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International Journal of Electronics
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Sliding mode control and stability
analysis of buck DC-DC converter
J.-F. Tsai a & Y.-P. Chen aa
Department of Electrical and Control Engineering , National Chiao Tung University , 1001 Ta Hsueh Road, Hsinchu, 300, ROC Published online: 26 Feb 2007.
To cite this article: J.-F. Tsai & Y.-P. Chen (2007) Sliding mode control and stability analysis of buck DC-DC converter, International Journal of Electronics, 94:3, 209-222, DOI: 10.1080/00207210601176692
To link to this article: http://dx.doi.org/10.1080/00207210601176692
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Vol. 94, No. 3, March 2007, 209–222
Sliding mode control and stability analysis of buck
DC-DC converter
J.-F. TSAI* and Y.-P. CHEN
Department of Electrical and Control Engineering,
National Chiao Tung University, 1001 Ta Hsueh Road, Hsinchu, 300, ROC
(Received 26 September 2004; in final form 14 January 2007)
This paper is proposed to deal with the voltage regulation of buck DC-DC converter based on sliding mode control (SMC) technology. A buck DC-DC converter with parasitic resistance is inherently a bilinear system possessing inevitable uncertainties, such as variable resistive load and input disturbance. First, the buck DC-DC converter is modified into an uncertain linear model. Then, SMC technology is adopted to suppress the input disturbance and reduce the effects from the load variation. In addition, the continuous conduction mode (CCM) for normal operation can be guaranteed by the design of sliding function. Finally, experimental results are included for demonstration.
Keywords: Buck DC-DC converter; Reaching and sliding region (RAS-region); Sliding mode control (SMC); Voltage regulation
1. Introduction
With the switching property, the sliding mode theory provides an intuitive way to control switching converters (Sira-Ramirez 1987, Sira-Ramirez and Ilic 1988). Compared to the state space average method (Mohan et al. 1995), the sliding mode theory leads to large signal stability. Besides, it is robust to uncertainties and much easier in implementation. Up to now, several control strategies for switching converters based on sliding mode theory have been proposed (Sira-Ramirez 1987, Sira-Ramirez and Ilic 1988, Carpita and Marchesoni 1996, Caceres and Barbi 1999, Escobar et al. 1999). Sira-Ramirez (1987) presented detailed analysis of bilinear switched networks and showed that the buck DC-DC converter could keep on the sliding regimes and achieve the constant regulation with indirect control for an exact system model. In addition, Carpita and Marchesoni (1996) successfully presented a robust SMC for a buck DC-DC converter with resistive-load variation and input disturbance; however, they did not consider how the system performance is affected by the choice of sliding function.
For simplicity, a buck DC-DC converter is conventionally modelled as a linear system by neglecting the unknown parasitic resistance, which usually results in a small uncertain deviation and then reduces the system precision. For improvement, the unknown parasitic resistance should be taken into consideration, which means *Corresponding author. Email: u9012803@cc.nctu.edu.tw
International Journal of Electronics
ISSN 0020–7217 print/ISSN 1362–3060 onlineß 2007 Taylor & Francis http://www.tandf.co.uk/journals
DOI: 10.1080/00207210601176692
the buck DC-DC converter is modelled as an uncertain bilinear system. However, it is not easy work to deal with such an uncertain bilinear model via control technologies. In this paper, the uncertain bilinear model is first changed into a linear form through suitable state variables transformation, and then the sliding mode control algorithm is employed for the buck DC-DC converter design.
In general, most of the researches base on the assumption that the buck DC-DC converter operates only in the continuous conduction mode (CCM). However, if the circuit components’ values are not appropriately selected, the buck DC-DC converter may operate in the discontinuous conduction mode (DCM). In this paper, with the sliding mode control algorithm, the buck DC-DC converter is finally driven to the sliding line and then stays on it, regardless of the existence of DCM during the transient.
The remainder of this paper is organized as follows. In x 2, the model of a buck DC-DC converter with parasitic resistance is first introduced as an uncertain bilinear form, and then modified into an uncertain linear model. Section 3 shows the design procedure of sliding mode control and the system stability analysis. Section 4 gives experimental results subject to load variation, input disturbance and the effects caused by different choices of sliding functions. Finally, conclusions are given in x 5.
2. Model description
The buck DC-DC converter with resistive load R is illustrated in figure 1, where E is the DC voltage source, L is the inductance, C is the capacitance, and rd is the
parasitic resistance. Note that R ¼ R0þR, where R0 is the nominal resistive
load and R varies in the range of [r1, r2]. Clearly, R 2 [Rmin, Rmax], where
Rmin¼R0þr1and Rmax¼R0þr2.
Assume the buck DC-DC converter is operating in continuous conduction mode (CCM), then, the state equation can be expressed as
_ IL _ VC " # ¼ 0 1 L 1 C 1 RC 2 6 6 4 3 7 7 5 IL VC " # þ rd L 0 0 0 2 4 3 5 IL VC " # u þ E L 0 2 4 3 5u ð1Þ
where VC is the capacitor voltage, IL is the inductor current, and u represents the
switching input with value 0 or 1. Note that (1) is linear in control and linear in state
E L R C rd IL VC + − IC
Figure 1. Buck DC-DC converter with resistive load.
variables VC and IL, but not jointly linear in control and state variables. That
means it is a bilinear system (Mohler 1991).
Let the desired output capacitor voltage be a constant VCd. Define VCVCd
and IC/C as the new state variables, x1and x2, then
_ x1 ¼ dðVCVCdÞ dt ¼ dVC dt ¼ IC C¼x2 ð2Þ
From figure 1, it is easy to obtain that IL¼ICþ
VC
R ¼Cx2þ
x1þVCd
R ð3Þ
Further differentiating (3) and using the first equation in (1), it leads to _ x2¼ 1 CLx1 1 CRx2þ E CLu rd CLILu VCd CL ð4Þ
Let x ¼ ½x1x2T, then (2) and (3) could be rewritten as
_x ¼ Ax þ ðB þ BðxÞÞu þ h ð5Þ where A ¼ 0 1 1 CL 1 CR " # ð6Þ B ¼ 0 E CL " # ð7Þ BðxÞ ¼ 0 rd CLIL " # ¼ 0 rd CLRx1 rd Cx2 rdVCd CLR " # ð8Þ h ¼ 0 VCd CL " # ð9Þ Significantly, the bilinear system (1) is changed into (5), which is a linear system with control input u and encounters the state-dependent uncertainty B(x) and external input h. Obviously, both B(x)u and h are matched disturbances and thus it is suitable to design the controller by using the sliding mode technique.
3. Sliding mode controller design and phase plane analysis 3.1 Design procedure of sliding mode controller
In general, there are two fundamental steps to design a sliding mode control. First, choose an appropriate sliding function s to guarantee the system stability in the sliding mode s ¼ 0. Second, derive the control algorithm such that the system trajectory reaches the sliding surface in a finite time and then stays thereafter.
Sliding mode control and stability analysis of buck DC-DC converter
However, unlike the conventional sliding mode control input, the control u in (5) only switches between 0 and 1, which makes the controller design more restrictive.
In the first step, the sliding function is chosen as
s ¼ x2þx1 ð10Þ
where (40) is a constant. It guarantees the system in the sliding mode s ¼ 0 is stable, i.e., x1(t) ! 0 as t ! 1. For the second step, the control algorithm is
purposely designed as
u ¼0:5ð1 signðsÞÞ ð11Þ
which obviously switches between 0 and 1 depending on the scalar sign of s. Most importantly, (11) must satisfy the following reaching and sliding condition
s _s50, 8s 6¼0 ð12Þ
such that the system trajectories could reach the sliding line in a finite time and then stay thereafter (Slotine and Li 1991, Utkin 1978). From (5) and (10), the derivative of the sliding function with respect to time is
_s ¼ 1 CLx1þ 1 CR x2þ E CLu rd CLILu VCd CL ð13Þ
Further substituting the control algorithm u in (11) leads to
_s ¼ 1 CLx1þ 1 CR x2þ E rdIL CL VCd CL for s50 1 CLx1þ 1 CR x2 VCd CL for s40 8 > > > < > > > : ð14Þ Clearly, if VCd CL E rdIL CL 5 1 CLx1þ 1 CR x25 VCd CL ð15Þ
then the reaching and sliding condition (12) is guaranteed. The region described in (15) is defined as the reaching and sliding region or RAS-region (Chen et al. 2000) in brief. By replacing IL given in (3), the inequality (15) can be rewritten into
ðR þ rdÞVCdER RCL 5 ðR þ rdÞ RCL x1þ 1 RC rd L x2 1 CLx1þ 1 RC x25 VCd CL ð16Þ
Clearly, the RAS-region is bounded by two lines, 1 and 2, with slopes
m1¼(R þ rd)/(LRC L RCrd) and m2¼R/(LRC L) respectively, which are
related to the unknown parasitic resistance rdand the value of chosen for sliding
function s in (10). The relationship between the slopes of 1, 2 and sliding line
are classified into six cases given in table 1 and shown in figure 2. Note that P1((RVCdþrdVCdER)/(R þ rd), 0) and P2(VCd, 0) are the points of 1 and 2
crossing x1-axis.
3.2 Phase plane analysis
In normal operation, the output voltage VC of the buck DC-DC converter is
non-negative, which implies
x1¼VCVCd VCd ð17Þ
and then the system trajectory should be inherently in the right-half plane x1 VCd. Moreover, when the buck DC-DC converter operates in the
CCM, described by (3), the inductor current IL should be greater than zero
and thus ICðtÞ4 VCðtÞ R ð18Þ x1 x1 x1 x1 x1 x1 s1 s1 s1 s1 s1 s2 s2 s2 s2 s2 s2 x2 x2 x2 s1 x2 x2 x2 P2 P1 P2 P1 P1 P1 P1 P1 P2 P2 P2 P2
Sliding line Sliding line Sliding line Sliding lin
e
Case A Case B
Case D Case E Case F
Case C
Sliding line Sliding line
Figure 2. Six cases of the relationship between the 1, 2and sliding line in the phase plane. Table 1. The relationship between the slopes of 1, 2and sliding line.
Case m1m2 A 05 1/R C R/L m150, m250, m2m15 (L/C4R2) B 1/R C R/L551/RC m150, m250, m25m15 C ¼1/RC m150, m2¼ 1, m155m2 D 1/RC551/RC þ rd/L m150, m240, m155m2 E ¼1/RC þ rd/L m1¼ 1, m240, 5m25m1 F 1/RC þ rd/L5 m140, m240, 5m25m1
Sliding mode control and stability analysis of buck DC-DC converter
If the components of the buck DC-DC converter are not carefully designed, it may fall into the DCM with IL¼0, governed as
ICðtÞ ¼
VCðtÞ
R ð19Þ
Since x1¼VCVCdand x2¼IC/C, (18) and (19) can be rewritten as
x1þRCx24 VCd ð20Þ
and
x1þRCx2¼ VCd ð21Þ
which is named as the drifting line in this paper and represents the DCM. According to (17), (20) and (21), the system trajectory of a buck DC-DC converter should be restricted to the following region
¼ x1 VCd
x1þRCx2 VCd
ð22Þ as shown in figure 3, where the system trajectories with different initial conditions are also included for u ¼ 0 (dashed-lines) and u ¼ 1 (solid-lines), which will converge to Q1(VCd, 0) and Q2(E VCd, 0), respectively. Later, dashed-lines and solid-lines will
denote the trajectories of u ¼ 0 and u ¼ 1, respectively in the phase plane. All the trajectories with u ¼ 0 are moving clockwise to reach the drifting line first and then approach to Q1. As regards the trajectories with u ¼ 1, they are moving spirally
clockwise to Q2.
According to table 1, the RAS-regions (12) could be mainly classified into two types: 1/RC as Type-I and 41/RC as Type-II. Clearly, Type-I consists of cases A, B and C and Type-II consists of cases D, E and F. For Type-I, is the
x1 = −VCd Q2 Q1 x1 x2 Drifting line W
Figure 3. System trajectories with different initial conditions.
RAS-region and separated into 0 and 1 by s ¼ 0 as depicted in figure 4. From
(16), 0and 1are bounded as
0¼ s40 250 x14 VCd x1þRCx24 VCd 8 > > > < > > > : for u ¼ 0 ð23Þ 1¼ s50 140 x14 VCd x1þRCx24 VCd 8 > > > < > > > : for u ¼ 1 ð24Þ
Two system trajectories related to J0in 0and J1in 1are also shown in figure 4,
where J0and J1represent the initial conditions. Since all the trajectories in 0and
1 satisfy the reaching condition (12), the trajectories starting from J0and J1will
approach the sliding line in a finite time and then maintain there to generate the desired sliding mode s ¼ 0. From (10), the system behaviour is exponentially stable in the sliding mode. Obviously, with the sliding mode controller (11), the reaching and sliding condition is globally satisfied for cases of Type-I. Moreover, it could be seen that none of the system portraits will enter the drifting line. Thus, if the following inequality is satisfied
5 1
RmaxC
ð25Þ then the buck DC-DC converter will operate only in CCM, which is achieved by (25) rather than the assumption as given in Sira-Ramirez (1987), Sira-Ramirez and Ilic (1988) and Carpita and Marchesoni (1996).
x1 = −VCd x1 P2 P1 s1 s2 x2 Dr ifting line W1 W Slidi ng line J1 J0
Figure 4. System trajectories for the cases of Type-I. Sliding mode control and stability analysis of buck DC-DC converter
As for the cases of Type-II, the system behaviour is more complicated than Type-I, due to the reaching and sliding condition, being only locally satisfied. Also, from (16), four sub-regions, 0, 00, 1and 01, shown in figure 5, are bounded as
0¼ s40 250 x14 VCd x1þRCx24 VCd 8 > > > < > > > : for u ¼ 0 ð26Þ 00¼ s40 240 for u ¼ 0 ð27Þ 1¼ s50 140 x14 VCd x1þRCx24 VCd 8 > > > < > > > : for u ¼ 1 ð28Þ 0 1¼ s50 150 x1þRCx24 VCd 8 < : for u ¼ 1 ð29Þ
where 0and 00 are related to u ¼ 0 and 1and 01 are related to u ¼ 1.
There are mainly five kinds of trajectories shown in figure 5 corresponding to different points J0, J1, J2, J3and J4to reach the sliding line. For the trajectories about
J0and J1, they are just like the cases of Type-I, which will reach the sliding line in
a finite time. For the trajectory about J2, it may come from 1 or start from 00.
x1 = −VCd x1 P2 P1 J2 J1 J3 S1 S2 J4 J0 s1 W′1 W′0 W1 W0 x2 Drifting lin e Slidi ng line W s2
Figure 5. System trajectories for the cases of Type-II.
From figure 5, the trajectory must get into the sub-region 0. For the trajectory
about J3, it may come from 0or starts from 01. Also, viewing from figure 5, the
trajectory has to enter the sub-region 1. Finally, for the trajectory about J4, it comes
from 0
0and then along the drifting line for u ¼ 0. Once the trajectory passes through
the sliding line, it will get into the sub-region 0
1 where the control input is changed
to be u ¼ 1. According to trend of these five trajectories, they all will be forced to reach the segment between S1and S2of the sliding line and then move along it to the
origin. Unlike the cases of Type-I, system portraits of Type-II may consist of both CCM and DCM during the transient, but they can be still successfully driven into the sliding line and kept on s ¼ 0 with a larger convergent rate comparing to those of Type-I.
From the above analysis, it is clear that the sliding mode controller (11) could drive the system, Type-I and Type-II, to reach the sliding line and then converge to the origin.
3.3 Robustness analysis
From (5), if varying resistive load leads to matched disturbances, it could be written as
_x ¼ ðA0þAÞx þ Bu þ BðxÞu þ h ð30Þ
where A0 ¼ 0 0 1 CL 1 CR0 2 4 3 5 ð31Þ
are related to the nominal parts of the system. Besides, Ax, B(x)u and h are all treated as matched disturbances since
Ax ¼ B RCL CR0ðR0þRÞE x2 ¼BdA ð32Þ BðxÞu ¼ B rd ðR0þRÞE x1 rdL E x2 rdVCd ðR0þRÞE u ¼ BdB ð33Þ h ¼ B VCd E ¼Bdh ð34Þ
Hence, (30) can be rearranged as
_x ¼ A0x þ Bu þ BðdAþdBþdhÞ ð35Þ
According to the sliding mode theory, all the matched disturbances BdA, BdB and
Bdhwill be eliminated once the system trajectory is restricted on sliding line. Thus,
the system behaviour is insensitive to load variation R, which will be demonstrated in experiment later.
Sliding mode control and stability analysis of buck DC-DC converter
4. Experimental results
A lab-prototype of buck DC-DC converter for theoretically verifying is fabricated, which consists of a current sensor LTS-6NP and a PC-based controller with NI-6024E. The maximum sampling rate is limited to 20 KHz and the parameters of the circuit are listed in table 2. Next, three examples are given to show the effect caused by different sliding functions and load variation.
Example 1: The experimental results in figure 6(a)–(c) are obtained for different sliding functions with ¼ 100, ¼ 500 and ¼ 1000. Figure 6(a) shows the system trajectories in the phase plane. Clearly, all the trajectories are successfully driven to the sliding lines and then approach to the origin. Note that the case of ¼ 100 is corresponding to the trajectory of Type-I in figure 4 passing or starting from J1. As
to the cases of ¼ 500 and 1000, their trajectories are of Type-II in figure 5 passing or starting from J1. From figure 6(b), it is easy to find that the system trajectories
will reach the sliding modes faster for smaller . However, it can also be seen from figure 6(c) that smaller results in slow convergent rate for the output voltage to the desired value. Moreover, from figure 6(c), the steady state errors are resulted from the inevitable measuring noise and they can be reduced when is increased. In this example, it can be concluded that with larger , the SMC can provide faster dynamics and tolerate larger measuring noise.
Example 2: Let ¼ 100, 3500 and 20000. The experimental results shown in figure 7(a)–(c) actually confirm the predicted dynamics in x 3. Figure 7(a) shows the system trajectories corresponding to J1for Type-I in figure 4 and corresponding to J2
and J4for Type-II in figure 5 with ¼ 100, 3500 and 20000, respectively. Note that
the trajectory for the case of ¼ 20000 contains DCM during the transient. From figure 7(b), it can be found that two of sliding functions for ¼ 3500 and ¼ 20000 do not always converge; however, they will finally converge to zero after a short time. Unlike example 1, figure 7(c) shows that larger results in larger overshoot, which may be undesirable in some applications.
This example shows that the system performance will be converged faster for larger . Besides, larger may also result in larger overshoot or drive the buck converter into DCM during the transient. More significantly, from figure 7(a)–(c), the SMC successfully drives the buck DC-DC converter into the sliding line and, finally, achieves the desired performance.
Example 3: Let ¼ 1000, 3300, 4000 and 20000, respectively and the buck converter is connected to two distinct resistive load as R ¼ 20.5 and R ¼ 6.9 . Figure 8 shows the experimental results of the output voltage error where 1and 2
symbol for the cases of R ¼ 20.5 and R ¼ 6.9 , respectively. It is shown that the system is robust to load variation. Moreover, it is obvious that with the same ,
Table 2. Parameters of the buck DC-DC converter.
Parameter E L C Lr R0 VCd
Value 12.28(V) 2.47(mH) 470(mF) 0.7() 15.35() 8(V)
0 1 2 3 4 5 ×10−3 −10000 −8000 −6000 −4000 −2000 0 2000 4000 λ=100 λ=500 λ=1000 Time (sec) S 0 0.01 0.02 0.03 0.04 0.05 −10 −8 −6 −4 −2 0 2 Time (sec) Error (volt) λ=1000 λ=500 λ=100 −10 −8 −6 −4 −2 (a) (b) (c) 0 2 −1000 0 1000 2000 3000 4000 5000 6000 7000 x1 x2 λ=1000 λ=500 λ=100
Figure 6. (a) The system trajectories in the phase plane with different ; (b) sliding functions with different ; (c) errors of the output voltage with different sliding function.
Sliding mode control and stability analysis of buck DC-DC converter
24 −10 −8 −6 −4 −2 (a) (b) 0 2 −2000 0 2000 4000 6000 8000 10000 x1 x2 λ=100 λ=3500 λ=20000 Drifting line 0 0.002 0.004 0.006 0.008 0.01 −1 −0.5 0 0.5 1 ×104 λ=100 λ=20000 Time (sec) S 0 0.01 0.02 0.03 0.04 0.05 −10 −8 −6 −4 −2 0 2 Time (sec) (c) Error (volt) λ=100 λ=20000 λ=3500 λ=3500
Figure 7. (a) The system trajectories in the phase plane with different ; (b) the sliding function with different ; (c) errors of the output voltage with different .
the error is reduced when the load R is changed from 20.5 to 6.9. It is because the signal-to-noise ratio is higher for smaller resistive load.
From above examples, one can assign a suitable to attain a desired performance, which is listed in table 3. If the overshoot is larger than the desired specification, it can be improved by decreasing the value of according to figure 7(c). If the output error is larger than the acceptable value, it can be reduced by increasing according to figure 6(c). While the convergent rate is too slow, it can be accelerated by increasing also viewing from figure 6(c).
5. Conclusion
An alternative model of buck DC-DC converter with uncertain bilinear terms is introduced in this paper. With the proposed SMC, the buck DC-DC converter is robust to resistive load variation. The RAS-region is analysed in detail with respect to different sliding functions. Significant system behaviours for Type-I and Type-II
1 2 3 4 5 −1 −0.75 −0.5 −0.25 0 0.25 0.5 Time (sec) Error (volt) 1 2 3 4 5 −1 −0.75 −0.5 −0.25 0 0.25 0.5 Time (sec) Error (volt) 1 2 3 4 5 −1 −0.75 −0.5 −0.25 0 0.25 0.5 Time (s ec ) Error (volt) 1 2 3 4 5 −1 −0.75 −0.5 −0.25 0 0.25 0.5 Time (sec) Error (volt) λ=20000 λ=4000 λ=3300 λ=1000 ψ1 ψ2 ψ1 ψ2 ψ2 ψ2 ψ1 ψ1 ψ2 ψ2 ψ1 ψ1 ψ1 ψ1 ψ1 ψ2 ψ2
Figure 8. Errors of the output voltage with unexpected load variation.
Table 3. Tune-up of in terms of the desired performance.
Desired performance
Reduce overshoot Decreased
Reduce output error Increased
Increase convergent rate Increased
Sliding mode control and stability analysis of buck DC-DC converter
are also depicted completely in phase plane. Finally, experimental results are given to verify the predicted system performances of the buck DC-DC converter by using the SMC in this paper.
Acknowledgement
The work of this paper was supported by a grant provided by Ministry of Education, Taiwan, ROC(EX-91-E-FA06-4-4).
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