## 3 Design of the Control System
### Nonlinear Disturbance Observer Design  
As detailed previously, the design of the controller necessitates considerations for electromagnetic force modeling errors and external disturbances. Given the minimal relative separation between satellites, the overall external disturbance impacting the formation can be considered as bounded forces [27]. Moreover, the modeling error for electromagnetic forces stays below 10% when the separation distance significantly surpasses the radius of the electromagnetic coils, and the intensity of the electromagnetic force is notably minor [8]. Consequently, it is justifiable to treat the effects of both external disturbances and electromagnetic force modeling errors as equivalent bounded total disturbances, denoted as \(\mathbf{w}\). Subsequently, this allows us to establish a necessary assumption for the discussions within this paper [28]. **Assumption 1**. : _The disturbance \(\mathbf{d}(t)=(d_{1},d_{2},d_{3})^{\mathsf{T}}\) and their derivatives \(\dot{\mathbf{d}}(t)\) are bounded, and the condition that as \(t\to\infty\), \(d_{i}(t)=0\) is satisfied, \(i=1,2,3\)._When considering system (1) with disturbances and incorporating the baseline feedback control law (31), we obtain the revised expression for system (1) as follows:  
\[\hat{\mathbf{x}}_{d}=(\mathbf{J}_{d}-\mathbf{R}_{d})\
abla H_{d}(\mathbf{x},\mathbf{x}^{*}(t))+\mathbf{g} \mathbf{u}_{DO}+\mathbf{g}\mathbf{d}(t) \tag{32}\]  
Due to the presence of disturbance \(\mathbf{d}(t)\) in system (32), which necessitate estimation and compensation for effective system control, a Nonlinear Disturbance Observer (NDOB), inspired by [28], is structured as follows  
\[\left\{ \begin{array}{l}\hat{\mathbf{M}}=-I(\mathbf{x})\mathbf{g}\mathbf{M}-I(\mathbf{x})[\mathbf{g} \mathbf{w}(\mathbf{x})+(\mathbf{J}_{d}-\mathbf{R}_{d})\
abla H_{d}(\mathbf{x},\mathbf{x}^{*}(t))+\mathbf{g} \mathbf{u}_{DO}]\\ \hat{\mathbf{d}}=\mathbf{M}+\mathbf{w}(\mathbf{x})\end{array} \right. \tag{33}\]  
where \(\hat{\mathbf{d}}\) represents the estimates of disturbances \(\mathbf{d}(t)\); the NDOB gain \(I(\mathbf{x})\) is defined as follows:  
\[I(\mathbf{x})=\frac{\partial\mathbf{w}(\mathbf{x})}{\partial\mathbf{x}} \tag{34}\]  
where the vector-valued function \(\mathbf{w}(\mathbf{x})\) is designed as \(\mathbf{w}(\mathbf{x})=\left[\mathbf{0}_{3},\mathbf{K}_{w}\mathbf{p}_{d}\right]^{\mathrm{T}}\), \(\mathbf{K}_{w}\) is the parameter of the function, and \(\mathbf{M}\) represents the state of the disturbance observer. Define the disturbance estimation error as \(\mathbf{e}_{d}=\hat{\mathbf{d}}-\mathbf{d}\), leading to the formulation of the disturbance estimation error system. \[\dot{\mathbf{e}}_{d}=-I(\mathbf{x})\mathbf{g}\mathbf{e}_{d}-\dot{\mathbf{d}} \tag{35}\]  
Substituting an error term into system (32), we obtain the following system  
\[\dot{\mathbf{x}}_{d}=(\mathbf{J}_{d}-\mathbf{R}_{d})\
abla H_{d}(\mathbf{x},\mathbf{x}^{*}(t))+\bm {g}\mathbf{u}_{DO}+\mathbf{g}\mathbf{\hat{d}}-\mathbf{g}\mathbf{e}_{d} \tag{36}\]  
In system (36), the disturbance compensation controller \(\mathbf{u}_{do}\) is intended to counterbalance the estimated disturbances \(\hat{\mathbf{d}}\), conforming to the equation specified below  
\[\mathbf{u}_{DO}=-\hat{\mathbf{d}} \tag{37}\]  
Thus, the disturbance rejection composite controller is designed as  
\[\mathbf{u}_{DRPBC}=\mathbf{u}_{FB}+\mathbf{u}_{DO} \tag{38}\]  
However, for EMFF satellites, the modulation of control force is accomplished through the adjustment of their magnetic dipole strength. It is coupled with the formation's position and the satellite's magnetic field strength. Therefore, the control force is not directly exerted on the satellite; rather, it is based on the current magnetic field strength, altering the satellite's own magnetic dipole strength to align with the required magnitude of the control force.