REPEATED ROOTS

When two or more of the roots are the same, then the expansion theorem given above fails. For then, after canceling one of the repeated factors from D(s) by the factor (s — aT) of the numerator, still another remains and becomes zero when s is set equal to ar. Some particular cases of equal roots are shown in Table 2.3, items 6, 7,11, and 12. The method of partial fractions, coupled with these entries in the table, suffices to deal conveniently with most cases encountered in stability and control work. However, for cases not conveniently handled in this way, a general formula is available for dealing with repeated roots. Equation (2.5,6) is used to find that part of the solution which corresponds to single roots. To this is added the solution corresponding to each multiple

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