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 1.5 1.5.5 w/o zero : 1/(.5s + 1) w/ stable zero : (.1s+1)/(.5s + 1) w/ unstable zero : (.1s+1)/(.5s + 1).5.1.15..5.3.35. time [s] Fig. 1: Difference of time response by location of zero.
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 () 3 3.1 () 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)
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-
Gain [db] 5 P P n.3 y d (t) M(s) r mm Phase[deg] 5 1 1 1 1 1 3 P P n 1 1 1 1 1 3 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 + 11.7)(s 13.3) s(s + 1.1)(s (1) +.7s + 3.9 1 ) 13.3 () = s + 9.9s 3 + 3.9 1 s +.3 1 s N p (s) = 7.71s + 11.31s + 7.9 1 N s (s) = 9.33 1 3 () s + 1 N u (s) = 9.1 1 3 s 1 K = 7.9 1. 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 = 5.19 1, 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).5 3..5.1 7(a) N u (s) = 1/(9.1 1 3 s+1) 7(b) [z] P ZP [z] N = 1 5. y d u F F P (s) N u (s) kanai
1 y d 3 5 u conv y conv y prop.5 e trconv e trprop u prop 3 Position[mm] Tracking error[mm] 1.5 1 Control Input[N] 1.5 3.5.1.15..5 (a) Reference y m and output y..5.5.1.15..5 (b) Tracking error e. Fig. 9: Simulation results (N = 5)..5.1.15..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 p3.....1.1 Gain[dB] Phase[deg] Gain[dB] Gain[dB] 3 (a) Impulse response of N u (s). P ZP (N=5) (N=1) 1 1 1 1 1 3 P ZP (N=5) (N=1) 1 1 1 1 1 3 (b) Frequency responses of (s) and P ZP (s). 15 1 5 Fig. 7: 1 1 1 1 15 1 5 (a) Proposed method. 1 1 1 1 (b) Conventional method. Pz Prop. Pz Conv. Fig. : Frequency responses u F F /y d and P (s) 9 9(a) 9(b).5.1.15..5 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).7 11 11(a) 11(b) 11(c) y d e u N =. s 9 ( ±1 [N] )
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