Output-Constrained Control of Non-affine Multi-agent Systems with Actuator Faults and Unknown Dead Zones


This paper presents the output-constrained control algorithm for non-affine multi-agent systems (MASs) with actuator faults and unknown dead zones. The error transformation method is employed to keep initial connectivity patterns in the non-affine MASs for consensus tracking control. The radial basis function neural networks are utilized to estimate the unknown nonlinear functions. Furthermore, the Nussbaum function is used to overcome partially unknown control direction problem. To address the problem of the constrained control, a state transformation technique is presented. In addition, the fault-tolerant consensus tracking protocol is designed to reduce the effects of actuator faults and dead zones. Furthermore, it is shown that the consensus tracking errors are cooperatively semi-globally uniformly ultimately bounded. Finally, the effectiveness of the proposed approach is illustrated by some simulation results.

This is a preview of subscription content, log in to check access.

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6


  1. 1.

    W. Chen, X. Li, R. Wei, C. Wen, Adaptive consensus of multi-agent systems with unknown identical control directions based on a novel Nussbaum-type function. IEEE Trans. Autom. Control 59(7), 1887–1892 (2014)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Z. Chen, Q. Han, Y. Yam, Z. Wu, How often should one update control and estimation: review of networked triggering techniques. Inf. Sci. 63(150201), 1–18 (2020)

    MathSciNet  Google Scholar 

  3. 3.

    Y. Dan, G. Yang, Adaptive fault-tolerant tracking control against actuator faults with application to flight control. IEEE Trans. Control Syst. Technol. 14(6), 1088–1096 (2006)

    Article  Google Scholar 

  4. 4.

    H. Du, H. Shao, P. Yao, Adaptive neural network control for a class of low-triangular-structured nonlinear systems. IEEE Trans. Neural Control. 17(2), 509–514 (2006)

    Article  Google Scholar 

  5. 5.

    P. Du, Y. Pan, H. Li, H. Lam, Nonsingular finite-time event-triggered fuzzy control for large-scale nonlinear systems. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.2992632

    Article  Google Scholar 

  6. 6.

    B. Fan, Q. Yang, S. Jagannathan, Y. Sun, Output-constrained control of non-affine multi-agent systems with partially unknown control directions. IEEE Trans. Autom. Control. (2019). https://doi.org/10.1109/TAC.2019.2892391

    Article  MATH  Google Scholar 

  7. 7.

    Y. Gao, S. Tong, Y. Li, Fuzzy adaptive output feedback dsc design for siso nonlinear stochastic systems with unknown control directions and dead-zones. Neurocomputing 167, 187–194 (2015)

    Article  Google Scholar 

  8. 8.

    M. Hu, J. Cao, A. Hu, Y. Yang, Y. Jin, A novel finite-time stability criterion for linear discrete-time stochastic system with applications to consensus of multi-agent system. Circuits Syst. Signal Process. 34(1), 41–59 (2015)

    Article  Google Scholar 

  9. 9.

    Y. Jiang, J. Liu, S. Wang, Consensus tracking algorithm via observer-based distributed output feedback for multi-agent systems under switching topology. Circuits Syst. Signal Process. 33(10), 3037–3052 (2014)

    MathSciNet  Article  Google Scholar 

  10. 10.

    J. Leng, H. Zhang, D. Yan, Q. Liu, X. Chen, D. Zhang, Digital twin-driven manufacturing cyber-physical system for parallel controlling of smart workshop. J. Ambient. Intell. Humaniz. Comput. 10(3), 1155–1166 (2019)

    Article  Google Scholar 

  11. 11.

    T. Li, M. Fu, L. Xie, J. Zhang, Distributed consensus with limited communication data rate. IEEE Trans. Autom. Control 56(2), 279–292 (2011)

    MathSciNet  Article  Google Scholar 

  12. 12.

    T. Li, Z. Li, D. Wang, C.L.P. Chen, Output-feedback adaptive neural control for stochastic nonlinear time-varying delay systems with unknown control directions. IEEE Trans. Neural Netw. Learn. Syst. 26(6), 1188–1201 (2015)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Y. Li, Z. Ma, S. Tong, Adaptive fuzzy fault-tolerant control of nontriangular structure nonlinear systems with error constraint. IEEE Trans. Fuzzy Syst. 26(4), 2062–2074 (2017)

    Article  Google Scholar 

  14. 14.

    Y. Li, X. Shao, S. Tong, Adaptive fuzzy prescribed performance control of nontriangular structure nonlinear systems. IEEE Trans. Fuzzy Syst. (2019). https://doi.org/10.1109/TFUZZ.2019.2937046

    Article  Google Scholar 

  15. 15.

    Y. Li, K. Sun, S. Tong, Observer-based adaptive fuzzy fault-tolerant optimal control for siso nonlinear systems. IEEE Trans. Cybern. 49(2), 649–661 (2018)

    Article  Google Scholar 

  16. 16.

    Y. Li, S. Tong, T. Li, Observer-based adaptive fuzzy tracking control of mimo stochastic nonlinear systems with unknown control directions and unknown dead zones. IEEE Trans. Fuzzy Syst. 23(4), 1228–1241 (2015)

    Article  Google Scholar 

  17. 17.

    Y. Li, G. Yang, Adaptive fuzzy decentralized control for a class of large-scale nonlinear systems with actuator faults and unknown dead zones. IEEE Trans. Syst. Man Cybern. Syst. 47(5), 729–740 (2017)

    Article  Google Scholar 

  18. 18.

    Z. Li, X. Liu, P. Lin, W. Ren, Consensus of linear multi-agent systems with reduced-order observer-based protocols. Syst. Control Lett. 60(7), 510–516 (2011)

    MathSciNet  Article  Google Scholar 

  19. 19.

    H. Liang, X. Guo, Y. Pan, T. Huang, Event-triggered fuzzy bipartite tracking control for network systems based on distributed reduced-order observers. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.2982618

    Article  Google Scholar 

  20. 20.

    H. Liang, L. Zhang, Y. Sun, T. Huang, Containment control of semi-markovian multiagent systems with switching topologies. IEEE Trans. Syst. Man Cybern. Syst. (2019). https://doi.org/10.1109/TSMC.2019.2946248

    Article  Google Scholar 

  21. 21.

    Q. Liu, J. Leng, D. Yan, D. Zhang, L. Wei, A. Yu, R. Zhao, H. Zhang, X. Chen, Digital twin-based designing of the configuration, motion, control, and optimization model of a flow-type smart manufacturing system. J. Manuf. Syst. (2020). https://doi.org/10.1016/j.jmsy.2020.04.012

    Article  Google Scholar 

  22. 22.

    Q. Liu, H. Zhang, J. Leng, X. Chen, Digital twin-driven rapid individualised designing of automated flow-shop manufacturing system. Int. J. Prod. Res. 57(12), 3903–3919 (2019)

    Article  Google Scholar 

  23. 23.

    Y. Liu, X. Liu, Y. Jing, X. Li, Annular domain finite-time connective control for large-scale systems with expanding construction. IEEE Trans. Syst. Man Cybern. Syst. (2019). https://doi.org/10.1109/TSMC.2019.2960009

    Article  Google Scholar 

  24. 24.

    R. Lu, Y. Xu, A. Xue, J. Zheng, Networked control with state reset and quantized measurements: observer-based case. IEEE Trans. Ind. Electron. 60(11), 5206–5213 (2012)

    Article  Google Scholar 

  25. 25.

    R. Lu, W. Yu, J. Lü, A. Xue, Synchronization on complex networks of networks. IEEE Trans. Neural Netw. Learn. Syst. 25(11), 2110–2118 (2014)

    Article  Google Scholar 

  26. 26.

    H. Ma, G. Yang, Adaptive fault tolerant control of cooperative heterogeneous systems with actuator faults and unreliable interconnections. IEEE Trans. Autom. Control 61(11), 3240–3255 (2016)

    MathSciNet  Article  Google Scholar 

  27. 27.

    R.D. Nussbaum, Some temarks on a conjecture in parameter adaptive control. Syst. Control Lett. 3(5), 243–246 (1983)

    Article  Google Scholar 

  28. 28.

    Y. Pan, G. Yang, Event-triggered fault detection filter design for nonlinear networked systems. IEEE Trans. Syst. Man Cybern. Syst. 48(11), 1851–1862 (2017)

    Article  Google Scholar 

  29. 29.

    K. Shahvali, Milad Shojaei, Distributed adaptive neural control of nonlinear multi-agent systems with unknown control directions. Nonlinear Dyn. 83(4), 2213–2228 (2016)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Y. Su, Cooperative global output regulation of second-order nonlinear multi-agent systems with unknown control direction. IEEE Trans. Autom. Control 60(12), 3275–3280 (2015)

    MathSciNet  Article  Google Scholar 

  31. 31.

    Y. Su, B. Chen, C. Lin, H. Wang, S. Zhou, Adaptive neural control for a class of stochastic nonlinear systems by backstepping approach. Inf. Sci. 369, 748–764 (2016)

    Article  Google Scholar 

  32. 32.

    M. Tong, W. Lin, X. Huo, Z. Jin, C. Miao, A model-free fuzzy adaptive trajectory tracking control algorithm based on dynamic surface control. Int. J. Adv. Robot. Syst. 17(1), 1–11 (2020)

    Article  Google Scholar 

  33. 33.

    S. Tong, X. Min, Y. Li, Observer-based adaptive fuzzy tracking control for strict-feedback nonlinear systems with unknown control gain functions. IEEE Trans. Cybern. (2020). https://doi.org/10.1109/TCYB.2020.2977175

    Article  Google Scholar 

  34. 34.

    H. Wang, D. Wang, G. Sun, Z. Peng, H. Zhang, Distributed model reference adaptive control for cooperative tracking of uncertain dynamical multi-agent systems. IET Control Theory Appl. 7(8), 1079–1087 (2013)

    MathSciNet  Article  Google Scholar 

  35. 35.

    W. Wang, H. Liang, Y. Pan, T. Li, Prescribed performance adaptive fuzzy containment control for nonlinear multiagent systems using disturbance observer. IEEE Trans. Cybern. (2020). https://doi.org/10.1109/TCYB.2020.2969499

    Article  Google Scholar 

  36. 36.

    W. Wei, W. Dan, Z. Peng, T. Li, Prescribed performance consensus of uncertain nonlinear strict-feedback systems with unknown control directions. IEEE Trans. Autom. Control 46(9), 1279–1286 (2016)

    Google Scholar 

  37. 37.

    C. Wu, L. Wu, J. Liu, Z. Jiang, Active defense-based resilient sliding mode control under denial-of-service attacks. IEEE Trans. Inf. Forensics Secur. 15(8), 237–249 (2019)

    Google Scholar 

  38. 38.

    F. Xiao, L. Wang, Asynchronous consensus in continuous-time multi-agent systems with switching topology and time-varying delays. IEEE Trans. Autom. Control 53(8), 1804–1816 (2008)

    MathSciNet  Article  Google Scholar 

  39. 39.

    G. Xie, L. Sun, T. Wen, X. Hei, F. Qian, Adaptive transition probability matrix-based parallel IMM algorithm. IEEE Trans. Syst. Man Cybern. Syst. (2019). https://doi.org/10.1109/TSMC.2019.2922305

    Article  Google Scholar 

  40. 40.

    D. Yao, H. Li, R. Lu, Y. Shi, Distributed sliding-mode tracking control of second-order nonlinear multiagent systems: an event-triggered approach. IEEE Trans. Cybern. (2020). https://doi.org/10.1109/TCYB.2019.2963087

    Article  Google Scholar 

  41. 41.

    S.J. Yoo, Connectivity-preserving consensus tracking of uncertain nonlinear strict-feedback multiagent systems: an error transformation approach. IEEE Trans. Neural Netw. Learn. Syst. 29(9), 4542–4548 (2018)

    Article  Google Scholar 

  42. 42.

    K. Zhang, G. Liu, B. Jiang, Robust unknown input observer-based fault estimation of leader-follower linear multi-agent systems. Circuits Syst. Signal Process. 36(2), 525–542 (2017)

    Article  Google Scholar 

  43. 43.

    L. Zhang, H. Lam, Y. Sun, H. Liang, Fault detection for fuzzy semi-Markov jump systems based on interval type-2 fuzzy approach. IEEE Trans. Fuzzy Syst. (2019). https://doi.org/10.1109/TFUZZ.2019.2936333

    Article  Google Scholar 

  44. 44.

    T. Zhang, S.S. Ge, Adaptive neural network tracking control of mimo nonlinear systems with unknown dead zones and control directions. IEEE Trans. Neural Netw. 20(3), 483–497 (2009)

    Article  Google Scholar 

  45. 45.

    T. Zhang, C.L. Philip, L. Chen, X. Wang, B. Hu, Design of highly nonlinear substitution boxes based on i-ching operators. IEEE Trans. Cybern. 48(12), 3349–3358 (2018)

    Article  Google Scholar 

  46. 46.

    T. Zhang, X. Wang, X. Xu, C. Chen, GCB-Net: graph convolutional broad network and its application in emotion recognition. IEEE Trans. Affect. Comput. (2019). https://doi.org/10.1109/TAFFC.2937768

    Article  Google Scholar 

  47. 47.

    W. Zhang, Q. Han, Y. Tang, Y. Liu, Sampled-data control for a class of linear time-varying systems. Automatica 103(8), 126–134 (2019)

    MathSciNet  Article  Google Scholar 

  48. 48.

    W. Zhang, Y. Tang, T. Huang, J. Kurths, Sampled-data consensus of linear multi-agent systems with packet losses. IEEE Trans. Neural Netw. Learn. Syst. 28(11), 2516–2527 (2016)

    MathSciNet  Article  Google Scholar 

  49. 49.

    D. Zhao, M. Chi, Z. Guan, Y. Wu, J. Chen, Distributed estimator-based fault detection for multi-agent networks. Circuits Syst. Signal Process. 37(1), 98–111 (2018)

    MathSciNet  Article  Google Scholar 

  50. 50.

    Q. Zhou, P. Du, H. Li, R. Lu, J. Yang, Adaptive fixed-time control of error-constrained pure-feedback interconnected nonlinear systems. IEEE Trans. Syst. Man Cybern. Syst. (2020). https://doi.org/10.1109/TSMC.2019.2961371

    Article  Google Scholar 

  51. 51.

    Q. Zhou, W. Wang, H. Liang, M. Basin, B. Wang, Observer-based event-triggered fuzzy adaptive bipartite containment control of multi-agent systems with input quantization. IEEE Trans. Fuzzy Syst. (2019). https://doi.org/10.1109/TFUZZ.2019.2953573

    Article  Google Scholar 

  52. 52.

    Q. Zhou, W. Wang, H. Ma, H. Li, Event-triggered fuzzy adaptive containment control for nonlinear multi-agent systems with unknown Bouc-Wen hysteresis input. IEEE Trans. Fuzzy Syst. (2019). https://doi.org/10.1109/TFUZZ.2019.2961642

    Article  Google Scholar 

  53. 53.

    S. Zhu, Y. Liu, Y. Lou, J. Cao, Stabilization of logical control networks: an event-triggered control approach. Sci. China Inf. Sci. 63(112203), 1–11 (2020)

    MathSciNet  Google Scholar 

  54. 54.

    Z. Zhu, Y. Pan, Q. Zhou, C. Lu, Event-triggered adaptive fuzzy control for stochastic nonlinear systems with unmeasured states and unknown backlash-like hysteresis. IEEE Trans. Fuzzy Syst. (2020). https://doi.org/10.1109/TFUZZ.2020.2973950

    Article  Google Scholar 

Download references


This work was partially supported by the National Natural Science Foundation of China (61703051), and the Project of Liaoning Province Science and Technology Program under Grant (2019-KF-03-13).

Author information



Corresponding author

Correspondence to Yingnan Pan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Li, S., Pan, Y. & Liang, H. Output-Constrained Control of Non-affine Multi-agent Systems with Actuator Faults and Unknown Dead Zones. Circuits Syst Signal Process (2020). https://doi.org/10.1007/s00034-020-01473-z

Download citation


  • Non-affine multi-agent systems
  • Actuator faults
  • Dead zones
  • Output-constrained control