# A Mechanical Feedback Classification of Linear Mechanical Control Systems
## 4 Classification of Uncontrollable Systems (2)  
In this section, we assume that (2) is not controllable (cf. Lemma 1), i.e.,  
\[\text{rank }\left(B,EB,\ldots,E^{n-1}B\right)=k<n.\]  
We will use the same class of mechanical transformations (7) and (8) to establish a canonical form for systems (2) that are not controllable and thus we will classify them (for the controllable case, Theorem 1 and form (15) provide a complete classification). **Theorem 2**. : _For given \(n,m,k\), the linear mechanical system (2) is LMF-equivalent to the following canonical form:_  
\[ \begin{array}{ll}\dot{x}_{c}=y_{c}&\dot{x}_{d}=y_{d}\\ \dot{y}_{c}=E_{F}x_{c}+B_{F}u&\dot{y}_{d}=E_{d}^{I}x_{d},\end{array}  \tag{18}\]  
_where \(\dim x_{c}=k\), and \(E_{F}\), \(B_{F}\) are block-diagonal matrices of dimension \(k\times k\), \(k\times m\), respectively, of the form (16) and (17), given by the mechanical half-indices \(\tilde{\rho}_{1},\ldots,\tilde{\rho}_{m}\) satisfying \(\sum_{j=1}^{m}\tilde{\rho}_{j}=k\), while \(E_{d}^{I}\) is in the Jordan form, that is, a block-diagonal \((n-k)\times(n-k)\) matrix whose diagonal blocks are of four possible forms:_  
1. _for a real eigenvalue_ \(\lambda_{j}\) _of_ \(E_{d}^{I}\)__ \[\text{either}\quad D_{j}^{\mathbb{R}}= \begin{pmatrix}\lambda_{j}&&&\\ &\ddots&&\\ &&\lambda_{j}&\end{pmatrix} \quad\text{or}\quad\quad J_{j}^{\mathbb{R}}= \begin{pmatrix}\lambda_{j}&1&&&\\ &\ddots&\ddots&\\ &&\lambda_{j}&1\\ &&&\lambda_{j}\end{pmatrix} ,\]
2. _for a complex eigenvalue_ \(\lambda_{j}=\alpha_{j}\pm\mathrm{i}\beta_{j}\) _of_ \(E_{d}^{I}\)__ \[\text{either}\quad D_{j}^{\mathbb{C}}= \begin{pmatrix}\tilde{C}_{j}&&&\\ &\ddots&&\\ &&C_{j}&\\ &&&C_{j}\end{pmatrix} \quad\text{or}\quad\quad J_{j}^{\mathbb{C}}= \begin{pmatrix} C_{j}&I_{2}&&\\ &\ddots&\ddots&\\ &&C_{j}&I_{2}\\ &&&C_{j}\end{pmatrix} ,\]  
_where_ \(C_{j}= \begin{pmatrix}\alpha_{j}&\beta_{j}\\ -\beta_{j}&\alpha_{j}\end{pmatrix} \) _and_ \(I_{2}= \begin{pmatrix}1&0\\ 0&1\end{pmatrix} \)_._  
Therefore any mechanical system (2) can be decomposed under LMF into two independent mechanical subsystems:  
1. a \(2k\)-dimensional controllable mechanical system (2) represented in \((x_{c},y_{c})\)-coordinates by the pair \((E_{F},B_{F})\), which is in the canonical form (15), with mechanical half-indices \(\tilde{\rho}_{1},\ldots,\tilde{\rho}_{m}\); the subindex \"\(c\)\" stands for controllable,
2. a dynamical linear mechanical system (a system without controls) represented in \((x_{d},y_{d})\)-coordinates by the second-order differential equation \(\dot{x}=E_{d}x_{d}\), where the matrix \(E_{d}\) can always be transformed into the Jordan form \(E_{d}^{I}\); the subindex \"\(d\)\" stands for dynamical. Proof. : The system (2) is uniquely defined by the pair \((E,B)\) on which an element \((\mathcal{T},F,G)\) of the group \(LMF\) acts according to the rule:  
\[(E,B)\overset{(\mathcal{T},F,G)}{\longmapsto}\Big{(}T(E+BF)T^{-1},TBG\Big{)},\]  
which is the same equivalence transformation as in [5], where a slightly different notation \((A=E,P=T^{-1},K=FT^{-1},Q=G)\) is used. By the result of [5], we bring the pair \((E,B)\) into the form:  
\[E= \begin{pmatrix}E_{F}&0\\ 0&E_{d}\end{pmatrix} ,\quad B= \begin{pmatrix}B_{F}\\ 0\end{pmatrix} ,\]  
where \(E_{d}\) is an \((n-k)\times(n-k)\) matrix and \((E_{F},B_{F})\) is in Brunovsky form. What remains to prove is to bring \(E_{d}\) into its canonical form \(E_{d}^{J}\). In order to do that, we use a change ofcoordinates \(x_{d}\mapsto T_{d}x_{d}\), \(y_{d}\mapsto T_{d}y_{d}\), where \(T_{d}:\mathbb{R}^{n-k}\to\mathbb{R}^{n-k}\) brings \(E_{d}\) into its real Jordan form \(E_{d}^{I}\), see e.g., [15], and apply \(T= \begin{pmatrix}I_{k}&0\\ 0&T_{d}\end{pmatrix} \). \(\qed\)  
Denote \(\mathcal{B}=\mathrm{Im}\;B\) and \(\mathcal{X}_{c}=\mathcal{B}+E\mathcal{B}+\ldots+E^{k-1}\mathcal{B}\).