A c b c c c ] := A cp b cp c c b cp A c c cp (9) (8) T u { xk ] = Asxk] b suk] yk] = c s xk] 3 () Moving ifference long sampling observer: N uk N] xk]

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IIC 9 6 Proposal of Moving Difference Long Sampling Observer for Quantization Error Reuction Koichi Sakata, Hiroshi Fujimoto (Yokohama National University) Abstract High resolution encoers are employe to inustrial equipment which is require accuracy, for example, NC machine tools, exposure systems, an so on. Generally, the ocity of machines is calculate by the ifference of the position given by the encoer. Although the high resolution encoer is employe, the quantization error an the half sample elay inuce by the ifference cannot be avoie. These are problems in the case of require high precise ocity. In this paper, we propose an observer which can estimate state variables with reucing the quantization error. Finally, simulations an experiment with experimental precision stages are performe to show the avantages of the propose observer. (precision stage, observer, quantization error, ea-time compensation ). () M/T () (3) (4) (5) (6) (7). v if k] = z zt u yk] () = ˆvk ] () y y real q yk] = y real k] qk] (3) N N v if k] = zn z N NT u yk] (4) = ˆvk N ] (5) 3. 3 { ẋp (t) = A cp x p (t) b cp (u(t) (t)) (6) y(t) = c cpx p(t) { ẋ (t) = A c x (t) (7) (t) = c c x (t) x = x T p, x T ] T { ẋ(t) = Ac x(t) b c u(t) y(t) = c cx(t) (8) /6

A c b c c c ] := A cp b cp c c b cp A c c cp (9) (8) T u { xk ] = Asxk] b suk] yk] = c s xk] 3 () Moving ifference long sampling observer: N uk N] xk] = Axk N] B. () uk ] A ] B := A N s A N s b s b s ] () ˆxi ] = Aˆxi] Bui] H(yi] c s ˆxi]) (3) i NT u NT u N T u yi] NT u uk N] ˆxk] = Aˆxk N] B. uk ] H(yk N] c s ˆxk N]) (4) k T u N { ξi ] = Âξi] ˆByi] Ĵui] ˆxi] = Ĉξi] ˆDyi] (5) i NT u yk-n] uk-n] uk-n] uk-n-] uk-n-] (k-n)t u (k-n-)t u (k-n)t u (k-)t u (k-n)t u (k-n-)t u (k-n)t u (i-)t y yk-n-] yk-n-] yk-n] yk-n] yk-n] yk-] (i-) T y Sampling time. uk-] yk-] (k-) T u uk-] yk] kt u it y uk] uk ] yk] (k) T u uk N] ξk] = Âξk N] ˆByk N] Ĵ. ˆxk] = Ĉξk] ˆDyk] (6) k T u 3 3 n o (8) n o yk n o ] n o ˆxk n o ] ξk n o ] = Âξk N n o] ˆByk N n o ] uk N n o ] Ĵ. uk n o ] ˆxk n o ] = Ĉξk n o] ˆDyk n o ] (7) ˆxk] uk n o ] ˆxk] = A ˆxk n o ] B. (8) A B ] := A n o s uk ] A n o s b s b s ] (9) 4. 4 3 /6

uk] z -N-no z --no z -no z - ^J ξ B^ k-no] z -N C^ z -N A^ ξ yk-no] D^ k-n] x^ A k-no] B x^k] Magnitue B] Phase eg] 5 3 4 Characteristic from y to v if ifference per Tu ifference per NTu 3 4 4 Frequency response of ifference in N = 4. Moving ifference long sampling observer with ea-time. 5 x 3 (ifference per Tu) (ifference per NTu) u(t) H (T ) u (t) uk] - e -sti bc y(t) x(t) cc e-st o S S (T u) Encoer yk] Ac e -sti k] c s A s x p k] x k] Observer xk]..4.6.8. (a) Estimate ocity A cp 3 Block iagram of the system. Specifications of plant. Mass M 4.3 kg Viscosity B Thrust coefficient K t Sampling perio T u Input ea-time T i.8 N/(m/s) 8.5 N/A / s T u Output ea-time T o T u Resolution. µm b cp c cp ] = M B M () ] ] A c = () c c u(t).5 Hz (t) s.5 4 4 y v 5 N = 4 NT u T u N 4 3 6 y ˆv ˆ N = 4 5 x (ifference per Tu) (ifference per NTu)..4.6.8. (b) Estimate ocity error 5 Time response by ifference in N = 4. Stanar eviation of ocity errors an isturbance errors in simulation. Velocity Disturbance Singlerate min-orer.5e m/s 4. mn Singlerate full-orer 9.6e m/s.6 mn 8.58e m/s. mn LPF LPF NT u 7 8 3σ 6 nm, 5 Hz 3/6

Magnitue B] Characteristic from y to v hat 5 singlerate min orer observer singlerate full orer observer 5 3 4 5 x 3 Phase eg] 3 4 (a) y to ˆv..4.6.8. (a) Estimate ocity Magnitue B] Phase eg] Characteristic from y to hat singlerate min orer observer singlerate full orer observer 3 4 3 3 4 (b) y to ˆ 6 Frequency response of observer in N = 4. Sensitivity characteristic x..4.6.8. x..4.6.8. x..4.6.8. (b) Estimate ocity error Magnitue B] 3 Singlerate min orer observer Singlerate full orer observer 3 4 7 Sensitivity characteristic of observer in N = 4. 9 6 5. nm. µm 3 3σ nm, 5 Hz Disturbance N] Disturbance error N]..5 Disturbance trajectory.5 hat (singlerate min orer observer) hat (singlerate full orer observer) hat ()...4.6.8....... (c) Estimate isturbance Disturbance error hat (singlerate min orer observer)..4.6.8. hat (singlerate full orer observer)..4.6.8. hat ()..4.6.8. () Estimate isturbance error 8 Time response by observer in N = 4. 4 4/6

3 5 x..4.6.8. 図 Experimental precision stage. x..4.6.8. 5..4.6.8. Frequency response of plant (from force to stage position) x Phase eg] Magnitue B] (a) Estimate ocity Measurement Moel x 図..4.6.8. Frequency response of experimental precision stage. (b) Estimate ocity error 5 x (ifference per Tu) (ifference per NTu) Disturbance trajectory. Disturbance N].5.5 hat (singlerate min orer observer) hat (singlerate full orer observer) hat ()...4.6.8...4.6.8. (a) Estimate ocity (c) Estimate isturbance 5 x (ifference per Tu) (ifference per NTu) hat. (singlerate min orer observer) Disturbance error N]...4.6.8. hat (singlerate full orer observer). Disturbance error...4.6.8 hat...4.6.8. (b) Estimate ocity error (). 図 Experimental results by ifference in N = 4....4.6.8. () Estimate isturbance error 図9 Time response by observer with sinusoial noise in N = 4. 6. ま と め 移動差分による速度導出では 最低でも半サンプルの遅 れは避けられず 量子化誤差低減のために差分周期を長く するとさらに遅れてしまう これに対し オブザーバを用 いた速度導出ではプラントモデルを必要とするがサンプル 遅れの問題は解消される オブザーバの更新を移動差分的 に行うことで シングルレート同一次元オブザーバおよび シングルレート最小次元オブザーバに比べて量子化誤差成 分を低減できるオブザーバを提案した さらに提案法は特 定周波数のノイズに対して非干渉化できることも特徴であ る 最後にシミュレーションおよび実験により 提案法の 5/6

5 x 5 x..4.6.8. (a) Estimate ocity..4.6.8. (a) Estimate ocity x..4.6.8. x..4.6.8. x..4.6.8. x..4.6.8. x x..4.6.8. (b) Estimate ocity error 3 Experimental results by observer in N = 4. 3 Stanar eviation of ocity errors in experiment. Singlerate min-orer Singlerate full-orer Velocity 9.6e m/s.3e m/s 8.7e m/s 868636, 6868 R. C. Kavanagh an J. M. D. Murphy, The effects of quantization noise an sensor nonieality on igital-ifferentiator-base ocity measurement, IEEE Trans. Instrumentation an Measurement, vol. 47, no. 6, pp. 457 463, 998. T. Ohmae, T. Matsua, K. Kamiyama, an M. Tachikawa, A microprocessor-controlle highaccuracy wie-range spee regulator for motor rives, IEEE Trans. Inustrial Electronics, vol. 9, no. 3, pp. 7, 98. 3 R C. Kavanagh, Improve igital tachometer with reuce sensitivity to sensor nonieality, IEEE Trans. Inustrial Electronics, vol. 47, vo. 4, pp. 89 897,. 4 T. Tsuji, T. Hashimoto, H. Kobayashi, M. Mizuochi, 4..4.6.8. (b) Estimate ocity error Experimental results by observer with sinusoial noise in N = 4. an K. Ohnishi, A wie-range ocity measurement metho for motion control, in IEEE Trans. Inustrial Electronics, vol. 56, no., pp. 5 59, 9. 5 N. Iiyama, K. Ohnishi, an T. Tsuji, An approach to estimate ocity for acceleration control, in Proc. th IEEE International Workshop on Avance Motion Control, pp. 687-69, 8. 6 K. Fujita an K. Sao, Instantaneous spee etection with parameter ientification for ac servo systems, IEEE Trans. Inustry Applications, vol. 8, no. 4, pp. 864 87, 99. 7 Y. Hori, T. Umeno, T. Uchia, an Y. Konno, An instantaneous spee observer for high performance control of c servo motor using DSP an low precision shaft encoer, in Proc. 4th European Conf. Power Electronics, vol. 3, pp. 647 65, 99. 8 Hiroshi Fujimoto an Yoichi Hori, Visual servoing base on multirate control an ea-time compensation, Journal of the Robotics Society of Japan, vol., no. 6, pp. 78 787, 4. (in Japanese) 6/6

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