3.3. ROBUST CONTROLLER DESIGN FOR QUADRATIC BOOST CONVERTER 95
e close-loop transfer function is:
H
.
s
/
D
K
p
s C K
i
s
n
3
s
3
C n
2
s
2
C n
1
s
1
C n
0
s
4
C d
3
s
3
C d
2
s
2
C d
1
s
1
C d
0
1 C
K
p
s C K
i
s
n
3
s
3
C n
2
s
2
C n
1
s
1
C n
0
s
4
C d
3
s
3
C d
2
s
2
C d
1
s
1
C d
0
: (3.9)
After simplification the close-loop denominator obtained as:
s
5
C
d
3
C n
3
K
p
s
4
C
d
2
C n
2
K
p
C n
3
K
i
s
3
C
d
1
C n
1
K
p
C n
2
K
i
s
2
C
d
0
C n
0
K
p
C n
1
K
i
s C n
0
K
i
: (3.10)
Minimum and maximum of these coefficients are shown in Table 3.5.
Table 3.5: Minimum and maximum of close-loop coefficients (Equation (3.10))
Coeffi cient Minimum Maximum
?
3
+ ?
3
?
?
3.8059 × 10
3
− 2.0219 × 10
4
?
?
3.9932 × 10
3
− 1.2205 × 10
3
?
?
?
2
+ ?
2
?
?
+ ?
3
?
?
1.2105 × 10
8
+ 3.8177 × 10
8
?
?
− 2.0219 × 10
4
?
?
1.2176 × 10
8
+ 7.1462 × 10
8
?
?
− 1.2205 × 10
3
?
?
?
1
+ ?
1
?
?
+ ?
2
?
?
2.7068 × 10
11
− 3.9968 ×
10
12
?
?
+ 3.8177 × 10
8
?
?
2.9240 × 10
11
+ 5.8382 ×
10
10
?
?
+ 7.1462 × 10
8
?
?
?
0
+ ?
0
?
?
+ ?
1
?
?
1.1910 × 10
14
+ 1.2446 × 10
16
?
?
− 3.9968 × 10
12
?
?
1.6801 × 10
14
+ 3.1724 ×
10
16
?
?
+ 5.8382 × 10
10
?
?
?
0
?
?
1.2446 × 10
16
?
?
3.1724 × 10
16
?
?
We use this table to form the Kharitonov’s polynomials. Since the closed-loop denom-
inator is a fifth-order polynomial, we need to stabilize only K
2
.s/; K
3
.s/, and K
4
.s/ (see
Lemma 2.8).
e following code finds the suitable values of Kp and Ki which stabilize the K
2
.s/; K
3
.s/,
and K
4
.s/. Acceptable region found by the code is shown in Fig. 3.14.
clc
kp_min =0;
kp_max =.06;
kp_delta =.001;
ki_min =0;
96 3. CONTROLLER DESIGNFOR DC-DC CONVERTERSUSINGKHARITONOV’S THEOREM
ki_max =30;
ki_delta =.01;
sol =[0 0]; %sol=[kp ki]
N=(kp_max - kp_min )/kp_delta *(ki_max -ki_min)/ki_delta;
n=0;
for kp=kp_min :kp_delta:kp_max
for ki=ki_min :ki_delta:ki_max
n=n+1;
disp(' percentage of work done ')
100*n/N
disp('---')
% 5 4 3 2 1
%denominator is :s +z4*s +z3*s +z2*s +z1*s +z0
%maxsimum is shown with p
%minimum is shown with n
%For example , z4_p shows the z4 maximum .
%z4 is s^4 coefficient .
%z4_n shows the z4 minimum . z4 is s^4 coefficient.
z4_p =3.9932e3 -1.2205 e3*kp;
z4_n =3.8059e3 -20.219 e3*kp;
z3_p =1.2176 e8 +7.1462e8*kp -1.2205e3*ki;
z3_n =1.2105 e8 +3.8177e8*kp -20.219e3*ki;
z2_p =2.924 e11 +5.8382 e10*kp +7.1462 e8*ki;
z2_n =2.7065e11 -3.9968 e12*kp +3.8177 e8*ki;
z1_p =1.6801 e14 +3.1724 e16*kp +5.8382 e10 *ki;
z1_n =1.191 e14 +1.2446 e16*kp -3.9968 e12*ki;
z0_p =3.1724 e16 *ki;
z0_n =1.2446 e16 *ki;
%Since the interval polynomial is of 5th order
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