42 2. OVERVIEW OF KHARITONOV’S THEOREM
Origin is shown with a red star in Fig. 2.3. Kharitonov’s rectangle does not contain the
origin, so the given polynomial is stable.
Lemma 2.11 (Zero Exclusion Condition) Suppose that P D fp
.
; q
/
W q 2 Qg is an interval
polynomial with invariant degree and has at least one stable member. P is robustly stable if origin
of complex plane, i.e., Z D .0; 0/, is excluded from the Kharitonov’s rectangles for all ! 0.
2.3 CONTROLLER DESIGN FOR INTERVAL PLANTS
Assume a transfer function that numerator and denominator coefficients are uncertain:
p
.
s; q; r
/
D
N.s; q/
D.s; r/
D
P
m
iD0
q
i
; q
i
C
s
i
s
n
C
P
n1
iD0
Œ
r
i
; r
i
C
s
i
: (2.11)
Assume that p.s; q; r/ is strictly proper, that is m < n. We call p
.
s; q; r
/
interval plant for
obvious reasons. Assume a control scheme like that shown in Fig. 2.4.
–
+
Controller Interval Plant
Figure 2.4: Feedback control of interval plant p
.
s; q; r
/
.
We want to find a controller C.s/ which stabilizes all the plants belong to p
.
s; q; r
/
. is
problem can be solved using two approaches: applying Kharitonov’s theorem to the close-loop
denominator and using the 16-plant theorem.
2.3.1 FIRST APPROACH: APPLYING KHARITONOV’S THEOREM TO
THE CLOSED LOOP DENOMINATOR
Assume a control loop like that shown in Fig. 2.4. e designer must choose a controller. Gen-
erally, a first-order controllers like PI, lead, or lag is selected. Close-loop transfer function (in
presence of selected controller) is calculated next. Controller parameters (i.e., proportional gain
and integral gain for PI controllers) are selected such that the close-loop denominator is ro-
bustly stable with respect to Kharitonov’s theorem. Selection of suitable control parameters is
done with the aid of loop commands in software environments.
For example, if interval plant is given by H
P
.
s
/
D
asCb
s
2
CcsCd
with 1 < a < 2, 3 < b < 4,
5 < c < 6, and 7 < d < 8 and the designer chooses a PI controller with general form of
2.3. CONTROLLER DESIGN FOR INTERVAL PLANTS 43
H
C
.
s
/
D K
p
C
K
I
S
, then the close-loop transfer function is:
H
CL
.
s
/
D
H
P
.s/ H
C
.s/
1 C H
P
.s/ H
C
.s/
D
as C b
s
2
C cs C d
K
p
C
K
I
s
1 C
as C b
s
2
C cs C d
K
p
C
K
I
s
D
.as C b/
.
K
I
C K
P
s
/
s
3
C
c C aK
p
s
2
C
.
d C aK
I
C bK
P
/
s C bK
I
:
(2.12)
e denominator of obtained transfer function is s
3
C
c C aK
p
s
2
C
.
d
C
aK
I
C bK
P
/
s C
bK
I
. K
P
and K
I
are design parameters and must be find. Table 2.1 shows the minimum and
maximum values of denominator coefficients with respect to uncertainty bounds given for a; b; c,
and d .
Table 2.1: Minimum and maximum values of denominator coefficients
Term Minimum Maximum
1?
3
1 1
(? + ??
?
)?
2
-5 + 1
×
?
?
6 + 2
×
?
?
(? + ??
?
+ ??
?
)?
1
7 + 1
×
?
?
+ 3
×
?
?
8 + 2
×
?
?
+ 4
×
?
?
??
?
?
0
3 × ?
?
4 × ?
?
Based on the minimum and maximum values of coefficients shown in Table 2.1, four
Kharitonov’s polynomials are formed:
K
1
.
s
/
D 3K
I
C
.
7 C K
I
C 3K
P
/
s C .6 C 2K
P
/s
2
C s
3
;
K
2
.
s
/
D 4K
I
C
.
8 C 2K
I
C 4K
P
/
s C .5 C 1K
P
/s
2
C s
3
;
K
3
.
s
/
D 4K
I
C
.
7 C 1K
I
C 3K
P
/
s C .6 C 2K
P
/s
2
C s
3
;
K
4
.
s
/
D 3K
I
C
.
8 C 2K
I
C 4K
P
/
s C .5 C 1K
P
/s
2
C s
3
:
(2.13)
Obtaining the suitable values for K
P
and K
I
is done with the aid of nested loop. First, a
reasonable range and step for K
P
and K
I
are determined. Using nested loops suitable values of
K
P
and K
I
are found.
Since in this example the denominator .s
3
C
c C aK
p
s
2
C
.
d C aK
I
C bK
P
/
s C
bK
I
/ is a third-order polynomial, it is enough to stabilize K
3
.
s
/
(see Lemma 2.8). e fol-
lowing pseudo-code shows this step:
for K
P
D K
P;min
W K
P;step
W K
P;max
for K
I
D K
I;min
W K
I;st ep
W K
I;max
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