Buck Converter bykiran_2012

5/20/2018 Buck Converter bykiran_2012

1/99

ERCS.MADHU KIRAN THOTA.PARTHA SARADHI

1

THESIS ON

CONTROL OF BUCK CONVERTER BY POLYNOMIAL,

PID AND PD CONTROLLERS.

BY

ERCS. Madhu Kiran 850923-1232

&

THOTA. Partha Saradhi 850715-6175

This Thesis is presented as a part of our Degree in Master of Science in Electrical

Engineering.

Blekinge Institute of Technology [BTH]

School of Engineering

Supervisor: Anders Hultgren

June - 2012


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2

Abstract

Contents:

Names: Page no:

1. Introduction...51.1 Thesis Background5

1.2 Thesis Aim..6

1.3 Thesis Scope...6

1.4 Outline of thesis..6

2. Basic Buck Converter Circuit Topology.........72.1 Switch....7

2.2 Inductor............82.3 Capacitor.......8

2.4 Diode......82.5 State of operation..................8

2.5.1 On State [Switch is closed]..............................9

2.5.2 Off State [Switch is open].....92.5.3 Modes of Operation......10

2.5.4 Ericsson BMR 450 Feature.......10

3. Modeling of Buck Converter............................13

3.1 Mathematical equations of the buck converter when the switch is in ON state...133.2 Controllers used to control the Buck ..............18

3.3 Get average system..................18

4. Polynomial Pole Placement Controller......214.1 Polynomial controller Simulation.........264.2 Building the controller blocks in Simulink.................27

4.3 Results of Polynomial controller.....27

4.3.1 Simulation results of the Continuous and discrete model ............28

4.3.2 Simulation result without disturbance and without noise..30

4.3.3 Simulation result without disturbance and with noise....31

4.3.4 Simulation results with disturbance and without noise......33

4.3.5 Simulation results with disturbance and with noise.......35

4.3.6 Simulation results with large load capacitor...35

5. PID Controller...37

5.1 Transfer Function of PID..375.2 Structure of the Digital PID Controller.42

5.3 Simulation results of the Continuous and discrete model...44

5.4 Simulation result without disturbance and without noise....46

5.5 Simulation result without disturbance and with noise.47

5.6 Simulation result with disturbance and without noise.49

5.7 Simulation result with disturbance and with noise..50


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3

6. PD Controller...536.1 structure of the digital PD controller.....53

6.2 calculation for finding the poles of PD controller of second order..55

6.3 simulation results of the continuous and discrete model of PD controller58

6.4 simulation results without disturbance and with noise61

6.5 simulation results with disturbance and without noise61

7. Conclusion....63

7.1 Further work.63

8. References..65

Appendix

Appendix 1: Mathematical equation of buck converter when switch is ON........67

Appendix 2: Solving the equation for obtaining the Polynomials .. 73

Appendix 3: Resulted graphs for determining the pole for Polynomial controller .75

Appendix 4: Solving the equation for obtaining the PID...77

Appendix 5: Resulted graphs for determining the pole for PID controller....79

Appendix 6: Solving the euation for obtaining the PD Controller.81

Appendix 7: Resulted graphs for determining the pole for PD controller.83

Appendix 8: Matlab code for the Polynomial Pole Placement Controller.87

Appendix 9: Matlab code for PID Controller...91

Appendix 10: Matlab code for PD Controller..95


4/99


4

T E B I

T BTH, L U LNU

P P P, PID P I D PD

.

T DCDC B . T

, E BMR450 PID,

POLNOMIAL PD , M

S ,

.

T . T

P, PID PD .

B C, DCDC C, PID, PD & POLNOMIAL C.


5/99


5

1.

O ,

. S,

. I , .

1.1

A ,

,

, . S

ON OFF ,

,

, . T

SMPS ,

1.

A B DCDC . B

, DCDC

.

F , B

2. F ,

5 3.3

I IC . H ,

.

B

3. T P W M PWM . I

4.


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6

T ,

. I ,

.

1.2

T

P PD PID C , F

PD PID 5, 6. T

, , .

1.3

T :

S B C.

D .

D P PD PID C.

S B C C SIMULINK MATLAB,

.

F .

R A.

1.4

T 6 , , F ,

. T S B

C . T B C T

. T P P P PID C

M S F F . T C

,

.


7/99


7

2.

T .., ON OFF . H

F 1.

F 1: B B C .

W VS= , D=, L= , C= , R= V0= ,

W , .

T . A

. W ,

, .

I ,

. T B B C

.

2.1

T . A , PWM

ON OFF . D ON,

.D OFF,

. B PWM ON OFF

.


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8

2.2

T . I OFF

,

F 2.

2.3

T

. C

.

2.4

W OFF, ,

. T

,

. I

. T

.

W ,

.W , . A

.

T ,

.

2.5

T DCDC , .., ON OFF .

T ON .


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9

2.5.1 ( )

D ON ,

V L, . I ,

, F 2.

F 2: B C ON .

2.5.2 ( )

D OFF , ,

. A ,

F 3.

F 3: B C OFF .

T F 4 ON OFF

, .


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10

F 4: D ,

0 .

2.5.3

T ..,

. I , .

B ,

. T

. I ,

.

2.5.4 450

T BMR450 E

DCDC BMR 451 453,

DCDC .


11/99


11

F 5: E BMR 450

D PWM ,

BMR450. O .

I: 4.514 V, O: 20 A/100W.

S 25.65 12.9 8.2 1.01 0.51 0.323 .

20 A .

4.5 V 14 V .

0.6 V 5.5 V .

H , . 96.8 % , 5 V , 3.3 V .

5 MTBF.

T .

PM B .

V// .

P .

N .

W .

S .

O .


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12

O .

ON/OFF .

O .

ISO 9001/14001 .


13/99


13

3.

T .

F 6: B .

3.1

T F 6, CS ,

F 7, ,

. N ,

: 7 1

7 M 1.

S C1 6,7 8

6 8

C1, 1 6 8 C1, 7

, 1 ,

0 . L

.., [ ]211 ,,,, CCRRLU iiiii .

L

RL

R1R2

C1 C2

DC

S


14/99


14

iRL 7 iR1 ic1

4 6

5 ic2

2 3 8

1

iu

F 7: W CS F 6.

1 2 3 4 5 6 7 8

=

1

2

1

1

2

1

11100000

01000000

00100000

11100000

01000000

R

RL

C

C

U

R

RL

C

C

U

u

u

u

u

u

i

i

i

i

i

M 1: T ,

F 7.

W, U = , C1 = C1,

C2= C2, RL = RL,

R1= R1,

N ,

KCL K C L ,


15/99


15

7

4

5 L1 6 8

2

1 L2 3 L3

F 8: W KCL K C L F 6.

H ,

L1, L2, L3 .., 8, L1

3 1, 2 5 1

,

0 L1 .

A L2, 1, 2, 4 5 1

0 M 2 . S

L3 .

=

I

L

R

I

L

R

i

i

i

u

u

u 22

00010010

00011011

00010110

M 2: KCL F 8.

W, R2 ( 22 RidtdR ) = R2,

L ( Lidt

dL ) = L,

I ( Iidt

dI ) = I,


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16

F M 1 &2, M 3 88

,

M 3: F 6.

T ,

U ( uudt

d

u ) = ,

C1 ( 11 cC udtd

)= C1,

C2 ( 22 cudt

dc )= C2,

RL ( RLudt

dRL )= RL,

R1 ( 11 Rudt

dR )= R1,

R2 ( 22 Ridt

dR )= R2,

=

I

L

R

R

RL

C

C

U

I

L

R

R

RL

C

C

U

i

i

i

u

u

u

uu

u

u

u

i

i

i

ii

2

1

2

1

2

1

2

1

00010010

00011011

00010110

11100000

01000000

00100000

1110000001000000


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17

L ( Lidt

dL )= L,

I ( Iidt

dI )= .

B 3, A, B, C D ,

.

A = .

= +

A ,

=

L

C

C

idt

d

udt

d

udt

d

2

1

+++++++

+++

LRRRRRLRRRLRRRLCRRRCRRCRR

CRRRCCRRCRR

L )/()/()/(/1)/()/(1)/(1

)/(/1)/(1)/(1

2111211211

2211221221

12111121121

+

+

+

++

LRRRRL

CRRR

CRRRC

)/(/1

)/(0

)/(/10

2121

2211

12111

A C D , ,

1Ri

dt

d= [ ]1)21/(211)21/(11)21/(11 RRRRRRRRRRRRR +++ +

[ ]1)21/(210 RRRRR +

T A, B, C D .

L

C

C

u

i

i

2

1

++++

+++

+++

)/(11)/()/(1

)/()/(1)/(1

)/(1)/(1)/(1

21211211

2112121

2112121

RRRRRRRRRRR

RRRRRRR

RRRRRRR

L

L

C

C

i

u

u

2

1

+

+

++

)/(1

)/(0

)/(10

2121

211

211

RRRR

RRR

RRR

Ii

v


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18

S:

S M:

++++

+++

+++

=

LRRRRRLRRRLRRRL

CRRRCRRCRR

CRRRCCRRCRR

A

L )/()/()/(/1

)/()/(1)/(1

)/(/1)/(1)/(1

2111211211

2211221221

12111121121

+

+

++

=

LRRRRL

CRRR

CRRRC

B

)/(/1

)/(0

)/(/10

2121

2211

12111

=C [ ]1)21/(211)21/(11)21/(11 RRRRRRRRRRRRR +++

[ ]1)21/(210 RRRRRD +=

3.2

T

. T .

I P C, P I

D PID C.

T

. T

,

.

3.3

T . T

, 1 2

+=

+=

DuCxy

BuAxx&


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19

. A ,

.

A T, 1.

T .

21 )1( sddssaverage +=

A ,

ON OFF .

S .

W B1= B2=

+

+

++

)/(0

)/(0

)/(10

2121

211

211

RRRR

RRR

RRR

A , ))(1()( 22211211 BBdBBdB +++=

N B11, B12, B21 B22 ,

B11= B12= B21= B22=

Substituting the above 22211211 ,, andBBBB , Matrixs in the above equation 19, then

B = d + +

+

+

++

LRRRRL

CRRR

CRRRC

)/(/1

)/(0

)/(/10

2121

2211

12111

0/1

00

00

L

+

+

++

LRRRR

CRRR

CRRRC

)/(0

)/(0

)/(/10

2121

2211

12111

00

00

00

+

+

++

)/(0

)/(0

)/(10

2121

211

211

RRRR

RRR

RRR

0/1

0000

L

+

+

++

LRRRR

CRRRCRRRC

)/(0

)/(0)/(/10

2121

2211

12111

0I

VS


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20

(1-d) +

=d + + (1-d) +

After multiplying the above matrix with d and [1-d], then simplifying the above equation, by

takingd adjacent matrix as common,

= d +

=

The averaged system is:

A ,

++

+

++

=

LRRRRL

CRRR

CRRRC

idt

d

udt

d

udt

d

L

C

C

)/(/1

)/(0

)/(/10

2121

2211

12111

2

1

L

C

C

i

u

u

2

1

+

++

+

++

LRRIRRLVs

CRRIR

CRRRCI

)/()(/

)/(0

)/(/0

21021

22101

121110

0I

d

S , ,

. H 0. S

0.

00

00

00

+

+

++

)/(0

)/(0

)/(10

2121

211

211

RRRR

RRR

RRR

0I

VS

0/

00

00

LVS

+

+

++

LRRIRR

CRRIR

CRRRCI

)/(0

)/(0

)/(/0

21021

22101

121110

00

00

00

+

+

++

LRRIRR

CRRIR

CRRRCI

)/(0

)/(0

)/(/0

21021

22101

121110

0/

00

00

LVS

+

+

++

LRRIRR

CRRIR

CRRRCI

)/(0

)/(0

)/(/0

21021

22101

121110

+

+

++

+

LRRIRRLV

CRRIR

CRRRCI

S

)/(/

)/(0

)/(/0

21021

22101

121110

0I

d

+=

+=

DuCxy

BuAxx&


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21

4.

T P C P P

. W ,

.

F 9: B C S H () P C 7

T ,

7.

T

. B

, 1z 7. A

,

5 ..,

.

T 9

W , 1z

)(zC )(zD

1021 ,,, ddcc 2d

2

2

1

11)(

++= zczczC

2

2

1

10)(

++= zdzddzD

)()()()()( zYzDzYKzUzC refr =

+

kr 1/C(z)

D(z)

H(z)=B(z)/A(z)

+

-

+

V(z)

Yref(z) U(z) Y(z)

Controller Plant


22/99


22

T ,

.

T 9, ,

W

T () ,

M 2 .

S , ,

T .

T .

N =

R ,

)2()1()()2()1()()( 21021 = kydkydkydkyckyckYKku refr

)()()(

)(

)(

)(

zDzHzC

zHK

zY

zYr

ref +=

)(

)(

)( zA

zB

zH =

)(zH

)()()()(

)(

)(

)(

zDzBzCzA

zBK

zY

zYr

ref +

=

)(zP

)(zP )()()()( zDzBzCzA +

)(

)(

)(

)(

zP

zBK

zY

zYr

ref

=


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23

T .

C . H 3

B .

P =321

32223

2

azazaz

bzbzb

+++

++=

321

321

3211

21

+++

++

zazaza

zzbzb

T =

S ,

)1()( 2

2

1

1

1 ++= zczczC

)()( 221

10

1 ++= zdzddzD

P() =

M RHS R H S

=

T . T

)(

)()(

zA

zBzH =

)(zP )()()()( zDzBzCzA +

)(zC )(zD

)()()()( zDzBzCzA +

54321 ,,,, zzzzz

543

21

)2323()13221322()031221

31221()02112112()0111(1

++++++++

++++++++++++

zdbcazdbdbcacazdbdbdb

acacazdbdbacaczdbac

54343

2321543

43232121

2313032212

0221110123133

2212221111211

++++

+++++++

+++++++++

zdbzdbzdbzdbzdb

zdbzdbzdbzdbzcazcaza

zcazcazazcazcazazczc

)210)(21()211)(32112132121321 ++++++++++ zdzddzzbzbzczczazaza


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24

B ,

.

F P1, P2, P3, P4 P5 and

F P1, P2, P3, P4, P5.

F .

F F, M N .

F A2, ,

4,3,2,1 qqqq 5q

54321

11111

543211

)51)(41)(31)(21)(11()(

+++++

==

zpzpzpzpzp

zqzqzqzqzqzP

123451 qqqqqP =

213141513242524353542 qqqqqqqqqqqqqqqqqqqqP +++++++++=

321421521521431531

5414325325425433

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqP

=

432153215421543154324 qqqqqqqqqqqqqqqqqqqqP ++++=

543215 qqqqqP =

=

=

2

1

0

2

1

,

30030

23023

12312

01211

00101

d

d

d

c

c

F

ba

bbaa

bbbaa

bba

b

M


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25

T

T 55 , M

. T 1, 2, 3, 4 5 0.5, 0.5, 0.5, 0.5

0.5 .

H 0.5

PID, PD P . A

P P P .

A 236

,

1 = 0.1, 2 = 0.2, 3 = 0.3, 4 = 0.4, 5 = 0.5.

1 = 0.3, 2 = 0.4, 3 = 0.5, 4 = 0.6, 5 = 0.7

1 = 0.5, 2 = 0.5, 3 = 0.5, 4 = 0.5, 5 = 0.5

A 236

, 1 = 0.5, 2 = 0.5, 3 = 0.5, 4 = 0.5, 5 = 0.5,

, PID, PD P

.

A 8,

. H

, b N, N

10 20 N = 20 b= 9,

.5 s 16.1

8,

++++

+

++++++++++

+

=

54321

43215321542154315432

)321421521521431531

541432532542543(3

213141513242524353542

)12345(1

qqqqq

qqqqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqa

qqqqqqqqqqqqqqqqqqqqa

qqqqqa

N

1021 ,,, ddcc 2d NMF = 1


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26

=

1 0.73

2 0.61

3 0.54

4 0.49

5 0.44

T 1: P P M .

=

bNh

2

W = ,

N = 20,

b = B 9.

A ,

=

kh

9*2*20

2

6

4

= eh

S 6

4 e , P

M 8,

3.3.

4.1

A P C,

S M. T

F 9. T , F 9, S F 10.


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27

4.2

H

F 10.

F 10: P C

T F 10, 3

, B()/A() () ()

2

2

1

11)(

++= zczczC , 221

10)( ++= zdzddzD .

4.3

T M

.

y1

To Workspace7

d1

To Workspace1

Step1

Kr

Gain2

dpoly(z)

1Discrete Filter2

Bdtf(z)

Adtf(z)

Discrete Filter1

1

cpoly(z)

Discrete FilterAdd


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28

4.3.1

F 11: D .

T . A ,

,

.

F F 11, ,

18

. T 3.3V

.

T ,

. F S , F

11.

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

4

X= 42

Y= 0.27488

Sampling sequence k

O

utputvoltage

Discrete model

X= 42

Y= 3.3037

Vout

d


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29

F 12: C

S 3.3V, . W

3.3V. T P C .

T 50

3.3V. I .

F 12.

F 13: C T M

0 0.5 1 1.5 2 2.5 3 3.5

x 10-4

0

0.5

1

1.5

2

2.5

3

3.5

4

X: 0.000129

Y: 3.304

Time s

Outputvoltage

Continuous model

X= 0.000129

Y= 0.27488

Vout

d

Bctf(s)

Actf(s)

Transfer Fcn

yc1

To Workspace7

t

To Workspace2

dc1

To Workspace1

Step1

Kr

Gain2

dpoly(z)

1

Discrete Filter2

1

cpoly(z)

Discrete Filter

Clock

Add


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30

4.3.2

T , . I

.

F 14: S , , .

I F 14 3.38V

.., 3.38.

A 14,

. A 0.2816 = 3.38/12

, 3.3V.

T ,

.

W ,

. A .

0 1 2 3 4 5 6

x 10-4

-1

0

1

2

3

4

5

Time

Outputvoltage,

Inductorcurr

ent,Controlleroutput

Simulation result, without disturbance, without noise

Vout

d

iL/10


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31

H F 14 F 15.

F 15: PWM M C

4.3.3

I , ,

.

H, ,

0.2 0.2, 33MH.T , F. 16. A

,

Vout_N

To Workspace9

dVg_N

To Workspace8

Vout

To Workspace7

Noise

To Workspace6

dVg

To Workspace5

pwm

To Workspace4

iL

To Workspace3

ts

To Workspace2

d_N

To Workspace10

d

To Workspace1

Switch

Step2

Step1

x' = Ax+Bu

y = Cx+Du

State-Space

Saturation

Repeating

Sequence


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32

F 16: N M

T , T . T . A

F 17.

0 1 2 3 4

x 10-4

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25Noise Magnitude

Time

N

oise


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33

F 17: S , .

I , . B

, . I 3.0V3.5V . T

.

F , 15.

4.3.4

I , ,

, .

T , P C

.

0 1 2 3 4 5 6

x 10-4

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time

MeasurementNoise

Output voltage measurement with noise, without disturbance

With Noise


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34

F 18: S , ,

T 3.309 V 3.3V;

.

A 60 180 5 15,

3.309V 2.883V 3.355V

10 .

T (3.3092.883)/3.309 = 12%,

. W ,

.

0 1 2 3 4 5 6

x 10-4

-1

0

1

2

3

4

5

Time

Outputvo

ltage,

Controlleroutput,PWMw

ave

simulation results,with disturbance and without noise

Vout

d

PWM


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35

4.3.5

A 4.3.4 ,

F 19.

F 19: S , D N

A , , . A

0.280.

4.3.6

I 1,RRL 2R 1003

,

P C,

F 20. W

. W .

0 1 2 3 4 5 6x 10

-4

-1

0

1

2

3

4

5

Time

Measurem

entNoise

Output voltage measurement with noise, with disturbance

With Noise


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36

F 20: S , .

A 20, . T

3.3V,

, .

B . T

.

0 1 2 3 4 5 6

x 10-4

-1

0

1

2

3

4

5

Time

Outputvoltage,

Inductorcurrent,Controlleroutpu

t


Vout

diL/10


37/99


37

5.

H PID P P, I I, D D. A

PID .

W sensorseterror VVV = (37)

H sensorV setV

. V , , I sensorV

setV .

T K, K K

.

5.1

Consider the continuous-time PID Controller transfer function [9] as shown below,

T :

/

V PID

C. T

, PID

C 8.

I ,

. W . I

9. A

. S, .

+

++=

sN

T

sT

sTKsH

d

d

i

PID

1

11)(

sTz /)1(1

s/1

)1/( 1zTs


38/99


38

T

B E , O

PID C :

+

+

+

+==

1

1

11

11

1

)1(

1

11

)(

)()(

zNTT

T

zNTT

NT

zT

TK

zS

zRzH

sd

d

sd

s

i

sPID

H R(1) S(1) C D

.

T . P R

S ,

F 1210 ,,, srrr 2s .

F

sd

d

NTT

TG

+=

11

11

= zT

T

sT i

s

i

)1( 1= zT

TsT

s

d

d

)210()( 211 ++= zrzrrzR

)211()( 211 ++= zszszS

11 1)1(1

1

1

1

+

+=

+

=

+ zNTT

T

NTT

NT

zNT

Ts

N

T

sd

d

sd

s

s

dd


39/99


39

T

( )

+

+

1

1

11

1

1

11

zG

zT

GNT

zT

TK d

s

i

s

( )( ) ( ) ( ) ( )

( )( )11

111

11

11

1111

++

=zGzT

T

zGTzTzGTzTTzGK

i

d

sis

ii

N D P ( )( )11 11 zGzTi

N S

sd

d

NTT

TG

+=

( )( )11 11 zGzTi = ( )

+

11 11 zNTT

TzT

sd

d

i

211

++

+= z

NTT

TTTzz

NTT

TTT

sd

id

i

sd

di

i

++

+=

2111 z

NTT

Tzz

NTT

TT

sd

d

sd

di

iT C

21 21

++

+

+= z

NTT

T

NTT

NTTz

sd

d

sd

sd

( )( )1112

2

1

1

21 1112

1 +=++=+

+

+

++= zszzszsz

NTT

T

NTT

NTTz

sd

d

sd

sd

N

sd

d

NTT

Tss

+== 21

+

+=

sd

sd

NTT

NTTs

21


40/99


40

N

sd

d

NTT

TG

+=

N ..

( )( ) ( ) ( ) ( )

++

d

sis

iiT

zGTzTzGTzTTzGK

11111 1111

+++

d

is

d

is

d

is

ssiiiiT

zTGNT

T

zTGNT

T

TGNTzGTTzGTzGTzTTK

211211 2

N 1z 2z

+++

+

d

is

si

d

is

i

d

is

siiT

TGNTTT

T

TGNTGTz

T

TGNTGTGTTzK 21

2

N iT

+++

+

d

s

i

s

d

s

d

s

i

s

i T

GNT

T

T

T

GNT

GzT

GNT

T

GT

GzKT 1

2

1

21

N S

sd

d

NTT

TG

+=

22

110

2

1

.

21.1

++=

+

+

+

+

+

++

+++

zrzrrT

NTNTT

TNTT

TzK

T

NT

NTT

T

T

T

NTT

T

NTT

TzK

T

NT

NTT

T

T

TK

d

s

sd

d

sd

d

d

s

sd

d

i

s

sd

d

sd

d

d

s

sd

d

i

s

W sd

d

NTT

Ts

+=1 .


41/99


41

2

2

1

10

11

2

111

1

1 211

++=

+

+

++++

+

zrzrr

T

NTssz

T

NTss

T

Tsz

T

NTs

T

T

K

d

s

d

s

i

s

d

s

i

s

F

+++=

d

s

sd

d

i

s

T

NT

NTT

T

T

TKr 10

Kr =1

++

+

d

s

i

s

sd

d

T

NT

T

T

NTT

T211

Kr =2

+

d

s

T

NTss 11 =

+

+ d

s

sd

d

T

NT

NTT

TK 1

sd

sd

NTT

NTTs

+

+=

21

sd

d

NTT

Ts

+=2

F 121,0 ,, srrr 2s di TTK ,, N

Td .

2

210

)1(

)2(

sd

d

sd

d

sd

d

NTT

T

rNTT

Tr

NTT

Tr

K

++

+

+

=

210

)1(

rrr

NTT

TK

TT sd

d

si++

+

=


42/99


42

3

20

1

+

++

+

+

=

sd

d

sd

d

sd

d

sd

NTT

TK

rNTT

Tr

NTT

T

TT

sd

d

s

sd

d

d

NTT

T

TNTT

T

N

T

+

+

=

1

5.2

T PID C 0, , , , PIDC.

T PID C

, . T

,

+ ( /) PID C.

F 21: E PID C.

F 22: E PID C.

)1( 1z

)11( 1 zs

B/A

Plant+

-

u(t) y(t)

T=R 1/S

R

r(t)


43/99


43

C D C:

321

321

)(

)()(

23

2

azazaz

bzbzb

zA

zBzH

+++

++==

)()()()()( 11111 += zRzBzSzAzP

)()( 221

10

1 ++= zrzrrzR

)1()( 221

1

1 ++= zszszS

))(21()1)(3211(

)()()()()(

22

110

32122

11

321

11111

++++++++++=

+=

zrzrrzzbzbzszszazaza

zRzBzSzAzP

T .

T 1, 2, 3, 4 5 0.5, 0.5, 0.5, 0.5 0.5

.

H 0.5 ,

PID P ..

T .

A 16

,

1 = 0.1, 2 = 0.2, 3 = 0.3, 4 = 0.4, 5 = 0.5.

1 = 0.3, 2 = 0.4, 3 = 0.5, 4 = 0.6, 5 = 0.7

1 = 0.5, 2 = 0.5, 3 = 0.5, 4 = 0.5, 5 = 0.5

A 16

, 1 = 0.5, 2 = 0.5, 3 = 0.5, 4 = 0.5, 5 = 0.5,

, . A .


44/99


44

5.3

T F 23,

F 23: D

W ,

. W 3.3 2.904, 0.3

.

T 20 . T

20

2.9V 3.3V.

T D PID C ;

. W ,

. A .

T D S PID C, .

0 20 40 60 80 100 120-15

-10

-5

0

5

10

15

20

25

30

X= 33

Y= 0.26569

Sampling sequence k

Outputvoltage

Discrete model

X= 33

Y= 3.1939

Vout

d


45/99


45

F 24: D T M S.

F 25: C

S 3.3V, . W

3V.

y1

To Workspace7

d1

To Workspace1

Step1

rpoly(z)

spoly(z)

Discrete Filter2

Bdtf(z)

Adtf(z)

Discrete Filter1Add

0 0.5 1 1.5 2 2.5 3 3.5

x 10-4

-15

-10

-5

0

5

10

15

20

25

X: 8.7e-005

Y: 2.904

Time s

O

utputvoltage

Continuous model

X= 8.7e-005

Y= 0.23905

Vout

d


46/99


46

T 50

3.3V.

T C S PID C, .

F 26: C T M S.

5.4 T . F

.

F 27: S .

Bctf(s)

Actf(s)

Transfer Fcn

yc1

To Workspace7

t

To Workspace2

dc1

To Workspace1

Step1

rpoly(z)

spoly(z)

Discrete Filter2

Clock

Add

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

-1

0

1

2

3

4

5

X: 5.84e-005

Y: 3.023

Time

Outputvoltage,

Inductorcurrent,Controlleroutput


X: 5.84e-005

Y: 0.1827

Vout

d

iL/10


47/99


47

I , 3

.. 3.3. A

0.182 = 3.023/12 .

T PWM S .

F 28: PWM S

5.5 I , ,

, . H,

.

T 33MH 0.2 0.2,

; F 30

, .

Vout_N

To Workspace9

dVg_N

To Workspace8

Vout

To Workspace7

Noise

To Workspace6

dVg

To Workspace5

pwm

To Workspace4

iL

To Workspace3

ts

To Workspace2

d_N

To Workspace10

d

To Workspace1

Switch

Step2

Step1

x' = Ax+Bu

y = Cx+Du

State-Space

Saturation

Repeating

Sequence


48/99


48

F 29: S

T , .., . S

.

T

F 30.

0 1 2 3

x 10-4

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25Noise Magnitude

Time

Noise


49/99


49

F 30: .

I , ,

. W

3.0V3.5V ,

.

5.6

I , ,

; .

T , PID C . S 515 F 30.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

-1

0

1

2

3

4

5

Time

MeasurementNoise


With Noise


50/99


50

F 31: S

T 3.018 V

3.3V; . T 5 15, 3.018V 2.617V.

I 3.001V 10

, .

5.7

A , 5.6

F 32.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

-1

0

1

2

3

4

5

X: 4.617e-005

Y: 3.018

Time

Output

voltage,

Inductorcurrent,Controlle

routput

Simulation result, with disturbance, with noise

X: 5.349e-005

Y: 2.617

X: 4.617e-005

Y: 0.1698

X: 7.154e-005

Y: 0.1617

X: 7.154e-005

Y: 3.001

Vout

d

iL/10


51/99


51

F 32: S

A , , .T

0.280 .

T , . F

, F 33.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

-1

0

1

2

3

4

5

Time

MeasurementNoise

Output voltage measurement with noise, with disturnance

With Noise


52/99


52

F 33: S , .

A , , 33

. W PID

3.3. W

.

0 0.5 1 1.5 2 2.5 3 3.5

x 10-4

0

0.5

1

1.5

2

2.5

3

3.5

Time

Outputvoltage,

Inductorcurrent,Controlleroutput

Simulation result, with disturbance, with noise

Vout

d

iL/10


53/99


53

6.

I C ,

sensorseterror VVV = PD

.

PD ,

.

P . T

. A,

.

D . T ,

,

.

6.1

T PD C 0, , & PD

C.

T PID C

, . T

,

+ ( /) PD C.

)1( 1z

)11( 1 zs


54/99


54

F 34: E PD C.

F 35: E PD C.

H PD C ,

+

+

sN

TsTk

d

d

1

)1(

A T .


55/99


55

+

=

1

1

1

12

z

z

hs

S ,

K

1

1

1

1

1

121

1

121

+

+

+

+

z

z

hN

T

z

z

hT

d

d

K

)1(

21

)1(2

1

11

11

++

++

zhN

Tz

zh

Tz

d

d

K1

1

)2

1()2

1(

)2

1()2

1(

++

++

zhN

T

hN

T

zh

Th

T

dd

dd

)1(1)(0)1(1)( = kydkydkucku

)10)(()11)((11

+=+ zddzyzczu

F , 1, 0, 1

PD .

6.2

Computation of the coefficients of the Digital Controller:

=

)(

)()(

1

11

zA

zBzH

++

+=

21

21

211

21

zaza

zbzb


56/99


56

T B (1

) & A (1

) PD

)()()()()( 11111 += zDzBzCzAzP

E

= )211(21

++ zaza ++ )11( 1zc )21( 21 + zbzb )10(

1+ zdd

C (1

) & D (1

) PD T .

( ) ( )322132211 1202110112211111 +++++++++ zdbzdbzdbzdbzcazazcazazc

I 1

, 2

, 3

,

( ) ( ) ( )1212021121101111 321 dbcazdbdbacazdbacz +++++++++=

F ,2,1 pp 3p ,2,1qq 3q

321

111

3211

)31)(21)(11()(

+++=

=

zpzpzp

zqzqzqzP

T ,2,1pp 3p

1231 qqqP =

2131322 qqqqqqP ++=

3213 qqqP =

I () .

W1 = 0111 dbac ++

2 = 0211211 dbdbaca +++

3 = 1212 dbca +

F 1, 2 3; 0, 1 1 .

0 =

1

111

b

acp, 1 =

2

123

b

cap 1 0 1


57/99


57

2 ,

2 = 0211211 dbdbaca +++

S .

++

+=

22

22

221211

31)11(2)22(211

babbba

pbpabapbbc

F .

T .

A 26

,

1 = 0.48, 2 = 0.46, & 3 = 0.47

1 = 0.21, 2 = 0.24, & 3 = 0.25

1 = 0.38, 2 = 0.36, & 3 = 0.37

A 26

, 1 = 0.38, 2 = 0.36, & 3 = 0.37,

, . A

.


58/99


58

6.3

F 36: D PD , .

I ,

. A

.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-5

0

2

4

6

8

10

12

Time

controlledinputvo

ltage,controlleroutput,PWMwave

Discrete PD controlled Buck, switched continuous model with saturation

dVg

d

PWM


59/99


59

F: 37 , .

I 5.4, 5. A

, PD .

T 1.3 . W ,

1.3

5V.

0 1 2

x 10-4

0

1

2

3

4

5

6

X= 0.000128

Y= 0.45416

Time s

Outputvoltage

Continuous modelX: 0.000126

Y: 5.425 Vout

d


60/99


60

F 38: PD .

I .

T 63 . W ,

5V.

T ,

. W ,

. A .

0 20 40 60 80 100 1200

1

2

3

4

5

6

Sampling sequence k

Outputvoltage

Discrete model

Vout

d


61/99


61

6.4

I , ,

, . H,

.

T 5.3 5.5, .

39: , .

6.5

H , ,

.

2 3 4 5 6 7 8 9 10 11

x 10-5

3.5

4

4.5

5

5.5

6

X: 4.98e-005

Y: 5.33

Time

MeasurementNoise


X: 9.284e-005

Y: 5.58

With Noise


62/99


62

T

. S 5A15A.

.

F 40: , .

T 5.44 V

5V, .

A 60 180 5 15,

, 5.4

.

0 1 2

x 10-4

0

2

4

6

8

10

12

X: 0.000147

Y: 1.906

Time

Outputvoltage,

Inductorc

urrent,Controlleroutput

Simulation result, with disturbance, without noise

X: 0.0001472

Y: 0.4818

X: 0.0001442

Y: 5.429

X: 2.18e-005

Y: 10.71

X: 2.384e-005

Y: 5.465

X: 2.188e-005

Y: 0.8957

Vout

d

iL/10


63/99


63

7.

I , E BMR450 DC

DC B C. A S

, ,

. H

, M

.

T .., P P P, PID

PD. W

B()/A() 1, 2, 0, 1 2. B PID ,

, . A

C D .., 0, 1, 2, 1 2 PID,

1, 2, 0, 1 2 PID 0, 1, 2, 1 2. A PD

,

1, 0 1.

A , PID PD .

T ,

PID PD

. T P

PID PD C.

I 4 , T

PID PD .

7.1

T RLS R L

S , LMS L M S . A PID, PD C

. T

.


64/99


64


65/99


65

8.

1 R W. E D M, F , 2 .,

N : K A P, 2004, . 12.

2 A D A G, "S ", EE, NIT, R. A: ://..//19037931/BC,

25J2012.

3 R W. E, DCDC P C, A W E E

E E. A:

4 C, N C, T K, "DCDC CA P

M LP E S", C A D I

C S, IEEE T, V. 26, . 1367 1381, A2007.

5 S , M , E E

E T, B I T BTH, ING/S

E.

://..//./042330900125600325988/6447896169

912578700388518!OD 25J2012.

6 K.A T.H; PID C T, D T, 1995;

T I S A S, ISBN10:1556175167.

7 B S; A ; 1988; S ; L,S; ISBN: 9144266014.

8 B L/ B T,A D C E

E

, , SISBN: 9144269412.. 138141.

9 L D. L, G ; ISBN10: 1846280559, 2006, .

106115.


66/99


66


67/99


67

1

.

M 4: D =A + B.

S 88, = A + B , M 4

33

C1, C2, L. A RL, R1, R2 M 4,

. T U I23

C121,28, C2 31,38, L71,78.

Li

uc

uc

2

1

= + +

= + +

=

B ,

L

C

C

00

00

00

2

1

001

000

100

L

C

C

i

u

u

2

1

011

100

100

2

1

R

R

R

i

u

u L

0

0

1

1

0

0

I

U

i

u

2

2

R

R

R

u

i

i L

011

100

100

L

C

C

i

u

u

2

1

010

100

000

2

1

R

R

R

i

u

u L

0

1

0

0

0

0

I

U

i

u

2

1

R

R

R

u

i

i L

2

1

00

0/10

00/1

R

R

RL

2

1

R

R

R

i

u

u L

=

I

L

R

R

RL

C

U

I

L

R

R

RL

C

C

U

i

i

i

u

u

u

u

u

u

u

u

i

i

i

i

i

2

1

2

1

2

1

2

1

00010010

00011011

00010110

11100000

01000000

00100000

11100000

01000000


68/99


68

= + +

S ,

=

= +

= +

= +

= + +

S , .

2

1

00

0/10

00/1

R

R

RL

2

1

R

R

R

i

u

u L

011

100

100

L

C

C

i

u

u

2

1

010

100

000

2

1

R

R

R

i

u

u L

0

1

0

0

0

0

I

U

i

u

010

100

000

2

1

00

0/10

00/1

R

R

RL

2

1

R

R

R

i

u

u L

011

100

100

L

C

C

i

u

u

2

1

0

1

0

0

0

0

I

U

i

u

2

1

10

1/10

00/1

R

R

RL

2

1

R

R

R

i

u

u L

011

100

100

L

C

C

i

u

u

2

1

0

1

0

0

0

0

I

U

i

u

2

1

R

R

R

i

u

u L

++

++

)/(1)/(0

)/()/(0

00

21211

2112121

RRRRR

RRRRRRR

RL

011

100

100

L

C

C

i

u

u

2

1

0

1

0

0

0

0

I

U

i

u

2

1

R

R

R

i

u

u L

+++

+++

)/()/(1)/(1

)/()/()/(

00

2112121

2121211211

RRRRRRR

RRRRRRRRRR

RL

L

C

C

i

u

u

2

1

+

+

)21/(10

)21/(210

00

RRR

RRRR

I

U

i

u

L

C

C

u

i

i

2

1

001

000

100

L

C

C

i

u

u

2

1

011

100

100

2

1

R

R

R

i

u

u L

0

0

1

1

0

0

I

U

i

u


69/99


69

= +

+ +

A

= + +

+

= +

L

C

C

u

i

i

2

1

001

000

100

L

C

C

i

u

u

2

1

011

100

100

+++

+++

)/()/(1)/(1

)/()/()/(

00

2112121

2121211211

RRRRRRR

RRRRRRRRRR

RL

L

C

C

i

u

u

2

1

+

+

)21/(10

)21/(210

00

RRR

RRRR

I

U

iu

0

0

1

1

0

0

I

U

i

u

+++

+++

)/()/(1)/(1

)/()/()/(

00

2112121

2121211211

RRRRRRR

RRRRRRRRRR

RL

+

+

)21/(10

)21/(210

00

RRR

RRRR

L

C

C

u

i

i

2

1

001

000

100

L

C

C

i

u

u

2

1

+++

+++

+++

)/(11)/()/(

)/()/(1)/(1

)/()/(1)/(1

21211211

2112121

2112121

RRRRRRRRRRR

RRRRRRR

RRRRRRR

L

L

C

C

i

u

u

2

1

+

+

+

)/(0

)/(0

)/(0

2121

211

211

RRRR

RRR

RRR

I

U

i

u

0

0

1

1

0

0

I

U

i

u

L

C

C

u

i

i

2

1

++++

+++

+++

)/(11)/()/(1

)/()/(1)/(1

)/(1)/(1)/(1

21211211

2112121

2112121

RRRRRRRRRRR

RRRRRRR

RRRRRRR

L

L

C

C

i

u

u

2

1

+

+

++

)/(1

)/(0

)/(10

2121

211

211

RRRR

RRR

RRR

I

U

i

u


70/99


70

T .

T

H UR1 UC1

= +

R1 = +

+ UC1

R1 = +

A = .

= +

2

1

R

R

R

i

u

u L

+++

+++

)/()/(1)/(1

)/()/()/(

00

2112121

2121211211

RRRRRRR

RRRRRRRRRR

RL

L

C

C

i

u

u

2

1

+

+

)21/(10

)21/(210

00

RRR

RRRR

I

U

i

u

[ ])/()/()/( 2121211211 RRRRRRRRRR +++

L

C

C

i

u

u

2

1

[ ])/(0 2121 RRRR +

I

U

i

u

[ ])/()/()/(1 2121211211 RRRRRRRRRR +++

L

C

C

i

u

u

2

1

[ ])/(0 2121 RRRR +

I

U

i

u

L

C

C

u

i

i

2

1

++++

+++

+++

)/(11)/()/(1

)/()/(1)/(1

)/(1)/(1)/(1

21211211

2112121

2112121

RRRRRRRRRRR

RRRRRRR

RRRRRRR

L

L

C

C

i

u

u

2

1

+

+

++

)/(1

)/(0

)/(10

2121

211

211

RRRR

RRR

RRR

Ii

v

110 CR uuu +=


71/99


71

S :

= +

O :

+

L

C

C

i

u

u

2

1

++++

+++

+++

LRRRRRLRRRLRRRL

CRRRCRRCRR

CRRRCCRRCRR

L )/(11)/()/(/1

)/()/(1)/(1

)/(/1)/(1)/(1

21211211

2211221221

12111121121

L

C

C

i

u

u

2

1

+

+

++

LRRRRL

CRRR

CRRRC

)/(/1

)/(0

)/(/10

2121

2211

12111

Ii

v

[ ])/()/()/(1 2121211211 RRRRRRRRRR +++

L

C

C

i

u

u

2

1

[ ])/(0 2121 RRRR +

I

U

i

u


72/99


72


73/99


73

2

.

R , AC+BD P.

S .

F

2

2

1

11)(

++= zczczC

23235

132213224

031221312213

021111222

01111

dbcaP

dbdbcacaP

dbdbdbacacaP

dbdbcaacP

dbacP

+=

+++=

+++++=

++++=

++=

=

2

1

0

2

1

30030

23023

12312

01211

00101

d

d

d

c

c

ba

bbaa

bbbaa

bba

b

++++

+

++++++++++

+

54321

43215321542154315432

)321421521521431531

541432532542543(3

213141513242524353542

)12345(1

qqqqq


qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqa


qqqqqa

=

=

2

1

0

21

,

30030

23023

12312

0121100101

d

d

d

cc

F

ba

bbaa

bbbaa

bbab

M


74/99


74

++++

+

++++++++++

+

=

54321

43215321542154315432

)321421521521431531

541432532542543(3

213141513242524353542

)12345(1

qqqqq


qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqa


qqqqqa

N


75/99


75

3

.

W 1 = 0.1, 2=0.2, 3=0.3, 4=0.4, 5=0.5. A S 66

.

F 34: 0.1, 0.2, 0.3, 0.4, 0.5

66

.

A 1 = 0.3, 2 = 0.4, 3 = 0.5, 4 = 0.6, 5 = 0.7 66

, 0.5 .

0 1 2 3 4 5 6

x 10-4

-2

0

2

4

6

8

10

Time

Outputvoltage,

Controllero

utput,PWM

wave

Switched continuous model with saturation

Vout

d

PWM


76/99


76

F 35: S 0.3, 0.4, 0.5, 0.6, 0.7

66

.

F F 34 35, ,

P P P . S

0.5, 0.5, 0.5, 0.5, 0.5 66

.

0 1 2 3 4 5 6

x 10-4

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time

Outputvoltage,

Controlleroutput,PWM

wave


Vout

d

PWM


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77

4 .

M ,

T .

F P1, P2, P3, P4, P5 1, 2, 2, 4 5.

T ,

R , AC+BD P.

54343

2321543

43232121

2313032212

0221110123133

2212221111211

++++

+++++++

+++++++++

zrbzrbzrbzrbzrb

zrbzrbzrbzrbzsazsaza

zsazsazazsazsazazszs

543

21

)2323()13221322()031221

31221()02112112()0111(1

++++++++

++++++++++++

zrbsazrbrbsasazrbrbrb

asasazrbrbasaszrbas

)543211

)51)(41)(31)(21)(11()(

54321

111111

+++++

==

zpzpzpzpzp

zqzqzqzqzqzP

123451 qqqqqP =

213141513242524353542 qqqqqqqqqqqqqqqqqqqqP +++++++++=

321421521521431531

5414325325425433

qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqP

=

432153215421543154324 qqqqqqqqqqqqqqqqqqqqP ++++=

543215 qqqqqP =


78/99


78

S .

T M I F*M=J => F= (M)*J, .

23235

132213224

031221312213

021111222

01111

rbsaP

rbrbsasaP

rbrbrbasasaP

rbrbsaasP

rbasP

+=

+++=

+++++=

++++=

++=

=

2

1

0

2

1

30030

23023

12312

01211

00101

r

r

r

s

s

ba

bbaa

bbbaa

bba

b

++++

+

++++++++++

+

54321

43215321542154315432

)321421521521431531

541432532542543(3

213141513242524353542

)12345(1

qqqqq


qqqqqqqqqqqqqqqqqq

qqqqqqqqqqqqqqqa


qqqqqa


79/99


79

5

.

W 1 = 0.1, 2=0.2, 3=0.3, 4=0.4, 5=0.5. A 16

.

F 36: PID 0.1, 0.2, 0.3, 0.4, 0.5

16

.

A 1 = 0.3, 2 = 0.4, 3 = 0.5, 4 = 0.6, 5 = 0.7 16

, 0.5 .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

-1

0

1

2

3

4

5

6

Time

Outputvoltage,

Controlleroutput,PWMwave


Vout

d

PWM


80/99


80

F 37: PID 0.3, 0.4, 0.5, 0.6, 0.7

16

.

F F 36 37, ,

PID . S 0.5, 0.5,

0.5, 0.5, 0.5 16

.

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

x 10-4

-1

0

1

2

3

4

5

Time

Outputvoltage,

Controlleroutput,PWM

wave


Vout

d

PWM


81/99


81

6

2 =

+

++

1

111

22

123

1211 b

acp

bb

cap

baca

2=

+++

21

212112121311221211 22222

bb

babcpbcabpbbbabbca

( ) 22222 2112311222211211212 bapbpbbbababbbacbbp +++=

+

+=

22

22

221211

)11(2311222211

babbba

pabpbbbapbbc

++

+=

22

22

221211

31)11(2)22(211

babbba

pbpabapbbc


82/99


82


83/99


83

7

.

I , 7.7 ,

0.48, 0.46 & 0.47 26

,

100

,

5.

0 1 2

x 10-4

0

1

2

3

4

5

6

7

8

Time

Outputvoltage

,Controlleroutput,PWMwave


Voutd

PWM


84/99


84

I 0.21, 0.24 & 0.25 26

,

0 20 40 60 80 100 1200

1

2

3

4

5

6

Sampling sequence k

Outputvoltage

Discrete model with saturation

Vout

d

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

x 10-5

0

2

4

6

8

10

12

Time

controlledinputvo

ltage,

controlleroutput,PWM

wave

Discrete polynomial controlled Buck, switched continuous model with saturation

dVg

d

PWM


85/99


85

I , ,

.

H 3.9,

.

T ,

, .

0 1 2

x 10-4

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

Time

Outputvoltage,

Controlleroutput,PWMwave


Vout

d

PWM


86/99


86

0 20 40 60 80 100 1200

0.5

1

1.5

2

2.5

3

3.5

4

Sampling sequence k

Outputvoltage

Discrete model with saturation

Vout

d


87/99


87

8

%% BMR450 Simple controller

clear all

close all

clc

%% DC-DC Converter System Design

L = 0.9e-6;

C1 = 150e-6; %150e-6;

RL = 10e-3;

R1 = 5e-3;

%Load resistor

R2 = 100e-3;

% Load capacitor.

C2=300e-3;

Vg = 12;

I_intial = 5; % Load Current; 10

I_final = 15;

% Sampling interval for the controller

h = 6e-6; %/100;%20

% pwm signal frequency

fs = 1/h;

% Samplinginterval for the pwm generator

hpwm = h/100;

% Set point

yRef = 3.3;

I_steptime =h;

%noise = 1e-10;

noise = 0;

% Model single input

A = [-1/(C1*(R1+R2)) 1/(C1*(R1+R2)) ((1/C1)-(R1/(C1*(R1+R2))))

;1/(C2*(R1+R2)) -1/(C2*(R1+R2)) R1/(C2*(R1+R2));(-1/L)+(R1/(L*(R1+R2))) -

R1/(L*(R1+R2)) (-RL)-((R1*R2)/(L*(R1+R2)))];

% Input signals duty cycle and load current

B1 = [0 (-1/C1) R1/(C1*(R1+R2));0 -R1/(C2*(R1+R2));1/L (R1*R2)/(L)*(R1+R2)];%

% With power supply

B0 = [ 0;0;0]; % Without power supply

B = B1*Vg; %averaged system, the control signal is duty cycle d

C = [(1-R1)/R1*(R1+R2)) (R1/R1*(R1+R2)) (R1+R2)/R1*(R1+R2)];D = [0 - (R1*R2)/R1*(R1+R2);];

%Model

A2[-1/(C1*(R1+R2)) 1/(C1*(R1+R2)) ((1/C1)-(R1/(C1*(R1+R2)))) ;1/(C2*(R1+R2))

-1/(C2*(R1+R2)) R1/(C2*(R1+R2));(-1/L)+(R1/(L*(R1+R2))) -R1/(L*(R1+R2)) (-

RL)-((R1*R2)/(L*(R1+R2)))];

B2=[0 (-1/C1) R1/(C1*(R1+R2));0 -R1/(C2*(R1+R2));1/L (R1*R2)/(L)*(R1+R2)];

C2=[(1-R1)/R1*(R1+R2)) (R1/R1*(R1+R2)) (R1+R2)/R1*(R1+R2)];

D2=[0 -(R1*R2)/R1*(R1+R2)];


88/99


88

% Initial values

uC0 = 0;

uC20 = 0;

iL0 = 0;

% Continuous system

Gcss = ss(A,B,C,D);

[Ac,Bc,Cc,Dc]=ssdata(Gcss)

Gctf=tf(Gcss);

[Bctf,Actf]=tfdata(Gctf(1),'v')

% Discrete system

Gdss=c2d(Gcss,h,'zoh');

[Ad,Bd,Cd,Dd]=ssdata(Gdss)

Gdtf=tf(Gdss);

[Bdtf,Adtf]=tfdata(Gdtf(1),'v')

% polynomial coefficients, index means # delays

Bd1 = Bdtf(2);

Bd2 = Bdtf(3);

Bd3 = Bdtf(4);

Ad0 = Adtf(1);

Ad1 = Adtf(2);

Ad2 = Adtf(3);

Ad3 = Adtf(4);

b1=Bd1;

b2=Bd2;

b3=Bd3;

a0=Ad0;

a1=Ad1;

a2=Ad2;

a3=Ad3;

q1=0.5;

q2=0.5;

q3=0.5;

q4=0.5;

q5=0.5;

% F is the MATRIX containing coeficents for c1,c2,d0,d1,d2

% J is the MAtrix contianing POLES and Constants

K=[1 0 b1 0 0;a1 1 b2 b1 0;a2 a1 b3 b2 b1; a1 a2 0 b3 b2;0 a3 0 0 b3];

L=[(-a1-(q1 + q2 + q3 + q4 + q5));(-a2-(-q1*q2 -q1*q3-q1*q4-q2*q3-q1*q5-

q2*q4-q2*q5-q3*q4-q3*q5-q4*q5));(-a3+(-q3*q4*q5-q2*q4*q5-q2*q3*q5-q2*q3*q4-

q1*q4*q5-q1*q3*q5-q1*q3*q4-q1*q2*q5-q1*q2*q4 - q1*q2*q3));(q2*q3*q4*q5 +

q1*q3*q4*q5 + q1*q2*q4*q5 + q1*q2*q3*q5 + q1*q2*q3*q4);(-q1*q2*q3*q4*q5)];

% I is the resultant matrix

M=inv(K)*L;


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89

%finding values using poleplacement

c1=M(1,1);

c2 =M(2,1);

d0 =M(3,1);

d1 = M(4,1);

d2 = M(5,1);

% C polynomial and D polynomial

cpoly = [ 1 c1 c2];

dpoly = [ d0 d1 d2];

% design of Stationary Gain

% K = D(1)+A(1)C(1)/B(1)=(d0+d1)+(1+a1+a2)(1+c1)/(b1+b2)

Kr = (d0+d1+d2)+(1+a1+a2+a3)*(1+c1+c2)/(b1+b2+b3);

% Simulation

%% SimulationN = 100; % 100 sample intervals

sim('pBuckswitch',N*h); % PWM model with saturationplot

% %

figure(6)

plot(ts,Vout,ts(1:length(d)),d,ts,iL/10)

legend('Vout','d','iL/10')

grid

title('Switched continuous model with saturation')

title('Simulation result, without disturbance, without noise')

xlabel('Time'),ylabel('Output voltage, Inductor current, Controller output')

figure(7)

plot(ts,Vout,ts(1:length(d)),d,ts,pwm)

legend('Vout','d','PWM')

grid

title(simulation results,with disturbance and without noise')

xlabel('Time'),ylabel('Output voltage, Controller output, PWM wave')

figure(8)

plot(ts(1:5001),dVg(1:5001)*12,ts(1:5001),d(1:5001),ts(1:5001),pwm(1:5001))

legend('dVg','d','PWM')

title('Discrete polynomial controlled Buck, switched continuous model with

saturation')

xlabel('Time'),ylabel('controlled input voltage, controller output, PWM

wave')

figure(9)plot(ts(1:2001),d(1:2001),ts(1:2001),pwm(1:2001))

legend('d','PWM')

title('Controller output & PWM wave')

xlabel('Time'),ylabel('Controller output, PWM wave')

figure(10)

plot(ts(1:10000),Noise(1:10000))

title('Noise Magnitude')

xlabel('Time'),ylabel('Noise')


90/99


90

%

figure(11)

plot(ts(1:10000),Vout_N(1:10000))

legend('Without Noise')

title('Output voltage measurement without noise, without disturbance')

xlabel('Time'),ylabel('Measurement Noise')


91/99


91

9

%% BMR450 Simple controller (Digital PID 1 controller)clear allclose allclc

%% DC-DC Converter System Design

L = 0.9e-6;C1 =30e-6 ;%30e-6;RL =2e-3 ;%2e-3;%50e-3;R1 = 10e-3; %10e-3;%50e-3;

%Load resistorR2 = 100e-3;%100e-3;%20e-3;

% Load capacitor.C2=2e-3;%2e-3;

Vg = 12;

I_intial = 5; % Load Current; 10I_final = 15;

% Sampling interval for the controllerh =1*1e-6; %1*1e-6; %/100;%20

% pwm signal frequencyfs = 1/h;

% Samplinginterval for the pwm generatorhpwm = h/100;

% Set pointyRef = 3.3;

I_steptime =h;%noise = 1e-10;

noise = 0;

% Model single inputA = [-1/C1*(R1+R2) (1/C1)*(R1+R2) (1/C1)-R1/C1*(R1+R2);

(1/C2)*(R1+R2) (-1/C2)*(R1+R2) (R1/C2)*(R1+R2);(-1/L)+R1/L*(R1+R2) (-R1/L)*(R1+R2) (-RL)-(R1*R2)/L*(R1+R2)];

% Input signals duty cycle and load currentB1 = [0 (-1/C1) + (R1/C1)*(R1+R2);

0 (-R1/C2)*(R1+R2);

1/L (R1*R2/L)*(R1+R2)];With power supply


92/99


92

B0 = [0; 0; 0]; % Without power supply

B = B1*Vg; %averaged system, the control signal is duty cycle d

C = [1-R1/R1*(R1+R2) R1/R1*(R1+R2) (R1+R1)/R1*(R1+R2)];

D = [0 - (R1*R2)/R1*(R1+R2)];

%ModelA2= [-1/C1*(R1+R2) (1/C1)*(R1+R2) (1/C1)-R1/C1*(R1+R2);

(1/C2)*(R1+R2) (-1/C2)*(R1+R2) (R1/C2)*(R1+R2);(-1/L)+R1/L*(R1+R2) (-R1/L)*(R1+R2) (-RL)-(R1*R2)/L*(R1+R2)];

B2= [0 (-1/C1) + (R1/C1)*(R1+R2);0 (-R1/C2)*(R1+R2);

1/L R1*R2/L*(R1+R2)];

C2= [1-R1/R1*(R1+R2) R1/R1*(R1+R2) (R1+R1)/R1*(R1+R2)];

D2= [0 - (R1*R2)/R1*(R1+R2)];

% Initial valuesuC0 = 0;uC20 = 0;iL0 = 0;

% Continuous systemGcss = ss(A,B,C,D);[Ac,Bc,Cc,Dc]=ssdata(Gcss)Gctf=tf(Gcss);[Bctf,Actf]=tfdata(Gctf(1),'v')

% Discrete systemGdss=c2d(Gcss,h,'zoh');[Ad,Bd,Cd,Dd]=ssdata(Gdss)Gdtf=tf(Gdss);[Bdtf,Adtf]=tfdata(Gdtf(1),'v')

% polynomial coefficients, index means # delays

Bd1 = Bdtf(2);

Bd2 = Bdtf(3);Bd3 = Bdtf(4);Ad0 = Adtf(1);Ad1 = Adtf(2);Ad2 = Adtf(3);Ad3 = Adtf(4);

b1=Bd1;b2=Bd2;


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93

b3=Bd3;a0=Ad0;a1=Ad1;a2=Ad2;a3=Ad3;

q1=0.5;q2=0.5;q3=0.5;q4=0.5;q5=0.5;

% F is the MATRIX containing coeficents for c1,c2,d0,d1,d2% J is the MAtrix contianing POLES and Constants

%J=[(-a1+1-(q1 + q2 + q3 + q4 + q5));(-a2+a1-(-q1*q2 -q1*q3-q1*q4-q2*q3-

q1*q5-q2*q4-q2*q5-q3*q4-q3*q5-q4*q5));(-a3+a2+(-q3*q4*q5-q2*q4*q5-q2*q3*q5-

q2*q3*q4-q1*q4*q5-q1*q3*q5-q1*q3*q4-q1*q2*q5-q1*q2*q4 -

q1*q2*q3));(+a3+q2*q3*q4*q5 + q1*q3*q4*q5 + q1*q2*q4*q5 + q1*q2*q3*q5 +

q1*q2*q3*q4)];F=[1 0 b1 0 0;a1 1 b2 b1 0;a2 a1 b3 b2 b1; a1 a2 0 b3 b2;0 a3 0 0 b3];

J=[(-a1-(q1 + q2 + q3 + q4 + q5));(-a2-(-q1*q2 -q1*q3-q1*q4-q2*q3-q1*q5-

q2*q4-q2*q5-q3*q4-q3*q5-q4*q5));(-a3+(-q3*q4*q5-q2*q4*q5-q2*q3*q5-q2*q3*q4-

q1*q4*q5-q1*q3*q5-q1*q3*q4-q1*q2*q5-q1*q2*q4 - q1*q2*q3));(q2*q3*q4*q5 +

q1*q3*q4*q5 + q1*q2*q4*q5 + q1*q2*q3*q5 + q1*q2*q3*q4);(-q1*q2*q3*q4*q5)];

M=inv(F)*J;

%finding values using poleplacements1 = M(1,1);

s2 = M(2,1);r0 = M(3,1);r1 = M(4,1);r2 = M(5,1);

% C polynomial and D polynomialspoly = [ 1 s1 s2];% spoly1=1;

rpoly = [ r0 r1 r2];

% Simulation

%% SimulationN = 1*100; % 100 sample intervals% sim('pidBuckd1',N); % Discrete time model without saturation% sim('pidBuckd2',N); % Discrete time model with saturation% sim('pidBuckc1',N*h); % Continuous time modelsim('pidBuckswitch',N*h); % PWM model with saturationplot

figure(6)plot(ts,Vout,ts(1:length(d)),d,ts,iL/10)legend('Vout','d','iL/10')


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gridtitle('Switched continuous model with saturation')title('Simulation result, with disturbance, with noise')xlabel('Time'),ylabel('Output voltage, Inductor current, Controller output')

figure(7)

plot(ts,Vout,ts(1:length(d)),d,ts,pwm)legend('Vout','d','PWM')gridtitle('Switched continuous model with saturation')xlabel('Time'),ylabel('Output voltage, Controller output, PWM wave')

figure(8)plot(ts(1:1001),dVg(1:1001)*12,ts(1:1001),d(1:1001),ts(1:1001),pwm(1:1001))legend('dVg','d','PWM')title('Discrete PID controlled Buck, switched continuous model with

saturation')xlabel('Time'),ylabel('controlled input voltage, controller output, PWM

wave')

figure(9)plot(ts(1:5001),d(1:5001),ts(1:5001),pwm(1:5001))legend('d','PWM')title('Controller output & PWM wave')xlabel('Time'),ylabel('Controller output, PWM wave')

figure(10)plot(ts(1:10000),Noise(1:10000))title('Noise Magnitude')xlabel('Time'),ylabel('Noise')

figure(11)plot(ts(1:10000),Vout_N(1:10000))legend('With Noise')title('Output voltage measurement with noise, without disturbance')xlabel('Time'),ylabel('Measurement Noise')


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10

%% BMR450 Simple controllerclear allclose allclc

%% DC-DC Converter System Design%%Circuit data%% Analysis of 3rd and 2nd order Buck POL converter

%%AH 2011 Dec 16

L=0.9e-6;RL=10e-3;C1=150e-6;R1=2e-3;C2=470e-6;%C2=1000e-6;R2=10e-3;

Vg=12;

A3 = [-1/(C1*(R1+R2)) 1/(C1*(R1+R2)) (1/C1)-(R1/(C1*(R1+R2)));...1/(C2*(R1+R2)) -1/(C2*(R1+R2)) R1/(C2*(R1+R2));...(-1/L)+(R1/(L*(R1+R2))) -R1/(L*(R1+R2)) (-RL/L)+((R1*R2)/(L*(R1+R2)))];

B3 = [0 -R2/(C1*(R1+R2));...0 -R1/(C2*(R1+R2));...Vg/L (R1*R2)/(L)*(R1+R2)];

C3 = [1-R1/(R1+R2) (R1/(R1+R2)) (R1*R2)/(R1+R2)];

D3= [0 (R1*R2)/(R1+R2)];figure(1)step(A3,B3(:,1),C3,D3(1))figure(2)bode (A3,B3(:,1),C3,D3(1))

R_L=60e-3;C_1=680e-6;%%%%%------------------here you can adapt-----------------%%%%%%R_1=35e-3;%%%%%------------------here you can adapt-----------------%%%%%%

Vg = 12; % Input Voltage

I_intial = 5; % Load Current; 10I_final = 15;% Sampling interval for the controllerh = 2e-6;% pwm signal frequencyfs = 1/h;% Samplinginterval for the pwm generatorhpwm = h/100;% Set pointyRef = 5;


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I_steptime = h;% noise = 1e-10;noise = 0;

A2 = [0 1/C_1;-1/L -(R_L+R_1)/L];

B2 = [0 -1/C_1;Vg/L R_1/L];

C2 = [ 1 R_1 ; 0 1 ];

D2 = [ 0 0 ; 0 0 ];% Model two input in state spaceAA = A2;BB = B2;CC = C2;DD = D2;

% Initial valuesuC0 = 0;iL0 = 0;

% Continuous systemGcss = ss(A2,B2(:,1),C2(1,:),D2(1,1));[Ac,Bc,Cc,Dc] = ssdata(Gcss)Gctf = tf(Gcss);[Bctf,Actf] = tfdata(Gctf,'v')

% Discrete systemGdss = c2d(Gcss,h,'zoh');[Ad,Bd,Cd,Dd] = ssdata(Gdss)Gdtf = tf(Gdss);[Bdtf,Adtf] = tfdata(Gdtf,'v')

% plot of step response% figure(1)% step(Gctf)% hold on% step(Gdtf)% hold off

% polynomial coefficients, index means # delaysBd0 = Bdtf(1);Bd1 = Bdtf(2);Bd2 = Bdtf(3);Ad0 = Adtf(1);Ad1 = Adtf(2);Ad2 = Adtf(3);

%% Polynomial Controller Design a la Schmidtbauerb1 = Bd1;


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b2 = Bd2;a1 = Ad1;a2 = Ad2;% Pole placementq1 = 0.38;q2 = 0.36;q3 = 0.37;

%polynomial controller derivation using method from Schmidtbauerp1 = -q1-q2-q3;p2 = q1*q2 + q2*q3 + q1*q3;p3 = -q1*q2*q3;

% p1 = a1 + c1 + b1*d0% p2 = a1*c1+ a2 + b1*d1 + b2*d0% p3 = a2*c1 + b2*d1

% polynomial coefficientsc1 = (-b1^2*p3 + b1*b2*(p2-a2) + b2^2*(p1-a1))/(a2*b1^2 + b2^2 + a1*b1*b2);d0 = (p1-a1-c1)/b1;

d1 = (p3-a2*c1)/b2;

a1a2b1b2

c1d0d1

% C polynomial and D polynomialcpoly = [ 1 c1 ];dpoly = [ d0 d1 ];

% design of Stationary Gain% Kr = D(1)+A(1)C(1)/B(1)=(d0+d1)+(1+a1+a2)(1+c1)/(b1+b2)Kr = (d0+d1)+(1+a1+a2)*(1+c1)/(b1+b2);

% Calculate the static gain% Definition of continuous system:Csys=ss(AA,BB,CC,DD);% Discretizing the system:Dsys=c2d(Csys,h,'zoh');% Definition of the transfer function for the controller:

Rsys=tf(dpoly,cpoly,h);% Calculation of the closed system:Msys=feedback(Dsys,Rsys,1,1,-1);% the closed system has two inputs (d and I0) and two outputs (u0 and IL).% Generation of all four step responsesstep(Msys)% calculation of all zeros, poles and static gains.[z,p,k,Ts]=zpkdata(Msys)% calculation of all transfer functionHz=tf(Msys)


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%% SimulationN = 100; % 100 sample intervalssim('PDCBuckd1',N); % Discrete time model without saturationsim('PDCBuckd2',N); % Discrete time model with saturation

sim('PDCBuckc1',N*h); % Continuous time modelsim('PDCBuckswitch',N*h); % PWM model with saturationplot

figure(2)stairs([y1 d1])legend('Vout','d')title('Discrete model')xlabel('Sampling sequence k'),ylabel('Output voltage')

figure(3)td=0:h:h*(length(y1)-1);stairs(td,y1)hold on, stairs(td,d1),hold off

legend('Vout','d')title('Discrete model without saturation')xlabel('Time s'),ylabel('Output voltage')

figure(4)stairs([y2 d2])legend('Vout','d')title('Discrete model with saturation')xlabel('Sampling sequence k'),ylabel('Output voltage')

figure(5)plot(t,yc1)dt=t(end)/(length(dc1)-1);

tt=0:dt:t(end);hold on,stairs(tt,dc1),hold offlegend('Vout','d')title('Continuous model')xlabel('Time s'),ylabel('Output voltage')%

figure(6)plot(ts,Vout,ts(1:length(d)),d,ts,iL/10)legend('Vout','d','iL/10')gridtitle('Switched continuous model with saturation')title('Simulation result, with disturbance and without noise')xlabel('Time'),ylabel('Output voltage, Inductor current and Controller

output')

figure(7)plot(ts,Vout,ts(1:length(d)),d,ts,pwm)legend('Vout','d','PWM')gridtitle('Switched continuous model with saturation')xlabel('Time'),ylabel('Output voltage, Controller output, PWM wave')

figure(8)


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plot(ts(1:1001),dVg(1:1001)*12,ts(1:1001),d(1:1001),ts(1:1001),pwm(1:1001))legend('dVg','d','PWM')title('Discrete PD controlled Buck, switched continuous model with

saturation')xlabel('Time'),ylabel('controlled input voltage, controller output, PWM

wave')

figure(9)plot(ts(1:2001),d(1:2001),ts(1:2001),pwm(1:2001))legend('d','PWM')title('Controller output & PWM wave')xlabel('Time'),ylabel('Controller output, PWM wave')

figure(10)plot(ts(1:10000),Noise(1:10000))title('Noise Magnitude')xlabel('Time'),ylabel('Noise')%

figure(11)plot(ts(1:10000),Vout_N(1:10000))legend('With Noise')title('Output voltage measurement with noise, without disturbance')xlabel('Time'),ylabel('Measurement Noise')

Buck Converter bykiran_2012

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