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Notations

In this document, matrices are denoted by boldface letters (e.g.,$A\in \mathbb {R} ^{n\times n}$), column matrices are denoted by lowercase boldface letters (e.g.,$x\in \mathbb {R} ^{n}$), scalars are denoted by simple letters (e.g.,γ $\in \mathbb {R}$), and operators are denoted by script letters (e.g., ${\mathcal {G}}$: ${\mathcal {L_{2e}}}$ →${\mathcal {L_{2e}}}$. The set of $n$ by $m$ real matrices is denoted as $\mathbb {R} ^{n\times m}$, the set of $n$ by $m$ complex matrices is denoted as $\mathbb {C} ^{n\times m}$, and the set of $nxn$ symmetric matrices is denoted as $\mathbb {S} ^{n}$. 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 $V\in \mathbb {C} ^{n\times m}$ is denoted by $V^{H}$. The notation He{·} is used as a shorthand in situations with limited space, where He{·} = (·) + (·) $^{H}$. The real and imaginary parts of the complex number $z\in \mathbb {C}$ are denoted as Re($z$) and Im($z$), respectively. The Kroenecker product of two matrices is denoted by ⊗.

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

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

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

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

$y(t)=Cx(t)+Du(t),$.
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

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

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

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

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

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

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

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

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