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Model Predictive Discrete-Time Sliding Mode Control of a Nanopositioning Piezostage Without Modeling Hysteresis Qingsong Xu, Member, IEEE, and Yangmin Li, Senior Member, IEEE

Abstract—This paper proposes an enhanced model predictive discrete-time sliding mode control (MPDSMC) with proportional-integral (PI) sliding function and state observer for the motion tracking control of a nanopositioning system driven by piezoelectric actuators. One distinct advantage of the proposed controller lies in that its implementation only requires a simple second-order model of the system, whereas it does not need to know neither the hysteresis model nor the bounds on system uncertainties. The unmodeled hysteresis is eliminated by the one-step delayed disturbance estimation technique and the neglected residual modes are suppressed by employing a properly-designed state observer. Moreover, the reasons why the model predictive control methodology and PI action can eliminate the chattering effects and produce a low level of tracking error are discovered in state-space framework. Experimental results demonstrate that the performance of the proposed MPDSMC controller is superior to both conventional PID and DSMC methods in motion tracking tasks. A precise tracking is achieved by the nanopositioning stage along with the hysteretic nonlinearity mitigated to a negligible level, which validates the feasibility of the proposed controller in the domain of micro-/nanomanipulation. Index Terms—Flexure mechanisms, hysteresis, nanopositioning, nonlinear control, piezoelectric actuators (PZTs).

I. INTRODUCTION

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ANOPOSITIONING stages actuated by piezoelectric actuators (PZTs) are popularly applied in a variety of situations which require an ultrahigh precision positioning, e.g., the scanning probe microscope [1], biological manipulation [2], and so on. The PZT endows the stages with such merits as nanometer-level resolution and rapid response. In addition, the piezostages usually employ a flexure structure [3], [4] to deliver motions by making use of elastic deformations of the material. The flexure mechanism enables a smooth output motion by eliminating the adverse effects such as clearance and backlash. Nevertheless, PZT introduces nonlinearity into the system due to the piezoelectric hysteresis and creep effects, and the

Manuscript received November 09, 2010; revised March 05, 2011; accepted May 04, 2011. Manuscript received in final form May 16, 2011. Date of publication June 23, 2011; date of current version May 22, 2012. Recommended by Associate Editor G. Cherubini. This work was supported in part by Macao Science and Technology Development Fund under Grant 016/2008/A1 and in part by the research committee of the University of Macau under Grant SRG006FST11-XQS. The authors are with the Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macao, 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.2011.2157345

stage has a low damping which causes the structural vibration problem. 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 10%–15% of the stage travel range. Thus, the hysteresis has to be suppressed in high precision applications. Extensive works have been done to model and suppress the hysteretic behavior in the literature, e.g., [5] and [6]. Insightful surveys of control approaches to nanopositioning are available in [7], [8]. The models of hysteresis are usually approached by Preisach model [9], Prandtl-Ishlinskii model [10], Duhem model [11], Bouc-Wen model [12], and Dahl model [13], etc. An inverse model is then constructed and utilized as an input “shaper” to suppress the hysteresis effect. However, it is very difficult to precisely capture the complicated hysteretic behavior, which depends not only on the amplitude but also on the frequency of input signals. Most of the models employ a great number of parameters to improve the modeling accuracy, which blocks their use in real-time control. Even worse, the hysteresis can hardly be compensated by utilizing a standalone inverse model [14]. In view of that, some approaches without modeling the hysteresis are exploited by considering the hysteresis as a disturbance. For instance, robust and adaptive control schemes have been implemented successfully [15]–[17]. More simply, a linear model can be identified for the plant, based on which, a high-gain feedback method [18] or nonlinear robust control is designed to suppress the disturbance [19], [20]. Owing to the low damping of the system, a number of vibration modes exist and a high-order model is commonly identified, that results in a high-order controller as a consequence. Some intelligent approaches have also been applied to the hysteresis control of PZT actuators [21]–[23]. From the practical point of view, a linear model of low order (e.g., second order) is very desirable for easy implementation on control strategies. However, the adoption of a low-order model means that the residual modes will not be considered in the controller (and observer) design. The neglect of residual modes may cause both control spillover and observation spillover [24]. Spillover is undesirable because it may induce system instability and performance degradation [25]. In view of the aforementioned difficulties, it is a major challenge in piezostate control by employing a low-order model while without modeling the complicated hysteresis effects. In the literature, some attempts have been pioneered to deal with the piezostage control problem by employing a low-order linear model. For example, by identifying a second-order model of a

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piezostage, a precise control is implemented in [26] based on the integral resonant control method, in which the charge actuation is employed to reduce piezoelectric hysteresis. Although the hysteresis becomes almost negligible if a PZT actuator is driven by a charge source, the stroke will be reduced as the cost. Hence, voltage actuation is still widely adopted in practice. An enhanced adaptive sliding mode motion tracking control is reported in [27] for a PZT-actuated system, which only uses a second-order dynamics model. However, the sliding mode control (SMC) is designed based on the continuous-time model, and the uncertainty bounds are required to implement the controller. Based on the second-order dynamics model of a PZT actuator, a model predictive sliding mode control is presented in [28], where a nonlinear energy-based model is employed to construct an inverse compensator of the hysteresis. The same model predictive-based control is also adopted in [29] for the force control of a PZT actuator. In the above works, a sliding mode controller is designed based on a proportional-type sliding function, whereas no stability analysis of the control system is given. Besides, the spillover effects are not treated explicitly. In this research, based on a second-order mass-spring-damper model of a nanopositioning system, we propose a new model predictive discrete-time sliding model control (MPDSMC) with proportional-integral (PI) type of sliding function, and provide stability analysis of the closed-loop system in detail. The observation spillover is suppressed by designing a proper state observer. It is known that SMC is a popular nonlinear control approach to deal with disturbance. The discrete-time sliding mode control (DSMC) is presented for the implementation on sampled-data systems [30], [31]. For a discrete-time system, the strategy of model predictive control (MPC) can be employed to predict the system performance in a specified time in the future and to produce an optimal control action with respect to a predefined cost criterion at each time step [32]. By combining the SMC and MPC techniques, model predictive sliding mode control has been recently proposed to achieve the advantages of both methods [33], [34]. Although the combined control has been adopted in a few of previous works [28], [29] for the control of PZT actuator by suppressing the chattering phenomenon, it still remains unclear why the MPC methodology can eliminate the chattering phenomenon in the sliding mode. In the current research, we will show that the proposed MPDSMC with the PI action drives the system state to slide in a vicinity of the [35], sliding surface with a boundary layer of thickness which is much lower than a commonly designed DSMC with boundary layer in the sliding mode. In addition, the state tracking error of the order is achieved with the presented control scheme. The theoretical analysis and the effectiveness of the PI action in the control scheme will be verified by experimental studies performed on a nanopositioning platform. The major objective of the current research is to compensate for the unmodeled hysteretic effects by using a secondorder linear model of the piezostage system. In the rest of the paper, the control problem for a nanopositioning stage is formulated in Section II, where an equivalent controller with a PI sliding function is proposed. A DSMC with common switching control is constructed in Section III. Then, the design proce-

dures of MPDSMC along with a state observer are outlined in Section IV, where the stability and tracking error bounds are analyzed. The designed controllers are verified in Section V via both simulation and experimental studies. Finally, Section VI concludes this research. II. PROBLEM FORMULATION A. Dynamics Modeling of a Nanopositioning System The entire dynamics model of a nanopositioning system with nonlinear hysteresis can be established as follows: (1) where is the time variable, parameters , , , and represent the mass, damping coefficient, stiffness, and -axis displacement of the nanopositioning stage, respectively; is the piezoelectric coefficient, denotes the input voltage, and describes the hysteretic effect of the system in terms of force which is not modeled in this research. The dynamics model (1) can be expressed in the state-space form (2) (3) where the state vector , the unmodeled hysteresis effect is considered as a disturbance to the system, and the four matrices are

(4) It is observed that the disturbance satisfies the matching , where is a scalar. condition, i.e., Using a sampling time , the continuous-time system model (2)–(3) can be discretized as (5) (6) where the notation trices are calculated by

and the corresponding ma-

(7) (8) which indicate that both

and

are

, i.e., the order of .

B. Sliding Mode Controller Design Most of the existing works construct a discrete-time sliding mode controller using a proportional sliding function defined using the system state only [30], [31], [36], which leads to the

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order reduction of the system. A few of previous studies employ the integral term in the sliding surface [35], [37]. In this research, we define a PI-type sliding surface in order to speed up the system response. By assigning a vector of state error (9)

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C. Gain Vectors Selection To implement the PI sliding function, the gain vectors and should be designed such that is invertible. Moreover, the following statement specifies how to select the two gain vectors to guarantee the stability of the closed-loop system. . Then, In the ideal sliding mode, one has (15) reduces to

where denotes the desired system state, a PI-type sliding function is defined as follows:

(19)

(10)

Substituting (19) into (13), results in the closed-loop state error dynamics

where (11) represents the sliding surface or sliding hyperplane, and the integral error vector is defined as (12)

(20) where

Besides, and are constant proportional and integral gain vectors, which are to be designed to assign appropriate eigenvalues for the system. In view of (9) and (5), one gets

(13) where denotes the overall disturbance. is the solution of Considering that the equivalent control [38], we can deduce that

(14) which allows the generation of an equivalent controller (15) where (16) and the disturbance is estimated by its one-step delayed value . Thus, (15) becomes (17) where (18) is derived by noting (13).

(21) (22) (23) with denoting an identity matrix. are the soluThe eigenvalues of the matrices , , and tions of , , and , respectively, which determine the closed-loop dynamic behaviors. To ensure the stability of the closed-loop system, the eigenvalues of the three matrices should lie inside the unit circle in the complex -plane. can be calculated as the For instance, the eigenvalues of solutions to the equation [39]

(24) at the origin. It is observed that there is one fixed pole The other pole lies on the real axis whose position can be adjusted by tuning the values of in order to make the pole locate inside the unit circle. A faster response can be produced by placing the pole close to 0 and a better robustness can be obtained by locating the pole and near 1 for the controller. Similarly, the values of should also be tuned to locate the eigenvalues of the other two matrices inside the unit circle as well. A case study will be given in Section V to show how to design the gains in detail. III. DSMC CONTROLLER DESIGN The equivalent controller designed in the previous section takes effect in the sliding mode when the system state trajectory is kept on the sliding surface. However, if the initial state of the system does not lie on the sliding surface or external disturbances occur during the sliding mode motion, the equivalent control alone cannot drive the state towards the . Thus,

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a switching control is necessary [6], [15]–[17], [40]. In the following discussions, a DSMC with switching control is designed.

Considering (32) and (33) together, yields for

(34)

A. Controller Design and Analysis A commonly used method is to augment the equivalent con, whose role is trol with a discontinuous switching control to force the system state to reach the sliding mode. Then, the overall control effort is assigned as

decreases monotonously, and the diswhich indicates that of steps. crete sliding mode is reached after a finite number It has been shown in [41] that the relationship (34) represents a sufficient condition for the existence of discrete sliding mode.

(25)

Remark 1: Theorem 1 gives a sufficient condition for the existence of the discrete sliding mode. Due to the discontinuity , chattering may occur in the conof the signum function trol input. To alleviate the chattering phenomenon, the boundary layer technique is adopted by replacing the signum function in (26) with the saturation function

Theorem 1: For the system (5) with the switching function (10), if the following controller (26) is employed, then the discrete sliding mode will occur after a finite number of steps:

if if

(26) where is a positive switching gain vector and the last righthand term represents the switching control action. Proof: In view of (14) and (16), the following expression can be derived: (27) Substituting (26) into (27), gives (28) where

(35)

where the positive constant denotes the boundary layer thickness, which ensures that is always bounded by . In selection of parameter , a tradeoff between the chattering and tracking error should be taken. Equation (31) reveals that the selection of the control gain depends on the upper bound of the disturbance estimation error . In classical SMC, this term is the upper bound of the disturbance instead. Generally, the disturbance estimation error is much lower than the disturbance value. Hence, a much smaller can be designed in comparison with switching gain vector the classical SMC. This is one of the major advantages of the constructed DSMC with one-step delayed disturbance estimation over SMC. B. Tracking Error Bound Analysis

(30)

Taking into account the relationship (28) and noting Property 2 in [37], we can deduce that has a magnitude of the order . It means that the ultimate state will stay in the neighbor. hood of with a boundary layer of the thickness Considering (10), the error dynamics (20) can be further expressed as follows after a necessary calculation:

is designed to meet

(36)

(29) is the estimation error vector of the disturbance, which has a magnitude of and is assumed to be bounded, i.e.,

If the switching control gain vector the condition

where (31) where is an arbitrary positive constant, then, in case of we have

(37) (38) (39)

,

and , one can deduce that , i.e., the ultimate state error is of the order . Due to the existence of unmodeled dynamics of the system including the actuator nonlinearities, sensor noises, and high-frequency dynamics, the switching control constructed above may still induce chattering phenomenon. Chattering is the oscillations of the system states around the sliding surface with a certain amplitude and frequency [6], [15]–[17]. As a result, chattering leads to poor system performance and reduces usable Since

(32) On the other hand, if

, we can derive that

(33)

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life of the actuator. In Section IV, the model predictive control scheme is employed to eliminate the chattering effect.

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Additionally, the matrices take on the following forms:

IV. MPDSMC CONTROLLER DESIGN In this section, an integrated MPC and DSMC control scheme is constructed to handle the nonlinearities and disturbances existing in the system. Specifically, the MPC is used to generate to drive the system state to the sliding an optimal control surface, and the equivalent control is employed to maintain the state trajectory on the sliding surface for the subsequent time. It will be demonstrated that the system state is forced to reach the sliding mode smoothly in an optimal manner without chattering effects.

.. .

.. .

.. .

.. .

..

.

.. .

..

.

.. .

(45)

(46)

.. .

Using the above notations, the cost function for minimization can be expressed by

A. Controller Design

(47)

Substituting the overall control action (40) into (27) and noting (17), gives

where is the weighting to limit the partial control effort. It approaches to zero as tends to zero. is noticeable that Thus, the cost function will vanish at the steady state. Substituting (43) into the cost function (47) and applying the , yields optimization criterion by setting (48)

(41) which describes the dynamics of the sliding mode and also represents a one-step prediction of the sliding mode dynamics. Based on (41), an -step prediction of the sliding mode dynamics can be obtained as follows:

where the future disturbance values in they are estimated by

are unknown. Thus,

(49) Since only the first value of the predicted control sequence is used, the optimal control effort can be obtained as

(42) where The form

is called the prediction horizon. prediction functions can be stacked together into the

(50) . where the vector Therefore, the total control action can be expressed as

(43) where the vectors for the sliding surface, future control, and disturbance estimation errors are (51) .. . B. Stability Analysis Theorem 2: For the system (5) with the switching function (10), if the disturbance change rate is limited, i.e., the inequality (30) is satisfied, then the controller (51) leads to a stable closedloop control system. Proof: Substituting (51) into (27), results in (41). Then, inserting (48) into the stacked (43), yields

.. .

.. .

(44) (52)

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where is defined in the cost function (47) to restrict the control input. For simplicity, it is assumed that there is no penalty for . the control effort, i.e., Then, (52) reduces to

represents the estimate of , and the where the notation in (5) is obtained by its one-step value of the disturbance delayed estimation (59)

(53) Considering only the first value of the predicted sliding hyperplane vector (53) and taking into account (44) and (49), we have (54) is of the which indicates that the ultimate magnitude of as shown in [35]. order Since the disturbance change rate is bounded as supposed is also in (30), it is reasonable to deduce that the rate of limited (55) Thus, we can generate that (56) is defined as the quasi-sliding mode band width. where According to Definition 1 and Definition 2 in [31], one can conclude that the system (5) satisfies the reaching condition of vicinity of the sliding surface the quasi-sliding mode in the in a finite number of steps. Therefore, the closed-loop control system is stable. Remark 2: The essence of the presented MPDSMC is to drive the system state to a quasi-sliding mode as shown in (56). One advantage of this quasi-sliding model [31] lies in that it does not require the system state to cross the sliding hyperplane in each successive control step as in [30]. Therefore, the chattering phenomenon is eliminated. This further results in a reduction of the control effort and improved control performance [31]. has an ultimate magnitude Remark 3: It is observed that as shown in (54). Therefore, the controller of the order (51) drives the system state to slide in the vicinity of with a , whose magnitude is much boundary layer of thickness lower than the thickness achieved by the DSMC controller designed in Section III. Therefore, a much lower tracking error is expected for the proposed MPDSMC controller.

Subtracting (57) from (5) allows the derivation (60) denotes the estimation error of the state where . It can be seen that the state observer is independent of the controller. To guarantee the stability of the observation unit, the observer gain should be designed to make all the eigenvalues locate inside the unit circle. of the matrix The convergence speed of the observer relies on the loca. The poles can be tion of the poles of the matrix arbitrarily placed by the gain vector . Generally, small pole value will give rapid convergence, and the convergence speed of the observer should be faster than the system response so that the observer dynamics is insignificant compared to the system dynamics. However, too small pole produces instability of the system, which comes from the noise and spillover effect of the neglected system dynamics. Hence, trial-and-error approach is needed to position the poles appropriately. D. Tracking and Estimation Error Bounds Analysis In order to evaluate the tracking error bound, we note that and . From (36), we can deduce that , i.e., the ultimate state error is of the order . , and are nonsingular, i.e., the number In case of of inputs is equal to the number of system states, the matrix as shown in (39) will vanish. Under such situation, (36) reduces to (61) as shown in (54) and Property 2 in Then, noting , which is much lower [37], we can derive that tracking error obtained with the DSMC designed than the in Section III. is of the order , Furthermore, considering that is . we can deduce from (60) that the ultimate bound of For the systems with relative degrees greater than 1, it has been shown in [37] that the state estimation error will be reduced to for the closed-loop system. V. EXPERIMENTAL INVESTIGATION

C. State Observer Design An insight into the controller (51) reveals that its implementafeedback of the system. However, tion needs the full state only partial states are available in practice. Thus, a state observer is designed to estimate the full state by making use of the only measurable position information of the nanopositioning system. Although various methods are available [42], a Luenbergerlike observer is adopted here owing to its simplicity. The state observer takes on the form (57) (58)

In this section, the controllers designed above are validated by a series of simulations and experimental studies. A. Experimental Setup The test plant employed in this research, i.e., an XYZ nanopositioning piezostage, is shown in Fig. 1. It was designed as a decoupled parallel-kinematic structure. Details about its working principle can be found in previous works [43] of the authors. The stage is fabricated from Al7075 alloy by the wire-EDM process. Driven by three PZTs with the stroke of (model P-840.20 produced by Physik Instrumente Co., 30

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Fig. 1. (a) CAD model and (b) prototype of a flexure-based nanopositioning piezostage which is driven by three stacked PZTs through lever displacement amplifiers and measured by three noncontact capacitive sensors.

Ltd.), the stage delivers amplified translational motions by making use of elastic deformations of the lever displacement amplifiers. The output positions are measured by three capacitive sensors (model: D-510.050 from the Physik Instrumente). A dSPACE DS1005 (from dSPACE GmbH) rapid prototyping system equipped with DS2001 A/D and DS2102 D/A modular boards are employed to implement the controller. The D/A board is used to produce the voltage control signal which is then amplified by a three-axis piezo voltage amplifier (model E-503.00 from the Physik Instrumente) to provide a voltage of 20–100 V for the drives of PZTs. Besides, the sensor output voltage signals are passed through a signal conditioner (model E-509.E03 from the Physik Instrumente) and then simultaneously acquired by the A/D board. Control algorithms are developed with MATLAB/Simulink software and downloaded to DS1005 PPC board to realize the real-time control.

Fig. 2. x-axis hysteresis loops which are obtained by applying a 0.5-Hz sine wave input. The maximum open-loop output hysteresis accounts for 16.0% of the motion range.

B. Plant Model Identification By applying a 0.5-Hz sinusoidal wave with the amplitude of 10 V to PZT-1, the hysteresis loop in the -axis is generated as shown in Fig. 2, which exhibits a nonlinear relationship between the input voltage and output displacement. The output hysteresis accounts of 16.0% of the travel range, which provides a big challenge to the controller design task. In this research, the hysteresis is treated as a bounded disturbance to the nanopositioning system, which is considered as a second-order linear system as described in (1). The linear plant model is identified by the swept-sine approach through experiments. Specifically, sine waves with the amplitude of 0.05 V and frequency range of 5–300 Hz are produced by the D/A board to drive one PZT. The position responses of the XYZ stage in the three working directions are simultaneously recorded using a sampling rate of 5 kHz. It is found that, with each PZT driven individually, the displacement responses in the passive axes are 20 dB lower than that in the dominant axis over the low frequency range (0–40 Hz). Thus, the three axial motions of the XYZ stage are well decoupled up to 40 Hz, and three single-input-single-output (SISO) controllers can be adopted for the three working axes of the nanopositioning stage in low frequency control. For the purpose of verifying the proposed control scheme, only the -axis motion control is handled in this research. The

Fig. 3. Plant frequency responses obtained by experiment (dashed line) and identified second-order transfer function (solid line) for the x-axis, which show a peak resonant mode around 82 Hz.

input-output data sets are used to identify the plant transfer function by estimating the model from the frequency response data. The frequency responses obtained from the experimental data and the identified model are compared in Fig. 3. The first resonant mode occurs around 82 Hz, and the identified second-order model matches the system dynamics well in the frequencies below 120 Hz. The identified transfer function is (62) It is noticeable that, in order to capture the high-frequency dynamics accurately, a much higher order model is required to be identified. Here, a simple second-order model is employed to demonstrate the effectiveness of the proposed control scheme. Furthermore, as the frequency of input command exceeding 40 Hz and closing to the resonant model, the cross coupling, i.e., the

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Fig. 4. (a) Desired position input; (b) simulation results of DSMC with and without the integral action in the sliding function; (c) the maximum tracking error of MPDSMC versus the prediction horizon ; (d) simulation results of MPDSMC with and without the integral action in the sliding function.

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passive motions (in - and -axes), become significant. Hence, the three axial motions of the stage, which present a resonant mode of 82 Hz, become coupled at frequencies higher than 40 Hz. C. Controller Parameters Design Comparing the transfer function of (1) with (62) yields the mass-spring-damper system parameters, which allows the calculation of the discrete-time system model (5) with a sampling time of 1 ms. That is

in the presence of unmodeled high-frequency dynamics. The . corresponding gains are obtained as Next, the sliding function is designed by selecting appropriate and proportional and integral gain vectors . It can be checked that the matrix has two constant and eigenvalues of 0 and 1. Each of the other two matrices has an eigenvalue fixed at 0. Thus, the gains are chosen to and of and locate guarantee that the free eigenvalues inside the unit circle, respectively. Noting that , the eigenvalues can be expressed by (64)

(63) First, the state observer is constructed by the pole placement technique for the closed-loop observer system. Simulations show that small pole values lead to rapid convergence in that a high bandwidth is achieved by the observer. However, experiments reveal that too small pole produces instability of the system, which arises from the sensor noises and spillover effects of the neglected dynamics. Thus, the poles are assigned as (0.8, 0.8), which are selected to guarantee the stability of the observer

are all constants. It is seen that the where , , , and and are related to the two eigenvalues, ratios and respectively. The eigenvalues are assigned as to ensure a quick response of the closed-loop control system. Then, by selecting arbitrarily, extensive simdoes not improve performance. It ulations show that results in a better tracking result, and is found that does not make more contribution. Thus, further increasing the gain vectors can be calculated as and . Besides, to generate better results, the weighting parameter is chosen for the MPDSMC after some trials, and

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parameters and are selected for the DSMC controller. In addition, the control effort is constrained between 2 to 10 V, that will be produced by the D/A board and then amplified by the piezo voltage amplifier (with a fixed gain of 10) to provide the driving voltage range of 20 to 100 V. D. Simulation Studies The performance of the implemented controllers are first tested by simulation studies1. For a sinusoidal reference input as shown in Fig. 4(a), the DSMC tracking results with and without the integral action in the sliding function are shown , the in Fig. 4(b). By defining the tracking error maximum tracking error is described as follows: (65) It is observed that the integral action significantly reduces by 81.9% compared to the P-type sliding function for the DSMC. Fig. 4(b) and (d) also indicate that the constructed state observer estimates the position accurately with the maximum error less than 0.30%. Concerning the MPDSMC, it is known that the larger the prediction horizon , the heavier the calculation burden. With the control parameters designed above, simulations are carried out to disclose the relationship between the maximum steady state tracking error and the prediction horizon . The results produces the are depicted in Fig. 4(c). It is found that is sebest results in terms of tracking error. Hence, lected in both simulations and the later experimental studies. The tracking errors of two types of MPDSMC controllers are compared in Fig. 4(d). The results reveal that the integral action in the sliding function substantially alleviates the maximum by 95.9%. error From the error magnitudes as plotted in Fig. 4(b) and (d), we observe that the MPDSMC relies more on the integral action. But both with the PI sliding function, the MPDSMC by 5.1% in comparison with the DSMC further reduces scheme. The simulation results demonstrate the effectiveness of proposed MPDSMC strategy. E. Experimental Test Results The performance test of the designed controllers are carried out by several experimental studies performed below. For comparisons, the root-mean-square (RMS) error is defined as follows:

(66)

First, in order to perform a comparison with simulation results, the 2-Hz reference signal with 20- m amplitude as shown in Fig. 5(a) is used. Moreover, a traditional PID controller is designed by tuning its parameters to produce a good tracking. The tracking results of PID, DSMC, and MPDSMC (both with 1Simulation program designed with MATLAB/Simulink is available by sending a message to [email protected] (Dr. Xu).

Fig. 5. (a) Reference position input; (b) experimental results of PID, DSMC, and MPDSMC tracking errors, and the state observer estimation errors.

PI sliding function) are illustrated in Fig. 5(b). It is observed that the maximum error and RMS error of PID tracking are 4.12% and 2.42%, respectively. In contrast, the DSMC produces 1.72% and 0.63%. The proposed MPDSMC scheme further reduces the errors to 1.20% and 0.44%, respectively. In comparison with PID result, both DSMC and MPDSMC alleviate the tracking errors significantly. In particular, the MPDSMC substantially reduces the maximum error by 70.9% and 30.2% and RMS error by 81.8% and 30.2% as compared with PID and DSMC schemes, respectively. Fig. 5(b) also indicates that the state observer estimates the position value with the maximum error below 0.58%. Compared with high-gain feedback methods presented in [18] which achieved positioning errors of 1.59% and 0.95% for a sinusoidal reference input with frequency of 1 Hz and amplitude of 50 m, the proposed MPDSMC scheme (for a 2-Hz 20- m command input) alleviates the maximum and RMS errors by 24.5% and 53.6%, respectively. The performance of the high-gain feedback approach was augmented by an inverse feedforward control [18]. Concerning the proposed MPDSMC scheme, its tracking accuracy may be improved by resorting to an iterative control scheme [44]. In addition, comparing the experimental results [see Fig. 5(b)] with simulation results [see Fig. 4(b) and (d)], we can observe that the actual tracking errors are worse than

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Fig. 7. Frequency responses of the closed-loop system with the proposed MPDSMC controller.

Fig. 6. Dynamic resolution test results of the nanopositioning stage using (a) a 1-Hz and (b) a 10-Hz sinusoidal wave inputs with 50 nm peak-to-peak amplitude.

simulated ones with both DSMC and MPDSMC controllers. The differences mainly arise from the high-frequency dynamics which cannot be captured by the second-order model used in the simulation studies. The presence of high-frequency dynamics degrades the performance of the designed controllers as indicated by the experimental results. Second, the dynamic resolution of the nanopositioning stage is tested. It is known that the resolution can be well presented using the sensitivity and noise spectrum (variance) of the sensing system [20], [45]. However, such approach is more suitable for evaluating static resolution of the system. In order to exhibit the dynamic resolution and demonstrate the effectiveness of the state observer, sinusoidal signals with small amplitudes are used as input in the experimental test. For instance, Fig. 6(a) and (b) depict the results which are obtained using reference inputs of 50-nm amplitude along with 1 and 10 Hz frequencies, respectively. The sine waves can be clearly identified even with a 10-Hz higher frequency input. It indicates that the resolution of the positioning system is better than 50 nm. Furthermore, the results also illustrate that the implemented state observer acts as a low-pass filter with two extra functions of filtering the noisy sensor readings and suppressing the residual modes of high-frequency dynamics. Next, with the proposed MPDSMC controller, the hysteresis effects are tested by applying a 0.5-Hz reference input with amplitude of 230 m. Scaling the output displacement with the

maximum input voltage value yields the results as illustrated in Fig. 2. It is observed that the closed-loop hysteresis is significantly suppressed to 1.7%. As compared with the open-loop hysteresis width (16.0%), it has been substantially reduced by 89.4% owing to the effectiveness of the MPDSMC strategy. Actually, the proposed scheme mitigates the tracking errors 1.01% and 0.56%, respectively. Compared to with the low-frequency hysteresis test in [18] where the maximum positioning error of 1.62% is achieved by high-gain feedback methods using a full-range (50 m) command input, the MPDSMC approach further reduces the maximum error by 37.7% using the full-range (230 m) input. Moreover, the bandwidth of the closed-loop system with the MPDSMC control is tested by applying a sinusoidal reference input with the amplitude of 0.2- m and varying frequencies of 4–200 Hz. The Bode diagram is plotted in Fig. 7. It is observed that the 3 dB bandwidth is 90.6 Hz, which is over the resonant frequency (82 Hz) of the piezostage system. However, the corresponding phase lag is as high as 90 degrees, which causes a large tracking error. In this research, the closed-loop control bandwidth is defined as the frequency at which the phase lag attains 6 degrees. Fig. 7 illustrates that the 6-degree-lag bandwidth is 31.4 Hz, which is equivalent to 38.3% of the first resonant frequency. This cutoff frequency corresponds to a magnitude of 0.74 dB, which means a 8.2% tracking error. It should be noted that the above bandwidth is tested using a small amplitude of the input command where the hysteresis can be neglected. As the motion range increases, the bandwidth decreases due to the effect of the hysteresis. Even so, it demonstrates the effectiveness of the implemented controller as compared with the open-loop test result depicted in Fig. 3 which is also obtained using low-amplitude input. F. Discussions on System Performance The aforementioned experimental studies confirm the feasibility and effectiveness of the proposed control strategy. Due to a not-high resonant mode (82 Hz) of the employed system, the bandwidth of the servo system is limited. The performance

XU AND LI: MPDSMC OF A NANOPOSITIONING PIEZOSTAGE WITHOUT MODELING HYSTERESIS

of the adopted piezostage is worse than most of the commercial stages (e.g., piezo system model P-562.3CD provided by Physik Instrumente Co.) in terms of bandwidth and resolution. In contrast, it has almost identical dynamic behavior in the three working axes owing to a special parallel-kinematics design [43]. Moreover, it is observed that the resonant mode is suppressed by the implemented MPDSMC control. The observation spillover is successfully eliminated by the state observer at the cost of a lower response speed as reflected by the assigned pole values. On the other hand, the noise in sensor readings requires a further treatment to enhance the positioning resolution of the system. By adopting a closed-loop sampling rate of 1 kHz (which is sufficient for the investigated plant), the above simulations and experiments demonstrate the effectiveness of the proposed control approaches. The results also reveal the superiority of the proposed MPDSMC with PI-type sliding function over a MPDSMC with only P-type one. The controller parameters are not optimally designed, and the current amplitudes of steadystate errors (see Fig. 5) indicate that the performance of the MPDSMC may be improved to further reduce the tracking error down to the positioning resolution. For instance, an iterative control [44] may be employed to compensate the periodic errors. While this control study is preliminary and there is plenty of room for performance improvement, the enhancement of positioning precision for the nanopositioning system over the PID and regular DSMC control elaborated by the conducted investigations demonstrates the effectiveness of the proposed MPDSMC control scheme with PI sliding function and displays great potential for the future research. In the future, experiments will be conducted to demonstrate the capability of the piezostage for nanomanipulation tasks. VI. CONCLUSION The results presented in this paper show that a nanopositioning stage with piezoelectric actuation can be precisely controlled by using only a second-order mass-spring-damper system model without modeling the complicated hysteresis effects. Such a simple framework is very attractive for its easy implementation characteristics, which are enabled by the proposed model predictive discrete-time sliding mode controller with proportional-integral sliding function. It is found that the MPC strategy forces the system state to slide in the vicinity of boundary layer in an optimal the sliding surface with an manner, which eliminates the chattering phenomenon. The PI action in the MPDSMC drives the state tracking error to a in the sliding mode, which indicates boundary layer of a much lower tracking error in comparison with a commonly designed DSMC controller. The experimental results confirm that the proposed controller improves the tracking performance as compared with both PID and DSMC approaches. Besides, the hysteresis has been substantially reduced to 1.7% and a 50-nm dynamic resolution is achieved by the nanopositioning stage. Since the implementation of the controller does not require any bounds on uncertainties and unknown parameters of the system, the proposed controller can be easily extended to the control of micro-/nanopositioning stages driven by other types of smart actuators as well.

993

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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, China, in 2004 and 2008, respectively. He is currently an Assistant Professor of electromechanical engineering with the University of Macau. His current research interests include parallel robots, micro-/nanorobotics, micro-/nanopositioning, computational intelligence, smart materials, and structures, etc.

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 Full Professor of electromechanical engineering with the University of Macau, Macao, China, where he also directs the Mechatronics Laboratory. He has authored about 240 scientific papers, and has served 100 international conference program committees. His research interests include micro/nanomanipulation, nanorobotics, micromanipulator, mobile robot, modular robot, multibody dynamics, and control. Dr. Li currently serves as Technical Editor of the IEEE/ASME TRANSACTIONS ON MECHATRONICS, Associate Editor of the IEEE TRANSACTIONS ON AUTOMATION SCIENCE ENGINEERING, 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. He is a Macao chapter co-chair of IEEE System, Man, and Cybernetics Society, he is also a representative of IEEE SMC Society in IEEE Nano Technology Council. He is a member of the American Society of Mechanical Engineers (ASME).

Model Predictive Discrete-Time Sliding Mode Control of ...

both simulation and experimental studies. Finally, Section VI concludes this research. II. PROBLEM FORMULATION. A. Dynamics Modeling of a Nanopositioning System. The entire dynamics model of a nanopositioning system with nonlinear hysteresis can be established as follows: (1) where isthetimevariable,parameters , ...

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