Abstract
To ease readability and minimize the need of external references, this chapter introduces all required mathematical preliminaries for the later analysis.
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- 1.
Let X be a non-empty set. The triple \((X, +, \cdot )\) with “inner” composition \(+\) (addition) and “outer” composition \(\mathbb {F}\times X \rightarrow X, \; (\alpha ,\, \varvec{v}) \mapsto \alpha \cdot \varvec{v}\) (scalar multiplication) is called vector space (over a field \(\mathbb {F}\)), if following axioms hold: (i) \((X,+)\) is an Abelian group [8, p. 57], (ii) \(\alpha \cdot (\varvec{v} + \varvec{w}) = \alpha \cdot \varvec{v} + \alpha \cdot \varvec{w}\) and \((\alpha + \beta )\cdot \varvec{v} = \alpha \cdot \varvec{v} + \beta \cdot \varvec{v}\) for all \(\alpha , \beta \in \mathbb {F}\) and \(\varvec{v},\varvec{w} \in X\) (distributivity) and (iii) \(\alpha \cdot (\beta \cdot \varvec{v}) = \alpha \, \beta \cdot \varvec{v}\) and \(1\cdot \varvec{v} = \varvec{v}\) with multiplicative identity \(1 \in \mathbb {F}\) for all \(\alpha , \beta \in \mathbb {F}\) and \(\varvec{v} \in X\) [8, p. 119].
- 2.
For a definition see [8, p. 73].
- 3.
A set \(C \subset \mathbb {R}^m\), \(m\in \mathbb {N}\), is compact, if and only if, C is closed and bounded (Heine-Borel theorem, see [8, Theorem 3.5] or [327, Theorems 2–6 I]).
- 4.
If X is any non-empty collection of elements \(x, y, z,\ldots \) and there exists a function \(d_X:X \times X \rightarrow \mathbb {R}_{\ge 0}\) with the properties (i) \(d_X(x, y) = d_X(y, x)\) (symmetry), (ii) \(d_X(x, z) \le d_X(x, y) + d_X(y, z)\) (triangular inequality) and (iii) \(d_X(x, y) = 0\) if and only if \(x=y\), then \((X, d_X)\) is called metric space with the metric \(d_X\) on X. \(d_X(x, y)\) is called the distance between x and y (see [327, pp. 111–112] or [8, pp. 142, 143]).
- 5.
A metric space \((X, d_X)\) with metric \(d_X:X \times X \rightarrow \mathbb {R}_{\ge 0}\) is complete, if every Cauchy sequence converges in X. A complete normed vector space is called Banach space (see [8, pp. 187, 188]).
- 6.
A vector space X over a field K with norm induced by the scalar (or inner) product \(\langle \cdot ,\cdot \rangle :\, X \times X \rightarrow K\) [8, p. 165] (e.g. the Euclidean norm \(\Vert \cdot \Vert _2 = \sqrt{\langle \cdot ,\cdot \rangle }\) is such a norm) is called inner product space. A complete inner product space is a Hilbert space [8, p. 189].
- 7.
A standard basis of the Euclidean space \(\mathbb {R}^n\) consists of unit vectors \(\varvec{e}_k = (\varvec{0}_{k-1}^{\top }, \, 1, \,\varvec{0}_{n-k}^{\top })^{\top }\) for all \(k \in \{1, \ldots , n\}\), where each unit vector \(\varvec{e}_k\) points in the direction of the k-th axis of the Cartesian coordinate system; e.g. for \(\mathbb {R}^3\), the basis vectors are given by \(\varvec{e}_1 = (1, 0, 0)^{\top }\), \(\varvec{e}_2 = (0, 1, 0)^{\top }\) and \(\varvec{e}_3 = (0, 0, 1)^{\top }\) (see [100, pp. 76, 81] or [8, Beispiel I.12.4]). The standard basis is orthonormal, since \(\Vert \varvec{e}_k \Vert = 1\) (having unit or normalized length) and \(\langle \varvec{e}_k,\, \varvec{e}_l\rangle = \varvec{e}_k^{\top }\varvec{e}_l = 0\) for all \(k \ne l \in \{1, \ldots , n\}\) (orthogonality) (see [100, p. 273]).
- 8.
An interval \(I\subseteq \mathbb {R}\) has the following property: \(x, z \in I:\, x<z \; \Rightarrow \; \forall \, x< y < z:y \in I\) [8, p. 107]. Let \(a, b \in \mathbb {R}\) with \(a<b\), then e.g. (a, b), [a, b), (a, b] or [a, b] and \(\emptyset \) are intervals.
- 9.
A subset \(I^n \subset \mathbb {R}^n\) is called interval of \(\mathbb {R}^n\), if there exist (line) intervals \(I_k \subset \mathbb {R}\) such that \(I^n = \prod _{k=1}^n \, I_k\). For \(\varvec{v}=(v_1, \ldots , v_n)^{\top }, \, \varvec{w}=(w_1, \ldots , w_n)^{\top } \in \mathbb {R}^n\) with \(v_k \le w_k\) for all \(k \in \{1, \ldots , n\}\), write \((\varvec{v}, \, \varvec{w}) := \prod _{k = 1}^n(v_k,\, w_k)\), \((\varvec{v}, \, \varvec{w}] := \prod _{k = 1}^n(v_k,\, w_k]\), \([\varvec{v}, \, \varvec{w}) := \prod _{k = 1}^n[v_k,\, w_k)\) or \([\varvec{v}, \, \varvec{w}] := \prod _{k = 1}^n[v_k,\, w_k]\). The interval \(I^n = (\varvec{v}, \, \varvec{w})\) is open, whereas the interval \(I^n = [\varvec{v}, \, \varvec{w}]\) is closed [10, p. 8].
- 10.
A collection \(\mathcal {B}(X)\) of subsets of X is called \(\sigma \)-algebra, if it has following properties: (i) \(X \in \mathcal {B}(X)\) (hence \(\mathcal {B}(X)\) is non-empty), (ii) \(B \in \mathcal {B}(X) \Rightarrow (X \setminus B) \in \mathcal {B}(X)\) and (iii) \(B_{n} \in \mathcal {B}(X) \text { for all } n \in \mathbb {N}\Rightarrow \bigcup _{n \in \mathbb {N}} B_{n} \in \mathcal {B}(X)\). Note that each \(\sigma \)-algebra includes also \(\emptyset \) [10, p. 3].
- 11.
Here \(\mathfrak {P}(X)\) denotes the power set of \(X \subseteq \mathbb {R}^n\). The power set \(\mathfrak {P}(X)\) of a set X consists of all subsets of X (e.g. \(\emptyset \in X \in \mathfrak {P}(X)\) [8, p. 10]).
- 12.
Let \((X, \mathcal {B}(X), \mu )\) be a measure space. A measure \(\mu \) is called non-increasing, if \(X_1 \subset X_2 \subseteq X\) implies \(\mu (X_1) \le \mu (X_2)\) [10, Satz IX.2.3(iii)]. A mapping \(\mu :\mathfrak {P}(X) \rightarrow [0,\, \infty ]\) is called \(\sigma \)-subadditive, if for every sequence \(\{A_k\}_{k \in \mathbb {N}}\) with \(A_k \in \mathfrak {P}(X)\) the following holds \(\mu (\bigcup _{k = 1}^{\infty } A_k) \le \sum _{k = 1}^{\infty } \mu (A_k)\) [10, p. 17].
- 13.
A proper interval \(I \subseteq \mathbb {R}\) has \(\mu _L(I)>0\) (it is non-empty and consists of more than one element) [8, p. 107].
- 14.
Let \((I,\mathcal {A},\mu _L)\) be a measurable space. A function \(\varvec{f}:I \rightarrow Y\) is said to be simple, if it has following properties: (i) \(\varvec{f}(I) := \{ \varvec{f}(x) \in Y \, \vert \, x \in I \; \}\) is finite, (ii) \(\varvec{f}^{-1}(\varvec{y}) \in \mathcal {A}\) for all \(\varvec{y} \in Y\) and (iii) \(\mu _L(\varvec{f}^{-1}(\mathbb {R}^n \setminus \{\varvec{0}\}))<\infty \) where \(\varvec{f}^{-1}(\mathbb {R}^n \setminus \{\varvec{0}\}) := \{ \; x \in I \, \vert \, \varvec{f}(x) \ne \varvec{0}\; \}\) [10, p. 65].
- 15.
For a definition of a Cauchy-sequence in the context of semi-norms, see [10, p. 86].
- 16.
Let X be a non-empty set and \(A \subset X\). Then the indicator (or characteristic) function [8, p. 18] of A is given by
$$ \chi _A:\, A \rightarrow \{0, 1\}, \qquad x \mapsto \chi _A(x) := {\left\{ \begin{array}{ll} 1, &{} x\in A \\ 0, &{} x\in X \setminus A. \end{array}\right. } $$ - 17.
For more details on Cauchy-Riemann integration, see [9, Abschnitt VI.3].
- 18.
A semi-norm has the properties (np\(_2\)) and (np\(_3\)) of a norm (see p. 33) whereas (np\(_1\)) is replaced by \(\Vert \varvec{x} \Vert \ge 0\) for all \(\varvec{x} \in Y\) (see [10, p. 85]).
- 19.
A set \(U \subset \mathbb {R}^n\), \(n \in \mathbb {N}\), is said to be a neighborhood of \(\varvec{x}_{0} \in \mathbb {R}^n\), if and only if, there exists \(\delta >0\) such that \(\mathbb {B}^n_{\delta }(\varvec{x}_{0}) \subset U\) (see [8, p. 144]). Clearly, for \(\varvec{x}_{0} \in \mathbb {R}^n\) and \(\delta >0\), \(\overline{\mathbb {B}}^n_{\delta }(\varvec{x}_{0})\) is a neighborhood around \(\varvec{x}_{0}\).
- 20.
- 21.
The Laplace transform of a function \(f(\cdot ) \in \mathcal {L}^{1}_{{{\mathrm{loc}}}}(\mathbb {R}_{\ge 0};\mathbb {R})\) is defined by \(f(s):=\mathscr {L}\left\{ f(t)\right\} :=\int _0^{\infty }f(t)\exp (-st)\mathrm{\, d} t \,\) or
with \(\mathfrak {R}(s) \ge \alpha \), if there exists \(\alpha \in \mathbb {R}\) such that \([t \mapsto \exp (-\alpha t)f(t)] \in \mathcal {L}^{1}(\mathbb {R}_{\ge 0};\mathbb {R})\) [149, S. 742]. The inverse of the Laplace transform is given by \(f(t)=\mathscr {L}^{-1}\left\{ f(s)\right\} \) or 
. \(\mathcal {L}^{p}_{({{\mathrm{loc}}})}(I;Y)\) is the space of measurable, (locally) p-integrable functions mapping \(I \rightarrow Y\) and \(\mathfrak {R}(s)\) denotes the real part of the complex variable \(s \in \mathbb {C}\). - 22.
Actually, all coefficients must have the same sign. The negative case is not considered. Note that any polynomial \(p_{-}(s)\in \mathbb {R}[s]\) with only negative coefficients can be rendered into a polynomial \(p_{+}(s)=-p_{-}(s)\) with only positive coefficients.
- 23.
Nowadays, it is common to use the term “high-frequency gain” for both system descriptions in the time (state space) and in the frequency (transfer function) domain. Formerly, “high-frequency gain” denoted the “leading coefficient” of the numerator of the transfer function, whereas “instantaneous gain” was equivalently used in the time domain (see [62]).
- 24.
For stable systems of form (5.44), the system matrix \(\varvec{A}\) is Hurwitz which implies \(\det (\varvec{A})\ne 0\). Hence, the inverse \(\varvec{A}^{-1}\) exists.
- 25.
Definition 5.83 (Comparison function classes \(\mathcal {K}\), \(\mathcal {K}_{\infty }\) and \(\mathcal {KL}\))
For \(0 <\bar{x} \le \infty \), \(\alpha (\cdot ) \in \mathcal {C}([0,\bar{x});\mathbb {R}_{\ge 0})\) and \(\beta (\cdot ,\cdot ) \in \mathcal {C}( \mathbb {R}_{\ge 0}\times [0,\bar{x});\mathbb {R}_{\ge 0})\), the classes of the comparison functions are defined as follows:
$$ \begin{array}{llll} \boxed {\text {class }\mathcal {K}:} &{} \alpha (\cdot ) \in \mathcal {K}&{} :\Longleftrightarrow &{} \left\{ \begin{array}{ll} \text {(i)} &{} \lim _{x \rightarrow 0}\alpha (x) = 0 \quad \wedge \quad \\ \text {(ii)} &{} \forall \, x\in (0,\bar{x}):\; \alpha (x)> 0 \quad \wedge \quad \\ \text {(iii)} &{} \forall \, x_2, x_1 \in (0,\bar{x}) :\; x_2 \ge x_1 \\ &{} \quad \;\Rightarrow \; \alpha (x_2) \ge \alpha (x_1)>0 \\ &{} (\alpha (\cdot ) \,\text {is monotonically increasing)}. \end{array}\right. \\ \boxed {\text {class } \mathcal {K}_{\infty }:} &{} \alpha (\cdot ) \in \mathcal {K}_{\infty }&{} :\Longleftrightarrow &{} \left\{ \begin{array}{ll} \text {(i)} &{} \alpha (\cdot ) \in \mathcal {K}\quad \wedge \quad \\ \text {(ii)} &{} \bar{x} = \infty \quad \wedge \quad \\ \text {(iii)} &{} \lim _{x \rightarrow \infty }\alpha (x) = \infty . \end{array}\right. \\ \boxed {\text {class } \mathcal {KL}:} &{} \beta (\cdot ,\cdot ) \in \mathcal {KL}&{} :\Longleftrightarrow &{} \left\{ \begin{array}{ll} \text {(i)} &{} \forall \, t \ge 0:\beta (t,\cdot ) \in \mathcal {K}\quad \wedge \quad \\ \text {(ii)} &{} \forall \, x_0 \in [0,\bar{x}):\\ &{} \text {(a)} \; \forall \, t_2 \ge t_1 \ge 0 \; \Rightarrow \; \beta (t_2, x_0) \le \beta (t_1, x_0) \\ &{} \quad (\beta (\cdot , x_0)\, \text {is monotonically decreasing)}\\ &{} \text {(b)} \; \lim _{t \rightarrow \infty } \beta (t, x_0) = 0. \end{array}\right. \end{array} $$For example, \(x \mapsto 1-e^{-x} \in \mathcal {K}\) but \(\notin \mathcal {K}_{\infty }\) or \((t, x) \mapsto \tfrac{1}{1 + t} \, x \in \mathcal {KL}\).
with 
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Hackl, C.M. (2017). Mathematical Preliminaries. In: Non-identifier Based Adaptive Control in Mechatronics. Lecture Notes in Control and Information Sciences, vol 466. Springer, Cham. https://doi.org/10.1007/978-3-319-55036-7_5
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