# Development and Simulation of Motion Control System for Small Satellites Formation
## 4 Control Law Design Based on Passivity Concept  
Let us introduce additional system outputs (14) as  
\[\mathbf{y}=\left[\begin{array}{cc}\alpha_{1}e_{x}+e_{x}&\alpha_{2}e_{y}+e_{y}& \alpha_{3}e_{z}+e_{z}\end{array}\right]^{\mathsf{T}}, \tag{15}\]  
where \(\alpha_{1},\alpha_{2},\alpha_{3}\) is a known positive-definite scaling gains. Now the system (14) takes the following general form:  
\[\dot{\mathbf{e}}=\bar{\mathbf{A}}(\mathbf{e},t)\mathbf{e}+\mathbf{B}\bar{ \mathbf{u}},\quad\mathbf{y}=\mathbf{C}\mathbf{e} \tag{16}\]  
where the matrices are as follows:  
\[\bar{\mathbf{A}}=\left[\begin{array}{cccccc}0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ \dot{\theta}^{2}-\frac{\mu}{\gamma}+\frac{3}{2}\frac{\mu}{r_{c}^{2}}\Big{(}2 +\frac{x}{r_{c}}\Big{)}\Psi&0&\dot{\theta}+\frac{3}{2}\frac{\mu}{r_{c}^{2}} \Psi&2\dot{\theta}&\frac{3}{2}\frac{\mu}{r_{c}^{2}}z\Psi&0\\ -\ddot{\theta}&-2\dot{\theta}&\dot{\theta}^{2}-\frac{\mu}{\gamma}&0&0&0\\ 0&0&0&0&-\frac{\mu}{\gamma}&0\end{array}\right] \tag{17}\]  
\[\mathbf{B}=\left[\begin{array}{cccccc}0&0&0&\frac{1}{m_{f}}&0&0\\ 0&0&0&0&\frac{1}{m_{f}}&0\\ 0&0&0&0&\frac{1}{m_{f}}\end{array}\right]^{\mathsf{T}},\quad\mathbf{C}=\left[ \begin{array}{cccccc}\alpha_{1}&0&0&1&0&0\\ 0&\alpha_{2}&0&0&1&0\\ 0&0&\alpha_{3}&0&0&1\end{array}\right] \tag{18}\]  
Let us show that the outputs (15) passify [33] system (16). To do this, it is necessary to show that the zero dynamics [31] of system (16) are uniformly asymptotically stable. To isolate the zero dynamics, it is necessary to find the constant matrices \(\mathbf{M}\in\mathbb{R}^{6\times 3}\) and \(\mathbf{N}\in\mathbb{R}^{3\times 6}\) which satisfy the relations [34]  
\[\mathbf{CM}=\mathbf{0}_{3}\quad\mathbf{NB}=\mathbf{0}_{3}\quad\mathbf{NM}= \mathbf{I}_{3} \tag{19}\]  
Then, the equation of zero dynamics is obtained in the form:  
\[\dot{\eta}=\mathbf{A}_{\eta}(\mathbf{e})\eta\quad\in\mathbb{R}^{3}, \tag{20}\]  
where \(\mathbf{A}_{\eta}(\mathbf{e})\in\mathbb{R}^{3\times 3}\) and is calculated by the formula:  
\[\mathbf{A}_{\eta}(\mathbf{e})=N\bar{\mathbf{A}}(\mathbf{e},t)M \tag{21}\]  
Matrices \(\mathbf{M}\) and \(\mathbf{N}\) satisfying conditions (19) have the form [13]:  
\[\mathbf{M}=\left[\begin{array}{cccc}1&0&0\\ 0&1&0\\ 0&0&1\\ -\alpha_{1}&0&0\\ 0&-\alpha_{2}&0\\ 0&0&-\alpha_{3}\end{array}\right],\quad\mathbf{N}=\left[\begin{array}{cccccc} 1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\end{array}\right] \tag{22}\]  
Then because of (21), one obtains  
\[\mathbf{A}_{\eta}=\left[\begin{array}{cccc}-\alpha_{1}&0&0\\ 0&-\alpha_{2}&0\\ 0&0&-\alpha_{3}\end{array}\right] \tag{23}\]Considering that the coefficients \(\alpha_{1},\alpha_{2},\alpha_{3}\) are positive, it can be argued that the system (20) is uniformly asymptotically stable. This immediately implies [33] that system (16) is strictly the minimal phase. Additionally, let us check the result of the product of matrices \(\mathbf{C}\mathbf{B}\):  
\[\mathbf{C}\mathbf{B}=\left[\begin{array}{cccccc}\alpha_{1}&0&0&1&0&0\\ 0&\alpha_{2}&0&0&1&0\\ 0&0&\alpha_{3}&0&0&1\end{array}\right]\left[\begin{array}{cccc}0&0&0&0\\ 0&0&0\\ \frac{1}{m_{f}}&0&0\\ 0&\frac{1}{m_{f}}&0\\ 0&0&\frac{1}{m_{f}}\end{array}\right]=\frac{1}{m_{c}}\mathbf{I}_{3} \tag{24}\]  
We obtain the result that \(\mathbf{C}\mathbf{B}\) is a symmetric positive definite matrix. Then, from the results of [34], it follows that there exists a constant matrix of feedback gains \(\mathbf{K}\) in outputs \(\mathbf{y}\), which makes the system (16) asymptotically stable, and control aim (12) is achieved. The passification-based control law has a form  
\[\mathbf{\bar{u}}(t)=-\mathbf{K}\mathbf{y}(t) \tag{25}\]  
where matrix \(\mathbf{K}\in\mathbb{R}^{3\times 3}\).