2.4. CASE STUDY: ROBUST CONTROL OF A POSITION CONTROL SYSTEM 51
where N
e
, N
o
, D
e
, and D
o
shows the even part of numerator, odd part of numerator, even part
of denominator and odd part of denominator, respectively.
e characteristic equation for the block diagram shown in Fig. 2.6 is .s/ D 1 C G
.
s
/
C.s/. Variable s is replaced with j! to obtain the proportional and integral gains (K
p
; K
i
> 0)
which stabilize the loop:
.
j!
/
D 1 C
N
.
j!
/
D
.
j!
/
C
.
j!
/
D 1 C
N
e
!
2
C j! N
o
!
2
D
e
.
!
2
/
C j! D
o
.
!
2
/
K
i
C jK
p
!
j!
D 0: (2.24)
.
j!
/
has two parts: a real part (
R
.
j!
/
) and an imaginary part (
I
.
j!
/
).
.
j!
/
D 0
means
R
.
j!
/
D
I
.
j!
/
D 0. is leads to the following system of equations:
!
2
N
o
!
2
K
p
C N
e
!
2
K
i
D !
2
D
o
!
2
N
e
!
2
K
p
C N
o
!
2
K
i
D D
e
.!
2
/:
(2.25)
Solution of the obtained system is:
K
p
D
!
2
N
o
!
2
D
o
!
2
C
N
e
!
2
D
e
!
2
N
e
2
.
!
2
/
C !
2
N
o
2
.
!
2
/
K
i
D
!
2
.N
e
!
2
D
o
!
2
N
o
!
2
D
e
!
2
/
N
e
2
.
!
2
/
C !
2
N
o
2
.!
2
/
:
(2.26)
Studying a numeric example is quite helpful.
For example, for
G
1
.
s
/
D
0:054
42:83 10
5
s
3
C 0:004s
2
C 0:0023s
;
G
1
.
j!
/
D
0:054
0:004!
2
C j!
.
42:83 10
5
!
2
C 0:0023
/
;
N
e
!
2
D 0:054;
N
o
!
2
D 0;
D
e
!
2
D 0:004!
2
;
D
o
!
2
D 42:83 10
5
!
2
C 0:0023;
K
p
D
0:054 0:004!
2
0:054
2
D 0:0741!
2
and
K
i
D
!
2
.0:054
42:83 10
5
!
2
C 0:0023
/
0:054
2
D 0:0079!
4
C 0:0426!
2
:
52 2. OVERVIEW OF KHARITONOV’S THEOREM
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.01
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Kp
Ki
Figure 2.8:
K
p
; K
i
D .0:0741!
2
, 0:0079!
4
C 0:0426!
2
) tupple for 0 < ! < 2:35
Rad
s
.
Figure 2.8, shows the
K
p
; K
i
D .0:0741!
2
,0:0079!
4
C 0:0426!
2
) tupple for 0 < ! <
2:35
Rad
s
. Figure 2.8 is produced with the aid of the following program.
%This program finds the acceptable range of Kp and Ki
%to stabilize G1
clc
%Desired frequency range
omega_min =0;
omega_step =.01;
omega_max =2.35;
%plant(G(s)) and its even( with subscrip e),
%odd(with subscrip o), decomposition .
%
2.4. CASE STUDY: ROBUST CONTROL OF A POSITION CONTROL SYSTEM 53
% N(jw) Ne(-w ^2)+jw.No(-w^2)
% G(jw)= ------- = ----------------------
% D(jw) De(-w ^2)+jw.Do(-w^2)
%
% -w^2. No.Do-Ne.De
% Kp = -------------------
% Ne ^2+w ^2.No ^2
%
% w^2.(Ne.Do -No.De)
% Ki = -------------------
% Ne ^2+w ^2.No ^2
syms w
N1 =.054;
D1 =[42.83e -5 .004 .0023 0];
G1=tf(N1,D1);
%even and odd decomposition of plant
Ne_G1 =.054;
No_G1 =0;
De_G1 =-0.004* w^2;
Do_G1 =0.0023 -42.83e -5*w^2;
%Kp and Ki
Kp_G1=-(w^2* No_G11*Do_G11 + Ne_G11 *De_G11)/(Ne_G11^2+ w^2*
No_G11 ^2);
Ki_G1=w^2*( Ne_G11*Do_G11 -No_G11*De_G11 )/( Ne_G11 ^2+w^2*
No_G11 ^2);
%KP_G1 is the acceptable proportional gains which stabilize
%the loop
%KI_G1 is the acceptable integral gains which stabilize the loop
%Next to lines are initializations
KP_G1 =0;
KI_G1 =0;
for omega=[omega_min: omega_step: omega_max]
KP_G1 =[KP_G1 ;subs(Kp_G1 ,w,omega)];
KI_G1 =[KI_G1 ;subs(Ki_G1 ,w,omega)];
end
%polotting
plot(KP_G1 ,KI_G1),hold on
54 2. OVERVIEW OF KHARITONOV’S THEOREM
grid minor
xlabel('Kp ')
ylabel('Ki ')
e curve shown in Fig. 2.8 divides the plane into two regions. Points that lie between
the curve and K
p
axis (see Fig. 2.9) are the stable region (i.e., closed-loop poles lie in LHP).
0.06
0.05
0.04
0.03
0.02
0.01
0
-0.01
0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45
Kp
Ki
Figure 2.9: Stabilzing region for G
1
.
s
/
D
0:054
42:8310
5
s
3
C0:004s
2
C0:0023s
.
Figures 2.10 and 2.11 show the stable region for eight of Kharitonov’s plants. e shaded
region shows the intersection of all Kharitonov’s plants. Obtained result is the same as Fig. 2.7.
Figure 2.10 is produced with the aid of the following program.
2.4. CASE STUDY: ROBUST CONTROL OF A POSITION CONTROL SYSTEM 55
0.5
0.4
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
Kp
Ki
Figure 2.10: Coefficients selected from the shaded region will stabilize eight of Kharitonov’s
plants.
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