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supposed to be located in the left half of the s-plane.
5.9.4  Proportional-Integraland Derivative Controller
      Suppose we want to improve the transient response as well as reduce the steady
state error, then we use the PID (proportional-integral and derivative) controller.
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVIEW       493
Fig. 531    Proportional-integral and derivative controller.
There are two ways of designing a PID controller:  1) design a PD controller first to
improve the transient response and then add a PI controller to improve the steady-
state error or 2) design a PI controller first and then add a PD controller. Both
methods are iterative because one affects the other.
The schematic diagram of a PID controller is shown in Fig. 5.31.
5.9.5  Feedback Compensation
    The desired transient response can also be obtained by feedback compensation.
With a proper choice of the feedback-loop transfer function, the root-locus can
be modified to obtain the desired transient response. This method offers an added
advantage that the parts of the system can be isolated for improvement in tran-
sient response prior to closing the major loop. This approach is also equivalent to
relocating the open-loop poles of the system so that the root-locus is reshaped to
obtain thf; desired closed-loop poles.                \
     Feedback compensation can be accomplished in two ways: 1) major-loop com-
pensation and 2) minor-loop compensation as schematically shown in Fig. 5.32.
    Major-/oop compensation.    Let Hc(s) = khs be the transfer function of the
major loop. This form of transfer function is representative of a tachometer or a
rate gyro whose input is a displacement and output is the time rate of change of
displacement or velocity. For example, if the input is bank angle, then the output
will be roll rate.
    The closed-loop transfer function is given by
  kG(s)
T(s)=l+kkG() .+~
(5.169)
The characteristic equation of this major-loop feedback compensated system is
I
 !
  j
494            PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
given by
a) Major-Ioop compensation
                                    -{8, r   
  ?-   L                                     
┏━━━━━━━━━━━━━━━━━━━━━┓
┃  Z:d-                                    ┃
┗━━━━━━━━━━━━━━━━━━━━━┛
Fig. 532    Feedback compensation.
                                          l+kkhG(s)(s + k  ) = 0                            (5.170)
Thus,in principle, the major-loop feedback compensation introduces a zeroinco the
system at s - -llkh so that the root-locus is reshaped to pass through the desired
operating point. By varying the parameter kh, we can vary the gain as well as the
location of this zero. Even though this concept is similar to the PD compensation,
there is a difference. The compensator zero in the case of a PD compensator is an
open  loop zero, whereas the zero introduced in major-loop feedback compensation
is not an open-loop zero.
    Minor-/oop compensation.    With Hc(s) : khs, the open-loop transfer func-
tion of the minor loop is Gc(s)khs. Thus, the addition of a zero at the origin of the
minor-loop root-locus considerably speeds up the response of the minor loop and
also has an effect on the overall system performance. Once the gain kh is adjusted
to obtain the desired performance of the minor loop, the outer loop is closed, and
the gain k is adjusted to obtain the specified overall system performance. This
　
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