# Roto-Translational Control of Spacecraft in Low Earth Orbit Using Environmental Forces and Torques
## 3 Controller Design
### Stability Analysis
: Let \(V\) be a candidate Lyapunov function defined as  
\[V(t)\triangleq\boldsymbol{X}^{T}P\boldsymbol{X}+\frac{1}{2}\widetilde{\mathbf{ \Theta}}_{2}{}^{T}\Gamma_{2}^{-1}\widetilde{\mathbf{\Theta}}_{2}, \tag{38}\]  
where  
\[\widetilde{\mathbf{\Theta}}_{2}\triangleq\mathbf{\Theta}_{2}-\widetilde{ \mathbf{\Theta}}_{2}. \tag{39}\]  
The candidate Lyapunov function can be bounded by  
\[\underline{\lambda}\|\boldsymbol{X}\|^{2}+\underline{\gamma}\leq V(t)\leq \overline{\lambda}\|\boldsymbol{X}\|^{2}+\overline{\gamma}, \tag{40}\]where \(\underline{\lambda}\), \(\overline{\lambda}\), \(\underline{\gamma}\), \(\overline{\gamma}\in\mathbb{R}_{>0}\) are known bounding constants. Let \(\mathbf{X}\triangleq A\mathbf{X}+\mathbf{B}u_{y}\in\mathbb{R}^{4}\) be the state space representation of (8) and (9), where \(A\in\mathbb{R}^{4\times 4}\) is a known constant matrix. In the roto-translational maneuver, the set of DMD surfaces that simultaneously generate the required cross-sectional area and environmental torques are computed using (35), producing a residual mismatch \(\chi_{1}\triangleq S_{i}-S_{d,i}\in\mathbb{R}\). **Assumption 7**. : _A numerical optimization algorithm can be used to find a suitable set of DMD surface lengths (i.e., \(L_{1},\ L_{2},\ L_{3}\), and \(\ L_{4}\)) provided (25), (28), (33) and (34). The residual \(\chi_{1}\) can be upper bounded by a constant for the entire maneuver such that \(|\chi_{1}|<\zeta_{1}\)._  
Taking the time derivative of (38), using (39), substituting (33) and rearranging terms yields  
\[ \begin{split}\dot{V}(t)=\mathbf{X}^{T}\Big{(}PA^{*}+A^{*T}P\Big{)}+2 \mathbf{X}^{T}PBY_{2}\widetilde{\mathbf{\Theta}}_{2}+\mathbf{X}^{T}PB\frac{\Big{(}\hat{ \rho}_{i}\widehat{C}_{\mathrm{D}}^{\dagger}\widehat{V}_{r,i}^{2}\Big{)}\chi_ {1}}{m_{i}}+\\ +\widetilde{\mathbf{\Theta}}_{2}^{T}\Gamma_{2}^{-1}\hat{\mathbf{\Theta}} -\widetilde{\mathbf{\Theta}}_{2}^{T}\Gamma_{2}^{-1}\hat{\mathbf{\Theta}}_{2},\end{split}  \tag{41}\]  
where \(A^{*}\triangleq A-\mathbf{B}\mathbf{K}_{LQR}\). Since the gain \(\mathbf{K}_{LQR}\) is obtained by solving a Linear Quadratic Regulator (LQR) problem, then \(A^{*}\) is Hurwitz and a symmetric positive-definite matrix \(Q_{1}\in\mathbb{R}^{4\times 4}\) can be determined so that \(PA^{*}+A^{*T}P=-Q_{1}\). Using (34) and Assumptions 6 and 7, (41) can be upper bounded as  
\[\dot{V}(t)\leq-\lambda_{1}\|\mathbf{X}\|^{2}+\zeta_{4}, \tag{42}\]  
where \(\lambda_{1}\triangleq\lambda_{min}\Big{\{}Q_{1}-\frac{1}{2}\Big{\}}\in \mathbb{R}\), \(\lambda_{min}\{\cdot\}\) denotes the minimum eigenvalue of \(\{\cdot\}\), \(\zeta_{4}\triangleq\zeta_{2}+\frac{\zeta_{3}^{2}}{\lambda}\in\mathbb{R}_{>0}\), and [PERSON]'s inequality as well as the upper bounds \(\widetilde{\mathbf{\Theta}}_{2}^{T}\Gamma_{2}^{-1}\hat{\mathbf{\Theta}}_{2}\leq\zeta_ {2}\) and \(\left\|PB\Big{(}\Big{(}\hat{\rho}_{i}\widehat{C}_{\mathrm{D}}^{i}\widehat{V}_{ r,i}^{2}\Big{)}\chi_{1}/m_{i}\Big{)}\right\|\leq\zeta_{3}\) were used. Provided that the control gains are selected such that \(\lambda_{1}>0\) and using (40), then (42) can be rewritten as  
\[\dot{V}\leq-\frac{\lambda_{1}}{\lambda}V(t)+\frac{\lambda_{1}\overline{\gamma} }{\overline{\lambda}}+\zeta_{4}. \tag{43}\]  
By invoking the Comparison Lemma from [32], the solution to (43) can be obtained as  
\[V\leq V(0)\text{exp}\left\{-\frac{\lambda_{1}}{\overline{\lambda}}t\right\}+ \frac{\overline{\lambda}}{\lambda_{1}}\bigg{(}\frac{\lambda_{1}\overline{ \gamma}}{\overline{\lambda}}+\zeta_{4}\bigg{)}\bigg{(}1-\text{exp}\left\{- \frac{\lambda_{1}}{\overline{\lambda}}t\right\}\bigg{)}. \tag{44}\]  
From (38), (40) and (44), then \(\mathbf{X}\in\mathcal{L}_{\infty}\). Since \(\widehat{\mathbf{\Theta}}\in\mathcal{L}_{\infty}\) by (34), then \(S_{d,i}\in\mathcal{L}_{\infty}\); therefore, \(S_{i}\in\mathcal{L}_{\infty}\) from Assumption 7.