Information Technology Reference
In-Depth Information
For the third Kharitonov polynomial p
3
.s/ D c
n
s
n
C c
n1
s
n1
CCc
1
s C c
0
the
coefficients are
c
2k
D c
min
if k is an even number;c
2kC1
D c
max
2kC1
if k is an even number
2k
(3.7)
c
2k
D c
max
if k is an odd number;c
2kC1
D c
min
2kC1
if k is an odd number
2k
For the fourth Kharitonov polynomial p4.s/ D c
n
s
n
C c
n1
s
n1
CCc
1
s C c
0
the coefficients are
c
2k
D c
max
if k is an even number;c
2kC1
D c
min
2kC1
if k is an even number
2k
(3.8)
c
2k
D c
min
if k is an odd number;c
2kC1
D c
max
2kC1
if k is an odd number
2k
3.2.2
An Application Example
As an example the following characteristic polynomial is considered p./ D
3
C
.aCb/
2
CabCK with parametric uncertainty given by K2Œ8;12, a2Œ2:5;3:5,
b2Œ3:8;4:2. The coefficients of the characteristic polynomial have the following
variation ranges: c
3
D 1, c
2
D .a C b/2Œ6:3;7:7, c1 D ab2Œ9:5;14:7 and c
0
D
K2Œ8;12. The associated Kharitonov polynomials are
p
1
./ D 12 C 14:7 C 6:3
2
C
3
p
2
./ D 8 C 14:7 C 7:7
2
C
3
(3.9)
p
3
./ D 8 C 9:5 C 7:7
2
C
3
p
4
./ D 12 C 9:5 C 6:3
2
C
3
Next, Routh-Hurwitz criterion is applied for each one of the four Kharitonov
polynomials. For polynomial p
1
./
3
j 1
14:7
2
j 6:3
12
(3.10)
1
j 12:795 0
0
j 12
Since there is no change of sign in the first column of the Routh matrix it can be
concluded that polynomial p
1
./ has stable roots. For polynomial p
2
./
3
j 1
14:7
2
j 7:7
8
(3.11)
1
j 13:6610 0
0
j 8
