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     Next, we add the PI controller. Select a pole at s = O and a zero at s - -0.5 so
that the transfer function of the PID controller is grven by
                                     G(s) = k(s + 0.5)(s + 15)(s + 19.9014)
                                  s~s+ ) s+3~s+9)
504             PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
Fig.538    PID controller for Example 5.11.
     Now we draw the root-locus of the PID system as shown in Fig. 5.38b and pick
the point corresponding to < :0.3. We get k = 6.1112 and closed-loop poles at
-3.6846 +  j12.2343, -11.2460, and -0.4969.
    The unit-step responses of the basjc, PD-, and PID-compensated systems are
shown in Fig. 5.38c. It may be observed that the PID-compensated system meets
the design requirements.
                                             Example 5.12
    For the system of Example 5.10, design a major-loop feedback to achieve the
sameperformance. .
     So/ution.    We have found in Example 5.10 that Ozc = 86.2498. With this, we
obtain the equrvalent pole location, zc - 2.9261, and kh = 1]zc = 0.3418. We then
LINEAR SYSTEMS, THEORY, AND DESIGN:A BRIEF REVIEW       505
plot the root-locus using MATLAB and obtain the value of the gain as 40.2971,
which is equal to kkh so that k -  117.8967. The reader may verify that this response
is identical to that of the PD controller of Example 5.10.
                                         Example 5.13
   For the control system shown in Fig. 5.39a, determine the gain kh so that the
minor loop operates with a damping ratio of 0.707 and the complete system has a
damping ratio of 0.4.
     So/ution.    Consider the minorloop.We draw the root-locus using MATLAB4
as shown in Fig. 5.39b and pick the point on the root-locus corresponding to < -
0.707. Then, we get kh = 14.1018 and p -.5.8471, -1.0765 :1:  j1.1194. Having
designed the minor loop and knowing tI:e value of kh, we can simplify the system
a) Controlsystem
Fig.539   Minor-Ioop design for Example 5.13.
7
 i
,ii
506           PERFORMANCE, STABILITY, DYNAMICS, AND CONTROL
en
x
<
o
tio
E
c) Outer-Ioop block diagram
                   Reaf Axis
d) Root-Iocus of outer loop
Fig.539   Minor-Ioop design for Example 5.13,continued.
LINEAR SYSTEMS, THEORY, AND DESIGN: A BRIEF REVIEW       507
block diagram as shown in Fig. 5.39c. We have
Gi(s) :
s3 + 8S2 + (kh + 15)s
For this system, we draw the root-locus as shown in Fig. 5.39d and obtain k -
69.5452 and p = -4.9495, -1.5202 1:  j3.4222 for operating at <:0.4. This
completes the design.
5.10  State-Space Analysis and Design
   The classical method of analyses discussed in the previous sections is called
the frequency-domain technique because it is based on system representat.ion in
the form of transfer function. The main advantage of tl:is approach is that the
governing' differential equation of the system is replaced by an algebraic transfer
function. However, a disadvantage of this classical method is that it is limited to
linear time-invariant systems with zero initial conditions. The modern state-space
approach is more general in nature because it can be used to represent nonlinear,
time-varying systems with nonzero initial conditions. The state-space method can
also handle multi-input-multi-output systems in a compact manner. Furthermore,
the state-space approach becomes very attractive because it is based on matrix
algebra, and powerful matrix analyses tools like MATLAB4 are commercially
　
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