[15 17],,. [18 20], ESO, ESO,, ESO,,.,,,,.,, ;,,. 2 (Spacecraft attitude dynamics),, I, ω I b, T d, u, I ω I b + (ωb) I Iω I b = u + T d, (1)

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1 DOI: /CTA Control Theory & Applications Vol. 30 No. 12 Dec. 2013,,, (, ) :,. (ADRC) (TD) (ESO) (AFC)3.,. (ESO),. ESO,.,,.,.,. : ; ; ; ; : V : A Active disturbance rejection control of attitude for spacecraft WU Zhong, HUANG Li-ya, WEI Kong-ming, GUO Lei (School of Instrumentation Science and Optoelectronics Engineering, Beihang University, Beijing , China) Abstract: In order to attenuate the effects of the parameter variations and disturbances of the spacecrafts on attitude control accuracy and stability, an active disturbance rejection controller (ADRC) is designed. ADRC consists of three parts: tracking differentiator (TD), extended state observer (ESO) and attitude feedback controller (AFC). TD smoothes the process of attitude maneuver and provides differential signal of the attitude; ESO estimates the attitude and the disturbances acting on the spacecraft by taking full advantage of the information of the gyros and the attitude sensors; AFC realizes the attitude control of the spacecraft by compensating the disturbances from the ESO. Compared with the relative results, ESO has better performance by adopting the composite measurement information to correct the estimate, and ADRC has simpler structure by adopting only one loop to realize attitude control and disturbance compensation. Simulation results of a certain spacecraft demonstrate that the ADRC of this paper is feasible. Key words: spacecraft; attitude control; active disturbance rejection controller; attitude sensors; rate gyros 1 (Introduction),,.,,,,,., (active disturbance rejection controller, ADRC),,, [1 4],.,,, [5 10]. (tracking differentiator, TD) (extended state observer, ESO) (nonlinear state error feedback, NLSEF)3.,,,., [11] [12],. [13 14],.,,. : ; : wuzhong@buaa.edu.cn; Tel.: : ( 973 (2012CB720003); ( ).

2 [15 17],,. [18 20], ESO, ESO,, ESO,,.,,,,.,, ;,,. 2 (Spacecraft attitude dynamics),, I, ω I b, T d, u, I ω I b + (ωb) I Iω I b = u + T d, (1) 0 ω 3 ω 2 ω 1 : (ωb) I = ω 3 0 ω 1, ωb I = ω 2. ω 2 ω 1 0 ω 3 (single gimbal control moment gyroscopes, SGCMG), SGCMG h, ḣ + (ωb) I h = u. (2) (modified Rodrigues parameters, MRP), p, ω o b, ṗ = G (p) ω o b, (3) : G(p) = 1 2 {E + p + pp T [ 1 2 (1 + pt p)]e}, E. 3 (ESO design) 3.1 (ESO based on attitude sensors and rate gyros) x 1 = p, x 2 = ṗ, (1) (2), {ẋ1 = x 2, (4) ẋ 2 = f (x 1, x 2 ) + Bτ, : f(x 1, x 2 ) = 4 (1 + p T p) 2 GT (ω I b C b o ω I o) + GI 1 [ (ω I b) (Iω I b + h) + T d ] G(C b o ω I o + Ċb o ω I o), ; B = G(p)I 1, ωo I, τ = ḣ SGCMG. x 3 = f(x 1, x 2 ), ẋ 3 = a(t), (4) ẋ 1 = x 2, ẋ 2 = x 3 + Bτ, (5) ẋ 3 = a.,., y p, y p = p + n p, (6) n p. y ω, n ω. y ω = ω I b + n ω, (7) ω I b = ω o b + C b o ω I o. (8) (2),, ṗ : ṗ = G(y p )(y ω C b o ω I o), (9) C b o., (5), (6) (7). x 1 z 1, x 2 z 2, x 3 z 3, e=α(y p z 1 )+β[g(y p )(y ω C b o ω I o) z 2 ], (10) : ż 1 = z 2 + K 1 e, ż 2 = z 3 + K 2 e + Bτ, ż 3 = K 3 e, (11) : K 1, K 2 K 3, α, β (ESO analysis),, e 1 = x 1 z 1, e 2 = x 2 z 2, e 3 = x 3 z 3, e e = αe 1 + βe 2. (12)

3 12 : 1619 (5)(11) (12), : ė 1 K 1 α 1 K 1 β 0 e 1 0 ė 2 = K 2 α K 2 β 1 e (13) ė 3 K 3 α K 3 β 0 e 3 a 3,., K 1, K 2 K 3,,. : K 1 = diag{k 11, K 12, K 13 }, K 2 = diag{k 21, K 22, K 23 }, K 3 = diag{k 31, K 32, K 33 }, e 1 = [e 11 e 12 e 13 ] T, e 2 = [e 21 e 22 e 23 ] T, e 3 = [e 31 e 32 e 33 ] T, a = [a 1 a 2 a 3 ] T, (13) : ė 11 = K 11 αe 11 + (1 K 11 β)e 21, ė 12 = K 12 αe 12 + (1 K 12 β)e 22, ė 13 = K 13 αe 13 + (1 K 13 β)e 23, ė 21 = K 21 αe 11 K 21 βe 21 + e 31, ė 22 = K 22 αe 12 K 22 βe 22 + e 32, ė 23 = K 23 αe 13 K 23 βe 23 + e 33, ė 31 = K 31 αe 11 K 31 βe 21 + a 1, ė 32 = K 32 αe 12 K 32 βe 22 + a 2, ė 33 = K 33 αe 13 K 33 βe 23 + a 3. (14) (14), 3, 2. (14), 2 ė 12 K 12 α 1 K 12 β 0 ė 22 = K 22 α K 22 β 1 ė 32 K 32 α K 32 β 0 e 12 e 22 e 32 (15), s 3 + (K 12 α + K 22 β) s a 2. (15) (K 22 α + K 32 β) s + K 32 α = 0. (16), α, β, K 12, K 22 K 32 0, (K 12 α + K 22 β)(k 22 α + K 32 β) > K 32 α,,., a 2,, [13 14].,. 3.3 (ESO parameter tuning) 3.2,. 2, λ 1, λ 2, λ 3, (s + λ 1 )(s + λ 2 )(s + λ 3 ) = 0. (17), (16), : λ 1 + λ 2 + λ 3 = K 12 α + K 22 β, λ 1 λ 2 + λ 1 λ 3 + λ 2 λ 3 = K 22 α + K 32 β, (18) λ 1 λ 2 λ 3 = K 32 α.,., α, β, 3 3 K 12, K 22 K 32. α = 1, β = 0, (11), ; α=1, β =1, (11),, ; 0 < α < 1, 0 < β < 1, (11),,,.,,. 4 (ADRC design) 3,,., (TD) (attitude feedback controller, AFC),, 1. 1 Fig. 1 Schematics of ADRC for spacecraft attitude

4 , TD, SGCMG. AFC ESO, TD ESO. 4.1 (TD design) p d, SGCMG,., TD,,., TD, : { v 1 = v 2, v 2 = 2rv 2 r 2 (19) (v 1 p d ), : r TD ; v 2, p d,, SGCMG. 4.2 (AFC design), { τ 0 = k 1 (v 1 z 1 ) + k 2 (v 2 z 2 ), τ = B 1 (20) (τ 0 z 3 ), k 1 k 2. 5 (Simulation and analysis), MATLAB/Simulink., 4 SGCMG. n = rad/s, I = kg m SGCMG, σ 0 = [ ] T, SGCMG 300Nms, sin(nt) sin(2nt) T d = sin(nt) sin(2nt) N m sin(nt) sin(2nt), ω(t 0 ) = [0 0 0] T ( )/s, [φ 0 θ 0 ψ 0 ] T = [0 0 0 ] T. ω d = [0 0 0] T ( )/s, [φ d θ d ψ d ] T = [0 0 0 ] T. ESO K 1 =diag{20, 20, 20}, K 2 =diag{100, 100, 100}, K 3 = diag{800, 800, 800}. k 1 =0.3, k 2 =10. TD r = ) ESO. ESO, ESO. ESO, (11) α=1, β =0. ESO, (11) α =1, β =0.5., ESO , ESO., ESO,. 2 ESO Fig. 2 Attitude of ESO based on attitude sensors 3 ESO Fig. 3 Attitude rate of ESO based on attitude sensors 4 ESO Fig. 4 Attitude of ESO based on composite measurement information

5 12 : ESO Fig. 5 Attitude velocity of ESO based on composite measurement information 2)., ESO, ,,,,. 8 SGCMG,. 9 SGCMG, 0.02 rad/s,. 11 SGCMG, SGCMG,.,,,,,. 8 Fig. 8 Desired control torque 9 SGCMG Fig. 9 SGCMG gimbal rates 10 SGCMG Fig. 10 SGCMG gimbal angle 6 Fig. 6 Attitude angle of spacecraft 11 SGCMG Fig. 11 SGCMG singularity measure 7 Fig. 7 Attitude angular velocity of spacecraft 6 (Conclusion),

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(L d L q ). i d = 0,, [8].,,,,, / (maximum torque per ampere, MTPA) [9],. MTPA [9] [10],,,. MTPA, MTPA.,,,. MTPA,,,.,. 2 (Model of PMSM) d-q, :

(L d L q ). i d = 0,, [8].,,,,, / (maximum torque per ampere, MTPA) [9],. MTPA [9] [10],,,. MTPA, MTPA.,,,. MTPA,,,.,. 2 (Model of PMSM) d-q, : 32 2 2015 2 DOI: 10.7641/CTA.2015.40361 Control Theory & Applications Vol. 32 No. 2 Feb. 2015 /,,, (, 266071) : (PMSM), / (MTPA). MTPA,.., MTPA.. : ; ; ; / : TP273 : A Adaptive sliding mode maximum torque