# Modeling and Control of Satellite Formations: A Survey
## 2 Multiple-Input-Multiple-Output Formation Architecture
The desired relative position of the satellites is given by the variables \(x_{c}\), \(y_{c}\), and \(z_{c}\). Appropriate sliding variables are formed, including an integral term to compensate for constant perturbations (see [71] (pp. 147-153)):  
\[e_{x}=x_{c}-x,\quad e_{y}=y_{c}-y,\quad e_{z}=z_{c}-z,\] \[\sigma_{i}=\dot{e}_{i}+c_{i,0}e_{i}+c_{i,-1}\int e_{i}(\tau) \mathrm{d}\tau \tag{28}\]  
with some parameters of the desired motion in the sliding mode \(c_{i,0}\), \(c_{i,-1}\), where \(i\in\{x,y,z\}\). For numerical analysis, the values \(c_{i,0}=1.8\), \(c_{i,-1}=1.0\) are taken. The following formation control algorithms are considered:  
(1) Sliding mode controller and sliding mode perturbation observer  
\[u_{i}=10\sigma_{i}+\hat{\sigma}_{i\,\mathrm{eq}},\quad\dot{\phi} _{i\,\mathrm{eq}}=20\big{(}-\hat{\sigma}_{i\,\mathrm{eq}}+2\,\mathrm{sign}(s_ {i})\big{)},\] \[s_{i}=\sigma_{i}+\int\big{(}u_{i}-2\,\mathrm{sign}(s_{i})\big{)} \,\mathrm{d}\tau,\quad i\in\{x,y,z\}. \tag{29}\]  
Here, \(\hat{\sigma}_{i\,\mathrm{eq}}\) is the \"equivalent control\", the result of passing signals \(v_{i}=(L_{i}+\rho_{i})\sigma(s_{i})\), where \(\rho_{i}>0\), through low-pass filters. (According to the authors of the review, \(\hat{\sigma}_{i\,\mathrm{eq}}\) is not exactly equivalent control, as defined in [71], since, unlike the indicated works, in (29) the filtering occurs in a _closed_ rather than _open_ control loop.) As noted in [68], the \(\hat{\sigma}_{i\,\mathrm{eq}}\) signals, after a finite transient time, are an accurate estimate of the matched reduced perturbations \(\psi_{i}^{0}(\cdot)\) (linear combinations of object output and \(e_{i}\) tracking error (see [68, 72] for details)). (2) The following second order sliding modes super-twisting algorithm, ensuring convergence of variables on the sliding surface and their derivatives to zero in a finite time  
\[u_{i}=3|\sigma_{i}|^{\frac{1}{2}}\,\mathrm{sign}(\sigma_{i})+\int\mathrm{ sign}(\sigma_{i})\,\mathrm{d}\tau,\quad i\in\{x,y,z\}. \tag{30}\]  
(3) The sliding modes continuous algorithm with an integral sliding surface, based on the Lyapunov method. This algorithm implements continuous control in a sliding mode, ensuring robust stabilization of \(\sigma_{i}\) with respect to matched perturbations. It uses two types of moving variables: \(\sigma_{i}\) and additional integral moving variables \(\eta_{i}\)[72]. It can be written as  
\[\eta_{i} =\sigma_{i}+\tilde{k}_{0i}s_{i},\quad\dot{s}_{i}=\mathrm{sign}( \sigma_{i}), \tag{31}\] \[\tilde{u}_{i} =\psi_{i}^{0}(\cdot)+(\tilde{\rho}_{i}/2^{a_{i}})|\eta_{i}|^{2a_ {i}-1}\,\mathrm{sign}(\eta_{i}), \tag{32}\]  
where \(\tilde{\rho}_{i}>0\), \(\tilde{k}_{0i}>0\), \(0.5<a_{i}<1.0\), are the constants, chosen by a designer; \(\tilde{u}=E(x)u\), where \(E(x)\) denotes the nonsingular transformation matrix to a new basis in which the original system is reduced to \(m\) independent subsystems. When \(\tilde{u}\) is found, the control is obtained by the inverse transformation \(u=E^{-1}(x)\bar{u}\). For the considered problem, (27), algorithm (31) and (32) read as  
\[u_{i}=\frac{\beta_{i}}{2^{a_{i}}}\cdot\left|\sigma_{i}+\tilde{k}_{0i}\int \operatorname{sign}(\sigma_{i})\,\mathrm{d}\tau\right|^{2a_{i}-1}\operatorname {sign}\left(\sigma_{i}+\tilde{k}_{0i}\int\operatorname{sign}(\sigma_{i})\, \mathrm{d}\tau\right), \tag{33}\]  
where \(a_{i}=0.9\), \(\tilde{k}_{0i}=0.5\), \(\hat{\rho}_{i}=10\), \(i\in\{x,y,z\}\). The control signals \(u_{i}\) in (29)-(33) are continuous, as discontinuous high frequency components are filtered or integrated. The comparative results of applying the proposed control algorithms are presented in [68]. [PERSON] et al. [10] explored the three types of reconfiguration maneuvers described in [73] such as resizing, reassigning, and reorienting. Resizing is the change in distance between spacecraft in the formation. In [10], a hybrid optimization strategy with a genetic algorithm and basis vector theory is used for fuel-optimal reconfiguration of a satellite group.