Rotating-Sliding-Line-Based Sliding-Mode Control for Single-Phase ...

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Abstract—A new method to the sliding-mode control of single- phase uninterruptible-power-supply inverters is introduced. The main idea behind this new ...

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 59, NO. 10, OCTOBER 2012

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Rotating-Sliding-Line-Based Sliding-Mode Control for Single-Phase UPS Inverters Hasan Komurcugil, Member, IEEE

Abstract—A new method to the sliding-mode control of singlephase uninterruptible-power-supply inverters is introduced. The main idea behind this new method is to utilize a timevarying slope in the sliding surface function. It is shown that the sliding line with the time-varying slope can be rotated in the phase plane in such a direction that the tracking time of the output voltage is improved during load variations. The adjustment of the time-varying slope is achieved dynamically by employing a simple function which involves the error variables of the system. This function is obtained from the input/output relationship of the single-input fuzzy logic controller operating on the error variables. When a newly computed slope value is applied to the system, the position of the representative point is changed so as to achieve the desired response. The performance of the proposed control method has been tested through computer simulations and experiments using a triac-controlled resistive load and a diode bridge rectifier load. The results of the proposed method are compared with a classical sliding mode controller and a standard controller. It has been shown that the proposed method is capable of shortening the tracking and sliding times, resulting in a smaller total harmonic distortion in the output voltage. Index Terms—Sliding mode control (SMC), time-varying slope, uninterruptible power supply (UPS).

I. I NTRODUCTION

T

HE demand for uninterruptible power supplies (UPSs) has increased in case of utility power failures. Their main function is to deliver a sinusoidal ac voltage with low total harmonic distortion (THD), whose magnitude and frequency can be controlled, to critical loads. In addition, a UPS system must also have a good disturbance rejection, a good voltage regulation, and a fast dynamic and transient response for sudden changes at the load. Various control techniques have been proposed to achieve these objectives [1]–[22], [29]–[32]. The discrete-time control methods proposed for single-phase UPS systems, such as repetitive-based control [1]–[3], sliding-mode control (SMC) [4], [5], and deadbeat control [6]–[8], provide satisfactory performance for the UPS systems at the cost of a relatively fast microprocessor which must perform pulsewidth calculations on a pulse-by-pulse basis. When the inverter feeds a nonlinear load, a larger pulsewidth is needed (due to high di/dt) to keep

Manuscript received December 12, 2010; revised April 6, 2011; accepted May 16, 2011. Date of publication June 13, 2011; date of current version April 27, 2012. The author is with the Department of Computer Engineering, Eastern Mediterranean University, Gazi Magusa, Mersin 10, Turkey (e-mail: [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/TIE.2011.2159354

the output voltage distortion at a low level. If the time for a given microprocessor to perform the pulsewidth calculations is longer, then the maximum obtainable pulsewidth will be limited which increases the distortion in the output voltage [9], [22]. The continuous-time control methods are much faster and give less distortion [10]–[14]. Although the methods presented in [10] and [11] exhibit good performance, the design of the feedback control loops is complicated. Furthermore, it is shown in [14] that the integral controller employed in the outer loop that is used for voltage control is redundant, and its elimination leads to zero output impedance of the inverter, with the result that the output voltage is unaffected by the load, provided that the inverter pulsewidth-modulation output is not saturated. Each of these methods has its particular advantages and disadvantages. In general, the feedback gains in these methods are selected that yield, as closely as possible, the desired system performance. Continuous-time control methods based on the SMC technique have also been proposed [13], [15]–[18]. The SMC introduced in [13] has the advantages of fixed switching frequency and no load current measurement, but the load current observer increases the complexity of the controller. In [15], control based on a two-level hysteresis-type switching function does not take into account the three-level nature of the inverter output voltage. The method in [16] requires the computation of the filter inductor’s current reference function. Furthermore, implementation of the three-level switching function is based on the computation of separate sliding surfaces for each inverter leg, with the associated hardware complexity. In [17], separate control loops are designed for the inductor current and the output voltage. In [18], control based on a three-level hysteresis function leads to a reduction in the switching frequency and a simplification in the controller compared to existing methods. Despite the very well known advantages of SMC, it suffers from the fixed coefficient (commonly termed as λ) utilized in the sliding surface function which limits the dynamic response of the closed-loop system. Note that the sliding surface function represents a sliding line with a slope equal to −λ in the sliding mode. If a small λ is chosen, the output voltage will take a long time to track its reference in case of a load current disturbance. On the other hand, if a large λ is chosen, the reaching time of the representative point (RP) to the sliding surface will be longer. In addition, large λ results in large poles of the sliding mode dynamics which may drive the inverter system into an unstable operation. Furthermore, if the value of λ is too high, it can cause an overshoot in the output voltage (see Section III). Since it is not desired to operate the inverter with a large λ at all times, then it is most desirable if an SMC approach

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The behavior of the system can be described by the following state-space form:          0 1 0 0 x1 x˙ 1 = + u + (4) V 1 s − LC D(t) x˙ 2 0 x2 LC where D(t) is the disturbance term defined as D(t) = − Fig. 1. Single-phase UPS inverter.

based on a time-varying slope could be developed for the UPS system. In [24]–[27], time-varying-slope-based SMC methods were proposed for the second-order systems. The methods presented in [24] and [25] have the ability to rotate and/or shift the sliding line so as to achieve a faster response. In [26], the rotation of the sliding surface is based on a 2-D fuzzy rule base. Recently, there has been an attempt to perform this rotation using a linear function obtained from the approximation of the input–output relationship of a 1-D fuzzy rule base [27]. In this paper, inspired by the control approach presented in [27], a rotating-slidingline-based SMC for single-phase UPS inverters is introduced. The main idea behind this method is to utilize a time-varying slope in the sliding surface function so that different slope values are applied to the system during the transient and steadystate operations. It is shown that the sliding line with a timevarying slope can be rotated in the phase plane when the system is subjected to a load variation. An important consequence of using a time-varying slope is that the inverter is operated with a large λ during the transient caused by a load current disturbance resulting in a faster voltage response. The value of λ is returned back to its original value and remains constant during the steady state. The adjustment of the time-varying slope is achieved dynamically by employing a simple first-order function which involves the error variables of the system. This function is a result of the approximation from the input/output relationship of the single-input fuzzy logic controller (SIFLC) operating on the error variables of the system. II. S YSTEM M ODELING Fig. 1 shows a single-phase UPS inverter. The state-space equation describing the operation of the inverter can be written as          d vo 0 1/C vo 0 −io /C = + u+ −1/L 0 Vs /L 0 iL dt iL (1) where u is the control input which takes the values in the finite set {−1, 0, 1}. Let us define the output voltage error x1 and its derivative (rate of change of the output voltage error) as x1 = vo − vo∗

(2)

x2 = x˙ 1 = v˙ o − v˙ o∗ = (iC − i∗C ) /C

(3)

where x˙ 1 denotes the derivative of x1 and vo∗ is the reference for the output voltage chosen as vo∗ = Vm sin(ωt).

1 ∗ dv˙ o∗ 1 dio − v − . C dt LC o dt

(5)

It should be noted that D(t) is not bounded due to the derivative of the load current. III. SMC W ITH F IXED S LIDING L INE Let a linear sliding surface function S be expressed as S = λx1 + x2 ,

λ>0

(6)

where λ is a real constant. The dynamic behavior of (6) without external disturbance on the sliding surface is S = λx1 + x˙ 1 = 0.

(7)

In the phase plane (x1 − x2 plane), S = 0 represents a line, called the sliding line, passing through the origin with a slope equal to −λ. The sliding mode (S = 0) is described by the following first-order equation: x˙ 1 = −λx1 .

(8)

During the sliding mode, the output voltage error is expressed as x1 (t) = x1 (0)e−λt .

(9)

It should be noted that λ must be a positive real constant for achieving system stability. This fact can be easily verified by substituting a negative λ quantity into (9) which results in x1 (t) moving away from zero. In general, the SMC exhibits two modes: the reaching mode and the sliding mode [28]. While in the reaching mode, a reaching control law is applied to drive the system states to the sliding line rapidly. When the system states are on the sliding line, the system is said to be in the sliding mode in which an equivalent control law is applied to drive the system states, along the sliding line, to the origin. In this paper, the control input is obtained from u = −sign(S) where the sign function is replaced by a three-level hysteresis function [18]. In [18], it is shown that such control input satisfies the existence condition for the stability of the reaching mode. When the system is in the sliding mode, the robustness of the inverter will be guaranteed, and the dynamics of the inverter will depend on the coefficient of the sliding line (λ). It is clear from (9) that, if a small value of λ is chosen, the output voltage will take a long time to track its reference (vo∗ ) because of the slow convergence speed on the sliding line. On the other hand, if a large value of λ is chosen, the tracking time will be shorter, but the reaching time of the RP to the sliding line will be longer. In order to show the influence of λ on the dynamic performance of the system, the

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Fig. 2. Responses of SMC-controlled inverter with different slopes in the positive cycle. (a) Output voltage. (b) Output voltage error. (c) Rate of change of output voltage error. (d) State trajectory in the phase plane.

sliding-mode-controlled inverter system, with the parameters given in [21], has been simulated by using two different fixed coefficients (λ1 = 5000 and λ2 = 30 000). The load was assumed to be a triac-controlled resistive load with RL = 3 Ω. Fig. 2 shows the responses of output voltage (vo ), output voltage error (x1 ), rate of change of output voltage error (x2 ) in the positive cycle and state trajectory in the phase plane in a complete cycle. It is obvious from Fig. 2(a) that the output voltage responds very slowly in tracking its reference when a small λ value is utilized in the sliding surface function. The response of output voltage can be made faster by employing larger λ value, but this time reaching mode lasts longer. Fig. 2(b) shows that x1 exponentially converges to zero during the sliding mode. As can be seen from Fig. 2(d), when the load current disturbance occurs, the state trajectory starts to move toward the sliding line in the reaching mode and then hits to the sliding line. Thereafter, the sliding mode starts, and the RP moves along the sliding line to the origin of the phase plane by a zigzag trajectory. From the simulation results presented in Fig. 2(b), one can easily find the approximate starting time of the sliding mode and the time to reach origin for both trajectories. The approximate times at which the sliding mode of the corresponding trajectory starts are found to be ts,λ1 = 0.0649 s and ts,λ2 = 0.065 s. Similarly, the trajectory obtained by λ1

reaches the origin at to,λ1 = 0.067 s, whereas the trajectory obtained by λ2 reaches the origin at to,λ2 = 0.06522 s. It is evident that the sliding and tracking times of the state trajectory obtained by λ2 are shorter than that obtained by λ1 . The region for attainable slopes preserving the system’s stability is bounded by λmax and λmin [21]. While the lower bound for λmin can be easily assigned, the choice of λmax is not arbitrary. The sliding line which has the largest slope that does not violate the reaching condition is called the maximum slope sliding line [23]. Its feasible value can be determined as follows. It is well known that, for global stability, the Lyapunov function V (t) = S 2 /2 has to have the minimum under the condition V˙ (t) < 0. Taking the time derivative of (6) and substituting the resulting equation into the equation of V˙ (t) give [21]   1 Vs ˙ V (t) = S λx2 − x1 + u + D(t) . (10) LC LC Substitution of u = −sign(S) into (10) yields   1 Vs ˙ V (t) =S λx2 − x1 − sign(S)+D(t) LC LC     1 Vs =|S| sign(S) λx2 − x1 +D(t) − . LC LC

(11)

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Fig. 4.

Simulated waveforms of vo and io for a very large λ value.

Fig. 5.

Input/output relationship of single-input FLC.

Fig. 3. Regions of existence of the sliding mode for different λ values (λ1  λ2 ) in the phase plane. (a) λ1 . (b) λ2 .

It is clear that V˙ (t) < 0 if the following condition holds:   1 Vs x1 + D(t) < . (12) sign(S) λx2 − LC LC From (12), we obtain the following inequalities:   l1 = −x2 + λ−1 ωo2 x1 − d1 > 0 for S < 0   l2 = −x2 + λ−1 ωo2 x1 − d2 < 0 for S > 0

(13) (14)

where ωo2 = 1/LC

(15)

d1 = ωo2

(16)

[Vs + LCD(t)]

d2 = ωo2 [−Vs + LCD(t)] .

(17)

Equations l1 = 0 and l2 = 0 define two lines in the phase plane with the same slope and different intersection points. The regions of existence of the sliding mode [18] for different λ values (λ1  λ2 ) are depicted in Fig. 3. It is clear that the sliding line splits the phase plane into two regions. Regions 1 and 2 represent (13) and (14), respectively. In each region, the state trajectory is directed toward the sliding line by an appropriate switching action. The sliding mode occurs only on the portion of the sliding line, S = 0, that covers both regions. This portion is within S1 and S2 . It can be seen from Fig. 3(b) that the large λ value (λ2 ) causes a reduction of the sliding mode existence region. When the state trajectory hits the sliding

line in a point outside the sliding mode existence region S1 S2 , it overshoots the sliding line which leads to an overshoot in the output voltage. In order to demonstrate this fact, the closedloop inverter system has been simulated with a very large λ value (λ = 9 × 105 ) under a triac-controlled resistive load (RL = 3 Ω). Fig. 4 shows the output voltage and the output current waveforms obtained from this simulation result. It can be observed that the output voltage has an oscillatory response with overshoots as expected. It should be noted that the system becomes unstable with an extremely large λ value. The combination of (13) and (14) gives   −1  2 ωo2 x1 − d1 x−1 2 < λ < ωo x1 − d2 x2 .



(18)

Then, the maximum feasible value (an upper bound) of λ can be determined by   λ < ωo2 x1 − d2 x−1 2 .

(19)

Clearly, the upper bound of λ depends on Vs , L, C, D(t), and x1 for the entire range of x2 . Since d2 is time dependent, it is difficult to obtain an accurate upper bound for λ. Here, an adjustment method based on x1 and x2 could be developed for λ.

KOMURCUGIL: ROTATING-SLIDING-LINE-BASED SLIDING-MODE CONTROL FOR SINGLE-PHASE UPS INVERTERS

Fig. 6.

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Block diagram of the sliding-mode-controlled UPS system with the proposed rotating sliding line.

IV. SMC W ITH ROTATING S LIDING L INE The adjustment of the time-varying slope can be done by using fuzzy logic rules which operate on the error variables x1 and x2 [21]. The output of the fuzzy logic controller is the timevarying coefficient λF L (t) that should be always positive due to the stable operation of the system. In [27], it is shown that a suitable 2-D rule base with 49 rules can be used to produce λF L (t). Although this rule base works fine, it involves 49 rules which increase the computation complexity. Again, it is shown in [27] that the 2-D rule base can be reduced to a 1-D rule base, which contains seven rules only, by using the fact that the 2-D rule base produces exactly the same output when at least one input changes its sign in the adjacent quadrant. Inspired from this fact, the input to the 1-D rule base can be written as [21], [27]     (20) Xd (t) = |X1 (t)| − X˙ 2 (t) where X1 = K1 x1 and X2 = K2 x2 . In both rule bases, seven triangular membership functions (MFs) are considered for x1 , x2 , and λF L (t). While input MFs are defined in the range [−1, +1], the output MFs are defined in [0, +1]. All MFs have an equal base and 50% overlap with the adjacent MF [27]. Gains K1 and K2 are used to scale the corresponding input into the universe of the discourse range [−1, +1]. The input/output relationship (control surface) of the SIFLC whose input is Xd (t) is shown in Fig. 5. One can easily see that this graph is actually a piecewise linear function that can be approximated as [21] ¯ F L (t) = −0.45Xd (t) + 0.5. λ

(21)

The steady-state value of the time-varying coefficient is FL λF L (t) = Ku λ (t) = 0.5. In order to get the correct coefficient value in the steady state, the output of the SIFLC is multiplied by a gain Ku whose value will be twice of what would be used in the SMC with a fixed coefficient (Ku = 2λ). The block diagram of the UPS system with the proposed rotating sliding line is depicted in Fig. 6. The selection of scaling gains K1 and K2 plays an important role on the performance of the system. Since λF L (t) > 0, then the upper bound of Xd (t) that makes the sliding mode stable can be easily determined as Xd (t) < (0.5/0.45).

(22)

Fig. 7. Photograph of the experimental setup.

Equation (22) can be used for tuning K1 and K2 to their optimal values. The piecewise linear approximation eliminates the need for using fuzzy rules in the SIFLC.

V. S IMULATION AND E XPERIMENTAL R ESULTS In order to demonstrate the feasibility of the proposed rotating-sliding-line approach, the sliding-mode-controlled UPS system has been tested both by simulations and experiments. Simulations are carried out using the SimPowerSystems toolbox of Simulink. Experimental results were obtained from a hardware setup in which the proposed control method is implemented in continuous time as analog. Fig. 7 shows the photograph of this setup constructed in the laboratory. The parameters of the system are Vm = 200 V, Vs = 300 V, L = 250 μH, and C = 100 μF. The gain Ku was selected as Ku = 2λ = 10 × 103 . The scaling gains were selected as K1 = 2 × 10−3 and K2 = 2.4 × 10−5 . Since the waveforms of vo , io , and the state trajectory obtained with the fixed-sliding-line-based SMC are presented in [21], they are not given in this paper due to the space limitation. Note that the gain used in the displayed experimental output voltage waveforms was 1/25. The expression of the overall

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Fig. 8. Simulated and experimental waveforms of vo and io for a triaccontrolled resistive load. (a) Simulation. (b) Experiment.

Fig. 9. Simulated and experimental waveforms of vo and S for the load case given in Fig. 5. (a) Simulated waveforms. (b) Experimental waveforms.

average switching frequency of the inverter which was derived in [18] is given by

1 ω 2 Vs 2 m − m2 fs,av = o (23) 2h π 2

line but changing with time in compliance with λF L (t). Fig. 9 shows the simulated and experimental waveforms of output voltage and the sliding surface function in one output cycle for the same load case. The output voltage is sinusoidal in shape, except the transient period, in both cases. In the positive cycle, the sliding surface function switches between zero and a negative value determined by the hysteresis width. Similarly, in the negative cycle, it switches between zero and the positive hysteresis width in compliance with u = −sign(S). Fig. 10 shows the state trajectories obtained by the fixedsliding-line-based SMC with λ = 5000 and the proposed method with K2 = 1.2 × 10−5 under a triac-controlled resistive load. It is observed that both trajectories enter the sliding mode almost at the same time; however, their total sliding times to reach the origin are found to be different (to,fixed = 0.067 s and to,rotating = 0.0656 s, respectively). This shows that the total sliding and tracking times of the proposed method are shorter than that obtained by the fixed-sliding-line-based SMC. It is evident that high λ results in high output current. Therefore, in a real application, a current-limiting scheme can be used to allow the operation on the maximum current. The simulation results obtained from the proposed SMC are compared with the results obtained from the standard control method introduced in [14] and the fixed-sliding-line-based SMC. Table I gives the harmonic analyses of the output voltages and the tracking and sliding times in the three methods for the triac-controlled resistive load case. The THD of the output voltage obtained by the proposed SMC is smaller than that obtained by the standard

where h is the hysteresis width and m is defined as m = (Vm /Vs )(1 − (ω 2 /ωo2 )). When these parameters are chosen as h = 0.056 V/μs and m = 0.665, the average switching frequency can easily be computed as 21.67 kHz. Fig. 8 shows the simulated and experimental waveforms of output voltage and load current for a triac-controlled resistive load (R = 3 Ω) in the positive cycle. As it is mentioned earlier, when the system is operated with a large coefficient such as λ2 , the poles of the system will be very large which are located far from the origin on the left half of the s-plane. In such a case, the system response becomes faster, but large-valued poles are not practicable. Therefore, the response obtained by λ1 will be considered in the comparisons. When the output voltages shown in Fig. 2(a) with λ1 and Fig. 8(a) are compared, the response of the rotating-sliding-line-based SMC is seen to be much faster than that of the conventional fixed-sliding-line-based SMC. Such fast response is the result of a high λ value applied to the system by the controller during this large step change in the load. The time-varying coefficient λF L (t) and the state trajectory in the phase plane for such a load case were presented in [21]. From these results, it can be easily seen that the state trajectory movement in the sliding mode is not following a straight

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Fig. 11. Experimental waveforms of vo and io for the diode rectifier load.

Fig. 10. State trajectories obtained by the fixed-sliding-line-based SMC and the proposed SMC. TABLE I C OMPARISONS OF THE T HREE C ONTROL M ETHODS FOR O UTPUT VOLTAGE

Fig. 12. Experimental waveforms of vo and S for the diode rectifier load.

control method and the conventional SMC. In addition, the output voltage produced by the proposed method tracks its reference in a shorter time during the load current change. Another advantage of the proposed approach is that it leads to a smaller steady-state error in the output voltage. Fig. 11 shows the experimental output voltage and the load current waveforms for a diode bridge rectifier connected to a 400-μF capacitor in parallel with a 60-Ω resistor. A simulation result of the same load case with different RC values was given in [21]. It is obvious that the output voltage is not affected by this nonlinear current which shows that the proposed SMC acts very fast in correcting the output voltage. Fig. 12 shows the output voltage and the sliding surface function (S) for the diode rectifier load case. VI. C ONCLUSION A new SMC for single-phase UPS inverters is introduced based on a rotating sliding line. It is shown that the sliding line can be rotated in the phase plane in such a direction that the tracking capability of the inverter is improved when it is subjected to load variations. The idea behind the proposed method

is to compute the coefficient of the sliding surface function continuously. The computation of this time-varying coefficient is done by using a linear function that is approximated from the input/output relationship of the SIFLC operating on the error variables of the system. Simulation and experimental results presented demonstrate that the proposed method not only improves the dynamic response of the UPS inverter subjected to disturbances in the load but also operates with a smaller coefficient in the steady state and a larger coefficient in the transient periods. R EFERENCES [1] Y.-Y. Tzou, R.-S. Ou, S.-L. Jung, and M.-Y. Chang, “High-performance programmable ac power source with low-harmonic distortion using DSPbased repetitive control technique,” IEEE Trans. Power Electron., vol. 12, no. 4, pp. 715–725, Jul. 1997. [2] K. Zhang, Y. Kang, J. Xiong, and J. Chen, “Direct repetitive control of SPWM inverter for UPS purpose,” IEEE Trans. Power Electron., vol. 18, no. 3, pp. 784–792, May 2003. [3] C. Rech, H. Pinheiro, H. A. Gründling, H. Leaes, and J. R. Pinheiro, “A modified discrete control law for UPS applications,” IEEE Trans. Power Electron., vol. 18, no. 5, pp. 1138–1145, Sep. 2003. [4] S.-L. Jung and Y.-Y. Tzou, “Discrete sliding-mode control of a PWM inverter for sinusoidal output waveform synthesis with optimal sliding curve,” IEEE Trans. Power Electron., vol. 11, no. 4, pp. 567–577, Jul. 1996. [5] T.-L. Tai and J.-S. Chen, “UPS inverter design using discrete-time slidingmode control scheme,” IEEE Trans. Ind. Electron., vol. 49, no. 1, pp. 67– 75, Feb. 2002. [6] A. Kawamura, R. Chuarayaratip, and T. Haneyoshi, “Deadbeat control of PWM inverter with modified pulse patterns for uninterruptible power supply,” IEEE Trans. Ind. Electron., vol. 35, no. 2, pp. 295–300, May 1988. [7] C. Hua, “Two-level switching pattern deadbeat DSP controlled PWM inverter,” IEEE Trans. Power Electron., vol. 10, no. 3, pp. 310–317, May 1995.

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Hasan Komurcugil (S’94–M’99) was born in Cyprus in 1965. He received the B.Sc., M.Sc., and Ph.D. degrees in electrical engineering from Eastern Mediterranean University (EMU), Gazi Magusa, Turkey, in 1989, 1991, and 1998, respectively. From 1993 to 1998, he was a Research Assistant with the Electrical and Electronic Engineering Department, EMU. In 1998, he joined the Computer Engineering Department, EMU, as an Assistant Professor, where he was promoted to Associate Professor and Professor positions in 2002 and 2008, respectively. From 2004 to 2010, he was the Head of Computer Engineering Department, EMU. In 2010, he played an active role in preparing the department’s first self-study report for the use of the Accreditation Board for Engineering and Technology. Since October 2010, he has been serving as the Board Member of higher education, planning, evaluation, accreditation, and coordination council (YODAK) in North Cyprus. His research interests include power electronics, control systems, digital system design, digital signal processing, and fuzzy logic control. Dr. Komurcugil is a member of the Chamber of Electrical Engineers in North Cyprus.