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and time consuming. A simpler alternative is to introduce a compensator into the
system that compensates for the deficiencies ofthe original plant so that the overall
system,including the compensator, meets the specifieds perfonnance requirements.
      A compensator is also called a controller. Compensators that, employ pure inte-
gration to improve steady-state error orpure differevntiation to speed up the transient
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW       487
Damping Ratio
Fig.5.24   Relation between phase margin and dampmg ratio.
response are called ideal compensators. However, a disadvantage ofideal compen-
sators is that their implementation requires active networks like operational am-
plifiers. It is possible to construct passrve networks involving resistors, inductors,
and capacitors and to achieve performances close to those ofideal compensators.
Such compensators 'are called either lead or lag or lead-lag or lag-lead compen-
sators depending on their type. We will not be dealing with the issues concerrung
hardware implemention ofthe compensator designs discussed here. The interested
readermayreferelsewhere.l-3 LT~
      In this section, we willdiscuss the design ofcompensators to obtain the specified
transient response or steady-state error or both for single-input-single-out~ut sys-
tems. Basically, there are two design methods:  1) the r~ot-locus luethod and 2) the
frequency-response method. The frequency-response method has the advantage
that the explicit knowledge of the pZnt transfer function is not needed. All that
is needed is the plant frequency response. However, the main disadvantage of the
frequency-response method is that the quantities one deals with are not directly re-
lated to the time-response parameters, which are specified as design requirements.
Hence, the design becomes more of trial and error, and the number of iterations
depends on the knowledge and experience of the designer. On the other hand,
the root-locus method has a clear advantage in that the quantities it deals with are
directly related to the design requirements. Furthermore, the correlation ofthe root-
locus with time response is quite good. Also, the effect of changing compensator
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488           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
parameters can be easily observed by studying the root-locus. With the a\railability
of tools like MATLAB,4 the root-locus method becomes ver}r attractive for con-
trol system design. However, a disadvantage of the root-locus method is that it
becomes more complex as the order of the system increases.
        Here, we will use the root-locus method for compensator design of single-input-
single-output systems. Readers interestedin using frequency-domain methods may
ref9r elsewhere.1.2 For multi-input-multi-output systems, the modern state-space
methods are quite convenient. We will discuss these approaches in Section 5.10.
5.9.1 ProporOonal-Inte'graICompensator
   .To understand the basic principles of designing an integral compensator, con-
sider a type "O" system with unity feedback as shown in Fig. 5.25a  The root-locus
for this system is sketched in Fig. 5.25b. Let us assume that the system is operating
at point A, having the desired transient response. Recall that the transient response
is characterized by the settling time Ts, time for peak amplitude Tp, and the rise
time Tr, all of which depend on the damping ratio < and frequency c.o. At point A,
a) Given unity feedback system
                                             
                              L              
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┃                          ┃-N            ┃
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