Skip to main content
SpringerLink
Log in

Adaptive neural inverse optimal control with predetermined tracking accuracy for nonlinear MIMO systems

  • Original Paper
  • Published:
Nonlinear Dynamics Aims and scope Submit manuscript

Abstract

In addition to stability, the system optimality has also received attention because the system is expected to achieve higher performance with lower energy consumption. In general, the conventional approach to achieve optimal control of nonlinear MIMO systems is to solve the Hamilton–Jacobi–Bellman equation directly, which is time-consuming and sometimes impossible. To address this issue, this paper proposes an adaptive neural inverse optimal control method for uncertain MIMO systems. The method is based on an improved design criterion for the inverse optimal controller, which avoids the need for constructing auxiliary systems and enables direct stability analysis of MIMO systems. Additionally, an adaptive one-parameter update strategy is proposed to reduce the computational effort, which avoids the need to update the entire neural network. The proposed scheme guarantees that the tracking errors of the MIMO system converge to a given domain while minimizing a family of meaningful loss functions. Finally, the effectiveness of the presented method is verified through simulations.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10
Fig. 11
Fig. 12
Fig. 13
Fig. 14

Similar content being viewed by others

Data availability statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Abbreviations

\(\alpha _{\text{ i },j}\) :

Virtual controller

\(\epsilon \left( {x}\right) \) :

Approximation error

\(\hat{\varrho }_{\text{ i }}\) :

Estimation of \({\varrho }_{\text{ i }}\)

\(\varPi \left( {x}\right) \) :

Basis function

\(\varTheta ^*\) :

Ideal weight vector of neural networks

\(\tilde{\varrho }_{\text{ i }}\) :

Estimated error

\(\zeta _{\text{ i },j},c_{\text{ i },j},\lambda _{\text{ i }}\) :

Positive design constants

\(f_{\text{ i },j}\) :

Unknown smooth function

J(u):

Cost function

u :

Auxiliary controller

\(u^*\) :

Optimal controller

V :

Lyapunov function

\(z_{\text{ i },j}\) :

Tracking error

x :

System state vector

\({y}^{(a)}\) :

The ath derivative of y

References

  1. Pan, H., Zhang, D., Sun, W., Yu, X.: Event-triggered adaptive asymptotic tracking control of uncertain mimo nonlinear systems with actuator faults. IEEE Trans. Cybern. 52(9), 8655–8667 (2022). https://doi.org/10.1109/TCYB.2021.3061888

  2. Boudjedir, C.E., Bouri, M., Boukhetala, D.: Model-free iterative learning control with nonrepetitive trajectories for second-order mimo nonlinear systems-application to a delta robot. IEEE Trans. Ind. Electron. 68(8), 7433–7443 (2021). https://doi.org/10.1109/TIE.2020.3007091

    Article  Google Scholar 

  3. Chen, G., Liu, Y., Qing, X., Ma, M., Lin, Z.: Principle and topology derivation of single-inductor multi-input multi-output dc-dc converters. IEEE Trans. Ind. Electron. 68(1), 25–36 (2021). https://doi.org/10.1109/TIE.2020.2965490

    Article  Google Scholar 

  4. Huang, C., Liu, Z., Chen, C.L.P., Zhang, Y.: Adaptive fixed-time neural control for uncertain nonlinear multiagent systems. IEEE Trans. Neural Netw. Learn. Syst. (2022). https://doi.org/10.1109/TNNLS.2022.3165836

    Article  PubMed  Google Scholar 

  5. Liu, D., Liu, Z., Chen, C.L.P., Zhang, Y.: Distributed adaptive neural fixed-time tracking control of multiple uncertain mechanical systems with actuation dead zones. IEEE Trans. Syst. Man Cybern. Syst. 52(6), 3859–3872 (2022). https://doi.org/10.1109/TSMC.2021.3075967

    Article  Google Scholar 

  6. Xu, Y., Li, T., Yang, Y., Shan, Q., Tong, S., Chen, C.L.P.: Anti-attack event-triggered control for nonlinear multi-agent systems with input quantization. IEEE Trans. Neural Netw. Learn. Syst. (2022). https://doi.org/10.1109/TNNLS.2022.3164881

    Article  PubMed  Google Scholar 

  7. Lin, Z., Liu, Z., Zhang, Y., Chen, C.P.: Adaptive neural consensus tracking control for multi-agent systems with unknown state and input hysteresis. Nonlinear Dyn. 105, 1625–1641 (2021)

    Article  Google Scholar 

  8. He, W., Kong, L., Dong, Y., Yu, Y., Yang, C., Sun, C.: Fuzzy tracking control for a class of uncertain mimo nonlinear systems with state constraints. IEEE Trans. Syst. Man Cybern. Syst. 49(3), 543–554 (2019). https://doi.org/10.1109/TSMC.2017.2749124

    Article  Google Scholar 

  9. Yu, J., Shi, P., Liu, J., Lin, C.: Neuroadaptive finite-time control for nonlinear mimo systems with input constraint. IEEE Trans. Cybern. 52(7), 6676–6683 (2022). https://doi.org/10.1109/TCYB.2020.3032530

    Article  PubMed  Google Scholar 

  10. Sui, S., Tong, S.: Finite-time fuzzy adaptive ppc for nonstrict-feedback nonlinear mimo systems. IEEE Trans. Cybern. 53(2), 732–742 (2023). https://doi.org/10.1109/TCYB.2022.3163739

    Article  PubMed  Google Scholar 

  11. Zhang, Y., Su, X., Liu, Z., Chen, C.L.P.: Event-triggered adaptive fuzzy tracking control with guaranteed transient performance for mimo nonlinear uncertain systems. IEEE Trans. Cybern. 51(2), 736–749 (2021). https://doi.org/10.1109/TCYB.2019.2894343

    Article  PubMed  Google Scholar 

  12. Huang, L., Li, Y., Tong, S.: Fuzzy adaptive output feedback control for mimo switched nontriangular structure nonlinear systems with unknown control directions. IEEE Trans. Syst. Man Cybern. Syst. 50(2), 550–564 (2020). https://doi.org/10.1109/TSMC.2017.2778099

    Article  Google Scholar 

  13. Wang, J., Pan, H., Zhang, D.: Event-triggered adaptive finite-time control for mimo nonlinear systems with actuator faults. IEEE Trans. Ind. Electron. 70(7), 7343–7352 (2023). https://doi.org/10.1109/TIE.2022.3201291

    Article  Google Scholar 

  14. Ye, H., Song, Y.: Prescribed-time tracking control of mimo nonlinear systems under non-vanishing uncertainties. IEEE Trans. Autom. Control (2022). https://doi.org/10.1109/TAC.2022.3194100

    Article  Google Scholar 

  15. Borrelli, F., Baotić, M., Bemporad, A., Morari, M.: Dynamic programming for constrained optimal control of discrete-time linear hybrid systems. Automatica 41(10), 1709–1721 (2005)

    Article  MathSciNet  Google Scholar 

  16. Xue, S., Luo, B., Liu, D., Gao, Y.: Event-triggered adp for tracking control of partially unknown constrained uncertain systems. IEEE Trans. Cybern. 52(9), 9001–9012 (2022). https://doi.org/10.1109/TCYB.2021.3054626

    Article  PubMed  Google Scholar 

  17. Wen, G., Chen, C.L.P., Ge, S.S.: Simplified optimized backstepping control for a class of nonlinear strict-feedback systems with unknown dynamic functions. IEEE Trans. Cybern. (2020). https://doi.org/10.1109/TCYB.2020.3002108

    Article  PubMed  Google Scholar 

  18. Li, Y., Fan, Y., Li, K., Liu, W., Tong, S.: Adaptive optimized backstepping control-based rl algorithm for stochastic nonlinear systems with state constraints and its application. IEEE Trans. Cybern. 52(10), 10542–10555 (2022). https://doi.org/10.1109/TCYB.2021.3069587

    Article  PubMed  Google Scholar 

  19. Chen, C., Modares, H., Xie, K., Lewis, F.L., Wan, Y., Xie, S.: Reinforcement learning-based adaptive optimal exponential tracking control of linear systems with unknown dynamics. IEEE Trans. Autom. Control 64(11), 4423–4438 (2019). https://doi.org/10.1109/TAC.2019.2905215

    Article  MathSciNet  Google Scholar 

  20. Zhao, B., Liu, D., Luo, C.: Reinforcement learning-based optimal stabilization for unknown nonlinear systems subject to inputs with uncertain constraints. IEEE Trans. Neural Netw. Learn. Syst. 31(10), 4330–4340 (2020). https://doi.org/10.1109/TNNLS.2019.2954983

    Article  MathSciNet  PubMed  Google Scholar 

  21. Liu, X., Zhao, B., Liu, D.: Fault tolerant tracking control for nonlinear systems with actuator failures through particle swarm optimization-based adaptive dynamic programming. Appl. Soft Comput. 97, 106766 (2020)

    Article  Google Scholar 

  22. Wang, H., Yang, C., Liu, X., Zhou, L.: Neural-network-based adaptive control of uncertain mimo singularly perturbed systems with full-state constraints. IEEE Trans. Neural Netw. Learn. Syst. (2021). https://doi.org/10.1109/TNNLS.2021.3123361

    Article  PubMed  PubMed Central  Google Scholar 

  23. Krstic, M., Li, Z.-H.: Inverse optimal design of input-to-state stabilizing nonlinear controllers. IEEE Trans. Autom. Control 43(3), 336–350 (1998). https://doi.org/10.1109/9.661589

    Article  MathSciNet  Google Scholar 

  24. Freeman, R.A., Kokotovic, P.V.: Inverse optimality in robust stabilization. SIAM J. Control Optim. 34(4), 1365–1391 (1996)

    Article  MathSciNet  Google Scholar 

  25. Li, Y.-M., Min, X., Tong, S.: Adaptive fuzzy inverse optimal control for uncertain strict-feedback nonlinear systems. IEEE Trans. Fuzzy Syst. 28(10), 2363–2374 (2020). https://doi.org/10.1109/TFUZZ.2019.2935693

    Article  Google Scholar 

  26. Li, Y., Min, X., Tong, S.: Observer-based fuzzy adaptive inverse optimal output feedback control for uncertain nonlinear systems. IEEE Trans. Fuzzy Syst. 29(6), 1484–1495 (2021). https://doi.org/10.1109/TFUZZ.2020.2979389

    Article  Google Scholar 

  27. Lu, K., Liu, Z., Chen, C.L.P., Wang, Y., Zhang, Y.: Inverse optimal design of direct adaptive fuzzy controllers for uncertain nonlinear systems. IEEE Trans. Fuzzy Syst. (2021). https://doi.org/10.1109/TFUZZ.2021.3064678

    Article  Google Scholar 

  28. Lu, K., Liu, Z., Yu, H., Chen, C.L.P., Zhang, Y.: Inverse optimal adaptive neural control for state-constrained nonlinear systems. IEEE Trans. Neural Netw. Learn. Syst. (2023). https://doi.org/10.1109/TNNLS.2023.3243084

    Article  PubMed  Google Scholar 

  29. Zeng, D., Liu, Z., Wang, Y., Chen, C., Zhang, Y., Wu, Z.: Adaptive fuzzy inverse optimal control of nonlinear switched systems. IEEE Trans. Fuzzy Syst. (2023). https://doi.org/10.1109/TFUZZ.2023.3244836

    Article  Google Scholar 

  30. Do, K.: Inverse optimal gain assignment control of evolution systems and its application to boundary control of marine risers. Automatica 106, 242–256 (2019). https://doi.org/10.1016/j.automatica.2019.05.020

    Article  MathSciNet  Google Scholar 

  31. Do, K.: Bounded and inverse optimal formation stabilization of second-order agents. Automatica 123, 109367 (2021). https://doi.org/10.1016/j.automatica.2020.109367

    Article  MathSciNet  Google Scholar 

  32. Do, K.: Inverse optimal control of stochastic systems driven by lévy processes. Automatica 107, 539–550 (2019). https://doi.org/10.1016/j.automatica.2019.06.016

    Article  MathSciNet  Google Scholar 

  33. Deng, H., Krstić, M.: Stochastic nonlinear stabilization-ii: inverse optimality. Syst. Control Lett. 32(3), 151–159 (1997)

    Article  MathSciNet  Google Scholar 

  34. Zhu, Z., Liang, H., Liu, Y., Xue, H.: Command filtered event-triggered adaptive control for mimo stochastic multiple time-delay systems. Int. J. Robust Nonlinear Control 32(2), 715–736 (2022)

    Article  MathSciNet  Google Scholar 

  35. Lai, G., Zhang, Y., Liu, Z., Chen, C.L.P.: Indirect adaptive fuzzy control design with guaranteed tracking error performance for uncertain canonical nonlinear systems. IEEE Trans. Fuzzy Syst. 27(6), 1139–1150 (2019). https://doi.org/10.1109/TFUZZ.2018.2870574

    Article  Google Scholar 

  36. Lu, K., Liu, Z., Lai, G., Chen, C.L.P., Zhang, Y.: Adaptive consensus tracking control of uncertain nonlinear multiagent systems with predefined accuracy. IEEE Trans. Cybern. 51(1), 405–415 (2021). https://doi.org/10.1109/TCYB.2019.2933436

    Article  PubMed  Google Scholar 

  37. Tong, S., Sui, S., Li, Y.: Fuzzy adaptive output feedback control of mimo nonlinear systems with partial tracking errors constrained. IEEE Trans. Fuzzy Syst. 23(4), 729–742 (2015). https://doi.org/10.1109/TFUZZ.2014.2327987

    Article  Google Scholar 

  38. Han, S.I., Lee, J.M.: Partial tracking error constrained fuzzy dynamic surface control for a strict feedback nonlinear dynamic system. IEEE Trans. Fuzzy Syst. 22(5), 1049–1061 (2014). https://doi.org/10.1109/TFUZZ.2013.2279543

  39. Chen, W., Saif, M.: Output feedback controller design for a class of mimo nonlinear systems using high-order sliding-mode differentiators with application to a laboratory 3-d crane. IEEE Trans. Ind. Electron. 55(11), 3985–3997 (2008). https://doi.org/10.1109/TIE.2008.2004384

    Article  Google Scholar 

  40. Xiao, B., Yang, X., Karimi, H.R., Qiu, J.: Asymptotic tracking control for a more representative class of uncertain nonlinear systems with mismatched uncertainties. IEEE Trans. Ind. Electron. 66(12), 9417–9427 (2019). https://doi.org/10.1109/TIE.2019.2893852

    Article  Google Scholar 

  41. Song, Y., Huang, X., Wen, C.: Robust adaptive fault-tolerant pid control of mimo nonlinear systems with unknown control direction. IEEE Trans. Ind. Electron. 64(6), 4876–4884 (2017). https://doi.org/10.1109/TIE.2017.2669891

    Article  Google Scholar 

Download references

Funding

This work was supported in part by the National Natural Science Foundation of China under Grant 62273103, in part by the National Key Research and Development Program of China under Project 2020AAA0108303, in part by Guangdong Province Universities and Colleges Pearl River Scholar Funded Scheme, in part by the Natural Science Foundation of Guangdong Province under Grant 2022A1515010151, in part by the National Natural Science Foundation of China under Grant 6210021076 and in part by the Guangzhou Municipal Science and Technology Project under Grant 202201010381.

Author information

Authors and Affiliations

  1. School of Automation, Guangdong University of Technology, Guangzhou, 510006, China

    Zhuangbi Lin, Zhi Liu, Yun Zhang & Zongze Wu

  2. College of Mathematics and Informatics, South China Agricultural University, Guangzhou, 510642, China

    Zhuangbi Lin

  3. Guangdong-Hong Kong-Macao Joint Laboratory for Smart Discrete Manufacturing, Guangzhou, 510006, China

    Zhi Liu, Yun Zhang & Zongze Wu

  4. School of Computer Science and Engineering, South China University of Technology, Guangzhou, 510006, China

    C. L. Philip Chen

Authors
  1. Zhuangbi Lin
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zhi Liu
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. C. L. Philip Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yun Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Zongze Wu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Zhi Liu.

Ethics declarations

Conflict of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

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

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Lin, Z., Liu, Z., Chen, C.L.P. et al. Adaptive neural inverse optimal control with predetermined tracking accuracy for nonlinear MIMO systems. Nonlinear Dyn 112, 4449–4464 (2024). https://doi.org/10.1007/s11071-023-09075-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11071-023-09075-5

Keywords

Search

Navigation