Root Locus Analysis of Parameter
Variations and Feedback Control!
Robert Stengel, Aircraft Flight Dynamics!
MAE 331, 2016
Learning Objectives
•! Effects of system parameter variations on
modes of motion
•! Root locus analysis
–! Evanss rules for construction
–! Application to longitudinal dynamic models
Reading:!
Flight Dynamics!
357-361, 465-467, 488-490, 509-514!
Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.
http://www.princeton.edu/~stengel/MAE331.html
http://www.princeton.edu/~stengel/FlightDynamics.html
1
Review Questions!
!!
!!
!!
!!
!!
!!
!!
!!
!!
What is a Laplace transform?!
What are some characteristics of Laplace transforms?!
What is the significance of (sI – F)–1?!
Why is the partial fraction expansion of (sI – F)–1
important?!
What is the dimension of the transfer function matrix
with 4 outputs and 3 inputs?!
Why must every complex root (or eigenvalue) be
accompanied by its complex conjugate?!
Why is it bad for roots to lie in the right half of the
s plane plot?!
Of what use is a Bode plot?!
What are some important rules for constructing a
Bode plot?!
2
Characteristic Equation
!x(s) = [ sI " F ]
"1
[ sI ! F]
!1
[ !x(0) + G !u(s) + L!w(s)]
Adj ( sI ! F ) CT ( s )
=
=
sI ! F
sI ! F
(n " n)
(1"1)
Characteristic equation defines the modes of motion
sI ! F = "(s) = s n + an !1s n !1 + ... + a1s + a0
= ( s ! #1 ) ( s ! #2 ) (...) ( s ! #n ) = 0
Recall: s is a complex variable
s = ! + j"
3
Real Roots
!! On real axis in s Plane
!! Represents convergent or
divergent time response
!! Time constant, ! = –1/" = –
1/µ, sec
s Plane = (! + j" ) Plane
!i = µi (Real number)
x ( t ) = x ( 0 ) e µt
4
Complex Roots
!! Occur only in complexconjugate pairs
!! Represent oscillatory modes
!! Natural frequency, #n, and
damping ratio, $, as shown
!1 = µ1 + j"1 = #$% n + j% n 1# $
" = cos#1 $
2
Stable
!2 = µ2 + j" 2 = µ1 # j"1 ! !1*
Unstable
= #$% n # j% n 1# $ 2
s Plane = (! + j" ) Plane
5
Natural Frequency, Damping Ratio,
and Damped Natural Frequency
( s ! "1 )( s ! "1* ) = $% s ! ( µ1 + j#1 )&' $% s ! ( µ1 ! j#1 )&'
= s 2 ! 2 µ1s + ( µ12 + #12 ) ! s 2 + 2() n s + ) n2
(µ
1
µ1 = !"# n = !1 Time constant
2
+ $ 12 )
12
= # n = Natural frequency
$1 = # n 1! " 2 ! # ndamped = Damped natural frequency
6
Longitudinal Characteristic Equation
How do the roots vary when we change M!?
! Lon (s) = s 4 + a3 s 3 + a2 s 2 + a1s + a0
(
= s + DV +
4
L"
)
VN # M q s
(
3
with Lq = Dq = 0
)
L
L
L
$
'
+ &( g # D" ) V V + DV " V # M q # M q " V # M " ) s 2
N
N
N
%
(
L
L
+ M q $( D" # g ) V V # DV " V ' + D" M V # DV M " s
&%
N
N)
(
L
L
+ g MV " V # M" V V = 0
N
N
{
(
)
}
7
… or pitch-rate damping, Mq?
(
! Lon (s) = s 4 + DV +
L"
)
3
VN # M q s
(
)
L
L
L
$
'
+ &( g # D" ) V V + DV " V # M q # M q " V # M " ) s 2
N
N
N
%
(
L
L
+ M q $( D" # g ) V V # DV " V ' + D" M V # DV M " s
&%
N
N)
(
L
L
+ g MV " V # M" V V = 0
N
N
{
(
)
}
8
Evanss Rules for
Root Locus Analysis!
Walter R. Evans
(1920-1999)
9
Root Locus Analysis of Parametric
Effects on Aircraft Dynamics
•! Parametric variations alter
eigenvalues of F
•! Graphical technique for
finding the roots as a
parameter (“gain”) varies
Locus: the set of all points whose
location is determined by stated
conditions
Short
Period
Variation with
±M !
Phugoid
Phugoid
Short
Period
s Plane
10
Parameters of
Characteristic Polynomial
! Lon (s) = s 4 + a3s 3 + a2 s 2 + a1s + a0
(
= ( s " #1 ) ( s " #2 ) ( s " # 3 ) ( s " # 4 )
)(
)
= s 2 + 2$ P% nP s + % n2P s 2 + 2$ SP% nSP s + % n2SP = 0
Break the polynomial into 2 parts to examine the
effect of a single parameter
! ( s ) = d(s) + k n(s) = 0
11
Effect of a0 Variation on
Longitudinal Root Location
Example: Root locus gain, k = a0
! Lon (s) = "# s 4 + a3s 3 + a2 s 2 + a1s $% + [ k ] ! d(s) + kn(s)
= ( s & '1 ) ( s & '2 ) ( s & ' 3 ) ( s & ' 4 ) = 0
d ( s ) : Polynomial in s, degree = n; there are n (= 4) poles
n ( s ) : Polynomial in s, degree = q; there are q (= 0) zeros
d(s) = s 4 + a3s 3 + a2 s 2 + a1s
= "# s ! ( 0 ) $% ( s ! & '2 ) ( s ! & '3 ) ( s ! & '4 ); four poles, one at 0
n(s) = 1; no zeros
12
Effect of a1 Variation on
Longitudinal Root Location
Example: Root locus gain, k = a1
! Lon (s) = s 4 + a3s 3 + a2 s 2 + ks + a0 ! d(s) + kn(s)
= ( s " #1 ) ( s " #2 ) ( s " # 3 ) ( s " # 4 ) = 0
d ( s ) : Polynomial in s, degree = n; there are n (= 4) poles
n ( s ) : Polynomial in s, degree = q; there are q (= 1) zeros
d(s) = s 4 + a3s 3 + a2 s 2 + a0
= ( s ! " '1 ) ( s ! " '2 ) ( s ! " '3 ) ( s ! " '4 ); four poles
n(s) = s; one zero at 0
13
Three Equivalent Equations for
Evaluating Locations of Roots
! ( s ) = d(s) + k n(s) = 0
1+ k
k
n(s)
=0
d(s)
n(s)
= !1 = (1)e± j" (rad ) = (1)e ± j180°
d(s)
14
Evan’s Root Locus Criterion
All points on the locus of roots must satisfy the equation
k
n(s)
= !1
d(s)
i.e., all points on root locus must have phase angle = ±180°
k
n(s)
= (1)e ± j180°
d(s)
15
Longitudinal Equation Example
Typical flight condition
! Lon (s) == s 4 + a3s 3 + a2 s 2 + a1s + a0
= s 4 + 2.57s 3 + 9.68s 2 + 0.202s + 0.145
2
2
= "# s 2 + 2 ( 0.0678 ) 0.124s + ( 0.124 ) $% "# s 2 + 2 ( 0.411) 3.1s + ( 3.1) $% = 0
Phugoid
Short Period
16
Effect of a0 Variation
! Lon (s) = s 4 + 2.57s 3 + 9.68s 2 + 0.202s + 0.145 = 0
k = a0
!(s) = s 4 + a3s 3 + a2 s 2 + a1s + a0
= s ( s 3 + a3s 2 + a2 s + a1 ) + k
= s ( s + 0.21) "# s 2 + 2.55s + 9.62 $% + k
Rearrange
k
= %1
s ( s + 0.21) !" s 2 + 2.55s + 9.62 #$
17
Effect of a1 Variation
k = a1
!(s) = s 4 + a3s 3 + a2 s 2 + a1s + a0
= s 4 + a3s 3 + a2 s 2 + ks + a0
= ( s 4 + a3s 3 + a2 s 2 + a0 ) + ks
= #$s 2 " 0.00041s + 0.015%&#$s 2 + 2.57s + 9.67%& + ks
Rearrange
ks
= !1
"# s ! 0.00041s + 0.015 $% "# s 2 + 2.57s + 9.67 $%
2
18
Origins of Roots (k = 0)
n poles of d(s)
!(s) = d(s) + kn(s) #k"0
##
" d(s)
k
= %1
s ( s + 0.21) !" s + 2.55s + 9.62 #$
2
ks
= !1
"# s 2 ! 0.00041s + 0.015 $% "# s 2 + 2.57s + 9.67 $%
19
Destinations of Roots as k
Becomes Large
1) q roots go to the zeros of n(s)
d(s) + kn(s) d(s)
=
+ n(s) #k!±"
##
! n(s) = ( s $ z1 ) ( s $ z2 )!
k
k
2) (n – q) roots go to infinity
! sn
$
$
! d(s) + kn(s) $ ! d(s)
+
k
)
)))
'
+
k
=
' s ( n*q ) ± R ' ±(
k'± R'±(
q
#
&
#
&
#
&
n(s)
%
"
% " n(s)
"s
%
20
Destinations of Roots for Large k
k
= %1
s ( s + 0.21) !" s 2 + 2.55s + 9.62 #$
ks
= !1
"# s 2 ! 0.00041s + 0.015 $% "# s 2 + 2.57s + 9.67 $%
R
R
21
(n – q) Roots Approach Asymptotes
as k –> ±"
Asymptote angles for positive k
! (rad) =
" + 2m"
, m = 0,1,...,(n # q) # 1
n#q
Asymptote angles for negative k
2m"
! (rad) =
, m = 0,1,...,(n # q) # 1
n#q
22
Origin of Asymptotes =
Center of Gravity
(Sum of real parts of poles minus sum of real parts of zeros)
÷ (# of poles minus # of zeros)
q
n
"c.g." =
$!
i =1
"i
# $! zj
j =1
n#q
23
Root Locus on Real Axis
•! Locus on real axis
–! k > 0: Any segment with odd number of
poles and zeros to the right on the axis
–! k < 0: Any segment with even number of
poles and zeros to the right on the axis
24
Roots for Positive and Negative
Variations of k = a0
k
= %1
s ( s + 0.21) !" s + 2.55s + 9.62 #$
2
25
Roots for Positive and Negative
Variations of k = a1
ks
= !1
"# s ! 0.00041s + 0.015 $% "# s 2 + 2.57s + 9.67 $%
2
26
Root Locus Analysis of
Simplified Longitudinal Modes!
27
Approximate Phugoid Model
2nd-order equation
!!x Ph
#
# !V! & % *DV
=%
( ) % LV
%$ !"! (' %
VN
$
*g & #
( !V
%
0 ( %$ !"
('
#
& % T+ T
( + % L+ T
(' %
VN
$
Characteristic polynomial
&
(
( !+ T
('
sI ! FPh = det ( sI ! FPh ) "
#(s) = s 2 + DV s + gLV / VN
= s 2 + 2$% n s + % n 2
! n = gLV / VN
"=
DV
2 gLV / VN
28
Effect of Airspeed on Natural
Frequency and Period
Neglecting compressibility effects
g
2g
=
mVN2
LV g
! #$C LN " N S %&
VN m
1
2g
2g 2
#
2 %
'$C LN 2 " NVN S (& = mV 2 [ mg ] = V 2
N
N
13.87
!n " 2 gV "
,rad/s
N
VN (m/s)
Period, T = 2! / " n
# 0.45VN (m/s), sec
29
Effect of L/D on Damping Ratio
Neglecting compressibility effects and thrust sensitivity
to velocity, TV
DV !
!
C DN " NVN S m
2 2 g VN
=
1
#C D " NVN S %&
m$ N
C DN " NVN2 S 2
2mg
Natural
Velocity Frequency
m/s
rad/s
50
0.28
100
0.14
200
0.07
400
0.035
=
Period
sec
23
45
90
180
!=
1 # C DN &
2 %$ C LN ('
!"
DV
2 gLV / VN
1
2 ( L / D )N
Damping
L/D Ratio
5
10
20
40
0.14
0.07
0.035
0.018
30
Effect of LV/VN Variation
k = gLV/VN
!(s) = ( s 2 + DV s ) + k
= "# s ( s + DV ) $% + [ k ]
Change in damped
natural frequency
! ndamped ! ! n 1" # 2
31
Effect of DV Variation
k = DV
(
!(s) = ( s 2 + gLV / VN ) + ks
= # s + j gLV / VN
$
)( s " j
)
gLV / VN % + [ ks ]
&
Change in damping ratio
!
32
Approximate Short-Period Model
Approximate Short-Period Equation (Lq = 0)
!!x SP
# Mq
# !q! & %
=%
()%
%$ !"! (' % 1
$
M"
*
L"
VN
&
# M+ E
( # !q & %
( % !" ( + % *L+ E
(' %
VN
(' %$
$
&
(
( !+ E
('
Characteristic polynomial
$L
'
$
L '
!(s) = s 2 + & " # M q ) s # & M " + M q " )
VN (
% VN
(
%
= s 2 + 2*+ n s + + n 2
$ L#
'
" Mq )
&
$
L '
% VN
(
! n = " & M # + M q # ); * =
V
$
%
N (
L '
2 "& M # + M q # )
VN (
%
33
Effect of M% on Approximate
Short-Period Roots
k = M%
$L
'
$
L '
!(s) = s 2 + & " # M q ) s # & M q " ) # k = 0
VN (
% VN
(
%
$
L '
= & s + " ) s # Mq # k = 0
VN (
%
(
)
Change in damped
natural frequency
34
Effect of Mq on Approximate
Short-Period Roots
k = Mq
Change primarily in
damping ratio
!(s) = s 2 +
$
L"
L '
s # M" # M q & s + " )
VN
VN (
%
2
2
0 *
-4 0 *
-4 $
$ L# '
$ L# '
L '
2 , L#
2 2 , L#
2
/
!(s) = 1 s "
+ &
+ M # 5 1s "
" &
+ M# /5 " k & s + # ) = 0
)
)
VN (
% 2VN (
% 2VN (
/ 2 2 , 2VN
/2 %
23 ,+ 2VN
.6 3 +
.6
(–)
(–)
35
Effect of L!/VN on Approximate
Short-Period Roots
k = L%/VN
•! Change primarily
in damping ratio
!(s) = s 2 " M q s " M # +
L#
s " Mq
VN
(
)
2
2
0 *M
-4 0 * M
-4
$ Mq '
$ Mq '
2 , q
22 , q
/
/2
= 1s +
" &
+
M
"
+
M
s
+
5
1
#
# 5 + k s " Mq = 0
)
&
)
% 2 (
% 2 (
/2 2 , 2
/2
23 ,+ 2
.6 3 +
.6
(
(–)
(–)
)
36
Flight Control Systems!
SAS = Stability Augmentation System
37
Effect of Scalar Feedback Control
on Roots of the System
K
H (s) =
kn(s)
d(s)
Block diagram algebra
!y(s) = H (s)!u(s) =
kn(s)
kn(s)
!u(s) =
K !" (s)
d(s)
d(s)
= K H (s)[ !yc (s) " !y(s)]
!y(s) = K H (s)!yc (s) " K H (s)!y(s)
38
Scalar Closed-Loop
Transfer Function
K
[1+ K H
H (s) =
kn(s)
d(s)
(s)] !y(s) = K H (s)!yc (s)
!y(s)
K H (s)
=
!yc (s) [1+ K H (s)]
39
Roots of the Closed-Loop
Control System
kn(s)
Kkn(s)
Kkn(s)
!y(s)
d(s)
=
=
=
kn(s) % [ d(s)+ Kkn(s)] ! closed ( s )
!yc (s) "
$1+ K
'
loop
d(s) &
#
K
Closed-loop roots are solutions to
! closed" (s) = d(s) + Kkn(s) = 0
loop
or
K
kn(s)
= !1
d(s)
40
Pitch Rate Feedback to Elevator
k"qE ( s # z"qE )
!q(s)
K H (s) = K
=K 2
= #1
2
!" E(s)
s + 2$ SP% nSP s + % nSP
!! # of roots = 2
!! Angles of asymptotes, &, for
!! # of zeros = 1
the roots going to "
!! K -> +": –180 deg
!! Destinations of roots (for k =
±"):
!! K -> –": 0 deg
!! 1 root goes to zero of n(s)
!! 1 root goes to infinite radius
41
Pitch Rate Feedback to Elevator
•! Center of gravity on real
axis
•! Locus on real axis
–! K > 0: Segment to the left of
the zero
–! K < 0: Segment to the right of
the zero
Feedback effect is analogous
to changing Mq
42
Next Time:!
Advanced Longitudinal Dynamics!
Flight Dynamics!
204-206, 503-525!
Learning Objectives
Angle-of-attack-rate aero effects
Fourth-order dynamics
Steady-state response to control
Transfer functions
Frequency response
Root locus analysis of parameter variations
Nichols chart
Pilot-aircraft interactions
43
Supplemental Material!
!
44
Corresponding 2nd-Order
Initial Condition Response
Same envelopes for displacement and rate
x1 ( t ) = Ae!"# nt sin %# n 1! " 2 t + $ '
&
(
x2 ( t ) = Ae!"# nt %# n 1! " 2 ' cos %# n 1! " 2 t + $ '
&
(
&
(
45
Multi-Modal LTI Responses Superpose
Individual Modal Responses
•! With distinct roots,
(n = 4) for example,
partial fraction
expansion for each
state element is
!xi ( s ) =
d1i
+
d2i
+
d3i
+
d4i
( s " #1 ) ( s " #2 ) ( s " #3 ) ( s " #4 )
, i = 1, 4
Corresponding 4th-order time response is
!xi ( t ) = d1i e"1t + d2i e"2t + d3i e"3t + d4i e"4t , i = 1, 4
46
Magnitudes of Roots Going to Infinity
R(+)
R(–)
s ( n!q ) = R e! j180° " # or
R e! j 360° " !#
s = R1 ( n!q ) e! j180° ( n!q ) " #
or
1 ( n!q )
R
e
! j 360° ( n!q )
" !#
Magnitudes of roots are the same for given k
Angles from the origin are different
47
Asymptotes of Roots (for k -> ±")
4 roots to infinite radius
Asymptotes = ±45°, ±135°
3 roots to infinite radius
Asymptotes = ±60°, –180°
48
Summary of Root Locus Concepts
Destinations
of Roots
Origins
of Roots
Center
of Gravity
Locus on
Real Axis
49
Effects of Airspeed, Altitude, Mass, and
Moment of Inertia on Fighter Aircraft
Short Period
Airspeed variation at constant altitude
Airspeed
m/s
91
152
213
274
Dynamic
Pressure
P
2540
7040
13790
22790
Angle of
Attack
deg
14.6
5.8
3.2
2.2
Natural
Frequency
rad/s
1.34
2.3
3.21
3.84
Period
sec
4.7
2.74
1.96
1.64
Damping
Ratio
0.3
0.31
0.3
0.3
Altitude variation with constant dynamic pressure
Airspeed
m/s
122
152
213
274
Altitude
m
2235
6095
11915
16260
Natural
Frequency
rad/s
2.36
2.3
2.24
2.18
Period
sec
2.67
2.74
2.8
2.88
Damping
Ratio
0.39
0.31
0.23
0.18
Mass variation at constant altitude
Mass
Variation
%
-50
0
50
Natural
Frequency
rad/s
2.4
2.3
2.26
Period
sec
2.62
2.74
2.78
Damping
Ratio
0.44
0.31
0.26
Moment of inertia variation at constant altitude
Moment of
Inertia
Variation
%
-50
0
50
Natural
Frequency
rad/s
3.25
2.3
1.87
Period
sec
1.94
2.74
3.35
Damping
Ratio
0.33
0.31
0.31
50
Wind Shear Encounter
•! Inertial Frames
–! Earth-Relative
–! Wind-Relative (Constant Wind)
Pitch Angle, #
Angle of Attack, !
•! Non-Inertial Frames
Flight Path Angle, "
–! Body-Relative
–! Wind-Relative (Varying Wind)
Earth-Relative Velocity
Wind Velocity
Air-Relative Velocity
51
Pitch Angle and Normal Velocity
Frequency Response to Axial Wind
•! Pitch angle resonance at phugoid natural frequency
•! Normal velocity (~ angle of attack) resonance at phugoid
and short period natural frequencies
!!
"# ( j$ )
"Vwind ( j$ )
! VN
MacRuer, Ashkenas, and Graham, 1973
!" ( j# )
!Vwind ( j# )
52
Pitch Angle and Normal Velocity
Frequency Response to Vertical Wind
•! Pitch angle resonance at phugoid and short
period natural frequencies
•! Normal velocity (~ angle of attack) resonance
at short period natural frequency
!
!" ( j# )
VN !$ wind ( j# )
=
!" ( j# )
!" wind ( j# )
MacRuer, Ashkenas, and Graham, 1973
53
Microbursts
1/2-3-km-wide
Jetimpinges on
surface
Ring vortex
forms in
outlow
High-speed outflow
from jet core
Outflow strong enough
to knock down trees
http://en.wikipedia.org/wiki/Microburst
54
The Insidious Nature of
Microburst Encounter
The wavelength of the phugoid mode and
the disturbance input are comparable
DELTA 191 (Lockheed L-1011)
https://www.youtube.com/watch?v=dKwyU1RwPto
Headwind
Downdraft
Tailwind
Landing Approach
http://en.wikipedia.org/wiki/Delta_Air_Lines_Flight_191
55
Importance of Proper Response
to Microburst Encounter
!! Stormy evening July 2, 1994
!! USAir Flight 1016, Douglas DC-9, Charlotte
!! Windshear alert issued as 1016 began descent along glideslope
!! DC-9 encountered 61-kt windshear, executed missed approach
!! Go-around procedure begun correctly -- aircraft's nose rotated up -but power was not advanced
!! Together with increasing tailwind, aircraft stalled
!! Crew lowered nose to eliminate stall, but descent rate increased,
causing ground impact
!! Plane continued to descend, striking trees and telephone poles
before impact
http://en.wikipedia.org/wiki/US_Airways_Flight_1016
56
Optimal Flight Path !
Through Worst JAWS Profile
•!
•!
•!
•!
Graduate research of Mark Psiaki
Joint Aviation Weather Study (JAWS)
measurements of microbursts (Colorado
High Plains, 1983)
Negligible deviation from intended path
using available controllability
Aircraft has sufficient performance
margins to stay on the flight path
Downdraft
Headwind
Airspeed
Angle of Attack
Pitch Angle
Throttle Setting
57
Optimal and 15° Pitch
Angle Recovery during!
Microburst Encounter
Graduate Research of Sandeep Mulgund
Altitude vs. Time
Airspeed vs. Time
Encountering
outflow
Angle of Attack vs. Time
Rapid arrest of
descent
FAA Windshear Training Aid, 1987, addresses proper
operating procedures for suspected windshear
58