buck converter bykiran_2012
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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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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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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
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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.
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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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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
,
.
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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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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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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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F 4: D ,
0 .
2.5.3
T ..,
. I , .
B ,
. T
. I ,
.
2.5.4 450
T BMR450 E
DCDC BMR 451 453,
DCDC .
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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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O .
ON/OFF .
O .
ISO 9001/14001 .
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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
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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 ,
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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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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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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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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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. 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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(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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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
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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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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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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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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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=
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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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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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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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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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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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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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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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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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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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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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
Simulation result, without disturbance, without noise
Vout
diL/10
-
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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
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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
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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
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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 .
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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++
+
=
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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)
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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 .
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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
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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
Simulation result, without disturbance, without noise
X: 5.84e-005
Y: 0.1827
Vout
d
iL/10
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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
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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
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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
Output voltage measurement with noise, without disturbance
With Noise
-
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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
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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
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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
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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
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F 34: E PD C.
F 35: E PD C.
H PD C ,
+
+
sN
TsTk
d
d
1
)1(
A T .
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+
=
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
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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
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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
.
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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
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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
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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
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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
Output voltage measurement with noise, without disturbance
X: 9.284e-005
Y: 5.58
With Noise
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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
.
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-
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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.
-
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66
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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
-
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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 +=
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqa
qqqqqqqqqqqqqqqqqqqqa
qqqqqa
=
=
2
1
0
21
,
30030
23023
12312
0121100101
d
d
d
cc
F
ba
bbaa
bbbaa
bbab
M
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
74
++++
+
++++++++++
+
=
54321
43215321542154315432
)321421521521431531
541432532542543(3
213141513242524353542
)12345(1
qqqqq
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqa
qqqqqqqqqqqqqqqqqqqqa
qqqqqa
N
-
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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
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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
Switched continuous model with saturation
Vout
d
PWM
-
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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 =
-
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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
qqqqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqqqq
qqqqqqqqqqqqqqqa
qqqqqqqqqqqqqqqqqqqqa
qqqqqa
-
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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
Switched continuous model with saturation
Vout
d
PWM
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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
Switched continuous model with saturation
Vout
d
PWM
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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
-
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82
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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
Switched continuous model with saturation
Voutd
PWM
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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
Switched continuous model with saturation
Vout
d
PWM
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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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
-
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ERCS.MADHU KIRAN THOTA.PARTHA SARADHI
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)];
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% 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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%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')
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%
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')
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%% 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
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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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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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%% 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')
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