Root Locus Analysis ofParameter Variations

Robert Stengel, Aircraft Flight Dynamics

MAE 331, 2010

Effects of parameter variationson modes of motion

Root locus analysis

Evans! rules for construction

Application to longitudinal

dynamic models

Copyright 2010 by Robert Stengel. All rights reserved. For educational use only.http://www.princeton.edu/~stengel/MAE331.html

http://www.princeton.edu/~stengel/FlightDynamics.html

Characteristic Equation: A Critical

Component of the Response!s

Laplace Transform

sI ! F[ ]!1=Adj sI ! F( )

sI ! F(n x n)

Characteristic equation defines the modes of motion

sI ! F = "(s) = sn+ a

n!1sn!1

+ ...+ a1s + a

0

= 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"

Eigenvalues (or Roots) ofthe Dynamic System

!(s) = s " #1( ) s " #2( ) ...( ) s " #n( ) = 0

! = cos"1#

Real roots

are confined to the real axis

represent convergent or divergent modes

time constant, ! = 1/"

Complex roots

occur only in complex-conjugate pairs

represent oscillatory modes

natural frequency, #n, and damping ratio,$, as shown s Plane = ! + j"( ) Plane

Roots are solutions of thecharacteristic equation

Example: LongitudinalCharacteristic Equation

!Lon(s) = s

4+ a

3s3+ a

2s2+ a

1s + a

0

= s4+ D

V+L"VN

# Mq( )s3

+ g # D"( )LV

VN

+ DV

L"VN

# Mq( ) # Mq L" V

N

# M"$

%&'

()s2

+ Mq

D" # g( )LV

VN

# DV

L"VN

$%&

'()+ D"MV # DVM"{ }s

+ g MV

L"VN

# M"LV

VN

( ) = 0

!Lon(s) = s

2+ 2"#

ns +#

n

2( )Ph

s2+ 2"#

ns +#

n

2( )SP

Typically factors into oscillatory phugoid and short-period modes

with Lq = Dq = 0

Evanss Rules forRoot Locus Analysis

Root Locus Analysis ofParametric Effects on

Aircraft Dynamics

Parametric variations altereigenvalues of F

Graphical technique forfinding the roots

Locus: the set of all points whoselocation is determined by stated

conditions

s Plane

Example: How do the roots vary whenwe change pitch-rate damping, Mq?

!Lon(s) = s

4+ D

V+L"VN

# Mq( )s3

+ g # D"( )LV

VN

+ DV

L"VN

# Mq( ) # Mq L" V

N

# M"$

%&'

()s2

+ Mq

D" # g( )LV

VN

# DV

L"VN

$%&

'()+ D"MV # DVM"{ }s

+ g MV

L"VN

# M"LV

VN

( ) = 0

Short-Period Roots

with Mq = 0

Phugoid Roots

with Mq = 0

Effect of Parameter

Variations on Root

Location

Let root locus gain = k = ai (just a notation change)

Option 1: Vary k and calculate roots for each new value

Option 2: Apply Evans!s Rules of Root Locus Construction

Walter R. Evans

!Lon(s) = s

4+ a

3s3+ a

2s2+ a

1s + a

0

= s " #1( ) s " #2( ) s " #3( ) s " #4( ) = s " #1( ) s " #1

*( ) s " #3( ) s " #3*( )

= s2+ 2$

P%

nPs +%

nP

2( ) s2 + 2$SP%nSP s +%nSP2( ) = 0

Effect of a0 Variation onLongitudinal Root Location

Example: k = a0

where

d(s) = s4+ a

3s3+ a

2s2+ a

1s

= s ! " '1( ) s ! " '2( ) s ! " '3( ) s ! " '4( )

n(s) = 1

!Lon(s) = s

4+ a

3s3+ a

2s2+ a

1s"# $% + k[ ] & d(s) + kn(s)

= s ' (1( ) s ' (2( ) s ' (3( ) s ' (4( ) = 0

d s( ) : Polynomial in s

n s( ) : Polynomial in s

Example: k = a1

where

d(s) = s4+ a

3s3+ a

2s2+ a

0

= s ! " '1( ) s ! " '2( ) s ! " '3( ) s ! " '4( )

n(s) = s

!Lon(s) = s

4+ a

3s3+ a

2s2+ ks + a

0" d(s) + kn(s)

= s # $1( ) s # $2( ) s # $3( ) s # $4( ) = 0

Effect of a1 Variation onLongitudinal Root Location

Three Equivalent Equations

for Defining Roots

1+ kn(s)

d(s)= 0

kn(s)

d(s)= !1 = (1)e! j" (rad ) = (1)e! j180(deg)

d(s) + k n(s) = 0

Longitudinal Equation Example

Original 4th-order polynomial

!Lon(s) = s

4+ 2.57s

3+ 9.68s

2+ 0.202s + 0.145

= s2+ 2 0.0678( )0.124s + 0.124( )

2"#

$% s

2+ 2 0.411( )3.1s + 3.1( )

2"#

$% = 0

!Lon(s) = s

4+ a

3s3+ a

2s2+ a

1s + a

0

= s " #1( ) s " #2( ) s " #3( ) s " #4( ) = s " #1( ) s " #1

*( ) s " #3( ) s " #3*( )

= s2+ 2$

P%

nPs +%

nP

2( ) s2 + 2$SP%nSP s +%nSP2( ) = 0

Typical flight condition

Example: Effect of a0 Variation

!(s) = s4 + a3s3+ a2s

2+ a1s + a0

= s4+ a3s

3+ a2s

2+ a1s( ) + k

= s s3+ a3s

2+ a2s + a1( ) + k

= s s + 0.21( ) s2 + 2.55s + 9.62"# $% + k

k

s s + 0.21( ) s2 + 2.55s + 9.62!" #$= %1

Example: k = a0

Original 4th-order polynomial

!Lon(s) = s

4+ 2.57s

3+ 9.68s

2+ 0.202s + 0.145 = 0

ks

s2 ! 0.00041s + 0.015"# $% s

2+ 2.57s + 9.67"# $%

= !1

Example: k = a1

!(s) = s4 + a3s3+ a2s

2+ a1s + a0

= s4+ a3s

3+ a2s

2+ ks + a0

= s4+ a3s

3+ a2s

2+ a0( ) + ks

= s2 " 0.00041s + 0.015#$ %& s

2+ 2.57s + 9.67#$ %& + ks

Example: Effect of a1 Variation

The Root Locus

Criterion All points on the locus of roots must satisfy the equation

k[n(s)/d(s)] = 1

Phase angle(1) = 180 deg

k = a0 : k

n(s)

d(s)= k

1

s4+ a

3s3+ a

2s2+ a

1s= !1

k = a1 : k

n(s)

d(s)= k

s

s4+ a

3s3+ a

2s2+ a

0

= !1

Number of roots = 4

Number of zeros = 0

(n q) = 4

Number of roots = 4

Number of zeros = 1

(n q) = 3

Number of roots (or poles) of the denominator = n

Number of zeros of the numerator = q

Spirule

Origins of Roots (for k = 0)

!(s) = d(s) + kn(s)k"0

# "## d(s)

Origins of the roots are the Poles of d(s)

Destinations of Roots (for k -> ")

q roots go to the zeros of n(s)

d(s) + kn(s)

k=d(s)

k+ n(s)

k!"# !## n(s)

No zeros when k = a0 One zero at origin when k = a1

Destinations of Roots (for k -> ")

d(s) + kn(s)

n(s)=d(s)

n(s)+ k

k!"# !## s

n$q( ) R! "

sn$q( )

= Re$ j180

!" or Re$ j360

! $"

s = Re$ j180 n$q( )

!" or Re$ j360 n$q( )

! $"

(n q) roots go to infinite radius from the origin

4 roots to infinite radius 3 roots to infinite radius

+R

R

(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

Origin of Asymptotes =

Center of Gravity

"c.g." =

!"i # ! z jj=1

q

$i=1

n

$

n # q

Root Locus on Real Axis Locus on real axis

k > 0: Any segment with odd numberof poles and zeros to the right

k < 0: Any segment with even numberof poles and zeros to the right

First Example: k = a0k

s s + 0.21( ) s2 + 2.55s + 9.62!" #$= %1

Second Example: k = a1ks

s2 ! 0.00041s + 0.015"# $% s

2+ 2.57s + 9.67"# $%

= !1

Summary of Root

Locus Concepts

Origins

of Roots

Destinations

of Roots

Center

of Gravity

Locus on

Real Axis

Root Locus Analysis ofSimplified Longitudinal Modes

Approximate Phugoid Model

Second-order equation

!!xPh =! !V! !"

#

$%%

&

'(()

*DV *g

LVVN

0

#

$

%%%

&

'

(((

!V!"

#

$%%

&

'((+

T+T

L+TVN

#

$

%%%

&

'

(((!+T

Characteristic polynomial

sI ! FPh = det sI ! FPh( ) " #(s) = s2+ DVs + gLV /VN

= s2+ 2$%ns +%n

2

gLV/V

N, D

V

Parameters

!(s) = s2+ D

Vs( ) + k

= s s + DV( ) + k

k = gLV/VN

Effect of LV or 1/VN Variation on

Approximate Phugoid Roots

Change indamped naturalfrequency

!n1"# 2

Effect of DV Variation on

Approximate Phugoid Roots

k = DV

!(s) = s2+ gLV /VN( ) + ks

= s + j gLV /VN( ) s " j gLV /VN( ) + ks

Change indamping ratio

!

Approximate Short-Period Model

Approximate Short-Period Equation (Lq = 0)

Characteristic polynomial

Parameters

!!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 ++

n

2

M!, M

q,

L!

VN

Effect of M% on ApproximateShort-Period Roots

k = M%

!(s) = s2 +L"

VN# Mq

$

%&'

()s # Mq

L"

VN

$

%&'

()# k = 0

= s +L"

VN

$

%&'

()s # Mq( ) # k = 0

Change indamped naturalfrequency

Effect of Mq on ApproximateShort-Period Roots

!(s) = s2 +L"

VN

s # M" # k s +L"

VN

$%&

'()

= 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

k = Mq

Change primarilyin damping ratio

Effect of L%/VN on Approximate

Short-Period Roots

!(s) = s2 " Mqs " M# + k s " 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 primarilyin damping ratio

Example: How do the roots vary whenwe change pitch-rate damping, Mq?

!Lon(s) = s

4+ D

V+L"VN

( )s3 + g # D"( )LV VN

+ DV

L"VN

( ) # M"$%&'

()s2

+ D"MV # DVM"{ }s + g MVL"VN

# M"LV

VN

( )# M

qs3 # D

VM

q( ) + MqL"VN

$%&

'()s2+ M

qD" # g( )

LV

VN

# DV

L"VN

$%&

'()s = 0

!Lon (s) = d s( ) " Mq s3+ DV +

L#VN

$%&

'()s2 " D# " g( )

LVVN

" DVL#VN

$%&

'()s{ }

= d s( ) " Mqs s2+ DV +

L#VN

$%&

'()s " D# " g( )

LVVN

" DVL#VN

$%&

'(){ }

= d s( ) + kn s( ) = 0

kn(s)

d(s)= !1

Separate Mq terms in the characteristic polynomial

Group Mq terms in the characteristic polynomial

Factor terms that are multiplied by Mq to find the 3 zeros 2 zeros near origin similar to approximate phugoid roots, effectively

canceling Mq effect on them

Next Time:Transfer Functions andFrequency Response