3.2. ROBUST CONTROLLER DESIGN FOR QUADRATIC BUCK CONVERTER 81
Table 3.2: Maximum and minimum of close loop coefficients (Equation (3.5))
Coeffi cient Minimum Maximum
?
3
+ ?
3
?
?
1.2452 × 10
3
+ 4.5778 × 10
3
?
?
2.8165 × 10
3
+ 4.6571 × 10
3
?
?
?
2
+ ?
2
?
?
+ ?
3
?
?
1.3134 × 10
8
+ 6.5189 × 10
8
?
?
+ 4.5778 × 10
3
?
?
1.3205 × 10
8
+ 6.6678 × 10
8
?
?
+ 4.6571 × 10
3
?
?
?
1
+ ?
1
?
?
+ ?
2
?
?
8.9629 × 10
10
+ 4.0827 ×
10
11
?
?
+ 6.5189 × 10
8
?
?
2.5067 × 10
11
+ 9.2743 ×
10
11
?
?
+ 6.6678 × 10
8
?
?
?
0
+ ?
0
?
?
+ ?
1
?
?
2.1678 × 10
15
+ 1.0069 × 10
17
?
?
+ 4.0827 × 10
11
?
?
2.1808 × 10
15
+ 1.0234 ×
10
17
?
?
+ 9.2743 × 10
11
?
?
?
0
?
?
1.0069 × 10
17
?
?
1.0234 × 10
17
?
?
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. 3.6. If we decrease the steps
(kp_delta=.0001;), we take a better view of acceptable region (Fig. 3.7).
%Robust PI controller design for quadratic buck converter
clear all
clc
kp_min =0;
kp_max =.015;
kp_delta =.001; %decrese it to .0001 if you want a take
%a better view
ki_min =0;
ki_max =10;
ki_delta =.01;
sol =[0 0]; %sol=[kp ki] keeps the acceptable gains for PI
%controller
N=(kp_max - kp_min )/kp_delta *(ki_max -ki_min)/ki_delta;
n=0;
82 3. CONTROLLER DESIGNFOR DC-DC CONVERTERSUSINGKHARITONOV’S THEOREM
nAcceptable =0; %nAcceptable keeps the number of acceptable
%solutions found.
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('---')
k1=roots ([1 2.8165e3+4.6571 e3*kp 1.3134 e8 +6.5189 e8*kp+
4.5778 e3*ki 8.9629 e10 +4.0827 e11 *kp +6.5189 e8*ki
2.1808 e15 +1.0234 e17*kp +9.2743 e11*ki 1.0234 e17*ki]);
k1_real=real (k1);
T1=sum(k1_real >0); %T1 keeps the number of unstable
%poles in k1
k2=roots ([1 2.8165e3+4.6571 e3*kp 1.3134 e8 +6.5189 e8*kp+
4.5778 e3*ki 2.5067 e11 +9.2743 e11 *kp +6.6678 e8*ki
2.1678 e15 +1.0069 e17*kp +4.0827 e11*ki 1.0234 e17*ki]);
k2_real=real (k2);
T2=sum(k2_real >0); %T2 keeps the number of unstable
%poles in k2
k3=roots ([1 1.2452e3+4.5778 e3*kp 1.3205 e8 +6.6678 e8*kp+
4.6571 e3*ki 8.9629 e10 +4.0827 e11 *kp +6.5189 e8*ki
2.1808 e15 +1.0234 e17*kp +9.2743 e11*ki 1.0069 e17*ki]);
k3_real=real (k3);
T3=sum(k3_real >0); %T3 keeps the number of unstable
%poles in k3
if ((T1+T2+T3)==0)
nAcceptable=nAcceptable +1;
disp('kp=')
kp
disp('ki=')
ki
disp(' ************** ')
3.2. ROBUST CONTROLLER DESIGN FOR QUADRATIC BUCK CONVERTER 83
sol=[sol ;[kp ki]];
end
end
end
sol(1,:) =[]; %first row is initialization [0 0]. So , we remove it.
plot(sol (:,2) ,sol(:,1),'.'), xlabel('Ki '), ylabel ('Kp '), grid
minor
%End of code
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0 1 2 3 4 5 6 7 8
Ki
Acceptable Ki and Kp
Kp
Figure 3.6: Acceptable PI coefficients (kp_delta=.001).
84 3. CONTROLLER DESIGNFOR DC-DC CONVERTERSUSINGKHARITONOV’S THEOREM
0.014
0.012
0.01
0.008
0.006
0.004
0.002
0
0 1 2 3 4 5 6 7 8
Ki
Kp
Figure 3.7: Acceptable PI coefficients (kp_delta=.0001).
e following scenario is used to test the close-loop system: input voltage’s value changes
from 42–54 V at t D 50 ms. Output load changes from 5–25 at t D 100 ms and controller
reference voltage changes from 12–20 V at
t
D
150
ms. e test scenario is summarized in
Table 3.3.
Table 3.3: Test scenario for quadratic buck converter
Change In Time Initial Value Final Value
Final – Initial
× 100
Initial
Vin 50 ms 42 54 28%
Rload 100
ms 5 25 400%
Vref 150 ms 12 20 66%
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