Trajectory tracking of quadrotor based on disturbance rejection control

33 11 216 11 DOI: 1.7641/CTA.216.6117 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] : 216 3 4; : 216 9 7.. E-mail: jbsu@sjtu.edu.cn. :. (6153312, 6152163). Supported by National Natural Science Foundation of China (6153312, 6152163).

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

1424 33 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)

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 5. 3.2 (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,

1426 33 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 2 2 + 2 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 2 3 +2ξ ξ T 3 P 3 B 3 d 3 ρ 2 s 2 2 η 2 s 2 λ min (Q 3 ) ξ 3 2 + 2 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 25 25 t)]t, d 3 =diag{2, 5sin( 2π 25 t)}.

11 : 1427 1 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 2 4. 4, t =s, t =1s t =5s, s 1 s 2,,., s 1 s 2. 5,,,,.

1428 33 5 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,,,

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