35
C H A P T E R 2
Overview of Kharitonov’s
eorem
2.1 INTRODUCTION
Kharitonov’s theorem addresses the stability problem of interval polynomials. It can be consid-
ered as a generalization of Routh–Hurwitz stability test. Routh–Hurwitz is concerned with an
ordinary polynomial, i.e., a polynomial with fixed coefficients, while Kharitonov’s theorem can
study the stability of polynomials with uncertain coefficients. Before presenting the theorem,
some definition must be introduced.
2.2 KHARITONOV’S THEOREM AND RELATED
MATHEMATICAL TOOLS
Before presenting the Kharitonov’s theorem, some definitions must be studied. Required defi-
nitions are given below.
Definition 2.1 (Stability). A fixed polynomial P .s/, i.e., with fixed coefficients, is said to be
stable (or Hurwitz) if all its roots lie in the strict Left Half Plane (LHP).
For example, p
.
s
/
D s
2
C 2s C 5 is stable since its roots .1 ˙ 2j /, lie in the LHP. De-
pendence of a transfer function on a vector of uncertain parameters q is shown with the aid of
p.s; q/ instead of p.s/.
Definition 2.2 (Robust Stability). A given family of polynomials P D fp
.
; q
/
W q 2 Qg is
said to be robustly stable if, for all q 2 Q, P .s; q/ is stable. at is, for all roots of P .s; q/ lie in
the strict LHP.
Definition 2.3 (D Stability). A polynomial is said to be D stable if all of its roots lie in D C .
In the above definition, D is sub-region of LHP. In fact, D stability is an attempt to restrict
the acceptable region of poles. Assume a system that needs a settling time of less than 5 sec and
36 2. OVERVIEW OF KHARITONOV’S THEOREM
Im
Re
Figure 2.1: Acceptable region for close loop poles.
damping greater than 0.707. e acceptable region of close loop poles for such a system is no
longer LHP. e acceptable region is shown in Fig. 2.1.
Definition 2.4 (Interval Polynomial). An interval polynomial is the family of all polynomials
p
.
s
/
D a
0
C a
1
s
1
C a
2
s
2
C C a
n
s
n
; (2.1)
where 8i; a
i
2
Œ
l
i
; u
i
and 0 …
Œ
l
n
; u
n
.
In the above definition, l
i
and u
i
denote the bounding interval (i.e., minimum and max-
imum) for the i-th coefficient. 0 …
Œ
l
n
; u
n
keeps the degree of polynomial family constant, i.e.,
all the members are polynomial of degree n.
For example, p
.
s; q
/
D
Œ
1; 2
s
5
C
Œ
3; 4
s
4
C
Œ
5; 6
s
3
C
Œ
7; 8
s
2
C
Œ
9; 10
s C Œ11; 12 is an
interval polynomial; it contains an infinite number of polynomials, for instance, p
.
s; q
/
D
1:2s
5
C
3:9s
4
C
5s
3
C
7:1s
2
C
9:13s
C
12
is a member of this interval polynomial.
Definition 2.5 (Invariant Degree). A family of polynomials given by P D fp
.
; q
/
W q 2 Qg
is said to have invariant degree if: given any q
1
; q
2
2 Q, it follows that:
deg p
s; q
1
D deg p
s; q
2
; (2.2)
2.2. KHARITONOV’S THEOREM AND RELATED MATHEMATICAL TOOLS 37
where deg shows the degree of polynomial, i.e., maximum power available in the polynomial.
Definition 2.6 (Kharitonov’s Polynomials). Associated with the interval polynomial
p
.
s; q
/
D
P
n
iD0
q
i
; q
i
C
s
i
are the for fixed polynomials:
K
1
.
s
/
D q
0
C
q
1
s
C
q
2
C
s
2
C q
3
C
s
3
C q
4
s
4
C q
5
s
5
C q
6
C
s
6
C : : :
K
2
.
s
/
D q
0
C
C q
1
C
s C q
2
s
2
C q
3
s
3
C q
4
C
s
4
C q
5
C
s
5
C q
6
s
6
C : : :
K
3
.
s
/
D q
0
C
C q
1
s C q
2
s
2
C q
3
C
s
3
C q
4
C
s
4
C q
5
s
5
C q
6
s
6
C : : :
K
4
.
s
/
D q
0
C q
1
C
s C q
2
C
s
2
C q
3
s
3
C q
4
s
4
C q
5
C
s
5
C q
6
C
s
6
C : : :
(2.3)
For example, if
p
.
s; q
/
D Œ1; 2s
5
C Œ3; 4s
4
C Œ5; 6s
3
C Œ7; 8s
2
C Œ9; 10s C Œ11; 12:
en:
K
1
.
s
/
D 11 C 9s C 8 s
2
C 6s
3
C 3s
4
C s
5
;
K
2
.
s
/
D 12 C 10s C 7s
2
C 5s
3
C 4s
4
C 2s
5
;
K
3
.
s
/
D 12 C 9s C 7s
2
C 6s
3
C 4s
4
C s
5
;
K
5
.
s
/
D 11 C 10s C 8s
2
C 5s
3
C 3s
4
C 2s
5
:
(2.4)
eorem 2.7 (Kharitonov’s eorem (1978a)). An interval polynomial family P with invariant
degree is robustly stable if and only if its four Kharitonov’s polynomials are stable.
What is somewhat surprising about Kharitonov’s theorem is that the stability problem of
an infinite number of polynomials is reduced to stability problem of four fixed polynomials. e
stability of these four polynomials can be tested via Routh–Hurwitz or any other method, i.e.,
using a software package.
For example, if
p
.
s; q
/
D Œ1; 2s
5
C Œ3; 4s
4
C Œ5; 6s
3
C Œ7; 8s
2
C Œ9; 10s C Œ11; 12: (2.5)
en,
K
1
.
s
/
D 11 C 9s C 8 s
2
C 6s
3
C 3s
4
C s
5
;
K
2
.
s
/
D 12 C 10s C 7s
2
C 5s
3
C 4s
4
C 2s
5
;
K
3
.
s
/
D 12 C 9s C 7s
2
C 6s
3
C 4s
4
C s
5
;
K
5
.
s
/
D 11 C 10s C 8s
2
C 5s
3
C 3s
4
C 2s
5
:
(2.6)
38 2. OVERVIEW OF KHARITONOV’S THEOREM
Using the classical Routh–Hurwitz table, it is easy to verify that all four polynomial have zero
in Right Half Plane (RHP). So, using the Kharitonov’s theorem, all the members of the afore-
mentioned interval polynomial are not stable.
Kharitonov’s theorem can be extended to polynomial with complex coefficients. Assume
q
i
and r
i
denote the uncertainty in the real and imaginary parts of the coefficients of s
i
, respec-
tively, i.e.,
p
.
s; q; r
/
D
n
X
iD0
.q
i
C jr
i
/s
i
: (2.7)
Assume Q and R as uncertainty bounding sets for q and r, respectively. P D
f
p
.
; q; r
/
W
q
2
Q; r
2
R
g
is a complex coefficient interval polynomial family. Like real coef-
ficient case q
i
q
i
q
i
C
; r
i
r
i
r
i
C
p
.
s; q; r
/
D
n
X
iD0
q
i
; q
i
C
C j
r
i
; r
i
C
s
i
: (2.8)
Lemma 2.8 e nth order interval polynomial p
.
s; q
/
D
P
n
iD0
q
i
; q
i
C
s
i
is robustly stable if
and only if the following is true.
• For n D 3; K
3
.s/ is stable.
• For n D 4; K
2
.s/ and K
3
.s/ are stable.
• For n D 5; K
2
.s/, K
3
.s/ and K
4
.s/ are stable.
• For n 6; .K
1
.s/, K
2
.s/, K
3
.s/ and K
4
.s/ are stable.
For example, testing the stability of p
.
s; q
/
D
Œ
1; 1
s
3
C
Œ
2; 3
s
2
C
Œ
0:5; 1
s C
Œ
1; 2
can
be done by testing the stability of K
3
.
s
/
D s
3
C 2s
2
C 0:5s C 2. Since K
3
.
s
/
D s
3
C 2s
2
C
0:5s C 2 is stable, the given family is stable.
Definition 2.9 (Complex Coefficient Kharitonov’s Polynomials). Associated with complex co-
efficient interval polynomials given as
p
.
s; q; r
/
D
n
X
iD0
q
i
; q
i
C
C j
r
i
; r
i
C
s
i
(2.9)
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