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1 38 1 Vol. 38, No ACTA AUTOMATICA SINICA January, , 2 1 1, 3 (Disturbance observer, DOB),,, H. H.,, Q,,.,,. DOI,, H,, /SP.J Systematic Design Method of Disturbance Observer Guaranteeing Closed-loop System s Robust Stability YIN Zheng-Nan 1, 2 SU Jian-Bo 1 GAO Xiu-Xing 1, 3 Abstract The sufficient condition on the robust stability against model uncertainties in the feedback system with disturbance observer (DOB) is studied. Weighting functions satisfying this condition are consequently selected such that robust stability to plant s parameters variation is guaranteed by the disturbance observer designed with standard H disturbance observer design method. Selection methods of frequency weighting functions are proposed concerning about the robustness design specifications and structural restrictions in designing the H DOB. Trade-off between peek magnitude in cut-off frequency of Q- filter and disturbance depressing performance is optimized in designing low-pass filter of disturbance observer via selections of the weighting functions. Therefore, the resultant disturbance observer has the optimal disturbance rejection performance with satisfactory peak magnitude. Experimental results show the proposed method could guarantee robust stability of closed-loop system while providing design freedom for other available design specifications. Key words Robust motion control, disturbance observer (DOB), H standard control problem, low-pass filter, two degrees of freedom control system (Disturbance observer, DOB) [1],,,, [2 6]. DOB, (Q ), DOB Manuscript received July 4, 2011; accepted September 20, 2011 ( ) Supported by Key Project of National Natural Science Foundation of China ( ) Recommended by Associate Editor JIA Ying-Min , , , 1. Electronic Information and Electric Engineering School, Shanghai Jiao Tong University, Shanghai , P. R. China 2. Electronics and Automation Faculty of Kim Il Sung University, Pyongyang , D. P. R. of Korea 3. Department of Automation of Kim Chaek Industrial University, Pyongyang , D. P. R. of Korea Q. Q, Butterworth [7 9]., [10 11],.,, H. H DOB H, Q. [12 14] H,. [12],,,. [13], [14], Q.

2 1 : 13,,. H DOB, H, DOB,.. [15] H,, Q.,., DOB, Q,., DOB,., H,,,. Q,,., : 1), DOB ; 2), DOB H ; 3). H, ; Q,. :, H DOB.,,.,, H. 1 H H DOB I II, I H ; II H. H H, Q.,,. 1.1 H (I ) 1, Pn 1 (s) Q(s) DOB C(s). (P n = P ), y = P n(s)c(s) 1 + P n (s)c(s) r + P n(s)(1 Q(s)) 1 + P n (s)c(s) d + P n (s)c(s) + Q(s) ξ (1) 1 + P n (s)c(s) r, d, ξ y. Fig. 1 1 DOB Feedback control system with DOB,, Q(s).,.. P (s) = P n (s)(i + (s)) (2) DOB [12, 16] DOB Q. 1, σ( (jω)t (jω)) < 1, ω (3), σ( (jω)) < W T (jω), ω W T (s) W T T < 1, T (s) = P n(s)c(s) + Q(s) 1 + P n (s)c(s) (4), T (s). (1), T (s),

3 DOB Q(s), C(s), H DOB., (1), d y, S(s) min W S S < 1, S(s) = Q(s) 1 Q(s) 1 + P n (s)c(s) W 1 S(jω) (1 + P n (jω)c(jω)) < W SD(jω), ω W SD (s), (5) (5) min W SD S DOB < 1, S DOB (s) = 1 Q(s) (6) Q(s), S DOB (s) DOB, (4) (6), DOB min Q(s) [ ] WSD (s)s DOB (s) W T (s)t (s) < 1 (7) 2 H (7) Q(s). H DOB LMI [12 14].,,,, Q. [12] (3) (4) Q. Fig. 2 2 DOB H H control problem for DOB design 1.2 H (II ) DOB, Q., d ξ y DOB S DOB (s) T DOB (s) = 1 S DOB (s) = Q(s)., (s) W TD (s)t DOB (s) < 1, W TD (s) σ( (jω)) < W TD (jω), ω., DOB [ ] γ WSD S DOB max γ, min < 1 (8) W TD T DOB Q(s) RH Q(s) Ω k { Ω k = F (s) F (s) = N(s) D(s), } deg(d) deg(n) k (9) Q k. [15] (8) H,, (8) : [ ] γ 1 WSD (1 + P K) max γ, min K(s) W TD P K(1 + 1 P K) < 1 (10), P (s), K(s) Q(s),, Q. DOB Q, I,, Q,.,,., II DOB,,, Q, I. 1. 1,, II I.,, II I, 1., DOB : I,

4 1 : 15 Table 1 1 I II H DOB Comparison of type I and type II design methods of H DOB I II DOB Q Q II, II. 2 I II. (7) (8), I S(s) T (s), II DOB S DOB (s) T DOB (s)., I II (7) DOB (8). DOB, I II,, I II. (4), : L(jω)+Q(jω) < W 1 T (jω)(1+l(jω)), ω (11), L(s) = P n (s)c(s)., L(jω) + Q(jω) < L(jω) + Q(jω), ω (12), (13), (11) Q(jω) < W 1 T (jω)(1 + L(jω)) L(jω), ω (13) Q, DOB, (13).., (13) 0, E(ω) = W 1 T (jω)(1 + L(jω)) L(jω) > 0, ω (14) :, (14)., DOB (14)., ( DOB ) : W T (s)t 0 (s) < 1 W T (jω)l(jω)(1 + L(jω)) 1 < 1, ω (15), T 0 (s) = L(s)(1 + L(s)) 1 DOB. (15) (14).,. (12), (16) C(ω) : C(ω) = L(jω) + Q(jω) L(jω) + Q(jω) (16) C(ω) ω 0 L(jω 0 ) Q(jω 0 ).,,, (Cut-off). 3 Q(s) L(s).,,., Q,. W TD (s) W 1 TD (jω) < E(ω), ω (17) (13) Q(jω) < W 1 TD (jω), ω W TD (s)q(s) < 1 (18), (18) II.

5 16 38 Fig. 3 3 (a) (a) Nyquist locuses (b) (b) Frequency-phase responses Q(s) L(s) Comparison of frequency-phase characteristics of Q(s) and L(s) (1), d S(s),, (5) (6) d ( ) S(s). : (5), (6) DOB., : max γ, min γ W SD (s)(1 Q(s)) 1 < 1 (19) Q(s) (18) (19), DOB : [ ] γ WSD (1 Q) max γ, min = Q(s) W TD Q [ ] (20) γ WSD S DOB max γ, min < 1 Q(s) W TD T DOB (20) (8), DOB,, I II. II DOB., DOB Q,, Q(s) Q(s) Ω k.., [17] II. 2.2, DOB : 1., C(s).,, 2, DOB. 2. DOB (7), I W SD (s) W T (s). 3. (14), E(ω). 4. (17) W TD (s), II (20)., DOB, Q(s) 1 Q(s) II, Q. 5, H, ( DGKF [18] ). Matlab (Robust control toolbox). [15],. 2.3 H,. DOB Q, W TD (s) (20) W SD (s), Q. DOB, Q : 1) Q(s)P 1 (s), Q(s) k (P 1 (s) ). 2) d = p q t q + p q 1 t q 1 +

6 1 : 17 + p 0, Q(s) q ( ) ( (21) ). 3) n. (9) Q., Q, : Q(s) Ω n,k,q (21), Ω n,k,q n k ( n) q ( m) : Ω n,k,q = { G(s) G(s) = M(s) N(s) n N(s) = a i s i, M(s) = i=0 m b j s j, j=0 a n 0, b m 0, m = n k, a l = b l, l = 0, 1,, q } (22) W TD (s) W SD (s) (10) (21). Q Q (s) (10) K (s) : Q (s) = L (s)(1 L (s)) 1 L (s) = P (s) K (s) (23) K (s) (10), W TD (s) W SD (s) P (s)., H, K (s) P (s), L (s) K (s). P (s) k, L (s) Q (s) k. n q W TD (s) W SD (s) W TD (s) W TD (s) Q(s). H, (10), W 1 TD (s) P (s)., k W 1 TD (s) = δ M TD (s), M TD(s) = k c i s i (24) i=0 δ c i (i = 0, 1,, k) (17)., W 1 TD (jω) ( ) E(ω). W TD (s), k (17),., W 1 TD (jω) E(ω), W SD (s) W TD (s), Q(s) n q W SD (s). (10) W SD (s) P (s), n = n W + k ( ). γ, 1, n = n W + k 1., n, W TD (s) n W n W = n k + 1. W SD (s) = M SD(s) N SD (s) N SD (s) = n k+1 i=0 d i s i, M SD (s) = n k+1 j=0 f j s j (25) (23), q a l = b l (l = 0, 1,, q), K (s) q + 1, (10), P (s), K (s) q + 1 W SD (s) q + 1, : W SD (s) = 1 MSD(s) sq+1 Ñ SD (s) Ñ SD (s) = n k q j=0 d j s j (26) q m ( : Q(s) m ), Ñ SD (s) = d 0, W SD (s) = d 1 0 s q+1 M SD(s) (27) H,. λ > 0 (26) W SD (s) = 1 (s + λ) q+1 M SD(s) (28) 1 (10) γ, d 0 1. M SD (s) f i (i = 0, 1,, n k + 1) W 1 SD (s) ( ) Q(s)., Q, (24) (28),, H W TD (s) W SD (s) M TD (s)

7 18 38 M SD (s). c i (i = 0, 1,, q) f j (j = 0, 1,, n k + 1) W 1 SD (s) W 1 TD (s), Q, Q, II.,. DOB S DOB (s) T DOB (s). :,., DOB [7]. II. 3.1,,., ( ). T (s) = 1 (σs) 2 + 2ξ(σs) + 1 (29) 4 T (s) S(s) = 1 T (s) ξ. S(s) T (s) 1,.,, S(s) T (s),., S(s) T (s). 4,, S(s) T (s).,,, ξ.,,. (7), I,, (7), DOB., (10), II,. 4 Fig. 4 T (s) S(s) Frequency responses of S(s) and T (s) at different damping rates 3.2, W SD (s) W TD (s),.,. ξ < 1 : W SD = s2 + 2ξs + 1 (s ) 2 W TD = s2 + 2ξs (30), II Q Q 1 = Q 2 = Q 3 = 1.349s s s s , ξ 1 = s s s s , ξ 2 = s s s s , ξ 3 = 0.7 (31) 5 S DOB (s) T DOB (s). 5, (14) E(ω), ξ E(ω),.. ξ = 0.2, (1 rad/s), ξ = ( 5 12 db ).

8 1 : 19 ), [12]. : P (s) = Kω 2 n1 s 2 + 2ζ 1 ω n1 s + ωn1 2 ω 2 n2 s 2 + 2ζ 2 ω n2 s + ω 2 n2 0.5T Ds T D s + 1 (34) 5 Fig. 5 S DOB(s) T DOB(s) Frequency responses of S DOB(s) and T DOB(s) at different damping rates 600 rad/s 0.3, rad/s (34), DOB. Kω 2 n1 P n (s) = s 2 + 2ζ 1 ω n1 s + ωn1 2 (35) E(ω) ( 6 (a) ) 0 db,.,,. E(ω)., (ξ > 1) ( 6 (a) W 1 TD (s) ): W SD = W TD = 1 (s ) 2 (s + 0.3)(s + 20) 50 Q Q (s) = 50s s s s (32) (33) 6 (a) Q (s) Q 1 (s), Q 2 (s) Q 3 (s).,.. 6 (b),, 1 Q (s).,,. 4, DOB,., ( ) ( Fig. 6 (a) (a) Robust stability analysis (b) (b) Comparison of disturbance attenuating 6 performances at low frequencies Robust stability analysis and comparison of disturbance suppression performances, (2) (s) = P (s)/ P n (s) 1. L(s)

9 rad/s 55, C(s) = 5 850( s + 1) s + 1 (36) (s), I (7) W SD1 (s) = 0.5(s ) s W T (s) = s(s ) 2 (s )(s ) 2 (37) 8, (s) W T (s). (28) (29), (14) E(ω),, (17) W TD (s) ( 7), W 1 1 SD (jω) < WSD1 (jω), ω W SD(s): W TD (s) = (s + ρ)(s2 + 2ξ Q ω Q s + ω 2 Q) W SD (s) = 0.5(s ) s (38) 8 Fig. 8 (39) Robust stability analysis of closed-loop system using (39) 8,.,. 9 1 Q (s), [12] ( 1 Q W (s) )., rad/s, 1 Q (s) 1 Q W (s) 6 db,,, 4., ρ = 100, ξ Q = 2, ω Q = rad/s., II Q Q (s) ( 7): Q (s) = s s s (39) 9 Fig. 9 (39) Q W (s) Comparison of disturbance attenuating performances in low frequencies between (39) and Q W (s) ρ = 2 000, ξ Q = 0.4 (< ), ω Q = rad/s., W TD (s) = s s s Fig. 7 7 E(ω), W 1 TD (s) Q (s) Frequency responses of E(ω), W 1 TD (s), and Q (s) (40) ( 10),

10 1 : 21,, 3.2. (32), Q Q (s) = s s s (41) 11,. 12,, rad/s, 1 Q (s) 1 Q W (s), (39) ( 3.2 )., 3.2 ξ > 1 ( (30)),,,,. Fig Comparison of disturbance attenuating performances at low frequencies 12 Fig. 12 (41) Q W (s) Comparison of disturbance attenuating performances at low frequencies between (41) and Q W (s) 5,, H.,..,, Q,,. References 1 Ohishi K, Murakami T. Advanced motion control in robotics. In: Proceedings of the 15th Annual Conference of IEEE Industrial Electronics Society. Philadelphia, USA: IEEE, Fig. 11 (41) Robust stability analysis of closed-loop system using (41) 2 Natori K, Tsuji T, Ohnishi K, Hace A, Jezernik K. Timedelay compensation by communication disturbance observer for bilateral teleoperation under time-varying delay. IEEE Transactions on Industrial Electronics, 2010, 57(3): Ohishi K. Realization of fine motion control based disturbance observer. In: Proceedings of the 10th IEEE International Workshop on Advanced Motion Control. Trento, Italy: IEEE, Nakata T, Tomizuka M. Robust engine torque control by iterative learning control. In: Proceedings of the American Control Conference. St. Louis, USA: IEEE, Tanaka H, Ohnishi K, Nishi H, Kawai T, Morikawa Y, Ozawa S, Furukawa T. Haptic endoscopic surgery robot utilizing FPGA. In: Proceedings of the 10th IEEE International Workshop on Advanced Motion Control. Trento, Italy: IEEE,

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Disturbance observer and feedforward design for a high-speed direct-drive positioning table.

11 Rahmam A A, Ohnishi K. Robust time delay control system based on communication disturbance observer with inner loop input. In: Proceedings of the 36th Annual Conference on IEEE Industrial Electronics Society. Glendale, USA: IEEE, Schrijver E, Dijk J V. Disturbance observers for rigid mechanical systems: equivalence, stability, and design. Journal of Dynamic Systems, Measurement, and Control, 2002, 124(4): Umeno T, Hori Y. Robust speed control of DC servomotors using modern two degrees-of-freedom controller design. IEEE Transactions on Industrial Electronics, 1991, 38(5): Kempf C J, Kobayashi S. Disturbance observer and feedforward design for a high-speed direct-drive positioning table. IEEE Transactions on Control Systems Technology, 1999, 7(5): Tan K K, Lee T H, Dou H F, Chin S J, Zhao S. Precision motion control with disturbance observer for pulsewidthmodulated-driven permanent-magnet linear motors. IEEE Transactions on Magnetics, 2003, 39(3): Noh I, Won S. Robust adaptive control using disturbance observer for system with actuator of first order time lag. In: Proceedings of the IEEE International Conference on Control and Automation. Christchurch, New Zealand: IEEE, Wang C C, Tomizuka M. Design of robustly stable disturbance observers based on closed loop consideration using H optimization and its applications to motion control systems. In: Proceedings of the American Control Conference. Boston, USA: IEEE, Thum C K, Du C, Lewis F L, Chen B M, Ong E H. H α disturbance observer design for high precision track following in hard disk drives. IET Control Theory and Applications, 2009, 3(12): Zhang G Z, Chen J, Li Z P. Analysis and design of H robust disturbance observer based on LMI. In: Proceedings of the 7th World Congress on Intelligent Control and Automation. Chongqing, China: IEEE, Yin Zheng-Nan, Su Jian-Bo, Liu Yan-Tao. Design of disturbance observer with robust performance based on H norm optimization. Acta Automatica Sinica, 2011, 37(3): (,,. H., 2011, 37(3): ) 16 Fan X, Tomizuka M. Robust disturbance observer design for a power-assist electric bicycle. In: Proceedings of the American Control Conference. Baltimore, USA: IEEE, Francis B A, Wonham W M. The internal model principle of control theory. Automatica, 1976, 12(5): Doyle J C, Glover K, Khargonekar P P, Francis B A. Statespace solutions to standard H 2 and H control problems. IEEE Transactions on Automatic Control, 1989, 34(8): yjn2914@yahoo.cn (YIN Zheng-Nan Ph. D. candidate in the Department of Automation, Shanghai Jiao Tong University. He received his bachelor and master degrees from Kim II Sung University, D. P. R. Korea in His research interest covers robot motion control and robust control. Corresponding author of this paper.).,,. jbsu@sjtu.edu.cn (SU Jian-Bo Professor in the Department of Automation, Shanghai Jiao Tong University. His research interest covers robot control, multi-sensor fusion, machine learning, and human-computer interaction.).,. ksh616@126.com (GAO Xiu-Xing Associate professor in the Department of Automation, Kim Chaek Industrial University, D. P. R. of Korea. His research interest covers robot control, intelligent control and its application.)

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,,.,, : 1),,,,, 2),,,,, 3),,,,,,,,,, [6].,,, ( ),, [9], : 1), 2),,,,, 3),,, 2.,, [10].,,,,,,,,, [11]. 2.1,, [12],, ;, ; Fig. 1 1 Granular hier 36 7 Vol. 36, No. 7 2010 7 ACTA AUTOMATICA SINICA July, 2010 1, 2 1, 2, 3 1, 2,,,,,,, DOI,,, 10.3724/SP.J.1004.2010.00923 Distributed Simulation System Hierarchical Design Model Based on Quotient Space