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Control Systems 0108400
Lect 03 Math Models, LT, Electric
Element, Transfer Function
Dr. M. N. Najem
Summer 2010-2011
May 18, 2011
Dr. Najem 2
Mathematical models
• In the previous lecture after obtaining a
schematic, the control engineer makes
simplifying assumptions in order to keep
the ensuing model manageable and still
approximate physical reality.
• Next step is to develop mathematical
models from schematics of physical
systems
Dr. Najem 3
Mathematical models …
• Transfer functions in frequency domain (classical approach).
• State-space in time domain (modern approach).
• In every case the first step in developing a mathematical model is to apply the fundamental physical laws of science an engineering:– In modeling electric networks: Ohm, KVL, KCL, Nodal,
Equivalent circuit, ..
– in modeling mechnical systems: Newton laws (forces, torques)
• From these equations we obtaing the relationship between system’s output and input
Dr. Najem 4
Mathematical models …
• Differential equations can describe the relationship between the input and output of a system.
• The form of the D.E and its coefficients are a formulation of the system.
• Although the D.E relates the system to its input and output, it is not a satisfying representation from a system perspective. Look at a LTI nth order D.E:
Dr. Najem 5
Mathematical models …
• The system parameters (the coefficients), the output, as well as the input appear through out the D.E.
)()(
...)()(
)()(
...)()(
011
1
1011
1
1 trbdt
tdrb
dt
trdb
dt
trdbtca
dt
tdca
dt
tcda
dt
tcdm
m
mm
m
mn
n
nn
n
• Prefer a mathematical representation such that the input, output, and system are distinct and separate parts; ( conveniently represent interconnection of several subsystems). Mathematical function that appears inside the block called the transfer function, denoted, H(s), or G(s).
SystemInput
r(t)
Output
c(t)
Dr. Najem 6
Review of Laplace transform
Dr. Najem 7
Laplace transform
• Addressed in MATH IV course
00
01)(
)()()(2
1)()]([
)()()]([
1
0
t
ttu
tutfdsesFj
tfsFL
dtetfsFtfL
j
j
st
st
• Multiplication f(t) by u(t) yields a time function that is zero
for t < 0.
Dr. Najem 8
Laplace transform
• Refer to Laplace transform in Math IV to
review the various operations and to relate
to Laplace transform tables.
• Inverse of Laplace transform
– Partial-fraction expansion
– Use of various theroems
• Also, refer to solving D.E.s using Laplace
transform.
Dr. Najem 9
LT examples
• Presented and discussed on board
Dr. Najem 10
Review of resistive electric
networks
Dr. Najem 11
Some elements
Independent sources: Resistor:
Dr. Najem 12
Superposition example
2kW1kW
2kW12V
I0
2mA
4mA
– +
• Given the following circuit
• Calculate I0 using superposition
Superposition example …
Dr. Najem 13
• Solve the problem by leaving one independent source at a time and
opening the other Independent current sources and short circuiting
the other independent voltage sources
2kW1kW
2kW
I’0
2mA
I’0 = -4/3 mA
Open current source
Short circuit
voltage source
Superposition example …
Dr. Najem 14
2kW1kW
2kW
I’’0
4mA
I’’0 = 0
Short circuit
Open current source
Superposition example …
Dr. Najem 15
2kW1kW
2kW12V
I’’’0
– +
I’’’0 = -4 mA
Final result:I’0 = -4/3 mA
I’’0 = 0
I’’’0 = -4 mA
I0 = I’0+ I’’0+ I’’’0 = -16/3 mA
Open current source
Open
current
source
Dr. Najem 16
Use of KVL and KCL
Loop1:
Node1:
Node1
Loop1
17
Equivalent circuits USE THEVENIN TO COMPUTE Vo
W kRTh 3/1024//2
0)(246
2
211
2
IIkkIV
mAI
circuitbuttonleftonAnalysisLoop
mAmAI
I3
5
6
26 21
][3/3243/20*2*4 21 VVIkIkVOC
VocCalculate V0
using Voltage
division
Dr. Najem 18
Nodal analysis and Mesh analysis
Mesh analysis
224111
213212111 )()(
SS
S
VIRIRV
IIRIIRIRVSolve for I1 and I2.
Dynamic networks
Dr. Najem 19
Dr. Najem 20
Characteristics of dynamic network
• Dynamic Elements Ohm’s Law: ineffective
• Inductor:
Dr. Najem 21
Characteristics of dynamic network ...
• Dynamic Elements Ohm’s Law: ineffective• Capacitor:
Dr. Najem 22
Characteristics of dynamic network ...
• Example:
t
tRidtiCdt
tdiLtv
0)()(
1)()(
then
If
v(t)
Dr. Najem 23
Dynamic relationships into
Algebraic operation LT
• Using Laplace Transform on previous circuit
)()(1
)()(
)()0()(1
))0()(()]([)()1(
sRIsICs
sILs
sRIs
i
s
sI
CissILtLsVs
• Then generalize resistor R into impedance Z
LssZ )(1Cs
sZ1
)(2
Dr. Najem 24
RsZsZ
sVsI
sRIsIsZsIsZsV
s
s
)()(
)()(
)()()()()()(
21
21
Dynamic relationships into
Algebraic operation LT …
Dr. Najem 25
Dynamic elements Laplace
transform models
• Capacitor LT model
Dr. Najem 26
• Inductor LT model
Dynamic elements Laplace
transform models
Dr. Najem 27
Resistor LT model
• V(s) = RI(s)
Sources as LT models
Dr. Najem 28
Laplace and analysis methods
• Laplace application to analysis techniques
analysismesh and analysis Nodal
circuit Equivalent
KCL and KVL
ionsuperposit
Law Ohms dGeneralize
These will be
applied as in
Resistive Networks
Key: Laplace transform models of (dynamic) elements.
Transfer Functions
Dr. Najem 29
Dr. Najem 30
System representations
• Mathematical form of representing (modeling) the system.
• In this course we are concerned with the following system representation methods:
– Differential equations
– Transfer functions
– State-space
Dr. Najem 31
Transfer Functions• System analysis: emphasize is on relation between
input and output (using blocks)
• Circuit analysis: detailed analysis (voltages, branch currents)
• System analysis: how the system processes the input to form the output, or how the system transforms the input into output.
• Output: variable to be controlled
• Input: variable used and to be adjusted to change or influence the output
Dynamic systemInput
r(t)
Output
c(t)
Dr. Najem 32
Transfer Function H(s) or G(s)
• Gives Quantitative Description of ‘ how
the system processes the input to form the
output’.
Dr. Najem 33
input to H(s) or G(s)
• Input is (t) impulse fucntion
• The resultant output y(t) is called: the unit impulse response.
• In this case: X(s) = L [ (t)] = 1 and
Y(s) = Laplace Transform of the unit impulse response
Therefore, H(s) = Y(s)/X(s) = Y(s)
Therefore, the transfer function of a system is the Laplace transform of the unit impulse response of the system
Dr. Najem 34
Facts on Transfer Functions• Independent of input, a property of the system structure
and parameters.
• Obtained with zero initial conditions.
• Generates a mathematical model of the system and algebraically relates the representation of the output to the representation of the input,.
• Applies only on LTI systems.
• Rational Function of s (Linear, lumped, fixed)
• H(s): Transfer function can be easily converted into frequency response function of the system H(jw) or H( j2f ). Simply replace s = jw.
• |H( j2f )| or |H( j )|: amplitude response function and H(j2f) or H( j ): Phase response function
Dr. Najem 35
Properties of Transfer Function
• Properties of Transfer Function for Linear, Lumped stable systems.
)(
)(
...
...)(
01
1
01
1
sD
sN
asasa
bsbsbsH
nn
nn
mm
mm
• Corresponding differential equations:
)(...)()(
)(...)()(
01
)1(
1
)(
01
)1(
1
)(
txbdt
txdb
dt
txdb
tyadt
tyda
dt
tyda
m
m
mm
m
m
n
n
nn
n
n
coef. ai, bj: all real! Why? Results from real system components.
Dr. Najem 36
• Differential equations do not separate the representation of input, output, and system; transfer functions do.
• Replacing D.Es. with algebraic equations simplifies the representation of individual subsystems and simplifies modeling interconnected subsystems.
• Roots of N(s), D(s): are real, or complex conjugate pairs.
• Zeros of the transfer function:= roots of N(s)
• Poles of the transfer function:= roots of D(s)
Properties of Transfer Function …
1,2:
)1)(2()()13()( 2
poles
sssDsssD
Dr. Najem 37
H(s) of BIBO stable system
H(s) = G(s) = N(s) / D(s)
• Degree of N(s) must be Degree of D(s).
– If degree N(s) > Degree D(s) then divide:
)(
)('...)( 01
sD
sNcscscsH k
k
• but, under a bounded-input x(t) = u(t) (unit step function) => L[u(t)] = X(s) = 1/s, then:
Dr. Najem 38
H(s) of BIBO stable system …
...)(...)(
)(
)('...)(
1
01
1
tcty
ssD
sN
s
ccscsY k
k
( )(t not bounded!
so y(t) unbounded)
Dr. Najem 39
H(s) of BIBO stable system …
• Poles: must lie in the left half of the s-
plan.
0)Re(
))...()(()( 21
j
nssssD
then:
• Question: Any restriction on zeros?
• Answer: No (for BIBO stable system)
WHY? …..
Dr. Najem 40
– Electric Networks Transfer Functions
– Translational Mechanical Systems Transfer Functions
– Rotational Mechanical Systems Transfer Functions
– Transfer Functions for Systems with Gears
– Electromechanical Systems Transfer Functions
– Electric/Mechanical Circuit Analogs
– You do linearization on your own
Next 2 lectures modeling of: