IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 22, NO. 1, JANUARY 2014

Model Reference Adaptive Control With Perturbation Estimation for a Micropositioning System Qingsong Xu, Member, IEEE, and Minping Jia

Abstract— This brief presents a scheme of model reference adaptive control with perturbation estimation (MRACPE) for precise motion control of a piezoelectric actuation micropositioning system. One advantage of the proposed scheme lies in the fact that the size of tracking error can be predesigned, which is desirable in practice. A second-order nominal system is assumed, and the unmodeled dynamics and nonlinearity effect are treated as a lumped perturbation, which is approximated by a perturbation estimation technique. A dead-zone modification of the adaptive rules is introduced to mitigate the parameter drifts and to speed up the parameter convergence. Moreover, the proposed MRACPE scheme employs the desired displacement trajectory rather than the voltage signal as the reference input. The stability of the closed-loop control system is proved through Lyapunov stability analysis. Experimental studies show that the MRACPE is superior to conventional proportional-integralderivative control in terms of positioning accuracy for both setpoint and sinusoidal positioning tasks, which is enabled by a significantly enlarged control bandwidth. Index Terms— Adaptive control, disturbance, hysteresis, micro/nanopositioning, motion control, piezoelectric actuators.

I. I NTRODUCTION

M

ICRO/NANOPOSITIONING systems with piezoelectric actuation are widely employed in diverse applications where an ultrahigh-precision motion within a microscale workspace is needed [1]. Piezoelectric stack actuators (PSAs) are usually adopted owing to their merits in terms of high resolution, fast response, and high force density. Hence, the aforementioned applications can benefit more from PSAs than other kinds of actuators. However, the main challenge of using piezoactuated systems arises from the piezoelectric hysteretic nonlinearity. Under an open-loop voltage-drive approach, the hysteresis can induce a positioning error which is greater than 15% of the stroke. To tackle this issue, various control techniques have been developed to ensure the robustness of the system in the presence of dynamics model and hysteresis uncertainties [2], [3]. Considering that hysteresis modeling is often a complicated Manuscript received July 4, 2012; revised January 20, 2013; accepted February 6, 2013. Manuscript received in final form February 16, 2013. Date of publication March 7, 2013; date of current version December 17, 2013. This work was supported by the Macao Science and Technology Development Fund under Grant 024/2011/A and the Research Committee of University of Macau under Grant MYRG083(Y1-L2)-FST12-XQS. Recommended by Associate Editor G. Cherubini. Q. Xu is with the Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Macao, China (e-mail: [email protected]). M. Jia is with the School of Mechanical Engineering, Southeast University, Nanjing 211189, China (e-mail: [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.2013.2248061

work, some approaches without modeling the hysteresis effect have been presented, such as the integral resonant control [1], H∞ robust control [4], [5], sliding mode control [6], [7], and iterative learning control [8]. In particular, as compared with robust control approach, the adaptive control does not require a prior information about the bounds on uncertain or timevarying items. Hence, adaptive control paves a more straightforward way to the precision control of micropositioning systems. Nevertheless, only limited work has been dedicated to the extension of adaptive controllers to micropositioning system control. In the literature, it has been shown that the hysteresis effect can be described by the Krasnosel’skii and Pokrovskii (KP) model and compensated for using the inverse KP model [9]. However, because of the accuracy limitation of the identified model, there always exist differences between the predicted and exact hysteresis. By considering the uncertainties caused by the inversion-model-based compensation as input disturbances into the system, an adaptive hysteresis model has been established in [9] for a model reference control of active material actuators. Alternatively, by treating the hysteresis nonlinearity as an unknown disturbance to the ideal doubleintegral plant model, an active disturbance rejection control was employed to actively reject the nonlinearity based on an estimate provided by an observer [10]. However, seldom can a micropositioning system be represented by a simple doubleintegral model. Moreover, to develop a controller without modeling the complicated nonlinear effect, an adaptive robust controller was devised in [11]. Yet, uncertainty bounds are required to realize the control system. In previous work [12], a model reference adaptive control (MRAC) strategy was reported to compensate for the hysteresis effect of a micropositioning stage. Even though the adaptive controller was realized without modeling the hysteresis effect nor acquiring the uncertainty bounds, a Prandtl–Ishlinskii hysteresis model was required to convert the desired motion trajectory into a voltage input. More recently, a MRAC scheme based on hyperstability theory was presented for a piezoactuated system [13]. Nonetheless, a Bouc–Wen hysteresis model was still employed to identify the dynamics equation of the system. From a practical point of view, it is preferable to develop a MRAC scheme without modeling the complicated nonlinear effects. By considering the nonlinearity as perturbations to the system, several perturbation estimation methods have been reported to be integrated with MRAC schemes. To name a few, a MRAC with disturbance rejection strategy was presented for the systems that can be represented by parabolic or hyperbolic partial differential equations along with known disturbance model or constant disturbance [14].

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However, these scenarios are different from the situation in a PSA-actuated system where the unmodeled disturbance involving hysteresis nonlinearity is not constant. In addition, an adaptive perturbation approximator was outlined to estimate the structured perturbations of second-order systems online [15], and a perturbation estimation mechanism was proposed to design a MRAC controller for a class of multi-input, multi-output (MIMO) dynamic systems [16]. The aforementioned two methods were implemented on the basis of the assumption that the plant states are all measurable. Nevertheless, this assumption does not always hold for micropositioning systems which typically only offer displacement feedback. Moreover, the reference signals used in the preceding literature are all given in terms of voltage. For practical applications, a desired motion trajectory instead of voltage signal is predefined. To this end, the motivation of this brief is to devise a new MRAC scheme to compensate for the unmodeled hysteresis effect of a class of PSA-actuated systems which possess a second-order nominal model. Specifically, by treating the uncertainties as a lumped perturbation to the nominal secondorder system, a scheme of MRAC with perturbation estimation (MRACPE) is developed and validated on a micropositioning system. As compared with existing works, the proposed scheme allows the predesign of the maximum tracking error. It is capable of estimating the unmodeled perturbation of the system. Moreover, the scheme adopts the desired displacement trajectory rather than voltage as the reference signal. In the rest of this brief, the dynamics model for a PSAdriven system and the perturbation estimation technique are presented in Section II. The design procedure of a MRACPE controller is then detailed in Section III, where a dead-zone modification scheme of adaptive parameters is constructed. Section IV describes the experimental setup and model identification process. The controller performances are investigated in Section V through several comparative studies with respect to the traditional proportional-integral-derivative (PID) control approach. Section VI concludes the brief. II. DYNAMICS M ODEL AND P ERTURBATION E STIMATION A. Dynamics Modeling Considering the micropositioning stage as a mass-springdamper system, the dynamics model which integrates the components of the stage and PSA can be developed as follows: M x(t) ¨ + B x(t) ˙ + K x(t) = Du(t) + P(t)

(1)

where t is the time variable, x denotes the output displacement, and M, B, K , and D represent the equivalent mass, damping parameter, output stiffness, and displacement-voltage coefficient, respectively. In addition, the perturbation term P(t) describes the lumped effect of hysteresis, model parameter uncertainties, unmodeled high-frequency dynamics, and other disturbances. It is assumed that P(t) is bounded. Equation (1) can be rewritten into the form ˙ + α0 x(t) = β0 u(t) + f (t) x(t) ¨ + α1 x(t)

(2)

where α1 = B/M, α0 = K /M, β0 = D/M, and f (t) = P(t)/M.

The dynamics model (2) indicates that the PSA-actuated system is represented by a second-order linear model, and the hysteresis effect is involved in the unknown perturbations f (t) to the nominal system. The unmodeled nonlinearity and disturbances need to be compensated for in order to achieve a precise control of the position x. In the next section, a perturbation estimation technique is employed for the online estimation of the unmodeled perturbation involving the hysteresis nonlinearity. B. Perturbation Estimation A short review of the perturbation estimation method is outlined here since more details can be found in [17]. Consider a general nonlinear system given by x(n) = f(X) + f(X) + [B(X) + B(X)]u(t) + d(t)

(3)

where the vector x(n) = [x 1(n1 ) , x 2(n2 ) , . . . , x m(nm ) ]T ∈ m with (n ) T T T x i i ∈ , the mglobal state vector X = [X1 , . . . , Xm ] ∈ r = (r = i=1 n i ) with the state sub-vector Xi [x i , x˙i , . . . , x i(ni −1) ]T ∈ ni , and x i (i = 1, . . . , m) denoting (n ) m independent coordinates. In addition, the superscript in x i i refers to the n i th order of time derivative. The bounded perturbations in (3) can be combined together to form a perturbation vector (t) = f + Bu(t) + d(t) = x(n) − f − Bu(t).

(4)

The estimation for the perturbation is approximated as est (t) = x(n) cal − f − Bu(t − T )

(5)

where x(n) cal denotes a calculated state vector since the measurement values of higher order states of the system are not always available. Additionally, T is the sampling time interval, and u(t − T ) represents the control input in the previous time step. In practice, the sampling frequency is selected high enough to ensure that u(t) ≈ u(t − T ). (n) To obtain the state vector xcal , various types of state observers can be employed, such as the Luenberger observer. In this brief, the state vector is computed based on a backward difference equation (n−1) x(n−1) cal (t) − xcal (t − T ) . (6) T The above approach is employed because of its advantage in terms of computational efficiency. In this brief, given (2), the perturbation f can be expressed by ˙ + α0 x(t) − β0 u(t). (7) f (t) = x(t) ¨ + α1 x(t) (n)

xcal (t) =

Based on the perturbation estimation strategy, f is estimated as follows: fˆ(t) = x(t) ¨ + α1 x(t) ˙ + α0 x(t) − β0 u(t − T ).

(8)

It is observed from (8) that the full states (x, x, ˙ and x) ¨ of the system are required to implement the perturbation estimation. In a typical positioning system, only the position information (x) is provided by the displacement sensors. Hence, the other

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states (x˙ and x) ¨ are estimated by the aforementioned state calculator (6). In practice, the dynamics model parameters (α1 , α0 , and β0 ) can only be estimated with some degree of uncertainty. In Section III, a new MRACPE control scheme is devised to achieve precise position control in the presence of model uncertainties. III. MRACPE C ONTROL D ESIGN The basic idea of MRACPE lies in the establishment of a closed-loop controller with parameters updated online to change the response of the system. The control error is defined as the deviation of the system output from the desired response of a reference model. Based on this error, the controller parameters are adaptively updated. The control objective is to drive the parameters to converge to ideal values for matching the reference model response. A. MRACPE Controller Design Using the estimated perturbation (8), (2) becomes ˙ + α0 x(t) = β0 u(t) + fˆ(t). x(t) ¨ + α1 x(t)

(9)

For the purpose of motion tracking control, a reference model is defined as x¨m (t) + a1 x˙ m (t) + a0 x m (t) = b0 u d (t)

(10)

where x m is the reference model output, u d is the input to the model, and a1 , a0 , and b0 are positive parameters. For brevity, the time variable t is omitted hereafter in this section. Defining the tracking error as e = x − xm

(11)

then, subtracting (10) from (9) leads to an error dynamics equation ˙ 0 e = β0 u −b0u d +(a1 −α1 )x˙ +(a0 −α0 )x + fˆ. (12) e¨ +a1e+a Assigning a vector of state error E = [e e˙]T , (12) can be expressed into the state-space form E˙ = AE + β0 Bu + where

(13)

0 1 0 0 A= , B= , = −a0 −a1 1 δ

(14)

with δ = (a1 − α1 )x˙ + (a0 − α0 )x − b0 u d + fˆ. It has been shown that, if all the eigenvalues of the matrix A have negative real parts, then all solutions of E to (13) will tend to zero as t → ∞ [18]. Moreover, (13) is asymptotically stable since, given any symmetric positive definite matrix Q, there exists a symmetric positive definite matrix P which is the unique solution to the linear equation AT P + PA = −Q.

(15)

In addition, an auxiliary control item is defined as follows: eˆ = ET PB.

(16)

Theorem 1: For the system described by (9) with a reference model (10), the tracking error e defined by (11) satisfies lim e(t) = 0 if the MRACPE control law is given by

t →∞

u = k0 u d + k1 x + k2 x˙ + k3 fˆ

(17)

where the adaptive laws for the control parameters k0 , k1 , k2 , and k3 are chosen as k˙0 k˙1 k˙2 k˙3

= −η0 eˆ u d = −η1 eˆ x

(18) (19)

= −η2 eˆ x˙ = −η3 eˆ fˆ

(20) (21)

where η0 , η1 , η2 , and η3 are all positive constants. Proof: To demonstrate the stability of the proposed controller, a Lyapunov candidate function is defined as follows: 1 1 V = ET PE + (β0 k0 − b0 )2 2 2η0 β0 1 + (β0 k1 + a0 − α0 )2 2η1 β0 1 + (β0 k2 + a1 − α1 )2 2η2 β0 1 + (β0 k3 + 1)2 . (22) 2η3 β0 In view of (13) and (16), the time derivative of the first item in (22) can be expressed as V˙1 = ET PE˙ = ET PAE + ET P(β0 Bu + ) 1 = ET (AT P + PA)E + ET PB(β0 u + δ) 2 1 ˆ 0 u + δ). (23) = − ET QE + e(β 2 Hence, the time derivative of the Lyapunov function is k˙0 1 ˆ 0 u + δ) + (β0 k0 − b0 ) V˙ = − ET QE + e(β 2 η0 k˙1 k˙2 + (β0 k1 + a0 − α0 ) + (β0 k2 + a1 − α1 ) η1 η2 k˙3 + (β0 k3 + 1). (24) η3 Next, substituting (17) into (24) and rearranging the items leads to k˙0 1 ˆ d+ V˙ = − ET QE + eu (β0 k0 − b0 ) 2 η0 k˙1 + ex ˆ + (β0 k1 + a0 − α0 ) η1 k˙2 (β0 k2 + a1 − α1 ) + eˆ x˙ + η2 k˙3 (25) + eˆ fˆ + (β0 k3 + 1). η3 Then, inserting the adaptive laws (18)–(21) into (25) and considering (15) allows the derivation 1 V˙ = − ET QE ≤ 0. (26) 2

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Fig. 2.

Fig. 1.

Block diagram of the proposed MRACPE.

Thus, the stability of the closed-loop control system is proved. B. Dead-Zone Modification of Adaptive Laws In practice, because of the noises in the tracking error, the control parameters (k0 , k1 , k2 , and k3 ) drift as demonstrated by the experiments carried out in Section V later. In order to alleviate the drift issue, a dead-zone modification scheme [19] is adopted to turn off the adaptation processes once the tracking error attains a predefined small value. Specifically, the adaptive laws (18)–(21) are revised as follows: ˙k0 = −η0 eˆ u d , if |e| > ε (27) 0, if |e| ≤ ε −η1 eˆ x, if |e| > ε (28) k˙1 = 0, if |e| ≤ ε −η2 eˆ x, ˙ if |e| > ε (29) k˙2 = 0, if |e| ≤ ε −η3 eˆ fˆ, if |e| > ε k˙3 = (30) 0, if |e| ≤ ε where ε is the assigned dead-zone size, which enables the restriction of the tracking error e. Generally, the smaller the ε, the lower the tracking errors. The ε value can be tuned by the trial-and-error approach through experimental studies. C. Overview of the Control Scheme Recalling the expression for the estimated perturbation in (8), the control law (17) can be further written as u(t) = k0 u d (t) + (k1 + k3 α0 )x(t) ˙ + k3 x(t) ¨ +(k2 + k3 α1 )x(t) −k3 β0 u(t − T ).

(31)

The block diagram of the control scheme is illustrated in Fig. 1. It is observed that the controller has two inputs, i.e., the desired control voltage u d and displacement x of the system; the output is the voltage u that will be used to drive the PSA actuator. By using a voltage signal as the reference input, several MRAC schemes have been realized in previous work [15], [16]. However, for practical applications, a desired displacement trajectory instead of voltage is predefined for the

Experimental setup of the PSA-driven micropositioning system.

stage. Thus, once a displacement is given, the corresponding desired voltage needs to be determined. In this brief, this mapping is implemented by resorting to an inverse model of the reference model (10), which is expressed by the following transfer function: 1 Ud (s) = (s 2 + a1 s + a0 ) (32) G −1 m (s) = X m (s) b0 where Ud and X m represent the Laplace transform of u d and x m , respectively. IV. E XPERIMENTAL S ETUP AND C ONTROLLER S ETUP In this brief, the effectiveness of the designed MRACPE control scheme is verified by conducting a series of experimental studies on a PSA-driven micropositioning system. A. Experimental Setup The experimental setup of a custom-built PSA-actuated micropositioning system is depicted in Fig. 2. Actually, a PSA and a voice coil motor are used to construct a dualstage micropositioning system as detailed in [20]. In this brief, the PSA-driven inner flexure stage is employed to verify the proposed control scheme. The PSA is connected to a high-voltage amplifier with an adjustable gain value of 10. The output position of the stage is measured by a laser displacement sensor (model: LK-H055, from Keyence Corp.). In addition, a Natural Instruments (NI) cRIO-9022 real-time controller combined with cRIO-9118 reconfigurable chassis is adopted to implement the control algorithm. The chassis is equipped with analog I/O modules (NI-9215 and NI-9263) for the acquisition of sensor readings and production of excitation voltage signals. The NI cRIO-9118 chassis contains a field-programmable gate array (FPGA) core, and the associated cRIO-9022 realtime controller communicates with a personal computer (PC) via Ethernet port. The control algorithm is programmed using the NI LabVIEW software. With the proposed MRACPE control algorithm, the control hardware enables a maximum closed-loop sampling frequency of 100 kHz. In this brief, a sufficient sampling frequency of 5 kHz is adopted. B. Statics Test and Dynamics Model Identification First, the motion range of the micropositioning system is tested by applying a sinusoidal voltage signal with the amplitude of 10 V and frequency of 0.2 Hz. The results are illustrated in Fig. 3, which shows that a motion range over 92 μm is obtained. In addition, the open-loop output

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Fig. 3. Open-loop hysteresis loop obtained with a 0.2-Hz sinusoidal reference input.

versus input curve exhibits a hysteresis width of 16.5%. The presence of significant hysteresis nonlinearity necessitates the development of a control technique in order to achieve precise positioning. Next, the dynamics model parameters α1 , α0 , and β0 of the nominal plant model (2) are identified by means of frequency response approach. Specifically, swept sine waves with the amplitude of 0.5 V and frequency range of 1–650 Hz are produced to drive the PSA via the high-voltage amplifier. Both the excitation voltage and displacement output signals are acquired to generate the frequency response of the system as shown in Fig. 4. From the experimental data, a secondorder model is identified, and the model parameters are given in Table I. Fig. 4 indicates that the second-order model is capable of approximating the frequency response up to 600 Hz, which covers the resonant frequency of 470 Hz. In order to capture the system dynamics at frequencies higher than 600 Hz, a model of higher order is needed. In this brief, a simple secondorder model is employed to demonstrate the effectiveness of the proposed control scheme, and the residual modes at higher frequencies are considered as disturbances to be suppressed by the MRACPE controller.

Fig. 4. Frequency responses obtained by experiment and the identified second-order model. TABLE I PARAMETERS OF THE P LANT M ODEL AND D ESIGNED C ONTROLLER Parameter α0 α1 β0 a0 a1 b0 η0 η1 η2 η3 ε Q

Value 8.721 × 106 2.953 × 102 4.404 × 107 8.721 × 106 1.477 × 103 4.404 × 107 1 × 10−2 1 × 10−4 1 × 10−9 1 × 10−17 0.7 diag{3 × 105 , 1 × 10−3 }

represent the desired and actual system output at the kth time step, and K p , K i , and K d denote the proportional, integral, and derivative gains, respectively. The PID gains are initially tuned by simulations using the Ziegler–Nichols (Z-N) method and then finely adjusted through experimental studies to eliminate the overshoot: K p = 0.01, K i = 0.0023, and K d = 0.029. A sampling time of T = 0.0002 s is adopted in both MRACPE and PID control.

C. Controller Setup For experimental investigations, the reference model is chosen with parameters (a0 , a1 , and b0 ) as described in Table I. The assigned parameters of the MRACPE controller are also shown in Table I. As depicted in Fig. 1, by passing the desired position trajectory x d through the reference model inverse G −1 m , a reference voltage signal u d is generated. Excited by u d , an output x m = x d is then produced by the reference model. For a comparative study, the popular PID controller is also implemented. A digital PID control strategy is employed as follows: u k = K p ek + K i

k

ek + K d (ek − ek−1 )

(33)

j =0

with the positioning error ek = x dk − x k , where x dk and x k

V. E XPERIMENTAL R ESULTS AND D ISCUSSION A. Set-Point Positioning Test First, the set-point positioning capability of the micropositioning system is examined. By commanding the output platform from home position to 20 μm, the positioning results using PID and MRACPE are depicted in Fig. 5. Both controllers are tuned to achieve a quick response without overshoot. For a quantitative comparison, the performances are shown in Table II. It is observed that the PID scheme produces a slightly quicker response since its 95% settling time is 0.006 s smaller than that of SMCPE. Even so, the MRACPE delivers a more accurate positioning, as it creates a steady-state root mean square (RMS) error which is 7% lower than the PID result.

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(a) Fig. 5. Experimental results of set-point positioning using PID and MRACPE controllers.

TABLE II S ET-P OINT AND S INUSOIDAL P OSITIONING R ESULTS OF PID AND MRACPE C ONTROLLERS Performance Set-point positioning (20 μm) 95% settling time (s) RMS steady-state error (μm)

PID

MRACPE

0.025 0.275

0.031 0.256

Sinusoidal positioning (10 Hz) Maximum steady-state error (%) RMS steady-state error (%)

39.20 25.60

6.51 2.96

(b) Fig. 7. (a) 10-Hz sinusoidal motion tracking results of PID and MRACPE (ε = 1.6) controllers. (b) Positioning errors.

B. Sinusoidal Positioning Test

Fig. 6. Adaptation processes of the MRACPE (ε = 0.7) control parameters for set-point positioning.

Additionally, the adaptation processes of the four control parameters of the MRACPE scheme are depicted in Fig. 6. The dead-zone parameter is assigned as ε = 0.7 to limit the positioning errors within ±0.7 μm, which is the noise level of the displacement sensor. It is seen that, as the response approaches the steady state, the controller parameters converge to the corresponding ideal values. It is found that a quicker response speed of MRACPE can be obtained by optimally tuning the control gains (η0 , η1 , η2 , and η3 ).

Then, a 10-Hz, 20-μm-amplitude sinusoidal positioning control is performed by employing the two controllers, and the results are shown in Fig. 7. It is found that the PID algorithm leads to identical responses in each cycle. Concerning the MRACPE, after the transient progress within the first few cycles, it gives identical responses afterwards. Additionally, the PID control produces the maximum and RMS steady-state positioning errors of 7.840 and 5.119 μm, which account for 39.20% and 25.60% of the positioning range, respectively. In contrast, the MRACPE results in the maximum and RMS steady-state errors of 1.302 and 0.591 μm, i.e., 6.51% and 2.96% of the positioning range, respectively. The sinusoidal positioning results are summarized in Table II. As compared with PID control, the proposed scheme improves the positioning accuracy by 83% and 88% in terms of the maximum and RMS errors, respectively. The sinusoidal positioning results are generated by the same controllers as used in the aforementioned set-point positioning tests, except that ε = 1.3 is reassigned in MRACPE. With the PID gains optimally tuned (see PID 2 in Fig. 7), the

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Fig. 8. Positioning errors of the MRACPE with different dead zones ε for sinusoidal positioning.

maximum and RMS errors of 25.96% and 14.44% are produced, respectively, which are still about four times worse than the MRACPE results. Hence, the superiority of MRACPE over PID control is evident from the experimental results. Furthermore, to discover the influence of the dead-zone selection on the MRACPE performance, the positioning results with three different dead zones (2.3, 1.6, and 1.3 μm) are examined. The MRACPE positioning errors and the parameter adaptation procedures are shown in Figs. 8 and 9, respectively. Fig. 8 indicates that the maximum steady-state positioning errors are restricted by the assigned dead-zone values. The smaller the dead zone, the better the positioning accuracy. In view of Fig. 9, it is found that the controller parameters k0 , k1 , k2 , and k3 arrive at the steady states quickly in two of the three case studies (ε = 2.3 and 1.6). Concerning the case with a smaller dead-zone value (ε = 1.3), the convergence speed of the controller parameters is slower as a result of the parameter drifts. Further experimental tests show that the parameters get saturated finally. Hence, a tradeoff between the tracking accuracy and parameter convergence speed is needed to assign the dead-zone value. C. Control Bandwidth Test In addition, the control bandwidth of the PSA-actuated system is tested by applying the 20-μm sinusoidal signal with the frequency varying from 1 to 100 Hz. The closed-loop frequency responses are shown in Fig. 10. It is observed that there are large phase lags (over 90°) within the ordinary 3-dB bandwidth, which leads to large tracking errors. Hence, in this brief, the closed-loop control bandwidth is defined as the frequency at which the phase is lagged 30°. With the PID and MRACPE control, the 30° lag bandwidth of 12.8 and 79.3 Hz are achieved, which are equivalent to 2.7% and 16.9% of the resonant frequency, respectively. These cutoff frequencies correspond to small errors of 0.43 and 1.72 dB for the magnitude responses of PID and MRACPE, respectively. As compared with the PID algorithm, the

Fig. 9. Adaptation processes of the control parameters of MRACPE with different dead zones ε for sinusoidal positioning.

proposed MRAPCE improves the control bandwidth by over six times. D. Discussion The aforementioned experimental results confirm the effectiveness of the proposed control scheme for a PSA-actuated micropositioning system. Since the scheme is implemented without modeling the hysteresis nonlinearity, the results demonstrate the efficiency of the proposed idea for disturbance suppression. Considering that the nominal plant model is used to develop the perturbation estimator for the estimation of unmodeled dynamics and uncertainties in the control design, the results also reveal the feasibility of the identified second-order model for the positioning application. Fig. 10 indicates that, with increasing reference input frequency, the performance of PID control drops more quickly than that of the proposed MRACPE control. This is one of the reasons why a 10-Hz sinusoidal reference input is adopted in the aforementioned comparative study. Actually, the MRACPE achieves an RMSE less than 10% of positioning range for a higher speed tracking of reference input with the frequency of 40 Hz. It is noticed that, although MRACPE produces a slightly lower transient response than PID control, it achieves more accurate positioning in both set-point and sinusoidal tracking tasks. Another advantage of the MRACPE is that its positioning error size can be predefined by assigning the deadzone (ε) value. Experimental results reveal that the smaller the dead zone, the slower the convergence speed of controller parameters. Even so, the convergence speed of the parameters does not affect the steady-state tracking error since the latter is restricted by the assigned ε value. The smaller the ε, the more accurate the positioning result. However, too small a ε (e.g., ε = 1.0) produces the maximum positioning error exceeding the ε value, or even causes instability of the system due to the drifts of controller parameters, which are mainly induced by the noises of the displacement sensor. Hence, a tradeoff between the tracking accuracy and stability is

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R EFERENCES

Fig. 10. Control bandwidth test results of the micropositioning system with PID and MRACPE controllers.

required to choose the dead-zone value. The tracking errors can be reduced by employing displacement sensors with lower noise level. Alternatively, a state observer may be adopted to estimate the state values. Nevertheless, the employment of state observer complicates the control design process at the same time. Besides, the control gains of MRACPE are not optimally tuned. An optimal tuning of the four control gains is a possible solution to further improve the positioning results. Hence, there is plenty of room for the performance improvement of the proposed controller. Even so, the enhancement of positioning accuracy for the micropositioning system over the traditional PID as elaborated by the conducted experiments validates the effectiveness of the proposed MRACPE scheme and exhibits a potential for the future work. Moreover, the robust controller design procedure [5] will be employed to achieve high positioning accuracy at high speeds. VI. C ONCLUSION The main contribution of this brief lies in the proposal of a new MRACPE control scheme for a PSA-actuated micropositioning system without modeling the hysteretic nonlinearity effect. The controller was designed based on Lyapunov stability analysis, and its performance was verified by a series of experimental studies. Results indicated that the MRACPE with dead-zone modification of the adaptive parameters is capable of further suppressing the disturbances and substantially mitigating the positioning errors as compared with conventional PID control. The MRACPE enables over six times increase of control bandwidth in comparison with PID algorithm. Moreover, the dead zone allows the specification of the maximum tracking error for the MRACPE scheme, which is attractive for practical applications. Due to an uncomplicated structure, the proposed control scheme can be extended to position control of other types of micro/nanopositioning systems as well.

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