A Variable-Sampling Controller for Brushless DC ... - Semantic Scholar
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 5, OCTOBER 2007
A Variable-Sampling Controller for Brushless DC Motor Drives With Low-Resolution Position Sensors Chung-Wen Hung, Cheng-Tsung Lin, Chih-Wen Liu, Senior Member, IEEE, and Jia-Yush Yen, Member, IEEE
Abstract—This paper proposes a variable-sampling variable structure controller (VSC) for brushless dc (BLDC) motor drives with low-resolution position sensors (Hall sensors). The variable-sampling characteristics arising from the situation where measurements depend on the path-embedded sensors will result in the uncertainty of the discrete-time models. To circumvent this issue, a modification of the conventional discrete-time VSC control law for BLDC motor drives with Hall sensors is derived to achieve the robustness of speed control. Three measurement error-mitigation methods are also presented to reduce the errors due to low-resolution position feedback. Both simulated and experimental results confirm the effectiveness of the proposed controller and error-mitigation techniques. Index Terms—Discrete-time sliding-mode control, variable structure control (VSC), variable-sampling systems.
I. I NTRODUCTION
V
ARIABLE-SAMPLING systems describe a class of systems whose sensor outputs are available only at some situations not specified by the sampling mechanism. For example, the situation arises when the measured variables are dependent on the marks along the trajectory. The type of systems is encountered in brushless dc (BLDC) motor drives with lowresolution position sensors, because the position measurements are not available at the fixed-sampling instance. The BLDC motor usually uses three or more Hall sensors to obtain rotor position and speed measurements. Without extra hardware, it would be necessary to inverse the time difference between two successive Hall-sensor signals to obtain reliable speed measurement. Notice that there are only a few sensors available to the motor; therefore, the measurement would be available only at discrete instances. Depending on the number of poles, there may be 6 or 12 sensor pulses per rotation. The sampling time, thus, becomes a variable according to the motor speed. Similar applications, such as the computer disk drives and the optical drives, all have to extract the motion information along the motion trajectory and result in speed-dependent sampling. These systems have uncertainty in
discrete-time model and require a servo controller that is not sensitive to the sampling period. Previous researches on the discrete-time variable structure control (DVSC) have dedicated on chattering-free in fixed-sampling rate [1], [2]. The other researches on the variable-sampling control have focused on multiple but also fixed-sampling-rate problems [3]–[10]. These controls offer improved performance if one considers the slow-sampling-rate system. More recent results have considered high-performance servo systems [11], [12] and have achieved smoother system responses that utilizes the fact that control efforts are updated more frequently than the measurements. Moore et al. [13] summarized these control strategies into an N-delay input/output control, and there is also a successful implementation [14]. However, the results, which are directly addressing variablesampling rate, are very limited. In 1993, Hori [15] published an interesting result where he considered a pure integrator system and was able to transform the speed–position relation into a time-invariant system. The result is very neat. However, it is valid only for the case of pure integrator system. The study in [16]–[19] have derived the control laws using variablesampling models. As an alternative, this paper investigates a DVSC for BLDC motors. The stability of the DVSC is proved in the study in [20] but the convergence seems slower. The studies showed that such approach is practical and beneficial. In [21] and [22], the multirate-sampling effect was considered as the disturbance to derive the control low, which is similar to what this paper proposes, and got a successful result. However, it did not consider the implementation of the BLDC motor. Following the direction, this paper proposes a modification of the DVSC proposed in the study in [20], [23], and [24] for the development of variable-sampling DVSC for BLDC motor applications. Three measurement error-mitigation methods are also presented to reduce the errors due to low-resolution position feedback. II. C ONTROL L AW AND M EASUREMENT E RROR M ITIGATION FOR BLDC M OTORS A. Control Law
Manuscript received March 15, 2007. This work was supported by the National Science Council of Taiwan, R.O.C., under Contract NSC-93-2213E-002-054. C.-W. Hung, C.-T. Lin, and C.-W. Liu are with the Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: [email protected]). J.-Y. Yen is with the Department of Mechanical Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2007.901303
We first consider a continuous-time single-input system x(t) ˙ = F x(t) + Gu(t). Using a fixed-sampling period of T , one has the fixed-rate-equivalent discrete-time system x(k + 1) = Φx(k) + Γu(k) (1) T where Φ = eF T , Γ = 0 eF λ Gdλ, T is the sampling period, and the input pair [Φ, Γ] is assumed controllable.
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HUNG et al.: VARIABLE-SAMPLING CONTROLLER FOR BLDC MOTOR DRIVES
The manipulated input u(k) is also known at every sampling instance. The VSC design for the continuous-time system depends on the choice of a desired sliding surface s. For stable control, the continuous-time VSC has to satisfy the sliding condition ss˙ < 0. A reaching law that satisfies the sliding condition is s(t) ˙ = −εsgn (s(t)) − q · s(t),
ε > 0,
q > 0.
Now, consider that the sampling time T changes with the motor speed, the simplified dynamics of a BLDC motor is described by J ω˙ + Dω = τe − τL τe = Kτ · i V =R · i + L
(2)
A discrete-time counterpart of the reaching law (2) can be expressed as s(k + 1) − s(k) = −qT s(k) − εT sgn (s(k)) ε > 0, q > 0, and 1 − qT > 0. (3) The DVSC is required to achieve the following performances [20]. 1) Starting from any initial state, the trajectory will move monotonically toward the switching plane and cross it in finite time. 2) Once the trajectory has crossed the switching plane, it will cross the plane again in every successive sampling period, resulting in a zigzag motion about the switching plane. 3) The size of each successive zigzag step is nonincreasing and the trajectory stays within a specified band. Let s(k) = C T x(k); one can transform system (1) into a normal form [20] x(k + 1) = Φx(k) + Γu(k)
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(4)
Φ11 Φ12 x1 , x= , ΓT = [0 0 · · · 1], Φ21 Φ22 x2 C T = [C1 1], and Φ22 and x2 are scalars. Note that if [Φ, Γ] is controllable then [Φ11 Φ12 ] is also controllable, thus maintaining the stability characteristics. From (4), s(k + 1) − s(k) = C T x(k + 1) − C T x(k) = C T Φx(k) + C T Γu(k) − C T x(k), and compare with (3) to obtain the control law for fixedsampling DVSC where Φ =
x˙ 1 = x2 = −˙ω 1 dτe dτL D ˙ − + x˙ 2 = − (−ω) . J J dt dt
s(k + 1) = (1 − qT )s(k) − εT sgn (s(k)) ,
1 − qT > 0. (6)
The first term on the right-hand side has a sign opposite to the second term in (6). From 2), the right-hand side of (6) should have an opposite sign to s(k). For fixed-sampling time system, it is possible to derive a band, QSMB for s(x) that achieves 2) [24] εT x |s(x)| < . (7) 2 − qT
(8-3)
(9) (10)
The input torque is determined by the coil current. Therefore, τe = Kτ · i, and 1 di dτL D ˙ − x˙ 2 = − (−ω) Kτ − J J dt dt Kτ di 1 dτL D ˙ − − = − (−ω) . (11) J J dt Kτ dt Define the manipulated input as u=
di dt
and let δ=
The reaching law (3) suggests a quasi-sliding mode band (QSMB) [20] that achieves the DVSC performance 2). From (3), we obtain
di +e dt
where ω is the motor speed, τe is the input torque from the windings, τL is the load torque, Kτ is the torque constant, e is the back-emf, L is the armature inductance, i is the armature current, V is the terminal voltage, and R is the terminal resistance. The pulsewidth-modulation (PWM) voltage-source inverter is used in the system. In order to simplify the VSC, the inverter is supposed to supply current as quickly as possible, so we neglect the dynamics of the current. In order to derive the system equation, take x = [x1 x2 ]T , x1 = ωref − ω, and x2 = dx1 /dt. The dynamic equations of BLDC motors can be rewritten as
u(k) = −(C T Γ)−1 C T Φx(k) − s(k) + qT s(k) + εT sgn(s(k))] . (5)
(8-1) (8-2)
1 dτL Kτ dt
then Kτ D ˙ − (u − δ). x˙ 2 = − (−ω) J J The “change of the load disturbance” is a matched disturbance. The constant offset, however, remains to be resolved. This is canceled by the integral action built-in to the system by the manipulated input, and the simulation is shown in next section. Then, the load τL is assumed to be zero temporarily, and the matrix of BLDC motors is F = continuous-time-system 0 1 0 ,G= . Set the time index to be k and 0 −D/J −Kτ /J hold the input u(k) to be constant. The state after time Tk is described by the following equation: x(k + 1) = Φ(Tk )x(k) + Γ(Tk )u(k)
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(12)
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Fig. 1. Variable-sampling VSC structure.
range of state x1 = ωref − ω when the system enters the steadystate band is as follows:
where Φ(Tk ) = eF Tk = L−1 Γ(Tk ) =
Γ1 (Tk ) Γ2 (Tk )
x+ 1 (k) =
Tk = (eF λ dλ) · G 0
=
Kτ D
J D
Kτ D
and −D J Tk
J −D e − Tk .
D e− J Tk − 1
s + D/J 0 a = D/J and take inverse Laplace transform
1/s 1/s(s + a) Φ(Tk ) = L 0 1/s + a 1 a1 1 − e−aTk = 0 e−aTk Φ11 Φ12 (Tk ) = . Φ21 Φ22 (Tk )
Z2 Q − Rβ x− 1 (k) = − 1 + Z1
Now, (sI − F )−1 = (1/s(s + D/J))
−1
Z2 Q − Rβ 1 + Z1
(13) 1 . Let s
Using (5), one has the control law −1 T C Φ(Tk )x(k) − C T x(k) u(k) = − C T Γ(Tk ) + qTk C T x(k) + εTk sgn C T x(k) = − (C1 Γ1 (Tk ) + Γ2 (Tk ))−1
π/6 2π/12 = Tk Tk
(17)
ω(k − 1) − ω(k) ∼ . −ω(k) ˙ = Tk
(18)
The DVSC rule is used to achieve 2). According to (15), the output of the controller would be integrated to obtain a current command i∗ (k). The proportional control Kp converts the difference between the command i∗ and the measured current i into a voltage command V for the PWM inverter i∗ (k) = Tk ·
× {[(C1 Φ11 + Φ21 )x1 (k)
k
u(n),
and
i∗ (0) = 0.
(19)
n=0
+ (C1 Φ12 (Tk ) + Φ22 (Tk ))x2 (k)] − s(k) + qTk s(k) + εTk sgn (s(k))} .
where Z1 = Φ11 −C1 Φ12 −RW1 +RW2 C1 , Z2 = Φ12 (Tk )− RW2 +Rα, α = 1−qTk , β = εTk , γ = sgn[s(k)], R = Γ1 (Tk )/ (Γ1 (Tk )C2 +Γ2 (Tk )), W1 = C1 Φ11 +Φ21 , and W2 = C1 Φ12 × (Tk ) + Φ22 (Tk ). The system block diagram is shown in Fig. 1. The sampling time and speed ω are measured at the rising edge of the signal coming out of the Hall sensors. Notice that the four-pole BLDC consists of 12 commutation signals per mechanical revolution. Speed and negative acceleration are measured by the following equations: ω(k) =
(14)
(16)
(15)
At each sampling instance, note that Tk is a variable which changes with the motor speed; thus, the computation of u(k) also requires of the matrix Φ(Tk ). From (7), the band still exists when T becomes variable; however, the range of the band has to cover the worst sampling period. Following an argument similar to the study in [20], the
The value of Kp is determined by choosing a high value but constrained by the maximum output voltage of the inverter. To achieve an effective i∗ command, a high-bandwidth proportional current loop is used with a system clock of 4 kHz. This is basically the same as the PWM carrier frequency. Here, ki , i = 0, 1, . . ., means the system clock during a sampling interval Tk . From our simulations and experiments, the measured current i(ki ) converges closely to i∗ (k) at every sampling interval Tk . As a result, the current loop running at the system clock does not affect the variable-sampling VSC.
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HUNG et al.: VARIABLE-SAMPLING CONTROLLER FOR BLDC MOTOR DRIVES
B. Measurement-Error-Mitigation Techniques Three methods in reducing the measurement errors due to misalignment of Hall sensors and quantized position feedback are proposed. As an example, the four-pole BLDC motor consists of 12 commutation signals per mechanical revolution. Therefore, there is a 30◦ mechanical degree separation between two commutation signals. Unfortunately, it is very difficult, in practice, for these Hall sensors to be located exactly 30◦ apart. The misalignment can arise from manufacturing tolerances and mechanical deflection in the Hall sensors and in the rotor pole placements. Errors can also arise from quantized position feedback. Therefore, error-mitigation technique is important for high-performance BLDC motor drives with low-resolution Hall sensors. First, a moving-average-filter method is proposed. To obtain more accurate feedback measurements, this method applies a moving average filter to the feedback before they are fed into the variable-sampling VSC. The advantage of this method is that it yields accurate steady-state response, but the transient response has highly oscillatory behavior. The oscillation is a result of the filter-phase delay; the system fails to respond to the true feedback in time and causes the output to fluctuate. This delay may cause instability and oscillation at low speeds. The different orders of moving average filters induce different effects. Normally, higher order filters achieve more accurate results but with longer delay. Extensive simulated results reveal that the negative acceleration x2 is influenced more seriously by the phase delay, but the speed error x1 is almost unaffected. The second method for error mitigation is to calibrate the angle difference among Hall sensors as a priori. When the motor is tuned to achieve open-loop steady-state operation, the angle between the two Hall sensors can be measured very accurately by calculating the time difference between the two Hall signals. Then, the calibrated angular positions can be used to calculate the more accurate speed. This method would work better in low speed and faster in transient mode but induces more serious noise in the higher speed. The hybrid method, combining the above two methods, becomes a logical approach to avoid the effects caused by the filter delay and to suppress the errors introduced by the misalignment of Hall sensors. When the speed is low, the speed feedback is calculated by the second method. On the other hand, near steady state, the negative acceleration x2 is filtered before it is fed back to the variable-sampling VSC. This hybrid method allows the system to respond faster in the transient state and would also achieve accurate steady-state response. All experimental results of these three methods are presented in Section III.
III. S IMULATED AND M EASURED R ESULTS The ratings of the BLDC motor under study are listed in Table I. The selection of parameters (C1 , ε, and q) for the variable-sampling VSC is important. For instance, larger C1 speeds up the transient response, but it is constrained by the maximum supply current. From (7), ε is directly proportional to the QSMB, and the system will overshoot when ε is too large. On the other hand, large ε could speed-up transient response.
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TABLE I MOTOR RATINGS
Fig. 2. Simulated voltage and current commands.
Fig. 3. Simulated comparison between variable-sampling and fixed-sampling VSCs (in T = 1/600 and 1/4000 s) at speed command = 200 r/min.
From (6), qT is required to be smaller than one, so q has to be smaller than 1/Tmax (Tmax is the maximum sampling time of the system), but large q could speed-up transient response. The parameters for the variable-sampling VSC of the studied BLDC motor are tuned to be C1 = 50, ε = 45, and q = 10 by the trial-and-error. Moreover, Kp is tuned to be 250, and the typical simulated input ∆i and output voltage are shown in Fig. 2. Due to the variable-sampling nature of the system, oscillatory responses may result using fixed-sampling controllers. Variable-sampling-control algorithm, however, can guarantee stable system response. The simulated results are shown in Fig. 3. The oscillation is caused by the asynchronization
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Fig. 6. Experimental structure includes a BLDC motor, power stage, and an embedded controller (TMS320F243).
Fig. 4. Simulated speed response of the variable-sampling VSC to load torque applied at t = 1 s (load torque = 0.4 N · m).
Fig. 7. Measured speed responses of the variable-sampling VSC with three moving average filters for measurement error mitigation (filter order: dot = 2, solid = 6, and dashed = 12). Fig. 5. Simulated speed response of the variable-sampling VSC when motor parameters drift.
between the controller-system clock and the actual sampling feedback, which is dependent on the motor speed. In Fig. 3, the rate for the fixed-sampling controller is set to 600 Hz (suitable for 3000 r/min) and 4 kHz (the same as the system clock) with the speed command set to 200 r/min. In order to show the robustness properties of the variablesampling VSC, this paper has included two simulated results. The first simulation is to show that the control is not affected by the load torque τL . As shown in Fig. 4, the speed command is set to be 3000 r/min, and a load (a load torque of 0.4 N · m) is added to the system 1 s after the motor is started. Second, Fig. 5 shows that the control also works well when the motor parameters have drifted for about 20%. In particular, the simulation sets J and D to drift into opposite signs and set the system-state parameters to drift for ±50%. The experimental results of variable-sampling VSC are shown in the following. The experimental setup includes a BLDC motor and an embedded-variable sampling controller for the motor driver, as shown in Fig. 6. The ratings of the motor are described in Table I, and the controller is implemented on a TI DSP TMS320F243 digitalsignal processor. TMS320F243 is a high-performance microcontroller designed for high-speed calculation and is equipped with many onchip peripherals for motor-control applications. The ADC unit built-in on the DSP is used to read the feedback
current signal. The onchip general purpose I/O and the capture units on the DSP are applied to capture signals of Hall sensors to generate precise firing commands for the power stage and calculate the motor speed. The PWM unit of the DSP creates the corresponding PWM waveforms to drive the BLDC motor according to the control command specified by the variablesampling VSC. A power stage block is used to provide a highvoltage source in driving the BLDC motor, and there is a current sensor in this block to measure the current fed to the BLDC motor. There is also a protection circuit block to protect the DSP from the high voltage or current from the power block. To save computation time and to avoid accumulation of truncation error, most computations required in (15) are carried out before hand and the results stored in a lookup table for actual implementation in the TMS320F243. Most terms in (15) are arranged as functions of the sampling time Tk , and the items in the tables are computed in advance in a personal computer. The Q15 mode is applied to all the processes, including the feedback filtering. The built-in 32-b accumulator in the DSP also helps suppress the truncation effects. 1) Variable-Sampling VSC With Different Order Filter for Measurement Error Mitigation: The experimental results are shown in Fig. 7. As aforementioned in Section II-B, the steadystate response is accurate, but the transient response exhibits highly oscillatory behavior. It also shows the effect of phase delay in low-speed mode, which causes the unstable and oscillatory response. Higher order filter amplifies the delay effect.
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HUNG et al.: VARIABLE-SAMPLING CONTROLLER FOR BLDC MOTOR DRIVES
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Fig. 10. Measured control error of the variable-sampling VSC with hybrid error-mitigation method in 3000-r/min command. Fig. 8. Measured speed response of the variable-sampling VSC with calibrated-sensor angular position.
when we use 12th-order filter in high-speed command, the steady-state error would be reduced to less than 0.5% according to our extensive experiments. IV. C ONCLUSION
Fig. 9. Measured speed response of the variable-sampling VSC with hybrid error-mitigation method.
2) Variable-Sampling VSC With Calibrated Sensor Angular Position: The experimental results are shown in Fig. 8 under different speed commands. Contrary to our expectation, there are more serious noise contaminations for the high-speed command, both in the transient and the steady-state responses. There are also larger dc offsets in the high-speed areas. These noise and offsets may arise from the fact that the autocalibration procedure could not completely cancel out the angle-position error. This is evident from the fact that there are larger negative acceleration x2 measurement errors at high speed than at low speed. 3) Variable-Sampling VSC With Hybrid Error-Mitigation Method: Fig. 9 shows the hybrid experimental results with the hybrid measurements, and Fig. 10 shows the control-error responses of the actual experiments. As expected, the hybrid method provides a faster response with accurate steady-state behavior. In Fig. 9, we combine the second-order moving average filter, and the motor response is less than 0.5 s. The steadystate error contains a ±1.5% fluctuation, and the offset is less than 0.5% for all the tested speed commands. It is worth noting that the motor works well in the low-speed command, even as low as 300 r/min. To achieve even more precise steady-state response, we could still choose higher order filters, but higher order filters sacrifice performance of low speed. Furthermore,
This paper proposes a variable-sampling VSC for BLDC motor drives with low-resolution position sensors (Hall sensors). A modification of the conventional DVSC control law for BLDC motor drives with Hall sensors is derived to achieve the robustness of speed control. Three measurement-error-mitigation methods are also presented to reduce the errors due to lowresolution position feedback. Both simulated and experimental results confirm the effectiveness of the proposed controller and error-mitigation techniques. The proposed variable-sampling VSC for sensorless BLDC motor drives is under investigation. R EFERENCES [1] K. Abidi and A. Sabanovic, “Sliding-mode control for high-precision motion of a Piezostage,” IEEE Trans. Ind. Electron., vol. 54, no. 1, pp. 629–637, Feb. 2007. [2] M. Sun, Y. Wang, and D. Wang, “Variable-structure repetitive control: A discrete-time strategy,” IEEE Trans. Ind. Electron., vol. 52, no. 2, pp. 610–616, Apr. 2005. [3] H. M. Al-Rahmani and G. F. Franklin, “Multirate control: A new approach,” Automatica, vol. 28, no. 1, pp. 35–44, Jan. 1992. [4] H. In and C. Zhang, “A multirate digital controller for model matching,” Automatica, vol. 30, no. 6, pp. 1043–1050, Jun. 1994. [5] P. Colaneri and G. De Nicolao, “Multirate LQG control of continuoustime stochastic systems,” Automatica, vol. 31, no. 4, pp. 591–596, Apr. 1995. [6] M. J. Er and B. D. O. Anderson, “Design of reduced-order multirate output linear functional observer-based compensator,” Automatica, vol. 31, no. 2, pp. 237–242, Feb. 1995. [7] M. De La Sen, “The reachability and observability of hybrid multirate sampling linear systems,” Comput. Math. Appl., vol. 31, no. 1, pp. 109–122, Jan. 1996. [8] K. G. Arvanitis and G. Kalogeropoulos, “Stability robustness of LQ optimal regulators based on multirate sampling of plant output,” J. Optim. Theory Appl., vol. 97, no. 2, pp. 299–337, May 1998. [9] M. Mizuochi, T. Tsuji, and K. Ohnishi, “Force sensing and force control using multirate sampling method,” in Proc. IEEE IECON, Nov. 6–10, 2005, pp. 1919–1924. [10] M. Mizuochi, T. Tsuji, and K. Ohnishi, “Force sensing and force control using multirate sampling method,” in Proc. IEEE ISIE, Dubrovnik, Croatia, Jun. 20–23, 2005, pp. 1629–1634. [11] T. Hara and M. Tomizuka, “Multi-rate controller for hard disk drive with redesign of state estimator,” in Proc. Amer. Control Conf., Philadelphia, PA, Jun. 24–26, 1998, pp. 3033–3037.
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[12] S. E. Baek and S. H. Lee, “Design of a multi-rate estimator and its application to a disk drive servo system,” in Proc. Amer. Control Conf., San Diego, CA, Jun. 1999, pp. 3640–3644. [13] K. L. Moore, S. P. Bhattacharyya, and M. Dahleh, “Capabilities and limitations of multirate control schemes,” Automatica, vol. 29, no. 41, pp. 941–995, 1993. [14] H. Fujimoto, A. Kawamura, and M. Tomizuka, “Generalized digital redesign method for linear feedback system based on N-delay control,” IEEE/ASME Trans. Mechatronics, vol. 4, no. 2, pp. 101–110, Jun. 8, 1999. [15] Y. Hori, “Robust and adaptive control of a servomotor using low precision shaft encoder,” in Proc. IEEE IECON, Lahaina, HI, Nov. 15–19, 1993, pp. 73–78. [16] A. M. Phillips, “Multi-rate and variable-rate estimation and control of systems with limited measurements with applications to information storage devices,” Ph.D. dissertation, Dept. Mech. Eng., Univ. California, Berkeley, CA, 1995. [17] A. M. Phillips and M. Tomizuka, “Multi-rate estimation and control under time-varying data sampling with applications to information storage devices,” in Proc. Amer. Control Conf., Seattle, WA, Jun. 1996, pp. 4151–4155. [18] J.-Y. Yen, Y.-L. Chen, and M. Tomizuka, “Variable sampling rate controller design for brushless dc motor,” in Proc. 41st IEEE Conf. Decision and Control, Las Vegas, NV, Dec. 2002, pp. 462–467. [19] L. Kovudhikulrungsri and T. Koseki, “Precise speed estimation from a low-resolution encoder by dual-sampling-rate observer,” IEEE/ASME Trans. Mechatron., vol. 11, no. 6, pp. 661–670, Dec. 2006. [20] W. B. Gao, Y. Wang, and A. Homaifa, “Discrete-time variable structure control system,” IEEE Trans. Ind. Electron., vol. 42, no. 2, pp. 117–122, Apr. 1995. [21] S. Janardhanan and B. Bandyopadhyay, “Output feedback-sliding-mode control for uncertain systems using fast output sampling technique,” IEEE Trans. Ind. Electron., vol. 53, no. 5, pp. 1677–1682, Oct. 2006. [22] S. Janardhanan and B. Bandyopadhyay, “Multirate output feedback based digital redesign of sliding mode control algorithms,” in Proc. Int. Workshop Variable Structure Syst., Sardinia, Italy, Jun. 2006, pp. 41–44. [23] W. B. Gao and J. C. Hung, “Variable structure control of nonlinear systems: A new approach,” IEEE Trans. Ind. Electron., vol. 40, no. 1, pp. 45–55, Feb. 1993. [24] A. Bartoszewicz, “Remark on ‘Discrete-time variable structure control system’,” IEEE Trans. Ind. Electron., vol. 43, no. 1, pp. 235–238, Feb. 1996.
Chung-Wen Hung was born in Chunghwa, Taiwan, R.O.C., in 1968. He received the B.S. degree in electrical engineering from Feng Chia University, Taichung, Taiwan, R.O.C., and the M.S. degree in electrical engineering from National Taiwan University, where he is currently working toward the Ph.D. degree in electrical engineering. From 1992 to 2002, he was an R&D Engineer at the Industrial Technology Research Institute, Hsinchu, Taiwan, R.O.C. He has been a Technical Marketing Manager of the IC Company for five years. His research interest is motor control.
Cheng-Tsung Lin was born in Chiayi, Taiwan, R.O.C., in 1978. He received the B.S. degree in electrical engineering from National Sun Yat-sen University, Kaohsiung, Taiwan, R.O.C., in 2001, and the M.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2003, where he is currently working toward the Ph.D. degree in electrical engineering. His current research interests include motor control, drive technologies, and field programmable gate-array design.
Chih-Wen Liu (S’93–M’96–SM’02) was born in Taiwan, R.O.C., in 1964. He received the B.S. degree in electrical engineering from National Taiwan University (NTU), Taipei, Taiwan, R.O.C., in 1987, and the M.S. and Ph.D. degrees in electrical engineering from Cornell University, Ithaca, NY, in 1992 and 1994, respectively. He is currently with NTU, where he is a Professor of Electrical Engineering. His main research interests include the application of computer technology to power-system monitoring, protection, and control. His research interests include motor control and power electronics.
Jia-Yush Yen (M’89) was born in Taipei, Taiwan, R.O.C., in 1958. He received the B.S. degree in mechanical engineering from National Tsing-Hwa University, Hsinchu, Taiwan, R.O.C., in 1980, the M.S. degree in mechanical engineering from the University of Minnesota, Minneapolis, in 1983, and the Ph.D. degree in mechanical engineering from the University of California, Berkeley, in 1989. During his studies at Berkeley, he received an IBM Graduate Fellowship in 1984–1985. Since 1989, he has been with National Taiwan University, Taipei, Taiwan, R.O.C., where he is currently an Associate Professor of Mechanical Engineering. His research interests are in the areas of modeling and control of electromechanical systems, particularly in precision control of computer peripherals, precision-measurement systems, and micromechanical systems. Dr. Yen currently serves as the Treasurer of the Control Systems Chapter of the IEEE Taipei Section. He is a member of the American Society of Mechanical Engineers.
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IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 54, NO. 5, OCTOBER 2007
A Variable-Sampling Controller for Brushless DC Motor Drives With Low-Resolution Position Sensors Chung-Wen Hung, Cheng-Tsung Lin, Chih-Wen Liu, Senior Member, IEEE, and Jia-Yush Yen, Member, IEEE
Abstract—This paper proposes a variable-sampling variable structure controller (VSC) for brushless dc (BLDC) motor drives with low-resolution position sensors (Hall sensors). The variable-sampling characteristics arising from the situation where measurements depend on the path-embedded sensors will result in the uncertainty of the discrete-time models. To circumvent this issue, a modification of the conventional discrete-time VSC control law for BLDC motor drives with Hall sensors is derived to achieve the robustness of speed control. Three measurement error-mitigation methods are also presented to reduce the errors due to low-resolution position feedback. Both simulated and experimental results confirm the effectiveness of the proposed controller and error-mitigation techniques. Index Terms—Discrete-time sliding-mode control, variable structure control (VSC), variable-sampling systems.
I. I NTRODUCTION
V
ARIABLE-SAMPLING systems describe a class of systems whose sensor outputs are available only at some situations not specified by the sampling mechanism. For example, the situation arises when the measured variables are dependent on the marks along the trajectory. The type of systems is encountered in brushless dc (BLDC) motor drives with lowresolution position sensors, because the position measurements are not available at the fixed-sampling instance. The BLDC motor usually uses three or more Hall sensors to obtain rotor position and speed measurements. Without extra hardware, it would be necessary to inverse the time difference between two successive Hall-sensor signals to obtain reliable speed measurement. Notice that there are only a few sensors available to the motor; therefore, the measurement would be available only at discrete instances. Depending on the number of poles, there may be 6 or 12 sensor pulses per rotation. The sampling time, thus, becomes a variable according to the motor speed. Similar applications, such as the computer disk drives and the optical drives, all have to extract the motion information along the motion trajectory and result in speed-dependent sampling. These systems have uncertainty in
discrete-time model and require a servo controller that is not sensitive to the sampling period. Previous researches on the discrete-time variable structure control (DVSC) have dedicated on chattering-free in fixed-sampling rate [1], [2]. The other researches on the variable-sampling control have focused on multiple but also fixed-sampling-rate problems [3]–[10]. These controls offer improved performance if one considers the slow-sampling-rate system. More recent results have considered high-performance servo systems [11], [12] and have achieved smoother system responses that utilizes the fact that control efforts are updated more frequently than the measurements. Moore et al. [13] summarized these control strategies into an N-delay input/output control, and there is also a successful implementation [14]. However, the results, which are directly addressing variablesampling rate, are very limited. In 1993, Hori [15] published an interesting result where he considered a pure integrator system and was able to transform the speed–position relation into a time-invariant system. The result is very neat. However, it is valid only for the case of pure integrator system. The study in [16]–[19] have derived the control laws using variablesampling models. As an alternative, this paper investigates a DVSC for BLDC motors. The stability of the DVSC is proved in the study in [20] but the convergence seems slower. The studies showed that such approach is practical and beneficial. In [21] and [22], the multirate-sampling effect was considered as the disturbance to derive the control low, which is similar to what this paper proposes, and got a successful result. However, it did not consider the implementation of the BLDC motor. Following the direction, this paper proposes a modification of the DVSC proposed in the study in [20], [23], and [24] for the development of variable-sampling DVSC for BLDC motor applications. Three measurement error-mitigation methods are also presented to reduce the errors due to low-resolution position feedback. II. C ONTROL L AW AND M EASUREMENT E RROR M ITIGATION FOR BLDC M OTORS A. Control Law
Manuscript received March 15, 2007. This work was supported by the National Science Council of Taiwan, R.O.C., under Contract NSC-93-2213E-002-054. C.-W. Hung, C.-T. Lin, and C.-W. Liu are with the Department of Electrical Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: [email protected]). J.-Y. Yen is with the Department of Mechanical Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2007.901303
We first consider a continuous-time single-input system x(t) ˙ = F x(t) + Gu(t). Using a fixed-sampling period of T , one has the fixed-rate-equivalent discrete-time system x(k + 1) = Φx(k) + Γu(k) (1) T where Φ = eF T , Γ = 0 eF λ Gdλ, T is the sampling period, and the input pair [Φ, Γ] is assumed controllable.
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HUNG et al.: VARIABLE-SAMPLING CONTROLLER FOR BLDC MOTOR DRIVES
The manipulated input u(k) is also known at every sampling instance. The VSC design for the continuous-time system depends on the choice of a desired sliding surface s. For stable control, the continuous-time VSC has to satisfy the sliding condition ss˙ < 0. A reaching law that satisfies the sliding condition is s(t) ˙ = −εsgn (s(t)) − q · s(t),
ε > 0,
q > 0.
Now, consider that the sampling time T changes with the motor speed, the simplified dynamics of a BLDC motor is described by J ω˙ + Dω = τe − τL τe = Kτ · i V =R · i + L
(2)
A discrete-time counterpart of the reaching law (2) can be expressed as s(k + 1) − s(k) = −qT s(k) − εT sgn (s(k)) ε > 0, q > 0, and 1 − qT > 0. (3) The DVSC is required to achieve the following performances [20]. 1) Starting from any initial state, the trajectory will move monotonically toward the switching plane and cross it in finite time. 2) Once the trajectory has crossed the switching plane, it will cross the plane again in every successive sampling period, resulting in a zigzag motion about the switching plane. 3) The size of each successive zigzag step is nonincreasing and the trajectory stays within a specified band. Let s(k) = C T x(k); one can transform system (1) into a normal form [20] x(k + 1) = Φx(k) + Γu(k)
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(4)
Φ11 Φ12 x1 , x= , ΓT = [0 0 · · · 1], Φ21 Φ22 x2 C T = [C1 1], and Φ22 and x2 are scalars. Note that if [Φ, Γ] is controllable then [Φ11 Φ12 ] is also controllable, thus maintaining the stability characteristics. From (4), s(k + 1) − s(k) = C T x(k + 1) − C T x(k) = C T Φx(k) + C T Γu(k) − C T x(k), and compare with (3) to obtain the control law for fixedsampling DVSC where Φ =
x˙ 1 = x2 = −˙ω 1 dτe dτL D ˙ − + x˙ 2 = − (−ω) . J J dt dt
s(k + 1) = (1 − qT )s(k) − εT sgn (s(k)) ,
1 − qT > 0. (6)
The first term on the right-hand side has a sign opposite to the second term in (6). From 2), the right-hand side of (6) should have an opposite sign to s(k). For fixed-sampling time system, it is possible to derive a band, QSMB for s(x) that achieves 2) [24] εT x |s(x)| < . (7) 2 − qT
(8-3)
(9) (10)
The input torque is determined by the coil current. Therefore, τe = Kτ · i, and 1 di dτL D ˙ − x˙ 2 = − (−ω) Kτ − J J dt dt Kτ di 1 dτL D ˙ − − = − (−ω) . (11) J J dt Kτ dt Define the manipulated input as u=
di dt
and let δ=
The reaching law (3) suggests a quasi-sliding mode band (QSMB) [20] that achieves the DVSC performance 2). From (3), we obtain
di +e dt
where ω is the motor speed, τe is the input torque from the windings, τL is the load torque, Kτ is the torque constant, e is the back-emf, L is the armature inductance, i is the armature current, V is the terminal voltage, and R is the terminal resistance. The pulsewidth-modulation (PWM) voltage-source inverter is used in the system. In order to simplify the VSC, the inverter is supposed to supply current as quickly as possible, so we neglect the dynamics of the current. In order to derive the system equation, take x = [x1 x2 ]T , x1 = ωref − ω, and x2 = dx1 /dt. The dynamic equations of BLDC motors can be rewritten as
u(k) = −(C T Γ)−1 C T Φx(k) − s(k) + qT s(k) + εT sgn(s(k))] . (5)
(8-1) (8-2)
1 dτL Kτ dt
then Kτ D ˙ − (u − δ). x˙ 2 = − (−ω) J J The “change of the load disturbance” is a matched disturbance. The constant offset, however, remains to be resolved. This is canceled by the integral action built-in to the system by the manipulated input, and the simulation is shown in next section. Then, the load τL is assumed to be zero temporarily, and the matrix of BLDC motors is F = continuous-time-system 0 1 0 ,G= . Set the time index to be k and 0 −D/J −Kτ /J hold the input u(k) to be constant. The state after time Tk is described by the following equation: x(k + 1) = Φ(Tk )x(k) + Γ(Tk )u(k)
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(12)
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Fig. 1. Variable-sampling VSC structure.
range of state x1 = ωref − ω when the system enters the steadystate band is as follows:
where Φ(Tk ) = eF Tk = L−1 Γ(Tk ) =
Γ1 (Tk ) Γ2 (Tk )
x+ 1 (k) =
Tk = (eF λ dλ) · G 0
=
Kτ D
J D
Kτ D
and −D J Tk
J −D e − Tk .
D e− J Tk − 1
s + D/J 0 a = D/J and take inverse Laplace transform
1/s 1/s(s + a) Φ(Tk ) = L 0 1/s + a 1 a1 1 − e−aTk = 0 e−aTk Φ11 Φ12 (Tk ) = . Φ21 Φ22 (Tk )
Z2 Q − Rβ x− 1 (k) = − 1 + Z1
Now, (sI − F )−1 = (1/s(s + D/J))
−1
Z2 Q − Rβ 1 + Z1
(13) 1 . Let s
Using (5), one has the control law −1 T C Φ(Tk )x(k) − C T x(k) u(k) = − C T Γ(Tk ) + qTk C T x(k) + εTk sgn C T x(k) = − (C1 Γ1 (Tk ) + Γ2 (Tk ))−1
π/6 2π/12 = Tk Tk
(17)
ω(k − 1) − ω(k) ∼ . −ω(k) ˙ = Tk
(18)
The DVSC rule is used to achieve 2). According to (15), the output of the controller would be integrated to obtain a current command i∗ (k). The proportional control Kp converts the difference between the command i∗ and the measured current i into a voltage command V for the PWM inverter i∗ (k) = Tk ·
× {[(C1 Φ11 + Φ21 )x1 (k)
k
u(n),
and
i∗ (0) = 0.
(19)
n=0
+ (C1 Φ12 (Tk ) + Φ22 (Tk ))x2 (k)] − s(k) + qTk s(k) + εTk sgn (s(k))} .
where Z1 = Φ11 −C1 Φ12 −RW1 +RW2 C1 , Z2 = Φ12 (Tk )− RW2 +Rα, α = 1−qTk , β = εTk , γ = sgn[s(k)], R = Γ1 (Tk )/ (Γ1 (Tk )C2 +Γ2 (Tk )), W1 = C1 Φ11 +Φ21 , and W2 = C1 Φ12 × (Tk ) + Φ22 (Tk ). The system block diagram is shown in Fig. 1. The sampling time and speed ω are measured at the rising edge of the signal coming out of the Hall sensors. Notice that the four-pole BLDC consists of 12 commutation signals per mechanical revolution. Speed and negative acceleration are measured by the following equations: ω(k) =
(14)
(16)
(15)
At each sampling instance, note that Tk is a variable which changes with the motor speed; thus, the computation of u(k) also requires of the matrix Φ(Tk ). From (7), the band still exists when T becomes variable; however, the range of the band has to cover the worst sampling period. Following an argument similar to the study in [20], the
The value of Kp is determined by choosing a high value but constrained by the maximum output voltage of the inverter. To achieve an effective i∗ command, a high-bandwidth proportional current loop is used with a system clock of 4 kHz. This is basically the same as the PWM carrier frequency. Here, ki , i = 0, 1, . . ., means the system clock during a sampling interval Tk . From our simulations and experiments, the measured current i(ki ) converges closely to i∗ (k) at every sampling interval Tk . As a result, the current loop running at the system clock does not affect the variable-sampling VSC.
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HUNG et al.: VARIABLE-SAMPLING CONTROLLER FOR BLDC MOTOR DRIVES
B. Measurement-Error-Mitigation Techniques Three methods in reducing the measurement errors due to misalignment of Hall sensors and quantized position feedback are proposed. As an example, the four-pole BLDC motor consists of 12 commutation signals per mechanical revolution. Therefore, there is a 30◦ mechanical degree separation between two commutation signals. Unfortunately, it is very difficult, in practice, for these Hall sensors to be located exactly 30◦ apart. The misalignment can arise from manufacturing tolerances and mechanical deflection in the Hall sensors and in the rotor pole placements. Errors can also arise from quantized position feedback. Therefore, error-mitigation technique is important for high-performance BLDC motor drives with low-resolution Hall sensors. First, a moving-average-filter method is proposed. To obtain more accurate feedback measurements, this method applies a moving average filter to the feedback before they are fed into the variable-sampling VSC. The advantage of this method is that it yields accurate steady-state response, but the transient response has highly oscillatory behavior. The oscillation is a result of the filter-phase delay; the system fails to respond to the true feedback in time and causes the output to fluctuate. This delay may cause instability and oscillation at low speeds. The different orders of moving average filters induce different effects. Normally, higher order filters achieve more accurate results but with longer delay. Extensive simulated results reveal that the negative acceleration x2 is influenced more seriously by the phase delay, but the speed error x1 is almost unaffected. The second method for error mitigation is to calibrate the angle difference among Hall sensors as a priori. When the motor is tuned to achieve open-loop steady-state operation, the angle between the two Hall sensors can be measured very accurately by calculating the time difference between the two Hall signals. Then, the calibrated angular positions can be used to calculate the more accurate speed. This method would work better in low speed and faster in transient mode but induces more serious noise in the higher speed. The hybrid method, combining the above two methods, becomes a logical approach to avoid the effects caused by the filter delay and to suppress the errors introduced by the misalignment of Hall sensors. When the speed is low, the speed feedback is calculated by the second method. On the other hand, near steady state, the negative acceleration x2 is filtered before it is fed back to the variable-sampling VSC. This hybrid method allows the system to respond faster in the transient state and would also achieve accurate steady-state response. All experimental results of these three methods are presented in Section III.
III. S IMULATED AND M EASURED R ESULTS The ratings of the BLDC motor under study are listed in Table I. The selection of parameters (C1 , ε, and q) for the variable-sampling VSC is important. For instance, larger C1 speeds up the transient response, but it is constrained by the maximum supply current. From (7), ε is directly proportional to the QSMB, and the system will overshoot when ε is too large. On the other hand, large ε could speed-up transient response.
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TABLE I MOTOR RATINGS
Fig. 2. Simulated voltage and current commands.
Fig. 3. Simulated comparison between variable-sampling and fixed-sampling VSCs (in T = 1/600 and 1/4000 s) at speed command = 200 r/min.
From (6), qT is required to be smaller than one, so q has to be smaller than 1/Tmax (Tmax is the maximum sampling time of the system), but large q could speed-up transient response. The parameters for the variable-sampling VSC of the studied BLDC motor are tuned to be C1 = 50, ε = 45, and q = 10 by the trial-and-error. Moreover, Kp is tuned to be 250, and the typical simulated input ∆i and output voltage are shown in Fig. 2. Due to the variable-sampling nature of the system, oscillatory responses may result using fixed-sampling controllers. Variable-sampling-control algorithm, however, can guarantee stable system response. The simulated results are shown in Fig. 3. The oscillation is caused by the asynchronization
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Fig. 6. Experimental structure includes a BLDC motor, power stage, and an embedded controller (TMS320F243).
Fig. 4. Simulated speed response of the variable-sampling VSC to load torque applied at t = 1 s (load torque = 0.4 N · m).
Fig. 7. Measured speed responses of the variable-sampling VSC with three moving average filters for measurement error mitigation (filter order: dot = 2, solid = 6, and dashed = 12). Fig. 5. Simulated speed response of the variable-sampling VSC when motor parameters drift.
between the controller-system clock and the actual sampling feedback, which is dependent on the motor speed. In Fig. 3, the rate for the fixed-sampling controller is set to 600 Hz (suitable for 3000 r/min) and 4 kHz (the same as the system clock) with the speed command set to 200 r/min. In order to show the robustness properties of the variablesampling VSC, this paper has included two simulated results. The first simulation is to show that the control is not affected by the load torque τL . As shown in Fig. 4, the speed command is set to be 3000 r/min, and a load (a load torque of 0.4 N · m) is added to the system 1 s after the motor is started. Second, Fig. 5 shows that the control also works well when the motor parameters have drifted for about 20%. In particular, the simulation sets J and D to drift into opposite signs and set the system-state parameters to drift for ±50%. The experimental results of variable-sampling VSC are shown in the following. The experimental setup includes a BLDC motor and an embedded-variable sampling controller for the motor driver, as shown in Fig. 6. The ratings of the motor are described in Table I, and the controller is implemented on a TI DSP TMS320F243 digitalsignal processor. TMS320F243 is a high-performance microcontroller designed for high-speed calculation and is equipped with many onchip peripherals for motor-control applications. The ADC unit built-in on the DSP is used to read the feedback
current signal. The onchip general purpose I/O and the capture units on the DSP are applied to capture signals of Hall sensors to generate precise firing commands for the power stage and calculate the motor speed. The PWM unit of the DSP creates the corresponding PWM waveforms to drive the BLDC motor according to the control command specified by the variablesampling VSC. A power stage block is used to provide a highvoltage source in driving the BLDC motor, and there is a current sensor in this block to measure the current fed to the BLDC motor. There is also a protection circuit block to protect the DSP from the high voltage or current from the power block. To save computation time and to avoid accumulation of truncation error, most computations required in (15) are carried out before hand and the results stored in a lookup table for actual implementation in the TMS320F243. Most terms in (15) are arranged as functions of the sampling time Tk , and the items in the tables are computed in advance in a personal computer. The Q15 mode is applied to all the processes, including the feedback filtering. The built-in 32-b accumulator in the DSP also helps suppress the truncation effects. 1) Variable-Sampling VSC With Different Order Filter for Measurement Error Mitigation: The experimental results are shown in Fig. 7. As aforementioned in Section II-B, the steadystate response is accurate, but the transient response exhibits highly oscillatory behavior. It also shows the effect of phase delay in low-speed mode, which causes the unstable and oscillatory response. Higher order filter amplifies the delay effect.
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HUNG et al.: VARIABLE-SAMPLING CONTROLLER FOR BLDC MOTOR DRIVES
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Fig. 10. Measured control error of the variable-sampling VSC with hybrid error-mitigation method in 3000-r/min command. Fig. 8. Measured speed response of the variable-sampling VSC with calibrated-sensor angular position.
when we use 12th-order filter in high-speed command, the steady-state error would be reduced to less than 0.5% according to our extensive experiments. IV. C ONCLUSION
Fig. 9. Measured speed response of the variable-sampling VSC with hybrid error-mitigation method.
2) Variable-Sampling VSC With Calibrated Sensor Angular Position: The experimental results are shown in Fig. 8 under different speed commands. Contrary to our expectation, there are more serious noise contaminations for the high-speed command, both in the transient and the steady-state responses. There are also larger dc offsets in the high-speed areas. These noise and offsets may arise from the fact that the autocalibration procedure could not completely cancel out the angle-position error. This is evident from the fact that there are larger negative acceleration x2 measurement errors at high speed than at low speed. 3) Variable-Sampling VSC With Hybrid Error-Mitigation Method: Fig. 9 shows the hybrid experimental results with the hybrid measurements, and Fig. 10 shows the control-error responses of the actual experiments. As expected, the hybrid method provides a faster response with accurate steady-state behavior. In Fig. 9, we combine the second-order moving average filter, and the motor response is less than 0.5 s. The steadystate error contains a ±1.5% fluctuation, and the offset is less than 0.5% for all the tested speed commands. It is worth noting that the motor works well in the low-speed command, even as low as 300 r/min. To achieve even more precise steady-state response, we could still choose higher order filters, but higher order filters sacrifice performance of low speed. Furthermore,
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Chung-Wen Hung was born in Chunghwa, Taiwan, R.O.C., in 1968. He received the B.S. degree in electrical engineering from Feng Chia University, Taichung, Taiwan, R.O.C., and the M.S. degree in electrical engineering from National Taiwan University, where he is currently working toward the Ph.D. degree in electrical engineering. From 1992 to 2002, he was an R&D Engineer at the Industrial Technology Research Institute, Hsinchu, Taiwan, R.O.C. He has been a Technical Marketing Manager of the IC Company for five years. His research interest is motor control.
Cheng-Tsung Lin was born in Chiayi, Taiwan, R.O.C., in 1978. He received the B.S. degree in electrical engineering from National Sun Yat-sen University, Kaohsiung, Taiwan, R.O.C., in 2001, and the M.S. degree in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 2003, where he is currently working toward the Ph.D. degree in electrical engineering. His current research interests include motor control, drive technologies, and field programmable gate-array design.
Chih-Wen Liu (S’93–M’96–SM’02) was born in Taiwan, R.O.C., in 1964. He received the B.S. degree in electrical engineering from National Taiwan University (NTU), Taipei, Taiwan, R.O.C., in 1987, and the M.S. and Ph.D. degrees in electrical engineering from Cornell University, Ithaca, NY, in 1992 and 1994, respectively. He is currently with NTU, where he is a Professor of Electrical Engineering. His main research interests include the application of computer technology to power-system monitoring, protection, and control. His research interests include motor control and power electronics.
Jia-Yush Yen (M’89) was born in Taipei, Taiwan, R.O.C., in 1958. He received the B.S. degree in mechanical engineering from National Tsing-Hwa University, Hsinchu, Taiwan, R.O.C., in 1980, the M.S. degree in mechanical engineering from the University of Minnesota, Minneapolis, in 1983, and the Ph.D. degree in mechanical engineering from the University of California, Berkeley, in 1989. During his studies at Berkeley, he received an IBM Graduate Fellowship in 1984–1985. Since 1989, he has been with National Taiwan University, Taipei, Taiwan, R.O.C., where he is currently an Associate Professor of Mechanical Engineering. His research interests are in the areas of modeling and control of electromechanical systems, particularly in precision control of computer peripherals, precision-measurement systems, and micromechanical systems. Dr. Yen currently serves as the Treasurer of the Control Systems Chapter of the IEEE Taipei Section. He is a member of the American Society of Mechanical Engineers.
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