Design of Iterative Sliding Mode Observer for ... - IEEE Xplore

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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY, VOL. 21, NO. 4, JULY 2013

Design of Iterative Sliding Mode Observer for Sensorless PMSM Control Hyun Lee and Jangmyung Lee

Abstract— This brief proposes an iterative sliding mode observer (ISMO) for the robust sensorless control of a permanent magnet synchronous motor with variable parameters. In the conventional SMO, a low-pass filter and an additional position compensator for the rotor are used to reduce the chattering coming from the switching by means of a signum function. It is shown that the chattering can be further reduced by using a sigmoid function as the switching function in the observer. This observer is faster in estimating the velocity and position of the rotor than the traditional adaptive SMO, since it does not include the integral operations for the low-pass filter. The proposed ISMO also improves the performance in estimating the motor speed and angle by reducing the estimation error in the back electromotive force by iteratively applying the observer in the sensorless operation. The stability of the proposed SMO is verified by the Lyapunov function in determining the observer gain, and the validity of the observer is demonstrated by simulations and experiments. Index Terms— Iterative sliding mode observer (ISMO), permanent magnet synchronous motor (PMSM), sensorless control, sigmoid function, SMO.

N OMENCLATURE v d and v q i d and i q Rs and L s ωr λf v α and v β i α and i β θ

Stator voltages for d and q axes, respectively. Stator currents in the synchronous frame. Stator resistance and inductance, respectively. Electrical-angular velocity. Back electromotive force (EMF) constant of PMSM motor. Stator voltages for d and q axes, respectively. Stator currents in the fixed coordinate system. Position of the electrical rotor. I. I NTRODUCTION

I

N CONVENTIONAL industries, dc motors, which allow for the independent control of the field and armature

Manuscript received April 4, 2011; revised March 8, 2012; accepted May 1, 2012. Manuscript received in final form May 9, 2012. Date of publication June 6, 2012; date of current version June 14, 2013. This work was supported in part by the Ministry of Knowledge Economy, Korea, under the Human Resource Development Program for Robotics Support Program, supervised by the National IT Industry Promotion Agency NIPA-2010-(C7000-1001-0009). Recommended by Associate Editor N. K. Kazantzis. The authors are with Pusan National University, Busan 609-735, Korea (e-mail: [email protected]; [email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TCST.2012.2199493

currents, have been used to enable the rapid acceleration/deceleration and precise velocity and position control in variable speed driving systems. Studies of these dc motors are required to ensure their safe operation in the face of the wearing of the brushes. Also, the brushes are limited by the installation space for high-power and high-speed operations. However, currently, there are several servo systems, which are used for industrial robots and high-precision machinery, that have high-precision and high-speed responses [1]. To replace these expensive servo systems, electronic switching is adopted for dc motors instead of mechanical brushes. With this electronic switching system, the motors are classified as ac motors, which have a smaller volume for the same power as dc motors, and have a lower maintenance cost, since there are no brushes. Recently, the use of ac motors has increased. Generally, ac motors are classified as induction motors or permanent magnet synchronous motors (PMSMs). AC motors are mechanically much stronger and easier to manufacture than dc motors. However, they are not suitable for industries where precise control is required because of their low efficiency, slow response, and low power density [2]. For the precise torque control of PMSMs, which are widely used as high-speed drives, the position of the rotor should be identified precisely. Generally, encoders and resolvers are attached to the rotor for the control algorithm of the velocity and torque [3]. However, attaching the sensor to the rotor causes some difficulties, depending on the working environment, for example, in a humid environment. Also, since this position detector is expensive and requires additional hardware, it is not efficient to use encoders and resolvers. In addition, the inertia of the rotor is increased by the attachment of the sensor to the rotor axis [3], [4]. To resolve these problems, several studies have been conducted to estimate the position and velocity of the rotor without the need for position and velocity sensors. This type of algorithm is referred to as a sensorless control algorithm and has attracted a great deal of attention recently [5]–[8]. There have been several studies on the estimation of the rotation angle of synchronous motors using the circuit equations from the estimated back electromotive force (EMF), which are collectively known as a sensorless control algorithm. Although several algorithms use adaptive state observers and Kalman filters to estimate the rotor angle, it is not easy to obtain the position of the rotor in real situations where large ripples in the current and parameter fluctuations exist, since the algorithms require large computations, which results in a long cycle time [9]. To reduce the chattering in the sliding mode observer (SMO), a sigmoid function is proposed to replace the signum function, and it shows relatively a good performance [10], [11].

1063-6536/$31.00 © 2012 IEEE

LEE AND LEE: DESIGN OF ITERATIVE SLIDING MODE OBSERVER FOR SENSORLESS PMSM CONTROL

Fig. 2.

Fig. 1. Relation between stator reference coordinates and synchronous rotating coordinates systems.

However, in the estimation of the back EMF, which is a major factor for the success of the observer, there exists a large ripple at the estimated back EMF within a cycle of the current control loop. To reduce the accumulative error in estimating the back EMF in the current control loop, it is proposed that the SMO be utilized multiple times within a single current control cycle. In this brief, an iterative sliding mode observer (ISMO) is proposed for a sensorless control algorithm to make it robust against disturbance and adaptive to parameter variations. In a current control cycle, the SMO estimates the back EMF several times to reduce the ripples in the estimation of the back EMF within a current control cycle. To reduce the unavoidable chattering in the sliding mode algorithm, a sigmoid function is adopted as a switching function. The estimated back EMF is used to calculate the position and velocity of the rotor. Therefore, the fast and precise estimation of the back EMF is required to reduce the chattering in the estimation of the rotor velocity. As a novel approach, in this brief the SMO is recursively utilized with different switching gains in a current control cycle to reduce the estimation error of the back EMF, so that the chattering in the estimation of the position and velocity of the rotor can be minimized. The stability of the proposed ISMO is proved, based upon the Lyapunov stability criterion [12], [13]. II. M ODELING OF PMSM The voltage equations of the stator are sufficient to analyze the operation of the PMSM, since the rotor of the PMSM is made of a permanent magnet which has a constant flux. The voltages in the three phases can be transformed into the synchronous coordinates in the two phases for vector control. In the two-phase synchronous coordinates system, the voltage equations of the PMSM stator are represented as follows [14]: d L s −ωr L s Rs + dt id 0 vd = + . (1) d vq iq ωr λ f ωr L s Rs + dt Ls In the fixed coordinate systems, the voltage equations are represented as follows: d Rs + dt iα e Ls 0 vα = + α (2) d vβ iβ eβ Ls 0 Rs + dt where eα = −λ f ωr sin θ and eβ = λ f ωr cos θ .

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PMSM sensorless control system with conventional SMO.

Equation (2) can be transformed into the state equations of the stator currents as follows: Rs 1 1 d iα = − iα − eα + vα (3a) dt Ls Ls Ls d Rs 1 1 iβ = − iβ − eβ + vβ . (3b) dt Ls Ls Ls The relation between the stator coordinates and the synchronous coordinates is graphically represented in Fig. 1. III. P ROPOSED ISMO A. Conventional SMO The sliding mode control changes the system states to ensure that those on the sliding surface are robust against parameter variations and disturbances. It is a sort of variable structure control for nonlinear control. The control inputs drive the system states to the sliding surface to make the system stable and robust against modeling uncertainties and external disturbances. With this SMO, a robust sensorless PMSM control system can be implemented with system nonlinearities and bounded disturbances. Fig. 2 represents the block diagram of the PMSM sensorless control system with the conventional SMO. There exists a severe chattering problem, since the Bang-Bang control with a signum switching function is a discontinuous control which is susceptible to chattering. To reduce the chattering, a lowpass filter is added at the Bang-Bang control output with cutoff frequency ωn . However, this low-pass filter causes a long time delay in estimating the position of the rotor. In the case of the conventional SMO, the cutoff frequency ωn for the low-pass filter is calculated as follows: ωc2 − ωc1 ωc2 − ωc1 ωn = ωn−1 + ω∗ − ω1 (4) ω2 − ω1 ω2 − ω1 where ωn = 2π f, ω∗ is the reference speed of the rotor, f is the cutoff frequency for the filter, and ωn−1 is the previous value of ωn . Also, ωc1 and ωc2 are the angular frequencies at rotor speeds of ω1 and ω2 , respectively. According to (4), it is recognized that the adjustment of the cutoff frequency is utilized to estimate the position and velocity of the rotor as precisely as possible in the conventional SMO. B. ISMO For the sensorless control of the PMSM, studies on the estimation of the position and velocity of the rotor have been performed for several years. The critical issue in such

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research is that the sensorless control algorithm needs to be implemented using the limited computing power of the microprocessor or DSP used for control. In this brief, an ISMO is proposed and implemented for the sensorless control system of the PMSM, adopting the SMO recursively. This system does not require any system modifications, and it is easy to design and to set up the control parameters. The state equations of the observer in (3) can be represented as Rs 1 dˆ vα − i α = − iˆα + dt Ls Ls d ˆ Rs 1 vβ − i β = − iˆβ + dt Ls Ls

1 k H (iˆα − i α ) Ls 1 k H (iˆβ − i β ) Ls

(5a) (5b)

where iˆx is used for the estimated value of i x , k is the gain constant of the observer, and H represents the sigmoid function which replaces the signum function and a low-pass filter. The sigmoid function is formulated as ⎤ ⎡ 2 − 1 ¯ H (i α ) 1+exp(−a i¯α ) ⎦ = ⎣ (6) 2 H (i¯β ) −1 1+exp(−a i¯β )

where a is positive for the slope of the sigmoid function, and i¯α = iˆα − i α and i¯β = iˆβ − i β represent the current errors of the stator current. The sliding surface is defined as

T

T sn = sα sβ = i¯α i¯β . (7) The sliding mode surface sn can be defined as the estimation error of the stator current. When the condition s˙n sn < 0 is satisfied, the sliding mode exists, and this implies that sn → 0 for t → ∞, sα = i¯α , and sβ = i¯β . To set up the existence condition of the sliding mode, the Lyapunov function candidate is defined as 1 2 1 (8) V = snT sn = sα + sβ2 . 2 2 From (3) and (5), the error equations are derived as Rs s˙¯α = i¯˙α = i˙ˆα − i˙α = − i¯α + Ls Rs s˙¯β = i¯˙β = iˆ˙β − i˙β = − i¯β + Ls

1 eα − Ls 1 eβ − Ls

1 k H (i¯α ) Ls 1 k H (i¯β ). Ls

To satisfy the existence condition of the sliding mode, snT s˙n < 0 should be satisfied, which is represented as Rs ¯ 2 ¯ 2 1 T ¯ ¯ ¯ (eα i α − k i α H (i α ) i + iβ ) + sn s˙n = − Ls α Ls 1 + (eβ i¯β − k i¯β H (i¯β )) < 0. Ls As a result, the observer condition is obtained as k ≥ max |eα |, eβ .

(9a) (9b) V˙ =

(10)

(11)

This sigmoid function requires an observer gain k, which is a constant value between −kmax and kmax , to satisfy the Lyapunov stability condition (10). Notice that when the signum function is used for the switching function, the value of k is either −1 or 1.

Fig. 3.

Improved SMO with sigmoid function.

With the predetermined observer gain k, the sliding mode may exist on the sliding surface as follows:

T

T

s˙α s˙β = sα sβ ≈ 00 . (12) Therefore, to satisfy the inequality condition (10), the following conditions are obtained: k H i¯α = eˆα (13a) ¯ (13b) k H i β = eˆβ . Using the sigmoid function, the sliding mode control becomes continuous, which reduces the chattering. The estimation of the back EMF in (13) can be used to estimate the position and velocity of the rotor as follows: eˆα θˆ = − tan−1 (14a) eˆβ d ωˆ = θˆ . (14b) dt Notice that eˆα and eˆβ are assumed to be nonzero to avoid the nonobservable situation which happens at the zero speed. Fig. 3 illustrates the improved SMO with the sigmoid function to reduce the chattering. The adopted sigmoid function can reduce the chattering caused by discontinuous switching. Using the sigmoid function, necessity of the integrator after the signum function has been eliminated. Therefore, the total computational cost with this sigmoid function becomes lower than the one with the addition of the signum function and an integrator. However, it still slows the system response and accumulates estimation errors due to the small gains near the switching boundary. Therefore, for a high-speed operation, a high gain is required for the switching function to compensate for the estimation errors. And the high switching gain results in the chattering in the estimation again, even though it may reduce the response time. Therefore, it is not desirable to keep the switching gain high to reduce the estimation error. When the switching gain is not properly selected, the observer cannot converse because of the phase delay. The proposed ISMO can minimize the ripples in the estimation of the back EMF, since it enables sensorless control to be achieved, even in the high-speed region with a small switching gain. In this brief, the gains are selected heuristically, based on the experimental data, in order to satisfy the sliding mode condition. Fig. 4 shows the concept of the proposed ISMO, and the observer structure is illustrated in detail in Fig. 5. Note that the switching gains are properly adjusted for each SMO process not to cause high chattering within the range of


Fig. 4.

Concept of ISMO.

Fig. 6.

Fig. 5.

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Structure of proposed ISMO.

satisfying the stability condition. This is a unique feature of the ISMO proposed in this brief. Fig. 4 illustrates the concept of the ISMO. The SMO is used four times in a current regulator period, which minimizes the error in the estimation of the back EMF. Fig. 5 illustrates the structure of the proposed ISMO where the SMO in the thick block has been used four times with different gains for a given reference value v s . Within a current control cycle, the SMO is executed recursively; in this system it is repeated four times, that is the maximum number allowed in one control cycle. This enables a smaller gain to be used for the sigmoid function and the ripples in the estimation of the back EMF to be reduced. The gain k for the SMO is adjusted to a lower value in each cycle to reduce the ripples in the estimation of the back EMF The current control cycle is limited not only by the performance of the microprocessor used in this system, but also by the switching devices in the space vector pulse width modulation (SVPWM). Therefore, instead of reducing the current control cycle itself, the SMO is used recursively to reduce the ripples in the estimation of the back EMF within the current control cycle. Note that since the control hardware is not changed at all and only an iteration loop for the SMO is added in the algorithm, there is no cost increase at all. IV. S YSTEM C ONFIGURATION The block diagram of the sensorless control system for the PMSM is shown in Fig. 6. The PMSM is modeled in three-phase stationary coordinates, and it is transformed into the (d, q) two-phase synchronous coordinates system for the vector control. In the control of the speed and current references, the PI control is used to effectively reduce the accumulative errors. The current is supplied to the stator of the motor through the SVPWM control in the form of a sinusoid.

Block diagram of proposed sensorless speed control system.

Using the estimated position and velocity of the rotor, the closed sensorless control of the motor was implemented. The sensorless controller of the PMSM is composed of a control arithmetic part and an intelligent power module (IPM) module. The Texas Instruments TMS320F2812 is used for the main processor of the controller part, and the PM300CSD060 of Mitsubishi is used as a switching device in the IPM module where the dead time is set to 2 μs. The inverter is formed by the PM30TPM diode module and a 4700-μF condenser to provide dc power to the IPM module. To minimize the effects of the gating noise on the control board, photocouplers are used. The control cycle of the current loop is set to 100 μs, while the velocity control cycle is kept at 1 ms to compute the control algorithms. The velocity control cycle can be reduced using a high-performance microprocessor. Since the reduction effect is not significant, the control cycle is kept at 1 ms. A CSMT-10 B (1 kW) SPMSM made by Samsung Rockwell was used for the experiments. The experimental PMSM has the following specifications: normal power = 1 kW, max. torque = 9.36 N·m, max. velocity = 3000 rpm, number of poles = 8, Rs = 0.25 , and L s = 1.3 mH. Also, the controller parameters are set as follows: control cycle for velocity = 1 ms, control cycle for current = 0.1 ms, and a = 0.05494. V. E XPERIMENTAL R ESULTS The proposed ISMO is compared to the conventional SMO in terms of its ability to control the PMSM by evaluating the response time, disturbance rejection characteristics, and back EMF. The response time is compared with the step input commanding 2000 rpm from the stationary status. While the performances are checked, the voltage control cycle is kept at 1 ms and the sampling period of the current control loop is at 100 μs. Fig. 7 compares the step responses of the conventional SMO and the proposed ISMO. For the 2000-rpm step command with 3.3% maximum overshoot, the settling time for the ISMO is 300 ms, while that of the conventional SMO is 450 ms. This fast response comes from the fact that the proposed ISMO does not include any integral operation for the lowpass filter, as in the case of the conventional SMO. It was verified by experiments that the estimation of θˆ and ωˆ is faster when the proposed ISMO is used than when the conventional

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(a)

Fig. 7.

Response times of conventional SMO and ISMO.

(b)

Fig. 8.

Comparison of responses against disturbances.

(c)

adaptive observer is used, where the sigmoid function is adopted in place of the signum function and the low-pass filter. The previous experiment checked only the speed performance of the sensorless control system. Therefore, to check the robustness of the proposed ISMO against disturbances while the motor is running, the disturbance rejection properties of the control system are checked. That is, while the motor is running at 2000 rpm, a sudden load is applied to the motor axis for 100 ms and the recovery characteristics are checked. These recovery characteristics are used to verify the robustness of the sensorless control system against unknown disturbances. Fig. 8 illustrates the experimental results of the disturbance responses for the conventional SMO and the proposed ISMO. While the motor is rotating at 2000 rpm, a load of 1.5 nm, which is about 1/6 of the standard torque, is applied to the motor axis for about 100 ms from 0.5 to 0.6 s. As shown in Fig. 10, the recovery time of the proposed ISMO is 80 ms, while that of the conventional SMO is about 125 ms. Note that the chattering exists in the velocity estimation in both cases. The ISMO exhibits excellent high-speed control in the estimation of the rotor position and velocity under sensorless control. This brief focuses on the high-speed tracking performance. Therefore. the fact that the observability of the PMSM is not guaranteed at zero speed is not considered seriously. As a future research work, the improvement of the slow-speed tracking performance can be achieved using an estimator/observer swapping system which allows the use of the observer at high speed and the estimator at low speed [15].

Fig. 9. Performance comparison of conventional SMO and ISMO for 2000-rpm motor control. (a) eˆα and eˆβ of SMO with signum function. (b) eˆα and eˆβ of SMO with sigmoid function. (c) eˆα and eˆβ of ISMO.

Fig. 9 compares the back EMF estimated by: 1) the conventional SMO; 2) the proposed SMO with the sigmoid function as a switching function; and 3) the ISMO for 2000 rpm speed control without any experimental load. The conventional SMO suffers from the large ripples in the back EMF in the highspeed control. As a result, the sensorless control of the PMSM becomes unstable in the case of the conventional SMO. The SMO with the sigmoid function exhibits good performance for stable sensorless control. However, there still exists a relatively large ripple in the back EMF, which can be reduced by the ISMO. Therefore, it is concluded that the ISMO is a good control algorithm for the sensorless control of PMSMs with small ripples in the estimated back EMF. Table I shows the peak-to-peak ripple voltages obtained with the three different algorithms: SMO with signum function, SMO with sigmoid function, and ISMO. The peak-to-peak ripple voltage is reduced by more than 50% by using the sigmoid function and is again further reduced by more than 60% by the ISMO. For the case of the 1.15-nm load, which is not known to the controller, the reduction effect is more significant. In addition to the loading effects, to reduce the effects of the variations of the stator resistance, the stator resistance can be estimated. To show the estimation performance of the stator resistance, R has been changed from 0.25 to 0.5 .


TABLE I

need to be adjusted automatically using intelligent algorithms such as fuzzy interferences.

P EAK - TO -P EAK R IPPLE V OLTAGES , V p− p , IN BACK EMF

XXX XXX algorithm SMO with signum XX Running condition XXX X function

SMO with sigmoid function

ISMO

2000 rpm

36 mV

15 mV

6.0 mV

2000 rpm (1.15-nm load)

33 mV

10 mV

2.0 mV

Resistance conversion

R

R

[ohm]

0.5

0.25 0 Fig. 10.

Time(s)

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1

Estimation performance of stator resistance variations.

The estimation performance is illustrated in Fig. 10, where the rising time of Rˆ is about 0.3 s, which is fast enough to compensate the change of the stator resistance since the stator resistance does not change abruptly [11]. This brief aims at minimizing the chattering at the highspeed control of the PMSM motor. To minimize the chattering, the control cycle needs to be minimized, which is achieved by using the sigmoid function in the SMO and by iteratively using a small gain SMO maximally instead of using a highgain SMO once in a current control loop. VI. C ONCLUSION In this brief, a sensorless control system for a PMSM was implemented by applying an ISMO. To make it robust against disturbances and parameter variations, the signum function used as a switching function in the conventional SMO was replaced by a sigmoid function. To further reduce the chattering and estimation error in a current control cycle, the SMO was applied four times repeatedly for the estimation of the back EMF. The proposed ISMO was robust and fast, so that the sensorless control system using this ISMO had a fast response and was robust against disturbances. The performance of the sensorless control system was verified with different velocities (500 and 2000 rpm) to ensure its fast response characteristics. To demonstrate the robustness of the system, the velocity characteristics were checked experimentally under a certain unknown load condition. In future works, the observer gains

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