## 4 Second Step: Regional Pole Placement
### Insights on the Reduced Model and on the Inclusion Principle  
In order to justify the procedure employed for gain computation, a discussion on the Inclusion Principle [23] is proposed for the case under investigation, i.e., on the fact that a \"part\" of the overall system behavior is reproduced by its contraction having smaller dimension. It should be noted that the formulation of the first-order model through a reduced state is beneficial to reduce the size of the matrix adopted to define the LMI. Hence, the numerical computation of the gains benefits of this system scale reduction. This is another important and novel feature of the proposed method.  
The following notation is introduced: let us denote \(\widetilde{\mathbf{V}}\) a matrix whose columns span \(\text{ker}(\mathbf{U})\), and \(\widetilde{\mathbf{U}}\) the unique left inverse of \(\widetilde{\mathbf{V}}\) such that \(\text{ker}\!\left(\widetilde{\mathbf{U}}\right)=\text{im}(\mathbf{V})\). Let us also define \(\mathbf{W}=\left[\mathbf{V}\;\widetilde{\mathbf{V}}\right]\), which is clearly invertible. We want to prove that:  
\[\mathbf{W}^{-1}\mathbf{A}_{\text{CL}}\mathbf{W}=\left[\begin{array}{cc}\hat {\mathbf{A}}_{\text{CL}}&\mathbf{U}\mathbf{A}_{\text{CL}}\widetilde{\mathbf{V}}\\ \widetilde{\mathbf{U}}\mathbf{A}_{\text{CL}}\mathbf{V}&\widetilde{\mathbf{U} }\mathbf{A}_{\text{CL}}\widetilde{\mathbf{V}}\end{array}\right]. \tag{23}\]  
First, it must be noted that \(\mathbf{V}\mathbf{U}+\widetilde{\mathbf{V}}\widetilde{\mathbf{U}}=\mathbf{I}\) (the proof is tedious but straightforward, so it will be omitted). Hence,  
\[\begin{array}{rl}\mathbf{A}_{\text{CL}}\mathbf{W}&=\left[\left(\mathbf{V} \mathbf{U}+\widetilde{\mathbf{V}}\widetilde{\mathbf{U}}\right)\mathbf{A}_{ \text{CL}}\mathbf{V}\right]\!\left[\left(\mathbf{V}\mathbf{U}+\widetilde{ \mathbf{V}}\widetilde{\mathbf{U}}\right)\mathbf{A}_{\text{CL}}\widetilde{ \mathbf{V}}\right]\\ &=\mathbf{W}\left[\begin{array}{cc}\mathbf{U}\mathbf{A}_{\text{CL}}\mathbf{V }&\mathbf{U}\mathbf{A}_{\text{CL}}\widetilde{\mathbf{V}}\\ \widetilde{\mathbf{U}}\mathbf{A}_{\text{CL}}\mathbf{V}&\widetilde{\mathbf{U} }\mathbf{A}_{\text{CL}}\widetilde{\mathbf{V}}\end{array}\right].\end{array} \tag{24}\]  
The claim is a consequence of the fact that \(\mathbf{U}\mathbf{A}_{\text{CL}}\mathbf{V}=\hat{\mathbf{A}}_{\text{CL}}\). The result of Equation (23) can be further simplified, by using the well-known fact that the null space of the transpose of a matrix is equal to the null space of the pseudoinverse. In our case, \(\text{ker}\!\left(\mathbf{V}^{T}\right)=\text{ker}(\mathbf{U})\), and then \(\mathbf{V}^{T}\overline{\mathbf{V}}=0\). Therefore,  
\[\mathbf{W}^{-1}\mathbf{A}_{\text{CL}}\mathbf{W}=\left[\begin{array}{cc}\hat {\mathbf{A}}_{\text{CL}}&\mathbf{U}\!\left(\mathbf{A}-\mathbf{B}\mathbf{k}_{0} ^{T}\right)\!\widetilde{\mathbf{V}}\\ \widetilde{\mathbf{U}}\!\left(\mathbf{A}-\mathbf{B}\mathbf{k}_{0}^{T}\right) \!\mathbf{V}&\widetilde{\mathbf{U}}\!\left(\mathbf{A}-\mathbf{B}\mathbf{k}_{0} ^{T}\right)\!\widetilde{\mathbf{V}}\end{array}\right]. \tag{25}\]  
Provided that:  
\[\mathbf{U}\!\left(\mathbf{A}-\mathbf{B}\mathbf{k}_{0}^{T}\right)\!\widetilde{ \mathbf{V}}=0, \tag{26}\]  
it is demonstrated that the set of closed-loop poles is constituted by the poles of \(\hat{\mathbf{A}}_{\text{CL}}\) (which are constrained into the LMI region \(\mathcal{D}\)) and the poles of \(\widetilde{\mathbf{U}}\!\left(\mathbf{A}-\mathbf{B}\mathbf{k}_{0}^{T}\right)\! \widetilde{\mathbf{V}}\), which are \(\mu_{1},\ldots,\mu_{p}\), as can be seen from the numerical results obtained in this work. Although a proof that Equation (26) always holds is not provided, it has been experienced in all the test cases studied (also including dozens of examples not shown in the paper, with random requirements and parameters) that such a condition is satisfied.