International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011

On backstepping controller Design in buck/boost DC-DC Converter Adel Zakipour, MahdiSalimi

Most of the papers presented in this field are on application of adaptive backstepping controller in buck DC to DC converters. For instance in [15] and [16] this control method is applied in a way that the controller can estimate load resistance. In [17] it is mentioned that more accurate system model can improve response of backstepping controller significantly. For this reason effect of Equivalent Series Resistance (ESR) of inductor is considered. However other parasitic elements such as power switch’s conduction loss has not been considered. Boost and buck/boost DC to DC converters are more affected by parasitic elements compared with buck converters. This is shown in figure (1). It’s obvious that consideration of these parasitic elements during modeling of buck/boost DC to DC converter can improve the designed controller’s response. However this subject is not mentioned in any of the previous papers. There are at least two main problems in applying adaptive backstepping technique to buck/boost DC to DC converters, considering parasitic elements: A) Parasitic elements’ value is highly depended on operation point, ambient temperature, etc. and it is impossible to consider a fixed value for parasitic elements of power switches. B)Adding effect of these elements complicates converter’s state-space model and classic adaptive backstepping controller design – which is used in papers [15]-[20] - is not applicable.

Abstract-In this paper, a new method for designing backstepping controller in buck/boost chopper is presented. This method is based on appropriate selection of converter’s state variable, which will simplify the designed equations and practical implementation of system. The result of simulation clearly illustrates that the designed controller has zero steady-state error and fast dynamic response and moreover in case of wide variations in load resistance or converter input voltage, the proposed controller can maintain stability. Keywords—DC to DC converter, Nonlinear and Adaptive Control, Modeling.

I. INTRODUCTION

D

Cto DC converters are widely used in renewable energy sources, industrial applications such as DC electric motors, computer systems and communication equipment [1][4]. According to the non-linear and time-variant nature of these converters, using linear control techniques is based on model linearization. Such model can describe converter’s behavior around the operation point properly. However, despite of feedback design simplicity in linear control, it’s not possible to control the system in wider range. Also presence of large disturbances might have a bad effect on response, even might cause instability [5]. Various non-linear controllers have been proposed to solve this problem. For example sliding mode controller, input/output feedback linearization, passivity based controller and adaptive backstepping [6]-[10]. Adaptive backstepping is presented by P.V.Kokotovic and his colleague in 1992 [10], and is successfully used in lots of non-linear systems, such as electric motors, auto-pilot, submarine, etc. [12]-[14]. This method is based on systematic and recursive design in feedback control of various systems. The most important capability of this technique is its capability to estimating uncertain parameters existing in system’s model. According to the presence of such uncertain parameters in DC to DC converters, using adaptive backstepping method for controlling these converters has been taken into consideration in recent years [15]-[19].

In this paper a new method for applying adaptive backstepping controller in buck/boost DC to DC converter considering parasitic elements effect, is presented. It is shown that by proper modeling, it is possible to estimate converter’s parasitic elements continuously and don’t assign constant value to these time variant elements. Also in designed equations, load resistance is assumed to be uncertain and the controller will estimate its variation.In second section buck/boost DC to DC converter state-space modeling, considering effect of energy dissipating elements is studied. In third section adaptive backstepping controller design is presented in details. Finally in order to investigate response of presented controller, buck/boost DC to DC converter is simulated by MATLAB/Simulink.

Adel Zakipour and Mahdi Salimiare with Department of electrical eng., Sarab Branch, Islamic Azad University, Sarab, Iran ([email protected] and [email protected])

II.

MODELING OF BUCK/BOOST DC TO DC CONVERTER CONSIDERING POWER LOSS

Generally there are several parasitic elements in DC to DC converter structure. Their presence cause power loss and also 147

International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011

(1) X = [x1 x2 ]T = [iL vout ]T When power switch (S) is on, according to equivalent circuit of figure (2-B), state-space equations can be written as following: − A1 = �

r S +r L

− A2 = �

r D +r L

L

Ẋ = A1 X + B1

0

v in

1 � B1 = �

L � 0 − 0 RC And when power switch is off:

Figure (1): parasitic elements effect in buck/boost DC to DC converter (𝑉𝑉𝑜𝑜 :average of output voltage, 𝑉𝑉𝑑𝑑 : average of input voltage and 𝐷𝐷 : duty cycle of power switch)

L

−

1

L 1

(2)

Ẋ = A2 X + B2

− � B2 = �

V DO

L � (3) 0 By combining equations (2) and (3), and considering the relations governing averaged state-space modeling in power electronic converters [1], more accurate buck/boost DC to DC converter can be modeled as below:

a)

1

C

−

RC

Ẋ = AX + B A = A1 u + A2 (1 − u) 1 1 − [rL + rS u + rD (1 − u) − (1 − u) L =� L � 1 1 (1 − u) − C RC 1 [vin u − VDO (1 − u)] B = B1 u + B2 (1 − u) = �L �(4) 0 In this equation, 𝑢𝑢 is duty cycle and usually is considered as control input of system. During steady-state dynamics of the system is zero and voltage gain of the converter could be formulized as following:

b) Power switch is on.

c) Power diode is on. Figure (2): buck/boost DC to DC converter in different operating conditions

from control view they may cause change in system behavior. This case in buck/boost DC to DC converter is more important. In figure (1) the response of a buck/boost DC to DC converter, considering the parasitic elements is compared to an ideal one which lossless. It is noticeable that these two characteristics are different specially for high (D) duty cycle values. This figure clearly illustrates necessity for more exact modeling during controller design process. Power loss in DC to DC has three main sources: A) Inductor’s loss due to presence of parasitic elements in winding which could be modeled with a resistor(r L ). B)Power loss in controlled power switch: according to high frequency nature of DC to DC converters and usual use of power MOSFETs, considering the linear behavior of these controlled switches in triode region, resistance(rs) can be used to model it. C)Voltage drop across the diode in forward bias which is modeled by two segment model . But of course the ESR of output capacitor is relatively small and will be neglected in this paper. Buck/ boost DC to DC with presence of parasitic elements are shown in figure (2). During state-space modeling of DC to DC converters, usually inductor’s current and capacitor’s voltage is used as state variables:

𝑉𝑉 𝐷𝐷𝐷𝐷

𝑢𝑢− 𝑉𝑉 (1−𝑢𝑢) 𝑉𝑉 0 𝑋𝑋̇ = � � ⇒ 𝑀𝑀 = 𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑟𝑟 𝐿𝐿 +𝑟𝑟 𝑆𝑆 𝑢𝑢 +𝑟𝑟 𝐷𝐷𝑖𝑖𝑖𝑖(1−𝑢𝑢 ) (5) 𝑉𝑉𝑖𝑖𝑖𝑖 0 +(1−𝑢𝑢) (1−𝑢𝑢 )𝑅𝑅

This equation clearly illustrates the influence of each parasitic element on the output voltage of converter. If we assume that parasitic elements in equation (5) are zero, the following equation will be obtained which is completely wellknown: 1T

u 1−u Main part of loss in DC to DC converters is due to inductor’s ESR. In figure (3) a simple circuit model is used for buck/boost DC to DC converter. In this circuit, only Rloss is energy dissipating element and power switches are assumed to be ideal. In a similar way averaged state-apace model of this converter can be written as below: M=

1T

Ẋ = Á X + B́ R

1

− (1 − u) − LOSS L L Á = �1 � 1 (1 − u) − C

148

RC

(6)

International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011

vin u B́ = � L � 0 And voltage gain of this one:

Step one: according to controlling purpose and above equation, first error variable z 2 is defined as below: R

1T

𝑉𝑉 𝑀𝑀́ = 𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑉𝑉𝑖𝑖𝑖𝑖

u

𝑧𝑧1 = 𝑥𝑥1 − 𝐼𝐼𝐿𝐿 (𝑟𝑟𝑟𝑟𝑟𝑟) ⇒ 𝑧𝑧̇1 = 𝑥𝑥̇ 1 − 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟) (10) By replacing the equation (9-A) in (10), we have: (10-A) 𝑧𝑧̇1 = 𝜃𝜃1 𝜔𝜔11 + 𝑙𝑙1 1 𝜔𝜔11 = − 𝑥𝑥1 (10-B) 𝐿𝐿 1 𝑣𝑣𝑖𝑖𝑖𝑖 𝑙𝑙1 = − 𝑥𝑥2 (1 − 𝑢𝑢) + 𝑢𝑢 − 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟) (10-C)

(7)

R LOSS +(1−u) (1−u )R

If 𝑀𝑀 = 𝑀𝑀́, it would be possible to assume that these two converters are equivalent and in this case we have: 1T

𝑅𝑅LOSS =

𝑉𝑉 𝑟𝑟 +𝑟𝑟 𝑢𝑢 +𝑟𝑟 𝐷𝐷 (1−𝑢𝑢 ) u� 𝐿𝐿 𝑆𝑆(1−𝑢𝑢 )𝑅𝑅 �+(1−𝑢𝑢)2 𝑉𝑉𝐷𝐷𝐷𝐷 𝑉𝑉 1 [𝑢𝑢− 𝐷𝐷𝐷𝐷 (1−𝑢𝑢)] (1−𝑢𝑢 )𝑅𝑅 𝑉𝑉 𝑖𝑖𝑖𝑖

𝐿𝐿

Since

𝑖𝑖𝑖𝑖

(8)

−1 (−𝜃𝜃�1 ̇ + 𝛾𝛾11 𝑧𝑧1 𝜔𝜔11 ) (14) 𝑉𝑉1̇ = 𝑧𝑧1 (𝜃𝜃�1 𝜔𝜔11 + 𝑙𝑙1 ) + �𝜃𝜃1 − 𝜃𝜃�1 �𝛾𝛾11 We know that, the time derivative of Lyapunov function must be negative in order to have a stable designed, this could happen if: (14-A) �𝜃𝜃�1 𝜔𝜔11 + 𝑙𝑙1 � = −𝑐𝑐1 𝑧𝑧1 ̇ −𝜃𝜃�1 + 𝛾𝛾11 𝑧𝑧1 𝜔𝜔11 = 0 (14-B) Which, in equation (14-A), 𝑐𝑐1 is a constant and positive coefficient.

2B

From historical view adaptive backstepping controller has been presented by P.V.Koktovic and his colleague in early 1990 [10]. At first this method was usable only in systems which their state equations form was PPF: Parametric pure form, but gradually devdeloped for PSF: Parametric Strict Form too [12]. Right now this method is usable in broad category which is given relatively comprehensive in [13].Considering that converters load resistance and R loss are uncertain, converter’s state space model– which is caculatedin[3] – are rewritten as below: 1 𝑣𝑣 (9-a) 𝑥𝑥̇ 1 = − [𝜃𝜃1 𝑥𝑥1 + (1 − 𝑢𝑢)𝑥𝑥2 ] + 𝑖𝑖𝑖𝑖 𝑢𝑢 4T

4T

R

𝐿𝐿

uncertain in these equations, it is possible to

parameter: 𝑧𝑧̇1 = 𝜃𝜃�1 𝜔𝜔11 + 𝑙𝑙1 + (𝜃𝜃1 − 𝜃𝜃�1 )𝜔𝜔11 (11) Now if we choose Lyapunov function as below: 1 1 −1 (𝜃𝜃1 − 𝜃𝜃�1 )2 (12) 𝑉𝑉1 = 𝑧𝑧12 + 𝛾𝛾11 2 2 −1 𝛾𝛾11 is a positive coefficient and is called parameter adaption gain. The reason of this naming will be clarified in next steps. Time derivative of Lyapunov function is expressed as below: −1 (13) �𝜃𝜃1 − 𝜃𝜃�1 �(−𝜃𝜃�1̇ ) 𝑉𝑉1̇ = 𝑧𝑧1 𝑧𝑧̇1 + 𝛾𝛾11 With replacing (11) in (13), we have:

III. CONTROLLER DESIGN

1

𝐿𝐿

θ1 is

rewrite (10-A) assuming θ1 as an estimated value for this

Calculating value of Rloss from equation (8) is not because values of parasitic elements are strongly depended on operating point of power switches, ambient temperature, etc. Moreover, load resistance (R) is an uncertain parameter and because of that adaptive method must be used to estimate Rloss. In this case if Rloss is estimated properly, circuits shown in figures (2) and (3) will be equivalent. According to this, designed controller for equivalent circuit of figure (3) can be used for main circuit of figure (2). 1T

R

R

Step two: generally equation (14-A) might not be correct, and the difference between two sides of this equation could be defined as below: 𝑧𝑧2 = �𝜃𝜃�1 𝜔𝜔11 + 𝑙𝑙1 � − (−𝑐𝑐1 𝑧𝑧1 ) = �𝜃𝜃�1 𝜔𝜔11 + 𝑙𝑙1 � + 𝑐𝑐1 𝑧𝑧1 (15) By replacing equations (10),(10-B) and (10-c) in equation (15) we will have: 1 𝑣𝑣 1 𝑧𝑧2 = −𝜃𝜃�1 𝑥𝑥1 − 𝑥𝑥2 (1 − 𝑢𝑢) + 𝑖𝑖𝑖𝑖 𝑢𝑢 − 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟) + 𝑐𝑐1 𝑥𝑥1 − 𝐿𝐿 𝐿𝐿 𝐿𝐿 𝑐𝑐1 𝐼𝐼𝐿𝐿 (𝑟𝑟𝑟𝑟𝑟𝑟) (16) Time derivative of z 2 is expressed as below. It worth noting that while simplifying the following equation, time derivatives of variables x 1 and x 2 are replaced from equations (9-A) and (9-B):

𝐿𝐿

(9-b) 𝑥𝑥̇ 2 = [(1 − 𝑢𝑢)𝑥𝑥1 − 𝜃𝜃2 𝑥𝑥2 ] 𝐶𝐶 1 (𝜃𝜃1 = 𝑅𝑅𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 , 𝜃𝜃2 = ) 𝑅𝑅 If we consider the capacitor’s voltage as an output, the resulted system would have non-minimum phase nature and for this reason adaptive backstepping control method cannot be applied [19]. This problem is easily solved by considering the inductor’s current as an output.We know that controller design purpose is calculation of control input in a way that inductor’s current become equivalent to reference current:𝑥𝑥2 = 𝐼𝐼𝐿𝐿 (𝑟𝑟𝑟𝑟𝑟𝑟)

R

R

𝑧𝑧̇2 = θ1 �𝜃𝜃�1

𝜃𝜃�1 1

𝐿𝐿𝐿𝐿

1

149

R

R

1

1

�1̇ 1 𝜃𝜃

𝐿𝐿

− 𝐼𝐼𝐿𝐿̈ (𝑟𝑟𝑟𝑟𝑟𝑟) −

𝑥𝑥1 − 𝑐𝑐1 𝑥𝑥1 � + θ2 � 𝑥𝑥2 (1 − 𝑢𝑢)� − 𝐿𝐿

𝐿𝐿

𝑣𝑣𝑖𝑖𝑖𝑖

𝑥𝑥2 (1 − 𝑢𝑢) − 𝜃𝜃�1 2 𝑢𝑢 − 𝐿𝐿 1 ̇ 𝑣𝑣 ̇ 𝑣𝑣 𝑥𝑥1 (1 − 𝑢𝑢)2 + 𝑥𝑥2 𝑢𝑢̇ + 𝑖𝑖𝑖𝑖 𝑢𝑢 + 𝑖𝑖𝑖𝑖 𝑢𝑢̇ 𝐿𝐿2

𝐿𝐿

1

𝑣𝑣𝑖𝑖𝑖𝑖

𝐿𝐿

𝑐𝑐1 𝑥𝑥2 (1 − 𝑢𝑢) + 𝑐𝑐1 𝑢𝑢 − 𝑐𝑐1 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟) 𝐿𝐿 𝐿𝐿 We can simplify this equation as below: ż 2 = θ1 ω12 + θ2 ω22 + l2 1 1 ω12 = θ�1 2 x1 − c1 x1 L

Figure (3): a simple circuit model for consideration of parasitic elements

1

𝐿𝐿2

R

R

L

𝐿𝐿

𝑥𝑥1 +

(17)

(18-A) (18-B)

International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011 1

ω22 = x2 (1 − u) (18-C) L ̇ 1 𝑣𝑣𝑖𝑖𝑖𝑖 𝜃𝜃�1 1 𝑥𝑥1 + 𝜃𝜃�1 2 𝑥𝑥2 (1 − 𝑢𝑢) − 𝜃𝜃�1 2 𝑢𝑢 𝑙𝑙2 = − 𝐿𝐿 𝐿𝐿 𝐿𝐿 ̇ 1 1 𝑣𝑣𝑖𝑖𝑖𝑖̇ 𝑣𝑣𝑖𝑖𝑖𝑖 − 𝑥𝑥1 (1 − 𝑢𝑢)2 + 𝑥𝑥2 𝑢𝑢̇ + 𝑢𝑢 + 𝑢𝑢̇ 𝐿𝐿 𝐿𝐿 𝐿𝐿𝐿𝐿 𝐿𝐿 1 𝑣𝑣𝑖𝑖𝑖𝑖 𝑢𝑢 − 𝐼𝐼𝐿𝐿̈ (𝑟𝑟𝑟𝑟𝑟𝑟) − 𝑐𝑐1 𝑥𝑥2 (1 − 𝑢𝑢) + 𝑐𝑐1 𝐿𝐿 𝐿𝐿 − 𝑐𝑐1 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟)

estimation rules for uncertain parameters. During simplification, equations (10-B), (18-B) and (18-C) are used. 1 1 1 𝜃𝜃�1̇ = 𝛾𝛾11 [− 𝑧𝑧1 𝑥𝑥1 + 𝑧𝑧2 �𝜃𝜃�1 2 𝑥𝑥1 − 𝑐𝑐1 𝑥𝑥1 �] (26-A) 𝐿𝐿 𝐿𝐿 𝐿𝐿 1 ̇ 𝜃𝜃� = 𝛾𝛾 [ 𝑥𝑥 (1 − 𝑢𝑢)𝑧𝑧 ] (26-B) 2

We rewrite the time derivative of z 2 by using equation (18-A) as below: 𝑧𝑧̇2 = 𝜃𝜃�1 𝜔𝜔12 + 𝜃𝜃�2 𝜔𝜔22 + 𝑙𝑙2 + �𝜃𝜃1 − 𝜃𝜃�1 �𝜔𝜔12 + �𝜃𝜃2 − 𝜃𝜃�2 �𝜔𝜔22 (19)

2

2

By considering equations (11) and (15) simultaneously, time derivative of z 1 can be written as below: 𝑧𝑧̇1 = −𝑐𝑐1 𝑧𝑧1 + 𝑧𝑧2 + (𝜃𝜃1 − 𝜃𝜃�1 )𝜔𝜔11 (22) By replacing z1 and z 2 respectively from equations (19) and

Table (1): simulation parameters fs=20e3; c1=44000*0.25; c2=25000*0.25; C=200e-6; L=10000e-6; R=10; RLOSS=0.2; teta1=RLOSS teta2=1/R Vin=10; gamma11=1e-7; gamma22=1e-7;

(22) in time derivative of Lyapunov function (equation (21)) and considering equations (10-B), (18-B) and (18-D), we have: 1 1 𝑉𝑉̇2 = −𝑐𝑐1 𝑧𝑧12 + 𝑧𝑧1 𝑧𝑧2 + 𝑧𝑧2 [−𝜃𝜃�1̇ 𝑥𝑥1 + 𝜃𝜃�1 2 𝑥𝑥2 (1 − 𝑢𝑢) − 𝐿𝐿 𝐿𝐿 1 1 ̇ 𝑣𝑣 ̇ 𝑣𝑣 𝑣𝑣 𝜃𝜃�1 𝑖𝑖𝑖𝑖2 𝑢𝑢 𝑥𝑥1 (1 − 𝑢𝑢)2 + 𝑥𝑥2 𝑢𝑢̇ + 𝑖𝑖𝑖𝑖 𝑢𝑢 + 𝑖𝑖𝑖𝑖 𝑢𝑢̇ − 𝐼𝐼𝐿𝐿̈ (𝑟𝑟𝑟𝑟𝑟𝑟) − 𝐿𝐿 𝐿𝐿 𝐿𝐿 𝐿𝐿𝐿𝐿 𝐿𝐿 1 1 𝑣𝑣 1 𝑐𝑐1 𝑥𝑥2 (1 − 𝑢𝑢) + 𝑐𝑐1 𝑖𝑖𝑖𝑖 𝑢𝑢 − 𝑐𝑐1 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟)+𝜃𝜃�12 2 𝑥𝑥1 − 𝑐𝑐1 𝜃𝜃�1 𝑥𝑥1 + 𝐿𝐿

𝐿𝐿

𝐿𝐿

1 −1 (−𝜃𝜃�1̇ + 𝛾𝛾11 𝑧𝑧1 𝜔𝜔11 + 𝜃𝜃�2 𝑥𝑥2 (1 − 𝑢𝑢)] + �𝜃𝜃1 − 𝜃𝜃�1 �𝛾𝛾11 𝐿𝐿 𝛾𝛾 𝑧𝑧 𝜔𝜔 )�𝜃𝜃 − 𝜃𝜃� �𝛾𝛾 −1 (−𝜃𝜃� ̇ + 𝛾𝛾 𝑧𝑧 𝜔𝜔 ) (23) 11 2

12

2

2

2

22

22 2

𝐿𝐿

22

V. CONCLUSION

In the above equation if we assume that the coefficient of z 2 equals to –c 2 z 2 and also assume that the coefficients of −1 −1 and �𝜃𝜃2 − 𝜃𝜃�2 �𝛾𝛾22 is zero: �𝜃𝜃1 − 𝜃𝜃�1 �𝛾𝛾11 2 2 𝑉𝑉̇2 = −𝑐𝑐1 𝑧𝑧1 + 𝑧𝑧1 𝑧𝑧2 + −𝑐𝑐2 𝑧𝑧2 (24)

A novel adaptive backstepping controller based on accurate averaged state-space model is proposed for buck/boost DC to DC converter to regulate the inductor current. In the design procedure load resistance and all of the parasitic elements assumed to be uncertain. Finally in order to verify accuracy of the proposed controller, buck/boost converter is simulated based on the designed controller via MATLAB/Simulink. Results clearly show that, in spite of large variations of load, the controller has good steady-state and dynamic response.

In the above equations c 2 is a constant and positive coefficient. It can be seen that if 4𝑐𝑐1 𝑐𝑐2 > 1, equation (24) is always negative and stability of the system is guaranteed.Buck/boost DC to DC converter’s control input is obtained by assuming that coefficient of z 2 is equals to –c 2 z 2 in equation (23): 𝐿𝐿 1 1 𝑣𝑣 𝑢𝑢̇ = {−𝜃𝜃�1̇ 𝑥𝑥1 + 𝜃𝜃�1 � 2 𝑥𝑥2 (1 − 𝑢𝑢) − 𝑖𝑖𝑖𝑖2 𝑢𝑢 −

𝑐𝑐1

1 𝐿𝐿 1

𝑥𝑥 2 +𝑣𝑣𝑖𝑖𝑖𝑖

𝑥𝑥1 � +𝜃𝜃�12

1

𝐿𝐿

𝐿𝐿

� 1 (1 − 𝑢𝑢) − 2 𝑥𝑥1 + 𝜃𝜃2 𝑥𝑥2

𝐿𝐿

𝐿𝐿 𝑣𝑣𝑖𝑖𝑖𝑖̇

𝑐𝑐1 𝑥𝑥2 (1 − 𝑢𝑢)+𝑐𝑐2 𝑧𝑧2 + 𝐿𝐿 𝐼𝐼𝐿𝐿̈ (𝑟𝑟𝑟𝑟𝑟𝑟)}

𝐿𝐿

𝑢𝑢 + 𝑐𝑐1

𝑣𝑣𝑖𝑖𝑖𝑖 𝐿𝐿

1

𝐿𝐿𝐿𝐿

2

In this part buck/boost DC to DC converter based on proposed adaptive backstepping controller is simulated by MATLAB/Simulink. General block diagram of the system is shown in figure (4). Also simulation parameters and other used elements are summarized in table (1).Choosingthese parametersare relatively straightforward and are given in details in [1].The response of designed controller to step reference is shown in figure (5). It obvious that this method has fast dynamic response and also has zero steady-state error.The response of the controller to load variation is given in figure (6). It’s obvious that adaptive controller is completely robust.

step we should choose Lyapunov function: 1 1 1 −1 1 −1 𝑉𝑉2 = 𝑧𝑧12 + 𝑧𝑧22 + 𝛾𝛾11 (𝜃𝜃1 − 𝜃𝜃�1 )2 + 𝛾𝛾22 (𝜃𝜃2 − 𝜃𝜃�2 )2 (20) 2 2 2 2 It’s time derivative is: −1 𝑉𝑉̇2 = 𝑧𝑧1 𝑧𝑧̇1 + 𝑧𝑧2 𝑧𝑧̇2 + 𝛾𝛾11 �𝜃𝜃1 − 𝜃𝜃�1 � �−𝜃𝜃�1̇ � +𝛾𝛾 −1 �𝜃𝜃 − 𝜃𝜃� �(−𝜃𝜃� ̇ ) (21) 2

2

IV. SIMULATION RESULTS

In equation above θ2 is estimated value for θ1 . Now in this

22

22 𝐿𝐿

Briefly, presented control system which is modeled based on equations (9-A) and (9-B), according to resulted control law (equation 25) and estimation rules ((26-A) and (26-B)) is stable because time derivative of Lyapunov function is negative.

𝐿𝐿

𝑥𝑥1 (1 − 𝑢𝑢)2 −

𝑢𝑢 − 𝑐𝑐1 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟) −

(25)

Similarly, if we consider that the coefficient of the expressions −1 −1 �𝜃𝜃1 − 𝜃𝜃�1 �𝛾𝛾11 and �𝜃𝜃2 − 𝜃𝜃�2 �𝛾𝛾22 is zero in (23), we can obtain 150

International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011

IV. ACKNOWLEDGMENT This research is supported completely by Islamic Azad University-Sarab Branch. REFERENCES [1] N.Mohan and et.al. “Power electronics: converters, applications and design” John Willey and Sons Inc, 3rd edition, 2004 [2] M.H.Rashid ”Power electronics: circuits, devices and applications” Prentice Hall, Englewood Cliffs, NJ, USA, 2nd edition 1993, [3] Y. He and F.L. Luo “Design and analysis of adaptive sliding-mode-like controller for DC–DC converters” IEE Proc.-Electr. Power Appl., Vol. 153, No. 3, May 2006 [4] B.K.Bose, “ Power electronics and AC drives” Prentice Hall, 1986 [5] M.salimi and et.al. “A Novel Method on Adaptive Backstepping Control of Buck Choppers” PEDSTC,Feb,2011 [6] J. G. Ciezki، R. W. Ashton،”The Design of Stabilizing Control for Shipboard DC-to-DC BUCK Choppers using Feedback Linearization Techniques”،0-7803-4489-8،IEEE،1998 [7] R. Leyva and et. Al, "Passivity-Based Integral Control of a Boost Converter for Large-Signal Stability IEEE Proc, Control theory Appl، Vol.153،No.2،March 2006 [8] G.Escobar and et.al. ”An Experimental Comparison of Several Nonlinear Controllers for Power Converters” 0272-1708/99/$10.00, IEEE, 1998 [9] Siew-Chong Tan, Member, IEEE, Y. M. Lai, Member, IEEE, and Chi K. Tse, Fellow, IEEE, “Indirect Sliding Mode Control of Power Converters Via Double Integral Sliding Surface” IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 2, MARCH 2008 [10] P.V.Kokotovic،”The Joy of Feedback: Nonlinear and Adaptive”،02721708/92،IEEE، 1992(Bode Prize Lecture) [11] ShuibaoGuo, Xuefang Lin-Shi, Bruno Allard, YanxiaGao, and Yi Ruan, “Digital Sliding-Mode Controller For High-Frequency DC/DC SMPS” IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 5, MAY 2010 [12] G.Cai,W.Tu “ Adaptive Backstepping Control of the Uncertain Unified Systems”, international journal of nonlinear science, vol.4, no.1, pp.1724, 2007 [13] M.Krstic and et. al. ” Nonlinear and Adaptive Control Design” John Wiley and Sons, Inc,1995 [14]HosseinAbootorabiZarchi, JafarSoltani, and Gholamreza Arab Markadeh “Adaptive Input–Output Feedback-Linearization-Based Torque Control of Synchronous ReluctanceMotorWithout Mechanical Sensor” IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 1, JANUARY 2010 [15] H.S.Ramirez and et. al.” Adaptive input-output linearization for PWM Regulation of DC-to-DC Converters” Preceding of American control conference, june, 1995 [16] L.K.Yi،et.al،”Adaptive Backstepping Sliding Mode Nonlinear Control for Buck DC/DC Switched Power Converter”،IEEE International Congruence on Control and Automation،Tune،2007 [17] S. C. Lin and C.C Tsai،”Adaptive Backstepping Control with Integral Action for PWM Buck DC-DC Converters”،Journal of the Chinese Institute of Engineering، Vol.28، No.6،PP.977،987،2005 [18] H. E. Fadil ، F. Giri،” Backstepping Based Control of PWM DC-DC Boost Power Converters”،Medwell، International Journal of Electrical and Power Engineering،479-485،2007 [19] H.ElFadil،and F.Giri،” Backstepping Based Control of PWM DC-DC Boost Converters”،1-4244-0755-9/07،IEEE،2007 [20]H.E.Fadil،and et.al. ”Nonlinear and Adaptive Control of Buck-Boost Power Converters” IEEE،2003

Figure (4) general block diagram of the adaptive backstepping controller in buck/boost DC to DC converter

Figure (5): The response of designed controller to step reference

Figure (6): The response of designed controller to load variation

151

On backstepping controller Design in buck/boost DC-DC Converter Adel Zakipour, MahdiSalimi

Most of the papers presented in this field are on application of adaptive backstepping controller in buck DC to DC converters. For instance in [15] and [16] this control method is applied in a way that the controller can estimate load resistance. In [17] it is mentioned that more accurate system model can improve response of backstepping controller significantly. For this reason effect of Equivalent Series Resistance (ESR) of inductor is considered. However other parasitic elements such as power switch’s conduction loss has not been considered. Boost and buck/boost DC to DC converters are more affected by parasitic elements compared with buck converters. This is shown in figure (1). It’s obvious that consideration of these parasitic elements during modeling of buck/boost DC to DC converter can improve the designed controller’s response. However this subject is not mentioned in any of the previous papers. There are at least two main problems in applying adaptive backstepping technique to buck/boost DC to DC converters, considering parasitic elements: A) Parasitic elements’ value is highly depended on operation point, ambient temperature, etc. and it is impossible to consider a fixed value for parasitic elements of power switches. B)Adding effect of these elements complicates converter’s state-space model and classic adaptive backstepping controller design – which is used in papers [15]-[20] - is not applicable.

Abstract-In this paper, a new method for designing backstepping controller in buck/boost chopper is presented. This method is based on appropriate selection of converter’s state variable, which will simplify the designed equations and practical implementation of system. The result of simulation clearly illustrates that the designed controller has zero steady-state error and fast dynamic response and moreover in case of wide variations in load resistance or converter input voltage, the proposed controller can maintain stability. Keywords—DC to DC converter, Nonlinear and Adaptive Control, Modeling.

I. INTRODUCTION

D

Cto DC converters are widely used in renewable energy sources, industrial applications such as DC electric motors, computer systems and communication equipment [1][4]. According to the non-linear and time-variant nature of these converters, using linear control techniques is based on model linearization. Such model can describe converter’s behavior around the operation point properly. However, despite of feedback design simplicity in linear control, it’s not possible to control the system in wider range. Also presence of large disturbances might have a bad effect on response, even might cause instability [5]. Various non-linear controllers have been proposed to solve this problem. For example sliding mode controller, input/output feedback linearization, passivity based controller and adaptive backstepping [6]-[10]. Adaptive backstepping is presented by P.V.Kokotovic and his colleague in 1992 [10], and is successfully used in lots of non-linear systems, such as electric motors, auto-pilot, submarine, etc. [12]-[14]. This method is based on systematic and recursive design in feedback control of various systems. The most important capability of this technique is its capability to estimating uncertain parameters existing in system’s model. According to the presence of such uncertain parameters in DC to DC converters, using adaptive backstepping method for controlling these converters has been taken into consideration in recent years [15]-[19].

In this paper a new method for applying adaptive backstepping controller in buck/boost DC to DC converter considering parasitic elements effect, is presented. It is shown that by proper modeling, it is possible to estimate converter’s parasitic elements continuously and don’t assign constant value to these time variant elements. Also in designed equations, load resistance is assumed to be uncertain and the controller will estimate its variation.In second section buck/boost DC to DC converter state-space modeling, considering effect of energy dissipating elements is studied. In third section adaptive backstepping controller design is presented in details. Finally in order to investigate response of presented controller, buck/boost DC to DC converter is simulated by MATLAB/Simulink.

Adel Zakipour and Mahdi Salimiare with Department of electrical eng., Sarab Branch, Islamic Azad University, Sarab, Iran ([email protected] and [email protected])

II.

MODELING OF BUCK/BOOST DC TO DC CONVERTER CONSIDERING POWER LOSS

Generally there are several parasitic elements in DC to DC converter structure. Their presence cause power loss and also 147

International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011

(1) X = [x1 x2 ]T = [iL vout ]T When power switch (S) is on, according to equivalent circuit of figure (2-B), state-space equations can be written as following: − A1 = �

r S +r L

− A2 = �

r D +r L

L

Ẋ = A1 X + B1

0

v in

1 � B1 = �

L � 0 − 0 RC And when power switch is off:

Figure (1): parasitic elements effect in buck/boost DC to DC converter (𝑉𝑉𝑜𝑜 :average of output voltage, 𝑉𝑉𝑑𝑑 : average of input voltage and 𝐷𝐷 : duty cycle of power switch)

L

−

1

L 1

(2)

Ẋ = A2 X + B2

− � B2 = �

V DO

L � (3) 0 By combining equations (2) and (3), and considering the relations governing averaged state-space modeling in power electronic converters [1], more accurate buck/boost DC to DC converter can be modeled as below:

a)

1

C

−

RC

Ẋ = AX + B A = A1 u + A2 (1 − u) 1 1 − [rL + rS u + rD (1 − u) − (1 − u) L =� L � 1 1 (1 − u) − C RC 1 [vin u − VDO (1 − u)] B = B1 u + B2 (1 − u) = �L �(4) 0 In this equation, 𝑢𝑢 is duty cycle and usually is considered as control input of system. During steady-state dynamics of the system is zero and voltage gain of the converter could be formulized as following:

b) Power switch is on.

c) Power diode is on. Figure (2): buck/boost DC to DC converter in different operating conditions

from control view they may cause change in system behavior. This case in buck/boost DC to DC converter is more important. In figure (1) the response of a buck/boost DC to DC converter, considering the parasitic elements is compared to an ideal one which lossless. It is noticeable that these two characteristics are different specially for high (D) duty cycle values. This figure clearly illustrates necessity for more exact modeling during controller design process. Power loss in DC to DC has three main sources: A) Inductor’s loss due to presence of parasitic elements in winding which could be modeled with a resistor(r L ). B)Power loss in controlled power switch: according to high frequency nature of DC to DC converters and usual use of power MOSFETs, considering the linear behavior of these controlled switches in triode region, resistance(rs) can be used to model it. C)Voltage drop across the diode in forward bias which is modeled by two segment model . But of course the ESR of output capacitor is relatively small and will be neglected in this paper. Buck/ boost DC to DC with presence of parasitic elements are shown in figure (2). During state-space modeling of DC to DC converters, usually inductor’s current and capacitor’s voltage is used as state variables:

𝑉𝑉 𝐷𝐷𝐷𝐷

𝑢𝑢− 𝑉𝑉 (1−𝑢𝑢) 𝑉𝑉 0 𝑋𝑋̇ = � � ⇒ 𝑀𝑀 = 𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑟𝑟 𝐿𝐿 +𝑟𝑟 𝑆𝑆 𝑢𝑢 +𝑟𝑟 𝐷𝐷𝑖𝑖𝑖𝑖(1−𝑢𝑢 ) (5) 𝑉𝑉𝑖𝑖𝑖𝑖 0 +(1−𝑢𝑢) (1−𝑢𝑢 )𝑅𝑅

This equation clearly illustrates the influence of each parasitic element on the output voltage of converter. If we assume that parasitic elements in equation (5) are zero, the following equation will be obtained which is completely wellknown: 1T

u 1−u Main part of loss in DC to DC converters is due to inductor’s ESR. In figure (3) a simple circuit model is used for buck/boost DC to DC converter. In this circuit, only Rloss is energy dissipating element and power switches are assumed to be ideal. In a similar way averaged state-apace model of this converter can be written as below: M=

1T

Ẋ = Á X + B́ R

1

− (1 − u) − LOSS L L Á = �1 � 1 (1 − u) − C

148

RC

(6)

International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011

vin u B́ = � L � 0 And voltage gain of this one:

Step one: according to controlling purpose and above equation, first error variable z 2 is defined as below: R

1T

𝑉𝑉 𝑀𝑀́ = 𝑜𝑜𝑜𝑜𝑜𝑜 = 𝑉𝑉𝑖𝑖𝑖𝑖

u

𝑧𝑧1 = 𝑥𝑥1 − 𝐼𝐼𝐿𝐿 (𝑟𝑟𝑟𝑟𝑟𝑟) ⇒ 𝑧𝑧̇1 = 𝑥𝑥̇ 1 − 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟) (10) By replacing the equation (9-A) in (10), we have: (10-A) 𝑧𝑧̇1 = 𝜃𝜃1 𝜔𝜔11 + 𝑙𝑙1 1 𝜔𝜔11 = − 𝑥𝑥1 (10-B) 𝐿𝐿 1 𝑣𝑣𝑖𝑖𝑖𝑖 𝑙𝑙1 = − 𝑥𝑥2 (1 − 𝑢𝑢) + 𝑢𝑢 − 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟) (10-C)

(7)

R LOSS +(1−u) (1−u )R

If 𝑀𝑀 = 𝑀𝑀́, it would be possible to assume that these two converters are equivalent and in this case we have: 1T

𝑅𝑅LOSS =

𝑉𝑉 𝑟𝑟 +𝑟𝑟 𝑢𝑢 +𝑟𝑟 𝐷𝐷 (1−𝑢𝑢 ) u� 𝐿𝐿 𝑆𝑆(1−𝑢𝑢 )𝑅𝑅 �+(1−𝑢𝑢)2 𝑉𝑉𝐷𝐷𝐷𝐷 𝑉𝑉 1 [𝑢𝑢− 𝐷𝐷𝐷𝐷 (1−𝑢𝑢)] (1−𝑢𝑢 )𝑅𝑅 𝑉𝑉 𝑖𝑖𝑖𝑖

𝐿𝐿

Since

𝑖𝑖𝑖𝑖

(8)

−1 (−𝜃𝜃�1 ̇ + 𝛾𝛾11 𝑧𝑧1 𝜔𝜔11 ) (14) 𝑉𝑉1̇ = 𝑧𝑧1 (𝜃𝜃�1 𝜔𝜔11 + 𝑙𝑙1 ) + �𝜃𝜃1 − 𝜃𝜃�1 �𝛾𝛾11 We know that, the time derivative of Lyapunov function must be negative in order to have a stable designed, this could happen if: (14-A) �𝜃𝜃�1 𝜔𝜔11 + 𝑙𝑙1 � = −𝑐𝑐1 𝑧𝑧1 ̇ −𝜃𝜃�1 + 𝛾𝛾11 𝑧𝑧1 𝜔𝜔11 = 0 (14-B) Which, in equation (14-A), 𝑐𝑐1 is a constant and positive coefficient.

2B

From historical view adaptive backstepping controller has been presented by P.V.Koktovic and his colleague in early 1990 [10]. At first this method was usable only in systems which their state equations form was PPF: Parametric pure form, but gradually devdeloped for PSF: Parametric Strict Form too [12]. Right now this method is usable in broad category which is given relatively comprehensive in [13].Considering that converters load resistance and R loss are uncertain, converter’s state space model– which is caculatedin[3] – are rewritten as below: 1 𝑣𝑣 (9-a) 𝑥𝑥̇ 1 = − [𝜃𝜃1 𝑥𝑥1 + (1 − 𝑢𝑢)𝑥𝑥2 ] + 𝑖𝑖𝑖𝑖 𝑢𝑢 4T

4T

R

𝐿𝐿

uncertain in these equations, it is possible to

parameter: 𝑧𝑧̇1 = 𝜃𝜃�1 𝜔𝜔11 + 𝑙𝑙1 + (𝜃𝜃1 − 𝜃𝜃�1 )𝜔𝜔11 (11) Now if we choose Lyapunov function as below: 1 1 −1 (𝜃𝜃1 − 𝜃𝜃�1 )2 (12) 𝑉𝑉1 = 𝑧𝑧12 + 𝛾𝛾11 2 2 −1 𝛾𝛾11 is a positive coefficient and is called parameter adaption gain. The reason of this naming will be clarified in next steps. Time derivative of Lyapunov function is expressed as below: −1 (13) �𝜃𝜃1 − 𝜃𝜃�1 �(−𝜃𝜃�1̇ ) 𝑉𝑉1̇ = 𝑧𝑧1 𝑧𝑧̇1 + 𝛾𝛾11 With replacing (11) in (13), we have:

III. CONTROLLER DESIGN

1

𝐿𝐿

θ1 is

rewrite (10-A) assuming θ1 as an estimated value for this

Calculating value of Rloss from equation (8) is not because values of parasitic elements are strongly depended on operating point of power switches, ambient temperature, etc. Moreover, load resistance (R) is an uncertain parameter and because of that adaptive method must be used to estimate Rloss. In this case if Rloss is estimated properly, circuits shown in figures (2) and (3) will be equivalent. According to this, designed controller for equivalent circuit of figure (3) can be used for main circuit of figure (2). 1T

R

R

Step two: generally equation (14-A) might not be correct, and the difference between two sides of this equation could be defined as below: 𝑧𝑧2 = �𝜃𝜃�1 𝜔𝜔11 + 𝑙𝑙1 � − (−𝑐𝑐1 𝑧𝑧1 ) = �𝜃𝜃�1 𝜔𝜔11 + 𝑙𝑙1 � + 𝑐𝑐1 𝑧𝑧1 (15) By replacing equations (10),(10-B) and (10-c) in equation (15) we will have: 1 𝑣𝑣 1 𝑧𝑧2 = −𝜃𝜃�1 𝑥𝑥1 − 𝑥𝑥2 (1 − 𝑢𝑢) + 𝑖𝑖𝑖𝑖 𝑢𝑢 − 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟) + 𝑐𝑐1 𝑥𝑥1 − 𝐿𝐿 𝐿𝐿 𝐿𝐿 𝑐𝑐1 𝐼𝐼𝐿𝐿 (𝑟𝑟𝑟𝑟𝑟𝑟) (16) Time derivative of z 2 is expressed as below. It worth noting that while simplifying the following equation, time derivatives of variables x 1 and x 2 are replaced from equations (9-A) and (9-B):

𝐿𝐿

(9-b) 𝑥𝑥̇ 2 = [(1 − 𝑢𝑢)𝑥𝑥1 − 𝜃𝜃2 𝑥𝑥2 ] 𝐶𝐶 1 (𝜃𝜃1 = 𝑅𝑅𝐿𝐿𝐿𝐿𝐿𝐿𝐿𝐿 , 𝜃𝜃2 = ) 𝑅𝑅 If we consider the capacitor’s voltage as an output, the resulted system would have non-minimum phase nature and for this reason adaptive backstepping control method cannot be applied [19]. This problem is easily solved by considering the inductor’s current as an output.We know that controller design purpose is calculation of control input in a way that inductor’s current become equivalent to reference current:𝑥𝑥2 = 𝐼𝐼𝐿𝐿 (𝑟𝑟𝑟𝑟𝑟𝑟)

R

R

𝑧𝑧̇2 = θ1 �𝜃𝜃�1

𝜃𝜃�1 1

𝐿𝐿𝐿𝐿

1

149

R

R

1

1

�1̇ 1 𝜃𝜃

𝐿𝐿

− 𝐼𝐼𝐿𝐿̈ (𝑟𝑟𝑟𝑟𝑟𝑟) −

𝑥𝑥1 − 𝑐𝑐1 𝑥𝑥1 � + θ2 � 𝑥𝑥2 (1 − 𝑢𝑢)� − 𝐿𝐿

𝐿𝐿

𝑣𝑣𝑖𝑖𝑖𝑖

𝑥𝑥2 (1 − 𝑢𝑢) − 𝜃𝜃�1 2 𝑢𝑢 − 𝐿𝐿 1 ̇ 𝑣𝑣 ̇ 𝑣𝑣 𝑥𝑥1 (1 − 𝑢𝑢)2 + 𝑥𝑥2 𝑢𝑢̇ + 𝑖𝑖𝑖𝑖 𝑢𝑢 + 𝑖𝑖𝑖𝑖 𝑢𝑢̇ 𝐿𝐿2

𝐿𝐿

1

𝑣𝑣𝑖𝑖𝑖𝑖

𝐿𝐿

𝑐𝑐1 𝑥𝑥2 (1 − 𝑢𝑢) + 𝑐𝑐1 𝑢𝑢 − 𝑐𝑐1 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟) 𝐿𝐿 𝐿𝐿 We can simplify this equation as below: ż 2 = θ1 ω12 + θ2 ω22 + l2 1 1 ω12 = θ�1 2 x1 − c1 x1 L

Figure (3): a simple circuit model for consideration of parasitic elements

1

𝐿𝐿2

R

R

L

𝐿𝐿

𝑥𝑥1 +

(17)

(18-A) (18-B)

International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011 1

ω22 = x2 (1 − u) (18-C) L ̇ 1 𝑣𝑣𝑖𝑖𝑖𝑖 𝜃𝜃�1 1 𝑥𝑥1 + 𝜃𝜃�1 2 𝑥𝑥2 (1 − 𝑢𝑢) − 𝜃𝜃�1 2 𝑢𝑢 𝑙𝑙2 = − 𝐿𝐿 𝐿𝐿 𝐿𝐿 ̇ 1 1 𝑣𝑣𝑖𝑖𝑖𝑖̇ 𝑣𝑣𝑖𝑖𝑖𝑖 − 𝑥𝑥1 (1 − 𝑢𝑢)2 + 𝑥𝑥2 𝑢𝑢̇ + 𝑢𝑢 + 𝑢𝑢̇ 𝐿𝐿 𝐿𝐿 𝐿𝐿𝐿𝐿 𝐿𝐿 1 𝑣𝑣𝑖𝑖𝑖𝑖 𝑢𝑢 − 𝐼𝐼𝐿𝐿̈ (𝑟𝑟𝑟𝑟𝑟𝑟) − 𝑐𝑐1 𝑥𝑥2 (1 − 𝑢𝑢) + 𝑐𝑐1 𝐿𝐿 𝐿𝐿 − 𝑐𝑐1 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟)

estimation rules for uncertain parameters. During simplification, equations (10-B), (18-B) and (18-C) are used. 1 1 1 𝜃𝜃�1̇ = 𝛾𝛾11 [− 𝑧𝑧1 𝑥𝑥1 + 𝑧𝑧2 �𝜃𝜃�1 2 𝑥𝑥1 − 𝑐𝑐1 𝑥𝑥1 �] (26-A) 𝐿𝐿 𝐿𝐿 𝐿𝐿 1 ̇ 𝜃𝜃� = 𝛾𝛾 [ 𝑥𝑥 (1 − 𝑢𝑢)𝑧𝑧 ] (26-B) 2

We rewrite the time derivative of z 2 by using equation (18-A) as below: 𝑧𝑧̇2 = 𝜃𝜃�1 𝜔𝜔12 + 𝜃𝜃�2 𝜔𝜔22 + 𝑙𝑙2 + �𝜃𝜃1 − 𝜃𝜃�1 �𝜔𝜔12 + �𝜃𝜃2 − 𝜃𝜃�2 �𝜔𝜔22 (19)

2

2

By considering equations (11) and (15) simultaneously, time derivative of z 1 can be written as below: 𝑧𝑧̇1 = −𝑐𝑐1 𝑧𝑧1 + 𝑧𝑧2 + (𝜃𝜃1 − 𝜃𝜃�1 )𝜔𝜔11 (22) By replacing z1 and z 2 respectively from equations (19) and

Table (1): simulation parameters fs=20e3; c1=44000*0.25; c2=25000*0.25; C=200e-6; L=10000e-6; R=10; RLOSS=0.2; teta1=RLOSS teta2=1/R Vin=10; gamma11=1e-7; gamma22=1e-7;

(22) in time derivative of Lyapunov function (equation (21)) and considering equations (10-B), (18-B) and (18-D), we have: 1 1 𝑉𝑉̇2 = −𝑐𝑐1 𝑧𝑧12 + 𝑧𝑧1 𝑧𝑧2 + 𝑧𝑧2 [−𝜃𝜃�1̇ 𝑥𝑥1 + 𝜃𝜃�1 2 𝑥𝑥2 (1 − 𝑢𝑢) − 𝐿𝐿 𝐿𝐿 1 1 ̇ 𝑣𝑣 ̇ 𝑣𝑣 𝑣𝑣 𝜃𝜃�1 𝑖𝑖𝑖𝑖2 𝑢𝑢 𝑥𝑥1 (1 − 𝑢𝑢)2 + 𝑥𝑥2 𝑢𝑢̇ + 𝑖𝑖𝑖𝑖 𝑢𝑢 + 𝑖𝑖𝑖𝑖 𝑢𝑢̇ − 𝐼𝐼𝐿𝐿̈ (𝑟𝑟𝑟𝑟𝑟𝑟) − 𝐿𝐿 𝐿𝐿 𝐿𝐿 𝐿𝐿𝐿𝐿 𝐿𝐿 1 1 𝑣𝑣 1 𝑐𝑐1 𝑥𝑥2 (1 − 𝑢𝑢) + 𝑐𝑐1 𝑖𝑖𝑖𝑖 𝑢𝑢 − 𝑐𝑐1 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟)+𝜃𝜃�12 2 𝑥𝑥1 − 𝑐𝑐1 𝜃𝜃�1 𝑥𝑥1 + 𝐿𝐿

𝐿𝐿

𝐿𝐿

1 −1 (−𝜃𝜃�1̇ + 𝛾𝛾11 𝑧𝑧1 𝜔𝜔11 + 𝜃𝜃�2 𝑥𝑥2 (1 − 𝑢𝑢)] + �𝜃𝜃1 − 𝜃𝜃�1 �𝛾𝛾11 𝐿𝐿 𝛾𝛾 𝑧𝑧 𝜔𝜔 )�𝜃𝜃 − 𝜃𝜃� �𝛾𝛾 −1 (−𝜃𝜃� ̇ + 𝛾𝛾 𝑧𝑧 𝜔𝜔 ) (23) 11 2

12

2

2

2

22

22 2

𝐿𝐿

22

V. CONCLUSION

In the above equation if we assume that the coefficient of z 2 equals to –c 2 z 2 and also assume that the coefficients of −1 −1 and �𝜃𝜃2 − 𝜃𝜃�2 �𝛾𝛾22 is zero: �𝜃𝜃1 − 𝜃𝜃�1 �𝛾𝛾11 2 2 𝑉𝑉̇2 = −𝑐𝑐1 𝑧𝑧1 + 𝑧𝑧1 𝑧𝑧2 + −𝑐𝑐2 𝑧𝑧2 (24)

A novel adaptive backstepping controller based on accurate averaged state-space model is proposed for buck/boost DC to DC converter to regulate the inductor current. In the design procedure load resistance and all of the parasitic elements assumed to be uncertain. Finally in order to verify accuracy of the proposed controller, buck/boost converter is simulated based on the designed controller via MATLAB/Simulink. Results clearly show that, in spite of large variations of load, the controller has good steady-state and dynamic response.

In the above equations c 2 is a constant and positive coefficient. It can be seen that if 4𝑐𝑐1 𝑐𝑐2 > 1, equation (24) is always negative and stability of the system is guaranteed.Buck/boost DC to DC converter’s control input is obtained by assuming that coefficient of z 2 is equals to –c 2 z 2 in equation (23): 𝐿𝐿 1 1 𝑣𝑣 𝑢𝑢̇ = {−𝜃𝜃�1̇ 𝑥𝑥1 + 𝜃𝜃�1 � 2 𝑥𝑥2 (1 − 𝑢𝑢) − 𝑖𝑖𝑖𝑖2 𝑢𝑢 −

𝑐𝑐1

1 𝐿𝐿 1

𝑥𝑥 2 +𝑣𝑣𝑖𝑖𝑖𝑖

𝑥𝑥1 � +𝜃𝜃�12

1

𝐿𝐿

𝐿𝐿

� 1 (1 − 𝑢𝑢) − 2 𝑥𝑥1 + 𝜃𝜃2 𝑥𝑥2

𝐿𝐿

𝐿𝐿 𝑣𝑣𝑖𝑖𝑖𝑖̇

𝑐𝑐1 𝑥𝑥2 (1 − 𝑢𝑢)+𝑐𝑐2 𝑧𝑧2 + 𝐿𝐿 𝐼𝐼𝐿𝐿̈ (𝑟𝑟𝑟𝑟𝑟𝑟)}

𝐿𝐿

𝑢𝑢 + 𝑐𝑐1

𝑣𝑣𝑖𝑖𝑖𝑖 𝐿𝐿

1

𝐿𝐿𝐿𝐿

2

In this part buck/boost DC to DC converter based on proposed adaptive backstepping controller is simulated by MATLAB/Simulink. General block diagram of the system is shown in figure (4). Also simulation parameters and other used elements are summarized in table (1).Choosingthese parametersare relatively straightforward and are given in details in [1].The response of designed controller to step reference is shown in figure (5). It obvious that this method has fast dynamic response and also has zero steady-state error.The response of the controller to load variation is given in figure (6). It’s obvious that adaptive controller is completely robust.

step we should choose Lyapunov function: 1 1 1 −1 1 −1 𝑉𝑉2 = 𝑧𝑧12 + 𝑧𝑧22 + 𝛾𝛾11 (𝜃𝜃1 − 𝜃𝜃�1 )2 + 𝛾𝛾22 (𝜃𝜃2 − 𝜃𝜃�2 )2 (20) 2 2 2 2 It’s time derivative is: −1 𝑉𝑉̇2 = 𝑧𝑧1 𝑧𝑧̇1 + 𝑧𝑧2 𝑧𝑧̇2 + 𝛾𝛾11 �𝜃𝜃1 − 𝜃𝜃�1 � �−𝜃𝜃�1̇ � +𝛾𝛾 −1 �𝜃𝜃 − 𝜃𝜃� �(−𝜃𝜃� ̇ ) (21) 2

2

IV. SIMULATION RESULTS

In equation above θ2 is estimated value for θ1 . Now in this

22

22 𝐿𝐿

Briefly, presented control system which is modeled based on equations (9-A) and (9-B), according to resulted control law (equation 25) and estimation rules ((26-A) and (26-B)) is stable because time derivative of Lyapunov function is negative.

𝐿𝐿

𝑥𝑥1 (1 − 𝑢𝑢)2 −

𝑢𝑢 − 𝑐𝑐1 𝐼𝐼𝐿𝐿̇ (𝑟𝑟𝑟𝑟𝑟𝑟) −

(25)

Similarly, if we consider that the coefficient of the expressions −1 −1 �𝜃𝜃1 − 𝜃𝜃�1 �𝛾𝛾11 and �𝜃𝜃2 − 𝜃𝜃�2 �𝛾𝛾22 is zero in (23), we can obtain 150

International Conference on Electrical, Electronics and Civil Engineering (ICEECE'2011) Pattaya Dec. 2011

IV. ACKNOWLEDGMENT This research is supported completely by Islamic Azad University-Sarab Branch. REFERENCES [1] N.Mohan and et.al. “Power electronics: converters, applications and design” John Willey and Sons Inc, 3rd edition, 2004 [2] M.H.Rashid ”Power electronics: circuits, devices and applications” Prentice Hall, Englewood Cliffs, NJ, USA, 2nd edition 1993, [3] Y. He and F.L. Luo “Design and analysis of adaptive sliding-mode-like controller for DC–DC converters” IEE Proc.-Electr. Power Appl., Vol. 153, No. 3, May 2006 [4] B.K.Bose, “ Power electronics and AC drives” Prentice Hall, 1986 [5] M.salimi and et.al. “A Novel Method on Adaptive Backstepping Control of Buck Choppers” PEDSTC,Feb,2011 [6] J. G. Ciezki، R. W. Ashton،”The Design of Stabilizing Control for Shipboard DC-to-DC BUCK Choppers using Feedback Linearization Techniques”،0-7803-4489-8،IEEE،1998 [7] R. Leyva and et. Al, "Passivity-Based Integral Control of a Boost Converter for Large-Signal Stability IEEE Proc, Control theory Appl، Vol.153،No.2،March 2006 [8] G.Escobar and et.al. ”An Experimental Comparison of Several Nonlinear Controllers for Power Converters” 0272-1708/99/$10.00, IEEE, 1998 [9] Siew-Chong Tan, Member, IEEE, Y. M. Lai, Member, IEEE, and Chi K. Tse, Fellow, IEEE, “Indirect Sliding Mode Control of Power Converters Via Double Integral Sliding Surface” IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 23, NO. 2, MARCH 2008 [10] P.V.Kokotovic،”The Joy of Feedback: Nonlinear and Adaptive”،02721708/92،IEEE، 1992(Bode Prize Lecture) [11] ShuibaoGuo, Xuefang Lin-Shi, Bruno Allard, YanxiaGao, and Yi Ruan, “Digital Sliding-Mode Controller For High-Frequency DC/DC SMPS” IEEE TRANSACTIONS ON POWER ELECTRONICS, VOL. 25, NO. 5, MAY 2010 [12] G.Cai,W.Tu “ Adaptive Backstepping Control of the Uncertain Unified Systems”, international journal of nonlinear science, vol.4, no.1, pp.1724, 2007 [13] M.Krstic and et. al. ” Nonlinear and Adaptive Control Design” John Wiley and Sons, Inc,1995 [14]HosseinAbootorabiZarchi, JafarSoltani, and Gholamreza Arab Markadeh “Adaptive Input–Output Feedback-Linearization-Based Torque Control of Synchronous ReluctanceMotorWithout Mechanical Sensor” IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 57, NO. 1, JANUARY 2010 [15] H.S.Ramirez and et. al.” Adaptive input-output linearization for PWM Regulation of DC-to-DC Converters” Preceding of American control conference, june, 1995 [16] L.K.Yi،et.al،”Adaptive Backstepping Sliding Mode Nonlinear Control for Buck DC/DC Switched Power Converter”،IEEE International Congruence on Control and Automation،Tune،2007 [17] S. C. Lin and C.C Tsai،”Adaptive Backstepping Control with Integral Action for PWM Buck DC-DC Converters”،Journal of the Chinese Institute of Engineering، Vol.28، No.6،PP.977،987،2005 [18] H. E. Fadil ، F. Giri،” Backstepping Based Control of PWM DC-DC Boost Power Converters”،Medwell، International Journal of Electrical and Power Engineering،479-485،2007 [19] H.ElFadil،and F.Giri،” Backstepping Based Control of PWM DC-DC Boost Converters”،1-4244-0755-9/07،IEEE،2007 [20]H.E.Fadil،and et.al. ”Nonlinear and Adaptive Control of Buck-Boost Power Converters” IEEE،2003

Figure (4) general block diagram of the adaptive backstepping controller in buck/boost DC to DC converter

Figure (5): The response of designed controller to step reference

Figure (6): The response of designed controller to load variation

151