# LMIs in Control/Applications/Problem of Space Rendezvous and LMI Approaches

LMIs in Control/Applications/Problem of Space Rendezvous and LMI Approaches
This is a Problem of Space Rendezvous and LMI Approaches
In Section 12.4 of their book LMIs in Control Systems: Analysis, Design, and Applications, Duan and Yu discuss the problem of space rendezvous and how it can be formulated into an LMI problem. Modeling and simulating space rendezvous is of importance because it is used for any cargo or passenger spacecraft traveling to and from earth-orbiting space stations and also for satellites servicing aging in-orbit satellites, and for potential missions to mine asteroids.

## The System

Though Duan and Yu first mention space rendezvous in Example 7.14 of their book. In this example, they show that the relative orbital dynamic model of spacecraft rendezvous can be described by the famous Clohessy-Wiltshire equations.
    ${\begin{aligned}m{\ddot {r_{x}}}-2m\omega _{0}{\dot {r_{y}}}-3m\omega _{0}^{2}r_{x}&=T_{x}+d_{x}\\m{\ddot {r_{y}}}+2m\omega _{0}{\dot {r_{x}}}&=T_{y}+d_{y}\\m{\ddot {r_{z}}}+m\omega _{0}^{2}r_{z}&=T_{z}+d_{z}\\\end{aligned}}$
where
* $r_{x},r_{y},r_{z}$ are the components of the relative position between chaser and target
* $\omega _{0}=\pi /12$ [rad/h] is the orbital angular velocity of the target satellite
* $m$ is the mass of the chaser
* $T_{i},(i=x,y,z)$ is the i-th component of the control input force acting on the relative motion dynamics
* $d_{i},(i=x,y,z)$ is the i-th component of the external disturbance
The C-W equations give a first-order approximation of the chaser's motion in a target-centered coordinate system and is often used in planning space rendezvous problems (ISS, Salyut, and Tiangong space stations are just some examples.)
With appropriate definitions of states and variables the dynamic equations of motion for space-rendezvous can be converted into standard state-space form for LMI optimization as follows:
    ${\begin{aligned}{\dot {x}}&=Ax+B_{1}u+B_{2}d\\y&=Cx\\\end{aligned}}$
where the vectors in the above state-space representation are defined as follows:
    ${\begin{aligned}x={\begin{bmatrix}r_{x}&r_{y}&r_{z}&{\dot {r_{x}}}&{\dot {r_{y}}}&{\dot {r_{z}}}\\\end{bmatrix}}^{T}\\\\y={\begin{bmatrix}r_{z}&r_{y}&r_{z}\end{bmatrix}}^{T}\\\\u={\begin{bmatrix}T_{x}&T_{y}&T_{z}\end{bmatrix}}^{T}\\\\d={\begin{bmatrix}d_{x}&d_{y}&d_{z}\end{bmatrix}}^{T}\\\end{aligned}}$
and the matrices in the above state-space representation are defined as follows:
    ${\begin{aligned}A={\begin{bmatrix}0&0&0&1&0&0\\0&0&0&0&1&0\\0&0&0&0&0&1\\3\omega _{0}^{2}&0&0&0&2\omega _{0}^{2}&0\\0&0&0&-2\omega _{0}&0&0\\0&0&-\omega _{0}^{2}&0&0&0\end{bmatrix}}\\\\B_{1}=B_{2}={\begin{bmatrix}0_{3x3}\\I_{3}\\\end{bmatrix}}\\\\C={\begin{bmatrix}I_{3}&0_{3x3}\\\end{bmatrix}}\\\\\end{aligned}}$

## The Data

The data required are the mass properties of both the target and chaser vehicles for space rendezvous. Also required is the orbital angular velocities of the target and chasers and measurements of relative kinematics between the two.

## The Optimization Problem

The optimization problem is trying to attenuate the disturbance to output transfer function using either the H-inf or H2 norm.
* Objective: Hinf or H2 norm
* Variables: Controller Gains
* Constraints: Relative Dynamics/Kinematics between Chaser and Target in Orbit

## The LMI: Space Rendezvous LMI Optimization

The space rendezvous problem can be approached with either H-inf or H-2 optimization formulations. Both formulations can achieve closed-loop stability which ensures that rendezvous occurs because the relative distance between target and chaser eventually approaches zero. The LMIs for the H-inf and H2 optimization problem are shown below which are easily solvable because the matrices for the space rendezvous problem are available above in standard form.
Duan and Yu approach the $H_{\infty }$. The minimum attenuation level from disturbance to output can be found by solving the following LMI optimization problem.
    ${\begin{aligned}\\&min\gamma _{\infty }\\\\&{\text{s.t. }}X>0\\\\&{\begin{bmatrix}(AX+B_{1}W)^{T}+AX+B_{1}W&B_{2}&(C_{1}X+D_{1}W)^{T}\\B_{2}^{T}&-\gamma _{H\infty }I&D_{2}^{T}\\C_{1}X+D_{1}W&D_{2}&-\gamma _{H\infty }I\end{bmatrix}}<0\\\end{aligned}}$
which is the same as Theorem 8.1 in Duan and Yu's Book, the solution to the $H_{\infty }$ problem.

## Conclusion:

The LMI for Space Rendezvous is a useful and interesting method to model and simulate practical problems in spacecraft engineering. Space Rendezvous usually requires very good vision-based navigation or an exceptional human operator that can close the gap for final mating of the two docking adapters.

## Implementation

A link to CodeOcean or other online implementation of the LMI

## Related LMIs

Links to other closely-related LMIs
* Optimal_Output_Feedback_Hinf_LMI
* H2OptimalOutputFeedback

## External Links

A list of references documenting and validating the LMI.
* LMI Methods in Optimal and Robust Control - A course on LMIs in Control by Matthew Peet.
* - LMIs in Control Systems: Analysis, Design, and Applications - Duan and Yu