2

Abstract—In order to increase the precision and simplify the process, a novel control strategy was proposed for the buck converter, which is based on the inverse system method aiming at the nonlinear identity of the mathematic model of the buck converter using the state-space average modeling method, a pseudo-linear model is given based on the inverse system method, then a Variable Structure Controller is designed to meet some performance index in terms of the pseudo-linear system. Through comparing the simulation performance with the method this paper proposed and the exited methods, it shows excellent starting-up response and strong robust to the disturbance of the input voltage and load, and the design process of the controller was simplified for a really application.

I. INTRODUCTION S a kind of DC-DC converters, Buck converter is a kind of typical switching nonlinear system. There are large limits using linear control theory on these systems, and

its dynamic response and control precision can not reach the desired result. So doing research into new non-linear control technology, it can solve the deficiency that the traditional linear control technology is for switching converter fundamentally [1-2].

To the non-linear control problems, generally we first change it into linear system, and then using linear control theory for system design and analysis. The differential geometry and inverse system are the two main branches. In recent years, the differential geometry for non-linear control system has rapid development, and it provides a feasible way to solve complex non-linear control system [3]. But this state feedback linearization method lead the problem into the geometry domain, using the lee derivatives and other complex abstract differential geometry method, the calculation is complex and the control method is difficult to achieve in engineer. As a new theory of new non-linear feedback linearization control theory, inverse system method have remarkable development and a series of theoretical results especially in the decouple control for the non-linear system [4]. Although essentially it still use the feedback linearization method to achieve the exact linearization of multi-variable, multi-linear, strong coupling system, it does not depend on the solution of nonlinear system or its stability analysis and it has no need to bring the issues into geometric domain for discussion. It is intuitive, simple and easy to understand.

Based on the considerations above, the paper adopts

Jianhua Wu. Electrical Engineering Department, northeastern university,

Shenyang 110819, China (e-mail: wujianhua@mail.neu.edu.cn). Yaqiong Liu. Electrical Engineering Department, northeastern university,

Shenyang 110819, China (e-mail: livezcc211@yahoo.cn).

inverse system method to make the original system to a pseudo-linear system, this will allow the subsequent design of control strategy become simply [5]. Also taking into account the variable structure has good robustness and fast-tracking capabilities, and the design of this controller is simple, the paper adopts variable structure control theory to design control law for the pseudo-linear system [6]. At last, establish the Simulink model, through comparing the simulation performance with the method this paper proposed and the method proposed in literature [7], it confirms the feasibility of the method, and is also shows good dynamic and static performance.

A. The CCM Buck Modeling At present for the DC-DC converter, the most widely used

method for modeling is the state-space average method. The state-space model for the CCM buck converter showed in Fig. 1 can be depicted as:

1

Lin

CL C

L

Cdi dUdtdu

i udt R C

L

C

u⎧⎪⎪⎨⎪⎪⎩

= − +

= − (1)

d is the duty cycle.

Fig. 1. Buck converter circuit

Select the state variable [ ]1 2[ , ] ,L Cx x x i u= = , the input variable u is the duty cycle d, the output variable 2y x= , through substitution, we can get:

2

( ) ( )( )

x f x g x uy h x x

= +⎧⎨ = =⎩

(2)

Where2

1 2

1

( ) 1 1

L

xLf xx x

C R C

⎡ ⎤−⎢ ⎥⎢ ⎥=⎢ ⎥−⎢ ⎥⎣ ⎦

, ( )0

inUg x L

⎡ ⎤⎢ ⎥=⎢ ⎥⎢ ⎥⎣ ⎦

.

B. The Construction of Pseudo-linear System This paper series the inverse system of the object before the

original system to get the pseudo-linear system, then it can be easy to design the controller. According to the inverse algorithm in literature [10], the inverse process of the buck converter is showed as:

Derivation for the output variable:

Variable Structure Control of Buck Converter Based on Inverse System

Jianhua Wu, Yaqiong Liu, Haixin Zhang, Jianhua Guo, Xinglong Zhao

A

iV

L

R u

+

−

PWM

C

609

Third International Workshop on Advanced Computational Intelligence August 25-27, 2010 - Suzhou, Jiangsu, China

978-1-4244-6337-4/10/$26.00 @2010 IEEE

1 21 1

Ly x x

C R C= − (3)

The equation above does not include the input variable, so derivation for the output variable again:

1 22 2 21 1 1( ) in

L L

Uy x x u

CL CLR C R C= − − − + (4)

The equation above include input variable clearly, it can get the inverse system of the buck converter:

2 12(1 )

LL

in

L LCLy x xR CR C

uU

+ − += (5)

In the equation (5), assume v y= , series the inverse system of the object before the original system to get a pseudo-linear system, it is described as Fig. 2:

Fig. 2. Linearization and decoupling of the buck converter

After the original system is linearized and decoupled, it is equivalent to a second-order integral inverse system.

C. Design the Variable Structure Controller After get the pseudo-linear system, it will allow the

subsequent design of control strategy become simply. Taking into account the variable structure has good robustness and fast-tracking capabilities, so this paper design the controller based on variable structure theory.

According to the variable structure method [9], Assume the design goal is *x x→ , *e x x= − and switching surface s ce e= + , using index reaching method

sgn( )s Ce e ks sε= + = − − , we can get the variable structure control law: * *( ) ( ) sgn( )v c k x ck x x cx cx xε= − + − − − − + (6)

II. SIMULATION RESULTS

A. Simulation Parameters In order to verify the correctness and validity of the control

strategy to design the buck converter controller, the paper establishes the simulink model of the buck converter, the simulation parameters are described as:

Input voltage inU =100 V, load resistance LR =10 Ω, inductance L =2 mH ， capacitor C = 10 Fμ , switching frequency sf =100 kHZ 。

Variable structure control parameters are: 4 42 10 , 2 10 , 10k c ε= × = × = .

The goal output voltage is 60 V. Using this method, the output of characteristics of the

system simulation is showed as in Fig. 3:

Fig. 3. Simulation block diagram

B. Simulation Waveform Comparing the simulation performance with the method

this paper proposed and the state feedback linearization control theory, the simulation results is shown as follows(the red line present the result of this paper ,the blue line present the result of the state feedback linearization control theory)

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-3

0

10

20

30

40

50

60

outp

ut

volta

ge

Fig. 4. Output response waveform

Though Fig. 4 it can get the risetime of the paper and the literature is 400 sμ and 600 sμ respectively.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-3

0

10

20

30

40

50

60

70

outp

ut v

olta

ge

load disturbance

Fig. 5. The output response waveforms when the lode resistance change from 10 Ω to 13Ω

Fig. 5 presents the load resistance change from 10 Ω to 13 Ω, the voltage up rush by the method in literature and this paper are both 1 V, recovery time is 280 sμ and 100 sμ , respectively.

610

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-3

0

10

20

30

40

50

60

outp

ut

volta

ge

Voltage disturbance

Fig. 6. The output response when the input voltage changes from 100V to 90V

Fig. 6 presents the output response when the input voltage changes from 100V to 90V at time 1000 sμ , it can show that the disturbance of the input voltage almost has no effect on output voltage by the method in literature and in this paper. These characteristics can be explained by the theory in the literature [7], in the expression of inverse system the input voltage inU is denominator, when there have disturbances in input voltage, the inverse system will change duty cycle according to the input voltage, so the output voltage can remain stable.

III. CONCLUSION Doing research into new non-linear control technology, it

can solve the deficiency that the traditional linear control technology is for switching converter fundamentally. Against the complexity of differential geometry control theory, the paper proposes using inverse system method to linearize the buck converter based on state-space average model. Considering the robustness and rapid tracking ability of the variable structure, then uses variable structure theory to design variable control law of buck converter. Through comparing the simulation performance with the method this paper proposed and the state feedback linearization control theory, it shows excellent starting-up response and strong robust to the disturbance of the input voltage and load. Then it verifies that the method is feasible.

REFERENCES [1] S. H. Tan, Y. Yu, “Adaptive fuzzy modeling of nonlinear dynamical

systems,” Automatica, vol. 32, no. 4, pp. 637~643, 1996. [2] D. He, R. M. Nelms, “Fuzzy logic average current-mode control for

DC-DC converters using an inexpensive 8-bitmicrocontroller,” IEEE Transaction Industry Applications, vol. 41, no. 6, pp. 1531-1538, 2005.

[3] D. Z. Chen, Geometric theory of nonlinear systems, science press, Beijing, 1982.

[4] J. H. Guo, L. H. Zhang, Inverse internal model control of three-phase voltage-type PWM rectifier, Coal Science and Technology, 2009.

[5] J. H. Choi, E. A. Misawa, “A study on sliding mode state estimation dynamic systems,” Measurement and Control, vol. 121, pp. 255~260, 1999.

[6] C. W. Li, Y. Z. Feng, Muti-variable nonlinear control based on Inverse system method, Tsinghua press, Beijing, 1991.

[7] D. X. Shuai, Y. X. Xie and X. G. Wang, “Optimal control of buck converter by state feedback linearization,” Chinese Journal of Electrical Engineering, vol. 28, no. 33, Nov. 25, 2008.

[8] J. A. Burton, S. I. Zinober, “Continuous approximations of variable structure control,” International Journal of Systems Science, vol. 17, no.6, pp. 252~259, 1986.

[9] V. I. Utkin, Sliding mode and their application in VSSs, Moscow, 1978. [10] X. Z. Dai, Muti-variable nonlinear system based on neural network

inverse system control method, science press, Beijing, 2005.

611