Stability analysis of linear/nonlinear switching active disturbance rejection control based MIMO continuous systems

doi:10.23919/JSEE.2021.000082

Journal of Systems Engineering and Electronics ›› 2021, Vol. 32 ›› Issue (4): 956-970.doi: 10.23919/JSEE.2021.000082

• CONTROL THEORY AND APPLICATION • Previous Articles Next Articles

Stability analysis of linear/nonlinear switching active disturbance rejection control based MIMO continuous systems

Hui WAN¹(), Xiaohui QI¹(), Jie LI^2,*()

¹ Department of Unmanned Aerial Vehicle Engineering, Army Engineering University of PLA, Shijiazhuang 050003, China
² The 32nd Research Institute, China Electronics Technology Group Corporation, Shanghai 201800, China

Received:2020-11-10 Online:2021-08-18 Published:2021-09-30
Contact: Jie LI E-mail:huiwan_0425@163.com;qi-xh@163.com;lijienewlife1234@163.com
About author:|WAN Hui was born in 1991. She received her B.E. and M.E. degrees from the Department of Unmanned Aerial Vehicle Engineering, the former Mechanical Engineering College, Shijiazhuang, China, in 2014 and 2016, respectively. She is currently working towards her Ph.D. degree in Army Engineering University. Her research interests include theory and application of active disturbance rejection control and flight control of unmanned aerial vehicle. E-mail: huiwan_0425@163.com||QI Xiaohui was born in1963. She received her B.E., M.E. and Ph.D. degrees from Nanjing University of Science and Technology, Nanjing, China, in 1983, 1988, 2010, respectively. She is currently a professor at the Department of Unmanned Aerial Vehicle Engineering, Army Engineering University of PLA. Her research interests include flight control theory and application for unmanned aerial vehicle, and adaptive control. E-mail: qi-xh@163.com||LI Jie was born in 1987. He received his B.E., M.E. and Ph.D. degrees from the Department of Unmanned Aerial Vehicle Engineering, the former Mechanical Engineering College, Shijiazhuang, China, in 2010, 2012, 2016, respectively. Currently, he is working towards his Ph.D. degree in Army Engineering University. His research interests include theory and application of active disturbance rejection control, particularly fully exerting advantages of the nonlinear mechanism. E-mail: lijienewlife1234@163.com
Supported by:
This work was supported by the Scientific Research Innovation Development Foundation of Army Engineering University ((2019)71)

Abstract

Abstract:

In this paper, a linear/nonlinear switching active disturbance rejection control (SADRC) based decoupling control approach is proposed to deal with some difficult control problems in a class of multi-input multi-output (MIMO) systems such as multi-variables, disturbances, and coupling, etc. Firstly, the structure and parameter tuning method of SADRC is introduced into this paper. Followed on this, virtual control variables are adopted into the MIMO systems, making the systems decoupled. Then the SADRC controller is designed for every subsystem. After this, a stability analyzed method via the Lyapunov function is proposed for the whole system. Finally, some simulations are presented to demonstrate the anti-disturbance and robustness of SADRC, and results show SADRC has a potential applications in engineering practice.

Key words: linear/nonlinear switching active disturbance rejection control (SADRC), multi-input multi-output (MIMO) continuous system, decoupling control, stability analysis

Hui WAN, Xiaohui QI, Jie LI. Stability analysis of linear/nonlinear switching active disturbance rejection control based MIMO continuous systems[J]. Journal of Systems Engineering and Electronics, 2021, 32(4): 956-970.

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References 37

1	HAN J Q, WANG W Nonlinear tracking-differentiator. Journal of System and Mathematical Science, 1994, 14 (3): 177- 183.
2	HAN J Q, YUAN L L The discrete form of tracking-differentiator. Journal of System and Mathematical Science, 1999, 19 (3): 268- 273.
3	HAN J Q Nonlinear state error feedback control law—NLSEF. Control and Decision, 1995, 10 (3): 221- 226.
4	HAN J Q The “extended state observer” of a class of uncertain systems. Control and Decision, 1995, 10 (1): 85- 88.
5	HAN J Q Auto-disturbance rejection controller and it’s applications. Control and Decision, 1998, 13 (1): 19- 23.
6	HAN J Q Linear and nonlinear in feedback systems. Control and Decision, 1988, 3 (2): 27- 32.
7	HAN J Q Control theory, is it a model analysis approach or a direct control approach. Journal of System and Mathematical Science, 1989, 9 (4): 328- 335.
8	ZHAO Z L, GUO B Z On convergence of nonlinear active disturbance rejection control for a class of nonlinear systems. Journal of Dynamical and Control Systems, 2016, 22 (2): 385- 412. doi: 10.1007/s10883-015-9304-5
9	GUO B Z, ZHAO Z L On convergence of the nonlinear active disturbance rejection control for MIMO systems. Journal on Control and Optimization, 2013, 51 (2): 1727- 1757. doi: 10.1137/110856824
10	ZHAO Z L, GUO B Z Active disturbance rejection control approach to stabilization of lower triangular systems with uncertainty. International Journal of Robust and Nonlinear Control, 2015, 26 (11): 2314- 2337.
11	WU D, CHEN K Frequency-domain analysis of nonlinear active disturbance rejection control via the describing function method. IEEE Trans. on Industrial Electronics, 2013, 60 (9): 3906- 3914. doi: 10.1109/TIE.2012.2203777
12	WU D, CHEN K Limit cycle analysis of active disturbance rejection control system with two nonlinearities. ISA Transactions, 2014, 53 (4): 947- 954. doi: 10.1016/j.isatra.2014.03.001
13	LI J, QI X H, XIA Y Q, et al Frequency domain stability analysis of nonlinear active disturbance rejection control system. ISA Transactions, 2015, 56 (5): 188- 195.
14	LI J, XIA Y Q, QI X H, et al Absolute stability analysis of nonlinear active disturbance rejection control for single-input-single-output systems via the circle criterion method. IET Control Theory & Applications, 2015, 9 (15): 2320- 2329.
15	QI X H, LI J, XIA Y Q, et al On the robust stability of active disturbance rejection control for SISO systems. Circuits, Systems, and Signal Processing, 2017, 35 (1): 65- 81.
16	LI J, XIA Y Q, QI X H Robust absolute stability analysis for interval nonlinear active disturbance rejection based control system. ISA Transactions, 2017, 69, 122- 130. doi: 10.1016/j.isatra.2017.04.017
17	WAN H. Stability and application for active-disturbance-rejection-controller. Beijing: Academy of Mathematics and Systems Science, 2001. (in Chinese)
18	LI J, QI X H, XIA Y Q, et al. On asymptotic stability for nonlinear ADRC based control system with application to the ball-beam problem. Proc. of American Control Conference, 2016: 4725–4730.
19	GAO Z Q. Scaling and bandwidth—parameterization based controller tuning. Proc. of American Control Conference, 2003: 4989–4996.
20	LI R H, CAO J H, LI T S Active disturbance rejection control design and parameters configuration for ship steering with wave disturbance. Control Theory & Applications, 2018, 35 (11): 1601- 1609.
21	YOO D, STEPHEN S T, GAO Z. On convergence of the linear ex-tended state observer. Proc. of the IEEE International Symposium on Intelligent Control, 2006: 1645–1650.
22	YANG X, HUANG Y. Capability of extended state observer for estimating uncertainties. Proc. of American Control Conference, 2009: 3700 – 3705.
23	SHAO L W, LIAO X Z, XIA Y Q, et al Stability analysis and synthesis of third order extended state observer. Information and Control, 2008, 37 (2): 135- 139.
24	YOO D, YAU S T, GAO Z Optimal fast tracking observer band-width of the linear extended state observer. International Journal of Control, 2007, 80 (1): 102- 111. doi: 10.1080/00207170600936555
25	HUANG Y, WANG J, SHI D. On convergence of extended state observers for discrete-time nonlinear systems. Proc. of the 34th Chinese Control Conference, 2015: 551 – 556.
26	ZHENG Q, GAO L Q, GAO Z. On stability analysis of active disturbance rejection control for nonlinear time-varying plants with unknown dynamics. Proc. of the 46th IEEE Conference on Decision and Control, 2007: 3501 – 3506.
27	CHEN Z Q, SUN M W, YANG R G On stability analysis of linear disturbance rejection control. Acta Automatica Sinica, 2013, 39 (5): 574- 580.
28	XUE W, HUANG Y. Comparison of the DOB based control, a special kind of PID control and ADRC. Proc. of American Control Conference, 2011: 4373 – 4379.
29	XUE W, HUANG Y. The active disturbance rejection control for a class of MIMO blocklower-triangular system. Proc. of the 31st Chinese Control Conference, 2011: 6362 – 6367.
30	XUE W, HUANG Y. On performance analysis of ADRC for non-linear uncertain systems with unknown dynamics and discontinuous disturbances. Proc. of the 32nd Chinese Control Conference, 2013: 465 – 470.
31	LI J, XIA Y Q, QI X, et al On the necessity, scheme and basis of the linear-nonlinear switching in active disturbance rejection control. IEEE Trans. on Industrial Electronics, 2017, 64 (2): 1425- 1435. doi: 10.1109/TIE.2016.2611573
32	LI J, QI X H, XIA Y Q, et al On linear/nonlinear active disturbance rejection control. Acta Automatica Sinica, 2016, 42 (2): 202- 212.
33	MOKHTARI R M, BRAHAM A C, CHERKI B Extended state observer based control for coaxial-rotor UAV. ISA Transactions, 2015, 11 (24): 1- 14.
34	SU S X, YANG H Z Active-disturbance-rejection dynamic nonlinear decoupling control for a class of multivariable systems. CIESC Journal, 2010, 61 (8): 1949- 1954.
35	LIU Q, LI D, TIAN W. Design of multi-loop ADRC controllers based on the effective open-loop transfer function method. Proc. of the 33rd Chinese Control Conference, 2014: 3649 – 3654.
36	LIU M, JI Y H, LI J F, et al Active disturbance rejection attitude control for quadrotor aircraft. Computer Simulation, 2016, 33 (3): 711- 75.
37	LI Y, CHEN Z Q, SUN M W, et al Attitude control for quadrotor helicopter based on discrete-time active disturbance rejection control. Control Theory & Applications, 2015, 32 (11): 1470- 1477.

Controller	y₁ channel	y₂ channel
LADRC^[35]	ξ = 1.83, w_o = 13, w_c = 1.1, b₀ = 10	ξ = 2.2, w_o = 2, w_c = 1.68, b₀ = ?14
NLADRC	r₁ = 10, h₁ = 0.45, c = 0.15, α₁ = 0.5, α₂ = 0.25, δ = 0.0005, β₀₁ = 1, β₀₂ = 1.11, β₀₃ = 0.1736	r₁ = 10, h₁ = 0.45, c = 0.15, α₁ = 0.5, α₂ = 0.25, δ = 0.0005, β₀₁ = 1, β₀₂ = 1.11, β₀₃ = 0.1736
SADRC	α₁ = 1, α₂ = 0.5, α₃ = 0.25, w_c = 1.1, w_o = 10, w_oN = 5, δ_s = 0.005, δ = 0.002, b₀ = 1, β₀₁ = 3 w_oN, β₀₂ = 3w_oN²/5, β₀₃ = w_oN³/9	α₁ = 1, α₂ = 0.5, α₃ = 0.25, w_c = 1.68, w_o = 10, w_o = 5, δ_s = 0.005, δ = 0.002, b₀ = 1, β₀₁ = 3w_oN, β₀₂ = 3w_oN²/5, β₀₃ = w_oN³/9

IAE	LADRC	NLADRC	SADRC
${{{\rm{IAE}}_y}_{_1}} $	2.66	2.73	2.73
${{{\rm{IAE}}_y}_{_2}} $	3.07	3.16	3.10
IAE	5.73	5.89	5.83

IAE	δ_s=0.003	δ_s=0.005	δ_s=0.008
${ { {\rm{IAE} }_y}_{_1} }$	2.80	2.73	2.80
$ { { {\rm{IAE} }_y}_{_2} } $	3.09	3.10	3.00
IAE	5.89	5.83	5.80

IAE	δ_s=0.003	δ_s=0.005	δ_s=0.008
$ { { {\rm{IAE} }_y}_{_1} } $	2.80	2.73	2.80
$ { { {\rm{IAE} }_y}_{_2} } $	3.09	3.10	3.00
IAE	5.89	5.83	5.80

Overshoot σ and IAE	LADRC	NLADRC	SADRC
${\sigma _{{y_{_1}}}} $	0	0	0
${\sigma _{ {y_{_2} } } } $	0	0	0
σ	0	0	0
${ { {\rm{IAE} }_y}_{_1} } $	1.60	1.65	1.62
${ { {\rm{IAE} }_y}_{_2} } $	2.33	2.47	2.42
IAE	3.93	4.12	4.04

Stability analysis of linear/nonlinear switching active disturbance rejection control based MIMO continuous systems

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Controller	$\phi $ channel	θ channel	ψ channel
LADRC^[36]	w_o = 28, w_c = 2.8, b₀ = 0.424	w_o = 30, w_c = 3, b₀ = 0.424	w_o = 30, w_c = 3.2, b₀ = 0.213
NLADRC^[37]	ESO: α₁ = 0.75, α₂ = 0.5, α₃ = 0.25, β₀₁ = 30, β₀₂ = 300, β₀₃ = 1000, b₀ = 0.9, δ = 0.006, NLESF: δ = 3, α₁ = 0. 5, α₂ = 0.05, β₁ = 150, β₂ = 120	ESO: α₁ = 0.75, α₂ = 0.5, α₃ = 0.25, β₀₁ = 30, β₀₂ = 300, β₀₃ = 1000, b₀ = 0.9, δ = 0.006, NLESF: δ = 3, α₁ = 0. 5, α₂ = 0.05, β₁ = 150, β₂ = 120	ESO: α₁ = 0.75, α₂ = 0.5, α₃ = 0.25, b₀ = 0.06, δ = 0.004, h = 0.0015, β₀₁ = 30, β₀₂ = 300, β₀₃ = 1000, NLESF: δ = 1, α₁ = 0. 5, α₂ = 0.05, β₁ = 300, β₂ = 180
SADRC	α₁ = 1, α₂ = 0.5, α₃ = 0.25, w_c = 2.8, w_o = 30, w_oN = 15, δ_s = 0.005, b₀ = 0.424, δ = 0.002, β₀₁ = 3w_oN, β₀₂ = 3w_oN²/5, β₀₃ = w_oN³/9	α₁ = 1, α₂ = 0.5, α₃ = 0.25, w_c = 3, w_o = 30, w_oN = 15, δ_s = 0.005, b₀ = 0.424, δ = 0.002, β₀₁ = 3w_oN, β₀₂ = 3w_oN²/5, β₀₃ = w_oN³/9	α₁ = 1, α₂ = 0.5, α₃ = 0.25, w_c = 3.2, w_o = 30, w_oN = 15, δ_s = 0.005, b₀ = 0.033, δ = 0.002, β₀₁ = 3w_oN, β₀₂ = 3w_oN²/5, β₀₃ = w_oN³/9

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[2]	Songtao Zhang, Xiaowei Zhao, and Jiantong Zhang. Stability and stabilization of discrete T-S fuzzy time-delay system based on maximal overlapped-rules group [J]. Systems Engineering and Electronics, 2016, 27(1): 201-.
[3]	Zhichao Zhou, Yang Xiao, and Dong Wang. Stability analysis of wireless network with improved fluid model [J]. Systems Engineering and Electronics, 2015, 26(6): 1149-1158.
[4]	Pan Xiong, Feng Wang, Xibin Cao, and Guangren Duan. Robust D-stability LMI conditions of matrix polytopes via affine parameter-dependent Lyapunov functions [J]. Journal of Systems Engineering and Electronics, 2013, 24(6): 984-991.
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[6]	Shi Wuxi. Indirect adaptive fuzzy control for a class of nonlinear discrete-time systems [J]. Journal of Systems Engineering and Electronics, 2008, 19(6): 1203-1207.
[7]	Lu Hongqian, Huang Xianlin, Gao X Z, Ban Xiaojun & Yin Hang. Stability analysis of the simplest Takagi-Sugeno fuzzy control system using circle criterion [J]. Journal of Systems Engineering and Electronics, 2007, 18(2): 311-319.

IAE	LADRC	NLADRC	SADRC
IAE_?	10.54	10.71	10.68
IAE_θ	10.61	11.13	10.84
IAE_ψ	13.31	13.93	13.30
IAE	34.46	35.77	34.82

IAE	δ_s=0.003	δ_s=0.005	δ_s=0.008
IAE_?	10.68	10.68	10.70
IAE_θ	10.81	10.84	10.82
IAE_ψ	13.30	13.30	13.25
IAE	34.79	34.82	34.77

IAE	δ_s=0.003	δ_s=0.005	δ_s=0.008
IAE_?	10.68	10.69	10.70
IAE_θ	10.82	10.85	10.83
IAE_ψ	13.31	13.31	13.29
IAE	34.81	34.85	34.82

Overshoot σ and IAE	LADRC	NLADRC	SADRC
σ_?	0	0	0
σ_θ	0	0	0
σ_ψ	0	0	0
σ	0	0	0
IAE_?	10.60	10.85	10.73
IAE_θ	10.62	10.81	10.71
IAE_ψ	12.42	13.86	13.36
IAE	33.64	35.52	34.80