This preview shows pages 1–2. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: ____—————————————— EEE3€57 ' Summary of Formulae ___—____——————— Some Laplace transforms 6(t) 1
10) l S 1 no) — $2
it"’1(t) 1 l
m! Sm+ exp(—at)l(t) S + a
sin(a)t)l(t) 2 ‘0 2 S + a)
cos(wt)l(t) 2 s 2 s M) f(t — T) <—> (“17(5)
e"””f(t) <—) F(s+a) Laplace transform theorems
If f(!) <) F(s) , then
‘ f(t) * go) H mm» if,le sF(s)—f<o—) 2 u
d if” e s2F(s)~sf(0—)f(0) F (s)
s :[f(t)dt e)
0 WT f(0+) = lim sF(s) s—no FVT lim f(1) = lingsF(s) provided all poles I—DG) of sF(s) have negative real part. Simpliﬁed form of Mason’s gain rule
2 path TF3 1 — 2 loop TFs For complex poles at s = ~0i jwd , (on =,/0'2+a)f,, g=0’/w,, 2
a=gwrn (0d:er lug G(s) = where a)” is the undamped natural frequency and
g is the damping ratio. The associated quadratic
factor is s2 + Zgwns + a): For the second order system with transfer
function (02 H(s) : .——’L—
$2 + 2gro,,s + mi ’ tr 3 ﬂ’ﬂ sec, where ﬂ = tan—'(wd/O')
V wd
t, = 72/0,,
Mp — {Gr/cod)” '(g/ 1'52)”
4 4
ts = — = 0' gwn A necessary condition for all roots of the
polynomial a(s) = aos" + 013"“ + (12'.t""'2 +...+an, 00' > 0
to have negative real part is a, > 0, all i . A necessary and suﬂicient condition is given by
the Routh array:  ' n S I do a; a4 a5 .....
s“' a, a3 a5 a7 s“ b, b2 b3 s"’3 c. c; (:3 s
1 *
where
= alaz — 0003 b2 = 01(14 — 0005 etc
9 $
01 01
_ bla3 — 01172 c 9135 ‘ “lbs etc
Ci " wb‘ , 2 — b1 ’ The number of sign changes in the ﬁrst column is
equal to the number of roots with real part _
positive. Special cases (1) When the ﬁrst entry in
a row is zero (and the entire row is not zero),
replace the zero with e = 0 + and (2) When an
entire row is zero, replace it with the derivative
of the row above. A unity feedback system with feedforward ~
transfer function G(s) is ‘type k’ if G(s) has k
poles at the origin. The static position, velocity and acceleration
error constants are KP = lim G(s), Kv =1ingsG(s), K“ = ling szG(s). S—DO
For a unit step, unit ramp and unit parabolic
input, the steady state error .955 is equal to 1 l 1 Type 0, type 1 and type 2 systems have a ﬁnite
as, in response to step, ramp and parabolic inputs respectively. For the characteristic equation 1+ G(s)H(s) = 1+ K A“) = o ,
B(s) where G(s)H(s) has n poles and m zeroes
(n 2 m ), the root locus is the locus of roots for
positive K. A point 3 = so is on the locus if and only if 40(s0)H(so) = 180° i [360° . The steps in drawing a root locus are:
1. Mark poles and zeroes of G(s)H(s) . 2. Draw the locus on the real axis to the left of
an odd number of real axis poles plus zeroes.
3. Draw the asymptotes. The centroid is given by .
ca = 2 Pi “ Z 2i
n — m '
and the angles of the asymptotes by
0 0
¢a =M’1 = 0, 12,."
n — m
4. Find breakaway and breakin points by
solving B(s)A'(s) = A(s)B'(s) 5. Calculate departure angles from poles and
arrival angles at zeroes by considering a
point on the' root locus inﬁnitesimally close
to the pole or zero. 6. Calculate imaginary axis crossings using the
characteristic equation and substituting
s = ja) or using the Routh criterion. The value ofK at a point s = so on the locus can
be found from [C(so)H(so) =1 . Lead and lag compensators have transfer
functions ofthe form T3+1=K 7'
aTs+l c 1 3+—
aT with 0 < a <1 for a lead compensator and a >1 for a lag compensator.  For a system with open loop transfer function , G(s)H(s) the Nyquist stability criterion is
Z=N+P where Z = number of zeroes of H G(s)H (s) in the RHP, N = number of clockwise encirclements
ofthe (l,0) point, and P = number of poles of G(s)H(s) in the RHP. Suppose 0(3) is the open loop transfer function
of a unity feedback system. The gain margin is
the inverse of lG(jw) at the phase crossover
frequency (when 1000) = —1 80° ). The phase
margin is the angle by which 4000;) exceeds
—1 80° at the gain crossover frequency (when
lKGQ‘w)‘ =1 or 0 dB). ' For the lead (lag) compensator Ts+l
s =_K
DU aTs+1 Kca ,a<l a) The max (min) phase ¢m occurs at frequency
where mu’ 1
a) =——
max shuns—"0‘, a ———1‘ST“¢W
1+a 1+sm¢max The state space equations for a LTI system are
x = Ax + Bu y = Cx + Du
and the transfer function is .
G(s) = C(sI — A)" B + D
The eigenvalues of A are the characteristic roots
(poles) of the system and are found by solving sI—A=0. ...
View
Full
Document
 Spring '08
 Staff

Click to edit the document details