Hierarchical Variable Structure Control for the Path Following and Formation Maintenance of Multi-agent Systems

Abstract

In this paper, a hierarchical variable structure control (HVSC) is proposed for path following and formation maintenance of multi-agent systems (MAS) based on the one leader and one follower (1L-1F) configuration. In terms of the leader, the path following of a nonholonomic mobile robot (NMR) can be regarded as tracking of a virtual reference NMR. Then, a feedback control law is used to attain path following based on kinematics of the NMR. Subsequently, a variable structure controller (VSC) with known upper bound of the disturbances is designed to achieve velocity control based on the dynamics of the NMR. Furthermore, as far as the formation maintenance is concerned, a method of feedback linearization is utilized to exponentially stabilize the relative distance and orientation between the leader and follower. Similarly, a VSC is also designed for complete velocity control such that the aforementioned formation is maintained. The proposed method with hierarchical structure simultaneously attains the path following of the leader and maintains the formation for the MAS. In addition, the robustness of the proposed scheme is guaranteed in spite of persistent disturbances and the stability analysis of the closed-loop system is proved via Lyapunov stability criteria. Finally, computer simulations are conducted to validate the feasibility and effectiveness of the proposed control scheme.

Appendix: (The proof of Theorem 1)

The Lyapunov function $V={\sigma _{n}^{T}} \sigma _{n} /2>0$ as ∥σ_n∥ ≠ 0 is defined. Taking its time derivative with substitution of Eqs. 8 and 23 with $k_{f_{n} } >\max \| \bar {{M}}_{n}^{-1} \bar {{\tau }}_{d_{n} } \|$ yields the following result:

$$\begin{array}{@{}rcl@{}} \dot{{V}}&=&{\sigma_{n}^{T}} \dot{{\sigma }}_{n} \\ &=&{\sigma_{n}^{T}} [C_{n} (\dot{{V}}_{d_{n} } -\dot{{V}}_{n} )] \\ &=&{\sigma_{n}^{T}} \{C_{n} [\dot{{V}}_{d_{n} } -\bar{{M}}_{n}^{-1} (\bar{{B}}_{n} \tau_{n} -\bar{{H}}_{n} V_{n} -\bar{{G}}_{n} -\bar{{\tau }}_{d_{n} } )]\} \\ &=&-\lambda_{n} \| \sigma_{n} \|^{2}-(\varepsilon_{n} +k_{f_{n} } )\| \sigma_{n} \|^{2}/(\| \sigma_{n} \| +\sigma_{n_{b} } )+{\sigma_{n}^{T}} \bar{{M}}_{n}^{-1} \bar{{\tau }}_{d_{n} } \\ &\le& -\lambda_{n} \| \sigma_{n} \|^{2}-(\varepsilon_{n} +k_{f_{n} } )\| \sigma_{n} \|^{2}/(\| \sigma_{n} \| +\sigma_{n_{b} })\\ &&+\| \bar{{M}}_{n}^{-1} \bar{{\tau }}_{d_{n} } \| \,\| \sigma_{n} \| \\ &=&\frac{\| \sigma_{n} \| }{(\| \sigma_{n} \| +\sigma_{n_{b} } )}[-\lambda_{n} \| \sigma_{n} \| (\| \sigma_{n} \| +\sigma_{n_{b} } )-(\varepsilon_{n} +k_{f_{n} } )\| \sigma_{n} \|\\ &&+\| \bar{{M}}_{n}^{-1} \bar{{\tau }}_{d_{n} } \| (\| \sigma_{n} \| +\sigma_{n_{b} } )] \\ &\le& \frac{-\| \sigma_{n} \| }{(\| \sigma_{n} \| +\sigma_{n_{b} } )}[\lambda_{n} \| \sigma_{n} \|^{2}+(\lambda_{n} \sigma_{n_{b} } +\varepsilon_{n} )\| \sigma_{n} \| -k_{f_{n} } \sigma_{n_{b} } ] \\ &=&\frac{-\| \sigma_{n} \| }{(\| \sigma_{n} \| +\sigma_{n_{b} } )}v_{\sigma } (\| \sigma_{n} \| ) \end{array} $$

where v_σ(∥σ_n∥) = v_σ2∥σ_n∥² + v_σ1∥σ_n∥− v_σ0, with v_σ2, v_σ1 and v_σ0 described in Eq. 23c. As ∥σ_n∥ > s_σ, the inequality v_σ(∥σ_n∥) ≥ 0 is satisfied. Thus, outside of the domain D_σ in Eq. 23 making $\dot {{V}}\le 0$ is obtained. According to the Lyapunov stability criteria, the sliding surface σ_n exponentially converges into the domain D_σ. Then, once the operating point reaches the stable sliding surface $\sigma _{n} ,E_{V_{n} } \to 0$ as $t\to \infty$.