Particle Swarm Optimization based Modeling and Compensation of Hysteresis of PZT Micro-actuator used in High Precision Dual-stage

Servo System Md. Arifur Rahman, Abdullah Al Mamun, Kui Yao and Yohanes Daud

Abstract—Piezoelectric micro-actuator made from PZT (Lead-Zirconium-Titanium) has been a popular choice as the secondary actuator of a dual-stage actuator system. However, the advantage gained by the precision of secondary actuator is somewhat lost by the inherent hysteresis nonlinearity of PZT actuator, if not compensated. This paper proposes a new rigorous technique for modeling and compensation of the hysteresis of PZT actuator used in dual-stage actuator system, which is established by artificial intelligence based heuristic optimization technique based on particle swarm optimization. In this paper, first the model parameters of Generalized Prandtl-Ishlinskii (GPI) model are identified off-line by PSO technique and corresponding inverse GPI is obtained as the hysteresis compensator. Then in case of parameter uncertainty, PSO based online tuning method is employed to adaptively adjust the parameters of inverse GPI model. For the linear controller of dual-stage, a simple design approach is followed. Simulation results show that the proposed technique can be efficiently used for the identification of hysteresis of PZT micro-actuator and the adaptive tuning the parameters of the effectiveness of the design. Keywords—Hysteresis; GPI; PSO; Inverse GPI

I. INTRODUCTION

High precision servo system has become an important interdisciplinary research topic which is involved in a vast range of applications across various fields such as semiconductor manufacturing, information technology and biology etc. Many of these systems require motion over large distance as well as fine precision. In order to meet both requirements, a dual-stage servo system consisting of a coarse actuator for large motion and a fine actuator for precise motion over short distance is often used. Large inertia of the coarse actuator and its high frequency flexible modes make it hard to achieve highly precise motion control using it alone. A lightweight micro-actuator, piggyback on the coarse actuator and capable of fast movement, is used to control the motion with precision. However, distance covered by the micro-actuator is very small and hence it is not capable of maneuvering over large distance. *Corresponding Author: Md. Arifur Rahman Email: a0092557@nus.edu.sg This work is supported by the Singapore National Research Foundation (NRF) under CRP Award No. NRF-CRP-4-2008-06 and IMRE/10-1C0107. Md. Arifur Rahman and Abdullah Al Mamun are with Department of Electrical & Computer Engineering, National University of Singapore, 4 Engineering Drive 3, Singapore 117576. Kui Yao is with the Institute of Materials Research and Engineering, A*STAR (Agency for Science, Technology and Research), 3 Research Link, Singapore 117602.

Synergy of the primary coarse actuator and the secondary micro-actuator makes it possible to achieve motion control over large distance with fine precision. Piezoelectric micro-actuator of high bandwidth is a popular choice as the secondary stage. However, the level of precision is compromised by hysteresis nonlinearity, an inherent property of PZT actuator, reducing the accuracy of the micro-actuator and thus limiting the achievable precision of the servomechanism. Hysteresis may also cause oscillations in the responses that may even lead to closed loop instability [1]. Such oscillations could be particularly detrimental in applications involving micro-positioning measurement and control [2-5]. Therefore, it is important to compensate the hysteresis nonlinearity of PZT micro-actuator. A number of hysteresis models have been proposed in the past to depict the hysteresis behavior of smart actuators and thereby designing the compensator of the hysteresis effects [6, 7]. These models include both physics-based models [6] and phenomenological models [8-10]. Phenomenological models are identified from input-output data obtained from experiment. Phenomenological hysteresis models published in the literature include Preisach model [8], the Krasnosel’skii–Pokrovskii model [10], the Prandtl–Ishlinskii model (PI) [5] etc. By applying a generalized play hysteresis operator to PI model in conjunction with density function and envelope function, it can be used to characterize accurate hysteresis nonlinearities in piezoelectric actuators. The PSO technique can generate a high-quality solution within shorter calculation time and stable convergence characteristics than other stochastic methods. Much research is still in progress for proving the potential of the PSO in solving complex problems. In control system, uncertainties are encountered both in the environment and within the system. These uncertainties can occur, for example, due to the simplified models or the equipment exposed to the environment of temperature and pressure. Then, the identified model parameters are fluctuated and the controllers need to be regularly retuned. In recent years, some research works were done in parameters tuning algorithm with particle swarm, by which the PID parameters are optimized [11-14]. Motivated by the previous works on hysteresis modeling and parameter tuning technique by PSO; in this paper, we propose a PSO based modeling technique where a GPI model is adopted for identification of hysteresis nonlinearity in PZT micro-actuator of dual-stage actuator system and corresponding inverse GPI model is identified as the hysteresis compensator. This identification of GPI

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Proceedings of 2014 IEEEInternational Conference on Mechatronics and Automation

August 3 - 6, Tianjin, China

and inverse GPI model is done offline. Then in case of parameter uncertainty, PSO based online tuning method is used to adaptively adjust the parameters of hysteresis compensator. In this paper, we focus on the hysteresis nonlinearity of the PZT micro-actuator used in a dual-stage HDD servo system. The paper is organized as follows. VCM and PZT frequency responses are presented in Section 2. Hysteresis analysis and identification are discussed in Section 3. Section 4 presents the simulation results. Conclusion of our findings and scope of future works are shown in Section 5.

II. FREQUENCY RESPONSE AND LINEAR MODELS

Our experimental setup for measuring the frequency response consists of a dynamic signal analyzer (DSA), VCM driver, PZT amplifier and the Laser Doppler Vibrometer (LDV) OFV 5001. The swept sine excitation signal generated by the DSA is applied to the driver/amplifier and actuator displacement is measured using the LDV. The resolution of the LDV is set to 50 nm/V. The experimental setup and the arrangements for model identification from frequency response experiment are shown in Figures 1(a) and 1(b). LabVIEW frequency response analyzer code generates the excitation signal (swept sine) and this signal and measured displacement are fed to two inputs of DSA. The analyzer produces the frequency response data and the result is saved for further processing. Frequency responses of the micro-actuator and VCM actuator are shown in Figure 2 and their linear models of are identified from the frequency response data. Method used in [15] is applied for model identification. Responses of the identified models are also shown in the same figures.

Remark: The measured frequency response relates an input signal measured in volts to an output signal measured in volts. So the transfer functions are unit-free. The output signal can be converted into displacement unit simply by multiplying by the scaling factor of 50 nm/V.

III. HYSTERESIS MODELING OF PZT MICRO ACTUATOR

A. Hysteresis Analysis

Hysteresis is a non-vanishing input-output loop at asymptotically low frequency which is an inherent nonlinear characteristic of the system. At high frequency, phase lag from the linear system adds to the response; therefore in order to analyze hysteresis, it is necessary to capture the time domain data at low frequency. In order to determine how low this frequency should be to precisely characterize hysteresis, data is captured at different low frequencies and the input-output loops are analyzed.

(a)

(b)

DAC

dSPACESD 1104

ADC

PZT micro-actuator Driver

PZT micro-actuator

(HDD)

LDV

InputDisplacement

MeasuredDisplacement

(c)

DSA(Dynamic Signal

Analyzer)

Input 1

Input 2

VCM or PZT micro-

actuator Driver

VCM or PZT micro-

actuator(HDD)

LDV

Sweep signal Displacement

MeasuredDisplacement

Figure 1: a) Experimental setup b) Frequency response experiment c)

Hysteresis identification experiment

(a)

(b) Figure 2: Frequency response a) PZT micro actuator b) VCM actuator

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The details of our hysteresis analysis were done in our previous work [16]. For capturing the hysteresis we have done an experiment. The sinusoidal signal generated by dSPACE card is sent to PZT driver through the DAC (digital-to-analog converter) channel and the signal from the LDV is acquired through the ADC (analog-to-digital converter) channel (Figure 1(c)). The sampling interval is set to 25 μs. Experimental results for different frequencies sinusoidal signals and corresponding simulated responses from linear model are shown in Figure 3 and 4 respectively. It is observed that phase lag of the linear model is negligible for 100 Hz and the input-output loop is almost vanished. Therefore, we use 100 Hz sinusoidal excitation to examine the hysteresis characteristics at different amplitudes (Figure 5).

Figure 3: Input-output loop at different frequencies (from experiment)

Figure 4: Input-output loop at different frequencies (from linear model)

Figure 5: Input-output loop for different amplitudes of the input sine

wave

B. Hysteresis Model

In this paper, GPI model is adopted for hysteresis identification. The details of hysteresis play operator and

GPI model have been discussed in [3]. The PI hysteresis models are limited to symmetric hysteresis loops, therefore generalized PI can be utilized to input-output relationships of the PZT actuators. The GPI model is in discrete form which makes use of generalized play operators and is described as

( ) = ( ( )) + ∑ [ ]( ) , (1)

( ( )) = ( ( )) if ̇ ( ) ≥ 0( ) if ̇ ( ) < 0 (2)

Where, v(k) is the discrete time input with k=[0,1,2…N] and N being the total number of discrete samples. In (1), n is the number of generalized play operator described as (0) = ( (0), 0), ( ) = [ ]( ) = ( ( ), [ ]( − 1)), (3)

with ( , ) = ( ( ) − , ( ( ) + , )), (4)

The density function and envelope functions and are = Where = and j = 1, 2, 3, ...n

= + ,

= + . (5)

The constants , , , , , , , are to be identified to complete the modeling of hysteresis and PSO is used to find these parameters.

C. Hysteresis Identification

In this paper, GPI hysteresis model parameters are identified through PSO which is a population based optimization technique inspired by the intelligence of swarms [17]. In PSO, a potential solution of the optimization problem is defined as one particle. A number of particles constituting the swarm move around in the search space for the optimum solution and each particle adjust its status according to its own experience as well as the experiences of other particles. Each particle is initialized by assigning its velocity and position randomly. At each iteration of the PSO algorithm, a particle keeps track of its own previous best position as well as the previous global best position among all the particles and adjusts its position by using the information of current position, current velocity, distance between the current position and previous own best position and distance between the current position and previous global best position. The velocity and position of each particle is changed as follows ( + 1) = + () − ( )+ ()( − ( ))

( + 1) = ( ) + ( + 1) (6)

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Where is the iteration number; and are the velocity and position of the particle respectively; and

are the best position of particle and best position of the swarm respectively and , , are the weight/weighing function, cognitive parameter and social parameter respectively. In our previous work [16], the details of the identification of GPI model parameters using PSO was discussed where different swarm size and different weights are used for the best fitting of the model with the experimental data. The model is identified by one set of input-output data obtained from experiment and validated by other sets of experimental data. In segmented parameterization, first we obtain all the parameters from one set of input-output data and pick half of the parameters from there and fix them. Next we obtain the remaining parameters from another set of input-output data. From here we obtain the parameters for the best fitted model with the experimental data (Figure 6).

Figure 6: Hysteresis in PZT micro-actuator a) GPI model identified by 8V sine input-output b) GPI model identified by 10V sine input-output

c) GPI model identified by segmented parameterization

IV. PSO BASED HYSTERESIS COMPENSATION

The parameters obtained for GPI model by using PSO are used for inverse GPI model, given below. ( ) = ( ( ) + ∑ ̂ [ ]( )), (7)

Where , denotes the inverse function of γ ( ) = ( ( ))if ̇ ( ) ≥ 0( ( ))if ̇ ( ) < 0 (8)

Where, ( ) is the discrete time input with =[0,1,2 … ] and being the total number of discrete samples. In (5), n is the number of generalized play operator described as (0) = ̂ [ ](0) = ̂ ( (0), 0),

( ) = ̂ [ ]( ) = ̂ ( ( ), ̂ [ ]( − 1), (9) With ̂ ( , ) = ( ( ) − , ( ( ) + , )), (10)

The parameters , and are obtained as

= 1 , = + ∑ ( − ) = − ( + ∑ )( + ∑ ) (11)

When the modeling is perfect, the exact inverse GPI model can compensate the effect of hysteresis. Figure 7 shows the compensation of hysteresis when the modeling is accurate, i.e. the same parameters identified from the experiment using off-line PSO technique are used for designing the hysteresis and compensator models. This simulation results are obtained for a multiple frequency reference trajectory. However in case of parameter uncertainty which is a common practical problem, the exact modeling is quite impossible and online parameter tuning becomes somewhat necessary. Here in this paper, we propose a PSO based parameter tuning technique for adaptively adjusting the parameters of inverse GPI model in case of any parameter change. Figure 8 shows the block diagram of the proposed technique in micro-actuator control system and dual-stage servo system. The flowchart in Figure 9 shows the proposed technique of PSO based hysteresis compensator tuning where the target cost function is defined as the

Target cost function, = ( )

Where, is the hysteresis-compensated response and is the actual response from micro-actuator loop.

IV. SIMULATION RESULTS

In this section, the obtained simulation results are presented and discussed. Numerical simulations are conducted using a system developed in Matlab and Simulink of MathWorks. It includes the dynamics of the system as well as the proposed controller. The following simulation scenarios are carried out: A. Scenario1: PZT micro-actuator control

Figure 8(a) shows this configuration. In this simulation, the PZT micro-actuator controller is designed as a simple PI controller and simulation is done in discrete time with a sampling frequency of 40 KHz. The parameters of hysteresis model are changed by 50% from the identified values and corresponding step responses are captured. The global minimum of cost function decreases to a minimum value after few iterations of tuning (Figure

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10(a)) and the step response becomes better with minimized overshoot and steady-state error which is approximately same as the hysteresis-compensated response (Figure 10(b)). B. Scenario2: Dual-stage control (discrete time)

Figure 8(b) shows this configuration where the PZT micro-actuator controller is designed as a PI controller and VCM controller is designed as a PID controller and the simulation is done in discrete time with a sampling frequency of 40 KHz. The parameters of the hysteresis model are increased by 50% from the identified values and corresponding step responses are captured (Figure 10(c) and 10(d)).

(a) (b)

(c) Figure 7: Hysteresis compensation for no parameter change

Refrence +Micro-

actuator Linear

controller

Notch Filter Nonlinear PZTMicro-actuator

+

_

Micro-actuator Loop

Hysteresis compensated model of PZT

Micro-actuator

Hysteresiscompensator

+

PSO based Parameter tuning ofInverse GPI model

e

+

_

yd

ym

(a)

Refrence +Micro-

actuator Linear

controller

Notch Filter Nonlinear PZTMicro-actuator

VCMLinear

controllerNotch Filter VCM

+ Decoupler++

+

+

+

+

_

Micro-actuator Loop

VCM Loop

Hysteresis compensated model of PZT

Micro-actuator

Hysteresiscompensator

+

PSO based Parameter tuning ofInverse GPI model

e

+

_

yd

ym

yv

(b)

Figure 8: PSO based parameters tuning of hysteresis compensator in a) PZT micro-actuator control b) Dual-stage Actuator system

C. Scenario3: Dual-stage control (continuous time)

Figure 8(b) shows this configuration where the PZT micro-actuator controller is designed as a lag filter and VCM controller is designed as a lead-lag controller and the simulation is done in continuous time. The parameters of the hysteresis model are decreased by 50% from the identified values and corresponding step responses are captured (Figure 10(e) and 10(f)).

End

Initialization of particles’ positions { ( ), ( ), ( ), ( ), ( ), ( ), ( ), ( )}

Initialization of particles’ velocities

Using particles’ positions calculate the target cost function for each particle

If ( ) < ( )

The local best position “Pbest” of each particle =The initial position of each particle

& The best global position “gbest” of the swarm

=The position of the particle which has the lowest “T”

If k=0

= = is that has the lowest “T” from

If <

= =

If desired target cost function is achieved

Update particles’ positions &velocities

1

2

1

2

Start

No

Yes

No Yes

No Yes

No

Yes

Figure 9: Flowchart of PSO based hysteresis compensator tuning

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(a) (b)

(c) (d)

(e) (f)

Figure 10: global minimum of cost function for a) scenario 1 c) scenario 2 e) scenario 3; step response for b) scenario 1 d) scenario 2 f) scenario 3

VI. CONCLUSION This paper proposes a new PSO based parameter tuning for compensation of hysteresis nonlinearity seen in PZT micro-actuator. Besides PSO technique with different parameters tuning results an efficient identification of nonlinear model of the micro-actuator. The proposed scheme is found to compensate successfully the hysteresis nonlinearity of PZT micro-actuator even in worst possible case. Resulting step response shows that in case of parameter change, the proposed technique results in negligible oscillation and overshoot and insignificant steady-state error in the output after a few iterations. Although this paper substantiates the application of proposed scheme to compensate for hysteresis behavior and improving the system performance of a dual-stage actuation system of a commercial HDD, this scheme can be applied to other applications where hysteresis behavior is encountered or to any kind of dual-stage system where piezoelectric actuator is used.

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