Unformatted text preview: t o u nity as well as
nonunity feedback systems, a nd a re more general. Steady-state-error specifications
in this case are t ranslated in terms of constraints on t he closed-loop transfer function
T he u nity feedback system in Fig. 6.36a is t ype 1 system. We have designed
this system e arlier t o meet t he following transient specifications: PO = 16%, 445 = 16% t r S 0.5 t. S 2 e. =0 To meet t he s teady-state specification we m ust have 8
K S 0.05 => K 2: 160 B ut Fig. 6.41 indic~tes t hat for K > 64, t he poles of T (s) move o ut of t he region
acceptable for transient performance. Clearly, we c an meet either t he t ransient o r
t he steady-st~te spe?ification b ut n ot both. In such case we m ust a dd some kind
~f compe~~at.lOn, ~hich will modify the root locus to meet all t he specifications. A
httle f amlhanty With root-locus techniques gives t he insight a nd j udgment needed to
choose a proper compensator transfer function. Figure 6.41 indicates t hat shiftin
t he r oot locus t o t he left will accomplish t he desired performance. We can p lace!
c ompensator of transfer function Gc(s) in series with G(s) (Fig. 6.45a) a nd select 6 Continuous-Time System Analysis Using the Laplace Transform 446 We shall now discuss a compensation which is primarily used t o improve thE
s teady-state performance. For unity feedback systems, t he s teady-state performancE
of a system is improved by placing an integrator in the forward p ath of G \ s)
T his procedure increases t he s ystem type, thus increasing K p ' K v, K a, etc. ThE
c ompensator in this case is Gc(s) = l /s.
In this scheme the compensator is a n ideal integrator. Hence, this scheme i~
known as i ntegral c ontrol. Design of a n ideal integrator necessitates elaborate and
expensive equipment. Hence, such a compensator is used where cost considerations
are not very important. For example, integrating gyroscopes are used for this
purpose in aircraft. In most cases a l ag c ompensator (described below) which
closely approximates t he a behavior of an integrator is used. A lag compensator
transfer function is given by 19.4 K =600 \0 - 30 o - 15 iO s +o'
s+(3 O'>!3 Gc(s) = - - - 19.4 K =600 44~ 6.7 Application to Feedback and Control S tate EqUi'l.tions and
Gc(O) = F ig. 6 .46 Root locus of the system in Fig. 6.36a after lead compensation. the poles and zeros of Gc(s) in such a way as t o shift the centroid of t~e r oot locus
t o t he left. I f Gc(s) has a single pole and a single zero, then choosm~ t he pole
farther t o t he left of the zero would shift the centroid to t he left accordmg to the
fourth rule of the root locus. Thus, we use ~ A lag compensator can be readily realized using a simple R C circuit depicted in
Fig. 6.47. For a unity feedback system, addition of a compensator Gc(s) causes
all t he e rror constants Kp, Kv, Ka, e tc., t o be multiplied by Gc(O). Thus, the
lag compensator increases Kp, Kv, K a" etc. by a factor (O'/!3), t hereby reducing
steady-state errors. ( 3)0'
s plane Such a c ompensator is called the l ead c ompensator, which is readily realized
using a simple R C circui...
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