MEC committee

連続時間不安定零点を持つ精密位置決めステージの軌道追従制御法―時間軸反転とマルチレートフィードフォワードによる安定逆系の設計―
Tracking control method for high-precision stage
with continuous time unstable zeros:
Stable inversion by time axis reversal and multirate feedforward

Wataru Ohnishi, Thomas Beauduin, and Hiroshi Fujimoto
The University of Tokyo
Table
Linear encoder

Linear motor
CarriageAir guide
Linear encoder

W. Ohnishi and H. Fujimoto, “Multirate Feedforward Control with State Trajectory Generation based on Time Axis Reversal for Plant
with Continuous Time Unstable Zeros,” IEEE International Conference on Advanced Intelligent Mechatronics, 2016, pp. 689–694.

1.1 Introduction －examples of the plant with unstable zeros－

Hard Disk Drive

High-precision stage

Motor and converter

Robot

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http://hflab.k.u-tokyo.ac.jp/

Atomic Force Microscope

Airplane
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1.2 Zeros in discrete time domain
𝑃𝑐 𝑠 =−(𝑠−140)(𝑠+100)𝑠(𝑠+2000)(𝑠+2)(𝑠 2 +20𝑠+40000)

𝑃𝑠 𝑧 =

𝐾(𝑧+3.547) (𝑧−1.014) (𝑧−0.9900) (𝑧+0.2543)(𝑧−1) (𝑧−0.9998) (𝑧−0.8187) (𝑧 2 − 1.998𝑧 + 0.998)

Discretization by
zero-order hold(𝑇𝑢 = 100𝜇s)• Intrinsic zeros (真性零点)
have counterpart in continuous time zeros
Generated from sensor and actuator collocation• Discretization zeros (離散化零点)
are generated by discretization
Approximated as Euler-Frobenius polynomial

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Relativeorder

Discretization zeros

2−1

3−2 − 3, −2 − 3≈ −3.7, −0.26Åström, K., Hagander, P. and Sternby, J.: Zeros of sampled systems, Automatica, Vol. 20, No. 1, pp. 31–38, 1984.
T. Hagiwara, T. Yuasa, and M. Araki, “Stability of the limiting zeros of sampled-data systems with zero-and first-orderholds,” Int. J. Control, vol. 58, no. 6, pp. 1325–1346, 1993.

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1.3 Introduction －problem of unstable zeros－• Its inversion system has unstable poles

𝐶𝑓𝑓 = 𝑃−1𝑠 (𝑠 + 84.65) (𝑠 2 + 22.21𝑠 + 1.151 × 104 )−1𝑃𝑐 𝑠 =−0.1034 (𝑠 + 231.6) (𝑠 − 93.65)
𝑃𝑠−1 𝑧(𝑧 − 1) (𝑧 − 0.9916) (𝑧 2 − 1.998𝑧 + 0.9978)=−5.175 × 10−10 (𝑧 + 1.001) (𝑧 − 1.009) (𝑧 − 0.9771)• UndershootTradeoffs between settling time 𝑡𝑠 and% Undershoot 𝑀𝑢(𝑏0 denotes unstable zeros)𝑀𝑢 >1𝑏0 𝑡𝑠Hoagg, J. and Bernstein, D.: Nonminimum-phase zeros – much to do about nothing - classical control - revisitedpart II, IEEE Control Systems, Vol. 27, No. 3, pp. 45–57 (2007)2016/12/06 Goodwin, G. C., Graebe, S. F. and Salgado, M. E.: Control System Design (2000).

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2. Approximated inverse based single rate feedforward

Approximated plant inversePlant• Plant approximated inverse is designed in discrete time
Discretization zeros and intrinsic zeros are dealt in same time
Stable part:
Unstable part:

andis approximated asis designed1. nonminimum-phase zeros ignore (NPZI) method [Butterworth, et al. 2012]
2. zero-phase-error tracking controller (ZPETC) method [Tomizuka, 1987]
3. zero-magnitude-error tracking controller (ZMETC) method [Wen & Potsaid, 2004]
Trade off between zero-phase-error and zero-magnitude-error characteristics
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3. Preactuation Perfect Tracking Control (PPTC) method[Ohnishi, Fujimoto, AIM2016]

State trajectory generation
with time axis reversal

Multirate feedforward(stable inversion for unstable intrinsic zeros)(stable inversion for unstable discretization zeros)

Stable part state trajectory generation

Plant

Unstable part state trajectory generation

1.

State trajectory generation
Stable inversion for continuous unstable zeros
by time axis reversal

2.

diverges whenis unstable→time axis reversal

Multirate feedforward
Stable inversion for unstable discretization zeros[Fujimoto, Hori, Kawamura, 2001]

Stable trajectory generation is a key to achieve PTC for NMP system 5 / 14

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3.1 Conventional PTC method –State trajectory generationState trajectory generation
with time axis reversal

Multirate feedforward(stable inversion for unstable intrinsic zeros)(stable inversion for unstable discretization zeros)

Stable part state trajectory generation[Fujimoto, Hori, Kawamura, 2001]
Plant

Unstable part state trajectory generation

Coeffs of numerator

Derivative relationship(Control Canonical Form)

Diverge for plant with continuous unstable zeros→Approximated in conv study [Fukushima, Fujimoto, et al., 2006]
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Position, velocity, acceleration…

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1.

Stable & unstable decomposition

Stabilized byimaginary axis reversal2.

State trajectory generation for stable part

3.

State trajectory generation for unstable part
Convolution between
time axis reversed reference
andimaginary axis reversed unstable zeros,and then time axis reversed again[Sogo, 2006]

4.
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Overall state trajectory
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3.2 Preactuation PTC based on multirate FF
Example2016/12/06

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3.3 Preactuation PTC based on multirate FF
State trajectory generation
with time axis reversal

Multirate feedforward(stable inversion for unstable intrinsic zeros)(stable inversion for unstable discretization zeros)

Stable part state trajectory generation

Plant

Unstable part state trajectory generation• Multirate feedforward
Single rate system[Fujimoto, Hori, Kawamura, 2001]

Multirate system

matrix become non-singular

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Stable inversion for discretization zeros!

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3.4 Experimental setup(measured)• High-precision stage
Guide: air guideMotor: Linear motor
Encoder: 1nm resolution × 2

Table
Linear encoder

Linear motor
CarriageAir guide
Linear encoder

Nano-sage I• Frequency response
From measured current to measured position

Unstable cont. zeros × 2

Frequency response of plant
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8th order model

Model of nano-sage I

Unstable intrinsic. zeros × 2
Unstable discretization zeros × 1

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3.4 Simulation and Experiment –FB controller–

Closed loop performance

Step response (simulation)

FB controller• PID by pole placement design(2 Hz)• Shaping filter for 30Hz NMP zeroFB Bandwidth: 2.3 HzPlant
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3.4 Simulation and Experiment

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SimSimSim

ExpExpExp

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3.4 Simulation and Experiment
SimSimSim

Preactuation

PPTC: Preactuation perfect tracking controlInput current from -0.4 [s]ExpExpExpPreactuation

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4. Conclusion
State trajectory generation
with time axis reversal

Multirate feedforward(stable inversion for unstable intrinsic zeros)(stable inversion for unstable discretization zeros)

Stable part state trajectory generation

Plant

Unstable part state trajectory generation

MethodMax error
[mm]

ZPETC

0.631

ZMETC

1.30

PPTC

0.0288-95.4%• Preactuation Perfect Tracking Control
achieves perfect tracking for plant with continuous unstable zeros
by infinite time preactuation
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