Q$, $M$, and $R$ where $Q$ and $M$ are self-adjoint. Then the following statements are equivalent:
      * $Q>0$ and $M-RQ^{-1}R^{*}>0$ both hold.
      * $M>0$ and $Q-R^{*}M^{-1}R>0$ both hold.
      * ${\begin{bmatrix}M&R\\R^{*}&Q\end{bmatrix}}>0$ is satisfied.
More concisely:
       ${\begin{bmatrix}M&R\\R^{*}&Q\end{bmatrix}}>0\iff {\begin{bmatrix}M&0\\0&Q-R^{*}M^{-1}R\end{bmatrix}}>0\iff {\begin{bmatrix}M-RQ^{-1}R^{*}&0\\0&Q\end{bmatrix}}>0$
WIP, additional references to be added

## 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/Apkarian Filter

LMIs in Control/pages/Apkarian Filter

## The System:

The number of LMI constraints needed to check quadratic stability is reduced if all the subsystems in the polytopic model has the same matrix $B$. This can be achieved by adding an Apkarian filter in the input of the system.

## The Optimization Problem:

Apkarian Filter
Let consider our TS-LIA model. This can be re written in linear form as:
       ${\dot {x}}=A(z(t))x+B(z(t))u$
The filter should be such that the equilibrium of the states are the input values and the dynamics should be fast, so we could assume the dynamics of the filter negligible (i.e. the input of the filter is equivalent to the input of the quadrotor). One possible filter is shown , where $A_{F}$ = −100$I_{4}$, $B_{F}$ = 100$I_{4}$ and $I_{4}$ ∈ $R^{4\times 4}$ is the identity matrix.
       ${\dot {x_{F}}}=A_{F}x_{F}+B_{F}u_{F};y_{f}=x_{f}$.
When applying the filter, we are imposing that the output of the filter is the new input of the TS-LIA model (i.e. $u$ = $y_{F}$ ). Then, the extended model is:
       ${\dot {x_{c}}}={\begin{bmatrix}A(z(t))&B(z(t))\\0&A_{F}\\\end{bmatrix}}x_{c}+{\begin{bmatrix}0\\B_{F}\end{bmatrix}}u_{F}=A_{e}(z(t))x_{e}+B_{e}u_{f};x_{e}={\begin{bmatrix}x\\x_{F}\end{bmatrix}}$
This prefiltering does not affect the procedure followed to obtain the TS-LIA model, so the premise variables, membership functions and activations functions remains the same.
State Feedback Controller Design
Let consider the state feedback control law for the extended TS-LIA model:${\dot {x_{e}}}=\sum _{i=1}^{32}h_{i}(z(t))[A_{ei}x_{e}+B_{ei}u_{F}]$, where the state feedback control laws are :$u_{F}=\sum _{i=1}^{32}h_{i}(z(t))K_{i}x(t)$, we get the closed loop system :${\dot {x_{e}}}=\sum _{i=1}^{32}\sum _{j=1}^{32}h_{j}(z(t))[A_{ei}x_{e}+B_{ei}K_{j}]x_{e}$

## The LMI:

The design of the controller is done by solving an LMI problem involving the quadratic stability constraints. In case we want D- stabilization, the following set of LMI constraints are needed:
       $L\otimes P+M\otimes P(A_{ei}+B_{e}K_{i})+MT\otimes (A_{e}i+B_{e}K_{i})^{T}P<0$ ∀i = 1, . . . , 32.

## Conclusion:

The LMI is feasible.

## References

* Control, A. (2016). Gain-scheduling Control of a Quadrotor Using the Takagi-Sugeno Approach.
# LMIs in Control/pages/Basic Matrix Theory
## Basic Matrix Notation

Consider the complex matrix $A\in \mathbb {C} ^{n\times m}$.
       $A={\begin{bmatrix}a_{11}&\dots &a_{1m}\\\vdots &\ddots &\vdots \\a_{n1}&\dots &a_{nm}\end{bmatrix}}\in \mathbb {C} ^{n\times m}$
Transpose of a Matrix
The transpose of $A$, denoted as $A^{T}$ or $A'$ is:
       $A^{T}={\begin{bmatrix}a_{11}&\dots &a_{n1}\\\vdots &\ddots &\vdots \\a_{1m}&\dots &a_{nm}\end{bmatrix}}\in \mathbb {C} ^{m\times n}.$
Adjoint of a Matrix
The adjoint or hermitian conjugate of $A$, denoted as $A^{*}$ is:
       $A^{*}={\begin{bmatrix}a_{11}^{*}&\dots &a_{n1}^{*}\\\vdots &\ddots &\vdots \\a_{1m}^{*}&\dots &a_{nm}^{*}\end{bmatrix}}\in \mathbb {C} ^{m\times n}.$
Where $a_{nm}^{*}$ is the complex conjugate of matrix element $a_{nm}$.
Notice that for a real matrix $A\in \mathbb {R} ^{n\times m}$, $A^{*}=A^{T}$.

## Important Properties of Matricies

Hermitian, Self-Adjoint, and Symmetric Matricies
A square matrix $A\in \mathbb {C} ^{n\times n}$ is called Hermitian or self-adjoint if $A=A^{*}$.
If $A\in \mathbb {R} ^{n\times n}$ is Hermitian then it is called symmetric.
Unitary Matricies
A square matrix $A\in \mathbb {C} ^{n\times n}$ is called unitary if $A^{*}=A^{-1}$ or $A^{*}A=I$.

## 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/Tools/Structured Singular Value

LMIs in Control/Tools/Structured Singular Value

## The System

    ${\begin{aligned}M{\text{ with transfer function }}{\hat {M}}(s)=C(sI-A)^{-1}B+D,&&{\hat {M}}\in H_{\infty }\end{aligned}}$

## The Data

The matrices $A,B,C,D$.

## The Optimization Problem

${\begin{aligned}{\text{There exists }}\Theta \in \Theta {\text{ such that }}||\Theta M\Theta ^{-1}||^{2}<\gamma .\end{aligned}}$

## The LMI: The Lyapunov Inequality

       ${\begin{aligned}{\text{Find}}\;&X>0:\\{\begin{bmatrix}A^{T}X+XA&XB\\B^{T}X&-\Theta \end{bmatrix}}+\gamma ^{-2}{\begin{bmatrix}C^{T}\\D^{T}\end{bmatrix}}\Theta {\begin{bmatrix}C&D\end{bmatrix}}<0\\\end{aligned}}$

## Conclusion:

${\text{The optimization problem and the LMI are equivalent. }}\gamma {\text{ must be optimized using bisection.}}$

## Implementation

https://github.com/mcavorsi/LMI

## Related LMIs

Eigenvalue Problem

## External Links

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

This will serve as the unlinked "hidden" page that keeps track of all the pages present in the wiki book LMIs in Control.
Pages with the prefix 'LMIs in Control/pages' in the '' and 'Talk' namespaces:
# LMIs in Control/pages/Apkarian Filter-and State Feedback

LMIs in Control/pages/Apkarian Filter-and State Feedback

## The System:

The number of LMI constraints needed to check quadratic stability is reduced if all the subsystems in the polytopic model has the same matrix $B$. This can be achieved by adding an Apkarian filter in the input of the system.

## The Optimization Problem:

Apkarian Filter
Let consider our TS-LIA model. This can be re written in linear form as:
       ${\dot {x}}=A(z(t))x+B(z(t))u$
The filter should be such that the equilibrium of the states are the input values and the dynamics should be fast, so we could assume the dynamics of the filter negligible (i.e. the input of the filter is equivalent to the input of the quadrotor). One possible filter is shown , where $A_{F}$ = −100$I_{4}$, $B_{F}$ = 100$I_{4}$ and $I_{4}$ ∈ $R^{4\times 4}$ is the identity matrix.
       ${\dot {x_{F}}}=A_{F}x_{F}+B_{F}u_{F};y_{f}=x_{f}$.
When applying the filter, we are imposing that the output of the filter is the new input of the TS-LIA model (i.e. $u$ = $y_{F}$ ). Then, the extended model is:
       ${\dot {x_{c}}}={\begin{bmatrix}A(z(t))&B(z(t))\\0&A_{F}\\\end{bmatrix}}x_{c}+{\begin{bmatrix}0\\B_{F}\end{bmatrix}}u_{F}=A_{e}(z(t))x_{e}+B_{e}u_{f};x_{e}={\begin{bmatrix}x\\x_{F}\end{bmatrix}}$
This prefiltering does not affect the procedure followed to obtain the TS-LIA model, so the premise variables, membership functions and activations functions remains the same.
State Feedback Controller Design
Let consider the state feedback control law for the extended TS-LIA model:${\dot {x_{e}}}=\sum _{i=1}^{32}h_{i}(z(t))[A_{ei}x_{e}+B_{ei}u_{F}]$, where the state feedback control laws are :$u_{F}=\sum _{i=1}^{32}h_{i}(z(t))K_{i}x(t)$, we get the closed loop system :${\dot {x_{e}}}=\sum _{i=1}^{32}\sum _{j=1}^{32}h_{j}(z(t))[A_{ei}x_{e}+B_{ei}K_{j}]x_{e}$

## The LMI:

The design of the controller is done by solving an LMI problem involving the quadratic stability constraints. In case we want D- stabilization, the following set of LMI constraints are needed:
       $L\otimes P+M\otimes P(A_{ei}+B_{e}K_{i})+MT\otimes (A_{e}i+B_{e}K_{i})^{T}P<0$ ∀i = 1, . . . , 32.

## Conclusion:

The LMI is feasible.

## Related LMIs

* LMI for Natural Frequency in State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Maximum_Natural_Frequency_in_State_Feedback#The_LMI%3A
* LMI for Minimum Decay Rate in State Feedback. https://en.wikibooks.org/wiki/LMIs_in_Control/pages/Minimum_Decay_Rate_in_State_Feedback#The_LMI%3A

## References

* Control, A. (2016). Gain-scheduling Control of a Quadrotor Using the Takagi-Sugeno Approach.
# LMIs in Control/page/Quadratic Polytopic Hinf- Optimal State Feedback Control/

LMIs in Control/page/Quadratic Polytopic Hinf- Optimal State Feedback Control/
The System:
Consider System with following state-space representation.
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{1}q(t)+B_{2}w(t)\\p(t)&=C_{1}x(t)+D_{11}q(t)+D_{12}w(t)\\z(t)&=C_{2}x(t)+D_{21}q(t)+D_{22}w(t)\\\end{aligned}}$
Add uncertainty to system matrices
    $A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12}$
New state-space representation
    ${\begin{aligned}{\dot {x}}(t)&=(A+A_{i})x(t)+(B_{1}+B_{i})q(t)+(B_{2}+B_{i})w(t)\\p(t)&=(C_{1}+C_{i})x(t)+(D_{11}+D_{i})q(t)+(D_{12}+D_{i})w(t)\\z(t)&=C_{2}x(t)+D_{21}q(t)+D_{22}w(t)\\\end{aligned}}$
The Optimization Problem:
Recall the closed-loop in state feedback is:
$S(P,K)=$
       ${\begin{aligned}{\begin{bmatrix}A+B_{2}2F&&B_{1}\\C_{1}+D_{12}F&&D_{11}\end{bmatrix}}\\\end{aligned}}$
This problem can be formulated as $H\infty$ optimal state-feedback, where K is a controller gain matrix.
The LMI: An LMI for Quadratic Polytopic $H\infty$ Optimal State-Feedback Control
$||S(P(\Delta ),K(0,0,0,F))||_{H\infty }\leq \gamma$
$Y>0$
       ${\begin{aligned}{\begin{bmatrix}Y(A+A_{i})^{T}+(A+A_{i})Y+Z^{T}(B_{2}+B_{1,i})^{T}+(B_{2}+B_{1,i})Z&&*^{T}&&*^{T}\\(B_{1}+B_{1,i})^{T}&&-\gamma I&&*^{T}\\(C_{1}+C_{1,i})Y+(D_{12}+D_{12,i})Z&&(D_{11}+D_{11,i})&&-\gamma I\end{bmatrix}}<0\end{aligned}}$
Conclusion:
The $H\infty$ Optimal State-Feedback Controller is recovered by $F=ZY^{-1}$
Implementation: https://github.com/JalpeshBhadra/LMI/tree/master
Related LMIs:
# LMIs in Control/page/systemzeros with feedthrough

Let's say we have a transfer function defined as a ratio of two polynomials: ${\begin{aligned}H(s)={\frac {N(s)}{D(s)}}\\\end{aligned}}$ Zeros are the roots of N(s) (the numerator of the transfer function) obtained by setting $N(s)=0$ and solving for s.The values of the poles and the zeros of a system determine whether the system is stable, and how well the system performs. Similarly, the system zeros are either real or appear in complex conjugate pairs. In the case of system zeros with feedthrough, we take $D$ as full rank.

## The System

Consider a continuous-time LTI system, $G$ , with minimal statespace representation $(A,B,C,D)$
       ${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\y(t)=Cx(t)+Du\end{aligned}}$

## The Data

The matrices needed as inputs are:
    ${\begin{aligned}A\in \mathbb {R} ^{n\times n}\\B\in \mathbb {R} ^{n\times m}\\N\in \mathbb {R} ^{p\times n}\\\end{aligned}}$
In this case, $m\leq p$

## The LMI: System Zeros with feedthrough

The transmission zeros of $G(s)=C(sI-A)^{-}1B+D$ are the eigenvalues of $A-B(D^{T}D)^{-1}D^{T}C$. Therefore , $G(s)$ is a minimum phase if and only if there exists $P\in \mathbb {S} ^{q}$, where $P>0$ such that
       ${\begin{aligned}P(A-B(D^{T}D)^{-1}D^{T}C)+(A-B(D^{T}D)^{-1}D^{T}C)^{T}P<0\\\end{aligned}}$

## Conclusion:

If P exists, it ensures non-minimum phase. Eigenvalues of $A-B(D^{T}D)^{-1}D^{T}C$ then gives the zeros of the system.

## Related LMIs

LMIs_in_Controls/pages/systemzeroswithoutfeedthrough

## Implementation

https://github.com/Ricky-10/coding107/blob/master/systemzeroswithfeedthrough

## 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.
* Control_Systems/Poles_and_Zeros