LMIs in Control/Click here to continue/Notation

Notations Notations Related to Subspaces 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 Notations Related to Vectors and Matrices 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 Other Notations Examples 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 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.