NON-LINEAR CONTROLLER APPLIED TO BOOST DC-DC CONVERTERS USING THE STATE SPACE AVERAGE MODEL

Luigi Galotto Jr., Carlos A. Canesin, Raimundo Cordero, Cristiano A. Quevedo, Rubenz Gazineu

Universidade Federal de Mato Grosso do Sul - Departamento de Engenharia Eltrica CEP 79070-900, C.P. 549, Campo Grande, MS, Brazil

luigi@batlab.ufms.br, canesin@dee.feis.unesp.br

Abstract - This work aims to present a methodology to use the large signal state space average model of the DC-DC converters for high performance controllers design. The design using classic control theories need the SISO transfer function. Often, these functions are obtained using linearization around the point of operation. It reduces the range of operation to the designed controller with the expected performance. Also, the presented controller showed to have high disturbance rejection dependent on the sensors used. The presented controller is described, simulated and experimentally confirmed.

Keywords Average model, DC-DC converter, state

space.

I. INTRODUCTION

The control of dynamic systems is a theme of extreme importance in the world, mainly in the modern automated machines times. Each system needs an adequate controller to reach the expected design requirements, which is designed using previous obtained system model. The models are essential for controller design in any dynamic system. Despite some systems are quite simple and slow in such way that they could have their controller tuned heuristically, there are others where strict designs are mandatory to get stability, safety and the expected performance. The power electronic converters generally are this kind of processes. Their dynamic responses are fast in relation to other physical systems and they usually lead with high amount of energy, increasing the importance and the risks. It is used the boost DC-DC converter to illustrate the importance of this state space design approach.

It is presented in [1-6] some methodologies used to power electronic converters modeling. The most accurate model is called as exact model [1], due to the fact to present the system dynamics in all its switching states. Even tough it is more accurate, this model it not appropriate for controller design, because it is not explicit the dynamic effect of the duty cycle. This model can also be dispensed when the switching effects are not focused. Usually, the linearized or small signal model in transfer function form is used to design a classical controller. Sometimes the non-linear behavior is meaningful and a different control strategy must be adopted as adaptive [7] and robust controllers [8]. State feedback [9-10] is also a widely used approach in many applications. Specifically for DC-DC converter there are also the ramptime current and sliding mode techniques [11-13] that act directly on switching. The presented technique is closest to the state feedback but it also includes some feedforward signals and the derivative control.

II. MODELS

A. Exact Model The exact model is obtained through a set of state space

equations as shown in (1), to each j switching state. uBxAx jj += (1) In this model, u is the vector with the external variables to

the converter and x is the vector with the internal states related with the energy stores.

There will have two switching states to the boost DC-DC converter in continuous conduction mode: switch on and diode on. The state space equations to these two states are represented in the equations (2) and (3) respectively.

+

=

o

i

i

o

i

o

IV

LC

IV

IV

0/1/10

0000

(2)

+

=

o

i

i

o

i

o

IV

LC

IV

LC

IV

0/1/10

0/1/10

(3)

It is observed that the vector u is composed in the case by

the input voltage Vi and by the output current Io, because these variables are dependent on the source and loads externally. The states are composed by the input current and output voltage, which are related with the inductor and the capacitor. It is also important to verify in (2) that the matrix is zero, because the diode is not conducting and there are two independent circuits. On the other hand, in (3) it is observed the relation between the output voltage and the input current.

In this model, it is not observed the influence to the duty cycle which is implicit in the duration of each equation.

B. Average Model Using the exact model, the average model [3] can be

easily obtained with the sum of each equation weighted by the time of contribution normally given by the duty cycle as shown in (4).

uBTt

xATt

x

ee B

N

jj

j

A

N

jj

j

==

+

=

11

(4)

978-1-4244-3370-4/09/$25.00 2009 IEEE 733

As an example to the studied converter, the average model will have the equivalent matrices Ae and Be as shown in (5).

+

=

o

i

i

o

i

o

IV

L

CIV

Ld

Cd

IV

01

10

0)1(

)1(0

(5)

In this equation, it is possible to observe the influence of

the duty cycle d, in equivalent matrix Ae. In this case the average model is non-linear, due to the duty cycle inside the matrix.

Although d is an internal parameter in the matrix, in the controller point of view it is an input. Equation (5) can be reorganized to allow the controllers design. The duty cycle d is considered as an input and the equation is linearized around the point of operation.

C. Small Signal Model The linearized model with the duty cycle in the input is

obtained for zero input voltage. An example of this model is presented in [2]. The equation is a SISO transfer function, which is ideal to classic controllers design. The transfer functions are rich of physical meaning. They can be analyzed with zeros and poles map, frequency responses and time responses. However, it has a worst accuracy compared to the other models.

III. COMPARING THE MODELS

The models were simulated in open loop, for different duty cycle steps in 1 second intervals, as shown in Fig. 1. Note that the linearized model is close to the switching model only to d = 0,5 which is its linearization point. The state space average mode, on the other hand, is close to the switching model in practically all points of operation in continuous conduction mode.

IV. STATE SPACE BASED CONTROLLER DESIGN.

Using the converter average model shown in (4) the controller is found so that its control variable is the derivative of the state instead of the duty cycle.

1 2 3 4 5 6 7 8-5

0

5

10

15

20

25

30

35

40

45

Time (s)

Outp

ut V

olta

ges

(V)

Open-loop Responses

Average ModelSwitching ModelLinearized Model

d = 0,2d = 0,1

d = 0,3

d = 0,4

d = 0,5

d = 0,6

d = 0,7

Fig. 1. Open loop responses.

In the case of the boost converter, the equations (6) and (7) are given from the (5) to determinate the relation between the duty cycle and the derivative of the states.

ioi VVdIL += ).1(. (6)

oio IIdVC = ).1(. (7)

Using these equations, it is possible to know the influence

of the inputs and the states in the system dynamics. The voltage error (Vo* Vo) is sent to the controller which

will determinate the necessary derivative to compensate the response ( *oV ). These desired derivative is inserted in equation (7) with the output current will determine the necessary value of (1 - d).Ii*. The current error (Ii* Ii) is then sent to the second controller which will determine the desired derivative of the current ( *iI ). Using (6), it is obtained the output d.

This described controller is illustrated in Fig. 2.B compared with the current model control strategy (Fig. 2.A).

It is important to emphasize that the use of 4 sensors is not mandatory. In the experimental results presented in section VI, only input current and output voltage were used while the output current and the input voltage are set to zero.

Fig. 2. Diagram for the Controller.

V. SIMULATED RESULTS

The objective of the simulated results is to make a comparison between the classic voltage and current mode controllers [3] and the non-linear controller presented.

The Table I shows the values used in to the boost converter model. These values have been chosen to keep the converter in continuous conduction mode.

TABLE I Simulated boost parameters

Parameter Value Inductance (L) 1.1 mH Capacitance (C) 100 F Ii initial condition 0 A Vo initial condition 24 V R (load) 24

Vo

Vo* + -

Ii

Ii* + - d

(A)Current Mode Control

C1(s)

Vo

Vo* + -

Ii

Ii* + -

d

(B)Proposed Controller

C1(s) C2(s)

C.dVo* dt (7) C2(s) (6)

L.dIi* dt

978-1-4244-3370-4/09/$25.00 2009 IEEE 734

0 0.1 0.2 0.3 0.4 0.5 0.60

10

20

30

40

50

60

Time (s)

Out

put V

olta

ge (V

)

ReferenceVoltage ModeCurrent ModeNon-Linear

Fig. 3. Closed loop responses.

The three controllers were applied in the same model

simultaneously to track the same output voltage reference steps. The Figure 3 shows the results to the three kind of controllers. Note that in the 18 to 24 volts step, all controllers were designed to present approximately the same performance. However, the non-linearity of the converter modifies the response performance dependent on the level of operation as expected to the boost converter.

It should be informed that all steps in different voltage levels were normalized and plotted together in the charts in Fig. 4, so that these results can be better visualized. Thus it is possible to verify how the response performance can vary to each controller.

The Figure 4.A shows the high variations of the dynamic response to the controller in voltage mode. The damping showed reduction increasing the voltage operation. Since the 36V the response goes unstable.

The variations in dynamics to the current mode controller are shown in Fig. 4.B. They were less intense in relation to the previous controller. However, there is still the influence of the point of operation.

In the case of the non-linear controller, the performance is practically independent, because the converter non-linearity is embedded inside of the controller, as shown in Fig. 4.C.

A. Input Current Loop Analysis Beyond the advantage of a constant step response, the

dynamic response may be less sensible to input voltage and load variations. The information about how the system states vary ((6) and (7)) is very useful. To illustrate how the system may become less sensible to input voltage or load variation, let be first considered the input current control.

Equation (8) derived from (6). In this case, oV~

is the average output voltage, which is added to normalize the equation and to keep the classic controller design unchanged, as shown in Fig. 2.

o

iio

VVILVd +=

..~1 (8)

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

5

10

Time (s)(A)

Norm

aliz

ed O

utp

ut Vo

ltage

Voltage Mode

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

5

10

Time (s)(B)

Norm

aliz

ed O

utpu

t Vol

tage

Current Mode

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.10

5

10

Time (s)(C)

Norm

aliz

ed O

utpu

t Vol

tage

Non-linear Controller

Fig. 4. Normalized closed loop responses.

Presuming that the first controller was designed to 50V for

the input voltage and 100V for the output voltage, and only the input current is measured, then (8) is simplified into (9). Therefore, just an offset is added to the controller output (x) and no changes will be observed in the control response.

xILd i +=+= 5.05.0.1 (9)

In this example, if the output voltage is measured, then (10) will be used. The additional information will make the duty cycle varies in function of the controller output and the output voltage simultaneously.

oV

xd 150.3001 += (10)

Table II shows the simulation results of Ii control. Each

row is a different controller with the same plant parameters and the same PI tuning. Equations (8), (9) and (10) are applied to the same Plant and PI in order to visualize the effect of the controllers. All simulations are subjected to the same disturbances shown in the last row.

If only one sensor is used, the application of (9) will have the exactly same dynamic response of the PI controller. In this case, there is no advantage in using this equation, but it is used here to demonstrate that there are no negative effects

978-1-4244-3370-4/09/$25.00 2009 IEEE 735

if non available sensors are replaced by constant average values.

In the third row of Table II, the output voltage is available and it is used in the loop in the form of (10). The best effect is observed in the load variation where the system becomes very robust. The reference tracking is faster and constant for different steps. A slightly depreciation of the Vi disturbance rejection is observed. This drawback may be overcome using an input voltage sensor as shown in the third row.

If the input voltage is also measured, the input current derivative will be very well controlled, because all causes of variations are known. Of course, some applications may not justify the use of three sensors in this case to control just the input current. However, it depends on problem criticity and performance requirements. Furthermore, for high power converter the cost is dependent only on the power circuit and the sensors are not the most expensive components.

Usually, the output voltage is already measured in current mode control and an improvement may be achieved changing the control algorithm.

B. Output Voltage Loop Analysis The equations used in the input current loop have been

derived from (6). Similarly, Equation (7) will be used to derive the equations for voltage loop. The output of the voltage loop controller is the output voltage derivative as shown in Fig. 2.

The objective of this function is to find the reference input current given the latest used duty cycle and the output

current. The average duty cycle d~ is used to normalize the function and to keep the same PI controller.

)1(

.).~1(d

IVCdI ooi

+=

(11)

The duty cycle is always available, and then there is only

one variation of (11), which is when the output current is not measured. In this case, the average operating current is used.

TABLE II

Simulated Input Current Control with different controllers Controller Vi Variation Load Variation Ii Reference Variation Sensors

PI

0 1 2 31

2

3

4

5

I i (A)

Time (s) 3 4 5 6 7

1

2

3

4

5

I i (A)

Time (s)7 8 9 10 11

1

2

3

4

5

6

I i (A)

Time (s)

1 (Ii)

PI and (9)

0 1 2 31

2

3

4

5

I i (A)

Time (s) 3 4 5 6 7

1

2

3

4

5

I i (A)

Time (s)7 8 9 10 11

1

2

3

4

5

6

I i (A)

Time (s)

1 (Ii)

PI and (10)

0 1 2 31

2

3

4

5

I i (A)

Time (s) 3 4 5 6 7

1

2

3

4

5

I i (A)

Time (s)7 8 9 10 11

1

2

3

4

5

6

I i (A)

Time (s)

2 (Vo, Ii)

PI and (8)

0 1 2 31

2

3

4

5

I i (A)

Time (s) 3 4 5 6 7

1

2

3

4

5

I i (A)

Time (s)7 8 9 10 11

1

2

3

4

5

6

I i (A)

Time (s)

3 (Vi, Vo, Ii)

Variations

0 1 2 30

50

100Vi

V i (V

)

Time (s) 3 4 5 6 7

0

100

200

300

400

500Rload

R (O

hms)

Time (s)7 8 9 10 11

1

2

3

4

5

6Ii Reference

I i (A)

Time (s)

978-1-4244-3370-4/09/$25.00 2009 IEEE 736

The simulation results are presented in the Table III. The controller is compared with a typical current mode control. The same tuning, plant and parameters are used in all simulations.

The use of the presented controller with the same number of sensors as the current mode slightly improves rejection of load disturbances and step responses. However, the effect of input voltage variation is amplified and it makes this controller good only for very well stabilized input voltage.

Adding the input voltage sensor, the input voltage disturbance rejection is highly improved.

For high performance control, four sensors would provide a constant dynamic in different points of operation and very good disturbance rejection.

Furthermore, it is possible to use, in some kind of applications, different artificial intelligence techniques in order to estimate the desirable variables, eliminating additional sensors.

VI. EXPERIMENTAL RESULTS

A boost converter has been designed using the presented controllers. It has been evaluated the current and voltage loops with the same plant and the same number of sensors.

A. Input Current Control Results

The Figures from 5 to 8 shows the current control results, using only two sensors (input current and output voltage). The channel 1 is the input current with 1A/100mV, channel 2 is the output voltage with 1:500 scale and the channel 3 is the duty cycle.

Figure 5 and 7 are the responses under load variation. Figure 6 and 8 are the responses under input voltage variation. It is experimentally confirmed that the load disturbance rejection is highly improved and the voltage disturbance effect is worst as expected.

TABLE III

Simulated Output Voltage Control with different controllers Controller Vi Variation Load Variation Ii Reference Variation Sensors

Current Mode

0 2 4 60

50

100

150

200

V o (V

)

Time (s) 6 8 10 12 14

0

50

100

150

200

V o (V

)

Time (s)14 16 18 20 22

0

100

200

300

400

V o (V

)Time (s)

2 (Ii and Vo)

Current Mode And

Equations (8) and (11)

0 2 4 60

50

100

150

200

V o (V

)

Time (s) 6 8 10 12 14

0

50

100

150

200

V o (V

)

Time (s)14 16 18 20 22

0

100

200

300

400

V o (V

)

Time (s)

2 (Ii and Vo)

Current Mode And

Equations (8) and (11)

0 2 4 60

50

100

150

200

V o (V

)

Time (s) 6 8 10 12 14

0

50

100

150

200

V o (V

)

Time (s)14 16 18 20 22

0

100

200

300

400

V o (V

)

Time (s)

3 (Ii, Vo, Vi)

Current Mode And

Equations (8) and (11)

0 2 4 60

50

100

150

200

V o (V

)

Time (s) 6 8 10 12 14

0

50

100

150

200

V o (V

)

Time (s)14 16 18 20 22

0

100

200

300

400

V o (V

)

Time (s)

4 (Ii, Vo, Vi, Io)

Variations

0 2 4 60

50

100Vi

V i (V

)

Time (s) 6 8 10 12 14

0

500

1000Rload

R (O

hms)

Time (s)14 16 18 20 22

0

100

200

300

400Ii Reference

V o (V

)

Time (s)

978-1-4244-3370-4/09/$25.00 2009 IEEE 737

Fig. 5. Ii PI control with load variation.

Fig. 6. Ii PI control with input voltage variation.

Fig. 7. Ii non-linear control with load variation.

Fig. 8. Ii non-linear control with input voltage variation.

B. Output Voltage Control Results The experimental results related with the output voltage

control are presented in Fig. 9 to 14. The channels are the same as in the previous results.

In the Figures 9 to 11 are presented results related with the typical current mode control. The non-linear controller with the same PI tuning resulted in the responses presented in the Fig. from 12 to 14.

Figures 9 and 12 show the responses under 50% load changes. The settling time for the non-linear controller response is far smaller than the current mode control response.

Figures 10 and 13 show the responses under input voltage variations. The disturbance overshoot is higher in the non-linear controller as expected from the experimental results.

But the settling time is also smaller in relation to the current mode.

Figures 11 and 14 show the responses for output voltage reference steps. The non-linear controller is faster than the current mode control for the same PI tuning. Moreover the dynamic response is more constant for different voltage steps comparing to the current mode.

This feature can be observed in Fig. 15. The Figure 15 shows the same normalized responses illustrated in Fig. 4. The normalized responses are the same data presented in Fig. 11 and 14. The responses are adjusted to the zero voltage level and the same step time.

In Figure 15 is clear to observe the difference among the transient responses.

978-1-4244-3370-4/09/$25.00 2009 IEEE 738

Fig. 9. Vo current mode control with load variation.

Fig. 10. Vo current mode control with Vi variation.

Fig. 11. Output Reference Voltage Variation, current mode control.

Fig. 12. Vo non-linear control with load variation.

Fig. 13. Vo non-linear control with Vi variation.

Fig. 14. Output Reference Voltage Variation, non-linear control.

978-1-4244-3370-4/09/$25.00 2009 IEEE 739

-1.5 -1 -0.5 0 0.5 10

5

10

15

20

25

30

Time (s)

vo

Step

(V

)

Current Mode Responses

-5.5 -5 -4.5 -4 -3.5 -30

10

20

30

40

Time (s)

vo

Step

(V

)

Non-linear Controller Responses

Fig. 15. Experimental normalized closed loop responses.

VII. CONCLUSIONS

This work presented a short introduction on modeling methods for DC-DC converters and a control strategy based on the state space average model to allow the operation in different voltage levels without relevant changes in dynamics performance. The state space average model is full of important information concerned to the converter dynamics in different points of operation. It allows a high performance controller and its computation complexity is not so far from the classic controllers.

Results of simulation and the examples were presented to the boost converter to illustrate the control strategy. Experimental results were also provided to confirm the controller capability to lead with the plant non-linearity.

The advantages of this controller basically are: adequate for different operating points and reduces the effects of load and input voltage disturbances. The drawbacks are: the number of sensors is considerable to reach the best results and for few sensors some features may be worst while others are improved.

The presented control strategy may use different number of sensors. More sensors will improve the dynamic response and the disturbance rejection. The application will determine the best strategy to be used and the adequate number of sensors. In spite of the cost, the use of more sensors will probably bring additional advantages such as fault tolerance and noise reduction.

Future studies may be done in order to explore more sensors, including its additional advantages, or, the application of estimation techniques in order to reduce the number of sensors with the same performance.

REFERENCES

[1] A. Merdassi, L. Gerbaud, S. Bacha, A new automatic average modeling tool for power electronics systems, in

Proc. Power Electronics Specialists Conference, pp. 34253431, 2008.

[2] R. Tymerski, V. Vorperian, F.C.Y. Lee, W.T. Baumann, Nonlinear modeling of the PWM switch, IEEE Transactions on Power Electronics, vol. 4, no. 2, pp. 225233, April 1989.

[3] V. B. Sriram, S. SenGupta, A. Patra, Indirect Current Control of a Single-Phase Voltage-Sourced Boost-Type Bridge Converter Operated in the Rectifier, IEEE Transactions on Power Electronics, vol. 18, no. 5, pp. 1130- 1137, September 2003.

[4] M.G. Giesselmann, Averaged and cycle by cycle switching models for buck, boost, buck-boost and Cuk converters with common average switch model, Energy Conversion Engineering Conference, in Proc. of the 32nd Intersociety IECEC, pp. 337341, 1997.

[5] Y. Ren, W. Kang, Z. Qian, A novel average model for single switch buck-boost DC-DC converter, in Proc. Power Electronics and Motion Control Conference, pp. 436439, 2000.

[6] M. U. Iftikhar, P. Lefranc, D. Sadarnac, C. Karimi, Theoretical and experimental investigation of averaged modeling of non-ideal PWM DC-DC converters operating in DCM, in Proc. of Power Electronics Specialists Conference, pp. 22572263, 2008.

[7] S.E. Beid, S. Doubabi, M. Chaoui, Adaptive Control of PWM Dc-to-Dc Converters Operating in Continuous Conduction Mode, in Proc. of Mediterranean Conference on Control and Automation, Athens Greece, pp. 1-5, 2007.

[8] W. Hernandez, Robust control of a buck-boost DC-DC switching regulator for the electronic systems of next-generation cars, in Proc. of Industrial Electronics Society, pp. 610, 2005.

[9] R. Leyva, L. Martnez-Salamero, H. Valderrama-Blavi, J. Maix, R. Giral, F. Guinjoan, Linear State-Feedback Control of a Boost Converter for Large-Signal Stability, IEEE Transactions on Circuits and Systems: fundamental theory and applications, vol. 48, no. 4, pp. 418-424, April 2001.

[10] H. Hinz, State Feedback Loop Design of a Boost Converter, in Proc. of Second International Conference on Power Electronics, Machines and Drives, pp. 192196, 2004.

[11] Y. Wu, S. Qiu, Y. Chen, Slide-mode control scheme with dynamic modifying errors for boost converter, Journal of Electronics, Springer, vol. 17, no. 4, pp. 370375, October 2000.

[12] P. Mattavelli, L. Rossetto, G. Spiazzi, P. Tenti, "General-purpose sliding-mode controller for DC/DC converter applications", in Proc. of IEEE Power Electronics Specialists Conference, pp. 609615, 1993.

[13] L.J. Borle; C.V. Nayar, Ramptime current control, in Proc. Applied Power Electronics Conference and Exposition, pp. 828834, 1996.

978-1-4244-3370-4/09/$25.00 2009 IEEE 740