## 3 Design of the Control System
### Baseline Feedback Controller Design  
Assuming that the function \(H\), within system (1), exhibits a strict minimum at the point \(\hat{\mathbf{x}}^{*}\) (where \(\hat{\mathbf{x}}^{*}\) represents an equilibrium point), it follows that \(H(\mathbf{x})\) may be locally regarded as a Lyapunov function. This, in turn, leads to the stability of the equilibrium point \(\hat{\mathbf{x}}^{*}\) with \(\mathbf{u}=\mathbf{0}\). In this regard, \(\hat{H}\leqslant 0\) due to the positive semi-definiteness of \(\mathbf{R}\). Nevertheless, if the targeted equilibrium point does not coincide with the minimum of the Hamiltonian function, it becomes necessary to suitably shape the function through the control signal \(\mathbf{u}\). Beyond adjusting the Hamiltonian function, it may also be requisite to alter the interconnection and damping matrices. This approach is encapsulated in the method known as Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC), which is elucidated in the subsequent theorem [25]. **Theorem 1**. : _(IDA-PBC) Suppose that there exist \(H_{d}:\mathbb{R}^{n}\to\mathbb{R},\mathbf{J}_{d}(\mathbf{x})=-J_{d}^{\mathrm{T}}(\mathbf{ x})\), \(\mathbf{R}_{d}(\mathbf{x})=\mathbf{R}_{d}^{\mathrm{T}}(\mathbf{x})\succcurlyeq 0\) satisfying the following equation for system (1)._  
\[\mathbf{g}^{\perp}(\mathbf{x})((\mathbf{J}(\mathbf{x})-\mathbf{R}(\mathbf{x}))\
abla H(\mathbf{x}))=\mathbf{g} ^{\perp}(\mathbf{x})((\mathbf{J}_{d}(\mathbf{x})-\mathbf{R}_{d}(\mathbf{x}))\
abla H_{d}(\mathbf{x})) \tag{21}\]  
_where \(\mathbf{g}^{\perp}\) is the full-rank left annihilator of \(\mathbf{g}\), \(\succcurlyeq\) represents the positive semi-definite. If \(\mathbf{x}^{*}\) is a minimum of \(H_{d}\), then there exists a controller_  
\[\mathbf{u}=\left(\mathbf{g}^{\mathrm{T}}(\mathbf{x})\mathbf{g}(\mathbf{x})\right)^{-1}\mathbf{g}^{ \mathrm{T}}(\mathbf{x})((\mathbf{J}_{d}(\mathbf{x})-\mathbf{R}_{d}(\mathbf{x}))\
abla H_{d}(\mathbf{x })-(\mathbf{J}(\mathbf{x})-\mathbf{R}(\mathbf{x}))\
abla H(\mathbf{x})) \tag{22}\]  
_that is capable of locally stabilizing system (1), rendering the closed-loop system as follows._  
\[\hat{\mathbf{x}}_{d}=(\mathbf{J}_{d}(\mathbf{x})-\mathbf{R}_{d}(\mathbf{x}))\
abla H_{d}(\mathbf{x}) \tag{23}\]  
In a recent study, the IDA-PBC method has been expanded to encompass a tracking design utilizing contraction analysis, as detailed by [PERSON] and [PERSON] [26]. The key findings are summarized as follows. **Theorem 2**. : _(tIDA-PBC) Consider the following system_  
\[\hat{\mathbf{x}}_{d}=(\mathbf{J}_{d}(\mathbf{x})-\mathbf{R}_{d}(\mathbf{x}))\
abla H_{d}(\mathbf{x}, \mathbf{x}^{*}(t)) \tag{24}\]\(H_{d}\) is a desired Hamiltonian function conforming to the subsequent conditions._  
\[ \begin{split}&(\mathbf{J}_{d}(\mathbf{x})-\mathbf{R}_{d}(\mathbf{x}))\
abla H_{d}( \mathbf{x},\mathbf{x}^{*}(t))|_{\mathbf{x}^{*}(t)}=\tilde{\mathbf{x}}^{*}(t),\\ &\alpha_{1}\mathbf{I}\prec\
abla^{2}H_{d}(\mathbf{x},\mathbf{x}^{*}(t))\prec \alpha_{2}\mathbf{I},\forall\mathbf{x}\in D_{\mathsf{T}}\end{split}  \tag{25}\]  
\(\alpha_{1}\)_, \(\alpha_{2}\) are positive constants and \(\alpha_{1}<\alpha_{2}\). \(D_{\mathsf{T}}\) is an open subset of \(\mathbb{R}^{n}\) which for all \(t\geqslant 0\) constitutes a sphere containing \(\mathbf{x}^{*}(t)\) and possessing a constant radius \(T\) centered around \(\mathbf{x}^{*}(t)\). Assuming that all eigenvalues of \(\mathbf{A}_{d}\triangleq\mathbf{J}_{d}-\mathbf{R}_{d}\) possess strictly negative real parts and that the ensuing matrix exhibits no eigenvalues on the imaginary axis for a given positive constant \(\epsilon\),_  
\[\mathbf{N}= \begin{bmatrix}\mathbf{A}_{d}&\eta\mathbf{A}_{d}\mathbf{A}_{d}^{\mathsf{T}}\\ -(\eta+\epsilon)\mathbf{I}&-\mathbf{A}_{d}^{\mathsf{T}}\end{bmatrix}