Appl. Math. Inf. Sci.9, No. 1L, 251-258 (2015) 251

Applied Mathematics & Information SciencesAn International Journal

http://dx.doi.org/10.12785/amis/091L32

Design and Implementation of Advanced Digital Controlsfor Piezo-Actuated Systems using Embedded ControlPlatformChi-Ying Lin∗ and Chien-Yao Li

Department of Mechanical Engineering, National Taiwan University of Science and Technology, Taipei, Taiwan

Received: 4 Dec. 2013, Revised: 4 Apr. 2014, Accepted: 5 Apr.2014Published online: 1 Feb. 2015

Abstract: This paper presents design of an advanced digital control method and implementation on an embedded control platformfor precision tracking of piezo-actuated systems. The proposed control method consisting of neural control and repetitive control aimsto achieve good performance for repetitive tracking tasks and robustness for uncertain parameters change. A FPGA basedembeddedcontroller, CompactRIO, is applied to realize such a complicated control algorithm under LabVIEW programming environment. Withcareful consideration of implementation issues involved in the controller design process, the experimental results demonstrate theeffectiveness of the method and verify the feasibility of using embedded hardware for advanced controls implementation.

Keywords: Piezo-Actuated Systems, Advanced Control, Embedded Control Platform

1 Introduction

In recent years, piezo-actuated systems have been widelyaccepted as a useful technology followed by promptdevelopment of materials and manufacturing processes.Various commercial products exist in diverse engineeringapplications including micro/nano positioning stages [1]ultra-precision machine tool [2] and measurement devices[3,4] adaptive structures [5,6] biomedical and consumerelectronics [7] and so on.

In the recent competitive market, developing specificsystems using embedded hardware such as Digital SignalProcessor (DSP) and Field Pro-grammable Gate Array(FPGA) is a very popular solution to compact thereal-time system design and optimize the maximumeconomic profits with less maintenance difficulties. Forpiezo-actuated systems Proportional-Integral-Derivative(PID) control has been extensively applied as a quickservo algorithm [3,4,8,9,10,11,12] forhardware-in-the-loop testing, due to its simplicity andeasy implementation. However, one vital drawback ofPID control is its limited ability to deal with nonlinearityand uncertainty commonly existed in piezo-actuatedsystems. For this reason a great number of advancedcontrol methods and experimental results based on

floating point control platforms [2,13,14,15] has beenpresented, to further improve the sys-tem performanceand robustness subject to a variety of industrialapplications. As recognized in the available literature,most studies aim to emphasize the effectiveness of theproposed control algorithms, but seldom discuss itspractical realization for optimal system performance.

Since most low cost embedded system plat-forms usefixed point microprocessors as CPU, the main concernthen comes from the fixed pointed arithmetical errors andin particular the instability occurred during real timeimplementation, together with an increased burden onstreamlining the code and not sacrificing muchperformance. Therefore, it is better to consider thesehardware constraints starting from the control designstage, so as to make appropriate trade-offs betweenfeasibility, high performance control and robustness.

To fully exploit the potential and benefits of advancedcontrols for piezo-actuated systems using embeddedcontroller, this study proposes a hybrid neural-repetitivecontrol method and investigates its implementation issueson a FPGA based embedded system platform. In thefollowing the applied embedded control platform is firstillustrated. The design and realization issues of the hybridcontrol are then discussed. Finally, repetitive tracking

∗ Corresponding author e-mail:chiying@mail.ntust.edu.tw

c© 2015 NSPNatural Sciences Publishing Cor.

252 C. Y. Lin, C. Y. Li: Design and Implementation of Advanced Digital Controls...

experi-ments performed on a piezoelectric actuatorsys-tem reveal the effectiveness of the proposed methodand its feasibility using embedded hardware.

2 FPGA-Based Embedded Control Platform

This study applied an embedded and programmablecontroller CompactRIO from National Instruments toinvestigate the advantages and issues occurred inimplementing advanced digital controls. To exploit thepractical merits by using embedded hardware, a fastpiezoelectric actuator control system is se-lected as theexperimental apparatus for controller performanceevaluation. As shown in Figure1, the CompactRIOcontroller comprises four main parts, including areal-time module, a FPGA module, a NI 9215 A/Dmodule and a NI 9263 D/A module. The real-timemodule is used to store data and communicate with theFPGA module, a reconfigurable FPGA based operatingenvironment with a maximum processing rate up to 40MHz. The control algorithm is calculated at the FPGAside using LabVIEW language. The control signal is sentto a piezo-driven actuator system (Piezomechanik Pst150/5/20 VS10) and the sensor signal from a stain gaugeamplifier is fed back to the FPGA module, constituting afeedback control system. Figure2 shows the photographof the experimental apparatus applied in this study.

Fig. 1: Schematic diagram of an embedded control platform fora piezo-actuated system.

3 Adcanced Digital Control via HybridNeural-Repetitive Control

A great number of works has been conducted forprecision tracking control of piezo-actuated systems. Themain challenge here is achieving desired trackingperformance subject to the dominant hysteresis effect anduncertain parameters change. As widely known, neuralcontrol is an effective method which can deal withnonlinear and uncertain systems. To attain highperformance and robustness of piezo tracking control atthe same time this study presents an advanced digital

Fig. 2: Photograph of the experimental setup.

C2

C1

G

Ĝ

e

+

-

+

+

++r yu¢

u1

u2

u

Fig. 3: Block diagram of the proposed advanced control for piezotracking control.

control method im-plemented using a FPGA basedembedded control platform. Besides the existing neuralcontrollers, a repetitive controller is added to improve theperiodic tracking control performance as explained in thefollowing section.

3.1 Control Design

Figure 3 illustrates the control block diagram of theproposed hybrid neural-repetitive control for piezotracking applications.

In Figure 3, C1 and C2 represent two differentcontrollers with control inputsu1 andu2, respectively.Gis the plant model ofG, which can be obtained usingstandard system modeling and identification techniques.The transfer function from referencer to tracking erroreis presented as follows:

e=1− GC2

1+GC1r. (1)

With the assumption thatG ≈ G, the magnitude of thesensitivity function for reference tracking is furtherreduced by introducing this control architecture.

c© 2015 NSPNatural Sciences Publishing Cor.

Appl. Math. Inf. Sci.9, No. 1L, 251-258 (2015) /www.naturalspublishing.com/Journals.asp 253

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1−1.5

−1

−0.5

0

0.5

1

1.5

Time (sec)

Ou

tpu

t (V

olt)

Input/Output Responses

InputOutput

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

−1.5

−1

−0.5

0

0.5

1

1.5

Time (Sec)

Ou

tpu

t (V

olt)

Model Validation

SimulationExperiment

Fig. 4: System identification based on embedded control platform: top plot: injected input and output responses; bottom plot: modelvalidation result.

In this paper,C1 is selected as a radial basis function(RBF) based neural network adaptive controller. Thecontrol goal of this controller is to stabilize the system asa performance baseline. To further improve the controlperformance, another neural controllerC2 containing arepetitive controller for periodic errors cancel-lation isadded. The design ofC2 is based on the techniquementioned in [16].

In this section we will illustrate how to construct aneural-repetitive controller using the idea of feedforwardcontrol with internal model constraints. Consider the caseof using controllerC2 alone, equation (1) then becomes

e= (1− GC2)r. (2)

The original feedback control problem has beentransformed to a feedforward control problem and thecontrol goal here is to minimize the tracking error with acost functionJ = (1− GC2). The simplest solution to thiscontrol problem is to find a stable plant inverse ofG [17].In practical applications we are interested in the case

when the system input contains some specific internalmodelsD, which is usually known a priori. For example,we may know the periodic components in the appliedtracking profile for precise positioning applications. Nowlet

1− GC2 = RD, (3)

whereR is a part of the controllerC2 and needs to bedesigned. The above equation is recognized as the famous“Bezout Identity”[18] with the assumption thatG andDare coprime. This equation has infinite solutions and oneway to represent these multiple solutions is using Youlaparameterization technique [18]. Given a pair of nominalsolution (R′

,C′2), we can express the solutions of

equation (3) asR= R′− GQ andC2 =C′2+DQ, whereQ

is a free design para-meter. Consider the performanceindex with 2-norm measure:

J = ||(R′− GQ)D||2. (4)

c© 2015 NSPNatural Sciences Publishing Cor.

254 C. Y. Lin, C. Y. Li: Design and Implementation of Advanced Digital Controls...

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−6

−4

−2

0

2

4

6

Time (sec)

Tra

ckin

g E

rro

r (µ

m)

PC basedFPGA based

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2−0.1

−0.05

0

0.05

0.1

Time (sec)

Tra

ckin

g E

rro

r (µ

m)

Fig. 5: Repetitive control tracking results: comparison of using PC based and FPGA based controllers.

Then the control goal of the neural controllerC2 is tominimize J and satisfy the deterministic constraintdescribed by the internal modelD simultaneously. Theselection of the internal model depends on the inputsignals we are dealing with. In this work the input signalis a periodic one for tracking control, soD is chosen asD = 1− z−N, whereN is the period of the deterministicperiodic signal. With above assumptions, we canrepresent this constrained optimization problem as

minQ∈RH∞

D=1−z−N

||(R′− GQ)D||2, (5)

whereRH∞ is a set of stable rational transfer functions. TominimizeJ and updateQadaptively, a RBF neural networkalgorithm [19] is applied in this paper.

To solve the Bezout identity equation (3) and updatethe adaptive neural controllerQ, this work adopted themethod using a zero-phase-error-tracking typefeedforward formula [17]. The procedure for obtainingthis particular solution is briefly listed as follows.

Consider a single input, single output (SISO) plantmodel G and factorizeG into two partsG = GoGi , inwhich Go is minimum phase andGi is non-minimumphase, respectively. Suppose

G=BA=

B+B−

A= GiGo,

G0 =B+

A, Gi = B−

,

(6)

whereA andB represent the denominator and numeratorof G, B+ andB− denote the stable and unstable parts ofB,respectively. Substituting equation (6) into equation (3),one can obtain

RD+C2Gi = 1

C2 =C2Go.(7)

c© 2015 NSPNatural Sciences Publishing Cor.

Appl. Math. Inf. Sci.9, No. 1L, 251-258 (2015) /www.naturalspublishing.com/Journals.asp 255

From the plant inversion idea presented in [17] onesolution pair (R′

,C′2) to solve equation (7) is given as

follows:

R′ =1

1− (1− γG∗i Gi)qz−N

C2 =γG∗

i qz−N

1− (1− γG∗i Gi)qz−N

C′2 =C′

2G−1o ,

(8)

whereγ is a learning parameter for performance tuning,Gi ∗ (z−1) = Gi(z), andq(z,z−1) is a zero phase low passfilter to suppress the instability caused by the high gainfeedback at undesired frequency ranges. Thisq filter isembedded within the internal modelD = 1−q(z,z−1)z−N. Althoughq is a discrete non-causalfilter, the causality condition still holds in equation (8)because of the long term delay cascaded withq. For adetailed parameter analysis of the controller, one mayrefer to [16].

3.2 Implementation Issues

Just like many embedded controllers, CompactRIO is afixed point based control platform which involves issuessuch as “quantization error” and “finite word lengtherror” in real implementation. All these inevitable errorsmay deteriorate the control performance and even systemstability if not carefully addressed. Therefore, saving asmany logic gates in FPGA as possible to optimize thestatic and dynamic arithmetical range and avoid overflow,or equivalently “program optimiza-tion”, is an importantdesign step especially when using complicated servocontrol algorithms or high sampling rates for fastdynamic systems.

As mentioned in the previous section, this studyapplied an advanced control consisting of an adaptiveneural controller and a repetitive controller. When usingan embedded control platform such as CompactRIO, eachcontrol action needs extra design efforts for successfulimplementation. To make these ideas clear, some issuesregarding the proposed controller implementation can besummarized as follows:

3.2.1 Model and controller order

If the control law is designed using a model based designmethod such as repetitive control, a low order plant modelis preferred. High order mathematical models usuallyneed high order controllers for successful control design.Reduction of controller order is one way to deal with thisproblem but may introduce more undesired errors. Forthis reason the model of the piezo-actuated system in thisstudy is identified based on a system identificationtechnique and fitted to a third order control system.

3 3.2 3.4 3.6 3.8 4 4.2 4.4 4.6 4.8 5−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

Tra

ckin

g E

rro

r (µ

m)

Input: 10Hz Sine Wave

Time (sec)

FPGA based (6bits)FPGA based (7bits)FPGA based (8bits)PC based

Fig. 6: Steady state tracking error of repetitive control by usingan embedded control platform: finite word length effects.

3.2.2 Realization of the neural network controller

The neuronsϕ applied in the neural controller areGaussian functions and can be represented as:

ϕ(||x− c||) = exp

[

−||x− c||2

2σ2

]

, (9)

wherec andσ denote the center and standard deviation ofthe Gaussian function, respectively. It is not difficult toimplement the above nonlinear function and achievedesired precision if using a floating point basedmicroprocessor. For fixed point based microprocessorsthe approximation of this exponential function can bemade by using a finite Taylor series expansion:

exp(x) =N

∑k=0

xk

k!. (10)

To reduce the computational burden this study applieda second order(N = 2) approximation to implement theneurons in the neural controller. Moreover, the appliedRBF network consists of an input layer, one hidden layerand output. The hidden layer has two neurons, meaningonly two sets of neuron parameters need to bedetermined. For simplicity, the values ofc andσ of theradial functionϕ are fixed.

c© 2015 NSPNatural Sciences Publishing Cor.

256 C. Y. Lin, C. Y. Li: Design and Implementation of Advanced Digital Controls...

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−5

0

5

Time (Sec)

Tra

ckin

g E

rro

r (µ

m)

Tracking a 10 Hz sine wave

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

−0.1

−0.05

0

0.05

0.1

0.15

Time (Sec)

Tra

ckin

g E

rro

r (µ

m)

RC(C2)

RC/NN(C2)

NN(C1) + RC(C

2)

NN(C1) + RC/NN(C

2)

Fig. 8: Experimental results of tracking a 4µm, 10 Hz sinusoidal input; top plot: time response in 1 second; bottom plot: zoom in; blueline: RC (C2); red line: RC/NN (C2); green line: NN (C1) + RC (C2); black line: NN (C1) + RC/NN (C2).

3.2.3 Realization of the repetitive controller

From the perspective of signal processing, the repetitivecontrol shown in equation (8) basically comprises threecomponents: Finite Impulse Response (FIR) filter, InfiniteImpulse Response (IIR) filter, and long term time delay.Because of the constraints coming from fixed pointoperations and limited memories, it is crucial tostreamline the FPGA code and maintain acceptableperformance with minimal errors caused by the embeddedhardware. For example, using a direct form II realizationof second-order IIR filter may reduce the number of usedregisters, but on the other hand it also requires moredynamic arithmetical range and memory space comparingto a direct form I realization method. In light of this, asimulation analysis is performed to determine the optimalrealization structures before implementation.

4 Implementation on a Piezo-ActuatedSystem

To demonstrate the effectiveness and investigate theimplementation issues of the proposed advanced controlon the embedded control platform, trajectory trackingexperiments were performed by giving a 4µm, 10 Hzsinusoidal reference input to the piezo-actuated system asdescribed in Section 2. For a high bandwidth piezoelectricactuator system, the sampling rate was selected as 10 kHzto avoid possible “inter-sample” errors.

4.1 System Identification

Because the repetitive control is a model based designmethod, a mathematical model is still required for thecontrol design. To this end this study applied a timedomain system identification (ID) method and injected aseries of square wave to the piezo actuator system asinput. Figure4 shows the system ID result based on theembedded controller CompactRIO. After filtering out the

c© 2015 NSPNatural Sciences Publishing Cor.

Appl. Math. Inf. Sci.9, No. 1L, 251-258 (2015) /www.naturalspublishing.com/Journals.asp 257

1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 1.5

−0.15

−0.1

−0.05

0

0.05

0.1

0.15

Time(sec)

Tra

ckin

g E

rro

r(µ

m)

Input: Sine Wave 10Hz, Sampling Rate: 10k Hz

PIDNNRC

Fig. 7: Steady state tracking error comparison: green line: PID;blue line: NN; red line: RC.

undesired noises and fixed point errors for the outputsignal, a third order discrete time model at 10 kHzsampling is obtained by using the system ID toolbox’ident’ in Matlab and can be represented as:

G(z) =0.1696z2−0.04588z

z3−0.8602z2+0.1239z+0.01492. (11)

A different input signal, frequency varying squarewave, was also injected to the control system to verify theaccuracy of the model, as shown in figure4.

4.2 Tracking Results

Figure 5 shows the tracking error results by usingrepetitive control (RC) alone (i.e., disablingC1 and theneural controller part inC2). To highlight the effects offixed point embedded controller, the result based on afloating point controller is also plotted for illustration.Ascan be seen, using a floating point based control platformassures the asymptotic error property because of the useof the repetitive controller. However, undesired rippleerrors occur at the steady state if the control isimplemented using a fixed point microcontroller. Forcomparison purpose the results using three different

resolution based arithmetic operations are also shown inFigure 6. It is clear that the finite word length errorsbecome more significant with a decreasing precision inthe embedded control system. The result presentedindicates that there exists an inherited conflict betweenhardware resources and performance in particular whenimplementing on a fixed point embedded controller.

Next, we present the experimental results of usingneural controller C1 by comparing with two othercontrollers, PID control and repetitive control. Note thatso far the hybrid control mechanism is still not activated.As shown in Figure7, the neural controller, although onlywith two neurons in the implementation, still achievesbetter performance than the fine-tuned PID controller.The obvious periodic errors shown in the plot thusmotivate the extra use of repetitive control for betterperformance over neural network control alone.

Figure 8 represents the tracking results by using ourproposed hybrid neural-repetitive control based on anembedded control platform. It is clear from the topsubplot that a large transient error occurs in the firstperiod cycles. Adding a neural controllerC1 considerablyreduces the transient error down to a± 0.15µm range. Inparticular, activating all the control actions inC2 furtherimproves the transient performance, as indicated in thebottom subplot. Although applying more control actionsseems giving the best tracking performance, it is noticedthat accumulated fixed point errors also lead to increasedsteady state errors (blue line v.s. black line).

5 Conclusion

In this paper, we present an advanced digital controlmethod which combines neural control with repetitivecontrol for tracking control of piezo-actuated systems.The developed control algorithm was implemented usinga FPGA based embedded control platform. Experimentalresults on periodic tracking control of a piezo-actuatedsystem demonstrate the effectiveness of the proposedmethod. From the results obtained, it is suggested to takeinto account the issues of error reduction, computationalburden, and programming efforts for best systemperformance when using embedded hardware foradvanced controls implementation.

Acknowledgement

This paper was financially supported by National ScienceCouncil, Taiwan, R.O.C., under grant number NSC97-2218-E-011-015.

References

[1] C. Y. Lin and P. Y. Chen, Precision Tracking Control of aBiaxial Piezo Stage Using Repetitive Control and Double-

c© 2015 NSPNatural Sciences Publishing Cor.

258 C. Y. Lin, C. Y. Li: Design and Implementation of Advanced Digital Controls...

Feedforward Compensation, Mechatronics. (2011),21, 239-249.

[2] B. S. Kim, J. Li, and T. C. Tsao, Two-Parameter RobustRepetitive Control With Application to a Novel dual-Stage Actuator for Noncircular Machining, IEEE/ASMETransactions on Mechatronics. (2004),9, 644-652.

[3] R. Liang, O. Jusko, F. L ¨udicke, and M. Neugebauer, A NovelPiezo Vibration Platform for Probe Dynamic PerformanceCalibration, Measurement Science and Technology. (2001),12, 1509-1514.

[4] O. G. Karhade, F. Levent Degertekin, and T. R. Kurfess,Active Control of Microinterferometers for Low-Noise Parallel Operation, IEEE/ASME Transactions onMechatronics. (2010),15, 1-8.

[5] R. Jha and J. Rower, Experimental Investigation of ActiveVibration Control Using Neural Networks and PiezoelectricActuators, Smart Materials and Structures. (2002),11, 115-121.

[6] J. Lin and W. Z. Liu, Experimental Evaluation of aPiezoelectric Vibration Absorber Using a Simplified FuzzyController in a Cantilever Beam, Journal of Sound andVibration. (2006),296, 567-582.

[7] K. K. Tan and S. C. Ng, Computer Controlled PiezoMicromanipulation System for Biomedical applications,Engineering Science and Education Journal. (2001),10, 249-256.

[8] K. Sang-Soon, U. Pinsopon, S. Cetinkunt, and S. Nakajima,Design, Fabrication, and Real-Time Neural NetworkControl of a Three-Degrees-of-Freedom Nanopositioner,IEEE/ASME Transactions on Mecha-tronics. (2000),5,273-280.

[9] C. Y. Hwang, Microprocessor-Based Fuzzy DecentralizedControl of 2-D Piezo-Driven Systems, IEEE Transactions onIndustrial Electronics. (2008),55, 1411-1420.

[10] G. Schitter and Nghi Phan, Field Programmable AnalogArray (FPAA) Based Control of an Atomic ForceMicroscope, American Control Conference. (2008), 2690-2695.

[11] C. Brecher, D. Lindermann, M. Merz, and C. Wenzel,FPGA-Based Control System for Highly Dynamic Axes inUltra-Precision Machining, American Society for PrecisionEngineering. (2010), 17-22.

[12] A. Hamed, E. Tisserand, Y. Berviller, and P. Schweitzer,Embedded System Design for Impedance Measurement ofMultiple-Piezo Sensor, Third International Conference onAdvances in Circuits, Electronics, and Micro-electronics.(2010), 6-11.

[13] H. C. Liaw, B. Shirinzadeh, and J. Smith., Sliding-ModeEnhanced Adaptive Motion Tracking Control of PiezoelectricActuation Systems for Micro/Nano Manipulation, IEEETransactions on Control Systems Technology. (2008),16,826-833.

[14] C. Y. Lin and Y. C. Liu, Precision TrackingControl and Constraint Handling of MechatronicServo Systems Using Model Predictive Control,IEEE/ASME Transactions on Mechatronics. (2011), 1-13. doi:10.1109/TMECH.2011.2111376.

[15] H. C. Liaw and B. Shirinzadeh, Neural network motiontracking control of piezo-actuated flexure-based mechanismsfor micro/nanomanipulation, IEEE/ASME Transactions onMechatronics. (2009),14, 517-527.

[16] C. Y. Lin and T. C. Tsao, Adaptive Control WithInternal Model for High Precision Motion Control, ASMEConference Proceedings. (2007),9, 1021-1029.

[17] M. Tomizuka, Zero Phase Error Tracking Algorithm forDigital Control, Journal of Dynamic Systems, Measurement,and Control. (1987),109, 65-68.

[18] S. Skogestad and I. Postlethwaite, Multivariable FeedbackControl: Analysis and Design, Wiley New York (1996).

[19] S. Haykin, Neural Networks: a Comprehensive Foundation,Prentice Hall PTR Upper Saddle River, NJ, USA, (1994).

Chi-Ying Linreceived the Ph.D. degreein mechanical engineeringfrom University of California,Los Angeles, USA, in 2008.He is currently an associateprofessor in the Departmentof Mechanical Engineering,National Taiwan Universityof Science and Technology,

Taiwan. His research interests include design and controlof precision positioning systems, active vibration control,intelligent robots, and mechatronics.

Chien-Yao Li receivedthe Master degree inmechanical engineering fromNational Taiwan Universityof Science and Technology,Taipei, Taiwan, in 2010. He iscurrently a research assistantin Institute of Physicsat Academia Sinica, Taiwan.His research interests include

nanopositioning system design and embedded controlsystems.

c© 2015 NSPNatural Sciences Publishing Cor.