Aircraft Flight Dynamics Robert F. Stengel Lecture14 RootLocus Analysis of Parameter Variations

Root Locus Analysis of
Parameter Variations

Robert Stengel, Aircraft Flight Dynamics

MAE 331, 2012"
•  Effects of system parameter
variations on modes of motion"
•  Root locus analysis"
–  Evanss rules for construction"
–  Application to longitudinal
dynamic models"
Copyright 2012 by Robert Stengel. All rights reserved. For educational use only.!
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Characteristic Equation: A Critical
Component of the Response’s
Laplace Transform "
sI − F
[ ]
−1
=
Adj sI − F
( )
sI − F
=
C
T
s
( )
sI − F
(n × n)
1×1

( )
•  Characteristic equation defines the modes of motion!
sI − F = Δ(s) = s
n
+ a
n −1
s
n −1
+ + a
1
s + a
0
= s −
λ
1
( )
s −
λ
2
( )

( )
s −
λ
n
( )
= 0
Δx(s) = sI − F
[ ]
−1

Δx( 0) + G Δu(s) + LΔw(s)
[ ]
•  Recall: s is a complex variable!
s =
σ
+ j
ω
Real Roots of the Dynamic System "
Δ(s) = s −
λ
1
( )
s −
λ
2
( )

( )
s −
λ
n
( )
= 0
•  Roots are solutions of the
characteristic equation"
λ
i
=
µ
i

(Real number)
x t
( )
= x 0
( )
e
µ
t
•  Real roots"
–  are confined to the real axis"
–  represent convergent or divergent
time response"
–  time constant,
τ
= –1/
λ
= –1/
µ
, sec
#
s Plane =
σ
+ j
ω
( )
Plane
Complex Roots of the Dynamic System "
€
δ
= cos

−1
ζ
•  Complex roots"
–  occur only in complex-conjugate pairs"
–  represent oscillatory modes"
–  natural frequency,
ω
n
, and damping ratio,
ζ
, as shown"
s Plane =
σ
+ j
ω
( )
Plane
λ
1
=
µ
1
+ j
ν
1
= −
ζω
n
+ j
ω

n
1−
ζ
2
– time constant = –1/μ = 1/ζω
n
"
Stable" Unstable"
– decay of exponential time-"
response envelope"
λ
2
=
µ
2
+ j
ν
2
=
µ
1
− j
ν
1
=
λ
1
*
= −
ζω

n
− j
ω
n
1−
ζ
2
Complex Roots, Damping Ratio,
and Damped Natural Frequency "

s −
λ
1
( )
s −
λ
1
*
( )
= s −
µ
1
+ j
ν
1
( )
$
%
&
'

s −
µ
1
− j
ν
1
( )
$
%
&
'
= s
2
−
µ
1
− j
ν
1
( )
+
µ
1
+ j
ν
1
( )
$
%
&

'
s +
µ
1
− j
ν
1
( )
µ
1
+ j
ν
1
( )
= s
2
− 2
µ
1
s +
µ
1
2
+
ν
1
2
( )
 s
2

+ 2
ζω
n
s +
ω
n
2

µ
1
= −
ζω
n
= −1 Time constant
ν
1
=
ω
n
1−
ζ
2

ω
n
damped
= Damped natural frequency
x
1
t

( )
= Ae
−
ζω
n
t
sin
ω
n
1−
ζ
2
t +
ϕ
%
&
'
(
x
2
t
( )
= Ae
−
ζω
n
t
ω
n
1−

ζ
2
%
&
'
(
cos
ω
n
1−
ζ
2
t +
ϕ
%
&
'
(
Identical exponentially
decaying envelopes for
both displacement and rate"
Corresponding Second-Order
Initial Condition Response"
General form of response"
Multi-Modal LTI Responses Superpose
Individual Modal Responses"
•  With distinct roots, (n = 4)
for example, partial
fraction expansion for
each state element is

(Flight Dynamics, p. 325)
"
Δx
i
s
( )
=
d
1
i
s −
λ
1
( )
+
d
2
i
s −
λ
2
( )
+
d
2
i
s −
λ
3
( )

+
d
2
i
s −
λ
4
( )
•  Corresponding 4
th
-order time response is"
Δx
i
t
( )
= d
1
i
e
λ
1
t
+ d
2
i
e
λ
2
t
+ d

3
i
e
λ
3
t
+ d
4
i
e
λ
4
t
•  Details in next lecture"
Evanss Rules for
Root Locus Analysis
Root Locus Example: "
4
th-
Order Longitudinal
Characteristic Equation"
Δ
Lon
(s) = s
4
+ a
3
s
3
+ a

2
s
2
+ a
1
s + a
0
= s
4
+ D
V
+
L
α
V
N
− M
q
( )
s
3
+ g − D
α
( )
L
V
V
N
+ D
V

L
α
V
N
− M
q
( )
− M
q
L
α
V
N
− M
α
$
%
&
'
(
)
s
2
+ M
q
D
α
− g
( )
L

V
V
N
− D
V
L
α
V
N
$
%
&
'
(
)
+ D
α
M
V
− D
V
M
α
{ }
s
+ g M
V
L
α
V

N
− M
α
L
V
V
N
( )
= 0
Δ
Lon
(s) = s
2
+ 2
ζω
n
s +
ω
n
2
( )
Ph
s
2
+ 2
ζω
n
s +
ω
n

2
( )
SP
•  Typically factors into oscillatory phugoid and short-period modes
"
€
with L
q
= D
q
= 0
Root Locus Analysis of
Parametric Effects on
Aircraft Dynamics
"
•  Parametric variations alter
eigenvalues of F"
•  Graphical technique for
finding the roots with a
new parameter value"
Locus: the set of all points whose
location is determined by stated
conditions"
s Plane!
Phugoid "
Roots"
Short Period"
Root"
Short Period"
Root"

Example: How do the roots vary when
we change pitch-rate damping, M
q
?"
Δ
Lon
(s) = s
4
+ D
V
+
L
α
V
N
− M
q
( )
s
3
+ g − D
α
( )
L
V
V
N
+ D
V
L

α
V
N
− M
q
( )
− M
q
L
α
V
N
− M
α
$
%
&
'
(
)
s
2
+ M
q
D
α
− g
( )
L
V

V
N
− D
V
L
α
V
N
$
%
&
'
(
)
+ D
α
M
V
− D
V
M
α
{ }
s
+ g M
V
L
α
V
N

− M
α
L
V
V
N
( )
= 0
•  M
q
could be changed by"
–  Variation in aircraft aerodynamic configuration"
–  Effect of feedback control, i.e., control of
pitching moment (via elevator) that is
proportional to pitch rate"
Effect of Parameter
Variations on Root
Location "
•  Let root locus gain = k = a
i
(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"
Δ
Lon
(s) = s
4
+ a

3
s
3
+ a
2
s
2
+ a
1
s + a
0
= s −
λ
1
( )
s −
λ
2
( )
s −
λ
3
( )
s −
λ
4
( )
= s −
λ
1

( )
s −
λ
1
*
( )
s −
λ
3
( )
s −
λ
3
*
( )
= s
2
+ 2
ζ
P
ω
n
P
s +
ω
n
P
2
( )
s

2
+ 2
ζ
SP
ω
n
SP
s +
ω
n
SP
2
( )
= 0
Effect of a
0
Variation on
Longitudinal Root Location "
•  Example: Root locus gain, k = a
0
!
where
d(s) = s
4
+ a
3
s
3
+ a
2

s
2
+ a
1
s
= s −
λ
'
1
( )
s −
λ
'
2
( )
s −
λ
'
3
( )
s −
λ
'
4
( )
n(s) = 1
Δ
Lon
(s) = s
4

+ a
3
s
3
+ a
2
s
2
+ a
1
s
"
#
$
%
+ k
[ ]
≡ d(s)+ kn(s)
= s −
λ
1
( )
s −
λ
2
( )
s −
λ
3
( )

s −
λ
4
( )
= 0
d s
( )
: Polynomial in s
n s
( )
: Polynomial in s
•  Example: Root locus gain, k = a
1
!
where
d(s) = s
4
+ a
3
s
3
+ a
2
s
2
+ a
0
= s −
λ
'

1
( )
s −
λ
'
2
( )
s −
λ
'
3
( )
s −
λ
'
4
( )
n(s) = s
Δ
Lon
(s) = s
4
+ a
3
s
3
+ a
2
s
2

+ ks + a
0
≡ d(s)+ kn(s)
= s −
λ
1
( )
s −
λ
2
( )
s −
λ
3
( )
s −
λ
4
( )
= 0
Effect of a
1
Variation on
Longitudinal Root Location "
Three Equivalent Equations
for Defining Roots "
1 + k
n(s)
d(s)
= 0

k
n(s)
d(s)
= −1 = (1)e
− j
π
(rad )
= (1)e
− j180(deg)
d(s) + k n(s) = 0
Longitudinal Equation Example"
•  Original 4
th
-order polynomial!
Δ
Lon
(s) = s
4
+ 2.57s
3
+ 9.68s
2
+ 0.202s + 0.145
= s
2
+ 2 0.0678
( )
0.124s + 0.124
( )
2

"
#
$
%
s
2
+ 2 0.411
( )
3.1s + 3.1
( )
2
"
#
$
%
= 0
Δ
Lon
(s) = s
4
+ a
3
s
3
+ a
2
s
2
+ a
1

s + a
0
= s −
λ
1
( )
s −
λ
2
( )
s −
λ
3
( )
s −
λ
4
( )
= s −
λ
1
( )
s −
λ
1
*
( )
s −
λ
3

( )
s −
λ
3
*
( )
= s
2
+ 2
ζ
P
ω
n
P
s +
ω
n
P
2
( )
s
2
+ 2
ζ
SP
ω
n
SP
s +
ω

n
SP
2
( )
= 0
•  Typical flight condition!
Phugoid" Short Period"
Example: Effect of a
0
Variation"
Δ(s) = s
4
+ a
3
s
3
+ a
2
s
2
+ a
1
s + a
0
= s
4
+ a
3
s
3

+ a
2
s
2
+ a
1
s
( )
+ k
= s s
3
+ a
3
s
2
+ a
2
s + a
1
( )
+ k
= s s + 0.21
( )
s
2
+ 2.55s +9.62
"
#
$
%

+ k
k
s s + 0.21
( )
s
2
+ 2.55s + 9.62
!
"
#
$
= −1
•  Example: k = a
0
!
•  Original 4
th
-order polynomial!
Δ
Lon
(s) = s
4
+ 2.57s
3
+ 9.68s
2
+ 0.202s + 0.145 = 0
•  Rearrange:!
ks
s

2
− 0.00041s + 0.015
"
#
$
%
s
2
+ 2.57s + 9.67
"
#
$
%
= −1
•  Example: k = a
1
!
Δ(s) = s
4
+ a
3
s
3
+ a
2
s
2
+ a
1
s +a

0
= s
4
+ a
3
s
3
+ a
2
s
2
+ ks + a
0
= s
4
+ a
3
s
3
+ a
2
s
2
+ a
0
( )
+ ks
= s
2
− 0.00041s + 0.015

#
$
%
&
s
2
+ 2.57 s + 9.67
#
$
%
&
+ ks
Example: Effect of a
1
Variation"
•  Rearrange:!
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 = a
0
: k
n(s)
d(s)
= k
1
s
4

+ a
3
s
3
+ a
2
s
2
+ a
1
s
= −1

k = a
1
: k
n(s)
d(s)
= k
s
s
4
+ a
3
s
3
+ a
2
s
2

+ a
0
= −1
= k
s − 0
( )
s
4
+ a
3
s
3
+ a
2
s
2
+ 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)"
Δ(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) = s − z
1
( )
s − z
2
( )

No zeros when k = a
0
" One zero at origin when k = a
1
"
Destinations of Roots (for k -> ±∞) "
d(s)+ kn(s)
n(s)
!
"

#
$
%
&
=
d(s)
n(s)
+ k
!
"
#
$
%
&
k→±R→±∞
) →)))
s
n
s
q
+ k
!
"
#
$
%
&
→ s
n−q
( )

± R →±∞
•  (n – q) roots go to infinite radius from the origin"
R(+)"
R(–)"
s
n−q
( )
= Re
− j180°
→ ∞ or Re
− j 360°
→ −∞
s = R
1 n−q
( )
e
− j180° n−q
( )
→ ∞ or R
1 n−q
( )
e
− j 360° n−q
( )
→ −∞
•  Asymptotes of the root loci are described by"
•  Magnitudes of roots are the same for given k"
•  Angles from the origin are different"
Destinations of Roots (for k -> ±∞) "
4 roots to infinite radius"

3 roots to infinite radius"
(n – q) Roots Approach Asymptotes
as k –> ±∞"
θ
(rad) =
π
+ 2m
π
n − q
, m = 0,1, ,(n − q) − 1
θ
(rad) =
2m
π
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." =
σ
λ
i
−
σ
z
j
j =1
q

∑
i =1
n
∑
n − q
Root Locus on Real Axis"
•  Locus on real axis"
–  k > 0: Any segment with odd number
of poles and zeros to the right"
–  k < 0: Any segment with even number
of poles and zeros to the right"
First Example: Positive and
Negative Variations of k = a
0
"
k
s s + 0.21
( )
s
2
+ 2.55s + 9.62
!
"
#
$
= −1
Second Example: Positive and
Negative Variations of k = a
1
"

ks
s
2
− 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 of
Simplified Longitudinal Modes
Approximate Phugoid Model "
•  Second-order equation"

Δ

x
Ph
=
Δ

V
Δ

γ
#
$
%
%
&
'
(
(
≈
−D
V
−g
L
V
V
N
0
#
$

%
%
%
&
'
(
(
(
ΔV
Δ
γ
#
$
%
%
&
'
(
(
+
T
δ
T
L
δ
T
V
N
#
$

%
%
%
&
'
(
(
(
Δ
δ
T
•  Characteristic polynomial"
sI − F
Ph
= det sI − F
Ph
( )
≡ Δ(s) = s
2
+ D
V
s + gL
V
/ V
N
= s
2
+ 2
ζω
n

s +
ω
n
2
gL
V
/ V
N
, D
V
•  Parameters"
Δ(s) = s
2
+ D
V
s
( )
+ k
= s s + D
V
( )
+ k
k = gL
V
/V
N
"
Effect of L
V
or 1/V

N
Variation on
Approximate Phugoid Roots "
•  Change in
damped natural
frequency"

ω
n
damped

ω
n
1−
ζ
2
Effect of D
V
Variation on
Approximate Phugoid Roots "
k = D
V
"
Δ(s) = s
2
+ gL
V
/ V
N
( )

+ ks
= s + j gL
V
/ V
N
( )
s − j gL
V
/ V
N
( )
+ ks
•  Change in
damping ratio"
ζ
Approximate Short-Period Model "
•  Approximate Short-Period Equation (L
q
= 0)"
•  Characteristic polynomial"
•  Parameters"

Δ

x
SP
=
Δ

q

Δ

α
#
$
%
%
&
'
(
(
≈
M
q
M
α
1 −
L
α
V
N
#
$
%
%
%
&
'
(
(

(
Δq
Δ
α
#
$
%
%
&
'
(
(
+
M
δ
E
−L
δ
E
V
N
#
$
%
%
%
&
'
(
(

(
Δ
δ
E
Δ(s) = s
2
+
L
α
V
N
− M
q
$
%
&
'
(
)
s − M
α
+ M
q
L
α
V
N
$
%
&

'
(
)
= s
2
+ 2
ζω
n
s +
ω
n
2
M
α
, M
q
,
L
α
V
N
Effect of M
α
on Approximate
Short-Period Roots "
k = M
α
"
Δ(s) = s
2

+
L
α
V
N
− M
q
$
%
&
'
(
)
s − M
q
L
α
V
N
$
%
&
'
(
)
− k = 0
= s +
L
α
V

N
$
%
&
'
(
)
s − M
q
( )
− k = 0
•  Change in damped
natural frequency"
Effect of M
q
on Approximate
Short-Period Roots"
Δ(s) = s
2
+
L
α
V
N
s − M
α
− k s +
L
α
V

N
$
%
&
'
(
)
= s −
L
α
2V
N
+
L
α
2V
N
$
%
&
'
(
)
2
+ M
α
*
+
,
,

-
.
/
/
0
1
2
3
2
4
5
2
6
2
s −
L
α
2V
N
−
L
α
2V
N
$
%
&
'
(
)

2
+ M
α
*
+
,
,
-
.
/
/
0
1
2
3
2
4
5
2
6
2
− k s +
L
α
V
N
$
%
&
'

(
)
= 0
k = M
q
"
•  Change primarily
in damping ratio"
Effect of L
!
/V
N
on Approximate
Short-Period Roots"
Δ(s) = s
2
− M
q
s − M
α
+ k s − M
q
( )
= s +
M
q
2
−
M
q

2
$
%
&
'
(
)
2
+ M
α
*
+
,
,
-
.
/
/
0
1
2
3
2
4
5
2
6
2
s +
M

q
2
−
M
q
2
$
%
&
'
(
)
2
+ M
α
*
+
,
,
-
.
/
/
0
1
2
3
2
4
5

2
6
2
+ k s − M
q
( )
= 0
k = L
α
/V
N
"
•  Change primarily
in damping ratio"
How do the 4
th
-order roots vary when we
change pitch-rate damping, M
q
?"
Δ
Lon
(s) = s
4
+ D
V
+
L
α
V

N
( )
s
3
+ g − D
α
( )
L
V
V
N
+ D
V
L
α
V
N
( )
− M
α
$
%
&
'
(
)
s
2
+ D
α

M
V
− D
V
M
α
{ }
s + g M
V
L
α
V
N
− M
α
L
V
V
N
( )
− M
q
s
3
− D
V
M
q
( )
+ M

q
L
α
V
N
$
%
&
'
(
)
s
2
+ M
q
D
α
− g
( )
L
V
V
N
− D
V
L
α
V
N
$

%
&
'
(
)
s = 0
•  Identify M
q
terms in the characteristic polynomial"
How do the 4
th
-order roots vary when we
change pitch-rate damping, M
q
?"
Δ
Lon
(s) = d s
( )
− M
q
s
3
+ D
V
+
L
α
V
N

$
%
&
'
(
)
s
2
− D
α
− g
( )
L
V
V
N
− D
V
L
α
V
N
$
%
&
'
(
)
s
{ }

= d s
( )
− M
q
s s
2
+ D
V
+
L
α
V
N
$
%
&
'
(
)
s − D
α
− g
( )
L
V
V
N
− D
V
L

α
V
N
$
%
&
'
(
)
{ }
= d s
( )
+ kn s
( )
= 0
•  Group M
q
terms in the characteristic polynomial"
k
n(s)
d(s)
= −1
How do the 4
th
-order roots vary when we
change pitch-rate damping, M
q
?"
−M
q

s s
2
+ D
V
+
L
α
V
N
( )
s − D
α
− g
( )
L
V
V
N
− D
V
L
α
V
N
#
$
%
&
'
(

{ }
s
4
+ D
V
+
L
α
V
N
( )
s
3
+ g − D
α
( )
L
V
V
N
+ D
V
L
α
V
N
( )
− M
α
#

$
%
&
'
(
s
2
+ D
α
M
V
− D
V
M
α
{ }
s + g M
V
L
α
V
N
− M
α
L
V
V
N
( )
)

*
+
+
,
+
+
-
.
+
+
/
+
+
= −1
•  Factor terms that are multiplied by M
q
to find the 3 zeros"
–  2 zeros near origin similar to approximate phugoid roots,
effectively canceling M
q
effect on them "

−M
q
s s − z
1
( )
s − z
2
( )

s
2
+ 2
ζ
P
ω
n
P
s +
ω
n
P
2
( )
s
2
+ 2
ζ
SP
ω
n
SP
s +
ω
n
SP
2
( )
= −1
s

2
− z
1
s
( )
 s
2
+ 2
ζ
P
ω
n
P
s +
ω
n
P
2
( )

−M
q
s
2
− z
1
s
( )
s − z
2

( )
s
2
+ 2
ζ
P
ω
n
P
s +
ω
n
P
2
( )
s
2
+ 2
ζ
SP
ω
n
SP
s +
ω
n
SP
2
( )


−M
q
s − z
2
( )
s
2
+ 2
ζ
SP
ω
n
SP
s +
ω
n
SP
2
( )
= −1
•  M
q
variation has virtually no effect on phugoid roots"
•  Effect on short-period roots is predicted by 2
nd
-order model"
Next Time:
Transfer Functions and
Frequency Response


Reading
Flight Dynamics, 342-355
Virtual Textbook, Part 15