Distributed cooperative adaptive tracking control for heterogeneous systems with hybrid nonlinear dynamics

Abstract

The cooperative leader-following tracking for a group of heterogeneous mechanical systems with nonlinear hybrid order dynamics is studied. The controlled systems are considered to be composed of followers (agents) with hybrid first- and second-order time-varying dynamics. The leader is an unknown nonautonomous nonlinear system and can only give the state information of position and velocity to its neighboring followers. The followers are linked by the directed graph with fixed communication topology. And, not all of them have the information path to the leader directly. The directed information topology graph is required to have at least one spanning tree for position and velocity, respectively. Distributed cooperative adaptive control protocols are developed for all followers with first- or second-order dynamics to achieve the ultimate synchronization to the leader. The control protocols are designed based on the neural networks and the adaptive estimation algorithm for unknown time-varying functions and control coefficients. The convergence and boundedness of the synchronization error is proved by the Lyapunov theory. The simulation example verifies the correctness of the developed distributed control protocols.

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References

  1. 1.

    Lewis, F.L., Zhang, H., Hengster-Movric, K., et al.: Cooperative Control of Multi-agent Systems: Optimal and Adaptive Design Approaches. Springer, Berlin (2013)

    Google Scholar 

  2. 2.

    Ren, W., Beard, R.W., Atkins, E.M.: A survey of consensus problems in multi-agent coordination. In: Proceedings of the American Control Conference, pp. 1859–1864 (2005)

  3. 3.

    Zhang, H., Lewis, F.L.: Adaptive cooperative tracking control of higher-order nonlinear systems with unknown dynamics. Automatica 48(7), 1432–1439 (2012)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Ren, W., Beard, R.W., Atkins, E.M.: Information consensus in multivehicle cooperative control. IEEE Trans. Control Syst. Technol. 27(2), 71–82 (2007)

    Article  Google Scholar 

  5. 5.

    Song, J.: Observer-based consensus control for networked multi-agent systems with delays and packet-dropouts. Int. J. Innov. Comput. Inf. Control 12(4), 1287–1302 (2016)

    Google Scholar 

  6. 6.

    Wang, W., Yu, Y.: Fuzzy adaptive consensus of second-order nonlinear multi-agent systems in the presence of input saturation. Int. J. Innov. Comput. Inf. Control 12(2), 533–542 (2016)

    Google Scholar 

  7. 7.

    Shen, Q., Shi, P.: Output consensus control of multiagent systems with unknown nonlinear dead zone. IEEE Trans. Syst. Man Cybern. Syst. 46(10), 1329–1337 (2016)

    Article  Google Scholar 

  8. 8.

    Lee, T.H., Wu, Z.G., Park, J.H.: Synchronization of a complex dynamical network with coupling time-varying delays via sampled-data control. Appl. Math. Comput. 219(3), 1354–1366 (2012)

    MathSciNet  MATH  Google Scholar 

  9. 9.

    Shen, Q., Shi, P.: Distributed command filtered backstepping consensus tracking control of nonlinear multiple-agent systems in strict-feedback form. Automatica 53, 120–124 (2015)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Lee, T.H., Park, J.H.: Improved criteria for sampled-data synchronization of chaotic Lur’e systems using two new approaches. Nonlinear Anal. Hybrid Syst. 24, 132–145 (2017)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Meng, Z., Ren, W., Cao, Y., et al.: Leaderless and leader-following consensus with communication and input delays under a directed network topology. IEEE Trans. Syst. Man Cybern. Part B Cybern. 41(1), 75–88 (2011)

    Article  Google Scholar 

  12. 12.

    Shen, Q., Jiang, B., Shi, P., et al.: Cooperative adaptive fuzzy tracking control for networked unknown nonlinear multiagent systems with time-varying actuator faults. IEEE Trans. Fuzzy Syst. 22(3), 494–504 (2014)

    Article  Google Scholar 

  13. 13.

    Shi, P., Shen, Q.: Cooperative control of multi-agent systems with unknown state-dependent controlling effects. IEEE Trans. Autom. Sci. Eng. 12(3), 827–834 (2015)

    Article  Google Scholar 

  14. 14.

    Cao, Y., Yu, W., Ren, W., et al.: An overview of recent progress in the study of distributed multi-agent coordination. IEEE Trans. Ind. Inform. 9(1), 427–438 (2013)

    Article  Google Scholar 

  15. 15.

    Ren, W., Beard, R.W.: Distributed Consensus in Multi-vehicle Cooperative Control. Springer, London (2008)

    Google Scholar 

  16. 16.

    Ren, W., Cao, Y.: Distributed Coordination of Multi-agent Networks: Emergent Problems, Models, and Issues. Springer, Berlin (2010)

    Google Scholar 

  17. 17.

    Qu, Z.: Cooperative Control of Dynamical Systems: Applications to Autonomous Vehicles. Springer, Berlin (2009)

    Google Scholar 

  18. 18.

    Zheng, Y., Zhu, Y., Wang, L.: Consensus of heterogeneous multi-agent systems. IET Control Theory Appl. 5(16), 1881–1888 (2011)

    MathSciNet  Article  Google Scholar 

  19. 19.

    Liu, C.L., Liu, F.: Stationary consensus of heterogeneous multi-agent systems with bounded communication delays. Automatica 47(9), 2130–2133 (2011)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Zheng, Y., Wang, L.: Consensus of heterogeneous multi-agent systems without velocity measurements. Int. J. Control 85(7), 906–914 (2012)

    MathSciNet  Article  Google Scholar 

  21. 21.

    Liu, C.L., Liu, F.: Dynamical consensus seeking of heterogeneous multic-systems under input delays. Int. J. Commun. Syst. 26(10), 1243–1258 (2013)

    Google Scholar 

  22. 22.

    Feng, Y., Xu, S., Lewis, F.L., et al.: Consensus of heterogeneous first-second-order multic-systems with directed communication topologies. Int. J. Robust Nonlinear Control 25(3), 362–375 (2015)

    Article  Google Scholar 

  23. 23.

    Zheng, Y., Wang, L.: Finite-time consensus of heterogeneous multi-agent systems with and without velocity measurements. Syst. Control Lett. 61(8), 871–878 (2012)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Das, A., Lewis, F.L.: Distributed adaptive control for synchronization of unknown nonlinear networked systems. Automatica 46(12), 2014–2021 (2010)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Das, A., Lewis, F.L.: Cooperative adaptive control for synchronization of second-order systems with unknown nonlinearities. Int. J. Robust Nonlinear Control 21(13), 1509–1524 (2011)

    MathSciNet  Article  Google Scholar 

  26. 26.

    Lewis, F.W., Jagannathan, S., Yesildirak, A.: Neural Network Control of Robot Manipulators and Non-linear Systems. CRC Press, Boca Raton (1998)

    Google Scholar 

  27. 27.

    Khoo, S., Xie, L., Man, Z.: Robust finite-time consensus tracking algorithm for multirobot systems. IEEE/ASME Trans. Mechatron. 14(2), 219–228 (2009)

    Article  Google Scholar 

  28. 28.

    Shivakumar, P.N., Chew, K.H.: A sufficient condition for nonvanishing of determinants. In: Proceedings of the American Mathematical Society, pp. 63–66 (1974)

  29. 29.

    Stone, M.H.: The generalized Weierstrass approximation theorem. Math. Mag. 21(5), 237–254 (1948)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Khalil, H.K.: Nonlinear Systems, vol. 9(4.2), 3rd edn. Prentice Hall, New Jersey (2002)

    Google Scholar 

  31. 31.

    Ge, S.S., Wang, C.: Adaptive neural control of uncertain MIMO nonlinear systems. IEEE Trans. Neural Netw. 15(3), 674–692 (2004)

    Article  Google Scholar 

Download references

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Appendix: Proof of Theorem 1

Appendix: Proof of Theorem 1

Define the Lyapunov function as

$$\begin{aligned} V=V_r +V_W +V_g +V_e \end{aligned}$$
(27)

where \(V_r =\frac{1}{2}r^\mathrm{T}Pr\), \(V_W =\frac{1}{2}{\tilde{W}}^\mathrm{T}F^{-1}{\tilde{W}}\), \(V_g =\frac{1}{2}{\tilde{g}}^\mathrm{T}\eta ^{-1}{\tilde{g}}\), \(V_e =\frac{1}{2}\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_p^{l_2 } \).

According to (26), by differentiating \(V_r \), we have

$$\begin{aligned} {\dot{V}}_r= & {} r^\mathrm{T}P{\dot{r}}=r_{n1}^T P^{l_1 }{\dot{r}}_{n1} +r_{n2}^T P^{l_2 }{\dot{r}}_{n2} \nonumber \\= & {} -r_{n1}^T P^{l_1 }(L+B)\left( \left( {{\tilde{W}}^{l_1 }} \right) ^\mathrm{T}\phi ^{_{l_1 } }+\varepsilon ^{l_1 }+{\tilde{g}}^{l_1 }u^{l_1 }\right. \nonumber \\&\left. +\,cr_{n1} +w^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} \right) \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left( \left( {{\tilde{W}}^{l_2 }} \right) ^\mathrm{T}\phi ^{_{l_2 } }+\varepsilon ^{l_2 }+cr_{n2} \right. \nonumber \\&\left. +\,{\tilde{g}}^{l_2 }u^{l_2 }+w^{l_2 }-{\underline{1}}^{l_2 }f_0 \right) \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v \nonumber \\= & {} -r^\mathrm{T}P(L+B)\left[ {{\tilde{W}}^\mathrm{T}\phi +cr+{\tilde{g}}u} \right] \nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v \nonumber \\= & {} -r^\mathrm{T}P(D+B){\tilde{W}}^\mathrm{T}\phi +r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi \nonumber \\&-\,r^\mathrm{T}P(L+B)cr-r^\mathrm{T}P(D+B){\tilde{g}}u \nonumber \\&+r^\mathrm{T}PA{\tilde{g}}u \nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v \end{aligned}$$
(28)

Considering the fact that \(x^\mathrm{T}y=\mathrm{tr}\left\{ {yx^\mathrm{T}} \right\} \), we have

$$\begin{aligned} {\dot{V}}_r= & {} -\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\phi r^\mathrm{T}P(D+B)} \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi -r^\mathrm{T}P(L+B)cr \nonumber \\&-\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}ur^{T}P(D+B)} \right\} +r^\mathrm{T}PA{\tilde{g}}u\nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v \end{aligned}$$
(29)

Differentiating \(V_W \), \(V_g \) and \(V_e \), we have

$$\begin{aligned} {\dot{V}}_W +{\dot{V}}_g +{\dot{V}}_e= & {} \mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}F^{-1}\dot{{\tilde{W}}}} \right\} \nonumber \\&+\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}\eta ^{-1}\dot{{\tilde{g}}}} \right\} +\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \nonumber \\= & {} -\,\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}F^{-1}\dot{\hat{{W}}}} \right\} -tr\left\{ {{\tilde{g}}^\mathrm{T}\eta ^{-1}\dot{\hat{{g}}}} \right\} \nonumber \\&+\,\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \end{aligned}$$
(30)

Combining (29) and (30), we have

$$\begin{aligned} {\dot{V}}= & {} {\dot{V}}_r +{\dot{V}}_W +{\dot{V}}_g +{\dot{V}}_e \nonumber \\= & {} -\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\phi r^\mathrm{T}P(D+B)} \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi -r^\mathrm{T}P(L+B)cr \nonumber \\&-\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}ur^{T}P(D+B)} \right\} +r^\mathrm{T}PA{\tilde{g}}u\nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v \nonumber \\&-\,\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}F^{-1}\dot{\hat{{W}}}} \right\} -tr\left\{ {{\tilde{g}}^\mathrm{T}\eta ^{-1}\dot{\hat{{g}}}} \right\} +\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \nonumber \\= & {} -\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\left[ {F^{-1}\dot{\hat{{W}}}+\phi r^\mathrm{T}P(D+B)} \right] } \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi -r^\mathrm{T}P(L+B)cr \nonumber \\&-\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}\left[ {\eta ^{-1}\dot{\hat{{g}}}+ur^\mathrm{T}P(D+B)} \right] } \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{g}}u-r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v +\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \end{aligned}$$
(31)

The error \({\tilde{g}}_i =g_i -{\underline{g}}_i \ge 0\) if \(\hat{{g}}_i ={\underline{g}}_i \). Then, substituting (24) and (25) into (31), we have

$$\begin{aligned} {\dot{V}}= & {} \mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\kappa \hat{{W}}} \right\} +r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi -r^\mathrm{T}P(L+B)cr \nonumber \\&+\,r^\mathrm{T}PA{\tilde{g}}u-r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v +\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \nonumber \\&+\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}\theta \hat{{g}}} \right\} \nonumber \\&-\left\{ {\begin{array}{ll} 0,&{}\quad \hbox {if }\hat{{g}}_i >{\underline{g}}_i \\ 0,&{}\quad \hbox {if }\hat{{g}}_i ={\underline{g}}_i \hbox { and }u_i r_i p_i (d_i +b_i )<0 \\ \mathrm{tr}\left\{ {\hbox {diag}\left\{ {{\tilde{g}}_i u_i r_i p_i (d_i +b_i )} \right\} } \right\} ,&{}\hbox { if }\hat{{g}}_i ={\underline{g}}_i \hbox { and }u_i r_i p_i (d_i +b_i )\ge 0 \\ \end{array}} \right. \nonumber \\\le & {} \mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\kappa \hat{{W}}} \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi -r^\mathrm{T}P(L+B)cr \nonumber \\&+\,\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}\theta \hat{{g}}} \right\} +r^\mathrm{T}PA{\tilde{g}}u\nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda e_v +\left( {e_p^{l_2 } } \right) ^\mathrm{T}e_v \ \end{aligned}$$
(32)

Then, according to Lemma 2, we have

$$\begin{aligned} {\dot{V}}\le & {} -\frac{1}{2}cr^\mathrm{T}Qr+\mathrm{tr}\left\{ {{\tilde{W}}^\mathrm{T}\kappa (W-{\tilde{W}})} \right\} \nonumber \\&+\,r^\mathrm{T}PA{\tilde{W}}^\mathrm{T}\phi +\mathrm{tr}\left\{ {{\tilde{g}}^\mathrm{T}\theta (g-{\tilde{g}})} \right\} +r^\mathrm{T}PA{\tilde{g}}u \nonumber \\&-\,r_{n1}^T P^{l_1 }(L+B)\left[ {w^{l_1 }+\varepsilon ^{l_1 }-{\underline{1}}^{l_1 }x_{v,0} } \right] \nonumber \\&-\,r_{n2}^T P^{l_2 }(L+B)\left[ {w^{l_2 }+\varepsilon ^{l_2 }-{\underline{1}}^{l_2 }f_0 } \right] \nonumber \\&+\,\left[ {\left( {e_p^{l_2 } } \right) ^\mathrm{T}+r_{n2}^T P^{l_2 }A\left( {D+B} \right) ^{-1}\Lambda } \right] \nonumber \\&\left( {r_{n2} -\Lambda e_p^{l_2 } } \right) \end{aligned}$$
(33)

Taking norm on (33), we have

$$\begin{aligned} {\dot{V}}\le & {} -\frac{1}{2}c{\underline{\sigma } }\left( Q \right) \left\| r \right\| ^{2}+\kappa W_M \left\| {{\tilde{W}}} \right\| _F -\kappa \left\| {{\tilde{W}}} \right\| _F^2 \nonumber \\&+\,\phi _M \bar{{\sigma }}\left( P \right) \bar{{\sigma }}\left( A \right) \left\| r \right\| \left\| {{\tilde{W}}} \right\| _F \nonumber \\&+\,\theta g_M \left\| {{\tilde{g}}} \right\| _F -\theta \left\| {{\tilde{g}}} \right\| _F^2 +u_M \bar{{\sigma }}\left( P \right) \bar{{\sigma }}\left( A \right) \left\| r \right\| \left\| {{\tilde{g}}} \right\| _F \nonumber \\&+\,\bar{{\sigma }}\left( P \right) \bar{{\sigma }}(L+B)B_M \left\| r \right\| -{\underline{\sigma } }\left( \Lambda \right) \left\| {e_p^{l_2 } } \right\| ^{2} \nonumber \\&+\,\frac{\bar{{\sigma }}\left( \Lambda \right) \bar{{\sigma }}\left( {P^{l_2 }} \right) \bar{{\sigma }}\left( A \right) }{{\underline{\sigma } }\left( {D+B} \right) }\left\| r \right\| ^{2}\nonumber \\&+\,\left[ {1+\frac{\bar{{\sigma }}\left( {\Lambda ^{2}} \right) \bar{{\sigma }}\left( {P^{l_2 }} \right) \bar{{\sigma }}\left( A \right) }{{\underline{\sigma } }\left( {D+B} \right) }} \right] \left\| {e_p^{l_2 } } \right\| \left\| r \right\| \end{aligned}$$
(34)

where \(B_M =\varepsilon _M +w_M +f_M +x_{0M} \).

Write (34) as

$$\begin{aligned} {\dot{V}}\le -z^\mathrm{T}Sz+K^\mathrm{T}z=-V_z (z) \end{aligned}$$
(35)

where \(z=\left[ {{\begin{array}{llll} {\left\| r \right\| }&{}\quad {\left\| {{\tilde{W}}} \right\| _F }&{}\quad {\left\| {e_p^{l_2 } } \right\| }&{}\quad {\left\| {{\tilde{g}}} \right\| _F } \\ \end{array} }} \right] ^\mathrm{T}\), \(K=\left[ {{\begin{array}{llll} {\bar{{\sigma }}\left( P \right) \bar{{\sigma }}(L+B)B_M }&{}\quad {\kappa W_M }&{}\quad 0&{}\quad {\theta g_M } \\ \end{array} }} \right] ^\mathrm{T}\),

$$\begin{aligned} S=\left[ {{\begin{array}{llll} {\frac{1}{2}c{\underline{\sigma } }\left( Q \right) -\psi }&{} \varsigma &{} \vartheta &{} \chi \\ \varsigma &{} \kappa &{} 0&{} 0 \\ \vartheta &{} 0&{} {{\underline{\sigma } }\left( \Lambda \right) }&{} 0 \\ \chi &{} 0&{} 0&{} \theta \\ \end{array} }} \right] . \end{aligned}$$

Then, \(V_z (z)\) is positive definite if \(S>0\) and \(\left\| z \right\| >\frac{\left\| K \right\| }{{\underline{\sigma } }(S)}\).

With Sylvester’s criterion, \(S>0\) if

$$\begin{aligned}&\frac{1}{2}c{\underline{\sigma } }\left( Q \right) -\psi>0 \nonumber \\&\quad \kappa \left( {\frac{1}{2}c{\sigma }\left( Q \right) -\psi } \right) -\varsigma ^{2}>0 \nonumber \\&\quad {\underline{\sigma } }\left( \Lambda \right) \left[ {\kappa \left( {\frac{1}{2}c{\underline{\sigma } }\left( Q \right) -\psi } \right) -\varsigma ^{2}} \right] -\kappa \vartheta ^{2}>0 \nonumber \\&\quad \theta \left\{ {{\underline{\sigma } }\left( \Lambda \right) \left[ {\kappa \left( {\frac{1}{2}c{\underline{\sigma } }\left( Q \right) -\psi } \right) -\varsigma ^{2}} \right] -\kappa \vartheta ^{2}} \right\} \nonumber \\&\qquad -\kappa \chi ^{2}{\underline{\sigma } }\left( \Lambda \right) >0 \end{aligned}$$
(36)

Solve the above inequalities, we can get the condition (22).

Define \(B_d \) as

$$\begin{aligned} B_d =\frac{\left\| K \right\| _1 }{{\underline{\sigma } }(S)}=\frac{B_M \bar{{\sigma }}(P)\bar{{\sigma }}(L+B)+\kappa W_M +\theta g_M }{{\underline{\sigma } }(S)} \nonumber \\ \end{aligned}$$
(37)

Then, if \(z\ge B_d \), \(\left\| z \right\|>\frac{\left\| K \right\| _1 }{{\underline{\sigma }}(S)}>\frac{\left\| K \right\| }{{\underline{\sigma }}(S)}\) holds. Under condition (22), we have \({\dot{V}}\le -V_z (z)\) with \(V_z (z)\) being positive definite.

By [3], we have

$$\begin{aligned} {\underline{\sigma }}(\Gamma )\left\| z \right\| ^{2}\le V\le \bar{{\sigma }}(\varphi )\left\| z \right\| ^{2} \end{aligned}$$
(38)

where \(\Gamma =\hbox {diag}\left( {\frac{{\underline{\sigma } }(P)}{2},\hbox { }\frac{1}{2\bar{{\sigma }}(F)},\frac{1}{2}\hbox { },\frac{1}{2\bar{{\sigma }}(\eta )}} \right) \) and \(\varphi =\hbox {diag}\left( {\frac{\bar{{\sigma }}(P)}{2},\hbox { }\frac{1}{2{\sigma }(F)},\frac{1}{2},\hbox { }\frac{1}{2{\sigma }(\eta )}} \right) \).

According to Theorem 4.18 in [30], we can conclude that for any \(z(t_0 )\) there exists a \(T_0 \) such that

$$\begin{aligned} \left\| {z(t)} \right\| \le \sqrt{\frac{\bar{{\sigma }}(\varphi )}{{\underline{\sigma } }(\Gamma )}}B_d ,\forall t\ge t_0 +T_0 \end{aligned}$$
(39)

Define \(d=\min _{\left\| z \right\| \ge B_d } V_z (z)\), then according to [30]

$$\begin{aligned} T_0 =\frac{V(t_0 )-\bar{{\underline{\sigma } }}(\phi )B_d^2 }{d} \end{aligned}$$
(40)

With z, (39) implies that the synchronization error r is ultimately bounded. Then, according to Lemma 1, \(\delta _p \) and \(\delta _v \) are CUUB and all nodes in \({{\mathcal {G}}}\) achieve the synchronization to the leader.

For part (2) of Theorem 1, the state \(x_{p,i} ,i\in N\) and\(_{ }x_{v,i} ,i\in l_2 \) are bounded \(\forall t\ge t_0 \). From (35), we have

$$\begin{aligned} {\dot{V}}\le -{\sigma }(S)\left\| z \right\| ^{2}+\left\| K \right\| \left\| z \right\| \end{aligned}$$
(41)

According to (38) and (41), we have

$$\begin{aligned} \frac{d}{dt}(\sqrt{V})\le -\frac{{\underline{\sigma } }(S)}{2\bar{{\sigma }}(\varphi )}\sqrt{V}+\frac{\left\| K \right\| }{2\sqrt{{\underline{\sigma } }(\Gamma )}} \end{aligned}$$
(42)

Then, we can conclude that V(t) is bounded for \(t\ge t_0 \) under Corollary 1.1 in [31]. Since (27) implies \(\left\| r \right\| ^{2}\le \frac{2V(t)}{{\underline{\sigma } }(P)}\), r(t) is bounded. And, \(x_0 \) is bounded by \(x_{0M} \) in Assumption 1. \(x_{p,i} \hbox { },i\in N\) and\(_{ }x_{v,i} ,i\in l_2 \) are bounded \(\forall t\ge t_0 \). The proof is done. \(\square \)

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Li, X., Shi, P. & Wang, Y. Distributed cooperative adaptive tracking control for heterogeneous systems with hybrid nonlinear dynamics. Nonlinear Dyn 95, 2131–2141 (2019). https://doi.org/10.1007/s11071-018-4681-4

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Keywords

  • Control synchronization
  • Consensus tracking
  • Neural networks
  • Heterogeneous multi-agent systems
  • Unknown nonlinear dynamics