# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## 4 A Multibond Graph in a Derivative Causality Assignment  
The information on the causality of a bond graph model has been a great tool to find easy and direct results in the physical domain. The assignment of integral causality to a single bond graph (BGI) and a multibond graph (MBGI) can determine the state spaces of the systems. Furthermore, the structural controllability and structural observability of a system in the physical domain are obtained through the causal paths from the sources and detectors to the storage elements, respectively. Hence, the linearization of systems modeled by single bond graphs or multibond graphs using causal paths have been presented [30]. In order to get the steady state of a system, a derivative causality for the multiport storage elements of a multibond graph is assigned (MBGD), which is shown in Figure 10.  
Figure 10 illustrates that all the multiport storage elements have a derivative causality assignment, and the multiport dissipation elements and multiport gyrators must have the  
Figure 10: Multibond graph in a derivative causality assignment. appropriate causality to get a correct multibond graph model. The relationships of the MBGD are obtained through the following Lemma. **Lemma 2**. : _Consider a multibond graph model in a preferred derivative causality assignment (MBGD) of an LTI system whose scheme is shown in Figure 10, where all the multiport storage elements have a derivative causality assignment and the multiport junction structure is defined by:_  
\[\left[ \begin{array}{c}\underline{z}(t)\\ \underline{z}_{d}(t)\\ \underline{l}_{\text{G}\text{Y}}^{d}(t)\\ \underline{D}_{\text{in}}^{d}(t)\\ \underline{y}(t)\end{array} \right]=\left[ \begin{array}{ccccc}\textbf{J}_{ \text{11}}^{\text{11}}&\textbf{J}_{\text{11}}^{\text{12}}&\textbf{J}_{\text{1 1}}^{\text{13}}&\textbf{J}_{\text{12}}^{\text{11}}&\textbf{J}_{\text{13}}^{ \text{11}}\\ \textbf{J}_{\text{11}}^{\text{21}}&\textbf{J}_{\text{11}}^{\text{23}}&\textbf{J }_{\text{11}}^{\text{22}}&\textbf{J}_{\text{13}}^{\text{22}}&\textbf{J}_{\text {13}}^{\text{11}}\\ \textbf{J}_{\text{11}}^{\text{31}}&\textbf{J}_{\text{11}}^{\text{32}}&\textbf{J }_{\text{11}}^{\text{33}}&\textbf{J}_{\text{12}}^{\text{31}}&\textbf{J}_{\text {13}}^{\text{31}}\\ \textbf{J}_{\text{11}}^{\text{11}}&\textbf{J}_{\text{12}}^{\text{12}}&\textbf{J }_{\text{12}}^{\text{13}}&\textbf{J}_{\text{22}}&\textbf{J}_{\text{23}}\\ \textbf{J}_{\text{31}}^{\text{11}}&\textbf{J}_{\text{31}}^{\text{12}}&\textbf{J }_{\text{31}}^{\text{13}}&\textbf{J}_{\text{32}}&\textbf{J}_{\text{33}}\\ \end{array} \right]\left[ \begin{array}{c}\overset{\bullet}{\textbf{x}}(t)\\ \underline{x}_{d}(t)\\ \underline{O}_{\text{G}\text{Y}}^{d}(t)\\ \underline{D}_{\text{out}}^{d}(t)\\ \underline{u}(t)\end{array} \right] \tag{22}\]  
_where the entries of \(\mathbf{J}\) take values inside the set \(\{\textbf{0},\,\pm\mathbf{I},\,\pm\mathbf{K}_{t}\}\), \(\mathbf{I}\) being the identity matrix and \(\mathbf{K}_{t}\) the multiport transformer module. The constitutive relations for the multiport dissipation and multiport gyrators are expressed by:_  
\[\underline{D}_{\text{out}}^{d}(t) = \mathbf{L}^{d}\underline{D}_{\text{in}}^{d}(t) \tag{23}\] \[\underline{O}_{\text{G}\text{Y}}^{d}(t) = \mathbf{X}_{\text{G}\text{Y}}^{d}\underline{I}_{\text{G}\text{Y} }^{d}(t) \tag{24}\]  
_The multiport storage elements are given by \((\ref{eq:1})\) and \((\ref{eq:2})\).