OpenLLM-France/wikimedia huggingface dataset:
# Japanese/Kanji/Lesson 10

| 南 Meaning: south | 音読み： なん | 南東 （なんとう） south-east |
| 南 Meaning: south | 訓読み： みなみ | 南風 （みなみかぜ） southern wind |
| 東 Meaning: east | 音読み： とう | 東京 （とうきょう） Tokyo |
| 東 Meaning: east | 訓読み： ひがし | 東野 （ひがしの） surname |
| 西 Meaning: west | 音読み： さい | 関西　（かんさい）Kansai (south-western half of Japan, including Osaka) |
| 西 Meaning: west | 訓読み： にし | |
| 外 Meaning: outside; foreign | 音読み： がい | 外国人：（がいこくじん）　foreigner 外国語: (がいこくご) foreign language |
| 外 Meaning: outside; foreign | 訓読み： そと | 外に出たい: (そとにでたい) I want to go out. |
| Meaning: name; well-known | 音読み： メイ, ミョウ | 名人 (めいじん) master, expert 有名 (ゆうめい) famous 名字 (みょうじ) surname |
| Meaning: name; well-known | 訓読み： な | 名前 (なまえ) name 仮名 (かな) Japanese syllabary character - 'kana' |
# Japanese/Kanji/Lesson 11

| 高 Meaning: high; tall | 音読み： コウ | 高地 (こうち) high ground 高年 (こうねん) old age |
| 高 Meaning: high; tall | 訓読み： たか | 高い (たかい) high; expensive |
| Meaning: small | 音読み： ショウ | 小説 (しょうせつ) novel, story, fiction 小学校 (しょうがっこう) elementary school |
| Meaning: small | 訓読み： ちい（さい）, こ〜, お〜, さ〜 | 小さい (ちいさい) small 小犬 (こいぬ) puppy 小川 (おがわ) brook, creek |
| Meaning: middle; center | 音読み： チュウ | 中国 (ちゅうごく) China 中年 (ちゅうねん) middle age |
| Meaning: middle; center | 訓読み： 〜ちゅう, 〜じゅう, ナカ, うち | 水中 (すいちゅう) underwater 年中 (ねんじゅう) all year, year-round Aの中に (Aのなかに) inside A 田中 (たなか) Tanaka (surname) |
| Meaning: big | 音読み： ダイ, タイ | 大学 (だいがく) university 重大 (じゅうだい) important, serious 大洋 (たいよう) ocean |
| Meaning: big | 訓読み： おお（きい）, おお（いなる）, おお〜, おお（いに） | 大きい (おおきい) big 大人 (おとな) adult |
| 長 Meaning: long; head, chief, director | 音読み： チョウ | 長男 (ちょうなん) eldest son 社長 (しゃちょう) company president |
| 長 Meaning: long; head, chief, director | 訓読み： なが（い）, なが（たらしい）, なが（らえる）, た（ける）, おさ, とこし（え） | 長い (ながい) long 長椅子 (ながいす) sofa, couch |
# High School Mathematics Extensions/Further Modular Arithmetic/Problem Set

* HSME
* Content
* Further Modular Arithmetic
* Multiplicative Group and Discrete Log
* Problems & Projects
* Problem Set
* Project
* Solutions
* Exercises Solutions
* Problem Set Solutions
* Misc.
* Definition Sheet
* Full Version
* PDF Version
1. Suppose in mod m arithmetic we know x ≠ y and
    $y^{2}\equiv x^{2}{\pmod {m}}\!$
find at least 2 divisors of m.
2. Derive the formula for the Carmichael function, λ(m) = smallest number such that a$^{λ(m)}$ ≡ 1 (mod m).
3. Let p be prime such that p = 2$^s$ + 1 for some positive integer s. Show that if g is not a square in mod p, i.e. there's no h such that h$^2$ ≡ g, then g is a generator mod p. That is g$^q$ ≠ 1 for all q < p - 1.
# High School Life Science/The Respiratory System

The respiratory system is a form of body cells that work together to bring about respiration.
# High School Life Science/What are Living Things?

Living things are organisms that can undertake all seven life processes or activities.
# Mechanisms/The Wheel

Wheel
A wheel is a circular component, intended to rotate about an axis to facilitate or simplify movement, transportation or performing labor in machines. It is one of the two main components of the Wheel and axle, one of the six simple mechanisms. It may have a tyre added to the outer rim to improve grip or cushion impacts.
# Mechanisms/Plain Bearings

A plain bearing is simply where two surfaces pass each other, either in rotation or sliding. Sometimes a special alloy such as "Muntz metal" is used for the bearing surfaces. Lubrication between the two surfaces is essential, to prevent them pressing against each other. The pressure within a correctly lubricated bearing produces a wedge of oil, that the rotating or sliding parts actually ride on!
It is called a "Bearing" because it has to bear the weight of the moving components, whilst still allowing free movement and not eroding too quickly!
# Mechanisms/Screw Thread

Screw Thread is a feature of one of the most used component in machinery which is fasteners. Fasteners comes under the category of Temporary joints which makes them versatile in application from carpentry to automobiles and from simple machine tools to large aerospace assemblies. The basic principle it works on is of pushing a load up to a slanted edge of a right angle triangle (just reference).
# Mechanisms/The Lever

A lever is a length of strong material like a pole or a pipe, that when moved a long way at one end, moves the other end a short distance, thus amplifying the force used to move the lever.
The point about which the lever moves is called "the Fulcrum". This needs to be a strong point that will not slip or crumble during use.
Levers can bend if they are not rigid enough, or even snap off if too much effort is used!
force$_1$ x distance$_1$ = force$_2$ x distance$_2$
# Mechanisms/Gears/Introduction and terminology

Gears are used to change speed of shafts in rotational movement. This is accomplished by successively engaging teeth. They rely on direct contact and do not use any intermediate link or connector.
# Mechanisms/Pistons

The piston, with its connecting rod and bearing, transfers the force of the combustion and expansion of the power stroke to the crankshaft. The piston itself, its rings, and the piston or gudgeon pin are together called the piston assembly. The cutaway shape on this piston allows it to clear the counterweights on this rotating crankshaft. The shape of the piston crown depends on the shape of its combustion chamber, and its compression ratio. In diesel engines, the combustion chamber may be formed totally or in part in the piston crown, depending on the method of injection, so they use pistons with different shapes. The piston crown may be flat, concave, dome, or recessed. The piston must stand up to great heat and pressure. It also must change direction from about 10 times a second to up to hundreds of times a second. In most engines, the weight of the pistons is important for engine balance. This is why pistons should only be replaced in matched sets. Some pistons are forged, while others are cast aluminum alloys. All pistons expand as they heat up. As there is more metal near the gudgeon pin, this area tends to expand the most. To allow for this, many pistons are machined into a slightly oval shape. This is called cam grinding. Then, as the piston heats up and expands, it becomes round. Other methods to control expansion include steel struts or ribs, expansion slots in the skirt, or slots called heat dams that restrict movement of the heat.
# Mechanisms/Ball Bearings

Types of bearings
* Thrust bearings
* Needle bearings
* Taper roller bearings
* Bush ball bearings
* Linear movement bearings
* Angular contact bearings
# Mechanisms/Cams

Cams are machine elements whose function is to control lateral movement through its rotational displacement. One application is the control of valve positions (they control inlet of air-fuel mixture as well as outlet of the exhaust gases e.g CO,CO2 & H2O). They are shaped to be eccentric, so that the axis of rotation is slightly offset, and to enable a follower to trace out the path that the make as they rotate. They need to be made of a material with a hard surface on the face that contacts the follower. The bearing surface is either a bush or a plain bearing drilled and ground in the material itself. They are often lubricated as they turn to try to minimise wear. Rotational speeds need to be quite low to maintain full contact of the follower wheel on the cam surface with out jumping. Cams have been superceeded in some machines by hydraulic or electric actuators that offer more flexibility in control of the displacement graph.
# Mechanisms/Gears

Gears are machine elements used for the transmission of rotary motion between two shafts. They can also be used to alter the speed and direction of motion. It is also usual that the transmitted motion be uniform, i.e. a constant motion of the driving gear produces constant motion of the driven gear It is this property which eventually determines much of the mathematics of the gear's profile.

## Types of Gears

* Introduction and terminology
* Spur gears
* Internal gears
* Bevel gears
* Worm gears
* Hypoid gears
* Planetary gears
* Helical gears
* Herringbone gears
HELICAL GEARS: it is the gear in which the teeth are inclined to the axis of rotation. the angle of inclination is called helix angle. the tooth surface generated in helical gear is called involute helicoid. this surface is generated when every point on the angular edge of the paper is cut in the shape of parallelogram wrapped around a base cylinder and unwound generate an involute curve.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/KYP Lemma with Feedthrough
## Definition

Consider a square, continuous-time LTI system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\longrightarrow {\mathcal {L}}_{2e}$, with minimal state-space realization ($A$, $B$, $C$, $D$), where $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{m\times n}$, and $D\in \mathbb {R} ^{m\times m}$.

## LMI 1.1

The system ${\mathcal {G}}$ is positive real (PR) under either of the following equivalent necessary and sufficient conditions.
* There exists $P\in \mathbb {S} ^{n}$, where $P>0$, such that ${\begin{bmatrix}PA+A^{T}P&PB-C^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$.
* There exists $Q\in \mathbb {S} ^{n}$, where $Q>0$, such that ${\begin{bmatrix}AQ+QA^{T}&B-QC^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$
This is a special case of the KYP Lemma for QSR dissipative systems with $Q=0$, $S={\frac {1}{2}}\cdot 1$, and $R=0$.

## LMIs 1.2

The system ${\mathcal {G}}$ is strictly positive real (SPR) under either of the following equivalent necessary and sufficient conditions.
* There exists $P\in \mathbb {S} ^{n}$, where $P>0$, such that ${\begin{bmatrix}PA+A^{T}P&PB-C^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$.
* There exists $Q\in \mathbb {S} ^{n}$, where $Q>0$, such that ${\begin{bmatrix}AQ+QA^{T}&B-QC^{T}\\*&-(D+D^{T})\end{bmatrix}}\leq 0$
This is a special case of the KYP Lemma for QSR dissipative systems with $Q=\epsilon 1$, $S={\frac {1}{2}}\cdot 1$, and $R=0$, where $\epsilon \in \mathbb {R} _{>0}$.

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A downloadable book on LMIs by Ryan James Caverly and James Richard Forbes.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Minimum Gain Lemma

LMIs in Control/Click here to continue/LMIs in system and stability Theory/Minimum Gain Lemma
Minimum Gain Lemma

## The System

Suppose there is a continuous-time LTI system where one must develop a controller for such system that is unstable. The minimum gain of a system would be obtained by taking the infimum of ratio between norms of the outputs and inputs over all nonzero inputs. With the Large Gain theorem expressing that if such an unstable system contains a finite value for the minimum gain of the system which is also non-zero, then any controller is capable of stabilizing the closed-loop feedback system, so long as the the controller also has a large minimum gain.
With this theorem led to the development of the Minimum Gain Lemma where applicable analysis can determine whether a closed loop system achieves a non-zero minimum gain value despite being inherently unstable in the open-loop case. By having an LTI system where the system $G$ corresponds to the following matrices $A\in \mathbb {R} ^{m\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, and $D\in \mathbb {R} ^{p\times m}$.

## The Data

A system of the plant $G$ will be required with the following matrices: $A\in \mathbb {R} ^{m\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, and $D\in \mathbb {R} ^{p\times m}$.

## The LMI: Minimum Gain Lemma

The following two LMI's are equivalent, with the same variable:
Suppose there exists $P\in \mathbb {S} ^{n}$, and $\upsilon \in \mathbb {R} _{\geq 0}$, Where $P\geq 0$ and:
       ${\begin{aligned}{\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}D\\(PB-C^{T}D)^{T}&\upsilon ^{2}I-D^{T}D\end{bmatrix}}&\leq 0\end{aligned}}$
Or (Obtained using Schur compliment):
       ${\begin{aligned}{\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}D&0\\(PB-C^{T}D)^{T}&-D^{T}D&\upsilon I\\0&\upsilon I&-I\end{bmatrix}}&\leq 0\end{aligned}}$

## Conclusion:

Solving this LMI will give the minimum gain from the LTI system. This minimum gain can then be used if the system has been made stable based on the value of $\upsilon$ obtained from the optimization.

## Implementation

* Minimum Gain Lemma Example

## 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.
* The minimum gain lemma - A paper about Minimum gain lemma by Bridgeman and Forbes
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Modified Minimum Gain Lemma

LMIs in Control/Click here to continue/LMIs in system and stability Theory/Modified Minimum Gain Lemma
Modified Minimum Gain Lemma

## The System

Suppose there is a continuous-time LTI system where one must develop a controller for such system that is unstable. The minimum gain of a system would be obtained by taking the infimum of ratio between norms of the outputs and inputs over all nonzero inputs. With the Large Gain theorem expressing that if such an unstable system contains a finite value for the minimum gain of the system which is also non-zero, then any controller is capable of stabilizing the closed-loop feedback system, so long as the the controller also has a large minimum gain.
With this theorem led to the development of the Minimum Gain Lemma where applicable analysis can determine whether a closed loop system achieves a non-zero minimum gain value despite being inherently unstable in the open-loop case. By having an LTI system where the system $G$ corresponds to the following matrices $A\in \mathbb {R} ^{m\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, and $D\in \mathbb {R} ^{p\times m}$.
Essentially the same concept as the Minimum Gain Lemma; however, this modification ensure that the system being optimized is also Lyapunov stable. This Essentially means that the variable $Q$ must ensure that the system is stable to this degree, otherwise the system itself is infeasible.

## The Data

A system of the plant $G$ will be required with the following matrices: $A\in \mathbb {R} ^{m\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, and $D\in \mathbb {R} ^{p\times m}$.

## The LMI: Modified Minimum Gain Lemma

The following two LMI's are equivalent, with the same variable:
Suppose there exists $Q\in \mathbb {S} ^{n}$, and $\upsilon \in \mathbb {R} _{\geq 0}$, where $Q>0$ such that:
       ${\begin{aligned}{\begin{bmatrix}AQ+QA^{T}&B-QC^{T}D\\(B-QC^{T}D)^{T}&\upsilon ^{2}I-D^{T}D\end{bmatrix}}&\leq 0\end{aligned}}$
Or (Obtained using Schur compliment):
       ${\begin{aligned}{\begin{bmatrix}AQ+QA^{T}&B-QC^{T}D&0\\(B-QC^{T}D)^{T}&-D^{T}D&\upsilon I\\0&\upsilon I&-I\end{bmatrix}}&\leq 0\end{aligned}}$

## Conclusion:

Solving this LMI will give the minimum gain from the LTI system. This minimum gain can then be used if the system has been made stable based on the value of $\upsilon$ obtained from the optimization. This system will also prove that the plant is Lyapunov stable.

## Implementation

* Minimum Gain Lemma Example

## 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.
* The minimum gain lemma - A paper about Minimum gain lemma by Bridgeman and Forbes
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Kharitonov-Bernstein-Haddad(KBH) Theorem

WIP, Description in progress

## Kharitonov-Bernstein-Haddad (KBH) Theorem

Consider the set of matrices
${\mathcal {A}}=\{{\textbf {A}}={\begin{bmatrix}{\textbf {0}}_{(n-1)\times 1}&&{\textbf {1}}_{(n-1)\times (n-1)}\\-a_{0}&\cdots &-a_{n-1}\end{bmatrix}}|{\underline {a_{j}}}\leq a_{j}\leq {\overline {a_{j}}},j=0,1,2,\cdots ,n-1\}$
Every matrix in the set ${\mathcal {A}}$ is Hurwitz if and only if there exist ${\textbf {P}}_{i}\in mathbb{S}^{n},i=1,2,3,4$ where ${\textbf {P}}_{i}>0,i=1,2,3,4,$ such that
${\textbf {P}}_{i}A_{i}+{\textbf {A}}_{i}^{T}{\textbf {P}}_{i}<0,\quad i=1,2,3,4,$
where
${\textbf {A}}_{i}={\begin{bmatrix}{\begin{bmatrix}{\textbf {0}}_{(n-1)\times 1}&{\textbf {1}}_{(n-1)\times (n-1)}\end{bmatrix}}\\{\textbf {a}}_{i}\end{bmatrix}},\quad i=1,2,3,4,$
${\textbf {a}}_{1}=-[{\underline {a}}_{0}\quad {\underline {a}}_{1}\quad {\overline {a}}_{2}\quad {\overline {a}}_{3}\quad \cdots \quad {\underline {a}}_{n-4}\quad {\underline {a}}_{n-3}\quad {\overline {a}}_{n-2}\quad {\overline {a}}_{n-1}],$
${\textbf {a}}_{2}=-[{\underline {a}}_{0}\quad {\overline {a}}_{1}\quad {\overline {a}}_{2}\quad {\underline {a}}_{3}\quad \cdots \quad {\underline {a}}_{n-4}\quad {\overline {a}}_{n-3}\quad {\overline {a}}_{n-2}\quad {\underline {a}}_{n-1}],$
${\textbf {a}}_{3}=-[{\overline {a}}_{0}\quad {\underline {a}}_{1}\quad {\underline {a}}_{2}\quad {\overline {a}}_{3}\quad \cdots \quad {\overline {a}}_{n-4}\quad {\underline {a}}_{n-3}\quad {\underline {a}}_{n-2}\quad {\overline {a}}_{n-1}],$
${\textbf {a}}_{3}=-[{\overline {a}}_{0}\quad {\underline {a}}_{1}\quad {\underline {a}}_{2}\quad {\overline {a}}_{3}\quad \cdots \quad {\overline {a}}_{n-4}\quad {\underline {a}}_{n-3}\quad {\underline {a}}_{n-2}\quad {\overline {a}}_{n-1}],$
${\textbf {a}}_{4}=-[{\overline {a}}_{0}\quad {\overline {a}}_{1}\quad {\underline {a}}_{2}\quad {\underline {a}}_{3}\quad \cdots \quad {\overline {a}}_{n-4}\quad {\overline {a}}_{n-3}\quad {\underline {a}}_{n-2}\quad {\underline {a}}_{n-1}].$
WIP, additional references to be added

## External Links

A list of references documenting and validating the LMI.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control

LMI For Stabilization Condition for Systems With Unsymmetrical Saturated Control
The LMI in this page gives the feasibility conditions which, if satisfied, imply that the correstponding system can be stabilized.

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+BSat(u(t)),\\x(0)&=x_{0},\\Sat(u)_{i}=,\end{aligned}}$
where $x\in \|R^{n}$ is the state, $u\in \|R^{m}$ is the control input.
For the system given as above, its symmetrical saturated control form can be derived by following the procedure in the original article. The new system will have the form:
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+{\tilde {B}}sat(z(t))+Ew\end{aligned}}$
where $w_{i}=u_{i}-{\frac {\alpha _{i}-\beta _{i}}{2}},z_{i}=w_{i}{\frac {2}{\alpha _{i}+\beta _{i}}}$

## The Data

The system matrices $(A,{\tilde {B}},E)$, the saturation bounds $(\alpha _{i},\beta _{i})$ of the control inputs. Positive scalars $\rho ,\eta$.

## The LMI: The Stabilization Feasibility Condition

       ${\begin{aligned}&{\text{Find}}\;X,Y,Z:\\&{\text{subj. to: }}X>0,\\&\quad {\begin{bmatrix}AX+{\tilde {B}}(D_{s}Y+{\hat {D}}_{s}^{-}Z)\end{bmatrix}}+{\begin{bmatrix}AX+{\tilde {B}}(D_{s}Y+{\hat {D}}_{s}^{-}Z)\end{bmatrix}}^{\top }+{\frac {\eta }{\rho }}X+{\frac {1}{\eta }}EE^{\top }<0,\\{\begin{bmatrix}\mu &Z_{i}\\*X\end{bmatrix}}>0,i=1,...,{\bar {m}}\end{aligned}}$
Here $D_{s}$ is a diagonal matrix with a component either 0 or 1, and $D_{s}+D_{s}^{-}={\frac {\Lambda +\mathrm {T} }{2}}$ and ${\hat {D}}_{s}^{-}=e_{f_{m}}\times D_{s}^{-}$

## Conclusion:

The feasibility of the given LMI implies that the system is stabilizable with control gains $K=YX^{-1},H=ZX^{-1}$.

## Implementation

A link to CodeOcean or other online implementation of the LMI

## External Links

* Stabilization of Systems with Unsymmetrical Saturated Control: An LMI Approach Link to the original article.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/LMI for Schur Stabilization

LMI for Schur Stabilization
Similar to the stability of continuous-time systems, one can analyze the stability of discrete-time systems. 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 and a linear time-invariant system with this property is called a Schur stable system.

## The System

We consider the following system:
${\begin{aligned}x(k+1)=Ax(k)+Bu(k)\end{aligned}}$
where the matrices $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times r}$, $x\in \mathbb {R} ^{n}$, and $u\in \mathbb {R} ^{r}$ are the state matrix, input matrix, state vector, and the input vector, respectively.
Moreover, $k$ represents time in the discrete-time system and $k+1$ is the next time step.
The state feedback control law is defined as follows:
${\begin{aligned}u(k)=Kx(k)\end{aligned}}$
where $K\in \mathbb {R} ^{n\times r}$ is the controller gain. Thus, the closed-loop system is given by:
${\begin{aligned}x(k+1)=(A+BK)x(k)\end{aligned}}$

## The Data

The matrices $A$ and $B$ are given.
We define the scalar as $\gamma$ with the range of $0<\gamma \leq 1$.

## 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:
${\begin{aligned}{\begin{bmatrix}-\gamma I&(A+BK)\\(A+BK)^{T}&-\gamma I\end{bmatrix}}<0\\\end{aligned}}$

## The LMI: LMI for Schur stabilization

The LMI for Schur stabilization can be written as minimization of the scalar, $\gamma$, in the following constraints:
${\begin{aligned}&{\text{min}}\quad \gamma \\&{\text{s.t.}}\quad {\begin{bmatrix}-\gamma I&(A+BK)\\(A+BK)^{T}&-\gamma I\end{bmatrix}}<0\\\end{aligned}}$

## Conclusion:

After solving the LMI problem, we obtain the controller gain $K$ and the minimized parameter $\gamma$. This problem is a special case of Intensive Disk Region Design (page 230 in [1]). This problem may not have a solution even when the system is stabilizable. In other words, once there exists a solution, the solution is robust in the sense that when there are parameter perturbations, the closed-loop system's eigenvalues are not easy to go outside of a circle region within the unit circle [1].

## Implementation

A link to Matlab codes for this problem in the Github repository:
https://github.com/asalimil/LMI-for-Schur-Stability

## Related LMIs

LMI for Hurwitz stability

## External Links

* - LMI in Control Systems Analysis, Design and Applications

## Return to Main Page

LMIs in Control/Tools