A Simple Robust Feedback Controller for Voltage Regulation ... · Cuk and Middlebrook Concluding remarks and future research 4 / 30 Distinguishing features of converters in DCM include

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A Simple Robust Feedback Controller for Voltage Regulation of a BoostConverter in Discontinous Conduction Mode

Aya Alawieh‡,Harish Pillai†, Romeo Ortega‡, Alessandro Astolfi¶, and Eric Berthelot §

‡ Laboratoire des Signaux et Systemes, Supelec, 91192 Gif-sur-Yvette, France.

† Department of Electrical Engineering, IIT Bombay, Mumbai 400076, India.

¶ Dept. of Electrical and Electronic Engineering, Imperial College, London, SW7 2AZ, UK

and DISP, University of Roma, “Tor Vergata”, Via del Politecnico 1, 00133 Rome, Italy.

§ Laboratoire de Genie Electrique de Paris, Supelec, 91192 Gif-sur-Yvette, France.

Inter GDR MACS-SEEDS: CSE, TELECOM ParisTech, 27/01/2011

Introduction

Introduction• DiscontinuousConduction Mode(DCM)

• Features ofconverters in DCM

• Existing techniques

• The differentperspective of this work

Problem formulation

A Robust SwitchingAlgorithm

Simulation andexperimental results

Approximate method ofCuk and Middlebrook

Concluding remarksand future research

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Discontinuous Conduction Mode (DCM)





Problem formulation





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• Ideal switches in power converters are typically implemented

using unidirectional semiconductor devices that may lead to anew operation mode generically called discontinuous

conduction mode (DCM).

• The DCM arises when the ripple is large enough to cause the

polarity of the signal (current or voltage) applied to the switch

to reverse, violating the unidirectionality assumptions made in

the realization of the switch.

• To achieve high performance some new converters are

purposely design to operate all the time in DCM.

• The existing techniques for controller analysis and design, in

particular, the approximations of averaging dynamics (valid

under fast switching) or small ripple, are not valid anymore.

Features of converters in DCM





Problem formulation





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Distinguishing features of converters in DCM include the following.

(i) The control is not a continuous signal, but directly the switches

positions, that take values in the binary set 0, 1, and decide

the commutation among the various converter topologies.

(ii) Due to technological considerations, the activation of the

switches is submitted to a minimal dwell time that has to be

taken into account in the design.

(iii) Besides the commutations induced by (designer selected)

switch positions, there are forced commutations due to the

aforementioned violation of the unidirectionality assumption,

e.g., the presence of diodes.

(iv) As the ripple cannot be neglected, the control objective is not

stabilization of an equilibrium but generation of a periodic orbit

(with “minimal amplitude”) around the desired operating point.

Existing techniques





Problem formulation





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• With the existing techniques, the converter is treated as a

continuous (possibly nonlinear) dynamical system, with theaction of the switches assimilated to continuous signals

ranging in some closed interval, e.g., [0, 1].

• The approximations in the existing techniques leads to below

par performances of the device (or, as shown here, even

instability). This situation makes the problem a challenging

new task for control systems theorists.

• A burgeoning literature on hybrid systems has emerged in the

control community in the last few years.

• The main thrust of the research has been towards the

development of general methodologies, mainly for analysis but

also for controller design, of classes of theoretically–motivated,

hybrid dynamical system.

The different perspective of this work





Problem formulation





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• In this work a different perspective is adopted, namely

development of a solution for a specific example of practical

relevance.

• We consider a boost converter operating in DCM, which

exhibits the four distinguishing features.

• Our main contribution is a simple robust algorithm that, in

contrast with current practice, gives explicit formulas for the

switching times without approximations.

• This device constitutes a basic building block in power

converter design, hence it is our belief that the main ideas maybe applicable to other converters.

Problem formulation

Introduction

Problem formulation• Boost convertertopologies

• Dynamics of thesystem

• Periodic orbit• Time evolution of x1

and x2





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Boost converter topologies

Introduction




and x2





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vinrL

x1

R0L

0

1C x2

vinrL

x1

R0

L

0

1C x2

vinrL

x1

R0

L

0

1C x2

Figure 1: Ideal representation of the boost converter in the topologies:

Ω1 (top), Ω2 (middle) and Ω3 (bottom).

Dynamics of the system

Introduction




and x2





9 / 30

The dynamics of the system is described by a piece–wise affine

model

x = Aix+ bivin, i = 1, 2, 3,

where the pairs (Ai, bi) for the three topologies are given by

Ω1 : A1 =

[ −rLL

−1L

1C

−1CR0

]

, b1 =

[

1L

0

]

Ω2 : A2 =

[

0 00 −1

CR0

]

, b2 =

[

00

]

Ω3 : A3 =

[ −rLL

00 −1

CR0

]

, b3 =

[

1L

0

]

Periodic orbit

Introduction




and x2





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The control objective is to generate an attractive limit cycle (of period

t1 + t2 + t3) contained in the band x2(t) ∈ [x∗2 − ε1, x∗2 + ε2],

where x∗2 ∈ R+ is the desired average value for x2 and—to

minimize the voltage ripple—the constants εi ∈ R+ are as small as

possible. That is, the derivation of a rule to compute the switchinginstants t3 and t2.

x1

x2

x01

x02 x12

t1

t2

t3

Figure 2: Typical periodic orbit in the phase plane.

Time evolution of x1 and x2

Introduction




and x2





11 / 30

• An obvious procedure to compute the switching times relies on

the solution of the differential equations along the periodicorbit, which leads to a set of nonlinear algebraic equations that

are very hard, if at all possible, to solve.

x1x2

x01

x02

x12

t1t1 t2t2 t3t3

tt

Figure 3: Time evolution of x1(t) and x2(t) along a periodic orbit.

A Robust Switching Algorithm

Introduction

Problem formulation


• Computation oft2 + t3

• Computation of t3

• Control algorithm

• Estimation of theTime Constants• Estimation of theTime Constants




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Computation of t2 + t3

Introduction

Problem formulation









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Define the time interval I1 := [t1, t1 + t2 + t3]. For all t ∈ I1 the

capacitor voltage evolves according to

x2 = −1

CR0x2,

whose solution is

x2(t) = e− 1

CR0(t−s)

x2(s), ∀t ≥ s, ∀t, s ∈ I1.

Hence, along the orbit depicted in Fig. 2 and from Fig. 3, one has

x02 = e− 1

CR0(t2+t3)

x12,

yielding

t2 + t3 = R0C ln

(

x12x02

)

. (1)

Computation of t3

Introduction

Problem formulation









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Define the time interval I2 := [t1 + t2, t1 + t2 + t3]. For all time

t ∈ I2 the inductor current evolves according to

x1 = −rL

Lx1 +

vin

L,

whose solution with zero initial conditions is

x1(t) =(

1− e−rLL

t) vin

rL, ∀t ∈ I2.

Hence, fixing x1(t3) = x01—again referring to Figs. 2 and 3—andsolving for t3 yields

t3 = −L

rLln

(

1−rLx

01

vin

)

. (2)

Control algorithm

Introduction

Problem formulation









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Step 1. Fix a point (x01, x02) ∈ R

2+, such that t3 in (2) yields t3 ≥ tD

and x02 < x∗2.

Step 2. At a time t0 ≥ 0 when x(t0) = (x01, x02) switch from u = 0

to u = 1.

Step 3. Wait (in mode Ω1) until x1(t) = 0 and measure the

corresponding x12.

Step 4. Compute t2 + t3 from (1). If t2 > 0 go to Step 5, else definet2 := tD − t1, then go to Step 5.

Step 5. Wait (in mode Ω2) for t2 units of time and then switch from

u = 1 to u = 0.

Step 6. Wait (in mode Ω3) for t3 units of time and then measure the

state, call it (x01, x02). Check whether, for the new (x01, x

02), (2) yields

t3 ≥ tD and x02 < x∗2. If so, go to Step 2, else wait for a longer time

until the state meets the requirements, then assign the value t3 to

this new time and go to Step 2.

Estimation of the Time Constants

Introduction

Problem formulation









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The computations involved in (1) and (2) depend on the time

constants rLL

and CR0.

• Estimation of the parameter CR0.

During the interval I1, x2(t) satisfies

x2 = −1

CR0x2,

Discretizing this equation with a (fast) sampling time Tf yields the

difference equation

x2(k) = θx2(k − 1), θ := e− 1

CR0Tf ,

where the standard notation

x2(k) = x2(t), ∀ t ∈ ((k − 1)Tf , kTf ],

Estimation of the Time Constants

Introduction

Problem formulation









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Now, sample x2(t) with the sampling rate Tf ∈ R+ and take N

samples. Define the N–dimensional vectors

X2 := col(x2(1), . . . , x2(N)),

Φ := col(x2(0), . . . , x2(N − 1)).

Since X2 = θΦ, it is clear that θ can be computed from

θ =1

|Φ|2Φ>X2, (3)

with | · | the Euclidean norm. Note that |Φ| is bounded away from

zero because x2(t) ∈ R+. From the knowledge of θ the timeconstant CR0 is directly obtained.

Simulation and experimentalresults

Introduction

Problem formulation



• Simulation results

• Experimental results



18 / 30

Simulation results

Introduction

Problem formulation







19 / 30

• The simulations were developed in two steps, the first was

devoted to illustrate the performance under nominal (ideal)

conditions. The second set of simulations were done with thecircuit toolbox Simpower systems, which includes realistic

models of the Mosfet and the diode.

• For the simulations the considered circuit parameters are

L = 0.0001H, R0 = 100Ω, rL = 0.001Ω, C = 0.00025F

73.98 74 74.02 74.04 74.06 74.08 74.1 74.12 74.140

1

2

3

4

5

6

7

x 1 [A]

x2 [V]

Figure 4: A periodic orbit obtained in the ideal case.

Simulation results

Introduction

Problem formulation







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0 1 2 3 4 5 6

x 10−4

−1

0

1

2

3

4

5

6

7

time (sec)

X1

(A)

0 1 2 3 4 5 6

x 10−4

73.98

74

74.02

74.04

74.06

74.08

74.1

74.12

74.14

74.16

X2

(V)

time (sec)

73.95 74 74.05 74.1 74.15 74.2−1

0

1

2

3

4

5

6

7

x 1 [A]

x2 [V]

Figure 5: Time evolution of x1(t) and x2(t) and the periodic orbit in

the phase plane for the realistic simulation.

Simulation results

Introduction

Problem formulation







21 / 30

0 1 2 3 4 5 6

x 10−4

−1

0

1

2

3

4

5

6

7

time (sec)

X1

(A)

0 1 2 3 4 5 6

x 10−4

73.98

74

74.02

74.04

74.06

74.08

74.1

74.12

74.14

74.16

74.18

time (sec)

X2

(V)

73.95 74 74.05 74.1 74.15 74.2−1

0

1

2

3

4

5

6

7

x2 [V]

x 1 [A]

Figure 6: Time evolution of x1(t) and x2(t) and the periodic orbit in

the phase plane for the realistic simulation with a 20% error in Ro.

Simulation results

Introduction

Problem formulation







22 / 30

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

73.95

74

74.05

74.1

74.15

74.2

74.25

74.3

time [sec]

x 2 [V]

Figure 7: Time evolution of x2(t) and the orbit in the phase plane for

the realistic simulation with a step variation in R0 .

Simulation results

Introduction

Problem formulation







23 / 30

0 0.5 1 1.5 2 2.5 3 3.5 4

x 10−3

73.95

74

74.05

74.1

74.15

74.2

74.25

time [sec]

x 2 [V]

73.95 74 74.05 74.1 74.15 74.2 74.25−1

0

1

2

3

4

5

6

7

x2 [V]

x 1 [A]

Figure 8: Time evolution of x2(t) and the orbit in the phase plane for

the realistic simulation in the case of the adaptive algorithm.

Experimental results

Introduction

Problem formulation







24 / 30

Experiments were carried out in a setup located in the Laboratoire

de Genie Electrique de Paris. The circuit parameters are

L = 0.00136H, R0 = 500Ω, C = 0.000094F, rL ≈ 0.

Figure 9: Schematic of the printed circuit board of the converter.

Experimental results

Introduction

Problem formulation







25 / 30

−1 0 1 2 3 4 5 6 7 8

x 10−4

−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

x 1 [A]

time [sec]−1 0 1 2 3 4 5 6 7 8

x 10−4

13.785

13.79

13.795

13.8

13.805

13.81

13.815

time [sec]

x 2 [V]

13.788 13.79 13.792 13.794 13.796 13.798 13.8 13.802 13.804 13.806 13.808−0.04

−0.02

0

0.02

0.04

0.06

0.08

0.1

0.12

x 1 [A]

x2 [V]

Figure 10: Time evolution of x1(t) and x2(t) and the orbit in the

phase plane (the experimental result)

Approximate method of Cuk andMiddlebrook

Introduction

Problem formulation




• Approximate method


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Approximate method

Introduction

Problem formulation






27 / 30

In this method Asynchronous mode with a fixed sampling time is

assumed and the duty ratios are computed fixing the maximum

current x01 and approximating

eAit ≈ I2 +Ait.

t1 and t3 are calculated by these equations:

t1 =

√

2LTM

R(M − 1), t3 =

√

M(M − 1)2LT

R,

Where M :=x02

vin

Approximate method

Introduction

Problem formulation






28 / 30

73.7 73.8 73.9 74 74.1 74.2−1

0

1

2

3

4

5

6

7

x2 [V]

x 1 [A]

Figure 11: Unstable behavior in the phase plane using the approxi-

mate method with initial condition (x01, x02) = (6.5, 74).

Concluding remarks and futureresearch

Introduction

Problem formulation




Concluding remarksand future research• Conclusion and futureresearch

29 / 30

Conclusion and future research

Introduction

Problem formulation




Concluding remarksand future research• Conclusion and futureresearch

30 / 30

• The key observation that allows to obtain explicit solutions is

that the existence of the periodic orbit can be guaranteedwithout the computation of the flow associated to topology Ω1,

but just looking at the intervals I1 and I2, where the inductor

and capacitor dynamics are decoupled.

• This fact, of course, stems from the basic operation principle of

the boost converter, where magnetic energy is stored in the

inductor while electric energy of the capacitor is transferred to

the load. In the DCM no magnetic energy is added to the

inductor, but the capacitor continues its discharge—with thesame time constant.

• Since this principle is ubiquitous to all power converters it isour belief that similar calculations can be done for more

complicated power circuits.

Documents

A Simple Robust Feedback Controller for Voltage Regulation ... · Cuk and Middlebrook Concluding remarks and future research 4 / 30 Distinguishing features of converters in DCM include