pole placement in LMI regions
$begingroup$
In this paper, the authors elegantly present a LMI region as a subset $mathcal{D}$ of the complex plane
$$ mathcal{D} = {z in mathbb{C} | L + zcdot M + bar{z}cdot M^T < 0 }$$ where $L = L^T$ and $M$ are some real matrices. Then, they proceed to say that a real matrix $A$ is $mathcal{D}$ stable (that is, it has all eigenvalues in $mathcal{D}$) iff there $exists$ a symmetric positive definite matrix $X$, such that
$$ L otimes X + M otimes (Xcdot A) + M^T otimes (A^Tcdot X) < 0$$ where $otimes$ denotes the Kronecker product. How can I prove this?
My ideea
Indeed, for $L = 0$ and $M = 1$ the LMI region is the left complex plane and indeed the above condition becomes the well known Lyapunov stability requirement for the dynamical system $dot{x} = Acdot x$ which is equivalent with the eigenvalues of $A$ being in the left complex plane! Am I suppose to see this as some sort of transformation of the imaginary axis in a convex region?
convex-optimization dynamical-systems lmis
asked Jan 21 at 12:46
C MariusC Marius
581210
$endgroup$
add a comment |
$begingroup$
In this paper, the authors elegantly present a LMI region as a subset $mathcal{D}$ of the complex plane
$$ mathcal{D} = {z in mathbb{C} | L + zcdot M + bar{z}cdot M^T < 0 }$$ where $L = L^T$ and $M$ are some real matrices. Then, they proceed to say that a real matrix $A$ is $mathcal{D}$ stable (that is, it has all eigenvalues in $mathcal{D}$) iff there $exists$ a symmetric positive definite matrix $X$, such that
$$ L otimes X + M otimes (Xcdot A) + M^T otimes (A^Tcdot X) < 0$$ where $otimes$ denotes the Kronecker product. How can I prove this?
My ideea
Indeed, for $L = 0$ and $M = 1$ the LMI region is the left complex plane and indeed the above condition becomes the well known Lyapunov stability requirement for the dynamical system $dot{x} = Acdot x$ which is equivalent with the eigenvalues of $A$ being in the left complex plane! Am I suppose to see this as some sort of transformation of the imaginary axis in a convex region?
convex-optimization dynamical-systems lmis
asked Jan 21 at 12:46
C MariusC Marius
581210
$endgroup$
$begingroup$
The proof can be found in the Appendix of the full version of the paper: researchgate.net/publication/…
$endgroup$
– shamisen
Jan 21 at 22:03
$begingroup$
@shamisen thank you!
$endgroup$
– C Marius
Jan 21 at 22:32
add a comment |
$begingroup$
In this paper, the authors elegantly present a LMI region as a subset $mathcal{D}$ of the complex plane
$$ mathcal{D} = {z in mathbb{C} | L + zcdot M + bar{z}cdot M^T < 0 }$$ where $L = L^T$ and $M$ are some real matrices. Then, they proceed to say that a real matrix $A$ is $mathcal{D}$ stable (that is, it has all eigenvalues in $mathcal{D}$) iff there $exists$ a symmetric positive definite matrix $X$, such that
$$ L otimes X + M otimes (Xcdot A) + M^T otimes (A^Tcdot X) < 0$$ where $otimes$ denotes the Kronecker product. How can I prove this?
My ideea
Indeed, for $L = 0$ and $M = 1$ the LMI region is the left complex plane and indeed the above condition becomes the well known Lyapunov stability requirement for the dynamical system $dot{x} = Acdot x$ which is equivalent with the eigenvalues of $A$ being in the left complex plane! Am I suppose to see this as some sort of transformation of the imaginary axis in a convex region?
convex-optimization dynamical-systems lmis
asked Jan 21 at 12:46
C MariusC Marius
581210
$endgroup$
In this paper, the authors elegantly present a LMI region as a subset $mathcal{D}$ of the complex plane
$$ mathcal{D} = {z in mathbb{C} | L + zcdot M + bar{z}cdot M^T < 0 }$$ where $L = L^T$ and $M$ are some real matrices. Then, they proceed to say that a real matrix $A$ is $mathcal{D}$ stable (that is, it has all eigenvalues in $mathcal{D}$) iff there $exists$ a symmetric positive definite matrix $X$, such that
$$ L otimes X + M otimes (Xcdot A) + M^T otimes (A^Tcdot X) < 0$$ where $otimes$ denotes the Kronecker product. How can I prove this?
My ideea
Indeed, for $L = 0$ and $M = 1$ the LMI region is the left complex plane and indeed the above condition becomes the well known Lyapunov stability requirement for the dynamical system $dot{x} = Acdot x$ which is equivalent with the eigenvalues of $A$ being in the left complex plane! Am I suppose to see this as some sort of transformation of the imaginary axis in a convex region?
convex-optimization dynamical-systems lmis
convex-optimization dynamical-systems lmis
asked Jan 21 at 12:46
C MariusC Marius
581210
asked Jan 21 at 12:46
C MariusC Marius
581210
edited Jan 21 at 20:57
C Marius
asked Jan 21 at 12:46
C MariusC Marius
581210
asked Jan 21 at 12:46
C MariusC Marius
581210
asked Jan 21 at 12:46
C MariusC Marius
581210
581210
$begingroup$
The proof can be found in the Appendix of the full version of the paper: researchgate.net/publication/…
$endgroup$
– shamisen
Jan 21 at 22:03
$begingroup$
@shamisen thank you!
$endgroup$
– C Marius
Jan 21 at 22:32
add a comment |
$begingroup$
The proof can be found in the Appendix of the full version of the paper: researchgate.net/publication/…
$endgroup$
– shamisen
Jan 21 at 22:03
$begingroup$
@shamisen thank you!
$endgroup$
– C Marius
Jan 21 at 22:32
$begingroup$
The proof can be found in the Appendix of the full version of the paper: researchgate.net/publication/…
$endgroup$
– shamisen
Jan 21 at 22:03
$begingroup$
The proof can be found in the Appendix of the full version of the paper: researchgate.net/publication/…
$endgroup$
– shamisen
Jan 21 at 22:03
$begingroup$
@shamisen thank you!
$endgroup$
– C Marius
Jan 21 at 22:32
$begingroup$
@shamisen thank you!
$endgroup$
– C Marius
Jan 21 at 22:32
add a comment |
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$begingroup$
The proof can be found in the Appendix of the full version of the paper: researchgate.net/publication/…
$endgroup$
– shamisen
Jan 21 at 22:03
$begingroup$
@shamisen thank you!
$endgroup$
– C Marius
Jan 21 at 22:32