## 3 Relative Motion Dynamics of Two Satellites in a Near-Circular Orbit  
For describing the dynamics of the relative motion of two satellites, moving in near-circular orbits at the Earth's central gravitational field, equations in the Local Vertical Local Horizontal (LVLH) reference system in relative coordinates according to the HCW model is used, see [80; 81; 82; 83; 84]. In the present paper, the LVLH reference system is employed, where the OZ axis is directed from the center of the Earth, the OY axis is directed normal to the orbital plane, and the OX axis complements the triple to the right coordinate frame, see Figure 1.  
Taking into account that the motion along the normal to the orbital plane (along OY axis) is isolated, and proceeding from the universal gravitation law and [PERSON]'s third law under the assumption that for distance \(\rho\) of the satellite to the center of the Earth inequalities \(\rho\gg x\), \(z\) are held, one obtains the following HCW equations of satellite motion in the LVLH local coordinate system:  
\[\ddot{x} =-2\omega\dot{z}+F_{x}, \tag{9}\] \[\ddot{z} =2\omega\dot{x}+3\omega z+F_{z}, \tag{10}\]  
where \(F_{x}\), \(F_{z}\) are the components of the vector of non-gravitational forces applied to the satellite, expressed in acceleration units. These forces include the aerodynamic drag force \(f_{x}\) along the OX axis, which is controlled by the satellite turning through the attack angle \(\alpha\). Based on the physical considerations it is valid that \(f_{x}=f_{x}(\alpha)\leq 0\), \(f_{x}(0)\leq|f_{x}|\leq f_{x}(\pi/2)\). Therefore it is natural considering angle \(\alpha\) within \(0\leq\alpha\leq\pi/2\). In (10), \(\omega\) denotes the averaged angular velocity of the spacecraft in orbit and satisfies the expression \(\omega=\sqrt{\mu/a^{3}}\) where \(\mu=GM\), \(G\) is the gravitational constant, \(M\) is the mass of the central body (for the Earth, \(\mu~{}=~{}398\),\(603~{}\times 10^{9}\) m\({}^{3}\)s\({}^{-2}\)), and \(a\) is the semi-major axis of the satellite orbit. HCW Equations (9) and (10) are derived under the assumptions: \(\sqrt{x^{2}+z^{2}}\) is small compared to \(\rho\); the aerodynamic force is small, the acceleration caused by it does not exceed \(1.7~{}\times 10^{-6}\) m/s\({}^{2}\); the eccentricity of the orbit is small, the orbit is very close to the circular one; and the orbital rate \(\omega\) is approximately constant. For the resulting system (9), (10) in [17], a non-degenerate coordinate transformation is performed to the real Jordan form [36], as a result of which the model (9), (10) is represented in the form of a double integrator and a harmonic oscillator, connected in parallel. For the double integrator, a speed-optimal control is created in the form of relay feedback with a quadratic switching function. A similar, but technically more complex approach is used for the harmonic oscillator. Since the control is scalar, it is not possible to apply both control laws simultaneously and independently to both the double integrator and oscillator. To resolve this contradiction, [PERSON] [17] developed and studied for different flight scenarios an algorithm for switching from one control law to another, based on the specifics of the rendezvous problem requirements. The linearized equations of the satellites relative motion in the OXZ plane (9), (10) in the state-space form read as (cf. [5; 21]):  
Figure 1: Coordinate system associated with reference point O, moving in a circular orbit. \[\dot{\chi} =A\chi+Bu,\quad y=C\chi, \tag{11}\] \[u =-\varphi(y), \tag{12}\]  
where \(\chi=\left[ \begin{array}{cccc}x_{12}&\dot{x}_{12}&z_{12}&\dot{z}_{12}\\ \end{array} \right]^{\mathrm{T}}\in\mathbb{R}^{4}\) is the system state vector, where \(x_{12}\) denotes the difference in coordinates of the second and first satellites along the OX axis, \(x_{12}=x_{2}-x_{1};z_{12}=z_{2}-z_{1}\); and \(u\) denotes the control action (difference of aerodynamic forces acting on satellites, in units of acceleration). It is limited in modulus by the value \(u_{\mathrm{max}}\).