HDD. Point to Point. Fig. 1: Difference of time response by location of zero.

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1 Trajectory Tracking Control using Two-degree-of-freedom Control Based on Zero-Phase-Minimum-Phase Function Factorization for Nonminimum-Phase Continuous-Time System T. Shiraishi and H. Fujimoto (The University of Tokyo) Abstract The purpose of this paper is development of high-precision trajectory tracking control for nonminimum-phase continuous time systems with unstable zeros. This paper proposes a two degree of freedom control system design method that is based on a novel factorization method for nonminimum-phase continuous time systems. First, nonminimumphase continuous time systems is factorized to minimum-phase system and zero-phase system in continuous time domain. The feedforward controller is constructed from inverse system of each factorized system. The inverse system of the minimum-phase system is designed by multi-rate perfect model following control theory, and the inverse system of zerophase system is designed by zero-phase FIR filter. Finally, This paper shows the effectiveness of proposed method by simulation and experimental results. Key Words: nonminimum-phase continuous-time system, tracking control, under-shoot, zero-phase minimum-phase function factorization 1 1) 3 ) 3) ) 1 1) kanai ) 5) HDD ) Point to Point Output w/o zero : 1/(.5s + 1) w/ stable zero : (.1s+1)/(.5s + 1) w/ unstable zero : (.1s+1)/(.5s + 1) time [s] Fig. 1: Difference of time response by location of zero.

2 P (s) P (s) = N p(s) = K N s(s)n u (s) N p (s) l n K N s (s) N u (s) m q N p (s) N s (s) N u (s) (1) N p (s) = b pl s l + b pl s l + + b p1 s + b p () N s (s) = b sm s m + b sm s m + + b s1 s + 1 (3) N u (s) = b uq s q + b uq s q + + b q1 s + 1 () () P (s) = K N s(s) N u (s) Nu(s) N u (s) N u (s) N u (s) P (s) = (5) K N s(s) N u (s) N u (s)n u (s) () P MP (s) = K N s(s) N u (s) (7) P ZP (s) = N u (s)n u (s) () P MP (s) P ZP (s)?? P ZP (s) P ZP (s) () δ t = Trajectory Generator Gn δ1 yn Feedforward Controller e CFB uff yd P Fig. : Proposed control system. δ δ3 Impulse response of Tu Nu ( s) Fig. 3: Impulse response of N u (s). 3. δ N etr y time () C F F (s) = P MP (s) P ZP (s) (9) P MP (s) 11) ZPETC P ZP (s) FIR ZPF IIR FIR FIR 3.3 P ZP (s) [z] FIR FIR ZPF FIR N G ZP F [z, z ] = α + α k (z k + z k ) (1) k=1 α n (n =, 1, N) N u (s) t = nt u δ n 3 α k α k = α k = N δ n δ n k (11) n=k α k α + ( α 1 + α + + α N ) (1) FIR P ZP (s)

3 Tu u pk[ k] u pi[ i] = u [ k + 1] pk u pk [k] Tr = Tu Tu u pk [ k +1] [ k + ] k k + 1 k + time i i + 1 Fig. : Multirate-sampling (case of n+q=). (1) [z] [z] = G ZP F [z, z ]z (N+1) (13) IIR FIR N 3. P MP (s) PTC P MP (s) (n + q) ẋ p (t) = A p x p (t) + b p u p (t) (1) T u x pk [k + 1] = A pk x pk [k] + b pk u pk [k] (15) P MP (s) n + q = x pi [i + 1] = A pi x pi [i] + b pi u pi [i] (1) A pi = A (n+q) pk [ ] b pi = A (n+q) pk b pk A pk b pk b pk x d [i] u pi [i] u pi [i] = b pi (I za pi)x d [i + 1] (17) x d [i] x d (t) t = it r y d (t) x d (t) = 1 [ ] 1 s s n+q T yd (t) (1) N s (s) (17) [ ] O I C MP [z] = b pi A pi b (19) pi u pk x v f x x 1 l C x w b M L µ θ,kθ Fig. 5: Physical model of high precision stage. θ g m Table 1: Stage parameters. M Carriage Mass [kg] 5.5 m Table Mass [kg] 7.7 b Widths of table [m].1 h Heights of table [m].1 L Length from pivot to CG [m].9 I Length from pivot to x [m] - µ θ Viscous coefficient of pivot [Nms/rad].1 µ θ Stiffness coefficient of pivot [Nms/rad] 133 C x Viscous coefficient of air bearing [Ns/m] T u µs FB.1 :7-9-1 XY 5 1 f x P (s) = x (s) f(s) = b s + b 1 s + b a s + a 3 s 3 + a s + a 1 s I y = m(b + h )/1, a = MmL + MI y, a 3 = (M + m)µ θ + (ml + I y )C x, a = (M + m)k θ mgl(m + m) + µ θ C x, z h x () a 1 = (k θ mgl)c x, b = ml + I y mll, b 1 = µ θ, b = k θ mgl 5 l x (s)/f(s) :7-9-

4 Gain [db] 5 P P n.3 y d (t) M(s) r mm Phase[deg] P P n Fig. : Frequency response of P (s) and P n (s). 1 l x 1 x x x x 1 x v w x = ( vx 1 + wx )(w v) P (s) = x (s)/f(s) P n (s) P n (s) P n (s) = 7.71(s )(s 13.3) s(s + 1.1)(s (1) +.7s ) 13.3 () = s + 9.9s s s N p (s) = 7.71s s N s (s) = () s + 1 N u (s) = s 1 K = FB C F B (s) PID C notch (s) C F B (s) = C P ID (s) C notch (s) (3) C P ID (s) = p s + p 1 s + p s(s + l 1 ) C notch (s) = s + Q 1 s + ω s + Q s + ω p =.9 1, p 1 =. 1 5, p = , l 1 = 9., Q 1 = 5., Q = 1.9, ω =. () (5) FB C F B (s) ( ) ωd M(s) =, ω d = π () s + [ ω d y d (t) = L M(s) r ], r =. (7) s. PTC 1) :--1 :--1 :--1 PTC PTC P n (s) = K N s(s) () n + q n () y n G n (s) G n (s) = K N s(s)n u (s) (9) y n N u (s) (a) N u (s) = 1/( s+1) 7(b) [z] P ZP [z] N = 1 5. y d u F F P (s) N u (s) kanai

5 1 y d 3 5 u conv y conv y prop.5 e trconv e trprop u prop 3 Position[mm] Tracking error[mm] Control Input[N] (a) Reference y m and output y (b) Tracking error e. Fig. 9: Simulation results (N = 5) (c) Control input u. Impulse respose 1 1 State variables 1 y prop b p x p1 b p1 x p b p x p Gain[dB] Phase[deg] Gain[dB] Gain[dB] 3 (a) Impulse response of N u (s). P ZP (N=5) (N=1) P ZP (N=5) (N=1) (b) Frequency responses of (s) and P ZP (s) Fig. 7: (a) Proposed method (b) Conventional method. Pz Prop. Pz Conv. Fig. : Frequency responses u F F /y d and P (s) 9 9(a) 9(b) Fig. 1: Trajectory of b p x 3, b p1 x, b p x 1, and y. 9(c) y m e u FIR N 5.1 s 9(a) 9(b) 9(c) x 3 = ẋ, x = ẋ 1 y(t) = b p x 3 (t) + b p1 x (t) + b p x 1 (t) (3) 1 (3) (a) 11(b) 11(c) y d e u N =. s 9 ( ±1 [N] )

6 5 1) A. Hara, K. Saiki, H. Fujimoto and K. Sakata : Basic Examination on Simultaneous Optimization of Mechanism and Control for High Precision Stage, Proc. of IIC7, Vol. 7, No. 11, pp. 5 7 (7) (in Japanese) ) M. Tomizuka : Zero phase error tracking algorithm for digital control, ASME Journal of Dynamic Systems, Meas and Control, Vol. 113, pp. 1 (197) 3) B.P. Rigney, L.Y. Pao, and D.A. Lawrence : Nonminimum phase dynamic inversion for settle time applications, Control Systems Technology, IEEE Transactions on, Vol. 17, No. 5, pp (9). ) H. Fujimoto, Y. Hori and A. Kawamura : Perfect Tracking Control Method Based on Multirate Feedforward Control, Transactions of SICE, Vol. 3, No. 9, pp () (in Japanese) 5) M. Miyoshi and Y. Kaneda : Active noise control using inverse filtering of a sound field in a room, J. Acoust. Soc. Jpn, Vol., No. 1, pp. 3 1 (199) (in Japanese) ) T. Tomaru and Y. Mori : A New Design of Feedfoward Control for Non Minimum-Phase Systems Based on the Extended Inverse Function. IEEJ Trans. EIS, Vol. 17, No., pp (7) (in Japanese) 7) T. Sogo : Approximation of continuous-time inverse systems with the sampled-data counterparts and its application to iterative learning control, Transactions of SICE, Vol. 3, No. 1, pp (7) (in Japanese) ) K. Fukushima, H. Fujimoto and S. Nakagawa : Short-Span Seeking Control of Hard Disk Drive with Vibration Suppression PTC, IEEJ Trans. IA, Vol. 1, No., pp () (in Japanese) 9) C.J. Kempf and S. Kobayashi. Disturbance observer and feedforward design for a high-speed direct-drive positioning table. Control Systems Technology, IEEE Transactions on, Vol. 7, No. 5, pp (1999) 1) H. Van Brussel, C.-H. Chen, and J. Swevers, Accurate motion controller design based on an extended pole placement method and a disturbance observer, Ann. CIRP, vol. 3, no. 1, pp , ) T. Shirasihi and H. Fujimoto : Design Method of Mode Switching Control System using Perfect Tracking Control, Proc. of IIC1, Vol. 1, No. 1, pp. 19 (1)(in Japanese) Position[mm] Tracking error[mm] Control Input[N] y d conventional proposed (a) Reference y m and output y. conventional proposed time[s] (b) Tracking error e. conventional proposed time[s] (c) Control input u. Fig. 11: Experimental results. 1), : --, 3(), p57-, ) :,, pp.95 9, 19.

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