OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/Click here to continue/Applications of Non-Linear Systems/Global Lyapunov Function Search for System with Polynomial Dynamics

[UNDER CONSTRUCTION] - CME
Find the global Lyapunov function for a system with known polynomial dynamics.

## Required Data

The system dynamics to be tested in the form ${\dot {x}}=f(x)$ where $f(x)$ is a polynomial function of the states $x$.

## Find the Lyapunov Function

Find a polynomial (our Lyapunov function) $V(x)$, supposing there exists scalar $\epsilon >0$ such that:
$V(x)-\epsilon x^{T}x\geq 0$
$-{\dot {V}}(x)=-\nabla V(x)^{T}f(x)\geq 0$

## Example

Code example in SOStools probably worthwhile!

## Reference

SOS tools manual is a good one
Tie in the reference from SOS tools? The Khalil one seems correct.

## Related LMIs

Add things here about SOS methods
Local Lyapunov Search
# LMIs in Control/Click here to continue/Applications of Non-Linear Systems/LMI
## The System

Consider a nonlinear, continuous-time system
${\dot {y}}=A(x)+B_{u}(x)u,$ $y=C_{y}(x)+D_{yu}(x)u,$ where $x\in \mathbb {R} ^{n}$ is the state vector, $u\in \mathbb {R} ^{n_{u}}$ is the input and $y\in \mathbb {R} ^{n_{y}}$ is the output.

## The Data

$x\in \mathbb {R} ^{n}$ is the state vector, $u\in \mathbb {R} ^{n_{u}}$ is the input and $y\in \mathbb {R} ^{n_{y}}$ is the output. $A,B_{u},C_{y},D_{yu}$ are multivariable functions of x. $A(0)=0$ (that is, 0 is an equilibrium point of the unforced system associated with the system). $C_{y}(0)=0$ and $B_{u},D_{yu}$ have no singularities at the origin.

#### Proof

* ${\begin{bmatrix}A(x)&B_{u}(x)\\C_{y}(x)&D_{yu}(x)\end{bmatrix}}$ = ${\begin{bmatrix}A&B_{u}\\C_{y}&D_{yu}\end{bmatrix}}$ + ${\begin{bmatrix}B_{p}\\D_{yp}\end{bmatrix}}\bigtriangleup (x){\begin{bmatrix}I-D_{qp}\bigtriangleup (x)\end{bmatrix}}6-1$${\begin{bmatrix}C_{q}&D_{qu}\end{bmatrix}}$
# LMIs in Control/Click here to continue/Applications of Non-Linear Systems/Lyapunov Stability of a System With Polynomial Dynamics

[UNDER CONSTRUCTION] - CME
Find the local Lyapunov function for a system with known polynomial dynamics.

## Required Data

The system dynamics to be tested in the form ${\dot {x}}=f(x)$ where $f(x)$ is a polynomial function of the states $x$.
A function or functions representing the local domain over which stability is to be determined, $X:=\{x:p_{i}(x)\geq 0,i=1,...,k\}$. Commonly this is a ball of specified radius such that $p(x)=R^{2}-x^{T}x$. [Is this assuming P-Compactness of the set?]

## Find the Lyapunov Function

Suppose there exists Lyapunov function $V(x)$, a scalar $\epsilon >0$, and $s_{0},s_{i},t_{0},t_{i}\in \Sigma _{s}$ (where $\Sigma _{s}$ is the set of Sum-of-Squares polynomials, all of which are positive) such that:
$V(x)=s_{0}(x)+\sum \limits _{i}s_{i}(x)p_{i}(x)+\epsilon x^{T}x$
$-{\dot {V}}(x)=-\nabla V(x)^{T}f(x)=t_{0}(x)+\sum \limits _{i}t_{i}(x)p_{i}(x)+\epsilon x^{T}x$
The Lyaponov function, if it exists, will be positive and always decreasing over the desired set $X$

## Example

Code example in SOStools probably worthwhile!

## Reference

SOS tools manual is a good one
Tie in the reference from SOS tools? The Khalil?

## Related LMIs

Add things here about SOS methods
Global Lyapunov Search
# LMIs in Control/Click here to continue/Controller synthesis

Controller synthesis
* Continuous Time
** State Feedback
*** Stabilization of Second-Order System
*** LQ Regulation via H2 Control
*** Controller to achieve the desired Reachable set; Polytopic uncertainty
*** Controller to achieve the desired Reachable set; Norm bound uncertainty
*** Controller to achieve the desired Reachable set; Diagonal Norm-bound uncertainty
** D-Stability
*** Continuous Time D-Stability Controller
** Stabilizability LMI
** LMI for the Controllability Grammian
** LMI for Decentralized Feedback Control
** LMI for Mixed H2 / Hinf Output Feedback Controller
** Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
** H-inf Optimal Quadratically Stabilizing Controllers with Parametric Norm-Bounded Uncertainty
** Stabilizing State-Feedback Controllers with Structured Norm-Bounded Uncertainty
** Optimal State-Feedback Controllers with Structured Norm-Bounded Uncertainty
** Hinf Optimal Output Controllability for Systems With Transients
** Quadratic Polytopic Stabilization
** Quadratic D-Stabilization
** Quadratic Polytopic Full State Feedback Optimal Hinf Control
** Quadratic Polytopic Full State Feedback Optimal H2 Control
** Continuous-Time Static Output Feedback Stabilizability
** Multi-Criterion LQG
** Inverse Problem of Optimal Control
** Nonconvex Multi-Criterion Quadratic Problems
** Static-State Feedback Problem
** Mixed H2 Hinf with desired pole location control
** Mixed H2 Hinf with desired pole location control for perturbed systems
** Robust H2 State Feedback Control
** LQ Regulation via H2 control
* Discrete Time
** Discrete Time Stabilizability
** Quadratic Schur Satbilization
* Design For Insensitive Region
** Generic Insensitive Strip Region Design
** Generic Insensitive Disk Region Design
** Design for Insensitive Strip Region
** Design for Insensitive Disk Region
* Takagi Sugeno Models for Polytopic Controller Design
** Quadratic Stability
** Apkarian Filter and State Feedback
** Minimum Decay Rate in State Feedback
** Maximum Natural Frequency in State Feedback
# LMIs in Control/Click here to continue/Applications of Non-Linear Systems/Control of Rational Systems using Linear-fractional Representations and Linear Matrix Inequalities*
## Introducation

An optimization-based methodology for the multiobjective control of a large class of nonlinear systems is performed.

## The System

Consider a nonlinear, continuous-time system
${\dot {y}}=A(x)+B_{u}(x)u,$ $y=C_{y}(x)+D_{yu}(x)u,$ where $x\in \mathbb {R} ^{n}$ is the state vector, $u\in \mathbb {R} ^{n_{u}}$ is the input and $y\in \mathbb {R} ^{n_{y}}$ is the output.

## The Data

$x\in \mathbb {R} ^{n}$ is the state vector, $u\in \mathbb {R} ^{n_{u}}$ is the input and $y\in \mathbb {R} ^{n_{y}}$ is the output. $A,B_{u},C_{y},D_{yu}$ are multivariable functions of x. $A(0)=0$ (that is, 0 is an equilibrium point of the unforced system associated with the system). $C_{y}(0)=0$ and $B_{u},D_{yu}$ have no singularities at the origin.

#### Proof

* ${\begin{bmatrix}A(x)&B_{u}(x)\\C_{y}(x)&D_{yu}(x)\end{bmatrix}}$ = ${\begin{bmatrix}A&B_{u}\\C_{y}&D_{yu}\end{bmatrix}}$ + ${\begin{bmatrix}B_{p}\\D_{yp}\end{bmatrix}}$$\bigtriangleup (x)$${\begin{bmatrix}C_{q}&D_{qu}\end{bmatrix}}$
For a given scalar \sigma > 0, we associate with the Linear differential inclusion, ${\dot {x}}=Ax+B_{u}u+B_{p}p,$ $q=C_{q}x+D_{qu}u+D_{qp}p,$ $y=C_{y}x+D_{yu}u+D_{yp}p,$ $p=\bigtriangleup (t)q,\|\bigtriangleup (t)\|\leq \sigma ^{-}1,\bigtriangleup (t)\in D(r),t\geq 0.$

## The LMI: Control of Rational Systems using Linear-fractional Representations and Linear Matrix Inequalities

For given \sigma > 0, the LDI system is quadratically stable if there exists P, S, and G such that the LMIs $P>0,S>0,G=-G^{T},S,G\in B(r),$
${\begin{bmatrix}A^{T}P+PA+C_{q}^{T}SC_{q}&PB_{p}+C{q}^{T}G+C_{q}^{T}SD_{q}p\\B_{p}^{T}P-GC_{q}+D_{qp}^{T}SC_{q}&D_{qp}^{T}SD_{qp}G-\sigma ^{2}S+D_{qp}^{T}G-GD_{q}p\end{bmatrix}}<0$
hold. Then, for every $\bigtriangleup \in D(r)$ such that $\|\bigtriangleup (t)\|\leq \sigma ^{-}1,$ $det(I-D_{pq}\bigtriangleup )^{-1}\neq 0,$
${\begin{bmatrix}y_{max}^{2}I&C_{y}\\C_{y}^{T}&P\end{bmatrix}}\geq 0.$
${\begin{bmatrix}AQ+QA^{T}+B_{p}TB_{p}^{T}+B_{u}Y+T^{T}B_{u}^{T}&QC_{q}^{T}+Y^{T}D_{qu}^{T}+B_{p}TD_{qp}^{T}+B_{p}H\\C_{q}Q+D_{qu}Y+D_{qp}TB_{p}^{T}-HB_{p}^{T}&D_{qp}TD_{qp}^{T}-\sigma ^{2}T+D_{qp}H-HD_{qp}^{T}\end{bmatrix}}$ < 0

## Conclusion

The above LMIs provide a unified setting, as well as an efficient computational procedure, for answering (possibly conservatively) several control problems pertaining to a quite generic class of nonlinear systems. This method makes an explicit and systematic connection (via LFRs and LMIs) between robust control methods and nonlinear systems.
# LMIs in Control/Click here to continue/Applications of Non-Linear Systems/Local Minimum of Polynomial via SOS Method

[UNDER CONSTRUCTION] - CME
The local minimum of a certain polynomial functions f(x) can be found using Sum-of-Squares (SOS) methods. Here we assume the local set has a P-Compact representation, and that this representation is used.

## Required Data

A polynomial function f(x) whose minimum is desired.
A P-Compact representation of the set.

## First Steps

Touch up on the Putinar Positivstellensatz and what it means for us.

## Why does it work

Based in S-Procedure, with the Positivstellensatz all about the right representation.

## Example

Code example in SOStools probably worthwhile!

## Reference

SOS tools manual is a good one

## Related LMIs

SOS Basics
Global Minimum
# LMIs in Control/Click here to continue/Applications of Non-Linear Systems/Stability of Switching Systems - Arbitrary Switching
## Stability of Switching Systems - Quadratic Stability Under Arbitrary Switching

For gain scheduled systems, stability of each subsystem {A$_1$,A$_2$} does not guarantee stability under arbitrary switching. Additionally, smart switching can stabilize two unstable systems.

## The System

The state space formulation of each subsystem is given as follows:
${\begin{aligned}{\dot {x}}(t)=A_{i}(t)+B_{i}u(t)\\y(t)=C_{i}x(t)+D_{i}u(t)\end{aligned}}$
Where i = 1,2,...,n for each of the subsystems in the switching system.

## The Data

For a switching system with multiple subsystems, the A matrix for each is defined by
$A_{i}=A_{1},A_{2},...A_{n}$

## The LMI:Quadratic Stability Under Arbitrary Switching

The switched system ${\dot {x}}(t)\in {A_{1}x(t),A_{2}x(t)}$ is stable under arbitrary switching if there exists some P > 0 such that
$A_{1}^{T}P+PA_{1}<0$ and
$A_{2}^{T}P+PA_{2}<0$

## Conclusion:

This implies that both A$_1$ and A$_2$ are Hurwitz.
There is not necessarily a common quadratic Lyapunov function for both A$_1$ and A$_2$.
This quadratic stability condition under arbitrary switching is a useful condition to use when designing controllers for switching systems. This LMI does not provide information on how the controller is designed, but is to be used as an additional condition to stabilize a switching system.

## Implementation

This implementation requires Yalmip and Sedumi.
Quadratic Stability Under Arbitrary Switching

## Related LMIs

Lyapunov Stability of a System with Polynomial Dynamics/
Global Minimum of Polynomial via SOS Method/
Local Minimum of Polynomial via SOS Method/
Global Lyapunov Function Search for System with Polynomial Dynamics/

## 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.

## Return to Main Page:

LMIs in Control: https://en.wikibooks.org/wiki/LMIs_in_Control
# LMIs in Control/Click here to continue/Applications of Non-Linear Systems/Global Minimum of Polynomial via SOS Method

[UNDER CONSTRUCTION] - CME
The global minimum of a certain polynomial functions $f(x)$ can be found using Sum-of-Squares (SOS) methods. This is a useful starting point to the more-useful but less straightforward issue of local minimums.

## Required Data

A polynomial function $f(x)$ whose minimum is desired.

## The Problem

To find the minimum value of $f(x)$, we suppose that there exists a scalar $p$ such that
$f(x)-p\geq 0$
This is equivalent to determining if $(f(x)-p)$ has a SOS representation, since a SOS polynomial is never negative. This becomes an optimization problem by attempting to find the largest possible $p$ such that a representation exists. In other words, the optimization problem becomes
$\min _{p}-p$ subject to $f(x)-p\geq 0$

## Example

Code example in SOStools probably worthwhile!

## Reference

SOS tools manual is a good one

## Related LMIs

SOS Basics
Local Minimum is a solid follow-up
# Instructional Technology/Learning Management Systems/Benefits

Learning Management systems (LMS) such as Blackboard and Moodle are excellent tools that can be used in the K-12 classroom to promote the school and its curriculum.
Benefits to Students and Parents
A learning management system allows students to download and upload files, participate in chat and discussion boards, as well as take assessments. Students are also able to view grades as well as contact the teacher and classmates. An LMS is a step above a classroom website which is a more static approach to giving classroom information. These systems utilize tools similar to those that tech savy students already use outside of the classroom. Although most educators think of an LMS as being limited to older students it can help younger students in the elementary grades to organize their work while simultaneously keeping parents in tune with what is going on in the classroom. An LMS allows teachers to tutor and build skills by placing reusable learning objects, documents, video, practice exercises, and assessments on the system. Using an LMS allows parents to review the course curriculum, classroom calendar and monitor their student's progress. If the instructor posts samples of student work and rubrics, students and parents can see how other classmates are performing and have a clearer idea of academic expectations. The use of a learning management system gives the teacher the opportunity to promote classroom activities and course expectations. However, once an LMS is implemented there is pressure on the instructor to keep the content up to date and located in such a way as to be easily found by the students and parents. Once parents and students have ready access to certain information, for example grades, that information must be kept up to date. An LMS system also allows for distance learning for students that are out of school due to illness or suspension.
Benefits to Teachers
The use of a learning management system also allows for teacher collaboration at grade level or by subject in the design of learning objects to post on the system. Given proper planning time teachers can design thematic units using reuseable learning objects and prescriptive activities for groups of students based on test data. The Amarillo Independent School District in Texas uses the Blackboard learning management system with a set of add-ons to support teacher collaboration and curriculum development.(Waters,2007) This allows teachers across the district to work independently and contribute to the team effort asynchronously without having to attend district wide meetings.
Waters, J (2007,March). Curriculum Unbound. T.H.E. Journal,34, 40-48.
# Instructional Technology/Learning Management Systems
## Learning Management Systems

This section was started by IT 7310, Learning Management Systems, in the Instructional Technology program at Wayne State University, Detroit, Michigan, USA
The instructor is Tim Spannaus. Student contributors will identify their sections.
While the WSU IT students will start this section, we welcome your comments and contributions.
K-12 Classroom
Learning Objects
Selecting a LMS Vendor
Knowledge Management
Universal Design
Reusable Learning Objects in Special Education
Learning Management Systems in Healthcare
A Non-Techies Guide to Selecting a LMS
Project Based Learning
Return to Instructional Technology.
# Instructional Technology/Learning Management Systems/Selecting a LMS Vendor

In this section we will discuss some of the important elements to selecting an LMS Vendor. Many organizations spend millions of dollars in trying to find the right company to customize a learning management solution.
There are several different sources online that tell you how to select an LMS Vendor. Throughout the search with my employer I found various resources that were helpful. These resources provide information on defining a Learning Management System and selecting a vendor. This section will deal primarily with selecting an LMS vendor.

## 10 Steps to Successful Vendor Selection

1. Determine and document your business requirements 2. Research Vendors 3. Conduct Vendor Demonstrations 4. Narrow Down your Potential Vendors 5. Conduct a LMS Needs Assessment 6. Develop and Distribute the Purchase Requisition 7. Prepare for Any Cultural Changes 8. Narrow Your List and Evaluate Proposals 9. Conduct Final Presentations 10. Negotiate the Contract

### Determining Business Requirements

Determining your business requirements is the most crucial step in selecting a LMS vendor. If you don't know your requirements you won't be able to effectively communicate them to vendors. Here's a few important points about your business requirements:
* Document them - don't try to go from memory on what you think you need.
* First began by documenting your current state - how you conduct business today.
* Next record your future state - where you would like to be.
Consider how an LMS will help bridge the gap and enable you to meet your goals.

### Researching Vendors

It is very easy to research vendors with the wealth of information that is online. However, just because you find a vendor online, how will you know that vendor will meet the needs of your company. Inside of blindly searching for a vendor, the best method is to leverage industry standards or other companies with similar goals and objectives.

### Conducting Vendor Demonstrations

One of the most time consuming aspects of working with vendors is the demonstrations and presentations. Below you can find some helpful information on making this process more efficient for you. http://www.brandonhall.com/public/publications/vendordemos/vendordemos.htm
Return to Instructional Technology.
Return to Learning Management Systems.
# MIPS Assembly/MIPS Emulation

This page is going to talk about using MIPS emulators to test MIPS code on a non-MIPS computer.
A common MIPS Emulator is Spim which is fully cross platform. Another one is MIPS Assembler and Simulator, which has been successfully deployed in and used in a few high schools. QEMU is another common emulator environment for MIPS. Yet another Emulator is MARS a cross-platform Java MIPS IDE.
Open Virtual Platforms (OVP) http://www.OVPworld.org includes the freely available simulator OVPsim, a library of models of processors, peripherals and platforms, and APIs which enable users to develop their own models. The models in the library are open source, written in C, and include the MIPS 4K, 24K and 34K cores. These models are created and maintained by Imperas http://www.imperas.com and in partnership with MIPS Technologies have been tested and assigned the MIPS-Verified(tm) mark. Sample MIPS-based platforms include both bare metal environments and platforms for booting unmodified Linux binary images. These platforms/emulators are available as source or binaries and are fast, free, and easy to use. OVPsim is developed and maintained by Imperas and is very fast (100s of million instructions per second), and built to handle multicore architectures. To download the MIPS OVPsim simulators/emulators visit http://www.OVPworld.org/mips.
# MIPS Assembly/Floating Point Instructions
## Arithmetic

Integer implementation of floating-point addition

## Further Reading

* Floating Point
# MIPS Assembly/Introduction

This page is going to serve as a general foreword about this book.

## What This Book is About

This book is going to discuss the MIPS assembly language. This book will cover not only the straightforward facet of this subject (how to program MIPS), but will also look deeper, and discuss MIPS from a very low level. In this way, this book should be useful both for people just learning to program MIPS, and also to people who are looking to do advanced tasks in MIPS (such as write a MIPS assembler program, or construct a low-level OS kernel in MIPS). This book will not talk about the specifics of MIPS hardware, however.

## Who This Book is For

This book is designed to be a reference for all people who are interested in MIPS. This book starts off with the basics behind the language, and discusses the various operations in such a manner that beginners will be able to get a handle of MIPS programming. However, this book also contains a number of advanced sections for experienced programmers who are looking at doing advanced projects with the MIPS architecture.

## How This Book is Organized

This book is organized in such a fashion that the most simple material is presented first, and the most advanced material is saved towards the end. The first section of the book is reserved for historical and interesting information about MIPS, and listings of real-world MIPS implementations. The second section is going to go into the MIPS assembly language, talking about each instruction individually, and explaining how to use them. The third section is going to talk about the topics of programming, assembling, and emulating MIPS code. Finally, the fourth section is going to talk about advanced topics, such as the internal machine code of MIPS machines, the nature of pseudoinstructions, some advanced system instructions, and handling exceptions.

## Where to Go From Here

This book will serve as a complete reference to programming MIPS. While there are no Wikimedia resources specifically designed to follow this text, the reader may benefit from reading about higher-level languages, or assemblers, or another advanced programming topic.
For more information about how to design MIPS and other types of microprocessor systems, see Microprocessor Design.
# MIPS Assembly/Exceptions

This page is going to talk about exception handling. We will also talk about what exceptions are, and what causes them.
# MIPS Assembly/Licensing

The text of this book is released under the following license:
# MIPS Assembly/MIPS Assemblers

This page is going to list some assemblers for programming in MIPS, and will discuss how to program using MIPS assemblers.
/edit by Till Fischer, University of Ulm (Germany)
Heres a link to a Download of the MARS, an emulation Assembler Program by the Missouri State University, we use that one in our Studies in order to learn Assembler: http://courses.missouristate.edu/KenVollmar/MARS/