# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## 2 Modeling in a Bond Graph
The different multiport elements that are part of a multibond graph model can be grouped into interconnected blocks. Therefore, a multiport system represented by a multibond graph model in a predefined integral causality assignment (MBGI) and their key vectors are shown in Figure 9.  
Figure 6: Multibond graph with efforts and flows. Figure 7: Multiport elements. The multiport elements of Figure 7 are as follows:  
* \(\left(\underline{MS_{\epsilon}},\underline{MS_{f}}\right)\) represent the effort and flow modulated multiport sources. * \(\left(\mathbb{C},\mathbb{I}\right)\) denote the multiport storage elements defined by multiport capacitance and inertia, respectively. * \(\mathbb{R}\) are the multiport dissipation elements that constitute the multiport resistors. * \(\left(\underline{0},\underline{1},\underline{MTF}\right)\) represent the multiport junction structure with \(\underline{0}\) and \(\underline{1}\) multiport junctions, and the multiport transformers are denoted by \(\underline{MTF}\). * \(\mathbb{G}\)Y is the multiport gyrator. * \(\left(\underline{DS_{\epsilon}},\underline{DS_{f}}\right)\) determine the detectors for the effort and flow, respectively. The multiport energy variables are \(\underline{p}(t)\) and \(\underline{q}(t)\) related to multiport elements \(\mathbb{I}\) and \(\mathbb{C}\), respectively. The key vectors in Figure 7 are as follows:  
* \(\underline{I_{GY}(t)}\in\Re^{\mathbb{R}}\) and \(\underline{O_{GY}(t)}\in\Re^{\mathbb{S}}\) denote the inputs and outputs of the multiport gyrators. * \(\underline{D_{int}}(t)\in\Re^{\mathbb{R}}\) and \(\underline{D_{out}}(t)\in\Re^{\mathbb{L}}\) represent the relationships between the multiport junction structure and multiport dissipation elements. * \(\underline{u}(t)\in\Re^{\mathbb{R}}\) and \(\underline{y}(t)\in\Re^{\mathbb{R}}\) determine the multiport inputs and outputs of the system. The state space representation of a multiport system based on a multibond graph model is described by the following Lemma:  
**Lemma 1**. : _Consider an LTI system modeled by a multibond graph with a preferred integral causality assignment (MBGI) whose multiport storage elements can have integral and derivative causality assignments. The relationships of the multiport key vectors are shown in Figure 9, and the multiport junction structure is determined by:_  
\[\left[ \begin{array}{c}\underline{\dot{x}}(t)\\ \underline{I_{GY}(t)}\\ \underline{D_{in}}(t)\\ \underline{y}(t)\\ \underline{z_{d}}(t)\end{array} \right]=\left[ \begin{array}{cccc}\mathbb{S} _{11}^{11}&\mathbb{S}_{11}^{12}&\mathbb{S}_{12}^{11}&\mathbb{S}_{13}^{11}& \mathbb{S}_{14}^{11}\\ \mathbb{S}_{11}^{21}&\mathbb{S}_{11}^{21}&\mathbb{S}_{12}^{21}&\mathbb{S}_{13}^ {21}&\mathbb{S}_{13}^{21}&\mathbb{0}\\ \mathbb{S}_{21}^{21}&\mathbb{S}_{21}^{21}&\mathbb{S}_{22}&\mathbb{S}_{23}& \mathbb{0}\\ \mathbb{S}_{31}^{11}&\mathbb{S}_{31}^{21}&\mathbb{S}_{32}&\mathbb{S}_{33}& \mathbb{0}\\ \mathbb{S}_{41}^{11}&\mathbb{0}&\mathbb{0}&\mathbb{0}&\mathbb{0}\\ \end{array} \right]\left[ \begin{array}{c}\underline{z}(t)\\ \underline{O_{GY}(t)}\\ \underline{U}(t)\\ \underline{\underline{\underline{\underline{\underline{\underline{\underline{ \underline{\underline{\underline{\underline{\underline{\underline{\underline{\underline{ \underline{\underline{\underline{\underline{ \underline{          \underline{ \cdotcdot}}}}}}}}}}}}}} (t) (t) (t)\\ \underline{U}(t)\\ \underline{U}(t)\\ \underline{U}(t)\\ \end{array} \right] \tag{1}\]  
_where the entries of \(\mathbb{S}\) take values inside the set \(\{\mathbf{0},\pm\mathbf{I},\ \pm\mathbf{K}_{t}\}\), \(\mathbf{I}\) being the identity matrix and \(\mathbf{K}_{t}\) the multiport transformer module.