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Adaptive Sliding Mode Control With Perturbation Estimation and PID Sliding Surface for Motion Tracking of a Piezo-Driven Micromanipulator Yangmin Li, Senior Member, IEEE, and Qingsong Xu, Member, IEEE

Abstract—This paper proposes an improved sliding mode control with perturbation estimation (SMCPE) featuring a PID-type sliding surface and adaptive gains for the motion tracking control of a micromanipulator system with piezoelectric actuation. One advantage of the proposed controller lies in that its implementation only requires the online estimation of perturbation and control gains without acquiring the knowledge of bounds on system uncertainties. The dynamic model of the system with Bouc–Wen hysteresis is established and identified through particle swarm optimization (PSO) approach, and the controller is designed based on Lyapunov stability analysis. A high-gain observer is adopted to estimate the full state from the only measurable position information. Experimental results demonstrate that the performance of proposed controller is superior to that of conventional SMCPE in both set-point regulation and motion tracking control. Moreover, a submicron accuracy tracking and contouring is achieved by the micromanipulator with dominant hysteresis compensated for a low magnitude level, which validates the feasibility of the proposed controller in the field of micro/nano scale manipulation as well. Index Terms—Flexure mechanism, hysteresis, micromanipulator, nonlinear system, piezoelectric actuation, robust control.

I. INTRODUCTION

M

ICROPOSITIONING stages with piezoelectric actuation are widely used in a variety of applications where an ultrahigh precision motion over a micro scale workspace is required [1]. Piezoelectric actuators (PZTs) are capable of delivering nanometer positioning resolution, large blocking force, high stiffness, and fast response and are hence commonly utilized to meet the specific requirements. Nevertheless, the major problem of piezo-driven stages comes from the nonlinearities in the PZT attributed to hysteresis, creep, and drift over extended periods of time. The hysteresis is a nonlinear relationship between the applied voltage and output displacement of the PZT and induces a severe open-loop positioning error as high as

Manuscript received May 22, 2009; revised July 05, 2009; accepted July 14, 2009. Manuscript received in final form July 25, 2009. First published October 09, 2009; current version published June 23, 2010. Recommended by Associate Editor S. Devasia. This work was supported in part by the research committee of the University of Macau under Grant UL016/08-Y2/EME/LYM01/FST and the Macao Science and Technology Development Fund under Grant 016/2008/A1. The authors are with the Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macao SAR, China (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2009.2028878

10%–15% of the stage travel range. Therefore, the hysteresis has to be suppressed in order to suit for high-precision applications. The successful compensation for hysteresis relies on the design of suitable control strategies [2]. Typically, the hysteresis is compensated by feedforward control resorting to an inverse of the hysteresis model such as the Preisach model, Duhem model, Maxwell model, and Bouc–Wen model, etc. [3]. Considering that most hysteresis models are only applicable with some particular input signal frequencies, a feedforward combined with feedback control is usually adopted for a precision motion tracking [4], [5]. Additionally, taking into account that modeling the hysteresis is a complicated procedure, some approaches based on plant model identification without the consideration of hysteresis have been exploited in terms of robust PID (proportional-integral-derivative) control [6], control [7]–[10], and inversion-based technique [11], [12], etc. Furthermore, model-reference and model-free intelligent controllers have been implemented as well [13], [14]. Besides, by considering the hysteresis as a disturbance or an uncertainty, sliding mode control (SMC) has been employed in the piezo-driven stage [15], [16] since SMC is an effective and simple way to deal with model imperfection and uncertainties for nonlinear systems [17]. SMC is a nonlinear control approach that drives the system’s state trajectory onto a specified sliding surface and maintains the trajectory on this surface for the subsequent time. However, in conventional SMC design, a priori knowledge of the bounds on system uncertainties has to be acquired. As a result, the controller based on the above knowledge trends to be overconservative, which may induce poor tracking performance and undesirable oscillations in control signal. To overcome this drawback, sliding mode control with perturbation estimation (SMCPE) is proposed in [18] as an enhanced version of the conventional SMC. It has been shown that the need to know the uncertainty bounds is removed in the SMCPE and is replaced with a strategy of online estimation of the perturbations [18]. In the literature, SMCPE has been implemented for a number of applications [19]–[21]. The existing SMCPE is featured with a PD-type sliding surface and constant control gains. However, the commonly used PD sliding surface produces a slow response speed. In the literature, an integral term is first introduced in the sliding variable definition in [22]. Then, it has been reported in [23] that the traditional SMC with PID sliding surface offers a faster transient response with less steady-state error. Hence, SMC with PID sliding surface has been adopted in various situations [24],

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in the transverse direction than that in working direction. Thus, the amplifier also acts as a decoupler with the roles of transmitting axial force of the actuator and preventing the actuator from suffering undesirable transverse motion and loads as well. Hence, the two actuators are well isolated and protected. Moreover, the ideal translation provided by compound parallelogram flexures allows the generation of decoupled output motion for the stage. Different from a common decoupled XY parallel stage with output motion decoupling, the presented one has both input and output decoupling in virtue of actuation isolation and decoupled output motion. This total decoupling property is necessary for the situations where the manipulator is underactuated and sensory feedback of end-effector positions is not allowed. More details about the working principle of the micromanipulator can be found in previous works of the authors [30]. B. Experimental Setup Fig. 1. Schematic diagram of the decoupled XY parallel micromanipulator.

[25]. In this paper, a PID-type sliding surface is introduced into the conventional SMCPE to establish a new SMCPE-PID controller with faster transient response. On the other hand, the fixed high control gains may amplify the oscillations in control signal and excite high-frequency unmodeled dynamics of the system, which are undesirable phenomena as well [17], [26]. To overcome this limitation, adaptive control methodology with control parameters updated online is a promising approach [27]–[29]. Hence, variable gains with adaptive rules are introduced to further develop an adaptive SMCPE-PID controller to change the values of both PID and switching gains online without changing the sliding surface. Moreover, the stability of the proposed controller is proved using the Lyapunov analysis and the tracking performance of the resulting control system as compared to that of a conventional SMCPE is demonstrated by experimental investigations on a piezo-driven micromanipulator system. In the rest of the paper, the experimental setup of an XY micromanipulator system is described in Section II, which serves as a testbed for the controller performance evaluation. Then, a dynamic model of the system with Bouc–Wen hysteresis is identified through particle swarm optimization (PSO) approach in Section III. Based on the established model, a conventional SMCPE controller design is outlined in Section IV, and the design of the improved SMCPE controller based on Lyapunov stability analysis is presented in Section V. Afterward, experimental studies with the designed controllers are carried out in Section VI, where a high-gain observer is employed to estimate the full state of the system. Finally, Section VII concludes this research. II. EXPERIMENTAL TEST BED DESCRIPTION A. Mechanical Architecture of the XY Stage As illustrated in Fig. 1, the XY parallel micromanipulator is constructed with four identical prismatic-prismatic limbs and actuated by two PZTs through integrated displacement amplifiers. It is known that PZT cannot bear transverse loads due to the risk of damage. The embedded displacement amplifier acts as an ideal prismatic joint and possesses larger ratio of stiffness

The experimental setup of the XY micromanipulator prototype is graphically shown in Fig. 2(a). The monolithic stage is fabricated from a piece of light material Al 7075-T651. Two 20- m-stroke PZTs (model PAS020 produced by Thorlabs, Inc.) are adopted to drive the XY stage, and the PZT is actuated with a voltage of 0–75 V through a two-axis piezo amplifier and driver (BPC002 from the Thorlabs). The displacements of the output platform are measured by two laser displacement sensors (Microtrak II, head model: LTC-025-02, from MTI Instrument, Inc.). The analog voltage outputs (within the range of 5 V) of the two sensors are connected to a PCI-based data acquisition (DAQ) board (PCI-6143 with 16-bit A/D convectors, from NI Corp.) through a shielded I/O connector block (SCB-68 from NI) with noise rejection. The digital outputs of the DAQ board are then read by a host computer simultaneously, and the connection between the instruments is illustrated in Fig. 2(b). Since the sensitivity of the laser sensor is 2.5 mm/10 V and the maximum value of 16-bit digital signal corresponds to 10 V, the resolution of the displacement detecting system can be calculated as (1) The resolution level step size of the sensor output can be identified from the home position data plotted in Fig. 3. However, due to a considerable level of noise, the resolution of the sensor is claimed as 0.1 m by the manufacturer. A preliminary open-loop test shows that the XY stage has a m m with the maxworkspace around the area of imum cross-talk of 1.5% between the two axes, which verifies the well-decoupled property of the micromanipulator. It should be noted that the decoupling in the real stage is also determined by the machining accuracy of the parallel mechanism. The presence of the cross-talk between the two axes motion will be eliminated by resorting to a proper control design carried out in the subsequent sections. III. DYNAMIC MODELING AND IDENTIFICATION In view of the fact that the smaller the system uncertainty, the better the motion tracking performance, the hysteresis is modeled in this research even though a sliding mode-based con-

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where the parameters , , , and represent the mass, damping coefficient, stiffness, and -axis displacement of the XY micromanipulator, respectively; is the piezoelectric coefficient, denotes the input voltage, and indicates the hysteretic loop in terms of displacement whose magnitude and shape are determined by parameters , , , and the order , where governs the smoothness of the transition from elastic to plastic response. is assigned in (3) For the elastic structure and material, as usual. In addition, denotes the overall perturbation of the system arising from model parameter uncertainties, unmodeled dynamics, and other unknown terms. B. Model Identification With PSO

Fig. 2. Experimental setup. (a) Photograph of the XY micromanipulator system. (b) Scheme of hardware connection.

Due to the nonlinearity introduced by the hysteresis effect , it is not easy to effectively identify the parameters of the dynamic model through conventional approaches. In this research, the seven parameters , , , , , , and are identified simultaneously based on particle swarm optimization (PSO). The PSO is adopted in the current problem due to its superiority of performance over other methods such as the direct search approach and genetic algorithm (GA) [32], [33]. It has been shown that the selection of the fitness function has a great effect on the identification results [34]. In the current research, the fitness function is chosen as follows: (4) with (5)

Fig. 3. Zero position of the micromanipulator in x-axis direction measured by laser displacement sensor.

troller can be designed without modeling the hysteresis. The Bouc–Wen model is considered in this work since it has fewer parameters and is easier to integrate with the rest of the model. It has already been verified that the Bouc–Wen model is suitable to describe the hysteresis loop of PZT [31]. Additionally, since the stage is well decoupled, the two axial motions can be treated independently. Thus, two single-input–single-output (SISO) controllers can be employed for the - and -axes of the micromanipulator, respectively. For the sake of brevity, only the treatment of -axis motion is presented in this paper. A. Dynamic Modeling The entire dynamic model of the micromanipulator system with nonlinear hysteresis can be established as follows: (2) (3)

is the where denotes the total number of samples and error of the th sample, which is calculated as the deviation of Bouc–Wen model output from experimental result . The model identification process is carried out offline as follows: 1) Data Collection: With a full-range input voltage signal (0–75 V) applied to the PZT, the output displacement of the stage is measured and recorded. 2) Model Implementation: The dynamic model with Bouc–Wen hysteresis is implemented with Matlab/ Simulink as shown in Fig. 4. The dynamic model output is generated by the simulation. 3) Model Identification: The PSO algorithm is performed for optimizing the model parameters to match the simulation result to experimental data. The optimization is carried out with a PSO toolbox [35] running in Matlab environment, and the identified model parameters are shown in Table I. With the obtained model, the comparison of model output and experimental result is illustrated in Fig. 5. Since the Bouc–Wen model cannot represent nonsymmetric hysteresis exactly, there exists an error between the identified model and the experimental result. It can be observed that the maximum deviation with respect to the travel range of the stage is as large as 6.6%, which occurs at the lower turning point of 4, 8, and 12 s as shown in Fig. 5(a).

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Fig. 4. Dynamics simulation model implemented with Matlab/Simulink; (a) The entire dynamic model for calculating the output displacement x from the input voltage u, (b) Bouc–Wen model for calculating the hysteresis term h from the input voltage u.

TABLE I IDENTIFIED DYNAMIC MODEL PARAMETERS OF THE MICROMANIPULATOR WITH BOUC–WEN HYSTERESIS Fig. 5. (a) A 0.25-Hz input voltage signal applied to the PZT. (b) Hysteresis loop obtained by experiment and dynamic model.

The bounded perturbations in the system (6) can be combined together to form a perturbation vector:

To compensate for the model errors and other uncertainties, a sliding mode controller is considered in this work since it offers robustness performance in the presence of model imperfection and uncertainties. In the following section, the conventional SMCPE strategy is designed first to control the micromanipulator system. IV. CONVENTIONAL SMCPE DESIGN A. Perturbation Estimation Technique A brief review of perturbation estimation strategy for the online estimation of model perturbations is outlined here since more details can be found in the literature [18], [19]. Consider a general nonlinear system given by

(7) The estimation for the perturbation

is approximated as (8)

where denotes a calculated state vector since the measurement of higher order states of the system (e.g., acceleration) cannot always be realized, is the sampling time interval, and represents the control input in the previous time-step. In practice, the sampling frequency is selected high enough to . ensure that Traditionally, the state vector is computed based on a backward difference equation

(6) where the vector , the global state vector with the state subvector

with

and deindependent coordinates. Besides, the superscript in noting refers to the th order time derivative.

(9) In spite of its simplicity, the above calculation has inherent limitations due to the accuracy and quantization noise, which restricts the achievable bandwidth of the feedback controller as stated in [36]. Alternatively, a closed-loop high-gain observer [37] without the above limitations is employed and implemented in Section VI in this paper.

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By taking the time derivative of both sides of (14), the following sliding dynamics can be generated:

(17) Fig. 6. Block diagram of the proposed adaptive SMCPE-PID robust controller.

Then, substituting (17) into (16) with the consideration of (15) yields

B. Conventional SMCPE Design Given the system model (2), the perturbation

is given by (10)

Based on the perturbation estimation strategy reviewed above, can be estimated as follows:

(18) If the gain

is designed to meet the condition (19)

(11) Then, the system model (2) becomes

where

is an arbitrary constant, then it follows that for

(12) (20)

represents an error between the where system’s real perturbation and its estimation. To design a SMCPE controller, an error coordinate is defined as

(21) (22) (23)

(13) Hence, considering (18) and (23), one can derive that represents the desired position trajectory, and dewhere notes the time variable that will be omitted in the rest of this section to have a concise manner of representation. In view of the linear second-order dynamic system, a first order PD-type sliding surface or switching function is selected as follows: (14) is a design parameter. where Theorem 1: For the system (12) with the sliding surface (14), if the the tracking error defined by (13) satisfies control law is given by

(15) represents the signum function and is a positive where switching gain. Proof: Considering a positive definite Lyapunov function , its first derivative can be obtained as candidate (16)

(24) which allows the conclusion that the states can reach the in finite time [38]. Besides, (24) also switching surface ensures that the states will be confined to the surface for all future time since leaving the surface requires to be positive, which is impossible as implied by the inequality. as . According to Thus, the switching variable the definition of in (14), we can conclude that the tracking and and that error satisfies and as . Therefore, the SMCPE controller guarantees a zero steady-state tracking error. Remark 1: Due to the discontinuity of the signum function , chattering may occur in the control input. To alleviate the chattering phenomenon, the boundary layer technique is adopted by replacing the signum function in (15) with saturation function with the notation for for where the positive constant thickness, which ensures that

(25)

represents the boundary layer is always bounded by . In the

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Fig. 7. Inverse dynamic model implemented with Matlab/Simulink for the calculation of the hysteresis term h from the given position x .

Theorem 2: For the system (12) with the sliding surface (26), if the the tracking error defined by (13) satisfies control law is given by

TABLE II PARAMETERS OF THE IMPLEMENTED CONTROLLERS

(27) represents the signum function and is a positive where switching gain. Proof: Defining a positive definite Lyapunov function can, the first derivative can be expressed by didate selection of parameter , a tradeoff between the chattering and tracking error should be taken. From (19), it can be observed that the selection of the control gain depends on the upper bound of the perturbation estimation error . In the classical SMC, this term is the upper bound of the actual perturbation instead. Generally, since the perturbation estimation error is much smaller than the actual perturbation, a much smaller control gain can be achieved when compared to the corresponding classical SMC. This is the major advantage of the SMCPE over SMC [19]. In practice, the gain value is usually tuned by the trial and error approach since no universal method is available yet. V. IMPROVED SMCPE CONTROLLER DESIGN

(28) Taking the time derivative of the sliding variable in (26), results in

(29) where the time variable is omitted for the clarity of representation. Substituting (29) into (28) with the consideration of the control law (27), leads to

In this section, the scheme of SMCPE with PID-type sliding surface is designed for the micromanipulation system. A. SMCPE With PID Sliding Surface In order to design an improved SMCPE controller, a sliding surface of PID type is defined as follows:

(30) Similarly, if the gain

is chosen to satisfy the condition:

(26)

and where the error coordinate is described by (13), and are positive design parameters. As far as the guideline for the selection of control gains and is concerned, the gains should be chosen such that the characteristic polynomial ( denotes a complex variable here) is strictly Hurwitz, i.e., a polynomial with roots locating strictly in the open left half of the complex plane.

(31) is an arbitrary constant, then it can be deduced where , the derivative of the positive definite from (30) that, for Lyapunov candidate . Based on the same idea as conducted in the proof of Theorem 1, it can be concluded that the as . According to the definition of sliding variable in (26), we can further conclude that the tracking error satisfies and and that and

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process. However, the constant gains may increase the oscillations in the control signal and induce the excitation of high frequency unmodeled dynamics of the system [17], [26], which are undesirable phenomena. To overcome this drawback, variable gains with adaptive rules are introduced into the SMCPE controller in this subsection to change the values of both PID gains and switching gains online. Theorem 3: For the system (12) with the sliding surface given below (33) the tracking error defined by (13) satisfies control law is given by

if the

Fig. 8. Set-point regulation control results of the designed controllers.

TABLE III CONTROLLER PERFORMANCE FOR CONTROL TYPE (a) SET-POINT REGULATION AND (b) SINUSOIDAL MOTION TRACKING

(34) where represents the signum function and the variable control gains , , and are estimated online according to the adaptive rules (35) (36) (37)

as . Therefore, the proposed SMCPE-PID controller ensures a zero steady-state tracking error. As stated in Remark 1, the chattering is alleviated by replacing the signum function in (27) with the saturation function described by (25). Moreover, substituting (25) and (11) into the control law (27), allows the generation

Proof: To evaluate the stability of the closed-loop system, an estimation error for the desired control parameter is introduced below (38) In addition, a positive definite Lyapunov function candidate is considered as (39) Differentiating (39) with respect to time, gives (40) Taking the time derivative of the sliding variable defined in (33) results in

(32)

B. SMCPE With PID Sliding Surface and Adaptive Gains In above designed SMCPE-PID controller, the control gains , , and are assigned as constants during the control Authorized licensed use limited to: Universidade de Macau. Downloaded on June 24,2010 at 11:43:16 UTC from IEEE Xplore. Restrictions apply.

(41)

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Fig. 9. Sinusoidal motion tracking results of (a)–(b) PID, (c)–(d) SMCPE, and (e)–(f) the proposed controller with 0.2-Hz input rate.

where the time variable is omitted for the clarity of representation. Substituting the control law (34) into (41) gives

As stated in (19) and (31), if the desired switching gain satisfies the condition (44)

(42) is an arbitrary constant, then we can conclude that from (43) for . In view of the same idea as conducted in the proof of Theorem 1, one can deduce that the as . According to the definition switching variable of in (33), we can conclude that the tracking error satisfies and and that and as . Therefore, the proposed adaptive SMCPE-PID controller guarantees a zero steady-state tracking error. As described in Remark 1, the chattering is alleviated by replacing the signum function in (34) with the saturation function (25). where

and substituting (35)–(37) Then, taking into account and (42) into (40), a necessary calculation produces

(43)

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Fig. 10. Velocity and acceleration calculated by (a)–(b) traditional method in (9) and (c)–(d) high-gain observer in (47) with   : .

= 02

Besides, substituting (11) and (25) into (34) allows the derivation of the control input

= 22,  = 8,  = 10, and

The performance of the controllers designed above is verified by experimental studies conducted in the subsequent section. VI. EXPERIMENTAL STUDIES AND DISCUSSIONS

(45) is restricted by the Remark 2: Although the parameter condition (44), it is not obviously associated with the controller (34). Actually, it can be considered as an approximate initial value for the control gain , which is associated with the concorresponds to the controller troller. Hence, the parameter indirectly. The implementation of the control law (45) does not require any bounds on the unknown parameters of the system. The control gains , , and are estimated online using the adaptive rules (35)–(37) with initial values. The gains vary with time until the tracking error ( and ) as well as the sliding variable reach zero. In theory, after the sliding surface is reached, the trajectory will remain on the surface thereafter, which guarantees that the gain varying with respect to the absolute value of the sliding variable will not tend to infinity as time evolves. However, in practice, a small deviation from the sliding surface may cause the increase of value [39], which may bring instability to the closed-loop system. To overcome this problem, the adaptive law for is modified as follows: for for where

A. Controller Implementation Issues An insight into (45) reveals that both full-state feedback and full-state trajectory are required to implement the proposed SMCPE-PID controller. Although the full-state trajectory can be generated by differentiating the desired position trajectory in advance, the velocity and acceleration feedbacks have to be estimated since only the position can be measured by the available displacement sensors. Generally, the full state can be estimated by resorting to the measured position using a backward difference equation (9). However, its limitations arise from the accuracy and quantization noise, which restricts the achievable bandwidth of the feedback controller in consequence. Alternatively, a closed-loop high-gain observer can be employed without the above limitations [36], [37]. Thus, a one-input three-output high-gain observer is realized to estimate the full state as follows: (47a) (47b) with

(46)

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(48)

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Fig. 11. Multifrequency motion tracking results of the proposed controller. (a) Desired and actual position trajectories. (b) Actual versus desired position. (c) Sliding surface variable. (d) Control effort of the controller.

where the measured position is the input to the observer, and the observer output is the full-state feedback, i.e., . The bandwidth of the observer depends on the design gains , , and , and the accuracy of the estimated state relies on the design parameter since the estimation becomes exact as approaches zero. Therefore, the robust controller is constructed by the above two components of the proposed SMCPE-PID controller and the high-gain observer. The block diagram of the control scheme is illustrated in Fig. 6. It can be observed that the input to the robust controller is the desired position trajectory, hysteresis term, and measured position, while the output is the voltage that will be applied to PZT actuator. Specifically, once the desired position is given, the velocity and acceleration trajectories trajectory ( and ) are obtained offline, and the hysteresis term is calculated in advance based on the inverse dynamic model implemented by Matlab/Simulink as shown in Fig. 7, where the details about the Bouc–Wen model are depicted in Fig. 4(b). Besides, for the purpose of comparison, an incremental-type PID controller is also implemented in the discrete-time form as follows:

(49a) (49b) where

denotes the sampling time interval.

The performance of the constructed model-free PID and model-based sliding mode controllers is tested by experiments carried out below with a sampling time of 0.02 s. B. Set-Point Regulation First, the set-point regulation capabilities of the controllers are examined by experiments. The designed controller parameters are tabulated in Table II. By commanding the end-effector from the home position (0, 0) to the position (55 m, 0) in the workspace, the regulation results are depicted in Fig. 8 and quantitatively tabulated in Table III for a clear comparison. It is observed that the traditional PID controller can produce a regulation result with the lowest overshoot magnitude and quickest transient response. Although the proposed controller gives the moderate response speed with about 10% overshoot, the response is over 2.5 times quicker than that of the conventional SMCPE. It is also found that much quicker response can be obtained by the conventional SMCPE controller with larger gain values. However, it is at the expense of clear chattering. Moreover, from the steady-state regulation results as magnified inside Fig. 8 and compared in Table III, we can observe that the deviations of the set-point regulation are maintained around m by the three types of feedback controllers, which are m as shown in Fig. 3, and worse than the noise level of the proposed controller gives a slight worse result than the other types of controllers. This displacement variation causes a deviation around 2.4 mV in the input voltage. The reason lies in the relatively high gains adopted in the controllers. The high feedback gains are chosen to improve the tracking accuracy, whereas

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Fig. 12. Linear contouring result along a 45 line with a feed rate of 20 m/s. (a) Tracking errors in the two axes. (b) Linear contouring error.

the noises from the displacement sensor are amplified simultaneously as an expense. C. Sinusoidal Motion Tracking In addition, a sinusoidal motion trajectory with 0.2 Hz frequency and 20 m peak-to-peak (p-p) amplitude is tracked with the three types of controllers, and the results are compared in Fig. 9. From the magnitude of the tracking errors as shown in Fig. 9(b), (d), and (f) and described in Table III, we can see that the performance of the proposed controller is superior to that of both conventional SMCPE and PID controllers in terms of p-p tracking errors. Compared with the PID controller which prom, the SMCPE controller can only duces a p-p error of m. Whereas the proposed suppress the error within m in controller substantially reduces the p-p error to contrast. Besides, with the proposed controller, the higher order states (velocity and acceleration) estimated by both traditional method (9) and using the high-gain observer (47) are illustrated in Fig. 10. It is obvious to see that the traditional method cannot predict the higher order states, especially the acceleration [see Fig. 10(b)] due to the noise present in the position data provided by the displacement sensor. On the contrary, the high-gain controller can be used to estimate the full state more precisely owing to its filtering effect. This explains the reason why the high-gain observer is recommended for the proposed controller alternatively. D. Multifrequency Motion Tracking With the proposed controller, the tracking of a multifrequency sinusoidal motion trajectory as shown in Fig. 11(a) is undertaken as well. The planned multifrequency motion covers the whole range (0–110 m) of the manipulator workspace. From the plot of the actual position versus desired position as depicted in Fig. 11(b), we can observe that the hysteresis effect has been compensated significantly since the width of the hysteresis loop has been reduced to a low level of 1.5% with comparison to 13.5% obtained by the open-loop test with the same input rate. In addition, the sliding variable plotted in Fig. 11(c) implies that the sliding surface is maintained within the boundary layer

Fig. 13. Linear contouring error bound versus feed rate. The square markers represent the mean errors.

(i.e., 18), which is an assigned value for the controller. Moreover, the control effort described in Fig. 11(d) indicates that the chattering is eliminated by the designed controller. E. Contour Tracking Other than the single-axis motion tracking tested above, the biaxial contour tracking can reflect the two-axis cooperative tracking performance for two-dimensional motion [9], [40]. Different from the trajectory tracking error, i.e., the difference between the desired position and the actual position, contouring error is defined as the minimum distance between the actual position and the desired trajectory along an orthogonal direction to the trajectory. By driving the two axes simultaneously, a linear contour along a 45 line is adopted to verify the contouring capability of the XY micromanipulator in this work. The manipulator is commanded to move from the home position (0, 0) to the workspace center point (55 m, 55 m) and then return to the home. For a feed rate of 20 m/s, the two axial tracking errors and contouring errors are shown in Fig. 12(a) and (b), respectively. It is observed that the tracking errors (with the maximum value of 2.458 m) of the two axes cancel each other and result in much lower magnitude of the contouring errors (with the maximum value of 0.773 m). This implies that the two axes of the decoupled XY micromanipulator have similar tracking capabilities.

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LI AND XU: ADAPTIVE SLIDING MODE CONTROL WITH PERTURBATION ESTIMATION AND PID SLIDING SURFACE

Moreover, as the increasing of the feed rate from 10 to 100 m/s, the linear contours are tested as well and the results are plotted in Fig. 13. It can be seen that the contouring error increases as the rising of the feed rate. In order to constrain the maximum contouring error below 1 m, the feed rate should be restricted within 30 m/s. F. Discussions on Controller Performance From above experimental results obtained by the three types of controllers, we can conclude that the PID controller is more suitable for set-point control since it enables a quick response without much overshoot regulation. On the other hand, the proposed adaptive SMCPE-PID controller is the most suitable controller for motion tracking due to its lowest peak-to-peak error. In particular, the proposed controller is superior to the conventional SMCPE scheme in both set-point regulation with a quicker response speed and trajectory tracking with a smaller tracking error. Concerning the implementation of the proposed controller, it is remarkable here that the initial values of the three estimated gains defined in (35), (36), and (46) can be assigned to any values (e.g., zero [39]) in theory since the gain values will be self-updated in accordance with the tracking error subsequently. Actually, in order to obtain a rapid transient response, instead of arbitrary selection, the three initial values are tuned manually by trial and error through experiments. As a result, the positive conin this case) so stant in (46) deserves a larger value ( as to avoid unnecessary update of control gain in the presence of small deviations of the sliding surface from zero. It is also noticeable that the tracking performance of the designed controllers is limited by the resolution of the displacement sensors. The sensors are the bottle-neck in ultra-precision positioning applications. Using sensors with a higher resolution, more accurate motion tracking can be achieved even with the same controller proposed in this work. VII. CONCLUSION In this paper, a new SMCPE controller with PID-type sliding surface and adaptive gains is proposed for the motion tracking of a piezoelectrically actuated micromanipulator system. The set-point regulation experimental results show that the proposed controller can substantially speed up the transient response as compared to conventional SMCPE controller. Further sinusoidal motion tracking experiments demonstrate that the proposed controller can improve the tracking performance with the smallest peak-to-peak error in comparison to both PID and SMCPE controllers. With the proposed controller, a submicron accuracy single-axis motion tracking and biaxial linear contouring have been achieved by the micromanipulator system and the hysteresis has been significantly compensated to a low level of magnitude, which validates the effectiveness of the designed controller as well. Since the implementation of the controller does not require any bounds on uncertainties and unknown parameters of the system while only the estimated initial values of the controller gains are needed, the proposed control can be widely used in the field of micro/nano scale manipulation.

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ACKNOWLEDGMENT The authors would like to thank the the Associate Editor and the anonymous reviewers for their valuable comments and kind suggestions to improve the quality of this paper. REFERENCES [1] J. J. Abbott, Z. Nagy, F. Beyeler, and B. J. Nelson, “Robotics in the small, part I: Microrobotics,” IEEE Robot. Autom. Mag., vol. 14, no. 2, pp. 92–103, Jun. 2007. [2] S. Devasia, E. Eleftheriou, and S. O. R. Moheimani, “A survey of control issues in nanopositioning,” IEEE Trans. Control Syst. Technol., vol. 15, no. 5, pp. 802–823, Sep. 2007. [3] Y.-C. Huang and D.-Y. Lin, “Ultra-fine tracking control on piezoelectric actuated motion stage using piezoelectric hysteretic model,” Asian J. Contr., vol. 6, no. 2, pp. 208–216, 2004. [4] G. Song, J. Zhao, X. Zhou, and J. De Abreu-Garcia, “Tracking control of a piezoceramic actuator with hysteresis compensation using inverse Preisach model,” IEEE/ASME Trans. Mechatronics, vol. 10, no. 2, pp. 198–209, Apr. 2005. [5] C.-J. Lin and S.-R. Yang, “Precise positioning of piezo-actuated stages using hysteresis-observer based control,” Mechatronics, vol. 16, no. 7, pp. 417–426, Sep. 2006. [6] H. G. Xu, T. Ono, and M. Esashi, “Precise motion control of a nanopositioning PZT microstage using integrated capacitive displacement sensors,” J. Micromech. Microeng., vol. 16, no. 12, pp. 2747–2754, Dec. 2006. [7] A. Sebastian and S. M. Salapaka, “Design methodologies for robust nano-positioning,” IEEE Trans. Control Syst. Technol., vol. 13, no. 6, pp. 868–876, Nov. 2005. [8] M. Boukhnifer and A. Ferreira, “ H loop shaping bilateral controller for a two-fingered tele-micromanipulation system,” IEEE Trans. Control Syst. Technol., vol. 15, no. 5, pp. 891–905, Sep. 2007. [9] J. Dong, S. M. Salapaka, and P. M. Ferreira, “Robust control of a parallel-kinematic nanopositioner,” J. Dyn. Sys., Meas., Control, vol. 130, no. 4, pp. 041007-1–041007-15, 2008. [10] T. W. Seo, H. S. Kim, D. S. Kang, and J. Kim, “Gain-scheduled robust control of a novel 3-DOF micro parallel positioning platform via a dual stage servo system,” Mechatronics, vol. 18, no. 9, pp. 495–505, Nov. 2008. [11] S. S. Aphale, S. Devasia, and S. O. R. Moheimani, “High-bandwidth control of a piezoelectric nanopositioning stage in the presence of plant uncertainties,” Nanotechnology, vol. 19, no. 12, pp. 125503-1–125503-9, Mar. 2008. [12] Y. Wu and Q. Zou, “Iterative control approach to compensate for both the hysteresis and the dynamics effects of piezo actuators,” IEEE Trans. Control Syst. Technol., vol. 15, no. 5, pp. 936–944, Sep. 2007. [13] K. Santa and S. Fatikow, “Control system for motion control of a piezoelectric micromanipulation robot,” Adv. Robot., vol. 13, no. 6, pp. 577–589, 2000. [14] F.-J. Lin, R.-J. Wai, K.-K. Shyu, and T.-M. Liu, “Recurrent fuzzy neural network control for piezoelectric ceramic linear ultrasonic motor drive,” IEEE Trans. Ultrason., Ferroelectr., Freq. Control, vol. 48, no. 4, pp. 900–913, Jul. 2001. [15] H. C. Liaw, B. Shirinzadeh, and J. Smith, “Sliding-mode enhanced adaptive motion tracking control of piezoelectric actuation systems for micro/nano manipulation,” IEEE Trans. Control Syst. Technol., vol. 16, no. 4, pp. 826–833, Jul. 2008. [16] J.-C. Shen, W.-Y. Jywe, C.-H. Liu, Y.-T. Jian, and J. Yang, “Slidingmode control of a three-degrees-of-freedom nanopositioner,” Asian J. Contr., vol. 10, no. 3, pp. 267–276, May 2008. [17] T.-C. Kuo, Y.-J. Huang, and S.-H. Chang, “Sliding mode control with self-tuning law for uncertain nonlinear systems,” ISA Trans., vol. 47, no. 2, pp. 171–178, Apr. 2008. [18] H. Elmali and N. Olgac, “Sliding mode control with perturbation estimation (SMCPE): A new approach,” Int. J. Control, vol. 56, pp. 923–941, 1992. [19] H. Elmali and N. Olgac, “Implementation of sliding mode control with perturbation estimation (SMCPE),” IEEE Trans. Control Syst. Technol., vol. 4, no. 1, pp. 79–85, Jan. 1996. [20] N.-I. Kim, C.-W. Lee, and P.-H. Chang, “Sliding mode control with perturbation estimation: Application to motion control of parallel manipulator,” Control Eng. Practice, vol. 6, no. 11, pp. 1321–1330, Nov. 1998.

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Yangmin Li (M’98–SM’04) received the B.S. and M.S. degrees from Jilin University, Changchun, China, in 1985 and 1988, respectively, and the Ph.D. degree from Tianjin University, Tianjin, China, in 1994, all in mechanical engineering. He is currently a Professor of Electromechanical Engineering at the University of Macau, Macao SAR, China, where he also directs the Mechatronics Laboratory. He has authored 205 scientific papers, and has served 70 international conference program committees. His research interests include micro/nanomanipulation, nanorobotics, micromanipulator, mobile robot, modular robot, multibody dynamics, and control. Dr. Li is a Member of the American Society of Mechanical Engineers (ASME). He currently serves as Technical Editor of the IEEE/ASME TRANSACTIONS ON MECHATRONICS, a Council Member and an Editor of the Chinese Journal of Mechanical Engineering, and a Member of Editorial Board of the International Journal of Control, Automation, and Systems.

Qingsong Xu (M’09) received the B.S. degree in mechatronics engineering (with honors) from Beijing Institute of Technology, Beijing, China, in 2002, and the M.S. and Ph.D. degrees in electromechanical engineering from the University of Macau, Macao SAR, China, in 2004 and 2008, respectively. He is currently a Post-Doctoral Fellow at the University of Macau. His current research interests include design, analysis, and control of parallel manipulators, compliant mechanisms, and micro-/nanomanipulators, with a particular emphasis on the field of micro-/nanomanipulation.

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