Abstract
This work studies the tracking issue of uncertain nonlinear systems. The existence of odd rational powers, multiple unknown parameters and the dead-zone input add many difficulties for control design. During procedures of the control design, by introducing an appropriate Lyapunov function, utilizing recursive control method and the inequality technique, some appropriate intermediate auxiliary control laws are designed under the hypothesis that nonlinear terms in the system are known. When those nonlinear terms are unknown, by employing the powerful approximation ability of fuzzy systems, the intermediate auxiliary control laws are approximated recursively and used to construct the virtual control. Finally, a new fuzzy adaptive tracking controller is constructed to ensure a small tracking error and the boundedness of all states. In this paper, the overparameterization problem is significantly avoided since only two adaptive laws are adopted. Numerical and practical examples are used to verify the raised theory.
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Khalil H K and Grizzle J W, Nonlinear Systems, Upper Saddle River, NJ: Prentice Hall, 2002.
Yu J, Shi P, and Zhao L, Finite-time command filtered backstepping control for a class of nonlinear systems, Automatica, 2018, 92: 173–180.
Yu J, Shi P, Chen X, et al., Finite-time command filtered adaptive control for nonlinear systems via immersion and invariance, Science China Information Sciences, 2021, https://doi.org/10.1007/s11432-020-3144-6.
Yu J, Zhao L, Yu H, et al., Barrier Lyapunov functions-based command filtered output feedback control for full-state constrained nonlinear systems, Automatica, 2019, 105: 71–79.
Swaroop D, Hedrick J K, Yip P P, et al., Dynamic surface control for a class of nonlinear systems, IEEE Transactions on Automatic Control, 2000, 45(10): 1893–1899.
Ding S, Park J H, and Chen C C, Second-order sliding mode controller design with output constraint, Automatica, 2020, 112: 108704.
Ding S, Liu L, and Zheng W X, Sliding mode direct yaw-moment control design for in-wheel electric vehicles, IEEE Transactions on Industrial Electronics, 2017, 64(8): 6752–6762.
Qian C and Lun W, A continuous feedback approach to global strong stabilization of nonlinear systems, IEEE Transactions on Automatic Control, 2001, 46(7): 1061–1079.
Zhao X, Shi P, Zheng X, et al., Adaptive tracking control for switched stochastic nonlinear systems with unknown actuator dead-zone, Automatica, 2015, 60: 193–200.
Zhao L, Liu G, and Yu J, Finite-time adaptive fuzzy tracking control for a class of nonlinear systems with full-state constraints, IEEE Transactions on Fuzzy Systems, 2020, https://doi.org/10.1109/TFUZZ.2020.2996387.
Cui G, Yu J, and Shi P, Observer-based finite-time adaptive fuzzy control with prescribed performance for nonstrict-feedback nonlinear systems, IEEE Transactions on Fuzzy Systems, 2020, https://doi.org/10.1109/TFUZZ.2020.3048518.
Cui G, Yu J, and Wang Q G, Finite-time adaptive fuzzy control for MIMO nonlinear systems with input saturation via improved command-filtered backstepping, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2020, https://doi.org/10.1109/TSMC.2020.3010642.
Zhao Y, Lu Q, Feng Z, et al., Delay differential equations under nonlinear impulsive control and applications to neural network models, Journal of Systems Science and Complexity, 2012, 25(4): 707–719.
Zhao X, Shi P, Zheng X, et al., Intelligent tracking control for a class of uncertain high-order nonlinear systems, IEEE Transactions on Neural Networks and Learning Systems, 2015, 27(9): 1976–1982.
Tong S, Liu C, and Li Y, Fuzzy-adaptive decentralized output-feedback control for large-scale nonlinear systems with dynamical uncertainties, IEEE Transactions on Fuzzy Systems, 2010, 18(5): 845–861.
Tong S, Zhang L, and Li Y, Observed-based adaptive fuzzy decentralized tracking control for switched uncertain nonlinear large-scale systems with dead zones, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2015, 46(1): 37–47.
Sun W, Su S F, Wu Y, et al., Adaptive fuzzy control with high-order barrier Lyapunov functions for high-order uncertain nonlinear systems with full-state constraints, IEEE Transactions on Cybernetics, 2019, 50(8): 3424–3432.
Sun Z Y, Liu C Y, Su S F, et al., Robust stabilization of high-order nonlinear systems with unknown sensitivities and applications in humanoid robot manipulation, IEEE Transactions on Systems, Man, and Cybernetics: Systems, 2019, https://doi.org/10.1109/TSMC.2019.2931768.
Sun W, Xia J, and Wu Y, Adaptive tracking control for mobile manipulators with stochastic disturbances, Journal of Systems Science and Complexity, 2019, 32(5): 1393–1403.
Li P and Yang G H, A novel adaptive control approach for nonlinear strict-feedback systems using nonlinearly parameterised fuzzy approximators, International Journal of Systems Science, 2011, 42(3): 517–527.
Zhao X, Zheng X, Niu B, et al., Adaptive tracking control for a class of uncertain switched nonlinear systems, Automatica, 2015, 52: 185–191.
Shi W, Observer-based indirect adaptive fuzzy control for SISO nonlinear systems with unknown gain sign, Neurocomputing, 2016, 171: 1598–1605.
Labiod S and Guerra T M, Indirect adaptive fuzzy control for a class of nonaffine nonlinear systems with unknown control directions, International Journal of Control, Automation and Systems, 2010, 8(4): 903–907.
Diao S, Sun W, and Yuan W, Adaptive fuzzy practical tracking control for flexible-joint robots via command filter design, Measurement and Control, 2020, 53(5–6): 814–823.
Chen B, Liu X, Liu K, et al., Direct adaptive fuzzy control of nonlinear strict-feedback systems, Automatica, 2009, 45(6): 1530–1535.
Wang N and Er M J, Direct adaptive fuzzy tracking control of marine vehicles with fully unknown parametric dynamics and uncertainties, IEEE Transactions on Control Systems Technology, 2016, 24(5): 1845–1852.
Lu K, Liu Z, Lai G, et al., Adaptive fuzzy tracking control of uncertain nonlinear systems subject to actuator dead zone with piecewise time-varying parameters, IEEE Transactions on Fuzzy Systems, 2018, 27(7): 1493–1505.
Xiao W, Cao L, Dong G, et al., Adaptive fuzzy control for pure-feedback systems with full state constraints and unknown nonlinear dead zone, Applied Mathematics and Computation, 2019, 343: 354–371.
Zhu B C, Zhang T P, and Wang F, Adaptive tracking control for a class of stochastic systems with dead-zone model, Journal of Systems Science and Complexity, 2012, 32(11): 1331–1342.
Sun Z Y, Shao Y, and Chen C C, Fast finite-time stability and its application in adaptive control of high-order nonlinear system, Automatica, 2019, 106: 339–348.
Liu J K, Robot Control System Design and Matlab Simulation, Tsinghua University Press, Beijing, 2008.
Yang Z and Zhang H, A fuzzy adaptive tracking control for a class of uncertain strick-feedback nonlinear systems with dead-zone input, Neurocomputing, 2018, 272: 130–135.
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This paper was supported by Scientific and Technological Innovation Programs of Higher Education Institutions in Shanxi (STIP) under Grant No. 2019L0011, and the Major Scientific and Technological Innovation Project in Shandong Province under Grant No. 2019JZZY011111.
This paper was recommended for publication by Editor YU Jinpeng.
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Liu, Z., Shi, Y. & Wu, Y. Tracking Control of Uncertain High-Order Nonlinear Systems with Odd Rational Powers and the Dead-Zone Input: A Direct Fuzzy Adaptive Control Method. J Syst Sci Complex 35, 1685–1699 (2022). https://doi.org/10.1007/s11424-022-1011-1
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DOI: https://doi.org/10.1007/s11424-022-1011-1