Active disturbance rejection control for precise position tracking of piezoelectric actuators①

Zheng Zhaoying (鄭兆瑛), Lu Qishuai, Zhang Sijiong

(*National Astronomical Observatories / Nanjing Institute of Astronomical Optics & Technology,Chinese Academy of Sciences, Nanjing 210042, P.R.China)(**Key Laboratory of Astronomical Optics & Technology, Nanjing Institute of Astronomical Optics & Technology,Chinese Academy of Sciences, Nanjing 210042, P.R.China)(***University of Chinese Academy of Sciences, Beijing 100049, P.R.China)

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Active disturbance rejection control for precise position tracking of piezoelectric actuators①

Zheng Zhaoying (鄭兆瑛)******, Lu Qishuai***, Zhang Sijiong②

(*National Astronomical Observatories / Nanjing Institute of Astronomical Optics & Technology,Chinese Academy of Sciences, Nanjing 210042, P.R.China)(**Key Laboratory of Astronomical Optics & Technology, Nanjing Institute of Astronomical Optics & Technology,Chinese Academy of Sciences, Nanjing 210042, P.R.China)(***University of Chinese Academy of Sciences, Beijing 100049, P.R.China)

Positioning with high precision piezoelectric actuators is widely used. To overcome positioning inaccuracy caused by hysteresis and creep of actuators, a precise tracking method for piezoelectric actuators using active disturbance rejection control (ADRC) has been proposed in this paper. This method, in real-time, actively estimates and compensates parameter uncertainties, nonlinear factors such as hysteresis, and external disturbances in the tracking system. Precise tracking of the piezoelectric actuator can be achieved without any form of feedforward compensations. The experimental results demonstrate that the active disturbance rejection controller can reduce tracking errors by over 90% comparing with those using the PID controller. Those features of the proposed control method are very suitable for applications in adaptive optics.

active disturbance rejection control (ADRC), piezoelectric actuators, position tracking

0 Introduction

Piezoelectric actuators can implement micrometers and even nanometers positioning. In recent years, piezoelectric actuators are increasingly applied in scanning probe microscopes[1], atomic force microscopes[2,3], micropositioning mechanisms[4,5], aerospace applications[6]and adaptive optics systems[7,8]. However, there are some disadvantages of piezoelectric actuators such as hysteresis, creep and so on. Those disadvantages of piezoelectric actuators result in positioning inaccuracy and limiting the whole system performances in which piezoelectric actuators are used as the correcting units. The control methods to improve the accuracy for the position tracking of piezoelectric actuators are needed.

To achieve highly precise positioning for piezoelectric actuators, a lot of researches on control algorithms for these devices have been done in recent years. In order to suppress the effect of hysteresis of piezoelectric actuators, the main nonlinear constituent, on their positioning, piezoelectric actuators can be driven by charge[9]. However, the implementation of the charge amplifier is complex and expensive. For this reason the piezoelectric actuators driven by voltage are widely chosen in many applications. For previous researches on suppressing the effect of hysteresis of piezoelectric actuators, a theoretical hysteresis model is firstly established, then the inversion of the hysteresis as a feedforward is set up in a control loop to compensate the positioning inaccuracy caused by the hysteresis[10]. Song, et al.[11]built up a classical Preisach model, and applied the inverse model and a feedback controller to eliminate the hysteresis for the micro position tracking control. However, this method only applies to low frequency input and is rate-dependent. Guo, et al.[12]proposed a real-time inverse hysteresis compensation method with a modified Prandtl-Ishlinskii model. Ru, et al.[13]applied a novel mathematical model obtained based on Prandtl-Ishlinskii operator to characterizing hysteresis and employed an adaptive inverse control algorithm to reduce hysteresis. Zhu, et al.[14]developed an ellipse-like model to describe the hysteresis of piezoelectric actuators and the accuracy of tracking control has been improved by a real-time feedforward controller with the inverse model. However, these methods are also rate-dependent. Eielsen, et al.[15]argued an online adaptive nonlinear hysteresis compensation method for certain periodic desired trajectories. However, the certain periodic trajectories are only suited for scanning applications of piezoelectric actuators, such as atomic force microscopes. Our main purpose for control piezoelectric actuators is to precisely regulate a tip/tilt mirror to achieve a high image quality. These rate-dependent control methods are not appropriate for the applications in adaptive optics. Li, et al.[16]proposed a fuzzy hysteresis model (FHM), and developed the enhanced adaptive hybrid controller to achieve high performance tracking. Although the algorithm is rate-independent, it is highly complex. Because atmospheric turbulences change rapidly, the control method must be simple and suitable for the real-time implementation in adaptive optics.

Active disturbance rejection control (ADRC)[17-19]is a control algorithm that satisfies the requirements of adaptive optics systems aforementioned. ADRC technique has been widely applied in many fields[20-22]. The gist of the ADRC algorithm is explained as follows. Taking the second order system as an example, the system is taken as a double integrator model, and an extended state observer (ESO) built up for this system model, in which there is an extra extended state variable used for estimating modeling errors, parameter uncertainties, internal and external disturbances in the system. The control algorithm actively compensates for all the factors mentioned above in real-time with updated estimation of those by the ESO so as to achieve precise position tracking of actuators. Moreover, the advantages of the ADRC algorithm are of low complexity and requiring a little prior information of the real system. To verify the effectiveness of the ADRC algorithm on position tracking of piezoelectric actuators, several cases of trajectories are designed to track in a series of experiments.

This paper is arranged as follows. In the first section, an experimental system is described in detail and a simple model of piezo platform is given. The second section presents the ADRC method. The experimental results are provided in Section 3. Finally, this paper ends with conclusions in Section 4.

1 System description

1.1 Experimental setup

An experimental system, as shown in Fig.1, has been established for the investigation of the ADRC control algorithm. The architecture of the experimental system is depicted in the sketch as shown in Fig.2. It consists of a piezo tip/tilt platform with inbuilt strain gauge sensors to measure its angle changes, a servo controller card, voltage amplifiers, a sensor processing module, and Zynq-7000 with a digital-to-analogue (D/A) board and an analog-to-digital (A/D) board.

Fig.1 Experimental platform

Fig.2 Sketch of the experimental architecture

The piezo tip/tilt platform employed is from PI (Physik Instrumente), which has the nominal 2 mrads angular displacement range corresponding to a range of operating voltage from 0 to 100V. The servo controller card can be switched servo status on and off. When the servo status is in the off mode, there’s only a slew rate limiter active. Voltage amplifiers with a fixed gain of 10 provide voltage ranging from -20V to +120V. The sensor processing module is connected with both the platform and Zynq-7000. The voltage conversion ranges of the D/A and A/D boards are from 0V to 3.3V with 12-bit resolution. Zynq-7000 includes the programmable logic (PL) and the processing system (PS). The D/A and A/D cards are driven by PL, and the real-time control algorithm is run on PS. In these experiments, the sampling frequency of the control loop is set at 5kHz.

1.2 Piezo tip/tilt platform

As shown in Fig.3, when control input voltage changes, the voltage of one PZT (Piezoelectric Ceramic Transducer) actuator of each pair increases and the voltage of the other decreases by the exact same magnitude. It may be simply modeled as a lightly damped second order system. The angular displacement of platform in tip axis is modeled as

(1)

M(t)=nU(t)+φ(U,t)

(2)

where n is the proportional coefficient between moment M(t) and the control voltage U(t), φ(U,t) is the moment caused by nonlinear factors such as hysteresis of piezoelectric actuators, uncertain disturbances and so on.

Fig.3 Working principle of one axis motion (from PI user manual)

2 Active disturbance rejection control (ADRC)

In this paper, linear active disturbance rejection control (ADRC) is applied to control the piezo tip/tilt platform. The dynamics of the platform in tip axis can be regarded as

(3)

where b=n/J.

Fig.4 Block diagram of the control system

2.1 ESO design

(4)

The Luenberger observer of this system expressed by Eq.(4) in state-form space is designed as

(5)

λ(s)=s3+l1s2+l2s+l3=(s+ω0)3

(6)

So the observer poles are all placed at -ωoin Laplace s-plane. This design can let the observer gain vector L of ESO be easily tuned by just changing the observer bandwidth ωo. Thus only adjusting the observer bandwidth ωo, the three components of the observer gain vector L can be tuned. The larger the ωois, the faster and the more accurate the observer is. However, a larger ωoalso increases noise sensitivity, and is limited by hardware constraints. Hence a proper ωoshould be tuned between the tracking performance and hardware constraints.

2.2 Control Algorithm

(7)

Disregarding the estimation error, the model of the platform in tip axis is reduced to a double integrator,

(8)

The original problem is simplified to a much simpler one, which can be dealt by a proportional-derivative (PD) controller

U0(t)=kp[θd(t)-z1(t)]-kdz2(t)

(9)

(10)

where ωcis the bandwidth of the controller. Obviously, the larger it is, the faster the response speed is. However, like ωo, ωcis tuned based on the competing requirements of tracking performance, noise sensitivity and stability margin.

3 Experimental results

3.1 Single frequency triangular trajectory

The frequency of the desired triangular trajectory in this experiment is 10Hz and the amplitude of that is 50μrad. To show the characteristics of the ADRC algorithm, the tracking results using ADRC are compared with ones using PID and hybrid controllers[16], shown in Fig.5. The hybrid control algorithm takes more than 1ms. Even if FHM is identified at first, the hybrid controller without updating FHM online still takes about 20μs. ADRC just takes less than 1μs. As shown in Fig.5, because of the precise position tracking of ADRC algorithm, the trajectory tracked by ADRC controller (the blue dash curve) nearly coincides with the desired one (the black solid curve). ADRC controller can obtain 1.01μrad maximum tracking error, and the root mean square (RMS) error is 0.29μrad, which is about the noise level. The RMS error with the PID controller is 3.29μrad, and that with hybrid controller is 1.14μrad. Compared with PID controller, the ADRC controller reduces RMS error by 91.2%. And ADRC controller reduces time cost significantly.

Fig.5 Tracking performances of ADRC comparing with those of servo card at 10Hz with 50μrad triangular waveform trajectory

3.2 Single frequency sinusoidal trajectory

Fig.6 shows the experimental result of the desired single sinusoidal trajectory, whose frequency is 50Hz and amplitude is 50μrad. The maximum error decreased from 23.60μrad with PID controller to 1.84μrad with ADRC controller, and reduced by 92%. The reduction of RMS error of angular displacement tracking is approximately 98% for the ADRC controller compared with that of the PID controller. More experimental results of the desired trajectories with other different frequencies and amplitudes are shown in Table 1.

Fig.6 Tracking performances of ADRC comparing with those of servo card at 50Hz with 50μrad sinusoidal waveform trajectory

Table 1 Tracking errors using ADRC and servo card as to several sinusoidal trajectories with different frequencies and amplitudes

Desiredtrajectory(μrad)ErrorMAX(μrad)PIDADRCErrorRMS(μrad)PIDADRC50@50Hz23.601.8415.400.3225@50Hz12.041.507.670.29100@50Hz45.563.8430.610.4850@100Hz40.213.6627.430.6050@30Hz15.551.489.990.3950@10Hz6.320.973.600.28

3.3 Multiple frequencies sinusoidal trajectory

To further demonstrate the advantages of ADRC controller method, the tracking experiment with a multiple frequencies sinusoidal trajectory is implemented. The desired trajectory is 300+60sin(40πt)+40sin(100πt)+20sin(200πt)μrad. The experimental result is shown in Fig.7. Using the PID controller, the maximum tracking error of angular displacement is 36.12μrad, and the RMS tracking error is 18.16μrad. However, the ADRC controller can obtain 3.48μrad maximum error, decreased by 90.37%, and 0.45μrad RMS error, decreased by 97.52%.

Fig.7 Tracking performances of ADRC comparing with those of servo card for multiple frequency sinusoidal trajectory

4 Conclusions

In this paper, a control algorithm, active disturbance rejection control, has been developed for piezoelectric actuators. The control algorithm is model independent and more tolerant to uncertain dynamics and unknown disturbances. The hysteresis and creep of piezoelectric actuators can be treated as parts of the unknown disturbances. The ADRC control algorithm actively estimates and compensates disturbances to control systems in real-time, such that precise tracking of the piezoelectric actuators without any hysteresis models can be implemented. This algorithm is of low complexity, and takes less than 1μs on the Zynq-7000 platform. So ADRC suits for real-time implementation. The experimental results clearly show the effectiveness of this algorithm. It provides a reduction of more than 90% tracking error compared with that of PID controller card. ADRC is suitable for controlling requirements of adaptive optics and other areas for precise positioning.

Acknowledgement

Many thanks to Dr. Li Changwei at Nanjing Institute of Astronomical Optics & Technology.

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Zheng Zhaoying, born in 1984, received his Ph.D. degree from the University of Chinese Academy of Sciences. Now he works in the Nanjing Institute of Astronomical Optics & Technology. His research focuses on high performance control in adaptive optics.

10.3772/j.issn.1006-6748.2015.03.014

①Supported by the National Natural Science Foundation of China (No. 11373048).

②To whom correspondence should be addressed. E-mail: sjzhang@niaot.ac.cn Received on Feb. 27， 2014***