Adaptive Neural Network-Based Control for a Class of Nonlinear Pure-Feedback Systems With Time-Varying Full State Constraints

Tingting Gao^{1}, Yan-Jun Liu^{1}, Lei Liu^{1}, Dapeng Li^{2}

1. College of Science, Liaoning University of Technology, Jinzhou 121000, China; 2. School of Electrical Engineering, Liaoning University of Technology, Jinzhou 121001, China

Abstract In this paper, an adaptive neural network (NN) control approach is proposed for nonlinear pure-feedback systems with time-varying full state constraints. The pure-feedback systems of this paper are assumed to possess nonlinear function uncertainties. By using the mean value theorem, pure-feedback systems can be transformed into strict feedback forms. For the newly generated systems, NNs are employed to approximate unknown items. Based on the adaptive control scheme and backstepping algorithm, an intelligent controller is designed. At the same time, time-varying Barrier Lyapunov functions (BLFs) with error variables are adopted to avoid violating full state constraints in every step of the backstepping design. All closedloop signals are uniformly ultimately bounded and the output tracking error converges to the neighborhood of zero, which can be verified by using the Lyapunov stability theorem. Two simulation examples reveal the performance of the adaptive NN control approach.

Fund:This work was supported in part by the National Natural Science Foundation of China (61622303, 61603164, 61773188), the Program for Liaoning Innovative Research Team in University (LT2016006), the Fundamental Research Funds for the Universities of Liaoning Province (JZL201715402), and the Program for Distinguished Professor of Liaoning Province.

Tingting Gao, Yan-Jun Liu, Lei Liu, Dapeng Li, "Adaptive Neural Network-Based Control for a Class of Nonlinear Pure-Feedback Systems With Time-Varying Full State Constraints," IEEE/CAA Journal of Automatica Sinica, vol. 5, no. 5, pp. 923-933, 2018.

[1] Q. L. Wei, D. R. Liu, Q. Lin, and R. Z. Song, "Discrete-time optimal control via local policy iteration adaptive dynamic programming," IEEE Trans. Cybern., vol. 47, no. 10, pp. 3367-3379, Oct. 2017. [2] C. C. Hua, X. P. Guan, and P. Shi, "Robust backstepping control for a class of time delayed systems," IEEE Trans. Autom. Control, vol. 50, no. 6, pp. 894-899, Jun. 2005. [3] Y. Q. Xia, M. Y. Fu, P. Shi, Z. J. Wu, and J. H. Zhang, "Adaptive backstepping controller design for stochastic jump systems," IEEE Trans. Autom. Control, vol. 54, no. 12, pp. 2853-2859, Dec. 2009. [4] M. Wang, X. Y. Wang, B. Chen, and S. C. Tong, "Robust adaptive fuzzy tracking control for a class of strict-feedback nonlinear systems based on backstepping technique," J. Control Theor. Technol., vol. 5, no. 3, pp. 317-322, Aug. 2007. [5] P. Li and G. H. Yang, "A novel adaptive control approach for nonlinear strict-feedback systems using nonlinearly parameterised fuzzy approximators," Int. J. Syst. Sci., vol. 42, no. 3, pp. 517-527, Mar. 2011. [6] T. S. Li, S. C. Tong, and G. Feng, "A novel robust adaptive-fuzzytracking control for a class of nonlinear multi-input/multi-output systems," IEEE Trans. Fuzzy Syst., vol. 18, no. 1, pp. 150-160, Feb. 2010. [7] J. P. Yu, P. Shi, W. J. Dong, and C. Lin, "Adaptive fuzzy control of nonlinear systems with unknown dead zones based on command filtering," IEEE Trans. Fuzzy Syst., vol. 26, no. 1, pp. 46-55, Feb. 2018. [8] Y. X. Li, G. H. Yang, and S. C. Tong, "Fuzzy adaptive distributed event-triggered consensus control of uncertain nonlinear multi-agent systems," IEEE Trans. Syst. Man Cybern.:Syst., doi:10.1109/TSMC.2018. 2812216. [9] M. Z. Hou, Z. Q. Deng, and G. R. Duan, "Adaptive control of uncertain pure-feedback nonlinear systems," Int. J. Syst. Sci., vol. 48, no. 10, pp. 2137-2145, Jul. 2017. [10] Q. L. Wei, D. R. Liu, and H. Q. Lin, "Value iteration adaptive dynamic programming for optimal control of discrete-time nonlinear systems," IEEE Trans. Cybern., vol. 46, no. 3, pp. 840-853, Mar. 2016. [11] Q. L. Wei, F. L. Lewis, D. R. Liu, R. Z. Song, and H. Q. Lin, "Discretetime local value iteration adaptive dynamic programming:convergence analysis," IEEE Trans. Syst. Man Cybern.:Syst., vol. 18, no. 1, pp. 150-160, Feb. 2018. [12] S. S. Ge and C. Wang, "Adaptive NN control of uncertain nonlinear pure-feedback systems," Automatica, vol. 38, no. 4, pp. 671-682, Apr. 2002. [13] B. B. Ren, S. S. Ge, K. P. Tee, and T. H. Lee, "Adaptive neural control for output feedback nonlinear systems using a Barrier Lyapunov function," IEEE Trans. Neur. Netw., vol. 21, no. 8, pp. 1339-1345, Aug. 2010. [14] F. Wang, B. Chen, C. Lin, J. Zhang, and X. Z. Meng, "Adaptive neural network finite-time output feedback control of quantized nonlinear systems," IEEE Trans. Cybern., vol. 48, no. 6, pp. 1839-1848, Jun. 2018. [15] W. Si and W. Zeng, "Adaptive neural output-feedback control for nonstrict-feedback stochastic nonlinear time-delay systems with hysteresis," IEEE/CAA J. of Autom. Sinica, 2017, doi:10.1109/JAS.2017. 7510451. [16] H. Wang, K. Liu, X. Liu, B. Chen, and C. Lin, "Neural-based adaptive output-feedback control for a class of nonstrict-feedback stochastic nonlinear systems," IEEE Trans. Cybern., vol. 45, no. 9, pp. 1977-1987, Sep. 2015. [17] C. Peng, Y. Bai, X. Gong, Q. J. Gao, C. J. Zhao, and Y. T. Tian, "Modeling and robust backstepping sliding mode control with adaptive RBFNN for a novel coaxial eight-rotor UAV," IEEE/CAA J. of Autom. Sinica, vol. 2, no. 1, pp. 56-64, Jan. 2015. [18] X. M. Sun and S. S. Ge, "Adaptive neural region tracking control of multi-fully actuated ocean surface vessels," IEEE/CAA J. of Autom. Sinica, vol. 1, no. 1, pp. 77-83, Jan. 2014. [19] Y. Yang and D. Yue, "Distributed tracking control of a class of multi-agent systems in non-affine pure-feedback form under a directed topology," IEEE/CAA J. of Autom. Sinica, vol. 5, no. 1, pp. 169-180, Jan. 2018. [20] R. W. Zuo, X. M. Dong, Y. Chen, Z. C. Liu, and C. Shi, "Adaptive neural control for a class of non-affine pure-feedback nonlinear systems," Int. J. Control, doi:10.1080/00207179.2017.1393106. [21] S. S. Ge and C. Wang, "Direct adaptive NN control of a class of nonlinear systems," IEEE Trans. Neur. Netw., vol. 13, no. 1, pp. 214-221, Jan. 2002. [22] L. Liu, Z. S. Wang, and H. G. Zhang, "Adaptive fault-tolerant tracking control for MIMO discrete-time systems via reinforcement learning algorithm with less learning parameters," IEEE Trans. Autom. Sci. Eng., vol. 14, no. 1, pp. 299-313, Jan. 2017. [23] H. Q. Wang, P. Shi, H. Y. Li, and Q. Zhou, "Adaptive neural tracking control for a class of nonlinear systems with dynamic uncertainties," IEEE Trans. Cybern., vol. 47, no. 10, pp. 3075-3087, Oct. 2017. [24] Y. J. Liu, S. Li, S. C. Tong, and C. L. P. Chen, "Neural approximationbased adaptive control for a class of nonlinear nonstrict feedback discrete-time systems," IEEE Trans. Neur. Netw. Learn. Syst., vol. 28, no. 7, pp. 1531-1541, Jul. 2017. [25] Z. Liu, G. Y. Lai, Y. Zhang, X. Chen, and C. L. P. Chen, "Adaptive neural control for a class of nonlinear time-varying delay systems with unknown hysteresis," IEEE Trans. Neur. Netw. Learn. Syst., vol. 25, no. 12, pp. 2129-2140, Dec. 2014. [26] M. Chen and S. S. Ge, "Direct adaptive neural control for a class of uncertain nonaffine nonlinear systems based on disturbance observer," IEEE Trans. Cybern., vol. 43, no. 4, pp. 1213-1225, Aug. 2013. [27] M. Chen, P. Shi, and C. C. Lim, "Adaptive neural fault-tolerant control of a 3-DOF model helicopter system," IEEE Trans. Syst. Man Cybern.:Syst., vol. 46, no. 2, pp. 260-270, Feb. 2016. [28] T. S. Li, Z. F. Li, D. Wang, and C. L. P. Chen, "Output-feedback adaptive neural control for stochastic nonlinear time-varying delay systems with unknown control directions," IEEE Trans. Neur. Netw. Learn. Syst., vol. 26, no. 6, pp. 1188-1201, Jun. 2015. [29] J. P. Yu, B. Chen, H. S. Yu, C. Lin, and L. Zhao, "Neural networksbased command filtering control of nonlinear systems with uncertain disturbance," Inf. Sci., vol. 426, pp. 50-60, Feb. 2018. [30] Z. Wang, Y. Xu, R. Q. Lu, and H. Peng, "Finite-time state estimation for coupled markovian neural networks with sensor nonlinearities," IEEE Trans. Neur. Netw. Learn. Syst., vol. 28, no. 3, pp. 630-638, Mar. 2017. [31] Z. Wang, R. Q. Lu, F. R. Gao, and D. R. Liu, "An indirect data-driven method for trajectory tracking control of a class of nonlinear discretetime systems," IEEE Trans. Ind. Electron., vol. 64, no. 5, pp. 4121-4129, May 2017. [32] Q. L. Wei, F. L. Lewis, Q. Y. Sun, P. F. Yan, and R. Z. Song, "Discretetime deterministic Q-learning:a novel convergence analysis," IEEE Trans. Cybern., vol. 47, no. 5, pp. 1224-1237, May 2017. [33] L. Liu, Y. J. Liu, and S. C. Tong, "Neural networks-based adaptive finitetime fault-tolerant control for a class of strict-feedback switched nonlinear systems," IEEE Trans. Cybern., doi:10.1109/TCYB.2018.2828308. [34] A. Bemporad, "Reference governor for constrained nonlinear systems," IEEE Trans. Autom. Control, vol. 43, no. 3, pp. 415-419, Mar. 1998. [35] R. Q. Lu, Y. Xu, and R. D. Zhang, "A new design of model predictive tracking control for networked control system under random packet loss and uncertainties," IEEE Trans. Ind. Electron., vol. 63, no. 11, pp. 6999-7007, Nov. 2016. [36] M. Sampei, H. Kiyota, and M. Ishikawa, "Control strategies for mechanical systems with various constraints-control of non-holonomic systems," in Proc. 1999 IEEE Int. Conf. Systems, Man, and Cybernetics, Tokyo, Japan, pp. 158-165. [37] Z. Liu, G. Y. Lai, Y. Zhang, and C. L. P. Chen, "Adaptive neural output feedback control of output-constrained nonlinear systems with unknown output nonlinearity," IEEE Trans. Neur. Netw. Learn. Syst., vol. 26, no. 8, pp. 1789-1802, Aug. 2015. [38] K. P. Tee, S. S. Ge, and E. H. Tay, "Barrier Lyapunov functions for the control of output-constrained nonlinear systems," Automatica, vol. 45, no. 4, pp. 918-927, Apr. 2009. [39] H. Y. Li, L. J. Wang, H. P. Du, and A. Boulkroune, "Adaptive fuzzy backstepping tracking control for strict-feedback systems with input delay," IEEE Trans. Fuzzy Syst., vol. 25, no. 3, pp. 642-652, Jun. 2017. [40] K. P. Tee and S. S. Ge, "Control of nonlinear systems with full state constraint using a Barrier Lyapunov function," in Proc. 48h IEEE Conf. Decision and Control (CDC) held jointly with 28th Chinese Control Conf., Shanghai, China, 2009, pp. 8618-8623. [41] Y. J. Liu and S. C. Tong, "Barrier Lyapunov functions for Nussbaum gain adaptive control of full state constrained nonlinear systems," Automatica, vol. 76, pp. 143-152, Feb. 2017. [42] K. P. Tee and S. S. Ge, "Control of nonlinear systems with partial state constraints using a Barrier Lyapunov function," Int. J. Control, vol. 84, no. 12, pp. 2008-2023, Dec. 2011. [43] Y. J. Liu, S. M. Lu, S. C. Tong, X. K. Chen, C. L. P. Chen, and D. J. Li, "Adaptive control-based Barrier Lyapunov functions for a class of stochastic nonlinear systems with full state constraints," Automatica, vol. 87, pp. 83-93, Jun. 2018. [44] Y. J. Liu, M. Z. Gong, S. C. Tong, C. L. P. Chen, and D. J. Li, "Adaptive fuzzy output feedback control for a class of nonlinear systems with full state constraints," IEEE Trans. Fuzzy Syst., doi:10.1109/TFUZZ.2018. 2798577. [45] D. J. Li, S. M. Lu, Y. J. Liu, and D. P. Li, "Adaptive fuzzy tracking control-based Barrier functions of uncertain nonlinear MIMO systems with full state constraints and applications to chemical process," IEEE Trans. Fuzzy Syst., doi:10.1109/TFUZZ.2017.2765627. [46] L. Bai, H. Y. Li, H. J. Liang, Q. Zhou, and L. J. Wang, "Adaptive fuzzy control for nonstrict-feedback stochastic nonlinear systems with full-state constraints and unknown dead zone," in Proc. 4th Int. Conf. Information, Cybernetics and Computational Social Systems (ICCSS), Dalian, China, 2017, pp. 26-31. [47] D. J. Li and D. P. Li, "Adaptive controller design-based neural networks for output constraint continuous stirred tank reactor," Neurocomputing, vol. 153, pp. 159-163, Apr. 2015. [48] B. S. Kim and S. J. Yoo, "Approximation-based adaptive control of uncertain non-linear pure-feedback systems with full state constraints," IET Control Theor. Appl., vol. 8, no. 17, pp. 2070-2081, Nov. 2014. [49] W. He, Y. H. Chen, and Z. Yin, "Adaptive neural network control of an uncertain robot with full-state constraints," IEEE Trans. Cybern., vol. 46, no. 3, pp. 620-629, Mar. 2016. [50] D. P. Li, Y. J. Liu, S. C. Tong, C. L. P. Chen, and D. J. Li, "Neural networks-based adaptive control for nonlinear state constrained systems with input delay," IEEE Trans. Cybern., doi:10.1109/TCYB.2018. 2799683. [51] Y. J. Liu, S. C. Tong, C. L. P. Chen, and D. J. Li, "Adaptive NN control using integral Barrier Lyapunov functionals for uncertain nonlinear block-triangular constraint systems," IEEE Trans. Cybern., vol. 47, no. 11, pp. 3747-3757, Nov. 2017. [52] Q. Zhou, L. J. Wang, C. W. Wu, H. Y. Li, and H. P. Du, "Adaptive fuzzy control for nonstrict-feedback systems with input saturation and output constraint," IEEE Trans. Syst. Man Cybern.:Syst., vol. 47, no. 1, pp. 1-12, Jan. 2017. [53] H. Y. Li, L. Bai, L. J. Wang, Q. Zhou, and H. Q. Wang, "Adaptive neural control of uncertain nonstrict-feedback stochastic nonlinear systems with output constraint and unknown dead zone," IEEE Trans. Syst. Man Cybern.:Syst., vol. 47, no. 8, pp. 2048-2059, Aug. 2017. [54] Y. J. Liu, S. M. Lu, and S. C. Tong, "Neural network controller design for an uncertain robot with time-varying output constraint," IEEE Trans. Syst. Man Cybern.:Syst., vol. 47, no. 8, pp. 2060-2068, Aug. 2017. [55] K. P. Tee, B. B. Ren, and S. S. Ge, "Control of nonlinear systems with time-varying output constraints," Automatica, vol. 47, no. 11, pp. 2511-2516, Nov. 2011. [56] B. S. Kim and S. J. Yoo, "Approximation-based adaptive tracking control of nonlinear pure-feedback systems with time-varying output constraints," Int. J. Control Autom. Syst., vol. 13, no. 2, pp. 257-265, Apr. 2015. [57] D. P. Li, D. J. Li, Y. J. Liu, S. C. Tong, and C. L. P. Chen, "Approximation-based adaptive neural tracking control of nonlinear MIMO unknown time-varying delay systems with full state constraints," IEEE Trans. Cybern., vol. 47, no. 10, pp. 3100-3109, Oct. 2017. [58] D. P. Li and D. J. Li, "Adaptive neural tracking control for an uncertain state constrained robotic manipulator with unknown time-varying delays," IEEE Trans. Syst. Man Cybern.:Syst., doi:10.1109/TSMC.2017. 2703921. [59] Y. J. Liu, S. M. Lu, D. J. Li, and S. C. Tong, "Adaptive controller design-based ABLF for a class of nonlinear time-varying state constraint systems,"IEEE Trans. Syst. Man Cybern.:Syst., vol. 47, no. 7, pp. 1546-1553, Jul. 2017. [60] S. M. Lu, D. P. Li, and Y. J. Liu, "Adaptive neural network control for uncertain time-varying state constrained robotics systems," IEEE Trans. Syst. Man Cybern.:Syst., doi:10.1109/TSMC.2017.2755377. [61] L. Ma and D. P. Li, "Adaptive neural networks control using Barrier Lyapunov functions for DC motor system with time-varying state constraints," Complexity, vol. 2018, pp. Article No. 5082401, Jan. 2018. [62] C. X. Wang, Y. Q. Wu, and J. B. Yu, "Barrier Lyapunov functions-based adaptive control for nonlinear pure-feedback systems with time-varying full state constraints," Int. J. Control Autom. Syst., vol. 15, no. 6, pp. 2714-2722, Dec. 2017.