# Geometric Methods and Computational Tools for Switching Structured Linear Systems
## 4 Control and Observation Problems
### Disturbance Decoupling by State Feedback  
The problem of disturbance decoupling by state feedback concerns a switching structured linear system whose state-space representation is  
\[\Sigma_{\sigma(t)}\equiv\left\{ \begin{array}{rcl}\dot{x}(t)&=&A_{\sigma(t)}\, x(t)+B_{\sigma(t)}\,u(t)+D_{\sigma(t)}\,d(t),\\ y(t)&=&C_{\sigma(t)}\,x(t),\end{array} \right. \tag{5}\]  
where \(u\in\mathbb{R}^{m}\) is the control input, \(d\in\mathbb{R}^{s}\) is the disturbance input, and \(y\in\mathbb{R}^{p}\) is the to-be-decoupled output (i.e., the output to be made insensitive to the disturbance). With the obvious meaning of the symbols, one can also refer to the graph representation  
\[\left((G,\mathcal{E}_{\sigma(t)}),\,G^{u}_{\sigma(t)}\cup G^{d}_{\sigma(t)},G^ {y}_{\sigma(t)}\right)\]  
and, therefore, adopt the unifying notation  
\[\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G^{u}_{\sigma}\cup G^{d}_{\sigma},G^ {y}_{\sigma}).\]  
Hence, the disturbance decoupling problem by state feedback is stated as follows. **Problem 1**. : _Let \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G^{u}_{\sigma}\cup G^{d}_{\sigma},G^ {y}_{\sigma})\) be a switching structured linear system affected by the disturbance input \(d\). The problem of disturbance decoupling by state feedback consists in finding a switching essential state feedback \(\mathcal{F}_{\sigma}\) such that the compensated switching structured linear system \(\Sigma_{\sigma}^{\mathcal{F}}((G,\mathcal{E}_{\sigma}\setminus\mathcal{F}_{ \sigma}),\,G^{u}_{\sigma}\cup G^{d}_{\sigma},G^{y}_{\sigma})\) does not exhibit any switching path \(P_{s}\) in \((G,\mathcal{E}_{\sigma}\setminus\mathcal{F}_{\sigma})\) whose tail and head are in \(G^{d}_{h}\) and \(G^{y}_{h^{\prime}}\), respectively, for any \(h,h^{\prime}\in\mathcal{H}\)._  
**Theorem 1**. : _The following statements are equivalent:_  
* _Problem_ 1 _is solvable;_
* \(G^{d}_{h}\subseteq V^{*}(\mathcal{E}_{\sigma},G^{u}_{\sigma},G\setminus(( \cup_{k\in\mathcal{H}}G^{y}_{k}))\quad\text{for all }h\in\mathcal{H}\)_._  
A proof of Theorem 1 was given in [14]. **Remark 1**. : _The formulation of disturbance decoupling given in Problem 1, which requires that there are no switching paths connecting the disturbance input to the output, is stronger (i.e., more demanding) than a formulation that requires that the output is insensitive to the disturbance for almost any instance of the parameters. In fact, the latter formulation includes the case where there may be different paths from the disturbance input to the output, but these paths are such that the effects on the output of the disturbance through each one of them compensate for one another. As will be shown in the following example, the condition of Theorem 1 is necessary and sufficient for the solvability of the disturbance decoupling problem as stated in Problem 1, while it is not necessary for the weaker formulation mentioned in this remark (the issue dealt with in this remark and in the following illustrative example has been clarified thanks to profitable discussions held with Prof. [PERSON])._  
**Example 1**. : _To illustrate the content of Remark 1 with an example, a classic (i.e., non-switching) structured linear system is considered and the notation is simplified consequently._  
_Let Problem 1 be stated for a structured linear system \(\Sigma((G,\mathcal{E}),G^{u}\cup G^{d},G^{y})\), where \(G=\{x_{1},x_{2},x_{3},x_{4}\}\), \(\mathcal{E}=\{(x_{1},x_{2}),  
_Hence, the condition of Theorem 1 is not satisfied and Problem 1 is not solvable. Indeed, from the direct inspection of the digraph of system \(\Sigma((G,\mathcal{E}),G^{u}\cup G^{d},G^{y})\) shown in Figure 8 (left), it is easy to see that there is no essential state feedback achieving the cancellation of any edge of the path \(P=\{(x_{1},x_{2}),(x_{2},x_{4})\}\) from \(d\) to \(y\)._  
_Nevertheless, considering the state space representation of \(\Sigma((G,\mathcal{E}),G^{u}\cup G^{d},G^{y})\), which is of the form (5) with_  
\[A=\left[ \begin{array}{cccc}0&0&0&0\\ a_{21}&0&0&0\\ 0&0&0&0\\ 0&a_{42}&a_{43}&0\end{array}