Notations
![{\displaystyle \mathbb {R} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/786849c765da7a84dbc3cce43e96aad58a5868dc)
|
set of all real numbers
|
![{\displaystyle \mathbb {R} ^{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/97dc5e850d079061c24290bac160c8d3b62ee139)
|
set of all positive real numbers
|
![{\displaystyle \mathbb {R} ^{-}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/158001a03e958f49f5885033776a420fc47b7267)
|
set of all negative real numbers
|
![{\displaystyle \mathbb {C} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/f9add4085095b9b6d28d045fd9c92c2c09f549a7)
|
set of all complex numbers
|
![{\displaystyle \mathbb {C} ^{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c79f3e776f6dfb76eb39c85ad01c12d06836b87a)
|
right-half complex plane
|
![{\displaystyle \mathbb {C} ^{-}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ef850c756b367d9e303ae4fa5f58209cd63c3541)
|
left-half complex plane
|
![{\displaystyle \mathbb {R} ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
|
set of all real vectors of dimension ![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
|
![{\displaystyle \mathbb {C} ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a53b4e76242764d1bca004168353c380fef25258)
|
set of all complex vectors of dimension ![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b) ![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
|
![{\displaystyle \mathbb {R} ^{m\times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9970da40c686dbff0db2dc8d134c56e860b7f492)
|
set of all real matrices of dimension ![{\displaystyle m\times n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d)
|
![{\displaystyle \mathbb {C} ^{m\times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a4686bb6c9ac6a18f9cdb4b4ed8141c7de64c4d7)
|
set of all complex matrices of dimension ![{\displaystyle m\times n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/12b23d207d23dd430b93320539abbb0bde84870d)
|
![{\displaystyle \mathbb {R} _{r}^{m\times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4c8e0822feea79a53b60578dd0b9537110e1ace3)
|
set of real matrices with rank ![{\displaystyle r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
|
![{\displaystyle \mathbb {C} _{r}^{m\times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dfbde12570c808309754689d89c79a4a3ab30abf)
|
set of complex matrices with rank ![{\displaystyle r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0d1ecb613aa2984f0576f70f86650b7c2a132538)
|
![{\displaystyle {\bar {\mathbb {C} }}^{+}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/db9414f67bff5c0f1dc4c9045e50f566a74124c2)
|
closed right-half complex plane, ![{\displaystyle {\bar {\mathbb {C} }}^{+}=\{s|s\in \mathbb {C} ,Re(s)\leq 0\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a87467b2425461bc77ce50fa5150ff48c05acad1)
|
ker![{\displaystyle (T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f9a82bdcb95d511ed6965b0534a3eda98ceb62) |
kernel of transformation or matrix ![{\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
|
Image![{\displaystyle (T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5f9a82bdcb95d511ed6965b0534a3eda98ceb62) |
image of transformation or matrix ![{\displaystyle T}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ec7200acd984a1d3a3d7dc455e262fbe54f7f6e0)
|
conv![{\displaystyle (\mathbb {F} )}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c51fe55cb5c68ef1bc12133604739fdce8f9f7c5) |
convex hull of set ![{\displaystyle \mathbb {F} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/573f72afae7df709959ab1a58cd643743466a187)
|
![{\displaystyle \mathbb {S} ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41c00c9d3635f5230e6ac11902a50f2323794ed6)
|
set of symmetric matrix in ![{\displaystyle \mathbb {R} ^{n\times n},\mathbb {S} ^{n}=\{M|M=M^{T},M\in \mathbb {R} ^{n\times n}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d028c5192ab5630fb7fbbd60e0f2ebb95ecd0601)
|
![{\displaystyle \mathbb {F} _{B}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7e28144928f479727b4573486f684a2005041d97)
|
boundary set of ![{\displaystyle \mathbb {F} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/573f72afae7df709959ab1a58cd643743466a187)
|
![{\displaystyle \mathbb {F} _{E}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f3beefbd511a2f8c300187c5158679e06492e9e)
|
set of all extreme points of ![{\displaystyle \mathbb {F} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/573f72afae7df709959ab1a58cd643743466a187)
|
![{\displaystyle 0_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a0b58ccf0ecd384736c60c50f81167d9d0807ab)
|
zero vector in ![{\displaystyle \mathbb {R} ^{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c510b63578322050121fe966f2e5770bea43308d)
|
![{\displaystyle 0_{m\times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6ca4405d59fa5eb254613e16eefefff3eae5587d)
|
zero matrix in ![{\displaystyle \mathbb {R} ^{m\times n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9970da40c686dbff0db2dc8d134c56e860b7f492)
|
![{\displaystyle I_{n}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aba34f081d776e30204f3458e4f50b403b09e5c6)
|
identity matrix of order ![{\displaystyle n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
|
![{\displaystyle A^{-1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/83ba3a7118652cffd5de466dc439ee9184371d50)
|
inverse matrix of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle A^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d978c0c4f22282a313e4107f97b9eda56315b258)
|
transpose of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle {}^{-}A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/497d89be8b120a4adffa59012da007571add590b)
|
complex conjugate of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle A^{H}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8904d54cf6b5808177fdfbd13c2b73966d8e0ab)
|
transposed complex conjugate of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
Re( ) |
real part of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
Im( ) |
imaginary part of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
det( ) |
determinant of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
Adj ) |
adjoint of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
trace( ) |
trace of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
rank( ) |
rank of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \kappa (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8447cd494704eefa4930beb224a7d35209ef1af0)
|
condition number of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \rho (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a9d232ff707be40f0398c038ee817fc0742b9c8b)
|
spectral radius of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle A>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/232a94dbd97f2ce670d4987c5ad8ad82b072861b)
|
is Hermite (symmetric) positive definite
|
![{\displaystyle A\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebee3e714a32508aa26658d5c88f6a09237c82dc)
|
is Hermite (symmetric) semi-positive definite
|
![{\displaystyle A>B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5791db2db54217e5290e070826a0bacc33e0388d)
|
![{\displaystyle A-B>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/267ffc8dd23d6ace245755c0121f36ae9fbe495a)
|
![{\displaystyle A\geq B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/11e0077e04b2950393ccf440a4e7129cfb0fb35d)
|
![{\displaystyle A-B\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7ddfd5da978adf6cfdd1d0be0244dd4fbd2ba153)
|
![{\displaystyle A<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4caa73f1ae386242ad1da3f3bdf913f522bd159)
|
is Hermite (symmetric) negative definite
|
![{\displaystyle A\geq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ebee3e714a32508aa26658d5c88f6a09237c82dc)
|
is Hermite (symmetric) semi-negative definite
|
![{\displaystyle A<B}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f7bff3eeb3cd93f23984f67cb4a10787de83596)
|
![{\displaystyle A-B<0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2fa6d5d9b3ef07817ad19abb820f3dd4f9220dd9)
|
![{\displaystyle A^{\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adba896b74b0d7c0b3cb0dcf4c8db251829450fb)
|
![{\displaystyle A-B\leq 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a510cededcea6dbe2ca9c036056f1d462aaf2c25)
|
![{\displaystyle A^{\frac {1}{2}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/adba896b74b0d7c0b3cb0dcf4c8db251829450fb)
|
matrix satisfying ![{\displaystyle Z^{T}Z=A>0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4db2e40b8488b97e9c4162ebb9a1078b79c4b098)
|
![{\displaystyle \lambda (A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/81c1311c400c28b9db66e593eb58084df580baff)
|
set of all eigenvalues of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \lambda _{i}(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3d2d037e41b4550dcc631bc0e36076b366b4c33b)
|
th eigenvalue of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \lambda _{max}(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8607b6b9676c0bce25705c2a26020d09d3d3f68c)
|
maximum eigenvalue of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \lambda _{min}(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52341b138f322aa31b716f054c0389ba4725c939)
|
minimum eigenvalue of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \sigma _{i}(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/41c8d1845766173614b87c5b9d7f5e2cce28b31f)
|
th singular value of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \sigma _{max}(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f3b9c564ebb3bb68de41e0b19d2cec9abb9d290)
|
maximum singular value of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \sigma _{min}(A)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2899c75100a78126e9a39555ea6c40c97decf80d)
|
minimum singular value of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \langle A\rangle _{s}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/54a620ac6e64265e1ac9a72199b566bb18874de5)
|
sum of matrix and its transpose, ![{\displaystyle \langle A\rangle _{s}=A+A^{T}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0ae42f9244b8ca7baff0e92465c4f799ee75265e)
|
![{\displaystyle \left\Vert A\right\|_{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b92e92795a96a2bc56171948678d0edb2e4efff4)
|
spectral norm of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \left\Vert A\right\|_{F}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d85e4830d22ba4a7e6a49f4daf6362a11e80070a)
|
Frobenius norm of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \left\Vert A\right\|_{1}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06ae52945a5866a701547f9f3b2f3ba97b42a659)
|
row-sum norm of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
![{\displaystyle \left\Vert A\right\|_{\infty }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/91a9ed31d4b6cc950247bc425b397190029e8f03)
|
column-sum norm of matrix ![{\displaystyle A}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7daff47fa58cdfd29dc333def748ff5fa4c923e3)
|
Consider the square matrix
. The eigenvalues of
are denoted by
. The matrix A is Hurwitz if all of its eigenvalues are in the open left-half complex plane
(i.e., Re
). 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.,
. If
, then the minimum eigenvalue of A is denoted by
and its maximum eigenvalue is denoted by
.
Consider the matrix B
. The minimum singular value of B is denoted by
(B) and its maximum singular value is denoted by
(B). The range and nullspace of B are denoted by
(B)
and
(B), respectively. The Frobenius norm of B is ||B|| =
.
A state-space realization of the continuous-time linear time-invariant (LTI) system
,
.
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
![{\displaystyle x_{k+1}=A_{d}x_{k}+B_{d}u_{k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c2c369a059f8d265103be6217755ecae4674e9cd)
![{\displaystyle y_{k}=C_{d}x_{k}+D_{d}u_{k},}](https://wikimedia.org/api/rest_v1/media/math/render/svg/955129a5ab98ec4dd968895ae9b964ca5b2d8a6e)
is often written compactly as
.
The
∞ norm of the LTI system
is denoted by ||
||∞ and the
norm of
is denoted by
||
||
.
The inner product spaces
for continuous-time signals are defined as follows.
The inner product sequence spaces ℓ2 and ℓ2e for discrete-time signals are defined as follows.
- 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.