Modeling General Concepts
Dr. Kevin Craig
Professor of Mechanical Engineering
Rensselaer Polytechnic Institute
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Modeling: General Concepts
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Classification of System Inputs
Pure and Ideal Elements vs. Real Devices; Ideal vs. Real Sources
Transfer Functions
Linearization of Nonlinear Physical Effects
Loading Effects
Block Diagram
Time Domain & Frequency Domain
– Step Response & Frequency Response
• State-Space Representation
• Poles and Zeros of Transfer Functions
• Sensitivity Analysis
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Classification of System Inputs
System Inputs
Initial Energy
Storage
Kinetic
External Driving
Potential
Input / System / Output
Concept:
Classification
of
System Inputs
Deterministic
Stationary
Transient
Periodic
Sinusoidal
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Random
Unstationary
"Almost
Periodic"
NonSinusoidal
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• Input – some agency which can cause a system to respond.
• Initial energy storage refers to a situation in which a
system, at time = 0, is put into a state different from some
reference equilibrium state and then released, free of
external driving agencies, to respond in its characteristic
way. Initial energy storage can take the form of either
kinetic energy or potential energy.
• External driving agencies are physical quantities which
vary with time and pass from the external environment,
through the system interface or boundary, into the system,
and cause it to respond.
• We often choose to study the system response to an
assumed ideal source, which is unaffected by the system to
which it is coupled, with the view that practical situations
will closely correspond to this idealized model.
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• External inputs can be broadly classified as deterministic or
random, recognizing that there is always some element of
randomness and unpredictability in all real-world inputs.
• Deterministic input models are those whose complete time
history is explicitly given, as by mathematical formula or a table
of numerical values. This can be further divided into:
– transient input model: one having any desired shape, but existing
only for a certain time interval, being constant before the
beginning of the interval and after its end.
– periodic input model: one that repeats a certain wave form over
and over, ideally forever, and is further classified as either
sinusoidal or non-sinusoidal.
– almost periodic input model: continuing functions which are
completely predictable but do not exhibit a strict periodicity, e.g.,
amplitude-modulated input.
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• Random input models are the most realistic input models
and have time histories which cannot be predicted before
the input actually occurs, although statistical properties of
the input can be specified.
• When working with random inputs, there is never any hope
of predicting a specific time history before it occurs, but
statistical predictions can be made that have practical
usefulness.
• If the statistical properties are time-invariant, then the input
is called a stationary random input. Unstationary random
inputs have time-varying statistical properties. These are
often modeled as stationary over restricted periods of time.
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Pure and Ideal Elements vs. Real Devices
• A pure element refers to an element (spring, damper,
inertia, resistor, capacitor, inductor, etc.) which has only
the named attribute.
• For example, a pure spring element has no inertia or
friction and is thus a mathematical model (approximation),
not a real device.
• The term ideal, as applied to elements, means linear, that
is, the input/output relationship of the element is linear, or
straight-line. The output is perfectly proportional to the
input.
• A device can be pure without being ideal and ideal without
being pure.
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• From a functional engineering viewpoint, nonlinear
behavior may often be preferable, even though it leads to
difficult equations.
• Why do we choose to define and use pure and ideal
elements when we know that they do not behave like the
real devices used in designing systems? Once we have
defined these pure and ideal elements, we can use these as
building blocks to model real devices more accurately.
• For example, if a real spring has significant friction and
mass, we model it as a combination of pure/ideal spring,
mass, and damper elements, which may come quite close
in behavior to the real spring.
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Physical Model of a Real Spring
Ks
f, x
M
B
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Ks
B
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Ideal vs. Real Sources
• External driving agencies are physical quantities which
pass from the environment, through the interface into the
system, and cause the system to respond.
• In practical situations, there may be interactions between
the environment and the system; however, we often use the
concept of ideal source.
• An ideal source (force, motion, voltage, current, etc.) is
totally unaffected by being coupled to the system it is
driving.
• For example, a “real” 6-volt battery will not supply 6 volts
to a circuit! The circuit will draw some current from the
battery and the battery’s voltage will drop.
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Transfer Functions
• Definition and Comments
– The transfer function of a linear, time-invariant,
differential equation system is defined as the ratio of
the Laplace transform of the output (response function)
to the Laplace transform of the input (driving function)
under the assumption that all initial conditions are zero.
– By using the concept of transfer function, it is possible
to represent system dynamics by algebraic equations in
s. The highest power of s in the denominator
determines the order of the system.
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• Transfer Function
2
d
x
2
D x 2
dt
x
∫ ⎡⎣ ∫ ( x ) dt ⎤⎦dt
2
D
dx
Dx dt
x
∫ (x)dt
D
– Differential Operator
– Consider the spring-mass-damper system
Kx
B(dx/dt)
B
K
+x
M
F(t)
M
+x
F(t)
Physical Model
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Free-Body Diagram
(x is measured from the
static equilibrium
position)
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Apply Newton’s
Second Law
Using the
differential
operator D we
can transform
the differential
equation to an
algebraic
equation and
then write the
transfer function
for the system.
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d2x
∑ Fx = M dt 2 = Mx
F ( t ) − Bx − Kx = Mx
Mx + Bx + Kx = F ( t )
Mx + Bx + Kx = F ( t )
Mathematical Model
Differential Equation
d2x
dx
MD x ≡ M 2 =Mx
BDx ≡ B = Bx
dt
dt
MD 2 x + MDx + Kx = F ( t ) Algebraic Equation
2
2
MD
( + BD + K ) x = F ( t )
x
1
=
F MD 2 + BD + K
Transfer Function
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– The transfer function is a property of a system itself,
independent of the magnitude and nature of the input or
driving function.
– The transfer function gives a full description of the dynamic
characteristics of the system.
– The transfer function does not provide any information
concerning the physical structure of the system; the transfer
functions of many physically different systems can be
identical.
– If the transfer function of a system is known, the output or
response can be studied for various forms of inputs with a
view toward understanding the nature of the system.
– If the transfer function of a system is unknown, it may be
established experimentally by introducing known inputs and
studying the output of the system.
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• Convolution Integral
– For a linear time-invariant system the transfer function
G(s) is
Y(s)
G(s) =
X(s)
where X(s) is the Laplace transform of the input and
Y(s) is the Laplace transform of the output, assuming
all initial conditions are zero.
– The inverse Laplace transform is given by the
convolution integral:
t
t
0
0
y(t) = ∫ x( τ)g(t − τ)dτ = ∫ g(τ)x(t − τ)dτ
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t<0
g(t) = 0
x(t) = 0
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• Impulse-Response Function
– The Laplace transform of the response of a system to a
unit-impulse input, when the initial conditions are zero, is
the transfer function of the system, i.e., Y(s) = G(s).
– The inverse Laplace transform of the system transfer
function, g(t), is called the impulse-response function. It is
the response of a linear system to a unit-impulse response
when the initial conditions are zero.
– The transfer function and the impulse-response function
of a linear, time-invariant system contain the same
information about the system dynamics.
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– Experimentally, one can excite a system at rest with an
impulse input (a pulse input of very short duration
compared with the significant time constants of the
system) and measure the response. This response is the
impulse-response function, the Laplace transform of
which is the transfer function of the system.
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• Three Basic Input-Output Relationships
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• Step Response and Impulse Response
– By a step input of any variable we mean a situation where
the system is “at rest” at time t = 0 and we instantly change
the input quantity, from wherever it was just before t = 0,
by a given amount, either positive or negative, and then
keep the input constant at this new value “forever.”
– The integral of a step input is a ramp and the derivative of a
step input is an impulse.
– An impulse has an infinite magnitude and zero duration and
is mathematical fiction and does not occur in physical
systems.
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Explanation
of the
Impulse Function
If the magnitude of a pulse
input to a system is very
large and its duration is very
short compared to the
system’s speed of response,
then we can approximate the
pulse input by an impulse
function.
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Step Responses
of the
Three Basic Elements
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• Frequency Response
– If the input to a linear system is a sine wave, the steadystate output (after the transients have died out) is also a sine
wave with the same frequency, but with a different
amplitude and phase angle. Both amplitude ratio and phase
angle change with frequency.
– The following plots show the frequency response of the
three basic elements.
– Note that a decibel dB = 20 log10 (amplitude ratio).
• 0 dB is an amplitude ratio of 1
• + 6 dB is an amplitude ratio of 2
• - 6 dB is an amplitude ration of ½
• + 20 dB is an amplitude ratio of 10
• - 20 dB is an amplitude ratio of 1/10.
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Linearization of Nonlinear Physical Effects
• Many real-world nonlinearities involve a “smooth”
curvilinear relation between an independent variable x and
y = f(x)
a dependent variable y:
• A linear approximation to the curve, accurate in the
neighborhood of a selected operating point, is the tangent
line to the curve at this point.
• This approximation is given conveniently by the first two
terms of the Taylor series expansion of f(x):
2
2
(x − x)
df
df
y = f (x) +
+"
(x − x) + 2
dx x = x
dx x = x
2!
df
−
≈
+
y
y
(x − x)
df
dx x = x
y≈ y+
(x − x)
dx x = x
yˆ = Kxˆ
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• For example, in liquid-level control systems, when the tank
is not prismatic, a nonlinear volume/height relationship
exists and causes a nonlinear system differential equation.
For a conical tank of height H and top radius R we would
have:
πR 2 3
V=
h
2
3H
πR 2 h 3 πR 2 h 2 ˆ
V≈
h
+
2
2
3H
H
• Often a dependent variable y is related nonlinearly to
several independent variables x1, x2, x3, etc. according to
the relation: y=f(x1, x2, x3, …).
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• We may linearize this relation using the multivariable form of
the Taylor series:
∂f
y ≈ f ( x 1 , x 2 , x 3 , ") +
∂x1
∂f
+
∂x 3
y ≈ y+
x1 ,x 2 ,x 3
∂f
( x1 − x 1 ) +
∂x 2
,"
( x2 − x2 )
x1 ,x 2 ,x 3 ,"
( x3 − x3 ) + "
x1 ,x 2 ,x 3 ,"
∂f
∂x1
xˆ 1 +
x1 ,x 2 ,x 3 ,"
∂f
∂x 2
xˆ 2 +
x1 ,x 2 ,x 3 ,"
∂f
∂x 3
xˆ 3 + "
x1 ,x 2 ,x 3 ,"
yˆ = K1xˆ 1 + K 2 xˆ 2 + K 3 xˆ 3 + "
The partial derivatives can be thought of as the sensitivity of the
dependent variable to small changes in that independent variable.
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• For example, in a ported gas-filled piston/cylinder where
gas mass, temperature, and volume are all changing, the
perfect gas law gives us for pressure p:
RTM
V
RTM RM
RT
RMT
p≈
T − T) +
M − M) −
V − V)
+
(
(
(
V
V
V
V
p=
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Example: Magnetic Levitation
System
Applications
include magnetic
bearings for
vacuum pumps,
conveyor systems in
clean rooms, highspeed levitated
trains, and
electromagnetic
automotive valve
actuators.
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Electromagnet
Phototransistor
Infrared LED
Levitated Ball
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Magnetic Levitation System
⎛ i2 ⎞
f ( x,i ) = C ⎜ 2 ⎟
⎝x ⎠
Equation of Motion:
⎛ i2 ⎞
mx = mg − C ⎜ 2 ⎟
⎝x ⎠
At Equilibrium:
⎛i ⎞
mg = C ⎜ 2 ⎟
⎝x ⎠
2
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Linearization:
⎛ i2 ⎞
⎛ i2 ⎞
⎛ 2i 2 ⎞
⎛ 2i
C ⎜ 2 ⎟ ≈ C ⎜ 2 ⎟ − C ⎜ 3 ⎟ xˆ + C ⎜ 2
⎝x ⎠
⎝x ⎠
⎝ x ⎠
⎝x
2
2
⎛
⎞
⎛
⎞
⎛ 2i
i
2
i
mxˆ = mg − C ⎜ 2 ⎟ + C ⎜ 3 ⎟ xˆ − C ⎜ 2
⎝x ⎠
⎝ x ⎠
⎝x
2
⎛
⎞
⎛ 2i
2
i
mxˆ = C ⎜ 3 ⎟ xˆ − C ⎜ 2
⎝ x ⎠
⎝x
⎞ˆ
⎟i
⎠
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⎞ˆ
⎟i
⎠
⎞ˆ
⎟i
⎠
Use of Experimental Testing in Multivariable Linearization
f m = f (i, y)
∂f
∂f
f m ≈ f ( i0 , y0 ) +
( y − y0 ) +
∂y i0 ,y0
∂i
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( i − i0 )
i0 ,y0
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Block Diagrams
• A block diagram of a system is a pictorial representation of
the functions performed by each component and of the
flow of signals. It depicts the interrelationships that exist
among the various components.
• It is easy to form the overall block diagram for the entire
system by merely connecting the blocks of the components
according to the signal flow. It is then possible to evaluate
the contribution of each component to the overall system
performance.
• A block diagram contains information concerning dynamic
behavior, but it does not include any information on the
physical construction of the system.
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• Many dissimilar and unrelated systems can be represented
by the same block diagram.
• A block diagram of a given system is not unique. A
number of different block diagrams can be drawn for a
system, depending on the point of view of the analysis.
• Closed-Loop System Block Diagram:
Σ
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Σ
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B(s)
= G c (s)G(s)H(s)
E(s)
C(s)
= G c (s)G(s)
E(s)
Open-Loop Transfer Function
Feedforward Transfer Function
Closed-Loop Transfer Functions
G c (s)G(s)
C(s)
=
R(s) 1 + G c (s)G(s)H(s)
G c (s)G(s)H(s) >> 1
C(s)
G(s)
=
D(s) 1 + G c (s)G(s)H(s)
G c (s)G(s)H(s) >> 1
G c (s)H(s) >> 1
C(s)
1
⇒
R(s)
H(s)
C(s)
⇒0
D(s)
G(s)
C(s) =
[G c (s)R(s) + D(s)]
1 + G c (s)G(s)H(s)
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• Blocks can be connected in series only if the output of one
block is not affected by the next following block. If there
are any loading effects between components, it is
necessary to combine these components into a single
block.
• In simplifying a block diagram, remember:
– The product of the transfer functions in the feedforward
direction must remain the same.
– The product of the transfer functions around the loop
must remain the same.
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Some Rules of Block Diagram Algebra
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Loading Effects
• The unloaded transfer function is an incomplete
component description.
• To properly account for interconnection effects one must
know three component characteristics:
– the unloaded transfer function of the upstream
component
– the output impedance of the upstream component
– the input impedance of the downstream component
• Only when the ratio of output impedance over input
impedance is small compared to 1.0, over the frequency
range of interest, does the unloaded transfer function give
an accurate description of interconnected system behavior.
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u
G1(s)
G2(s)
y
⎡
⎤
Y(s) ⎢
1 ⎥
= ⎢G1 (s)
⎥ G 2 (s)
Z
U(s) ⎢
1 + o1 ⎥
Zi2 ⎦⎥
⎣⎢
Only if
Zo1
<< 1
Zi2
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for the frequency range of interest will
loading effects be negligible.
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• In general, loading effects occur because when analyzing
an isolated component (one with no other component
connected at its output), we assume no power is being
drawn at this output location.
• When we later decide to attach another component to the
output of the first, this second component does withdraw
some power, violating our earlier assumption and thereby
invalidating the analysis (transfer function) based on this
assumption.
• When we model chains of components by simple
multiplication of their individual transfer functions, we
assume that loading effects are either not present, have
been proven negligible, or have been made negligible by
the use of buffer amplifiers.
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Analog Electronics
Example:
Loading Effects
⎡ Vin ⎤ ⎡ RCs + 1 −R ⎤ ⎡ Vout ⎤
⎢ i ⎥ = ⎢ Cs
⎥⎢i ⎥
1
−
⎦ ⎣ out ⎦
⎣ in ⎦ ⎣
Vout
1
1
when iout = 0
=
=
Vin RCs + 1 τs + 1
Zout =
Zin =
Vout
i out
Vin
iin
=
Vin =0
=
iout =0
R
RCs + 1
RCs + 1
Cs
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Resistor 15 KΩ
Vin
Vout
Capacitor 0.01 μF
RC Low-Pass Filter
Output Impedance
Input Impedance
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Resistor 15 KΩ
Vin
Capacitor 0.01 μF
Resistor 15 KΩ
Capacitor 0.01 μF
Vout
2 RC Low-Pass Filters in Series
Vout
⎛ 1 ⎞⎛ 1 ⎞
≠ G(s)1−unloaded G(s) 2−unloaded = ⎜
⎟⎜
⎟
Vin
⎝ RCs + 1 ⎠⎝ RCs + 1 ⎠
Vout
= G(s)1−loaded G(s) 2−unloaded
Vin
⎛
1
⎛ 1 ⎞ ⎜⎜
=⎜
⎟
+
RCs
1
⎝
⎠ ⎜ 1 + Zout −1
⎜
Zin −2
⎝
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Only if
Zout-1 << Zin-2
for the frequency
range of interest will
loading effects be
negligible.
⎞
⎟⎛ 1 ⎞
1
⎟⎜
=
⎟
2
+
RCs
1
⎟⎝
⎠ ( RCs + 1) + RCs
⎟
⎠
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Time Domain & Frequency Domain
• Time domain and frequency domain are two ways of
looking at the same dynamic system. They are
interchangeable, i.e., no information is lost in changing
from one domain to another.
• They are complementary points of view that lead to a
complete, clear understanding of the behavior of a dynamic
engineering system.
• Roughly speaking, in the time domain we measure how
long something takes, whereas in the frequency domain we
measure how fast or slow it is.
• These are two ways of viewing the same thing!
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– When you hear music and see color, you are experiencing the
frequency domain. It is all around you, just like the time
domain.
– The frequency domain is a kind of hidden companion to our
everyday world of time. We describe what happens in the
time domain as temporal and in the frequency domain as
spectral.
– Most signals and processes involve both fast and slow
components happening at the same time. Frequency domain
analysis separates these components and helps to keep track of
them.
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• Mechanical Spectrum
–
–
–
–
Second hand of a clock: 1 rpm
Audio CDs: 200 to 500 rpm
Dentist’s drill: 400,000 rpm
Two-foot diameter tire on a car traveling at 60 mph: 840
rpm
– Earth’s rotation: 0.00069 rpm (1000 mph at the equator!)
– 1 Hz = 60 rpm = 2π rad/sec = 6.28 rad/sec. All have
dimensions 1/time.
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Mechanical
Spectrum
Electromagnetic
Spectrum
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• Electromagnetic Spectrum
– Electromagnetic effects can be described in the frequency
domain as well.
– Electromagnetic waves travel at the speed of light c, which
depends on the medium (fastest in a vacuum, slower in other
media).
– The frequency of vibration f depends on the wavelength λ of
the electromagnetic phenomenon and the speed of
propagation c of the medium according to f = c/λ.
– Long-wavelength electromagnetic waves are radio waves (see
spectrum diagram). Frequencies range from a few kHz to 300
GHz.
– Higher frequencies are emitted by thermal motion, which we
call infrared radiation.
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– Frequencies of visible light range from 440 THz (red
light) to 730 THz (violet light).
– Humans perceive different frequencies within the visible
light spectrum as different colors. Unlike the ear, the
eye has a nonlinear response to combinations of
frequencies.
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Electromagnetic Spectrum
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Radio Spectrum
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• Time Domain
– The time domain is a record of the response of a dynamic
system, as indicated by some measured parameter, as a
function of time. This is the traditional way of observing the
output of a dynamic system.
– An example of time response is the displacement of the mass
of the spring-mass-damper system versus time in response to
the sudden placement of an additional mass (here 50% of the
attached mass) on the attached mass. The resulting response
is the step response of the system due to the sudden
application of a constant force to the attached mass equal to
the weight of the additional mass. Typically when we
investigate the performance of a dynamic system we use as
the input to the system a step input.
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• Frequency Domain
– Over one hundred years ago, Jean Baptiste Fourier
showed that any waveform that exists in the real world
can be generated by adding up sine waves.
– By picking the amplitudes, frequencies, and phases of
these sine waves, one can generate a waveform
identical to the desired signal.
– While the situation presented on the next page is
contrived, it does illustrate the idea. On the left is a
“real-world” signal and on the right are three signals,
the sum of which is the same as the “real-world” signal.
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– A more convincing example is to observe that a square
wave can be represented by a series of sine waves of
different amplitudes, frequencies, and phase angles. In
the diagram below, a square wave has been
approximated with only two sine waves. As more sine
waves are added to the series, the approximation
becomes better and better.
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– Any real-world signal can be broken down into a sum
of sine waves and this combination of sine waves is
unique. Any real-world signal can be represented by
only one combination of sine waves.
– In the diagram, a waveform is represented as the sum of
two sine waves.
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– In figure (a) is a three-dimensional graph of this
addition of sine waves. The three axes are time,
amplitude, and frequency. The time and amplitude axes
are familiar from the time domain. The third axis,
frequency, allows us to visually separate the sine waves
that add to give us the complex waveform.
– If we view this three-dimensional graph along the
frequency axis, we get the view shown in figure (b).
This is the time-domain view of the sine waves.
Adding them together at each instant of time gives the
original waveform.
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– Now view the three-dimensional graph along the time
axis, as in figure (c). Here we have axes of amplitude
versus frequency. This is what is called the frequency
domain. Every sine wave we separated from the input
appears as a vertical line. Its height represents its
amplitude and its position represents its frequency. We
know each line represents a sine wave and so we have
uniquely characterized our input signal in the frequency
domain. This frequency domain representation of our
signal is called the spectrum of the signal. Each sine
wave line of the spectrum is called a component of the
total signal.
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– It is most important to understand that we have neither
gained nor lost information, we are just representing it
differently.
– You can now see why a sine wave is the second
important signal, the step input being the other, used to
excite a dynamic system.
– Since any real-world signal can be represented by the
sum of sine waves, if we can predict the response of a
system to a sine wave input of varying frequency,
amplitude, and phase angle, then we can predict the
response of the system to any real-world signal once we
know the frequency spectrum of that real-world signal.
Modeling General Concepts
K. Craig
61
The Three Basic Element Input-Output Relationships
Resistor
Damper
Inductor
Mass
Capacitor
Spring
Modeling General Concepts
K. Craig
62
qin = i, v
qout = e, f
Resistor, Damper
1
1
i= e v= f
R
B
e = Ri f = Bv
Capacitor, Spring
de
i = C = CDe
dt
1
e=
i
CD
Inductor, Mass
di
e = L = LDi
dt
1
i=
e
LD
Modeling General Concepts
1 df 1
v=
= Df
K dt K
K
f= v
D
dv
f =M
= MDv
dt
1
v=
f
MD
K. Craig
63
Step Response
Modeling General Concepts
K. Craig
64
Step Response
Modeling General Concepts
K. Craig
65
Step Response
Modeling General Concepts
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66
• Step Response and Impulse Response
– By a step input of any variable we mean a situation where
the system is “at rest” at time t = 0 and we instantly change
the input quantity, from wherever it was just before t = 0,
by a given amount, either positive or negative, and then
keep the input constant at this new value “forever.”
– The integral of a step input is a ramp and the derivative of a
step input is an impulse.
Modeling General Concepts
K. Craig
67
The impulse function is explained by the
figure, where we approximate the step function
by a terminated ramp and then let the rise time
of the ramp approach zero. As we let the ramp
get steeper and steeper, the magnitude of de/dt
approaches infinity, and its duration approaches
zero, but the area under it will always be es. If
es = 1 (a unit step function), its derivative is
called the unit impulse function with an area or
strength equal to one unit. The step function is
the integral of the impulse function, or
conversely, the impulse function is the
derivative of the step function. When we
multiply the impulse function by some number,
we increase the “strength of the impulse”, but
“strength” now means area, not height as it
does for “ordinary” functions.
Modeling General Concepts
K. Craig
68
– An impulse that has an infinite magnitude and zero
duration is mathematical fiction and does not occur in
physical systems. If, however, the magnitude of a pulse
input to a system is very large and its duration is very short
compared to the system’s speed of response, then we can
approximate the pulse input by an impulse function. The
impulse input supplies energy to the system in an
infinitesimal time.
– The step response of a component or system is the time
response to a step input of some magnitude. The impulse
response of a system is the derivative of the step response
and is the time response to an impulse input of some
strength.
Modeling General Concepts
K. Craig
69
• Frequency Response
– Linear ODE with Constant Coefficients
dn qo
d n −1q o
dq o
+ a 0q o =
a n n + a n −1 n −1 + " + a1
dt
dt
dt
d mqi
d m−1q i
dq i
+ b0qi
b m m + b m−1 m−1 + " + b1
dt
dt
dt
– qo is the output (response) variable of the system
– qi is the input (excitation) variable of the system
– an and bm are the physical parameters of the system
Modeling General Concepts
K. Craig
70
– If the input to a linear system is a sine wave, the steadystate output (after the transients have died out) is also a sine
wave with the same frequency, but with a different
amplitude and phase angle.
q i = Qi sin(ωt)
– System Input:
q o = Qo sin(ωt + φ)
– System Steady-State Output:
– Both amplitude ratio, Qo/Qi , and phase angle, φ, change
with frequency, ω.
– The frequency response can be determined analytically from
the Laplace transfer function:
G(s)
s = iω
Modeling General Concepts
Sinusoidal
Transfer Function
M(ω)∠φ(ω)
K. Craig
71
– A negative phase angle is called phase lag, and a
positive phase angle is called phase lead.
– If the system being excited were a nonlinear or timevarying system, the output might contain frequencies
other than the input frequency and the output-input
ratio might be dependent on the input magnitude.
– Any real-world device or process will only need to
function properly for a certain range of frequencies;
outside this range we don’t care what happens.
Modeling General Concepts
K. Craig
72
System
Frequency
Response
(linear scales used)
Modeling General Concepts
K. Craig
73
Analog Electronics
Example
RC Low-Pass Filter
Time Response &
Frequency Response
Time Response
Vout
1
=
Vin RCs + 1
Resistor 15 KΩ
Vin
Capacitor 0.01 μF
Vout
Time Constant τ = RC
Modeling General Concepts
K. Craig
74
Frequency
Response
Bandwidth = 1/τ
Vout
K
=
( iω ) =
Vin
iωτ + 1
Modeling General Concepts
K∠0D
( ωτ ) + 1 ∠ tan ωτ
2
2
−1
=
K
2
ωτ
+
1
( )
2
∠ − tan −1 ωτ
K. Craig
75
Amplitude Ratio = 0.707 = -3 dB
Response to Input 1061 Hz Sine Wave
1
Input
Phase Angle = -45°
0.8
0.6
0.4
amplitude
0.2
0
-0.2
Output
-0.4
-0.6
-0.8
-1
0
Modeling General Concepts
0.5
1
1.5
2
time (sec)
2.5
3
3.5
4
x 10
-3
K. Craig
76
– When one has the frequency-response curves for any
system and is given a specific sinusoidal input, it is an
easy calculation to get the sinusoidal output.
– What is not obvious, but extremely important, is that the
frequency-response curves are really a complete
description of the system’s dynamic behavior and allow
one to compute the response for any input, not just sine
waves.
– Every dynamic signal has a frequency spectrum and if
we can compute this spectrum and properly combine it
with the system’s frequency response, we can calculate
the system time response.
Modeling General Concepts
K. Craig
77
– The details of this procedure depend on the nature of
the input signal; is it periodic, transient, or random?
– For periodic signals (those that repeat themselves over
and over in a definite cycle), Fourier Series is the
mathematical tool needed to solve the response
problem.
– Although a single sine wave is an adequate model of
some real-world input signals, the generic periodic
signal fits many more practical situations.
– A periodic function qi(t) can be represented by an
infinite series of terms called a Fourier Series.
Modeling General Concepts
K. Craig
78
a0 2 ∞ ⎡
⎛ 2πn ⎞
⎛ 2πn ⎞ ⎤
q i ( t ) = + ∑ ⎢a n cos ⎜
t ⎟ + b n sin ⎜
t ⎟⎥
T T n =1 ⎣
⎝ T ⎠
⎝ T ⎠⎦
T
2
⎛ 2πn ⎞
a n = ∫ q i ( t ) cos ⎜
t ⎟ dt
T ⎠
⎝
T
−
2
T
2
⎛ 2πn ⎞
b n = ∫ q i ( t ) sin ⎜
t ⎟ dt
T ⎠
⎝
T
−
Fourier
Series
2
Modeling General Concepts
K. Craig
79
q i(t)
1.5
Consider the Square Wave:
-0.01
+0.01
t
-0.5
0
a0
=
T
∫
0.01
−0.5dt +
−0.01
∫ 1.5dt
0
0.02
0
= 0.5 = average value
⎛ 2πn ⎞
a n = ∫ −0.5cos ⎜
t ⎟ dt +
⎝ 0.02 ⎠
−0.01
0.01
∫
0
⎛ 2πn ⎞
1.5cos ⎜
t ⎟ dt = 0
⎝ 0.02 ⎠
1 − cos ( nπ )
⎛ 2πn ⎞
⎛ 2πn ⎞
b n = ∫ −0.5sin ⎜
t ⎟ dt + ∫ 1.5sin ⎜
t ⎟ dt =
50nπ
⎝ 0.02 ⎠
⎝ 0.02 ⎠
−0.01
0
4
4
q i ( t ) = 0.5 + sin (100πt ) + sin ( 300πt ) + "
π
3π
0
Modeling General Concepts
0.01
K. Craig
80
– The term for n = 1 is called the fundamental or first
harmonic and always has the same frequency as the
repetition rate of the original periodic wave form (50
Hz in this example); whereas n = 2, 3, … gives the
second, third, and so forth harmonic frequencies as
integer multiples of the first.
– The square wave has only the first, third, fifth, and so
forth harmonics. The more terms used in the series, the
better the fit. An infinite number gives a “perfect” fit.
Modeling General Concepts
K. Craig
81
2
1.5
1
amplitude
Plot of the
Fourier
Series for
the square
wave
through
the third
harmonic
0.5
0
-0.5
-1
-0.01 -0.008 -0.006 -0.004 -0.002
0
0.002 0.004 0.006 0.008
time (sec)
0.01
4
4
q i ( t ) = 0.5 + sin (100πt ) + sin ( 300πt )
3π
π
Modeling General Concepts
K. Craig
82
– For a signal of arbitrary periodic shape (rather than the
simple and symmetrical square wave), the Fourier
Series will generally include all the harmonics and both
sine and cosine terms.
– We can combine the sine and cosine terms using:
A cos ( ωt ) + Bsin ( ωt ) = Csin ( ωt + α )
C = A 2 + B2
−1 A
α = tan
B
– Thus
q i ( t ) = A i0 + A i1 sin ( ω1t + α1 ) + A i2 sin ( 2ω1t + α 2 ) +"
Modeling General Concepts
K. Craig
83
– A graphical display of the amplitudes (Aik) and the
phase angles (αk) of the sine waves in the equation for
qi(t) is called the frequency spectrum of qi(t).
– If a periodic qi(t) is applied as input to a system with
sinusoidal transfer function G(iω), after the transients
have died out, the output qo(t) will be in a periodic
steady-state given by:
q o ( t ) = A o0 + A o1 sin ( ω1t + θ1 ) + A o2 sin ( 2ω1t + θ2 ) + "
A ok = A ik G ( iωk )
θk = α k + ∠G ( iωk )
– This follows from superposition and the definition of
the sinusoidal transfer function.
Modeling General Concepts
K. Craig
84
Frequency Response
Modeling General Concepts
K. Craig
85
1
KD
t
q out
q in = A sin ( ωt )
1
1
=
q in = ∫ q in dt + ( q out )initial
KD
K0
Frequency Response
1 t
A sin ( ωt )
K ∫0
A
A
cos ( ωt ) +
= ( q out )initial −
Kω
Kω
A
A
π
sin ⎛⎜ ωt − ⎞⎟ +
= ( q out )initial +
Kω ⎝
2 ⎠ Kω
q out = ( q out )initial +
q out
Modeling General Concepts
K. Craig
86
Frequency Response
Modeling General Concepts
K. Craig
87
State-Space Representation
• Conventional Control Theory (root-locus and frequency
response analysis and design) is applicable to linear, timeinvariant, single-input, single-output systems. This is a
complex frequency-domain approach. The transfer
function relates the input to output and does not show
internal system behavior.
• Modern Control Theory (state-space analysis and design)
is applicable to linear or nonlinear, time-varying or timeinvariant, multiple-input, multiple-output systems. This is
a time-domain approach. This state-space system
description provides a complete internal description of the
system, including the flow of internal energy.
Modeling General Concepts
K. Craig
88
– A state-determined system is a special class of lumpedparameter dynamic system such that: (i) specification of
a finite set of n independent parameters, state variables,
at time t = t0 and (ii) specification of the system inputs
for all time t ≥ t0 are necessary and sufficient to
uniquely determine the response of the system for all
time t ≥ t0.
– The state is the minimum amount of information
needed about the system at time t0 such that its future
behavior can be determined without reference to any
input before t0.
Modeling General Concepts
K. Craig
89
– The state variables are independent variables capable
of defining the state from which one can completely
describe the system behavior. These variables
completely describe the effect of the past history of the
system on its response in the future.
– Choice of state variables is not unique and they are
often, but not necessarily, physical variables of the
system. They are usually related to the energy stored in
each of the system's energy-storing elements, since any
energy initially stored in these elements can affect the
response of the system at a later time.
Modeling General Concepts
K. Craig
90
– State variables do not have to be physical or measurable
quantities, but practically they should be chosen as such
since optimal control laws will require the feedback of
all state variables.
– The state-space is a conceptual n-dimensional space
formed by the n components of the state vector. At any
time t the state of the system may be described as a
point in the state space and the time response as a
trajectory in the state space.
– The number of elements in the state vector is unique,
and is known as the order of the system.
Modeling General Concepts
K. Craig
91
– Since integrators in a continuous-time dynamic system
serve as memory devices, the outputs of integrators can
be considered as state variables that define the internal
state of the dynamic system. Thus the outputs of
integrators can serve as state variables.
– The state-variable equations are a coupled set of firstorder ordinary differential equations where the
derivative of each state variable is expressed as an
algebraic function of state variables, inputs, and
possibly time.
Modeling General Concepts
K. Craig
92
G G G
G
x(t) = f (x, u, t)
G G G
G
y(t) = g(x, u, t)
G•
G
G
x(t) = A(t)x(t) + B(t)u(t)
G
G
G
y(t) = C(t)x(t) + D(t)u(t)
Non-Linear, Time-Varying
Linear, Time-Varying
•
D(t)
Direct Transmission Matrix
Input Matrix
u(t)
Inputs
B(t)
+
+
State Variables
•
Σ
x(t)
∫ dt
State Matrix
x(t)
Outputs
+
C(t)
+
Σ
y(t)
Output Matrix
A(t)
Modeling General Concepts
K. Craig
93
State-Space to Transfer Function
G•
G
G
x(t) = Ax(t) + Bu(t)
G
G
G
y(t) = Cx(t) + Du(t)
Laplace Transform
Y(s)
= G(s)
U(s)
sX(s) − x(0) = AX(s) + BU(s)
Y(s) = CX(s) + DU(s)
sX(s) − AX(s) = BU(s)
Zero Initial Conditions
[sI − A ] X(s) = BU(s)
−1
X(s) = [sI − A ] BU(s)
−1
Y(s) = ⎡⎣C [sI − A ] B + D ⎤⎦ U(s)
−1
Y(s) = [ CΦ (s)B + D ] U(s)
Define: Φ (s) = [sI − A ]
Y(s)
= [ CΦ (s)B + D ] = G(s)
U(s)
Modeling General Concepts
K. Craig
94
• The poles of the transfer function are the eigenvalues of
the system matrix A.
sI − A = 0
Characteristic Equation
• A zero of the transfer function is a value of s that satisfies:
sI − A − B
C
D
=0
• The transfer function can be written as:
sI − A − B
C
D
G(s) =
sI − A
Modeling General Concepts
K. Craig
95
Poles and Zeros of Transfer Functions
• Definition of Poles and Zeros
– A pole of a transfer function G(s) is a value of s (real,
imaginary, or complex) that makes the denominator of
G(s) equal to zero.
– A zero of a transfer function G(s) is a value of s (real,
imaginary, or complex) that makes the numerator of
G(s) equal to zero.
K(s + 2)(s + 10)
– For Example:
G(s) =
s(s + 1)(s + 5)(s + 15) 2
Poles: 0, -1, -5, -15 (order 2)
Zeros: -2, -10, ∞ (order 3)
Modeling General Concepts
K. Craig
96
• Collocated Control System
– All energy storage elements that exist in the system
exist outside of the control loop.
– For purely mechanical systems, separation between
sensor and actuator is at most a rigid link.
• Non-Collocated Control System
– At least one storage element exists inside the control
loop.
– For purely mechanical systems, separating link between
sensor and actuator is flexible.
Modeling General Concepts
K. Craig
97
• Physical Interpretation of Poles and Zeros
– Complex Poles of a collocated control system and those
of a non-collocated control system are identical.
– Complex Poles represent the resonant frequencies
associated with the energy storage characteristics of the
entire system.
– Complex Poles, which are the natural frequencies of the
system, are independent of the locations of sensors and
actuators.
– At a frequency of a complex pole, even if the system
input is zero, there can be a nonzero output.
Modeling General Concepts
K. Craig
98
– Complex Poles represent the frequencies at which
energy can freely transfer back and forth between the
various internal energy storage elements of the system
such that even in the absence of any external input,
there can be nonzero output.
– Complex Poles correspond to the frequencies where the
system behaves as an energy reservoir.
– Complex Zeros of the two control systems are quite
different and they represent the resonant frequencies
associated with the energy storage characteristics of a
sub-portion of the system defined by artificial
constraints imposed by the sensors and actuators.
Modeling General Concepts
K. Craig
99
– Complex Zeros correspond to the frequencies where the
system behaves as an energy sink.
– Complex Zeros represent frequencies at which energy being
applied by the input is completely trapped in the energy
storage elements of a sub-portion of the original system such
that no output can ever be detected at the point of
measurement.
– Complex Zeros are the resonant frequencies of a subsystem
constrained by the sensors and actuators.
– If n is the number of poles and m is the number of zeros, the
system is said to have n-m zeros at infinity if m < n because
the transfer function approaches zero as s approaches infinity.
If the zeros at infinity are also counted, the system will have
the same number of poles and zeros. No physical system can
have n < m; otherwise , it would have infinite response at ω =
∞.
Modeling General Concepts
K. Craig
100
Transfer Function Pole-Zero Example
Two-Mass,
Three-Spring,
Motor-Driven
Dynamic
System
(shown with
optical encoders
instead of
infrared position
sensors)
Modeling General Concepts
K. Craig
101
Physical System Schematic
Infrared Position
Sensor
Motor with
Encoder
Springs
Infrared Position
Sensor
Rack and Pinion
M1
M2
Connecting Bar
Linear Bearings
Guideway
Two-Mass Three-Spring Dynamic System
Modeling General Concepts
K. Craig
102
Diagram of Physical Model
Jmotor Bmotor
Tfriction Jpinion
X1
Tm
θ
rp
K
X2
K
M1
B1
Ff1
K
M2
B2
Ff2
Two-Mass Three-Spring Dynamic System
Physical Model
Modeling General Concepts
K. Craig
103
Mathematical Model:
Transfer Functions and State Space Equations
X1 (s)
3.2503s 2 + 4.5887s + 2518.7
= 4
Vin (s) s + 2.3449s3 + 1265.7s 2 + 1414.1s + 284460
X 2 (s)
1259.3
= 4
Vin (s) s + 2.3449s3 + 1265.7s 2 + 1414.1s + 284460
1
0
0 ⎤
⎡ 0
⎢ −489.45 −0.93313 244.72
0 ⎥
⎥
A=⎢
0
0
1 ⎥
⎢ 0
⎢ 387.45
⎥
0
774.90
1.4118
−
−
⎣
⎦
⎡1 0 0 0 ⎤
⎡0 ⎤
C=⎢
D= ⎢ ⎥
⎥
⎣0 0 1 0 ⎦
⎣0 ⎦
Modeling General Concepts
⎡ 0 ⎤
⎢3.2503⎥
⎥
B= ⎢
⎢ 0 ⎥
⎢ 0 ⎥
⎣
⎦
K. Craig
104
Mathematical Model:
Poles and Zeros
Poles:
Zeros:
−0.536 ± 17.1i
⇒
−0.637 ± 31.2i ⇒
X1 ( s )
⇒ −0.706 ± 27.8i
Vin ( s )
X2 (s)
⇒
Vin ( s )
Modeling General Concepts
ω = 17.1 rad/s ζ =0.0313
ω=31.2 rad/s ζ =0.0204
⇒
ω = 27.8 rad/s
ζ =0.0254
None
K. Craig
105
Frequency Response Plots:
Analytical
Bode Diagrams
X1
Vin
-20
-30
Phase (deg); Magnitude (dB)
-40
-50
-60
-70
0
-50
-100
-150
10
0
10
1
10
2
Frequency (rad/sec)
Modeling General Concepts
K. Craig
106
Frequency Response Plots:
Analytical
X2
Vin
Bode Diagrams
-20
-40
Phase (deg); Magnitude (dB)
-60
-80
0
-100
-200
-300
10
0
10
1
10
2
Frequency (rad/sec)
Modeling General Concepts
K. Craig
107
Sensitivity Analysis
• Consider the function y = f(x). If the parameter x changes
by an amount Δx, then y changes by the amount Δy. If Δx
is small, Δy can be estimated from the slope dy/dx as
follows:
dy
Δy =
Δx
dx
• The relative or percent change in y is Δy/y. It is related to
the relative change in x as follows:
Δy dy Δx ⎛ x dy ⎞ Δx
=
=⎜
⎟
y dx y ⎝ y dx ⎠ x
Modeling General Concepts
K. Craig
108
• The sensitivity of y with respect to changes in x is given
by:
x dy dy / y d(ln y)
Sxy =
• Thus
y dx
=
dx / x
=
d(ln x)
Δy
y Δx
= Sx
y
x
• Usually the sensitivity is not constant. For example, the
function y = sin(x) has the sensitivity function:
x cos ( x )
x dy x
x
S =
= cos ( x ) =
=
y dx y
sin ( x )
tan ( x )
y
x
Modeling General Concepts
K. Craig
109
• Sensitivity of Control Systems to Parameter
Variation and Parameter Uncertainty
– A process, represented by the transfer function G(s), is
subject to a changing environment, aging, ignorance of
the exact values of the process parameters, and other
natural factors that affect a control process.
– In the open-loop system, all these errors and changes
result in a changing and inaccurate output.
– However, a closed-loop system senses the change in the
output due to the process changes and attempts to
correct the output.
– The sensitivity of a control system to parameter
variations is of prime importance.
Modeling General Concepts
K. Craig
110
– Accuracy of a measurement system is affected by
parameter changes in the control system components
and by the influence of external disturbances.
– A primary advantage of a closed-loop feedback control
system is its ability to reduce the system’s sensitivity.
– Consider the closed-loop system shown. Let the
disturbance D(s) = 0.
D(s)
R(s)
+
E(s)
Σ
B(s)
Modeling General Concepts
+
+ Σ
G c (s)
G(s)
C(s)
H(s)
K. Craig
111
– An open-loop system’s block diagram is given by:
C(s)
R(s)
G c (s)
G(s)
– The system sensitivity is defined as the ratio of the
percentage change in the system transfer function T(s)
to the percentage change in the process transfer
function G(s) (or parameter) for a small incremental
change:
C(s)
T(s) =
R(s)
∂T / T ∂T G
T
=
SG =
∂G / G ∂G T
Modeling General Concepts
K. Craig
112
– For the open-loop system
C(s)
T(s) =
= G c (s)G(s)
R(s)
G(s)
∂T / T ∂T G
T
=
= G c (s)
=1
SG =
∂G / G ∂G T
G c (s)G(s)
– For the closed-loop system
G c (s)G(s)
C(s)
=
T(s) =
R(s) 1 + G c (s)G(s)H(s)
∂T / T ∂T G
=
S =
∂G / G ∂G T
1
G
1
=
=
(1 + G c GH) 2 G c G
G c (1 + G c GH )
1 + G c GH
T
G
Modeling General Concepts
K. Craig
113
– The sensitivity of the system may be reduced below
that of the open-loop system by increasing GcGH(s)
over the frequency range of interest.
– The sensitivity of the closed-loop system to changes in
the feedback element H(s) is:
G c (s)G(s)
C(s)
T(s) =
=
R(s) 1 + G c (s)G(s)H(s)
∂T / T ∂T H
S =
=
∂H / H ∂H T
−(G c G) 2
−G c GH
H
=
=
2
G cG
(1 + G c GH)
(1 + G cGH )
1 + G c GH
T
H
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K. Craig
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– When GcGH is large, the sensitivity approaches unity
and the changes in H(s) directly affect the output
response. Use feedback components that will not vary
with environmental changes or can be maintained
constant.
– As the gain of the loop (GcGH) is increased, the
sensitivity of the control system to changes in the plant
and controller decreases, but the sensitivity to changes
in the feedback system (measurement system) becomes
-1.
– Also the effect of the disturbance input can be reduced
by increasing the gain GcH since:
G (s)
C (s) =
D (s)
1 + Gc (s) G (s) H (s)
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K. Craig
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• Therefore:
– Make the measurement system very accurate and
stable.
– Increase the loop gain to reduce sensitivity of the
control system to changes in plant and controller.
– Increase gain GcH to reduce the influence of external
disturbances.
• In practice:
– G is usually fixed and cannot be altered.
– H is essentially fixed once an accurate measurement
system is chosen.
– Most of the design freedom is available with respect to
Gc only.
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K. Craig
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• It is virtually impossible to achieve all the design
requirements simply by increasing the gain of Gc. The
dynamics of Gc also have to be properly designed in order
to obtain the desired performance of the control system.
• Very often we seek to determine the sensitivity of the
closed-loop system to changes in a parameter α within the
transfer function of the system G(s). Using the chain rule
we find:
T
T G
Sα = SGSα
• Very often the transfer function T(s) is a fraction of the
form:
N(s, α)
T(s, α) =
D(s, α)
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– Then the sensitivity to α (α0 is the nominal value) is
given by:
STα =
∂T / T ∂ ln T ∂ ln N
∂ ln D
=
=
−
= SαN − SαD
∂α / α ∂ ln α ∂ ln α α0 ∂ ln α α0
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