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Section 4.3  Sensitivity of Control Systems to Parameter Variations   239

        4.3  SENSITIVITY  OF CONTROL SYSTEMS TO        PARAMETER     VARIATIONS

                         A  process, represented  by the transfer  function  G(s), whatever its nature, is subject
                         to a changing environment, aging, ignorance  of the exact values of the process para-
                         meters, and other natural factors that  affect  a control process. In the open-loop sys-
                         tem,  all  these  errors  and  changes  result  in  a  changing  and  inaccurate  output.
                         However, a closed-loop  system senses the change  in the output  due to the  process
                         changes and attempts to correct the output. The sensitivity of a control system to pa-
                        rameter  variations  is  of  prime  importance. A  primary  advantage  of  a  closed-loop
                        feedback  control system is its ability to reduce the system's sensitivity [1^4,18].
                            For the closed-loop case, if G c(s)G(s) »  1  for all complex frequencies  of inter-
                         est, we can use Equation  (4.2) to obtain (letting T d(s)  =  0 and N(s)  =  0)

                                                       Y(s)  ^  R(s).
                        The output is approximately equal to the input. However, the condition G c($)G(s)  >>>  1
                        may cause the system response to be highly oscillatory and even unstable. But the fact
                        that increasing the magnitude  of the loop gain reduces the effect  of G(s) on the output
                        is an exceedingly  useful  result. Therefore, the  first  advantage  of a feedback  system is
                        that the effect  of the variation of the parameters of the process, G(s), is reduced.
                            Suppose the process (or plant) G(s) undergoes a change such that the true plant
                        model is G(s)  +  AG(s). The change in the plant may be due to a changing external
                        environment  or  natural  aging, or  it  may just  represent  the  uncertainty  in  certain
                        plant  parameters. We consider  the  effect  on  the  tracking  error  E(s)  due to  AG(s).
                        Relying on the principle  of superposition, we can let T d(s)  = N(s)  =  0 and consid-
                        er only the reference  input R(s), From Equation  (4.3), it follows that


                                                      1  +  G c(s){G{s) +  AG{s))

                        Then the change in the tracking error is
                                      ,                 -G c(s)  AG(s)
                                 AE(   =
                                    {S)                                              {S)
                                         (1 +  G c(s)G(s) +  G c(s) AG(s))(l  + G c(s)G(s))  '
                        Since we usually find  that G c(s)G(s) »  G c(s) AG(s),  we have

                                                        -G c(s)  AG(s)  n/  x
                                                AE(s)  «           ^-R(s).
                                                   W              2
                                                         (1 +  L(s))
                        We  see  that  the  change  in  the  tracking  error  is  reduced  by  the  factor  1 +  L(s),
                        which is generally greater than 1 over the range of frequencies  of interest.
                            For large L(s), we have  1 +  L(s)  ~  L(s),  and  we can approximate  the  change
                        in the tracking error by

                                                           1   AG(J)
                                                A£W x                m                       (49)
                                                        ~W)      m -
                        Larger magnitude L{s) translates into smaller changes in the tracking error  (that is,
                        reduced  sensitivity  to  changes  in  AG(s)  in  the  process). Also, larger  L{s)  implies
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