OpenLLM-France/wikimedia huggingface dataset:
# Minecraft/Netherite

Minecraft
Blocks | Items | Mobs
Structures
Miscellaneous
Guides
The strongest mineral in Minecraft.

## Obtaining

To obtain it, you can smelt ancient debris to netherite scraps, and craft 4 netherite scraps with 4 golden ingots.

## Use

While it cannot be crafted with to make tools on a crafting table, you can use a smithing table to combine it with diamond tools, weapons, and armour.
You can also use 9 netherite ingots to craft a netherite block, or 1 netherite ingot to craft a lodestone.

## Notes

* Netherite gear, weapons, and armor (the Netherite toolset) cannot be destroyed with lava or fire. However, they can still be destroyed by a lowly cactus.
# Minecraft/Panda

Pandas are passive mobs that spawn in jungles. They can be bred with bamboo. They can drop bamboo and up to three xp.
# Minecraft/Pig
## Pig

Minecraft
Blocks | Items | Mobs
Structures
Miscellaneous
Guides
Your pink friends! Looks like bacon sandwich is for lunch today!

## Abilities and strengths/weaknesses

Other
This mob has no special abilities or disabilities.

### Drops

* 1 - 2 raw porkchops
* 1 - 3 exp. points

## Description

You can equip them with a saddle, and ride on them. To control their movement, you need a carrot-on-stick. They can then be encouraged a small speed boost by pressing "use", but this costs durability of the carrot-on-stick.
Can be bred with carrots, potatoes or beetroot.

## Achivements
### When pigs fly

Ride a pig off a cliff.

## Strategies

None

## Mobs

* Animals
       Sheep
       Pig
       Cow
       Chicken
       Bat
       Cat
       Ocelot
       Wolf
       Fox
       Horse
       Donkey
       Mule
       Llama
       Bunny
       Frog
       Axolotl
       Tadpole
       Panda
       Mooshroom
* Monsters
       Creeper
       Zombie
       Skeleton
       Stray
       Witch
       Slime
       Phantom
       Warden
* Ocean
       Cod
       Pufferfish
       Salmon
       Tropical Fish
       Squid
       Glow Squid
       Dolphin
       Turtle
       Guardian
       Elder Guardian
* Nether
       Blaze
       Ghast
       Piglin
       Zombified Piglin
       Hoglin
       Strider
       Wither Skeleton
* End
       Enderman
       Endermite
       Shulker
* Illager/Villagers
       Villager
       Wandering Trader
       Illager
       Pillager
       Evoker
       Vex
       Ravanger
* Utility
       Snow Golem
       Iron Golem
       Allay
* Bosses
       Wither
       Ender Dragon
* Secret mobs
       Npc
# Minecraft/Pillager Raid

Minecraft
Blocks | Items | Mobs
Structures
Miscellaneous
Guides
A pillager raid is an event that occurs in villages after killing a pillager captain and obtaining the "bad omen" debuff.

## What happens

Up to seven waves of increasing difficulty occurs. Each wave, enemies spawn from one direction to attack the village. To progress to the next wave, you must kill all the enemies.
If you manage to beat every wave, you will get the "hero of the village" buff, which decreases the emerald cost of all villagers' items by up to 5 emeralds, but cannot decrease the cost to below 1.

## Enemies

The following enemies can spawn in a raid:
* Pillager
* Illager
* Evoker
* Witch
* Ravanger (only on wave 5 +)

## Notes

* By drinking milk, you can remove the bad omen and hero of the village effects, avoiding the raid or removing the villager discounts.

## Achivements
### We need a hero!

Beat a pillager raid.
# Minecraft/Npc
## Npc

Minecraft
Blocks | Items | Mobs
Structures
Miscellaneous
Guides
Mobs summoned with cheats, used for adventure maps.

## Abilities and strengths/weaknesses

Other
This mob has no special abilities or disabilities.

### Drops

Nothing

## Description

They can be given text, buttons with commands and run commands when the ui is opened or closed. You can also change their appearance.
Summon by typing the following in the chat window: /summon npc ~~~.
Npcs can be modified by changing commands and dialog. When giving it commands, it may run them when a button is pressed or when the dialog is opened or closed. The commands can be separated with a new line, unlike when using command blocks. A new selector becomes available: @initiator. This selector selects the player that interacted with the npc.

## Strategies

None

## Mobs

* Animals
       Sheep
       Pig
       Cow
       Chicken
       Bat
       Cat
       Ocelot
       Wolf
       Fox
       Horse
       Donkey
       Mule
       Llama
       Bunny
       Frog
       Axolotl
       Tadpole
       Panda
       Mooshroom
* Monsters
       Creeper
       Zombie
       Skeleton
       Stray
       Witch
       Slime
       Phantom
       Warden
* Ocean
       Cod
       Pufferfish
       Salmon
       Tropical Fish
       Squid
       Glow Squid
       Dolphin
       Turtle
       Guardian
       Elder Guardian
* Nether
       Blaze
       Ghast
       Piglin
       Zombified Piglin
       Hoglin
       Strider
       Wither Skeleton
* End
       Enderman
       Endermite
       Shulker
* Illager/Villagers
       Villager
       Wandering Trader
       Illager
       Pillager
       Evoker
       Vex
       Ravanger
* Utility
       Snow Golem
       Iron Golem
       Allay
* Bosses
       Wither
       Ender Dragon
* Secret mobs
       Npc
# Minecraft/Other

Minecraft
Blocks | Items | Mobs
Structures
Miscellaneous
Guides

## The basics

* Health
* Hunger
* Crafting
* Armour

## General

* Experience
* Enchantments
* Brewing Reference
* Buffs
* Debuffs
* minerals

## Mobs

* Breeding
* Pillager Raid
* Zombie Seige
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-time strong stabilizability
## The System

Consider the continous-time LTI system, $G:L_{2e}\rightarrow L_{2e}$ with state-space realization ($A_{d},B_{d},C_{d},0$)
    ${\begin{aligned}{\dot {x}}(k)&=A_{d}x(k)+B_{d}u(k),\\y(k)&=C_{d}x(k),\end{aligned}}$
where $A_{d}\in \mathbb {R} ^{n\times n}$, $B_{d}\in \mathbb {R} ^{n\times m}$, $C_{d}\in \mathbb {R} ^{p\times n}$, and it and it is assumed that ($A_{d},B_{d}$),is stabilizable, ($A_{d},C_{d}$) is detectable, and the transfer matrix $G(s)=C_{d}(s1-A_{d})^{-1})B_{d}$ has no poles on the imaginary axis.

## The Data

The matrices $A_{d},B_{d},C_{d}$.

## The Optimization Problem

The system G is strongly stabilizable if there exist $P\in \mathbb {S} ^{n}$, $Z\in \mathbb {R} ^{n\times p}$, and $\gamma \in \mathbb {R} _{>0}$, where $P>0$, such that
    ${\begin{aligned}{\displaystyle {\begin{aligned}{\begin{bmatrix}A_{d}^{T}PA_{d}-P-A_{d}^{T}ZC_{d}-C_{d}^{T}Z^{T}A_{d}&&C_{d}^{T}Z^{T}\\*&&-P\\*&&*&&-\gamma I\end{bmatrix}}<0\end{aligned}}}\\{\displaystyle {\begin{aligned}{\begin{bmatrix}N_{11}&&(A_{d}+B_{d}F)^{T}Z&&XB_{d}&&C_{d}^{T}Z^{T}\\*&&-\gamma I&&0&&Z^{T}\\*&&*&&-\gamma I&&0\\*&&*&&*&&-P\end{bmatrix}}<0\end{aligned}}}\\\end{aligned}}$

## Conclusion:

where $N_{11}=(A_{d}+B_{d}F)^{T}P(A_{d}+B_{d}F)-P+(A_{d}+B_{d}F)^{T}ZC_{d}^{T}Z^{T}(A_{d}+B_{d}F),F=-B_{d}^{T}X,X=Y$ and $X\in S_{n}$ , $X\geq 0$ is the solution to the discrete-time Lyapunov equation given by
    ${\begin{aligned}A_{d}XA_{d}^{T}+X-B_{d}B_{d}^{T}X=0\end{aligned}}$
Moreover, a controller that strongly stabilizes G is given by the state-space realization
    ${\begin{aligned}{\dot {x}}_{c,k+1}=(A_{d}+B_{d}F+P^{-1}ZC_{d})x_{k}-P^{-1}Zu_{k}\\y_{c,k}=-B_{d}^{T}Xx_{k}\end{aligned}}$

## Implementation

* -example code

## Related LMIs

* https://en.wikibooks.org/wiki/LMIsinControl/StabilityAnalysis/ContinuousTime/StrongStabilizability - Continuous Time Strong Stabilizability

## External Links

* http://control.asu.edu/MAE598_frame.htm LMI Methods in Optimal and Robust Control- A course on LMIs in Control by Matthew Peet.\\
* https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory] - A List of LMIs by Ryan Caverly and James Forbes.\\
* https://web.stanford.edu/~boyd/lmibook/ LMIs in Systems and Control Theory- A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/General LMI Region D-Admissibility
## The System

Consider $A,E\in \mathbb {R} ^{n\times n}$. The pair ($E$,$A$) is D-admissible if it is regular and causal, and the eigenvalues of ($E$,$A$) lie within the LMI region D of the complex plane, which is defined as
$D=\{z\in \mathbb {C} :f_{D}(z)<0}\$, where
$f_{D}(z):=\Lambda +z\Phi +{\bar {z}}\Phi ^{T}=[\lambda _{kl}+\Phi _{kl}z+\Phi _{lk}{\bar {z}}]_{1\leq k,l\leq m},$
$\Lambda \in \mathbb {S} ^{m},\Phi \in \mathbb {R} ^{m\times m}$, and ${\bar {z}}$ is the complex complex conjugate of $z$.

## Conditions

The pair ($E$,$A$) is D-admissible if and only if any of the following equivalent conditions are satisfied.
* There exist $P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},$ $U,V\in \mathbb {R} ^{n\times (n-n_{e})},$ where $n_{e}=$ rank$(E),{\mathcal {R}}(U)={\mathcal {N}}(E^{T}),{\mathcal {R}}(V)={\mathcal {N}}(E),$and $P>0,$ satisfying $[\lambda _{kl}EPE^{T}+\phi _{kl}E^{T}PA^{T}+AVSU^{T}+US^{T}V^{T}A^{T}]_{1\leq k,l\leq m}<0,$
* There exist $P,Q\in \mathbb {S} ^{n},$ where $P>0,$ satisfying $E^{T}QE\geq 0$ and $[\lambda _{kl}EPE^{T}+\phi _{kl}APE+\phi _{lk}E^{T}PA^{T}+A^{T}QA]_{1\leq k,l\leq m}<0,$
* There exist $P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},U\in \mathbb {R} ^{n\times (n-n_{e})},$ where $n_{e}=$ rank$(E),UE=0,$ and $P>0,$ satisfying $[\lambda _{kl}EPE^{T}+\phi _{kl}APE+\phi _{lk}E^{T}PA^{T}+A^{T}U^{T}SUA]_{1\leq k,l\leq m}<0,$
* There exist $P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},$$U,V\in \mathbb {R} ^{n\times (n-n_{e})},$ where $n_{e}=$ rank$(E),{\mathcal {R}}(U)={\mathcal {N}}(E^{T}),{\mathcal {R}}(V)={\mathcal {N}}(E),$ and $P>0,$ satisfying $\Lambda \otimes EPE^{T}+\Phi \otimes (APE)+\Phi ^{T}\otimes (EPA^{T})+1_{mm}\otimes (AVSU^{T}+US^{T}V^{T}A^{T})<0,$ where $\otimes$ is the Kroenecker product and $1_{mm}$ is an $m\times m$ matrix filled with ones.
* There exist $P,Q\in \mathbb {S} ^{n},$ where $P>0,$ satisfying $E^{T}QE\geq 0$ and $\Lambda \otimes EPE^{T}+\Phi \otimes (APE)+\Phi ^{T}\otimes (EPA^{T})+1_{mm}\otimes (A^{T}QA)<0,$ where $\otimes$ is the Kroenecker product and $1_{mm}$ is an $m\times m$ matrix filled with ones.
* There exist $P\in \mathbb {S} ^{n},S\in \mathbb {R} ^{(n-n_{e})\times (n-n_{e})},U\in \mathbb {R} ^{n\times (n-n_{e})},$ where $n_{e}=$ rank$(E),UE=0,$ and $P>0,$ satisfying $\Lambda \otimes EPE^{T}+\Phi \otimes (APE)+\Phi ^{T}\otimes (EPA^{T})+1_{mm}\otimes (A^{T}U^{T}SUA)<0,$ where $\otimes$ is the Kroenecker product and $1_{mm}$ is an $m\times m$ matrix filled with ones.

## Reference

Caverly, Ryan; Forbes, James (2021). LMI Properties and Applications in Systems, Stability, and Control Theory.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Dissipativity of Affine Parametric Varying Systems
## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{w}w(t),\\z(t)&=C_{z}(\theta )x(t)+D_{zw}(\theta )w(t),\end{aligned}}$
where $C_{z}$ and $D_{zw}$ depend affinely on parameter $\theta \in \mathbb {R} ^{p}$.

## The Data

The matrices $A,B_{w},C_{z}(.),D_{zw}(.)$.

## The Optimization Problem:

Solve the following semi-definite program
       ${\begin{aligned}&\min _{\{P\succ 0,\gamma ,\theta \}}\gamma \\&\quad s.t.\quad {\begin{bmatrix}A^{\top }P+PA&PB_{w}-C_{z}(\theta )^{\top }\\B_{w}^{\top }P-C_{z}(\theta )&2\gamma I-D_{zw}(\theta )-D_{zw}(\theta )^{\top }\end{bmatrix}}\preceq 0.\end{aligned}}$

## Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/b6cd6b81f75be4a2052ba3fa76cad1a2f9c49caa

## Conclusion

The dissipativity of $H_{\theta }$ (see [Boyd,eq:6.59]) exceeds ${\gamma }$ if and only if the above LMI holds and the value function returns the minimum provable dissipativity.

## Remark

It is worth noticing that passivity corresponds to zero dissipativity.

## 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/Click here to continue/LMIs in system and stability Theory/Entropy Bond for Affine Parametric Varying Systems
## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{w}w(t),\\z(t)&=C_{z}(\theta )x(t)+D_{zw}(\theta )w(t),\end{aligned}}$
where $C_{z}$ and $D_{zw}$ depend affinely on parameter $\theta \in \mathbb {R} ^{p}$.

## The Data

The matrices $A,B_{w},C_{z}(.),D_{zw}(.)$.

## The Optimization Problem:

Solve the following semi-definite program
       ${\begin{aligned}&\min _{\{P\succ 0,\gamma ^{2},\lambda ,\theta \}}\gamma ^{2}\\&\quad s.t.\quad D_{zw}(\theta )=0,{\begin{bmatrix}A^{\top }P+PA&PB_{w}&C_{z}(\theta )^{\top }\\B_{w}^{\top }P&-\gamma ^{2}I&0\\C_{z}(\theta )&0&-I\end{bmatrix}}\preceq 0,\quad {\rm {{Tr}(B_{w}^{\top }PB_{w})\leq \lambda .}}\end{aligned}}$

## Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/02f31a2d7a22b2464dfe9212eb76409bda9439b1

## Conclusion

The value function of the above semi-definite program returns a bound for $\gamma$-entropy of the system, which is defined as
       ${\begin{aligned}I_{\gamma }(H_{\theta })\triangleq {\begin{cases}{\frac {-\gamma ^{2}}{2\pi }}\int _{-\infty }^{\infty }\log \det(I-\gamma ^{2}H_{\theta }(i\omega )H_{\theta }(i\omega )^{*})d\omega ,\quad {\text{if}}\ \|H_{\theta }\|_{\infty }<\gamma \\\infty ,\quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad \quad {\text{otherwise.}}\end{cases}}\end{aligned}}$

## Remark

When it is finite, $I_{\gamma }(H_{\theta })$ is given by ${\rm {{Tr}(B_{w}^{\top }PB_{w})}}$ where $P$, is asymmetric matrix with the smallest possible maximum singular value among all solutions of a Riccati equation.

## 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/Click here to continue/LMIs in system and stability Theory/Discrete-time Negative Imaginary Lemma
## The System

Given a square, discrete-time LTI system G: L$_{2e}$ --> L$_{2e}$ with state-space realization (A$_d$, B$_d$, C$_d$, D$_d$) where
$A_{d}\in \mathbb {R} ^{nxn}$, $B_{d}\in \mathbb {R} ^{nxm}$, $C_{d}\in \mathbb {R} ^{mxn}$, and $D_{d}\in \mathbb {R} ^{mxm}$.
In this system, $C_{d}(z1-A_{d})^{-1}B_{d}+D_{d}=B_{d}^{T}(z1-A_{d}^{T})^{-1}C_{d}^{T}+D_{d}^{T}$ and $det(1+A)\neq 0$ and $det(1-A)\neq 0$.

## The Data

$A_{d}\in \mathbb {R} ^{nxn}$, $B_{d}\in \mathbb {R} ^{nxm}$, $C_{d}\in \mathbb {R} ^{mxn}$, and $D_{d}\in \mathbb {R} ^{mxm}$

## The LMI:

The system G posed above is considered to be negative imaginary under either of the sufficient and necessary conditions:
* There exists $P\in \mathbb {S} ^{n}$, where P > 0 such that
$A_{d}^{T}PA_{d}-P\leq 0,$
$C_{d}+B_{d}^{T}(A_{d}^{T}-1)^{-1}P(A_{d}+1)=0$
2. There exists $Q\in \mathbb {S} ^{n}$, where Q > 0 such that
$A_{d}QA_{d}^{T}-Q\leq 0,$
$B_{d}+(A_{d}-1)^{-1}Q(A_{d}^{T}+1)C_{d}^{T}=0$

## Conclusion

By using the LMI described above, a discrete LTI system can be evaluated for the negative imaginary condition.

## Implementation

This LMI can be implemented in any LMI solver such as YALMIP, using an algorithmic solver like MOSEK.

## Related LMIs

* Negative Imaginary Lemma

## External Links

* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete - time Bounded Real Lemma

Discrete-Time Bounded Real Lemma
A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.
Discrete-Time Bounded Real Lemma or the H∞ norm can be found by solving a LMI.

## The System

Discrete-Time LTI System with state space realization $(A_{d},B_{d},C_{d},D_{d})$ ${\begin{aligned}&A_{d}\in {\bf {{R^{n*n}},}}&B_{d}\in {\bf {{R^{n*m}},}}&C_{d}\in {\bf {{R^{p*n}},}}&D_{d}\in {\bf {{R^{p*m}}\;}}\\\end{aligned}}$

## The Data

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

## The Optimization Problem

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

## The LMI:

Discrete-Time Bounded Real Lemma
The LMI formulation
H∞ norm < $\gamma$
${\begin{aligned}P\in {S^{n}};\gamma \in {R_{>0}}\;\\&P>0,\\{\begin{bmatrix}A_{d}^{T}PA_{d}-P&A_{d}^{T}PB_{d}&C_{d}^{T}\\*&B_{d}^{T}PB_{d}-\gamma I&D_{d}^{T}\\*&*&-\gamma I\end{bmatrix}}&<0,\end{aligned}}$

## Conclusion:

The H∞ norm is the minimum value of $\gamma \in {R_{>0}}$that satisfies the LMI condition. If $(A_{d},B_{d},C_{d},D_{d})$ is the minimal realization then the inequalities can be non-strict.

## Implementation

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

## Related LMIs

- Continuous time KYP_Lemma_(Bounded_Real_Lemma)

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete-time Stabilizability

Discrete-Time Stabilizability
A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.
Discrete-Time LTI systems can be made stable using controller gain K, which can be found using LMI optimization, such that the close loop system is stable.

## The System

Discrete-Time LTI System with state space realization $(A_{d},B_{d},C_{d},D_{d})$ ${\begin{aligned}&A_{d}\in {\bf {{R^{n*n}},}}&B_{d}\in {\bf {{R^{n*m}},}}&C_{d}\in {\bf {{R^{p*n}},}}&D_{d}\in {\bf {{R^{p*m}}\;}}\\\end{aligned}}$

## The Data

The matrices: System $(A_{d},B_{d},C_{d},D_{d}),P,W$.

## The Optimization Problem

The following feasibility problem should be optimized:
Maximize P while obeying the LMI constraints. Then K is found.

## The LMI:

Discrete-Time Stabilizability
The LMI formulation
${\begin{aligned}P\in {S^{n}};W\in {R^{m*n}}\;\\&P>0\\{\begin{bmatrix}P&A_{d}P+B_{d}W\\*&P\end{bmatrix}}&>0,\\K_{d}=WP^{-1}\end{aligned}}$

## Conclusion:

The system is stabilizable iff there exits a $P$, such that $P>0$. The matrix $A_{d}+B_{d}K_{d}$ is Schur with $K_{d}=WP^{-1}$

## Implementation

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

## Related LMIs

- Continuous Time Stabilizability

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Discrete - time H2 Norm

Discrete-Time H2 Norm
A discrete time system operates on a discrete time signal input and produces a discrete time signal output. They are used in digital signal processing, such as digital filters for images or sound. The class of discrete time systems that are both linear and time invariant, known as discrete time LTI systems.
Discrete-Time LTI systems' H2 norm can be found by solving a LMI.

## The System

Discrete-Time LTI System with state space realization $(A_{d},B_{d},C_{d},0)$ ${\begin{aligned}A_{d}\in {\bf {{R^{n*n}},\ B_{d}\in {\bf {{R^{n*m}},\ C_{d}\in {\bf {R^{p*n}}}}}}}\end{aligned}}$

## The Data

The matrices: System $(A_{d},B_{d},C_{d},0),P,Z$.

## The Optimization Problem

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

## The LMI:

Discrete-Time Bounded Real Lemma
The LMI formulation
H2 norm < $\mu$
${\begin{aligned}P\in {S^{n}};Z\in {S^{p}};\mu \in {R_{>0}}\;\\&P>0,&Z>0\\{\begin{bmatrix}P&A_{d}P&B_{d}\\*&P&0\\*&*&I\end{bmatrix}}&>0,\\{\begin{bmatrix}Z&C_{d}P\\*&P\end{bmatrix}}&>0,\\trZ<\mu ^{2}\end{aligned}}$

## Conclusion:

The H2 norm is the minimum value of $\mu \in {R_{>0}}$that satisfies the LMI condition.

## Related LMIs

- Continuous time H2 norm.

## 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.
* - MATLAB function for the H2 norm.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Exterior Conic Sector Lemma
## The Concept

The conic sector theorem is a powerful input-output stability analysis tool, providing a fine balance between generality and simplicity of system characterisations that is conducive to practical stability analysis and robust controller synthesis.

## The System

Consider a square, contiuous-time linear time-invariant (LTI) system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (A, B, C, D), where ${\mathcal {E,A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},{\mathcal {C}}\in {\mathcal {R}}^{p\times n},$ and ${\mathcal {D}}\in {\mathcal {R}}^{p\times m}$.
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)+Du(t)\\\end{aligned}}$

## The Data

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

## LMI : Exterior Conic Sector Lemma

The system ${\mathcal {G}}$ is in the exterior cone of radius r centered at c (i.e. ${\mathcal {G}}\in$excone$_r$(c)), where $r\in {\mathcal {R}}_{>0}$ and $\in {\mathcal {R}}$, under either of the following equivalent necessary and sufficient conditions.
    1. There exists P $\in {\mathcal {S}}^{n}$, where P $\geq 0$, such that
       ${\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}(D-CI)\\(PB-C^{T}(D-CI))^{T}&r^{2}I-(D-cI)^{T}(D-cI)\end{bmatrix}}\leq 0.$
    2. There exists P $\in {\mathcal {S}}^{n}$, where P $\geq 0$, such that
       ${\begin{bmatrix}PA+A^{T}P-C^{T}C&PB-C^{T}(D-CI)&0\\(PB-C^{T}(D-CI))^{T}&-(D-cI)^{T}(D-cI)&rI\\0&(rI)^{T}&-I\end{bmatrix}}\leq 0.$
Proof, Applying the Schur complement lemma to the $r^{2}I$ terms in (1) gives (2).

## Conclusion:

If there exist a positive definite $P$ matrix satisfying above LMIs then the system ${\mathcal {G}}$ is in the exterior cone of radius r centered at c.

## Implementation

Code for implementation of this LMI using MATLAB. https://github.com/VJanand25/LMI

## Related LMIs

KYP Lemma
State Space Stability

## References

1. J. C. Willems, “Dissipative dynamical systems - part I: General theory,” Archive Rational Mechanics and Analysis, vol. 45, no. 5, pp. 321–351, 1972.
2. D. J. Hill and P. J. Moylan, “The stability of nonlinear dissipative systems,” IEEE Transactions on Automatic Control, vol. 21, no. 5, pp. 708–711, 1976.
3. LMI Properties and Applications in Systems, Stability, and Control Theory, by Ryan James Caverly1 and James Richard Forbes2
4. Bridgeman, Leila Jasmine, and James Richard Forbes. "The exterior conic sector lemma." International Journal of Control 88.11 (2015): 2250-2263.