LMIs in Control/Click here to continue/Notation

Notations

Notations Related to Subspaces

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

Notations Related to Vectors and Matrices

$0_{n}$	zero vector in $\mathbb {R} ^{n}$
$0_{m\times n}$	zero matrix in $\mathbb {R} ^{m\times n}$
$I_{n}$	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
$A\geq 0$	$A$ is Hermite (symmetric) semi-positive definite
$A>B$	$A-B>0$
$A\geq B$	$A-B\geq 0$
$A<0$	$A$ is Hermite (symmetric) negative definite
$A\geq 0$	$A$ is Hermite (symmetric) semi-negative definite
$A<B$	$A-B<0$
$A^{\frac {1}{2}}$	$A-B\leq 0$
$A^{\frac {1}{2}}$	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 $A$
$\langle A\rangle _{s}$	sum of matrix $A$ and its transpose, $\langle A\rangle _{s}=A+A^{T}$
$\left\Vert A\right\|_{2}$	spectral norm of matrix $A$
$\left\Vert A\right\|_{F}$	Frobenius norm of matrix $A$
$\left\Vert A\right\|_{1}$	row-sum norm of matrix $A$
$\left\Vert A\right\|_{\infty }$	column-sum norm of matrix $A$

Examples

Consider the square matrix $A\in \mathbb {R} ^{n\times n}$. 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 $A\in \mathbb {S} ^{n}$, then the minimum eigenvalue of A is denoted by $\lambda (A)$ and its maximum eigenvalue is denoted by ${\bar {\lambda }}(A)$.

Consider the matrix B $\in \mathbb {R} ^{n\times m}$. The minimum singular value of B is denoted by ${\underline {\sigma }}$ (B) and its maximum singular value is denoted by ${\bar {\sigma }}$(B). The range and nullspace of B are denoted by $\mathbb {R}$(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 ${\mathcal {H}}$∞ norm of the LTI system ${\mathcal {G}}$ is denoted by ||${\mathcal {G}}$||∞ and the ${\mathcal {H}}_{2}$ norm of ${\mathcal {G}}$ is denoted by ||${\mathcal {G}}$||$_{2}$.

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

$\{{\mathcal {L}}_{2}\}=\left\{x:\mathbb {R} _{\geq 0}\rightarrow \mathbb {R} ^{n}|||x||_{2}^{2}=\int _{0}^{\infty }x^{T}(t)x(t)dt<\infty \right\},$

$\{{\mathcal {L}}_{2e}\}=\left\{x:\mathbb {R} _{\geq 0}\rightarrow \mathbb {R} ^{n}|||x||_{2T}^{2}=\int _{0}^{T}x^{T}(t)x(t)dt<\infty ,\forall T\in \mathbb {R} _{\geq 0}\right\}.$

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

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.