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Control System Design Introduction K. Craig 1
Control System Design:
An Introduction
Electrical-ElectronicsEngineer
Controls Engineer
Mechatronic System Design
MechanicalEngineer
ComputerSystemsEngineer
Electro-Mechanics
SensorsActuators
EmbeddedControl
Modeling &Simulation
Control System Design Introduction K. Craig 2
Topics
• Control System Design Overview
– Fundamental Concepts
• System Inputs
• Step and Sine Inputs
• Transfer Functions and Analogies
• Poles and Zeros of Transfer Functions
• Block Diagrams and Loading Effects
• Time Domain and Frequency Domain
• State-Space Representation
• Linearization of Nonlinear Effects
Control System Design Introduction K. Craig 3
– Open-Loop Control
• Basic and Feedforward Control
– Closed-Loop Control
• Stability and Performance
• Sensitivity Analysis
• Feedback Control Design Procedure
• PID Control and Digital Implementation
• Pulse Width Modulation
• Parasitic Effects
• Sensor Fusion
• Observers for Measurement and Control
• Advanced Control: Adaptive, Fuzzy Logic
• Trade-Offs & Control Design Performance Limitations
Control System Design Introduction K. Craig 4
Control System Design Overview
• Classical Control Design (root-locus and frequency
response analysis and design, i.e., transform methods) is
applicable to linear, time-invariant, 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 Design (state-space analysis and
design) is applicable to linear or nonlinear, time-varying or
time-invariant, 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.
Control System Design Introduction K. Craig 5
• The aim of both techniques is to find a compensation
Gc(s) that satisfies the design specifications.
• Knowledge of both approaches, modern and
classical, is essential to produce the best designs.
Feedback Control System
Gc(s)
H(s)
C(s)
R(s) E(s)
B(s)
M(s)
D(s)
+
+
_+
G(s)A(s)
V(s)
+
+
N(s)
Control System Design Introduction K. Craig 6
Fundamental Concepts
• System Inputs
• Step and Sine Inputs
• Transfer Functions and Analogies
• Poles and Zeros of Transfer Functions
• Block Diagrams and Loading Effects
• Time Domain and Frequency Domain
• State-Space Representation
• Linearization of Nonlinear Effects
Control System Design Introduction K. Craig 7
System Inputs
Control System Design Introduction K. Craig 8
System Inputs
Initial Energy
StorageExternal Driving
PotentialKinetic Deterministic Random
Stationary Unstationary
Transient Periodic"Almost
Periodic"
SinusoidalNon-
Sinusoidal
Input / System / Output
Concept:
Classification of
System Inputs
Control System Design Introduction K. Craig 9
• 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.
Control System Design Introduction K. Craig 10
• 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.
Control System Design Introduction K. Craig 11
• 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.
Control System Design Introduction K. Craig 12
Step and Sine Inputs
• Engineers typically use two inputs to evaluate
dynamic systems: a step input and a sinusoidal input.
• Step Input
– By a step input of any variable, we will always
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. This leads to a transient response called
the step response of the system.
Control System Design Introduction K. Craig 13
• Sine Input
– When the input to the system is a sine wave, the
steady-state response of the system, after all the
transients have died away, is called the frequency
response of the system.
• These two input types lead to the two views of dynamic
system response: time response and frequency response.
• Why only use these two types of input to evaluate a
dynamic system?
– The practical difficulty is that precise mathematical
functions for actual real-world inputs will not generally
be known in practice. Therefore the random nature of
many practical inputs makes difficult the development
of performance criteria based on the actual inputs
experienced by real system.
Control System Design Introduction K. Craig 14
– It is thus much more common to base
performance evaluation on system response to
simple "standard" inputs – step input and sine
wave input. This approach has been successful
for several reasons:
• Experience with the actual performance of various
classes of systems has established a good correlation
between the response of systems to these standard
inputs and the capability of the systems to accomplish
their required tasks.
• Design is much concerned with comparison of
competitive systems. This comparison can often be
made nearly as well in terms of standard inputs as for
real inputs.
• Simplicity of form of standard inputs facilitates
mathematical analysis and experimental verifications.
Control System Design Introduction K. Craig 15
Transfer Functions & Analogies
• 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 the Laplace variable s, or the differential
operator D. The highest power of s or D in the
denominator determines the order of the system.
Control System Design Introduction K. Craig 16
22
2
2
dx d xDx D x
dt dt
x x(x)dt (x)dt dt
D D
2
x 2
d xF M Mx
dt
F(t) Bx Kx Mx
Mx Bx Kx F(t)
+x
F(t)
M
KxB(dx/dt)
M
K
F(t)
B
+x
Physical Model
Free-Body Diagram(x is measured from the
static equilibrium
position)
Differential Operator D
Mass-Spring-Damper
Physical Model
Newton’s 2nd Law
D ↔ s
Laplace Variable s
Control System Design Introduction K. Craig 17
22
2
2
2
2
Differential Equation
Algebraic Equation
Tra
Mx Bx Kx F(t)
d x dxMD x M =Mx BDx B Bx
dt dt
MD x MDx Kx F(t)
(MD BD K)x F(t)
xnsf
1
F MD BDer Functio
Kn
Using the differential operator D we can transform the
differential equation to an algebraic equation and then write
the transfer function for the system.
Control System Design Introduction K. Craig 18
– 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.
Control System Design Introduction K. Craig 19
Basic Component
Equations
(Constitutive Equations)
in out
out
e e iR
dei C
dt
Kirchhoff’s Current Node Law
R C out
R C
R C
in out out
i i i
i i 0
i i
e e deC
R dt
outout in
outout in
out out in
out
in
deRC e e
dt
dee Ke
dt
De e Ke
e K
e D 1
K 1
RC
Cein eout
iin iout
R
RC Low-Pass Filter
Control System Design Introduction K. Craig 20
K
fi
B
+v
fo
oo i
dfBf f
K dt
Large Reservoir
Constant Height H
HFlow
Resistance
R
h
Tank
(Area A)
Sp constant gH (bottom of reservoir)
fluid density
g acceleration due to gravity
supply pressure
pS
tankp gh q = volume flow rate
dhRC h H
dt
AC
g
outout in
dee Ke
dt
Analogies
Control System Design Introduction K. Craig 21
+
-
∫Σ
1
Voltage
eout
External
Voltage
ein
Gain Block
Gain Block
Summation
BlockIntegration
Block
Output
Block
Input
Block
outde
dtK1
Gain Block
oute
outout in
outin out
dee Ke
dt
de 1Ke e
dt
1st – Order System Block Diagram
RC Electrical System
RC K 1
Simulation Block Diagram
Control System Design Introduction K. Craig 22
The Three Basic Element Input-Output Relationships
Resistor
Damper
Capacitor
Spring
Inductor
Mass
Control System Design Introduction K. Craig 23
Resistor, Damper
1 1i e v f
R B
e Ri f Bv
qin = i, v
qout = e, f
de 1 df 1i C CDe v Df
dt K dt K
1 Ke i f v
CD D
di dve L LDi f M MDv
dt dt
1 1i e v f
LD MD
Capacitor, Spring
Inductor, Mass
Control System Design Introduction K. Craig 24
• Step Response and Impulse Response
– 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.
– 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.
Control System Design Introduction K. Craig 25
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.
Control System Design Introduction K. Craig 26
Step Responses
of the
Three Basic Elements
Control System Design Introduction K. Craig 27
• Frequency Response
– If the input to a linear system is a sine wave, the
steady-state 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.
Control System Design Introduction K. Craig 28
Frequency Response
Control System Design Introduction K. Craig 29
Frequency Response
Control System Design Introduction K. Craig 30
qin qout1
KD t
out in in out initial
0
1 1q q q dt q
KD K
in
t
out out initial 0
out out initial
out initial
q Asin t
1q q Asin t
K
A Aq q cos t
K K
A Aq sin t
K 2 K
Frequency Response
Control System Design Introduction K. Craig 31
Analogies
• Analogies Give Engineers Insight!
– Insight based on fundamentals is the key to
innovative multidisciplinary problem solving.
– A person trying to explain a difficult concept will often say
“Well, the analogy is …” The use of analogies in everyday
life aids in understanding and makes everyone better
communicators. Mechatronic systems depend on the
interactions among mechanical, electrical, magnetic, fluid,
thermal, and chemical elements, and most likely
combinations of these. They are truly multidisciplinary and
the designers of mechatronic systems are from diverse
backgrounds. Knowledge of physical system analogies can
give design teams a significant competitive advantage.
Control System Design Introduction K. Craig 32
Electrical – Mechanical Analogies
• A signal, element, or system which exhibits mathematical
behavior identical to that of another, but physically
different, signal, element, or system is called an
analogous quantity or analog.
• Let’s explore the common electrical-mechanical analogy.
– These systems are modeled using combinations of pure (only
have the characteristic for which they are named) and ideal
(linear in behavior) elements: resistor (R), capacitor (C), and
inductor (L) for electrical systems and damper (B), spring (K), and
mass (M) for mechanical systems. The variables of interest are
voltage (e) and current (i) for electrical systems and force (f) and
velocity (v) for mechanical systems.
Control System Design Introduction K. Craig 33
• Force causes velocity, just as voltage causes current.
• A damper dissipates mechanical energy into heat, just as
a resistor dissipates electrical energy into heat.
• Springs and masses store energy in two different ways
(potential energy and kinetic energy), just as capacitors
and inductors store energy in two different ways (electric
field and magnetic field).
• The product (f)(v) represents instantaneous mechanical
power; (e)(i) represents instantaneous electrical power.
2 2 22 2
2 2
1 1 (Kx) 1 f 1 1 qKx Ce
2 2 K 2 K 2 2 C
1 1Mv Li
2 2
Spring
Potential Energy
Mass
Kinetic Energy
Capacitor
Electric Field
Energy
Inductor
Magnetic Field
Energy
Control System Design Introduction K. Craig 34
Control System Design Introduction K. Craig 35
Control System Design Introduction K. Craig 36
force f voltage e
velocity v current i
damper B resistor R
spring K capacitor 1/C
mass M inductor L
Resistor e Ri Damper f Bv
di dvInductor e L Mass f M
dt dt
1Capacitor e idt Spring f K vdt
C
Electrical – Mechanical
Analogies
Control System Design Introduction K. Craig 37
RC Electrical System Spring-Damper Mechanical System
K
fi
B
+v
fo
C
ein eout
i
iR
in R C
in out
outin out
outout in
out
in
e e e 0
e iR e 0
dee C R e 0
dt
deRC e e
dt
e 1
e RCD 1
i B K
i
i o
oi o
o o i
o
i
f f f 0
f Bv Kx 0
f Bv f 0
ff B f 0
K
Bf f f
K
f 1
BfD 1
K
RC
B
K
Control System Design Introduction K. Craig 38
Reineout
i
L
i
in L R
in out
outin out
outout in
out
in
e e e 0
die L e 0
dt
ede L e 0
dt R
deLe e
R dt
e 1
LeD 1
R
LR Electrical System Mass-Damper Mechanical System
fi
B
+v
fo M
i B M
i
oi o
o o i
o
i
f f f 0
f Bv M v 0
ff f M 0
B
Mf f f
B
f 1
MfD 1
B
L
R M
B
Control System Design Introduction K. Craig 39
in L R C
in out
out outin out
2
out outout in2
out S
22in
2
n n
e e e e 0
die L Ri e 0
dt
de dede L C R C e 0
dt dt dt
d e deLC RCdt e e
dt dt
e K1=
1 2e LCD RCD 1D D 1
fi
B
+v
M
K
fo
LRC Electrical System
Mass-Spring-Damper
Mechanical System
Cein i
LR
eout
i K B M
i
o oi o
o o o i
o S
2 2i2
n n
f f f f 0
f Kx Bv M v 0
f ff f B M 0
K K
M Bf f f f
K K
f K1=
M B 1 2fD D 1 D D 1
K K
n S
1 R CK 1
LC 2 L
n S
K B 1K 1
M 2 KM
Control System Design Introduction K. Craig 40
• We can use this analogy to explain the flow of current and the
changes in voltages in a LC (inductor-capacitor) electrical circuit
– difficult to envision for most mechanical engineers and even
for some electrical engineers – by comparing it to a spring-mass
mechanical system.
– The diagrams on the next two slides are color-coded: green, blue,
purple, and orange diagrams for each system correspond to each
other, as do the vertical lines on the graph indicating capacitor
voltage and inductor current at the four specific instances. By
comparing the motion of the mass – its changing potential energy
corresponding to energy stored in the electric field of the capacitor
and its changing kinetic energy corresponding to energy stored in
the magnetic field of the inductor – one can better understand how
electrical capacitors and inductors function.
• For enhanced multidisciplinary engineering system design and
better communication and insight among the design team
members, the use of analogies is a powerful addition to an
engineer’s toolbox.
Control System Design Introduction K. Craig 41
eL
i
eC
CL
eL
i = 0
eC
CL
eL
i
eC
CL
i = 0
eC
CL
eL
eL
i
eC
CL
imax
eC = 0CL
eL = 0
eL
i
eC
CL
imax
eC = 0CL
eL = 0
M
K
v = 0
x = +max
M
K
v = max
x = 0
M
K
v = max
x = 0
M
K
v = 0
x = -max
Inductor-Capacitor (LC) ↔ Mass-Spring (MK) Oscillations
Control System Design Introduction K. Craig 42
eL
i
eC
CL
eL
i = 0
eC
CL
eL
i
eC
CL
i = 0
eC
CL
eL
eL
i
eC
CL
imax
eC = 0CL
eL = 0
eL
i
eC
CL
imax
eC = 0CL
eL = 0
Control System Design Introduction K. Craig 43
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.
– For Example:2
K(s 2)(s 10)G(s)
s(s 1)(s 5)(s 15)
Poles: 0, -1, -5, -15 (order 2)
Zeros: -2, -10, (order 3)
Control System Design Introduction K. Craig 44
• Colocated 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.
• Noncolocated Control System
– At least one storage element exists inside the
control loop.
– For purely mechanical systems, separating link
between sensor and actuator is flexible.
Control System Design Introduction K. Craig 45
m0 m1
K
x0 x1
Frictionless Surface
F
1
t 0 1 e
0 1
1 1m m m m
m m
2
0 10 2 2
t e
11 2 2
t e
x (s) m s KG (s)
F(s) m s (m s K)
x (s) KG (s)
F(s) m s (m s K)
G0(s) – Colocated System
G1(s) – Noncolocated System
e
s 0 s 0
Ks i
m
1
Ks i
m
Open-Loop Poles
Open-Loop ZerosColocated System:
Noncolocated System: No Zeros
Rigid Body Mode
Flexible Mode
Control System Design Introduction K. Craig 46
2
2
2 2
x (s) Ms 2KG(s)
F(s) (Ms 3K)(Ms K)
Colocated Transfer Function
Complex Conjugate Poles1 1
3 3
Ki
M
3Ki
M
Complex Conjugate Zeros2 1
2Ki
M
1 3
2
M MK K K
x2x1
F(t)
Frictionless Surface
Control System Design Introduction K. Craig 47
M MK K K
M MK K K
M MK K K
1
K
M
3
3K
M
2
2K
M
fixed
undeflected
node
Mode Shapes
Control System Design Introduction K. Craig 48
• Physical Interpretation of Poles and Zeros
– Complex Poles
• Complex Poles of a colocated control system and
those of a noncolocated 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.
• Complex Poles correspond to the frequencies where
the system behaves as an energy reservoir. Energy
can freely transfer back and forth between the
various internal energy storage elements of the
system.
Control System Design Introduction K. Craig 49
– Complex Zeros
• 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.
• 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.
Control System Design Introduction K. Craig 50
Block Diagrams & Loading Effects
• 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.
Control System Design Introduction K. Craig 51
• 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.
• 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.
Control System Design Introduction K. Craig 52
Some Rules of Block Diagram Algebra
Control System Design Introduction K. Craig 53
• 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 Zo over
input impedance Zi 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.
Control System Design Introduction K. Craig 54
G1(s)u yG2(s)
1 2o1
i2
Y(s) 1G (s) G (s)
ZU(s)1
Z
o1
i2
Z1
Z
Only if this is true for the frequency
range of interest will loading effects
be negligible.
G1(s) and G2(s) are Unloaded Transfer Functions
Control System Design Introduction K. Craig 55
• 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.
Control System Design Introduction K. Craig 56
Passive
RC Low-Pass Filter
Loading Effects Example
Cein eout
iin iout
R
outin
outin
outout
in
ee RCs 1 R
ii Cs 1
e 1 1 when i 0
e RCs 1 s 1
Control System Design Introduction K. Craig 57
2 RC Low-Pass Filters in Series
Cein
iin
RC eout
iout
R
out
2
in
e 1
e RCs 1 RCs
out1 unloaded 2 unloaded
in
e 1 1G(s) G(s)
e RCs 1 RCs 1
Analysis of Complete Circuit:
Control System Design Introduction K. Craig 58
in
out
outout
out e 0
inin
in i 0
e RZ
i RCs 1
e RCs 1Z
Output Impedance
Input Impeda i Cs
nce
Cein eout
iin iout
R
Control System Design Introduction K. Craig 59
out1 loaded 2 unloaded
in
out 1
in 2
2
eG(s) G(s)
e
1 1 1
ZRCs 1 RCs 11
Z
1
RCs 1 RCs
Only if Zout-1 << Zin-2
for the frequency range of interest
will loading effects be negligible.
Control System Design Introduction K. Craig 60
Time & Frequency Domains
• We have all witnessed how engineers from different
backgrounds describe the same concepts using different
language and different points of view which often can lead
to confusion and ultimately design errors. Being able to
describe concepts, with clarity and insight, in a variety of
ways is essential.
• 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.
Control System Design Introduction K. Craig 61
• 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.
Control System Design Introduction K. Craig 62
Physical System Model
Physical Model Step Response
Time Domain
Control System Design Introduction K. Craig 63
• 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 below 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.
Real-World Signal Three Component Signals
Control System Design Introduction K. Craig 64
Real-World Signal Three Component Signals
Any real-world signal can be broken down into a sum of sine waves
and this combination of sine waves is unique.
Every dynamic signal has a frequency spectrum and if we can
compute this spectrum and properly combine it with the system
frequency response, we can calculate the system time response.
Frequency Domain
Control System Design Introduction K. Craig 65
• Most signals and processes involve both fast and slow
components happening at the same time. In the time
domain (temporal) we measure how long something
takes, whereas in the frequency domain (spectral) we
measure how fast or slow it is. No one domain is always
the best answer, so the ability to easily change domains is
quite valuable and aids in communicating with other team
members.
• To show how the time and frequency domains are the
same, the figure on the next slide shows three axes: 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. Note that
phase information is not represented here.
Control System Design Introduction K. Craig 66
Relationship between
Time & Frequency Domains
Control System Design Introduction K. Craig 67
State Space Representation
– A state-determined system is a special class of
lumped-parameter 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.
Control System Design Introduction K. Craig 68
– 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.
– 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.
Control System Design Introduction K. Craig 69
– 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.
– 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.
Control System Design Introduction K. Craig 70
x(t) A(t)x(t) B(t)u(t)
y(t) C(t)x(t) D(t)u(t)
Linear, Time-Varying
B(t)
A(t)
D(t)
C(t)dt
+
++
+
Direct Transmission Matrix
Input Matrix
State Matrix
Output Matrix
x(t)
x(t) y(t)
u(t)Outputs
Inputs
The state-variable equations are a coupled set of first-order ordinary
differential equations. The derivative of each state variable is expressed as
an algebraic function of state variables, inputs, and possibly time.
Control System Design Introduction K. Craig 71
2
Mx Bx Kx F t
x 1
F MD BD K
x v
1v F Bv Kx
M
0 1 0x x
FK B 1v v
M M M
2
2
1Kx
2
1Mv
2
Spring Potential Energy
Mass Kinetic Energy
Control System Design Introduction K. Craig 72
L
RC eoutein
iRiout = 0
iL
iC
2
out outout in2
d e deLC RC e e
dt dt
out
2
in
e 1
e LCD RCD 1
2
C
2
L
1Ce
2
1Li
2
Capacitor Electric
Field Energy
Inductor Magnetic
Field EnergyR L C out C
in out
out
out
i i i i e e
die Ri L e
dt
1e idt
C
de i
dt C
in
CC
R 11
ii L LeL
e1e00
C
Control System Design Introduction K. Craig 73
Linearization of Nonlinear Effects
• Many real-world nonlinearities involve a “smooth”
curvilinear relation between an independent variable x
and a dependent variable y: y = f(x)
• 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
2
x x x x
x x
df d f (x x)y f (x) (x x)
dx dx 2!
dfy y (x x)
dx
x x
dfy y (x x)
dx
ˆ ˆy Kx
Control System Design Introduction K. Craig 74
• 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, …).
• We may linearize this relation using the multivariable form
of the Taylor series:
1 2 3 1 2 3
1 2 3
1 2 3 1 2 3 1 2 3
1 2 3 1 1 2 2
1 2x ,x ,x , x ,x ,x ,
3 3
3 x ,x ,x ,
1 2 3
1 2 3x ,x ,x , x ,x ,x , x ,x ,x ,
1 1 2 2 3 3
f fy f (x ,x ,x , ) (x x ) (x x )
x x
f(x x )
x
f f fˆ ˆ ˆy y x x x
x x x
ˆ ˆ ˆ ˆy K x K x K x
The partial derivatives can be thought of as the sensitivity of the dependent
variable to small changes in that independent variable.
Control System Design Introduction K. Craig 75
Electromagnet
Infrared LEDPhototransistor
Levitated Ball
ExampleMagnetic Levitation System
Applications
include magnetic
bearings for
vacuum pumps,
conveyor systems
in clean rooms,
high-speed
levitated trains,
and
electromagnetic
automotive valve
actuators.
Control System Design Introduction K. Craig 76
2
2
imx mg C
x
At Equilibrium:
Equation of Motion:
2
2
img C
x
2 2 2
2 2 3 2
i i 2 i 2 i ˆˆC C C x C ix x x x
2 2
2 3 2
i 2 i 2 i ˆˆ ˆmx mg C C x C ix x x
2
3 2
2 i 2 i ˆˆ ˆmx C x C ix x
Linearization:
Magnetic Levitation System
+x
i
mg
Electromagnet
Ball (mass m)
2
2
if x,i C
x
Control System Design Introduction K. Craig 77
Use of Experimental Testing in Multivariable Linearization
0 00 0
m
m 0 0 0 0
i ,yi ,y
f f (i, y)
f ff f i , y y y i i
y i
Control System Design Introduction K. Craig 78
Control System Types
Everything Needs Controls
for Optimum Functioning!
• Process or Plant
• Process Inputs
‒ Manipulated Inputs
‒ Disturbance Inputs
• Response Variables
Control systems are an integral part
of the overall system and not
after-thought add-ons!
The earlier the issues of control are
introduced into the design process, the
better!
Why Controls?
• Command Following
• Disturbance Rejection
• Parameter Variations
Plant
Manipulated
Inputs
Disturbance
Inputs
Response
Variables
Control System Design Introduction K. Craig 79
• Classification of Control System Types– Open-Loop
• Basic
• Input-Compensated Feedforward
– Disturbance-Compensated
– Command-Compensated
– Closed-Loop (Feedback)
• Classical
– Root-Locus
– Frequency Response
• Modern (State-Space)
• Advanced
– e.g., Adaptive, Fuzzy Logic
Control System Design Introduction K. Craig 80
Basic Open-Loop Control System
Satisfactory if:
• disturbances are not too great
• changes in the desire value are not too severe
• performance specifications are not too stringent
Plant
Control
Director
Control
Effector
Desired Value
of
Controlled Variable
Controlled
Variable
Plant Disturbance Input
Plant
Manipulated
Input
Flow of Energy
and/or Material
Control System Design Introduction K. Craig 81
Open-Loop Input-Compensated Feedforward Control:
Disturbance-Compensated
• Measure the disturbance
• Estimate the effect of the disturbance on the
controlled variable and compensate for it
Plant
Control
Director
Control
Effector
Desired Value
of
Controlled Variable
Controlled
Variable
Plant Disturbance Input
Plant
Manipulated
Input
Flow of Energy
and/or Material
Disturbance
Sensor
Disturbance
Compensation
Control System Design Introduction K. Craig 82
• Disturbance-Compensated Feedforward
Control
– Basic Idea: Measure important load variables and
take corrective action before they upset the
process.
– In contrast, a feedback controller, as we will see,
does not take corrective action until after the
disturbance has upset the process and generated
an error signal.
– There are several disadvantages to disturbance-
compensated feedforward control:
• The load disturbances must be measured on
line. In many applications, this is not feasible.
Control System Design Introduction K. Craig 83
• The quality of the feedforward control
depends on the accuracy of the process
model; one needs to know how the
controlled variable responds to changes in
both the load and manipulated variables.
• Ideal feedforward controllers that are
theoretically capable of achieving perfect
control may not be physically realizable.
Fortunately, practical approximations of
these ideal controllers often provide very
effective control.
Control System Design Introduction K. Craig 84
Open-Loop Input-Compensated Feedforward Control:
Command-Compensated
Based on the
knowledge of plant
characteristics, the
desired value input is
augmented by the
command
compensator to
produce improved
performance.
Plant
Control
Director
Control
Effector
Desired Value
of
Controlled Variable
Controlled
Variable
Plant Disturbance Input
Plant
Manipulated
Input
Flow of Energy
and/or Material
Command
Compensator
Control System Design Introduction K. Craig 85
• Comments:
– Open-loop systems without disturbance or command
compensation are generally the simplest, cheapest,
and most reliable control schemes. These should be
considered first for any control task.
– If specifications cannot be met, disturbance and/or
command compensation should be considered next.
– When conscientious implementation of open-loop
techniques by a knowledgeable designer fails to yield
a workable solution, the more powerful feedback
methods should be considered.
Control System Design Introduction K. Craig 86
Closed-Loop (Feedback)
Control System
Open-Loop Control System
is converted to a
Closed-Loop Control System
by adding:
• measurement of the controlled variable
• comparison of the measured and desired values of the
controlled variable
Plant
Control
Director
Control
Effector
Desired Value
of
Controlled Variable
Controlled
Variable
Plant Disturbance Input
Plant
Manipulated
Input
Flow of Energy
and/or Material
Controlled
Variable
Sensor
Control System Design Introduction K. Craig 87
• Basic Benefits of Feedback Control
– Cause the controlled variable to accurately follow the
desired variable; corrective action occurs as soon as
the controlled variable deviates from the command.
– Greatly reduces the effect on the controlled variable
of all external disturbances in the forward path. It is
ineffective in reducing the effect of disturbances in
the feedback path (e.g., those associated with the
sensor), and disturbances outside the loop (e.g.,
those associated with the reference input element).
– Is tolerant of variations (due to wear, aging,
environmental effects, etc.) in hardware parameters
of components in the forward path, but not those in
the feedback path (e.g., sensor) or outside the loop
(e.g., reference input element).
Control System Design Introduction K. Craig 88
– Can give a closed-loop response speed much
greater than that of the components from which
they are constructed.
• Inherent Disadvantages of Feedback Control
– No corrective action is taken until after a deviation
in the controlled variable occurs. Thus, perfect
control, where the controlled variable does not
deviate from the set point during load or set-point
changes, is theoretically impossible.
– It does not provide predictive control action to
compensate for the effects of known or
measurable disturbances.
Control System Design Introduction K. Craig 89
– It may not be satisfactory for processes with large
time constants and/or long time delays. If large
and frequent disturbances occur, the process may
operate continually in a transient state and never
attain the desired steady state.
– In some applications, the controlled variable
cannot be measured on line and, consequently,
feedback control is not feasible.
• For situations in which feedback control by itself is
not satisfactory, significant improvements in control
can be achieved by adding feedforward control.
Control System Design Introduction K. Craig 90
Gc(s)
H(s)
C(s)
R(s) E(s)
B(s)
M(s)
D(s)
+
+
_+
G(s)A(s)
V(s)
+
+
N(s)
Feedback Control System Block Diagram
c
B(s)G (s)G(s)H(s)
E(s)
c
C(s)G (s)G(s)
E(s)
c
c
G (s)G(s)C(s)
R(s) 1 G (s)G(s)H(s)
c
C(s) G(s)
D(s) 1 G (s)G(s)H(s)
c
c
G (s)G(s)H(s)C(s)
N(s) 1 G (s)G(s)H(s)
Closed
Loop
Open Loop
Feedforward
Control System Design Introduction K. Craig 91
Stability and Performance
• If a system in equilibrium is momentarily excited by command
and/or disturbance inputs and those inputs are then removed,
the system must return to equilibrium if it is to be called
absolutely stable.
• If action persists indefinitely after excitation is removed, the
system is judged absolutely unstable.
• If a system is stable, how close is it to becoming unstable?
Relative stability indicators are gain margin and phase margin.
• If we want to make valid stability predictions, we must include
enough dynamics in the system model so that the closed-loop
system differential equation is at least third order.
– An exception to this rule involves systems with dead times,
where instability can occur when the dynamics (other than
the dead time itself) are zero, first, or second order.
Control System Design Introduction K. Craig 92
• The analytical study of
stability becomes a
study of the stability of
the solutions of the
closed-loop system’s
differential equations.
• A complete and general
stability theory is based
on the locations in the
complex plane of the
roots of the closed-loop
system characteristic
equation, stable
systems having all of
their roots in the LHP.
Control System Design Introduction K. Craig 93
• Results of practical use to engineers are mainly limited to
linear systems with constant coefficients, where an exact
and complete stability theory has been known for a long
time.
• Exact, general results for linear time-variant and
nonlinear systems are nonexistant. Fortunately, the
linear time-invariant theory is adequate for many practical
systems.
• For nonlinear systems, an approximation technique
called the describing function technique has a good
record of success.
• Digital simulation is always an option and, while no
general results are possible, one can explore enough
typical inputs and system parameter values to gain a high
degree of confidence in stability for any specific system.
Control System Design Introduction K. Craig 94
• Two general methods of determining the presence of
unstable roots without actually finding their numerical
values are:
– Routh Stability Criterion
• This method works with the closed-loop
system characteristic equation in an algebraic
fashion.
– Nyquist Stability Criterion
• This method is a graphical technique based on
the open-loop frequency response polar plot.
• Both methods give the same results, a statement of
the number (but not the specific numerical values) of
unstable roots. This information is generally
adequate for design purposes.
Control System Design Introduction K. Craig 95
• This theory predicts excursions of infinite magnitude for
unstable systems. Since infinite motions, voltages,
temperatures, etc., require infinite power supplies, no
real-world system can conform to such a mathematical
prediction, casting possible doubt on the validity of our
linear stability criterion since it predicts an impossible
occurrence.
• What actually happens is that oscillations, if they are to
occur, start small, under conditions favorable to and
accurately predicted by the linear stability theory. They
then start to grow, again following the exponential trend
predicted by the linear model. Gradually, however, the
amplitudes leave the region of accurate linearization,
and the linearized model, together with all its
mathematical predictions, loses validity.
Control System Design Introduction K. Craig 96
• Since solutions of the now nonlinear equations are usually
not possible analytically, we must now rely on experience
with real systems and/or nonlinear computer simulations
when explaining what really happens as unstable
oscillations build up.
• First, practical systems often include over-range alarms
and safety shut-offs that automatically shut down
operation when certain limits are exceeded. If certain
safety features are not provided, the system may destroy
itself, again leading to a shut-down condition. If safe or
destructive shut-down does not occur, the system usually
goes into a limit-cycle oscillation, an ongoing,
nonsinusoidal oscillation of fixed amplitude. The wave
form, frequency, and amplitude of limit cycles is governed
by nonlinear math models that are usually analytically
unsolvable.
Control System Design Introduction K. Craig 97
• Most of our discussion of performance will involve
rather specific mathematical performance criteria
whereas the ultimate success of a controlled process
generally rests on economic considerations which are
difficult to calculate.
• This rather nebulous connection between the
technical criteria used for system design and the
overall economic performance of the manufacturing
unit results in the need for much exercise of judgment
and experience in decision making at the higher
management levels.
• Control system designers must be cognizant of these
higher-level considerations but they usually employ
rather specific and relatively simple performance
criteria when evaluating their designs.
Control System Design Introduction K. Craig 98
• Control System Objective
– C follow desired value V and ignore disturbances
– Technical performance criteria must have to do
with how well these two objectives are attained
• Performance depends both on system characteristics
and the nature of V, D, and N.
Gc(s)
H(s)
C(s)
R(s) E(s)
B(s)
M(s)
D(s)
+
+
_+
G(s)A(s)
V(s)
+
+
N(s)
Basic Linear Feedback System
Control System Design Introduction K. Craig 99
• The practical difficulty is that precise mathematical functions
for V, D, and N will not generally be known in practice.
• Therefore the random nature of many practical commands
and disturbances makes difficult the development of
performance criteria based on the actual V, D, and N
experienced by real system.
• It is thus much more common to base performance
evaluation on system response to simple "standard" inputs
such as steps, ramps, and sine waves.
• This approach has been successful for several reasons:
– In many areas, experience with the actual performance of
various classes of control systems has established a
good correlation between the response of systems to
standard inputs and the capability of the systems to
accomplish their required tasks.
Control System Design Introduction K. Craig 100
– Design is much concerned with comparison of
competitive systems. This comparison can often
be made nearly as well in terms of standard
inputs as for real inputs.
– Simplicity of form of standard inputs facilitates
mathematical analysis and experimental
verifications.
– For linear systems with constant coefficients,
theory shows that the response to a standard
input of frequency content adequate to exercise
all significant system dynamics can then be used
to find mathematically the response to any form of
input.
Control System Design Introduction K. Craig 101
• Standard performance criteria may be classified as
falling into two categories:
– Time-Domain Specifications: Response to steps,
ramps, and the like, e.g., step response criteria
rise time, peak time, percent overshoot, settling
time, decay ratio, and steady-state error.
– Frequency-Domain Specifications: Concerned
with certain characteristics of the system
frequency response, e.g., bandwidth, peak
amplitude ratio, gain margin, and phase margin.
Control System Design Introduction K. Craig 102
• Both time-domain and frequency-domain design
criteria generally are intended to specify one or the
other of:
– speed of response
– relative stability
– steady-state errors
• Both types of specifications are often applied to the
same system to ensure that certain behavior
characteristics will be obtained.
• All performance specifications are meaningless
unless the system is absolutely stable.
Control System Design Introduction K. Craig 103
• It is important to realize that, because of model
uncertainties, it is not merely sufficient for a system to
be stable, but rather it must have adequate stability
margins.
• Stable systems with low stability margins work only
on paper; when implemented in real time, they are
frequently unstable.
• The way uncertainty has been quantified in classical
control is to assume that either gain changes or
phase changes occur. Typically, systems are
destabilized when either gain exceeds certain limits
or if there is too much phase lag (i.e., negative phase
associated with unmodeled poles or time delays).
• The tolerances of gain or phase uncertainty are the
gain margin and phase margin.
Control System Design Introduction K. Craig 104
• Consider the following design problem:
– Given a plant transfer function G2(s), find a
compensator transfer function G1(s) which yields
the following:
• stable closed-loop system
• good command following
• good disturbance rejection
• insensitivity of command following to modeling
errors (performance robustness)
• stability robustness with unmodeled dynamics
• sensor noise rejection
Control System Design Introduction K. Craig 105
• Without closed-loop stability, a discussion of
performance is meaningless. It is critically important
to realize that the compensator is actually designed
to stabilize a nominal open-loop plant. Unfortunately,
the true plant is different from the nominal plant due
to unavoidable modeling errors.
• Knowledge of modeling errors should influence the
design of the compensator.
• We assume here that the actual closed-loop system
is absolutely stable.
Control System Design Introduction K. Craig 106
Desired Shape for Open-Loop
Transfer Function
Smooth transition from the low to high-frequency range, i.e., -20 dB/decade
slope near the gain crossover frequency
Frequencies for good
command following,
disturbance reduction,
sensitivity reduction
Sensor noise,
unmodeled high-
frequency dynamics
are significant here.
Gain below this level
at high frequencies
Gain above this level
at low frequencies
• stable closed-loop system
• good command following
• good disturbance rejection
• insensitivity of command
following to modeling
errors (performance
robustness)
• stability robustness with
unmodeled dynamics
• sensor noise rejection
Open-Loop Shaping
Linear, Time-Invariant Systems
Control System Design Introduction K. Craig 107
Instability in Feedback Control Systems
• All feedback systems can become unstable if
improperly designed.
• In all real-world components there is some kind of
lagging behavior between the input and output,
characterized by ’s and n’s.
• Instantaneous response is impossible in the real
world!
• Instability in a feedback control system results from
an improper balance between the strength of the
corrective action and the system dynamic lags.
Control System Design Introduction K. Craig 108
Example• Liquid level C in a tank is
manipulated by controlling
the volume flow rate M by
means of a three-position
on/off controller with error
dead space EDS.
• Transfer function 1/As
between M and C
represents conservation of
volume between volume
flow rate and liquid level.
• Liquid-level sensor
measures C perfectly but
with a data transmission
delay dt. Tank Liquid-Level Feedback Control System
Area A
+M -M
EDS
C(t)
Control System Design Introduction K. Craig 109
MatLab / Simulink Block Diagram
M = 5, A = 2, tau_dt = 0.2 : unstable
M = 3, A = 2, tau_dt = 0.1 : stable
Tank Level Feedback Control System
Three-PositionOn-Off Controller
Transport Delay
SumStep Input
Sign
s
1/A
Plant
M
M
M
Flow RateDead Zone
C
C
B
B
Strength of corrective action is represented by M (also by 1/A).
System dynamic lag is represented by dt.
Control System Design Introduction K. Craig 110
0 0.5 1 1.5 2 2.5 30
0.5
1
1.5
2
2.5
3
3.5
time (sec)
signal C: solid
signal B: dotted
signal 0.1*M: dashed
Stable Behavior of the Tank Liquid-Level
Feedback Control System
C
B
M
Control System Design Introduction K. Craig 111
0 0.5 1 1.5 2 2.5 3-0.5
0
0.5
1
1.5
2
2.5
3
3.5
time (sec)
signal C: solid
signal B: dotted
signal 0.1*M: dashed
Unstable Behavior of the Tank Liquid-Level
Feedback Control System
C
B
M
Control System Design Introduction K. Craig 112
Sensitivity Analysis
• Consider the function y = f(x). If the parameter x
changes by an amount x, then y changes by the
amount y. If x is small, y can be estimated from
the slope dy/dx as follows:
• The relative or percent change in y is y/y. It is
related to the relative change in x as follows:
dyy x
dx
y dy x x dy x
y dx y y dx x
Control System Design Introduction K. Craig 113
• The sensitivity of y with respect to changes in x is
given by:
• Thus
• Usually the sensitivity is not constant. For example,
the function y = sin(x) has the sensitivity function:
y
x
x dy dy / y d(ln y)S
y dx dx / x d(ln x)
y
x
y xS
y x
y
x
x dy x xcos(x) xS cos(x)
y dx y sin(x) tan(x)
Control System Design Introduction K. Craig 114
• 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.
Control System Design Introduction K. Craig 115
– 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.
Gc(s)
H(s)
C(s)R(s) E(s)
B(s)
M(s)
D(s)
+
+
_+
G(s)
Control System Design Introduction K. Craig 116
– An open-loop system’s block diagram is given by:
– 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:
T
G
C(s)T(s)
R(s)
T / T T GS
G / G G T
C(s)R(s)Gc(s) G(s)
Control System Design Introduction K. Craig 117
– For the open-loop system
– For the closed-loop system
c
T
G c
c
C(s)T(s) G (s)G(s)
R(s)
T / T T G G(s)S G (s) 1
G / G G T G (s)G(s)
c
c
T
G
2cc c c
c
G (s)G(s)C(s)T(s)
R(s) 1 G (s)G(s)H(s)
T / T T GS
G / G G T
1 G 1
G G(1 G GH) G 1 G GH
1 G GH
Control System Design Introduction K. Craig 118
– 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:
c
c
T
H
2
c c
2cc c
c
G (s)G(s)C(s)T(s)
R(s) 1 G (s)G(s)H(s)
T / T T HS
H / H H T
(G G) G GHH
G G(1 G GH) 1 G GH
1 G GH
Control System Design Introduction K. Craig 119
– 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:
c
G(s)C(s) D(s)
1 G (s)G(s)H(s)
Control System Design Introduction K. Craig 120
• 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.
Control System Design Introduction K. Craig 121
• 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
within the transfer function of the system G(s). Using
the chain rule we find:
• Very often the transfer function T(s) is a fraction of
the form:
T T G
GS S S
N(s, )T(s, )
D(s, )
Control System Design Introduction K. Craig 122
– Then the sensitivity to (0 is the nominal value)
is given by:
0 0
T N DT / T ln T ln N ln DS S S
/ ln ln ln
Control System Design Introduction K. Craig 123
Feedback Control System Design
Procedure
• Control Engineering is an important part of the design
process of most dynamic systems.
• The deliberate use of feedback can:
– Stabilize an otherwise unstable system
– Reduce the error due to disturbance inputs
– Reduce the tracking error while following a command
input
– Reduce the sensitivity of a closed-loop transfer
function to small variations in internal system
parameters
Control System Design Introduction K. Craig 124
• Remember that the purpose of control is to aid the
product or process – the mechanism, the robot, the
chemical plant, the aircraft, or whatever – to do its
job.
• Engineers must take into account early in their plans
the contribution of control to the design process!
More and more systems are being designed so that
they will not work without feedback!
• Control system design begins with a proposed
product or process whose satisfactory dynamic
performance depends on feedback for:
– Stability
– Disturbance Regulation
– Tracking Accuracy
– Reduction of the Effects of Parameter Variations
Control System Design Introduction K. Craig 125
• Having a general step-by-step approach for feedback
control system design serves two purposes:
– It provides a useful starting point for any real-
world controls problem.
– It provides meaningful checkpoints once the
design process is underway.
System
Design
System Dynamics
&
Control Structure
Control System Design Introduction K. Craig 126
Other Components
Communications
ComputationSoftware, Electronics
Operator
InterfaceHuman Factors
ActuationPower Modulation
Energy Conversion
Physical SystemMechanical, Fluid, Thermal,
Chemical, Electrical,
Biomedical, Civil, Mixed
InstrumentationEnergy Conversion
Signal Processing
Modern
Multidisciplinary
Engineering
System
Simultaneous
Optimization
of all
System Components
Control System Design Introduction K. Craig 127
Electrical-ElectronicsEngineer
Controls Engineer
Multidisciplinary System Design
MechanicalEngineer
ComputerSystemsEngineer
Electro-Mechanics
SensorsActuators
EmbeddedControl
Modeling &Simulation
Social Scientists
&
Non-Technical Experts
Business
Experts
Problem-
Specific
Engineers
Physicists, Chemists,
Mathematicians, &
Computer Scientists
Multidisciplinary Engineering System Design Team
Control System Design Introduction K. Craig 128
• Sequence of Steps for Feedback Control System
Design
1. Understand the process and translate dynamic
performance requirements into time, frequency, or pole-
zero specifications.
– What is the system and what is it supposed to do?
– The importance of understanding the process cannot
be overemphasized!
– Do not confuse the approximation with the reality!
– You must be able to:
• Use the simplified model for its intended purpose
• Return to an accurate model or the actual physical
system to really verify the design performance
Control System Design Introduction K. Craig 129
Examples of Dynamic Performance Requirements
Time Response Frequency Response
Pole-Zero
Control System Design Introduction K. Craig 130
2. Select the types and number of sensors considering
location, technology, functional performance,
physical properties, quality factors, and cost.
– If you can’t observe it, you can’t control it!
– Which variables are important to control?
– Which variables can physically be measured?
– Select sensors that indirectly allow a good
estimate to be made of the critical unmeasurable
variables.
– It is important to consider sensors for the
disturbances, e.g., in chemical processes, it is
often beneficial to sense a load disturbance
directly because improved performance can be
obtained if this information is fed forward to the
controller.
Control System Design Introduction K. Craig 131
3. Select the types and number of actuators
considering location, technology, functional
performance, physical properties, quality factors,
and cost.
– In order to control a dynamic system, you must be
able to influence the response. The actuator does
this.
– Before choosing a specific actuator, consider
which variables can be influenced.
– The actuators must be capable of driving the
system so as to meet the required performance
specifications.
Control System Design Introduction K. Craig 132
4. Make a linear model of the process, actuator, and
sensor.
– Take the best choice for process, actuator, and sensor.
– Identify the equilibrium point of interest.
– Construct a small-signal dynamic model valid over the
range of frequencies included in the performance
specifications.
– Validate this model with experimental measurements
where possible.
– Express the model in many forms: state-variable, pole-
zero, and frequency-response forms.
– Simplify and reduce the order of the model, if necessary.
– Quantify model uncertainty.
Control System Design Introduction K. Craig 133
5. Make a simple trial design based on concepts of
lead-lag compensation or PID control.
– To form an initial estimate of the complexity of the
design problem, sketch a frequency-response
(Bode) plot and a root-locus plot with respect to
plant gain.
– If the plant-actuator-sensor model is stable and
minimum phase, the Bode plot will probably be
the most useful; otherwise, the root locus shows
very important information with respect to
behavior in the right-half plane.
– Try to meet specifications with a simple controller
of the lead-lag, PID variety.
– Do not overlook feedforward of the disturbances.
– Consider the effect of sensor noise.
Control System Design Introduction K. Craig 134
6. Consider modifying the plant itself for improved closed-
loop control.
– Based on the simple control design, evaluate the source
of the undesirable characteristics of system
performance.
– Reevaluate the specifications, the physical configuration
of the process, and the actuator and sensor selections in
light of the preliminary design. Return to step 1 if
improvement seems necessary or feasible.
– It may be much easier to meet specifications by altering
the process than to meet them by control strategies
alone!
– Consider all parts of the design, not only the control
logic, to meet the specifications in the most cost-
effective way.
Control System Design Introduction K. Craig 135
7. Make a trial pole-placement design based on optimal
control or other criteria.
– If the trial-and-error compensators do not give entirely
satisfactory performance, consider a design based on
optimal control.
– Select the location for your control poles that balance
system performance and control effort.
– Select the location for the estimator poles that
represent a compromise between sensor and process
noise.
– Plot the corresponding open-loop frequency response
and the root locus to evaluate the stability margins of
this design and its robustness to parameter changes.
– Compare this optimal design with the transform-
method design and select the better of the two.
Control System Design Introduction K. Craig 136
8. Build a computer model and simulate the performance of
the design.
– After reaching the best compromise among process
modification, actuator and sensor selection, and controller
design choice, run a simulation of the system.
– Include important nonlinearities, parasitic effects, and
parameter variations you expect to find during operation.
– Design iterations should continue until the simulation
confirms acceptable stability and robustness.
– As the design progresses, more complete and detailed
models (“truth models”) will be used.
– Digital control implementation should be taken into account.
– If the performance is not satisfactory, return to step 1 and
repeat. Consider modifying the plant itself for improved
closed-loop control.
Control System Design Introduction K. Craig 137
9. Build a prototype and test it.
– The proof of the pudding is in the eating!
– Simulation without experimental verification is at
best questionable and at worst useless!
– At this point you verify the quality of the model,
discover unexpected effects, and consider ways to
improve the design.
– Implement the controller using an embedded
software/hardware.
– Tune the controller, if necessary.
– After these tests, you may want to reconsider the
sensor, actuator, and process and return to step 1.
Control System Design Introduction K. Craig 138
• This outline is an approximation of good practice.
• One very important consideration (Step 6) was for
changing the plant itself to make the control problem
easier and provide maximum closed-loop
performance.
– In many cases, proper plant modifications can
provide additional damping or increase the
stiffness, change in mode shapes, reduction of
system response to disturbances, reduction of
Coulomb friction, change in thermal capacity or
conductivity, and so on.
– Designing the system and “throwing it over the
wall” to the control group is inefficient and flawed!
– System design and control design must be done
simultaneously!
Control System Design Introduction K. Craig 139
Digital Implementation of PID Control
Anti-Aliasing
FilterSensor
Plant /
ProcessActuator
A/D
Converter
D/A
Converter
Digital
Computer
Sampling
System
Digital Set Point
Sampled &
Quantized
Measurement
Sampled & Quantized
Control Signal
Sampling
Switch
Power Domain
Information Domain
Control System Design Introduction K. Craig 140
• Advantages of Digital Control
– The current trend toward using dedicated,
microprocessor-based, and often decentralized
(distributed) digital control systems in industrial
applications can be rationalized in terms of the major
advantages of digital control:
• Digital control is less susceptible to noise or
parameter variation in instrumentation because
data can be represented, generated, transmitted,
and processed as binary words, with bits
possessing two identifiable states.
Control System Design Introduction K. Craig 141
• Very high accuracy and speed are possible
through digital processing. Hardware
implementation is usually faster than software
implementation.
• Digital control can handle repetitive tasks
extremely well, through programming.
• Complex control laws and signal conditioning
methods that might be impractical to
implement using analog devices can be
programmed.
• High reliability can be achieved by minimizing
analog hardware components and through
decentralization using dedicated
microprocessors for various control tasks.
Control System Design Introduction K. Craig 142
• Large amounts of data can be stored using
compact, high-density data storage methods.
• Data can be stored or maintained for very long
periods of time without drift and without being
affected by adverse environmental conditions.
• Fast data transmission is possible over long
distances without introducing dynamic delays,
as in analog systems.
• Digital control has easy and fast data retrieval
capabilities.
• Digital processing uses low operational
voltages (e.g., 0 - 12 V DC).
• Digital control has low overall cost.
Control System Design Introduction K. Craig 143
Digital Signals are:
• discrete in time
• quantized in amplitude
You must understand the effects of:
• sample period
• quantization size
Discrete
in
Time
Continuous
in
Time
Discrete
in
Amplitude
D-D
D-C Continuous
in
Amplitude
C-D
C-C
Control System Design Introduction K. Craig 144
• In a real sense, the problems of analysis and design
of digital control systems are concerned with taking
into account the effects of the sampling period, T,
and the quantization size, q.
• If both T and q are extremely small (i.e., sampling
frequency 50 or more times the system bandwidth
with a 16-bit word size), digital signals are nearly
continuous, and continuous methods of analysis and
design can be used.
• It is most important to understand the effects of all
sample rates, fast and slow, and the effects of
quantization for large and small word sizes.
• It is worthy to note that the single most important
impact of implementing a control system digitally is
the delay associated with the D/A converter, i.e., T/2.
Control System Design Introduction K. Craig 145
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
0.2
0.4
0.6
0.8
1
1.2
1.4
time (sec)
am
plit
ud
e:
co
ntin
uo
us a
nd
qu
an
tize
d
Simulation of Continuous and Quantized Signal
Control System Design Introduction K. Craig 146
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
0.6
0.8
1
time (sec)
am
plit
ud
e
Continuous Output and D/A Output
Control System Design Introduction K. Craig 147
• Aliasing
– The analog feedback signal coming from the sensor
contains useful information related to controllable
disturbances (relatively low frequency), but also may
often include higher frequency "noise" due to
uncontrollable disturbances (too fast for control
system correction), measurement noise, and stray
electrical pickup. Such noise signals cause
difficulties in analog systems and low-pass filtering is
often needed to allow good control performance.
Control System Design Introduction K. Craig 148
– In digital systems, a phenomenon called aliasing
introduces some new aspects to the area of noise
problems. If a signal containing high frequencies is
sampled too infrequently, the output signal of the
sampler contains low-frequency ("aliased")
components not present in the signal before
sampling. If we base our control actions on these
false low-frequency components, they will, of course,
result in poor control. The theoretical absolute
minimum sampling rate to prevent aliasing is 2
samples per cycle; however, in practice, rates of
about 10 are more commonly used. A high-
frequency signal, inadequately sampled, can produce
a reconstructed function of a much lower frequency,
which can not be distinguished from that produced by
adequate sampling of a low-frequency function.
Control System Design Introduction K. Craig 149
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2-0.5
-0.4
-0.3
-0.2
-0.1
0
0.1
0.2
0.3
0.4
0.5
time (sec)
am
plit
ud
e:
an
alo
g a
nd
sa
mp
led
sig
na
ls
Simulation of Continuous and Sampled Signal
Control System Design Introduction K. Craig 150
PID Control & Digital Implementation
• A digital controller differs from an analog controller in that
the signals must be sampled and quantized.
• A signal to be used in digital logic needs to be sampled
first; then the samples need to be converted by an analog-
to-digital (A/D) converter into a quantized digital number.
• Once the digital computer has calculated the proper next
control signal value, this value needs to be converted into a
voltage and held constant or otherwise extrapolated by a
digital-to-analog converted (D/A) in order to be applied to
the actuator of the process.
• The control signal is not changed until the next sampling
period.
Control System Design Introduction K. Craig 151
• As a result of the sampling, there are more strict limits on
the speed or bandwidth of a digital controller than on
analog devices.
• A reasonable rule of thumb for selecting the sampling
period is that during the rise time of the response to a
step, the input to the discrete controller should be
sampled approximately 6 times. By adjusting the
controller for the effects of sampling, the sampling can
be adjusted to 2 to 3 times per rise time. This
corresponds to a sampling frequency that is 10 to 20
times the system’s closed-loop bandwidth.
• The quantization of the controller signals introduces an
equivalent extra noise into the system, and to keep this
interference at an acceptable level, the A/D converter
usually has an accuracy of 10 to 12 bits.
Control System Design Introduction K. Craig 152
• We will consider a simplified technique for finding a
discrete (sampled, but not quantized) equivalent to a
given continuous controller.
• The method depends on the sampling period Ts being
short enough that the reconstructed control signal is
close to the signal that the original analog controller
would have produced.
• We also assume that the numbers used in the digital
logic have enough accurate bits so that the quantization
implied in the A/D and D/A processes can be ignored.
• Finding a discrete equivalent to a given analog controller
is equivalent to finding a recurrence equation for the
samples of the control which will approximate the
differential equation of the controller.
Control System Design Introduction K. Craig 153
• The assumption is that we have the transfer function of an
analog controller and wish to replace it with a discrete
controller that will accept samples of the controller input
e(kTs), from a sampler and, using past values of the control
signal, u(kTs), and present and past values of the input,
e(kTs), will compute the next control signal to be sent to the
actuator.
• Let’s consider the PID controller, as an example. The
proportional-integral-derivative (PID) controller is the most
widely used controller in use today. It can stabilize a
system, increase the speed of response of a system, and
reduce steady-state errors of a system. t
P I D0
IP D
de(t)u(t) K e(t) K e( )d K
dt
KU(s) K K s E(s)
s
Control System Design Introduction K. Craig 154
• Proportional Control
– Virtually all controllers have a large proportional gain.
While we will see that derivative gain can provide
incremental improvements at high frequencies, and
integral gain improves performance at lower
frequencies, the proportional gain is the primary actor
across the entire frequency range of operation.
– Here the manipulating variable U is directly
proportional to the actuating signal E.
– The corrective effort is made proportional to system
"error"; large errors engender a stronger response
than do small ones. We can vary in a continuous
fashion the energy and/or material sent to the
controlled process.
Control System Design Introduction K. Craig 155
– Proportional control exhibits nonzero steady-state
errors for even the least-demanding commands and
disturbances.
• Why is this so? Suppose for an initial equilibrium
operating point xc = xv and steady-state error is
zero. Now ask xc to go to a new value xvs. It
takes a different value for the manipulated input U
to reach equilibrium at the new xc. When the
manipulated input U is proportional to the
actuating signal E, a new U can only be achieved
if E is different from zero which requires xc xv;
thus, there must be a steady-state error.
P P
P s s P s s
u (t) K e(t)
u (kT T ) K e(kT T )
Control System Design Introduction K. Craig 156
• Integral Control
– When a proportional controller can use large loop
gain and preserve good relative stability, system
performance, including those on steady-state error,
may often be met.
– However, if difficult process dynamics such as
significant dead times prevent use of large gains,
steady-state error performance may be
unacceptable.
– When human process operators notice the existence
of steady-state errors due to changes in desired
value and/or disturbance they can correct for these
by changing the desired value ("set point") or the
controller output bias until the error disappears. This
is called manual reset.
Control System Design Introduction K. Craig 157
– Integral control is a means of removing steady-state
errors without the need for manual reset. It is
sometimes called automatic reset.
– If the value of e(t) is doubled, then the value of u(t)
varies twice as fast.
– For e(t) = 0, u(t) remains stationary.
– We have seen why proportional control suffers from
steady-state errors. We need a control that can
provide any needed steady output (within its design
range, of course) when its input (system error) is
zero.
II
t
I0
Kdu(t) U(s)K e(t)
dt E(s) s
u(t) K e( )d
Control System Design Introduction K. Craig 158
Integral control has the undesirable side effects of
reducing response speed and degrading stability.
Proportional vs. Integral Control
Control System Design Introduction K. Craig 159
– Although integral control is very useful for removing
or reducing steady-state errors, it has the undesirable
side effect of reducing response speed and
degrading stability.
– Why? Reduction in speed is most readily seen in the
time domain, where a step input (a sudden change)
to an integrator causes a ramp output, a much more
gradual change.
– Stability degradation is most apparent in the
frequency domain (Nyquist Criterion) where the
integrator reduces the phase margin by giving an
additional 90 degrees of phase lag at every
frequency, rotating the (B/E)(i) curve toward the
unstable region near the -1 point.
Control System Design Introduction K. Craig 160
– Occasionally an integrating effect will naturally
appear in a system element (actuator, process, etc.)
other than the controller.
– These gratuitous integrators can be effective in
reducing steady-state errors. Although controllers
with a single integrator are most common, double
(and occasionally triple) integrators are useful for the
more difficult steady-state error problems, although
they require careful stability augmentation.
– Conventionally, the number of integrators between E
and C in the forward path has been called the system
type number.
Control System Design Introduction K. Craig 161
In addition to the number of
integrators, their location (relative
to disturbance injection points)
determines their effectiveness in
removing steady-state errors.
In Figure (a) the integrator gives
zero steady-state error for a step
command but not for a step
disturbance.
By relocating the integrator as in
Figure (b), either or both step
inputs Vs and Us can be
"canceled" by M without requiring
E to be nonzero.
Integrators must be located upstream
from disturbance-injection points if they
are to be effective in removing steady-
state errors due to disturbances.
Location is not significant for steady-
state errors caused by commands.
Control System Design Introduction K. Craig 162
– Integral control can be used by itself or in combination
with other control modes. Proportional + Integral (PI)
Control is the most common mode.
– Integral gain provides DC and low-frequency stiffness.
When a DC error occurs, the integral gain will move to
correct it. The higher the gain, the faster the correction.
Fast correction implies a stiffer system.
– Don’t confuse DC stiffness with dynamic stiffness. A
system can be quite stiff at DC and not stiff at all at high
frequencies! Higher integral gains will provide higher
DC stiffness but will not substantially improve stiffness
at or above the loop bandwidth.
Control System Design Introduction K. Craig 163
– PI controllers are more complicated to implement than P
controllers. Saturation becomes more complicated, as
integral wind-up must be avoided. In analog controllers,
clamping diodes must be added, and in digital
controllers, saturation algorithms must be coded.
– Integral gain can cause instability. In the open loop, the
integral, with its 90º phase lag, reduces phase margin.
In the time domain, the common result of adding integral
gain is overshoot and ringing. As a result, larger
integral gains usually reduce bandwidth.
Control System Design Introduction K. Craig 164
s s
s s s
s
kT T
I s s I0
kT kT T
I I0 kT
I s I
sI s I s s s
u (kT T ) K e( )d
K e( )d K e( )d
u (kT ) K [area under e( )over one period]
Tu (kT ) K e(kT T ) e(kT )
2
Graphical Interpretation
of Numerical
Integration:
Area of Trapezoid
Control System Design Introduction K. Craig 165
• Derivative Control
– Proportional and integral control actions can be used
as the sole effect in a practical controller.
– But the various derivative control modes are always
used in combination with some more basic control law.
This is because the derivative mode produces no
corrective effect for any constant error, no matter how
large, and therefore would allow uncontrolled steady-
state errors.
– One of the most important contributions of derivative
control is in system stability augmentation. If absolute
or relative stability is the problem, a suitable derivative
control mode is often the answer.
– The stabilization or "damping" aspect can easily be
understood qualitatively from the following discussion.
Control System Design Introduction K. Craig 166
– Invention of integral control may have been
stimulated by the human process operators’ desire to
automate their task of manual reset. Derivative
control hardware may first have been devised as a
mimicking of human response to changing error
signals. Suppose a human process operator is given
a display of system error E and has the task of
changing manipulated variable M (say with a control
dial) so as to keep E close to zero.
Control System Design Introduction K. Craig 167
– If you were the operator, would you produce the same
value of M at t1 as at t2? A proportional controller would
do exactly that.
– A stronger corrective effect seems appropriate at t1 and
a lesser one at t2 since at t1 the error E is E1,2 and
increasing, whereas at t2 it is also E1,2 but decreasing.
– The human eye and brain senses not only the ordinate
of the curve but also its trend or slope. Slope is clearly
dE/dt, so to mechanize this desirable human response
we need a controller sensitive to error derivative.
– Such a control can, however, not be used alone since it
does not oppose steady errors of any size, as at t3, thus
a combination of proportional + derivative control, for
example, makes sense.
Control System Design Introduction K. Craig 168
– The relation of the general concept of derivative
control to the specific effect of viscous damping in
mechanical systems can be appreciated from the
figure below.
– Here an applied torque T tries to control position of
an inertia J. The damper torque on J behaves
exactly like a derivative control mode in that it always
opposes velocity d/dt with a strength proportional to
d/dt making motion less oscillatory.
Control System Design Introduction K. Craig 169
– Derivatives of E, C, and almost any available signal
in the system are candidates for a useful derivative
control mode.
– First derivatives are most common and easiest to
implement.
– The noise-accentuating characteristics of derivative
operations may often require use of approximate
(low-pass filtered) derivative signals.
– Derivative signals can sometimes be realized better
with sensors directly responsive to the desired value,
rather than trying to differentiate an available signal.
– In addition to stability augmentation, derivative
modes may also offer improvements in speed of
response and steady-state errors.
Control System Design Introduction K. Craig 170
– The derivative gain advances the phase of the loop
by virtue of the 90º phase lead of a derivative. Using
derivative gain will usually allow the system
responsiveness to increase, allowing the bandwidth
to nearly double in some cases.
– Derivative gain has high gain at high frequencies. So
while some derivative gain does help the phase
margin, too much hurts the gain margin by adding
gain at the phase crossover frequency, typically a
high frequency. This makes the derivative gain
difficult to tune. The designer sees overshoot
improve because of increased phase margin, but a
high-frequency oscillation, which comes from
reduced gain margin, becomes apparent.
Control System Design Introduction K. Craig 171
– Derivatives are also very sensitive to noise. The
derivative gain needs to followed by a low-pass filter
to reduce noise content. However, the lower break
frequency of the filter, the less benefit can be gained
from the derivative gain.
– Proportional + Derivative Control
– Derivative control has an anticipatory character,
however, it can never anticipate any action that has
not yet taken place.
– Derivative control amplifies noise signals and may
cause a saturation effect in the actuator.
P D
P D
de(t)u(t) K e(t) K
dt
U(s)K K s
E(s)
Control System Design Introduction K. Craig 172
– In the derivative term, the roles of u and e are
reversed from integration and a consistent
approximation can be written down at once.
sI s s I s I s s s
Tu (kT T ) u (kT ) K e(kT T ) e(kT )
2
Integration
sD s s D s D s s s
Tu (kT T ) u (kT ) K e(kT T ) e(kT )
2
Differentiation
Control System Design Introduction K. Craig 173
• Z Operator
– The Laplace Transform variable s is a differential
operator. The Z Transform variable z is a prediction
operator or a forward-shift operator.
– Consider the integral term.
s
s s
U(z) is the transfrom of u(kT )
zU(z) is the transform of u(kT T )
sI s s I s I s s s
sI I I
sI I
Tu (kT T ) u (kT ) K e(kT T ) e(kT )
2
TzU (z) U (z) K zE(z) E(z)
2
T z 1U (z) K E(z)
2 z 1
Control System Design Introduction K. Craig 174
– The derivative term is the inverse of the integral term.
– The complete discrete PID controller is thus
described by:
– The effect of the discrete approximation in the z-
domain is as if everywhere in the analog transfer
function the operator s has been replaced by the
composite operator
– The discrete equivalent to Da(s) is
D D
s
2 z 1U (z) K E(z)
T z 1
sP I D
s
T z 1 2 z 1U(z) K K K E(z)
2 z 1 T z 1
s
2 z 1
T z 1
d a
s
2 z 1D (z) D
T z 1
Control System Design Introduction K. Craig 175
• Example Problem
– The closed-loop system has a rise time of about 0.2
seconds and an overshoot of about 20%.
– What is the discrete equivalent of this controller?
Compare the step responses and control signals of
the two systems. Consider a sample period of 0.07
seconds (about three samples per rise time) and a
sample period of 0.035 seconds (about 6 samples
per rise time).
Y 45G s Plant Transfer Function
U (s 9)(s 5)
U s 6D s 1.4 PI Controller Transfer Function
E s
Control System Design Introduction K. Craig 176
d s
1.21z 0.79D (z) 1.4 for T 0.07
z 1
u(k 1) u(k) 1.4 1.21e(k 1) 0.79e(k)
d s
1.105z 0.895D (z) 1.4 for T 0.035
z 1
u(k 1) u(k) 1.4 1.105e(k 1) 0.895e(k)
Control System Design Introduction K. Craig 177
Output Response Control Signals
Control System Design Introduction K. Craig 178
Pulse Width Modulation
• Pulse width modulation (PWM) is a technique in which a
series of digital pulses is used to control an analog
circuit. The length and frequency of these pulses
determines the total power delivered to the circuit. PWM
signals are most commonly used to control DC motors,
but have many other applications ranging from controlling
valves or pumps to adjusting the brightness of an LED.
• The digital pulse train that makes up a PWM signal has a
fixed frequency and varies the pulse width to alter the
average power of the signal. The ratio of the pulse width
to the period is referred to as the duty cycle of the signal.
Control System Design Introduction K. Craig 179
– For example, if a PWM signal has a 10 ms period
and its pulses are 2 ms long, that signal is said to
have a 20 percent duty cycle.
• PWM can be used to reduce the total amount of
power delivered to a load without losses normally
incurred when a power source is limited by resistive
means. This is because the average power delivered
is proportional to the modulation duty cycle. With a
sufficiently high modulation rate, passive electronic
filters can be used to smooth the pulse train and
recover an average analog waveform.
Control System Design Introduction K. Craig 180
• High frequency PWM power control systems are easily
realizable with semiconductor switches. The discrete
on/off states of the modulation are used to control the
state of the switch(es) which correspondingly control
the voltage across or current through the load.
• The major advantage of this system is the switches
are either off and not conducting any current, or on
and have (ideally) no voltage drop across them. The
product of the current and the voltage at any given
time defines the power dissipated by the switch, thus
(ideally) no power is dissipated by the switch.
Realistically, semiconductor switches are non-ideal
switches, but high efficiency controllers can still be
built.
Control System Design Introduction K. Craig 181
A PWM signal is generated by comparing a triangle wave
signal with a DC signal.
This 3-Op-Amp Circuit produces a triangular wave and a
variable-pulse-width output.
U1A and U1B form a triangle-wave generator. U1B is a
comparator.
Control System Design Introduction K. Craig 182
Waveforms created
by the 3-Op-Amp
Circuit
U1A is configured as an integrator and U1B as a
comparator with hysteresis. At power up, the
comparator’s output voltage is assumed to be zero.
Control System Design Introduction K. Craig 183
Using PWM to Generate Analog Output
• PWM can be used as an inexpensive digital-to-
analog (D/A) converter. A wide variety of
microcontroller applications exist that need analog
output but do not require high-resolution D/A
converters.
• Conversion of PWM waveforms to analog signals
involves the use of analog low-pass filters.
• In a typical PWM signal, the frequency is constant,
but the pulse width (duty cycle) is a variable. The
pulse width is directly proportional to the amplitude of
the original unmodulated signal.
Control System Design Introduction K. Craig 184
Typical PWM Waveform
Frequency Spectrum
of a
PWM Signal
External Low-Pass Filter
PWM BWF K F
K 1
Control System Design Introduction K. Craig 185
Signal BW
4 kHZ
PWM
K = 5
PWM BWF K F 20kHz
Choose the -3 dB point
at 4 kHz
51RC 3.98(10 )
2 f
R 4.0k C 0.01 F
10
2
f 20kHz
1dB 20log 14.2dB
2 f RC 1
If the -14 dB attenuation will not suffice, a higher-order active low-
pass filter may be necessary or a higher PWM frequency.
Control System Design Introduction K. Craig 186
1/tau = 10 Hz = 62.8 rad/s
PWM Frequency K = 5, 10, 15, 20
PWM Low -Pass Filter Demonstration
Transfer FcnRC Low-Pass Filter
1
0.0159s+1Analog
PWM
Start
0
PWM
Amplitude
Frequency (hz)
Start Time (s)
Duty %
PWM Output
Frequency (Hz)
200
Duty (%)
50
Amplitude
1
Simulink Simulation
Control System Design Introduction K. Craig 187
0 0.05 0.1 0.15 0.2 0.250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (sec)
am
plit
ude
Response of Low-Pass Filter (1/tau = 10 Hz) to PWM Signal (50% Duty, 50 Hz)
0 0.05 0.1 0.15 0.2 0.250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (sec)
am
plit
ude
Response of Low-Pass Filter (1/tau = 10 Hz) to PWM Signal (50% Duty, 100 Hz)
Control System Design Introduction K. Craig 188
0 0.05 0.1 0.15 0.2 0.250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (sec)
am
plit
ude
Response of Low-Pass Filter (1/tau = 10 Hz) to PWM Signal (50% Duty, 150 Hz)
0 0.05 0.1 0.15 0.2 0.250
0.1
0.2
0.3
0.4
0.5
0.6
0.7
time (sec)
am
plit
ude
Response of Low-Pass Filter (1/tau = 10 Hz) to PWM Signal (50% Duty, 200 Hz)
Control System Design Introduction K. Craig 189
1/tau = 91 Hz = 571 rad/s
PWM LR Circuit Demonstration
Transfer FcnLR Circuit
R = 0.04 OhmsL = 70 micro-henries
1/.04
0.00175s+1
Transfer FcnLR Circuit
R = 0.04 OhmsL = 70 micro-henries
1/.04
0.00175s+1
Current_Step
Current_PWM
PWM
Step50% of 1 V
Start
0
PWM
Amplitude
Frequency (hz)
Start Time (s)
Duty %
PWM Output
Frequency (Hz)
30000
Duty (%)
50
Amplitude
1
Simulink Simulation
Control System Design Introduction K. Craig 190
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
2
4
6
8
10
12
14
time (sec)
curr
ent
(am
ps)
Response of LR Circuit (tau = 0.00175) to Step Input 0.5 volts
Control System Design Introduction K. Craig 191
What should the PWM switching frequency be so that the
current waveform is within P% of the step response?
TR
t Rt 2
L L
step
T TR R
2 2
L L
TR
2
L
R PI i Ie Ie Ie 1 I
e 100
P PIe 1 I Ie 1 I
100 100
P 2L Pe 1 T ln 1
100 R 100
Rf
P2Lln 1
100
Percentage Frequency
1 28.4 kHz
5 5.6 kHz
10 2.7 kHz
20 1.3 kHz
50 412 Hz
Control System Design Introduction K. Craig 192
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
2
4
6
8
10
12
14
time (sec)
curr
ent
(am
ps)
Response of LR Circuit (tau = 0.00175) to Step Input 0.5 volts and PWM 2.7 kHz
Control System Design Introduction K. Craig 193
0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.020
2
4
6
8
10
12
14
time (sec)
curr
ent
(am
ps)
Response of LR Circuit (tau = 0.00175) to Step Input 0.5 volts and PWM 28.4 kHz
Control System Design Introduction K. Craig 194
Parasitic Effects
• Parasitic effects are present in all real-world systems
and are troublesome to account for when the
systems are designed. They are rarely disabling
alone, but are debilitating if not dealt with effectively.
• These effects include:Coulomb Friction
Time Delay
Saturation
Compliance / Resonance
Backlash
Nonlinearity
Noise
Quantization
Control System Design Introduction K. Craig 195
• Questions:
– Are they significant?
• While individually they may not be debilitating, in
combination they might be.
• Also, implementing solutions to any of these
effects might exacerbate other effects.
– What to do about them?
• Approaches:
– Ignore them and hope for the best! Murphy’s Law says
ignore them at your own peril.
– Include the parasitic effects that you think may be
troublesome in the truth model of the plant and run
simulations to determine if they are negligible.
– If they are not negligible and can adversely affect your
system, you need to do something – but what?
Control System Design Introduction K. Craig 196
• General Remedies:
– Alter the design to reduce the effective loop gain
of the controller, especially at high frequencies
where the effects of parasitics are often
predominant. This generally entails sacrifice in
performance.
– Techniques specifically intended to enhance
robustness of the design are also likely to be
effective, but may entail use of a more
complicated control algorithm.
Control System Design Introduction K. Craig 197
Sensor Fusion
• When measuring a particular variable, a single type
of sensor for that variable may not be able to meet all
the required performance specifications.
• We sometimes combine several sensors into a
measurement system that utilizes the best qualities of
each individual device.
• Thus, sensors complement each other, giving rise to
the name complementary filtering. Another name is
sensor fusion and a more advanced version of a
similar idea is called Kalman filtering.
Control System Design Introduction K. Craig 198
• Basic Concept
– If a time-varying signal is applied to both a low-pass
filter and a high-pass filter, and if the two filter output
signals are summed, the summed output signal is
exactly equal to the input signal.
1
s 1
s
s 1
qi qi
+
+
Low-Pass Filter
High-Pass Filter
Control System Design Introduction K. Craig 199
-60
-50
-40
-30
-20
-10
0
Magnitu
de (
dB
)
10-2
10-1
100
101
102
-90
-45
0
45
90
Phase (
deg)
Bode Diagram
Frequency (rad/sec)
Control System Design Introduction K. Craig 200
Control System Design Introduction K. Craig 201
– The high-pass filter and the low-pass filter do not
have to be the simple filters shown. An example of
“stronger” filters would be:
• Mechatronics Example: Absolute Angle
Measurement
– The two basic sensors used are a micro-electro-
mechanical (MEMS) rate gyro using piezoelectric
tuning forks (no spinning wheel) and an inclinometer.
2
3 2
3s 3s 1Low Pass Filter
s 3s 3s 1
3
3 2
sHigh Pass Filter
s 3s 3s 1
Control System Design Introduction K. Craig 202
– The inclinometer measures tilt angle relative to gravity
vertical by immersing two circular sector capacitance
plates in a dielectric liquid. Angular tilting causes one
pair of plates to increase capacitance and the other to
decrease. These capacitance changes cause a
frequency change in an oscillator, which is then
converted to a pulse-width-modulated (PWM) signal.
By low-pass filtering the PWM signal, a DC voltage
proportional to tilt angle is obtained.
– A rate gyro gives a DC voltage output proportional to
angular velocity, with a flat frequency response to
about 50 Hz. Op-amp analog integration would give us
angular position, but the bias error in the rate gyro,
when integrated, quickly gives an unacceptable, ever-
increasing drift of the position signal.
Control System Design Introduction K. Craig 203
– The inclinometer does not suffer from a drift problem
(no integration is involved) and can thus be used to
correct for the gyro drift problem. It cannot, however,
be used by itself for angle measurements in
applications that require a fast response (like
measuring vehicle or robotic motions) since it is a
first-order instrument with low bandwidth, typically 0.5
Hz to 6 Hz, too slow for many applications.
– The two sensors are thus good candidates for a
complementary-filtering application, giving both
angular position and angular velocity data over about
a 50-Hz bandwidth with negligible drift.
Control System Design Introduction K. Craig 204
– While the configuration of the separate high-pass and
low-pass filters is most useful for explaining the basic
concept of complementary filtering, the practical
implementation uses instead a feedback type of
configuration that produces identical differential
equations and transfer functions.
– Also, realistic sensor models should be used for
analysis and simulation purposes. The inclinometer is
modeled as a first-order system (e.g., Ki =1, time
constant = 0.3). The rate gyro is modeled as a second-
order system (e.g., Kg = 1, damping ratio = 0.5, and
natural frequency = 50 Hz).
g g
gsensor sensori
2gactual i actual
2
n n
K sKInclinometer Rate Gyro
2 sss 11
Control System Design Introduction K. Craig 205
– The gyro bias error is taken as a constant (e.g., 0.005
rad/s) and the inclinometer noise is taken as a small
random signal.
– The complementary filter has two adjustments: ωn
which we take to be 0.2 rad/s and ζ which we take to
be 0.7. The major effect is that of ωn; larger values
correct bias effects more quickly but filter noise
effects less effectively.
– To test out this algorithm, we will take the input angle
to be zero for the first 20 seconds to see how the
system “fights out” the gyro bias and attenuates the
inclinometer noise. At 20 seconds, the input angle
steps up to 1.0 radian, so we can see the response to
sudden changes.
Control System Design Introduction K. Craig 206
Control System Design Introduction K. Craig 207
– Analyzing this block diagram results in the following
equation:
– This is how the Watson Vertical Reference System is
implemented. The description of that system is
shown on the next page.
2
2
n nm rg rg _ b inc inc _ n2 2
2 2
n n n n
s 2 s1
s 2 s s 2 s1 1
High-Pass Filter Low-Pass Filter
Control System Design Introduction K. Craig 208
Control System Design Introduction K. Craig 209
Sensor Fusion
theta_rg
UncorrectedGyro Angle
StepInputAngle
Sine Wave
Num_rg(s)
Den_rg(s)
Rate Gyro TF
theta_inc
NoisyInclinometer
Angle
theta_m
MeasuredAngle
Manual Switch
1/s
1/s
1/s
Num_inc(s)
Den_inc(s)
Inclinometer TF
2*zeta*omega_n
(omega_n) 2̂
gyro_bias
Constant
Band-LimitedWhite Noise
theta_actual
ActualAngle
Simulink Simulation
Control System Design Introduction K. Craig 210
0 5 10 15 20 25 30 35 40 45 50-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
time (sec)
angle
(ra
dia
ns)
Comparison: Integrated Rate-Gyro Angle and Corrected Angle (Noisy Inclinometer Angle Not Shown)
Corrected Angle
Uncorrected Angle
Control System Design Introduction K. Craig 211
Observers
• Controls and Sensors
– Sensors measure the quantity under control.
– We often assume the availability of near-perfect
feedback signals. This assumption is often
invalid.
• Four Common Problems Caused By Sensors
– Sensors and associated cabling are expensive.
– Sensors plus associated cabling reduce control
system reliability.
Control System Design Introduction K. Craig 212
– Some signals are impractical to measure or are
inaccessible.
– Sensors usually induce significant errors, e.g.,
noise, cyclical errors, limited responsiveness.
• Observers
– Augment or replace sensors in a control system.
– They are algorithms that combine sensed signals
with other knowledge of the control system to
produce observed signals which can be more
accurate, less expensive to produce, and more
reliable than sensed signals.
– They are an alternative to adding new sensors or
upgrading existing ones.
Control System Design Introduction K. Craig 213
• Observed Signals, compared to sensed signals, can be:
– More accurate
– Less expensive to produce
– More reliable
• Observer Advantages:
– Remove sensors which reduces cost and improves
reliability.
– Improve the quality of signals that come from sensors
allowing performance enhancement.
• Observer Disadvantages:
– Can be complicated to implement.
– Expend computational resources.
– Because observers form software control loops, they
can become unstable under certain conditions.
Control System Design Introduction K. Craig 214
• Design Issues
– Select observer technique for a given system
– How much will observer improve performance?
– Additional cost?
– Limitations of observers?
– Will observer be useful and what are required
resources?
• Implementation Issues
– Installation of observers
– Tune an observer
– Recognize effects of changing system parameters
on observer performance
Control System Design Introduction K. Craig 215
• Observers can combine knowledge of plant, power
converter device output, and feedback device to
extract a feedback signal that is superior to that
which can be obtained by using a feedback device
alone.
• Principle of an Observer
– By combining a measured feedback signal with
knowledge of the control-system components
(plant + feedback system) the behavior of the
plant can be known with greater accuracy and
precision than by using the feedback alone.
Control System Design Introduction K. Craig 216
• Role of an Observer in a Control System
– Observer augments sensor output and provides a
feedback signal.
Control System Design Introduction K. Craig 217
• Observer can be used to enhance system performance
– More accurate than sensor
– Reduce phase lag inherent in sensor
– Can provide observed disturbance signals which can
be used to improve disturbance response
– Can reduce system cost by augmenting performance of
low-cost sensor
– Can eliminate a sensor altogether
• Observers
– Not a panacea
– Add complexity to a system
– Requires computational resources
– May be less robust than physical sensors especially
when parameters change substantially during operation
Control System Design Introduction K. Craig 218
ControllerYPCR C
+
Power
ConverterPlant Sensor
Observer
Controller
Plant
Model
Sensor
Model
_
+
+ _
YO
CO
+
Σ
Σ Σ
Physical System
Modeled System
Observer
An observer is a mathematical structure that
combines sensor output and plant excitation
signals with models of the plant and sensor.
Control System Design Introduction K. Craig 219
• An observer provides feedback signals that are
superior to the sensor output alone.
• There are 5 Elements of an Observer:
– Sensor output
– Power converter output (plant excitation)
– Model (estimation) of the plant
– Model of the sensor
– PI or PID observer compensator
Control System Design Introduction K. Craig 220
• Key Guidelines for using an Observer in a Motion
System
– Performance Requirements
• Machines that demand rapid response to command
changes, stiff response to disturbances, or both will
likely benefit from an observer.
• The observer can reduce phase lag in the servo
loop, allowing higher gains, which improve command
and disturbance response.
– Available Computational Resources
• Observers almost universally rely on digital control.
• If the actual or planned control system is executed
on a high-speed processor, an observer can be
added without significant cost burden.
Control System Design Introduction K. Craig 221
• If digital control techniques are already
employed, the additional design effort to
implement an observer is relatively small.
– Controls Expertise in the User Base
• Observers require some level of controls
expertise for installation and configuration.
• The user base must be capable of
understanding the features of an observer if it
is to provide benefit.
– Sensor Noise
• Observers are most effective when the
position sensor produces limited noise.
• Sensor noise is often a problem in motion-
control systems.
Control System Design Introduction K. Craig 222
• Noise in servo systems comes from two major
sources: EMI generated by power converters and
transmitted to the control section of the servo
system, and resolution limitations in sensors,
especially in the feedback sensor.
• EMI can be reduced through appropriate wiring
practices and through the selection of
components that limit noise generation.
• Noise form sensors is difficult to deal with as
observers often exacerbate sensor-noise
problems. Lowering observer bandwidth will
reduce noise susceptibility, but it also reduces the
ability of the observer to improve the system, e.g.,
reducing observer bandwidth reduces the
accuracy of the observed disturbance signal.
Control System Design Introduction K. Craig 223
• The availability of high-resolution feedback sensors
raises the likelihood that an observer will
substantially improve system performance.
– Phase Lag in Motion Control Systems
• The two predominant sensors in motion-control
systems are incremental encoders and resolvers.
• Incremental encoders respond to position change
without substantial phase lag.
• Resolver signals are commonly processed with a
tracking loop, which generates substantial phase
lag, their presence makes it more likely that an
observer will substantially improve system
performance. The same applies to a sine encoder.
Control System Design Introduction K. Craig 224
• Independent of a feedback sensor, most motion-
control systems generate phase lag in the control
loop when they derive velocity from position.
Velocity is commonly derived from position using
simple differences. It is well known to inject a
phase lag of half the sample time. The phase lag
also provides an opportunity for the observer to
improve system performance.
– Summary: Five Key Guidelines for using an Observer
in a Motion System
• Need for high performance in the application
• Availability of computational resources in the
controller
Control System Design Introduction K. Craig 225
• Ability of the average user to install and configure
the system
• Availability of highly resolved position feedback
signal
• Presence of phase lag in the position or velocity
feedback signals
– The more of these guidelines that an application
meets, the more likely the observer can substantially
improve system performance.
Control System Design Introduction K. Craig 226
Adaptive Control
• Normal Control Design Procedure
– Develop a hierarchy of models (both physical and
mathematical) of the plant, ranging from a truth model, the
most realistic model developed, to a design model, one
simple enough for design purposes while still capturing the
essential characteristics of the actual system.
– Validate the models through comparisons of predicted
responses with actual measured responses.
– Design a controller on the basis of the plant design model.
– Test the control design by simulation on both the design
model and the truth model.
– Implement the control design on the actual plant.
– Tune the controller after installation.
Control System Design Introduction K. Craig 227
• Robust Control System
– When we use a model of the plant as the basis of a control
system design, we are assuming tacitly that this model is a
reasonable representation of the plant.
– Although the design model always differs from the true
plant in some details, we are confident that these details
are not important enough to invalidate the design.
– Also, ordinary feedback attempts to reduce the effects of
plant uncertainty and disturbances.
– A two-degrees-of-freedom control system (feedforward +
feedback) results in a robust control system where the
following key issues can be addressed effectively.
• command tracking and disturbance rejection
• insensitivity to modeling errors and to sensor noise
• insensitivity to unmodeled high frequency dynamics
• stability margins
Control System Design Introduction K. Craig 228
GC1(s)
H(s)
GC2(s)
Y(s)R(s) E(s)
D(s)
+_
+GP(s)
N(s)
+
+
++
B(s)
+
U(s)
Two-Degrees-of-Freedom
Control System
Control System Design Introduction K. Craig 229
• Need for Adaptation
– There are many applications, however, for which
a design model cannot be developed with any
reasonable degree of confidence. For example:
• Processes in which the underlying physical
principles are not understood well enough for
physical and mathematical modeling.
• Processes for which the physical principles are
understood but which have parameters that cannot
be measured or accurately estimated.
– Moreover, most dynamic processes change with
time. Parameters may vary because of plant
load, normal wear, aging, breakdown, and
changes in the environment in which the process
operates.
Control System Design Introduction K. Craig 230
• For example, the mass of the object being moved by a
robot manipulator will have a considerable effect on the
dynamics of the closed-loop system, and will mean that
a controller which is well tuned for an intermediate value
of the mass will be less well tuned if an extreme value is
used, and may even result in an unstable system.
– The feedback mechanism provides some degree of
immunity to discrepancies between the physical plant
and the model that is used for the design of the control
system.
– But sometimes that is not enough. A control system
designed on the basis of a nominal design model may
not behave as well as expected, because the design
model does not adequately represent the process in
its operating environment.
Control System Design Introduction K. Craig 231
– How can one deal with processes that are prone to
large changes, or for which adequate design models
are not available?
• One approach is brute force, i.e., high loop gain: as the
loop gain becomes infinite, the output of the process
tracks the input with vanishing error. Brute force rarely
works, however, for well-known reasons: dynamic
instability, control saturation, and susceptibility to noise
and other extraneous inputs.
• Modern robust control design techniques tolerate
substantial variation in one or more parameters, but
often with a sacrifice in performance. Moreover, these
techniques are not readily applicable to processes for
which no design model is available.
Control System Design Introduction K. Craig 232
• Adaptive Control
– Adaptive control may provide a solution to the
problem. The basic idea is to have the control law
adapt its own behavior, as it learns about the
process it is designed to control, or as the process
changes with its environment.
– An adaptive controller is a controller that can
modify its behavior in response to changes in
process dynamics and disturbance
characteristics; it is a controller with adjustable
parameters and a mechanism for adjusting the
parameters.
– The controller becomes nonlinear because of the
parameter-adjustment mechanism, but with a
special structure.
Control System Design Introduction K. Craig 233
An adaptive control system can be thought of as having
two loops:
• normal feedback loop with the process and the
controller
• parameter-adjustment loop, which is often slower than
the normal feedback loop
Controller Plant
Command
Signal
Parameter
Adjustment
Output
Control
Signal
Adaptive Control System
Controller Parameters
Control System Design Introduction K. Craig 234
– Note that there are a number of solutions to the
problem of keeping a controller in tune, as the
parameters of the system it is controlling vary.
• They range from common-sense approaches to
much more mathematical ones.
• The extra complexity of the more mathematical
approaches is often justified by lesser hardware
requirements and more reliable operation.
However, it is very difficult to prove the stability
properties of controllers whose parameters can
vary as time passes, therefore only fairly
restricted adaptation may be allowed in some
applications.
Control System Design Introduction K. Craig 235
– Two basic viewpoints on adaptive control have
emerged over the years.
• The first assumes that a design model is available,
but that the parameters of the model are either not
known or subject to a wide variation.
– Gain Scheduling
– Self Tuning
• The second assumes that no such design model is
available.
– Model Reference
• Direct Methods (gain scheduling and model-
reference) change control parameters directly while
Indirect Methods (self tuning) change control
parameters based on a solution to a design problem.
Control System Design Introduction K. Craig 236
– First Viewpoint: Design Model is Available
• The control system design depends explicitly upon the
parameters that are subject to variation, and then uses
measurements or estimates of these parameters as inputs
to the control system for the purpose of tuning it during the
course of its operation.
• If the parameters can be measured directly, the control
system gains can be scheduled for these measured
parameters. This approach is often called gain scheduling.
• If it is not possible to measure the uncertain parameters
directly, the relationships between these parameters and
the other measurable quantities in the system might
feasibly be exploited to estimate the parameters. A design
based on estimates of parameters may be called a self-
tuning controller, since it uses its own operating data to
estimate parameters in real time and thereby tune its own
behavior.
Control System Design Introduction K. Craig 237
Operating
Condition
Controller Plant
Command
Signal
Gain
Schedule
OutputControl
Signal
Adaptive Control System
Gain Scheduling
Controller Parameters
In gain scheduling there is an inner loop composed of the process and
the controller and an outer loop that adjusts the controller parameters on
the basis of the operating conditions. Gain scheduling can be regarded
as a mapping from process parameters to controller parameters. It can
be implemented as a function or a table lookup. Gain scheduling is thus
a very useful technique for reducing the effects of parameter variations.
Control System Design Introduction K. Craig 238
Controller Plant
Command
Signal
Estimation
Output
Control
Signal
Adaptive Control System
Self-Tuning
Controller Parameters
Process
Parameters
Controller
Design
Specification
In self-tuning, the inner loop consists of the process and an ordinary
feedback controller. The parameters of the controller are adjusted in real
time by the outer loop, which is composed of an estimator and a design
calculation. The process model and the control design are updated at each
sampling period. The block labeled “controller design” represents an on-
line solution to a design problem for a system with known parameters.
Control System Design Introduction K. Craig 239
– Second Viewpoint: Design Model is Unavailable
• If the process is not amenable to being modeled with any
reasonable accuracy, the control law must be designed
with little or no knowledge of the process. Model-
reference adaptive control (MRAC) takes this approach.
Known only is how the closed-loop system is required to
behave in response to the command signal. The desired
behavior is represented by an ideal or reference model.
• The key problem with MRAC is to determine the
adjustment mechanism so that a stable system, which
brings the performance error to zero, is obtained. This
parameter vector is typically a set of controller gains and
has nothing to do with the physical parameters of the
process being controlled. In principle, the operation of
the tuner is indifferent to the dynamics of the plant.
Control System Design Introduction K. Craig 240
Controller Plant
Command
Signal
Adjustment
Mechanism
Output
Control
Signal
Adaptive Control System
Model-Reference Adaptive
Controller Parameters
Model
In MRAC there is an inner feedback loop composed of the process and the
controller and an outer loop that adjusts the controller parameters. The reference
input is applied to both the real closed-loop plant and the ideal model. If the
performance error (the difference between the process output and the model
output) is sufficiently small, the closed-loop system is operating as desired and left
alone. But if the error is appreciable, it becomes the input to an adjustment
mechanism that adjusts controller parameters.
Control System Design Introduction K. Craig 241
– Summary
• In adaptive control, the process is controlled by a controller
that has adjustable gains. It is assumed that there exists
some kind of design procedure that makes it possible to
determine a controller that satisfies some design criteria if
the process and its environment are known. This is called
the underlying design problem. The adaptive control
problem is then to find a method of adjusting the controller
when the characteristics of the process and its environment
are unknown or changing.
• Gain scheduling and model-reference adaptive control are
called direct methods, because the controller parameters
are changed directly without the characteristics of the
process and its disturbances first being determined. The
self-tuning controller is called an indirect method, as the
controller parameters are obtained from a solution to a
design problem using the estimated process parameters
and possibly disturbance characteristics.
Control System Design Introduction K. Craig 242
• The construction of an adaptive controller thus contains
the following steps:
– Characterize the desired behavior of the closed-
loop system.
– Determine a suitable control law with adjustable
parameters.
– Find a mechanism for adjusting the parameters.
– Implement the control law.
• The key factors in choosing adaptive control are:
– Variations in process dynamics
– Variations in the character of the disturbances
– Engineering efficiency and ease of use
• Use of an adaptive controller will not replace good
process knowledge, which is still needed to choose
specifications, the structure of the controller, and the
design method.
Control System Design Introduction K. Craig 243
Fuzzy Logic Control
• Intelligent control is the discipline in which control
algorithms are developed by emulating certain
characteristics of intelligent biological systems. The
emergence of intelligent control has been fueled by
advancements in computing technology. Examples
include:
– Expert Systems, computer programs that emulate
the actions of a human who is proficient at some
task, are being used to construct expert
controllers that seek to automate the actions of a
human operator who controls a system.
Control System Design Introduction K. Craig 244
– Fuzzy Systems, rule-based systems that use fuzzy
logic for knowledge representation and inference,
are being used to automate the perceptual,
cognitive, and action-taking characteristics of
humans who perform control tasks.
– Artificial Neural Networks emulate biological neural
networks and have been used to learn how to control
systems by observing the way that a human
performs a control task and to learn in an on-line
fashion how best to control a system by taking
control actions, rating the quality of the responses
achieved when these actions are used, and then
adjusting the recipe used for generating control
actions so that the response of the system improves.
Control System Design Introduction K. Craig 245
– Genetic Algorithms, either on line or off line, are
being used to evolve controllers by maintaining a
population of controllers and using “survival of the
fittest” principles where “fittest” is defined by the
quality of the response achieved by the controller.
• The trend in the field of control is to integrate the
functions of intelligent systems with conventional control
systems to form highly autonomous systems that have
the capability to perform complex control tasks
independently with a high degree of success.
• These intelligent controllers are not mystical; they are
simply nonlinear, often adaptive controllers, and there
seems to be an existing conventional control approach
that is analogous to every new intelligent control
approach that has been introduced.
Control System Design Introduction K. Craig 246
• Fuzzy Control
– A fuzzy controller can be designed to roughly
emulate the human deductive process (i.e., the
process whereby we successfully infer
conclusions from our knowledge). As shown in
the figure on the next slide, the fuzzy controller
consists of four main parts.
• The rule base holds a set of if-then rules that
are quantified via fuzzy logic and used to
represent the knowledge that human experts
may have about how to solve a problem in
their domain of expertise.
Control System Design Introduction K. Craig 247
Inference
Mechanism
Rule
Base
Process
Fuzzific
atio
n
Fuzzy Controller
Reference
Input Inputs
Outputs
y(t)
r(t)u(t)
Fuzzy Controller Architecture
Control System Design Introduction K. Craig 248
– The fuzzy inference mechanism successively decides
what rules are the most relevant to the current
situation and applies the actions indicted by the rules.
– The fuzzification interface converts numeric inputs
into a form that the fuzzy inference mechanism can
use to determine which knowledge in the rule base is
most relevant at the current time.
– The defuzzification interface combines the
conclusions reached by the fuzzy inference
mechanism and provides a numeric value as an
output.
• Overall, the fuzzy control design methodology, which
primarily involves the specification of the rule base,
provides a heuristic technique to construct nonlinear
controllers.
Control System Design Introduction K. Craig 249
Trade-Offs & Performance Limitations
• Feedback control is at the heart of every mechatronic system.
• Changes cannot be effected instantaneously in a dynamical
system, and a correct control decision applied at the wrong
time could result in catastrophe.
• Control systems must be safe and robust, and guaranteed to
be so, before any thoughts of performance are considered.
• Robustness achieved through feedback control is subject to
limits; there is a robustness tradeoff present in all feedback
systems.
• An understanding of fundamental limitations, in practical,
physical terms rather than abstract, mathematical terms, is an
essential element in all engineering.
Control System Design Introduction K. Craig 250
• The ability to identify performance trade-offs and
fundamental performance limitations (at both the
component and system level), their sources, and quantify
their impact on performance is essential in mechatronics
design and control.
• The ability of most mechatronic systems to deliver
exceptional performance relies mainly on the dynamic
interaction between its components and on the
performance of its control system.
• Mechatronic systems are typically composed of
components such as sensors, actuators, communication
hardware, electronics, and signal conversion hardware.
Each of these components or subsystems have their own
performance characteristics, trade-offs, and limitations that
would impact the overall performance of the system.
Control System Design Introduction K. Craig 251
+
-
C(s)V(s)
+
+K(s) G(s)
D(s)
Σ Σ
K(s)G(s) C(s)T(s)
1 K(s)G(s) V(s)
1 C(s)S(s)
1 K(s)G(s) D(s)
T(s) S(s) 1
Control System Design Introduction K. Craig 252
• Fundamental Limitations in Feedback Control
Systems
– These fundamental limitations typically take three
forms.
– Some are in the form of algebraic equations, e.g.,
T(s) + S(s) = 1 holds at all frequencies, where
T(s) and S(s) are the complementary and
sensitivity transfer functions, respectively.
• This relation can be regarded as a constraint
on design, preventing independent choices
being made in regard to command-following
and disturbance-rejection performance.
Control System Design Introduction K. Craig 253
– Other constraints take the form of a frequency-domain
integral on a closed-loop transfer function such as S(s).
Hendrik Bode observed that there is a fundamental
limitation on the achievable sensitivity function S(s)
(sensitivity to disturbances and modeling errors) for a
feedback system.
• The log of the magnitude of the sensitivity function of a LTI
(linear, time-invariant), SISO (single-input, single-output)
feedback system, integrated over frequency, is conserved
under the action of feedback – it is zero for stable plant /
compensator pairs and is some fixed positive value for
unstable ones.
• Sensitivity improvements in one frequency range must be
paid for with sensitivity deteriorations in another frequency
range, and the price is higher if the plant is open-loop
unstable.
0log S(i ) d 0
Control System Design Introduction K. Craig 254
Sensitivity reduction at low frequency unavoidably leads
to sensitivity increase at higher frequencies.
Control System Design Introduction K. Craig 255
• However, physical systems do not exhibit good
frequency response fidelity beyond a certain
bandwidth. This is due to uncertain or unmodeled
dynamics in the plant, to digital control
implementations, to power limits, and to
nonlinearities, for example. So the integration is
performed over a finite frequency range, a
constraint imposed by the physical hardware we
use in the control loop. All the action of the
feedback design, the sensitivity improvements as
well as the sensitivity deteriorations, must occur
within a finite frequency range. So reducing the
sensitivity of a system to disturbances at one range
of frequencies by feedback control will amplify
transients and oscillations at other frequencies.
Control System Design Introduction K. Craig 256
– Example: Stick-Balancing (Inverted Pendulum
Problem)
• The difference between balancing long sticks and
short sticks has to do with the location of the
unstable mode. As ℓ gets smaller, the unstable
pole magnitude gets larger.
F
ℓ
2
2 2
1s
2sM m M mF
s s g3 12 2
2
2 2
gs
X 3 2sM m M mF
s s g3 12 2
Control System Design Introduction K. Craig 257
• A long rigid stick is easy to balance, but as it
becomes shorter, balancing it becomes more
difficult. Handicaps to human control include
reaction time, neuromuscular lags, limb
inertias, and other uncertainties. So while the
stick is rigid, the compensator frequency range
might be good for a frequency range up to
about 2 Hz. The control strategy is to keep the
sensitivity as small as possible over that
range. However, a dramatic increase in
sensitivity occurs as the stick becomes shorter
and even minor imperfections in the
implementation will cause instability. This is
the reason humans have trouble balancing
short sticks.
Control System Design Introduction K. Craig 258
– The third type takes the form of time-domain
integral constraints on a system signal such as
the feedback error.
• If the open-loop transfer function has two
poles at the origin (a double integrator) and
the closed-loop system is stable, then the error
e(t) following the application of a unit step
input applied at t = 0 must satisfy the relation:
• Equal areas of positive and negative error
must result.
0e(t)dt 0