Secure Implementation of Distributed Control for Multiple Spacecraft Against Malicious Eavesdrop** Attacks

Abstract

This paper addresses the secure distributed control problem of spacecraft formation flying systems, which are susceptible to malicious eavesdrop** attacks from hostile adversaries. A novel secure implementation framework of distributed control is proposed. Firstly, the uniform quantizer is decomposed, and an encoder is developed to convert system states and intermediate control parameters into integers. Then, the sensitive system states are pre-encrypted using the Paillier cryptosystem. Subsequently, an encrypted distributed control scheme is presented, incorporating the encoded and encrypted data. Notably, all data transmitted via the communication network are encrypted. Moreover, the encrypted distributed control law is directly evaluated using the received information from neighboring spacecraft, which maintains data security even when the controller devices are wiretapped. Following this secure implementation framework, the proposed encrypted distributed control law protects the data security of each spacecraft, while enabling effective control coordination. Simulation results verify the effectiveness of the proposed secure implementation framework.

Appendix

Choose the following Lyapunov function candidate

$$\begin{aligned} V=\frac{1}{2}\sum _{i}^{n}\boldsymbol{s}_i^\top \boldsymbol{M}_i \boldsymbol{s}_i \end{aligned}$$

(13)

where $\boldsymbol{M}_i=\textrm{diag}\{m_i,m_i,m_i\}\in \mathbb {R}^3$

$$\begin{aligned} \begin{aligned} \dot{V}&=\frac{1}{2}\sum _{i}^{n}\boldsymbol{s}_i^\top \dot{\boldsymbol{M}}_i\boldsymbol{s}_i+\sum _{i}^{n}\boldsymbol{s}_i^\top m_i\dot{\boldsymbol{s}}_i\\ &=\frac{1}{2}\sum _{i}^{n}\boldsymbol{s}_i^\top (\dot{\boldsymbol{M}}_i-2\boldsymbol{C}_i)\boldsymbol{s}_i+\sum _{i}^{n}\boldsymbol{s}_i^\top (m_i\ddot{\boldsymbol{\rho }}_i+m_i\dot{\boldsymbol{q}}_{ri}+\boldsymbol{C}_i\dot{\boldsymbol{\rho }}_i+\boldsymbol{C}_i\boldsymbol{q}_{ri})\\ &=\sum _{i}^{n}\boldsymbol{s}_i^\top (-\boldsymbol{f}_i+\boldsymbol{u}_i+m_i\dot{\boldsymbol{q}}_{ri}+\boldsymbol{C}_i\boldsymbol{q}_{ri}) \end{aligned} \end{aligned}$$

(14)

Substituting the distributed control law (11) in to (14), results in

$$\begin{aligned} \dot{V}=-\sum _{i}^{n}\boldsymbol{s}_i^\top K_i\boldsymbol{s}_i\le -\frac{2K_\textrm{min}}{M_\textrm{max}} V \end{aligned}$$

(15)

where $K_\textrm{min}$ denotes the minimum of the control gain sequence $K_i, (i=1,\cdots ,n)$ and $M_\textrm{max}$ is the mass maximum of the formation spacecraft. Clearly, it is from (15) inferred that $\boldsymbol{s}_i$ converges to zero exponentially. Recalling the definition of $\boldsymbol{s}_i$ in (10), it is concluded that $\dot{\boldsymbol{\rho }}_i\rightarrow 0$ and $\boldsymbol{\rho }_i-\boldsymbol{\rho }_j\rightarrow 0$, as $t\rightarrow \infty$. This completes the proof of Theorem 1.