Bifurcations and Limit Cycles in Models of Biological Systems - Advanced Models of Neural Networks - page 55

polynomial with coefficients that vary in intervals is equivalent to the study of the
stability of four polynomials with crisp coefficients computed from the boundaries
of the aforementioned intervals.

3.2 Generalization of the Routh-Hurwitz Criterion

with Kharitonov's Theorem

3.2.1 Application of the Routh Criterion to Systems

with Parametric Uncertainty

The following characteristic polynomial is considered

p./ D c n n C c n1 n1

CCc 1 C c 0 (3.1)

The polynomial's coefficients vary within intervals, that is

c min
i

c i c i max i D 0;1;2; ;n

(3.2)

The study of the stability of the above characteristic polynomial is now transferred
to the study of the stability of four Kharitonov polynomials, where each polynomial
is written as

p i ./ D c n n C c n1 n1 CCc 1 C c 0 i D 1; ;4

(3.3)

The associated coefficients are written as

c 2k ;c 2kC1 k D m;m 1; ;0;1

(3.4)

where m D n=2 if n is an even number and m D .n 1/=2 if n is an odd number.
For the first Kharitonov polynomial p 1 .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 max

2kC1 if k is an even number 2k

(3.5)

c 2k D c min if k is an odd number;c 2kC1 D c min

2kC1 if k is an odd number

2k

For the second Kharitonov polynomial p 2 .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 min

2kC1 if k is an even number 2k

(3.6)

c 2k D c max

2k if k is an odd number;c 2kC1 D c max 2kC1 if k is an odd number