Analysis of a Sliding-Mode Controlled SEPIC Converter*Paper

Javier Hernanz (Departament d'Enginyeria Electronica, U. P. C, Barcelona, Spain)

Luis Martinez (Departament d'Enginyeria Electronica, U. R. V., Tarragona, Spain)

Alberto Poveda (Departament d'Enginyeria Electronica, U. P. C, Barcelona, Spain)

Enric Fossas (Dept. de Matematica Aplicada i Telematica, U. P. C., Barcelona, Spain)

In this paper, different sliding surfaces are analyzed for the SEPIC converter . The simplest discontinuity surface leading to a stable equilibrium point corresponds to the constant input current . A fast transient response is obtained when a linear combination of the input and the output voltage is used to

control the converter. Other sliding surfaces providing adaptative control schemes are also investigated to

attenuate perturbation effects in the input voltage or in the output load .

1. Introduction

The search for new dc-to-dc converter structures is highly constrained by the existing trade-off between simplicity and performances of the resulting topologies. In this sense, there is nowadays a practical limit located in the fourth order converters, where the most representative elements are the Cuk and the SEPIC switching cells. Due to its significant properties, the Cuk converter has been extensively used and a

considerable amount of research has been devoted to the modelling, analysis and applications of that structure. Originally developed for high output voltage converters, the Single-Ended Primary Inductance Converter (SEPIC) features several interesting

properties. Namely, a multiple-output extension is straightforward and simpler than that for a Cuk converter, requiring only a winding, . a diode and a capacitor for each additional output.

However, the controller design following a conventional approach based on the use of a pulse width modulator (PWM) and a linear low frequency characterization of the converter, is difficult in this case

since the control to output transfer function has four

poles and one right half plane zero. In addition, the corresponding small-signal model cannot predict the regulator behavior when a large-signal perturbation

appears in the output load or in the input voltage.On the other hand, since switching converters

constitute an important case of variable structure systems, different sliding-mode strategies to control this

class of circuits have been reported in the last years3'6The sliding-mode control is a time-domain

technique suitable for large-signal applications and

exhibits a fast transient response with negligible overshoot and low-sensitivity to external perturbations. The main antecedent in the use of sliding-mode control in the SEPIC converter can be found in the work of P. Mattavelli et al.7's, where a general sliding surface is empirically derived and the corresponding parameters are experimentally adjusted.

A PWM nonlinear control suitable for large-signal applications can also be implemented. The resulting controller operates at constant switching frequency and

provides the same performances than a sliding-mode approach, whose main drawback is the absence of an external clock to fix the switching frequency. However since the PWM nonlinear technique is based on the implementation of the equivalent control of the sliding-mode approach, which is a nonlinear function of the state variables, it requires the use of nonlinear analog blocks such as dividers and multipliers9 whose adjustement is tedious and complicated. In a clear-cut contrast, the only nonlinear block required in the sliding-mode approach is a hysteresis comparator.

In this paper, we analyze the sliding-mode control of the ideal SEPIC converter shown in Fig. 1. Obtaining the sliding surfaces, their corresponding equilibrium points and the equivalent control is the first aim of the analysis. Finding the most appropriate surfaces to attenuate the disturbances in the input voltage or in the output load are also objectives of this work.

Fig. 1. SEPIC converter equivalent circuit.

* This work has been partially sponsored by Ministerio de Educacidn y Ciencia-Spain, TAP-94-0552 C03-03 and Universitat Politecnica de Catn1nnvn PR-951

1140 T.IEE Japan, Vol. 116-D, No.11, ’96

Analysis of a Sliding-Mode Controlled SEPIC Converter

2. Converter State Equations

In the continuous conduction mode the converter

has only one structural change during a switching

period. Considering u=1 during the transistor conduction subinterval and u=0 during the diode

conduction subinterval, the converter dynamics can be

described as follows

(1)

(2)

(3)

(4)

where

is the converter state vector corresponding to the

primary side equivalent circuit shown in Fig. 2, and

(5)

Fig. 2. Primary side equivalent circuit of the SEPIC converter,

3. Sliding-mode control

Fig. 3 shows the scheme of the sliding-mode controlled SEPIC converter analyzed in this paper. The switching surface s(x) is a linear combination of the converter state variables and an external reference. The hysteresis comparator provides the control law and the

pulse amplifier acts as a driver to turn on and off the power transistor Q.

Defining the equivalent control ueQ of the sliding regime as a continuous function of the state variables and the input voltage, we can write

(6)

In equilibrium (superscript*) we will have

(7)

After solving the equilibrium equations, we get the

coordinates of the equilibrium point

(8a)

where

(8b)

Considering as a switching surface the hyperplane

(9)

we can write

(10)

which corresponds to the equilibrium point.

Fig. 3. Sliding-mode controlled non-isolated SEPIC converter.

From (8) and (10) we obtain

(11)

or

(12)

4. Sliding Surfaces for Ideal Load Regulation

If K and a are both zero, the coordinate v2* of the

output voltage will be given by

電学論D,116巻11号,平 成8年 1141

(13)

Expression (13) will not depend on the load R if

(3=0. Therefore, the switching surface for ideal load regulation will be

(14)

Similarly, if a#0 and K=0, the coordinate v2µ

will not depend on R if y=6=0. In that case the

switching surface for ideal load regulation will be

expressed as

(15)

Following a similar procedure, we obtain two

additional switching surfaces that result in a coordinate

v2* independent of R when K # 0. Such surfaces are

expressed as

(16)

(17)

If we take equation (14) as sliding surface,

applying the Routh's criterion to the characteristic

polynomial of the linearized state equations around the

equilibrium point, it can be shown that the system will

be unstable in most practical cases.

As in the previous case, the stability conditions of

the sliding surface (15), are not fulfilled in most

practical cases.

Finally, if we consider the sliding surfaces y vi+

6v2=K and V2=K, we obtain an unstable equilibrium

point in both cases.

5. Sliding Surfaces for Ideal Line Regulation

If a and ƒÁ are both zero in equation (9), the auxiliar

variable t will be given by

(18)

and the coordinate v2* of the output voltage becomes

(19)

Expression (19) is independent of the input

voltage. Therefore, the discontinuity surface for ideal line regulation will be

(20)

From equations (19) and (20) we conclude that

both ideal line and load regulation can be obtained if a

=(3=ƒÁ=0. In that case, the sliding surface will be v2=

K which leads to an unstable equilibrium point.

Similarly, if a=y= 6=0 the sliding surface will be i2

=K which results in an unstable point.

Applying the Routh's criterion to the characteristic

polynomial of the linearized state equations around the

equilibrium point, it can be shown that the system will

be unstable.

6. Sliding Surfaces of the Form x,=K

There•@ are 4 sliding surfaces of the form x;= K

which satisfy the transversality condition. In the cases

i2=K, v1= Kand V2=K, the obtained equilibrium

points are unstable.

In the case i1=K, the coordinates (i1*, i2*, vl*, V2*) of the equilibrium point are

(21)

From (21) we conclude that the sliding surface s=it-K=0 will not provide ideal line regulation nor ideal load regulation. Applying the Routh's criterion to

the characteristic polynomial of the linearized state equations, it can be shown that the system will be stable if the following condition is satisfied

(22)

If, for design purposes, we choose Cl= 2, condition (22) will be always satisfied in the step-up voltage applications of the SEPIC converter.

7. Sliding Surfaces of the Form k;x;+kjxj=K

In addition to the sliding surfaces previously analyzed for ideal load or line regulation, we have

studied the following discontinuity surfaces which are a linear combination of two state variables

(23), (24)

In the first case, the analysis of the equilibrium

point shows that K cannot be an arbitrary constant. Specifically

(25)

The characteristic polynomial corresponding to the

linearized state equations will be given by

(26)

1142 T.IEE Japan, Vol. 116-D, No. 11,'96

Analysis of a Sliding-Mode Controlled SEPIC Converter

where

(27)

(28)

From equation (26) we can depict the root locus diagram of the linearized system around the equilibrium point as illustrated in Fig. 4. We have considered the set of parameters

Fig. 4. Root locus diagram corresponding to the linearized system

around the equilibrium point when the switching surface is

ai1+ƒÀi2=K.

Note that k=0 (with k= a/ƒÀ) results in an

unstable equilibrium point corresponding to the

switching surface i2-K=0, whereas k=•‡ leads to a

stable equilibrium point which corresponds to the

switching surface i1-K=0.

In the case of the sliding surface ai1+8v2=K, the

characteristic polynomial representing the linearized

state equations will be given by

The corresponding root locus diagram is shown in

Fig. 5, where we have considered the same set of

parameters as in Fig. 4. Note that k=•‡ (with k=a/ƒÂ)

leads to a stable equilibrium point corresponding to the

discontinuity surface i1=K, while k=0 results in a

unstable equilibrium point which corresponds to the

switching surface v2=K. Finally, the equilibrium point

will be stable if the following condition is satisfied

(30)

Fig. 5. Root locus diagram corresponding to the linearized system

around the equilibrium point when the switching surface is ai1+Sv2=K.

8. Examples

The following examples correspond to the

converter parameters previously defined.

Example 1

Fig. 6 shows a HSPICE simulation of the output

voltage when the sliding control uses the surface it = K.

The input current reaches the averaged reference value

it*=150mA after 19ms approximately resulting in a

peak-to-peak ripple of 28.6mA and a switching frequency of 32KHz. The width of the conparator

hysteresis is 0.02 volts..

Fig. 6. Output voltage for the sliding surface i1=150mA.

(29)

電学論D,116巻11号,平 成8年 1143

Example 2

A significant improvement in the transient

response can be achieved if the sliding control uses the

surface ait+Sv2=K as depicted in figure 7. The output

voltage reaches the expected average value in steady-

state in 3ms. The constants a and ƒÀ are respectively 20

and 1, and the width of the comparator hysteresis is 0.2

v. The resulting switching frequency is 31KHz.

Finally, we conclude that the surface i1=K does not

provide ideal load regulation and behaves as a current

source, whereas the surface ai1+Sv2 = K results in a

good load regulation behaviour for values of a/ƒÀ = 20.

Fig. 7. Output voltage for the sliding surface 20i1+v2=18.

9. Conclusions

Complex converters such as the SEPIC converter

can be controlled by means of sliding techniques

offering a fast transient response and a low sensitivity

to external perturbations. Different sliding surfaces

have been analyzed. The simplest surface is it=K,

which leads the converter to a stable equilibrium point

without load regulation after a fast transient-state.

Adding a term proportional to voltage v2 results is the

sliding surface ai1+Sv2=K, which combines a good

load regulation for small values of a/ƒÂ with a fast

transient response in the output voltage.

The obtained results can be used in a practical

design since other important requirements are also met

by the control technique reported in this paper. The in

rush current, for exemple, has no overshoots and

exhibits a first-order dynamic behavior until it reaches

the•@ steady-state value of current i1 . Avoiding•@ core

saturation can be also carried out by inluding current

limiters in the feedback paths of inductor currents i1

and i2. Finally, most power losses are originated in the

power stage due to switching and conduction

phenomena. The contribution of the control loop to the

global power losses is negligible, due to the

implementation simplicity and the small level of the

processed power in the feedback path.

(Manuscript received September 11, '95, revised May 7, '96)

References

(1) R. P. Massey and E. C. Snyder, "High voltage single-ended dc-dc converter" IEEE PESC Record 1977 pp 156-159.

(2) V. I. Utkin, "Sliding Modes and Their Applications in Variable Structure Systems", MIR, Moscow, 1978.

(3) F. Bilanovic, 0. Music and A. Sabanovic, "Buck converter Regulator Operating in Sliding Mode", Proceedings of PCI, 1983, pp 331-340.

(4) H. Sira-Ramirez, "Sliding Motions in Switched Bilinear Networks", IEEE Transactions on Circuits and Systems, vol. CAS-34, n° 8, pp 919-923, August 1987.

(5) R. Venkataraman, A. Sabanovic and S. Cuk, "Sliding Mode Control of DC-to-DC Converters", Proceedings of IECON'85, pp251-258.

(6) E. Fossas, L. Martinez and J. Ordines,"Sliding Mode Control Reduces Audiosusceptibility and load Perturbation in the Cuk Converter", IEEE Transactions on Circuits and Systems, vol. 39, n° 10, October 1992, pp 847-849.

(7) P. Mattavelli, L. Rosetto, G. Spiazzi and P. Tenti, "General-Purpose Sliding-Mode Controller for DC-DC converter applications", Proceedings of PESC'93, pp 609-615.

(8) P. Mattavelli, L. Rosetto, G. Spiazzi and P. Tenti, "Sliding mode control of SEPIC converters", Proceedings of ESPC93,

pp173-178.(9) J. Maj6, L. Martinez, A. Poveda, L. Garcia de Vicuna, F.

Guinjoan, A. F. SSnchez, J. C. Marpinard and M. Valentin, "Nonlinear continuous time control of a bidirectional coupled-inductor Cuk converter". IEICE Trans. on Communications. Vol. e75-B, No 11, Nov. 1992, pp 1134-1141.

Javier Hernanz (non-member) received the degree of Ingeniero de Telecomunicacidn from

the Universitat PolitBcnica de Catalunya, Barcelona, Spain in 1993.

He is working currently towards the Ph. D. degree in Electronics at the Universitat

PolitBcnica de Catalunya, where he is an instructor. His research interests include control

theory and power electronics.

Luis Martinez (non-member) received the Ingeniero de Telecomunicaci6n and Doctor

Ingeniero degrees form the Universitat PolitBcnica de Catalunya, Barcelona, Spain, in

1978 and 1984, respectively.He is presently Professor at the Universitat

Rovira i Virgili. His research interests are in the field of control theory and power-conditioning

systems.

Alberto Poveda (non-member) received the Ingeniero de Telecomunicacidn and Doctor

Ingeniero degrees form the Universitat Politc'cnica de Catalunya, Barcelona, Spain, in

1978 and 1988, respectively.He is employed as associate professor at the

Universitat PolitBcnica de Catalunya.

Enric Fossas (non-member) received the Graduate and Ph. D. degrees in Mathematics

from Universitat de Barcelona, Spain, in 1981 and 1986, respectively.

Since 1981 he has taugh mathematics at the Universitat de Barcelona and at the Universitat

PolitBcnica de Catalunya, where he is employed

I as associate professor. His research interests are in the field of system theory, control and differential geometry.

1144 T.IEE Japan, Vol. 116-D, No.11,'96