AC Servo Motor Position Sensorless Control using Mechanical Springs Akira Shimada
Yu Kishiwada
Michiyo Arimura
Department of Electrical System Engineering Polytechnic University Hashimotodai 4-1-1 Sagamihara,Kanagawa 229-1196,Japan Email:
[email protected]
Department of Electrical System Engineering Polytechnic University Hashimotodai 4-1-1 Sagamihara,Kanagawa 229-1196,Japan
Toyo Denki Seizo K.K. 2-9-2 Kyobashi Chuo-ku Tokyo,104-0031,Japan
Abstract— This paper describes a position sensorless control technique on AC servo motor position control systems. We had previously presented a paper on the DC servo motor position sensorless control technique using mechanical springs. It was based on a point of view that mechanical springs form the key components for the observability. On the basis of the result obtained from the successful experiment, we assumed that the AC servo motor position sensorless control system applying the vector control method would be identical as in the case of the DC motor. Using vector control, the controller needs the data of the magnetic pole position on the rotor of the AC servo motor.@By using the AC servo motor perfect position sensorless control technique, the controller should estimate both the magnetic pole position and mechanical position. In this paper, we demonstrate the estimation method for the latter as an initial step in the new control technology.
I. I NTRODUCTION Sensorless control technique essentially is a technique that does not reduce the number of sensors employed but controls objects employing estimations. This technique estimates variables that cannot be measured with the help of the other measured physical variables. Hence, It is a control technique and is always used as an application technique in sensors. Shimada et.al. presented a technique that controls the position and velocity of a mechanical system [1], [2]. This mechanical system comrises a cart, a DC servo motor, and a position sensor. The cart was driven by the motor via a timing belt. The position sensor is not used to control the mechanical system but is used to evaluate the control system. The feature of the control system is that it does not use a position or velocity sensor for control and requires the detection of the electrical driving voltage and current of a DC servo motor. The controller controls the position of the cart by estimating its position and velocity. Further, the system requires the useage of mechanical springs for the estimation thar has already been realized by theoretical analysis and certain experiments. The technique can be applied to inexpensive robot hands, etc. However, as yet, it has been considered only for application in DC servo systems. In the light of these fact, this paper
0-7803-9511-5/06/$20.00 ©2006 IEEE
presents a sensorless control technique using AC servo motors that constitutes the other appllication of the control technique. In the past studies, sensorless control techniques for AC servo motors virtually implied techniques that estimated the angle of magnetic poles and controlled the angle velocity based on vector control methodology [3], [4]. However, these technique on the whole could not estimate the angle position over 360 degrees of the electric angle, since it escentially employed the small ranged angle control method within 360 degrees of the electric angle. Therefore, when the system requests the initial position data of mechanical system connected to AC servo motors or it has error larger than 360 degreees of electric angle by electrical noises, it cannot fix the error problem. Incidentally, it is well known that AC servo motors can be considered in the same manner as DC servo motors when they are controlled by the vector control method or equivalent sine wave drive method. Further, this paper focuses attention on the feature of the AC servo motors that employs the vector control method and introduces a position sensorless control method of the mechanical systems driven by such AC servo motors. Currently, two types of application methods can be employed. One is a sensorless control method including estimation of the angle of the magnetic poles [3], [4]: it controls the d-axis electrical current so that it is brought to 0 amperes and the q-axis electrical current so that it is brought to the intended value based on the vector control method. Further, a sensorless control technique is installed to the mechanical system of which the cart is supported by single or double mechanical springs that are connected to the cart from a side or eather side of it. The other method is that in which the sensorless control is based on vector control but it uses rotary encoders installed on AC servo motors in order to only detect the angles of the magnetic poles. The first method is termed the perfect sensorless control: however this paper introduces the second method since it is deemed the first step in the development process.
559
AMC’06-Istanbul, Turkey
Fig. 3.
Fig. 1.
Equivalent system to DC servo motor system
Block diagram of the AC servo motor and mechanical plant
Fig. 4. Fig. 2.
Current feedback system
View of the experimental apparatus
II. E LECTRICAL C URRENT CONTROL OF AC SERVO MOTOR The AC servo motors studied in this paper comprise threephase permanent magnet synchronous motors that are employed for servo control. In order to perform an analysis, a three-phase equivalent circuit model of the motor should be converted to a two-phase model and transformed to the dq transformation model. This model is shown in Fig. 1, wherein ”⊗” implies multiplication. Further, it is assumed that the control plant has a degree of freedom in the linear mechanical system of which the main portion is a cart driven by the AC servo motor via the timing belt and pulley of radius r. Moreover, the principal purpose of this research is position control of the cart that is shown in Fig.2. The cart is tightened by both the right- and -left mechanical springs or either of them. When the motor is powered off, the cart stops at an equilibrium point. At the equilibrium pint, assume that the position of the cart is z, the velocity is v, the angle velocity of the rotor is ωm , and that the total mass is M , which includes the mass of the cart m and the mass equivalent of the moment of inertia of the rotor and the pulley J. Further, let the coefficient of the toatl viscous friction be D, the elastic coefficient of the spring, k; the number of springs n; the driving force, f ; the driving torque of the motor, τ ; the self-inductance for the single phase, La ; the resistance for the single phase Ra ; the number of the poles pair, p; the maximum magnetic flux, φf a . Next, a normal type vector controller includes two electrical current feedback loops related to the d-axis and q-axis and a decoupling function. As shown in Fig.1, when the control voltages vd , vq are expressed as vd = ud − vdd , vq = uq + vqd , the transfer functions from ud , uq to id , iq can be expressed
as 1/(sLa + Ra ). However, if the controller perfectly cancels the voltage vqd observed on the q-qxis at the beginning, the equivalent control plant becomes unobservable. In order to be observable, the controller leaves the leaves the pass with the velocity v to the voltage vqd . Therefore, the control voltages vd , vq should be expressed as follows: vd
=
ud +
pLa · iq · v r
(1)
pLa · id · v (2) r At that juncture, the system becomes equivalent to the one represented in Fig.3. For further analysis, in order to employ PI control algorithms, for exaple, the voltages ud , uq can be expressed as Equation (3),(4) in order to make the values of id , iq equal to their ientended values. Further the system is represented in Fig.4 wherein Equation (4) includes the . decoupling voltage input vqd vq
=
uq + uq2 = uq +
ud
= Kpi (idr − id ) + Kii
uq
= Kpi (iqr − iq ) + Kii
t
0t 0
(idr − id )dt
(3)
(iqr − iq )dt + vqd (4)
When the value of idr is equal to 0 ampere, the gains of the above equations are suitably selected, and the gain block 1/(pφf a ) is inserted before the point of reference of the electrical current of the q-axis, the control system then becomes equivalent to the system repesented in Fig.5. This implies that the control system is expressed as Equation (5),(6), when we focus on the inside of the q-axis electrical current feedback loop. Moreover, the current feedback loops
560
TABLE I P HISICAL PARAMETERS OF THE PLANT
exhibit high bandwidth performance and the control system is similar to that expressed as Equations (7),(8). The state equation related to x1 = [z, v, iq ]T , u1 = uq , y1 = iq is expressed as follows: x˙ 1 y1
⎡
=
A1 x1 + B1 u1
(5)
=
C1 x1
(6)
⎤
Variables Mass of the cart M Coefficient of viscosity Dz Radius of pulley r Coef. of elasticity k Number of spring n Armature resistance R Armature self inductance L E.M.F. constant Ke Torque constant KT Rotor inertia J Coef. of rotor viscosity Dm Current control gain Kpi Current control integral gain Kii Maximum magnetic flux φf a Number of pole pairs p
0 1 0 pφf a ⎥ ⎢ − nk D −M where A1 = ⎣ M Mr ⎦ , pφ a 0 − Lafra − R La ⎤ ⎡ 0 B1 = ⎣ 0 ⎦ , C1 = [0, 0, 1], M = J/r2 + m, D=
1 La r Dm /r2
+ Dz .
TABLE II C ONTROL PARAMETERS (F OR BOTH SPRINGS )
The state equation related to x2 = [z, v]T , u2 = τ, y2 = z is expressed as follows.
where A2 =
x˙ 2 y2 0 − nk M
Variables Weight function Qd Weight function Rd State feedback gain Fb Integral gain Ki Observer gain Hd Feedfoward gain Kp
= =
A2 x2 + B2 u2 (7) C2 x2 (8)
1 0 , B2 = , C2 = [1, 0]. 1 D −M Mr
Variables Weight function Qd Weight function Rd State feedback gain Fb Integral gain Ki Observer gain Hd Feedforward gain Kp
If the variables n, k is n, k > 0, Equations (5),(6) are observable and the observer related to those equations can be realized[1]. In practice, in order to design a digital observer, the zero-order hold function is assumed to be the input for Equation (5), and a differential equation is designed. Finally, the state observer can thus be designed. Subsequently, the state
Equivalent control plant
Values of parameters Diag(106 , 105 , 102 ) 1 [58.6278,19.3741] 0.6126 [0.0001,0.01,0.149,-0.1778] 380
TABLE III C ONTROL PARAMETERS (F OR ONE SPRINGS )
III. D ESIGN OF O BSERVER AND C ONTROL SYSTEM
Fig. 5.
Values of the parameters 0.16 [kg] 0.0355 [N · s/m] 0.008825 [m] 160 [N/m] 1 or 2 3.6 [Ω] 0.9 [mH] 0.35 [Vs/rad] 0.35 [Ns/rad] 4.7 × 10−6 [kgm2 ] 2.8 ×10−5 [Nms/rad] 500 400 0.175 4
Values of the parameters Diag(106 , 105 , 102 ) 1 [65.6948,19.3744] 19.3712 [0.0001,0.01,0.149,-0.1778] 225
variables position z, velocity v, and current iq are estimated. Further, the position estimation zˆ is treated as the real output z. Next, a digital position control system using Equation (7) and (8) can be designed. In Fig.6, the block diagram related to the d-axis is dropped. In this paper, this control system is assumed to be implemented as a digital control system with a sampling time of 1 ms. Moreover, for the controller, the current feedback controller including the decoupling controller is implemented with a sampling time of 0.1 ms. On the other hand, the observer may be implemented as a digital system with 0.1 ms of sampling time, or may be realized as the digital system with a sampling time of 1 ms by using decimation of the sampled data of uq , iq . At the time of simulation, we adopted a sampling time of 1 ms. IV. S IMULATION RESULTS
Fig. 6.
Block diagram of control system
The simulation results related to the presented control method under the conditions listed in Tables I,II, and III are shown in Fig7 and Fig.8.
561
The simulation results appear to indicate good performances. V. C ONCLUSION In this paper, a sensorless position control using AC servo motors employing mechanical springs are presented. This presentation aims to extend the sensorless control technique using DC servo motors to the one using AC servo motors. In the future, the experiments related to the presented technique will be carried out and improved upon by using sensorless techniques [3], [4] that estimate the angle of the magnetic poles. R EFERENCES [1] Akira Shimada, Koji Enomoto : Realization of Position Sensorless Control System Using Spring, Trans.of IEEJ IA, Vol.125,No.12, pp.12681273, 2004 [2] Shimada, Kishiwada, Fujita, Arimura: Sensorless Control based on Observerbility Using Mechanical Spring, The 47th Automatic Control Annual Joint Conference, 912,2004 [3] Ichikawa, Chen, Tomita, Doki, Okuma: Sensorless Controls of SalientPole Permanent Magnet Syncronous Motors Using Extended Electromotive Force Models,Trans.IEEJ IA,122-12,pp.1088-1096, 2002 [4] Shinnaka, Toba, Zhang: Control Technologies for Sensorless Drive of Permanent Magnet Synchronous Motors,IEEJ IA Conference,Symposium,1-S15-4,I-107-112,2004 [5] Sugimoto, Koyama, Tamai: Practice of Theory and Design on AC Servo System, Sougosyuppansha, 1990
Fig. 7.
Result of the control simulation (Both spring)
Fig. 8.
Result of the control simulation (One spring)
562
