# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## 2 Modeling in a Bond Graph
The constitutive relationships for the multiport elements are given by:_  
\[\underline{z}(t) = \mathbf{F}\underline{x}(t) \tag{2}\] \[\underline{z_{d}}(t) = \mathbf{F}_{d}\underline{x}(t)\] (3) \[\underline{O_{GY}}(t) = \mathbf{X}_{GY}\underline{I_{GY}}(t)\] (4) \[\underline{D_{out}}(t) = \mathbf{L}\underline{D_{int}}(t) \tag{5}\]  
_Furthermore, the properties of the submatrices of \(\mathbb{S}\) are: (1) \(S_{11}^{11}\), \(S_{11}^{22}\), and \(S_{22}\) are skew-symmetric; (2) \(S_{11}^{12}=-\left(\mathbf{S}_{11}^{21}\right)^{\mathsf{T}}\), \(S_{12}^{11}=-\left(\mathbf{S}_{21}^{11}\right)^{\mathsf{T}}\), \(S_{14}^{11}=-\left(\mathbf{S}_{41}^{11}\right)^{\mathsf{T}}\), and \(\mathbf{S}_{12}^{21}=-\left(\mathbf{S}_{21}^{12}\right)^{\mathsf{T}}\), then a state space representation of the multiport system is defined by:_  
\[\left[ \begin{array}{c}\underline{\dot{x}}(t)\\ \underline{y}(t)\end{array} \right]=\left[ \begin{array}{cc}\mathbf{A}& \mathbf{B}\\ \mathbf{C}&\mathbf{D}\end{array} \right]\left[ \begin{array}{c}\underline{x} (t)\\ \underline{u}(t)\end{array} \right] \tag{6}\]  
_where the corresponding matrices are:_  
\[\mathbf{A} = \mathbf{E}^{-1}\Big{[}\mathbf{S}_{11}^{11}+\mathbf{S}_{11}^{12} \mathbf{Q}_{X}\mathbf{S}_{XA}+\mathbf{S}_{12}^{11}\mathbf{Q}_{L}\mathbf{S}_{ LA}\Big{]}\mathbf{F} \tag{7}\] \[\mathbf{B} = \mathbf{E}^{-1}\Big{[}\mathbf{S}_{13}^{11}+\mathbf{S}_{11}^{12} \mathbf{Q}_{X}\mathbf{S}_{XB}+\mathbf{S}_{12}^{11}\mathbf{Q}_{L}\mathbf{S}_{ LB}\Big{]}\] (8) \[\mathbf{C} = \Big{[}\mathbf{S}_{31}^{11}+\mathbf{S}_{31}^{12}\mathbf{Q}_{X} \mathbf{S}_{XA}+\mathbf{S}_{32}\mathbf{Q}_{L}\mathbf{S}_{LA}\Big{]}\mathbf{F}\] (9) \[\mathbf{D} = \mathbf{S}_{33}+\mathbf{S}_{31}^{12}\mathbf{Q}_{X}\mathbf{S}_{XB}+ \mathbf{S}_{32}\mathbf{Q}_{L}\mathbf{S}_{LB} \tag{10}\]  
_with:_  
\[\mathbf{E}=\mathbf{I}-\mathbf{S}_{14}^{11}\mathbf{F}_{d}^{-1}\mathbf{S}_{41}^{11} \mathbf{F} \tag{11}\]  
_and the algebraic loops matrices are described by:_  
\[\mathbf{Q}_{X} = \mathbf{X}_{GY}\Big{[}\mathbf{I}-\Big{(}\mathbf{S}_{11}^{22}+ \mathbf{S}_{12}^{21}\mathbf{M}_{L}\mathbf{S}_{21}^{12}\Big{)}\mathbf{X}_{GY} \Big{]}^{-1} \tag{12}\] \[\mathbf{Q}_{L} = \mathbf{L}\Big{[}\mathbf{I}-\Big{(}\mathbf{S}_{22}+\mathbf{S}_{21}^ {12}\mathbf{M}_{X}\mathbf{S}_{12}^{21}\Big{)}\mathbf{L}\Big{]}^{-1} \tag{13}\]\[\mathbf{M}_{X} = \mathbf{X}_{CY}\Big{(}\mathbf{I}-\mathbf{S}_{11}^{22}\mathbf{X}_{CY} \Big{)}^{-1} \tag{14}\] \[\mathbf{M}_{L} = \mathbf{L}(\mathbf{I}-\mathbf{S}_{22}\mathbf{L})^{-1}\] (15) \[\mathbf{S}_{XA} = \mathbf{S}_{11}^{21}+\mathbf{S}_{12}^{21}\mathbf{M}_{L}\mathbf{S }_{21}^{11}\] (16) \[\mathbf{B} = \mathbf{E}^{-1}\Big{[}\mathbf{S}_{13}^{11}+\mathbf{S}_{11}^{12} \mathbf{Q}_{X}\mathbf{S}_{XB}+\mathbf{S}_{12}^{11}\mathbf{Q}_{L}\mathbf{S}_{ LB}\Big{]}\]0 (17) \[\mathbf{B} = \mathbf{E}^{-1}\Big{[}\mathbf{S}_{13}^{11}+\mathbf{S}_{11}^{12} \mathbf{Q}_{X}\mathbf{S}_{XB}+\mathbf{S}_{12}^{11}\mathbf{Q}_{L}\mathbf{S}_{ LB}\Big{]}\]1 (18) \[\mathbf{B} = \mathbf{E}^{-1}\Big{[}\mathbf{S}_{13}^{11}+\mathbf{S}_{11}^{12} \mathbf{Q}_{X}\mathbf{S}_{XB}+\mathbf{S}_{12}^{11}\mathbf{Q}_{L}\mathbf{S}_{ LB}\Big{]}\]2  
The proof of Lemma 1 is presented in Appendix A. The state equation described by Lemma 1 determines a system with linearly independent and dependent state variables according to the causality of the multiport storage elements. Hence, matrix \(\mathbf{E}\) gives the relationships between these state variables. When the dynamic period of a stable system has finished, the steady state response can be obtained.