Phase margin of linear time invariant systems from Routh array

摘要

A simple method to determine the gain margin of linear time invariant systems (LTIS) employing Routh stability criterion was developed in an earlier paper. The term 'stability ratio' was defined connecting the marginal gain K/sub m/ determined from the Routh array and the actual gain K of the system. This established a direct link between the Routh criterion and the gain margin. It was also shown that the phase cross-over frequency omega /sub cp/ can be found from the auxiliary equation formed from the elements of the s/sup 2/-row of the Routh array. Empirical formulae for computing the gain cross-over frequency omega /sub cg/ of type 0 systems and of higher-type systems in terms of system gain K and the marginal gain K/sub m/ are suggested. The gain cross-over frequency and phase margin calculated using the empirical formulae are in close agreement with exact values, justifying the use of the empirical formulae. It is shown that the Routh stability criterion can be used to measure the relative stability of LTIS.