linearized equations and modes of motion.pdf

8/11/2019 Linearized Equations and Modes of Motion.pdf

1/15

Linearized Equationsand Modes of Motion

Robert Stengel, Aircraft Flight DynamicsMAE 331, 2012

Linearization of nonlinear dynamic models Nominal ight path

Perturbations about the nominal ight path

Modes of motion Longitudinal

Lateral-directional

Copyright 2012 by Robert Stengel. A ll rights reserved. For education al use only.http://www.princeton.edu/~stengel/MAE331.html

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

The Mystery Airplane: Convair XF-92A (1948)

! Precursor to F-102, F-106, B-58, and F2Y ! M = 1.05 in a dive; under-powered, pre-area rule ! Landed at ! = 45, V = 67 mph (108 km/h) ! Violent pitchup during high-speed turns, alleviated by

wing fences ! Poor high-speed and good low-speed handling qualities

Nominal and ActualFlight Paths

Nominal and Actual Trajectories Nominal (or reference) trajectory and

control history

x N (t ), u N (t ), w N (t ){ } for t in [ t o , t f ]

Actual trajectory perturbed by Small initial condition variation, xo(t o ) Small control variation, u( t )

x (t ), u (t ), w (t ){ } for t in [ t o , t f ]= x

N (t ) + ! x (t ), u N (t ) + ! u (t ), w N (t ) + ! w (t ){ }

x : dynamic stateu : control input w : disturbance input


2/15

Both Paths Satisfy theDynamic Equations

Dynamic models for the actual and thenominal problems are the same

!x N (t ) = f [x N (t ), u N (t ), w N (t )], x N t o( ) given

!x (t ) = f [x (t ), u (t ), w (t )], x t o( ) given

!x (t ) = !x N (t ) + ! !x (t )x (t ) " x N (t ) + ! x (t )

#$%

&%

'(%

)% in t o , t f *+ ,-

! x (t o ) = x (t o ) " x N (t o )

! u (t ) = u (t ) " u N (t )! w (t ) = w (t ) " w N (t )

#$%

&%

'(%

)% in t o , t f *+ ,-

Differences in initialcondition and forcing ...

... perturb rate ofchange and the state

Approximate NeighboringTrajectory as a Linear Perturbation

to the Nominal Trajectory

!x (t ) = !x N (t ) + ! !x (t )

" f [x N (t ), u N (t ), w N (t ), t ] +# f

# x! x (t ) +

# f

# u! u (t ) +

# f

# w! w (t )

Approximate the new trajectory as the sum of thenominal path plus a linear perturbation

!x N (t ) = f [x N (t ), u N (t ), w N (t ), t ]!x (t ) = !x N (t ) + ! !x (t ) = f [x N (t ) + ! x (t ), u N (t ) + ! u (t ), w N (t ) + ! w (t ), t ]

Linearized Equation ApproximatesPerturbation Dynamics

Solve for the nominal and perturbation trajectories

separately

!x N (t ) = f [x N (t ), u N (t ), w N (t ), t ], x N t o( ) given

! !x (t ) " # f

# x x = x N ( t )u = u N ( t )w = w N ( t )

! x (t )$

%

&&&

'

(

)))

+# f

# u x = x N ( t )u = u N ( t )w = w N ( t )

! u (t )$

%

&&&

'

(

)))

+# f

# w x = x N ( t )u = u N ( t )w = w N ( t )

! w (t )$

%

&&&

'

(

)))

" F (t )! x (t ) + G (t )! u (t ) + L (t )! w (t ), ! x t o( ) given

dim( x ) = n ! 1

dim( u ) = m ! 1

dim( w ) = s ! 1

dim( ! x ) = n " 1

dim( ! u ) = m " 1

dim( ! w ) = s " 1

Nominal Equation

Perturbation Equation

Jacobian Matrices Express SolutionSensitivity to Small Perturbations

Sensitivity to state perturbations: stability matrix, F, is square

F (t ) =! f

! x x = x N ( t )u = u N ( t )w = w N ( t )

=

! f 1 ! x1! ! f 1 ! xn

! ! !! f n

! x1! ! f n

! xn

"

#

$$$$$

%

&

'''''

x = x N ( t )u = u N ( t )w = w N ( t )

G (t ) =! f

! u x = x N ( t )u = u N ( t )

w=

w N ( t )

L (t ) =! f

! w x = x N ( t )u = u N ( t )w = w N ( t )

Sensitivity to control and disturbance perturbations issimilar, but matrices, G and L, may not be square

dim( F ) = n ! n

dim( G ) = n ! m dim( L ) = n ! s


3/15

Linear and nonlinear, time-varying and time-invariant dynamic models Numerical integration ( time domain )

Linear, time-invariant (LTI) dynamic models Numerical integration ( time domain ) State transition (

time domain ) Transfer functions ( frequency domain )

How Is SystemResponse Calculated?

Numerical Integration :MATLAB Ordinary Differential Equation

Solvers *

Explicit Runge-KuttaAlgorithm

Numerical DifferentiationFormula

* http://www.mathworks.com/access/helpdesk/help/techdoc/index.html?/access/helpdesk/help/techdoc/ref/ode23.html. Shampine, L. F. and M. W. Reichelt, "The MATLAB ODE Suite," SIAM Journal on Scientic Computing , Vol. 18, 1997, pp 1-22.

Adams-Bashforth-MoultonAlgorithm

Modied RosenbrockMethod

Trapezoidal Rule

Trapezoidal Rule w/BackDifferentiation

MATLAB Simulation of Linear andNonlinear Dynamic Systems

MATLAB Main Script

% Nonlinear and Linear Examples

clear tspan = [0 10];xo = [0, 10]; [t1,x1 = ode23('NonLin',tspan,xo); xo = [0, 1]; [t2,x2] = ode23('NonLin',tspan,xo);xo = [0, 10]; [t3,x3] = ode23('Lin',tspan,xo); xo = [0, 1]; [t4,x4] = ode23('Lin',tspan,xo);

subplot(2,1,1) plot(t1,x1(:,1),'k',t2,x2(:,1),'b',t3,x3(:,1),'r',t4,x4(:,1),'g') ylabel('Position'), grid subplot(2,1,2) plot(t1,x1(:,2),'k',t2,x2(:,2),'b',t3,x3(:,2),'r',t4,x4(:,2),'g') xlabel('Time'), ylabel('Rate'), grid

Linear System

function xdot = Lin(t,x) % Linear Ordinary Differential Equation % x(1) = Position % x(2) = Rate xdot = [x(2)

-10*x(1) - x(2)];

Nonlinear System

function xdot = NonLin(t,x) % Nonlinear Ordinary Differential Equation % x(1) = Position % x(2) = Rate xdot = [x(2)

-10*x(1) + 0.8*x(1)^3 - x(2)];

x 1( t ) = x 2 ( t )

x 2 ( t ) = " 10 x 1( t ) " x 2 ( t )

x 1( t ) = x 2 ( t )

x 2 ( t ) = " 10 x 1( t ) + 0.8 x 13 ( t ) " x 2 ( t )

Comparison of DampedLinear and Nonlinear Systems

! x 1 (t ) = x 2 (t )

! x 2 (t ) = ! 10 x 1 (t ) ! 10 x 1

3 (t ) ! x 2 (t )

Linear plus Stiffening Cubic Spring

Linear plus Weakening Cubic Spring

! x 1 (t ) = x 2 (t )

! x 2 (t ) = ! 10 x 1(t ) + 0.8 x 1

3 (t ) ! x 2 (t )

! x 1 (t ) = x 2 (t )

! x 2 (t )=

! 10 x 1 (t ) ! x 2 (t )

Linear Spring Spring Force vs.Displacement

Spring Damper

Displacement

Rate of Change


4/15

Linear and StiffeningCubic Springs: Small and

Large Initial Conditions

Linear and nonlinear responses are indistinguishable with small initial condition

Linear and Weakening Cubic Springs: Small andLarge Initial Conditions

Linear, Time-Varying (LTV)Approximation of

Perturbation Dynamics

Stiffening Linear-Cubic SpringExample

! Nonlinear, time-invariant (NTI) equation

! x1 (t ) = f 1 = x2 (t )

! x2 (t ) = f 2 = ! 10 x1 (t ) ! 10 x13 (t ) ! x2 (t )! Integrate equations to produce nominal path

x1 N (0)

x2 N (0)

!

"##

$

%&&'

f 1 N f 2 N

!

"##

$

%&&dt '

0

t f

( x1 N (t )

x2 N (t )

!

"##

$

%&&

in 0, t f !" $%

! Analytical evaluation of partial derivatives

! f 1! x1

= 0; ! f 1

! x2= 1

! f 2!

x1= " 10 " 30 x1 N

2 (t ); ! f 2

! x2

= " 1

! f 1! u

= 0; ! f 1

! w = 0

! f 2! u

= 0; ! f 2

! w = 0


5/15

Nominal (NTI) and Perturbation(LTV) Dynamic Equations

!x N (t ) = f [x N (t )], x N (0) given

! x1 N (t ) = x2 N (t )! x2 N (t ) = ! 10 x1 N (t ) ! 10 x1 N

3 (t ) ! x2 N (t )

! !x (t ) = F (t )! x (t ), ! x (0) given

! ! x1(t )! ! x2 (t )

"

#$$

%

&''

=0 1

( 10 + 30 x1 N 2 (t )( ) ( 1

"

#$$

%

&''

! x1(t )! x2 (t )

"

#$$

%

&''

x1 N (0)

x2 N

(0)

!

"##

$

%&&

=09

!"#

$%&

! Nonlinear, time-invariant (NTI) nominal equation

! Perturbations approximated by linear, time-varying (LTV) equation

! x 1 (0)! x 2 (0)

"

#$$

%

&''

=01

"

#$

%

&'

Example

Example

Comparison of Approximate and Exact Solutions

x N (t )

! x (t )

x N (t ) + ! x (t )

x (t )

Initial Conditions

x2 N (0) = 9

! x2 (0) = 1

x2 N (t ) + ! x 2 (t ) = 10

x 2 (t ) = 10

!x N (t )

! !x (t )!x

N (t ) + ! !x (t )!x (t )

Suppose Nominal InitialCondition is Zero

!x

N (t )

= f [

x N (

t )],

x N (0)

=

0, x

N (t )

=

0 in 0,!

[ ]

Nominal solution remains at equilibrium

Perturbation equation is linear and time-invariant (LTI)

! ! x 1 (t )! ! x 2 (t )

"

#$$

%

&''

=

0 1

( 10 ( 30 0( )"# %& ( 1"

#$$

%

&''

! x 1 (t )! x 2 (t )

"

#$$

%

&''

Separation of theEquations of Motion intoLongitudinal and Lateral-

Directional Sets


6/15

Rigid-Body Equations ofMotion (Scalar Notation)

Rate of change ofTranslational Position

Rate of change ofAngular Position

Rate of change ofTranslational Velocity

Rate of change ofAngular Velocity(I xy = I yz = 0)

x1

x 2

x 3

x 4

x 5

x 6

x 7

x 8

x 9

x10

x11

x12

!

"

##################

$

%

&&&&&&&&&&&&&&&&&&

=

u

vw

x

y

z

p

q

r

'

( )

!

"

################

$

%

&&&&&&&&&&&&&&&&

State Vector

!u = X / m ! g sin " + rv ! qw!v = Y / m + g sin # cos " ! ru + pw!w = Z / m + g cos # cos " + qu ! pv

! x I = cos ! cos " ( )u + # cos $ sin " + sin $ sin ! cos " ( )v + sin $ sin " + cos $ sin ! cos " ( )w! y I = cos ! sin " ( )u + cos $ cos " + sin $ sin ! sin " ( )v + # sin $ cos " + cos $ sin ! sin " ( )w! z I = # sin ! ( )u + sin $ cos ! ( )v + cos $ cos ! ( )w

!! = p + q sin ! + r cos ! ( )tan " !" = q cos ! # r sin ! !$ = q sin ! + r cos ! ( )sec "

Grumman F9F

! p = I zz L + I xz N ! I xz I yy ! I xx ! I zz( ) p + I xz2 + I zz I zz ! I yy( )"# $%r{ }q( ) I xx I zz ! I xz2( )!q = M ! I xx ! I zz( ) pr ! I xz p 2 ! r 2( )"# $% I yy!r = I xz L + I xx N ! I xz I yy ! I xx ! I zz( )r + I xz2 + I xx I xx ! I yy( )"# $% p{ }q( ) I xx I zz ! I xz2( )

Reorder the State Vector

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

!

"

###

###############

$

%

&&&

&&&&&&&&&&&&&&&

new

=

x Lon

x Lat ' Dir

!

"##

$

%&&

=

uw

x zq

( v y

p

r)

*

!

"

##

##############

$

%

&&

&&&&&&&&&&&&&&

First six elements ofthe state arelongitudinal variables

Second six elements of

the state are lateral-directional variables

x1

x 2

x 3

x 4

x 5

x 6

x 7

x 8

x 9

x10

x11

x12

!

"

##################

$

%

&&&&&&&&&&&&&&&&&&

=

u

v

w

x

y

z

p

q

r

'

( )

!

"

################

$

%

&&&&&&&&&&&&&&&&

Longitudinal Equations of Motion Dynamics of position, velocity, angle, and

angular rate in the vertical plane!u = X / m ! g sin " + rv ! qw = ! x1 = f 1!w = Z / m + g cos # cos " + qu ! pv = ! x2 = f 2! x I = cos " cos $ ( )u + ! cos # sin $ + sin # sin " cos $ ( )v + sin # sin $ + cos # sin " cos $ ( )w = ! x3 = f 3! z I = ! sin " ( )u + sin # cos " ( )v + cos # cos " ( )w = ! x4 = f 4!q = M ! I xx ! I zz( ) pr ! I xz p 2 ! r 2( )%& '( I yy = ! x5 = f 5!" = q cos # ! r sin # = ! x6 = f 6

!x Lon (t ) = f [x Lon (t ), u Lon (t ), w Lon (t )]

!v = Y / m + g sin ! cos " # ru + pw = ! x7 = f 7! y I = cos " sin $ ( )u + cos ! cos $ + sin ! sin " sin $ ( )v + # sin ! cos $ + cos ! sin " sin $ ( )w = ! x8 = f 8! p = I zz L + I xz N # I xz I yy # I xx # I zz( ) p + I xz2 + I zz I zz # I yy( )%& '( r{ }q( ) I xx I zz # I xz2( ) = ! x9 = f 9!r = I xz L + I xx N # I xz I yy # I xx # I zz( )r + I xz2 + I xx I xx # I yy( )%& '( p{ }q( ) I xx I zz # I xz2( ) = ! x10 = f 10!! = p + q sin ! + r cos ! ( )tan " = ! x11 = f 11!$ = q sin ! + r cos ! ( )sec " = ! x12 = f 12

Lateral-DirectionalEquations of Motion

Dynamics of position, velocity, angle, andangular rate out of the vertical plane

!x LD (t ) = f [x LD (t ), u LD (t ), w LD (t )]


7/15

Sensitivity to Small Motions (12 x 12) stability matrix for the entire system

F (t ) =

! f 1! x1

! f 1! x2

... ! f 1! x12

! f 2! x1

! f 2! x2

... ! f 2! x12

... ... ... ...! f 12

! x1! f 12

! x2... ! f 12 ! x12

"

#

$$$$$$$$

%

&

''''''''

=

! !u! u

! !u! w ...

! !u!(

! !w! u

! !w! w ...

! !w!(

... ... ... ...! !(

! u! !(

! w ... ! !(

!(

"

#

$$$$

$$$

%

&

''''

'''

Four (6 x 6) blocks distinguish longitudinal and lateral-directional effects

F =F

Lon F

Lat ! Dir Lon

F Lon

Lat ! Dir F Lat ! Dir

"

#$$

%

&''

Effects of longitudinal perturbationson longitudinal motion

Effects of longitudinal perturbationson lateral-directional motion

Effects of lateral-directionalperturbations on longitudinal motion

Effects of lateral-directional perturbationson lateral-directional motion

Sensitivity toControl Inputs

Control input vector and perturbation

Four (6 x 3) blocks distinguish longitudinal and lateral-directionalcontrol effects

G =G

Lon G

Lat ! Dir Lon

G Lon

Lat ! Dir G Lat ! Dir

"

#$$

%

&''

Effects of longitudinal controls onlongitudinal motion

Effects of longitudinal controls onlateral-directional motion

Effects of lateral-directional controlson longitudinal motion

Effects of lateral-directional controls onlateral-directional motion

u(t ) =

! E (t )! T (t )

! F (t )! A(t )! R(t )

! SF (t )

"

#

$$

$$$$$$

%

&

''

''''''

Elevator , deg or rad

Throttle ,%

Flaps , deg or rad Ailerons , deg or rad

Rudder , deg or rad

SideForcePanels , deg or rad

! u (t ) =

! " E (t )! " T (t )

! " F (t )! " A (t )! " R (t )

! " SF (t )

#

$

%%

%%%%%%

&

'

((

((((((

Decoupling Approximationfor Small Perturbationsfrom Steady, Level Flight

Restrict the Nominal Flight Pathto the Vertical Plane

Nominallongitudinalequations reduce to

Nominal lateral-directional motions are zero x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

!

"

#################

#

$

%

&&&&&&&&&&&&&&&&&

& N

=

x Lon

x Lat ' Dir

!

"##

$

%&&

N

=

u N w N x N z N q N (

N

0

0

0

0

0

0

!

"

################

$

%

&&&&&&&&&&&&&&&&

!u N = X / m ! g sin " N ! q N w N !w N = Z / m + g cos " N + q N u N ! x I N = cos " N ( )u N + sin " N ( )w N ! z I N = ! sin " N ( )u N + cos " N ( )w N !q N =

M

I yy!" N

= q N

!x Lat ! Dir N

= 0

x Lat ! Dir N

= 0

Nominal State Vector

BUT, Lateral-directional

perturbations neednot be zero

! !x Lat " Dir N

# 0

! x Lat " Dir N # 0


8/15

Restrict the Nominal Flight Path toSteady, Level Flight

Calculate conditions for trimmed (equilibrium) ight See Flight Dynamics and FLIGHT program for a

solution method

0 = X / m ! g sin " N ! q N w N 0 = Z / m + g cos " N + q N u N V N = cos " N ( )u N + sin " N ( )w N 0 = ! sin "

N ( )u N + cos " N ( )w N 0 =

M

I yy0 = q N

Trimmed State Vector isconstant

Specify nominal airspeed (V N ) and altitude (h N = z N )

uw x zq!

"

#

$$$$$$$

%

&

'''''''

Trim

=

uTrimwTrim

V N t ( t 0( ) z N

0

! Trim

"

#

$$$$$$$$

%

&

''''''''

Small Perturbation Effects areUncoupled in Steady, Symmetric,

Level Flight Assume the airplane is symmetric and its nominal path

is steady, level ight Small longitudinal and lateral-directional perturbationsare approximately uncoupled from each other

(12 x 12) system is block diagonal constant, i.e., linear, time-invariant (LTI) decoupled into two separate (6 x 6) systems

F =F

Lon 0

0 F Lat ! Dir

"

#$$

%

&''

G =G

Lon 0

0 G Lat ! Dir

"

#$$

%

&''

L =L

Lon 0

0 L Lat ! Dir

"

#$$

%

&''

! !x Lon (t ) = F Lon ! x Lon (t ) + G Lon ! u Lon (t ) + L Lon ! w Lon (t )

! x Lon =

! x1! x2! x3! x4! x5! x6

"

#

$$$$$$$$

%

&

''''''''

Lon

=

! u! w! x! z! q! (

"

#

$$$$$$$

%

&

'''''''

(6 x 6) LTI LongitudinalPerturbation Model

! u Lon =! " T ! " E ! " F

#

$

%%%

&

'

(((

! w Lon =! uwind ! wwind ! qwind

"

#

$$$

%

&

'''

Dynamic Equation

State Vector ControlVector

Disturbance Vector

(6 x 6) LTI Lateral-Directional Perturbation Model

! !x Lat " Dir (t ) = F Lat " Dir ! x Lat " Dir (t ) + G Lat " Dir ! u Lat " Dir (t ) + L Lat " Dir ! w Lat " Dir (t )

! x Lat " Dir =

! x1! x2! x3! x4! x5! x6

#

$

%%%%%%%%

&

'

((((((((

Lat " Dir

=

! v! y! p! r! ) ! *

#

$

%%%%%%%%

&

'

((((((((

! u Lat " Dir =! # A! # R

! # SF

$

%

&&&

'

(

)))

! w Lon =! vwind ! pwind ! rwind

"

#

$$$

%

&

'''

Dynamic Equation

State Vector ControlVector

Disturbance Vector


9/15

Frequency DomainDescription of LTISystem Dynamics

Fourier Transform of aScalar Variable

Transformation from time domain to frequency domain

F ! x(t )[ ]= ! x( j " ) = ! x(t )e# j " t #$

$

% dt , " = frequency , rad / s

! x(t ) : real variable! x( j " ) : complex variable

= a (" ) + jb (" )

= A(" )e j # (" ) A : amplitude! : phase angle

j ! : Imaginary operator, rad/s

Fourier Transform of aScalar Variable

! x (t )

! x( j " ) = a (" ) + jb(" )

Laplace Transform ofa Scalar Variable

Laplace transformation from time domain to frequency domain

L ! x(t )[ ]= ! x( s ) = ! x(t )e " st 0

#

$ dt

s = ! + j "

= Laplace (complex) operator, rad/s

! x(t ) : real variable! x(s) : complex variable

= a (s ) + jb (s)

= A(s)e j " ( s )


10/15

Laplace Transformation is aLinear Operation

L ! x1 (t ) + ! x 2 (t )[ ]= L ! x1 (t )[ ]+ L ! x 2 (t )[ ]= ! x1 (s ) + ! x 2 ( s )

L a ! x (t )[ ]= aL ! x (t )[ ]= a ! x ( s )

Sum of Laplace transforms

Multiplication by a constant

Laplace Transforms ofVectors and Matrices

Laplace transform of a vector variable

L ! x (t )[ ]= ! x ( s ) =

! x1 (s )

! x 2 ( s )...

"

#

$$$

%

&

'''

Laplace transform of a matrix variable

L A (t )[ ]= A (s ) =a

11 (s ) a 12 (s ) ...

a21 (s ) a 22 ( s ) ...

... ... ...

!

"

###

$

%

&&&

Laplace transform of a time-derivative

L ! !x (t )[ ]= s ! x ( s ) " ! x (0)

Laplace Transform ofa Dynamic System

! !x ( t ) = F ! x ( t ) + G ! u ( t ) + L ! w (t )

System equation

Laplace transform of system equation

s ! x ( s ) " ! x (0) = F ! x ( s ) + G ! u ( s ) + L ! w ( s )

dim( ! x ) = (n " 1)

dim( ! u ) = (m " 1)

dim( ! w ) = ( s " 1)

Laplace Transform ofa Dynamic System

Rearrange Laplace transform of

dynamic equation

s ! x ( s ) " F ! x ( s ) = ! x (0) + G ! u ( s ) + L ! w ( s )

s I ! F[ ]" x ( s ) = " x (0) + G " u ( s ) + L " w ( s )

! x ( s ) = s I " F[ ]" 1

! x (0) + G ! u ( s ) + L ! w ( s )[ ]

F to left, I.C. to right

Combine terms

Multiply both sides by inverse of (s I F)


11/15

Matrix Inverse

A[ ]! 1 = Adj A( )

A=

Adj A( )det A

(n " n)(1 " 1)

=

C T

det A; C = matrix of cofactors

Cofactors aresigned minors of A

ij th minor of A is thedeterminant of A with the i th row and

j th column removed

y = Ax ; x = A! 1y

dim( x) = dim( y) = (n ! 1)

dim( A ) = (n ! n )

Forward Inverse

Numerator is a square matrix Denominator is a scalar

Matrix Inverse Examples

A =a

11 a

12

a21

a22

!

"##

$

%&&

; A ' 1 =

a22 ' a 21

' a 12 a 11

!

"##

$

%&&

T

a11

a22 ' a 12 a 21

=

a22 ' a 12

' a 21 a 11

!

"##

$

%&&

a11

a22 ' a 12 a 21

A =

a11

a12

a13

a21

a22

a23

a31

a32

a33

!

"

###

$

%

&&&

; A ' 1 =

a22

a33 ' a 23 a 32( ) ' a 21 a 33 ' a 23 a 31( ) a 21 a 32 ' a 22 a 31( )

' a 12 a 33 ' a 13 a 32( ) a 11 a 33 ' a 13 a 31( ) ' a 11 a 32 ' a 12 a 31( )a

12a

23 ' a 13 a 22( ) ' a 11 a 23 ' a 13 a 21( ) a 11 a 22 ' a 12 a 21( )

!

"

####

$

%

&&&&

T

a11

a22

a33

+ a12

a23

a31

+ a13

a21

a32 ' a 13 a 22 a 31 ' a 12 a 21 a 33 ' a 11 a 23 a 32

=

a22

a33 ' a 23 a 32( ) ' a 12 a 33 ' a 13 a 32( ) a 12 a 23 ' a 13 a 22( )

' a 21 a 33 ' a 23 a 31( ) a 11 a 33 ' a 13 a 31( ) ' a 11 a 23 ' a 13 a 21( )a

21a

32 ' a 22 a 31( ) ' a 11 a 32 ' a 12 a 31( ) a 11 a 22 ' a 12 a 21( )

!

"

####

$

%

&&&&

a11

a22

a33

+ a12

a23

a31

+ a13

a21

a32 ' a 13 a 22 a 31 ' a 12 a 21 a 33 ' a 11 a 23 a 32

dim( A ) = (2 ! 2)

dim( A ) = (3 ! 3)

A = a ; A! 1

=

1

a

dim( A ) = (1 ! 1)

Modes of Motion

Characteristic Polynomialof a LTI Dynamic System

sI ! F[ ]! 1

=

Adj sI ! F( )sI ! F

(n x n)

Characteristic polynomial of the system is a scalar denes the system s modes of motion

s I ! F = det s I ! F( )" # ( s )= s

n+ a

n ! 1s

n ! 1+ ... + a 1 s + a 0

! x ( s ) = s I " F[ ]" 1 ! x (0) + G ! u ( s ) + L ! w ( s )[ ] Inverse of characteristic matrix


12/15

Eigenvalues (or Roots) ofa Dynamic System

! ( s ) = s n + a n " 1 sn " 1

+ ... + a 1 s + a 0 = 0

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

Characteristic equation of the system

... where i are the eigenvalues of F or the roots of the characteristic polynomial

Eigenvalues are real or complex numbersthat can be plotted in the s plane

s Plane

! i

= " i

+ j # i

Real root

! *

i = "

i # j $ i

Positive real partrepresents instability

Complex roots occur in conjugate pairs

! i

= " i

Rigid-Body Motion of a Linear,Time-Invariant Aircraft Model

! ( s ) = s 12 + a 11 s11

+ ... + a 1s + a 0 = 0

= s " # 1( ) s " # 2( ) ...( ) s " # 12( )= 0

12 th -order system of LTI equations 12 eigenvalues of the stability matrix, F 12 roots of the characteristic equation

Characteristic equation of the system

Up to 12 modes of motion

In steady, level ight, longitudinal and lateral-directionalLTI models are uncoupled

! (s) = s " # 1( )! s " # 6( )$% &' long s " # 1( )! s " # 6( )$% &' lat "dir = 0

Longitudinal Modes of Motion inSteady, Level Flight

! Lon (s) = s " # 1( ) s " # 2( ) ...( ) s " # 6( )= 0

= s " # range( ) s " # height ( ) s " # phugoid ( ) s " # * phugoid ( ) s " # short period ( ) s " # *short period ( )

! !x Lon (t ) = F Lon ! x Lon (t ) + G Lon ! u Lon (t ) + L Lon ! w Lon (t )

! Lon (s ) = s " # ran( ) s " # hgt ( ) s2 + 2$ P% nP s + % nP2( ) s2 + 2$ SP% nSP s + % nSP2( )= 0

6 roots of the longitudinal characteristic equation

4 modes of motion (typical)

Real Complex

Range Height Phugoid Short Period

Real Complex Complex Complex

Complex Conjugate Roots Form aSingle Oscillatory Mode of Motion

s ! " phugoid ( )

s ! " * phugoid ( )

= s ! # phugoid + j $ phugoid ( )%& '( s ! # phugoid ! j $ phugoid ( )%& '(= s2 + 2 ) P$ nP s + $ nP

2( )

s ! " short period ( ) s ! " *short period ( )= s ! # short period + j $ short period ( )%& '( s ! # short period ! j $ short period ( )%& '(

= s2 + 2 ) SP

$ nSP

s + $ nSP

2

( )

Short Period Roots

Phugoid Roots

! n

: Natural frequency , rad/s ! : Damping ratio , -


13/15

Longitudinal Modes of Motion Eigenvalues determine the damping and natural

frequencies of the linear system

s modes of motion

! ran : range mode " 0

! hgt : height mode " 0

# P ,$ nP( ) : phugoid mode# SP ,$ nSP( ) : short - period mode

Longitudinal characteristicequation has 6 eigenvalues 4 eigenvalues normally

appear as 2 complex pairs Range and height modes

usually inconsequential

Lateral-Directional Modes ofMotion in Steady, Level Flight

! LD (s ) = s " # 1( ) s " # 2( ) ...( ) s " # 6( )= 0

= s " # crossrange( ) s " # heading( ) s " # spiral( ) s " # roll( ) s " # Dutch roll( ) s " # * Dutch roll( )

! !x Lat " Dir (t ) = F Lat " Dir ! x Lat " Dir (t ) + G Lat " Dir ! u Lat " Dir (t ) + L Lat " Dir ! w Lat " Dir (t )

! LD ( s ) = s " # cr( ) s " # head ( ) s " # S ( ) s " # R( ) s 2 + 2$ DR % n DR s + % n DR2( )= 0

Roots of the lateral-directional characteristic equation

5 modes of motion (typical)

Crossrange Heading Spiral Roll Dutch Roll

Lateral-Directional Modes of Motion

Lateral-directional characteristic

equation has 6 eigenvalues 2 eigenvalues normally appear asa complex pair

Crossrange and heading modesusually inconsequential

! cr : crossrange mode " 0

! head : heading mode " 0! S : spiral mode! R : roll mode

# DR ,$ n DR( ) : Dutch roll mode

Next Time:

Longitudinal Dynamics Reading

Flight Dynamics ,452-464, 482-486

Virtual Textbook , Part 11


14/15

Supplementalaterial

Sensitivity to Small Control andDisturbance Perturbations

Control-effect matrix

G (t ) =! f

! u x = x N ( t )u = u N ( t )w = w N ( t )

=

! f 1! u1

! f 1! u 2

... ! f 1

! um! f 2

! u1! f 2

! u2...

! f 2! um

... ... ... ...! f n

! u1! f n

! u2...

! f n! um

"

#

$$$$$$$$

%

&

'''''''' x = x N ( t )

u = u N ( t )w = w N ( t )

F (t ) =! f

! x x = x N ( t )u = u N ( t )w = w N ( t )

; G (t ) =! f

! u x = x N ( t )u = u N ( t )w = w N ( t )

; L (t ) =! f

! w x = x N ( t )u = u N ( t )w = w N ( t )

Disturbance-effect matrix

L (t ) =! f

! w x = x N ( t )u = u N ( t )w = w

N ( t )

=

! f 1! w1

! f 1! w2

... ! f 1

! ws! f 2

! w1! f 2

! w2...

! f 2! ws

... ... ... ...! f n

! w1! f n

! w2...

! f n! ws

"

#

$$$$$$$$

%

&

'''''''' x = x N ( t )

u = u N ( t )w = w

N ( t )

How Do We Calculate the PartialDerivatives?

Numerically First differences in f(x,u,w)

Analytically Symbolic evaluation of analytical

models of F, G, and L

F(t )

=

! f

! x x = x N ( t )u = u N ( t )w = w N ( t )

G(t )

=

! f

! u x = x N ( t )u = u N ( t )w = w N ( t )

L(t )

=

! f

! w x = x N ( t )u = u N ( t )w = w N ( t )

Numerical Estimation of theJacobian Matrices

F (t ) !

f 1

x1 + " x1( ) x2! xn

#

$

%%%%%

&

'

(((((

) f 1

x1 ) " x1( ) x2! xn

#

$

%%%%%

&

'

(((((

2" x1

f 1

x1

x2 + " x2( )! xn

#

$

%%%%%

&

'

(((((

) f 1

x1

x2 + " x2( )! xn

#

$

%%%%%

&

'

(((((

2 " x2!

f 1

x1

x2!

xn + " xn( )

#

$

%%%%%

&

'

(((((

) f 1

x1

x2!

xn + " xn( )

#

$

%%%%%

&

'

(((((

2" xn

f 2

x1 + " x1( ) x2! xn

#

$

%%%%%

&

'

(((((

) f 2

x1 ) " x1( ) x2! xn

#

$

%%%%%

&

'

(((((

2" x1

f 2

x1 x2 + " x2( )

! xn

#

$

%%%%%

&

'

(((((

) f 2

x1 x2 + " x2( )

! xn

#

$

%%%%%

&

'

(((((

2 " x2! !

! ! ! !

f n

x1 + " x1( ) x2! xn

#

$

%%%%%

&

'

(((((

) f n

x1 ) " x1( ) x2! xn

#

$

%%%%%

&

'

(((((

2" x1! !

f n

x1 x2!

xn + " xn( )

#

$

%%%%%

&

'

(((((

) f n

x1 x2!

xn + " xn( )

#

$

%%%%%

&

'

(((((

2" xn

#

$

%

%%%%%%%%%%%%%%%%%%%%%%%%%

&

'

(

(((((((((((((((((((((((((x = x N (t )

u = u N (t )

w = w N (t )


15/15

Sensitivity toDisturbance Inputs

Disturbance input vector and perturbation

Four (6 x 3) blocks distinguish longitudinal and lateral-directionaleffects

L =L

Lon L

Lat ! Dir Lon

L Lon

Lat ! Dir L Lat ! Dir

"

#$$

%

&''

Effects of longitudinal disturbanceson longitudinal motion

Effects of longitudinal disturbanceson lateral-directional motion

Effects of lateral-directionaldisturbances on longitudinal motion

Effects of lateral-directional disturbanceson lateral-directional motion

w (t ) =

uw (t )

ww(t )

qw (t )

vw (t )

pw(t )

rw (t )

!

"

########

$

%

&&&&&&&&

Axial wind, m / s

Normal wind, m / s

Pitching wind shear, deg / s or rad / s

Lateral wind, m / s

Rolling wind shear, deg / s or rad / s

Yawing wind shear, deg / s or rad / s

! w (t ) =

! u w (t )! w w (t )! q w (t )! vw (t )! pw (t )! rw (t )

"

#

$$$$$$$$

%

&

''''''''

Integration Algorithms Rectangular (Euler) Integration

x ( t k ) = x ( t k " 1) + # x ( t k " 1, t k )

$ x ( t k

"1) + f x ( t

k "

1), u ( t

k "

1), w ( t

k "

1)

[ ]# t , # t = t

k

" t k

"1

Trapezoidal (modied Euler) Integration (~MATLAB s ode23 )

x ( t k ) " x ( t k # 1) +12

$ x1

+ $ x2[ ]

where

$ x1

= f x ( t k # 1), u ( t k # 1), w ( t k # 1)[ ]$ t $ x

2= f x ( t k # 1) + $ x 1,u ( t k ), w ( t k )[ ]$ t

See MATLAB manual for descriptions of ode45 and ode15s

linearized equations and modes of motion.pdf

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