Adaptive terminal sliding mode control strategy for DC–DC buck converters

1. Adaptive terminal sliding-mode control strategy for DC–DC buck converters Hasan Komurcugil n Computer Engineering Department, Eastern Mediterranean University, Gazi Magusa, North Cyprus, via Mersin 10, Turkey a r t i c l e i n f o Article history: Received 2 January 2012 Received in revised form 22 June 2012 Accepted 17 July 2012 Available online 9 August 2012 This paper was recommended for publica- tion by Jeff Pieper Keywords: Sliding-mode control Terminal sliding-mode control Finite time convergence DC–DC buck converter a b s t r a c t This paper presents an adaptive terminal sliding mode control (ATSMC) strategy for DC–DC buck converters. The idea behind this strategy is to use the terminal sliding mode control (TSMC) approach to assure finite time convergence of the output voltage error to the equilibrium point and integrate an adaptive law to the TSMC strategy so as to achieve a dynamic sliding line during the load variations. In addition, the influence of the controller parameters on the performance of closed-loop system is investigated. It is observed that the start up response of the output voltage becomes faster with increasing value of the fractional power used in the sliding function. On the other hand, the transient response of the output voltage, caused by the step change in the load, becomes faster with decreasing the value of the fractional power. Therefore, the value of fractional power is to be chosen to make a compromise between start up and transient responses of the converter. Performance of the proposed ATSMC strategy has been tested through computer simulations and experiments. The simulation results of the proposed ATSMC strategy are compared with the conventional SMC and TSMC strategies. It is shown that the ATSMC exhibits a considerable improvement in terms of a faster output voltage response during load changes. & 2012 ISA. Published by Elsevier Ltd. All rights reserved. 1. Introduction DC–DC converters are power electronics devices which are widely used in many applications including DC motor drives, communication equipments, and power supplies for personal computers [1]. The buck type DC–DC converters are used in applications where the required output voltage is smaller than the input voltage. Since buck converters are inherently nonlinear and time-varying systems due to their switching operation, the design of high performance control strategy is usually a challen- ging issue. The main objective of the control strategy is to ensure system stability in arbitrary operating condition with good dynamic response in terms of rejection of input voltage changes, load variations and parameter uncertainties. Nonlinear control strategies are deemed to be better candidates in DC–DC converter applications than other linear feedback controllers. Various nonlinear control strategies for the buck converters have been proposed to achieve these objectives [2–15]. Among these control strategies, the sliding mode control (SMC) has received much attention due to its major advantages such as guaranteed stability, robustness against parameter variations, fast dynamic response and simplicity in implementation [2–4,6–8,10,11,14]. The design of an SMC consists of two steps: design of a sliding surface and design of a control law [8]. Once a suitable sliding surface and a suitable control law are designed, the system states can be forced to move toward the sliding surface and slide on the surface until the equilibrium (origin) point is reached. The SMC introduced in [3] has the advantages of separate switching action and the sliding action, but the computation requirement of the inductor’s current reference function increases the complexity of the controller. A simple and systematic approach to the design of practical SMC has been presented in [6]. The adaptive feedforward and feedback based SMC strategy introduced in [7] has the advantages of adjusting the hysteresis width according to the input voltage change and the sliding coefficient according to the load change. The indirect SMC via double integral sliding surface strategy introduced in [10] reduces the steady-state error in the output voltage at the expense of having additional two states in the sliding surface function. In [13], a time-optimal based SMC has been introduced aiming to improve the output voltage regulation of the converter subjected to any disturbance. The SMC strategy in [14] is based on the alternative model of the buck converter with bilinear terms. In most SMC strategies proposed for the buck converters so far, the most commonly used sliding surface is the linear sliding surface which is based on linear combination of the system states by using an appropriate time-invariant coefficient (commonly termed as l). The use of such coefficient makes the sliding line static during load variations resulting in a poor transient response in the output voltage. Despite the transient response can be made Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/isatrans ISA Transactions 0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.isatra.2012.07.005 n Tel.: þ90 392 6301363; fax: þ90 392 3650711. E-mail address: [email protected] ISA Transactions 51 (2012) 673–681

2. faster by utilizing a larger valued coefficient in the linear sliding surface function, the system states cannot converge to the equilibrium point in finite time. Different from the conventional SMC, the terminal sliding mode control (TSMC) has a nonlinear sliding surface function [18]. The nonlinear sliding surface function has the ability to provide a terminal convergence (finite-time convergence) of the output voltage error from an initial point to the equilibrium point. In this paper, an adaptive terminal sliding mode control (ATSMC) strategy is proposed for the DC–DC buck converters. The idea behind this strategy is to use the TSMC for assuring finite time convergence of the output voltage error to the equilibrium point and integrate an adaptive law to the TSMC strategy so as to make the sliding line dynamic during the load variations. Differ- ent from the linear sliding surface function in conventional SMC, the output voltage error (x1) in the nonlinear sliding surface function has a fractional power (commonly termed as g) in the TSMC. Therefore, the convergence time of the output voltage depends on the parameters l and g. The influence of the fractional power on the start up and the transient responses of the output voltage, the inductor current and the state trajectory is investigated. Since the performance of the TSMC with fixed l (i.e. static sliding line) during load variations does not exhibit the desired response, an adaptive terminal sliding mode control (ATSMC) strategy has been employed in which sliding line is made dynamic by making l load dependent. Finally, simulation and experimental results are presented for verification. The rest of this paper is organized as follows. In Section 2, the dynamic model of buck converter is given. Section 3 reviews the conventional sliding mode control method for the buck converter. In Section 4, the proposed adaptive terminal sliding mode control was described for the buck converter. In Section 5, the simulation and experimental results are presented and compared with the conventional sliding mode control method. Finally, the conclusions are addressed in Section 6. 2. Dynamic model of the DC-DC buck converter Fig. 1 shows a DC–DC buck converter. It consists of a DC input voltage source (Vin), a controlled switch (Sw), a diode (D), a filter inductor (L), filter capacitor (C), and a load resistor (R). Equations describing the operation of the converter can be written for the switching conditions ON and OFF, respectively, as diL dt ¼ 1 L ðVinÀvoÞ ð1Þ dvo dt ¼ 1 C iLÀ vo R ð2Þ and diL dt ¼ À vo L ð3Þ dvo dt ¼ 1 C iLÀ vo R ð4Þ Combining (1), (2), (3) and (4) gives diL dt ¼ 1 L ðuVinÀvoÞ ð5Þ dvo dt ¼ 1 C iLÀ vo R ð6Þ where u is the control input which takes 1 for the ON state of the switch and 0 for the OFF state. Let us define the output voltage error x1 and its derivative (rate of change of the output voltage error) as x1 ¼ voÀVref ð7Þ x2 ¼ _x1 ¼ _voÀ _V ref ¼ _vo ð8Þ where _x1 denotes the derivative of x1, and Vref is the DC reference for the output voltage. By taking the time derivative of (6), the voltage error x1 and the rate of change of voltage error x2 dynamics can be expressed as _x1 ¼ x2 ð9Þ _x2 ¼ À x2 RC Ào2 ox1 þo2 oðuVinÀVref Þ ð10Þ where o2 o ¼ 1=LC. 3. Conventional sliding mode control Let a linear sliding surface function S be expressed as S ¼ lx1 þx2, l40 ð11Þ where l is a time-invariant sliding coefficient. The dynamic behavior of (11) without external disturbance on the sliding surface is [16] S ¼ lx1 þ _x1 ¼ 0 ð12Þ In the phase-plane (x1 Àx2 plane), S¼0 represents a line, called sliding line, passing through the origin with a slope equal to m¼ Àl. The sliding mode (S¼0) is described by the following first-order equation: _x1 ¼ Àlx1 ð13Þ During the sliding mode, the output voltage error is expressed as x1ðtÞ ¼ x1ð0ÞeÀlt ð14Þ It should be noted that l must be positive for achieving the system stability. This fact can be easily verified by substituting a negative l quantity into (14) which results in x1(t) moving away from zero. In general, the SMC exhibits two modes: the reaching mode and the sliding mode. 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. When the state trajectory is above the sliding line, u¼0 (Sw is OFF) must be applied so as to direct the trajectory towards the sliding line. Conversely, when the state trajectory is below the sliding line, u¼1 (Sw is ON) must be applied so that the trajectory is directed towards the sliding line. The control law that adopts + C L D + SW – – RvoVin iL iRiC Fig. 1. DC–DC buck converter. H. Komurcugil / ISA Transactions 51 (2012) 673–681674

3. such switching can be defined as u ¼ 1 2 ð1ÀsignðSÞÞ ¼ 1 if So0 0 if S40 ð15Þ However, direct implementation of this control law causes the converter to operate at an uncontrollable infinite switching frequency which is not desired in practice. Hence, it is required to suppress the switching frequency into an acceptable range. In order to accomplish this, a hysteresis modulation (HM) method employing a hysteresis band with suitable switching boundaries around the switching line is used as [6] u ¼ 1 when SoÀh 0 when S4h ( ð16Þ where h is the hysteresis bandwidth. When S4h, switch Sw will turn OFF. Conversely, it will turn ON when So Àh. Such operation limits the operating frequency of the switch. When the system is in the sliding mode, the robustness of the converter will be guaranteed and the dynamics of the converter will depend on l. In order to ensure that the movement of the error variables is maintained on the sliding line, the following existence condition must be satisfied S_S o0 ð17Þ The time derivative of (11) can be written as _S ¼ l_x1 þ _x2 ð18Þ Substituting (9) and (10) into (18) yields the following inequalities [6]: l1 ¼ lÀ 1 RC x2 þo2 oðVinÀVref Àx1Þ40 for So0 when u ¼ 1 ð19Þ l2 ¼ lÀ 1 RC x2Ào2 oðVref þx1Þo0 for S40 when u ¼ 0 ð20Þ Equations l1 ¼0 and l2 ¼0 define two lines in the phase-plane with the same slope passing through points P1 ¼(Vin ÀVref,0) and P2 ¼(ÀVref,0) on the x1 axis, respectively. The slope of these lines is given by m1 ¼ m2 ¼ o2 o ðlÀ1=RCÞ ð21Þ The regions of existence of the sliding mode for different l values (l41/RC and lo1/RC) are depicted in Fig. 2. It is clear that the sliding line splits the phase-plane into two regions. In each region, the state trajectory is directed towards the sliding line by an appropriate switching action. The sliding mode occurs only on the portion of the sliding line, S¼0, that covers both regions. This portion is within S1 and S2. It can be seen from Fig. 2(a) that the large l value causes a reduction of sliding mode existence region. When the state trajectory hits the sliding line in a point outside the sliding mode existence region S1S2, it overshoots the sliding line which leads to an overshoot in the output voltage. The x2 intercepts of the lines l1 and l2 are Y1 ¼ o2 oðVref ÀVinÞ=ðlÀ1=RCÞ and Y2 ¼ o2 oVref =ðlÀ1=RCÞ, respectively. On the other hand, when l is small, the state trajectory hits the sliding line in a point inside the sliding mode existence region S1S2, and it moves toward the origin as seen in Fig. 2(b). Note that when lo1/RC, the slope of these lines is negative which changes the x2 intercepts of the lines l1 and l2 as Y2 ¼ o2 oVref =ðlÀ1=RCÞ and Y1 ¼ o2 oðVref ÀVinÞ=ðlÀ1=RCÞ, respectively. It is evident from (14) and Fig. 2 that the dynamic response of the buck converter depends on the value of l. In order to ensure that l is large enough for fast dynamic response and low enough to retain a large existence region, it has been proposed in [6] to set l as l ¼ 1 RC ð22Þ However, despite the dynamic response can be made faster by utilizing a large l in the sliding surface function, the system states still cannot converge to the equilibrium point in finite time. 4. Adaptive terminal sliding mode control for buck converter Let a nonlinear sliding surface function for the buck converter system defined in (9) and (10) be defined as Sn ¼ lx g 1 þ _x1 ð23Þ where l40, and 0o(g¼q/p)o1 where p and q are positive odd integers satisfying p4q. When the system is in the terminal sliding mode (Sn ¼0), its dynamics can be determined by the following nonlinear differential equation: _x1 ¼ Àlx g 1 ð24Þ Note that Eq. (24) reduces to _x1 ¼ Àlx1 for g¼1, which is the form of conventional SMC. It has been shown in [17] that x1 ¼0 is the terminal attractor of the system defined in (1). The term ‘‘terminal’’ is referred to the equilibrium which can be reached in finite time. Note that Eq. (24) can also be written as dt ¼ À dx1 lx g 1 ð25Þ Taking integral of both sides of (25) and evaluating the resulting equation on the closed interval (x1(0)a0, x1(ts)¼0) Fig. 2. Regions of existence of the sliding mode for: (a) l41/RC and (b) lo1/RC. H. Komurcugil / ISA Transactions 51 (2012) 673–681 675

4. gives the following equation [18]: ts ¼ À 1 l Z 0 x1ð0Þ dx1 x g 1 ¼ x1ð0Þ

8. 1Àg lð1ÀgÞ ð26Þ Eq. (26) means that when the system enters to the terminal sliding mode at t¼tr with initial condition x1(0)a0, the system state x1 converges to x1(ts)¼0 in finite time and stay there for tZts. In other words, when the state trajectory hits the sliding surface at time tr, the system state cannot leave the sliding line meaning that the state trajectory will belong to the sliding line for tZtr. However, it is obvious from (26) that the convergence time, ts, still depends on the parameters g and l. Therefore, these parameters must be carefully selected to ensure the desired response. 4.1. Selection of g When x1 is near the equilibrium point (9x19o1), the fractional power g¼q/p leads to 9x g 1949x19. In such a case, the system state with the nonlinear term x g 1 converges toward equilibrium point more faster than the linear term x1. Especially, when g is too small near the equilibrium point, the state trajectory moves toward the equilibrium point as if it follows the bottom of a bowl rather than following a straight line which results in a distorted inductor current during load variations. In order to show the influence of parameter g on the dynamic performance of the TSMC, a sample simulation study has been performed by using the parameters Vin ¼10 V, Vref¼5 V, L¼1 mH, C¼1000 mF, R¼10 O, and l¼1/RC¼100. Fig. 3 shows the start-up responses of the output voltage, the inductor current and the state trajectories of the converter with different g values. It is clear from Fig. 3(a) and (b) that the output voltage and the inductor current responses become faster with increasing the value of g. The main reason of this comes from the fact that the slope of the sliding line with g4 ¼0.8182 is greater than all other sliding line slopes as shown in Fig. 3(c) resulting in a faster response as pointed out in (14). All the state trajectories start from the point ÀVref¼ À5 V on the x1 axis (as vo(0)¼0 V), which agrees well with the intersection point of l2 shown in Fig. 2. When the trajectory reaches the sliding line, it starts to slide along it by making a zigzag movement. However, when it approaches the equilibrium point (9x19o1) it changes its direction and makes a movement similar to the bottom of a bowl. This direction change is inversely proportional with the g value. This means that any change in the direction becomes smaller when g values get larger. This fact is clearly visible in Fig. 3(c). Fig. 4 shows the responses of the output voltage and the inductor current for a step change in R from 10 O to 2 O which are obtained by the SMC method with g¼1 and the TSMC method with different g values. Unlike the start up case, it is interesting to note that the output voltage and the inductor current responses become faster with decreasing the value of g. Therefore, g value is to be chosen to make a compromise between start up and transient responses of the converter. It is well known that the current dynamics is faster than the voltage dynamics. Since the transient responses of the inductor current for different g values are super- imposed, then only two responses are presented in Fig. 4(b). 4.2. Selection of l It has been discussed before that l is usually set to 1/RC for a good performance of the system. However, the TSMC with constant l cannot exhibit the same transient performance when the converter is subjected to a load change. The main reason of this performance degrade is due to the static sliding line irrespective of the operating point change caused by the load change. It is clear from (21) that l is inversely proportional to R. Therefore, instead of fixing it at l¼1/RC with nominal load resistance, its value should be adaptively changed when a load change occurs. However, it is not possible to measure the load resistance directly in a practical implementation. Therefore, the instantaneous value of the load resistance can be estimated by measuring the output voltage across and the current passing through the load resistance as ^R ¼ vo iR ð27Þ 0 Fig. 3. Simulated start up responses of the output voltage, the inductor current, and the state trajectories obtained by the TSMC method with different g values: (a) vo, (b) iL, and (c) state trajectories. H. Komurcugil / ISA Transactions 51 (2012) 673–681676

9. Since the value of R is used to vary l in the TSMC, such control results in an adaptive terminal sliding mode control (ATSMC). The adaptation comes from the time variation of the l term in the nonlinear terminal sliding mode. The block diagram of the DC-DC buck converter with the proposed ATSMC method is depicted in Fig. 5. 4.3. Stability analysis and sliding mode dynamics The sufﬁcient condition for the existence of the terminal sliding mode is given by Sn _Sn o0 ð28Þ If a control input is designed which ensures (28), then the system will be forced towards the sliding surface and remains on it until origin is reached asymptotically. Now, let VðtÞ ¼ 1 2S2 n ð29Þ be a Lyapunov function candidate for the system described in (9) and (10). The time derivative of (29) can be written as _V ðtÞ ¼ Sn _Sn o0 ð30Þ Differentiating (23) with respect to time and using in (30), one can obtain _V ðtÞ ¼ Sn _Sn ¼ Snðlgx gÀ1 1 x2 þ _x2Þo0 ð31Þ Substituting (10) into (31) and solving for u gives ueq ¼ 1 o2 oVin x2 RC þo2 oðVref þx1ÞÀlgx gÀ1 1 x2 ð32Þ Eq. (32) is the equivalent control needed to keep the system motion on the sliding surface under ideal terminal sliding mode. However, the equivalent control may not be able to move the system states from reaching mode to the sliding mode. Therefore, an additional control action known as the switching control is needed that should be applied to the system together with the equivalent control. Hence, the total control input can be written as u ¼ 1 o2 oVin x2 RC þo2 oðVref þx1ÞÀlgx gÀ1 1 x2ÀKsignðSnÞ ð33Þ Fig. 4. Simulated transient responses of the output voltage and the inductor current due to a step change in R from 10 O to 2 O obtained by the SMC and the TSMC methods with different g values: (a) vo and (b) iL. Fig. 5. Block diagram of the DC–DC buck converter with the proposed ATSMC method. Fig. 7. Switching frequency for different input voltages. Fig. 6. Sliding function with hysteresis band. H. Komurcugil / ISA Transactions 51 (2012) 673–681 677

10. where K denotes the switching control gain. Rewriting (10) as _x2 ¼ À x2 RC Ào2 ox1 þo2 oðuVinÀVref ÞþdðtÞ ð34Þ and substituting (33) into (34) and the resulting equation into (31) yields _V ðtÞ ¼ Sn _Sn ¼ ðÀKSnsignðSnÞþSndðtÞÞo0 ð35Þ where satisfies K49d(t)9, and d(t) denotes the disturbances caused by parametric variations in the system. Clearly, for both Sn 40 and Sn o0, _V ðtÞ is always negative for all values of the system states which means that the system has a finite-time convergent stability irrespective of the disturbances in the system. This means that the ATSMC offers a strong robustness against the variations in input voltage and reference output voltage during the sliding mode. It should be noted that the ATSMC method does not offer a strategy to estimate the domain of attraction around the equilibrium point. Although, it is very difficult to estimate the domain of attraction analytically, there are some recent works that try to estimate it [19,20]. Substituting (9) and (10) into _Sn ¼ lgx gÀ1 1 _x1 þ _x2 and using Eq. (15) give the following inequalities that satisfy the existence condition given in (28): lgx gÀ1 1 x2À x2 RC þo2 oðVinÀVref Àx1Þ40 for So0 when u ¼ 1 ð36Þ lgx gÀ1 1 x2À x2 RC Ào2 oðVref þx1Þo0 for S40 when u ¼ 0 ð37Þ Eqs. (36) and (37) describe the sliding mode dynamics of the closed-loop system.Fig. 8. Switching frequency for different output voltage references. SMC SMC TSMC ATSMC TSMC and ATSMC 1.66ms TSMC and ATSMC ATSMC State trajectory due to the step change State trajectory due to the start up State trajectory due to the step change State trajectory due to the start up Fig. 9. Simulated start up and transient responses (due to a step change in R from 10 O to 2 O) of the output voltage, and the state trajectory obtained by the SMC, the TSMC and the ATSMC methods with g¼0.2: (a) vo, (b) magnified response of vo and iL obtained by ATSMC, (c) state trajectory obtained by the SMC and (d) state trajectory obtained by the TSMC and the ATSMC. H. Komurcugil / ISA Transactions 51 (2012) 673–681678

11. 4.4. Switching frequency The frequency at which a practical buck converter can be switched is limited by such factors as hysteresis bandwidth, input voltage, output voltage, and LC filter. Therefore, an estimate of the switching frequency would be helpful in designing the system. Consider a typical trajectory of the sliding function with hysteresis band depicted in Fig. 6. From the geometry of Fig. 6, we can write the ON and OFF periods of switch Sw as TON ¼ 2h _S þ n ð38Þ TOFF ¼ À2h _S À n ð39Þ where _S þ n and _S À n are the time derivatives of Sn for ON and OFF states of the switch Sw, respectively. The time derivative of (23) can be written as _Sn ¼ lgx gÀ1 1 x2À x2 RC þo2 oðuVinÀVref Àx1Þ ð40Þ Assuming that the errors x1 and x2 are negligible in the steady- state, Eqs. (38) and (39) can be written as TON ¼ 2h o2 oðVinÀVref Þ ð41Þ TOFF ¼ 2h o2 oVref ð42Þ Hence, the expression for the switching frequency can be obtained as fs ¼ 1 TON þTOFF ¼ o2 oVref 2h 1À Vref Vin ð43Þ Clearly, the switching frequency is inversely proportional to the hysteresis bandwidth (h). The hysteresis bandwidth is chosen so as to obtain a suitable switching frequency given in (43). In order to investigate the influence of other parameters, the converter’s switching frequency is computed by using (43) with L¼1 mH, C¼1000 mF, and h¼100 for different input voltages and different output voltage references. Figs. 7 and 8 show the switching frequency (fs) for different input voltages (Vin) at Vref¼5 V and different output voltage references (Vref) at Vin ¼10 V, respectively. It can be seen that fs increases with increasing Vin. As for output voltage reference variation, it is observed that fs decreases while Vref increases. Note that by changing the hysteresis bandwidth, it is possible to keep the switching frequency constant against input voltage variation at the expense of additional control complexity requirements. 5. Simulation and experimental results In order to demonstrate the performance of the proposed ATSMC strategy, the DC–DC buck converter system has been tested by simulations and experiments. Simulations are carried out using Simulink of Matlab with a step size of 2 ms. Experi- mental results were obtained from a hardware setup constructed in the laboratory. The parameters of the system are Vin ¼10 V, Vref¼5 V, h¼80, L¼1 mH, and C¼1000 mF. Fig. 9 shows the simulated start up and transient responses of the output voltage and the state trajectories obtained by SMC, TSMC and ATSMC strategies. The value of g used in the TSMC and Fig. 10. Experimental responses of the output voltage and the inductor current for start up and a step change in R from 10 O to 2 O: (a) start up and (b) step change. Fig. 11. Experimental responses of the inductor current and the control input for the step change in the load resistance from 10 O to 2 O: (a) responses of iL and u, (b) magnified responses of iL and u for R¼10 O and (c) magnified responses of iL and u for R¼2 O. H. Komurcugil / ISA Transactions 51 (2012) 673–681 679

12. ATSMC strategies was 0.2 (q¼1 and p¼5). The load resistor was set to R¼10 O during the start up. The transient responses are due to a step change in R from 10 O to 2 O at t¼0.08 s. It is obvious from Fig. 9(a) that the output voltage tracks its reference successfully in all cases. The start up response of the output voltage obtained by the SMC is faster than that of obtained by the TSMC and ATSMC. Conversely, the transient responses of the output voltage obtained by the TSMC and ATSMC are faster than that of obtained by the SMC. The ATSMC offers the fastest transient response which shows that the controller acts very fast in correcting the output voltage. Note that the TSMC and ATSMC exhibit exactly the same response from the start up until the step change occurs in the load resistance. The main reason of this comes from the fact that both methods employ the same l value until the load change takes place. After the load change occurs, while the TSMC continues to operate with the same l value, the ATSMC makes use of the adaptively computed l value. The magnified response of the output voltage, due to the step change in the load, together with the inductor current obtained by the ATSMC method is shown in Fig. 9(b). The output voltage takes approximately 1.66 ms to track its reference. Fig. 9(c) and (d) shows the state trajectories that correspond to the simulation case shown in Fig. 9(a). Note that both trajectories start from the point ÀVref¼ À5 V on the x1 axis, since vo(0)¼0 V. In the start up, the slope of the sliding line in Fig. 9(d) is smaller than that of in Fig. 9(c). However, after the load is changed, the slope of the sliding line in Fig. 9(d) becomes greater than that of in Fig. 9(c) leading to a faster response in the output voltage. Experimental results were also obtained for testing the dynamic performance of the proposed ATSMC method under step changes in the load, in the input voltage and in the reference output voltage. Fig. 10 shows the experimental responses of the output voltage and the inductor current for the start up and the step change in the load resistance from 10 O to 2 O. It can be seen from Fig. 10(a) that the start up responses of the output voltage and inductor current are in good agreement with the simulation results shown in Fig. 9(a) and Fig. 3(b), respectively. Fig. 10(b) shows the experimental result that corresponds to the simulation result presented in Fig. 9(b). It took about 2 ms for the controller to correct the output voltage at 5 V. The small discrepancy between the simulation and experimental results comes from the component tolerances, and non-ideal effects (e.g., finite time delay) in the practical system which cause the output voltage to make slower transitions. Fig. 11 shows the response of the inductor current and the control input for the step change in the load resistance from 10 O to 2 O. Fig. 11(a) shows the magnified response of the inductor current that is shown in Fig. 10(b) and the control input. The magnified responses of the control input for R¼10 O and R¼2 O are shown in Fig. 11(b) and (c), respectively. The switching frequency of the converter is measured approximately as fs ¼ 1/64 ms¼15.625 kHz. Note that one can obtain the same result by using (43) with Vin¼10 V, Vref¼5 V, h¼80, L¼1 mH, and C¼1000 mF. This shows that the switching frequency computation is fairly accurate. Fig. 12 shows the response of the output voltage for step changes in the reference output voltage (Vref) from 5 V to 7 V and the input voltage (Vin) from 10 V to 8 V. It is clear from Fig. 12(a) that the output voltage tracks its reference faster and successfully. Also, as shown in Fig. 12(b), the output voltage is almost not affected for the step change in the input voltage. The results presented in Fig. 12 show that the ATSMC method is very robust against input and reference voltage variations. Fig. 13 shows the state trajectory in the steady-state. 6. Conclusions An adaptive terminal sliding mode control (ATSMC) strategy, which assures finite time convergence of the output voltage error to the equilibrium point, has been proposed for DC–DC buck converters. The influence of the controller parameters on the performance of closed-loop system is investigated. It is observed that the start up response of the output voltage becomes faster when the value of the fractional power is increased. On the other hand, the transient response of the output voltage caused by the step change in the load resistance becomes faster when the value of the fractional power is decreased. Therefore, one should consider a compromise between start up and transient responses when choosing the value of the fractional power. Simulation and experimental results show that the ATSMC strategy, as compared to the conventional SMC and TSMC strategies, is quite successful in obtaining very fast output voltage responses to load disturbances. Fig. 12. Experimental responses of the output voltage for step changes in the reference output voltage and the input voltage: (a) response of vo for a step change in Vref from 5 V to 7 V and (b) response of vo for a step change in Vin from 10 V to 8 V. Fig. 13. Experimental state trajectory in the steady-state. H. Komurcugil / ISA Transactions 51 (2012) 673–681680

13. References [1] Rashid M. Power electronics: circuits, devices, and applications. Prentice- Hall; 1993. [2] Nguyen VM, Lee CQ. Tracking control of buck converter using sliding-mode with adaptive hysteresis. In: Proceedings of the IEEE power electronics specialists conference; 1995. p. 1086–93. [3] Nguyen VM, Lee CQ. Indirect implementations of sliding-mode control law in buck-type converters. In: Proceedings of the IEEE applied power electronics conference; 1996. p. 111–5. [4] Perry AG, Feng G, Liu YF, Sen PC. A new sliding mode like control method for buck converter. In: Proceedings of the IEEE power electronics specialists conference; 2004. p. 3688–93. [5] Leung KKS, Chung HSH. Derivation of a second-order switching surface in the boundary control of buck converters. IEEE Power Electronics Letters 2004;2(2) 63–7. [6] Tan SC, Lai YM, Cheung MKH, Tse CK. On the practical design of a sliding mode voltage controlled buck converter. IEEE Transactions on Power Electro- nics 2005;20(2):425–37. [7] Tan SC, Lai YM, Tse CK, Cheung MKH. Adaptive feedforward and feedback control schemes for sliding mode controlled power converters. IEEE Transac- tions on Power Electronics 2006;21(1):182–92. [8] Tan SC, Lai YM, Tse CK. A unified approach to the design of PWM-based sliding-mode voltage controllers for basic DC–DC converters in continuous conduction mode. IEEE Transactions on Circuits and Systems Part I 2006;53(8):1816–27. [9] Leung KKS, Chung HSH. A comparative study of boundary control with first- and second-order switching surfaces for buck converters operating in DCM. IEEE Transactions on Power Electronics 2007;22(4):1196–209. [10] Tan SC, Lai YM, Tse CK. Indirect sliding mode control of power converters via double integral sliding surface. IEEE Transactions on Power Electronics 2008;23(2):600–11. [11] Tan SC, Lai YM, Tse CK. General design issues of sliding-mode controllers in DC–DC converters. IEEE Transactions on Industrial Electronics 2008;55(3): 1160–74. [12] Babazadeh A, Maksimovic D. Hybrid digital adaptive control for fast transient response in synchronous buck DC–DC converters. IEEE Transactions on Power Electronics 2009;24(11):2625–38. [13] Jafarian MJ, Nazarzadeh J. Time-optimal sliding-mode control for multi- quadrant buck converters. IET Power Electronics 2011;4(1):143–50. [14] Tsai JF, Chen YP. Sliding mode control and stability analysis of buck DC–DC converter. International Journal of Electronics 2007;94(3):209–22. [15] Truntic M, Milanovic M, Jezernik K. Discrete-event switching control for buck converter based on the FPGA. Control Engineering Practice 2011;19:502–12. [16] Eker I. Second-order sliding mode control with experimental application. ISA Transactions 2010;49:394–405. [17] Zak M. Terminal attractors for addressable memory in neural network. Physics Letters A 1988;133(1–2):18–22. [18] Man ZH, Paplinski AP, Wu HR. A robust MIMO terminal sliding mode control scheme for rigid robotic manipulator. IEEE Transactions on Automatic Control 1994;39(12):2464–9. [19] Aguilar-Ibanez C, Sira-Ramirez H. A linear differential flatness approach to controlling the Furuta pendulum. IMA Journal of Mathematical Control and Information 2007;24(1):31–45. [20] Aguilar-Ibanez C, Sira-Ramirez H, Suarez-Castanon MS, Martinez-Navarro E, Moreno-Armendariz MA. The trajectory tracking problem for an unmanned four-rotor system: flatness-based approach. International Journal of Control 2012;85(1):69–77. H. Komurcugil / ISA Transactions 51 (2012) 673–681 681

Adaptive terminal sliding mode control strategy for DC–DC buck converters

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Adaptive terminal sliding mode control strategy for DC–DC buck converters