# A Mechanical Feedback Classification of Linear Mechanical Control Systems
## 3 Classification of Controllable Systems (2)
\begin{split} r_{0}=&\text{rank }\hat{B}=\text{rank }B= \bar{r}_{0}\\ r_{1}=&\text{rank }\big{(}\hat{B},\hat{A}\hat{B} \big{)}-\text{rank }\hat{B}=(\text{rank }B+\text{rank }B)-\text{rank }B=\bar{r}_{0}.\end{split} \]Assume that \((r_{0},r_{1},\ldots,r_{2i-2},r_{2i-1})=(\bar{r}_{0},\bar{r}_{0},\ldots,\bar{r}_{i- 1},\bar{r}_{i-1})\). Then  
\[r_{2i} =\text{rank }\left(\hat{B},\ldots,\hat{A}^{2i}\hat{B}\right)-\text{ rank }\left(\hat{B},\ldots,\hat{A}^{2i-1}\hat{B}\right)\] \[=\text{rank }\left(B,\ldots,E^{i}B\right)+\text{rank }\left(B,\ldots,E^{i-1}B\right)-2\text{ rank }\left(B,\ldots,E^{i-1}B\right)\] \[=\text{rank }\left(B,\ldots,E^{i}B\right)-\text{rank }\left(B,\ldots,E^{i-1}B\right)=\bar{r}_{i},\] \[r_{2i+1} =\text{rank }\left(\hat{B},\ldots,\hat{A}^{2i+1}\hat{B}\right)- \text{rank }\left(\hat{B},\ldots,\hat{A}^{2i}\hat{B}\right)\] \[=2\text{ rank }\left(B,\ldots,E^{i}B\right)-\left(\text{rank }\left(B,\ldots,E^{i}B\right)+\text{rank }\left(B,\ldots,E^{i-1}B\right)\right)\] \[=\text{rank }\left(B,\ldots,E^{i}B\right)-\text{rank }\left(B,\ldots,E^{i-1}B\right)=\bar{r}_{i}.\]  
By an induction argument, the sequence of \(n\) integers \(\tilde{\mathcal{R}}(E,B)=(\bar{r}_{0},\bar{r}_{1},\ldots,\bar{r}_{n-1})\) and that of \(2n\) integers \(\mathcal{R}(\hat{A},\hat{B})=(r_{0},\ldots,r_{2n-1})\) satisfy the desired relation \((i)\). Using (12) and \((i)\), calculate, for \(1\leq j\leq m\),  
\[\rho_{j}=\text{card}(r_{i}\geq j:0\leq i\leq 2n-1)=2\,\text{card}(\bar{r}_{i} \geq j:0\leq i\leq n-1)=2\bar{p}_{j},\]  
which proves \((ii)\). The following theorem asserts that mechanical half-indices form a set of complete invariants \(\bar{p}_{j}\) of linear controllable mechanical systems (2), as do controllability indices \(\rho_{j}\) and indices \(\bar{r}_{i}\). Theorem 1: _The following statements are equivalent, for fixed \(n\) and \(m\):_  
1. _Two controllable systems (_2_) and_ \((\widehat{\mathcal{LMS}})\)_, represented by pairs_ \((E,B)\) _and_ \((\hat{E},\tilde{B})\)_, respectively, are LMF-equivalent,_
2. \(\tilde{\mathcal{R}}(E,B)=\tilde{\mathcal{R}}(\tilde{E},\tilde{B})\)_,_
3. \(\tilde{\mathcal{P}}(E,B)=\tilde{\mathcal{P}}(\tilde{E},\tilde{B})\)_, i.e., the mechanical half-indices coincide,_
4. \(\mathcal{P}(\hat{A},\hat{B})=\mathcal{P}(\hat{\hat{A}},\hat{B})\)_, i.e., the controllability indices coincide,_  
_where \(\hat{A},\hat{B}\) and \(\hat{A},\hat{B}\) are of the form (4)._  
Proof: Equivalence of \((ii)\) and \((iii)\) follows from the definition. \((i)\Leftrightarrow(iii)\). We associate with (2), given by the pair \((E,B)\), a virtual linear (first-order) control system  
\[\hat{x}=Ex+Bv, \tag{13}\]  
and similarly with \((\widehat{\mathcal{LMS}})\), given by \((\hat{E},\tilde{B})\), we associate  
\[\hat{x}=E\hat{x}+B\bar{v}. \tag{14}\]  
Now we directly use the Brunovsky classification theorem [1] to prove that (13) and (14) are equivalent under a transformation \(\hat{x}=Tx\) and feedback \(v=Fx+G\bar{v}\), if and only if their controllability indices coincide. Note that the controllability indices of (13), respectively of (14), coincide with the mechanical half-indices of associated (2) (respectively \((\widehat{\mathcal{LMS}})\)). Now notice that \(\hat{x}=Tx\) and \(v=Fx+G\bar{v}\) establish feedback equivalence between (13) and (14) if and only if \(\hat{x}=Tx\), \(\bar{y}=Ty\), and \(u=Fx+G\bar{u}\) establish LMF-equivalence between (2) and \((\widehat{\mathcal{LMS}})\). Therefore \((i)\) is equivalent to \((iii)\). Equivalence of \((iii)\) and \((iv)\) follows immediately from Proposition 2. **Remark 3**. : _Notice that the general feedback group acting on systems of the form (3) by \(\hat{A}\mapsto S(\hat{A}+\hat{B}\hat{F})S^{-1}\), \(\hat{B}\mapsto S\hat{B}\hat{G}\) (where \(S:\mathbb{R}^{2n}\to\mathbb{R}^{2n}\) is any, not necessarily of the form (7)) is much bigger than the mechanical feedback group (7) and (8).