LMIs in Control/Click here to continue/Notation
Notations
Notations Related to Subspaces
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set of all real numbers |
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set of all positive real numbers |
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set of all negative real numbers |
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set of all complex numbers |
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right-half complex plane |
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left-half complex plane |
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set of all real vectors of dimension |
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set of all complex vectors of dimension |
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set of all real matrices of dimension |
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set of all complex matrices of dimension |
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set of real matrices with rank |
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set of complex matrices with rank |
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closed right-half complex plane, |
ker |
kernel of transformation or matrix |
Image |
image of transformation or matrix |
conv |
convex hull of set |
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set of symmetric matrix in |
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boundary set of |
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set of all extreme points of |
Notations Related to Vectors and Matrices
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zero vector in |
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zero matrix in |
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identity matrix of order |
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inverse matrix of matrix |
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transpose of matrix |
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complex conjugate of matrix |
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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 |
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condition number of matrix |
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spectral radius of matrix |
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is Hermite (symmetric) positive definite |
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is Hermite (symmetric) semi-positive definite |
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is Hermite (symmetric) negative definite |
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is Hermite (symmetric) semi-negative definite |
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matrix satisfying |
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set of all eigenvalues of matrix |
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th eigenvalue of matrix |
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maximum eigenvalue of matrix |
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minimum eigenvalue of matrix |
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th singular value of matrix |
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maximum singular value of matrix |
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minimum singular value of matrix |
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sum of matrix and its transpose, |
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spectral norm of matrix |
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Frobenius norm of matrix |
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row-sum norm of matrix |
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column-sum norm of matrix |
Notations of Relations and Manipulations
editOther Notations
editExamples
editConsider 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
edit- 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.