LMIs in Control/Stability Analysis/D-Stability

Definition

Consider $A\in \mathbb {R} ^{n\times n}$. The matrix $A$ is ${\mathcal {D}}$-stable if and only if there exists $P\in \mathbb {S} ^{n}$, where $P>0$, such that

$[\lambda _{kl}P+\phi _{kl}AP+\phi _{lk}PA^{T}]_{1<k,l<m}<0$,

or equivalent

$\Lambda \otimes P+\Phi \otimes (AP)+\Phi ^{T}\otimes (PA^{T})<0$,

where $\otimes$ is the Kroenecker product,

The eigenvalues of a ${\mathcal {D}}$-stable matrix lie within the LMI region ${\mathcal {D}}$, which is defined as

${\mathcal {D}}=\{z\in \mathbb {C} :f_{D}(z)<0\}$, 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 conjugate of $z$.

Conic Sector Region Stability via the Dilation Lemma

Consider $A\in \mathbb {R} ^{n\times n}$ and $k\in \mathbb {R} _{>0}$.

The matrix $A$ satisfies $\lambda (A)\subset {\mathcal {P}}(k)$, where ${\mathcal {P}}(k):=\{\lambda \in \mathbb {C} :\left\vert Im(\lambda )\right\vert <k\left\vert Re(\lambda )\right\vert \}$, if and only if there exist $X\in \mathbb {S} ^{n}$ and $\epsilon \in \mathbb {R} _{>0}$, where $X>0$, such that

${\begin{bmatrix}k(AX+XA^{T})&AX-XA^{T}\\*&k(AX+XA^{T})\end{bmatrix}}<0$.

Equivalently, the matrix $A$ satisfies $\lambda (A)\subset {\mathcal {P}}(k)$ if and only if there exist $X\in \mathbb {S} ^{n}$ and $\epsilon \in \mathbb {R} _{>0}$, and $F\in \mathbb {R} ^{n\times n}$, where $X>0$, such that

${\begin{bmatrix}0&-kX&X&0\\*&0&0&-X\\*&*&0&-kX\\*&*&*&0\end{bmatrix}}+He{\begin{Bmatrix}{\begin{bmatrix}A&0\\1&0\\0&1\\0&A\end{bmatrix}}{\begin{bmatrix}F&0\\0&F\end{bmatrix}}{\begin{bmatrix}k1&-\epsilon k1&\epsilon 1&1\\-1&-\epsilon 1&\epsilon k1&k1\end{bmatrix}}\end{Bmatrix}}<0$.

Moreover, for every $X$ that satisfies

${\begin{bmatrix}k(AX+XA^{T})&AX-XA^{T}\\*&k(AX+XA^{T})\end{bmatrix}}<0$,

$X$ and $F=-\epsilon ^{-1}(A-\epsilon ^{-1}1)^{-1}X$ are solutions to

${\begin{bmatrix}0&-kX&X&0\\*&0&0&-X\\*&*&0&-kX\\*&*&*&0\end{bmatrix}}+He{\begin{Bmatrix}{\begin{bmatrix}A&0\\1&0\\0&1\\0&A\end{bmatrix}}{\begin{bmatrix}F&0\\0&F\end{bmatrix}}{\begin{bmatrix}k1&-\epsilon k1&\epsilon 1&1\\-1&-\epsilon 1&\epsilon k1&k1\end{bmatrix}}\end{Bmatrix}}<0$

α-Region Stability via the Dilation Lemma

Consider $A\in \mathbb {R} ^{n\times n}$ and $\alpha \in \mathbb {R} _{>0}$. The matrix $A$ satisfies $\lambda (A)\subset {\mathcal {H}}(\alpha )$, where ${\mathcal {H}}(\alpha ):=\{\lambda \in \mathbb {C} :Re(\lambda )<-\alpha \}$ if and only if there exist $X\in \mathbb {S} ^{n}$ and $\epsilon \in \mathbb {R} _{>0}$, where $X>0$, such that

$AX+XA^{T}+2\alpha X<0$.

Equivalently, the matrix $A$ satisfies $\lambda (A)\subset {\mathcal {H}}(\alpha )$ if and only if there exist $X\in \mathbb {S} ^{n}$, $\epsilon \in \mathbb {R} _{>0}$, and $F\in \mathbb {R} ^{n\times n}$, where $X>0$, such that

${\begin{bmatrix}0&-X&X\\*&0&0\\*&*&-{\frac {1}{2}}\alpha ^{-1}X\end{bmatrix}}+He{\begin{Bmatrix}{\begin{bmatrix}A\\1\\0\end{bmatrix}}F{\begin{bmatrix}1&-\epsilon 1&\epsilon 1\end{bmatrix}}\end{Bmatrix}}<0$.

Moreover, for every $X$ that satisfies

$AX+XA^{T}+2\alpha X<0$

$X$ and $F=-\epsilon ^{-1}(A-\epsilon ^{-1}1)^{-1}X$ are solutions to

${\begin{bmatrix}0&-X&X\\*&0&0\\*&*&-{\frac {1}{2}}\alpha ^{-1}X\end{bmatrix}}+He{\begin{Bmatrix}{\begin{bmatrix}A\\1\\0\end{bmatrix}}F{\begin{bmatrix}1&-\epsilon 1&\epsilon 1\end{bmatrix}}\end{Bmatrix}}<0$.

Circular Region Stability via the Dilation Lemma

Consider $A\in \mathbb {R} ^{n\times n}$, $r\in \mathbb {R} _{>0}$, and $c\in \mathbb {R} _{<0}$, where $c<-r$. The matrix $A$ satisfies $\lambda (A)\subset {\mathcal {G}}(c,r)$, where ${\mathcal {G}}(c,r):=\{\lambda \in \mathbb {C} :\left\vert \lambda -c\right\vert <r\}$, if and only if there exist $X\in \mathbb {S} ^{n}$ and $\epsilon \in \mathbb {R} _{>0}$, where $X>0$, such that

$AX+XA^{T}-{\frac {c^{2}-r^{2}}{c}}X-{\frac {1}{c}}AXA^{T}<0$.

Equivalently, the matrix $A$ satisfies $\lambda (A)\subset {\mathcal {G}}(c,r)$ if and only if there exist $X\in \mathbb {S} ^{n}$, $\epsilon \in \mathbb {R} _{>0}$, and $F\in \mathbb {R} ^{n\times n}$, where $X>0$, such that

${\begin{bmatrix}0&-X&X&0\\*&0&0&-X\\*&*&{\frac {c}{c^{2}-r^{2}}}X&0\\*&*&*&cX\end{bmatrix}}+He{\begin{Bmatrix}{\begin{bmatrix}A\\1\\0\\0\end{bmatrix}}F{\begin{bmatrix}1&-\epsilon 1&\epsilon 1&1\end{bmatrix}}\end{Bmatrix}}<0$

Moreover, for every $X$ that satisfies

$AX+XA^{T}-{\frac {c^{2}-r^{2}}{c}}X-{\frac {1}{c}}AXA^{T}<0$

$X$ and $F=-\epsilon ^{-1}(A-\epsilon ^{-1}1)^{-1}X$ are solutions to

${\begin{bmatrix}0&-X&X&0\\*&0&0&-X\\*&*&{\frac {c}{c^{2}-r^{2}}}X&0\\*&*&*&cX\end{bmatrix}}+He{\begin{Bmatrix}{\begin{bmatrix}A\\1\\0\\0\end{bmatrix}}F{\begin{bmatrix}1&-\epsilon 1&\epsilon 1&1\end{bmatrix}}\end{Bmatrix}}<0$

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.