3. ConventionsΒΆ

1. We use the notation $$a \triangleq b$$ to denote that $$a$$ is equal to $$b$$ by definition.

2. We denote by $$\Re$$ the set of real numbers, by $$\mathbb Z$$ the set of integers and by $$\mathbb N \triangleq \{0, \, 1, \, \ldots \}$$ the set of natural numbers. The set $$\mathbb N_+$$ is defined as $$\mathbb N_+ \triangleq \{1, \, 2, \, \ldots \}$$.

3. $$f(\cdot)$$ denotes a function where $$(\cdot)$$ stands for the undesignated variables. $$f(x)$$ denotes the value of $$f(\cdot)$$ at the point $$x$$. $$f\colon A \rightarrow B$$ indicates that the domain of $$f(\cdot)$$ is in the space $$A$$ and its range in the space $$B$$.

4. We say that a function $$f \colon \Re^n \to \Re$$ is once continuously differentiable if $$f(\cdot)$$ is defined on $$\Re^n$$, and if $$f(\cdot)$$ has a continuous derivative on $$\Re^n$$.

5. For $$f \colon \Re \to \Re$$, we denote by $$f^{(N)}(\cdot)$$ its $$N$$-th derivative.

6. For $$f \colon \Re \to \Re$$ and $$t \in \Re$$, we denote by $$f(t^-) \triangleq \lim_{s \uparrow t} f(s)$$ the limit from below.

7. For $$s \in \Re$$, we define the ceiling function as $$\lceil s \rceil \triangleq \arg \min\{ k \in \mathbb Z \ | \ s \le k \}$$.

8. For $$t_R \in \Re$$ and $$t_I \in \mathbb N$$, we write $$t^+ \triangleq (t_R, \, t_I)^+$$ for the right limit at $$t$$. It holds that $$(t_R, \, t_I)^+ \Leftrightarrow (\lim_{\epsilon \to 0} (t_R+\epsilon), t_{I_{max}})$$, where $$I_{max}$$ is the largest occurring integer of superdense time. Similarly, we write $$\mathbin{^-t}$$ for the left limit at $$t$$, for which it holds that $$\mathbin{^-(t_R, \, t_I)} \Leftrightarrow (\lim_{\epsilon \to 0} (t_R-\epsilon), 0)$$.

9. We write a requirement shall be met if it must be fulfilled. If the feature that implements a shall requirement is not in the final system, then the system does not meet this requirement. We write a requirment should be met if it is not critical to the system working, but is still desirable.