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Non-singular terminal sliding-mode control of DC–DC buck converters
Control Engineering Practice 21 (2013) 321–332
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Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Non-singular terminal sliding-mode control of DC–DC buck converters Hasan Komurcugil n Computer Engineering Department, Eastern Mediterranean University, Famagusta, North Cyprus, Via Mersin 10, Turkey
a r t i c l e i n f o
abstract
Article history: Received 9 December 2011 Accepted 17 November 2012 Available online 11 December 2012
This paper presents a non-singular terminal sliding mode control (NTSMC) method for DC–DC buck converters. The NTSMC method eliminates the singularity problem which arises in the terminal sliding mode due to the fractional power and assures the finite time convergence of the output voltage error to the equilibrium point during the load changes. It is shown that the NTSMC method has the same finite time convergence as that of the terminal sliding mode control (TSMC) method. The influence of the fractional power on the state trajectory of the converter is investigated. It is observed that the slope of the sliding line becomes larger with decreasing value of the fractional power which leads to a faster transient response of the output voltage during the load changes. The theoretical considerations have been verified both by numerical simulations and experimental measurements from a laboratory prototype. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Sliding-mode control Terminal sliding-mode control Non-singular terminal sliding-mode control DC–DC buck converter
1. Introduction 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 non-linear and time-varying systems due to their switching operation, the design of high performance control strategy is usually a challenging 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. Non-linear 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 (Babazadeh & Maksimovic, 2009; Jafarian & Nazarzadeh, 2011; Leung & Chung, 2004, 2007; Nguyen & Lee, 1996, 1995; Perry, Feng, Liu, & Sen, 2004; Ramos, ˜ o, 2012; Sira-Ramı´rez, Luviano-Jua´rez, & Biel, Fossas, & Grin Corte´s-Romero, 2012; Tse & Adams, 1992; Tan, Lai, Cheung, & Tse, 2005; Tan, Lai, Tse, & Cheung, 2006a; Tan, Lai, & Tse, 2006b; Tan, Lai, & Tse, 2008a; Tan, Lai, & Tse, 2008b; Tsai & Chen, 2007; Truntic, Milanovic, & Jezernik, 2011; Wang, Xu, & Bao, 2011 ). 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 (Nguyen & Lee, 1995; Nguyen & Lee, 1996; Perry et al., 2004; Tan et al., 2005;
n
Tel.: þ90 392 630 1363; fax: þ90 392 365 0711. E-mail address: [email protected]
0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2012.11.006
Tan et al., 2006a, 2006b; Tan et al., 2008a, 2008b; Tsai & Chen, 2007; Ramos et al., 2012; Sira-Ramı´rez et al., 2012). The design of an SMC consists of two steps: design of a sliding surface and design of a control law (Utkin (1978); Tan et al., 2006a). 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 Nguyen and Lee (1996) 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 Tan et al. (2005). The adaptive feed forward and feedback based SMC method introduced in Tan et al. (2006a) 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 method introduced in Tan et al. (2008a) reduces the steady-state error in the output voltage at the expense of having additional two states in the sliding surface function. In Jafarian and Nazarzadeh (2011), a time-optimal based SMC has been introduced aiming to improve the output voltage regulation of the converter subjected to any disturbance. The SMC method in Tsai and Chen (2007) is based on the alternative model of the buck converter with bilinear terms. In most SMC methods 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. The use of such coefficient makes the sliding line static during load
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H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
variations resulting in a poor transient response in the output voltage. Despite the transient response can be made faster by utilizing a higher 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) employs a non-linear sliding surface function (Man, Paplinski, & Wu, 1994) which has the ability to converge the system states to the equilibrium point in finite time. Although the finite time convergence of the system states is achieved successfully, there exists a singularity problem due to negative fractional power in the TSMC design. A remedy to this problem is to develop a new TSMC that would avoid the singularity. In this paper, the non-singular terminal sliding mode control (NTSMC) method, introduced in Feng, Yu and Man (2002), has been adopted for controlling the DC–DC buck converter. The idea behind this method is to form a new non-linear sliding surface function that causes no singularity points and assures a finite time convergence of the output voltage error to the equilibrium point. It is shown that the NTSMC method with the new nonlinear sliding surface function has the same finite time convergence as that of the TSMC method. The singularity problem can be eliminated by utilizing the output voltage error (first system state) in the new non-linear sliding surface function with no fractional power. The influence of the fractional power on the state trajectory of the converter is investigated. It is observed that the slope of the sliding line becomes larger with decreasing value of the fractional power which leads to a faster transient response of the output voltage during the load changes. The theoretical considerations have been verified both by numerical simulations and experimental measurements from a laboratory prototype. The rest of this paper is organized as follows. In Section 2, the model of buck converter is given. Section 3 reviews the sliding and terminal sliding mode control methods for the buck converter. In Section 4, the non-singular terminal sliding mode control is described for the buck converter. In Section 5, the simulation and experimental results are presented and discussed. Finally, the conclusions are addressed in Section 6.
dvo 1 vo iL ¼ C dt R Combining (1)–(4) gives diL 1 ¼ ðuV in vo Þ L dt
ð5Þ
dvo 1 vo iL ¼ C dt R
ð6Þ
where u is the control input which takes 1 for the ON state of the switch and 0 for the OFF state. The output voltage error and its derivative (rate of change of the output voltage error) can be defined as x1 ¼ vo V ref
ð7Þ
x2 ¼ x_ 1 ¼ v_ o V_ ref ¼ v_ o
ð8Þ
where x_ 1 denotes the derivative of x1 , and V ref 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 x_ 1 ¼ x2 x_ 2 ¼
ð9Þ x_ 1 þ o2o uV in V ref x1 RC
Fig. 1 shows a DC–DC buck converter. It consists of a DC input voltage source (V in ), a controlled switch (Sw), a diode (D), a filter inductor (L), a filter capacitor (C), and a load resistor (R). The equations describing the operation of the converter can be written for the switching conditions ON and OFF, respectively as diL 1 ¼ ðV in vo Þ L dt
ð1Þ
3. Sliding and terminal sliding mode control methods for buck converter 3.1. Sliding mode control A linear sliding surface function can be expressed as S ¼ lx1 þ x2 ,
l 40
dvo vo iL ¼ C dt R
ð2Þ
and diL vo ¼ dt L
ð3Þ L
Sw
iL
+ Vin
+ D
C
vo –
– Fig. 1. DC–DC buck converter.
iR R
ð11Þ
where l is a real sliding coefficient. The dynamic behavior of (11) without external disturbance on the sliding surface is ð12Þ
In the phase-plane (x1 –x2 plane), S ¼ 0 represents a line, called sliding line, passing through the origin with a slope equal to l. The sliding mode (S ¼ 0) is described by the following first-order equation x_ 1 ¼ lx1
ð13Þ
During the sliding mode, the output voltage error is expressed as x1 ðtÞ ¼ x1 ð0Þelt
1
ð10Þ
where o2o ¼ 1=LC.
S ¼ lx1 þ x_ 1 ¼ 0 2. Modeling the DC–DC buck converter
ð4Þ
14Þ
It should be noted that l must be a positive real constant for achieving system stability. This fact can be easily verified by substituting a negative l quantity into (14) which results in x1 ðtÞ 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, u ¼ 0 (Sw is OFF) must be applied so as to direct the trajectory toward the sliding line. Conversely, when the state trajectory is below the sliding line, u ¼ 1 (Sw is ON) must be applied so that the trajectory is directed toward the sliding line. The equation of the
H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
323
x2
control input can be written as 1 u ¼ ½sgnðSÞ1 2
ð15Þ
—Vref
S=0
However, direct implementation of this control input causes the converter to operate at an uncontrollable infinite switching frequency which is not desired in practice. Hence, the following hysteresis modulation (HM) method employing a hysteresis band with the boundary layer is used to solve this very high frequency operation ( 1 when S o h u¼ ð16Þ 0 when S 4 h
Overshoot
S<0 l1 > 0
l1 = 0
Y2
S2
u=1
x1 S<0 l2 > 0
S1
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 the coefficient of the sliding line (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 SS_ o 0
l2 = 0
Vin— Vref
Y1
vo = Vin
vo = 0
ð17Þ
u=1
Hysteresis band
The time derivative of (11) can be written as S_ ¼ lx_ 1 þ x_ 2
x2
l2 = 0
ð18Þ
Substituting (9),(10) into (18) yields the following inequalities 1 x2 þ o2o V in V ref x1 40 for S o0 when u ¼ 1 l1 ¼ l RC
l1 = 0
Vin—Vref
-Vref
ð19Þ 1 x2 o2o V ref þ x1 o0 l2 ¼ l RC
for
S 40
when
u¼0
ð20Þ
S=0
Equations l1 ¼ 0 and l2 ¼ 0 define two lines in the phase-plane with the same slope passing through points P 1 ¼ V in V ref ,0 and P 2 ¼ V ref ,0 on the x1 axis, respectively. The slope of these lines is given by m1 ¼ m2 ¼
o2o
l1=RC
1 RC
x1 S1
ð21Þ
The regions of existence of the sliding mode for different l values (l 41=RC and l o 1=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 toward the sliding line by an appropriate switching action. The sliding mode occurs only on the portion of the sliding line, S ¼ 0, 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 S1 S2 , it overshoots the sliding line which leads to an overshoot in the output voltage. of l2 are Y1 ¼ The2 x2 intercepts the lines l1 and oo V ref V in = l1=RC and Y 2 ¼ o2o V ref = l1=RC , 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 S1 S2 , and it moves toward the origin as seen in Fig. 2(b). Note that when l o 1=RC, the slope of these lines is negative which changes the x2 intercepts of the lines l1 andl2 as Y 2 ¼ o2o V ref = l1=RC and Y 1 ¼ o2o ðV ref V in Þ= l1=RC , 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 Tan et al. (2005) to set l as follows:
l¼
S2
vo = 0
vo = Vin
Fig. 2. Regions of existence of the sliding mode for: (a) l 4 1=RC and (b) l o 1=RC.
3.2. Terminal sliding mode control A non-linear sliding surface function for the buck converter system defined in (9) and (10) can be defined as g St ¼ lx1 þ x_ 1
ð23Þ where l 40, and 0 o g ¼ q=p Þ o 1 where p and q are positive odd integers satisfying p 4 q. When the system is in the terminal sliding mode (St ¼ 0), its dynamics can be determined by the following non-linear differential equation g x_ 1 ¼ lx1
ð24Þ
Note that Eq. (24) reduces to x_ 1 ¼ lx1 for g ¼ 1, which is the form of conventional SMC. It has been shown in Zak (1988) that x1 ¼ 0 is the terminal attractor of the system defined in (24). The term ‘‘terminal’’ is referred to the equilibrium which can be reached in finite time. Rearranging (24) yields
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.
Y2
dt ¼
dx1 lxg1
ð25Þ
Taking integral of both sides of (25) and evaluating the resulting equation on the time interval (x1 ð0Þ a 0, x1 ðt s Þ ¼ 0) gives the
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H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
following equation (Man et al., 1994) Z 1g 9x1 ð0Þ9 1 0 dx1 t TSMC ¼ ¼ g s l x1 ð0Þ x1 l 1g
ð26Þ
Eq. (26) means that when the system enters to the terminal sliding mode at t ¼ t r with initial condition x1 ð0Þ a0, the system state x1 converges to x1 ðt s Þ ¼ 0 in finite time and stay there for t Zt s . In order to ensure that the movement of the error variables is maintained on the sliding line, the following existence condition must be satisfied St S_ t o0
ð27Þ
Taking time derivative of (23) and using it in (27) gives g1 St S_ t ¼ Sn lgx1 x_ 1 þ x_ 2
ð28Þ
Substituting (9),(10) into (28) and solving for u gives 1 x2 g1 uTSMC þ o2o ðV ref þ x1 Þlgx1 x2 ¼ 2 eq oo V in RC
ð29Þ
Eq. (29) is the equivalent control required to maintain the error variables on the sliding line under ideal terminal sliding mode. It is obvious from Eq. (29) that the fractional power g 1 in the third g1 term x1 x2 containing may result in a singularity if x2 a0 when x1 ¼ 0 due to g 1 o0 Feng et al. (2002). As a result of singularity, the TSMC cannot guarantee a bounded control input which may adversely affect the stability of the buck converter. A remedy to this problem is to develop a new TSMC that would avoid the singularity. In this paper, the non-singular terminal sliding mode control (NTSMC) method introduced in Feng et al. (2002) has been adopted for controlling the DC–DC buck converter.
4. Non-singular terminal sliding mode control A non-linear sliding surface function for the buck converter system defined in (9) and (10) can be defined as
Substituting (9),(10) into (33) and making use of (32) results in the following inequalities i l0 ðð1=gÞ1Þ h x2 o2o ðV in V ref x1 Þ 40 for Sn o0 when u ¼ 1 x2 x2 g RC ð35Þ i l0 ðð1=gÞ1Þ h x2 þ o2o ðV ref þ x1 Þ o 0 x2 x2 g RC
for
Sn 4 0
when
u¼0 ð36Þ
Combining (35),(36) gives i i l0 ðð1=gÞ1Þ h x2 l0 ðð1=gÞ1Þ h x2 o2o ðV in V ref x1 Þ o x2 o x2 þ o2o ðV ref þ x1 Þ x g 2 RC g RC ð37Þ It can easily be shown that the existence condition (32) is guaranteed if the following condition holds i g 2ð1=gÞ h x2 þ o2o ðV ref þ x1 Þ o 0 ð38Þ o2o V in o 0 x2 RC l Note that it is very difficult to defined by the inequality in (38). the sliding mode exists around x2 ffi 0), if the following condition
find the regions of existence However, it can be shown that the equilibrium point (x1 ffi 0, is satisfied
o2o V in o o2o V ref o0
ð39Þ
The block diagram of the DC–DC buck converter with the NTSMC method is shown in Fig. 3.
5. Simulation and experimental results In order to demonstrate the performance of the NTSMC approach, the DC–DC buck converter system has been tested both by simulations and experiments. The specifications of the buck converter are given in Table 1. The parameters of the controller are h ¼ 0:02, l ¼ 100, and g ¼ 0:6. Simulations are carried out using the Simulink of Matlab with a step size of 0.2 ms. The Simulink model
0 1=g
Sn ¼ x1 þ l x_ 1
ð30Þ 1=g . When the system is in where g is defined in (23) and l ¼ 1=l the terminal sliding mode (Sn ¼ 0), its dynamics can be determined by the following non-linear differential equation 1=g x_ 1
¼
1
l0
x1
ð31Þ
L
Sw
0
+
Vin
D
iL
iC
+
C
vo
u
–
iR R
–
vo
a=b
) It should be noted that by making use of property y ¼ x yb ¼ xa , (31) can be rewritten which is equivalent to (24). Therefore, the finite time taken to reach the equilibrium point of the NTSMC system is the same as the one of the TSMC system as given in (26). In order to ensure that the movement of the error variables is maintained on the sliding line, the following existence condition must be satisfied Sn S_ n o 0
ð32Þ
Taking time derivative of (30) and using it in (32) gives l0 ð1=gÞ1 Sn S_ n ¼ Sn x_ 1 þ x_ 1 x_ 2
g
Substituting (9),(10) into (33) and solving for u gives 0 g l x2 o2o l0 2ð1=gÞ þ ¼ ð V þ x Þx uNTSMC 1 ref eq 2 g o2o V in l0 gRC
ð33Þ
ð34Þ
In Eq. (34), it is obvious that as long as 1o ð1=gÞ o2 2ð1=gÞ (i.e., q op o2q), the term x2 is always non-singular . Therefore, the singularity problem will not occur in the NTSMC.
Hysteresis modulation
x1
1/ C
+
Vref
–
(...)1/γ
x1 γ
(1 / λ)1/ Sn
+
+
NTSMC
Fig. 3. Block diagram of the buck converter with the NTSMC method.
Table 1 Specifications of buck converter. Description
Parameter
Nominal value
Input voltage Inductance Capacitance Load resistance Reference output voltage
V in L C R V ref
10 V 1 mH 1000 mF 10 O 5V
H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
of buck converter with the proposed NTSMC is shown in Fig. 4. The block named ‘‘Hysteresis Modulation’’ implements Eq. (16). Experimental results were obtained from a hardware setup constructed in the laboratory.
325
Fig. 5 shows the simulated start-up response of the equivalent control input (uTSMC ), the output voltage (vo ), the output voltage eq error (x1 ), and the rate of change of the output voltage error (x2 ) obtained by TSMC method for R ¼ 2 O. The value of l was set to
Fig. 4. Simulink model of buck converter with the NTSMC method.
0.8
1
0.7
−1 0.5
x1 (V)
Equivalent control, ueq
0 0.6
0.4 0.3
−2
−3
0.2
−4 0.1
−5
0 0
0.05
0.1
0.15
0
0.2
0.05
1500
5
1000
4
500
x2 (V/s)
Output voltage, vo (V)
6
3
−500
1
−1000
0
0.05
0.1
Time (s)
0.15
0.2
0
2
0
0.1
Time (s)
Time (s)
0.15
0.2
−1500 0
0.05
0.1
0.15
0.2
Time (s)
Fig. 5. Simulated response of equivalent control input (uTSMC ), output voltage (vo ), the output voltage error (x1 ), and the rate of change of the output voltage error (x2 ) eq obtained by TSMC method: (a) uTSMC , (b) vo , (c) x1 , and (d) x2 . eq
0.6
1
0.5
0
0.4
−1 x1 (V)
Equivalent control, ueq
Fig. 6. Experimental response of x1 , x2 , and vo obtained by TSMC method: (a) x1 and x2 , and (b) vo .
0.3
−2
0.2
−3
0.1
−4
0
0
0.05
0.1
0.15
−5
0.2
0
0.05
Time (s)
0.15
0.2
0.15
0.2
300
6
250
5
200 4
x2 (V/s)
Output voltage, vo (V)
0.1 Time (s)
3
150 100
2 50 1
0
0
0
0.05
0.1 Time (s)
0.15
0.2
−50
0
0.05
0.1 Time (s)
Fig. 7. Simulated response of uNTSMC , vo , x1 , and x2 obtained by NTSMC method: (a) uNTSMC , (b) vo , (c) x1 , and (d) x2 . eq eq
Fig. 8. Experimental response of x1 , x2 , and vo obtained by NTSMC method: (a) x1 and x2 , and (b) vo .
5.05
7.5 SM-PID
NTSMC
5 7
SM-PID
Output voltage, vo (V)
Output voltage, vo (V)
4.95 SMC
4.9 4.85
NTSMC
4.8
SMC
6.5
6
5.5
4.75 5
4.7 4.65 0.25
0.3
0.35
0.4
4.5 0.25
0.3
Time (s)
0.35
0.4
Time (s)
5.001
5
Output voltage, vo (V)
4.999 NTSMC
4.998 4.997 4.996
SM-PID
4.995 4.994 4.993
SMC
4.992 4.991 0.25
0.3
0.35
0.4
Time (s)
Fig. 9. Simulated output voltage responses due to the step changes in R from 10 O to 2 O,V ref from 5 V to 7 V and V in from 10 V to 9 V obtained by SMC, SM-PID and NTSMC methods: (a) Step change in R from 10 O to 2 O, (b) Step change in V ref from 5 V to 7 V and (c) Step change in V in from 10 V to 9 V.
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H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
Table 2 Comparisons of three control methods for output voltage.
NTSMC SM-PID SMC
Settling time (ms)
Tracking time (ms)
Steady state error (mV)
(Due to R: 10 O-2 O) 13.8 33.3 40.2
(Due to V ref : 5 V-7 V) 25 50 50
(Due to V in : 10 V-9 V) 3.2 4.5 9
Fig. 10. Experimental 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 method (vo: 1 V/div, iL:1 A/div).
6
4.0 3.5
5
4 2.5 3
2.0 1.5
2
Inductor current, iL (A)
Output voltage, vo (V)
3.0
1.0 1 0.5 0.0
0 0.080
0.085
0.090
0.095
0.100
0.105
0.110
0.115
0.120
Time (s)
Fig. 11. Simulated and experimental 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 NTSMC method: (a) Simulation, and (b) Experiment (vo: 1 V/div, iL:0.5 A/div).
H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
100. On the other hand, the value of g was 0:6 (q ¼ 3 and p ¼ 5) which satisfies the condition q o p o 2q. Note that the steady state value of the control input must be 0.5 due to the fact that vO ¼ uV in for buck converters. It can be seen from Fig. 5(a) that the steady state value of uTSMC is 0.5 on the average. The eq occurrence of singularity is clearly visible on the control input resulting in unwanted oscillations on the output voltage shown in Fig. 5(b). The responses of x1 and x2 are shown in Fig. 5(c) and (d), respectively. Note that the singularity causes oscillations on the output voltage which start when x1 ¼ 0 and x2 a 0. Fig. 6 shows the experimental responses of x1 , x2 , and vo for the case presented in Fig. 5. It is clear that the Fig. 6(a) and (b) are in good agreement with the simulation results shown in Fig. 5(c), (d) and (b), respectively.
329
Fig. 7 shows the simulated start-up responses of uNTSMC , vo , x1 , eq and x2 obtained by NTSMC method for R ¼ 2 O using l ¼ 100 and g ¼ 0:6. It can be seen from Fig. 7(a) that the occurrence of singularity is avoided by the NTSMC method which does not cause any oscillations on the output voltage (see Fig. 7(b)). As explained in Section 4, the main reason of this avoidance comes from the fact that the fractional power (2 1=g ) in Eq. (34) is always positive for 1o ð1=gÞ o 2. Fig. 8 shows the experimental responses of x1 , x2 , and vo for the case presented in Fig. 7. One can easily see that Fig. 8(a) and (b) are in good agreement with the simulation results shown in Fig. 7(c), (d) and (b), respectively. Fig. 9 shows the simulated output voltage responses due to the step changes in R from 10 O to 2 O (Fig. 9(a)),V ref from 5 V to 7 V
a 2
3.0
2.0
1
1.5
1.0
Control input, u
Inductor current, iL (A)
2.5
0
0.5
0.0 0.0990
0.0995
0.1000 Time (s)
0.1005
-1 0.1010
b
Fig. 12. Simulated and experimental responses of the inductor current and the control input due to a step change in R from 10 O to 2 O obtained by the NTSMC method: (a) Simulation, and (b) Experiment (iL:0.5 A/div).
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H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
The experimental response of the output voltage and the inductor current obtained by SMC method for a step change in R from 10 O to 2 O is shown in Fig. 10. The output voltage takes about 50 ms to track its reference which agrees well with the simulation result of the same case shown in Fig. 9(a). Fig. 11 shows the simulated and experimental responses of the output voltage and the inductor current obtained by NTSMC method for a step change in R from 10 O to 2 O at t ¼ 0:1 s. As a result of this change in the load resistance, the inductor current changes from 0.5 A to 2.5 A. The drop in the output voltage went down by 0.3 V. It took about 4 ms for the NTSMC method to correct the output voltage at 5 V. It can be observed that Fig. 11(b) is in good agreement with the simulation result in Fig. 11(a). When Figs. 10 and 11(b) are compared, it can be seen that NTSMC method is much faster than SMC method. Fig. 12 shows the simulated and experimental responses of the inductor current and control input obtained by NTSMC method for a step change in R from 10 O to 2 O at t ¼ 0:1 s. It is clear that
(Fig. 9(b)) and V in from 10 V to 9 V (Fig. 9(c)) obtained by the SMC (Tan et al., 2005), sliding mode implementation of PID control (SM-PID) introduced in Tan et al. (2006a, 2006b) and the NTSMC methods. In order to avoid the infinite switching frequency, a hysteresis modulation (see Eq. 16) method employing a hysteresis band h ¼ 0:02 has been used in each method. Note that for the sake of a fair comparison, the value of l was set to 100 in all cases. Also, the value of g used in the NTSMC method was 0.6. It is clear from Fig. 9(a) and (b) that the NTSMC method acts faster than the SMC and SM-PID methods in correcting the output voltage at desired levels 5 V and 7 V, respectively. On the other hand, it is obvious from Fig. 9(c) that all methods result in a slight steadystate error in the output voltage when a step change takes place in V in . However, the NTSMC method leads to lesser steady-state error than the SMC and SM-PID methods. Table 2 gives the settling time, tracking time and steady-state error of the output voltage in three methods. It can be seen that the NTSMC outperforms all others in all cases.
1
0 .333
500
0 .6
2
x2 (V/s)
x1 (V) -0.40
-0.30
-0.20 3
0 0.00
-0.10
0.10
0 .8182 -500
-1000
-1500
-2000
1000 1
x2 (V/s)
1500 2
1000 500
x1 (V) -0.40
-0.35
-0.30
-0.25
-0.20
-0.15 3
500
-0.10
-0.05
0 0.00
0.05
-500
-1000
-1500
-2000 Fig. 13. Simulated state trajectories of the converter due to a step change in R from 10 O to 2 O obtained by the NTSMC method: (a) State trajectories with different g when l is constant, and (b) State trajectories with different l when g is constant.
H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
when the step change occurs, the switch stays ON until the inductor current reaches to its desired steady-state value. Then, when the desired steady-state is reached, the switch is turned OFF and ON, continuously, keeping the converter in the new operating point. The experimental result shown in Fig. 12(b) is in close agreement with the simulation result shown in Fig. 12(a). The simulated state trajectories of the converter, due to the step change in R from 10 O to 2 O, are shown in Fig. 13. The trajectory in Fig. 13(a) was obtained with l ¼ 100 and various values of g. It can be seen that when the value of g is high, the slope of the sliding line is small resulting in a slower dynamic response. When the value of g is lower, the slope of the sliding line is greater resulting in a faster response of controller. On the other hand, the trajectory in Fig. 13(b) was obtained with g ¼ 0:8182 and various values of l. It can be seen that when the value of l is low, the slope of the sliding line is small resulting in a slower dynamic response. When the value of l is higher, the slope of the sliding line is greater resulting in a faster response of controller. However, when the value of l is too high, an overshoot may occur on the output voltage. Fig. 14 shows the simulated and experimental responses of the output voltage and the inductor current to step load resistance
331
changes in R from 2 O to 10 O, from 10 O to 2 O, and from 2 O to 10 O, respectively. At t ¼ 0:6 s, the first step load change occurs and as a result of this, the inductor current changes from 2.5 A to 0.5 A and the output voltage exhibits an overshoot. The second load change occurs at t ¼ 1:2 s which causes a change in the inductor current from 0.5 A to 2.5 A and a drop in the output voltage. The third load change exhibits the identical behavior mentioned in the first load change. It is worth noting that the experimental result shown in Fig. 14(b) is in close agreement with the simulation result shown in Fig. 14(a). These results clearly show the performance of the NTSMC method.
6. Conclusions A non-singular terminal sliding mode control (NTSMC) method, which eliminates the singularity problem existing in the terminal sliding mode control (TSMC) and assures finite time convergence of the output voltage error to the equilibrium point, has been presented for DC–DC buck converters. It is shown that the NTSMC method has the same finite time convergence as that of the TSMC method. The influence of the fractional power, g, and
Output voltage, vo (V) and Inductor current, iL (A)
6
5
4
3
2
1
0
0.5
1
1.5
2
Time (s)
Fig. 14. Simulated and experimental responses of the output voltage and the inductor current to step load resistance changes in R from 2 O to 10 O, from 10 O to 2 O, and from 2 O to 10 O. (a) Simulation, and (b) Experiment (vo: 1 V/div, iL:1 A/div).
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the sliding coefficient, l, on the dynamic response of the converter is investigated. It is worth noting that when l is kept constant, the slope of the sliding line increases when the value of g is decreased which leads to a faster transient response of the output voltage during the load changes. On the other hand, it is observed that when g is kept constant, the slope of the sliding line increases when the value of l is increased leading to a faster transient response of the output voltage during the load changes. When the simulation results of the NTSMC method are compared with the conventional SMC and sliding mode implementation of PID control (SM-PID), it is seen that the NTSMC method exhibits a considerable improvement in terms of a faster output voltage response during load changes. The theoretical considerations have been verified both by numerical simulations and by experimental measurements from a laboratory prototype. References Babazadeh, A, & Maksimovic, D (2009). Hybrid digital adaptive control for fast transient response in synchronous buck DC–DC converters. IEEE Transactions on Power Electronics, 24, 2625–2638. Feng, Y, Yu, X, & Man, Z (2002). Non-singular terminal sliding mode control of rigid manipulators. Automatica, 38, 2159–2167. Jafarian, M J, & Nazarzadeh, J (2011). Time-optimal sliding-mode control for multi-quadrant buck converters. IET Power Electronics, 4, 143–150. Leung, K K S, & Chung, H S H (2004). Derivation of a second-order switching surface in the boundary control of buck converters. IEEE Power Electronics Letters, 2, 63–67. Leung, K K S, & Chung, H S H (2007). A comparative study of boundary control with first- and second-order switching surfaces for buck converters operating in DCM. IEEE Transactions on Power Electronics, 22, 1196–1209. Man, Z H, Paplinski, A P, & Wu, H R (1994). A robust MIMO terminal sliding mode control scheme for rigid robotic manipulator. IEEE Transactions on Automatic Control, 39, 2464–2469. Nguyen, V M, Lee, C Q (1995). Tracking control of buck converter using slidingmode with adaptive hysteresis. In Proceedings of the IEEE power electronics specialists conference, PESC’95 (pp.1086–1093), Atlanta.
Nguyen, V M, Lee, C Q (1996). Indirect implementations of sliding-mode control law in buck-type converters. In Proceedings of the IEEE applied power electronics conference and exposition, APEC’96 (pp.111–115), San Jose. Perry, A G, Feng, G, Liu, Y F, Sen, P C (2004). A new sliding mode like control method for buck converter. In: Proceedings of the IEEE power electronics specialists conference, PESC’04 (pp.3688–3693), Aachen. ˜ o, R (2012). Sliding mode controlled multiphase Ramos, R, Biel, D, Fossas, E, & Grin buck converter with interleaving and current equalization. Control Engineering Practice Sira-Ramı´rez, H, Luviano-Jua´rez, A, & Corte´s-Romero, J (2012). Robust input–output sliding mode control of the buck converter. Control Engineering Practice Tan, S C, Lai, Y M, Cheung, M K H, & Tse, C K (2005). On the practical design of a sliding mode voltage controlled buck converter. IEEE Transactions on Power Electronics, 20, 425–437. Tan, S C, Lai, Y M, & Tse, C K (2006b). A unified approach to the design of PWMbased sliding-mode voltage controllers for basic DC–DC converters in continuous conduction mode. IEEE Transactions on Circuits Systems-I, 53, 1816–1827. Tan, S C, Lai, Y M, & Tse, C K (2008a). Indirect sliding mode control of power converters via double integral sliding surface. IEEE Transactions on Power Electronics, 23, 600–611. Tan, S C, Lai, Y M, & Tse, C K (2008b). General design issues of sliding-mode controllers in DC–DC converters. IEEE Transactions on Industrial Electronics, 55, 1160–1174. Tan, S C, Lai, Y M, Tse, C K, & Cheung, M K H (2006a). Adaptive feedforward and feedback control schemes for sliding mode controlled power converters. IEEE Transactions on Power Electronics, 21, 182–192. Truntic, M, Milanovic, M, & Jezernik, K (2011). Discrete-event switching control for buck converter based on the FPGA. Control Engineering Practice, 19, 502–512. Tsai, J F, & Chen, Y P (2007). Sliding mode control and stability analysis of buck DC–DC converter. International Journal of Electronics, 94, 209–222. Tse, C K, & Adams, K M (1992). Quasi-linear analysis and control of DC–DC converters. IEEE Transactions on Power Electronics, 7, 315–323. Utkin, V I (1978). Sliding modes and their applications in variable structure systems. Moscow: Mir-Publisher. Wang, J, Xu, J, & Bao, B (2011). Analysis of pulse bursting phenomenon in constanton-time-controlled buck converter. IEEE Transactions on Industrial Electronics, 58, 5406–5410. Zak, M (1988). Terminal attractors for addressable memory in neural network. Physics Letters A, 133, 18–22.
Contents lists available at SciVerse ScienceDirect
Control Engineering Practice journal homepage: www.elsevier.com/locate/conengprac
Non-singular terminal sliding-mode control of DC–DC buck converters Hasan Komurcugil n Computer Engineering Department, Eastern Mediterranean University, Famagusta, North Cyprus, Via Mersin 10, Turkey
a r t i c l e i n f o
abstract
Article history: Received 9 December 2011 Accepted 17 November 2012 Available online 11 December 2012
This paper presents a non-singular terminal sliding mode control (NTSMC) method for DC–DC buck converters. The NTSMC method eliminates the singularity problem which arises in the terminal sliding mode due to the fractional power and assures the finite time convergence of the output voltage error to the equilibrium point during the load changes. It is shown that the NTSMC method has the same finite time convergence as that of the terminal sliding mode control (TSMC) method. The influence of the fractional power on the state trajectory of the converter is investigated. It is observed that the slope of the sliding line becomes larger with decreasing value of the fractional power which leads to a faster transient response of the output voltage during the load changes. The theoretical considerations have been verified both by numerical simulations and experimental measurements from a laboratory prototype. & 2012 Elsevier Ltd. All rights reserved.
Keywords: Sliding-mode control Terminal sliding-mode control Non-singular terminal sliding-mode control DC–DC buck converter
1. Introduction 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 non-linear and time-varying systems due to their switching operation, the design of high performance control strategy is usually a challenging 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. Non-linear 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 (Babazadeh & Maksimovic, 2009; Jafarian & Nazarzadeh, 2011; Leung & Chung, 2004, 2007; Nguyen & Lee, 1996, 1995; Perry, Feng, Liu, & Sen, 2004; Ramos, ˜ o, 2012; Sira-Ramı´rez, Luviano-Jua´rez, & Biel, Fossas, & Grin Corte´s-Romero, 2012; Tse & Adams, 1992; Tan, Lai, Cheung, & Tse, 2005; Tan, Lai, Tse, & Cheung, 2006a; Tan, Lai, & Tse, 2006b; Tan, Lai, & Tse, 2008a; Tan, Lai, & Tse, 2008b; Tsai & Chen, 2007; Truntic, Milanovic, & Jezernik, 2011; Wang, Xu, & Bao, 2011 ). 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 (Nguyen & Lee, 1995; Nguyen & Lee, 1996; Perry et al., 2004; Tan et al., 2005;
n
Tel.: þ90 392 630 1363; fax: þ90 392 365 0711. E-mail address: [email protected]
0967-0661/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conengprac.2012.11.006
Tan et al., 2006a, 2006b; Tan et al., 2008a, 2008b; Tsai & Chen, 2007; Ramos et al., 2012; Sira-Ramı´rez et al., 2012). The design of an SMC consists of two steps: design of a sliding surface and design of a control law (Utkin (1978); Tan et al., 2006a). 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 Nguyen and Lee (1996) 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 Tan et al. (2005). The adaptive feed forward and feedback based SMC method introduced in Tan et al. (2006a) 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 method introduced in Tan et al. (2008a) reduces the steady-state error in the output voltage at the expense of having additional two states in the sliding surface function. In Jafarian and Nazarzadeh (2011), a time-optimal based SMC has been introduced aiming to improve the output voltage regulation of the converter subjected to any disturbance. The SMC method in Tsai and Chen (2007) is based on the alternative model of the buck converter with bilinear terms. In most SMC methods 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. The use of such coefficient makes the sliding line static during load
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H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
variations resulting in a poor transient response in the output voltage. Despite the transient response can be made faster by utilizing a higher 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) employs a non-linear sliding surface function (Man, Paplinski, & Wu, 1994) which has the ability to converge the system states to the equilibrium point in finite time. Although the finite time convergence of the system states is achieved successfully, there exists a singularity problem due to negative fractional power in the TSMC design. A remedy to this problem is to develop a new TSMC that would avoid the singularity. In this paper, the non-singular terminal sliding mode control (NTSMC) method, introduced in Feng, Yu and Man (2002), has been adopted for controlling the DC–DC buck converter. The idea behind this method is to form a new non-linear sliding surface function that causes no singularity points and assures a finite time convergence of the output voltage error to the equilibrium point. It is shown that the NTSMC method with the new nonlinear sliding surface function has the same finite time convergence as that of the TSMC method. The singularity problem can be eliminated by utilizing the output voltage error (first system state) in the new non-linear sliding surface function with no fractional power. The influence of the fractional power on the state trajectory of the converter is investigated. It is observed that the slope of the sliding line becomes larger with decreasing value of the fractional power which leads to a faster transient response of the output voltage during the load changes. The theoretical considerations have been verified both by numerical simulations and experimental measurements from a laboratory prototype. The rest of this paper is organized as follows. In Section 2, the model of buck converter is given. Section 3 reviews the sliding and terminal sliding mode control methods for the buck converter. In Section 4, the non-singular terminal sliding mode control is described for the buck converter. In Section 5, the simulation and experimental results are presented and discussed. Finally, the conclusions are addressed in Section 6.
dvo 1 vo iL ¼ C dt R Combining (1)–(4) gives diL 1 ¼ ðuV in vo Þ L dt
ð5Þ
dvo 1 vo iL ¼ C dt R
ð6Þ
where u is the control input which takes 1 for the ON state of the switch and 0 for the OFF state. The output voltage error and its derivative (rate of change of the output voltage error) can be defined as x1 ¼ vo V ref
ð7Þ
x2 ¼ x_ 1 ¼ v_ o V_ ref ¼ v_ o
ð8Þ
where x_ 1 denotes the derivative of x1 , and V ref 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 x_ 1 ¼ x2 x_ 2 ¼
ð9Þ x_ 1 þ o2o uV in V ref x1 RC
Fig. 1 shows a DC–DC buck converter. It consists of a DC input voltage source (V in ), a controlled switch (Sw), a diode (D), a filter inductor (L), a filter capacitor (C), and a load resistor (R). The equations describing the operation of the converter can be written for the switching conditions ON and OFF, respectively as diL 1 ¼ ðV in vo Þ L dt
ð1Þ
3. Sliding and terminal sliding mode control methods for buck converter 3.1. Sliding mode control A linear sliding surface function can be expressed as S ¼ lx1 þ x2 ,
l 40
dvo vo iL ¼ C dt R
ð2Þ
and diL vo ¼ dt L
ð3Þ L
Sw
iL
+ Vin
+ D
C
vo –
– Fig. 1. DC–DC buck converter.
iR R
ð11Þ
where l is a real sliding coefficient. The dynamic behavior of (11) without external disturbance on the sliding surface is ð12Þ
In the phase-plane (x1 –x2 plane), S ¼ 0 represents a line, called sliding line, passing through the origin with a slope equal to l. The sliding mode (S ¼ 0) is described by the following first-order equation x_ 1 ¼ lx1
ð13Þ
During the sliding mode, the output voltage error is expressed as x1 ðtÞ ¼ x1 ð0Þelt
1
ð10Þ
where o2o ¼ 1=LC.
S ¼ lx1 þ x_ 1 ¼ 0 2. Modeling the DC–DC buck converter
ð4Þ
14Þ
It should be noted that l must be a positive real constant for achieving system stability. This fact can be easily verified by substituting a negative l quantity into (14) which results in x1 ðtÞ 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, u ¼ 0 (Sw is OFF) must be applied so as to direct the trajectory toward the sliding line. Conversely, when the state trajectory is below the sliding line, u ¼ 1 (Sw is ON) must be applied so that the trajectory is directed toward the sliding line. The equation of the
H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
323
x2
control input can be written as 1 u ¼ ½sgnðSÞ1 2
ð15Þ
—Vref
S=0
However, direct implementation of this control input causes the converter to operate at an uncontrollable infinite switching frequency which is not desired in practice. Hence, the following hysteresis modulation (HM) method employing a hysteresis band with the boundary layer is used to solve this very high frequency operation ( 1 when S o h u¼ ð16Þ 0 when S 4 h
Overshoot
S<0 l1 > 0
l1 = 0
Y2
S2
u=1
x1 S<0 l2 > 0
S1
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 the coefficient of the sliding line (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 SS_ o 0
l2 = 0
Vin— Vref
Y1
vo = Vin
vo = 0
ð17Þ
u=1
Hysteresis band
The time derivative of (11) can be written as S_ ¼ lx_ 1 þ x_ 2
x2
l2 = 0
ð18Þ
Substituting (9),(10) into (18) yields the following inequalities 1 x2 þ o2o V in V ref x1 40 for S o0 when u ¼ 1 l1 ¼ l RC
l1 = 0
Vin—Vref
-Vref
ð19Þ 1 x2 o2o V ref þ x1 o0 l2 ¼ l RC
for
S 40
when
u¼0
ð20Þ
S=0
Equations l1 ¼ 0 and l2 ¼ 0 define two lines in the phase-plane with the same slope passing through points P 1 ¼ V in V ref ,0 and P 2 ¼ V ref ,0 on the x1 axis, respectively. The slope of these lines is given by m1 ¼ m2 ¼
o2o
l1=RC
1 RC
x1 S1
ð21Þ
The regions of existence of the sliding mode for different l values (l 41=RC and l o 1=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 toward the sliding line by an appropriate switching action. The sliding mode occurs only on the portion of the sliding line, S ¼ 0, 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 S1 S2 , it overshoots the sliding line which leads to an overshoot in the output voltage. of l2 are Y1 ¼ The2 x2 intercepts the lines l1 and oo V ref V in = l1=RC and Y 2 ¼ o2o V ref = l1=RC , 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 S1 S2 , and it moves toward the origin as seen in Fig. 2(b). Note that when l o 1=RC, the slope of these lines is negative which changes the x2 intercepts of the lines l1 andl2 as Y 2 ¼ o2o V ref = l1=RC and Y 1 ¼ o2o ðV ref V in Þ= l1=RC , 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 Tan et al. (2005) to set l as follows:
l¼
S2
vo = 0
vo = Vin
Fig. 2. Regions of existence of the sliding mode for: (a) l 4 1=RC and (b) l o 1=RC.
3.2. Terminal sliding mode control A non-linear sliding surface function for the buck converter system defined in (9) and (10) can be defined as g St ¼ lx1 þ x_ 1
ð23Þ where l 40, and 0 o g ¼ q=p Þ o 1 where p and q are positive odd integers satisfying p 4 q. When the system is in the terminal sliding mode (St ¼ 0), its dynamics can be determined by the following non-linear differential equation g x_ 1 ¼ lx1
ð24Þ
Note that Eq. (24) reduces to x_ 1 ¼ lx1 for g ¼ 1, which is the form of conventional SMC. It has been shown in Zak (1988) that x1 ¼ 0 is the terminal attractor of the system defined in (24). The term ‘‘terminal’’ is referred to the equilibrium which can be reached in finite time. Rearranging (24) yields
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.
Y2
dt ¼
dx1 lxg1
ð25Þ
Taking integral of both sides of (25) and evaluating the resulting equation on the time interval (x1 ð0Þ a 0, x1 ðt s Þ ¼ 0) gives the
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H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
following equation (Man et al., 1994) Z 1g 9x1 ð0Þ9 1 0 dx1 t TSMC ¼ ¼ g s l x1 ð0Þ x1 l 1g
ð26Þ
Eq. (26) means that when the system enters to the terminal sliding mode at t ¼ t r with initial condition x1 ð0Þ a0, the system state x1 converges to x1 ðt s Þ ¼ 0 in finite time and stay there for t Zt s . In order to ensure that the movement of the error variables is maintained on the sliding line, the following existence condition must be satisfied St S_ t o0
ð27Þ
Taking time derivative of (23) and using it in (27) gives g1 St S_ t ¼ Sn lgx1 x_ 1 þ x_ 2
ð28Þ
Substituting (9),(10) into (28) and solving for u gives 1 x2 g1 uTSMC þ o2o ðV ref þ x1 Þlgx1 x2 ¼ 2 eq oo V in RC
ð29Þ
Eq. (29) is the equivalent control required to maintain the error variables on the sliding line under ideal terminal sliding mode. It is obvious from Eq. (29) that the fractional power g 1 in the third g1 term x1 x2 containing may result in a singularity if x2 a0 when x1 ¼ 0 due to g 1 o0 Feng et al. (2002). As a result of singularity, the TSMC cannot guarantee a bounded control input which may adversely affect the stability of the buck converter. A remedy to this problem is to develop a new TSMC that would avoid the singularity. In this paper, the non-singular terminal sliding mode control (NTSMC) method introduced in Feng et al. (2002) has been adopted for controlling the DC–DC buck converter.
4. Non-singular terminal sliding mode control A non-linear sliding surface function for the buck converter system defined in (9) and (10) can be defined as
Substituting (9),(10) into (33) and making use of (32) results in the following inequalities i l0 ðð1=gÞ1Þ h x2 o2o ðV in V ref x1 Þ 40 for Sn o0 when u ¼ 1 x2 x2 g RC ð35Þ i l0 ðð1=gÞ1Þ h x2 þ o2o ðV ref þ x1 Þ o 0 x2 x2 g RC
for
Sn 4 0
when
u¼0 ð36Þ
Combining (35),(36) gives i i l0 ðð1=gÞ1Þ h x2 l0 ðð1=gÞ1Þ h x2 o2o ðV in V ref x1 Þ o x2 o x2 þ o2o ðV ref þ x1 Þ x g 2 RC g RC ð37Þ It can easily be shown that the existence condition (32) is guaranteed if the following condition holds i g 2ð1=gÞ h x2 þ o2o ðV ref þ x1 Þ o 0 ð38Þ o2o V in o 0 x2 RC l Note that it is very difficult to defined by the inequality in (38). the sliding mode exists around x2 ffi 0), if the following condition
find the regions of existence However, it can be shown that the equilibrium point (x1 ffi 0, is satisfied
o2o V in o o2o V ref o0
ð39Þ
The block diagram of the DC–DC buck converter with the NTSMC method is shown in Fig. 3.
5. Simulation and experimental results In order to demonstrate the performance of the NTSMC approach, the DC–DC buck converter system has been tested both by simulations and experiments. The specifications of the buck converter are given in Table 1. The parameters of the controller are h ¼ 0:02, l ¼ 100, and g ¼ 0:6. Simulations are carried out using the Simulink of Matlab with a step size of 0.2 ms. The Simulink model
0 1=g
Sn ¼ x1 þ l x_ 1
ð30Þ 1=g . When the system is in where g is defined in (23) and l ¼ 1=l the terminal sliding mode (Sn ¼ 0), its dynamics can be determined by the following non-linear differential equation 1=g x_ 1
¼
1
l0
x1
ð31Þ
L
Sw
0
+
Vin
D
iL
iC
+
C
vo
u
–
iR R
–
vo
a=b
) It should be noted that by making use of property y ¼ x yb ¼ xa , (31) can be rewritten which is equivalent to (24). Therefore, the finite time taken to reach the equilibrium point of the NTSMC system is the same as the one of the TSMC system as given in (26). In order to ensure that the movement of the error variables is maintained on the sliding line, the following existence condition must be satisfied Sn S_ n o 0
ð32Þ
Taking time derivative of (30) and using it in (32) gives l0 ð1=gÞ1 Sn S_ n ¼ Sn x_ 1 þ x_ 1 x_ 2
g
Substituting (9),(10) into (33) and solving for u gives 0 g l x2 o2o l0 2ð1=gÞ þ ¼ ð V þ x Þx uNTSMC 1 ref eq 2 g o2o V in l0 gRC
ð33Þ
ð34Þ
In Eq. (34), it is obvious that as long as 1o ð1=gÞ o2 2ð1=gÞ (i.e., q op o2q), the term x2 is always non-singular . Therefore, the singularity problem will not occur in the NTSMC.
Hysteresis modulation
x1
1/ C
+
Vref
–
(...)1/γ
x1 γ
(1 / λ)1/ Sn
+
+
NTSMC
Fig. 3. Block diagram of the buck converter with the NTSMC method.
Table 1 Specifications of buck converter. Description
Parameter
Nominal value
Input voltage Inductance Capacitance Load resistance Reference output voltage
V in L C R V ref
10 V 1 mH 1000 mF 10 O 5V
H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
of buck converter with the proposed NTSMC is shown in Fig. 4. The block named ‘‘Hysteresis Modulation’’ implements Eq. (16). Experimental results were obtained from a hardware setup constructed in the laboratory.
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Fig. 5 shows the simulated start-up response of the equivalent control input (uTSMC ), the output voltage (vo ), the output voltage eq error (x1 ), and the rate of change of the output voltage error (x2 ) obtained by TSMC method for R ¼ 2 O. The value of l was set to
Fig. 4. Simulink model of buck converter with the NTSMC method.
0.8
1
0.7
−1 0.5
x1 (V)
Equivalent control, ueq
0 0.6
0.4 0.3
−2
−3
0.2
−4 0.1
−5
0 0
0.05
0.1
0.15
0
0.2
0.05
1500
5
1000
4
500
x2 (V/s)
Output voltage, vo (V)
6
3
−500
1
−1000
0
0.05
0.1
Time (s)
0.15
0.2
0
2
0
0.1
Time (s)
Time (s)
0.15
0.2
−1500 0
0.05
0.1
0.15
0.2
Time (s)
Fig. 5. Simulated response of equivalent control input (uTSMC ), output voltage (vo ), the output voltage error (x1 ), and the rate of change of the output voltage error (x2 ) eq obtained by TSMC method: (a) uTSMC , (b) vo , (c) x1 , and (d) x2 . eq
0.6
1
0.5
0
0.4
−1 x1 (V)
Equivalent control, ueq
Fig. 6. Experimental response of x1 , x2 , and vo obtained by TSMC method: (a) x1 and x2 , and (b) vo .
0.3
−2
0.2
−3
0.1
−4
0
0
0.05
0.1
0.15
−5
0.2
0
0.05
Time (s)
0.15
0.2
0.15
0.2
300
6
250
5
200 4
x2 (V/s)
Output voltage, vo (V)
0.1 Time (s)
3
150 100
2 50 1
0
0
0
0.05
0.1 Time (s)
0.15
0.2
−50
0
0.05
0.1 Time (s)
Fig. 7. Simulated response of uNTSMC , vo , x1 , and x2 obtained by NTSMC method: (a) uNTSMC , (b) vo , (c) x1 , and (d) x2 . eq eq
Fig. 8. Experimental response of x1 , x2 , and vo obtained by NTSMC method: (a) x1 and x2 , and (b) vo .
5.05
7.5 SM-PID
NTSMC
5 7
SM-PID
Output voltage, vo (V)
Output voltage, vo (V)
4.95 SMC
4.9 4.85
NTSMC
4.8
SMC
6.5
6
5.5
4.75 5
4.7 4.65 0.25
0.3
0.35
0.4
4.5 0.25
0.3
Time (s)
0.35
0.4
Time (s)
5.001
5
Output voltage, vo (V)
4.999 NTSMC
4.998 4.997 4.996
SM-PID
4.995 4.994 4.993
SMC
4.992 4.991 0.25
0.3
0.35
0.4
Time (s)
Fig. 9. Simulated output voltage responses due to the step changes in R from 10 O to 2 O,V ref from 5 V to 7 V and V in from 10 V to 9 V obtained by SMC, SM-PID and NTSMC methods: (a) Step change in R from 10 O to 2 O, (b) Step change in V ref from 5 V to 7 V and (c) Step change in V in from 10 V to 9 V.
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Table 2 Comparisons of three control methods for output voltage.
NTSMC SM-PID SMC
Settling time (ms)
Tracking time (ms)
Steady state error (mV)
(Due to R: 10 O-2 O) 13.8 33.3 40.2
(Due to V ref : 5 V-7 V) 25 50 50
(Due to V in : 10 V-9 V) 3.2 4.5 9
Fig. 10. Experimental 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 method (vo: 1 V/div, iL:1 A/div).
6
4.0 3.5
5
4 2.5 3
2.0 1.5
2
Inductor current, iL (A)
Output voltage, vo (V)
3.0
1.0 1 0.5 0.0
0 0.080
0.085
0.090
0.095
0.100
0.105
0.110
0.115
0.120
Time (s)
Fig. 11. Simulated and experimental 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 NTSMC method: (a) Simulation, and (b) Experiment (vo: 1 V/div, iL:0.5 A/div).
H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
100. On the other hand, the value of g was 0:6 (q ¼ 3 and p ¼ 5) which satisfies the condition q o p o 2q. Note that the steady state value of the control input must be 0.5 due to the fact that vO ¼ uV in for buck converters. It can be seen from Fig. 5(a) that the steady state value of uTSMC is 0.5 on the average. The eq occurrence of singularity is clearly visible on the control input resulting in unwanted oscillations on the output voltage shown in Fig. 5(b). The responses of x1 and x2 are shown in Fig. 5(c) and (d), respectively. Note that the singularity causes oscillations on the output voltage which start when x1 ¼ 0 and x2 a 0. Fig. 6 shows the experimental responses of x1 , x2 , and vo for the case presented in Fig. 5. It is clear that the Fig. 6(a) and (b) are in good agreement with the simulation results shown in Fig. 5(c), (d) and (b), respectively.
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Fig. 7 shows the simulated start-up responses of uNTSMC , vo , x1 , eq and x2 obtained by NTSMC method for R ¼ 2 O using l ¼ 100 and g ¼ 0:6. It can be seen from Fig. 7(a) that the occurrence of singularity is avoided by the NTSMC method which does not cause any oscillations on the output voltage (see Fig. 7(b)). As explained in Section 4, the main reason of this avoidance comes from the fact that the fractional power (2 1=g ) in Eq. (34) is always positive for 1o ð1=gÞ o 2. Fig. 8 shows the experimental responses of x1 , x2 , and vo for the case presented in Fig. 7. One can easily see that Fig. 8(a) and (b) are in good agreement with the simulation results shown in Fig. 7(c), (d) and (b), respectively. Fig. 9 shows the simulated output voltage responses due to the step changes in R from 10 O to 2 O (Fig. 9(a)),V ref from 5 V to 7 V
a 2
3.0
2.0
1
1.5
1.0
Control input, u
Inductor current, iL (A)
2.5
0
0.5
0.0 0.0990
0.0995
0.1000 Time (s)
0.1005
-1 0.1010
b
Fig. 12. Simulated and experimental responses of the inductor current and the control input due to a step change in R from 10 O to 2 O obtained by the NTSMC method: (a) Simulation, and (b) Experiment (iL:0.5 A/div).
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The experimental response of the output voltage and the inductor current obtained by SMC method for a step change in R from 10 O to 2 O is shown in Fig. 10. The output voltage takes about 50 ms to track its reference which agrees well with the simulation result of the same case shown in Fig. 9(a). Fig. 11 shows the simulated and experimental responses of the output voltage and the inductor current obtained by NTSMC method for a step change in R from 10 O to 2 O at t ¼ 0:1 s. As a result of this change in the load resistance, the inductor current changes from 0.5 A to 2.5 A. The drop in the output voltage went down by 0.3 V. It took about 4 ms for the NTSMC method to correct the output voltage at 5 V. It can be observed that Fig. 11(b) is in good agreement with the simulation result in Fig. 11(a). When Figs. 10 and 11(b) are compared, it can be seen that NTSMC method is much faster than SMC method. Fig. 12 shows the simulated and experimental responses of the inductor current and control input obtained by NTSMC method for a step change in R from 10 O to 2 O at t ¼ 0:1 s. It is clear that
(Fig. 9(b)) and V in from 10 V to 9 V (Fig. 9(c)) obtained by the SMC (Tan et al., 2005), sliding mode implementation of PID control (SM-PID) introduced in Tan et al. (2006a, 2006b) and the NTSMC methods. In order to avoid the infinite switching frequency, a hysteresis modulation (see Eq. 16) method employing a hysteresis band h ¼ 0:02 has been used in each method. Note that for the sake of a fair comparison, the value of l was set to 100 in all cases. Also, the value of g used in the NTSMC method was 0.6. It is clear from Fig. 9(a) and (b) that the NTSMC method acts faster than the SMC and SM-PID methods in correcting the output voltage at desired levels 5 V and 7 V, respectively. On the other hand, it is obvious from Fig. 9(c) that all methods result in a slight steadystate error in the output voltage when a step change takes place in V in . However, the NTSMC method leads to lesser steady-state error than the SMC and SM-PID methods. Table 2 gives the settling time, tracking time and steady-state error of the output voltage in three methods. It can be seen that the NTSMC outperforms all others in all cases.
1
0 .333
500
0 .6
2
x2 (V/s)
x1 (V) -0.40
-0.30
-0.20 3
0 0.00
-0.10
0.10
0 .8182 -500
-1000
-1500
-2000
1000 1
x2 (V/s)
1500 2
1000 500
x1 (V) -0.40
-0.35
-0.30
-0.25
-0.20
-0.15 3
500
-0.10
-0.05
0 0.00
0.05
-500
-1000
-1500
-2000 Fig. 13. Simulated state trajectories of the converter due to a step change in R from 10 O to 2 O obtained by the NTSMC method: (a) State trajectories with different g when l is constant, and (b) State trajectories with different l when g is constant.
H. Komurcugil / Control Engineering Practice 21 (2013) 321–332
when the step change occurs, the switch stays ON until the inductor current reaches to its desired steady-state value. Then, when the desired steady-state is reached, the switch is turned OFF and ON, continuously, keeping the converter in the new operating point. The experimental result shown in Fig. 12(b) is in close agreement with the simulation result shown in Fig. 12(a). The simulated state trajectories of the converter, due to the step change in R from 10 O to 2 O, are shown in Fig. 13. The trajectory in Fig. 13(a) was obtained with l ¼ 100 and various values of g. It can be seen that when the value of g is high, the slope of the sliding line is small resulting in a slower dynamic response. When the value of g is lower, the slope of the sliding line is greater resulting in a faster response of controller. On the other hand, the trajectory in Fig. 13(b) was obtained with g ¼ 0:8182 and various values of l. It can be seen that when the value of l is low, the slope of the sliding line is small resulting in a slower dynamic response. When the value of l is higher, the slope of the sliding line is greater resulting in a faster response of controller. However, when the value of l is too high, an overshoot may occur on the output voltage. Fig. 14 shows the simulated and experimental responses of the output voltage and the inductor current to step load resistance
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changes in R from 2 O to 10 O, from 10 O to 2 O, and from 2 O to 10 O, respectively. At t ¼ 0:6 s, the first step load change occurs and as a result of this, the inductor current changes from 2.5 A to 0.5 A and the output voltage exhibits an overshoot. The second load change occurs at t ¼ 1:2 s which causes a change in the inductor current from 0.5 A to 2.5 A and a drop in the output voltage. The third load change exhibits the identical behavior mentioned in the first load change. It is worth noting that the experimental result shown in Fig. 14(b) is in close agreement with the simulation result shown in Fig. 14(a). These results clearly show the performance of the NTSMC method.
6. Conclusions A non-singular terminal sliding mode control (NTSMC) method, which eliminates the singularity problem existing in the terminal sliding mode control (TSMC) and assures finite time convergence of the output voltage error to the equilibrium point, has been presented for DC–DC buck converters. It is shown that the NTSMC method has the same finite time convergence as that of the TSMC method. The influence of the fractional power, g, and
Output voltage, vo (V) and Inductor current, iL (A)
6
5
4
3
2
1
0
0.5
1
1.5
2
Time (s)
Fig. 14. Simulated and experimental responses of the output voltage and the inductor current to step load resistance changes in R from 2 O to 10 O, from 10 O to 2 O, and from 2 O to 10 O. (a) Simulation, and (b) Experiment (vo: 1 V/div, iL:1 A/div).
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the sliding coefficient, l, on the dynamic response of the converter is investigated. It is worth noting that when l is kept constant, the slope of the sliding line increases when the value of g is decreased which leads to a faster transient response of the output voltage during the load changes. On the other hand, it is observed that when g is kept constant, the slope of the sliding line increases when the value of l is increased leading to a faster transient response of the output voltage during the load changes. When the simulation results of the NTSMC method are compared with the conventional SMC and sliding mode implementation of PID control (SM-PID), it is seen that the NTSMC method exhibits a considerable improvement in terms of a faster output voltage response during load changes. The theoretical considerations have been verified both by numerical simulations and by experimental measurements from a laboratory prototype. References Babazadeh, A, & Maksimovic, D (2009). Hybrid digital adaptive control for fast transient response in synchronous buck DC–DC converters. IEEE Transactions on Power Electronics, 24, 2625–2638. Feng, Y, Yu, X, & Man, Z (2002). Non-singular terminal sliding mode control of rigid manipulators. Automatica, 38, 2159–2167. Jafarian, M J, & Nazarzadeh, J (2011). Time-optimal sliding-mode control for multi-quadrant buck converters. IET Power Electronics, 4, 143–150. Leung, K K S, & Chung, H S H (2004). Derivation of a second-order switching surface in the boundary control of buck converters. IEEE Power Electronics Letters, 2, 63–67. Leung, K K S, & Chung, H S H (2007). A comparative study of boundary control with first- and second-order switching surfaces for buck converters operating in DCM. IEEE Transactions on Power Electronics, 22, 1196–1209. Man, Z H, Paplinski, A P, & Wu, H R (1994). A robust MIMO terminal sliding mode control scheme for rigid robotic manipulator. IEEE Transactions on Automatic Control, 39, 2464–2469. Nguyen, V M, Lee, C Q (1995). Tracking control of buck converter using slidingmode with adaptive hysteresis. In Proceedings of the IEEE power electronics specialists conference, PESC’95 (pp.1086–1093), Atlanta.
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