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M.D. Bryant ME 344 notes 03/25/08
1
Laplace Transforms & Transfer Functions
Laplace Transforms: method for solving differential equations, converts differential equations in time t into algebraic equations in complex variable s Transfer Functions: another way to represent system dynamics, via the s representation gotten from Laplace transforms, or excitation by est
M.D. Bryant ME 344 notes 03/25/08
2
Laplace Transforms Purpose: Converts linear differential equation in t into algebraic equation in s
Forward transform, t s: f(t) F(s)
F(s) = t= 0
f(t) e st dt = L {f(t); t s }
Convergence/existence of integral: f(t) < M e at , a + Re(s) > 0
so that e[a + Re(s)]t finite as t
Inverse transform, s t: F(s) f(t)
f(t) = 12j
= c j
c+j F(s) e st ds = L1{F(s); s t }
Integral in complex plane, rarely do
M.D. Bryant ME 344 notes 03/25/08
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Procedure:
Convert Linear Differential Equations
!
x = Ax + f (t)
!
dn
x
dtn
+ an"1
dn"1
x
dtn"1
+!+ a2
d2
x
dt2+ a
1
d x
dt+ a
0x = f (t)
in time t into algebraic equations in complex variable s, via forward Laplace transforms:
X(s)=L {x(t); t s }, F(s)=L {f(t); t s }
Solve resulting algebraic equations in s,
for solution X = X(s)
Convert solution X(s) into time function x(t), via inverse Laplace transform:
x(t) = L1{X(s); s t }
M.D. Bryant ME 344 notes 03/25/08
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Example: Solve initial value problem, differential equation with initial condition:
x + 2 n x + 2n x = Ho us(t)
x(0) = xo , x (0) = x1
Apply Laplace transform
L= t= 0
{ . } e st dt to all terms in equation:
!
x (t)e" st
t= 0
#
$ dt + 2%&n x (t)e
" st
t= 0
#
$ dt + &n
2
x(t)e" st
dtt= 0
#
$
=
t= 0
Ho us(t) e st dt
factor constants and apply definition for us(t) :
t= 0
d2x
dt2 e st dt + 2n
t= 0
dxdt e st dt +
2n
t= 0
x e st dt
= Ho t= 0
e st dt
M.D. Bryant ME 344 notes 03/25/08
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Define X(s) = t= 0
x(t) e st dt
Integrate by parts
dxdt e stt=0 +s
t= 0
dxdt e st dt +2n {x(t) e stt=0
+s t= 0
x(t) e st dt } +
2n X(s) =
!
HO
("s) e st
t=0
Rearrange, note
!
limt"#e
$ st
= 0
 x (0) + s ( x(0) + s X(s) ) + 2n { x(0) +
s X(s) } + 2n X(s) =
Ho s
Note: L {dx/dt} =  x(0) + s X(s)
L {d2x/dt2} = L {d
!
x /dt} =
!
x (0) + sL {dx/dt}
Result: algebraic equation in s
!
(s2
+ 2"#n
s+#n
2
)X(s) =H
O
s+ x
1+ (s+ 2"#
n
)x0
Note: x (0) = x1 , x(0) = xo
M.D. Bryant ME 344 notes 03/25/08
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Solve
X(s) =
Ho s + x1 + ( s + 2 n) xo
s2 + 2 n s + 2n
Note:
o characteristic equation in denominator o terms from initial conditions x1 ,
xo grouped with excitation term
!
HO
s
M.D. Bryant ME 344 notes 03/25/08
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Apply inverse Laplace: x(t) = L1{X(s); s t }
x(t) = Ho
2n
L1{
2n
s (s2 + 2 n s + 2n)
}
+ xo
2n
L1{s
2n
s2 + 2 n s + 2n
}
+ x1 + 2 n xo
2n
L1{
2n
s2 + 2 n s + 2n
}
o Use Laplace transform tables for L1:
x(t) = Ho
2n
{1  e n t sin(n 1  2 t + arccos )
1  2 }
+ xo
2n
{ 
2n e n t sin(n 1  2 t  arccos )
1  2 }
+ x1 + 2 n xo
2n
{ n e n t sin(n 1  2 t )
1  2 }
M.D. Bryant ME 344 notes 03/25/08
8
Transfer Functions Method to represent system dynamics, via
s representation from Laplace transforms. Transfer functions show flow of signal through a system, from input to output.
Transfer function G(s) is ratio of output x
to input f, in sdomain (via Laplace trans.):
!
G(s) =X(s)
F(s) Method gives system dynamics
representation equivalent to
Ordinary differential equations State equations
Interchangeable: Can convert transfer function to differential equations
M.D. Bryant ME 344 notes 03/25/08
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Transfer Function G(s) Describes dynamics in operational sense Dynamics encoded in G(s) Ignore initial conditions (I.C. terms are
transient & decay quickly) Transfer function, for inputoutput
operation, deals with steady state terms
h(t)
H(s)
x(t)
X(s)
G(s)
Example: Speed of automobile
o Output: speed x(t) expressed as X(s) o Output determined by input, system
dynamics & initial conditions o Input: gas pedal depression h(t) o Dynamics: fuel system delivery,
motor dynamics & torque, transmission, wheels, car translation, mass, wind drag, etc.
o Initial conditions (initial speed) influence on current speed diminishes over time, thus ignore
M.D. Bryant ME 344 notes 03/25/08
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Transfer Function Example: 2nd order system
Differential equation at steady state (can ignore initial conditions)
x + 2 n x + 2n x = h(t)
Apply Laplace transform, define
X(s) =L{x(t); t s}, H(s) =L{h(t); t s }
Result: algebraic equation
(s2 + 2n s + 2n )X(s) = H(s)
Transfer function G(s): ratio of output
X(s) to input H(s)
G(s) = X(s)H(s) = 1
s2 + 2 n s + 2n
Note: o characteristic equation in denominator o denominator roots => eigenvalues o transfer functions eigenvalues called
poles
M.D. Bryant ME 344 notes 03/25/08
11
Example: First order system
!
" x + x = f (t)
Apply Laplace transform, define
X(s) =L{x(t); t s}, F(s) =L{f(t); t s }
L{ x ; t s} +L{x(t); t s} =L{f(t); t s }
Result: algebraic equation sX(s) + X(s) = F(s)
Transfer function
G(s) =X(s)/F(s) =
!
1
"s+1
Note: o characteristic equation in denominator o transfer functions eigenvalues = poles o p1 = 1 =  1/
M.D. Bryant ME 344 notes 03/25/08
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Transfer Function from State Equations
Matrix form
!
x = Ax + f (t)
Explicit equation form
!
x k= a
jkj=1
n
" xj+ f
k(t)
Define
Xk(s)=L {xk(t); t s }, Fk(s)=L {fk(t); t s }
Apply Laplace transform
!
sX(s) = AX(s)+F(s)
or
!
sXk(s) = a
jkj=1
n
" Xj(s)+F
k(s)
M.D. Bryant ME 344 notes 03/25/08
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Rearrange matrix form:
!
sI " A[ ]X(s) = F(s)
Solve, via Cramers rule:
!
Xk(s) =
det sI " A[ ]kth column replaced by F ( s )
{ }det sI " A[ ]
Transfer function, define which output
Xk(s) and which input Fj(s):
!
Gkj(s) =
Xk(s)
Fj(s)
Solve, via previous result with all
components of F(s) zero, except Fj(s)
M.D. Bryant ME 344 notes 03/25/08
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Example: differential equations from transfer function:
!
G(s) =X(s)
F(s)=
2s+ 5
s4
+ 3s3
+ 2s2
+ 2s+1
Cross multiply ratios:
!
s4 + 3s 3 + 2s 2 + 2s+1( )X(s) = 2s+ 5( )F(s)
!
s4
X + 3s3
X + 2s2
X + 2sX + X = 2sF + 5F Treat: s => d/dt
!
d4x
dt4
+ 3d3x
dt3
+ 2d2x
dt2
+ 2dx
dt+ x = 2
df
dt+ 5 f (t)
M.D. Bryant ME 344 notes 03/25/08
15
Block Diagrams
h(t)
H(s)
x(t)
X(s)
G(s)
Another way to represent system dynamics pictorially
Weakness: lacks causality information
Shows signal flow through system
Transfer function G(s) inside block
Output:
X(s) = G(s) H(s)
transfer function times input H(s)
Can assemble blocks into system model:
M.D. Bryant ME 344 notes 03/25/08
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Cascaded Blocks
H(s) X1(s)
G1(s)X2(s)
G2(s)X(s)
G3(s)
Output = input to next Models stringing of components
M.D. Bryant ME 344 notes 03/25/08
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Example: stereo system
CD player amplifier
speaker
CD player amplifier speakers
CD player:
!
G1(s) =
X1(s)
H (s)
Input: H(s) from CD laser reader
Output: CD voltage X1(s)
Power amplifier:
!
G2(s) =
X2(s)
X1(s)
Input: CD output voltage X1(s) Output: amp voltage X2(s)
Speakers:
!
G3(s) =
X(s)
X2(s)
Input: amp voltage X2(s) Output: sound, acoustic
pressure X(s)
M.D. Bryant ME 344 notes 03/25/08
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H(s) X1(s)
G1(s)X2(s)
G2(s)X(s)
G3(s)
Overall transfer function is product of block transfer functions:
!
G(s) =X(s)
H (s)=X(s)
X2(s)
X2(s)
X1(s)
X1(s)
H (s)=G
1(s)G
2(s)G
3(s)
Note:
!
G1(s) =
X1(s)
H (s),
!
G2(s) =
X2(s)
X1(s)
,
!
G3(s) =
X(s)
X2(s)
M.D. Bryant ME 344 notes 03/25/08
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Example H(s) X(s)3
s+1
2s !1
s2
+ 4s+1
Transfer function:
!
G(s) =X(s)
H (s)=G
1(s)G
2(s) =
3
s+1"2s #1
s2
+ 4s+1
!
G(s) =3(2s "1)
(s+1)(s2
+ 4s+1)=
6s " 3
s3
+ 5s2
+ 5s+1
Poles = roots of denominator (values of s
such that transfer function becomes infinite)
p1 = 1, p2,p3 = 2 3
Zeros = roots of numerator (values of s such that transfer function becomes 0) z1 = 1/2
M.D. Bryant ME 344 notes 03/25/08
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Summer (Summing Junction)
X1(s)
X2(s)X(s)+
+

X3(s)
Output = sum of inputs Sign on input => sign in equation Output X(s) =  X1(s) + X2(s) + X3(s)
M.D. Bryant ME 344 notes 03/25/08
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Example: Feedback Control System
R(s)X(s)
G(s)
H(s)
E(s)+

Goal: Closed loop transfer function
!
Gcl(s) =
X(s)
R(s)
Formulate:
!
E(s) = R(s)"H (s)X(s)
!
X(s) =G(s)E(s) Eliminate E(s):
!
X(s) =G(s)E(s) =G(s) R(s)"H (s)X(s)[ ]
Solve for closed loop X(s)/R(s):
!
Gcl(s) =
X(s)
R(s)=
G(s)
1+G(s)H (s)
M.D. Bryant ME 344 notes 03/25/08
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Example: Feedback Controller with Disturbance
Gc(s)R(s)X(s)
H(s)
E(s)+

Gp(s)++
D(s)
Closed loop transfer function, reference
input (temporarily set D(s) = 0 )
!
Gcl(s) =
X(s)
R(s)=
Gc(s)G
p(s)
1+Gc(s)G
p(s)H (s)
Transfer function, disturbance (set R(s) = 0)
!
Gd(s) =
X(s)
D(s)=
Gp(s)
1+Gc(s)G
p(s)H (s)
M.D. Bryant ME 344 notes 03/25/08
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Linear system, with both, sum outputs:
!
X(s) =Gcl(s)R(s)+G
d(s)D(s)
!
X(s) =G
c(s)G
p(s)
1+Gc(s)G
p(s)H (s)
R(s)
!
+G
p(s)
1+Gc(s)G
p(s)H (s)
D(s)