Trajectory tracking of quadrotor based on disturbance rejection control

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1 DOI: /CTA Control Theory & Applications Vol. 33 No. 11 Nov. 216, (,, 224) :,,,.,,, ;,... : ; ; ; ; ; : TP273 : A Trajectory tracking of quadrotor based on disturbance rejection control WU Chen, SU Jian-bo (Department of Automation; Key Laboratory of System Control and Information Processing, Ministry of Education, Shanghai Jiao Tong University, Shanghai 224, China) Abstract: This paper studies trace tracking control problem of quadrotor aircraft. To deal with the model uncertainties and external disturbances in quadrotor flying process, a novel sliding mode control based on extended state observer is proposed. System uncertainties are depressed and trace tracking errors are uniformly ultimately bounded under the proposed control law. Firstly, system model is divided into two subsystem models according to the coupling relationship between state variables. Extended state observers are designed to estimate mismatched uncertainties in the subsystem, and estimated values are added into switching function. Disturbances in the reconstructed system are observed by extended state observers designed based on the switching function, and they are rejected in the controller design procedure. Finally, system stability is proven through Lyapunov theory. The simulation results demonstrate the efficiency of the proposed control scheme for quadrotor trace tracking and the efficiency for depressing chattering phenomenon. Key words: quadrotor aircraft; trace tracking control; mismatched uncertainties; extended state observer; switching function; sliding mode control 1 (Introduction),.,.,,.,.,. PID [1] [2], ; [3] [4], ; [5] [6] [7] : ; : jbsu@sjtu.edu.cn. :. ( , ). Supported by National Natural Science Foundation of China ( , ).

2 11 : 1423,.,..,., [8] [9] [1 11].,,,..,,, [12 13]., [14], [15]. [16] [17 18] [19] [2].,., [21] [22] [23].,,.,,,.,., ;,,.. 2 (System model and problem formulation) 1,. 2 4, 1 3, ( )1, 3 ( )2, 4. φ, θ, ψ, ẍ = 1 m (C φs θ C ψ + S φ S ψ )u 1 K 1ẋ m, ÿ = 1 m (C φs θ S ψ S φ C ψ )u 1 K 2ẏ m, z = 1 m (C φc θ )u 1 g K 3ż m, φ = θ ψ I yy I zz + l u 2 K 4l φ, I xx I xx I xx θ = ψ φ I zz I xx + l u 3 K 5l θ, I yy I yy I yy ψ = φ θ I xx I yy I zz + 1 I zz u 4 K 6l I zz ψ, (1) : m, I xx, I yy, I zz x, y, z, K 1 K 6, l, C ( ) S ( ) cos( ) sin( ) ; u 1, u 2, u 3, u 4 4, 4 F 1, F 2, F 3, F 4 u 1 = F 1 + F 2 + F 3 + F 4, u 2 = l( F 2 + F 4 ), (2) u 3 = l( F 1 + F 3 ), u 4 = C( F 1 + F 2 F 3 + F 4 ), C. 1 Fig. 1 Structure of quadrotor aircraft, x, y, z x d, y d, z d. ψ

3 ψ d,, ψ ψ d., Σ 1 Σ 2., Σ 1 ẋx 1 = x 2, ẋx 2 = N 2 x 2 + g 2 x 3 + d 1, Σ 1 : (3) ẋx 3 = g 3 x 4, ẋx 4 = N 4 x 4 + W 4 U 1 + d 2, : d 1, d 2, [ ] [ ] [ ] [ ] x ẋ S x 1 =,x 2 =,x 3 = θ θ,x 4 =, y ẏ S φ φ g 2 = u [ ] [ ] 1 C φ C ψ S ψ C,g 3 = θ, ˆm C φ S ψ C ψ C φ ˆK 1 N 2 = ˆm ˆK, 2 ˆm ˆK 5ˆl Î zz Îxx ψ Î N 4 = yy Î yy Î yy Îzz ψ ˆK 4ˆl, Î xx Î xx Î xx ˆl [ ] [ ] Î W 4 = yy ˆl,d Δ 1 = x Δ,d 2 = θ. Δ y Δ φ (4), Σ 2 {ẋx 5 = x 6, Σ 2 : (5) ẋx 6 = N 6 x 6 + g 6 U 2 + f 6 + d 3, d 3 Σ 2, x 5 = x 6 = [ ] z ψ [ ˆK 3 ˆm,N 6 = ˆK 6ˆl, ˆm g φ θ Îxx Îyy, ] ż,f 6 = ψ Î zz [ ] C φ C θ Δf d 3 = z,g 6 = ˆm Δf ψ ˆl. Î zz (6) 3 (Control system design),,, x, y, z. 3.1 (Design of switching function) : 1), ; 2),. Σ 1 Σ 2, s 1 s 2,,. s 1 d 1 ˆd 1,. s 1 e 1, e 2, e 3 e 4 e 1 = x 1 x 1d, e 2 = x 2 ẋx 1d, e 3 = N 2 x 2 + g 2 x 3 ẍx 1d + ˆd 1, e 4 = N 2 2 x 2 + N 2 g 2 x 3 + dg 2 dt x 3 + g 2 g 3 x 4... x 1d + N 2ˆd 1 + ˆd 1dot, (7) : ˆd 1 d 1, ˆd 1dot. ˆd 1 ˆd 1dot 2.2. e 1, e 2, e 3, e 4 d 1, d 2 ėe 1 = e 2, ėe 2 = e 3 d 1, ėe 3 = e 4 N 2 d 1 d 1dot, ėe 4 = N 3 2 x 2 + N 2 dg 2 x 3 2 g 2 x 3 + N 2 + dt d 2 g 2 d 2 t x 3 + g 2 g 3 (N 4 x 4 + W 4 U 1 ) x (4) 1d + N 2 2 d 1 + g 2 g 3 d 2 + dg 2 dt g 3x 4 + dg 2 g 3 x 4 + N 2ˆd 1dot + ˆd 1dot = dt F 1 (x 2,x 3,x 4 )+M 1 (x 3,x 4 )U 1 + N 2 2 d 1 + g 2 g 3 d 2 + N 2ˆd 1dot + ˆd 1dot. e 1, e 2, e 3, e 4 Σ 1 (8)

4 11 : 1425 s 1 = c 1 e 1 + c 2 e 2 + c 3 e 3 + e 4, (9) c 1, c 2 c 3. c 1, c 2 c 3 (24) Hurwitz, s 1, e 1. Σ 2, s 1. Σ 2 e 5 e 6 { e5 = x 5 x 5d, (1) e 6 = x 6 ẋx 5d. e 5 e 6 d 3 {ėe 5 = e 6, ėe 6 = g 6 U 2 + f 6 + d 3. e 5, e 6 Σ 2 (11) s 2 = c 5 e 5 + e 6, (12) c 5. c 5, s 2, e (Design of extended state observer) (ESO),. d 1, ESO, d 2 d 3 ESO. ESO, ESO,. Σ 1 d 1, ˆx 2 = N 2ˆx 2 + g 2 x 3 + ˆd 1 + h 1 (x 2 ˆx 2 ), ˆd 1 = ˆd 1dot + h 2 (x 2 ˆx 2 ), (13) ˆd 1dot = h 3 (x 2 ˆx 2 ), : ˆx 2 x 2, ˆd 1 d 1, ˆd 1dot d 1, h 1,h 2 h 3 ESO. Σ 1, s 1 d 2 d 1, d 2s. s 1 ŝs 1 = c 1 e 2 + c 2 e 3 + c 3 e 4 + F 1 (x 2,x 3,x 4 )+ M 1 (x 3,x 4 )U 1 + ˆd 2s + h 4 (s 1 ŝs 1 ), ˆd 2s = h 5 (s 1 ŝs 1 ), (14) : ŝs 1 s 1, ˆd 2s d 2s, h 4, h 5 ESO. Σ 2, d 3, s 2 ŝs 2 =c 5 e 6 +g 6 U 2 +f 6 +ˆd 3 +h 6 (s 2 ŝs 2 ), (15) ˆd 3 =h 7 (s 2 ŝs 2 ), : ŝs 2 s 2, ˆd 3 Σ 2 d 3, h 6, h 7 ESO. 3.3 (Controller design) ˆd 1, ˆd 2s ˆd 3,, U 1, U 2. Σ 1 : ṡs 1 = ρ 1 s 1 η 1 sgn s 1. (16),,,.,. (9)(14)(16), Σ 1 U 1 = M 1 1 (x 3,x 4 )[ρ 1 s 1 + η 1 sgn s 1 + c 1 e 2 + c 2 e 3 + c 3 e 4 + F 1 (x 2,x 3,x 4 )+ˆd 2s ]. (17) Σ 1, Σ 2 : ṡs 2 = ρ 2 s 2 η 2 sgn s 2. (18) (12)(15)(18), Σ 2 U 2 = g 1 6 [ρ 2 s 2 + η 2 sgn s 2 + c 5 e 6 + f 6 + ˆd 3 ]. (19) 4 (Stability analysis), Lyapunov. Σ 1 (17) e 1. Σ 1 d 1 [24] { ξ 1 = A 1 ξ 1 + B 1 d 1, (2) d 1 = C 1 ξ 1 + D 1 d 1, : h 3 A 1 = I 2 h 2 I 2 h 1 + N 2,

5 h 3 B 1 =,C 1 =[ I 2 ],D 1 =I 2. I 2. A 1 Hurwitz, Q 1, P 1 P 1 A 1 + A T 1 P 1 = Q 1. Σ 1 s 1 d 2s { ξ 2 = A 2 ξ 2 + B 2 d 2s, (21) d 2s = C 2 ξ 2 + D 2 d 2s, : [ ] h5 A 2 =, I 2 h 4 [ ] h5 B 2 =,C 2 =[ I 2 ],D 2 = I 2. A 2 Hurwitz, Q 2, P 2 P 2 A 2 + A T 2 P 2 = Q 2. s 1 ξ 1 ξ 2 Lyapunov V Σ1 V Σ1 = 1 2 st 1 s 1 + ξ T 1 P 1 ξ 1 + ξ T 2 P 2 ξ 2. (22) (22) Lyapunov V Σ1 V Σ1 = ρ 1 s T 1 s 1 η 1 s T 1 sgn s 1 ξ T 1 Q 1 ξ 1 + s T d 1 2s + 2ξ ξ T 1 P 1 B 1 d 1 ξ T 2 Q 2 ξ 2 +2ξ ξ T 2 P 2 B 2 d 2s ρ 1 s 1 2 η 1 s 1 + s 1 d 2s λ min (Q 1 ) ξ 1 2 λ min (Q 2 ) ξ P 1 B 1 ξ 1 d 1 +2 P 2 B 2 ξ 2 d 2s. (23) d 1 d 2s, (23) (17), s 1 d 1 d 2s. (8) (9), e 2, e 3, e 4 ėe 2 1 e 2 d 1 ėe 3 = 1 e 3 + N 2 d 1 d 1dot, ėe 4 c 1 c 2 c 3 e 4 R( d 1, d 1dot, d 2s,s 1 ) (24) R( d 1, d 1dot, d 2s,s 1 )=c 2 d 1 +c 3 N 2 d 1 +c 3 d 1dot ρ 1 s 1 η 1 sgn s 1 d 2s. c 1, c 2 c 3 Hurwitz, e 2, e 3, e 4. s 1, (9) e 1. Σ 2 (19). Σ 2 d 3 { ξ 3 = A 3 ξ 3 + B 3 d 3, (25) d 3 = C 3 ξ 3 + D 3 d 3, : [ ] h7 A 3 =, I 2 h 6 [ ] h7 B 3 =,C 3 =[ I 2 ],D 3 = I 2. A 3 Hurwitz, Q 3, P 3 P 3 A 3 + A T 3 P 3 = Q 3. s 2 ξ 2 Lyapunov V Σ2 V Σ2 = 1 2 st 2 s 2 + ξ T 3 P 3 ξ 3. (26) V Σ2 = ρ 2 s T 2 s 2 η 2 s T 2 sgn s 2 ξ T 3 Q 3 ξ 3 + s T d ξ ξ T 3 P 3 B 3 d 3 ρ 2 s 2 2 η 2 s 2 λ min (Q 3 ) ξ P 3 B 3 ξ 3 d 3 + s 2 d 3. (27) d 3, (19), s 2 d 3. (12), s 2, e 5. 5 (System simulation), MATLAB.. 1, x d (t) =1sin( 2π 5 t), y d(t) = 1 cos( 2π 5 t), z d (t) = 1 6 t.,.7 1.3,, t =1s, t=5 s. d 1, d 2 d 3 d 1 =[2 2] T,d 2 =[5sin( 2π t) 5sin(2π t)]t, d 3 =diag{2, 5sin( 2π 25 t)}.

6 11 : Table 1 Nominal parameters of quadrotor ˆm 1.2 kg g 9.81 m/s 2 ˆl.5 m Î xx.82 kg m 2 Î yy.82 kg m 2 Î zz.149 kg m 2 ˆK 1 ˆK 3.5 kg/s ˆK 4 ˆK 6.5 kg/rad 2 Fig. 2 Tracking error comparison figure,. c 1 = 324, c 2 = 666, c 3 =45,c 5 =5, ρ 1 =15,ρ 2 =5,η 1 =,η 2 =. h 1 =6I 2,h 2 =12I 2,h 3 =8I 2, h 4 =6I 2,h 5 =9I 2,h 6 =6I 2,h 7 =9I 2. 2, 2, 1 s, e y,e z,,,, s 1, s 2,. 1 5 s,,, ;,,. φ, θ, ψ 3,,. 3 Fig. 3 Attitude angles comparison figure s 1 s , t =s, t =1s t =5s, s 1 s 2,,., s 1 s 2. 5,,,,.

7 Fig. 5 System input comparison figure 4 Fig. 4 Switching function value comparison figure 6 Fig. 6 Tracking error figure under traditional sliding mode control,.,. c 1 = 225, c 2 = 675, c 3 =5,c 5 =5, ρ 1 =25,ρ 2 =15,η 1 =5,η 2 =5. η 1 η 2. 6,, e x,e y,e z,. 7,,,

8 11 : 1429,. 8,,. 8 Fig. 8 System input value under traditional sliding mode control 6 (Conclusions),,..,.,,.,,,. (References): 7 Fig. 7 Switching function value under traditional sliding mode control [1] MILHIM A, ZHANG Y M. Gain scheduling based PID controller for fault tolerant control of a quad-rotor UAV [C] //American Institute of Aeronautics and Astronautics Infotech@Aerospace 21. Atlanta, GA: AIAA, 21: [2] BOUSBDALLAH S, NOTH A, SIEGWART R. PID vs LQ control techniques applied to an indoor micro quadrotor [C] //Proceedings of IEEE/RSJ International Conference on Intelligent Robots and Systems. Sendal: IEEE, 24: [3] DAS A, LEWIS F, SUBBARAO K. Backstepping approach for controlling a quadrotor using lagrange form dynamics [J]. Journal of Intelligent and Robotic Systems, 29, 56(1/2): [4] VOOS I. Nonlinear control of a quadrotor micro-uav using feedback-linearization [C] //Proceedings of IEEE International Conference on Mechatronics.Málaga: IEEE, 29: 1 6. [5] DYDEK Z T, ANNASWAMY A M, LAVRETSKY E. Adaptive control of quadrotor UAVs: a design trade study with flight evaluations [J]. IEEE Transactions on Control Systems Technology, 213, 21(4): [6] DIERKS T, JAGANNATHAN S. Neural network output feedback control of a quadrotor UAV [C] //Proceedings of IEEE Conference on Decision and Control. Cancun, Mexico: IEEE, 28: [7] SUMANTRI B, UCHIYAMA N, SANO S, et al. Robust tracking control of a quad-rotor helicopter utilizing sliding mode control with a nonlinear sliding surface [J]. Journal of System Design and Dynamics, 213, 7(2): [8] FENG Y, YU X, MAN Z. Non-singular terminal sliding mode control of rigid manipulators [J]. Automatica, 22, 38(12):

9 [9] CHWA D. Sliding-mode tracking control of nonholonomic wheeled mobile robots in polar coordinates [J]. IEEE Transactions on Control Systems Technology, 24, 12(4): [1] XU R, ÖZGÜNER Ü. Sliding mode control of a quadrotor helicopter [C] //Proceedings of IEEE Conference on Decision and Control. San Diego: IEEE, 26: [11] XIONG J J, ZHENG E H. Position and attitude tracking control for a quadrotor UAV [J]. ISA Transactions, 214, 53(3): [12] HAN Jingqing. Auto-disturbance-rejection controller and it s applications [J]. Control and Decision, 1998, 13(1): (. [J]., 1998, 13(1): ) [13] HAN J. From PID to active disturbance rejection control [J]. IEEE transactions on Industrial Electronics, 29, 56(3): [14] YANG X, HUANG Y. Capabilities of extended state observer for estimating uncertainties [C] //Proceedings of American Control Conference. St. Louis: IEEE, 29: [15] GAO Z. Scaling and bandwidth-parameterization based controller tuning [C] //Proceedings of the American Control Conference. Minneapolis, MN: IEEE, 26: [16] YAO J, JIAO Z, MA D. Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping [J]. IEEE Transactions on Industrial Electronics, 214, 61(11): [17] WANG Lipeng, ZHANG Huaguang, LIU Xiuchong, et al. Backstepping controller based sliding mode variable structure for speed control of SPMSM with extended state observer [J]. Control and Decision, 211, 26(4): (,,,. SPMSM [J]., 211, 26(4): ) [18] LIU H, LI S. Speed control for PMSM servo system using predictive functional control and extended state observer [J]. IEEE Transactions on Industrial Electronics, 212, 59(2): [19] SU J, MA H, QIU W, et al. Task-independent robotic uncalibrated hand-eye coordination based on the extended state observer [J]. IEEE Transactions on Systems, Man, and Cybernetics, 24, 34(4): [2] TALOLE S E, KOLHE J P, PHADKE S B. Extended state observer based control of flexible-joint system with experimental validation [J]. IEEE Transactions on Industrial Electronics, 21, 57(4): [21] XIA Y, ZHU Z, FU M, et al. Attitude tracking of rigid spacecraft with bounded disturbances [J]. IEEE Transactions on Industrial Electronics, 211, 58(2): [22] WANG J, LI S, YANG J, et al. Extended state observer-based sliding mode control for PWM-based DC DC buck power converter systemsl with mismatched disturbances [J]. IET Control Theory & Applications, 215, 9(4): [23] GINOYA D, SHENDGE P D, PHADKE S B. Sliding mode control for mismatched uncertain systems using an extended disturbance observer [J]. IEEE Transactions on Industrial Electronics, 214, 61(4): [24] WANG Lu, SU Jianbo. Attitude tracking of aircraft based on disturbance rejection control [J]. Control Theory & Applications, 213, 3(12): (,. [J]., 213, 3(12): ) : (199 ),,,, wuchensjtu@126.com; (1969 ),,,, jbsu@sjtu.edu.cn.

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