# A Mechanical Feedback Classification of Linear Mechanical Control Systems
## 2 Problem Statement
### Linear Mechanical Feedback Transformations  
In this subsection, we introduce mechanical feedback transformations under which we classify linear mechanical systems, namely mechanical changes of coordinates and mechanical feedback.  
A linear mechanical transformation is given by a linear transformation of the following form \(\tilde{z}=\mathcal{T}z\), where \(\mathcal{T}=\left(\begin{array}{cc}T&0\\ 0&T\end{array}\right),\ z=(x,y)\) and \(\tilde{z}=(\tilde{x},\tilde{y})\), i.e.,  
\[\tilde{x}=Tx,\qquad\tilde{y}=Ty, \tag{7}\]where \(T\) is an invertible \(n\times n\) real matrix. This linear transformation preserves configurations, i.e., maps \(x\)-coordinates into \(\hat{x}\)-coordinates. Moreover, since the derivatives of configurations are velocities, it induces the linear transformation \(\hat{y}=Ty\) (given by the same \(T\)) on velocities that maps the equation \(\hat{x}=y\) into \(\dot{\hat{x}}=\hat{y}\).  
The linear mechanical feedback is  
\[u=Fx+G\hat{u}, \tag{8}\]  
where \(F\) is an \(m\times n\) matrix and \(G\) is an \(m\times m\) invertible matrix. The linear mechanical system (2) transformed by the transformations (7) and (8) reads  
\[\dot{\hat{x}} =Ty=\tilde{y} \tag{9}\] \[\hat{y} =T(E+BF)T^{-1}\hat{x}+TBG\tilde{u}=\tilde{E}\hat{x}+\tilde{B}\tilde {u}.\]  
**Definition 1**.: _Two systems (2), given by \(\dot{x}=y\), \(\dot{y}=Ex+Bu\), and \((\widetilde{\mathcal{L}\mathcal{M}\mathcal{S}})\), given by \(\dot{\hat{x}}=\hat{y}\), \(\dot{y}=\tilde{E}\hat{x}+\tilde{B}\tilde{u}\), are called linear mechanical feedback equivalent, briefly LMF-equivalent, if there exist mechanical transformations (7) and (8) such that \(\tilde{E}=T(E+BF)T^{-1}\) and \(\tilde{B}=TBG\)._  
It is natural to consider mechanical transformations (7) and (8), since the class of linear mechanical control systems (2) is closed under those transformations, i.e., the transformed system (9) is linear and mechanical. However, in case of the subclass of Lagrangian systems (6), the mechanical transformations need not preserve the Lagrangian structure (see Section 5 for a detailed analysis).  
The group of linear mechanical transformations \(LMF\), consisting of triplets \((\mathcal{T},F,G)\), preserves trajectories, that is, any element \((\mathcal{T},F,G)\) of that group maps the trajectories of (2) into those of its LMF-equivalent system \((\widetilde{\mathcal{L}\mathcal{M}\mathcal{S}})\), given by (9). Indeed, if \(z(t,z_{0},u(t))\) is a trajectory of (2), passing through \(z_{0}=(x_{0},y_{0})\) and corresponding to a control \(u(t)\), then \(\tilde{z}(t,z_{0},\tilde{u}(t))=\mathcal{T}z(t,z_{0},u(t))\) is a trajectory of (9), passing through \(\tilde{z}_{0}=\mathcal{T}z_{0}=(Tx_{0},Ty_{0})\) and corresponding to \(\tilde{u}(t)\), where \(u(t)=Fx(t)+G\tilde{u}(t)\). Moreover, via \(T:\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\), it establishes a correspondence between trajectories in the configuration space \(\mathbb{R}^{n}\), i.e., \(\tilde{x}(t,z_{0},\tilde{u}(t))=Tx(t,z_{0},u(t))\), making the following diagram commutative (notice, however, that \(\pi(z(t,z_{0},u))=x(t,z_{0},u)\) depends on \(z_{0}=(x_{0},y_{0})\) consisting of an initial configuration \(x_{0}\)_and_ initial velocity \(y_{0}\)):  
where \(\pi:\mathbb{R}^{n}\times\mathbb{R}^{n}\rightarrow\mathbb{R}^{n}\) is the canonical projection \(\pi(x,y)=x\), which assigns to the pair \((x,y)\) the point \(x\) at which the velocity \(y\) is attached.