A sensorless approach for tracking control problem of tubular linear synchronous motor

Nguyen Hong Quang1, Nguyen Phung Quang2, Nguyen Van Lanh3 1Faculty of Mechanical, Electrical, Electronics Technology, Thai Nguyen University of Technology, Thai Nguyen City, Vietnam 2Institute for Control Engineering and Automation, Hanoi University of Science and Technology, Hanoi, Vietnam 3School of Electrical and Data Engineering, Faculty of Engineering and Information Technology, University of Technology Sydney, Sydney, Australia


INTRODUCTION
During the past few years, tubular linear synchronous motors (TLSM) have been intensively applied to various applications including robotics, transportation, and vehicle systems because of the absence of mechanical reduction and transmission devices (gears and lead screws) permits to obtain higher precision and reduced dimensions in comparison with rotary motors. Recently, the merits of using TLSM have been pointed out by [1]- [3], which can be listed as low cost, the durable structure, reliable operation. In addition, there has been a large number of researches devoted to applications of TLSM such as active vehicle suspension [4], the planar magnet array [5], jetting dispenser [6], and two-dimensional nanopositioning [7]. More recently, the progress of modern control engineering has been toward the tracking problem of TLSM, which can be noted as thrust optimization [8], model predictive control [9], [10], and fuzzy control [11], [12].
Tubular linear synchronous motors whose structure as shown in Figure 1 contains a tube (slider) with mounted drive magnets, and three phases winding in stator placed differently 120 o of electrical angle. Without auxiliary reducer or transmission, TLSM is capable of operating effectively by eliminating mechanical hysteresis. That however also rises the sensitiveness on the movement of slider due to frictional force, load variation, and non-sinusoidal flux. These unexpected forces shrink performance of the motion system both the transver- In recent years, sensorless control for linear permanent magnet motor have received a great deal of attention from industrial applications including: linear tubular motors [12], [18], [19], position sensorless control [4], [20], [21], end-effects [11], [22], [23]. Most of the researches on sensorless approach for linear permanent magnet motors has taken advantage of back electromotive (EMF) [19], [24], currents, and voltages into account to observe the velocity. Unfortunately, the main drawbacks of these methods are that the measurement of currents and voltage usually contain unexpected noise due to the impact of a power converter [25]. Moreover, at low speed or stopping operation, the value of EMF could be unreliable or vanish, which leads to the difficult implementation of the sensorless approach. Further, the occurrence of uncertainty in parameters (inductance and resistance) of the TLSM may result in noticeable estimation errors. Besides, to compensate impacts on control performance of the uncertainties and modeling imprecision (e.g. friction, parameter variation, and load disturbance), the adaptive method [26], [27] is employed to determine the required thrust force. The studies however have not been concerned with the sensorless control problem. It is worth noting that the position sensor is usually mounted on TLSM and the ordinary control scheme of TLSM is illustrated by Figure 2. However, obtaining velocity from the position by taking a derivative can lead to inaccurate results due to the noise contained in a position measurement.
From the issues pointed out above, this paper concerns tracking control problems of TLSM subjected to unknown loads. By employing a novel observer-based control approach, our work aims to solve a practical problem that the velocity of TLSM can not be measured directly and differential calculation from the measured position is inaccurate. To outline, our contributions can be highlighted: -As mentioned above, the use of sensorless approach involves difficulties in dealing with noise measurements and electrical parametric uncertainties. To alleviate these concerns, this paper provides a novel observer which requires the measured position to estimate the velocity of TLSM. This observer ensures that the velocity error exponentially converge to zero. Moreover, the method of choosing proper parameters for the observer is presented. -In addition, the multiple-loop control design which includes both a position-velocity controller and a current controller is provided to improve performance of the tracking control problem. By using the Lyapunov direct method, the position the position and velocity tracking errors are ensured to converge to small arbitrary values. The organization of this paper includes 5 parts as follows. The next presents a dynamic model of the TLSM in d−q axis and shows the main problems in sensorless control of TLSM. In section 3, a unique velocity observer is introduced, and asymptotic convergence of observer errors are proofed. After that, in section 4 the position-velocity tracking control system for TLSM is proposed. Later, the verification of whole system is demonstrated by simulation results in section 5. Finally, conclusions are summed up in section 6.

PROBLEM STATEMENT
As aforementioned, the TLSM has three separated wingdings a, b, c which contain three AC currents as: where ω is electrical velocity of the motor. Notice that the electrical angular position of TLSM can be expressed with respect to primary position as (1) By using Clarke and Park transform combine with (1), the dynamic model of tubular linear motor is expressed in d − q as (2)- (5): where i d (t), i q (t) are the stator current projected in d, q axis; the linear velocity and position of rotor is denoted as v(t), x(t); ψ p is the flux of the permanent magnet; R s , L stand for, respectively, stator's resistance and inductance; m, τ represent for the slider's mass and pole step length. The inputs voltage of system in d − q axis are denoted as u d (t) and u q (t). The disturbance consists two factor, the first one is the disturbance load f ℓ (t), the other one is the force generate by inductance fluctuation [24] combines with the detent force [28] which is represented as f m (t).
To be specific, the reference position and velocity of the TLSM are denoted as x r (t), v r (t). The goal of this research is to control both position and velocity of the TLSM, by which the actual position and velocity follow the desired trajectory, x r (t) and v r (t), with desired small errors and robust against the load variations. In fact, rotation motors can be easily setup velocity measurements by attaching a encoder or resolver, that of ❒ ISSN: 2088-8708 linear motors however is challenging due to mount, high cost and sensitive with external factor like humidity, temperature, vibration. To alleviate these concerns, a novel observer is designed to estimate the velocity from available data of the position and current sensor. In addition to concern reluctance effects and force ripple reduction, i d (t) should be regulated to the reference i dr (t) = 0.

VELOCITY OBSERVER DESIGN
. Furthermore, the disturbance is assumed to be bounded such that where F m and dF m are given positive constants. The following observer plays a key role in the derivation of our approach in which ρ x , ρ v , γ are real positive constants, then the observer errors dynamics can be obtained by the help of (2), (3) and (7) as (8) ˙ The following theorem provides a choice of observer parameters ρ x , ρ v , γ by which the observer (8) are wellposed.
the system (8) is exponentially stable. Moreover, there exist c 1 , c 2 > 0 such that Proof of Theorem 1: Denoting (8) can be rewritten as (11)ζ Examine the function as (12) And recalling that γ > F m , we have γ|ζ According to solution of (11), let us take the time derivative of (12) as (13)

4.
CONTROLLER DESIGN Using a cascade control strategy, we separate the TLSM system as presented in (2)-(5) into two subsystems which are position-velocity (outer subsystem) and current subsystem (inner subsystem). It should be noted that the time response of inner subsystem is much faster than that of the outer subsystem. The two control loops are present as follows.

Velocity-position controller
For simplicity sake, The desired velocity and acceleration is respectively denoted as v r (t) =ẋ r (t), a r (t) =v r (t). Further, we define these following symbols And follow the position-velocity subsystem in (2) In which i * q stands for the reference quadrature current which is assigned to inner control loop. Apply the assumption that i q (t) simultaneously track i * q . Then, replaced i q with i * q (t) in (17). The controller for outer loop is provided as (18) Theorem 2: Consider If k x , k v ∈ R ++ are large enough constants, then, the outer loop (17) is stable, and e x (t), e v (t) converge to arbitrary small values.

❒ ISSN: 2088-8708
Proof of Theorem 2: Using notations in (16), it lead toê v (t) = e v (t) − v(t). Then, (17) can be rewritten as (19)ė On studying the control performance and stability of the closed loop system, a Lyapunov candidate function is applied as By using (19), it establishes thaṫ From (15), it is clear thatV By applying the inequalities as where ϵ x , ϵ v , ϵ f > 0, from (20) we obtain thaṫ Accordingly, by choosing It implies that the tracking errors (e v (t), e x (t)) enter E in finite time due toV 2 (t) < 0. By choosing ϵ f large enough, the tracking errors can converges to arbitrary small values. Intuitively, the theorem 2 is proved.

Current controller
From the fact that dynamics of the current loop is always much faster than that of outer loop, the reference i * q can be assumed to be unvarying in inner-loop control process. Additionally, the inconstancy in inductance cause by end-effect phenomenon can be neglected. Continuously, the following notations are provided In what follows, let us establish a modified PI controller which cooperates with the velocity observer in section 3. The current loop can be consider as two parallel current system and therefore can be controlled by two PI-like controller as (24), (25)  In which k d , K p , k id , k iq are positive constants. From that, the inner closed-loop is deriveḋ Using the same method which presented in subsection 4.1, by applying the Lyapunov candidate function as in (28) It is clear that the tracking errors of both quadrature and direct current converge to zero. In concluded, the control scheme whole system is describe in Figure 3.

SIMULATION RESULT AND ANALYSIS
TLSM is used in this simulation has the parameters as follow: R s = 10.3 (Ohm); m = 0.171 (kg); L = 1.4 (mH); n p = 1; τ = 10 (mm); ψ p = 0.035 (W b). This motor is mounted a position sensor in which output data are sin(θ) and cos(θ). The control performances of the sensorless approach are verified by two difference simulation scenarios. On one hand, the first scenario verifies the effectiveness of the control system in the case of no measurement noise affect the TLSM. In the other hand, the other test is done with the impact of measurement noise to the position feedback signal. In both scenarios, the using controller, observer and disturbance load are the same.

None measurement noise case
As mention above, the feedback position from sensor is assumed perfectly accurate, the initial errors Although the error still converges to zero in less than 0.1 s Figure 5(c), the TLSM is affected by the disturbance load as in Figure 5(d). During this interval, the motor stay still, which is the advantage of propose velocity observer compare with the other EMF-based techniques. Accordingly, theorem 1 is verified. Furthermore, under the disturbance load, the position tracking is still maintained. Figure 5(a) depicts that the actual quadrature current i q (t) follows the desired signal i * q . With the high precision tracking and quick response time, the simulated results confirm the preformance of the propose control system for TLSM.

Presence of measurement noise case
In this case, opposition from the first case, The data from position sensor is assumed to contain measurement noise as (29) x meas (t) = x(t) + n(t), (29) in which x meas (t) denotes the measured signal from the position sensor, and n(t) is a noise measurement which is normally modelled by a white process. Difference from the first case, the initial error of observed position is not equal to zero due to the measurement noise. Therefore, the initial errors of observed position and velocity are chosen as x(0) = 5.10 −3 , v(0) = 0.1. With the measurement signal as depicted in Figure 6(a), the widely used method which consist low-pass filter combine with derivatives can not obtain the accurate velocity. The voltage value is illustrated in Figure 6 Remark 2: Under the impact of measurement noise n(t), the traditional method [31] which obtain the velocity via a differential calculation of measured position can not estimate the actual velocity. It should be noted that employing low-pass filters on measured signal seems to be ineffective because the measurement noise often is not considered a deterministic signal. Additionally, the use of Kalman filter [32], [33] in this case possibly results in the large velocity estimation error due to the effects of unknown disturbance load in velocity dynamics. Further, at low speed or stopping operation, the value of EMF could be unreliable or vanish, the EMF approach [19], [24] leads to the difficult and inaccurate velocity estimation. From the above analyses of previous approaches, the proposed method shows advantages in the velocity estimation in the occurrence of measurement noise n(t).

CONCLUSION
This note has provided a novel technical solution for the sensorless tracking control problem of TLSM under the lack of velocity sensors and unknown disturbance loads. The main contributions of our method have based on the proposed velocity-observer, which ensures asymptotic the convergence of observer errors. By cooperating with the observer, the position-velocity tracking controller and current controllers have been constructed by using Lyapunov direct method. These controllers have ensured that the position and velocity error converges to arbitrarily small values by choosing properly control parameters. In later work, the current sensorless control will be taken into account with no further sensor requirement in control TLSM.