Analog and Digital
Control Systems!
Robert Stengel!
Robotics and Intelligent Systems MAE 345,
Princeton University, 2015
Frequency Response
Transfer Functions
Bode Plots
Root Locus
Proportional-IntegralDerivative (PID) Control
•! Discrete-time Dynamic Models
•!
•!
•!
•!
•!
Copyright 2015 by Robert Stengel. All rights reserved. For educational use only.
http://www.princeton.edu/~stengel/MAE345.html
1
Analog vs. Digital Signals
•! Signal: A physical indicator that conveys information,
e.g., position, voltage, temperature, or pressure
•! Analog signal: A signal that is continuous in time,
with infinite precision and infinitesimal time spacing
between indication points
•! Discrete-time signal: A signal with infinite precision
and discrete spacing between indication points
•! Digital signal: A signal with finite precision and
discrete spacing between indication points
2
Analog vs. Digital Systems
•! System: Assemblage of parts with structure, connectivity, and
behavior that responds to input signals and produces output
signals
•! Analog system: A system that operates continuously, with infinite
precision and infinitesimal time spacing between signaling points
•! Discrete-time system: A system that operates continuously, with
infinite precision and discrete spacing between signaling points
•! Digital system: A system that operates continuously, with finite
precision and discrete spacing between signaling points
3
Frequency Response of a ContinuousTime Dynamic System
! < !n
! = !n
! > !n
Output
Angle, deg
Output Rate,
deg/s
Input
Angle, deg
! : Input frequency, rad/s
! n : Natural frequency of the system, rad/s
Hz :
Hertz, frequency in cycles per sec
! ( Hz ) =
! ( rad / s )
2"
•! Sinusoidal command input (i.e., Desired rotational angle)
•! Long-term output of the dynamic system:
–! Sinusoid with same frequency as input
–! Output/input amplitude ratio dependent on input frequency
–! Output/input phase shift dependent on input frequency
•! Bandwidth: Input frequency below which output amplitude
4
and phase angle variations are negligibly small
Amplitude-Ratio and Phase-Angle
Frequency Response of a 2nd-Order System
Straight-line
asymptotes at high
and low frequency
Output
Angle, deg
Straight-line
asymptotes at high
and low frequency
Output Rate,
deg/s
20log AR and ! vs. log "
5
Asymptotes of Frequency
Response Amplitude Ratio
20 log10 (! n ) has slope of
20n dB/decade on Bode plot
Natural
Frequency
20 log10 H CLi (j! ) =
= 20 log10 #$ ARi (! ) e j"i (! ) %&
= 20 #$ log10 ARi + log10 e j!i (" ) %&
= 20 log10 ARi + j!i (" ) ( 20 log10 e )
•! Logarithm of transfer function
–! Amplitude ratio and phase
angle are orthogonal
–! Decibels (dB): 20 log AR
–! Decade: power of 10
6
Fourier Transform of a
Scalar Variable
F [ x(t)] = x( j! ) =
#
$ x(t)e
" j! t
dt, ! = frequency, rad / s
j ! i ! !1
"#
x(t) : real variable
x( j! ) : complex variable
= a(! ) + jb(! )
x(t)
= A(! )e j" (! )
x( j! ) = a(! ) + jb(! )
A : amplitude
! : phase angle
7
Laplace Transforms of
Scalar Variables
•! Laplace transform of a scalar variable is a complex number
•! s is the Laplace operator, a complex variable
"
L [ x(t)] = x(s) = # x(t)e! st dt , s = $ + j%
0
x(t) : real variable
x(s) : complex variable
= a(! ) + jb(! )
= A(! )e j" (! )
Multiplication by a constant
L [ a x(t)] = a x(s)
Sum of Laplace transforms
L [ x1 (t) + x2 (t)] = x1 (s) + x2 (s)
8
Laplace Transforms of
Vectors and Matrices
Laplace transform of a
vector variable
! x1 (s) $
&
#
L [ x(t)] = x(s) = # x2 (s) &
# ... &
%
"
Laplace transform of a
matrix variable
! a11 (s) a12 (s) ... $
&
#
L [ A(t)] = A(s) = # a21 (s) a22 (s) ... &
# ...
...
... &%
"
Laplace transform of a derivative w.r.t. time
! dx(t) $
L#
= sx(s) ' x(0)
" dt &%
Laplace transform of an integral over time
#
& x(s)
L % " x(! )d! ( =
s
$
'
9
Laplace Transforms of the
System Equations
Time-Domain System Equations
x! (t) = F x(t) + G u(t)
y(t) = H x x(t) + H u u(t)
Dynamic Equation
Output Equation
Laplace Transforms of System Equations
sx(s) ! x(0) = F x(s) + G u(s)
y(s) = H x x(s) + H u u(s)
Dynamic Equation
Output Equation
10
Example of System Transformation
DC Motor Equations
! x!1 (t) $ ! 0 1 $ ! x1 (t) $ ! 0 $
&=#
&+#
#
&#
& u(t)
#" x!2 (t) &% " 0 0 % #" x2 (t) &% " 1 / J %
! y1 (t) $ ! 1 0 $ ! x1 (t)
&=#
#
&#
#" y2 (t) &% " 0 1 % #" x2 (t)
Dynamic Equation
$ ! 0 $
! x1 (t) $
&+#
&
& u(t) = #
&% " 0 %
#" x2 (t) &%
Output Equation
Laplace Transforms of Motor Equations
" sx1 (s) ! x1 (0)
$
$# sx2 (s) ! x2 (0)
% " 0 1 % " x1 (s) % " 0 %
'=$
'+$
'$
' u(s)
'& # 0 0 & $# x2 (s) '& # 1 / J &
" y1 (s) % " 1 0 % " x1 (s)
'=$
$
'$
$# y2 (s) '& # 0 1 & $# x2 (s)
Dynamic Equation
% " 0 %
" x1 (s) %
'+$
'
' u(s) = $
'& # 0 &
$# x2 (s) '&
Output Equation
11
Laplace Transform of State Response to
Initial Condition and Control
Rearrange
sx(s) ! F x(s) = x(0) + G u(s)
[ sI ! F ] x(s) = x(0) + G u(s)
!1
x(s) = [ sI ! F ] [ x(0) + G u(s)]
Matrix inverse
[ sI ! F ]!1 =
Adj ( sI ! F )
(n x n)
sI ! F
Adj ( sI ! F ) : Adjoint matrix (n " n) : Transpose of matrix of cofactors
sI ! F = det ( sI ! F ) : Determinant
(1" 1)
12
Characteristic Polynomial of a
Dynamic System
Characteristic matrix
"
$
$
( sI ! F ) = $
$
$
#
( s ! f11 )
! f21
...
! fn1
! f12
( s ! f22 )
...
! fn2
...
...
...
...
! f1n
! f2n
...
( s ! fnn )
%
'
'
'
'
'
&
(n x n)
Characteristic polynomial
sI ! F = det ( sI ! F )
" #(s) = s n + an!1s n!1 + ...+ a1s + a0
13
Eigenvalues (or Roots) of the
Dynamic System
Characteristic equation
!(s) = s n + an"1s n"1 + ...+ a1s + a0 = 0
= ( s " #1 ) ( s " #2 ) (...) ( s " #n ) = 0
!i are eigenvalues of F or roots of the characteristic polynomial, " ( s )
14
2 x 2 Eigenvalue Example
Characteristic matrix
" (s ! f )
11
( sI ! F ) = $
$ ! f21
#
! f12
( s ! f22 )
%
'
'
&
Determinant of characteristic matrix
sI ! F =
( s ! f11 )
! f21
! f12
( s ! f22 )
= ( s ! f11 ) ( s ! f22 ) ! f12 f21
= s 2 ! ( f11 + f22 ) s + ( f11 f22 ! f12 f21 )
Factors of the 2ndDegree Characteristic
Equation
x
! ( s ) = s 2 " ( f12 + f21 ) s + ( f11 f22 " f12 f21 )
=0
15
x
x
! = cos "1 #
!(s) = ( s " #1 ) ( s " #2 ) = 0
•! Solutions of the equation are
eigenvalues of the system, either
–! 2 real roots, or
–! Complex-conjugate pair
!1 = " 1 , !2 = " 2
x
s Plane
!1 = " 1 + j# 1 , !2 = " 1 $ j# 1
! ( s ) = s 2 + 2"# n s + # n2
16
Eigenvalues Determine the
Stability of the LTI System
Positive real part
represents instability
x
Envelope of time response
converges or diverges
x
!x1 (t) = e"#$ nt cos &$ n 1" # 2 t + %1 ( !x1 (0)
'
)
s Plane
Same criterion for real roots
!x(t) = eat !x(0)
17
Numerator of the Matrix Inverse
Matrix Inverse
[ sI ! F ]!1 =
Adj ( sI ! F )
(n x n)
sI ! F
Adjoint matrix is the transpose of the matrix of cofactors*
Adj ( sI ! F ) = CT
2 x 2 example
" (s ! f )
11
( sI ! F ) = $
$ ! f21
#
! f12
( s ! f22 )
%
'
'
&
(n x n)
" (s ! f )
22
Adj ( sI ! F ) = $
$
f12
#
( s ! f11 )
" (s ! f )
22
=$
$
f21
#
%
'
'
&
f12
( s ! f11 )
* Cofactors = signed minor determinants of the matrix
f21
%
'
'
&
18
T
Analog Control!
19
Transfer Function Matrix
Laplace Transform of Output Vector
from control input to system output
(neglect initial condition, and Hu = 0)
y(s) = H x x(s) = H x ( sI ! F ) G u(s) ! H (s)u(s)
!1
Transfer Function Matrix
H (s) ! H x ( sI ! F ) G
!1
(r x m)
20
Open-Loop Dynamic
Equation of Direct-Current
Motor with Load Inertia, J
! x!1 (t) $ ! 0 1 $ ! x1 (t) $ ! 0 $
&=#
&+#
#
&#
& u(t)
#" x!2 (t) &% " 0 0 % #" x2 (t) &% " 1 / J %
Same as simplified
pitching dynamics for
quadcopter, with J = Iyy
21
Open-Loop Transfer Function Matrix
for DC Motor
Transfer Function Matrix
H OL (s) = H x [ sI ! FOL ] G
!1
where
! 0 1 $
FOL = #
&;
" 0 0 %
(r x m)
! s '1 $
&
" 0 s %
[ sI ' F ] = #
OL
! 0 $
! 1 0 $
&
#
=
I
Hx = #
;
G
=
& 2
1 &
0
1
#
"
%
" J %
Matrix Inverse
" s 1 %
$
'
Adj ( sI ! FOL ) " s !1 %
!1
# 0 s & = " 1 / s 1 / s2 %
sI
!
F
=
=
=
[
$
'
$
'
OL ]
s2
sI ! FOL
1/ s &
# 0 s &
# 0
!1
22
Open-Loop Transfer Function Matrix
for DC Motor
H OL (s) = H x [ sI ! FOL ]
!1
Dimension = 2 x 1
Adj ( sI ! FOL )
G = Hx
G
sI ! FOL
" 1 0 % " 1 / s 1 / s 2 % " 0 % " 1 0 % " 1 / Js 2 %
=$
'$
'
'$
'=$
'$
1 / s & # 1 / J & # 0 1 & # 1 / Js &
# 0 1 &# 0
! 1
#
Js 2
#
=
# 1
# Js
"
$ !
& #
&=#
& #
& #
% #"
y1 (s)
u(s)
y2 (s)
u(s)
$
&
&
&
&
&%
Angle = Double integral of input torque, u(s)
Angular rate = Integral of input torque, u(s)
23
Closed-Loop System
Closed-loop dynamic equation
! x!1 (t) $ !
0
&=#
#
#" x!2 (t) &% #" 'c1 / J
$ ! x1 (t) $ ! 0
&+#
&#
'c2 / J & # x2 (t) & # c1 / J
%"
% "
1
$
& yC
&%
x! (t) = FCL x(t) + G u(t)
24
Closed-Loop Transfer Function Matrix
H CL (s) = H x [ sI ! FCL ] G = H x
!1
" (s + c J ) 1 %
2
'
$
$ !c1 J
s '
&
=#
c1 +
( 2 c2
*) s + s + -,
J
J
Adj ( sI ! FCL )
G
sI ! FCL
" 0
$
$# c1 / J
Dimension = 2 x 1
%
'
'&
Closed-loop transfer functions
Scalar components differ only in numerators
!
#
#
#
#
#"
! c1 J $
y1 (s) $
&
#
&
c
J
s
&
#
(
)
yC (s) &
% =
" 1
=
&
c1 *
' 2 c2
y2 (s)
& )( s + s + ,+
J
J
yC (s) &
%
! n (s) $
&
# 1
# n2 ( s ) &
%
"
- (s)
25
Frequency Response Functions
Substitute s = j! in transfer functions
y ( j! )
c1 J
Output Angle
= 1
=
2
Commanded Angle yC ( j! ) ( j! ) + ( c2 J ) ( j! ) + c1 J
=
! n2
"! 2 + 2#! n ( j! ) + ! n 2
! a1 (! ) + jb1 (! ) $ AR1 (! ) e j%1 (! )
Output Rate
y ( j! )
( j! ) c1 J
= 2
=
2
Commanded Angle yC ( j! ) ( j! ) + ( c2 J ) ( j! ) + c1 J
j! )! n 2
(
=
$ AR2 (! ) e j% (! )
"! 2 + 2#! n ( j! ) + ! n 2
2
Natural Frequency: ! n = c1 J
Damping Ratio:
! = ( c2 J ) 2" n = ( c2 J ) 2 c1 J
26
Bode Plot Depicts
Frequency Response
Hendrick Bode
% Frequency Response of DC Motor
Angle Control
F1 = [0 1;-1 0];
G1 = [0;1];
F1a = [0 1;-1 -1.414];
F1b = [0 1;-1 -2.828];
Hx = [1 0;0 1];
Sys1
Sys2
Sys3
20log AR and ! vs. log "
= ss(F1,G1,Hx,0);
= ss(F1a,G1,Hx,0);
= ss(F1b,G1,Hx,0);
w = logspace(-2,2,1000);
bode(Sys1,Sys2,Sys3,w), grid
27
Root (Eigenvalue) Locus
!
#
#
#
#
#"
! -2 $
! c1 J $
! 1 $
y1 (s) $
# n &
&
#
&
#
&
# - n2 s &
# ( c1 J ) s &
yC (s) &
% =
%
"
"
" s %
=
=
c1 * ( s 2 + 2.- n s + - n2 ) ( s 2 + 1.414- n s + 1)
y2 (s) & ' 2 c2
& )( s + s + ,+
J
J
yC (s) &
%
•! Variation of roots as scalar gain, k, goes from 0 to "
•! With nominal gains, c1 and c2,
!n =
c1
c J
= 1 rad / s, " = 2 = 0.707
J
2! n
% Root Locus of DC Motor Angle Control
F = [0 1;-1 -1.414];
G = [0;1];
Hx1 = [1 0]; % Angle Output
Hx2 = [0 1]; % Angular Rate Output
Sys1
Sys2
= ss(F,G,Hx1,0);
= ss(F,G,Hx2,0);
rlocus(Sys1), grid
figure
rlocus(Sys2), grid
28
Root (Eigenvalue) Locus
Increase Angle Feedback Gain
Initial
Root
Increase Rate Feedback Gain
Initial
Root
Zero
Initial
Root
Initial
Root
29
Classical Control System
Design Criteria
Step Response
Frequency Response
Eigenvalue Location
30
Single-Axis Angular
Control of Non-Spinning
Spacecraft
Pitching motion (about the y axis) is to be controlled
" !!(t) % " 0
$
'=$
!
$# q(t)
'& # 0
" ! (t)
$
$# q(t)
1 % " ! (t) % " 0
'+$
'$
0 & $# q(t) '& $# 1 / I yy
%
'=
'&
%
' M y (t)
'&
" Pitch Angle %
'
$
Pitch
Rate
'&
$#
Identical to the DC Motor control problem
31
Proportional-IntegralDerivative (PID) Controller
Control Error
Control Command
Control Law Transfer
Function
(w/common denominator)
e(s) = ! C (s) " ! (s)
e(s)
u(s) = cP e(s) + cI
+ cD se(s)
s
u(s) cP s + cI + cD s 2
=
e(s)
s
•! Proportional term weights control error directly
•! Integrator compensates for persistent (bias) disturbance
•! Differentiator produces rate term for damping
32
Proportional-IntegralDerivative (PID) Controller
Forward-Loop Angle
Transfer Function
gA = Actuator Gain
! (s) " cI + cP s + cD s 2 % " gA %
=
' $ I s2 '
e(s) $#
s
& $# yy '&
33
Closed-Loop Spacecraft Control Transfer
Function w/PID Control
Closed-Loop
Angle Transfer Function
Ziegler-Nichols PID Tuning Method
http://en.wikipedia.org/wiki/Ziegler–
Nichols_method
" c I + cP s + c D s 2 %
! (s)
gA '
$
I yy s 3
! (s)
e(s)
#
&
=
=
2
!
(s)
! c (s) 1+
" c I + cP s + c D s
%
g
1+
'
$
A
3
e(s)
I yy s
#
&
2
c D s + cP s + c I
=
3
I yy s gA + cD s 2 + cP s + cI
34
Closed-Loop Frequency
Response w/PID Control
! (s)
c I + cP s + c D s 2
=
! c (s) cI + cP s + cD s 2 + I yy s 3 gA
Let s = j! . As ! " 0
As ! " #
! ( j" )
c
# I =1
! c ( j" )
cI
! ( j" )
$cD" 2
c
jc
#
g = D gA = $ D gA
3 A
! c ( j" ) $ jI yy"
jI yy"
I yy"
AR #
cD
gA ; % # $90 deg
I yy"
Steady-state output =
desired steady-state input
High-frequency
response rolls off
and lags input
35
Digital Control!
Stengel, Optimal Control and Estimation,
1994, pp. 79-84, E-Reserve
36
From Continuous- to DiscreteTime Systems
•! Continuous-time systems are described by
differential equations, e.g.,
!!x(t) = F!x(t) + G!u(t)
•! Discrete-time systems are described by
difference equations, e.g.,
!x(t k +1 ) = "!x(t k ) + #!u(t k )
•! Discrete-time systems that are meant to
describe continuous-time systems are called
sampled-data systems
! = fcn ( F ); " = fcn ( F,G )
37
Sampled-Data (Digital)
Feedback Control
Digital-to-Analog
Conversion
Analog-to-Digital
Conversion
38
Integration of Linear, Time-Invariant
(LTI) Systems
!!x(t) = F!x(t) + G!u(t)
t
t
0
0
!x(t) = !x(0) + # !!x(" )d" = !x(0) + # [ F!x(" ) + G!u(" )] d"
Homogeneous solution (no input), #u = 0
t
!x(t) = !x(0) + # [ F!x(" )] d"
0
= eF(t$0) !x(0) = % ( t,, 0 ) !x(0)
! ( t,0 ) = eF(t"0) = eFt = State transition matrix from 0 to t
39
Equivalence of 1st-Order Continuousand Discrete-Time Systems
Example demonstrates
stability (green) and instability
(red), defined by sign of a
! = ax(t) , x(0) given
x(t)
t
!
= eat x(0)
x(t) = x(0) + ! x(t)dt
0
Choose small time interval, #t
Propagate state to time, t + n"t
40
Equivalence of 1st-Order
Continuous- and
Discrete-Time Systems
x0 = x(0)
x1 = x(!t) = ea!t x0 = " (!t) x0
x2 = x(2!t) = e2a!t x0 = ea!t x1 = " (!t) x1
.......
xk = x(k!t) = e ka!t x0 = ea!t xk#1 = " (!t) xk#1
.......
xn = x(n!t) = x(t) = ena!t x0 = " (!t) xn#1
41
Initial Condition Response of
Continuous-and Discrete-Time Models
Identical response at the sampling instants
!x(t k+1 ) = !x(t k ) + "
t k+1
tk
F!x(t)dt = eF(tk+1 #tk )!t !x(t k ) = $ ( !t ) !x(t k )
42
Input Response of LTI System
Inhomogeneous (forced) solution
t
!x(t) = e !x(0) + ( $% eF(t"# )G!u (# ) &' d#
Ft
0
t
= ! ( t " 0 ) #x(0) + ! ( t " 0 ) ) %& ! ( t " $ ) G#u ($ ) '( d$
0
43
Sampled-Data System
(Stepwise application of prior results)
Assume (tk – tk+1) = #t = constant
! ( t k +1 " t k ) = eF(tk+1 "tk ) = eF(#t ) = ! ( #t )
Assume #u = constant from tk to tk+1
!t
!x(t k+1 ) = " ( !t ) !x(t k ) + " ( !t ) ) %& " ( !t # $ ) G '( d$ !u k
0
Evaluate the integral
!x(t k+1 ) = " ( !t ) !x(t k ) + " ( !t ) $% I # " #1 ( !t ) &' F #1G!u ( t k )
( " ( !t ) !x(t k ) + ) ( !t ) !u ( t k )
44
Digital Flight Control
•! Historical notes:
–! NASA Fly-By-Wire F-8C (Apollo GNC system, 1972)
•! 1st conventional aircraft with digital flight control
•! NASA Dryden Research Center
–! Princeton's Variable-Response Research Aircraft (Z-80
microprocessor, 1978)
•! 2nd conventional aircraft with digital flight control
•! Princeton Universitys Flight Research Laboratory
http://www.princeton.edu/~stengel/FRL.pdf
45
Next Time:!
Sensors and Actuators!
46
SupplementaL
Material
47
Eigenvalues, Damping
Ratio, and Natural
Frequency
% Eigenvalues, Damping Ratio,
and Natural Frequency of Angle Control
F1 = [0 1;-1 0];
G1 = [0;1];
F1a = [0 1;-1 -1.414];
F1b = [0 1;-1 -2.828];
Hx = [1 0;0 1];
Sys1
Sys2
Sys3
= ss(F1,G1,Hx,0);
= ss(F1a,G1,Hx,0);
= ss(F1b,G1,Hx,0);
eig(F1)
eig(F1a)
eig(F1b)
damp(Sys1)
damp(Sys2)
Eigenvalues
!1 , !2
Damping Ratio, Natural Frequency
!
0 + 1i
0 - 1i
0
-0.707 + 0.707i
-0.707 - 0.707i
0.707
-0.414
-2.414
Overdamped
!n
1
1
48
Bode Plots of Proportional, Integral,
and Derivative Compensation
H( j" ) = 1
H( j" ) = 10
H( j" ) = 100
•! Amplitude ratio, AR(dB) =
constant
Phase Angle, ! ( deg ) = 0
H( j" ) =
1
j"
H( j" ) =
10
j"
H( j" ) = j"
H( j" ) = 10 j"
•! Slope of AR(dB) =
–20 dB/decade
•! Slope of AR(dB) =
+20 dB/decade
Phase Angle, ! = "90 deg
Phase Angle, ! = +90 deg
Single-Input/Single-Output
Sampled-Data Control
•! Sampler is an analog-to-digital (A/D) converter
•! Reconstructor is a digital-to-analog (D/A) converter
•! Sampling of input and feedback signal could occur
separately, before the summing junction
•! Sampling may be periodic or aperiodic
D/A
Plant
50
49
Equilibrium Response of
Linear Discrete-Time Model
•! Dynamic model
"x k +1 = #"x k + $"uk + %"wk
•! At equilibrium
"x k +1 = "x k = "x* = constant
•! Equilibrium response
%#w *
( I ! " ) #x* = $#u * +%
!1
#x* = ( I ! " ) ( $#u * +%
%#w *)
(.)* = constant
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Factors That Complicate
Precise Control!
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Error, Uncertainty, and Incompleteness
May Have Significant Effects
•!
•!
•!
•!
Scale factors and biases:
Disturbances:
Measurement error:
Modeling error
–! Parameters:
–! Structure:
•!
Saturation (limiting)
•!
Threshold
•!
Preload
•!
Friction
•!
Rectifier
•!
Hysteresis
•!
Higher-degree terms
y = kx + b
Constant/variable forces
Noise, bias
e.g., Inertia
e.g., Higher-order modes
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Nonlinearity
Complicates
Response
–! Displacement limit,
maximum force or rate
–! Dead zone, slop
–! Breakout force
–! Sliding surface resistance
–! Hard constraint, absolute
value (double)
–! Backlash, slop
–! Cubic spring, separation
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Discrete-Time and
Sampled-Data Models
Linear, time-invariant, discrete-time model
"x k +1 = #"x k + $"uk + %"wk
Discrete-time model is a sampled-data model if it represents
a continuous-time system
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Sampling Effect on Continuous Signal
•! Two waves, different frequencies, indistinguishable
in periodically sampled data
•! Frequencies above the sampling frequency aliased
to appear at lower frequency (frequency folding)
•! Solutions: Either
–! Sampling at higher rate or
–! Analog low-pass filtering
Folding
Frequency
Aliased
Spectrum
Actual
Spectrum
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Quantization and Delay Effects
•! Continuous signal sampled with finite
precision (quantized)
–! Solution: More bits in A/D and D/A conversion
•! Effective delay of sampled signal
–! Described as phase shift or lag
–! Solution:
•! Higher sampling rate or
•! Lead compensation
Analog-to-Digital Converter
rh illustrates zero-order hold effect
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Time Delay Effect
•! Control command delayed by ! sec
•! Laplace transform of pure time delay
L #$u ( t ! " ) %& = e!" s u ( s )
•! Delay introduces frequency-dependent phase lag with no
change in amplitude
AR e! j"# = 1
(
)
$ ( e! j"# ) = !#
•! Phase lag reduces closed-loop stability
58
Effect of Closed-Loop Time
Delay on Step Response
•! With no delay ! n = 1 rad / s, " = 0.707
•! With varying delays
59
Princeton's VariableResponse Research
Aircraft
60
Discrete-Time Frequency Response Can be
Evaluated Using the z Transform
!x k+1 = " !x k + # !u k
z transform is the Laplace transform of a periodic sequence
System z transform is
z!x(z) " !x(0) = # !x(z) + $! u(z)
which leads to
!x(z) = ( zI " # ) "1[ !x(0) + $! u(z)]
Further details are beyond the present scope
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Second-Order Example:
Airplane Roll Motion
# !p & # Roll rate, rad/s &
%
(=%
(
%$ !" (' %$ Roll angle, rad ('
!) A = Aileron deflection, rad
L p : roll damping coefficient
L! A : aileron control effect
Rolling motion of an airplane, continuous-time
# !!p & # L
%
(=% p
%$ !"! (' %$ 1
0 & # !p & # L) A &
(+%
(%
( !) A
0 (' %$ !" (' $% 0 ('
Rolling motion of an airplane, discrete-time
L T
#
ep
%
# !pk &
( = % e L pT ) 1
%
!
"
%$
k (
' %
Lp
%$
(
)
0 &
( # !pk)1
(%
1 ( %$ !"k)1
('
& # ~L T &
*A
(+%
( !* Ak)1
0
(' %$
('
T = sampling interval, s
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