Root Locus Analysis of Parameter Variations and Feedback Control!
Robert Stengel, Aircraft Flight Dynamics!MAE 331, 2016
•! Effects of system parameter variations on modes of motion
•! Root locus analysis–! Evans s rules for construction–! Application to longitudinal dynamic models
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
Reading:!Flight Dynamics!
357-361, 465-467, 488-490, 509-514!
1
Learning Objectives
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
sI!F[ ]!1 =Adj sI!F( )sI!F
=CT s( )sI!F
(n " n)1"1( )
Characteristic equation defines the modes of motion
sI ! F = "(s) = sn + an!1sn!1 + ...+ a1s + a0
= s ! #1( ) s ! #2( ) ...( ) s ! #n( ) = 0
!x(s) = sI " F[ ]"1 !x(0) +G!u(s) + L!w(s)[ ]
Recall: s is a complex variable
s = ! + j"3
Real Roots
!i = µi (Real number)x t( ) = x 0( )eµt
!! On real axis in s Plane!! Represents convergent or
divergent time response!! Time constant, !! = –1/"" = –
1/µµ, secs Plane = ! + j"( ) Plane
4
Complex Roots
" = cos#1$
!! Occur only in complex-conjugate pairs
!! Represent oscillatory modes!! Natural frequency, ##n, and
damping ratio, $$, as shown
s Plane = ! + j"( ) Plane
!1 = µ1 + j"1 = #$%n + j%n 1#$ 2
Stable Unstable
!2 = µ2 + j"2 = µ1 # j"1 ! !1*
= #$% n # j% n 1#$ 2
5
Natural Frequency, Damping Ratio, and Damped Natural Frequency
s ! "1( ) s ! "1*( ) = s ! µ1 + j#1( )$% &' s ! µ1 ! j#1( )$% &'= s2 ! 2µ1s + µ1
2 +#12( ) ! s2 + 2() ns +) n
2
µ1 = !"# n = !1 Time constant
µ12 +$1
2( )1 2=# n = Natural frequency
$1 =# n 1!" 2 !# ndamped= Damped natural frequency
6
Longitudinal Characteristic Equation
!Lon (s) = s4 + a3s
3 + a2s2 + a1s + a0
= s4 + DV +L"VN
# Mq( )s3+ g # D"( )LV VN + DV
L"VN
# Mq( ) # MqL"VN
# M"$%&
'()s2
+ Mq D" # g( )LV VN # DVL"VN
$%&
'()+ D"MV # DVM"{ }s
+ g MVL"VN
# M"LVVN( ) = 0
with Lq = Dq = 0
7
How do the roots vary when we change M!?
… or pitch-rate damping, Mq?
!Lon (s) = s4 + DV +
L"VN
# Mq( )s3+ g # D"( )LV VN + DV
L"VN
# Mq( ) # MqL"VN
# M"$%&
'()s2
+ Mq D" # g( )LV VN # DVL"VN
$%&
'()+ D"MV # DVM"{ }s
+ g MVL"VN
# M"LVVN( ) = 0
8
Evans s Rules for Root Locus Analysis!
9
Walter R. Evans(1920-1999)
Short Period
Short Period
PhugoidPhugoid
Root Locus Analysis of Parametric Effects on Aircraft Dynamics
•! Parametric variations alter eigenvalues of F
•! Graphical technique for finding the roots as a parameter (“gain”) varies
Locus: the set of all points whose location is determined by stated
conditions
s Plane
10
Variation with ±M!
Parameters of Characteristic Polynomial!Lon (s) = s
4 + a3s3 + a2s
2 + a1s + a0= s " #1( ) s " #2( ) s " #3( ) s " #4( )
= s2 + 2$ P% nPs +% nP
2( ) s2 + 2$ SP% nSPs +% nSP
2( ) = 0
11
Break the polynomial into 2 parts to examine the effect of a single parameter
! s( ) = d(s)+ k n(s) = 0
Effect of a0 Variation on Longitudinal Root Location
Example: Root locus gain, k = a0
d(s) = s4 + a3s3 + a2s
2 + a1s= s ! 0( )"# $% s ! & '2( ) s ! & '3( ) s ! & '4( ); four poles, one at 0
n(s) = 1; no zeros
!Lon (s) = s4 + a3s3 + a2s
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) polesn s( ) : Polynomial in s, degree = q; there are q (= 0) zeros
12
Example: Root locus gain, k = a1
d(s) = s4 + a3s3 + a2s
2 + a0
= s ! " '1( ) s ! " '2( ) s ! " '3( ) s ! " '4( ); four polesn(s) = s; one zero at 0
!Lon (s) = s4 + a3s
3 + a2s2 + ks + a0 ! d(s)+ kn(s)
= s " #1( ) s " #2( ) s " #3( ) s " #4( ) = 0
Effect of a1 Variation on Longitudinal Root Location
13
d s( ) : Polynomial in s, degree = n; there are n (= 4) polesn s( ) : Polynomial in s, degree = q; there are q (= 1) zeros
Three Equivalent Equations for Evaluating Locations of Roots
1+ k n(s)d(s)
= 0
k n(s)d(s)
= !1= (1)e± j" (rad ) = (1)e± j180°
! s( ) = d(s)+ k n(s) = 0
14
Evan’s Root Locus Criterion
15
k n(s)d(s)
= !1
k n(s)d(s)
= (1)e± j180°
All points on the locus of roots must satisfy the equation
i.e., all points on root locus must have phase angle = ±180°
Longitudinal Equation Example
!Lon (s) == s4 + a3s
3 + a2s2 + a1s + a0
= s4 + 2.57s3 + 9.68s2 + 0.202s + 0.145
= s2 + 2 0.0678( )0.124s + 0.124( )2"# $% s2 + 2 0.411( )3.1s + 3.1( )2"# $% = 0
Typical flight condition
Phugoid Short Period
16
Effect of a0 Variation
!(s) = s4 + a3s3 + a2s
2 + a1s + a0= s s3 + a3s
2 + a2s + a1( ) + k= s s + 0.21( ) s2 + 2.55s + 9.62"# $% + k
ks s + 0.21( ) s2 + 2.55s + 9.62!" #$
= %1
k = a0
!Lon (s) = s4 + 2.57s3 + 9.68s2 + 0.202s + 0.145 = 0
Rearrange
17
kss2 ! 0.00041s + 0.015"# $% s2 + 2.57s + 9.67"# $%
= !1
k = a1
!(s)= s4 +a3s3 +a2s
2 +a1s+a0= s4 +a3s
3 +a2s2 + ks+a0
= s4 +a3s3 +a2s
2 +a0( )+ ks= s2 "0.00041s+0.015#$ %& s
2 +2.57s+9.67#$ %&+ ks
Effect of a1 Variation
Rearrange
18
Origins of Roots (k = 0)
!(s) = d(s)+ kn(s) k"0# "## d(s)n poles of d(s)
19
ks s + 0.21( ) s2 + 2.55s + 9.62!" #$
= %1 kss2 ! 0.00041s + 0.015"# $% s2 + 2.57s + 9.67"# $%
= !1
Destinations of Roots as k Becomes Large
20
1) q roots go to the zeros of n(s)
d(s)+ kn(s)k
= d(s)k
+ n(s) k!±"# !## n(s) = s $ z1( ) s $ z2( )!
2) (n – q) roots go to infinity
d(s)+ kn(s)n(s)
!"#
$%&= d(s)
n(s)+ k!
"#$%& k'±R'±() '))) sn
sq+ k!
"#
$
%&' s n*q( ) ± R' ±(
21
ks s + 0.21( ) s2 + 2.55s + 9.62!" #$
= %1
R
kss2 ! 0.00041s + 0.015"# $% s2 + 2.57s + 9.67"# $%
= !1
R
Destinations of Roots for Large k
(n – q) Roots Approach Asymptotes as k –> ±"
!(rad) = " + 2m"n # q
, m = 0,1,...,(n # q) #1
!(rad) = 2m"n # q
, m = 0,1,...,(n # q) #1
Asymptote angles for positive k
Asymptote angles for negative k
22
Origin of Asymptotes = Center of Gravity
"c.g." =!"i
# ! z jj=1
q
$i=1
n
$n # q
23
(Sum of real parts of poles minus sum of real parts of zeros) ÷ (# of poles minus # of zeros)
Root Locus on Real Axis
•! Locus on real axis–! k > 0: Any segment with odd number of
poles and zeros to the right on the axis–! k < 0: Any segment with even number of
poles and zeros to the right on the axis
24
Roots for Positive and Negative Variations of k = a0
ks s + 0.21( ) s2 + 2.55s + 9.62!" #$
= %1
25
Roots for Positive and Negative Variations of k = a1
kss2 ! 0.00041s + 0.015"# $% s2 + 2.57s + 9.67"# $%
= !1
26
Root Locus Analysis of Simplified Longitudinal Modes!
27
Approximate Phugoid Model 2nd-order equation
!!xPh =! !V! !"
#
$%%
&
'(()
*DV *gLVVN
0
#
$
%%%
&
'
(((
!V!"
#
$%%
&
'((+
T+TL+T
VN
#
$
%%%
&
'
(((!+T
Characteristic polynomialsI! FPh = det sI! FPh( ) "#(s) = s2 + DVs + gLV /VN
= s2 + 2$% ns +% n2
28
! n = gLV /VN
" = DV
2 gLV /VN
Effect of Airspeed on Natural Frequency and Period
! n " 2 gVN" 13.87VN (m/s)
,rad/sPeriod,T = 2! /" n
# 0.45VN (m/s), sec
29
Neglecting compressibility effects
g LVVN
! gm
CLN"NS#$ %&
= 2gmVN
2 CLN
12"NVN
2S#$'
%&(= 2gmVN
2 mg[ ] = 2g2
VN2
Effect of L/D on Damping Ratio
VelocityNatural Frequency Period L/D
Damping Ratio
m/s rad/s sec50 0.28 23 5 0.14100 0.14 45 10 0.07200 0.07 90 20 0.035400 0.035 180 40 0.018
30
Neglecting compressibility effects and thrust sensitivity to velocity, TV
DV ! 1m
CDN"NVNS#$ %& ! = DV
2 gLV /VN
! " 12 L /D( )N
!CDN
"NVNS m2 2 g VN
=CDN
"NVN2S 2
2mg= 1
2CDN
CLN
#
$%&
'(
!(s) = s2 + DVs( ) + k= s s + DV( )"# $% + k[ ]
k = gLV/VN
Effect of LV/VN Variation
Change in damped natural frequency
!ndamped
!!n 1"# 2
31
Effect of DV Variation
k = DV
!(s) = s2 + gLV /VN( ) + ks= s + j gLV /VN( ) s " j gLV /VN( )#$
%& + ks[ ]
Change in damping ratio
!
32
Approximate Short-Period Model Approximate Short-Period Equation (Lq = 0)
Characteristic polynomial
!!xSP =! !q! !"
#
$%%
&
'(()
Mq M"
1 * L"VN
#
$
%%%
&
'
(((
!q!"
#
$%%
&
'((+
M+E
*L+EVN
#
$
%%%
&
'
(((!+E
!(s) = s2 + L"
VN#Mq
$%&
'()s # M" +Mq
L"
VN
$%&
'()
= s2 + 2*+ ns ++ n2
33
!n = " M# +MqL#VN
$
%&
'
(); * =
L#VN
"Mq
$
%&
'
()
2 " M# +MqL#VN
$
%&
'
()
Effect of M%% on Approximate Short-Period Roots
k = M%%
!(s) = s2 + L"VN
# Mq
$
%&'
()s # Mq
L"VN
$
%&'
()# k = 0
= s + L"VN
$
%&'
()s # Mq( ) # k = 0
Change in damped natural frequency
34
Effect of Mq on Approximate Short-Period Roots
!(s) = s2 + L"
VNs #M" #Mq s + L"
VN
$%&
'()
k = Mq
Change primarily in damping ratio
35
!(s) = s " L#
2VN+ L#
2VN
$%&
'()
2
+M#
*
+
,,
-
.
//
012
32
452
62s " L#
2VN" L#
2VN
$%&
'()
2
+M#
*
+
,,
-
.
//
012
32
452
62" k s + L#
VN
$%&
'()= 0
(–) (–)
Effect of L!/VN on Approximate Short-Period Roots
!(s) = s2 "Mqs "M# +L#
VNs "Mq( )
= s +Mq
2"
Mq
2$%&
'()
2
+M#
*
+,,
-
.//
012
32
452
62s +
Mq
2"
Mq
2$%&
'()
2
+M#
*
+,,
-
.//
012
32
452
62+ k s "Mq( ) = 0
k = L%%/VN
•! Change primarily in damping ratio
36(–) (–)
Flight Control Systems!
SAS = Stability Augmentation System 37
Effect of Scalar Feedback Control on Roots of the System
!y(s) = H (s)!u(s) = kn(s)
d(s)!u(s) = kn(s)
d(s)K!"(s)
Block diagram algebra
H (s) = kn(s)
d(s)
= KH (s) !yc(s)" !y(s)[ ]!y(s) = KH (s)!yc(s)" KH (s)!y(s)
K
38
Scalar Closed-Loop Transfer Function
1+ KH (s)[ ]!y(s) = KH (s)!yc(s)
!y(s)!yc(s)
= KH (s)1+ KH (s)[ ]
39
H (s)= kn(s)d(s)K
Roots of the Closed-Loop Control System
Closed-loop roots are solutions to
!closed"loop
(s) = d(s)+ Kkn(s) = 0
or
K kn(s)d(s)
= !1
!y(s)!yc(s)
=K kn(s)d(s)
1+K kn(s)d(s)
"
#$
%
&'
=Kkn(s)
d(s)+Kkn(s)[ ]=Kkn(s)!closedloop
s( )
40
Pitch Rate Feedback to Elevator
KH s( ) = K !q(s)
!"E(s)= K
k"Eq s # z"E
q( )s2 + 2$ SP% nSP
s +% nSP2 = #1
!! # of roots = 2!! # of zeros = 1!! Destinations of roots (for k =
±"):!! 1 root goes to zero of n(s)!! 1 root goes to infinite radius
!! Angles of asymptotes, &&, for the roots going to "!! K -> +": –180 deg!! K -> –": 0 deg
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!
43
Angle-of-attack-rate aero effectsFourth-order dynamics
Steady-state response to controlTransfer functions
Frequency responseRoot locus analysis of parameter variations
Nichols chartPilot-aircraft interactions
Learning Objectives
Flight Dynamics!204-206, 503-525!
Supplemental Material!!
44
Same envelopes for displacement and rate
Corresponding 2nd-Order Initial Condition Response
45
x1 t( ) = Ae!"#nt sin # n 1!" 2 t +$%&
'(
x2 t( ) = Ae!"#nt # n 1!" 2%&
'(cos # n 1!" 2 t +$%
&'(
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( ) = d1is " #1( ) +
d2is " #2( ) +
d3is " #3( ) +
d4is " #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 = 1,446
R(+) R(–)
s n!q( ) = Re! j180° "# or Re! j360° "!#
Magnitudes of roots are the same for given kAngles from the origin are different
47
s = R1 n!q( ) e! j180° n!q( ) "#
or R1 n!q( ) e! j360° n!q( ) "!#
Magnitudes of Roots Going to Infinity
Asymptotes of Roots (for k -> ±") 4 roots to infinite radius 3 roots to infinite radius
48Asymptotes = ±45°, ±135° Asymptotes = ±60°, –180°
Summary of Root Locus Concepts
Origins of Roots
Destinations 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 AltitudeNatural Frequency Period
Damping Ratio
m/s m rad/s sec -122 2235 2.36 2.67 0.39152 6095 2.3 2.74 0.31213 11915 2.24 2.8 0.23274 16260 2.18 2.88 0.18
Mass Variation
Natural Frequency Period
Damping Ratio
% rad/s sec --50 2.4 2.62 0.440 2.3 2.74 0.3150 2.26 2.78 0.26
Moment of Inertia Variation
Natural Frequency Period
Damping Ratio
% rad/s sec --50 3.25 1.94 0.330 2.3 2.74 0.3150 1.87 3.35 0.31
AirspeedDynamic Pressure
Angle of Attack
Natural Frequency Period
Damping Ratio
m/s P deg rad/s sec -91 2540 14.6 1.34 4.7 0.3152 7040 5.8 2.3 2.74 0.31213 13790 3.2 3.21 1.96 0.3274 22790 2.2 3.84 1.64 0.3
Airspeed variation at constant altitude
Altitude variation with constant dynamic pressure
Mass variation at constant altitude
Moment of inertia variation at constant altitude
50
Wind Shear Encounter•! Inertial Frames
–! Earth-Relative–! Wind-Relative (Constant Wind)
•! Non-Inertial Frames–! Body-Relative–! Wind-Relative (Varying Wind)
Earth-Relative Velocity Wind Velocity
Air-Relative Velocity
Angle of Attack, !!
Flight Path Angle, ""
Pitch Angle, ##
51
Pitch Angle and Normal Velocity Frequency Response to Axial Wind
! !
"# j$( )"Vwind j$( )
! VN!" j#( )!Vwind j#( )
MacRuer, Ashkenas, and Graham, 1973
•! Pitch angle resonance at phugoid natural frequency•! Normal velocity (~ angle of attack) resonance at phugoid
and short period natural frequencies
52
Pitch Angle and Normal Velocity Frequency Response to Vertical Wind
MacRuer, Ashkenas, and Graham, 1973
•! 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#( )
53
Microbursts1/2-3-km-wide
Jet impinges on surface
High-speed outflow from jet core
Ring vortex forms in outlow
Outflow strong enough to knock down trees
54http://en.wikipedia.org/wiki/Microburst
The Insidious Nature of Microburst Encounter
The wavelength of the phugoid mode and the disturbance input are comparable
55http://en.wikipedia.org/wiki/Delta_Air_Lines_Flight_191
DELTA 191 (Lockheed L-1011)https://www.youtube.com/watch?v=dKwyU1RwPto
Headwind Downdraft Tailwind
Landing Approach
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
56
!! 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
Optimal Flight Path !Through Worst JAWS Profile
Downdraft
Headwind
Airspeed
Angle of Attack
Pitch Angle
Throttle Setting
57
•! 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
Optimal and 15° Pitch Angle Recovery during !
Microburst EncounterGraduate Research of Sandeep Mulgund
Airspeed vs. Time
Encountering outflow
Rapid arrest of descent
FAA Windshear Training Aid, 1987, addresses proper operating procedures for suspected windshear 58
Altitude vs. Time
Angle of Attack vs. Time
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