Modern Control System Theory and Design, 2nd Edition

6.15. ROOT LOCUS OF TIME-DELAY FACTORS [26–28]

We previously considered time-delay factors in Section 6.7 when we discussed the Bode-diagram approach. The transfer function of a time-delay factor was defined in Eq. (6.95), and an example was solved using the Bode-diagram approach for a unity feedback control system whose forward transfer function was defined in Eq. (6.103). How would we draw the root locus of a control system containing a time-delay factor?

To answer this question, let us consider a unity-feedback control system whose forward transfer function, which contains a time-delay factor, is given by the following

and is illustrated in Figure 6.63.

Figure 6.63   Control system containing a time-delay factor.

The characteristic equation of the control system shown in Figure 6.63 is given by

or

Therefore, the angular condition is given by

The angular condition of e^−Ts is given by

Since the term e^−Tσ is real, its angular value is zero. Therefore,

as shown in Eq. (6.96). Therefore, ...