OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/D-stability Settling time poles

LMI for Settling Time Poles
The following LMI allows for the verification that poles of a system will fall within a settling time constraint. This can also be used to place poles for settling time when the system matrix includes a controller, such as in the form A+BK.

## The System

We consider the following system:
${\begin{aligned}{\dot {x}}(t)&=Ax\end{aligned}}$
or the matrix $A\in \mathbb {R} ^{n\times n}$, which is the state matrix.

## The Data

The data required is the matrix A and the settling time $t_{s}$ you wish to verify.

## The Optimization Problem

To begin, the constraint of the pole locations is as follows: ${(z+z^{*}) \over 2}+{4.6 \over t_{s}}{\leq }0$, where z is a complex pole of A. We define $2Re(z){\leq }-\alpha$. The goal of the optimization is to find a valid P > 0 such that the following LMI is satisfied.

## The LMI: LMI for Settling Time Poles

The LMI problem is to find a matrix P > 0 satisfying:
${\begin{aligned}AP+(AP)^{T}+\alpha P&<0\\\end{aligned}}$

## Conclusion:

If the LMI is found to be feasible, then the pole locations of A, represented as z, will meet the settling time specification of ${(z+z^{*}) \over 2}+{4.6 \over t_{s}}{\leq }0$, and the poles of A satisfy the previously defined constraint.

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/maxwellpeterson99/MAE509Code

## Related LMIs

- D-stabilization
- D-stability Controller
- D-stability Observer

## External Links

- LMI in Control Systems Analysis, Design and Applications
- A course on LMIs in Control by Matthew Peet

## Return to Main Page

-Matrix and LMI Properties and Tools
# LMIs in Control/Click here to continue/Fundamentals of Matrix and LMIs/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.
# Introduction to Programming Languages/Definition and Examples

Some programming languages treat functions as first class values. In other words, functions in these languages can be assigned to variables, and can be passed as parameters to and returned from other functions. This capacity opens up a vast horizon of possibilities to program developers. In this chapter we will explore some of these possibilities.

### Definition and Examples

A fundamental notion that we will use in this chapter is the concept of high-order functions, which we define, recursively, as follows:
* A function that does not receive other functions as parameters or return functions has order zero.
* A function that receives or returns a function of order n - 1 has order n.
As an example, the function below, implemented in SML, receives a polymorphic list of type 'a list, plus another function of type 'a -> 'b, and returns a new list of type 'b list. We obtain this new list by applying the mapping function to every element of the input list:
Our map function is very reusable. We can reuse the same algorithm with many different types of mapping functions and input lists. Below we have some examples:
In the previous example, map is a function of order one, because it receives a function of order zero as a parameter. High order functions are very common among functional languages, such as SML, Haskell, ocaml, erlang, and F#. However, even languages that have a more imperative semantics, such as C#, Python and Lua provide this functionality. In fact, almost every modern programming language has some support to high-order functions. Below we see our initial example, the map function, coded in Python:
It is also possible to use high-order functions in C, by passing pointers to functions as parameters of other functions. For instance, the code below implements our original map example. We point, however, that this coding style is not very typical of C, a fact that might justify the relatively verbose program.
Closures →
# Introduction to Programming Languages/Concepts of Functional Languages
### Concepts of Functional Languages

Functional programming is a form of declarative programming, a paradigm under which the computation of a program is described by its essential logic. This approach is in contrast to imperative programming, where specific instructions describe how a computation is to be performed. The only form of computation in a strictly functional language is a function. As in mathematics, a function establishes, by means of an equation, a relation between certain input and output. To the functional programmer, the exact implementation underneath a given computation is not visible. In this sense, we tend to say that functional programming allows one to focus on what is to be computed.
While functional programming languages have not traditionally joined the success of imperative languages in the industry, they have gained more traction in the recent years. This increasing popularity of the functional programming style is due to numerous factors. Among the virtues of typical functional languages like Haskell is the absence of mutable data and associated side-effects, a characteristic referred to as purity, which we shall study further. Together with the introduction of novel parallel architectures, we have seen an accompanying growth of concurrent programming techniques. Because functional languages are agnostic of global state, they provide a natural framework for implementations free of race conditions. In addition, they ease the design fault-tolerant functions, since only local data is of concern.
Another reason that has contributed to the emergence of the functional style is the appearance of the so-called hybrid languages. A prominent example among those is the Scala programming language. Scala offers a variety of features from the functional programming world like high-order functions and pattern matching, yet with a closely resembling object-oriented appearance. Even imperative constructs such as for loops can be found in Scala, a characteristic that helps reducing the barrier between functional and imperative programming. In addition, Scala compiles to bytecode of the Java Virtual Machine (JVM), enabling easy integration between the two languages and, consequently, allowing programmers to gradually move from one to the other. The same approach of compiling to the JVM was followed by Closure, a relatively new general-purpose language. Finally, both Java and C++, which were originally designed under strong imperative principles, feature nowadays lambdas and other functional programming constructs.
# Introduction to Programming Languages/Cost Models

Programming language constructs might have particular asymptotic complexities, which programmers should know, to produce efficient algorithms. In other words, even though an algorithm has some specific, well-known, computational cost, its implementation can have higher complexity, due to an implicit cost model that is part of that languages runtime environment. Furthermore, some programming languages miss features that might complicate the implementation of efficient algorithms. For instance, it will be hard to implement Bucket Sort without random access data structures. These minutia concern an aspect of programming languages' implementation that we call cost models.
In this chapter we will look into four cost models. The first concerns the use of lists in programming languages that follow the Lisp model. The second cost model concerns the invocation of functions in languages that support recursion. After that, we look into Prolog's unification model, trying to reason about the factors that cause unification to be more or less efficient. Finally, we discuss languages that support arrays, which are random access data structures.
← Programming Language Paradigms · Lists →
# Introduction to Programming Languages/Coercion
### Coercion

Many programming languages support the conversion of a value into another of a different data type. This kind of type conversions can be implicitly or explicitly made. Implicit conversion, which is also called coercion, is automatically done. Explicit conversion, which is also called casting, is performed by code instructions. This code treats a variable of one data type as if it belongs to a different data type. The languages that support implicit conversion define the rules that will be automatically applied when primitive compatible values are involved. The C code below illustrates implicit and explicit coercion. In line 2 the int constant 3 is automatically converted to double before assignment (implicit coercion). An explicit coercion is performed by involving the destination type with parenthesis, which is done in line 3.
A function is considered a polymorphic one when it is permited to perform implicit or explicit parameter coercion. If the same is valid for operands, the related operator is considered a polymorphic operator. Below, a piece of C++ code exemplifies these polymorphic expressions.
← Overloading · Parametric Polymorphism →
# Kdenlive

    Hint: The kdenlive manual has moved to the KDE Userbase which has translation support. All documentation efforts should go there from now on (since 2011-06-09).

## Chapters

* Introduction: what Kdenlive is
* Getting and installing Kdenlive
* Quickstart
* Digital Video and Non-Linear Editing
* Using Kdenlive
* Video and Audio Effects
* Troubleshooting and Known Bugs
* Frequently asked questions
* Deprecated

## Development

* KDE4 porting page
* Contributor/Developer HOWTO
* Bug reporting guide
# Kdenlive/Basics of Non-Linear Video Edition

Before editing video on computer was possible, most editing was done on linear editors --- these work by having essentially, two video recorders. The first would hold the video that you are editing together, and the second would be used to find the next clip that you want to record. Find a clip, record it, find the next clip, record it. This is called linear editing because the medium of tape forced things to be edited sequentially. If you have edited 10 minutes worth of footage together, and decide that you made a mistake after five minutes, you can't just "push" the last five minutes around to add space for new clips.
Non-linear editing is by far a more flexible way of editing. You can place clips anywhere within your final video (in Kdenlive, this is represented via the timeline), and you can move them around as much as you want, rearrange them in a completely different order. Best of all, at any point during this you can press play and (with a sufficiently powerful computer) watch back what you have done there and then.
# Kdenlive/Digital Video

Digital video is a generic term for pictures, with motion or not, that were changed to a format of sequenced numbers (digitized) for being stored in a computer file or a distribution medium, like a DVD, in a way that it can be read by a computer. The digital video files can be made from an analog source, such as a 35mm film movie or a VHS videocassete tape; or captured from a camera or device which does the analog/digital conversion internally and already outputs the video in digital format; or it can be completely generated in a computer, which is the case of an animation, a slideshow, or simply a text to be shown on-screen.
After a video is recorded and digitized, it can be edited with specialized electronic equipment or with a computer, and the latter allows the use of a simple and efficient method --- non-linear editing.
* Basic concepts of Non-Linear video editing
* Frames, NTSC, PAL, etc: basic info about video systems
* File formats and compression methods for digital video
* File formats and compression methods for audio in digital video
* Conversion among the various formats of digital video
# Kazakh/Vowels

| Latin | Cyrillic | Arabic | IPA | English Example |
| a | a | ا | ɑ | father |
| ä | ә | ٵ | æ | cat |
| e | е | | | yes |
| io | ё | | | yo-yo |
| i | и | | | eat |
| o | о | | | bold |
| ö | ө | | | German "schon" |
| u | у | | | boot |
| ū | ұ | | | upper |
| ü | ү | | | German "Fuhrer" |
# Kdenlive/CommandReference

Using Kdenlive < Index > Frequently Asked Questions

# Chapter 4. Command Reference
## The File Menu

File->New (Ctrl-n)
File->Open (Ctrl-o)
File->Open Recent
File->Save (Ctrl-s)
File->Save As...
File->Quit (Ctrl-q)

## The Edit Menu

Edit->Undo (Ctrl-z)
Edit->Redo (Ctrl-Shift-z)
Edit->Cut (Ctrl-x)
Edit->Copy (Ctrl-c)
Edit->Paste (Ctrl-v)

## The Project Menu

Project->Add Clips
Project->Delete Clips
Project->Clean Project
Project->Clip Properties

## The Timeline Menu

Timeline->Move/Resize Tool (q)
Timeline->Razor Tool (w)
Timeline->Spacer Tool (e)
Timeline->Snap To Frames
Timeline->Snap To Border

## The Render Menu

Render->Export Timeline

## The Layout Menu

Layout Menu->Load Layout 1 (F9)
Layout Menu->Load Layout 2 (F10)
Layout Menu->Load Layout 3 (F11)
Layout Menu->Load Layout 4 (F12)
Layout Menu->Save Layout 1 (Ctrl-Shift-F9)
Layout Menu->Save Layout 2 (Ctrl-Shift-F10)
Layout Menu->Save Layout 3 (Ctrl-Shift-F11)
Layout Menu->Save Layout 4 (Ctrl-Shift-F12)

## The Settings Menu

Settings->Show Toolbar
Settings->Show Statusbar
Settings->Configure Shortcuts...
Settings->Configure Toolbars...
Settings->Configure Kdenlive...

## The Help Menu

Help->Kdenlive Handbook
Help->What's This? (Shift-F1)
Help->Report Bug...
Help->About Kdenlive
Help->About KDE
# Kdenlive/Effects

Effects change the way that an audio or a video clip is shown in your video. They can be used for some little improvements, like filtering color blurs in the image or hum interference in the sound, or for surreal, impressive modifications like turning an image to its negative form, changing colors, adding a "vinyl-like" sound to your clips, and so on. They can be combined in a multitude of ways. Once the effects are provided by the editor software, its use depends only on the user creativity --- ok, some processing power too :)
In the following pages you can see in details which effects are available in Kdenlive, with descriptions of what they do and the meaning of their settings:
* Video effects
* Audio effects
# Kdenlive/Audio effects

Audio clips can be treated with a multitude of effects in Kdenlive, and the available audio effects are listed in this page.
Unless otherwise stated, the audio effects used in Kdenlive are based on LADSPA, an open-source architecture for sound plug-ins. They will be available only if MLT was compiled with support for LADSPA, and for achieving this the packages ladspa-sdk and libjack-dev and swh-plugins must be installed. However, this should not be an issue for people not interested in compiling Kdenlive locally.
* Mute : Hides the audio from a clip. This is a MLT effect.
* Volume : Adjust the variation of audio volume in a clip between two selected frames. This effect is provided by the MLT framework.
* Declipper : Remove from the soundtrack clicks caused by clipping, an artifact caused when the audio recording level was too high. If you can have a graphical visualization of the sound, in a software like Audacity, you will see the audio "overflowing" and having its peaks cut off --- and these peaks cause the clip artifact.
* Vinyl : Emulate the sound of a vinyl record player.
* Limiter : Limit the audio volume.
* Equalizer : A stereo equalizer, useful for adjusting playback levels separately for different tones, like treble, mid and bass.
* Phaser : Very commonly used by guitar players, this effect causes an inversion in the phase of a sound wave in regular periods of time. In a sound wave made from a constant tone, a phaser will make it sound like a "woouunnwooouunn...". Well, it is surely easier to understand it by hearing than by reading :)
* Pitch Scaler : Allows you to modify the tone modulation of a sound --- for example, to make a male voice sound like a female voice and vice-versa.
* Pitch Shifter :
* Rate Scaler :
* Reverb : Reverberation is what happens, for example, when you sing in a cathedral, by the sound reflecting in the walls.
* Room Reverb : Same effect as in reverb but with shorter delays, like in a bathroom.
# LMIs in Control/Click here to continue/Controller synthesis/Static-State Feedback Problem

LMIs in Control/Click here to continue/Controller synthesis/Static-State Feedback Problem
We are attempting to stabilizing The Static State-Feedback Problem

## The System

Consider a continuous time Linear Time invariant system
    ${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\\end{aligned}}$

## The Data

$A,B$ are known matrices

## The Optimization Problem

The Problem's main aim is to find a feedback matrix such that the system
    ${\begin{aligned}{\dot {x}}(t)=Ax(t)+Bu(t)\\\end{aligned}}$
and
    ${\begin{aligned}u(t)=Kx(t)\\\end{aligned}}$
is stable Initially we find the $K$ matrix such that $(A+BK)$ is Hurwitz.

## The LMI: Static State Feedback Problem

This problem can now be formulated into an LMI as Problem 1:
    $X(A+BK)+(A+BK)^{T}X<0$
From the above equation $X>0$ and we have to find K
The problem as we can see is bilinear in $K,X$
* The bilinear in X and K is a common paradigm
* Bilinear optimization is not Convex. To Convexify the problem, we use a change of variables.
Problem 2:
    $AP+BZ+PA^{T}+Z^{T}B^{T}<0$
where $P>0$ and we find $Z$ $K=ZP^{-1}$
The Problem 1 is equivalent to Problem 2

## Conclusion

If the (A,B) are controllable, We can obtain a controller matrix that stabilizes the system.

## Implementation

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

## Related LMIs

Hurwitz Stability

## 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/Controller synthesis/Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty

LMIs in Control/Click here to continue/Controller synthesis/Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty
${\begin{aligned}{\text{The system is quadratically stable if and only if there exists some }}\Theta \in P\Theta ,P>0,{\text{ and }}Z{\text{ such that the LMI is feasible.}}{\text{ Furthermore, there exists a quadratically stabilizing controller with }}u(t)=Kx(t).\end{aligned}}$

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)+Mp(t),&&p(t)=\Delta (t)q(t),\\q(t)&=Nx(t)+Qp(t)+D_{12}u(t),&&\Delta \in {\bf {{\Delta }\;,||\Delta ||\leq 1}}\\\end{aligned}}$

## The Data

The matrices $A,B,M,N,Q,D_{12}$.

## The LMI:

       ${\begin{aligned}{\text{Find}}\;&P>0,Z:\\{\begin{bmatrix}AP+BZ+PA^{T}+Z^{T}B^{T}&PN^{T}+Z^{T}D_{12}^{T}\\NP+D_{12}Z&0\end{bmatrix}}+{\begin{bmatrix}M\Theta M^{T}&M\Theta Q^{T}\\Q\Theta M^{T}&Q\Theta Q^{T}-\Theta \end{bmatrix}}<0\\\end{aligned}}$

## Conclusion:

If the LMI is feasible, the controller, K, is calculated by $K=ZP^{-1}$.

## Implementation

https://github.com/mcavorsi/LMI

## Related LMIs

Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
H-infinity Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
# LMIs in Control/Click here to continue/Controller synthesis/Quadratic Schur Satbilization

LMI for Quadratic Schur Stabilization
A discrete-time system is said to be stable if all roots of its characteristic equation lie in the open unit disk. This provides a condition for the stability of discrete-time linear systems with polytopic uncertainties and a linear time-invariant system with this property is called a Schur stable system.

## The System

Consider discrete time system
    ${\begin{aligned}x_{k+1}=Ax_{k}+Bu_{k},\\\end{aligned}}$
where $x_{k}\in \mathbb {R} ^{n}$, $u_{k}\in \mathbb {R} ^{m}$, at any $t\in \mathbb {R}$.
The system consist of uncertainties of the following form
    ${\begin{aligned}\Delta _{A(t)}=A_{1}\delta _{1}(t)+....+A_{k}\delta _{k}(t)\\\Delta _{B(t)}=B_{1}\delta _{1}(t)+....+B_{k}\delta _{k}(t)\\\end{aligned}}$
where $x\in \mathbb {R} ^{m}$,$u\in \mathbb {R} ^{n}$,$A\in \mathbb {R} ^{mxm}$ and $B\in \mathbb {R} ^{mxn}$

## The Data

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

## The LMI:

There exists some X > 0 and Z such that
       ${\begin{aligned}{\begin{bmatrix}X&&AX+BZ\\(AX+BZ)^{T}&&X\end{bmatrix}}+{\begin{bmatrix}0&&A_{i}X+B_{i}Z\\(A_{i}X+B_{i}Z)^{T}&&0\end{bmatrix}}>0\quad i=1,......,k\end{aligned}}$

## The Optimization Problem

The optimization problem is to find a matrix ${\begin{aligned}K\in \mathbb {R} ^{r\times n}\end{aligned}}$ such that:
${\begin{aligned}||A+BK||_{2}<\gamma \end{aligned}}$
According to the definition of the spectral norms of matrices, this condition becomes equivalent to:
${\begin{aligned}(A+BK)^{T}(A+BK)<\gamma ^{2}I\end{aligned}}$
Using the Lemma 1.2 in LMI in Control Systems Analysis, Design and Applications (page 14), the aforementioned inequality can be converted into:
${\begin{aligned}{\begin{bmatrix}-\gamma I&(A+BK)\\(A+BK)^{T}&-\gamma I\end{bmatrix}}<0\\\end{aligned}}$

## Conclusion:

The Controller gain matrix is extracted as $F=ZX^{-1}$
$u_{k}=Fx_{k}$
    ${\begin{aligned}x_{k+1}=Ax_{k}+Bu_{k},\\\quad \quad \quad =Ax_{k}+BFx_{k}\\\quad \quad =(A+BF)x_{k}\end{aligned}}$
It follows that the trajectories of the closed-loop system (A+BK) are stable for any $\,\Delta \,\in \,C_{0}(\Delta _{1},...,\Delta _{k})$

## Implementation

https://github.com/JalpeshBhadra/LMI/blob/master/quadratic_schur_stabilization.m

## Related LMIs

Schur Complement Schur Stabilization

## 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.
* LMI in Control Systems Analysis, Design and Applications