A Proof-of-Weighted-Planned-Behavior Consensus for Efficient and Reliable Cyber-Physical Systems

Abstract

Recently, blockchain was introduced into the cyber-physical systems, which provides services of privacy and trust. However, reliability and system performance issues exist when blockchain and cyber-physical systems are integrated. In this paper, we design a blockchain-enabled cyber-physical system, where a new blockchain consensus is used to solve the problems of reliability and system performance. Firstly, an autonomous consensus mechanism called Proof-of-Weighted-Planned-Behavior is established based on the theory of planned behavior. Then, the behavior of consensus participants gets further explained by introducing credit evaluation and vulnerable node analysis. Moreover, considering the Jain fairness index, a dynamic authorizer group mechanism that coordinates reliability and decentralization is developed. By optimizing the credit threshold of the authorization group, the security and reliability of our designed mechanism are guaranteed. Finally, the experimental simulation results prove that compared with the traditional consensus, our proposed consensus improves the reliability and the system performance of the blockchain-enabled cyber-physical systems.

Appendix

1.1 Proof of Theorem 1

Considering three credit evaluations, assume $\eta _i$ the hypothetical part $\eta _i^+$ is given a positive credit evaluation, $\eta _i$ the other part $\eta _i^0$ is given a moderate credit evaluation, and $\eta _i$ the other part $\eta _i^-$ is given a negative credit evaluation. Further, assume that the distribution of credit assessments is random, with $\{\eta _i^+\}_{i=1}^N\sim N(\eta _i^+,{\sigma ^+}^2)$,$\{\eta _i^0\}_{i=1}^N\sim N(\eta _i^0,{\sigma ^0}^2)$,$\{\eta _i^-\}_{i=1}^N\sim N(\eta _i-,{\sigma ^-}^2)$. Then, the expectation of $C_i$ reference ([3]) can be given by

$$\begin{aligned} \bar{C_i}=K(\eta _i^+C^+ +\eta _i^+C^0 +\eta _i^+C^-). \end{aligned}$$

(13)

The distribution of $\{\bar{C_i}\}_{i=1}^N$ can be obtained by $\{\bar{C_i}\}_{i=1}^N\sim N(\mu ,\sigma ^2)$, where $\mu$ and $\sigma ^2$ are defined by Eq. (7) and Eq. (8), respectively. Using $\bar{C_i}$ in Eq. (13) instead of $C_i$ in Eq. (6), the normal probability distribution function can be derived by Eq. (6). Finally, the credit threshold $\varepsilon$ can be solved by Eq. (6).