LMIs in Control/Stability Analysis/D-Stability

Definition

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

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

or equivalent

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

where $\otimes$ is the Kroenecker product,

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

$𝒟=\{z\in ℂ:f_D(z)<0\}$, where

$f_D(z):=\mathrm{\Lambda }+z\mathrm{\Phi }+\overline{z}{\mathrm{\Phi }}^T=\{{\lambda }_{kl}+{\varphi }_{kl}z+{\varphi }_{lk}\overline{z}{\}}_{1\le k,l\le m}$,

$\mathrm{\Lambda }\in {𝕊}^m$, $\mathrm{\Phi }\in {ℝ}^{m\times m}$, and $\overline{z}$ is the complex conjugate of $z$.

Conic Sector Region Stability via the Dilation Lemma

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

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

$\left[\begin{array}{cc}k(AX+X{A}^T)& AX-X{A}^T\\ *& k(AX+X{A}^T)\end{array}\right]<0$.

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

$\left[\begin{array}{cccc}0& -kX& X& 0\\ *& 0& 0& -X\\ *& *& 0& -kX\\ *& *& *& 0\end{array}\right]+He\left\{\begin{array}{c}\left[\begin{array}{cc}A& 0\\ 1& 0\\ 0& 1\\ 0& A\end{array}\right]\left[\begin{array}{cc}F& 0\\ 0& F\end{array}\right]\left[\begin{array}{cccc}k1& -ϵk1& ϵ1& 1\\ -1& -ϵ1& ϵk1& k1\end{array}\right]\end{array}\right\}<0$.

Moreover, for every $X$ that satisfies

$\left[\begin{array}{cc}k(AX+X{A}^T)& AX-X{A}^T\\ *& k(AX+X{A}^T)\end{array}\right]<0$,

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

$\left[\begin{array}{cccc}0& -kX& X& 0\\ *& 0& 0& -X\\ *& *& 0& -kX\\ *& *& *& 0\end{array}\right]+He\left\{\begin{array}{c}\left[\begin{array}{cc}A& 0\\ 1& 0\\ 0& 1\\ 0& A\end{array}\right]\left[\begin{array}{cc}F& 0\\ 0& F\end{array}\right]\left[\begin{array}{cccc}k1& -ϵk1& ϵ1& 1\\ -1& -ϵ1& ϵk1& k1\end{array}\right]\end{array}\right\}<0$

α-Region Stability via the Dilation Lemma

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

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

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

$\left[\begin{array}{ccc}0& -X& X\\ *& 0& 0\\ *& *& -\frac{1}{2}{\alpha }^{-1}X\end{array}\right]+He\left\{\begin{array}{c}\left[\begin{array}{c}A\\ 1\\ 0\end{array}\right]F\left[\begin{array}{ccc}1& -ϵ1& ϵ1\end{array}\right]\end{array}\right\}<0$.

Moreover, for every $X$ that satisfies

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

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

$\left[\begin{array}{ccc}0& -X& X\\ *& 0& 0\\ *& *& -\frac{1}{2}{\alpha }^{-1}X\end{array}\right]+He\left\{\begin{array}{c}\left[\begin{array}{c}A\\ 1\\ 0\end{array}\right]F\left[\begin{array}{ccc}1& -ϵ1& ϵ1\end{array}\right]\end{array}\right\}<0$.

Circular Region Stability via the Dilation Lemma

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

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

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

$\left[\begin{array}{cccc}0& -X& X& 0\\ *& 0& 0& -X\\ *& *& \frac{c}{c^2-r^2}X& 0\\ *& *& *& cX\end{array}\right]+He\left\{\begin{array}{c}\left[\begin{array}{c}A\\ 1\\ 0\\ 0\end{array}\right]F\left[\begin{array}{cccc}1& -ϵ1& ϵ1& 1\end{array}\right]\end{array}\right\}<0$

Moreover, for every $X$ that satisfies

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

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

$\left[\begin{array}{cccc}0& -X& X& 0\\ *& 0& 0& -X\\ *& *& \frac{c}{c^2-r^2}X& 0\\ *& *& *& cX\end{array}\right]+He\left\{\begin{array}{c}\left[\begin{array}{c}A\\ 1\\ 0\\ 0\end{array}\right]F\left[\begin{array}{cccc}1& -ϵ1& ϵ1& 1\end{array}\right]\end{array}\right\}<0$

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.