- Email: [email protected]
Characteristic model of travelling wave ultrasonic motor
Ultrasonics 54 (2014) 725–730
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Characteristic model of travelling wave ultrasonic motor Shi Jingzhuo ⇑, You Dongmei School of Electrical Engineering, Henan University of Science and Technology, Luoyang 471023, China
a r t i c l e
i n f o
Article history: Received 12 May 2013 Received in revised form 13 August 2013 Accepted 10 September 2013 Available online 18 September 2013 Keywords: Travelling wave ultrasonic motor Characteristic model Identification
a b s t r a c t In general, the design and analysis of ultrasonic motor and motor’s control strategy are based on mathematical model. The academic model is widely used in the analysis of traveling wave ultrasonic motor (TWUSM). But the dispersive characteristic of piezoelectric ceramics and other complicated process, such as the friction, make the model’s precision not so accurate. On the other hand, identification modeling method, which is built based on the tested data, has obtained increasing application in the study of ultrasonic motor’s control technology. Based on the identification model, many control strategies can be designed easily. But the identification model is an approximate model, so if a more accurate model of ultrasonic motor can be obtained, the analysis and design of motor control system will be more effective. Characteristic model is a kind of identification model which can accurately describe the characteristics of TWUSM. Based on the tested data, this paper proposes the modeling method of ultrasonic motor’s characteristic model. The paper also makes a comparison of the effectiveness of different identification algorithms. Aiming at the speed control of ultrasonic motor, the influence of the parameter’s initial values on the precision of model is discussed. The calculating results indicate the availability of this characteristic model. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction The mathematical model of traveling wave ultrasonic motor (TWUSM) is an important foundation for the analysis, design and performance evaluation of the motor’s control system [1,2]. In order to improve the performance of TWUSM’s control system and study more reasonable control strategies, we must make sure that the mathematical model is suitable for control applications [3–5]. Currently, there are three kinds of modeling methods of TWUSM: theoretical modeling, neural network modeling and identification modeling. The theoretical modeling mainly includes equivalent circuit modeling, finite element analysis modeling and mechanism analysis modeling. Mainly due to the characteristics of complexity, time-varying and dispersion of electromechanical energy conversion and transmission mechanism, it is difficult to obtain mathematical model which can precisely describe dynamic operating characteristics of TWUSM using theoretical modeling [6– 9]. Therefore, theoretical modeling is not suitable for the modeling of the motor. When neural network modeling is used, more incentive function and various learning rules are required. In addition, neural network model is more complex and not suitable for the design of the controller due to complex topological structure, thus this model is not commonly used [10–12]. The identification method based on experimental data has become the main method of the ⇑ Corresponding author. Tel.: +86 379 64231757; fax: +86 379 64231910. E-mail address: [email protected] (S. Jingzhuo). 0041-624X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2013.09.012
establishment of the control model of ultrasonic motor [13,14] because of the complex operation mechanism of TWUSM. But it is generally believed that the identification model is only an approximate model under the least squares criterion, and it cannot describe the overall perspective. That is, identification model can describe the main operating characteristics of the ultrasonic motor, but it is not an accurate description. This makes the design of ultrasonic motor controller based on the identification model must be adapted to this model deviation. There is no proof that ultrasonic motor can be described by linear differential equation with constant coefficients. It is well known that ultrasonic motors have various physical nonlinearities related to contact and frictional phenomena. Characteristic model proposed by Wu Hongxin is a kind of model based on experimental data [15], which can be classified into the scope of the identification model. But the difference is that the existing theory has proved that the real-time correction of the low order (two or three-order) characteristic model is an accurate description of high order real object. Based on the characteristic model, it is possible to obtain a relatively simple structure of the controller. This paper proposes the modeling method of ultrasonic motor’s characteristic model and puts forward three methods to determine identification algorithm initial parameters, which is determined by comparing the errors and online recursion algorithm for ultrasonic motor modeling and control. The calculating results of calculation manifest the availability of the characteristic model.
726
S. Jingzhuo, Y. Dongmei / Ultrasonics 54 (2014) 725–730
where
2. Characteristic model of ultrasonic motor The theories and concepts of characteristic model provide a feasible approach for the design of low-order and intelligent controller. For ultrasonic motor, the characteristic model is used in the speed and rotating position closed-loop control. The varied difference equation can described by using characteristic model, the form of three-order or two-order are available respectively. In general, all kinds of ultrasonic motors can be represented with high order transfer function as follows
GðsÞ ¼
bm sm þ bm1 sm1 þ þ b1 s þ b0 sn þ an1 sn1 þ þ a1 s þ a0
ð1Þ
where ai, bj are the molecules and denominator coefficients of transfer function respectively i = 0, . . . , n 1, j = 0, . . . , m. Eq. (1) can be decomposed into the sum of finite rational functions which is not higher than two order and consisting of inertia, oscillation, integral and multiple-root items m h X k kpþi kv X ki pþi GðsÞ ¼ 2 þ þ þ kpþi s s þ k s þ k s þ i pþi i¼1 i¼1
þ
q X
!
kxi 2
i¼1
ðs þ xi Þ
f2 ðkÞ ¼ a1 ðkÞa2 ðkÞ ¼ f1 T½a1 ðyÞ þ a2 ðyÞ þ T 2 ½a1 ðyÞa2 ðyÞg g 0 ðkÞ ¼ b1 ðkÞ þ b2 ðkÞ ¼ TðK 1 þ K 2 Þ g 1 ðkÞ ¼ ½a1 ðkÞb2 ðkÞ þ a2 ðkÞb1 ðkÞ ¼ TðK 1 þ K 2 Þ þ T 2 ½K 1 a2 ðyÞ þ K 2 a1 ðyÞ where ar(k) = 1 Tar(y), here k is point-in-time ,which describes the sequential relationship between current moment and former moment, while ar(y) describes the dependent relationship between other variables and itself. And then, we consider integral items and multiple-root items based on the proof of the first step. Single integral term corresponding to the input–output equation is equivalent to zero real roots in the first step. With the same kind of derivation, we have
yv x ðk þ 2Þ ¼ 2ð1 T x1 Þyv x ðk þ 1Þ ð1 2T x1 Þyv x ðkÞ þ T 2 ðkv þ kx1 Þuðk þ 1Þ
ð2Þ
ð5Þ
Combining Eqs. (4) and (5), and make y⁄(k) = y(k) + yvx(k), we have
Theorem 1. For any linear time-invariant n-order objects G(s), it can be decomposed as Eq. (2). Under the proper conditions of sampling period T, when we control position maintaining or position tracking, we can describe the characteristic model by using the following two-order varying difference equation, yðk þ 1Þ ¼ f1 ðkÞyðkÞ þ f2 ðkÞyðk 1Þ þ g 0 ðkÞuðkÞ þ g 1 ðkÞuðk 1Þ
f1 ðkÞ ¼ a1 ðkÞ þ a2 ðkÞ ¼ 2 T½a1 ðyÞ þ a2 ðyÞ
ð3Þ
Here, y(k) and u(k) are discretized output and input variable quantity of the moment k of system respectively, f1(k), f2(k), g0(k), g1(k) are the time-varying coefficients, If the object G(s) is stable or contains integral items, then: (1) f1(k), f2(k), g0(k), g1(k) are the coefficients of the Eq. (3), which are slowly time varying coefficients. (2) The value range of the coefficients can be determined beforehand. (3) When the system is in dynamic process, under the conditions of same input, the output of the characteristic model is equivalent to the actual output (appropriately ensure to select the sampling period T within the output error allowed). When the system is in a steady state, the static gain of Eq. (3) D0 is equal to the static gain of Eq. (1) D = a0/b0, i.e. that is to say the outputs of the system are equal under the steady-state. (4) When static gain D = 1, in the steady state the sum of all coefficients is equal to 1, i.e.
f1 ð1Þ þ f2 ð1Þ þ g 1 ð1Þ þ g 2 ð1Þ ¼ 1 (5) If there is an integral part in object, then,
f1 ð1Þ þ f2 ð1Þ ¼ 1 Here, the so-called ‘‘sampling period satisfies certain conditions’’, that is satisfying the sampling theorem, so as to make the system still satisfies the requirements of controllability and control accuracy after discretizing from the original continuous system. Theorem 1 can be proved by two steps. First, disregarding the integral items and multiple-root items in Eq. (2), i.e. kv = 0, kxi = 0, we have
yðk þ 1Þ ¼ f1 ðkÞyðkÞ þ f2 ðkÞyðk 1Þ þ g 0 ðkÞuðkÞ þ g 1 ðkÞuðk 1Þ ð4Þ
y ðk þ 2Þ ¼ ½a1 ðkÞ þ a2 ðkÞy ðk þ 1Þ a1 ðkÞa2 ðkÞy ðkÞ þ ½T 2 ðkv þ kx1 Þ þ b1 ðkÞ þ b2 ðkÞuðk þ 1Þ ½a2 ðkÞb1 ðkÞ þ a1 ðkÞb2 ðkÞuðkÞ þ ½e1 yv x ðk þ 1Þ e2 yv x ðkÞ
ð6Þ
where ai and bi are defined the same as in Eq. (4).
e1 ¼ 2ð1 T x1 Þ ½a1 ðkÞ þ a2 ðkÞ ¼ T½a1 ðyÞ þ a2 ðyÞ 2x1 e2 ¼ ð1 2T x1 Þ a1 ðkÞa2 ðkÞ ¼ T½a1 ðyÞ þ a2 ðyÞ 2x1 T 2 a1 ðyÞa2 ðyÞ Eq. (6) can still be written as Eq. (4),
yðk þ 1Þ ¼ f1 ðkÞyðkÞ þ f2 ðkÞyðk 1Þ þ g 0 ðkÞuðkÞ þ g 1 ðkÞuðk 1Þ ð7Þ f1 ðkÞ ¼ a1 ðkÞ þ a2 ðkÞ ¼ 2 T½a1 ðyÞ þ a2 ðyÞ
ð8Þ
f2 ðkÞ ¼ a1 ðkÞa2 ðkÞ ¼ f1 T½a1 ðyÞ þ a2 ðyÞ þ T 2 ½a1 ðyÞa2 ðyÞg ð9Þ g 0 ðkÞ ¼ T 2 ðkv þ kx1 Þ þ b1 ðkÞ þ b2 ðkÞ ¼ T 2 ðkv þ kx1 Þ þ TðK 1 þ K 2 Þ
ð10Þ
g 1 ðkÞ ¼ ½a1 ðkÞb2 ðkÞ þ a2 ðkÞb1 ðkÞ ¼ TðK 1 þ K 2 Þ þ T 2 ½K 1 a2 ðyÞ þ K 2 a1 ðyÞ
ð11Þ
Now we can gain the following results. (1) In general, the change of ar(k) is very small in each step. It is about 0.001. Therefore, we know that the coefficients f1(k), f2(k), g0(k) and g1(k) a are slowly time-varying according to Eqs. (8)–(11). (2) If there is not integral items and no multiple-root items, the range of f1(k) and f2(k) can be determined beforehand. Considering their maximum range, we can choose
f1 ðkÞ 2 ð1; 2;
f 2 ðkÞ 2 ½1; 0Þ
S. Jingzhuo, Y. Dongmei / Ultrasonics 54 (2014) 725–730
Since g0(k) and g1(k) not only depend on T and ar(y), but also depend on Kr, that is to say they are related to static gain D. If the range of the maximum value of static gain is known by input transformation, we can get g0(k), and g1(k) 1. If there is not integral items and/or no multiple-root items, f1(k) is less than but approximate to 2, while f2(k) is larger than but approaches 1. (3) If integral items and no multiple-root items do not exist, each step of the sampling period is less than 0.001, this may result in output error in dynamic state. When integral items and multiple-root items exist, there also exists a small error in dynamic state. In steady state, static gain D(1) can be expressed as follows:
Dð1Þ ¼
n X g 0 ð1Þ þ g 1 ð1Þ ki b0 ¼ ¼ 1 f1 ð1Þ f2 ð1Þ i¼1 ki a0
ð12Þ
It means that the static gain of the characteristic model D(1) is equal to that of the original plant b0/a0. (4) Due to Eq. (12) it is clear that when D = b0/a0 = 1, we have
f1 ð1Þ þ f2 ð1Þ þ g 1 ð1Þ þ g 2 ð1Þ ¼ 1 This means when static gain D = 1, the sum of all coefficients of the characteristic model is equal to one in a steady state. (5) If an integral item exists, it is easy to derive that in a steady state of position control
characteristic model by comparing identification methods. Third, the fixed parameters model is used to realize error comparison with the characteristic model. System identification is based on the input and output data to determine a mathematical model, which is equivalent with the studied system from a set of model categories. And system identification methods are varied. 3.1. Experimental data Identification method is based on the measured input and output data. Self-designed experimental system [3] block diagram is shown in Fig. 1, and the system architecture described in paper [3]. Experimental motor is the Shinsei USR60, which is a two-phase traveling wave ultrasonic motor. The photoelectric encoder E (Encoder) rigid connection with the motor’s shaft provides speed feedback signal. The driving circuit is the two-phase H-bridge structure constituted by MOSFET, and the control circuit, which takes the low-cost DSP chip DSP56F801 as a core, in supplemented by the PWM (Pulse Width Modulation) signal generator with phase shift realizes the control of the MOSFET switch state in the driving circuit. In the experimental data, we can choose 10 sets data as modeling data. Besides, we elect 5 sets as validation data to verify the validity of the model and help to select identification algorithm, with the reference speed 120, 100, 90, 80, 30 rpm. 3.2. Choice and comparison of identification methods In general, the identification model can be described by the following difference equation
f1 ð1Þ þ f2 ð1Þ ¼ 1 Until now, the proof of the Theorem 1 is completed.
Aðz1 ÞyðkÞ ¼ zd Bðz1 ÞuðkÞ þ eðkÞ
Corollary 1. For a linear constant high-order plant G(s) that satisfied the condition of Theorem 1, when velocity control or acceleration control is required, its characteristic model can be expressed with a three-order time varying difference equation, which is shown below.
yðk þ 1Þ ¼ f1 ðkÞyðkÞ þ f2 ðkÞyðk 1Þ þ f3 ðkÞyðk 2Þ þ g 0 ðkÞuðkÞ þ g 1 ðkÞuðk 1Þ þ g 2 ðkÞuðk 2Þ
727
ð13Þ
Compared with the speed control, the object is added an integral part in position control. Therefore, the design of the controller is also different. When the object G(s) is stable, then (1) In dynamic process, with the same input, the output of the characteristic model is equal to that of the practical object. In a steady state, the two outputs are equal, which means their static gains are equal D0 = D. (2) When the static gain D = 1, the sum of all the coefficients is equals to one in a steady state, i.e.
f1 ð1Þ þ f2 ð1Þ þ f3 ð1Þ þ g 0 ð1Þ þ g 1 ð1Þ þ g 2 ð1Þ ¼ 1 3. Fixed parameters model of the ultrasonic motor To establish the characteristic model of ultrasonic motor, we need to establish its fixed parameters model. There are three reasons. First, we need to prepare for the choice of the structure of the characteristic model. That is to identify the model structure (described by values of parameters na, nb and d in Eq. (14)) with a small sum of squared errors. Second, we determine the appropriate identification algorithm to identify initial parameters of the
ð14Þ
where
(
Aðz1 Þ ¼ 1 þ a1 z1 þ a2 z2 þ þ ana zna Bðz1 Þ ¼ b0 þ b1 z1 þ b2 z2 þ þ bnb znb
Here, y(k) and u(k) are the current speed and frequency control respectively. d is order delay. a1 ; . . . ; ana ; b0 ; . . . ; bnb are the parameters to be identify. e(k) is the white noise. z is time delay. na and nb are the order of A(z1) and B(z1) respectively. na, nb and d are determined by least squares model structure identification, which is based on experimental data. The paper selected 90 rpm experimental data to determine the structure of the model in order to select more suitable order. The calculation results show that when na = 3, nb = 2, d = 0 the loss function of the model and the final prediction error is minimum. This order is used in the following identification of the ultrasonic motor. In order to determine a suitable identification method for ultrasonic motor’s modeling, Least Squares algorithm (LS), Recursive Least Squares (RLS), Recursive Extended Least Squares method (RELS) and Recursive Maximum Likelihood (RML) are respectively used to identify model’s parameters, and the identification method is determined by comparing the sum of squared errors of each method. The calculation results are shown in Table 1. View from the sum of squared error of modeling data, the Least Squares algorithm and Recursive Maximum Likelihood method achieved a relatively small value of 229.1345 and 229.3930. Verify the results from the validation data, the Least Squares and Recursive Maximum Likelihood algorithms are similar too. Since Recursive Maximum Likelihood algorithm is relatively complex, the Least Squares algorithm is selected to identify the initial parameters. Using the Least Squares algorithm to establish ultrasonic motor’s model, the comparison of model output and measured data based on verification data shown in Fig. 2 shows that the model
728
S. Jingzhuo, Y. Dongmei / Ultrasonics 54 (2014) 725–730
− + −
Uref + Nref
Frequency adjustment
Speed controller
+
Voltage amplitude controller of phase A and B
Pulse width adjustment
Phase-shift PWM generator
H-bridge driver
Ultrasonic motor
− Encoder signal processor
E
Fig. 1. Structure of the experimental system for speed control.
Table 1 Error comparison of identification algorithms. Sum of squared errors
LS
RLS
RELS
RML
Modeling data Validation data
229.1345 251.996
356.4345 372.1168
346.4442 365.4386
229.3930 252.6143
140
Speed /(rpm)
120 100 80
Model output
…Tested data
60
Nref=120rpm
40 20
0.05
0.10
0.15
0.20
0.25
Time/s Fig. 2. Comparison between model output and tested data (Nref = 120 rpm).
output is consistent with the measured data, and the maximum absolute error is 6.577 rpm. 4. Establishment of the characteristic model According to the establish process of fixed-parameter model and comparing the sum of square error of different model structure, the structure of characteristic model is determined three order, i.e. na = 3, nb = 2. The least squares algorithm is chose to identify the characteristic model’s initial parameters by comparing the effect of the recognition algorithm. Thus, the characteristic model of ultrasonic motor can be written in the form as follows
yðkÞ ¼ a1 ðkÞyðk 1Þ þ a2 ðkÞyðk 2Þ þ a3 ðkÞyðk 3Þ þ b0 ðkÞuðkÞ þ b1 ðkÞuðk 1Þ þ b2 ðkÞuðk 2Þ
ð15Þ
where y(k-n) and u(k-n) are the speed and frequency control of former n moment respectively. a1(k), a2(k), a3(k), b0(k), b1(k), b2(k) are the time-varying characteristic model’s parameters. After determining the structure of the model, the initial value of identification algorithm and online recursive identification are needed to be determined. 4.1. Determination of identification’s initial value This section is to determine the initial value of h(k) and P(k), i.e. h(0) and P(0). Identification methods are used to estimate the model parameters hðkÞ ¼ ½ a1 ðkÞ a2 ðkÞ a3 ðkÞ b0 ðkÞ b1 ðkÞ b2 ðkÞ . Based on
the idea of identification, h(k) (estimated parameters for the current moment) is equivalent to the sum of h(k 1) (estimated parameters for the former moment) and correction term. Here, correction term describes the gaps between the estimated parameters and the truth-value, which is the product of every step gain matrix and the error. The error is the difference between the actual measurement data at current moment and predicted output data based on the model which is determined by the former estimated parameter. With the increase of the recursion steps, the difference becomes smaller and the estimated parameters at the current moment approximate the former. Then model parameters h(k) approach to the truth-value. Considering the operating characteristics of ultrasonic motor, three kinds of methods are proposed to determine the initial value of h(0) (the three kinds of methods hereinafter referred to as Method 1, Method 2, and Method 3). (1) According to the analysis of Section 3.2, we make a comparison of the sum of squared errors of several identification methods which has fixed-parameters and choose the method’s parameters as initial parameters which has the minimum sum of squared errors. (2) The motor system start running from zero, and the initial values mainly work at the starting stage of the control, which is low-speed area operation starting from zero speed. So the initial value can be set to the off-line identification parameters obtained in the low-speed case. (3) For different reference speed, the Least Squares algorithm is used to obtain a three-order model. Every speed corresponds to a set of model parameters which are the initial value of identification parameters. If there are more than one set of data of the same reference speed, the initial parameters are the combined recognition result of these multiple sets of data. The state estimation error covariance matrix P(k) is directly related to the correction effect of model parameters. The covariance is greater, and this means the difference is between the estimated value and truth-value greater. At the same time, the gain vector will be amplified, and the correction effect on the model parameter will be weighted too. There are two methods to determine P(0). One is make P(0) = aI, here I is a unity-matrix. The other one is the off-line identification method. 4.2. Determination of identification algorithm If the initial value is determined, the online recursive identification of the characteristic model is to be determined next. In order to obtain a suitable identification method, LS, RLS, RELS and RML are respectively used to recursive calculation. The sum of squared errors is calculated during recursive process according to the timevarying parameters of characteristic model. That is to say, the sum
729
S. Jingzhuo, Y. Dongmei / Ultrasonics 54 (2014) 725–730
Speed /(rpm)
30
θ
P
25 20
Model output
… Tested data
15
Nref =30rpm
k
10 0.1
0.2
0.3
0.4
0.5
0.6
Time/s Fig. 5. Comparison between model output and tested data (Nref = 30 rpm).
θ
k 3
P k Speed error /(rpm)
2 1 0 -1 -2 0.04
0.08
0.12
0.16
0.20
0.24
Time/s Fig. 6. Speed error in identify process (Nref = 120 rpm). Fig. 3. MATLAB program flow chart of RELS.
0.6
140
Speed error /(rpm)
Speed /(rpm)
120 100 80
Model output 60
… Tested data
40 20
0.4 0.2 0.0 -0.2 -0.4
Nref =120rpm
0.0 0.05
0.10
0.15
0.20
0.1
0.2
0.25
0.3
0.4
0.5
0.6
Time/s
Time/s
Fig. 7. Speed error in identify process (Nref = 30 rpm).
Fig. 4. Comparison between model output and tested data (Nref = 120 rpm).
of squared errors is calculated at every sampling period. And the sum of squared errors of every method is compared to determine the appropriate identification method. MATLAB is used in the calculation. And the calculation results are shown in Table 2. In Table 2, through the survey of RLS, RELS and RML, the sum of squared errors of Method 3 is less than the former two methods
whether viewing from the result of modeling data or validation data. Then, compare the Method 3 among RLS, RELS and RML, the sum of squared errors of RELS and RML are less than RLS, and the sum of squared errors of RML is a little bit bigger than RELS. Taking all factors into consideration, the RELS is chosen as the online recursive identification to build the characteristic
Table 2 Comparison of different identification algorithms. RLS
Modeling data Validation data
RELS
RML
Method 1
Method 2
Method 3
Method 1
Method 2
Method 3
Method 1
Method 2
Method 3
231.19 243.06
231.19 243.06
121.79 74.027
227.99 237.18
227.99 237.18
110.97 33.2096
227.92 237.27
227.8 237.02
112.12 33.3089
730
S. Jingzhuo, Y. Dongmei / Ultrasonics 54 (2014) 725–730
Table 3 Comparison of sum of squared errors between fixed parameters and characteristic models based on different identification algorithms. RLS
Modeling data Validation data
RELS
RML
Fixed parameters
Characteristic model
Fixed parameters
Characteristic model
Fixed parameters
Characteristic model
356.43 372.12
121.79 74.027
346.44 365.44
110.97 33.2096
229.39 252.61
112.12 33.3089
Table 4 Error comparison of validation data.
Fixed parameters Characteristic model
Reference speed (rpm)
Sum of squared error
Maximum absolute error (rpm)
Maximum relative error (%)
Average error (rpm)
120 120 30
72.7402 22.042 1.8128
6.577 2.62 0.56
11.24 3.65 2.93
1.4851 0.9871 0.1665
model; meanwhile, Method 3 is used to determine the initial parameters. The program flow chart of RELS is shown in Fig. 3. Based on characteristic model Eq. (15), Method 3 is used to online recursive identification and to establish ultrasonic motor’s model by Using RELS, the comparison of model output and measured data based on verification data is shown in Figs. 4 and 5. The reference speed values are 120 rpm and 30 rpm. Compared to Fig. 2, the maximum absolute error of Fig. 4 is only 2.62 rpm. The error of the model improved obviously. The maximum absolute error of Fig. 5 is 0.56 rpm, it is clear that the model output is very close to the measured data. The errors during identify process are shown in Figs. 6 and 7. From the figure we can see that the value of error fluctuates around zero, and no far deviate from zero. That is, the error of speed is small. Characteristic model’s output is considerably close to the measured data. And characteristic model can accurately describe the characteristics of object. 4.3. Comparison between fixed parameters and characteristic models The modeling process of fixed parameters model and characteristic are shown in Table 3. Comparing the sum of squared errors between fixed parameters and characteristic models based on different identification, and almost all algorithms show that the squared errors of characteristic model is smaller than that of fixed parameters model whether from the modeling data or validation data. In addition, according to the error comparison in Table 4, the error of characteristic model’s output is significantly smaller than that of fixed parameters model when the reference speed is 120 rpm. And the error of characteristic model is not large when the reference speed is 30 rpm. It is obvious that characteristic model’s output accurately approximate to the measured data. And the characteristic model is a more complete representation of the actual system. 5. Conclusions Characteristic model is a kind of identification model which is suitable for the design of control strategy of ultrasonic motor. A good model is the foundation of analysis and design of ultrasonic motor control system. Characteristic model is the accurate descrip-
tion of the actual object such as ultrasonic motor which is different from the traditional identification model. The modeling method of ultrasonic motor’s characteristic model is proposed in this paper. The accuracy of characteristic model is also theoretically proved. The comparison of the tested and calculated data shows that the characteristic model can describe the characteristics of ultrasonic motor better than the traditional identification model. And the proposed modeling method provides a feasible means for control modeling of traveling wave ultrasonic motor. References [1] H. Zhu, Z. Li, C. Zhao, An efficient approach to optimize the vibration mode of bar-type ultrasonic motors, Ultrasonics 50 (4–5) (2010) 491–495. [2] W. Chen, S. Shi, Y. Liu, P. Li, A new traveling wave ultrasonic motor using thick ring stator with nested PZT excitation, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57 (5) (2010) 1160–1168. [3] Y.X. Liu, J.K. Liu, W.S. Chen, A cylindrical traveling wave ultrasonic motor using a circumferential composite transducer, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58 (11) (2011) 2397–2404. [4] W.S. Chen, S.J. Shi, Y.X. Liu, P. Li, A new traveling wave ultrasonic motor using thick ring stator with nested PZT excitation, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57 (5) (2010) 1160–1168. [5] B. Radi, A.E. Hami, The study of the dynamic contact in ultrasonic motor, Appl. Math. Model. 34 (12) (2010) 3767–3777. [6] H. Mojallali, R. Amini, R.I. Zamanabadi, A.A. Jalali, Systematic experimental based modelling of a rotary piezoelectric ultrasonic motor, ISA Trans. 46 (1) (2007) 31–40. [7] T. Senjyu, M. Nakamura, N. Urasaki, et al., Mathematical model of ultrasonic motors for speed control, Electr. Power Comp. Syst. 36 (6) (2008) 637–648. [8] X.L. Lu, J.H. Hu, C.S. Zhao, Analyses of the temperature field of traveling-wave rotary ultrasonic motors, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58 (12) (2011) 2708–2719. [9] D.A. Stepanenko, V.T. Minchenya, Development and study of novel non-contact ultrasonic motor based on principle of structural asymmetry, Ultrasonics 52 (7) (2012) 866–872. [10] T.C. Chen, C.H. Yu, Generalized regression neural-network-based modelling approach for traveling-wave ultrasonic motors, Electr. Power Comp. Syst. 37 (6) (2009) 645–657. [11] L. Faajeng, W. Rongjong, H. Chunming, Identification and control of rotary traveling-wave type ultrasonic motor using neural networks, IEEE Trans. Control Syst. Technol. 9 (4) (2001) 672–680. [12] L. Faajeng, H. Yingchih, C. Syuanyi, Field-programmable gate array-based intelligent dynamic sliding-mode control using recurrent wavelet neural network for linear ultrasonic motor, IET Control Theory Appl. 4 (9) (2010) 1511–1532. [13] X.L. Zhang, Y.H. Tan, Modelling of ultrasonic motor with dead-zone based on Hammerstein model structure, J. Zhejiang Univ. Sci. A 9 (1) (2008) 58–64. [14] M. Shannchyi, O. Minsun, Establishing the dynamic transfer function of a shaft-driving type ultrasonic motor by system identification, in: Proc. World Congr. Intelligent Control Autom., Hangzhou, China, Jun 2004, pp. 302–305 [15] H.X. Wu, Y.W. Liu, Z.H. Liu, Y.C. Xie, Characteristic modeling and the control of flexible structure, Sci. China 44 (4) (2001) 278–291.
Contents lists available at ScienceDirect
Ultrasonics journal homepage: www.elsevier.com/locate/ultras
Characteristic model of travelling wave ultrasonic motor Shi Jingzhuo ⇑, You Dongmei School of Electrical Engineering, Henan University of Science and Technology, Luoyang 471023, China
a r t i c l e
i n f o
Article history: Received 12 May 2013 Received in revised form 13 August 2013 Accepted 10 September 2013 Available online 18 September 2013 Keywords: Travelling wave ultrasonic motor Characteristic model Identification
a b s t r a c t In general, the design and analysis of ultrasonic motor and motor’s control strategy are based on mathematical model. The academic model is widely used in the analysis of traveling wave ultrasonic motor (TWUSM). But the dispersive characteristic of piezoelectric ceramics and other complicated process, such as the friction, make the model’s precision not so accurate. On the other hand, identification modeling method, which is built based on the tested data, has obtained increasing application in the study of ultrasonic motor’s control technology. Based on the identification model, many control strategies can be designed easily. But the identification model is an approximate model, so if a more accurate model of ultrasonic motor can be obtained, the analysis and design of motor control system will be more effective. Characteristic model is a kind of identification model which can accurately describe the characteristics of TWUSM. Based on the tested data, this paper proposes the modeling method of ultrasonic motor’s characteristic model. The paper also makes a comparison of the effectiveness of different identification algorithms. Aiming at the speed control of ultrasonic motor, the influence of the parameter’s initial values on the precision of model is discussed. The calculating results indicate the availability of this characteristic model. Ó 2013 Elsevier B.V. All rights reserved.
1. Introduction The mathematical model of traveling wave ultrasonic motor (TWUSM) is an important foundation for the analysis, design and performance evaluation of the motor’s control system [1,2]. In order to improve the performance of TWUSM’s control system and study more reasonable control strategies, we must make sure that the mathematical model is suitable for control applications [3–5]. Currently, there are three kinds of modeling methods of TWUSM: theoretical modeling, neural network modeling and identification modeling. The theoretical modeling mainly includes equivalent circuit modeling, finite element analysis modeling and mechanism analysis modeling. Mainly due to the characteristics of complexity, time-varying and dispersion of electromechanical energy conversion and transmission mechanism, it is difficult to obtain mathematical model which can precisely describe dynamic operating characteristics of TWUSM using theoretical modeling [6– 9]. Therefore, theoretical modeling is not suitable for the modeling of the motor. When neural network modeling is used, more incentive function and various learning rules are required. In addition, neural network model is more complex and not suitable for the design of the controller due to complex topological structure, thus this model is not commonly used [10–12]. The identification method based on experimental data has become the main method of the ⇑ Corresponding author. Tel.: +86 379 64231757; fax: +86 379 64231910. E-mail address: [email protected] (S. Jingzhuo). 0041-624X/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.ultras.2013.09.012
establishment of the control model of ultrasonic motor [13,14] because of the complex operation mechanism of TWUSM. But it is generally believed that the identification model is only an approximate model under the least squares criterion, and it cannot describe the overall perspective. That is, identification model can describe the main operating characteristics of the ultrasonic motor, but it is not an accurate description. This makes the design of ultrasonic motor controller based on the identification model must be adapted to this model deviation. There is no proof that ultrasonic motor can be described by linear differential equation with constant coefficients. It is well known that ultrasonic motors have various physical nonlinearities related to contact and frictional phenomena. Characteristic model proposed by Wu Hongxin is a kind of model based on experimental data [15], which can be classified into the scope of the identification model. But the difference is that the existing theory has proved that the real-time correction of the low order (two or three-order) characteristic model is an accurate description of high order real object. Based on the characteristic model, it is possible to obtain a relatively simple structure of the controller. This paper proposes the modeling method of ultrasonic motor’s characteristic model and puts forward three methods to determine identification algorithm initial parameters, which is determined by comparing the errors and online recursion algorithm for ultrasonic motor modeling and control. The calculating results of calculation manifest the availability of the characteristic model.
726
S. Jingzhuo, Y. Dongmei / Ultrasonics 54 (2014) 725–730
where
2. Characteristic model of ultrasonic motor The theories and concepts of characteristic model provide a feasible approach for the design of low-order and intelligent controller. For ultrasonic motor, the characteristic model is used in the speed and rotating position closed-loop control. The varied difference equation can described by using characteristic model, the form of three-order or two-order are available respectively. In general, all kinds of ultrasonic motors can be represented with high order transfer function as follows
GðsÞ ¼
bm sm þ bm1 sm1 þ þ b1 s þ b0 sn þ an1 sn1 þ þ a1 s þ a0
ð1Þ
where ai, bj are the molecules and denominator coefficients of transfer function respectively i = 0, . . . , n 1, j = 0, . . . , m. Eq. (1) can be decomposed into the sum of finite rational functions which is not higher than two order and consisting of inertia, oscillation, integral and multiple-root items m h X k kpþi kv X ki pþi GðsÞ ¼ 2 þ þ þ kpþi s s þ k s þ k s þ i pþi i¼1 i¼1
þ
q X
!
kxi 2
i¼1
ðs þ xi Þ
f2 ðkÞ ¼ a1 ðkÞa2 ðkÞ ¼ f1 T½a1 ðyÞ þ a2 ðyÞ þ T 2 ½a1 ðyÞa2 ðyÞg g 0 ðkÞ ¼ b1 ðkÞ þ b2 ðkÞ ¼ TðK 1 þ K 2 Þ g 1 ðkÞ ¼ ½a1 ðkÞb2 ðkÞ þ a2 ðkÞb1 ðkÞ ¼ TðK 1 þ K 2 Þ þ T 2 ½K 1 a2 ðyÞ þ K 2 a1 ðyÞ where ar(k) = 1 Tar(y), here k is point-in-time ,which describes the sequential relationship between current moment and former moment, while ar(y) describes the dependent relationship between other variables and itself. And then, we consider integral items and multiple-root items based on the proof of the first step. Single integral term corresponding to the input–output equation is equivalent to zero real roots in the first step. With the same kind of derivation, we have
yv x ðk þ 2Þ ¼ 2ð1 T x1 Þyv x ðk þ 1Þ ð1 2T x1 Þyv x ðkÞ þ T 2 ðkv þ kx1 Þuðk þ 1Þ
ð2Þ
ð5Þ
Combining Eqs. (4) and (5), and make y⁄(k) = y(k) + yvx(k), we have
Theorem 1. For any linear time-invariant n-order objects G(s), it can be decomposed as Eq. (2). Under the proper conditions of sampling period T, when we control position maintaining or position tracking, we can describe the characteristic model by using the following two-order varying difference equation, yðk þ 1Þ ¼ f1 ðkÞyðkÞ þ f2 ðkÞyðk 1Þ þ g 0 ðkÞuðkÞ þ g 1 ðkÞuðk 1Þ
f1 ðkÞ ¼ a1 ðkÞ þ a2 ðkÞ ¼ 2 T½a1 ðyÞ þ a2 ðyÞ
ð3Þ
Here, y(k) and u(k) are discretized output and input variable quantity of the moment k of system respectively, f1(k), f2(k), g0(k), g1(k) are the time-varying coefficients, If the object G(s) is stable or contains integral items, then: (1) f1(k), f2(k), g0(k), g1(k) are the coefficients of the Eq. (3), which are slowly time varying coefficients. (2) The value range of the coefficients can be determined beforehand. (3) When the system is in dynamic process, under the conditions of same input, the output of the characteristic model is equivalent to the actual output (appropriately ensure to select the sampling period T within the output error allowed). When the system is in a steady state, the static gain of Eq. (3) D0 is equal to the static gain of Eq. (1) D = a0/b0, i.e. that is to say the outputs of the system are equal under the steady-state. (4) When static gain D = 1, in the steady state the sum of all coefficients is equal to 1, i.e.
f1 ð1Þ þ f2 ð1Þ þ g 1 ð1Þ þ g 2 ð1Þ ¼ 1 (5) If there is an integral part in object, then,
f1 ð1Þ þ f2 ð1Þ ¼ 1 Here, the so-called ‘‘sampling period satisfies certain conditions’’, that is satisfying the sampling theorem, so as to make the system still satisfies the requirements of controllability and control accuracy after discretizing from the original continuous system. Theorem 1 can be proved by two steps. First, disregarding the integral items and multiple-root items in Eq. (2), i.e. kv = 0, kxi = 0, we have
yðk þ 1Þ ¼ f1 ðkÞyðkÞ þ f2 ðkÞyðk 1Þ þ g 0 ðkÞuðkÞ þ g 1 ðkÞuðk 1Þ ð4Þ
y ðk þ 2Þ ¼ ½a1 ðkÞ þ a2 ðkÞy ðk þ 1Þ a1 ðkÞa2 ðkÞy ðkÞ þ ½T 2 ðkv þ kx1 Þ þ b1 ðkÞ þ b2 ðkÞuðk þ 1Þ ½a2 ðkÞb1 ðkÞ þ a1 ðkÞb2 ðkÞuðkÞ þ ½e1 yv x ðk þ 1Þ e2 yv x ðkÞ
ð6Þ
where ai and bi are defined the same as in Eq. (4).
e1 ¼ 2ð1 T x1 Þ ½a1 ðkÞ þ a2 ðkÞ ¼ T½a1 ðyÞ þ a2 ðyÞ 2x1 e2 ¼ ð1 2T x1 Þ a1 ðkÞa2 ðkÞ ¼ T½a1 ðyÞ þ a2 ðyÞ 2x1 T 2 a1 ðyÞa2 ðyÞ Eq. (6) can still be written as Eq. (4),
yðk þ 1Þ ¼ f1 ðkÞyðkÞ þ f2 ðkÞyðk 1Þ þ g 0 ðkÞuðkÞ þ g 1 ðkÞuðk 1Þ ð7Þ f1 ðkÞ ¼ a1 ðkÞ þ a2 ðkÞ ¼ 2 T½a1 ðyÞ þ a2 ðyÞ
ð8Þ
f2 ðkÞ ¼ a1 ðkÞa2 ðkÞ ¼ f1 T½a1 ðyÞ þ a2 ðyÞ þ T 2 ½a1 ðyÞa2 ðyÞg ð9Þ g 0 ðkÞ ¼ T 2 ðkv þ kx1 Þ þ b1 ðkÞ þ b2 ðkÞ ¼ T 2 ðkv þ kx1 Þ þ TðK 1 þ K 2 Þ
ð10Þ
g 1 ðkÞ ¼ ½a1 ðkÞb2 ðkÞ þ a2 ðkÞb1 ðkÞ ¼ TðK 1 þ K 2 Þ þ T 2 ½K 1 a2 ðyÞ þ K 2 a1 ðyÞ
ð11Þ
Now we can gain the following results. (1) In general, the change of ar(k) is very small in each step. It is about 0.001. Therefore, we know that the coefficients f1(k), f2(k), g0(k) and g1(k) a are slowly time-varying according to Eqs. (8)–(11). (2) If there is not integral items and no multiple-root items, the range of f1(k) and f2(k) can be determined beforehand. Considering their maximum range, we can choose
f1 ðkÞ 2 ð1; 2;
f 2 ðkÞ 2 ½1; 0Þ
S. Jingzhuo, Y. Dongmei / Ultrasonics 54 (2014) 725–730
Since g0(k) and g1(k) not only depend on T and ar(y), but also depend on Kr, that is to say they are related to static gain D. If the range of the maximum value of static gain is known by input transformation, we can get g0(k), and g1(k) 1. If there is not integral items and/or no multiple-root items, f1(k) is less than but approximate to 2, while f2(k) is larger than but approaches 1. (3) If integral items and no multiple-root items do not exist, each step of the sampling period is less than 0.001, this may result in output error in dynamic state. When integral items and multiple-root items exist, there also exists a small error in dynamic state. In steady state, static gain D(1) can be expressed as follows:
Dð1Þ ¼
n X g 0 ð1Þ þ g 1 ð1Þ ki b0 ¼ ¼ 1 f1 ð1Þ f2 ð1Þ i¼1 ki a0
ð12Þ
It means that the static gain of the characteristic model D(1) is equal to that of the original plant b0/a0. (4) Due to Eq. (12) it is clear that when D = b0/a0 = 1, we have
f1 ð1Þ þ f2 ð1Þ þ g 1 ð1Þ þ g 2 ð1Þ ¼ 1 This means when static gain D = 1, the sum of all coefficients of the characteristic model is equal to one in a steady state. (5) If an integral item exists, it is easy to derive that in a steady state of position control
characteristic model by comparing identification methods. Third, the fixed parameters model is used to realize error comparison with the characteristic model. System identification is based on the input and output data to determine a mathematical model, which is equivalent with the studied system from a set of model categories. And system identification methods are varied. 3.1. Experimental data Identification method is based on the measured input and output data. Self-designed experimental system [3] block diagram is shown in Fig. 1, and the system architecture described in paper [3]. Experimental motor is the Shinsei USR60, which is a two-phase traveling wave ultrasonic motor. The photoelectric encoder E (Encoder) rigid connection with the motor’s shaft provides speed feedback signal. The driving circuit is the two-phase H-bridge structure constituted by MOSFET, and the control circuit, which takes the low-cost DSP chip DSP56F801 as a core, in supplemented by the PWM (Pulse Width Modulation) signal generator with phase shift realizes the control of the MOSFET switch state in the driving circuit. In the experimental data, we can choose 10 sets data as modeling data. Besides, we elect 5 sets as validation data to verify the validity of the model and help to select identification algorithm, with the reference speed 120, 100, 90, 80, 30 rpm. 3.2. Choice and comparison of identification methods In general, the identification model can be described by the following difference equation
f1 ð1Þ þ f2 ð1Þ ¼ 1 Until now, the proof of the Theorem 1 is completed.
Aðz1 ÞyðkÞ ¼ zd Bðz1 ÞuðkÞ þ eðkÞ
Corollary 1. For a linear constant high-order plant G(s) that satisfied the condition of Theorem 1, when velocity control or acceleration control is required, its characteristic model can be expressed with a three-order time varying difference equation, which is shown below.
yðk þ 1Þ ¼ f1 ðkÞyðkÞ þ f2 ðkÞyðk 1Þ þ f3 ðkÞyðk 2Þ þ g 0 ðkÞuðkÞ þ g 1 ðkÞuðk 1Þ þ g 2 ðkÞuðk 2Þ
727
ð13Þ
Compared with the speed control, the object is added an integral part in position control. Therefore, the design of the controller is also different. When the object G(s) is stable, then (1) In dynamic process, with the same input, the output of the characteristic model is equal to that of the practical object. In a steady state, the two outputs are equal, which means their static gains are equal D0 = D. (2) When the static gain D = 1, the sum of all the coefficients is equals to one in a steady state, i.e.
f1 ð1Þ þ f2 ð1Þ þ f3 ð1Þ þ g 0 ð1Þ þ g 1 ð1Þ þ g 2 ð1Þ ¼ 1 3. Fixed parameters model of the ultrasonic motor To establish the characteristic model of ultrasonic motor, we need to establish its fixed parameters model. There are three reasons. First, we need to prepare for the choice of the structure of the characteristic model. That is to identify the model structure (described by values of parameters na, nb and d in Eq. (14)) with a small sum of squared errors. Second, we determine the appropriate identification algorithm to identify initial parameters of the
ð14Þ
where
(
Aðz1 Þ ¼ 1 þ a1 z1 þ a2 z2 þ þ ana zna Bðz1 Þ ¼ b0 þ b1 z1 þ b2 z2 þ þ bnb znb
Here, y(k) and u(k) are the current speed and frequency control respectively. d is order delay. a1 ; . . . ; ana ; b0 ; . . . ; bnb are the parameters to be identify. e(k) is the white noise. z is time delay. na and nb are the order of A(z1) and B(z1) respectively. na, nb and d are determined by least squares model structure identification, which is based on experimental data. The paper selected 90 rpm experimental data to determine the structure of the model in order to select more suitable order. The calculation results show that when na = 3, nb = 2, d = 0 the loss function of the model and the final prediction error is minimum. This order is used in the following identification of the ultrasonic motor. In order to determine a suitable identification method for ultrasonic motor’s modeling, Least Squares algorithm (LS), Recursive Least Squares (RLS), Recursive Extended Least Squares method (RELS) and Recursive Maximum Likelihood (RML) are respectively used to identify model’s parameters, and the identification method is determined by comparing the sum of squared errors of each method. The calculation results are shown in Table 1. View from the sum of squared error of modeling data, the Least Squares algorithm and Recursive Maximum Likelihood method achieved a relatively small value of 229.1345 and 229.3930. Verify the results from the validation data, the Least Squares and Recursive Maximum Likelihood algorithms are similar too. Since Recursive Maximum Likelihood algorithm is relatively complex, the Least Squares algorithm is selected to identify the initial parameters. Using the Least Squares algorithm to establish ultrasonic motor’s model, the comparison of model output and measured data based on verification data shown in Fig. 2 shows that the model
728
S. Jingzhuo, Y. Dongmei / Ultrasonics 54 (2014) 725–730
− + −
Uref + Nref
Frequency adjustment
Speed controller
+
Voltage amplitude controller of phase A and B
Pulse width adjustment
Phase-shift PWM generator
H-bridge driver
Ultrasonic motor
− Encoder signal processor
E
Fig. 1. Structure of the experimental system for speed control.
Table 1 Error comparison of identification algorithms. Sum of squared errors
LS
RLS
RELS
RML
Modeling data Validation data
229.1345 251.996
356.4345 372.1168
346.4442 365.4386
229.3930 252.6143
140
Speed /(rpm)
120 100 80
Model output
…Tested data
60
Nref=120rpm
40 20
0.05
0.10
0.15
0.20
0.25
Time/s Fig. 2. Comparison between model output and tested data (Nref = 120 rpm).
output is consistent with the measured data, and the maximum absolute error is 6.577 rpm. 4. Establishment of the characteristic model According to the establish process of fixed-parameter model and comparing the sum of square error of different model structure, the structure of characteristic model is determined three order, i.e. na = 3, nb = 2. The least squares algorithm is chose to identify the characteristic model’s initial parameters by comparing the effect of the recognition algorithm. Thus, the characteristic model of ultrasonic motor can be written in the form as follows
yðkÞ ¼ a1 ðkÞyðk 1Þ þ a2 ðkÞyðk 2Þ þ a3 ðkÞyðk 3Þ þ b0 ðkÞuðkÞ þ b1 ðkÞuðk 1Þ þ b2 ðkÞuðk 2Þ
ð15Þ
where y(k-n) and u(k-n) are the speed and frequency control of former n moment respectively. a1(k), a2(k), a3(k), b0(k), b1(k), b2(k) are the time-varying characteristic model’s parameters. After determining the structure of the model, the initial value of identification algorithm and online recursive identification are needed to be determined. 4.1. Determination of identification’s initial value This section is to determine the initial value of h(k) and P(k), i.e. h(0) and P(0). Identification methods are used to estimate the model parameters hðkÞ ¼ ½ a1 ðkÞ a2 ðkÞ a3 ðkÞ b0 ðkÞ b1 ðkÞ b2 ðkÞ . Based on
the idea of identification, h(k) (estimated parameters for the current moment) is equivalent to the sum of h(k 1) (estimated parameters for the former moment) and correction term. Here, correction term describes the gaps between the estimated parameters and the truth-value, which is the product of every step gain matrix and the error. The error is the difference between the actual measurement data at current moment and predicted output data based on the model which is determined by the former estimated parameter. With the increase of the recursion steps, the difference becomes smaller and the estimated parameters at the current moment approximate the former. Then model parameters h(k) approach to the truth-value. Considering the operating characteristics of ultrasonic motor, three kinds of methods are proposed to determine the initial value of h(0) (the three kinds of methods hereinafter referred to as Method 1, Method 2, and Method 3). (1) According to the analysis of Section 3.2, we make a comparison of the sum of squared errors of several identification methods which has fixed-parameters and choose the method’s parameters as initial parameters which has the minimum sum of squared errors. (2) The motor system start running from zero, and the initial values mainly work at the starting stage of the control, which is low-speed area operation starting from zero speed. So the initial value can be set to the off-line identification parameters obtained in the low-speed case. (3) For different reference speed, the Least Squares algorithm is used to obtain a three-order model. Every speed corresponds to a set of model parameters which are the initial value of identification parameters. If there are more than one set of data of the same reference speed, the initial parameters are the combined recognition result of these multiple sets of data. The state estimation error covariance matrix P(k) is directly related to the correction effect of model parameters. The covariance is greater, and this means the difference is between the estimated value and truth-value greater. At the same time, the gain vector will be amplified, and the correction effect on the model parameter will be weighted too. There are two methods to determine P(0). One is make P(0) = aI, here I is a unity-matrix. The other one is the off-line identification method. 4.2. Determination of identification algorithm If the initial value is determined, the online recursive identification of the characteristic model is to be determined next. In order to obtain a suitable identification method, LS, RLS, RELS and RML are respectively used to recursive calculation. The sum of squared errors is calculated during recursive process according to the timevarying parameters of characteristic model. That is to say, the sum
729
S. Jingzhuo, Y. Dongmei / Ultrasonics 54 (2014) 725–730
Speed /(rpm)
30
θ
P
25 20
Model output
… Tested data
15
Nref =30rpm
k
10 0.1
0.2
0.3
0.4
0.5
0.6
Time/s Fig. 5. Comparison between model output and tested data (Nref = 30 rpm).
θ
k 3
P k Speed error /(rpm)
2 1 0 -1 -2 0.04
0.08
0.12
0.16
0.20
0.24
Time/s Fig. 6. Speed error in identify process (Nref = 120 rpm). Fig. 3. MATLAB program flow chart of RELS.
0.6
140
Speed error /(rpm)
Speed /(rpm)
120 100 80
Model output 60
… Tested data
40 20
0.4 0.2 0.0 -0.2 -0.4
Nref =120rpm
0.0 0.05
0.10
0.15
0.20
0.1
0.2
0.25
0.3
0.4
0.5
0.6
Time/s
Time/s
Fig. 7. Speed error in identify process (Nref = 30 rpm).
Fig. 4. Comparison between model output and tested data (Nref = 120 rpm).
of squared errors is calculated at every sampling period. And the sum of squared errors of every method is compared to determine the appropriate identification method. MATLAB is used in the calculation. And the calculation results are shown in Table 2. In Table 2, through the survey of RLS, RELS and RML, the sum of squared errors of Method 3 is less than the former two methods
whether viewing from the result of modeling data or validation data. Then, compare the Method 3 among RLS, RELS and RML, the sum of squared errors of RELS and RML are less than RLS, and the sum of squared errors of RML is a little bit bigger than RELS. Taking all factors into consideration, the RELS is chosen as the online recursive identification to build the characteristic
Table 2 Comparison of different identification algorithms. RLS
Modeling data Validation data
RELS
RML
Method 1
Method 2
Method 3
Method 1
Method 2
Method 3
Method 1
Method 2
Method 3
231.19 243.06
231.19 243.06
121.79 74.027
227.99 237.18
227.99 237.18
110.97 33.2096
227.92 237.27
227.8 237.02
112.12 33.3089
730
S. Jingzhuo, Y. Dongmei / Ultrasonics 54 (2014) 725–730
Table 3 Comparison of sum of squared errors between fixed parameters and characteristic models based on different identification algorithms. RLS
Modeling data Validation data
RELS
RML
Fixed parameters
Characteristic model
Fixed parameters
Characteristic model
Fixed parameters
Characteristic model
356.43 372.12
121.79 74.027
346.44 365.44
110.97 33.2096
229.39 252.61
112.12 33.3089
Table 4 Error comparison of validation data.
Fixed parameters Characteristic model
Reference speed (rpm)
Sum of squared error
Maximum absolute error (rpm)
Maximum relative error (%)
Average error (rpm)
120 120 30
72.7402 22.042 1.8128
6.577 2.62 0.56
11.24 3.65 2.93
1.4851 0.9871 0.1665
model; meanwhile, Method 3 is used to determine the initial parameters. The program flow chart of RELS is shown in Fig. 3. Based on characteristic model Eq. (15), Method 3 is used to online recursive identification and to establish ultrasonic motor’s model by Using RELS, the comparison of model output and measured data based on verification data is shown in Figs. 4 and 5. The reference speed values are 120 rpm and 30 rpm. Compared to Fig. 2, the maximum absolute error of Fig. 4 is only 2.62 rpm. The error of the model improved obviously. The maximum absolute error of Fig. 5 is 0.56 rpm, it is clear that the model output is very close to the measured data. The errors during identify process are shown in Figs. 6 and 7. From the figure we can see that the value of error fluctuates around zero, and no far deviate from zero. That is, the error of speed is small. Characteristic model’s output is considerably close to the measured data. And characteristic model can accurately describe the characteristics of object. 4.3. Comparison between fixed parameters and characteristic models The modeling process of fixed parameters model and characteristic are shown in Table 3. Comparing the sum of squared errors between fixed parameters and characteristic models based on different identification, and almost all algorithms show that the squared errors of characteristic model is smaller than that of fixed parameters model whether from the modeling data or validation data. In addition, according to the error comparison in Table 4, the error of characteristic model’s output is significantly smaller than that of fixed parameters model when the reference speed is 120 rpm. And the error of characteristic model is not large when the reference speed is 30 rpm. It is obvious that characteristic model’s output accurately approximate to the measured data. And the characteristic model is a more complete representation of the actual system. 5. Conclusions Characteristic model is a kind of identification model which is suitable for the design of control strategy of ultrasonic motor. A good model is the foundation of analysis and design of ultrasonic motor control system. Characteristic model is the accurate descrip-
tion of the actual object such as ultrasonic motor which is different from the traditional identification model. The modeling method of ultrasonic motor’s characteristic model is proposed in this paper. The accuracy of characteristic model is also theoretically proved. The comparison of the tested and calculated data shows that the characteristic model can describe the characteristics of ultrasonic motor better than the traditional identification model. And the proposed modeling method provides a feasible means for control modeling of traveling wave ultrasonic motor. References [1] H. Zhu, Z. Li, C. Zhao, An efficient approach to optimize the vibration mode of bar-type ultrasonic motors, Ultrasonics 50 (4–5) (2010) 491–495. [2] W. Chen, S. Shi, Y. Liu, P. Li, A new traveling wave ultrasonic motor using thick ring stator with nested PZT excitation, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57 (5) (2010) 1160–1168. [3] Y.X. Liu, J.K. Liu, W.S. Chen, A cylindrical traveling wave ultrasonic motor using a circumferential composite transducer, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58 (11) (2011) 2397–2404. [4] W.S. Chen, S.J. Shi, Y.X. Liu, P. Li, A new traveling wave ultrasonic motor using thick ring stator with nested PZT excitation, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 57 (5) (2010) 1160–1168. [5] B. Radi, A.E. Hami, The study of the dynamic contact in ultrasonic motor, Appl. Math. Model. 34 (12) (2010) 3767–3777. [6] H. Mojallali, R. Amini, R.I. Zamanabadi, A.A. Jalali, Systematic experimental based modelling of a rotary piezoelectric ultrasonic motor, ISA Trans. 46 (1) (2007) 31–40. [7] T. Senjyu, M. Nakamura, N. Urasaki, et al., Mathematical model of ultrasonic motors for speed control, Electr. Power Comp. Syst. 36 (6) (2008) 637–648. [8] X.L. Lu, J.H. Hu, C.S. Zhao, Analyses of the temperature field of traveling-wave rotary ultrasonic motors, IEEE Trans. Ultrason. Ferroelectr. Freq. Control 58 (12) (2011) 2708–2719. [9] D.A. Stepanenko, V.T. Minchenya, Development and study of novel non-contact ultrasonic motor based on principle of structural asymmetry, Ultrasonics 52 (7) (2012) 866–872. [10] T.C. Chen, C.H. Yu, Generalized regression neural-network-based modelling approach for traveling-wave ultrasonic motors, Electr. Power Comp. Syst. 37 (6) (2009) 645–657. [11] L. Faajeng, W. Rongjong, H. Chunming, Identification and control of rotary traveling-wave type ultrasonic motor using neural networks, IEEE Trans. Control Syst. Technol. 9 (4) (2001) 672–680. [12] L. Faajeng, H. Yingchih, C. Syuanyi, Field-programmable gate array-based intelligent dynamic sliding-mode control using recurrent wavelet neural network for linear ultrasonic motor, IET Control Theory Appl. 4 (9) (2010) 1511–1532. [13] X.L. Zhang, Y.H. Tan, Modelling of ultrasonic motor with dead-zone based on Hammerstein model structure, J. Zhejiang Univ. Sci. A 9 (1) (2008) 58–64. [14] M. Shannchyi, O. Minsun, Establishing the dynamic transfer function of a shaft-driving type ultrasonic motor by system identification, in: Proc. World Congr. Intelligent Control Autom., Hangzhou, China, Jun 2004, pp. 302–305 [15] H.X. Wu, Y.W. Liu, Z.H. Liu, Y.C. Xie, Characteristic modeling and the control of flexible structure, Sci. China 44 (4) (2001) 278–291.