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Combining Distributed Consensus with Robust H_{∞}-Control for Satellite Formation Flying

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Coordinate System

#### 2.2. Linear Model of Satellite Dynamics

#### 2.3. Graph Theory Fundamentals

#### 2.3.1. Graph

#### 2.3.2. Directed Graph

#### 2.3.3. Path

#### 2.3.4. Reachability

#### 2.3.5. Neighbor

#### 2.3.6. Adjacency Matrix

#### 2.3.7. Degree Matrix

#### 2.3.8. Laplacian Matrix

#### 2.4. Distributed Control

#### 2.4.1. Connection to Graph Theory

#### 2.4.2. General State-Space Representation of a Distributed LTI System

#### 2.4.3. Decentralized and Distributed Control

#### 2.4.4. Distributed Consensus Approach

#### 2.4.5. Reference Tracking

#### 2.4.6. Fixing Coordinate Origin to an Agent

#### 2.5. Robust Control

#### 2.5.1. ${H}_{\infty}$ Control

#### 2.6. Distributed Robust Control

#### 2.6.1. Mixed Sensitivity Closed-Loop System with Distributed Controller Interconnections

#### 2.6.2. Obtaining Generalized Plant for Single Agent

#### 2.6.3. Computation of the CLTF

#### 2.6.4. Obtaining the Generalized Plant for the Overall System

## 3. Results

#### 3.1. Scenario Definition

#### 3.1.1. Network Topology

#### 3.1.2. Dynamically Decoupled State Definition

#### 3.1.3. Distributed Robust Consensus Approach

#### 3.2. Simulation Results

#### Fixing Coordinate Origin to an Agent

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AILRC | adaptive iterative learning reliable control |

ATV | automated transfer vehicle |

CLTF | closed-loop transfer function |

ESA | European Space Agency |

ISS | International Space Station |

LEO | low Earth orbit |

LFT | linear fractional transformation |

LQR | linear quadratic regulator |

LTI | linear time-invariant |

LVLH | local-vertical, local-horizontal |

MMS | Magnetospheric Multiscale Mission |

MPC | model predictive control |

NASA | National Aeronautics and Space Administration |

NetSat | networked pico-satellite distributed system control |

PCO | projected circular orbit |

SFF | satellite formation flying |

TF | transfer function |

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**Figure 1.**Local-vertical, local horizontal (LVLH) coordinate system with x-axis tangential to flight direction along the satellite’s orbit, z-axis towards Earth center and y axis completing the right-handed coordinate system.

**Figure 3.**Directed graph with its vertices are labeled with numbers and its edges with letters. Edges show arrows to visualize their direction, e.g., a is the edge from vertex 1 to vertex 2.

**Figure 5.**Two plants ${P}_{1}$ and ${P}_{2}$ with their controllers ${K}_{1}$ and ${K}_{2}$. The plants are not coupled, but the controllers are. Here, ${y}_{1}$ influences ${K}_{2}$ and ${y}_{2}$ influences ${K}_{1}$.

**Figure 6.**Exemplary block diagram of three distributed systems with their controllers coupled via the consensus approach. The interconnection topology is shown as a graph in the upper right corner.

**Figure 7.**Schematic drawing of three agents in an arbitrary coordinate system. Their states are labeled ${\overrightarrow{s}}_{1}$, ${\overrightarrow{s}}_{2}$ and ${\overrightarrow{s}}_{3}$. Relative references are labeled according to Equation (21) with ${\overrightarrow{r}}_{ji}$ being the reference vector from agent i to agent j. For readability, the inverse references ${\overrightarrow{r}}_{ij}=-{\overrightarrow{r}}_{ji}$ are omitted.

**Figure 8.**General control configuration ([27], p. 442).

**Figure 10.**Artificial drawing of ESA’s Darwin mission study. (Image: ESA/Darwin, 2002, http://www.esa.int/spaceinimages/Images/2002/11/Darwin_will_combine_light_from_four_or_five_telescopes_and_send_it_down_to_Earth).

**Figure 11.**Interconnection graph of five satellites in a spaceborne telescope example. Interconnections between satellites are assumed to be bidirectional.

**Figure 12.**3D plot of the trajectories of the five satellites subject to the presented consensus-based controller starting on v-bar ($-20,-10,0,10,20$ m).

**Figure 13.**The x–y motion of the trajectories of the five satellites subject to the presented consensus-based controller starting on v-bar ($-20,-10,0,10,20$ m).

**Figure 14.**The x–z motion of the trajectories of the five satellites subject to the presented consensus-based controller starting on v-bar ($-20,-10,0,10,20$ m).

**Figure 15.**The y–z motion of the trajectories of the five satellites subject to the presented consensus-based controller starting on v-bar ($-20,-10,0,10,20$ m).

**Figure 16.**Accelerations acting on satellite 1 during simulation time, namely disturbance and control acceleration.

**Figure 17.**3D plot of the trajectories of the five satellites subject to the presented robust consensus-based controller starting on v-bar ($-20,-10,10,20,0$ m). One satellite is fixed at the origin and uncontrolled. Still, the other satellites adjust their states to form the desired formation.

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**MDPI and ACS Style**

Scharnagl, J.; Kempf, F.; Schilling, K.
Combining Distributed Consensus with Robust *H*_{∞}-Control for Satellite Formation Flying. *Electronics* **2019**, *8*, 319.
https://doi.org/10.3390/electronics8030319

**AMA Style**

Scharnagl J, Kempf F, Schilling K.
Combining Distributed Consensus with Robust *H*_{∞}-Control for Satellite Formation Flying. *Electronics*. 2019; 8(3):319.
https://doi.org/10.3390/electronics8030319

**Chicago/Turabian Style**

Scharnagl, Julian, Florian Kempf, and Klaus Schilling.
2019. "Combining Distributed Consensus with Robust *H*_{∞}-Control for Satellite Formation Flying" *Electronics* 8, no. 3: 319.
https://doi.org/10.3390/electronics8030319