# 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.