Sliding Mode Control for DC/DC Converters: A New Sliding Surface N. VmQUEZ',

Student, IEEE,

c. HERNANDEZ', Member, IEEE, J. fiv-z3, and J. -U4,

Senior, IEEE,

knior,E~~

Instituto Tecnologico de Celaya, Celaya, Mexico, e-mail : (nvazquez', chdz' )@itc.mx Centro de Investigacion y Estudios Avanzados del IF", DF, Mexico, e-mail: [email protected] Centro Nacional de Investigacion y Desarrollo Tecnologico, Cuernavaca, Mkxico, e-mail: [email protected]

Abstract-In this paper a controller based on the variable structure theory is analyzed: the sliding mode control. The sliding mode control was proposed to control dc/dc converters, due to the good characteristics introduced to the complete system. In order to regulate the output voltage, the sliding mode control require a current loop in order to assure the stability of the system, the inclusion of this extra loop increases the cost. This paper presents an alternative methad to obtain the goad characteristics of the control strategy without the use of current sensors. The analysis, design, simulation and experimental results are presented Index Terms-Variable structure theory, dddc converters, Current sensorless.

I. INTRODUCTION The DC/DC converters are widely used in many applications, when they are used as regulated voltage source a fast regulation of the output voltage is necessary. In order to introduce a good dynamic response different papers have been proposed [I-71, in those schemes current loops are used, additionally to the output voltage loop, and some times more voltage loops. In [8] was proposed a sensor less current mode control, this technique no uses current loops, but three voltage loops are used; an observer-based technique is used. A control scheme proposed in the literature is the sliding mode control [6-71. The sliding mode control is based on the theory of variable structure, this technique permits to obtain the desired characteristics at the output voltage, additionally to the robustness to large load and input voltage variations. To regulate the output voltage the sliding mode control require a current loop, this is because one of the design steps (stability condition) needs to be fulfilled. and with the current loop is done. This extra loop increase the cost due to the current sensor and a rather complex implementation, this could be a reason for disregarding the use of the control scheme. In [9] was proposed an option to eliminate the use of the current sensor, but with a low response, because This work was partially sponsored by the Mexican Council of the Technological Education System (CoSNET) under Project No. 6.59.02-P

0-7803-7912-8/03/$17.00 0 2003 IEEE 422

an integrator is used to eliminate the use of a current reference generation. In this paper is proposed the sliding mode control with no current sensors; this is made with a new sliding surface that eliminate the use of the current sensor and current reference generation, also permits to fulfill all the controller design steps. The proposed method eliminates the generation of the current reference and only two voltage loops are needed to regulate the converter. The sliding mode control has not been discussed when the DC/DC converter is working in discontinuous conduction mode, a brief discussion about this is also made. In the next section the sliding mode control is commented. In section three the proposed controller is addressed, the design steps are also presented. In section four the simulation and experimental results are presented. And finally the conclusions are discussed.

11. SLIDING MODECONTROL The sliding mode control is based on the variable structure theory [ 10-121, and introduces to the complete system a good dynamic response and also robustness to large load and input voltage variations. The sliding mode control operates in a simplified way as follows: a sliding surface is defined with the equilibrium point, and the system is forced to be held into the sliding surface (existence condition), and then the system must reach the equilibrium point (stability) In order to assure that the controlled system operates properly the existence condition and stability must be verified. These are the summarized controller design steps; but also the system modeling could be considered as a design step. A. Systeiii i?iodeliiig

Traditionally the system to be controlled is modeled as if the DC/DC converter was operated in continuous conduction mode (CCM), but this is not always fulfilled. If the converter is designed to operate in CCM, at light loads the converter operates in discontinuous conduction mode (DCM); even worst the converter could be designed to op-

4; vo

T

- 1 Fig 1. DCDC boost converter

a) When SWI is turned on

erate in DCM as in high power factor pre-regulators. The behavior of the converter in CCM is different to the DCM operation. So in order to verify the operation of the converter also in DCM also a model is obtained for this case.

LI

The DC/DC boost converter is analyzed since this converter is equal or relatively more complicated to be controlled than others. The boost converter is shown in Fig 1; the sub-circuits of the converter are shown in Fig 2. The sub-circuit in Fig 2.a is obtained when the semiconductor SW, is turned on. The sub-circuit in Fig 2.b is obtained when the semiconductor SW, is turned off and the inductor current IL is positive. The sub-circuit in Fig 2.c is obtained when the semiconductor is turned off and the inductor current is zero or discharged; this last subcircuit is obtained when the converter is operating in DCM due to the diode D. The model of the complete system in matrix form considering the DCM operation is:

#c) When SWI is turned off and XI

=0

Fig2 Subcircuits of the dcldc boost converter

[6-71. The traditional sliding surface and control law are respectively: (3) o = s,(XI - x,,) + (x2- xz,) = 0

The current loop is necessary in order to assure the stability condition, because if sI is considered as zero in order to only feedback the voltage, the system becomes unstable. The existence condition for the boost D C D C converter with (1) and ( 2 ) as the sliding surface and control law can be obtained easily; an approximation to determine the existence condition is fulfilling the next inequality: n = Virz/fi,

b =I o / Z

The model of the converter considering that always is operating in CCM is:

S , is chosen properly to assure the existence condition and stability, so it is chosen positive. Notice that ( 5 ) is a simplified inequality (approximation).

111. PROPOSEDSLIDING MODE CONTROLLER B. Traditional sliding suiface

The traditional sliding surface used to regulate the output voltage of a DC/DC converter is a linear combination of the error states variables and some times to eliminate the steady state error an integrator of the error voltage is used

423

The proposed sliding mode controller permits to eliminate the use of the current sensor. This is made modifying the traditional sliding surface. This surface permits to fulfill the existence and stability conditions without the use of the inductor current.

C. Stability analysis

A. Sliding surface

The stability analysis of the controller is made with the equivalent control, the equivalent control is substituted into the system model, and is verified under that condition.

The proposed sliding surface is similar to the traditional, but the current error is emulated with a differential equation: CT = sleXl+ (xz - x Z r )= 0 (6) +V&

e,, = a -

(1- U ) -kieX, 2

The equivalent control is the control law when the system is into the sliding surface, and it is obtained from 6 = 0 , but changing if fo the equivalent control ueq:

(7)

As can be observer in (6) and (7) the proposed sliding surface not uses the inductor current, so it is not necessary to sense it. The model of the converter in continuous conduction mode motivates the differential equation (7); the differential equation that determines the inductor current behavior in the CCM model ( 2 ) is:

XI

=a-

WO.Y2(I - U )

This analysis is beyond of the purpose of this communication, so it is not included. but the analysis results in fulfilling the inequality:

(8)

2

To determine (7) only is added a term that stabilize the differential equation and changed x, by e,, . The inequality ( 1 5 ) is an approximation, but determines an region where the system is stable.

The control law used is (4).

B. Existence of the sliding mode In order to verify the existence condition the following condition must be fulfilled [ 121:

06 < O

(9)

To fulfill (9) , the values of the control law (4) is taken into account and also (9), that is:

If If

ii=+l-+oO

= -1 -+ CT > 0; theri d- < 0

IV. EXTENSION TO O T H E R DC/DC CONVERTERS The controller proposed in this paper can be extrapolated to other dc/dc converters. In this section only the buck, boost and buck boosi dc/dc converters are discussed, but also to other converter can be extrapolated. The generalized control Ian and sliding surface have the following form:

(10)

"={

I

I/OO

Using (21, ( 6 ) and (7) is obtained:

In table 1 the ten71 i n 18) is defined depending on the converter. Each tern1 is obtained according to the model in

With (10) and (1 1) existence condition are: Y > O

- M~o{SIX2-SI}+ r < 0

where: r = s,cl - b + kies, - .t2, Buck dddc converter

An approximation of (1 2 ) is:

- \i',,.\V? +a-

(1+1f)

2

Boost dc/dc converter

As can be observed the existence condition can be fulfilled like the traditional sliding surface was fulfilled, because the existence condition is the same.

424

Buck-boost dddc converter

(1-U)

-\i'(,s2- + a -

2

(]+U)

2

CCM of the converter, the model for each converter: the model of the buck dc/dc converter is:

like a logic gate.

v. SIMULATION AND EXPERIMENTAL RESULTS A D C D C boost converter was designed and built. The parameters are Po= 300W, Vin= 50V, Vo= 150V, fs=20KHz. Some simulation results of the converter are shown in Figs 3 and 4. The converter operation in CCM is shown in Fig 3, the converter operation at light loads (DCM operation) is shown if Fig 4. An integral term of the error voltage needs to be used to eliminate the steady state error in DCM, the integral term it is strictly necessary in DCM operation to eliminate the steady state error.

The model for the buck-boost dddc converter is:

The experimental results of the system are show in Fig 5 and 6. A load variation is made, as can be observed the system has a fast response. In this Fig. 5 it is shown the output voltage and current, In Fig 6 i t is shown the emulated current error, the output voltage and the output current; the emulated current is the solution of the differential equation (7).

The model of the boost dddc converter was obtained previously in ( 2 ) . In all the models:

The implementation of the term in (18) is relatively easy; it is not necessary a multiplier, only two diodes are used, this is because the function (1-u)/2 or (l+u)/2 acts

VI. CONCLUSIONS In this paper a controller based on the variable structure

Ouput Voltw

:

:

70

: 50

0 001

0 002

0 003

0 on1

0 002

0 003

0UP

Irductor C m n t

4,

3 2

1

0 l 0

0 001

i

Emdated C m n t E m r

i

08 1 0 2.0 I1-

I

J

0 001

0 002

0 003

0 00

Fig S . Experimental results: Test under load variation. Top to down: Load Current, Output voltage

Fig 3 Simulation results Converter operating in CCM

i

1

0 00'

Emulated Cunent Ermr 044

0

0 001

0 002

0 003

I

Fig 6. Experimental results: Test under load variation. Top to down: Emulated current error. Output Voltage. Output Current

0 00

Fig 4. Simulation results: Converter operating in DCM

42 5

theory is analyzed: the sliding mode control. The sliding mode control was proposed as a good technique to control dc/dc converters, due to the characteristics introduced to the complete system. In order to regulate the output voltage, the sliding mode control requires a current loop in order to assure the stability of the system, the inclusion of this extra loop increases the cost of the system. This paper presents an alternative method to obtain the good characteristics of the control strategy without the use of current sensors, this is made modifying the traditional sliding surface, and only two voltage are sensed: the input voltage and output voltage. This fact produces a cost reduction, simplifies the implementation, and increases the overall system reliability. The analysis, simulation and experimental results are presented.

VII. REFERENCES [I]

2. Yang, P.C. Sen. “Dc to Dc Buck Converter with novel current mode control”, IEEE Power Electronics Specialists Conference 1999 pp 1158-1 164.

426

K. Smedley, S. Cuk. “OneCycle Control of Switching Converters”, IEEE Transactions on Power Electronics, Vol 10, No 6, November 1995. pp 625-633. W. Tang, F.C. Lee,R.B. Ridley, I. Cohen. “Charge Control: Modeling, [3] Analysiis, and Design” IEEE Transactions on Power Electronics, Vol 8, No4, October 1997. pp 396403 [4] S. Hiti, D. Borojevic. “Robust Nonlinear Control for Boost Converter” IEEE Transactions on Power Electronics, Vol 10, No 6, November 1995. pp 651-658 [ 5 ] P. Mattavelli, L. Rossetto, G. Spiaui, P. Tenti. “General-purpose Fuvy Controller for DC-DC Converters”, IEEE Transactions on Power Electronics, Vol 12, No 1, January 1997. pp 79-86 [6] P. Mattavelli. L. Rossetto, G . Spiaui. “General Purpose Sliding Mode Controller for D C D C Converter Applications”, IEEE Power Elecuonics Specialists Conference PESC‘93. IEEE CODE pp 609-615 [7] L. Malesani, L. Rossetto, G . Spiazzi, P. tenti. “Performance Optimizacion of Cuk Converter by sliding-Mode Control”, IEEE Transactions on Power Electronics, Vol 10. No 3. May 1995. pp 302-309 [8] P. Midya, P.T. Krein, M.F. Greuel. “Sensorless Current Mode Control: An Observer-based Technique for DC-DC Converters’.. IEEE Transactions on Power Electronics. Vol 16. No 4, July 2001. pp 522-526 [9] H . Sira Ram’rez. R. Mirquez. M. Fliess. “Generalizad PI Sliding Mode Control of D C E Converters”, IFAC Symposioum on System Structure and Control. Prague, Agosto 29-3 1,2001 [IO] R. A. DeCarlo, S. Zali, G. P. Matthews, “Variable Structure Control of Nonlinear Multivariable Systems: A Tutorial”, Proceedings of the IEEE, vol. 76 NO. 3. March 1988, pp. 212 - 232. [ I I ] J. Y. Hung, W. Gao, J. C. Hung. “Variable Structure Control: A Survey”, IEEE Transactions on Industrial Electronics, vol. 40, No. 1, Feb. 1993, pp. 2 - 18. [I21 V.1 Utkin, Sliding Modes And Their. Applicariori Zri Variable Sriucmre Sysrerns, MIR Publishers, Moscow, 1974.

[2]

Student, IEEE,

c. HERNANDEZ', Member, IEEE, J. fiv-z3, and J. -U4,

Senior, IEEE,

knior,E~~

Instituto Tecnologico de Celaya, Celaya, Mexico, e-mail : (nvazquez', chdz' )@itc.mx Centro de Investigacion y Estudios Avanzados del IF", DF, Mexico, e-mail: [email protected] Centro Nacional de Investigacion y Desarrollo Tecnologico, Cuernavaca, Mkxico, e-mail: [email protected]

Abstract-In this paper a controller based on the variable structure theory is analyzed: the sliding mode control. The sliding mode control was proposed to control dc/dc converters, due to the good characteristics introduced to the complete system. In order to regulate the output voltage, the sliding mode control require a current loop in order to assure the stability of the system, the inclusion of this extra loop increases the cost. This paper presents an alternative methad to obtain the goad characteristics of the control strategy without the use of current sensors. The analysis, design, simulation and experimental results are presented Index Terms-Variable structure theory, dddc converters, Current sensorless.

I. INTRODUCTION The DC/DC converters are widely used in many applications, when they are used as regulated voltage source a fast regulation of the output voltage is necessary. In order to introduce a good dynamic response different papers have been proposed [I-71, in those schemes current loops are used, additionally to the output voltage loop, and some times more voltage loops. In [8] was proposed a sensor less current mode control, this technique no uses current loops, but three voltage loops are used; an observer-based technique is used. A control scheme proposed in the literature is the sliding mode control [6-71. The sliding mode control is based on the theory of variable structure, this technique permits to obtain the desired characteristics at the output voltage, additionally to the robustness to large load and input voltage variations. To regulate the output voltage the sliding mode control require a current loop, this is because one of the design steps (stability condition) needs to be fulfilled. and with the current loop is done. This extra loop increase the cost due to the current sensor and a rather complex implementation, this could be a reason for disregarding the use of the control scheme. In [9] was proposed an option to eliminate the use of the current sensor, but with a low response, because This work was partially sponsored by the Mexican Council of the Technological Education System (CoSNET) under Project No. 6.59.02-P

0-7803-7912-8/03/$17.00 0 2003 IEEE 422

an integrator is used to eliminate the use of a current reference generation. In this paper is proposed the sliding mode control with no current sensors; this is made with a new sliding surface that eliminate the use of the current sensor and current reference generation, also permits to fulfill all the controller design steps. The proposed method eliminates the generation of the current reference and only two voltage loops are needed to regulate the converter. The sliding mode control has not been discussed when the DC/DC converter is working in discontinuous conduction mode, a brief discussion about this is also made. In the next section the sliding mode control is commented. In section three the proposed controller is addressed, the design steps are also presented. In section four the simulation and experimental results are presented. And finally the conclusions are discussed.

11. SLIDING MODECONTROL The sliding mode control is based on the variable structure theory [ 10-121, and introduces to the complete system a good dynamic response and also robustness to large load and input voltage variations. The sliding mode control operates in a simplified way as follows: a sliding surface is defined with the equilibrium point, and the system is forced to be held into the sliding surface (existence condition), and then the system must reach the equilibrium point (stability) In order to assure that the controlled system operates properly the existence condition and stability must be verified. These are the summarized controller design steps; but also the system modeling could be considered as a design step. A. Systeiii i?iodeliiig

Traditionally the system to be controlled is modeled as if the DC/DC converter was operated in continuous conduction mode (CCM), but this is not always fulfilled. If the converter is designed to operate in CCM, at light loads the converter operates in discontinuous conduction mode (DCM); even worst the converter could be designed to op-

4; vo

T

- 1 Fig 1. DCDC boost converter

a) When SWI is turned on

erate in DCM as in high power factor pre-regulators. The behavior of the converter in CCM is different to the DCM operation. So in order to verify the operation of the converter also in DCM also a model is obtained for this case.

LI

The DC/DC boost converter is analyzed since this converter is equal or relatively more complicated to be controlled than others. The boost converter is shown in Fig 1; the sub-circuits of the converter are shown in Fig 2. The sub-circuit in Fig 2.a is obtained when the semiconductor SW, is turned on. The sub-circuit in Fig 2.b is obtained when the semiconductor SW, is turned off and the inductor current IL is positive. The sub-circuit in Fig 2.c is obtained when the semiconductor is turned off and the inductor current is zero or discharged; this last subcircuit is obtained when the converter is operating in DCM due to the diode D. The model of the complete system in matrix form considering the DCM operation is:

#c) When SWI is turned off and XI

=0

Fig2 Subcircuits of the dcldc boost converter

[6-71. The traditional sliding surface and control law are respectively: (3) o = s,(XI - x,,) + (x2- xz,) = 0

The current loop is necessary in order to assure the stability condition, because if sI is considered as zero in order to only feedback the voltage, the system becomes unstable. The existence condition for the boost D C D C converter with (1) and ( 2 ) as the sliding surface and control law can be obtained easily; an approximation to determine the existence condition is fulfilling the next inequality: n = Virz/fi,

b =I o / Z

The model of the converter considering that always is operating in CCM is:

S , is chosen properly to assure the existence condition and stability, so it is chosen positive. Notice that ( 5 ) is a simplified inequality (approximation).

111. PROPOSEDSLIDING MODE CONTROLLER B. Traditional sliding suiface

The traditional sliding surface used to regulate the output voltage of a DC/DC converter is a linear combination of the error states variables and some times to eliminate the steady state error an integrator of the error voltage is used

423

The proposed sliding mode controller permits to eliminate the use of the current sensor. This is made modifying the traditional sliding surface. This surface permits to fulfill the existence and stability conditions without the use of the inductor current.

C. Stability analysis

A. Sliding surface

The stability analysis of the controller is made with the equivalent control, the equivalent control is substituted into the system model, and is verified under that condition.

The proposed sliding surface is similar to the traditional, but the current error is emulated with a differential equation: CT = sleXl+ (xz - x Z r )= 0 (6) +V&

e,, = a -

(1- U ) -kieX, 2

The equivalent control is the control law when the system is into the sliding surface, and it is obtained from 6 = 0 , but changing if fo the equivalent control ueq:

(7)

As can be observer in (6) and (7) the proposed sliding surface not uses the inductor current, so it is not necessary to sense it. The model of the converter in continuous conduction mode motivates the differential equation (7); the differential equation that determines the inductor current behavior in the CCM model ( 2 ) is:

XI

=a-

WO.Y2(I - U )

This analysis is beyond of the purpose of this communication, so it is not included. but the analysis results in fulfilling the inequality:

(8)

2

To determine (7) only is added a term that stabilize the differential equation and changed x, by e,, . The inequality ( 1 5 ) is an approximation, but determines an region where the system is stable.

The control law used is (4).

B. Existence of the sliding mode In order to verify the existence condition the following condition must be fulfilled [ 121:

06 < O

(9)

To fulfill (9) , the values of the control law (4) is taken into account and also (9), that is:

If If

ii=+l-+oO

= -1 -+ CT > 0; theri d- < 0

IV. EXTENSION TO O T H E R DC/DC CONVERTERS The controller proposed in this paper can be extrapolated to other dc/dc converters. In this section only the buck, boost and buck boosi dc/dc converters are discussed, but also to other converter can be extrapolated. The generalized control Ian and sliding surface have the following form:

(10)

"={

I

I/OO

Using (21, ( 6 ) and (7) is obtained:

In table 1 the ten71 i n 18) is defined depending on the converter. Each tern1 is obtained according to the model in

With (10) and (1 1) existence condition are: Y > O

- M~o{SIX2-SI}+ r < 0

where: r = s,cl - b + kies, - .t2, Buck dddc converter

An approximation of (1 2 ) is:

- \i',,.\V? +a-

(1+1f)

2

Boost dc/dc converter

As can be observed the existence condition can be fulfilled like the traditional sliding surface was fulfilled, because the existence condition is the same.

424

Buck-boost dddc converter

(1-U)

-\i'(,s2- + a -

2

(]+U)

2

CCM of the converter, the model for each converter: the model of the buck dc/dc converter is:

like a logic gate.

v. SIMULATION AND EXPERIMENTAL RESULTS A D C D C boost converter was designed and built. The parameters are Po= 300W, Vin= 50V, Vo= 150V, fs=20KHz. Some simulation results of the converter are shown in Figs 3 and 4. The converter operation in CCM is shown in Fig 3, the converter operation at light loads (DCM operation) is shown if Fig 4. An integral term of the error voltage needs to be used to eliminate the steady state error in DCM, the integral term it is strictly necessary in DCM operation to eliminate the steady state error.

The model for the buck-boost dddc converter is:

The experimental results of the system are show in Fig 5 and 6. A load variation is made, as can be observed the system has a fast response. In this Fig. 5 it is shown the output voltage and current, In Fig 6 i t is shown the emulated current error, the output voltage and the output current; the emulated current is the solution of the differential equation (7).

The model of the boost dddc converter was obtained previously in ( 2 ) . In all the models:

The implementation of the term in (18) is relatively easy; it is not necessary a multiplier, only two diodes are used, this is because the function (1-u)/2 or (l+u)/2 acts

VI. CONCLUSIONS In this paper a controller based on the variable structure

Ouput Voltw

:

:

70

: 50

0 001

0 002

0 003

0 on1

0 002

0 003

0UP

Irductor C m n t

4,

3 2

1

0 l 0

0 001

i

Emdated C m n t E m r

i

08 1 0 2.0 I1-

I

J

0 001

0 002

0 003

0 00

Fig S . Experimental results: Test under load variation. Top to down: Load Current, Output voltage

Fig 3 Simulation results Converter operating in CCM

i

1

0 00'

Emulated Cunent Ermr 044

0

0 001

0 002

0 003

I

Fig 6. Experimental results: Test under load variation. Top to down: Emulated current error. Output Voltage. Output Current

0 00

Fig 4. Simulation results: Converter operating in DCM

42 5

theory is analyzed: the sliding mode control. The sliding mode control was proposed as a good technique to control dc/dc converters, due to the characteristics introduced to the complete system. In order to regulate the output voltage, the sliding mode control requires a current loop in order to assure the stability of the system, the inclusion of this extra loop increases the cost of the system. This paper presents an alternative method to obtain the good characteristics of the control strategy without the use of current sensors, this is made modifying the traditional sliding surface, and only two voltage are sensed: the input voltage and output voltage. This fact produces a cost reduction, simplifies the implementation, and increases the overall system reliability. The analysis, simulation and experimental results are presented.

VII. REFERENCES [I]

2. Yang, P.C. Sen. “Dc to Dc Buck Converter with novel current mode control”, IEEE Power Electronics Specialists Conference 1999 pp 1158-1 164.

426

K. Smedley, S. Cuk. “OneCycle Control of Switching Converters”, IEEE Transactions on Power Electronics, Vol 10, No 6, November 1995. pp 625-633. W. Tang, F.C. Lee,R.B. Ridley, I. Cohen. “Charge Control: Modeling, [3] Analysiis, and Design” IEEE Transactions on Power Electronics, Vol 8, No4, October 1997. pp 396403 [4] S. Hiti, D. Borojevic. “Robust Nonlinear Control for Boost Converter” IEEE Transactions on Power Electronics, Vol 10, No 6, November 1995. pp 651-658 [ 5 ] P. Mattavelli, L. Rossetto, G. Spiaui, P. Tenti. “General-purpose Fuvy Controller for DC-DC Converters”, IEEE Transactions on Power Electronics, Vol 12, No 1, January 1997. pp 79-86 [6] P. Mattavelli. L. Rossetto, G . Spiaui. “General Purpose Sliding Mode Controller for D C D C Converter Applications”, IEEE Power Elecuonics Specialists Conference PESC‘93. IEEE CODE pp 609-615 [7] L. Malesani, L. Rossetto, G . Spiazzi, P. tenti. “Performance Optimizacion of Cuk Converter by sliding-Mode Control”, IEEE Transactions on Power Electronics, Vol 10. No 3. May 1995. pp 302-309 [8] P. Midya, P.T. Krein, M.F. Greuel. “Sensorless Current Mode Control: An Observer-based Technique for DC-DC Converters’.. IEEE Transactions on Power Electronics. Vol 16. No 4, July 2001. pp 522-526 [9] H . Sira Ram’rez. R. Mirquez. M. Fliess. “Generalizad PI Sliding Mode Control of D C E Converters”, IFAC Symposioum on System Structure and Control. Prague, Agosto 29-3 1,2001 [IO] R. A. DeCarlo, S. Zali, G. P. Matthews, “Variable Structure Control of Nonlinear Multivariable Systems: A Tutorial”, Proceedings of the IEEE, vol. 76 NO. 3. March 1988, pp. 212 - 232. [ I I ] J. Y. Hung, W. Gao, J. C. Hung. “Variable Structure Control: A Survey”, IEEE Transactions on Industrial Electronics, vol. 40, No. 1, Feb. 1993, pp. 2 - 18. [I21 V.1 Utkin, Sliding Modes And Their. Applicariori Zri Variable Sriucmre Sysrerns, MIR Publishers, Moscow, 1974.

[2]