# On Invertibility of an Interconnected System Composed of Two Dynamic Subsystems
## 4 On Computation of the Dynamics Inverse of the Interconnected System  
After verifying the invertibility of the interconnected system, it is capable of recovering the original inputs uniquely from the global measurement. It implies that each original local input affects the global output distinguishably. In fact, if a system is invertible, there are already structure algorithms that allow one to express the input as a function of the output, its derivatives, and possibly some states (for example, in [4; 21; 23; 33]). A methodology given in Theorem 1 is now capable of checking the invertibility of a nonlinear system. Considering the interconnected input-output system \(\sum\) with two subsystems \(\sum_{L}\) and \(\sum_{H}\) from inputs \(\mathcal{U}\) into outputs \(\mathcal{Y}\), its composition input-output map is \(\mathcal{M}_{L}\circ\mathcal{M}_{H}\). If the interconnected system is left invertible, there exists an input-output system \(\sum^{-1}\) from inputs \(\mathcal{Y}\) into outputs \(\mathcal{U}\), and the inverse composition map is defined as \((\mathcal{M}_{L}\circ\mathcal{M}_{H})^{-1}\), such that the cascade system \(\sum\ \sum^{-1}:\ \mathcal{U}\rightarrow\mathcal{Y}\rightarrow\mathcal{U}\) is the identity. In our mainly algebraic setting, it was supposed that \(\mathcal{U}\), \(\mathcal{Y}\) are a good class of functions equipped with an algebraic structure; for example, they are differential vector space. Then the inverse of the interconnected system is defined as in Theorem 3. **Theorem 4**. : _Consider the interconnected system \(\sum\) that consists of two subsystems: the low-level \(\sum_{L}\) and high-level \(\sum_{H}\) subsystems, and an input-output set \((\mathcal{U},\mathcal{Y})\). If the interconnected system is strongly invertible at \((x_{H0},\ x_{L0})\) over \((\mathcal{U}_{L},\mathcal{Y})\), then the inverse interconnected system can also be an interconnected system with the input-output set \((\mathcal{Y},\ \mathcal{U})\), as follows:_  
\[(\mathcal{M}_{L}\circ\mathcal{M}_{H})^{-1}=\mathcal{M}_{H}^{-1}\circ\mathcal{M} _{L}^{-1} \tag{6}\]  
**Proof. ** Supposed that both \(\mathcal{M}_{L}\) and \(\mathcal{M}_{H}\) are invertible, then \(\mathcal{M}_{L}^{-1}\circ\mathcal{M}_{H}=\mathrm{i}_{U}\), \(\mathcal{M}_{H}^{-1}\circ\mathcal{M}_{H}=\mathcal{M}_{L}\circ\mathcal{M}_{L} ^{-1}=\mathrm{i}_{U}\) and \(\mathcal{M}_{H}\circ\mathcal{M}_{H}^{-1}=\mathrm{i}_{V}\). \(\square\)  
Thus, for any \(y\in\mathcal{Y}\), one can obtain  
\[\Big{[}(\mathcal{M}_{L}\circ\mathcal{M}_{H})\circ\Big{(}\mathcal{ M}_{H}^{-1}\circ\mathcal{M}_{L}^{-1}\Big{)}\Big{]}(y)=\Big{[}\mathcal{M}_{L} \circ\Big{(}\mathcal{M}_{H}\circ\mathcal{M}_{H}^{-1}\Big{)}\circ\mathcal{M}_ {L}^{-1}\Big{]}(u)\] \[=\ \mathcal{M}_{L}\circ\mathrm{i}_{U}\Big{(}\mathcal{M}_{L}^{-1}(u )\Big{)}\] \[=\ \mathcal{M}_{L}\circ\mathcal{M}_{L}^{-1}(u)\] \[=\ \mathrm{i}_{U}\ (u)\]  
It follows that \((\mathcal{M}_{H}\circ\mathcal{M}_{L})\circ\Big{(}\mathcal{M}_{L}^{-1}\circ \mathcal{M}_{H}^{-1}\Big{)}=\ \mathrm{i}_{V}\). Similarly, one can show that \(\Big{(}\mathcal{M}_{L}^{-1}\circ\mathcal{M}_{H}^{-1}\Big{)}\circ(\mathcal{M}_ {H}\circ\mathcal{M}_{L})=\ \mathrm{i}_{U}\). Therefore, \(\mathcal{M}_{L}\circ\mathcal{M}_{H}\) is invertible with inverse \(\mathcal{M}_{H}^{-1}\circ\mathcal{M}_{L}^{-1}\). It can be seen from Figure 3 that the inverse system of an interconnected system can also be regarded as an interconnected system, and its components are the inverse subsystems of each subsystem. In the interconnected inverse system, the global output is the original input of the interconnected system. This implies that it is capable of distinguishing the impacts of each local input on the global output. To compute the inverse of an input affine interconnected system, the structure algorithm allows us to express the input as a function of the output, its derivatives, and possibly some states, as shown in [4]. For the high-level subsystem depicted in (2), the expression of its inverse dynamics can be realized as from (7):  
\[\mathcal{M}_{H}^{-1}:\ \left\{