# On Invertibility of an Interconnected System Composed of Two Dynamic Subsystems
## 2 Inversion of Nonlinear Interconnected System
### Inverse of Interconnected System
#### 2.2.2 Inverse of Nonlinear Interconnected System  
As shown in Figure 2, the following problems are worth discussing: given the initial state and the corresponding output \(y\) produced by the input \(u\), under what conditions can the subsystems of the interconnected system uniquely recover the local input \(u\)? This problem is similar to the classical invertibility problem of a non-interconnected and nonlinear system. Considering the input-output map of the high-level subsystem as \(\mathcal{M}_{\rm H}:\mathcal{U}_{\rm L}\rightarrow\mathcal{Y}\), the input function space is generated by \(\mathcal{U}_{\rm L}\) and the corresponding output function space is generated by \(\mathcal{Y}\), then \(\mathcal{M}_{\rm H}\) maps an input \(u_{\rm L}(. )\) to the output \(y(. )\) generated by the system driven by \(u_{\rm L}(. )\) with an initial condition \(x_{\rm H0}\). In addition, the input-output map of the low-level subsystem is \(\mathcal{M}_{\rm L}:\ \mathcal{U}\rightarrow\mathcal{U}_{\rm L}\) for some input function space \(\mathcal{U}\) and the corresponding output \(\mathcal{U}_{\rm L}\). \(\mathcal{M}_{\rm L}\) maps an input \(u(. )\) to the output \(u_{\rm L}(. )\) is generated by the system driven by \(u(. )\) with an initial condition \(x_{\rm L0}\). Define composition maps \(\mathcal{M}_{\rm L}\circ\mathcal{M}_{\rm H}:\mathcal{U}\rightarrow\mathcal{Y}\) as the input-output map of the studied interconnected system, given initial conditions \((x_{\rm L0},\ x_{\rm H0})\), the composition \(\mathcal{M}_{\rm L}\circ\mathcal{M}_{\rm H}\) maps an input \(u(. )\) to the output \(y(. )\) driven by \(u(.)\). The interconnected system is said to be invertible if driven input \(u(t)\) can be recovered by the output \(y(. )\), part of the state and the initial state \((x_{\rm L0},\ x_{\rm H0})\). It is to extend the definition of invertibility to the cascade interconnected system by Definition 2 as follows:  
**Definition 2**. : _Fix an output set \(\mathcal{Y}\) and consider an arbitrary interval \([t_{0},\ T)\), the interconnected system described in (1) and (2) is invertible at a point \((x_{\rm L0},\ x_{\rm H0})=\ x_{H}(t_{0})\in\mathcal{X}\) over \(\mathcal{Y},x_{L}(t_{0})\in\mathcal{X}_{L}(t_{0})\) over \(\mathcal{U}_{\rm L}\), if for every \(y_{[t_{0},\ T)}\in\mathcal{Y}\), the equality \((\mathcal{M}_{L}\circ\mathcal{M}_{H})_{(x_{\rm L0},\ x_{\rm H0})}\big{(}u_{1[ t_{0},\ T)}\big{)}=(\mathcal{M}_{L}\circ\mathcal{M}_{H})_{(x_{\rm L0},\ x_{\rm H0})}\big{(}u_{2[t_{0},\ T)}\big{)}=y_{[t_{0},\ T)}\) implies that \(\exists e>0\), such that \(u_{1[t_{0},\ t_{0}+e)}=u_{2[t_{0},\ t_{0}+e)}\). The system is strongly invertible at a point \((x_{\rm L0},\ x_{\rm H0})\) if it is invertible for each \(x_{\rm L}\in\mathcal{N}_{L}(x_{\rm L0}\ )\), \(x_{H}\in\ \mathcal{N}_{H}(x_{\rm H0})\), where \((\mathcal{N}_{L},\mathcal{N}_{H})\) is some open neighborhood of \((x_{\rm L0},\ x_{\rm H0})\). The system is strongly invertible if there exists an open and dense sub-manifold \(\mathcal{M}_{L}of\ \mathcal{X}_{L}\). \(\mathcal{M}_{H}of\ \mathcal{X}_{H}\), such that \(\forall(x_{\rm L0}\,x_{\rm H0})\in(\mathcal{M}_{L},\mathcal{M}_{H})\), the system is strongly invertible at \((x_{\rm L0},\ x_{\rm H0})\)._  
The invertibility proposed in Definition 2 may not hold in two cases: on one hand, if there are two different inputs \(u_{1},\ u_{2}\) produce two equal outputs \(u_{L1}=u_{L2}\), while for the high-level subsystem, \(u_{\rm L1},u_{\rm L2}\) are its inputs, clearly, these equal inputs will definitely yield equal outputs \(y_{1}=y_{2}\) at the global terminal; in this case, invertibility of the interconnected system fails to hold; on the other hand, even if the two different inputs \(u_{1},\ u_{2}\) generate two different local outputs \(u_{L1},u_{L2}\) at the low-level subsystem, these two \(u_{L1},u_{L2}\) yield equal output \(y_{1}=y_{2}\) at the global level, and again, the invertibility of the interconnected system cannot be guaranteed.