46 2. OVERVIEW OF KHARITONOV’S THEOREM
–
+
Controller Interval Plant Fictitious Block
?(?) ? (?) ??
-j
φ
Figure 2.5: Feedback control with fictitious block Ae
j'
.
2.4 CASE STUDY: ROBUST CONTROL OF A POSITION
CONTROL SYSTEM
Figure 2.6 shows the block diagram of a position control system. A DC motor is used as an
actuator.
–
+
Controller DC Motor
?(?)
?(?) ?(?) ? (?)
?(?)
Figure 2.6: Block diagram of a position control system.
e DC motor transfer function is given by:
G
.
s
/
D
K
m
s
.
Js C b
/
.L
f
s C R
f
/
D
K
m
L
f
Js
3
C
bL
f
C JR
f
s
2
C bR
f
s
; (2.20)
where R
f
, L
f
, K
m
, b, and J show the armature resistance and inductance, motor torque con-
stant, coefficient of viscouse friction and inertia of the rotor, respectively. Consider a DC motor
with the following values in Table 2.2.
Table 2.2: Specification of the DC motor in Fig. 2.6
Nominal value Variation (%) Variation
R
f
1.5
Ω 10
%
[1.35 Ω,1.65 Ω]
L
f
15 mH 20 %
[12 mH,18 mH]
K
m
60 × 10
-3
N.m
A
10 %
[54 × 10
-3
N.m
, 66 × 10
-3
N.m
]
A A
b 2 × 10
-3
N.m.s
15 %
[1.7 × 10
-3
N.m.s, 2.3 × 10
-3
N.m.s]
J
1.7 × 10
-3
kg.m
2
40 %
[1.02 × 10
-3
kg.m
2
, 2.38 × 10
-3
kg.m
2
]
2.4. CASE STUDY: ROBUST CONTROL OF A POSITION CONTROL SYSTEM 47
e nominal transfer function is:
G
0
.
s
/
D
0:06
2:55
10
5
s
3
C
0:00258s
2
C
0:003s
: (2.21)
e lower and upper bounds of coefficients is shown in Table 2.3.
Table 2.3: Lower and upper bounds of coefficients
Lower Bound Upper Bound
K
m
54
× 10
-3
66
× 10
-3
L
f
J
12.24 × 10
-5
42.83 × 10
-5
bL
f
+ JR
f
0.0014
0.004
bR
f
0.0023
0.0038
Interval plant model of the DC motor is:
G
.
s
/
D
q
0
p
3
s
3
C p
2
s
2
C p
1
s C p
0
; (2.22)
where
q
0
2
54 10
3
; 66 10
3
;
p
3
2
12:24 10
3
; 42:83 10
3
;
p
2
2
Œ
0:0014; 0:004
;
p
1
2
Œ
0:0023; 0:0038
;
p
0
2
Œ
0; 0
:
Since the numerator is a simple constant (it has only two different representations namely q
0
and
q
0
C
), the number of Kharitonov plants associated with Equation (2.22) reduces to 8. Table 2.4
shows the Kharitonov plants.
We want to design a robust PI controller for the block diagram shown in Fig. 2.4. e
following program implements the pseudo code shown in Section 2.3.2. Figure 2.7, shows the
founded acceptable proportional and integral gains.
48 2. OVERVIEW OF KHARITONOV’S THEOREM
Table 2.4: e Kharitonov plants associated with Equation (2.22)
G
1
(s)
=
0.054
42.83 × 10
-5
s
3
+ 0.004s
2
+ 0.0023s
G
2
(s)
=
0.054
12.24 × 10
-5
s
3
+ 0.004s
2
+ 0.0038s
G
3
(s)
=
0.054
42.83 × 10
-5
s
3
+ 0.0014s
2
+ 0.0023s
G
4
(s)
=
0.054
12.24 × 10
-5
s
3
+ 0.0014s
2
+ 0.0038s
G
5
(s)
=
0.066
42.83 × 10
-5
s
3
+ 0.004s
2
+ 0.0023s
G
6
(s)
=
0.066
12.24 × 10
-5
s
3
+ 0.004s
2
+ 0.0038s
G
7
(s)
=
0.066
42.83 × 10
-5
s
3
+ 0.0014s
2
+ 0.0023s
G
8
(s)
=
0.066
12.24 × 10
-5
s
3
+ 0.0014s
2
+ 0.0038s
%This program finds the proportional gain and integral gain
%values to stabilize all the 8 Kharitonov plant
clc
%Search interval
Kpmin =0;
Kpstep =.001;
Kpmax =.16;
Kimin =0;
Kistep =.0005;
Kimax =.05;
%Kharitonov plants
G1=tf (.054 ,[42.83e -5 .004 .0023 0]);
G2=tf (.054 ,[12.24e -5 .004 .0038 0]);
G3=tf (.054 ,[42.83e -5 .0014 .0023 0]);
2.4. CASE STUDY: ROBUST CONTROL OF A POSITION CONTROL SYSTEM 49
G4=tf (.054 ,[12.24e -5 .0014 .0038 0]);
G5=tf (.066 ,[42.83e -5 .004 .0023 0]);
G6=tf (.066 ,[12.24e -5 .004 .0038 0]);
G7=tf (.066 ,[42.83e -5 .0014 .0023 0]);
G8=tf (.066 ,[12.24e -5 .0014 .0038 0]);
%Acceptable values of proportional gain and integral gains are
%stored in variables KP and KI respectively.
KP =0;
KI =0;
%n is used to show the progress of the loop to the user.
%N is the total number of iterations
n=0;
N=length ( Kpmin :Kpstep :Kpmax )*length(Kimin : Kistep:Kimax);
for Kp=Kpmin : Kpstep:Kpmax
for Ki=Kimin : Kistep:Kimax
n=n+1;
disp(' percentage of work done: ')
disp (100* n/N)
C=tf([ Kp Ki ],[1 0]);
P11=pole(feedback(C*G1 ,1));
P12=pole(feedback(C*G2 ,1));
P13=pole(feedback(C*G3 ,1));
P14=pole(feedback(C*G4 ,1));
P21=pole(feedback(C*G5 ,1));
P22=pole(feedback(C*G6 ,1));
P23=pole(feedback(C*G7 ,1));
P24=pole(feedback(C*G8 ,1));
NumberOfUnstablePoles =sum(real (P11) >0)+sum(real(P12) >0) ...
+sum( real(P13) >0)+sum(real(P14) >0)+sum(real(P21) >0) ...
+sum( real(P22) >0)+sum(real(P23) >0)+sum(real(P24) >0);
if NumberOfUnstablePoles ==0
KP=[KP;Kp];
KI=[KI;Ki];
50 2. OVERVIEW OF KHARITONOV’S THEOREM
end
end
end
plot(KP ,KI ,'.'),grid minor
xlabel('Kp ')
ylabel('Ki ')
0.04
0.035
0.03
0.025
0.02
0.015
0.01
0.005
0
0 0.02 0.04 0.06 0.08 0.1 0.12
Kp
Ki
Figure 2.7: Acceptable proportional and integral gains.
We can obtain the same result using another technique. We calculate the stabilizing region
for each of the Kharitonov plants separately and intersect the obtained results to obtain the
stabilizing region.
A transfer function G
.
s
/
D
N.s/
D.s/
can be transformed into the following form [Bhat-
tacharyya et al., 1995]:
G
.
j!
/
D
N.j!/
D.j!/
D
N
e
!
2
C j! N
o
!
2
D
e
.
!
2
/
C j! D
o
.
!
2
/
; (2.23)
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