# On Invertibility of an Interconnected System Composed of Two Dynamic Subsystems
## 3 On the Condition of Invertibility of the Interconnected System  
The system under consideration is an interconnected system, resulting in the unavailability of classical inversion technologies. It is therefore necessary to investigate a new toolset for guaranteeing the invertibility of interconnected systems. The problem of the inverse of interconnected systems can be regarded as a combination of invertible mappings and individual input recovery. Therefore, the basic idea of solving the invertibility  
Figure 2: The inverse of an interconnected system. problem is to utilize the relationship between the output and the state of the subsystem to synthesize the mapping and then use the nonlinear structure algorithm to recover the input of the corresponding subsystem. It can be concluded from Definition 2 that the interconnected system cannot be invertible, no matter whether the high-level or low-level subsystem is invertible or not. Therefore, the necessary and sufficient conditions for invertibility of interconnected systems on set \(\mathcal{U}\), \(\mathcal{U}_{L}\), and \(\mathcal{Y}\) can be given from the view of the individual subsystem. **Theorem 1. **_Consider the interconnected system \(\sum\) which consists of two subsystems: low-level \(\sum_{L}\) and high-level \(\sum_{H}\) subsystems depicted by (1) and (2), and an output set \(\mathcal{Y}\). The interconnected system is invertible at \((x_{H0},\ x_{L0})\) over \(\mathcal{Y}\), if and only if each subsystem, the low-level \(\sum_{L}\) and the high-level \(\sum_{H}\), is invertible at \(x_{L0}\) over \(\mathcal{U}_{L}\), and \(x_{H0}\) over \(\mathcal{Y}\), respectively._  
Proof. : Consider \(\mathcal{M}_{\mathrm{L}}\) as the input-output mapping of low-level \(\sum_{\mathrm{L}}\) subsystem, while \(\mathcal{M}_{\mathrm{H}}\) is the input-output mapping of high-level \(\sum_{\mathrm{H}}\) subsystem. Then the composition \(\mathcal{M}_{\mathrm{L}}\circ\mathcal{M}_{\mathrm{H}}\) can be considered as the input-output mapping of the interconnected system. In order to confirm the theorem, it needs to provide a condition from both sufficiency and necessity aspects. To begin with the sufficiency condition, the invertibility of a dynamic system refers to the bijective of the input-output mapping. Since both subsystems are invertible, the corresponding mapping \(\mathcal{M}_{\mathrm{L}}\) and mapping \(\mathcal{M}_{\mathrm{H}}\) are bijective mappings. Moreover, the composition of two bijective mappings is a bijective mapping, so input-output mapping \(\mathcal{M}_{\mathrm{L}}\circ\mathcal{M}_{\mathrm{H}}\) of the cascade system is bijective. Thus, the cascade interconnected system is invertible. The next task is to produce a necessary condition. It can be achieved by the non-invertibility of either subsystem. That is to say if any of the subsystems is not invertible at \((x_{L0},\ x_{H0})\), then the interconnected system \(\sum\) is also not invertible. On one hand, suppose that the high-level subsystem \(\sum_{\mathrm{H}}\) is not invertible, then no matter whether the low-level subsystem \(\sum_{\mathrm{L}}\) is invertible or not, for the low-level subsystem described in (1), fix an output set \(\mathcal{U}_{L}\) and consider an arbitrary interval \([t_{0},\ T)\). Two distinct inputs exist for \(\exists\ \varepsilon>0\ u_{1}\
eq u_{2}\) on \([t_{0},\ t_{0}+\varepsilon)\), and it is possible to generate two different outputs \(\mathcal{M}_{L(x_{L0})}\left(u_{1[t_{0},\ T)}\right)=u_{L[t_{0},\ T)}\), \(\mathcal{M}_{L(x_{L0})}\left(u_{2[t_{0},\ T)}\right)=u_{L[t_{0},\ T)}\), \(u_{L[t_{0},\ T)}\
eq u_{L[t_{0},\ T)}\). However, for the high-level subsystem in (2), even if the invertibility of this subsystem is guaranteed, fix an output set \(\mathcal{Y}\). These distinguishable inputs \(u_{L1}\
eq u_{L2}\) on \([t_{0},\ t_{0}+\varepsilon)\) may generate two equal outputs \(\mathcal{M}_{H(x_{H0})}\Big{(}u_{L1[t_{0},\ T)}\Big{)}=\mathcal{M}_{H(x_{H0}) }\Big{(}u_{L2[t_{0},\ T)}\Big{)}=y_{1[t_{0},\ T)}=y_{2[t_{0},\ T)}=y_{[t_{0},\ T )}\).