# Steady State Response of Linear Time Invariant Systems Modeledby Multibond Graphs
## Appendix C Proof of Theorem
\right]\underline{u}(t) \tag{20}\]  
The first line of (20) is expressed by:  
\[\overset{\bullet}{\underline{x}}(t)=\left(\mathbf{A}_{11}^{*}\right)^{-1} \underline{x}(t)-\left(\mathbf{A}_{11}^{*}\right)^{-1}\mathbf{A}_{12}^{*} \overset{\bullet}{\underline{x}}_{d}(t)-\left(\mathbf{A}_{11}^{*}\right)^{-1 }\mathbf{B}_{1}^{*}\underline{u}(t) \tag{21}\]  
Substituting (21) into the second line of (20):  
\[\underline{x}_{d}(t)=\mathbf{A}_{21}^{*}\left[(\mathbf{A}_{11}^{*})^{-1} \underline{x}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{A}_{12}^{*}\underline{x}_ {d}^{*}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{B}_{1}^{*}\underline{u}(t) \right]+\mathbf{A}_{22}^{*}\underline{x}_{d}^{*}(t)+\mathbf{B}_{2}^{*} \underline{u}(t) \tag{22}\]  
and reducing:  
\[\underline{x}_{d}(t)=\mathbf{A}_{21}^{*}(\mathbf{A}_{11}^{*})^{-1}\underline{ x}(t)+\left[\mathbf{A}_{22}^{*}-\mathbf{A}_{21}^{*}(\mathbf{A}_{11}^{*})^{-1} \mathbf{A}_{22}^{*}\right]\underline{\underline{x}}(t)+\left[\mathbf{B}_{2}^{* }-\mathbf{A}_{21}^{*}(\mathbf{A}_{11}^{*})^{-1}\mathbf{B}_{1}^{*}\right] \underline{u}(t) \tag{23}\]  
the state variables \(\underline{x}_{d}\) are linearly dependent, then:  
\[\mathbf{A}_{22}^{*}-\mathbf{A}_{21}^{*}(\mathbf{A}_{11}^{*})^{-1} \mathbf{A}_{12}^{*} = 0\] \[\mathbf{B}_{2}^{*}-\mathbf{A}_{21}^{*}(\mathbf{A}_{11}^{*})^{-1} \mathbf{B}_{1}^{*} = 0\]  
The expression (23) is reduced to:  
\[\underline{x}_{d}(t)=\mathbf{A}_{21}^{*}(\mathbf{A}_{11}^{*})^{-1}\underline{ x}(t)\]  
and deriving with respect to time:  
\[\underline{\underline{x}}_{d}^{*}(t)=\mathbf{A}_{21}^{*}(\mathbf{A}_{11}^{*})^{ -1}\underline{\underline{x}}(t) \tag{24}\]  
From the first line of (20) with (24):  
\[\underline{x}(t)=\left[\mathbf{A}_{11}^{*}+\mathbf{A}_{12}^{*}\mathbf{A}_{21}^ {*}(\mathbf{A}_{11}^{*})^{-1}\right]\underline{\underline{x}}(t)+\mathbf{B}_ {1}^{*}\underline{u}(t)\]in a compact form:  
\[\underline{x}(t)=\widetilde{\mathbf{A}^{*}}\underline{\underline{x}}(t)+ \mathbf{B}_{1}^{*}\underline{u}(t)\]  
where:  
\[\widetilde{\mathbf{A}^{*}}=\mathbf{A}_{11}^{*}+\mathbf{A}_{12}^{*}\mathbf{A}_{2 1}^{*}(\mathbf{A}_{11}^{*})^{-1}\]  
in terms of state equation:  
\[\underline{x}_{d}(t)=\mathbf{A}_{21}^{*}\left[(\mathbf{A}_{11}^{*})^{-1} \underline{x}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{A}_{12}^{*}\underline{x}_ {d}^{*}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{B}_{1}^{*}\underline{u}(t) \right]+\mathbf{A}_{22}^{*}\underline{x}_{d}^{*}(t)+\mathbf{B}_{2}^{*} \underline{u}(t) \tag{22}\]0  
Comparing (101) with \(\mathbf{E}_{\underline{x}}^{\bullet}=\mathbf{A}\underline{x}+\mathbf{B} \underline{u}\) given in (6):  
\[\underline{x}_{d}(t)=\mathbf{A}_{21}^{*}\left[(\mathbf{A}_{11}^{*})^{-1} \underline{x}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{A}_{12}^{*}\underline{x}_ {d}^{*}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{B}_{1}^{*}\underline{u}(t) \right]+\mathbf{A}_{22}^{*}\underline{x}_{d}^{*}(t)+\mathbf{B}_{2}^{*} \underline{u}(t) \tag{22}\]1  
and:  
\[\underline{x}_{d}(t)=\mathbf{A}_{21}^{*}\left[(\mathbf{A}_{11}^{*})^{-1} \underline{x}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{A}_{12}^{*}\underline{x}_ {d}^{*}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{B}_{1}^{*}\underline{u}(t) \right]+\mathbf{A}_{22}^{*}\underline{x}_{d}^{*}(t)+\mathbf{B}_{2}^{*} \underline{u}(t) \tag{22}\]2  
Then:  
\[\underline{x}_{d}(t)=\mathbf{A}_{21}^{*}\left[(\mathbf{A}_{11}^{*})^{-1} \underline{x}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{A}_{12}^{*}\underline{x}_ {d}^{*}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{B}_{1}^{*}\underline{u}(t) \right]+\mathbf{A}_{22}^{*}\underline{x}_{d}^{*}(t)+\mathbf{B}_{2}^{*} \underline{u}(t) \tag{22}\]3  
It is known that the steady state for an LTI system is:  
\[\underline{x}_{d}(t)=\mathbf{A}_{21}^{*}\left[(\mathbf{A}_{11}^{*})^{-1} \underline{x}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{A}_{12}^{*}\underline{x}_ {d}^{*}(t)-(\mathbf{A}_{11}^{*})^{-1}\mathbf{B}_{1}^{*}\underline{u}(t) \right]+\mathbf{A}_{22}^{*}\underline{x}_{d}^{*}(t)+\mathbf{B}_{2}^{*} \underline{u}(t) \tag{22}\]4  
and substituting (102) into (103), the equation (45) is proven.