Aircraft Control Devices and Systems

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

–! Evans

s 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

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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

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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

{

(

)

}

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Evans

s Rules for Root Locus Analysis!

Walter R. Evans (1920-1999)

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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

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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

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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

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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)

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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

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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

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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

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(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

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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

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Roots for Positive and Negative Variations of k = a0 k = %1 s ( s + 0.21) !" s + 2.55s + 9.62 #$ 2

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Roots for Positive and Negative Variations of k = a1 ks = !1 "# s ! 0.00041s + 0.015 $% "# s 2 + 2.57s + 9.67 $% 2

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Root Locus Analysis of Simplified Longitudinal Modes!

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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

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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

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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

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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 ( %

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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

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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 (–) (–)

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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

(

(–)

(–)

)

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Flight Control Systems!

SAS = Stability Augmentation System

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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)

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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

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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

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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

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Supplemental Material! !

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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°

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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

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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

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