By Krzysztof Bartecki
This monograph specializes in the mathematical modeling of disbursed parameter platforms during which mass/energy delivery or wave propagation phenomena happen and that are defined via partial differential equations of hyperbolic kind. The case of linear (or linearized) 2 x 2 hyperbolic platforms of stability legislation is taken into account, i.e., structures defined by way of coupled linear partial differential equations with variables representing actual amounts, reckoning on either time and one-dimensional spatial variable.
Based on useful examples of a double-pipe warmth exchanger and a transportation pipeline, regular configurations of boundary enter indications are analyzed: collocated, in which either signs have an effect on the approach on the related spatial element, and anti-collocated, during which the enter indications are utilized to the 2 assorted finish issues of the system.
The result of this e-book emerge from the sensible event of the writer won in the course of his stories performed within the experimental set up of a warmth alternate middle in addition to from his examine adventure within the box of mathematical and laptop modeling of dynamic structures. The booklet offers important effects referring to their state-space, move functionality and time-domain representations, which are precious either for the open-loop research in addition to for the closed-loop layout.
The e-book is basically meant to assist execs in addition to undergraduate and postgraduate scholars concerned about modeling and automated keep an eye on of dynamic systems.
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Extra info for Modeling and Analysis of Linear Hyperbolic Systems of Balance Laws
It is interpreted as an equation in the space X−1 where the state operator A naturally extends to the state operator A−1 in X−1 . 62) ˙ being the factor control vector (see also Zołopa and Grabowski 2008). However, according to the Grabowski-Calier theory, d is not being counted from Eq. 62) as it would require Salamon-Weiss theory, but from the so-called factor control equation (see Grabowski and Callier 2001a). 2 Resolvent Operator This section introduces the notion of the resolvent of the system state operator A considered in the previous section.
2 Transportation Pipeline A transportation pipeline of length L and diameter Dp with a reservoir in the upstream section and valve in the downstream section is depicted in Fig. 2. Pressure value in the inlet point (l = 0) of the pipeline is influenced by the variable fluid level h(t) and equals pi (t). Similarly, the fluid flow qo (t) at the pipeline outlet (l = L) can vary due to the changes in the valve position. 3 Examples of 2 × 2 Systems 15 Fig. 2 Schematic of a transportation pipeline. p(l, t)—fluid pressure; q(l, t)—fluid flow; pi (t) = p(0, t)—inlet fluid pressure; qo (t) = q(L, t)—outlet fluid flow; L—pipeline length; Dp —pipeline diameter In order to develop a mathematical model of the fluid flow in the pipeline, the following assumptions are made: • The flow is one-dimensional—mass flow q and pressure p depend only on time t and on the geometrical variable l.
Similarly, the boundary/observation operator Z ∈ L(D(M), W) in Eq. 32) can stand both for the operator Z + for the case of U = U + from Eq. 17), and for the operator Z ± with U = U ± from Eq. 18). The above state/signal representation of the boundary control system might seem to be only a trivial mathematical manipulation. However, as shown in (Arov et al. , the notion of the passivity or conservativity of a dynamical system, are much simpler to formulate and analyze in this framework than in its input/space/output counterpart given by Eqs.