iTanks_abg.tex

MTT command:

mtt iTanks abg tex

Figure 9.1: System iTanks: acausal bond graph

Figure 9.1 (on page ) shows the bond graph of a two-tank system superimposed on a schematic diagram. The two C components corresponds to the fluid storage and how it relates to the pressure at the base of the tanks. In this case, for simplicity, each tank ($i=1$ or $i=2$) is assumed to have a unity constitutive relationship:

$$= p_i = v_i =$$ (9.1)

The volumetric flow rate into the first, and out of the second, tank is represented by the two unlabelled R components. Again, each is assumed to have a unit constitutive relationship:

$$= f_i = \Delta_i =$$ (9.2)

The volumetric flow rate between the first and the second tanks is represented R component labelled $k$. The constitutive relationship is assumed linear of the form:

$$= f = k \Delta =$$ (9.3)

The system has two inputs:

$$\begin{aligned}u_1 &= \text{input pressure at left-hand pipe} \\ u_2 &= \text{input pressure at right-hand pipe} \end{aligned}$$

and two outputs:

$$\begin{aligned}y_1 &= p_1 = \text{pressure at left-hand tank} \\ y_2 &= p_2 = \text{pressure at right-hand tank} \end{aligned}$$

The system transfer-function matrix is given by:

$$\begin{aligned}G_{11} = G_{22} &= \frac{(s + k + 1)}{(s^2 + 2... ...{21} &= \frac{k}{(s^2 + 2 s {(k + 1)} + 2 k + 1)} \end{aligned}$$

However, Figure 9.1 (on page ) shows the causality of the SS components to invert the system with respect to its inputs and outputs. Figure (on page ) shows the causally complete bond graph; this system has no dynamic components in integral causality - the inverse has no poles and therefore the system has no zeros.

Some further representations of the inverse appear in the following sections.