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      5.8278
                                                -. -
                              = Jr6~
                                           -- 6.3587
                                Crdc _ -4'COnc
                                                                     - -0.4 * 6.3587
                                                                                                                                                                .
                        - _2.5435
Here, crdc and tOdc are the real and imaginary parts for the dominant second-order
poles.
       The transfer function of the PD compensator is given by
                                                Gc(s) : s + zc
Now we have to determine the location of the compensating zero zc so that the
root-locus passes through the point (oZtc, COeLc). The value of zc is determined by
502           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
the angle condition of Eq. (5.124), which in this case leads to
                                              Ozc - (OI  + o2 + 03) = (2n + 1)180
      Referring to Fig. 5.36b (ignoring the pole at s  =  pc), we find Oi  = 113.5736 deg,
82 = 85.5274 deg, and 03  - 67.1488 deg.Choosing n -. -l,we obtain Ozc - 86.2498
deg and zc = 2.9261. Then, the transfer function of the PD-compensated system
is given by
                   2.9261)  .
                                Gc(s) = k( ++32~+5~
                                                                    ~
      Then we draw the root-locus using MATLAB4 as shown in Fig. 5.36c and obtain
k = 40.2971  and  p = -2.5435 +_  j5.8317 and -2.9130 for operating at < = 0.4.
    Lead-Iag compensrUor. The transfer function of the lead.compensator is given
　
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