LMIs in Control/Click here to continue/Notation

Notations



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set of all real numbers

 

set of all positive real numbers

 

set of all negative real numbers

 

set of all complex numbers

 

right-half complex plane

 

left-half complex plane

 

set of all real vectors of dimension  

 

set of all complex vectors of dimension   

 

set of all real matrices of dimension  

 

set of all complex matrices of dimension  

 

set of   real matrices with rank  

 

set of   complex matrices with rank  

 

closed right-half complex plane,  

ker 

kernel of transformation or matrix  

Image 

image of transformation or matrix  

conv 

convex hull of set  

 

set of symmetric matrix in  

 

boundary set of  

 

set of all extreme points of  


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zero vector in  

 

zero matrix in  

 

identity matrix of order  

 

inverse matrix of matrix  

 

transpose of matrix  

 

complex conjugate of matrix  

 

transposed complex conjugate of matrix  

Re( )

real part of matrix  

Im( )

imaginary part of matrix  

det( )

determinant of matrix  

Adj )

adjoint of matrix  

trace( )

trace of matrix  

rank( )

rank of matrix  

 

condition number of matrix  

 

spectral radius of matrix  

 

  is Hermite (symmetric) positive definite

 

  is Hermite (symmetric) semi-positive definite

 

 

 

 

 

  is Hermite (symmetric) negative definite

 

  is Hermite (symmetric) semi-negative definite

 

 

 

 

 

matrix   satisfying  

 

set of all eigenvalues of matrix  

 

 th eigenvalue of matrix  

 

maximum eigenvalue of matrix  

 

minimum eigenvalue of matrix  

 

 th singular value of matrix  

 

maximum singular value of matrix  

 

minimum singular value of matrix  

 

sum of matrix   and its transpose,  

 

spectral norm of matrix  

 

Frobenius norm of matrix  

 

row-sum norm of matrix  

 

column-sum norm of matrix  

Notations of Relations and Manipulations

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Other Notations

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Examples

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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

 

 

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.
 
 

Reference

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  • 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.