As noted in the foregoing, the stability of a linear/invariant system is determined by the roots of the characteristic equation. A characteristic mode will be divergent if its real part is positive, and. convergent if the real part is negative, the latter denoting asymptotic stability. It is not necessary, however, actually to solve the characteristic equation in order to find whether the roots have positive real parts. This can be determined from its coefficients alone. The conditions on the coefficients that must be satisfied were first stated by Routh (ref. 3.5), who derived them from a theorem of Cauchy. Let the characteristic equation be
<Vn + W*"1 + • ■ • + c0 = 0 (e. > 0) (3.3,55)
The coefficient cn can always be made positive by changing signs throughout, so the requirement cn > 0 is not restrictive. The necessary and sufficient condition for asymptotic stability (i. e. that no root of the equation shall be zero or have a positive real part) is that each of a series of test functions shall be positive. The test functions are constructed by the simple scheme shown below. Write the coefficients of (3.3,55) in two rows as follows:
n vn-2 °n-4
Cn-1 CM-3 CM-5 ‘ ‘ ‘
Mow construct additional rows by cross-multiplication:
P*. Р32 P33 * ‘ *
P41 P42 P43 ‘ ‘ ’
P31 = Cn-lCn-2 ~~ cncn-З’ Р32 = Cn-lcn-4 cncn-5> e^c-
Р41 = РзіСп-3 Рз2Сп-1> P42 = Р’ЯСп-5 Cn-1^33’ e^°’
Р51 = ^44^32 Р? лРцг ef°-
The required test functions F0 • ■ ■ Fn are then the elements of the first column, cn, cn_v Pm ■ ■ • Pm+lil. If they are all positive, then there are no unstable roots. The number of test functions is n + 1, and the last one, Fn, always contains the product c0Fn_1. Duncan (ref. 3.6, Sec. 4.10) has shown that the vanishing of c0 and of Fn_x represent significant critical cases. If the system is stable, and some design parameter is then varied in such a way as to lead to instability, then the following conditions hold:
(a) If only c0 changes from + to —, then one real root changes from negative to positive; i. e. one divergence appears in the solution (Fig. 3.6).
(b) If only Fn_x changes from + to —, then the real part of one complex pair of roots changes from negative to positive; i. e. one divergent oscillation appears in the solution (Fig. 3.6).
Thus the conditions c0 = 0 and Fn_^ = 0 define boundaries between stability and instability. The former is the boundary between stability and static instability, and the latter is the boundary between stability and a divergent oscillation.