Abstract
In this paper, an adaptive prescribed performance controller, consisting of a novel scalarly virtual parameter adaptation (SVPA) technique, is developed for a class of single-input and single-output high-order nonlinear pure-feedback systems in the presence of model uncertain yet locally Lipschitz nonlinearities. The objective of this work is to improve the transient and steady performance of pioneering prescribed performance control (PPC) by incorporating a single SVPA mechanism into the virtual and actual controllers, therein, the unknown yet bounded parameters are defined with respect to proper composite system and virtual functions, bringing the gap between pioneering PPC and linearly parameterized approximator-based PPC schemes (including neural networks, fuzzy logic systems, etc.), that is, the computational complexity of proposed method exceeds PPC with one level (caused by introduced single adaptive law) yet maintains low level with comparison to linearly parametrized approximator-based PPC. It is guaranteed that both virtual and actual tracking errors converge transiently to small residual sets characterized by prescribed performance functions and control parameters simultaneously and ultimately converge to zero, which is also proved by rigorously mathematical analysis using Lyapunov stability theorem. The closed-loop signals are kept globally ultimately uniformly bounded, and comparative simulation results are presented to demonstrate the effectiveness and advantages of the theoretical findings.
Similar content being viewed by others
References
An, H., Guo, Z., Wang, G., Wang, C.: Low-complexity hypersonic flight control with asymmetric angle of attack constraint. Nonlinear Dyn. (2020). https://doi.org/10.1007/s11071-020-05531-8
Bechlioulis, C., Rovithakis, G.: A low-complexity global approximation-free control scheme with prescribed performance for unknown pure feedback systems. Automatica 50(4), 1217–1226 (2014)
Bechlioulis, C.P., Rovithakis, G.: Robust adaptive control of feedback linearizable MIMO nonlinear systems with prescribed performance. IEEE Trans. Autom. Control 53(9), 2090–2099 (2008)
Bechlioulis, C.P., Rovithakis, G.A.: Adaptive control with guaranteed transient and steady state tracking error bounds for strict feedback systems. Automatica 45(2), 532–538 (2009)
Bechlioulis, C.P., Rovithakis, G.A.: Prescribed performance adaptive control for multi-input multi-output affine in the control nonlinear systems. IEEE Trans. Autom. Control 55(5), 1220–1226 (2010)
Bechlioulis, C.P., Rovithakis, G.A.: Robust partial-state feedback prescribed performance control of cascade systems with unknown nonlinearities. IEEE Trans. Autom. Control 56(9), 2224–2230 (2011)
Bechlioulis, C.P., Rovithakis, G.A.: A priori guaranteed evolution within the neural network approximation set and robustness expansion via prescribed performance control. IEEE Trans. Neural Netw. Learn. Syst. 23(4), 669–675 (2012)
Bu, X., Wu, X., Huang, J., Wei, D.: A guaranteed transient performance-based adaptive neural control scheme with low-complexity computation for flexible air-breathing hypersonic vehicles. Nonlinear Dyn. 84(4), 2175–2194 (2016)
Choi, Y.H., Yoo, S.J.: Decentralized approximation-free control for uncertain large-scale pure-feedback systems with unknown time-delayed nonlinearities and control directions. Nonlinear Dyn. 85(2), 1053–1066 (2016)
Ferrara, A., Giacomini, L.: Control of a class of mechanical systems with uncertainties via a constructive adaptive/second order VSC approach. J. Dyn. Syst. Meas. Control 122(1), 33–39 (2000)
Gao, S., Dong, H., Ning, B.: Neural adaptive dynamic surface control for uncertain strict-feedback nonlinear systems with nonlinear output and virtual feedback errors. Nonlinear Dyn. 90(4), 2851–2867 (2017)
Ge, S.S., Hang, C.C., Zhang, T.: Adaptive neural network control of nonlinear systems by state and output feedback. IEEE Trans. Syst. Man Cybern. Part B (Cybern.) 29(6), 818–828 (1999)
Han, S.I., Lee, J.M.: Fuzzy echo state neural networks and funnel dynamic surface control for prescribed performance of a nonlinear dynamic system. IEEE Trans. Ind. Electron. 61(2), 1099–1112 (2013)
Hashemi, M.: Adaptive neural dynamic surface control of MIMO nonlinear time delay systems with time-varying actuator failures. Int. J. Adapt. Control Signal Process. 31(2), 275–296 (2017)
Hashemi, M., Askari, J., Ghaisari, J.: Adaptive control of uncertain nonlinear time delay systems in the presence of actuator failures and applications to chemical reactor systems. Eur. J. Control 29, 62–73 (2016)
Hashemi, M., Askari, J., Ghaisari, J.: Adaptive decentralised dynamic surface control for non-linear large-scale systems against actuator failures. IET Control Theory Appl. 10(1), 44–57 (2016)
Hashemi, M., Shahgholian, G.: Distributed robust adaptive control of high order nonlinear multi agent systems. ISA Trans. 74, 14–27 (2018)
Jia, F., Wang, X., Zhou, X.: Robust adaptive prescribed performance control for a class of nonlinear pure-feedback systems. Int. J. Robust Nonlinear Control 29(12), 3971–3987 (2019)
Kostarigka, A.K., Rovithakis, G.A.: Adaptive dynamic output feedback neural network control of uncertain MIMO nonlinear systems with prescribed performance. IEEE Trans. Neural Netw. Learn. Syst. 23(1), 138–149 (2012)
Li, Y., Tong, S.: Prescribed performance adaptive fuzzy output-feedback dynamic surface control for nonlinear large-scale systems with time delays. Inf. Sci. 292, 125–142 (2015)
Li, Y., Tong, S.: Adaptive neural networks prescribed performance control design for switched interconnected uncertain nonlinear systems. IEEE Trans. Neural Netw. Learn. Syst. 29(7), 3059–3068 (2017)
Li, Y., Tong, S., Li, T.: Adaptive fuzzy output feedback dynamic surface control of interconnected nonlinear pure-feedback systems. IEEE Trans. Cybern. 45(1), 138–149 (2014)
Li, Y., Tong, S., Liu, L., Feng, G.: Adaptive output-feedback control design with prescribed performance for switched nonlinear systems. Automatica 80, 225–231 (2017)
Li, Y., Yang, G.: Model-based adaptive event-triggered control of strict-feedback nonlinear systems. IEEE Trans. Neural Netw. Learn. Syst. 29(4), 1033–1045 (2017)
Liu, Y.J., Tong, S.: Barrier Lyapunov functions-based adaptive control for a class of nonlinear pure-feedback systems with full state constraints. Automatica 64, 70–75 (2016)
Lu, X., Jia, Y.: Adaptive coordinated control of uncertain free-floating space manipulators with prescribed control performance. Nonlinear Dyn. 97, 1541–1566 (2019)
Na, J., Chen, Q., Ren, X., Guo, Y.: Adaptive prescribed performance motion control of servo mechanisms with friction compensation. IEEE Trans. Ind. Electron. 61(1), 486–494 (2013)
Park, J., Huh, S., Kim, S., Seo, S., Park, G.: Direct adaptive controller for nonaffine nonlinear systems using self-structuring neural networks. IEEE Trans. Neural Netw. 16(2), 414–422 (2005)
Rovithakis, G.A.: Prescribed performance adaptive control of uncertain nonlinear systems: state-of-the-art and open issues. PAMM 18(1), e201800134 (2018)
Song, H., Tao, Z., Zhang, G., Lu, C.: Robust dynamic surface control of nonlinear systems with prescribed performance. Nonlinear Dyn. 76(1), 599–608 (2014)
Song, Q., Song, Y., Cai, W.: Dealing with traction/braking failures in high speed trains via virtual parameter based adaptive fault-tolerant control method. In: 2012 American Control Conference (ACC). IEEE, pp. 362–367 (2012)
Song, Y., Song, Q., Cai, W.: Fault-tolerant adaptive control of high-speed trains under traction/braking failures: a virtual parameter-based approach. IEEE Trans. Intell. Transp. Syst. 15(2), 737–748 (2013)
Sontag, E.D.: Mathematical Control Theory: Deterministic Finite Dimensional Systems, vol. 6. Springer Science & Business Media, Cham (2013)
Sui, S., Tong, S., Li, Y.: Observer-based fuzzy adaptive prescribed performance tracking control for nonlinear stochastic systems with input saturation. Neurocomputing 158, 100–108 (2015)
Zhang, L., Sui, S., Li, Y., Tong, S.: Adaptive fuzzy output feedback tracking control with prescribed performance for chemical reactor of MIMO nonlinear systems. Nonlinear Dyn. 80(1–2), 945–957 (2015)
Zhang, L., Tong, S., Li, Y.: Prescribed performance adaptive fuzzy output-feedback control of uncertain nonlinear systems with unmodeled dynamics. Nonlinear Dyn. 77(4), 1653–1665 (2014)
Zhang, T., Xia, M., Yi, Y.: Adaptive neural dynamic surface control of strict-feedback nonlinear systems with full state constraints and unmodeled dynamics. Automatica 81, 232–239 (2017)
Zhang, Z., Song, Y., Li, P., Wang, W., Qin, M.: Adaptive and robust variable-speed control of wind turbine based on virtual parameter approach. In: Proceedings of The 32nd Chinese Control Conference. IEEE, pp. 8886–8890 (2013)
Zuo, Z., Wang, C.: Adaptive trajectory tracking control of output constrained multi-rotors systems. IET Control Theory Appl. 8(13), 1163–1174 (2014)
Acknowledgements
This work is supported jointly by the National Natural Science Foundation of China under Grants 61790573 and 62073027, the Beijing Natural Science Foundation under Grant 4192046, and State Key Laboratory of Rail Traffic Control and Safety under Grant RCS2020ZZ003, Beijing Jiaotong University.
Author information
Authors and Affiliations
Corresponding author
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
About this article
Cite this article
Wu, C., Gao, S. & Dong, H. Adaptive prescribed performance control for nonlinear pure-feedback systems: a scalarly virtual parameter adaptation approach. Nonlinear Dyn 102, 2597–2615 (2020). https://doi.org/10.1007/s11071-020-06051-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11071-020-06051-1