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Unformatted text preview: 78 Chapter 5 The RootLocus Design Method " For a locus drawn with rlocus(sys L), the parameter and corresponding roots can
be found with [K, p] = rlocﬁnd(sysL) or with rttool.
‘9 Lead compensation, given by S+z
D(s)= , z< ,
s+p p approximates proportional—derivative (PD) control. For a ﬁxed error coefﬁcient,
it generally moves the locus to the left and improves the system damping.
° Lag compensation, given by S
Ds=~~~a Z>,
() HP in approximates proportional~integral (PI) control. It generally improves the
steady—state error for ﬁxed speed of response by increasing the lowfrequency
gain and typically degrades stability. ° The root locus can be used to analyze successive loop closures by studying two
(or more) parameters in succession. ° The root locus can be used to approximate the effect of time delay. REVIEW GU ESTIGNS Give two deﬁnitions for the root locus.
Deﬁne the negative root locus. Where are the sections of the (positive) root locus on the real axis? P959? What are the angles of departure from two coincident poles at s : —a on the real axis?
There are no poles or zeros to the right of ~a. 5. What are the angles of departure from three coincident poles at s = —a on the real axis?
There are no poles or zeros to the right of ~a. 6. What is the principal effect of a lead compensation on a root locus? 7. What is the principal effect of a lag compensation on a root locus in the vicinity of the
dominant closedloop roots? 8. What is the principal effect of a lag compensation on the steady—state error to a
polynomial reference input? 9. Why is the angle of departure from a pole near the imaginary axis especially important?
10. Deﬁne a conditionally stable system. 11. Show, with a root—locus argument, that a system having three poles at the origin MUST
be conditionally stable. PRQSLEMS Problems for Section 5.] : Root Locus ofa Basic Feedback System 5.1 Set up the listed characteristic equations in the form suited to Evans’s root—locus method.
Give L(s), (1(5), and b(s) and the parameter K in terms of the original parameters in each
case. Be sure to select K so that a(s) and [9(3) are monic in each case and the degree of
b(s) is not greater than that of (1(3). <w;aeeemwwuwwmwmu Figure 5.51
Pole~zero maps Problems 279 (a) s + (l/t) : 0 versus parameter 1:
(b) 52 + cs + c + l = 0 versus parameter c
(c) (s + o3 +A(Ts +1) : o (i) versus parameter A, (ii) versus parameter T, (iii) versus the parameter c, if possible. Say why you can or cannot. Can a plot of the roots be drawn versus c for given constant values of A and T by any means
at all? ((1) 1 + [k,, + + G(s) = 0. Assume that G(s) = 14%, where C(s) and (1(5)
are monic polynomials with the degree of d(s) greater than that of C(s).
(i) versus kp
(ii) versus k1
(iii) versus kD (iv) versus 1' Problems for Section 5.2: Guidelines for Sketching a Root Locus 5.2 Roughly sketch the root loci for the pole~zero maps as shown in Fig. 5.5l without the
aid of a computer. Show your estimates of the center and angles of the asymptotes. a
rough evaluation of arrival and departure angles for complex poles and zeros, and the loci for positive values of the parameter K . Each pole—zero map is from a characteristic
equation of the form where the roots of the numerator [7(5) are shown as small circles o and the roots of the denominator a(s) are shown as X’s on the s~plane. Note that in Fig. 5 .5 l(c) there are
two poles at the origin. X
A , n g m 2 +
X
(a) (b) (c)
X
X
X
X
(‘1) (e) (n :80 Chapter 5 The Root—Locus Design Method 5.3 5.4 5.5 5.6 For the characteristic equation K __._._._:0
1+ s(s+1)(s+5) (3) Draw the realaxis segments of the corresponding root locus.
(b) Sketch the asymptotes of the locus for K —> 00. (c) Sketch the locus?
((1) Verify your sketch with a MATLAB plot. Real poles and zeros. Sketch the root locus with respect to K for the equation .l—l—
KL(s) = O and the listed choices for L(s). Be sure to give the asymptotes, and the arrival
and departure angles at any complex zero or pole. After completing each hand sketch,
verify your results using MATLAB. Turn in your hand sketches and the MATLAB results
on the same scales. (5  2)
(a) L“) = S(s + l)(s +5)(s+ 10) 1
(b) “5) : s(s + l)(s ~ 5)(s +10) C ~ 2)(3 + 6)
(C) L“) : s —— 2 s + 4
(d) L“) = s(s +< l)(s 355m 1 10)
Complex poles and zeros. Sketch the root locus with respect to K for the equation
1 + KL(S) = O and the listed choices for L(s). Be sure to give the asymptotes and
the arrival and departure angles at any complex zero or pole. After completing each
hand sketch, verify your results using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales. (a) L(s) = thzﬂﬁéjﬁ
2
mLm=ﬁg%;%
2
@Lm=%i%§%
2
(e) L(s) = ﬁll);
2
mtm=£ﬁ% Multiple poles at the origin. Sketch the root locus with respect to K for the equation
1 + KL(S) = O and the listed choices for L(s). Be sure to give the asymptotes and
the arrival and departure angles at any complex zero or pole. After completing each
hand sketch, verify your results using MATLAB. Turn in your hand sketches and the MATLAB results on the same scales. _ l
(a) MS) # s2(3 + 8)
a» m = ,«mnwwmwzwerwmmww % 5.7 5.8 5.9 Problems 281 ML®=Ei§
@Lm=ﬁi%
mtm=ﬁi%
mLm=%g%
@Lm=ﬁgﬁ% Mixed real and complex poles. Sketch the root locus with respect to K for the equation
l + KL(s) = 0 and the listed choices for L(s). Be sure to give the asymptotes and
the arrival and departure angles at any complex zero or pole. After completing each
hand sketch, verify your results using MATLAB. Turn in your hand sketches and the
MATLAB results on the same scales. V ... (s + 2)
L _ __..___
(a) (5) s(s + io)(s2 + 2s + 2) _ (s + 2)
b L ,_ WM
( ) (S) 52(5 —— 10)(s2 — 65 + 25) _ o+mz
L _.
m m 3:3ﬁiaiﬁ _ (s+ 2)(s2 +4s+ 68)
d L _ m
( ) (S) 5% —— 10)(s2 n 43 + 85) _ [(er DZ +1'
L ._ W..—
(e) (S) 52 (s w 2) (s + 3) RHP and zeros. Sketch the root locus with respect to K for the equation 1 + KL(s) = O
and the listed choices for L(s). Be sure to give the asymptotes and the arrival and
departure angles at any complex zero or pole. After completing each hand sketch, verify
your results using MATLAB. Turn in your hand sketches and the MATLAB results on
the same scales; (a) L(s) = 3+—2—l~—; the model for a case of magnetic levitation with lead
5 + 10 S2 _ 1
compensation. _ s + 2 ___1__. . . . _ a .
(b) L(S) —— 3(3 + 10) (32 a 1), the magnetic levrtation system With integral control
and lead compensation. (c) L(s) = S S31
2 S +2S+1
( ) s(s + 20)2(32 M 2s + 2)
the damping ratio of the stable complex roots on this locus? . What is the largest value that can be obtained for _ (MD
@Lmua:3ag?
mLm= 1 (s — l)[(s + 2)2 + 3] Put the characteristic equation of the system shown in Fig, 5.52 in root—locus form With «««« nae en «LA “(Hanan/«Av. A. nun] blmﬂeiﬁ. an AA2_.A._A,“J:_2 1 /,\ A /,\ 0,, l1 1 / \ (“WA ,1, ...
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 Spring '07
 Balakrishnan
 Digital Signal Processing, Signal Processing

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