Identification and Compensation of Piezoelectric ... - Semantic Scholar
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 60, NO. 9, SEPTEMBER 2013
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Identification and Compensation of Piezoelectric Hysteresis Without Modeling Hysteresis Inverse Qingsong Xu, Member, IEEE
Abstract—This paper presents a new approach for hysteresis identification and compensation of piezoelectric actuators by resorting to an intelligent hysteresis model. In particular, a least squares support vector machine (LSSVM)-based hysteresis model is developed and used for both purposes of hysteresis identification and hysteresis compensation. By this way, the hysteresis inverse is not needed in the feedforward hysteresis compensator since the hysteresis model is directly used. To establish the LSSVM model, the problem of how to select input variables to convert the multivalued mapping into a single-valued one is addressed. The effectiveness of the presented idea is validated by a series of experimental studies on a piezoactuated system. Results show that the proposed approach is superior to the Bouc–Wen-model-based one in terms of both hysteresis modeling and compensation. The reported method is more computational effective than existing model-based hysteresis compensation approaches, and it is extensible to other smart actuator systems as well. Index Terms—Hysteresis model, motion control, piezoelectric actuator.
micro-/nanopositioning,
I. I NTRODUCTION
S
MART ACTUATORS based on smart materials (e.g., piezoelectric materials, shape memory alloys, magnetostrictive materials, etc.) are popularly employed for actuation in various precision systems dedicated to micropositioning, micromanipulation, and microassembly. Particularly, as a typical smart actuator, the piezoelectric actuator is attractive in both fields of academia and industry due to the merits of (sub)nanometer positioning resolution, rapid response speed, and self-sensing capability. Although numerous previous works have concentrated on research and applications of piezoelectric actuators [1]–[4], the nonlinear piezoelectric effect (in terms of hysteresis and creep) still remains a challenging problem nowadays. The hysteresis effect can be greatly alleviated by using a charge-driven approach or a capacitor insertion method [5], [6]; however, it is at the cost of stroke reduction. Hence, voltage actuation is widely adopted in practice. In order to identify
Manuscript received December 30, 2011; revised March 5, 2012 and April 25, 2012; accepted June 18, 2012. Date of publication July 6, 2012; date of current version May 2, 2013. This work was supported in part by the Macao Science and Technology Development Fund under Grant 024/2011/A and in part by the Research Committee of the University of Macau under Grant MYRG083(Y1-L2)-FST12-XQS. The author is with the Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Taipa, Macao, 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/TIE.2012.2206339
the hysteresis behavior, the establishment of a suitable hysteresis model is required. For instance, hysteresis identifications using Preisach model [7], [8], Prandtl–Ishlinskii model [9], Bouc–Wen model [10], etc., have been widely carried out. On the other hand, to fulfill the requirements of ultrahigh-precision positioning, the piezoelectric hysteresis has to be suppressed by implementing an appropriate control scheme. Generally, the existing hysteresis compensation schemes fall into two categories in terms of hysteresis model-free and hysteresis model-based methods. Concerning the first category, its main property lies in that no hysteresis model is required. The unmodeled hysteresis is considered as an uncertainty or a disturbance [11] to the nominal system, which is tolerated by an advanced robust or adaptive controller. For instance, the applications of slidingmode control [12]–[15], H∞ robust control [16], [17], fuzzy logic control [18]–[20], and neural network control [21] have been successfully reported. Generally, in the second category, a hysteresis model (e.g., Preisach model) is identified, and an inverse hysteresis model is constructed to implement a feedforward (FF) compensator [22], [23]. It has been shown that the inverse-model-based compensation can achieve an accurate positioning, whereas the result is very sensitive to the model accuracy [24], [25]. Fortunately, the combination of FF with feedback (FB) control can be adopted to suppress the hysteresis as well as creep effects [26]. Although the compensation of hysteresis nonlinearity without modeling the hysteresis effect is realizable by designing advanced FB controllers, extensive knowledge on control engineering is needed to design such kinds of control strategies. It is commonly believed that an FF compensator based on a simple hysteresis model (with fewer model parameters) in combination with a simple FB controller (e.g., PID control) makes it more feasible to suppress the hysteresis nonlinearity. The reason lies in that the latter allows the relief of burden on developing complicated modern controllers. For example, it has been shown that, by modeling the hysteresis with the Bouc–Wen model, an FF compensator combined with a PID FB controller is capable of compensating the nonlinear hysteresis effectively [27]. A review of model-based hysteresis compensation approaches reveals that most of the existing works compensate for the hysteresis effect by employing an inverse hysteresis model. Thus, both a hysteresis model and an inverse hysteresis model are required for the hysteresis identification and compensation purposes. More recently, it has been shown that it is possible to mitigate the hysteresis effect by adopting a Bouc–Wen hysteresis model directly while without using the hysteresis inverse [28]. However, only the quasi-static (low frequency) hysteresis was treated in previous work [28].
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Moreover, it has been recognized that the piezoelectric hysteresis effect is rate dependent [29]. The hysteresis behavior is dependent not only on the amplitude but also on the frequency of input voltage signals. Previous works reported that the artificial neural network (ANN) provides an efficient way to model the hysteresis nonlinearity [30], [31]. Since support vector machine (SVM) is superior to ANN in terms of global optimization and higher generalization capability, SVM is widely employed to estimate nonlinear system models accurately [32]. As a variant of SVM, least squares SVM (LSSVM) simplifies the regression to a problem that can be easily solved from a set of linear equations [33]. Thus, LSSVM has a relatively low complexity and is more computationally efficient than the standard SVM. Another advantage of LSSVM lies in that it has fewer parameters to tune. Typically, LSSVM only treats the problem of single-valued mapping between the input and output. However, due to the hysteresis nonlinearity, an input voltage corresponds to multiple position outputs for a piezoactuated system. Hence, one of the challenges in identifying the hysteresis behavior with LSSVM is how to convert the multivalued mapping problem into a single-valued one. Previously, a one-to-one mapping was constructed by introducing the current input value and input variation rate as one data set [30], [34]. Nevertheless, in the case that the input data are accompanied with noises, the variation rate is not smooth and will induce modeling error. Moreover, it is unknown how many orders of the variation rates are sufficient to establish the mapping. In addition, a regression model of hysteresis was developed in [35] by employing the current and previous inputs and previous outputs as exogenous inputs to transform the multivalued mapping into a single-valued one. However, the foregoing research took the position and voltage as output and input variables and developed both direct and inverse hysteresis models between the variables, which indicates a time-consuming procedure. The goal of this paper is to identify and compensate the rate-dependent piezoelectric hysteresis by using an intelligent hysteresis model while without modeling the hysteresis inverse. Particularly, an LSSVM-based hysteresis model is established, and an FF compensator is developed based on the sole model, which provides a computational effective way in hysteresis compensation. The effectiveness of the proposed idea is verified by a series of experimental studies. It is shown that the established LSSVM model is more effective than Bouc–Wen model in terms of hysteresis identification as well as hysteresis compensation performances. The rest of this paper is organized as follows. The dynamics modeling of piezoactuated systems with hysteresis behavior is outlined in Section II, where the idea of establishing a hysteresis model for both purposes of identification and compensation is introduced. In Section III, the hysteresis behavior is identified based on the LSSVM technique by formulating a nonlinear regression problem. Several experimental studies are carried out in Section IV to demonstrate the effectiveness of the presented approach in comparison with the Bouc–Wen-model-based one. Based on the identified models, hysteresis compensation is implemented in Section V along with comparative studies. Section VI concludes this paper.
II. M ODELING OF DYNAMICS W ITH H YSTERESIS B EHAVIOR A. Dynamics Modeling With Bouc–Wen Hysteresis Owing to a fewer number of parameters, the Bouc–Wen model has been extensively applied in piezoelectric hysteresis modeling. The entire dynamics model of a piezoactuated system can be established as follows [10], [36]: M y¨(t) + B y(t) ˙ + Ky(t) = K [Du(t) − H(t)]
(1)
˙ H(t) = αDu(t) ˙ − β |u(t)| ˙ H(t) − γ u(t) ˙ |H(t)|
(2)
where t is the time variable; parameters M , B, K, and y represent the mass, damping coefficient, stiffness, and displacement response of the piezoactuated system, respectively; D is the piezoelectric coefficient; u denotes the input voltage; and H indicates the hysteretic loop in terms of displacement whose magnitude and shape are determined by parameters α, β, and γ. Once the dynamics parameters M , B, and K are generated, the four parameters (D, α, β, and γ) of the Bouc–Wen model can be identified by minimizing an objective function f (D, α, β, γ) =
N 1 (yi − yBWi )2 N
(3)
I=1
where N denotes the total number of samples and yi − yBWi represents the residual error of the ith sample which is the discrepancy between Bouc–Wen model output (yBWi ) and experimental result (yi ). Given the input voltage u and Bouc–Wen hysteresis model H, the output displacement of the system is governed by model (1). In contrast, for a given desired displacement value yd , the required input voltage can be obtained from (1) uFF (t) =
1 [M y¨d (t) + B y˙ d (t) + Kyd (t) + KH(t)] . KD (4)
It is observed that the FF control signal (4) is generated by using the hysteresis term H while without solving the inverse hysteresis model. A block diagram implemented with Matlab and Simulink software is given in [10]. B. Dynamics Modeling With Intelligent Hysteresis Model Inspired by the hysteresis compensation using Bouc–Wen model which does not model the hysteresis inverse, an intelligent model is proposed hereinafter. First, the dynamics model (1) of the system is rewritten into the form ˙ + ωn2 y(t) = du(t) + h(t) y¨(t) + 2ξωn y(t)
(5)
where ξ and ωn denote the damping ratio and natural frequency of the piezoactuated system, respectively. d is a positive parameter, and h represents the hysteresis effect in terms of acceleration.
XU: IDENTIFICATION AND COMPENSATION OF PIEZOELECTRIC HYSTERESIS
Then, in consideration of (5), the hysteresis term can be derived h(t) = y¨(t) + 2ξωn y(t) ˙ + ωn2 y(t) − du(t).
(6)
However, the system parameters and output positions are not always available. Under such situations, it is necessary to establish an intelligent model to identify and compensate for the hysteresis term h. ˆ is obOnce an intelligent model of the hysteresis term h tained, an FF hysteresis compensator can be constructed to produce a desired output position yd 1 ˆ y¨d (t) + 2ξωn y˙ d (t) + ωn2 yd (t) − h(t) uFF (t) = (7) d which uses the hysteresis model directly while without solving the inverse hysteresis model. In the following section, a hysteresis model is established based on LSSVM technique. III. H YSTERESIS M ODELING U SING LSSVM It has been reported that the hysteresis has multivalued and nonsmooth features [13]. However, the introduced LSSVM is capable of accurately estimating high-dimensional smooth functions. Thus, only the multivalued nonlinearity of the hysteresis is treated in this paper. Under the hypothesis that the input command is smooth, an LSSVM regression model is established hereinafter to describe the piezoelectric hysteresis behavior. A. Establishment of Regression Model
(8)
with xk = [uk , . . . , uk−m , yk , . . . , yk−n , hk−1 , . . . , hk−l ]
with the given training data set {xk , hk }N k=1 where N represents the number of training data set, xk ∈ Rm+n+l+2 is an input vector as shown in (9), and hk ∈ R are the output data. Additionally, w is a weight vector, ϕ(·) denotes a nonlinear mapping from the input space to a higher dimensional feature space, and b is the bias. The LSSVM approach formulates the regression as an optimization problem in the primal weight space. Then, the conditions for optimality are obtained by solving a series of partial derivatives, which are used to construct the dual formulation as
0 1N
1T N Ω + Γ−1 IN
(9)
ˆ k denotes the hysteresis term predicted by LSSVM at where h the current time instant k. uk−1 , yk−1 , and hk−1 represent the input voltage, output position, and hysteresis term at previous time instant k − 1, respectively. In addition, m ≥ 0, n ≥ 0, and l ≥ 1 define the order of the model. The selection of the three orders is discussed later in Section IV-D.
It is known that the LSSVM maps the input data into a highdimensional feature space and constructs a linear regression function therein [34]. The unknown hysteresis function is approximated by ˆ h(x) = wT ϕ(x) + b
(10)
b 0 = α h
Ωkj = ϕ(xk )T ϕ(xj ) = K(xk , xj ),
(11)
k, j = 1, 2, . . . , N (12)
where K(·) is a predefined kernel function. The role of the kernel function is to avoid explicit computation of the map ϕ(·) in dealing with the high-dimensional feature space. Once b and α are calculated from (11), the solution for the regression problem can be obtained as N
αk K(x, xk ) + b
(13)
k=1
where K(·) is the kernel function satisfying Mercer’s condition, xk is the training data, and x denotes the new input data. By adopting the radial basis function as kernel function x − xk 2 K(x, xk ) = exp − σ2
(14)
where σ > 0 denotes the width parameter (which specifies the kernel sample variance σ 2 ) and · represents the Euclidean distance, the LSSVM model for the hysteresis model estimation becomes ˆ h(x) =
N k=1
B. LSSVM Modeling
where α = [α1 , α2 , . . . , αN ]T is called the support vector. The support values are αk = Γek with Γ ∈ R denoting the regularization factor. In addition, 1N = [1, 1, . . . , 1]T , h = [h1 , h2 , . . . , hN ]T , and IN is an identity matrix. Moreover, the kernel trick is employed to derive
ˆ h(x) =
Different from the scenario in [35], it is observed from (6) that the output variable is the hysteresis term h, whereas both input voltage u and output position y appear as input variables. By employing LSSVM, a nonlinear regression model is formulated to capture the hysteresis behavior ˆ k = f (xk ) h
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x − xk 2 αk exp − σ2
+ b.
(15)
With the assigned regularization parameter Γ and kernel parameter σ, the objective of the training process is to determine the support values αk and the bias b. The high generalization ability of the LSSVM model relies on the appropriate tuning of the two hyperparameters (Γ and σ). In this paper, the leave-oneout cross-validation approach is adopted to infer the values of the hyperparameters.
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TABLE I K EY PARAMETERS OF THE F OUR -L AYER P IEZOELECTRIC B IMORPH
Fig. 1. Experimental setup of a piezoactuated system. Analog voltage signal is produced by one DAC channel of NI USB-6259 board and then amplified by the voltage amplifier to drive the piezoelectric bimorph. The output position of the bimorph is measured by a laser sensor head. The analog output voltage signal of the sensor is conditioned by the conditioner and then acquired by one ADC channel of USB-6259 board. The board is connected to a computer (running LabVIEW software) via a universal serial bus (USB) interface. A brass-reinforced inner layer is embedded in the four-layer bimorph.
IV. E XPERIMENTAL S TUDIES ON H YSTERESIS I DENTIFICATION In this section, the hysteresis identifications based on Bouc–Wen model and LSSVM approach are carried out experimentally on a test bed. A. Experimental Setup The experimental setup is shown in Fig. 1. A four-layer piezoelectric bimorph actuator (model: T434-A4-201, from Piezo Systems, Inc.) with the dimension of 28 × 5 × 0.86 mm3 is selected. The actuator is fabricated from the piezoelectric material of industry type 5A (Navy Type II), and its key parameters are shown in Table I. A USB-6259 board (from National Instruments (NI) Corporation) with 16-b channels of digital-toanalog converter (DAC) and analog-to-digital converter (ADC) is adopted to produce an analog voltage, which is then amplified by a high-voltage amplifier (model: EPA-104, from Piezo Systems, Inc.) to provide a voltage of ±200 V for the drives of the piezoelectric actuator. The output displacement at the end point of piezoelectric bimorph is measured by a laser displacement sensor (model: LK-H055, from Keyence Corporation). The analog output voltage of the sensor signal conditioner is acquired by a PC through one ADC channel of the USB-6259 board. Moreover, LabVIEW software is employed to implement a real-time control of the piezoactuated system. B. Dynamics Model Identification Before developing the hysteresis models, a linear dynamics model of the system plant is identified by the swept-sine approach. Specifically, sine waves with the amplitude of 0.5 V and the frequency range of 1–1000 Hz are produced to drive the piezoelectric bimorph. The position responses of the piezoelectric actuator are recorded using a sampling rate of 2 kHz.
Fig. 2. Frequency response of the system plant obtained by experiment and identified second-order model.
The 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 by experiments and the identified model (16) are compared in Fig. 2, which shows that the first resonant mode occurs around 404 Hz G(s) =
1.247 × 108 . s2 + 1.847s + 6.477 × 106
(16)
It is observed that the identified second-order model matches the system dynamics well in terms of magnitude and phase at frequencies below 600 and 100 Hz, respectively. The fact that the second-order model cannot properly describe the phase behavior about 100 Hz indicates that a model of much higher order is required to capture the high-frequency dynamics accurately. Here, a simple second-order model is employed to demonstrate the effectiveness of the proposed hysteresis identification scheme. C. Bouc–Wen Model Results Concerning the dynamics parameters, the mass is taken as the nominal value M = 1.8 × 10−3 kg as shown in Table I. Additionally, by neglecting the nonlinear term H in (1), taking the Laplace transform and comparing the parameters with (16), the damping coefficient and stiffness values are calculated as B = 3.3 × 10−3 N · s/m and K = 1.1659 × 104 N/m, respectively. Other parameters are identified as follows. 1) Bouc–Wen Model Identification: To identify the hysteresis model, various types of signals can be utilized. In this
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Fig. 3. Results of the identified Bouc–Wen model. (a) Input voltage. (b) Experimental result and Bouc–Wen model output. (c) Bouc–Wen model output errors. (d) Displacement–voltage hysteresis loops.
Fig. 4. Testing Result 1 of the Bouc–Wen model. (a) Input voltage. (b) Experimental result and Bouc–Wen model output. (c) Bouc–Wen model output errors. (d) Displacement–voltage hysteresis loops.
paper, to illustrate the proposed idea, an input voltage signal [see Fig. 3(a)] is chosen hereinafter for the purpose of model identification
u(t) = 5e−0.13t cos(3πte−0.09t − 3.15) + 1.0 .
(17)
By applying the signal to the high-voltage amplifier, the position output is generated as shown in Fig. 3(b). By setting a time interval of 0.02 s, 500 training data sets are obtained as shown in Fig. 3(a) and (b). Then, the Bouc–Wen
model is identified by optimizing the four parameters to minimize the fitness function (3). Specifically, the particle swarm optimization is adopted for the function minimization, and the optimum parameters are obtained as follows: D = 2.4373 × 10−5 m/V, α = 0.1947, β = 3.3626, and γ = −2.8526. It is noticeable that the optimized piezoelectric coefficient (D = 24.373 μm/V) is different from the value D = 2.91 μm/V calculated from Table I. Instead, the optimized value is consistent with the coefficient calculated from experimental data as shown in Fig. 3(d). The reason lies in that the input
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Fig. 5. Testing Result 2 of the Bouc–Wen model. (a) Input voltage. (b) Experimental result and Bouc–Wen model output. (c) Bouc–Wen model output errors. (d) Displacement–voltage hysteresis loops.
Fig. 6. Training result of LSSVM model III. (a) Experimental result and LSSVM model output for hysteresis term h. (b) h versus input voltage. (c) Experimental result and LSSVM model result for displacement y. (d) y versus input voltage.
voltage as shown in Fig. 3(a) (as well as Figs. 4(a), 5(a), and 6–8) is the signal applied to the high-voltage amplifier. Through the amplifier, the analog voltage is amplified by ten times and then applied to the piezoelectric bimorph. Due to an offset of the laser measuring point away from the end (see close-up view in Fig. 1), the actual length of the bimorph is less than the nominal value (28 mm). Thus, the optimized D is slightly less than the “nominal” value of 2.91 × 10 μm/V.
2) Modeling Results: The experimental result and the simulated Bouc–Wen model output are compared in Fig. 3(b)–(d). The plots indicate that the Bouc–Wen model cannot exactly describe the complicated hysteresis behavior of the piezoactuated system. A relatively large error exists between the model output and experimental result as shown in Fig. 3(c). Specifically, the maximum model error is 15.56 μm, which accounts for 6.6% of the travel range of the piezoelectric actuator. Based on
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Fig. 7. Testing Result 1 of LSSVM model III. (a) Experimental result and LSSVM model output for hysteresis term h. (b) h versus input voltage. (c) Experimental result and LSSVM model result for displacement y. (d) y versus input voltage.
Fig. 8. Testing Result 2 of LSSVM model III. (a) Experimental result and LSSVM model output for hysteresis term h. (b) h versus input voltage. (c) Experimental result and LSSVM model result for displacement y. (d) y versus input voltage.
the displacement error e = yd − y, the root-mean-square (rms) error (rmse) is defined as follows: Nt 1 e2i rmse = Nt I=1
where Nt = 500 is the number of test data sets.
(18)
Concerning the identified Bouc–Wen model, the rmse is 4.43 μm, which accounts for 1.9% of the travel range of the piezoelectric actuator. It is observed that smaller model error is obtained when the input has lower magnitude and frequency. Hence, the model errors vary greatly at different amplitudes and frequencies of the input signal, which indicates that the Bouc–Wen model cannot capture the rate dependence of the hysteresis precisely.
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TABLE II T RAINING AND T ESTING R ESULTS OF B OUC –W EN M ODEL AND T HREE LSSVM M ODELS
3) Generalization Study: To test the generalization capability of the Bouc–Wen model, two new input signals [see Figs. 4(a) and 5(a)] are selected. For the Test Signal 1, the model output is shown in Fig. 4(b)–(d). The discrepancy between the model result and the actual output (yd ) obtained by experiments is shown in Fig. 4(c). It is observed that the Bouc–Wen model produces an rmse of 6.78 μm, which accounts for 2.8% of the motion range. Thus, as compared with the training error (1.9%), the Bouc–Wen model gives a much worse testing result even with a simple test signal. With a more complex Test Signal 2, the model output is described in Fig. 5(b)–(d). It is found that the Bouc–Wen model creates an rmse of 4.73 μm, i.e., 1.9% of the motion range. Hence, in comparison with the training result, the Bouc–Wen model obtains a slightly larger testing error for the test signal 2. D. LSSVM Model Results By taking the Laplace transform of (5) and comparing its parameters with (16), it can be deduced that ω = 2.5450 × 103 rad/s, ξ = 3.6287 × 10−4 , and d = 1.247 × 102 m/s2 V. 1) LSSVM Modeling: For the training of LSSVM model, the same exciting voltage signal as shown in (17) [see Fig. 3(a)] is employed. In order to accurately identify the hysteresis behavior based on LSSVM model, a suitable input vector (9) is required to be determined. By selecting three types of input variables (with different values of m, n, and l), three LSSVM models are trained by using corresponding input and output variables. The output variable is taken as the hysteresis term as shown in Fig. 6(a) which is generated by resorting to (6). Specifically, the LSSVM Model I only takes the voltage and hysteresis term as input variables by selecting m = 2 and l = 2. Similarly, LSSVM Model II only considers the position and hysteresis term as input variables by setting n = 2 and l = 2. In contrast, by choosing the voltage, position, and hysteresis term as input variables, LSSVM model III has m = 2, n = 2, and l = 2. It is verified that further increase of the orders leads to no significant improvement in the modeling accuracy. For illustration, the training results of LSSVM Model III are shown in Fig. 6. With the Test Signal 1 [see Fig. 4(a)] and Test Signal 2 [see Fig. 5(a)], the testing results are shown in Figs. 7 and 8, respectively. In addition, the training and testing results of Bouc–Wen model and three LSSVM models are summarized in Table II. It is observed that each LSSVM model achieves better identification result than the Bouc–Wen model. By comparing the results of Models I and II, it is found that Model II produces much better testing results than model I for
both test signals. This indicates that the position is preferable to voltage as input variables. Moreover, among the three LSSVM models, the LSSVM model III creates the best results in terms of lower modeling errors for both hysteresis term h and output position y. Specifically, the Model III produces a negligible training rmse for the hysteresis term h, which leads to a percent rmse of 1.05% for the output position y. With the Test Signal 1, the LSSVM Model III gives an rmse of 0.27% for h, which results in a 0.35% rmse for the output position y. Regarding the Test Signal 2, it creates 0.15% and 0.50% rmses for the hysteresis term h and output position y, respectively. Thus, LSSVM Model III is selected for a further comparison study with the identified Bouc–Wen model. It is derived that the LSSVM model has reduced the testing errors of output position by 87.4% and 74.3% in comparison with the Bouc–Wen model for Test Signals 1 and 2, respectively. Therefore, the effectiveness of LSSVM model is evident from the hysteresis identification results. V. E XPERIMENTAL S TUDIES ON H YSTERESIS C OMPENSATION In this section, different control schemes based on the developed Bouc–Wen and LSSVM models are realized to compensate for the hysteresis effect. In experimental studies, the real-time control is implemented with a sampling time of 0.004 s. A. FF Compensation In order to compensate the hysteresis nonlinearity, an FF control (4) based on the Bouc–Wen model is first implemented. Moreover, the block diagram of LSSVM-model-based FF control is shown in Fig. 9, where the control signal uFF is obtained by (7). It is seen that only the hysteresis model is needed while no inverse hysteresis model is required to implement the FF compensation. To demonstrate the performances of the Bouc–Wen and LSSVM models in terms of hysteresis compensation, a series of experimental studies has been performed. For instance, concerning a 2-Hz desired sinusoidal position trajectory as shown in Fig. 10(a), the FF control results of Bouc–Wen model and LSSVM model are shown in Fig. 10(a) and (c), and the tracking errors are compared in Fig. 10(b). Moreover, the control efforts in terms of voltage are shown in Fig. 10(d). Table III summarizes the control results, where the hysteresis width is defined as the ratio of the maximum discrepancy between the ascending and descending curves of hysteresis loop [see Fig. 10(c)] to the output motion range.
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Fig. 9.
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Block diagram of LSSVM hysteresis model-based control schemes.
Fig. 10. Motion tracking results of the FF compensation. (a) Reference and experimental results of Bouc–Wen and LSSVM models. (b) Tracking errors. (c) Actual–reference displacement hysteresis loops. (d) Control actions.
TABLE III H YSTERESIS C OMPENSATION R ESULTS OF D IFFERENT C ONTROL S CHEMES
superiority of LSSVM over Bouc–Wen model in FF hysteresis compensation is validated by the experimental results. It is further observed that the LSSVM model requires 7.2% higher control action than Bouc–Wen model for FF compensation. B. FF Plus FB Compensation To further suppress the residual tracking errors of the FF compensation, an FF plus FB hybrid control is realized. Due to the popularity, a digital PID FB control is employed as follows:
It is observed that the Bouc–Wen FF control gives an rmse of 6.76 μm (i.e., 5.0% of motion range), and the LSSVM FF control produces a 2.55-μm rmse (i.e., 1.9% of motion range). As compared with Bouc–Wen model, the LSSVM model is capable of suppressing the tracking error furthermore by 62%, which leads to a 58% reduction of hysteresis width. Hence, the
uFBk = Kp ek + Ki
k
ej + Kd (ek − ek−1 )
(19)
j=0
where the displacement error ek = ydk − yk with ydk and yk representing the desired and actual system outputs at the kth
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Fig. 11. Motion tracking results of the FF plus FB compensation. (a) Reference trajectory. (b) Tracking errors. (c) Actual–reference displacement hysteresis loops. (d) Control actions.
time step, respectively. In addition, Kp , Ki , and Kd denote the proportional, integral, and derivative gains, respectively. The control gains are tuned by Ziegler–Nichols method through experiments. With the LSSVM hysteresis model, three control schemes including FF, FB, and FF+FB can be easily switched as shown in Fig. 9. Concerning the same position reference trajectory as shown in Fig. 10(a), the tracking results of Bouc–Wen-modeland LSSVM-model-based FF+FB control schemes are shown in Fig. 11(a)–(c). Additionally, the stand-alone PID FB tracking error is also shown in Fig. 11(b), and the control actions of the three schemes are shown in Fig. 11(d). For a clear comparison, the control results are summarized in Table III. It is observed that, although the Bouc–Wen-model-based FF+FB control reduces the tracking error to 1.55 μm (i.e., 1.14% of motion range) and hysteresis width to 2.23%, the LSSVM-model-based one further suppresses the rmse and hysteresis to negligible levels of 0.62 μm (i.e., 0.46% of motion range) and 0.45%, respectively. Thus, the LSSVM-model-based hybrid control has improved the tracking accuracy and hysteresis compensation by 60% and 80%, respectively. Taking into account the control action, it is found that the LSSVM-model-based FF+FB control needs 0.17% lower control effort than the Bouc–Wen-model-based one. The slight lower control effort of the former is contributed by the employed PID FB control, which produces a 2.9% lower control action than the LSSVM-model-based FF control approach as depicted in Table III. The aforementioned experimental results clearly reveal the effectiveness of LSSVM model over Bouc–Wen model in terms of hysteresis compensation. It is noticeable that the performance of the LSSVM model can be improved to achieve a better generalization ability by selecting more comprehensive input signal to excite the piezoelectric actuator for the model training. Moreover, repetitive control or iterative learning con-
trol algorithms can be employed to further reduce the tracking error for periodic reference inputs. Although there is plenty of room for performance improvement of the proposed approach, the enhancement of hysteresis identification accuracy and compensation efficiency for the piezoactuated system over the traditional Bouc–Wen model elaborated by the conducted investigations validates the effectiveness of the proposed LSSVM-based scheme and displays great potential for the future research.
VI. C ONCLUSION This paper has concentrated on hysteresis identification and compensation of the piezoelectric actuator. It is found that the nonlinear hysteresis behavior can be well identified by resorting to a developed LSSVM-based intelligent hysteresis model. It has been shown that the position information is more eligible than voltage information as input variables in developing the hysteresis model. Moreover, the model can be directly employed to compensate the hysteresis effect without modeling the inverse hysteresis. This indicates that the established single hysteresis model can be used for both hysteresis identification and compensation, which is more computational effective than the existing approaches where both a hysteresis model and an inverse hysteresis model are employed. Experimental results demonstrate that the LSSVM model is superior to the Bouc–Wen model in terms of both hysteresis identification accuracy and hysteresis compensation effectiveness. It is noticeable that, although a specific kind of input command is adopted to identify the model parameters in the current investigation, the proposed approach is applicable to more cases with different commands. Moreover, the presented idea is more general and can be easily extended for hysteresis identification and compensation of other smart actuators as well.
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R EFERENCES [1] H.-J. Shieh, F.-J. Lin, P.-K. Huang, and L.-T. Teng, “Adaptive displacement control with hysteresis modeling for piezoactuated positioning mechanism,” IEEE Trans. Ind. Electron., vol. 53, no. 3, pp. 905–914, Jun. 2006. [2] H. C. Liaw and B. Shirinzadeh, “Robust adaptive constrained motion tracking control of piezo-actuated flexure-based mechanisms for micro/nano manipulation,” IEEE Trans. Ind. Electron., vol. 58, no. 4, pp. 1406–1415, Apr. 2011. [3] S. Polit and J. Dong, “Development of a high-bandwidth XY nanopositioning stage for high-rate micro-/nanomanufacturing,” IEEE/ASME Trans. Mechatron., vol. 16, no. 4, pp. 724–733, Aug. 2011. [4] P. A. Sente, F. M. Labrique, and P. J. Alexandre, “Efficient control of a piezoelectric linear actuator embedded into a servo-valve for aeronautic applications,” IEEE Trans. Ind. Electron., vol. 59, no. 4, pp. 1971–1979, Apr. 2012. [5] Y. K. Yong, K. Liu, and S. O. R. Moheimani, “Reducing cross-coupling in a compliant XY nanopositioner for fast and accurate raster scanning,” IEEE Trans. Control Syst. Technol., vol. 18, no. 5, pp. 1172–1179, Sep. 2010. [6] J. Minase, T.-F. Lu, B. Cazzolato, and S. Grainger, “A review, supported by experimental results, of voltage, charge and capacitor insertion method for driving piezoelectric actuators,” Precis. Eng., vol. 34, no. 4, pp. 692– 700, 2010. [7] X. Tan and J. S. Baras, “Adaptive identification and control of hysteresis in smart materials,” IEEE Trans. Autom. Control, vol. 50, no. 6, pp. 827– 839, Jun. 2005. [8] M. Ruderman, F. Hoffmann, and T. Bertram, “Modeling and identification of elastic robot joints with hysteresis and backlash,” IEEE Trans. Ind. Electron., vol. 56, no. 10, pp. 3840–3847, Oct. 2009. [9] K. Kuhnen, “Modeling, identification and compensation of complex hysteretic nonlinearities: A modified Prandtl–Ishlinskii approach,” Eur. J. Control, vol. 9, no. 4, pp. 407–421, 2003. [10] Y. Li and Q. Xu, “Adaptive sliding mode control with perturbation estimation and PID sliding surface for motion tracking of a piezo-driven micromanipulator,” IEEE Trans. Control Syst. Technol., vol. 18, no. 4, pp. 798–810, Jul. 2010. [11] J. Yi, S. Chang, and Y. Shen, “Disturbance observer-based hysteresis compensation for piezoelectric actuators,” IEEE/ASME Trans. Mechatron., vol. 14, no. 4, pp. 456–464, Aug. 2009. [12] J.-X. Xu and K. Abidi, “Discrete-time output integral sliding-mode control for a piezomotor-driven linear motion stage,” IEEE Trans. Ind. Electron., vol. 55, no. 11, pp. 3917–3926, Nov. 2008. [13] X. Chen and T. Hisayama, “Adaptive sliding-mode position control for piezo-actuated stage,” IEEE Trans. Ind. Electron., vol. 55, no. 11, pp. 3927–3934, Nov. 2008. [14] X. Yu and O. Kaynak, “Sliding mode control with soft computing: A survey,” IEEE Trans. Ind. Electron., vol. 56, no. 9, pp. 3275–3285, Sep. 2009. [15] Q. Xu and Y. Li, “Micro-/nanopositioning using model predictive output integral discrete sliding mode control,” IEEE Trans. Ind. Electron., vol. 59, no. 2, pp. 1161–1170, Feb. 2012. [16] Y. Wu and Q. Zou, “Robust-inversion-based 2-DOF control design for output tracking: Piezoelectric actuator example,” IEEE Trans. Control Syst. Technol., vol. 17, no. 5, pp. 1069–1082, Sep. 2009. [17] R. Wang, G.-P. Liu, W. Wang, D. Rees, and Y.-B. Zhao, “H∞ control for networked predictive control systems based on the switched Lyapunov function method,” IEEE Trans. Ind. Electron., vol. 57, no. 10, pp. 3565– 3571, Oct. 2010. [18] C.-L. Hwang, “Microprocessor-based fuzzy decentralized control of 2-D piezo-driven systems,” IEEE Trans. Ind. Electron., vol. 55, no. 3, pp. 1411–1420, Mar. 2008. [19] H.-P. Huang, J.-L. Yan, and T.-H. Cheng, “Development and fuzzy control of a pipe inspection robot,” IEEE Trans. Ind. Electron., vol. 57, no. 3, pp. 1088–1095, Mar. 2010. [20] C.-H. Wang and D.-Y. Huang, “A new intelligent fuzzy controller for nonlinear hysteretic electronic throttle in modern intelligent automobiles,” IEEE Trans. Ind. Electron., vol. 60, no. 6, pp. 2332–2345, Jun. 2013.
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[21] T. Orlowska-Kowalska, M. Dybkowski, and K. Szabat, “Adaptive slidingmode neuro-fuzzy control of the two-mass induction motor drive without mechanical sensors,” IEEE Trans. Ind. Electron., vol. 57, no. 2, pp. 553– 564, Feb. 2010. [22] W. T. Ang, P. K. Khosla, and C. N. Riviere, “Feedforward controller with inverse rate-dependent model for piezoelectric actuators in trajectorytracking applications,” IEEE/ASME Trans. Mechatron., vol. 12, no. 2, pp. 134–142, Apr. 2007. [23] M. Al Janaideh, S. Rakheja, and C. Su, “An analytical generalized Prandtl–Ishlinskii model inversion for hysteresis compensation in micropositioning control,” IEEE/ASME Trans. Mechatron., vol. 16, no. 4, pp. 734–744, Aug. 2011. [24] K. K. Leang, Q. Zou, and S. Devasia, “Feedforward control of piezoactuators in atomic force microscope systems: Inversion-based compensation for dynamics and hysteresis,” IEEE Control Syst. Mag., vol. 29, no. 1, pp. 70–82, Feb. 2009. [25] S. W. John, G. Alici, and C. D. Cook, “Inversion-based feedforward control of polypyrrole trilayer bender actuators,” IEEE/ASME Trans. Mechatron., vol. 15, no. 1, pp. 149–156, Feb. 2010. [26] J.-C. Shen, W.-Y. Jywe, H.-K. Chiang, and Y.-L. Shu, “Precision tracking control of a piezoelectric-actuated system,” Precis. Eng., vol. 32, no. 2, pp. 71–78, Apr. 2008. [27] Y. Li and Q. Xu, “A novel piezoactuated XY stage with parallel, decoupled, and stacked flexure structure for micro-/nanopositioning,” IEEE Trans. Ind. Electron., vol. 58, no. 8, pp. 3601–3615, Aug. 2011. [28] M. Rakotondrabe, “Bouc–Wen modeling and inverse multiplicative structure to compensate hysteresis nonlinearity in piezoelectric actuators,” IEEE Trans. Autom. Sci. Eng., vol. 8, no. 2, pp. 428–431, Apr. 2011. [29] M. A. Janaideh, S. Rakheja, and C.-Y. Su, “Experimental characterization and modeling of rate-dependent hysteresis of a piezoceramic actuator,” Mechatronics, vol. 19, no. 5, pp. 656–670, Aug. 2009. [30] S. Yu, G. Alici, B. Shirinzadeh, and J. Smith, “Sliding mode control of a piezoelectric actuator with neural network compensating rate-dependent hysteresis,” in Proc. IEEE Int. Conf. Robot. Autom., Barcelona, Spain, 2005, pp. 3641–3645. [31] R. Dong, Y. Tan, H. Chen, and Y. Xie, “A neural networks based model for rate-dependent hysteresis for piezoelectric actuators,” Sens. Actuator A, Phys., vol. 143, no. 2, pp. 370–376, 2008. [32] J. A. K. Suykens, “Support vector machines: A nonlinear modeling and control perspective,” Eur. J. Control, vol. 7, no. 2/3, pp. 311–327, 2001. [33] J. A. K. Suykens and J. Vandewalle, “Least squares support vector machine classifiers,” Neural Process. Lett., vol. 9, no. 3, pp. 293–300, 1999. [34] Q. Xu and P.-K. Wong, “Hysteresis modeling and compensation of a piezostage using least squares support vector machines,” Mechatronics, vol. 21, no. 7, pp. 1239–1251, Oct. 2011. [35] P.-K. Wong, Q. Xu, C.-M. Vong, and H.-C. Wong, “Rate-dependent hysteresis modeling and control of a piezostage using online support vector machine and relevance vector machine,” IEEE Trans. Ind. Electron., vol. 59, no. 4, pp. 988–2001, Apr. 2012. [36] C.-J. Lin and S.-Y. Chen, “Evolutionary algorithm based feedforward control for contouring of a biaxial piezo-actuated stage,” Mechatronics, vol. 19, no. 6, pp. 829–839, 2009.
Qingsong Xu (M’09) received the B.S. degree (Hons.) in mechatronics engineering from the Beijing Institute of Technology, Beijing, China, in 2002, and the M.S. and Ph.D. degrees in electromechanical engineering from the University of Macau, Taipa, Macau, in 2004 and 2008, respectively. He was a Visiting Scholar with the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland. He is currently an Assistant Professor of electromechanical engineering with the University of Macau. He currently serves as an Editorial Board Member of the International Journal of Advanced Robotic Systems. His current research interests include microelectromechanical systems devices, micro-/nanorobotics, micro-/nanomanipulation, smart materials and structures, and computational intelligence. Dr. Xu is a member of the American Society of Mechanical Engineers.
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Identification and Compensation of Piezoelectric Hysteresis Without Modeling Hysteresis Inverse Qingsong Xu, Member, IEEE
Abstract—This paper presents a new approach for hysteresis identification and compensation of piezoelectric actuators by resorting to an intelligent hysteresis model. In particular, a least squares support vector machine (LSSVM)-based hysteresis model is developed and used for both purposes of hysteresis identification and hysteresis compensation. By this way, the hysteresis inverse is not needed in the feedforward hysteresis compensator since the hysteresis model is directly used. To establish the LSSVM model, the problem of how to select input variables to convert the multivalued mapping into a single-valued one is addressed. The effectiveness of the presented idea is validated by a series of experimental studies on a piezoactuated system. Results show that the proposed approach is superior to the Bouc–Wen-model-based one in terms of both hysteresis modeling and compensation. The reported method is more computational effective than existing model-based hysteresis compensation approaches, and it is extensible to other smart actuator systems as well. Index Terms—Hysteresis model, motion control, piezoelectric actuator.
micro-/nanopositioning,
I. I NTRODUCTION
S
MART ACTUATORS based on smart materials (e.g., piezoelectric materials, shape memory alloys, magnetostrictive materials, etc.) are popularly employed for actuation in various precision systems dedicated to micropositioning, micromanipulation, and microassembly. Particularly, as a typical smart actuator, the piezoelectric actuator is attractive in both fields of academia and industry due to the merits of (sub)nanometer positioning resolution, rapid response speed, and self-sensing capability. Although numerous previous works have concentrated on research and applications of piezoelectric actuators [1]–[4], the nonlinear piezoelectric effect (in terms of hysteresis and creep) still remains a challenging problem nowadays. The hysteresis effect can be greatly alleviated by using a charge-driven approach or a capacitor insertion method [5], [6]; however, it is at the cost of stroke reduction. Hence, voltage actuation is widely adopted in practice. In order to identify
Manuscript received December 30, 2011; revised March 5, 2012 and April 25, 2012; accepted June 18, 2012. Date of publication July 6, 2012; date of current version May 2, 2013. This work was supported in part by the Macao Science and Technology Development Fund under Grant 024/2011/A and in part by the Research Committee of the University of Macau under Grant MYRG083(Y1-L2)-FST12-XQS. The author is with the Department of Electromechanical Engineering, Faculty of Science and Technology, University of Macau, Taipa, Macao, 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/TIE.2012.2206339
the hysteresis behavior, the establishment of a suitable hysteresis model is required. For instance, hysteresis identifications using Preisach model [7], [8], Prandtl–Ishlinskii model [9], Bouc–Wen model [10], etc., have been widely carried out. On the other hand, to fulfill the requirements of ultrahigh-precision positioning, the piezoelectric hysteresis has to be suppressed by implementing an appropriate control scheme. Generally, the existing hysteresis compensation schemes fall into two categories in terms of hysteresis model-free and hysteresis model-based methods. Concerning the first category, its main property lies in that no hysteresis model is required. The unmodeled hysteresis is considered as an uncertainty or a disturbance [11] to the nominal system, which is tolerated by an advanced robust or adaptive controller. For instance, the applications of slidingmode control [12]–[15], H∞ robust control [16], [17], fuzzy logic control [18]–[20], and neural network control [21] have been successfully reported. Generally, in the second category, a hysteresis model (e.g., Preisach model) is identified, and an inverse hysteresis model is constructed to implement a feedforward (FF) compensator [22], [23]. It has been shown that the inverse-model-based compensation can achieve an accurate positioning, whereas the result is very sensitive to the model accuracy [24], [25]. Fortunately, the combination of FF with feedback (FB) control can be adopted to suppress the hysteresis as well as creep effects [26]. Although the compensation of hysteresis nonlinearity without modeling the hysteresis effect is realizable by designing advanced FB controllers, extensive knowledge on control engineering is needed to design such kinds of control strategies. It is commonly believed that an FF compensator based on a simple hysteresis model (with fewer model parameters) in combination with a simple FB controller (e.g., PID control) makes it more feasible to suppress the hysteresis nonlinearity. The reason lies in that the latter allows the relief of burden on developing complicated modern controllers. For example, it has been shown that, by modeling the hysteresis with the Bouc–Wen model, an FF compensator combined with a PID FB controller is capable of compensating the nonlinear hysteresis effectively [27]. A review of model-based hysteresis compensation approaches reveals that most of the existing works compensate for the hysteresis effect by employing an inverse hysteresis model. Thus, both a hysteresis model and an inverse hysteresis model are required for the hysteresis identification and compensation purposes. More recently, it has been shown that it is possible to mitigate the hysteresis effect by adopting a Bouc–Wen hysteresis model directly while without using the hysteresis inverse [28]. However, only the quasi-static (low frequency) hysteresis was treated in previous work [28].
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Moreover, it has been recognized that the piezoelectric hysteresis effect is rate dependent [29]. The hysteresis behavior is dependent not only on the amplitude but also on the frequency of input voltage signals. Previous works reported that the artificial neural network (ANN) provides an efficient way to model the hysteresis nonlinearity [30], [31]. Since support vector machine (SVM) is superior to ANN in terms of global optimization and higher generalization capability, SVM is widely employed to estimate nonlinear system models accurately [32]. As a variant of SVM, least squares SVM (LSSVM) simplifies the regression to a problem that can be easily solved from a set of linear equations [33]. Thus, LSSVM has a relatively low complexity and is more computationally efficient than the standard SVM. Another advantage of LSSVM lies in that it has fewer parameters to tune. Typically, LSSVM only treats the problem of single-valued mapping between the input and output. However, due to the hysteresis nonlinearity, an input voltage corresponds to multiple position outputs for a piezoactuated system. Hence, one of the challenges in identifying the hysteresis behavior with LSSVM is how to convert the multivalued mapping problem into a single-valued one. Previously, a one-to-one mapping was constructed by introducing the current input value and input variation rate as one data set [30], [34]. Nevertheless, in the case that the input data are accompanied with noises, the variation rate is not smooth and will induce modeling error. Moreover, it is unknown how many orders of the variation rates are sufficient to establish the mapping. In addition, a regression model of hysteresis was developed in [35] by employing the current and previous inputs and previous outputs as exogenous inputs to transform the multivalued mapping into a single-valued one. However, the foregoing research took the position and voltage as output and input variables and developed both direct and inverse hysteresis models between the variables, which indicates a time-consuming procedure. The goal of this paper is to identify and compensate the rate-dependent piezoelectric hysteresis by using an intelligent hysteresis model while without modeling the hysteresis inverse. Particularly, an LSSVM-based hysteresis model is established, and an FF compensator is developed based on the sole model, which provides a computational effective way in hysteresis compensation. The effectiveness of the proposed idea is verified by a series of experimental studies. It is shown that the established LSSVM model is more effective than Bouc–Wen model in terms of hysteresis identification as well as hysteresis compensation performances. The rest of this paper is organized as follows. The dynamics modeling of piezoactuated systems with hysteresis behavior is outlined in Section II, where the idea of establishing a hysteresis model for both purposes of identification and compensation is introduced. In Section III, the hysteresis behavior is identified based on the LSSVM technique by formulating a nonlinear regression problem. Several experimental studies are carried out in Section IV to demonstrate the effectiveness of the presented approach in comparison with the Bouc–Wen-model-based one. Based on the identified models, hysteresis compensation is implemented in Section V along with comparative studies. Section VI concludes this paper.
II. M ODELING OF DYNAMICS W ITH H YSTERESIS B EHAVIOR A. Dynamics Modeling With Bouc–Wen Hysteresis Owing to a fewer number of parameters, the Bouc–Wen model has been extensively applied in piezoelectric hysteresis modeling. The entire dynamics model of a piezoactuated system can be established as follows [10], [36]: M y¨(t) + B y(t) ˙ + Ky(t) = K [Du(t) − H(t)]
(1)
˙ H(t) = αDu(t) ˙ − β |u(t)| ˙ H(t) − γ u(t) ˙ |H(t)|
(2)
where t is the time variable; parameters M , B, K, and y represent the mass, damping coefficient, stiffness, and displacement response of the piezoactuated system, respectively; D is the piezoelectric coefficient; u denotes the input voltage; and H indicates the hysteretic loop in terms of displacement whose magnitude and shape are determined by parameters α, β, and γ. Once the dynamics parameters M , B, and K are generated, the four parameters (D, α, β, and γ) of the Bouc–Wen model can be identified by minimizing an objective function f (D, α, β, γ) =
N 1 (yi − yBWi )2 N
(3)
I=1
where N denotes the total number of samples and yi − yBWi represents the residual error of the ith sample which is the discrepancy between Bouc–Wen model output (yBWi ) and experimental result (yi ). Given the input voltage u and Bouc–Wen hysteresis model H, the output displacement of the system is governed by model (1). In contrast, for a given desired displacement value yd , the required input voltage can be obtained from (1) uFF (t) =
1 [M y¨d (t) + B y˙ d (t) + Kyd (t) + KH(t)] . KD (4)
It is observed that the FF control signal (4) is generated by using the hysteresis term H while without solving the inverse hysteresis model. A block diagram implemented with Matlab and Simulink software is given in [10]. B. Dynamics Modeling With Intelligent Hysteresis Model Inspired by the hysteresis compensation using Bouc–Wen model which does not model the hysteresis inverse, an intelligent model is proposed hereinafter. First, the dynamics model (1) of the system is rewritten into the form ˙ + ωn2 y(t) = du(t) + h(t) y¨(t) + 2ξωn y(t)
(5)
where ξ and ωn denote the damping ratio and natural frequency of the piezoactuated system, respectively. d is a positive parameter, and h represents the hysteresis effect in terms of acceleration.
XU: IDENTIFICATION AND COMPENSATION OF PIEZOELECTRIC HYSTERESIS
Then, in consideration of (5), the hysteresis term can be derived h(t) = y¨(t) + 2ξωn y(t) ˙ + ωn2 y(t) − du(t).
(6)
However, the system parameters and output positions are not always available. Under such situations, it is necessary to establish an intelligent model to identify and compensate for the hysteresis term h. ˆ is obOnce an intelligent model of the hysteresis term h tained, an FF hysteresis compensator can be constructed to produce a desired output position yd 1 ˆ y¨d (t) + 2ξωn y˙ d (t) + ωn2 yd (t) − h(t) uFF (t) = (7) d which uses the hysteresis model directly while without solving the inverse hysteresis model. In the following section, a hysteresis model is established based on LSSVM technique. III. H YSTERESIS M ODELING U SING LSSVM It has been reported that the hysteresis has multivalued and nonsmooth features [13]. However, the introduced LSSVM is capable of accurately estimating high-dimensional smooth functions. Thus, only the multivalued nonlinearity of the hysteresis is treated in this paper. Under the hypothesis that the input command is smooth, an LSSVM regression model is established hereinafter to describe the piezoelectric hysteresis behavior. A. Establishment of Regression Model
(8)
with xk = [uk , . . . , uk−m , yk , . . . , yk−n , hk−1 , . . . , hk−l ]
with the given training data set {xk , hk }N k=1 where N represents the number of training data set, xk ∈ Rm+n+l+2 is an input vector as shown in (9), and hk ∈ R are the output data. Additionally, w is a weight vector, ϕ(·) denotes a nonlinear mapping from the input space to a higher dimensional feature space, and b is the bias. The LSSVM approach formulates the regression as an optimization problem in the primal weight space. Then, the conditions for optimality are obtained by solving a series of partial derivatives, which are used to construct the dual formulation as
0 1N
1T N Ω + Γ−1 IN
(9)
ˆ k denotes the hysteresis term predicted by LSSVM at where h the current time instant k. uk−1 , yk−1 , and hk−1 represent the input voltage, output position, and hysteresis term at previous time instant k − 1, respectively. In addition, m ≥ 0, n ≥ 0, and l ≥ 1 define the order of the model. The selection of the three orders is discussed later in Section IV-D.
It is known that the LSSVM maps the input data into a highdimensional feature space and constructs a linear regression function therein [34]. The unknown hysteresis function is approximated by ˆ h(x) = wT ϕ(x) + b
(10)
b 0 = α h
Ωkj = ϕ(xk )T ϕ(xj ) = K(xk , xj ),
(11)
k, j = 1, 2, . . . , N (12)
where K(·) is a predefined kernel function. The role of the kernel function is to avoid explicit computation of the map ϕ(·) in dealing with the high-dimensional feature space. Once b and α are calculated from (11), the solution for the regression problem can be obtained as N
αk K(x, xk ) + b
(13)
k=1
where K(·) is the kernel function satisfying Mercer’s condition, xk is the training data, and x denotes the new input data. By adopting the radial basis function as kernel function x − xk 2 K(x, xk ) = exp − σ2
(14)
where σ > 0 denotes the width parameter (which specifies the kernel sample variance σ 2 ) and · represents the Euclidean distance, the LSSVM model for the hysteresis model estimation becomes ˆ h(x) =
N k=1
B. LSSVM Modeling
where α = [α1 , α2 , . . . , αN ]T is called the support vector. The support values are αk = Γek with Γ ∈ R denoting the regularization factor. In addition, 1N = [1, 1, . . . , 1]T , h = [h1 , h2 , . . . , hN ]T , and IN is an identity matrix. Moreover, the kernel trick is employed to derive
ˆ h(x) =
Different from the scenario in [35], it is observed from (6) that the output variable is the hysteresis term h, whereas both input voltage u and output position y appear as input variables. By employing LSSVM, a nonlinear regression model is formulated to capture the hysteresis behavior ˆ k = f (xk ) h
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x − xk 2 αk exp − σ2
+ b.
(15)
With the assigned regularization parameter Γ and kernel parameter σ, the objective of the training process is to determine the support values αk and the bias b. The high generalization ability of the LSSVM model relies on the appropriate tuning of the two hyperparameters (Γ and σ). In this paper, the leave-oneout cross-validation approach is adopted to infer the values of the hyperparameters.
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TABLE I K EY PARAMETERS OF THE F OUR -L AYER P IEZOELECTRIC B IMORPH
Fig. 1. Experimental setup of a piezoactuated system. Analog voltage signal is produced by one DAC channel of NI USB-6259 board and then amplified by the voltage amplifier to drive the piezoelectric bimorph. The output position of the bimorph is measured by a laser sensor head. The analog output voltage signal of the sensor is conditioned by the conditioner and then acquired by one ADC channel of USB-6259 board. The board is connected to a computer (running LabVIEW software) via a universal serial bus (USB) interface. A brass-reinforced inner layer is embedded in the four-layer bimorph.
IV. E XPERIMENTAL S TUDIES ON H YSTERESIS I DENTIFICATION In this section, the hysteresis identifications based on Bouc–Wen model and LSSVM approach are carried out experimentally on a test bed. A. Experimental Setup The experimental setup is shown in Fig. 1. A four-layer piezoelectric bimorph actuator (model: T434-A4-201, from Piezo Systems, Inc.) with the dimension of 28 × 5 × 0.86 mm3 is selected. The actuator is fabricated from the piezoelectric material of industry type 5A (Navy Type II), and its key parameters are shown in Table I. A USB-6259 board (from National Instruments (NI) Corporation) with 16-b channels of digital-toanalog converter (DAC) and analog-to-digital converter (ADC) is adopted to produce an analog voltage, which is then amplified by a high-voltage amplifier (model: EPA-104, from Piezo Systems, Inc.) to provide a voltage of ±200 V for the drives of the piezoelectric actuator. The output displacement at the end point of piezoelectric bimorph is measured by a laser displacement sensor (model: LK-H055, from Keyence Corporation). The analog output voltage of the sensor signal conditioner is acquired by a PC through one ADC channel of the USB-6259 board. Moreover, LabVIEW software is employed to implement a real-time control of the piezoactuated system. B. Dynamics Model Identification Before developing the hysteresis models, a linear dynamics model of the system plant is identified by the swept-sine approach. Specifically, sine waves with the amplitude of 0.5 V and the frequency range of 1–1000 Hz are produced to drive the piezoelectric bimorph. The position responses of the piezoelectric actuator are recorded using a sampling rate of 2 kHz.
Fig. 2. Frequency response of the system plant obtained by experiment and identified second-order model.
The 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 by experiments and the identified model (16) are compared in Fig. 2, which shows that the first resonant mode occurs around 404 Hz G(s) =
1.247 × 108 . s2 + 1.847s + 6.477 × 106
(16)
It is observed that the identified second-order model matches the system dynamics well in terms of magnitude and phase at frequencies below 600 and 100 Hz, respectively. The fact that the second-order model cannot properly describe the phase behavior about 100 Hz indicates that a model of much higher order is required to capture the high-frequency dynamics accurately. Here, a simple second-order model is employed to demonstrate the effectiveness of the proposed hysteresis identification scheme. C. Bouc–Wen Model Results Concerning the dynamics parameters, the mass is taken as the nominal value M = 1.8 × 10−3 kg as shown in Table I. Additionally, by neglecting the nonlinear term H in (1), taking the Laplace transform and comparing the parameters with (16), the damping coefficient and stiffness values are calculated as B = 3.3 × 10−3 N · s/m and K = 1.1659 × 104 N/m, respectively. Other parameters are identified as follows. 1) Bouc–Wen Model Identification: To identify the hysteresis model, various types of signals can be utilized. In this
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Fig. 3. Results of the identified Bouc–Wen model. (a) Input voltage. (b) Experimental result and Bouc–Wen model output. (c) Bouc–Wen model output errors. (d) Displacement–voltage hysteresis loops.
Fig. 4. Testing Result 1 of the Bouc–Wen model. (a) Input voltage. (b) Experimental result and Bouc–Wen model output. (c) Bouc–Wen model output errors. (d) Displacement–voltage hysteresis loops.
paper, to illustrate the proposed idea, an input voltage signal [see Fig. 3(a)] is chosen hereinafter for the purpose of model identification
u(t) = 5e−0.13t cos(3πte−0.09t − 3.15) + 1.0 .
(17)
By applying the signal to the high-voltage amplifier, the position output is generated as shown in Fig. 3(b). By setting a time interval of 0.02 s, 500 training data sets are obtained as shown in Fig. 3(a) and (b). Then, the Bouc–Wen
model is identified by optimizing the four parameters to minimize the fitness function (3). Specifically, the particle swarm optimization is adopted for the function minimization, and the optimum parameters are obtained as follows: D = 2.4373 × 10−5 m/V, α = 0.1947, β = 3.3626, and γ = −2.8526. It is noticeable that the optimized piezoelectric coefficient (D = 24.373 μm/V) is different from the value D = 2.91 μm/V calculated from Table I. Instead, the optimized value is consistent with the coefficient calculated from experimental data as shown in Fig. 3(d). The reason lies in that the input
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Fig. 5. Testing Result 2 of the Bouc–Wen model. (a) Input voltage. (b) Experimental result and Bouc–Wen model output. (c) Bouc–Wen model output errors. (d) Displacement–voltage hysteresis loops.
Fig. 6. Training result of LSSVM model III. (a) Experimental result and LSSVM model output for hysteresis term h. (b) h versus input voltage. (c) Experimental result and LSSVM model result for displacement y. (d) y versus input voltage.
voltage as shown in Fig. 3(a) (as well as Figs. 4(a), 5(a), and 6–8) is the signal applied to the high-voltage amplifier. Through the amplifier, the analog voltage is amplified by ten times and then applied to the piezoelectric bimorph. Due to an offset of the laser measuring point away from the end (see close-up view in Fig. 1), the actual length of the bimorph is less than the nominal value (28 mm). Thus, the optimized D is slightly less than the “nominal” value of 2.91 × 10 μm/V.
2) Modeling Results: The experimental result and the simulated Bouc–Wen model output are compared in Fig. 3(b)–(d). The plots indicate that the Bouc–Wen model cannot exactly describe the complicated hysteresis behavior of the piezoactuated system. A relatively large error exists between the model output and experimental result as shown in Fig. 3(c). Specifically, the maximum model error is 15.56 μm, which accounts for 6.6% of the travel range of the piezoelectric actuator. Based on
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Fig. 7. Testing Result 1 of LSSVM model III. (a) Experimental result and LSSVM model output for hysteresis term h. (b) h versus input voltage. (c) Experimental result and LSSVM model result for displacement y. (d) y versus input voltage.
Fig. 8. Testing Result 2 of LSSVM model III. (a) Experimental result and LSSVM model output for hysteresis term h. (b) h versus input voltage. (c) Experimental result and LSSVM model result for displacement y. (d) y versus input voltage.
the displacement error e = yd − y, the root-mean-square (rms) error (rmse) is defined as follows: Nt 1 e2i rmse = Nt I=1
where Nt = 500 is the number of test data sets.
(18)
Concerning the identified Bouc–Wen model, the rmse is 4.43 μm, which accounts for 1.9% of the travel range of the piezoelectric actuator. It is observed that smaller model error is obtained when the input has lower magnitude and frequency. Hence, the model errors vary greatly at different amplitudes and frequencies of the input signal, which indicates that the Bouc–Wen model cannot capture the rate dependence of the hysteresis precisely.
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TABLE II T RAINING AND T ESTING R ESULTS OF B OUC –W EN M ODEL AND T HREE LSSVM M ODELS
3) Generalization Study: To test the generalization capability of the Bouc–Wen model, two new input signals [see Figs. 4(a) and 5(a)] are selected. For the Test Signal 1, the model output is shown in Fig. 4(b)–(d). The discrepancy between the model result and the actual output (yd ) obtained by experiments is shown in Fig. 4(c). It is observed that the Bouc–Wen model produces an rmse of 6.78 μm, which accounts for 2.8% of the motion range. Thus, as compared with the training error (1.9%), the Bouc–Wen model gives a much worse testing result even with a simple test signal. With a more complex Test Signal 2, the model output is described in Fig. 5(b)–(d). It is found that the Bouc–Wen model creates an rmse of 4.73 μm, i.e., 1.9% of the motion range. Hence, in comparison with the training result, the Bouc–Wen model obtains a slightly larger testing error for the test signal 2. D. LSSVM Model Results By taking the Laplace transform of (5) and comparing its parameters with (16), it can be deduced that ω = 2.5450 × 103 rad/s, ξ = 3.6287 × 10−4 , and d = 1.247 × 102 m/s2 V. 1) LSSVM Modeling: For the training of LSSVM model, the same exciting voltage signal as shown in (17) [see Fig. 3(a)] is employed. In order to accurately identify the hysteresis behavior based on LSSVM model, a suitable input vector (9) is required to be determined. By selecting three types of input variables (with different values of m, n, and l), three LSSVM models are trained by using corresponding input and output variables. The output variable is taken as the hysteresis term as shown in Fig. 6(a) which is generated by resorting to (6). Specifically, the LSSVM Model I only takes the voltage and hysteresis term as input variables by selecting m = 2 and l = 2. Similarly, LSSVM Model II only considers the position and hysteresis term as input variables by setting n = 2 and l = 2. In contrast, by choosing the voltage, position, and hysteresis term as input variables, LSSVM model III has m = 2, n = 2, and l = 2. It is verified that further increase of the orders leads to no significant improvement in the modeling accuracy. For illustration, the training results of LSSVM Model III are shown in Fig. 6. With the Test Signal 1 [see Fig. 4(a)] and Test Signal 2 [see Fig. 5(a)], the testing results are shown in Figs. 7 and 8, respectively. In addition, the training and testing results of Bouc–Wen model and three LSSVM models are summarized in Table II. It is observed that each LSSVM model achieves better identification result than the Bouc–Wen model. By comparing the results of Models I and II, it is found that Model II produces much better testing results than model I for
both test signals. This indicates that the position is preferable to voltage as input variables. Moreover, among the three LSSVM models, the LSSVM model III creates the best results in terms of lower modeling errors for both hysteresis term h and output position y. Specifically, the Model III produces a negligible training rmse for the hysteresis term h, which leads to a percent rmse of 1.05% for the output position y. With the Test Signal 1, the LSSVM Model III gives an rmse of 0.27% for h, which results in a 0.35% rmse for the output position y. Regarding the Test Signal 2, it creates 0.15% and 0.50% rmses for the hysteresis term h and output position y, respectively. Thus, LSSVM Model III is selected for a further comparison study with the identified Bouc–Wen model. It is derived that the LSSVM model has reduced the testing errors of output position by 87.4% and 74.3% in comparison with the Bouc–Wen model for Test Signals 1 and 2, respectively. Therefore, the effectiveness of LSSVM model is evident from the hysteresis identification results. V. E XPERIMENTAL S TUDIES ON H YSTERESIS C OMPENSATION In this section, different control schemes based on the developed Bouc–Wen and LSSVM models are realized to compensate for the hysteresis effect. In experimental studies, the real-time control is implemented with a sampling time of 0.004 s. A. FF Compensation In order to compensate the hysteresis nonlinearity, an FF control (4) based on the Bouc–Wen model is first implemented. Moreover, the block diagram of LSSVM-model-based FF control is shown in Fig. 9, where the control signal uFF is obtained by (7). It is seen that only the hysteresis model is needed while no inverse hysteresis model is required to implement the FF compensation. To demonstrate the performances of the Bouc–Wen and LSSVM models in terms of hysteresis compensation, a series of experimental studies has been performed. For instance, concerning a 2-Hz desired sinusoidal position trajectory as shown in Fig. 10(a), the FF control results of Bouc–Wen model and LSSVM model are shown in Fig. 10(a) and (c), and the tracking errors are compared in Fig. 10(b). Moreover, the control efforts in terms of voltage are shown in Fig. 10(d). Table III summarizes the control results, where the hysteresis width is defined as the ratio of the maximum discrepancy between the ascending and descending curves of hysteresis loop [see Fig. 10(c)] to the output motion range.
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Fig. 9.
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Block diagram of LSSVM hysteresis model-based control schemes.
Fig. 10. Motion tracking results of the FF compensation. (a) Reference and experimental results of Bouc–Wen and LSSVM models. (b) Tracking errors. (c) Actual–reference displacement hysteresis loops. (d) Control actions.
TABLE III H YSTERESIS C OMPENSATION R ESULTS OF D IFFERENT C ONTROL S CHEMES
superiority of LSSVM over Bouc–Wen model in FF hysteresis compensation is validated by the experimental results. It is further observed that the LSSVM model requires 7.2% higher control action than Bouc–Wen model for FF compensation. B. FF Plus FB Compensation To further suppress the residual tracking errors of the FF compensation, an FF plus FB hybrid control is realized. Due to the popularity, a digital PID FB control is employed as follows:
It is observed that the Bouc–Wen FF control gives an rmse of 6.76 μm (i.e., 5.0% of motion range), and the LSSVM FF control produces a 2.55-μm rmse (i.e., 1.9% of motion range). As compared with Bouc–Wen model, the LSSVM model is capable of suppressing the tracking error furthermore by 62%, which leads to a 58% reduction of hysteresis width. Hence, the
uFBk = Kp ek + Ki
k
ej + Kd (ek − ek−1 )
(19)
j=0
where the displacement error ek = ydk − yk with ydk and yk representing the desired and actual system outputs at the kth
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Fig. 11. Motion tracking results of the FF plus FB compensation. (a) Reference trajectory. (b) Tracking errors. (c) Actual–reference displacement hysteresis loops. (d) Control actions.
time step, respectively. In addition, Kp , Ki , and Kd denote the proportional, integral, and derivative gains, respectively. The control gains are tuned by Ziegler–Nichols method through experiments. With the LSSVM hysteresis model, three control schemes including FF, FB, and FF+FB can be easily switched as shown in Fig. 9. Concerning the same position reference trajectory as shown in Fig. 10(a), the tracking results of Bouc–Wen-modeland LSSVM-model-based FF+FB control schemes are shown in Fig. 11(a)–(c). Additionally, the stand-alone PID FB tracking error is also shown in Fig. 11(b), and the control actions of the three schemes are shown in Fig. 11(d). For a clear comparison, the control results are summarized in Table III. It is observed that, although the Bouc–Wen-model-based FF+FB control reduces the tracking error to 1.55 μm (i.e., 1.14% of motion range) and hysteresis width to 2.23%, the LSSVM-model-based one further suppresses the rmse and hysteresis to negligible levels of 0.62 μm (i.e., 0.46% of motion range) and 0.45%, respectively. Thus, the LSSVM-model-based hybrid control has improved the tracking accuracy and hysteresis compensation by 60% and 80%, respectively. Taking into account the control action, it is found that the LSSVM-model-based FF+FB control needs 0.17% lower control effort than the Bouc–Wen-model-based one. The slight lower control effort of the former is contributed by the employed PID FB control, which produces a 2.9% lower control action than the LSSVM-model-based FF control approach as depicted in Table III. The aforementioned experimental results clearly reveal the effectiveness of LSSVM model over Bouc–Wen model in terms of hysteresis compensation. It is noticeable that the performance of the LSSVM model can be improved to achieve a better generalization ability by selecting more comprehensive input signal to excite the piezoelectric actuator for the model training. Moreover, repetitive control or iterative learning con-
trol algorithms can be employed to further reduce the tracking error for periodic reference inputs. Although there is plenty of room for performance improvement of the proposed approach, the enhancement of hysteresis identification accuracy and compensation efficiency for the piezoactuated system over the traditional Bouc–Wen model elaborated by the conducted investigations validates the effectiveness of the proposed LSSVM-based scheme and displays great potential for the future research.
VI. C ONCLUSION This paper has concentrated on hysteresis identification and compensation of the piezoelectric actuator. It is found that the nonlinear hysteresis behavior can be well identified by resorting to a developed LSSVM-based intelligent hysteresis model. It has been shown that the position information is more eligible than voltage information as input variables in developing the hysteresis model. Moreover, the model can be directly employed to compensate the hysteresis effect without modeling the inverse hysteresis. This indicates that the established single hysteresis model can be used for both hysteresis identification and compensation, which is more computational effective than the existing approaches where both a hysteresis model and an inverse hysteresis model are employed. Experimental results demonstrate that the LSSVM model is superior to the Bouc–Wen model in terms of both hysteresis identification accuracy and hysteresis compensation effectiveness. It is noticeable that, although a specific kind of input command is adopted to identify the model parameters in the current investigation, the proposed approach is applicable to more cases with different commands. Moreover, the presented idea is more general and can be easily extended for hysteresis identification and compensation of other smart actuators as well.
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Qingsong Xu (M’09) received the B.S. degree (Hons.) in mechatronics engineering from the Beijing Institute of Technology, Beijing, China, in 2002, and the M.S. and Ph.D. degrees in electromechanical engineering from the University of Macau, Taipa, Macau, in 2004 and 2008, respectively. He was a Visiting Scholar with the Swiss Federal Institute of Technology (ETH), Zurich, Switzerland. He is currently an Assistant Professor of electromechanical engineering with the University of Macau. He currently serves as an Editorial Board Member of the International Journal of Advanced Robotic Systems. His current research interests include microelectromechanical systems devices, micro-/nanorobotics, micro-/nanomanipulation, smart materials and structures, and computational intelligence. Dr. Xu is a member of the American Society of Mechanical Engineers.
