LMIs in Control/Click here to continue/Notation

Notations

Notations Related to Subspaces

$ℝ$	set of all real numbers
${ℝ}^{+}$	set of all positive real numbers
${ℝ}^{-}$	set of all negative real numbers
$ℂ$	set of all complex numbers
${ℂ}^{+}$	right-half complex plane
${ℂ}^{-}$	left-half complex plane
${ℝ}^n$	set of all real vectors of dimension $n$
${ℂ}^n$	set of all complex vectors of dimension $n$$n$
${ℝ}^{m\times n}$	set of all real matrices of dimension $m\times n$
${ℂ}^{m\times n}$	set of all complex matrices of dimension $m\times n$
${ℝ}_r^{m\times n}$	set of $m\times n$ real matrices with rank $r$
${ℂ}_r^{m\times n}$	set of $m\times n$ complex matrices with rank $r$
${\overline{ℂ}}^{+}$	closed right-half complex plane, ${\overline{ℂ}}^{+}=\{s|s\in ℂ,Re(s)\le 0\}$
ker$(T)$	kernel of transformation or matrix $T$
Image$(T)$	image of transformation or matrix $T$
conv$(𝔽)$	convex hull of set $𝔽$
${𝕊}^n$	set of symmetric matrix in ${ℝ}^{n\times n},{𝕊}^n=\{M|M=M^T,M\in {ℝ}^{n\times n}\}$
${𝔽}_B$	boundary set of $𝔽$
${𝔽}_E$	set of all extreme points of $𝔽$

Notations Related to Vectors and Matrices

$0_n$	zero vector in ${ℝ}^n$
$0_{m\times n}$	zero matrix in ${ℝ}^{m\times n}$
$I_n$	identity matrix of order $n$
$A^{-1}$	inverse matrix of matrix $A$
$A^T$	transpose of matrix $A$
${}^{-}A$	complex conjugate of matrix $A$
$A^H$	transposed complex conjugate of matrix $A$
Re($A$)	real part of matrix $A$
Im($A$)	imaginary part of matrix $A$
det($A$)	determinant of matrix $A$
Adj$A$)	adjoint of matrix $A$
trace($A$)	trace of matrix $A$
rank($A$)	rank of matrix $A$
$\kappa (A)$	condition number of matrix $A$
$\rho (A)$	spectral radius of matrix $A$
$A>0$	$A$ is Hermite (symmetric) positive definite
$A\ge 0$	$A$ is Hermite (symmetric) semi-positive definite
$A>B$	$A-B>0$
$A\ge B$	$A-B\ge 0$
$A<0$	$A$ is Hermite (symmetric) negative definite
$A\ge 0$	$A$ is Hermite (symmetric) semi-negative definite
$A<B$	$A-B<0$
$A^{\frac{1}{2}}$	$A-B\le 0$
$A^{\frac{1}{2}}$	matrix $Z$ satisfying $Z^{T}Z=A>0$
$\lambda (A)$	set of all eigenvalues of matrix $A$
${\lambda }_i(A)$	$i$th eigenvalue of matrix $A$
${\lambda }_{max}(A)$	maximum eigenvalue of matrix $A$
${\lambda }_{min}(A)$	minimum eigenvalue of matrix $A$
${\sigma }_i(A)$	$i$th singular value of matrix $A$
${\sigma }_{max}(A)$	maximum singular value of matrix $A$
${\sigma }_{min}(A)$	minimum singular value of matrix $A$
$⟨A{⟩}_s$	sum of matrix $A$ and its transpose, $⟨A{⟩}_s=A+A^T$
${\Vert A\Vert }_2$	spectral norm of matrix $A$
${\Vert A\Vert }_F$	Frobenius norm of matrix $A$
${\Vert A\Vert }_1$	row-sum norm of matrix $A$
${\Vert A\Vert }_{\mathrm{\infty }}$	column-sum norm of matrix $A$

Examples

Consider the square matrix $A\in {ℝ}^{n\times n}$. The eigenvalues of $A$ are denoted by ${\lambda }_i(A),i=1,2,...,n$. The matrix A is Hurwitz if all of its eigenvalues are in the open left-half complex plane (i.e., Re ${\lambda }_i(A),i=1,2,...,n$ ). A matrix is Schur if all of its eigenvalues are strictly within a unit disk centered at the origin of the complex plane (i.e., $|{\lambda }_i(A)|<1,i=1,...,n)$. If $A\in {𝕊}^n$, then the minimum eigenvalue of A is denoted by $\lambda (A)$ and its maximum eigenvalue is denoted by $\overline{\lambda }(A)$.

Consider the matrix B $\in {ℝ}^{n\times m}$. The minimum singular value of B is denoted by $\underset{\_}{\sigma }$ (B) and its maximum singular value is denoted by $\overline{\sigma }$(B). The range and nullspace of B are denoted by $ℝ$(B) and $\mathbb{N}$(B), respectively. The Frobenius norm of B is ||B|| = $\sqrt{tr(B^{H}B)}$.

A state-space realization of the continuous-time linear time-invariant (LTI) system

$\dot{x}(t)=Ax(t)+Bu(t)$,

$y(t)=Cx(t)+Du(t),$.
is often written compactly as (A, B,C,D) in this document. The argument of time is often omitted in continuous-time state-space realizations, unless needed to prevent ambiguity. A state-space realization of the discrete-time LTI system

$x_{k+1}=A_d{x}_k+B_d{u}_k,$

$y_k=C_d{x}_k+D_d{u}_k,$

is often written compactly as $(A_d,B_d,C_d,D_d)$.

The $\mathscr{H}$∞ norm of the LTI system $𝒢$ is denoted by ||$𝒢$||∞ and the ${\mathscr{H}}_2$ norm of $𝒢$ is denoted by ||$𝒢$||${}_2$.

The inner product spaces ${ℒ}_{2}and{ℒ}_{2e}$ for continuous-time signals are defined as follows.

$\{{ℒ}_2\}=\{x:{ℝ}_{\ge 0}\to {ℝ}^n|||x|{|}_2^2=\underset{0}{\overset{\mathrm{\infty }}{\int }}{x}^T(t)x(t)dt<\mathrm{\infty }\},$

$\{{ℒ}_{2e}\}=\{x:{ℝ}_{\ge 0}\to {ℝ}^n|||x|{|}_{2T}^2=\underset{0}{\overset{T}{\int }}{x}^T(t)x(t)dt<\mathrm{\infty },\mathrm{\forall }T\in {ℝ}_{\ge 0}\}.$

The inner product sequence spaces ℓ2 and ℓ2e for discrete-time signals are defined as follows.
$\{{𝓁}_2\}=\{x:{\mathbb{Z}}_{\ge 0}\to {ℝ}^n|||x|{|}_2^2=\sum _{k=0}^{\mathrm{\infty }}{x}_k^T{x}_k<\mathrm{\infty },\}.$
$\{{𝓁}_{2e}\}=\{x:{\mathbb{Z}}_{\ge 0}\to {ℝ}^n|||x|{|}_{2N}^2=\sum _{k=0}^N{x}_k^T{x}_k<\mathrm{\infty },\mathrm{\forall }N\in {\mathbb{Z}}_{\ge 0}\}.$

Reference

• LMIs in Control Systems: Analysis, Design and Applications - by Guang-Ren Duan and Hai-Hua Yu, CRC Press, Taylor & Francis Group, 2013
• 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.