IIC-11-8 A Method of Trajectory Tracking Control for Nonminimum Phase Continuous Time Systems Takayuki Shiraishi, iroshi Fujimoto (The University of Tokyo) Abstract The purpose of this paper is achievement of high-precision trajectory tracking control for non-minimum phase continuous time systems (NMPCTS). This paper proposes a two degree of freedom (2DOF) control system design method that is based on a novel factorization method for NMPCTS. First, NMPCTS is factorized to minimum phase system and zero phase system in continuous time domain. The feedforward controller of 2DOF control is designed from inverse system of factorized systems. The inverse system of the minimum phase system is designed by multi-rate perfect model following control theory, and the inverse system of zero phase system is designed by zero phse type FIR filter. Finally, This paper shows the effectiveness of proposed method by simulation and experimental results., (nonminimum phase continuous-time systems, unstable-zeros, trajectory tracking control, under-shoot, zero-phase system ) 1. 2 2 FF 2 (1) (2) (3) 1 (4) FIR Delayed inverse () (6) 2 DD (7) Point to Point (8) FF Output 2 1. 1.. w/o zero : D p = (.2s+1) 3, N s = N u = 1 w/ stable zero : D p = (.2s+1) 3, N s =.1s + 1,N u = 1 w/ unstable zero : D p = (.2s+1) 3, N s = 1, N u =.1 s +1 1..1.1.2.2.3.3 Time [s] 1 Fig. 1. undershoot response by unstable zeros. 1/6
2 α + α Fig. 2. α + α Nu ( 1 γ j γ Nu ( α + α P AP ( Nu ( Poles-zeros location of P AP ( γ j γ Nu( α + α Re Re u[k] Plant u[k] FF r(t) Fig. 4. ( r T ) 4 P( PMP( Factorization PZP( N s( Nu( Nu( Dp( N u( CMP[z] Inverse N ( Nu( Dp( s S y[k] ( Ty) M CZP[z] Inverse S ( y T ) FF y[k] uff[k] Factorization of plant and feedforward controller. 3 Fig. 3. P ZP ( Poles-zeros location of P ZP ( 2 FF 2. P ( P ( = N s(n u ( (1) D p ( D p ( n N s ( N u ( m q N u ( N u( = (s γ)(s ω )(s ω ) (2) γ ω ω ω = α + ω = α γ, α, β > (1) P ( = N s( N u ( D p( = N s( N u ( D p ( Nu( N u ( (3) P AP ( (4) N u ( N u ( N u( = (s + γ)(s + ω )(s + ω ) () P AP ( 1 2 P AP ( N s ( P ( = D p( N N u (N u ( (6) u( = P MP ( P ZP ( (7) P MP ( P ZP ( 3 P ZP ( P ZP ( s = jω P ZP (jω) = (ω 2 γ 2 )(ω 2 ω 2 )(ω 2 ω 2 ) (8) P ZP (jω) (6) 3. 3 1 FF (6) C F F ( = Dp( N u( N s( 1 N u (N u ( (9) = P MP ( 1 P ZP ( 1 (1) P MP ( 1 P ZP ( 1 FIR (9) IIR FIR FIR 4 FF 2/6
r(t) ( Tr ) uff [k] u[k] y(t) CFF P( [z] Pn[z] y[ k] M [ z] Fig.. CFB [z] ufb[k] e[k] ym[k] osed control system. y[k] S ( T y ) etr[k] FF C MP [z] FIR C ZP [z] FF C F F [z] = C MP [z] C ZP [z] FF C F F [z] C F B (FB) P n M y u F F FF y m e tr T u = T y T r = (n + q)t u FB y y e y m y FB FF C F F [z] (7) (7) G( Z [ G( ] Z[ P ( ] Z[ P MP ( ] Z[ P ZP ( ] Z[ P ( ] FF Z[ P ( ] Z[ P MP ( ] Z[ P ZP ( ] 3 2 FF P ZP ( FF C ZP [z] FIR (9) FIR N G ZP F [z, z 1 ] = α + α k (z k + z k ) (11) k=1 α n(n =, 1, N) IIR N u ( 1 t = nt u δ n α k δ t = α k = α k = δ1 δ2 6 δ3 Impulse response of Tu Nu ( 1 Nu () N Fig. 6. Impulse response of Nu( 1. Tu u pk[ k] u pi[ i] = u [ k + 1] pk u pk [k] Tr = 2Tu Tu k k + 1 k + 2 i i + 1 7 Fig. 7. u δ N u pk [ k +1] [ k + 2] u pk (n=2) time time Multirate-sampling (case of n=2). ( 1 N δ n δ n k (12) n=k α k α + 2( α 1 + α 2 + + α N ) (13) FIR P ZP ( 1 N (11) C ZP [z] C ZP [z] = G ZP F [z, z 1 ]z (2N+1) (14) 3 3 FF P MP ( FF (1) P MP ( M( (n + q) ẋ p (t) = A p x p (t) + b p u p (t) (1) ẋ m (t) = A m x m (t) + b m r m (t) (16) M( M ( 1 M( = M ( N (17) s( ( ) ωm (n+q m) M ( = (18) s + ω m 3/6
N s ( P MP ( P MP ( M ( T u P MP ( M( x mk [k + 1] = A mk x mk [k] + b mk r mk [k] (19) x pk [k + 1] = A pk x pk [k] + b pk u pk [k] (2) P MP ( n + q = 2 7 u pi[i] x mi [i + 1] = A mi x mi [i] + b mi r mi [i] (21) A mi = A (n+q) mk (22) b mi = [ A (n+q 1) mk b mk A mk b mk b mk ] (23) x pi [i + 1] = A pi x pi [i] + b pi u pi [i] (24) A pi = A (n+q) pk (2) b pi = [ A (n+q 1) pk b pk A pk b pk b pk ] (26) FF C MP [z] [ ] C MP [z] = A mi b 1 pi (A mi A pi ) b mi b 1 pi b mi (27) e i [i] = x mi [i] x pi [i] e i[] T r (18) M ( 4. (7) 4 1 (8) XY 8 1 2 f x 2 Fig. 8. f 8 x 2 v x 2 x 1 l C x w b M L θ µ θ,kθ g m z h Physical model of high precision stage. 1 Table 1. Stage parameters. M Carriage Mass [kg]. m Table Mass [kg] 7.7 b Widths of table [m].1 h eights of table [m].14 L Length from pivot to CG [m].9 I Length from pivot to x 2 [m] - µ θ Viscous coefficient of pivot [Nms/rad].21 µ θ Stiffness coefficient of pivot [Nms/rad] 1336 C x Viscous coefficient of air bearing [Ns/m] 24 P ( = x 2( f( = b 22 s 2 + b 21 s + b 2 a 4 s 4 + a 3 s 3 + a 2 s 2 + a 1 s (28) I y = m(b 2 + h 2 )/12 (29) a 4 = MmL 2 + MI y (3) a 3 = (M + m)µ θ + (ml 2 + I y )C x (31) a 2 = (M + m)k θ mgl(m + m) + µ θ C x (32) a 1 = (k θ mgl)c x (33) b 22 = ml 2 + I y mll (34) b 21 = µ θ (3) b 2 = k θ mgl (36) 8 l x 2(/f( l =.2 x 1 x 2 x 2 8 x 2 x 1 x 2 v w x 2 = ( vx 1 + wx 2 )(w v) 9 P ( = x 2(/f( x 4/6
Gain [db] Phase[deg] 1 P P n 1 1 1 1 1 2 1 3 Frequency[z] 2 4 P P n Gain [db] Phase [deg] 1 1 2 1 1 1 1 1 1 2 Frequency [z] 1 IIRLPF ZPF IIRLPF ZPF 6 1 1 1 1 2 1 3 Frequency[z] Fig. 9. 9 P ( P n( Frequency response of P ( and P n (. 2 1 2 1 1 1 1 1 1 2 Frequency [z] Fig. 1. 1 P ZP ( 1 C ZP ( Frequency response of P ZP ( 1 and C ZP (. P n ( P n ( P n ( = 7.71(s 13.3)(s + 11.7) s(s + 1.818)(s 2 + 8.74s + 3.92 1 4 ) (37) 192.3 D p( = s(s + 1.818)(s 2 + 8.74s + 3.92 1 4 ) (38) N s ( = s + 11.7 (39) N u ( = s 13.3, Nu ( = s + 13.3 (4) k = 7.7, n = 4, m = 1, q = 1 (41) 4 2 FB FB FB C F B ( PID C notch ( C F B ( = C P ID ( C notch ( (42) C P ID( = 4.69 14 s 2 + 8.26 1 s +.19 1 6 (43) s 2 + 98.68s C notch ( = s2 +.48s + 4.8 1 4 s 2 + 21.9s + 4.8 1 (44) 4 FB C F B ( T u 2[µs] M ( ω m r(t) ( ±1 [N] ) ω m = 12[rad/s] r(t) = 8[mm] N u () N u ( 1 = 13.3/(s + 13.3) 1 C ZP [z] P ZP ( 1 FIR N 4 3 (3) C F F [z] P ZP ( = N s(n u () (4) D p ( FF C F F [z] 4 4 11 11(a) 11(b) 11(c) y m e u NT u 11(a) 11(b) 11(c) 4 12 12(a) 12(b) 12(c) y m e u 11. 2 /6
y, y [mm] 2 1 1 Ref 1.1.2.3.4. Error [mm].1.2.3.4. Force [N] 2 1 1 1 1 2.1.2.3.4. (a) Reference y m and output y. (b) Tracking error e. 11 Fig. 11. Simulation results. (c) Control input u. y, y [mm] 2 1 1 Ref Error [mm] 1 Force [N] 2 1 1 1 1.1.2.3.4..6 1.1.2.3.4..6 1.1.2.3.4..6 (a) Reference y m and output y. (b) Tracking error e. 12 Fig. 12. Experimental results. (c) Control input u. FIR ( 1868636 268628) 9J9 1 M. Tomizuka. Zero phase error tracking algorithm for digital control. ASME Journal of Dynamic Systems, Meas and Control, Vol. 113, pp. 6 1, 1987. 2 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., pp. 989 1, 29. 3,,.., Vol. 36, No. 9, pp. 766 772, 2-9-3. 4,.., Vol. 46, No. 1, pp. 3 1, 1989-12-2.. z., Vol. 43, No. 1, pp. 37 4, 27. 6,... C,, Vol. 127, No. 8, pp. 1228 1233, 27-8-1. 7,,. PTC.. D,, Vol. 126, No. 6, pp. 76 712, 26-6-1. 8,,,... IIC,, Vol. 27, No. 116, pp. 6 7, 27-9-14. 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., pp. 13 26, September 1999. 1,... IIC,, Vol. 21, No. 16, pp. 19 24, 21-8-31. 6/6