# Geometric Methods and Computational Tools for Switching Structured Linear Systems
## 4 Control and Observation Problems
### Disturbance Decoupling by Output Feedback  
The problem of disturbance decoupling by output feedback is considered last, since, as will be shown, the necessary and sufficient solvability condition involves both a maximal switching controlled invariant subset and a minimal switching conditioned invariant subset--namely, both of the geometric objects that, so far, have been separately considered in disturbance decoupling and unknown-input state observation, respectively. The problem of disturbance decoupling by output feedback concerns a switching structured linear system whose state-space representation is of the form  
\[\Sigma_{\sigma(t)}\equiv\left\{ \begin{array}{rcl}\dot{x}(t)&=&A_{\sigma(t)} \,x(t)+B_{\sigma(t)}\,u(t)+D_{\sigma(t)}\,d(t),\\ y(t)&=&C_{\sigma(t)}\,x(t),\\ z(t)&=&E_{\sigma(t)}\,x(t),\end{array} \right.\]  
where \(u\in\mathbb{R}^{m}\) is the control input, \(d\in\mathbb{R}^{s}\) is the disturbance input, \(y\in\mathbb{R}^{p}\) is the measurable output, and \(\ z\in\mathbb{R}^{r}\) is the to-be-decoupled output. With the obvious meaning of the symbols, one can also refer to the graph representation  
\[\left((G,\mathcal{E}_{\sigma(t)}),G_{\sigma(t)}^{u}\cup G_{\sigma(t)}^{d},G_{ \sigma(t)}^{y}\cup G_{\sigma(t)}^{z}\right)\]  
and, therefore, adopt the unifying notation  
\[\Sigma_{\sigma}\big{(}(G,\mathcal{E}_{\sigma}),G_{\sigma}^{u}\cup G_{\sigma}^ {d},G_{\sigma}^{y}\cup G_{\sigma}^{z}\big{)}.\]Hence, the disturbance decoupling problem by output feedback is formulated as follows. **Problem 3**. : _Let \(\Sigma_{\sigma}((G,\mathcal{E}_{\sigma}),G_{\sigma}^{u}\cup G_{\sigma}^{d},G_{ \sigma}^{y}\cup G_{\sigma}^{z})\) be a switching structured linear system affected by the disturbance input \(d\), with measurable output \(y\) and to-be-decoupled output \(z\). The problem of disturbance decoupling by output feedback consists in finding a switching essential output feedback \(\mathcal{L}_{\sigma}\) such that the compensated switching structured linear system \(\Sigma_{\sigma}^{\mathcal{L}}((G,\mathcal{E}_{\sigma}\setminus\bigcup\mathcal{ L}_{\sigma}),G_{\sigma}^{u}\cup G_{\sigma}^{d},G_{\sigma}^{y}\cup G_{\sigma}^{z})\) does not exhibit any switching path \(P_{s}\) in \((G,\mathcal{E}_{\sigma}\setminus\bigcup\mathcal{L}_{\sigma})\) whose tail and head are in \(G_{h}^{d}\) and \(G_{h^{\prime}}^{z}\), respectively, for any \(h,h^{\prime}\in\mathcal{H}\)._  
**Theorem 3**. : _The following statements are equivalent:_  
* _Problem_ 3 _is solvable;_
* \(S^{*}(\mathcal{E}_{\sigma},G_{\sigma}^{y},\cup_{h\in\mathcal{H}}G_{h}^{d})=V ^{*}(\mathcal{E}_{\sigma},G_{\sigma}^{u},G\setminus\cup_{h\in\mathcal{H}}G_{ h}^{z})\)_._  
Proof. : The condition of the theorem is equivalent to the fact that any switching path \(P_{s}\) in \((G,\mathcal{E}_{\sigma})\), whose tail and head are in \(G_{h}^{d}\) and \(G_{h^{\prime}}^{z}\), respectively, for some \(h,h^{\prime}\in\mathcal{H}\), contains at least one edge \((x_{j},x_{i})\in\mathcal{E}_{h^{\prime\prime}}\), for some \(h^{\prime\prime}\in\mathcal{H}\), whose tail and head are in \(G_{h^{\prime\prime}}^{y}\) and \(G_{h^{\prime\prime}}^{u}\). Then, a solution of the problem consists in a switching essential output feedback that removes all the edges of this kind. **Example 2**.