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