mm3 response of dynamic systemshomes.et.aau.dk/yang/de5/cc/mm3.pdf · mm2:ode model a general ode...
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9/8/2011 Classical Control 1
MM3 Response of Dynamic Systems
Readings: • Section 3.3 (response & pole locations, p.118-126);• Section 3.4 (time-domain specifications, p.126-131)•Section 3.6 (numerical simulation, p.138-143)
What have we talked in MM2?
• ODE models• Laplace transform• Block diagram transformation
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MM2: ODE Model A general ODE model:
SISO, SIMO, MISO, MIMO models Linear system, Time-invance, Linear Time-Invarance (LTI) Solution of ODE is an explicit description of dynamic behavior Conditions for unique solution of an ODE Solving an ODE:
Time-domain method, e.g., using exponential function Complex-domain method (Laplace transform) Numerical solution – CAD methods, e.g., ode23/ode45
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MM2: Block diagram Rules
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MM2: Simulink Block diagram System build-up
Using TF block Using nonlinear blocks Using math blocks
Creat subsystems Top-down Bottom-up
Usage of ode23 & ode45
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Goals for this lecture (MM3) Time response analysis Typical inputs 1st, 2nd and higher order systems
Performance specification of time response Transient performance Steady-state performance
Numerical simulation of time response
9/8/2011 Analog and Digital Control 7
d(t)
u(t)
(explicit) Dynamic: y(t)=f(d(t), u(t), y(0))
ODE model TF model
eaa
tm
a
etmm Tv
RK
RKKbJ
...)(
))(()(
1)(
))(()(
)(
22
21
etam
a
a
etm
e
m
etam
t
a
etm
a
t
a
m
KKbsRJsR
sR
KKbsJTSG
KKbsRJsK
sR
KKbsJ
RK
VSG
System Models: ODE/TF
y(t)
)0(,)0( mm
System (dynamic) feature description
Time response!
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Time Response Calculation (I)
d(t)=0
u(t)
y(t)
ODE model
aa
tm
a
etmm v
RK
RKKbJ
...)(
)0(,)0( mm
),,),0(),0(( feammm TTvf
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Time Response Calculation (II)
d(t)=0
u(t)
y(t)
TF model
)0(,)0( mm
),,)0(),0(( feammm TTvf
))(()(
etam
t
a
m
KKbsRJsK
VS
Laplace Trans Inv Laplace T.
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Time Response Calculation (II)
d(t)=0
u(t)
y(t)
TF model
)0(,)0( mm
),,),0(),0(( feammm TTvf
))(()(
etam
t
a
m
KKbsRJsK
VS
Laplace Trans Inv Laplace T.
)()()()()()(
sVsGSsUsGsY
am
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Time Response Calculation (III)
d(t)=0
u(t)
y(t)
TF model in MatlabNum= [K]; Den=[JR b+ktKf 0]; lsim(tf(Num,Den),u, T,X0)
)0(,)0( mm
))(()(
etam
t
a
m
KKbsRJsK
VS
Time Response Analysis Objective:
Based on TF model, Can we predict some key features of a dynamic response regarding to some typical input, without detail calculation of the solution? In another word, to get a rough sketch of the dynamic with all key (concerned) features kept
Typical inputs Corresponding responses Key features
Time Response Analysis – Typical Inputs
Supercomposition principle LTI system Typical representations
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Time Response Analysis – Impulse Signal Impulse signal
Features Convolution integration Approximation
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)(*)()()()(
1)(0
0
ttfdtftf
dtt
Time Response – First-order System (I)
First-order systemExamples: (mm2)
Motor speed controlCruise controlRC cuircuit One-tank level control
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tpt ectgketg
scsG
pp
psksG
1
)(or )(
:domain time
:constant time,1:pole,1
)(
1 :constant time,:pole,)(
Time Response – First-order System (II)
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First-order System - Impulse Response
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The coefficients of impulse response can be estimated by free-response
First-order System - Step Response
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First-order System - Ramp Response
9/8/2011 Classical Control 19Type of systems (mm4)
Time Response – Second-order System (I)
Second-order system
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Time Response – Second-order System (II)
Second-order system
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Second-order System – Pole (Root) Analysis
Second-order system
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Check execise one for MM2 – pendulumModel analysis )
Second-order System – Impulse Response
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The coefficients of impulse response can be estimated by free-response
Example: Impulse Response Estimation
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Second-order System – Step Response (I)
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Second-order System – Step Response (II)
9/8/2011 Classical Control 26Control specification!
Second-order System – Ramp Response
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Second-order System – Damping Effect
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Time Response – High-Order System
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See page 32-37 of the extra readings for detail explanation
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Goals for this lecture (MM3) Time response analysis Typical inputs 1st, 2nd and higher order systems
Performance specification of time response Transient performance Steady-state performance
Numerical simulation of time response
Performance Specification Objective: evaluation of control design Platform: based on step response of a typical 2nd-
order system
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Specification – Transient Performance (I)
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(See FC 126-131...) Rise time tr
Settling time ts
Overshoot Mp
Peak time tp
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21,
3.0%,355.0%,16
7.0%,5
6.46.4
8.1
ndd
p
p
ns
nr
t
M
t
t
Specification – Transient Performance (II)
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Specification – Steady-state Performance
)(lim)(lim0
ssFtfst
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Goals for this lecture (MM3) Time response analysis Typical inputs 1st, 2nd and higher order systems
Performance specification of time response Transient performance Steady-state performance
Numerical simulation of time response
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Numerical Simulation of Time Response Impulse response: impulse(sys) Step response: step(sys) ltiview(sys) Decomposition of transfer function to be a configuration based
on integor and gain elements (in order to simulate initial response in simulink)
EXAMPLE:sys1: Sys2:
num1=[1]; num2=[1 2];den1=[1 2 1]; den2=[1 2 3];impulse(tf(num1,den1),'r',tf(num2,den2),'b')step(tf(num1,den1),'r',tf(num2,den2),'b')