Practical Bipartite Consensus for Networked Lagrangian Systems in Cooperation-Competition Networks

Abstract

This paper gains an insight into the practical bipartite consensus problem of networked Lagrangian systems (NLSs) in the context of dynamics. The distributed leaderless and leader-following practical bipartite consensus control laws are proposed in cooperation-competition networks with uncertain parameters, effectively extending the theoretical framework in cooperation networks for NLSs. In the leaderless case, the symmetric regions are newly established to facilitate the final convergence analysis. By employing the structurally balanced networked topologies, a novel symmetric positive definite matrix is exploited to derive the gain selection criteria in the leader-following case. Finally, numerical simulations of a network of revolute joint manipulators are illustrated to verify the theoretical results under positive-negative and all-negative topologies.

Appendix

Proof of Lemma 2

Proof

Without loss of generality, one only needs to verify that the i th follower applies. According to Eq. 19, the conclusion can be obtained if the following formula holds

$$ \begin{array}{cc}[(-|a_{i1}|\phi_{i}+a_{i1}\phi_{1})+(-|a_{i2}|\phi_{i}+a_{i2}\phi_{2})+\cdots \\ \cdots+(-|a_{in}|\phi_{i}+a_{in}\phi_{n})-b_{i}\phi_{i}]d+b_{i}\phi_{i}d=0. \end{array} $$

(23)

In fact, one only needs to demonstrate that when the followers i and j are in the positive-negative or all-negative structurally balanced network, −|a_ij|ϕ_i + a_ijϕ_j = 0 arises, i,j ∈{1, 2,⋯ ,n}. For simplicity, the all-negative case is considered here. According to the results of [27], if i and j are in the same subset $\mathcal V_{l}, l=1, 2$, implying that a_ij = 0, thus Eq. 1 holds. If i and j are in the different subsets, ϕ_i = −ϕ_j. Consequently, a_ij ≤ 0, resulting in −|a_ij| = a_ij. Accordingly, −|a_ij|ϕ_i = −a_ijϕ_j, so that Eq. 1 still holds. Therefore, Lemma 2 is proven. □

Proof of Lemma 3

Proof

It follows from the structures of L and Φ that ΦLΦ is a traditional Laplacian matrix. Then $\widehat H= ({\varPhi }\widehat {P}+\zeta I_{n})[({{ {\varPhi } L{\varPhi }}})^{\mathrm {T}}+({{ {\varPhi } L{\varPhi }}})]+D$ is a symmetric matrix, where D is a nonnegative diagonal matrix with at least one positive element due to $\gamma >\max \limits ^{n}_{i=1}\{|b_{i}|\}$. As the results of Ref. [51], combining Assumptions 3 with 4, it stands to reason that $\widehat H$ is symmetric positive definite. □

Fig. 18

Velocity evolution of the eight two-link revolute joint manipulators under $\widehat {\mathcal {G}}_{1}$

Fig. 19

Position error evolution of the eight two-link revolute joint manipulators under $\widehat {\mathcal {G}}_{1}$

Fig. 20

Velocity error evolution of the eight two-link revolute joint manipulators under $\widehat {\mathcal {G}}_{1}$

Fig. 21

Position error evolution between the virtual leader and the eight two-link revolute joint manipulators under $\widehat {\mathcal {G}}_{1}$ (first coordinate)

Fig. 22

Position error evolution between the virtual leader and the eight two-link revolute joint manipulators under $\widehat {\mathcal {G}}_{1}$ (Second coordinate)

Fig. 23

Communication topology graph $\widehat {\mathcal {G}}_{2}$ of the all-negative network with a virtual leader

Fig. 24

Position evolution of the eight two-link revolute joint manipulators under $\widehat {\mathcal {G}}_{2}$

Fig. 25

Velocity evolution of the eight two-link revolute joint manipulators under $\widehat {\mathcal {G}}_{2}$

Fig. 26

Position error evolution of the eight two-link revolute joint manipulators under $\widehat {\mathcal {G}}_{2}$

Fig. 27

Velocity error evolution of the eight two-link revolute joint manipulators under $\widehat {\mathcal {G}}_{2}$

Fig. 28

Position error evolution between the virtual leader and the eight two-link revolute joint manipulators under $\widehat {\mathcal {G}}_{2}$ (first coordinate)

Fig. 12

Position error evolution between the virtual leader and the eight two-link revolute joint manipulators under $\widehat {\mathcal {G}}_{2}$ (Second coordinate)