Notations

In this document, matrices are denoted by boldface letters (e.g.,${\displaystyle A\in \mathbb {R} ^{n\times n}}$), column matrices are denoted by lowercase boldface letters (e.g.,${\displaystyle x\in \mathbb {R} ^{n}}$), scalars are denoted by simple letters (e.g.,γ ${\displaystyle \in \mathbb {R} }$), and operators are denoted by script letters (e.g., ${\displaystyle {\mathcal {G}}}$: ${\displaystyle {\mathcal {L_{2e}}}}$${\displaystyle {\mathcal {L_{2e}}}}$. The set of ${\displaystyle n}$ by ${\displaystyle m}$ real matrices is denoted as ${\displaystyle \mathbb {R} ^{n\times m}}$, the set of ${\displaystyle n}$ by ${\displaystyle m}$ complex matrices is denoted as ${\displaystyle \mathbb {C} ^{n\times m}}$, and the set of ${\displaystyle nxn}$ symmetric matrices is denoted as ${\displaystyle \mathbb {S} ^{n}}$. The identity matrix is written as 1 and a matrix filled with zeros is written as 0. The dimensions of 1 and 0 are specified when necessary. Repeated blocks within symmetric matrices are replaced by for brevity and clarity. The conjugate transpose or Hermitian transpose of the matrix ${\displaystyle V\in \mathbb {C} ^{n\times m}}$ is denoted by ${\displaystyle V^{H}}$. The notation He{·} is used as a shorthand in situations with limited space, where He{·} = (·) + (·) ${\displaystyle ^{H}}$. The real and imaginary parts of the complex number ${\displaystyle z\in \mathbb {C} }$ are denoted as Re(${\displaystyle z}$) and Im(${\displaystyle z}$), respectively. The Kroenecker product of two matrices is denoted by ⊗.

Consider the square matrix ${\displaystyle A\in \mathbb {R} ^{n\times n}}$. The eigenvalues of ${\displaystyle A}$ are denoted by ${\displaystyle \lambda _{i}(A),i=1,2,...,n}$. The matrix A is Hurwitz if all of its eigenvalues are in the open left-half complex plane (i.e., Re ${\displaystyle \lambda _{i}(A),i=1,2,...,n}$ ). 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., ${\displaystyle |\lambda _{i}(A)|<1,i=1,...,n)}$. If ${\displaystyle A\in \mathbb {S} ^{n}}$, then the minimum eigenvalue of A is denoted by ${\displaystyle \lambda (A)}$ and its maximum eigenvalue is denoted by ${\displaystyle {\bar {\lambda }}(A)}$.

Consider the matrix B ${\displaystyle \in \mathbb {R} ^{n\times m}}$. The minimum singular value of B is denoted by ${\displaystyle {\underline {\sigma }}}$ (B) and its maximum singular value is denoted by ${\displaystyle {\bar {\sigma }}}$(B). The range and nullspace of B are denoted by ${\displaystyle \mathbb {R} }$(B) and ${\displaystyle \mathbb {N} }$(B), respectively. The Frobenius norm of B is ||B|| = ${\displaystyle {\sqrt {tr(B^{H}B)}}}$.

A state-space realization of the continuous-time linear time-invariant (LTI) system

${\displaystyle {\dot {x}}(t)=Ax(t)+Bu(t)}$,

${\displaystyle y(t)=Cx(t)+Du(t),}$.
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

${\displaystyle x_{k+1}=A_{d}x_{k}+B_{d}u_{k},}$

${\displaystyle y_{k}=C_{d}x_{k}+D_{d}u_{k},}$

is often written compactly as ${\displaystyle (A_{d},B_{d},C_{d},D_{d})}$.

The ${\displaystyle {\mathcal {H}}}$∞ norm of the LTI system ${\displaystyle {\mathcal {G}}}$ is denoted by ||${\displaystyle {\mathcal {G}}}$||∞ and the ${\displaystyle {\mathcal {H}}_{2}}$ norm of ${\displaystyle {\mathcal {G}}}$ is denoted by ||${\displaystyle {\mathcal {G}}}$||${\displaystyle _{2}}$.

The inner product spaces ${\displaystyle {\mathcal {L}}_{2}and{\mathcal {L}}_{2e}}$ for continuous-time signals are defined as follows.

${\displaystyle \{{\mathcal {L}}_{2}\}=\left\{x:\mathbb {R} _{\geq 0}\rightarrow \mathbb {R} ^{n}|||x||_{2}^{2}=\int _{0}^{\infty }x^{T}(t)x(t)dt<\infty \right\},}$

${\displaystyle \{{\mathcal {L}}_{2e}\}=\left\{x:\mathbb {R} _{\geq 0}\rightarrow \mathbb {R} ^{n}|||x||_{2T}^{2}=\int _{0}^{T}x^{T}(t)x(t)dt<\infty ,\forall T\in \mathbb {R} _{\geq 0}\right\}.}$

The inner product sequence spaces ℓ2 and ℓ2e for discrete-time signals are defined as follows.
${\displaystyle \{{\mathcal {l}}_{2}\}=\left\{x:\mathbb {Z} _{\geq 0}\rightarrow \mathbb {R} ^{n}|||x||_{2}^{2}=\sum _{k=0}^{\infty }x_{k}^{T}x_{k}<\infty ,\right\}.}$
${\displaystyle \{{\mathcal {l}}_{2e}\}=\left\{x:\mathbb {Z} _{\geq 0}\rightarrow \mathbb {R} ^{n}|||x||_{2N}^{2}=\sum _{k=0}^{N}x_{k}^{T}x_{k}<\infty ,\forall N\in \mathbb {Z} _{\geq 0}\right\}.}$