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Unformatted text preview: find t he s tep a nd r amp r esponse for a given value o f K , we first c reate a n
mfiIe c64a.m as r. a nd e o (s)  o + 4)2 4 +4  + t + ~e8t cos (8t + 36.87 tTypical percentage values used are 2% to 5% for t •. 433 T his response, sketched in Fig. 6.36d, shows t hat t here is a s teadystate error e r =
0.1 radian. In many cases such a small steadystate error may be tolerable. If~
however, a zero steadystate error to a r amp i nput is reauired, t his system in its
present form is unsatisfactory. We m ust add some form of compensator t o t he
system. 16 16
e o(s) = s (s2 6.7 Application t o Feedback a nd Control S tate E quations 0
)] u(t) ( c64a.m) To plot t he s tep r esponse (as in Fig. 6.36c), we c reate a nother file c64b.m as:
K =7;c64a; h old o n,
K =16;c64aj K =80jc64a
T he u nit r amp r esponse o f t his s ystem is t he s ame a s t he u nit s tep r esponse o f a
s ystem with transfer function T (s)/s. Hence, we c an use t he file c64a t o find t he r amp
r esponse. To plot t he r amp r esponse, for K = 80, we c reate c64c.m file as follows: K =80j c 64aj
N umTFr=NumTFj D enTFr=conv([O 1 O J,DenTF)j
p rintsys(NumTFr,DenTFr);
s tep(NumTFr,DenTFr) 0 Design Specifications T he above discussion has given t he r eader some idea o f t he various specifications
a control system might require. In general we m ay be required to design a control
system t o m eet some o r all o f t he following specifications:
1 . T ransient R esponse
(a) Specified overshoot t o s tep input.
(b) Specified rise time t r a nd lor delay time t d.
(c) Specified settling time t s. 2. S teadyState E rror
Specified ~te.adystate e rror t o c ertain expected inputs such as step, ramp,
o r p~rabohc Inputs. ~n control systems, t he t ransient response is generally
specIfied for t he s tep Input. T he reason is t hat t he s tep i nput represents a
sudden j ump discontinuity. Hence if a system has a n a cceptable transient
response for s tep i nput, i t is likely t o have acceptable transient response for
most o.f th~ p ractical inputs. Steadystate errors, however, must be specified
for tYPIcal I nputs o f t he system. For t he given system one must determine what
kind of i nput ( step, ramp, etc) are likely to occur, a nd t hen specify acceptable
steadystate requirements for these inputs. 434 6 C ontinuousTime S ystem Analysis Using t he Laplace Transform 3. S ensitivity
T he s ystem s hould satisfy a specified sensitivity specifications t o some system
p arameter v ariations, or t o c ertain d isturbances. Sensitivity analysis will n ot
b e considered here. 6.7 Application t o Feedback a nd Control S tate E quations 435 . T he n ature of this response for t he u nderdamped case ( ( < 1) is i llustrated in
FIg. 6.38. T he response decays exponentially as e (wnt. Hence, t he t ime constant
o~ response is 1 /(w n . I t takes four time constants for an exponential t o decay t o
slIghtly less t han 2% of its initial value. Hence, t he s ettling Lime t . is given by 6.72 Analysis o f a SecondOrder System
T he t ransient...
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 Spring '13
 Bayliss
 Signal Processing, The Land

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