ABSTRACT
The response transform X2(s) has the general form given by Equation 7.32, which is repeated here in a slightly modified form. X1(s) is the driving transform [2-5]:
X s P sQ s X s
P s X s b s b s b s b sn n n n n n
( ) ( )( ) ( )
( ) ( )
=
=
+ + + + + −
− b0 (9.1)
Sections 7.6, 7.7, and 7.10 describe the methods used to evaluate the inverse transform L−1[F(s)] = f(t). However, before the inverse transformation can be performed, the characteristic polynomial Q(s) must be factored. CAD programs are readily available for obtaining the roots of a polynomial [1,6]. Section 7.13 shows that stability of the response x2(t) requires that all zeros of Q(s) have negative real parts. Since it is usually not necessary to find the exact solution when the response is unstable, a simple procedure to determine the existence of zeros with positive real parts is needed. If such zeros of Q(s) with positive real parts are found, the system is unstable and must be modified. Routh’s criterion is a simple method of determining the number of zeros with positive real parts without actually solving for the zeros of Q(s). Note that the zeros of Q(s) are poles of X2(s). The characteristic equation is
Q s b s b s b s b s bn n n n s n( ) = + + + + + =− − − −1 1 2 2 1 0 0 (9.2)
If the b0 term is zero, divide by s to obtain the equation in the form of Equation 9.2. The b’s are real coefficients, and all powers of s from sn to s0 must be present in the characteristic equation. A necessary but not sufficient condition for stable roots is that all the coefficients in Equation 9.2 must be positive. If any coefficients other than b0 are zero, or if all the coefficients do not have the same sign,
then there are pure imaginary roots or roots with positive real parts and the system is unstable. In that case, it is unnecessary to continue if only stability or instability is to be determined. When all the coefficients are present and positive, the system may or may not be stable because there still may be roots on the imaginary axis or in the right-half s plane (RHP). Routh’s criterion is mainly used to determine stability. In special situations, it may be necessary to determine the actual number of roots in the RHP. For these situations, the procedure described in this section can be used.
