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Advanced Control TheoryChibum Lee -Seoultech
Introduction to control
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Advanced Control TheoryChibum Lee -Seoultech
Control System
Control system: An interconnection of components
forming a system configuration that will provide a
desired system response
Relay/SCR
TemperatureSensor
Target Temperature
Heater
ControllerRood
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Advanced Control TheoryChibum Lee -Seoultech
Example of Control
Automobile driving
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Advanced Control TheoryChibum Lee -Seoultech
Types of controller
Typical controllers
• Embedded Controller :
Specially designed controller for specified systems
MPU, MCU, DSP, FPGA Cost for development useful for mass production
• PC Controller
Easy development, costly, easy access
• PLC (Programmable Logic Controller)
Widely used in general automation systems, sequence control
Use Ladder diagram, Modular composition
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Advanced Control TheoryChibum Lee -Seoultech
Control System Terminology
Systems
• A physical set of components that takes a signal, and
produces a signal
Signals
• A function representing some variable that contains some
information about the behavior of a system.
Systemsignal A signal B
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Advanced Control TheoryChibum Lee -Seoultech
Control System Terminology
Systems
• Plant: The physical object to be controlled.
Terminology comes from chemical process plants
like oil refineries or power plants.
• Sensor: The device that allows you to measure variables for
plant monitoring and for a variety of other purpose
In mechanical systems, it measures pressure, force, speed,
position, etc.
• Cf) Actuator: The device that causes the process to provide
the output. The device that provides the motive power to
the process
7
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Advanced Control TheoryChibum Lee -Seoultech
Control System Terminology
Systems
• Controller: The device or operation that gives commands to
the plant.
These commands are usually based on the current performance
of the plant compared to the desired performance of the plant.
The controller may or may not include the actuator, the device
that enforces the controllers commands on the plant.
Plant
Sensor
Controller
8
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Advanced Control TheoryChibum Lee -Seoultech
Control System Terminology
Signals
• Reference: What you’d like the measured output of your
controlled plant to be.
e.g. speed setting for your cruise control.
• Feedback (Measured output): Variables that are coming out
of your plant that are being compared to your reference.
e.g. speed measurement of your car from wheel speed sensors
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Advanced Control TheoryChibum Lee -Seoultech
Control System Terminology
Signals
• (Feedback) Error: Difference between your reference and
your feedback signals.
e.g. 10 km/h difference between reference and feedback.
• (Plant) Input: The signal that is passed from the controller
to the plant to affect some action.
e.g. Throttle variation.
• (Plant) Output: all the signals coming out of the plant.
Note: not all outputs are used in feedback
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Advanced Control TheoryChibum Lee -Seoultech
Control System Terminology
Systems and Signals
• Plant, Controller, and Sensor all systems.
usually have some dynamic or differential equation description.
transfer function representation for these systems.
• Signals carry information between these elements.
Plant
Sensor
Controllerreference error input output
feedback
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Advanced Control TheoryChibum Lee -Seoultech
Control System Terminology
Disturbance
• Disturbance: An unwanted signal that is not accessible to
the controller and tends to adversely affect the systems
output
System or signal ???
Plant
Sensors
Controller
disturbance
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Advanced Control TheoryChibum Lee -Seoultech
Control system diagram with disturbance and noise
Control system diagram with inner and outer loop
Control System Terminology
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Advanced Control TheoryChibum Lee -Seoultech
Closed vs. Open loop control
Closed loop control
• The difference of reference and
feedback are used as a means
of control.
Open loop control
• The output has no effect on control action
• usually simpler with fewer components
• Examples would be timers; e.g. toaster.
Gc Gp
Gc Gp
Closed-loop (with feedback)
Open-loop (without feedback)
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Advanced Control TheoryChibum Lee -Seoultech
Open-loop example
Closed vs. Open loop control
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Advanced Control TheoryChibum Lee -Seoultech
Closed-loop example
Closed vs. Open loop control
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Advanced Control TheoryChibum Lee -Seoultech
Closed vs. Open loop control
Comparison
Open-loop system Closed-loop system
Cost no sensors simple and
less expensive
Stabilitya stable plant cannot go unstable
a stable plant can go unstable, but an unstable plant can go stable
Performance
Disturbances or modeling mismatch can cause errors recalibration are
needed
more robust to disturbances and uncertainty They usually have better
performance
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Advanced Control TheoryChibum Lee -Seoultech
Closed vs. Open loop control
•Open loop •Closed loop
+
-
+
+
+
+
10% error 0.01% error
+
-
+
+
+
+
with model uncertainty
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Advanced Control TheoryChibum Lee -Seoultech
1. Study the plant and the control objectives.
2. Model the plant (if necessary, simplify the model.)
3. Decide the variables to be controlled (controlled outputs), the
measurements(sensor) and manipulated variables (actuators)
4. Select the control configuration and the type of controller.
5. Decide on performance specifications
6. Design a controller.
7. Analyze the resulting controlled system
8. Simulate the resulting controlled system
9. Choose hardware and software and implement the controller.
10. Test and validate the control system Hardware
labs
Control design procedure
Software
labs
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Advanced Control TheoryChibum Lee -Seoultech
Laplace transform
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Advanced Control TheoryChibum Lee -Seoultech
Why Laplace transform?
• Powerful for solving linear ODEs with initial values
• Easy to deal step and Dirac delta functions
• Comfortable for discontinued or complex periodic functions
Laplace Transforms
Easy
Difficult
AE Solver
Laplace Transform
InverseLaplace
Transform
DE SolverOriginal Prob.(differential eqn.)
Solution to original Prob.
S-domain Prob.(algebraic eqn.)
Solution to S-domain Prob.
T-domain
S-domain
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Advanced Control TheoryChibum Lee -Seoultech
Definition
Laplace transform of a function f(t)
• Integral transform
Inverse Laplace transform of F(s)
j
j
stdsF(s)eπj2
1
0
stF s f e f t dt
1 F f t
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Advanced Control TheoryChibum Lee -Seoultech
Laplace transform
The Laplace transform is a linear operation.
• Obeys the principle of superposition.
)]([)]([ )()(
))()(()()(
2211
0
22
0
11
0
22112211
txLatxLadttxaedttxae
dttxatxaetxatxaL
stst
st
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Advanced Control TheoryChibum Lee -Seoultech
• Shifting (s-domain)
• Shifting (t-domain)
• Differentiation
• Integration
• Final value theorem
• Initial values theorem )(lim)0(
)(lim)(
)(])([
)0(][][
)0()(][
)()]([
)()]([
0
0
2
2
ssFf
ssFf
s
sFdfL
fdt
dfsL
dt
fdL
fssFdt
dfL
sFeTtfL
asFtfeL
s
s
t
sT
at
Properties of Laplace transform
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Advanced Control TheoryChibum Lee -Seoultech
Laplace Transform usage in control system
2 primary uses for Laplace Transforms.
Compact representation of:
• (a) Signals
• (b) Systems
Plant
Sensors
Controller
disturbance
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Advanced Control TheoryChibum Lee -Seoultech
Signals
Step
Ramp
0
1
t au t a
t a
ase
u t as
2000
000
11
)()(
sdte
sdt
s
e
s
et
dts
edt
s
etdttesF
ststst
ststst
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Advanced Control TheoryChibum Lee -Seoultech
Signals
Sinusoid
Cf)
sin(t)ejt e jt
2 j
Euler’s formula
22
0
0
11
2
1
2
sin][sin)(
sjsjsj
dtej
ee
dtettLsF
sttjtj
st
)sin()( ttf
)cos()( ttf 22
11
2
1][cos)(
s
s
jsjstLsF
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Advanced Control TheoryChibum Lee -Seoultech
Signals
Impulse function
0,
0,0)()(
t
tttf
1)(
dtt
)}(1)(1{
1lim)(
0
ttt
1lim1
lim011
lim
11lim)(1)(1{
1lim
)}(1)(1{1
lim
)(lim)]([)(
000
000
00
00
|
s
se
s
e
s
e
s
s
e
sdtetdtet
dtett
dtettLsF
sss
ststst
st
st
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Advanced Control TheoryChibum Lee -Seoultech
Laplace transform
Signals
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Advanced Control TheoryChibum Lee -Seoultech
Systems
Systems are abstract representations of
dynamical phenomena.
1st order
Consider a free response would be considering
just the system.
systemaffectingsignalsystem
Faxx
1)0( ;0 txaxx
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Advanced Control TheoryChibum Lee -Seoultech
Signals and Systems
How signals affect systems and how systems
generate signals Convolution
Convolution
• Commutative Law:
• Distributive Law:
• Associative Law:
•
• Unusual Properties of Convolution:
• Convolution Theorem
0
t
f g t f g t d
f g g f
1 2 1 2f g g f g f g
f g v f g v
000 ff
1f f
f g f g
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Advanced Control TheoryChibum Lee -Seoultech
Signals and Systems
How does signal f affect system g?
signalsystemsignal
nconvolutio
system
)( )()()( sFsGtftg
output)(sG
)(tfinput
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Advanced Control TheoryChibum Lee -Seoultech
Convolution
1st order example again
In the time domain
systemaffectingsignalsystem
Faxx
systemaffectingsignal
systemofdynamics
responsesignalsystem
sFas
sX )( 1
)(
t
ta dfetx0
)()( Convolution
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Advanced Control TheoryChibum Lee -Seoultech
Ex. Solve
• LHS:
• RHS:
• Output:
'' , 0 1, ' 0 1y y t y y
Signals and Systems
output1
1)(
2
ssG
ttf )(Ramp input
,1)0'(,1)0( yy
System with IC
ttet sinh
1
1)( :TF)0'()0(
2
2
ssGYysyYs ,1)0'(,1)0( yy
2
1)()(
ssFttf
response IC))()(()( 1 sFsGLty
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Advanced Control TheoryChibum Lee -Seoultech
Signals and Systems
Visual explanation
• Example 1
• Example 2
system)( signal,)( sFsG
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Advanced Control TheoryChibum Lee -Seoultech
Modeling of Control systems
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Advanced Control TheoryChibum Lee -Seoultech
Mathematical Models
Models are key elements in the design and analysis of
control systems qualitative mathematical model
We must make a compromise b/w the simplicity of the
model vs. the accuracy of the results of analysis
???)t(u )t(y
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Advanced Control TheoryChibum Lee -Seoultech
Linear vs. nonlinear system
Linear system: the principle of superposition holds
• Linearity in mathematics
Let V and W be vector spaces over the same field K.
A function f: V → W is said to be a linear map if for any 2 vectors x and y
in V and any scalar α in K, the following conditions are satisfied:
• Linearity in system
A general system can be described by operator H, that maps an input
x(t) as a function of t to an output y(t) a type of black box description.
Linear systems satisfy the properties of superposition and homogeneity.
additivity
homogeneity
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Advanced Control TheoryChibum Lee -Seoultech
Linear Time Invariant System
Linear Time Invariant = Linear & time invariant
A time-invariant (TIV) system is one whose output does not depend explicitly on time.• If the input signal x(t) produces an output y(t), then
any time shifted input, x(t+), results in a time-shifted output y(t+ )
• Time invariant means that the coefficients in the differential equations are constant and don’t change with respect to time.
We can apply impulse response & Laplace transform in LTI system
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Advanced Control TheoryChibum Lee -Seoultech
Time-Varying Models
A time-varying system is a system that is not
time invariant its output depend explicitly
upon time
• Eg. a spacecraft control system.
The mass of fuel consumption changes due to fuel
consumption
Dynamic system
Linear Nonlinear
LinearTime
Invariant
LinearTime
Varying
Our focus
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Advanced Control TheoryChibum Lee -Seoultech
Transfer Functions
• Assuming zero initial conditions, take the Laplace
Transform of both sides
)()(
)()()()(
2
2
sFsYkbsms
sFskYsbsYsYms
)(1
)(
)(2
sGkbsmssF
sY
output
input
G(s)input output
TransferFunction
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Advanced Control TheoryChibum Lee -Seoultech
Transfer Functions
Transfer Function: the ratio of the Laplace
transform of the input and output of a linear
time-invariant system with zero initial conditions
and zero-point equilibrium.
Rational function in the complex variables
• Let x(t) : input , y(t) : output
xbxbxbxb
yayayaya
m
m
m
m
n
n
n
n
01
)1(
1
)(
01
)1(
1
)(
(n > m)
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Advanced Control TheoryChibum Lee -Seoultech
Transfer Functions
n-th order system since the highest power in the
denominator is n.
Note:
• limited to time-invariant, differential equation
• independent of the input magnitude. (homogeneity)
• no information on physically structure. (MKS and RLC)
01
1
1
01
1
1
ICs zero
)(
)(
][
][)( :TF
asasasa
bsbsbsb
sX
sY
inputL
outputLsG
n
n
n
n
m
m
m
m
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Advanced Control TheoryChibum Lee -Seoultech
System Poles and Zeros
• Roots of N(s)=0 : the system zeros z1, z2, …, zm
• Roots of D(s)=0 : the system poles p1, p2, …, pn
Note
• (System) Poles and zeros: real or either complex conjugate pairs
))(())((
))(())((
)(
)(
)(
121
121
01
1
1
01
1
1
nn
mm
n
n
n
n
m
m
m
m
pspspsps
zszszszsK
sD
sN
asasasa
bsbsbsbsG
numerator
denominator
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Advanced Control TheoryChibum Lee -Seoultech
Feedback Systems
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Advanced Control TheoryChibum Lee -Seoultech
Feedback Systems
Compared to open loop system, feedback control has
the following advantages:
• Decreased sensitivity of the system to variations in the parameters
of the process
• Improved rejection of the disturbance
• Improved measurement noise attenuation
• Improved reduction of the state-state error of the system
• Easy control and adjustment of the transient response of the
system
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Advanced Control TheoryChibum Lee -Seoultech
Feedback Systems
Closed-loop system subject to a disturbance and
a measurement noise
• Assume LTI system
(Mostly H(s) 1)
U(s) V(s)
Ym(s)
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Advanced Control TheoryChibum Lee -Seoultech
System Transfer Function
Define Tracking error:
If we consider input signals separately and use
principle of superposition, the output is given by
)()()( sYsRsE
)()()()(1
)()()()(
)()()(1
)(
)()()()(1
)()()(
sNsHsGsG
sHsGsGsD
sHsGsG
sG
sRsHsGsG
sGsGsY
pc
pc
pc
p
pc
pc
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Advanced Control TheoryChibum Lee -Seoultech
System Transfer Function
Assuming H(s)=1 , internal signals are given by
• Output
• Measured output
• Control input
)()()(1
)()()(
)()(1
)()(
)()(1
)()()( sN
sGsG
sGsGsD
sGsG
sGsR
sGsG
sGsGsY
pc
pc
pc
p
pc
pc
)()()(1
1)(
)()(1
)()(
)()(1
)()()( sN
sGsGsD
sGsG
sGsR
sGsG
sGsGsY
pcpc
p
pc
pc
m
)()()(1
)()(
)()(1
)()()(
)()(1
)()( sN
sGsG
sGsD
sGsG
sGsGsR
sGsG
sGsU
pc
c
pc
pc
pc
c
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Advanced Control TheoryChibum Lee -Seoultech
System Transfer Function
Gang of 4
Why “sensitivity” transfer function?
)()(/)(
)(/)(sS
sGsG
sTsT
pp
: Complimentary sensitivity transferfunction
: Sensitivity transfer function
)()(1
1
)()(1
)(
)()(1
)(
)()(1
)()(
sGsGsGsG
sG
sGsG
sG
sGsG
sGsG
pcpc
c
pc
p
pc
pc
SsGsG
TsGsG
sGsG
pc
pc
pc
)()(1
1
)()(1
)()(
the ratio of the change in the system TFto the change of a plant TF
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Advanced Control TheoryChibum Lee -Seoultech
Performance Specification
Good control tracking error small
• Good reference tracking
• Good disturbance rejection
• Good noise attenuation
)()()(1
)()()(
)()(1
)()(
)()(1
1)(
)()()()(
sNsGsG
sGsGsD
sGsG
sGsR
sGsGsE
sT
pc
pc
sSsG
pc
p
sS
pc
p
1)()()(1
1
)(
)(
sS
sGsGsR
sE
pc
1)()()()(1
)(
)(
)(
sSsG
sGsG
sG
sD
sEp
pc
p
1)()()(1
)()(
)(
)(
sT
sGsG
sGsG
sN
sE
pc
pc
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Advanced Control TheoryChibum Lee -Seoultech
Performance Specifications
Achieving good reference tracking, disturbance rejection,
noise attenuation simultaneously is not possible
• Algebraic limitation
Choice between Good reference tracking and noise attenuation
Solution?
• Separate the frequency components in reference, disturbance,
and noise
1)()(1
)()(
)()(1
1)()(
sGsG
sGsG
sGsGsTsS
cp
cp
cp
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Advanced Control TheoryChibum Lee -Seoultech
Time Domain Analysis
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Advanced Control TheoryChibum Lee -Seoultech
Time Domain Analysis
After the mathematical model of the system is obtained,
analysis of system performance is needed.
• Input signal to a control system is unknown
but what if to use known test input signal
• A correlation b/w the response characteristics of a system to a
typical test input signal and the capability of the system to cope
with actual input signals
SystemInput r(t) Output y(t)
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Advanced Control TheoryChibum Lee -Seoultech
Time Domain Analysis
Time response
• Transient response: goes from IC to final state
• Steady state response: the system output behaves as t
approaches infinity
)()()( tytyty sstr
SystemInput(t) Output(t)
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Advanced Control TheoryChibum Lee -Seoultech
1st Order Systems
Standard form ryyT
1
1
)(
)(
)()()1(
TssR
sY
sRsYTs
TF of 1st order system
Y(s) Y(s)
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Advanced Control TheoryChibum Lee -Seoultech
1st Order Systems –Step response
Unit step response w/ zero initial conditions
Ts
s
Ts
T
ssTssY
1
11
1
1
1
1
1)(
ssR
tUtr s
1)(
)()(
0t,1)( / Ttety
1
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Advanced Control TheoryChibum Lee -Seoultech
1st Order Systems –Step response
Characteristic feature of a 1st order system:
• At t=T,
ie. the response y(t) is 63.2% of its total change
• At t=0, the slope of the response
632.0 368.00.1 1)()( 1 eTyty
Te
Tdt
dy
t
Tt
t
11
0
/
0
T : time constant
y(t)
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Advanced Control TheoryChibum Lee -Seoultech
1st Order Systems –Ramp response
Unit ramp response w/ zero initial conditions
1
1
1
1
1)(
2
2
2
Ts
T
s
T
s
sTssY
0 t,)( / TtTeTtty
2
1)(
)()(
ssR
ttUtr s
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Advanced Control TheoryChibum Lee -Seoultech
1st Order Systems –Ramp response
The error between the reference & the output.
• Smaller T smaller ess
ess= T
Te(t)e)e(
)eT(
TeTtt
y(t)r(t)e(t)
tss
t/T
t/T
lim
1
y(t)
y(t)
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Advanced Control TheoryChibum Lee -Seoultech
1st Order Systems –Impulse response
Unit impulse response
1
1)(
TssY
0t,1
)( / TteT
ty
1)()()( sRttr
y(t)
y(t)
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Advanced Control TheoryChibum Lee -Seoultech
1st Order Systems –Impulse response
Compare
• Time response to an impulse reference signal is identical to
an initial condition response with zero reference.
TyyyT
1)0( w/ 0
1
1)(
1)()1(
0)()0()(
1
sTsY
sYsT
sYYTssTY
T
0)0( w/ (0) yyyT
1
1)(
1)()1(
1)()(
sTsY
sYsT
sYssTY
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Advanced Control TheoryChibum Lee -Seoultech
2nd Order System
Closed-loop
22
2
22
2
)(
)(
nn
n
ss
J
Ks
J
Bs
J
K
KBsJs
K
sR
sY
Y(s) (=(s))
Standard form of 2nd order system
JK
B
J
Kn
2,
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Advanced Control TheoryChibum Lee -Seoultech
2nd Order System
Solve the following differential equation
• Case 1.
• Case 2.
0)0(,0)0(,)(2 2 xxtxxx nn
1)0(,0)0(,02 2 xxxxx nn
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Advanced Control TheoryChibum Lee -Seoultech
2nd Order System
Dynamic behavior of 2nd order system can be described
in terms of damping ratio and natural frequency n
• Characteristic equation (denominator of TF=0)
• Poles (root of characteristic equation)
22
2
2)(
)(
nn
n
sssR
sY
02 22 nnss
2
21 1, nn jss
10
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Advanced Control TheoryChibum Lee -Seoultech
2nd Order System
Pole location
1, ,1 If
, ,1 If
1, ,10 If
, ,0 If
2
21
21
2
21
21
nn
n
nn
n
ss
ss
jss
jss
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Advanced Control TheoryChibum Lee -Seoultech
2nd Order System-Step response
For underdamped case,
22222
22
2
)(1)(
1
1
2)(
dn
d
dn
n
nn
n
ss
s
s
ssssY
10
)1
(tan re whe)sin(1
1
sin1
cos1)(
21
2
2
te
tetety
d
t
d
t
d
t
n
nn
21 nd
1)(lim
tyt
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Advanced Control TheoryChibum Lee -Seoultech
2nd Order System-Step response
For no damped case,
2222
2 11)(
nn
n
s
s
ssssY
0
tty ncos1)(
1)(lim
tyt
y(t)
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Advanced Control TheoryChibum Lee -Seoultech
2nd Order System-Step response
For critically damped case,
ssssssY
n
n
nn
n 1
)(
1
2)(
2
2
22
2
1
)1(1)( tety n
tn
1)(lim
tyt
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Advanced Control TheoryChibum Lee -Seoultech
2nd Order System-Step response
For overdamped case,
)1()1(
1
1
)1)(1()(
22
22
2
nn
nnnn
n
sss
ssssY
212
)1(
22
)1(
22
21
22
121
)1(12
1
)1(12
11)(
s
e
s
e
eety
tsts
n
tt nn
1)(lim
tyt
,1
1, 2
21 nnss
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Advanced Control TheoryChibum Lee -Seoultech
2nd Order System-Step response
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Advanced Control TheoryChibum Lee -Seoultech
2nd Order System-Impulse response
Impulse response or Initial condition response
For
For
22
2
2)(
nn
n
sssY
tety d
tn n
sin
1)(
2
0for 0)(lim
tyt
10
1
t
nntety
2)(
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Advanced Control TheoryChibum Lee -Seoultech
Higher Order Systems
Unit step response
01
1
1
01
1
1
)(
)(
asasasa
bsbsbsb
sR
sYn
n
n
n
m
m
m
m
)2(2
1)(
1)(
1
response ord 2nd
22
2
1
response ord1st
01
1
1
01
1
1
nrqss
csb
ps
a
s
a
sasasasa
bsbsbsbsY
r
k kkk
kkkkjkq
j j
j
n
n
n
n
m
m
m
m
r
k
kk
t
kkk
t
k
q
j
tp
j tectebeaaty kjkjj
1
22
1
1sin1cos)(
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Advanced Control TheoryChibum Lee -Seoultech
Higher Order Systems
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Advanced Control TheoryChibum Lee -Seoultech
Example – Dominant pole
Ex.
)25.6)(256(
)5.2(5.62
)(
)(2
sss
s
sR
sY
)256(
)5.2(10
)(
)(2
ss
s
sR
sY
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Advanced Control TheoryChibum Lee -Seoultech
Transient-Response specification (2nd ord. sys.)
Transient Response Characteristics to a unit step input
• Delay time, td : time required for reach 50%
• Rise time, tr : time required for rise from 10% to 90% or
from 0% to 100%
• Peak time, tp : time required for reach peak value
• Maximum overshoot, Mp :
• Settling time, ts: time required for reach
and stay 2% or 5% of
final value
y(t)
)()( yty p
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Advanced Control TheoryChibum Lee -Seoultech
Transient-Response specification (2nd ord. sys.)
Rise time:
tr
y(t)
2
2
2
1tansin
1cos
)sin1
(cos11)(
rdrdrd
rdrd
t
r
ttt
ttety rn
tetety d
t
d
t nn
sin1
cos1)(2
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Advanced Control TheoryChibum Lee -Seoultech
Transient-Response specification (2nd ord. sys.)
01
tan1
01
tan01
10For
1tan
1
21
21
2
21
d
r
d
r
t
t
???
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Advanced Control TheoryChibum Lee -Seoultech
Transient-Response specification (2nd ord. sys.)
• From system pole
22
2
2)(
)(
nn
n
sssR
sY
2
21 1, nn jss
)(11
tan1
2
1
dd
rt
n
nsangle2
1
1
1tan)(
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Advanced Control TheoryChibum Lee -Seoultech
y(t)
Transient-Response specification (2nd ord. sys.)
Peak Time
tp
,3,2,,00)sin(0)sin(
0)sin()cos()cos()sin(
0
)sin()cos()cos()sin()(
pdpdd
d
nnpd
t
pddpdnpdpd
d
nn
t
pddpd
d
ndt
pdpd
d
nt
np
ttte
tttte
ttettety
pn
pn
pnpn
tetety d
t
d
t nn
sin1
cos1)(2
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Advanced Control TheoryChibum Lee -Seoultech
y(t)
tetety d
t
d
t nn
sin1
cos1)(2
Transient-Response specification (2nd ord. sys.)
Maximum
Overshoot
Mp
%100)(
)()(PO)Overshoot(%
)sin(1
)cos(
1)(1)(
2211
2
y
yty
eee
ytyM
p
d
pp
n
n
dn
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Advanced Control TheoryChibum Lee -Seoultech
y(t)
Transient-Response specification (2nd ord. sys.)
Settling
Time
ts
• Approximation (comes from envelop function )
criterion 5%for 5
criterion 2%for 4
n
s
n
s
t
t
tetety d
t
d
t nn
sin1
cos1)(2
211
tne
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Advanced Control TheoryChibum Lee -Seoultech
Transient-Response specification (2nd ord. sys.)
y(t)
)1
(tan re whe)sin(1
1
sin1
cos1)(
21
2
2
te
tetety
d
t
d
t
d
t
n
nn
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Advanced Control TheoryChibum Lee -Seoultech
Parameter Selection –example1
Y(s)
Y(s)
J
K
KJ
KKb
ssKsKKbJs
K
sR
sY
nh
nn
n
h
,2
2)(
)(22
2
2
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Advanced Control TheoryChibum Lee -Seoultech
Parameter Selection –example1
criterion) (5% sec 86.13
, criterion) (2% sec 48.24
sec 65.01
1tan
sec 178.02
m,N 5.12,2
53.31
11
456.061.11
2.0
2
21
2
22
2
1 2
n
s
n
s
nd
r
hnnh
n
nd
p
p
tt
t
K
BKJKJK
J
K
KJ
KKb
t
eM
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Advanced Control TheoryChibum Lee -Seoultech
Steady-State Errors in Feedback Systems
Unity feedback system
)()()(1
lim)(lim)(lim
)()()(1
1)()()(
)()()(1
)()()(
00sR
sGsG
sssEtee
sRsGsG
sYsRsE
sRsGsG
sGsGsY
pcsst
ss
pc
pc
pc
number) type:())(())((
))(())(()()(
121
121 Npspspspss
zszszszsKsGsG
ll
N
mmpc
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Advanced Control TheoryChibum Lee -Seoultech
Unit step input
Steady-State Errors in Feedback Systems
0or ,1For
1
1
)()(
)()(lim)()(lim ,0For
constant)error position :(1
1
)()(lim1
1
1
)()(1lim)(lim)(lim
1
0
1
00
0
00
ssp
p
ss
l
m
spc
sp
p
ppcs
pcsst
ss
eKN
Ke
pspss
zszsKsGsGKN
KKsGsG
ssGsG
sssEtee
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Advanced Control TheoryChibum Lee -Seoultech
Unit ramp input
Steady-State Errors in Feedback Systems
0or)()(
)()(lim ,2For
1
)()(
)()(lim ,1For
or
0)()(
)()(lim)()(lim ,0For
constant)error velocity :(1
)()(lim
1
)()(
1lim
1
)()(1lim)(lim)(lim
1
1
0
1
1
1
0
1
0
1
00
0
0
200
ss
l
N
m
sv
v
ss
l
m
sv
ss
l
m
spc
sv
v
vpcs
pcs
pcsst
ss
epspss
zszsKsKN
Ke
pspss
zszsKsKN
e
pspss
zszsKssGssGKN
KKsGssGsGssGs
ssGsG
sssEtee
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Advanced Control TheoryChibum Lee -Seoultech
Steady-State Errors in Feedback Systems
Type 1 system response to a ramp input
y(t)
y(t)
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Advanced Control TheoryChibum Lee -Seoultech
Unit parabolic(acceleration) input
Steady-State Errors in Feedback Systems
0or)()(
)()(lim ,3For
1
)()(
)()(lim ,2For
or
0)()(
)()(lim)()(lim ,1For
constant)error on accelerati:(1
)()(lim
1
)()(
1lim
1
)()(1lim)(lim)(lim
1
12
0
1
2
12
0
1
12
0
2
0
2
0
220
300
ss
l
N
m
sa
a
ss
l
m
sa
ss
l
N
m
spc
sa
a
apcs
pcs
pcsst
ss
epspss
zszsKsKN
Ke
pspss
zszsKsKN
e
pspss
zszsKssGsGsKN
KKsGsGssGsGss
ssGsG
sssEtee
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Advanced Control TheoryChibum Lee -Seoultech
Type 2 system response to a parabolic input
Steady-State Errors in Feedback Systems
y(t)
y(t)
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Advanced Control TheoryChibum Lee -Seoultech
Steady-State Errors in Feedback Systems