OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/pages/Polytopic stability

An important result to determine the stability of the system with uncertainties

## The System:

Consider the system with Affine Time-Varying uncertainty (No input)
    ${\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta A(t))x(t)\\\end{aligned}}$
where
    ${\begin{aligned}\Delta A(t)=A_{1}\delta _{1}(t)+....+A_{k}\delta _{k}(t)\\\end{aligned}}$
where $\delta _{i}(t)$ lies in either the intervals
    ${\begin{aligned}\delta _{i}\in [\delta _{i}^{-},\delta _{i}^{+}]\\\end{aligned}}$
or the simplex
    ${\begin{aligned}\delta (t)\in {\delta :\Sigma \alpha _{i}=1,\alpha \geq 0}\end{aligned}}$
where $x\in \mathbb {R} ^{m}$ and $A\in \mathbb {R} ^{mxm}$

## The Data

The matrix A and $\Delta _{A(t)}$ are known

## The Optimization

The Definitions: Quadratic Stability for Dynamic Uncertainty
The system
    ${\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta A(t))x(t)\\\end{aligned}}$
is Quadraticallly Stable over $\Delta$ if there exists a P > 0
Theorem $(A+\Delta ,{\boldsymbol {\Delta }})$ is quadratically stable over ${\boldsymbol {\Delta }}:=Co(A_{1},...,A_{k})$ if and only if there exists a P > 0 such that
    ${\begin{aligned}(A+A_{i})^{T}P+P(A+A_{i})<0\quad for\quad all\quad i=1,....,k\end{aligned}}$
The theorem says the LMI only needs to hold at the EXTREMAL POINTS or VERTICES of the polytope.
* Quadratic Stability MUST be expressed as an LMI

## The LMI

    ${\begin{aligned}(A+\Delta )^{T}P+P(A+\Delta )<0\quad for\quad all\quad \Delta \in {\boldsymbol {\Delta }}\end{aligned}}$

## Conclusion:

Quadratic Stability Implies Stability of trajectories for any $\Delta$ with $\Delta \in {\boldsymbol {\Delta }}$ for all $t\geq 0$
Quadratic Stability is CONSERVATIVE.
There are Stable System which are not Quadratically stable. Quadratic Stability is sometimes referred to as an "infinite-dimensional LMI"
* Meaning it represents an infinite number of LMI constraints.
* One for each possible value $\Delta$ with $\Delta \in {\boldsymbol {\Delta }}$
* Also called a parameterized LMI
* Such LMIs are not tractable.
* For polytopic sets, the LMI can be made finite.

## Implementation

A link to implementation of the LMI
https://github.com/JalpeshBhadra/LMI/blob/master/polytopicstability.m

## Related LMIs

* Parametric Norm Bounded Uncertain System Quadratic Stability

## 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/pages/Positive Real Lemma

Positive Real Lemma
The Positive Real Lemma is a variation of the Kalman–Popov–Yakubovich (KYP) Lemma. The Positive Real Lemma can be used to determine if a system is passive (positive real).

## The System

       ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t)\\y(t)&=Cx(t)+Du(t)\\x(0)&=x_{0}\end{aligned}}$
where $x(t)\in \mathbb {R} ^{n}$, $y(t)\in \mathbb {R} ^{m}$, $u(t)\in \mathbb {R} ^{q}$, at any $t\in \mathbb {R}$.

## The Data

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

## The LMI: The Positive Real Lemma

Suppose ${\hat {G}}(s)(A,B,C,D)$ is the system. Then the following are equivalent.
       $1)\quad G\;{\text{is passive, i.e.}}\;\left\langle u,Gu\right\rangle _{L_{2}}\geq 0\;({\hat {G}}(s)+{\hat {G}}(s)^{*}\geq 0)$
       $2)\quad {\text{There exists a}}\;X>0\;{\text{such that}}$
             ${\begin{bmatrix}A^{T}X+XA&XB-C^{T}\\B^{T}X-C&-D^{T}-D\end{bmatrix}}\leq 0$

## Conclusion:

The Positive Real Lemma can be used to determine if the system $G$ is passive. Note from the (1,1) block of the LMI we know that $A$ is Hurwitz.

## Implementation

This implementation requires Yalmip and Sedumi. https://github.com/eoskowro/LMI/blob/master/Positive_Real_Lemma.m

## Related LMIs

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/pages/Projection Lemma

WIP, Description in progress
A condition for eliminating a variable in an LMI using orthogonal complements is presented.

## Definition 1: Orthogonal Complements

Let $A\in \mathbb {R} ^{m\times n}$, Then, $M_{a}$ is called a left orthogonal complement of $A$ if it satisfies
$M_{a}A=0,\quad {\text{rank}}(M_{a})=m-{\text{rank}}(A)$;
and $N_{a}$ is called a right orthogonal complement of $A$ if it satisfies
$AN_{a}=0,\quad {\text{rank}}(N_{a})=n-{\text{rank}}(A)$.
Using the definition of orthogonal complements, we have the following projection lemma:

## Projection Lemma

Let $P$, $Q$ and $H=H^{T}$ be given matrices of appropriate dimensions, $N_{p}$ and $N_{q}$ be the right orthogonal complement of $P$ and $Q$, respectively. Then, there exists $X$ such that
$H+P^{T}X^{T}Q+Q^{T}XP<0$
if and only if
$N_{p}^{T}P^{T}=0,\quad N_{q}^{T}Q^{T}=0$.
WIP, additional references to be added

## External Links

A list of references documenting and validating the LMI.
# LMIs in Control/pages/Quadratic Polytopic Hinf- Optimal State Feedback Control
## Quadratic Polytopic Full State Feedback Optimal $H_{\infty }$ Control

For a system having polytopic uncertainties, Full State Feedback is a control technique that attempts to place the system's closed-loop system poles in specified locations based off of performance specifications given. $H_{\infty }$ methods formulate this task as an optimization problem and attempt to minimize the $H_{\infty }$ norm of the system.

## The System

Consider System with following state-space representation.
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{1}q(t)+B_{2}w(t)\\p(t)&=C_{1}x(t)+D_{11}q(t)+D_{12}w(t)\\z(t)&=C_{2}x(t)+D_{21}q(t)+D_{22}w(t)\\\end{aligned}}$
where $x\in \mathbb {R} ^{m}$ , $q\in \mathbb {R} ^{n}$ , $w\in \mathbb {R} ^{g}$, $A\in \mathbb {R} ^{mxm}$, $B_{1}\in \mathbb {R} ^{mxn}$, $B_{2}\in \mathbb {R} ^{mxg}$, $p\in \mathbb {R} ^{p}$ , $C_{1}\in \mathbb {R} ^{pxm}$, $D_{11}\in \mathbb {R} ^{pxn}$, $D_{12}\in \mathbb {R} ^{pxg}$, $z\in \mathbb {R} ^{s}$, $C_{2}\in \mathbb {R} ^{sxm}$, $D_{21}\in \mathbb {R} ^{sxn}$ , $D_{22}\in \mathbb {R} ^{sxg}$ for any $t\in \mathbb {R}$.
Add uncertainty to system matrices
    $A,B_{1},B_{2},C_{1},C_{2},D_{11},D_{12}$
New state-space representation
    ${\begin{aligned}{\dot {x}}(t)&=(A+A_{i})x(t)+(B_{1}+B_{i})q(t)+(B_{2}+B_{i})w(t)\\p(t)&=(C_{1}+C_{i})x(t)+(D_{11}+D_{i})q(t)+(D_{12}+D_{i})w(t)\\z(t)&=C_{2}x(t)+D_{21}q(t)+D_{22}w(t)\\\end{aligned}}$

## The Optimization Problem:

Recall the closed-loop in state feedback is:
$S(P,K)=$
       ${\begin{aligned}{\begin{bmatrix}A+B_{2}F&&B_{1}\\C_{1}+D_{12}F&&D_{11}\end{bmatrix}}\\\end{aligned}}$
This problem can be formulated as $H\infty$ optimal state-feedback, where K is a controller gain matrix.

## The LMI:

An LMI for Quadratic Polytopic $H\infty$ Optimal State-Feedback Control $||S(P(\Delta ),K(0,0,0,F))||_{H\infty }\leq \gamma$
$Y>0$
       ${\begin{aligned}{\begin{bmatrix}Y(A+A_{i})^{T}+(A+A_{i})Y+Z^{T}(B_{2}+B_{1,i})^{T}+(B_{2}+B_{1,i})Z&&*^{T}&&*^{T}\\(B_{1}+B_{1,i})^{T}&&-\gamma I&&*^{T}\\(C_{1}+C_{1,i})Y+(D_{12}+D_{12,i})Z&&(D_{11}+D_{11,i})&&-\gamma I\end{bmatrix}}<0\end{aligned}}$

## Conclusion:

The $H\infty$ Optimal State-Feedback Controller is recovered by $F=ZY^{-1}$
Controller will determine the bound $\gamma$ on the $H_{\infty }$ norm of the system.

## Implementation:

https://github.com/JalpeshBhadra/LMI/tree/master

## Related LMIs

Full State Feedback Optimal $H_{\infty }$ Controller

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/pages/Output Energy Bounds for Lure System
## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+B_{p}p(t)+B_{w}w(t),\\z(t)&=C_{z}x(t)\\p_{i}(t)&=\phi _{i}(q_{i}(t)),i=1,\dots ,n_{p}\\q(t)&=C_{q}x(t),\\0&\leq \sigma \phi _{i}(\sigma )\leq \sigma ^{2}\ \forall \sigma \in \mathbb {R} \end{aligned}}$

## The Data

The matrices $A,B_{p},B_{w},C_{q},C_{z},x(0)$.

## The Optimization Problem:

The following optimization problem should be to find the tightest upper bound for the output energy of the above Lur'e system.
       ${\begin{aligned}&\min _{P\succ 0,\Lambda =diag(\lambda _{1},\dots ,\lambda _{n_{p}})\succeq 0,T=diag(\tau _{1},\dots ,\tau _{n_{p}})\succeq 0}x^{\top }(0)(P+C_{q}^{\top }\Lambda C_{q})x(0)\\&\quad \quad \quad \quad \quad \quad \quad {\begin{bmatrix}A^{\top }P+PA&PB_{p}+A^{\top }C_{q}^{\top }\Lambda +C_{q}^{\top }T\\B_{p}^{\top }P+\Lambda C_{q}A+TC_{q}&\Lambda C_{q}B_{p}+B_{p}^{\top }C_{q}^{\top }\Lambda -2T\end{bmatrix}}\preceq 0\\\end{aligned}}$

## Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/blob/master/LMIs%20for%20Output%20Energy%20Bounds%20of%20Lure's%20Systems

## Conclusion

The value function returns the the lowest bound for the energy function of the Lure's systems, i.e., $J=\int _{0}^{\infty }z^{\top }z\ dt$ with initial conditions $x(0)$.

## Remark

The key step in the proof is to satisfy ${\frac {d}{dt}}V(x)+z^{\top }z\leq 0$, where $V(.)$ is Lyapunov function in a special form.

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMI Properties and Applications in Systems, Stability, and Control Theory - A List of LMIs by Ryan Caverly and James Forbes.
* LMIs in Systems and Control Theory - A downloadable book on LMIs by Stephen Boyd.
# LMIs in Control/pages/Quadratic Schur Stabilization

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
# LMIs in Control/Fundamentals of Matrix and LMIs

Fundamentals of Matrix and LMIs
Fundamentals of Matrix and LMIs
* Basic Matrix Theory
* Notion of Matrix Positivity
* Matrix Inequalities and LMIs
* Convexity of LMIs
* Concatenation of LMIs
Properties of LMIs
* Change of Subject(Variable)
* Congruence Transformation
* Young’s Relation (Completion of the Squares)
** Young’s Relation-Based Properties
** Special Cases of Young’s Relation
** Iterative Convex Overbounding
* Projection Lemma (Matrix Elimination Lemma)
** Strict Projection Lemma
** Nonstrict Projection Lemma
** Reciprocal Projection Lemma
** Projection Lemma-Based Properties
* Ellipsoidal Inequality
* Continuous Time Properties
** Schur Complement
*** Strict and Nonstrict Schur Complement
*** Schur Complement Lemma-Based Properties
** Eigenvalue related Problems
*** LMI for Eigenvalue Minimization
*** Eigenvalue Problem
*** LMI for Generalized Eigenvalue Problem
** LMI for Matrix Norm Minimization
** LMI for Generalized Eigenvalue Problem
** LMI for Linear Programming
** LMI for Feasibility Problem
** Structured Singular Value
** LMI for Minimizing Condition Number of Positive Definite Matrix
** Continuous Quadratic Stability
* Discrete Time Properties
** Discrete Time Minimum Gain Lemma
** Discrete Time Modified Minimum Gain Lemma
* Finsler’s Lemma
* Dilation
* Tangential Nevanlinna Pick
* Nevanlinna Pick Interpolation with Scaling
# LMIs in Control/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
# LMIs in Control/Discrete-Time Algebraic Riccati Inequality (DARE)

Template:Discrete-Time Algebraic Riccati Inequality

## The System

Consider a Discrete-Time LTI system
    ${\begin{aligned}x_{k+1}=A_{d}x_{k}+B_{d}u_{k}\end{aligned}}$
    ${\begin{aligned}y_{k}=C_{d}x_{k}\end{aligned}}$
Consider $A_{d}\in \mathbb {R} ^{n\times n};B_{d}\in \mathbb {R} ^{n\times m}$

## The LMI: Discrete-Time Algebraic Riccati Inequality (DARE)

An algebraic Riccati equation is a type of nonlinear equation that arises in the context of infinite-horizon optimal control problems in continuous time or discrete time
The Discrete-Time Algebraic Riccati Inequality is given by
    ${\begin{aligned}A_{d}^{T}PA_{d}-A_{d}^{T}PB_{d}(R+B_{d}^{T}PB_{d})^{-1}B_{d}^{T}PA_{d}+Q-P\geq 0\end{aligned}}$
$P,Q\in \mathbb {S} ^{n}$ and $R\in \mathbb {S} ^{m}$ where $P>0,Q\geq 0,R>0$.
$P$ is the unknown n by n symmetric matrix and $A,B,Q,R$ are known real coefficient matrices.
The above equation can be rewritten using the Schur Complement Lemma as:
    ${\begin{bmatrix}A_{d}^{T}PA_{d}-P+Q&A_{d}^{T}PB_{d}\\B_{d}^{T}PA_{d}&R+B_{d}^{T}PB_{d}\end{bmatrix}}$${\begin{aligned}\geq 0\end{aligned}}$

## The Data

The Matrices $A_{d},B_{d},C_{d},Q,R$ are given
$Q$ and $R$ should necessarily be Hermitian matrices.
A square matrix is Hermitian if it is equal to its complex conjugate transpose.

## The Optimization Problem

Our aim is to find
$P$ - Unique solution to the discrete-time algebraic Riccati equation, returned as a matrix.
$K$ - State-feedback gain, returned as a matrix.
The algorithm used to evaluate the State-feedback gain is given by
    ${\begin{aligned}K=(R+B_{d}^{T}PB_{d})^{-1}B_{d}^{T}PA_{d}\end{aligned}}$
$L$ - Closed-loop eigenvalues, returned as a matrix.

## Conclusion:

Algebraic Riccati Inequalities play a key role in LQR/LQG control, H2- and H∞ control and Kalman filtering. We try to find the unique stabilizing solution, if it exists. A solution is stabilizing, if controller of the system makes the closed loop system stable.
Equivalently, this Discrete-Time algebraic Riccati Inequality is satisfied under the following necessary and sufficient condition:
    ${\begin{bmatrix}Q&0&A_{d}^{T}P&P\\0&R&B_{d}^{T}P&0\\PA_{d}&PB_{d}&P&0\\P&0&0&P\end{bmatrix}}$${\begin{aligned}\geq 0\end{aligned}}$

## Implementation

($X$ in the output corresponds to $P$ in the LMI)
A link to the Matlab code for a simple implementation of this problem in the GitHub repository:
https://github.com/yashgvd/ygovada

## Related LMIs

LMI for Continuous-Time Algebraic Riccati Inequality
LMI for Schur Stabilization

## External Links

A list of references documenting and validating the LMI.
* - LMI in Control Systems Analysis, Design and Applications
# LMIs in Control/KYP Lemma

LMIs in Control/KYP Lemma

## The System

    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)\\x(0)&=x_{0}\end{aligned}}$

## The Data

A List of inputs needed for the problem to be well-defined.

## The Optimization Problem

A description of the Problem to be solved, if appropriate.

## The LMI: Title

Title and mathematical description of the LMI formulation.
       ${\begin{aligned}{\text{Find}}\;X:&\\{\begin{bmatrix}X\end{bmatrix}}&>0\\{\begin{bmatrix}A^{T}X+XA\end{bmatrix}}&<0\end{aligned}}$

## Conclusion:

Interpretation of the results of the LMI

## Implementation

A link to CodeOcean or other online implementation of the LMI

## Related LMIs

Links to other closely-related LMIs

## 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/Discrete-Time Systems/Discrete-Time Lyapunov Stability

Discrete-Time Lyapunov Stability

## The System

Discrete-Time System
    ${\begin{aligned}x(t)_{k+1}&=A_{d}x(t)_{k},&A_{d}\in {\bf {{R^{n*n}}\;}}\\\end{aligned}}$

## The Data

The matrices $SystemA_{d},P$.

## The Optimization Problem

The following feasibility problem should be solved:
Find P obeying the LMI constraints.

## The LMI:

Discrete-Time Lyapunov Stability
The LMI formulation
       ${\begin{aligned}P\in {\bf {S^{n}}}\\Find\;&P>0,\\{\begin{bmatrix}A_{d}^{T}PA_{d}-P\end{bmatrix}}&\leq 0\end{aligned}}$

## Conclusion:

If there exists a $P\in {\bf {S^{n}}}$ satisfying the LMI then, $|\lambda _{i}(A_{d})|\leq 1,\forall i=1,2,...,n;$ and the equilibrium point ${\bar {x}}=0$ of the system is Lyapunov stable.

## Implementation

* - MATLAB implementation of the LMI.

## Related LMIs

Continuous_Time_Lyapunov_Inequality - Lyapunov_Inequality

## 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/H2Optimalobserver

State observer is a system that provides estimates of internal states of a given real system, from measurements of the inputs and outputs of the real system.The goal of $H_{2}$ -optimal state estimation is to design an observer that minimizes the $H_{2}$ norm of the closed-loop transfer matrix from w to z. Kalman filter is a form of Optimal Observer.

## The System

Consider the continuous-time generalized plant $P$ with state-space realization
    ${\begin{aligned}{\dot {x}}&=Ax+B_{1}w(t),\\y&=C_{2}x+D_{21}w\\\end{aligned}}$

## The Data

The matrices needed as input are $A,B,C,D$.

## The Optimization Problem

The task is to design an observer of the following form:
    ${\begin{aligned}{\dot {\hat {x}}}=A{\hat {x}}+L(y-{\hat {y}}),\\{\hat {y}}=C_{2}{\hat {x}}\\\end{aligned}}$

## The LMI: $H_{2}$ Optimal Observer

LMIs in the variables $P,G,Z,\nu$ are given by:
       ${\begin{aligned}{\begin{bmatrix}PA+A^{T}P-GC_{2}-{C_{2}}^{T}G^{T}&&PB_{1}-GD_{21}\\\star &&-1\end{bmatrix}}<0\\trZ<\nu \end{aligned}}$

## Conclusion:

The $H_{2}$ -optimal observer gain is recovered by $L=P^{-1}G$ and the $H_{2}$ norm of T(s) is $\mu ={\sqrt {\nu }}$

## Implementation

https://github.com/Ricky-10/coding107/blob/master/H2%20Optimal%20Observer

# 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.
* https://onlinelibrary.wiley.com/doi/abs/10.1002/rnc.1310
# LMIs in Control/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/Discrete-Time Systems
## Discrete-Time systems

      * Discrete-Time Lyapunov Stability
      * Discrete-Time H2-Optimal Observer
      * Discrete-Time H∞-Optimal Observer
      * Discrete-Time Mixed H2-H∞-Optimal Observer
      * Discrete-Time Algebraic Riccati Inequality (DARE)