# On Invertibility of an Interconnected System Composed of Two Dynamic Subsystems
## 4 On Computation of the Dynamics Inverse of the Interconnected System
\begin{array}{c}\mathrm{y}_{\mathrm{1}}^{(\mathrm{r_{1}})}\\ \vdots\\ \mathrm{y}_{\mathrm{m}}^{(\mathrm{r_{m}})}\end{array} \right]\quad-\quad \left[ \begin{array}{c}\mathrm{L}_{I_{H}}^{\mathrm{r_{1}}}\mathrm{h}_{ \mathrm{H}\mathrm{I}}(\mathrm{x}_{H})\\ \vdots\\ \mathrm{L}_{I_{H}}^{\mathrm{r_{m}}}\mathrm{h}_{\mathrm{H}\mathrm{H}}(\mathrm{x}_ {H})\end{array} \right]\right) \tag{18}\]  
In this situation, there will be no internal dynamics, and all the results will be finite time in nature, see reference [34]. However, normally, the total relative degree is assumed:  
\[\mathrm{r}=\sum_{\mathrm{i=1}}^{\mathrm{m}}\mathrm{r_{i}}<\mathrm{n} \tag{19}\]  
In this case, the system given by (2) can be presented on a new basis that is introduced as follows:  
Define the following change of the coordinates:  
\[\xi_{\mathrm{H}}=\left[\xi_{\mathrm{H}}^{1},\ \xi_{\mathrm{H}}^{2},\ \ldots,\xi_{\mathrm{H}}^{n}\right]^{\mathrm{T}}\] \[=\left[\mathrm{\phi}_{\mathrm{H}}^{1}(\mathrm{x}_{H}),\ \mathrm{\phi}_{\mathrm{H}}^{2}(\mathrm{x}_{H}),\ \ldots,\mathrm{\phi}_{\mathrm{H}}^{n}(\mathrm{x}_{H})\right]^{\mathrm{T}}\] \[=\left[\mathrm{h}_{\mathrm{H}\mathrm{I}}(\mathrm{x}_{H}),\ \mathrm{L}_{I_{H}} \mathrm{h}_{\mathrm{H}\mathrm{H}}(\mathrm{x}_{H}),\ldots,\mathrm{L}_{I_{H}}^{ \mathrm{r_{1}}-1}\mathrm{h}_{\mathrm{H}\mathrm{H}}(\mathrm{x}_{H})\ \right]^{\mathrm{T}}\ \mathrm{i}=1,\ldots,\mathrm{m}\] \[\xi_{\mathrm{H}}=\left[\xi_{\mathrm{H}\mathrm{I}},\xi_{\mathrm{H} \mathrm{I}},\ldots,\ \xi_{\mathrm{H}\mathrm{m}}\right]\] \[=\left[\mathrm{\phi}_{\mathrm{H}\mathrm{I}}(\mathrm{x}_{H}),\ \mathrm{\phi}_{\mathrm{H} \mathrm{I}}(\mathrm{x}_{H}),\ \ldots,\ \mathrm{\phi}_{\mathrm{H}\mathrm{m}}(\mathrm{x}_{H})\ \right]\] \[\zeta_{H}=\left[\mathrm{\phi}_{\mathrm{H}\mathrm{(r+1)}}(\mathrm{x }_{H}),\ \mathrm{\phi}_{\mathrm{H}\mathrm{(r+2)}}(\mathrm{x}_{H}),\ \ldots,\ \phi_{\mathrm{H}\mathrm{(n} \mathrm{x}_{H})}\ \right]^{\mathrm{T}}\] \[\mathrm{y}=\left[\xi_{\mathrm{H}\mathrm{I}}^{1},\xi_{\mathrm{H} \mathrm{I}}^{1},\ldots,\ \xi_{\mathrm{H}\mathrm{m}}^{1}\right]\]By applying the new local coordinates transformation proposed in [4], if the system holds the assumption of relative degree, it is always possible to find the function \(\phi_{\rm H(r+1)}({\rm x}_{H})\), \(\phi_{\rm H(r+2)}({\rm x}_{H})\),..., \(\phi_{\rm Hn}({\rm x}_{H})\), thus,  
\[\phi_{\rm H}({\rm x}_{H})=\left[\phi_{\rm H1}({\rm x}_{H}),\;\phi_{\rm H2}({\rm x }_{H}),\;\ldots,\;\phi_{\rm Hm}({\rm x}_{H}),\phi_{\rm H(r+1)}({\rm x}_{H}),\; \ldots,\;\phi_{\rm Hn}({\rm x}_{H})\right] \tag{20}\]  
The mapping \(\phi_{\rm H}({\rm x}_{H})\) is a local diffeomorphism, which means  
\[{\rm x}_{H}=\phi_{H}^{-1}(\xi_{\rm H},\zeta_{H}) \tag{21}\]  
Furthermore, according to [5], if the assumption is satisfied,  
**Assumption 1**. : _The distribution \(\Gamma=span\;\left\{ \begin{array}{cc}g_{H1}&g_{H2}&\cdots&g_{Hm}\end{array} \right\}\) is involutive, then, it is always possible to identify the function \(\phi_{H(r+1)}(x_{H})\), \(\phi_{H(r+2)}(x_{H})\),..., \(\phi_{Hn}(x_{H})\) in such a way that_  
\[ \begin{array}{c}{\rm L}_{\rm g_{H}}\phi_{\rm Hi}({\rm x}_{H})=0,\;{\rm i}={ \rm r}+1,\ldots,{\rm n},\;{\rm j}=1,\ldots,{\rm m}\\ \dot{\zeta}_{H}={\rm q}(\xi_{\rm H},\zeta_{H})\end{array} \]  
Then, the input vector \(u_{L}\) can be obtained by means of the output vector \(y\) and its derivatives:  
\[{\rm u}_{\rm L}=\Delta\Big{(}\phi_{H}^{-1}(\xi_{\rm H},\zeta_{H})\Big{)}^{-1} \left( \begin{array}{c}\left[\begin{array}{c}\xi_{\rm H1}^{(r_{1})}\\ \vdots\\ \xi_{\rm Hm}^{(r_{\rm m})}\end{array}\right]&-\;\;\left[\begin{array}{c}{\rm L }_{t_{H}}^{\rm r_{1}}{\rm h}_{\rm H1}\Big{(}\phi_{H}^{-1}(\xi_{\rm H},\zeta_ {H})\Big{)}\\ \vdots\\ {\rm L}_{t_{H}}^{\rm m}{\rm h}_{\rm Hm}\Big{(}\phi_{H}^{-1}(\xi_{\rm H},\zeta_ {H})\Big{)}\end{array}\right]\end{array} \right) \tag{22}\]  
Fortunately, along with the discussion of this paper, linear and nonlinear problems can be treated in parallel with each other.