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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 $$ n x n $$ 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.

$$ \{\mathcal{L}_2\} = \left\{x: \mathbb{R}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_2 = \int_{0}^\infty x^T(t)x(t)dt<\infty \right\}, $$

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

The inner product sequence spaces ℓ2 and ℓ2e for discrete-time signals are defined as follows. $$ \{\mathcal{l}_2\} = \left\{x: \mathbb{Z}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_{2} = \sum_{k=0}^\infty x^T_k x_k <\infty, \right\}. $$ $$ \{\mathcal{l}_{2e}\} = \left\{x: \mathbb{Z}_{\geq 0} \rightarrow \mathbb{R}^{n} | ||x||^2_{2N} = \sum_{k=0}^N x^T_k x_k <\infty, \forall N \in \mathbb{Z}_{\geq 0} \right\}. $$