OpenLLM-France/wikimedia huggingface dataset:
# LMIs in Control/Matrix and LMI Properties and Tools/Quadratic Inequalities
## Weighted Norm
### Required data

Consider $W\in \mathbb {S} ^{n}$, $x,y\in \mathbb {R} ^{n}$, and $\gamma \in \mathbb {R} _{\geqslant 0}$, where $W>0$.

### Matrix inequality

The inequality $(x-y)^{T}W(x-y)\leq \gamma$ is equivalent to the matrix inequality given by
${\begin{bmatrix}\gamma &(x-y)^{T}\\*&W^{-1}\end{bmatrix}}\geq 0$.

## Quadratic Inequality
### Required data

Consider $W\in \mathbb {S} ^{n}$, $A\in \mathbb {R} ^{n\times m}$, $x$, $c\in \mathbb {R} ^{m}$, $b\in \mathbb {R} ^{n}$, and $d\in \mathbb {R}$, where $W>0$.

### Matrix inequality

The quadratic inequality $(Ax+b)^{T}W(Ax+b)-c^{T}x-d\leq 0$ with $W>0$ is equivalent to the matrix inequality given by
${\begin{bmatrix}W^{-1}&Ax+b\\*&c^{T}x+d\end{bmatrix}}\geq 0$.

## 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/Matrix and LMI Properties and Tools/Set minimum theorem

In set theory and number theory, the set minimum theorem states that if S is a well-defined, constant finite set of positive integers, then for all sufficiently large positive integers x, the set S has at least one element that is smaller than x. Since S is a non-changing subset of positive integers, it has a minimum and that minimum is a constant; since it’s a finite constant, x simply has to exceed it. For example, if S = {12, 7, 9, 16, 13, 21}, a set with six terms, then min(S) = 9, so the statement “S has at least one term that is smaller than x” is false if x is between 1 and 8, inclusive, but it is true for x > 9. “Less than” excludes the edge case of x = 9, but “less than or equal to” includes when x = 9.
Another theorem is that if a number T is less than the smallest element of S, then T is not an element of S. The set minimum theorem puts a rigid ceiling on the set of positive integers that can be denied using that other theorem.
Note that sufficiently large is a special case of “almost all”. In the example given in the first paragraph, the statement is true for all x > 9. For the first 13 integers, the statement is true for 10, 11, 12, and 13; seems not too solid as 4/13 = 30.769%, which is less than a third. But if we extend our number line to cover the first 1000 natural numbers, the statement is true for 991 of those numbers, or 99.1% of the first 1000 natural numbers. If we go to a million, there are 999,991 valid solutions or 99.9991%. You get the point. The set of initial exceptions remains the same no matter how far we go.
# LMIs in Control/Matrix and LMI Properties and Tools/Strict Projection Lemma
## Introduction

Projection Lemma is also known as Matrix Elimination Lemma. Strict Projection Lemma is one of the characteristics of the Projection Lemma.

## Strict Projection Lemma

Consider $\Psi \in \mathbb {S} ^{n}$, $G\in \mathbb {R} ^{n\times m}$, $\Lambda \in \mathbb {R} ^{m\times p}$, and $H\in \mathbb {R} ^{n\times p}.$ There exists $\Lambda$ such that:
${\begin{aligned}\ \Psi +G\Lambda H^{T}+H\Lambda ^{T}G^{T}<0,\end{aligned}}$
if and only if,
${\begin{aligned}\ N_{G}^{T}\Psi N_{G}<0\end{aligned}}$,
${\begin{aligned}\ N_{H}^{T}\Psi N_{H}<0\end{aligned}}$,
where ${\mathcal {R}}(N_{G})={\mathcal {N}}(G^{T})$ and ${\mathcal {R}}(N_{H})={\mathcal {N}}(H^{T})$.

## External Links

A list of references documenting and validating the LMI.
# LMIs in Control/Matrix and LMI Properties and Tools/Non Strict Projection Lemma
## Introduction

Projection Lemma is also known as Matrix Elimination Lemma. Strict Projection Lemma is one of the characteristics of the Projection Lemma.

## Non Strict Projection Lemma

Consider $\Psi \in \mathbb {S} ^{n}$, $G\in \mathbb {R} ^{n\times m}$, $\Lambda \in \mathbb {R} ^{m\times p}$, and $H\in \mathbb {R} ^{n\times p}.$, where ${\mathcal {R}}(G)$ and ${\mathcal {R}}(H)$ is linear;y independent. There exists $\Lambda$ such that:
${\begin{aligned}\ \Psi +G\Lambda H^{T}+H\Lambda ^{T}G^{T}leq0,\end{aligned}}$
if and only if,
${\begin{aligned}\ N_{G}^{T}\Psi N_{G}\leq 0\end{aligned}}$,
${\begin{aligned}\ N_{H}^{T}\Psi N_{H}\leq 0\end{aligned}}$,
where ${\mathcal {R}}(N_{G})={\mathcal {N}}(G^{T})$ and ${\mathcal {R}}(N_{H})={\mathcal {N}}(H^{T})$.

## External Links

A list of references documenting and validating the LMI.
# LMIs in Control/Matrix and LMI Properties and Tools/Passivity and Bounded Realness

This section deals with passivity of a system.

## The System

Given a state-space representation of a linear system
    ${\begin{aligned}\ {\dot {x}}=Ax+Bu\\\ y=Cx+Du\\\end{aligned}}$
$x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{m},u\in \mathbb {R} ^{r}$ are the state, output and input vectors respectively.

## The Data

$A,B,C,D$ are system matrices.

## Definition

The linear system with the same number of input and output variables is called passive if
    ${\begin{aligned}\int \limits _{0}^{T}u^{T}y(t)dt\geq 0\\\end{aligned}}$         (1)
hold for any arbitrary $T\geq 0$, arbitrary input $u(t)$, and the corresponding solution $y(t)$ of the system with $x(0)=0$. In addition, the transfer function matrix
    ${\begin{aligned}G(s)&=C(sI-A)^{-1}B+D\\\end{aligned}}$         (2)
of system is called is positive real if it is square and satisfies
    ${\begin{aligned}\ G^{H}(s)+G(s)\geq 0\forall s\in \mathbb {C} ,Re(s)>0\\\end{aligned}}$         (2)

## LMI Conditon

Let the linear system be controllable. Then, the system is passive if an only if there exists $P>0$ such that
    ${\begin{aligned}\ {\begin{bmatrix}A^{T}P+PA&PB-C^{T}\\B^{T}P-C&-D^{T}-D\end{bmatrix}}\leq 0\\\end{aligned}}$         (3)

## Implementation

This implementation requires Yalmip and Mosek.
* https://github.com/NoobMasterLoki/LMI--master/blob/main/Passivity.m

## Conclusion:

Thus, it is seen that passivity and positive-realness describe the same property of a linear system, one gives the time-domain feature and the other provides frequency-domain feature of this property.

## 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.
* LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & amp; Francis Group, 2013
# LMIs in Control/Matrix and LMI Properties and Tools/Schur Stabilizability

LMI for Schur Stabilizability
Schur Stabilization is one method of ensuring that a controller can be made to stabilize a system. The following LMI is one that determines whether or not a system is indeed Schur Stabilizable, or having the property of being able to be Schur Stabilized.

## The System

We consider the following system:
${\begin{aligned}{\dot {x}}(t)&=Ax+Bu\end{aligned}}$
or the matrix pair (A,B). In both cases, 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.

## The Data

The data required is both the matrices A and B as seen in the form above.

## The Optimization Problem

The goal of the optimization is to find a valid symmetric P such that the following LMI is satisfied.

## The LMI: LMI for Schur stabilizability

The LMI problem is to find a symmetric matrix P and a matrix W satisfying:
${\begin{aligned}{\begin{bmatrix}-P&AP+BW\\(AP+BW)^{T}&-P\end{bmatrix}}<0\\\end{aligned}}$
Another LMI with the same result of finding Schur Stabilizability is to find a symmetric matrix P such that:
${\begin{aligned}{\begin{bmatrix}-P&PA^{T}\\AP&-P-\gamma BB^{T}\end{bmatrix}}<0,\gamma \leq 1\\\end{aligned}}$

## Conclusion:

If the one of the above LMIs is found to be feasible, then the system is Schur Stabilizable and the Schur Stabilization LMI will always give a feasible result as well, in addition to a controller K that will Schur Stabilize the system.

## Implementation

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

## Related LMIs

- Schur Stabilization

## External Links

- LMI in Control Systems Analysis, Design and Applications

## Return to Main Page

-Matrix and LMI Properties and Tools
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Interval Quadratic Stability

An LMI to determine the quadratic stability of a system with parametric, interval uncertainties.

## The System

Consider the system with Affine Time-Varying uncertainty
    ${\begin{aligned}{\dot {x}}(t)&=(A_{0}+\Delta (t))x(t)\\\end{aligned}}$
where
    ${\begin{aligned}\Delta (t)=A_{1}\delta _{1}(t)+...+A_{k}\delta _{k}(t)\\\end{aligned}}$
where $\delta _{i}(t)$ lies in the intervals
    ${\begin{aligned}\delta _{i}(t)\in [\delta _{i}^{-},\delta _{i}^{+}]\\\end{aligned}}$
where $x\in \mathbb {R} ^{m}$ and $A\in \mathbb {R} ^{mxm}$.

## The Data

The matrices A and $\Delta$ are known.

## The Optimization

This optimization problem ensures quadratic stability of the system with k interval uncertainties using $2^{k}$ LMI constraints.
$\delta (t)$ lies in the hypercube. The vertices of the hypercube define the vertices of the uncertainty set
    $V:={\Biggl \A_{0}+\sum _{i}A_{i}\delta _{i},\ \delta _{i}\in \{\delta _{i}^{-},\delta _{i}^{+}\}{\Biggl \}}$
$(A+\Delta ,{\boldsymbol {\Delta }})$ is quadratically stable over ${\boldsymbol {\Delta }}:=Co(V)$ if and only if there exists a P > 0 such that
    ${\begin{aligned}{\BigglA_{0}+\sum _{i}A_{i}\delta _{i}{\Biggl )}^{T}P+P{\BigglA_{0}+\sum _{i}A_{i}\delta _{i}{\Biggl )}<0\quad for\ every\quad \delta _{i}\in \{\delta _{i}^{-},\delta _{i}^{+}\}^{k}\end{aligned}}$

## The LMI

${\begin{aligned}P>0,\quad {\BigglA_{0}+\sum _{i}A_{i}\delta _{i}{\Biggl )}^{T}P+P{\BigglA_{0}+\sum _{i}A_{i}\delta _{i}{\Biggl )}<0\quad for\ every\quad \delta _{i}\in \{\delta _{i}^{-},\delta _{i}^{+}\}^{k}\end{aligned}}$

## Conclusion

Quadratic Stability Implies Stability of trajectories for any $\Delta$ with $\Delta \in {\boldsymbol {\Delta }}$ for all $t\geq 0$
Quadratic stability is conservative.
Stability does not imply quadratic stability.
Interval uncertainty is a special case of polytopic uncertainty.

## Implementation

Example of implementation of the LMI
https://github.com/MichaelDobos/LMI/blob/main/intervalquadraticstability.m

## Related LMIs

* Polytopic Quadratic Stability
* Discrete-Time Quadratic Stability

## External Links

* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* LMIs in Control Systems: Analysis, Design, and Applications - A textbook on LMIs in control by Duan and Yu.
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/KYP Lemma Without Feedthrough
## The Concept

It is assumed in the Lemma that the state-space representation (A, B, C, D) is minimal. Then Positive Realness (PR) of the transfer function C(SI − A)$^{-1}$B + D is equivalent to the solvability of the set of LMIs given in this page. Consider now the following scalar example, where (A, B, C, D)=(−α, 0, 0, 1), with α > 0. The transfer function is H(s) = 0 that is PR

## The System

Consider a contiuous-time LTI system, ${\mathcal {G}}:{\mathcal {L}}_{2e}\rightarrow {\mathcal {L}}_{2e}$, with minimal state-space relization (A, B, C, 0), where ${\mathcal {A}}\in {\mathcal {R}}^{n\times n},{\mathcal {B}}\in {\mathcal {R}}^{n\times m},$ and ${\mathcal {C}}\in {\mathcal {R}}^{m\times n},$.
    ${\begin{aligned}{\dot {x}}(t)&=Ax(t)+Bu(t),\\y(t)&=Cx(t)\\\end{aligned}}$

## The Data

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

## LMI : KYP Lemma without Feedthrough

The system ${\mathcal {G}}$ is positive real (PR) under either of the following equivalet necessary and sufficient conditions.
    1. There exists $P\in {\mathcal {S}}^{n},$ where $p>0$ such that
       ${\begin{aligned}PA+A^{T}P\geq 0\\PB=C^{T}\end{aligned}}$
    2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that
       ${\begin{aligned}AQ+QA^{T}\geq 0\\B=QC^{T}\end{aligned}}$
This is a special case of the KYP Lemma for QSR dissipative systems with Q = 0, Q = 0.5 and R = 0.
The system ${\mathcal {G}}$ is strictly positive real (SPR) under either of the following equivalet necessary and sufficient conditions.
    1. There exists $P\in {\mathcal {S}}^{n},$ where $p>0$ such that
       ${\begin{aligned}PA+A^{T}P<0\\PB=C^{T}\end{aligned}}$
    2. There exists $Q\in {\mathcal {S}}^{n},$ where $Q>0$ such that
       ${\begin{aligned}AQ+QA^{T}<0\\B=QC^{T}\end{aligned}}$
This is a special case of the KYP Lemma for QSR dissipative systems with Q = ε1, Q = 0.5 and R = 0. where ε $\in {\mathcal {R}}_{>0}.$

## Conclusion:

If there exist a positive definite $P$ for the the selected Q,S and R matrices then the system ${\mathcal {G}}$ is Positive Real.

## Implementation

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

## Related LMIs

KYP Lemma
State Space Stability Discrete Time KYP Lemma with Feedthrough

## 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 Transac- tions 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. Brogliato B., Maschke B., Lozano R., Egeland O. (2007) Kalman-Yakubovich-Popov Lemma. In: Dissipative Systems Analysis and Control. Communications and Control Engineering. Springer, London
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Generalized H2 Norm
## Generalized $H_{2}$ Norm

The $H_{2}$ norm characterizes the average frequency response of a system. To find the H2 norm, the system must be strictly proper, meaning the state space represented $D$ matrix must equal zero. The H2 norm is frequently used in optimal control to design a stabilizing controller which minimizes the average value of the transfer function, $G$ as much as possible. This optimal control problem is also called the Linear Quadratic Gaussian.

## The System

Consider a continuous-time, linear, time-invariant system $G:L_{2e}\to L_{2e}$ with state space realization $(A,B,C,0)$ where $A\in \mathbb {R} ^{n\times n}$, $B\in \mathbb {R} ^{n\times m}$, $C\in \mathbb {R} ^{p\times n}$, amd $A$ is Hurwitz. The generalized $H_{2}$ norm of $G$ is:
$\left\|G\right\|_{2,{\infty }}=\sup _{u\in L_{2},{u}\neq {\text{zero}}}$ ${\frac {\left\|Gu\right\|_{\infty }}{\ \left\|u\right\|_{2}}}$

## The Data

The transfer function $G$, and system matrices $A$, $B$, $C$ are known and $A$ is Hurwitz.

## The LMI: Generalized $H_{2}$ Norm LMIs

The inequality $\left\|G\right\|_{2,{\infty }}<\mu$ holds under the following conditions:
1. There exists $P\in \mathbb {S} ^{n}$ and $\mu \in \mathbb {R} _{>0}$ where $P>0$ such that:
       ${\begin{bmatrix}A^{T}P+PA&PB\\*&-\mu 1\end{bmatrix}}<0$.
       ${\begin{bmatrix}P&C^{T}\\*&\mu 1\end{bmatrix}}>0$.
2. There exists $Q\in \mathbb {S} ^{n}$ and $\mu \in \mathbb {R} _{>0}$ where $Q>0$ such that:
       ${\begin{bmatrix}QA^{T}+AQ&B\\*&-\mu 1\end{bmatrix}}<0$.
       ${\begin{bmatrix}Q&QC^{T}\\*&\mu 1\end{bmatrix}}>0$.
3. There exists $P\in \mathbb {S} ^{n},V\in \mathbb {R} ^{n\times n}$ and $\mu \in \mathbb {R} _{>0}$ where $P>0$ such that:
       ${\begin{bmatrix}-(V+V^{T})&V^{T}A+P&V^{T}B&V^{T}\\*&-P&0&0\\*&**-\mu 1&0\\*&*&*&-P\end{bmatrix}}<0$.
       ${\begin{bmatrix}P&C^{T}\\*&\mu 1\end{bmatrix}}>0$.

## Conclusion:

The generalized $H_{2}$ norm of $G$ is the minimum value of $\mu \in \mathbb {R} _{>0}$ that satisfies the LMIs presented in this page.

## Implementation

This implementation requires Yalmip and Sedumi.
Generalized $H_{2}$ Norm - MATLAB code for Generalized $H_{2}$ Norm.

## Related LMIs

LMI for System H_{2} Norm

## 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/LMIs in system and stability Theory/Hankel Norm of Affine Parameter 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
       ${\begin{aligned}&\min _{\{Q\succeq 0,\gamma ^{2},\theta \}}\gamma ^{2}\\&\quad s.t.\quad D_{zw}(\theta )=0,\quad A^{\top }Q+QA+C_{z}(\theta )C_{z}(\theta )\preceq 0,\quad \gamma ^{2}I-W_{c}^{1/2}QW_{c}^{1/2}\succeq 0,\end{aligned}}$
where $W_{c}$ is the controllability Gramian, i.e., $W_{c}\triangleq \int _{0}^{\infty }e^{At}B_{w}B_{w}^{\top }e^{A^{\top }t}dt$.

## Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/0faedcdd9fba92bc27a318d80159c04a0b342f35

## Conclusion

The Hanakel norm (i.e., the square root of the maximum eigenvalue) of $H_{\theta }$ is less than ${\gamma }$ if and only if the above LMI holds and the value function returns the maximum provable Hankel norm.

## Remark

$D_{zw}$ is assumed to be zero.

## 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/Hurwitz Stabilizability

This section studies the stabilizability properties of the control systems.

## The System

Given a state-space representation of a linear system
    ${\begin{aligned}\ \rho x=Ax+Bu\\\ y=Cx+Du\\\end{aligned}}$
Where $\rho$ represents the differential operator ( when the system is continuous-time) or one-step forward shift operator ( Discrete-Time system). $x\in \mathbb {R} ^{n},y\in \mathbb {R} ^{m},u\in \mathbb {R} ^{r}$ are the state, output and input vectors respectively.

## The Data

$A,B,C,D$ are system matrices.

## Definition

The system , or the matrix pair $(A,B)$ is Hurwitz Stabilizable if there exists a real matrix $K$ such that $(A+BK)$ is Hurwitz Stable. The condition for Hurwitz Stabilizability of a given matrix pair (A,B) is given by the PBH criterion:
    ${\begin{aligned}\ rank{\begin{bmatrix}sI-A&B\end{bmatrix}}&=n,\forall s\in \mathbb {C} ,Re(s)\geq 0\\\end{aligned}}$         (1)
The PBH criterion shows that the system is Hurwitz stabilizable if all uncontrollable modes are Hurwitz stable.

## LMI Condition

The system, or matrix pair $(A,B)$ is Hurwitz stabilizable if and only if there exists symmetric positive definite matrix $P$ and $W$ such that:
    ${\begin{aligned}AP+PA^{T}+BW+W^{T}B^{T}<0\\\end{aligned}}$         (2)
Following definition of Hurwitz Stabilizability and Lyapunov Stability theory, the PBH criterion is true if and only if , a matrix $K$ and a matrix $P>0$ satisfying:
    ${\begin{aligned}(A+BK)P+P(A+BK)^{T}<0\\\end{aligned}}$         (3)
Letting
    ${\begin{aligned}W=KP\\\end{aligned}}$         (4)
Putting (4) in (3) gives us (2).

## Implementation

This implementation requires Yalmip and Mosek.
* https://github.com/ShenoyVaradaraya/LMI--master/blob/main/Hurwitz_Stabilizability.m

## Conclusion

Compared with the second rank condition, LMI has a computational advantage while also maintaining numerical reliability.

## References

* 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.
* LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013
# LMIs in Control/Click here to continue/LMIs in system and stability Theory/Generalized Lyapunov Stability

WIP, Description in progress
The theorem can be viewed as a true essential generalization of the well-known continuous- and discrete-time Lyapunov theorems.

## Kronecker Product

The Kronecker Product of a pair of matrices $A\in \mathbb {R} ^{m\times n}$ and $B\in \mathbb {R} ^{p\times q}$ is defined as follows:
$A\otimes B={\begin{bmatrix}a_{11}B&a_{12}B&\cdots &a_{1n}B\\a_{21}B&a_{22}B&\cdots &a_{2n}B\\\vdots &\vdots &\ddots &\vdots \\a_{m1}B&a_{m2}B&\cdots &a_{mn}B\end{bmatrix}}\in \mathbb {R} ^{mp\times nq}$.

## Lemma 1: Manipulation Rules of Kronecker Product

Let $A,B,C$ be matrices with appropriate dimensions. Then, the Kronecker product has the following properties:
* $1\otimes A=A$;
* $(A+B)\otimes C=A\otimes C+B\otimes C$
* $(A\otimes B)(C\otimes D)=(AC)\otimes (BD)$
* $(A\otimes B)^{T}=A^{T}\otimes B^{T}$
* $(A\otimes B)^{-1}=A^{-1}\otimes B^{-1}$
* $\lambda (A\otimes B)={\lambda _{i}(A)\lambda _{j}{B}}$

## Theorem

In terms of Kronecker products, the following theorem gives the $\mathbb {D}$-stability condition for the general LMI region case: Let $\mathbb {D} =\mathbb {D} _{L,M}$ be an LMI region, whose characteristic function is
$F_{\mathbb {D} }=L+sM+{\overline {s}}M^{T}$
Then, a matrix $A\in \mathbb {R} ^{n\times n}$ is $\mathbb{D}_{L,M}$-stable if and only if there exists symmetric positive definite matrix $P$ such that
$R_{\mathbb {D} }(A,P)=L\otimes P+M\otimes (AP)+M^{T}\otimes (AP)^{T}<0$,
where $\otimes$ represents the Kronecker product.

## Lemma 2

Given two LMI regions $\mathbb {D} _{1}$ and $\mathbb {D} _{2}$, a matrix $A$ is both $\mathbb {D} _{1}$-stable and $\mathbb {D} _{2}$-stable if there exists a positive definite matrix $P$ , such that $R_{\mathbb {D} _{1}}(A,P)<0$ and $R_{\mathbb {D} _{2}}(A,P)<0$.
$$ 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/H infinity Norm 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 affinity 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
       ${\begin{aligned}&\min _{\{P\succ 0,\gamma \geq 0\}}\gamma \\&\quad s.t.{\begin{bmatrix}A^{\top }P+PA&PB_{w}\\B_{w}^{\top }P&-\gamma ^{2}I\end{bmatrix}}+{\begin{bmatrix}C_{z}^{\top }(\theta )\\D_{zw}^{\top }(\theta )\end{bmatrix}}{\begin{bmatrix}C_{z}(\theta )&D_{zw}(\theta )\end{bmatrix}}\preceq 0.\end{aligned}}$

## Implementation

https://github.com/mkhajenejad/Mohammad-Khajenejad/commit/5462bc1dc441bc298d50a2c35075e9466eba8355

## Conclusion

The value function of the above semi-definite program returns the ${\mathcal {H}}_{\infty }$ norm of the system.

## Remark

It is assumed that $A$ is stable and $(A,B_{w})$ is controllable and the semi-infinite convex constraint $\|H_{\theta }(j\omega )\|<\gamma$ for all $\omega \in \mathbb {R}$, is converted into a finite-dimensional convex LMI.

## 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.