Abstract
This paper presents an active disturbance rejection control (ADRC) method for a class of second-order nonlinear uncertain systems with guaranteed transient and steady state tracking error bounds. To deal with the tracking error constraint, an output error transformation and sliding surface technique are introduced to transform the constrained second-order system into an equivalent unconstrained first-order one. Then, an ADRC method is developed to achieve output tracking of the transformed uncertain system. The author proves that the closed-loop system is semi-globally uniformly bounded and it is sufficient to guarantee the tracking error constraint for the original system. Simulation results of a system of two inverted pendulums connected by a spring and a damper demonstrate the effectiveness of the proposed control method.
Similar content being viewed by others

References
Han J Q, From PID to active disturbance rejection control, IEEE Transactions on Industrial Electronics, 2009, 56(3): 900–906.
Gao Z Q, On the centrality of disturbance rejection in automatic control, ISA Transactions, 2014, 53(4): 850–857.
Huang Y and Xue W C, Active disturbance rejection control: Methodology and theoretical analysis, ISA Transactions, 2014, 53(4): 963–976.
Chen W H, Yang J, Guo L, et al., Disturbance-observer-based control and related methods — An Overview, IEEE Transactions on Automatic Control, 2016, 63(2): 1083–1095.
Wu Z H, Zhou H C, Guo B Z, et al., Review and new theoretical perspectives on active disturbance rejection control for uncertain finite-dimensional and infinite-dimensional systems, Nonlinear Dynamics, 2020, 101(12), DOI: https://doi.org/10.1007/s11071-020-05845-7.
Xue W C and Huang Y, Performance analysis of active disturbance rejection tracking control for a class of uncertain LTI systems, ISA Transactions, 2015, 58(8): 133–154.
Xue W C and Huang Y, Performance analysis of 2-DOF tracking control for a class of nonlinear uncertain systems with discontinuous disturbances, International Journal of Robust and Nonlinear Control, 2018, 28(4): 1456–1473.
Chen S, Bai W Y, Hu Y, et al., On the conceptualization of total disturbance and its profound implications, Science China Information Sciences, 2020, 63(2), 129201.
Chen S and Chen Z X, On active disturbance rejection control for a class of uncertain systems with measurement uncertainty, IEEE Transactions on Industrial Electronics, 2021, 68(2): 1475–1485.
Zhao Z L and Guo B Z, A novel extended state observer for output tracking of MIMO systems with mismatched uncertainty, IEEE Transactions on Automatic Control, 2018, 63(1): 211–218.
Zhao Z L and Guo B Z, On convergence of nonlinear active disturbance rejection control for SISO nonlinear systems, Journal of Dynamical and Control Systems, 2016, 22(2): 385–412.
Guo B Z and Zhao Z L, On convergence of the nonlinear active disturbance rejection control for mimo systems, SIAM Journal on Control and Optimization, 2013, 51(2): 1727–1757.
Xue W C, Zhang X C, Sun L, et al., Extended state filter based disturbance and uncertainty mitigation for nonlinear uncertain systems with application to fuel cell, IEEE Transactions on Industrial Electronics, 2020, 67(12): 10682–10692.
Sun L, Jin Y H, and You F Q, Active disturbance rejection temperature control of open-cathode proton exchange membrane fuel cell, Applied Energy, 2020, 261(9): 114381.
Zheng Q L, Ping Z, Soares S, et al., Active disturbance rejection temperature control of open-cathode proton exchange membrane fuel cell, Control Engineering Practice, 2018, 79(6): 154–169.
Bechlioulis C P and Rovithakis G A, Robust adaptive control of feedback linearizable mimo nonlinear systems with prescribed performance, IEEE Transactions on Automatic Control, 2008, 53(9): 2090–2099.
Bechlioulis C P and Rovithakis G A, Robust partial-state feedback prescribed performance control of cascade systems with unknown nonlinearities, IEEE Transactions on Automatic Control, 2011, 56(9): 2224–2230.
Bechlioulis C P and Rovithakis G A, Prescribed performance control for discontinuous output reference tracking, IEEE Transactions on Automatic Control, 2021, 66(9): 4409–4416.
Lee J, Mukherjee R, and Khalil H K, Output feedback stabilization of inverted pendulum on a cart in the presence of uncertainties, Automatica, 2015, 54: 146–157.
Ramirez-Neria M, Sira-Ramirez H, Garrido-Moctezuma R, et al., Linear active disturbance rejection control of underactuated systems: The case of the Furuta pendulum, ISA Transactions, 2014, 53(4): 920–928.
Shojaei K, Kazemy A, and Chatraei A, An observer-based neural adaptive PID2 controller for robot manipulators including motor dynamics with a prescribed performance, IEEE/ASME Transactions on Mechatronics, 2021, 26(3): 1689–1699.
Wang X, Wang Q, and Sun C, Prescribed performance fault-tolerant control for uncertain nonlinear MIMO system using actor-critic learning structure, IEEE Transactions on Neural Networks and Learning Systems, 2021, (99): 1–12, DOI: https://doi.org/10.1109/TNNLS.2021.3057482.
Freidovich L B and Khalil Hassan K, Performance recovery of feedback-linearization-based designs, IEEE Transactions on Automatic Control, 2008, 53(10): 2324–2334.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by the Key Laboratory of Systems and Control, Chinese Academy of Sciences.
Rights and permissions
About this article
Cite this article
Chen, Z. Active Disturbance Rejection Control of Second-Order Nonlinear Uncertain Systems with Guaranteed Transient and Steady State Tracking Error Bounds. J Syst Sci Complex 35, 1293–1309 (2022). https://doi.org/10.1007/s11424-022-1010-2
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-022-1010-2