ThD2-4 Adaptive compensation control for a piezoelectric actuator exhibiting rate-dependent hysteresis Y. Ueda, F. Fujii (Yamaguchi Univ. ) Abstract For realization of the precise positioning control of the bimorph type piezoelectric actuator including hysteresis and structural dynamics, this paper presents the identification of a modified hysteresis model. It has the ARMA model in which the Preisach model is embedded. This paper also proposes a design method for the compensation of hysteresis by using a novel inverse Preisach model. This model is formed in reference to a stop model. Identified model and controller exhibits good performance at 1[Hz] sinusoidal input, and the result of the numerical experiment of the feedforward compensation shows reasonable performance. Key Words: Hysteresis, Piezoelectric actuator, Preisach model, System identification, Compensation 1 1) 2) Preisach Prandtl-Ishlinskii operator of play-type(piop) Prandtl-Ishlinskii operator of stop-type(pios) Fig.1 1, 2, 3) 4) Preisach 5) Preisach FIR 2 (Affine Projection Algorithm; APA) 6) FIR Fig. 1: Feedforward compensation of the hysteresis. 2 1 1 7) Preisach 2 2.1 Preisach Preisach F (t) = K(α, β)γ α,β [u(t)]dαdβ, (α β) (1) T u(t) γ α,β [u(t)] Fig.2 u(t) α 1 1 u(t) β 1 1 K(α, β) Fig.3 αβ Preisach T K(α, β) T u(t) γ α,β [u(t)] Fig.4 γ α,β [u(t)] Preisach [No.16-14] 第 59 回自動制御連合講演会 (216.11.1-12 北九州市 ) 57
Fig. 2: Relay operator γ α,β [u(t)]. Fig. 3: Preisach plane T with α and β of relay operators. Fig.5 Preisach ARMA y m (k) = 2 a n y(k n)+ n=1 2 b n wn T x[u(k n)] (7) n= u(k) y(k) u(k) [ 1, 1] w n (n =, 1, 2) k, k 1,k 2 2 ARMA (2,2) ŵ n = b n w n, (n =, 1, 2) (8) Fig. 4: State of magnetization of T as input u(t) changes. Preisach α, β u(t) γ α,β [u(t)] α, β [ u max,u max ] N [u n 1,u n ](u = u max,u N = u max,u n = u n 1 +2u max /N, n =1,,N) K(α, β) K ij = uj u j 1 ui u i 1 K(α, β)dαdβ, (i =1,,N,j =1,,i) (2) K ij T (i, j) Γ ij [u(t)] (1) F (t) = N i=1 j=1 i K ij Γ ij [u(t)] (3) 3 N(N + 1)/2 w = [ K 11,K 21,K 22,...,K N1,...,K NN ] T (4) x[u(t)] = [ Γ 11, Γ 21, Γ 22,...,Γ N1,...,Γ NN ] T (5) F (t) =w T x[u(t)] (6) 2.2 1 (7) θ =[a 1,a 2, ŵ T, ŵ T 1, ŵ T 2 ] T (9) ϕ(k) =[ y(k 1), y(k 2), x[u(k)] T, x[u(k 1)] T, x[u(k 2)] T ] T (1) y m (k) =θ T ϕ(k) (11) y(k) y m (k) θ 5) (APA) [ ] ϕ(k) R = T ϕ(k) ϕ(k) T ϕ(k 1) ϕ(k 1) T ϕ(k) ϕ(k 1) T (12) ϕ(k 1) e(k) =y(k) y m (k) (13) [ ] e(k) ε = (14) {1 μ}e(k 1) [ ] q1 =(R + δi) 1 ε (15) q 2 θ(k +1)=θ(k)+μ{q 1 ϕ(k)+q 2 ϕ(k 1)}, ( <μ<2) (16) θ APA ŵ n N(N+1)/2 b n = ŵ n,i = ŵ n T x[u max ] i=1 (ŵ n =[ŵ n,1,, ŵ n,n(n+1)/2 ] T ) (17) 571
Fig. 5: Proposed model of hysteresis which incorporates the ARMA model structure and the Preisach model. F n (k) =w n x[u(k)] (18) u(k) =u max, u max F n (k) =1 3 3.1 Preisach 1, 3) Preisach Preisach Preisach F (t) = K(ᾱ, β) γᾱ, β[u(t)]dᾱd β (19) T γᾱ, β[u(t)] s r [u(t)] = max{min{u(t) u(t k )+s r [u(t k )],r}, r} (t =,t k <t<t k+1,k =, 1, ) (2) γᾱ, β[u(t)] = γ s r +r,s r r [u(t)] { 1 {(s = r,r); r s r s r [u(t)]} (21) 1 {(s r,r); s r [u(t)] <s r r} Preisach 2, 7) Preisach Preisach s r [u(t)] Fig.6 u(t) [ u max,u max ] r [,u max ] s r [u(t)] (2) [ r, r] sr Fig. 6: s r [u(t)]. Stop operator Fig. 7: Possible range of variations of s and r. Fig.7 Preisach T Fig.3 (2) s r [u(t)] (21) γᾱ, β[u(t)] Fig.8 T u(t) γᾱ, β[u(t)] Preisach r [,u max ] r 1 = u max /N, r N = u max,r n+1 = r n + u max /N, (n = 1,,N 1) N s r [ r, r] s rn,1 = u max (1 n)/n, s rn,n = u max (n 1)/N, s rn,m+1 = s rn,m + 2u max /N, (m =1,,n 1) s r Γ ij [u(t)] s ri [u(t)] = max{min{u(t) u(t k )+s ri [u(t k )],r i }, r i } (i =1,,N) (22) { 1 {(i, j); r i s ri,j s ri [u(t)]} Γ ij [u(t)] = 1 {(i, j); s ri [u(t)] <s ri,j r i } (23) (4) (5) Γ ij [u(t)] K ij c Γ ij [u(t)] x c Preisach F (t) =c T x c [u(t)] (24) Fig.8 T u(t) ᾱ, β 3.2 2 F n (k) c Fig.9 F n (k) u(k) u m (k) = 1 3 ct (x c [F (k)]+x c [F 1 (k)]+x c [F 2 (k)]) (25) θ = c (26) ϕ(k) = 1 3 (x c[f (k)] + x c [F 1 (k)] + x c [F 2 (k)]) (27) e(k) =u(k) u m (k) (28) 572
Fig. 8: State of magnetization of T as input u(t) changes. Fig. 1: Configuration of experimental system. Fig. 9: How the control input is calculated for the compensator of hysteresis. APA (25) (7) w n u(k) F n (k) [ 1, 1] c 4 4.1 Fig.1 FDK (PZBA 3) ART-Linux 3.2 GHz CPU(AMD Phenom II X4 955 Processor) PC PC 16 bit (LPC-361116) (As-94-15B) ±5[V] ±75[V] MESS-TEK (M-2218) MESS-TEK (TRA12-1K-V3) Displacement [mm].1.8.6.4.2 -.4 -.6 -.8 y y m Error -.1.5 1 1.5 2 Time [sec] Fig. 11: Actuator and model responses, and their difference for 1[Hz] sinusoidal input. 1[μm/V] 4.2 PC u 1.[ms] 1[Hz] 1 (7) c MATLAB Fig.11 y (7) y m y m y 1/1 Fig.12 c (24) 1 (24) H 1 (7) H Fig.1 y d Fig.13 Fig.14 d d(k) d(k) = max{y(k)} y d (k) (29) 573
.6.5 Compensated Uncompensated.4 Tracking error.3.2.1 -.1 Fig. 12: An example of inverse hysteresis calculated with eq.(24). Response.1.8.6.4.2 -.4 -.6 -.8 Compensated Uncompensated Desired -.1-1 -.5.5 1 y d Fig. 13: Comparison of the hysteresis loop of the compensated and the uncompensated system. 5 Preisach 1[Hz] 1) S. Xiao, Y. Li, Modeling and high dynamic compensating the rate-dependent hysteresis of piezoelectric actuators via novel modified inverse Preisach model, IEEE Transactions on control systems technology, 21-5, 1549/1557 (213) 2) K. Kuhnen, P. Krejci, Compensation of complex hysteresis and creep effects in piezoelectrically actuated systems A new Preisach modeling approach, IEEE -.3.5 1 1.5 2 Time (sec) Fig. 14: Comparison of the control performance of the compensated and the uncompensated system. Transactions on automatic control, 54-3, 537/55 (29) 3) K. Kuhnen, H. Janocha, Adaptive inverse control of piezoelectric actuators with hysteresis operators, IEEE European control conference, 1/6 (1999) 4) M. Tsai, J. Chen, Robust tracking control of a piezoactuator using a new approximate hysteresis model, Journal of dynamic systems, measurement, and control, 125-1, 96/12, (23) 5),,,,, 81-83, 15-221, (215) 6),,,, J67-A 126/132, (1984) 7),,,,, C( ), 123-11, 1958/1968, (23) 8) M. Brokate, J. Sprekels, Hysteresis and phase transitions, 1/355, Springer-Verlag, (1996) 574