Linearized Equationsand Modes of Motion

Robert Stengel, Aircraft Flight Dynamics

MAE 331, 2010

Linearization of nonlinear dynamic models

Nominal flight path

Perturbations about the nominal flight path

Modes of motion

Longitudinal

Lateral-directional

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

Nominal and ActualFlight Paths

Nominal and Actual

Trajectories

Nominal (or reference)trajectory and control history

xN (t), uN (t),wN (t){ } for t in [to,t f ]

Actual trajectory perturbed by Small initial condition variation, !xo(to)

Small control variation, !u(t)

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

= xN (t) + !x(t), uN (t) + !u(t),wN (t) + !w(t){ }

x : dynamic state

u : control input

w : disturbance input

Both Paths Satisfy theDynamic Equations

Dynamic models for the actual and thenominal problems are the same

!xN

(t) = f[xN

(t),uN

(t),wN

(t)], xNto( ) given

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

!x(t) = !xN (t) + !!x(t)

x(t) " xN (t) + !x(t)

#$%

&%

'(%

)% in to,t f*+ ,-

!x(to ) = x(to ) " xN (to )

!u(t) = u(t) " uN (t)

!w(t) = w(t) " wN (t)

#$%

&%

'(%

)% in to,t f*+ ,-

Differences in initialcondition and forcing ...

... perturb rate ofchange and the state

Approximate Neighboring

Trajectory as a Linear Perturbationto the Nominal Trajectory

!x(t) = !xN(t) + !!x(t)

" f[xN(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

!xN(t) = f[x

N(t),u

N(t),w

N(t),t]

!x(t) = !xN(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 trajectoriesseparately

... where the Jacobian matrices of the linear modelare evaluated along the nominal trajectory

!xN

(t) = f[xN

(t),uN

(t),wN

(t),t], xNto( ) given

!!x(t) "#f

#xt( )!x(t) +

#f

#ut( )!u(t) +

#f

#wt( )!w(t)

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

F(t) =!f

!x x=xN (t )u=u

N(t )

w=wN(t )

; G(t) =!f

!u x=xN (t )u=u

N(t )

w=wN(t )

; L(t) =!f

!w x=xN (t )u=u

N(t )

w=wN(t )

dim(x) = n !1

dim(u) = m !1

dim(w) = s !1

dim(!x) = n "1

dim(!u) = m "1

dim(!w) = s "1

to < t < t f

Jacobian Matrices Express the Solution

Sensitivity to Small Perturbations

Sensitivity to state perturbations: stability matrixis square

F(t) =!f

!x x=xN (t )u=uN (t )w=wN (t )

=

! f1

!x1

!! f

1

!xn

! ! !

! fn!x

1

!! fn

!xn

"

#

$$$$$

%

&

'''''x=xN (t )u=uN (t )w=wN (t )

G(t) =!f

!u x=xN (t )u=u

N(t )

w=wN(t )

L(t) =!f

!w x=xN (t )u=u

N(t )

w=wN(t )

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

dim(F) = n ! n

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

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 System

Response Calculated?

Numerical Integration of Ordinary

Differential Equations Given

Initial condition, control, and disturbance histories

x(t0)

u(t),w(t) for t in [to,t f ]

Path is approximated by executing a numerical algorithm

Nonlinear equations of motion

x (t) = f[x(t),u(t),w(t), t] , x(t0) given

Linear, time-invariant equations of motion

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

Integration Algorithms Rectangular (Euler) Integration

x(tk) = x(t

k!1) + "x(t

k!1,tk)

# x(tk!1) + f x(t

k!1),u(t

k!1),w(t

k!1)[ ] "t , "t = tk ! tk!1

Trapezoidal (modified Euler) Integration (~MATLAB!s ode23)

x(tk) ! x(t

k"1) +1

2#x

1+ #x

2[ ]

where

#x1

= f x(tk"1),u(t

k"1),w(t

k"1)[ ] #t

#x2

= f x(tk"1) + #x

1,u(t

k),w(t

k)[ ] #t

See MATLAB manual for descriptions of ode45 and ode15s

Linear Approximation ofPerturbation Dynamics

Stiffening Linear-Cubic Spring

Example Nonlinear, time-invariant (NTI) equation

!x1(t) = f

1= x

2(t)

!x2(t) = f

2= !10x

1(t) !10x

1

3(t) ! x

2(t)

Integrate equations to produce nominal path

x1(0)

x2(0)

!

"

##

$

%

&&'

f1N

f2N

!

"

##

$

%

&&dt'

0

t f

(x1N(t)

x2N(t)

!

"

##

$

%

&&

in 0,t f!" $%

Analytical evaluation of time-varying partial derivatives

! f1

!x1

= 0;! f

1

!x2

= 1

! f2

!x1

= "10 " 30x1N

2(t);

! f2

!x2

= "1

! f1

!u= 0;

! f1

!w= 0

! f2

!u= 0;

! f2

!w= 0

Nominal (NTI) and Perturbation

(LTV) Dynamic Equations

!xN(t) = f[x

N(t)], x

N(0) given

!x1N(t) = x

2N(t)

!x2N(t) = !10x

1N(t) !10x

1N

3(t) ! x

2N(t)

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

!!x1(t)

!!x2(t)

"

#

$$

%

&

''=

0 1

( 10 + 30x1N

2(t)( ) (1

"

#

$$

%

&

''

!x1(t)

!x2(t)

"

#

$$

%

&

''

x1N

(0)

x2N

(0)

!

"

##

$

%

&&=

0

9

!

"#

$

%&;

'x1(0)

'x2(0)

!

"

##

$

%

&&=

0

1

!

"#

$

%&

NTI

LTV

Initial

Conditions

Comparison of Approximate and Exact Solutions

xN(t)

!x(t)

xN(t) + !x(t)

x(t)

Initial Conditions

x2N

(0) = 9 !x2(0) = 1

x2N

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

2(t) = 10

!xN(t)

!!x(t)

!xN(t) + !!x(t)

!x(t)

Suppose xN (0) = 0

!xN

(t) = f[xN

(t)], xN

(0) = 0, xN

(t) = 0 in 0,![ ]

Nominal solution remains at equilibrium

Perturbation equation is linear and time-invariant (LTI)

!!x1(t)

!!x2(t)

"

#

$$

%

&

''=

0 1

(10 (1

"

#$

%

&'

!x1(t)

!x2(t)

"

#

$$

%

&

''

Separation of theEquations of Motion intoLongitudinal and Lateral-

Directional Sets

Rigid-Body Equations ofMotion (Scalar Notation)

Rate of change of Translational Position

Rate of change of Angular Position

Rate of change of Translational Velocity

Rate of change of Angular Velocity (Ixy and Iyz = 0)

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

!

"

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

$

%

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

=

u

v

w

x

y

z

p

q

r

'

()

!

"

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

$

%

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

State Vector

!u = X / m ! gsin" + rv ! qw

!v = Y / m + gsin# cos" ! ru + pw

!w = Z / m + gcos# cos" + qu ! pv

!xI= cos! cos"( )u + # cos$ sin" + sin$ sin! cos"( )v + sin$ sin" + cos$ sin! cos"( )w

!yI= cos! sin"( )u + cos$ cos" + sin$ sin! sin"( )v + # sin$ cos" + cos$ sin! sin"( )w

!zI= # sin!( )u + sin$ cos!( )v + cos$ cos!( )w

!! = p + qsin! + r cos!( ) tan"!" = qcos! # r sin!

!$ = qsin! + r cos!( )sec"

Grumman F9F

!p = IzzL + I

xzN ! I

xzIyy! I

xx! I

zz( ) p + Ixz2 + Izz Izz ! Iyy( )"# $%r{ }q( ) Ixx Izz ! Ixz2( )!q = M ! I

xx! I

zz( ) pr ! Ixz p2 ! r2( )"# $% Iyy

!r = IxzL + I

xxN ! I

xzIyy! I

xx! I

zz( )r + Ixz2 + Ixx Ixx ! Iyy( )"# $% p{ }q( ) Ixx Izz ! Ixz2( )

Rearrange the State VectorLongitudinal, Lateral-Directional

State Vector

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

!

"

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

$

%

&&&&&&&&&&&&&&&&&&new

=xLon

xLat'Dir

!

"##

$

%&&=

u

w

x

z

q

(v

y

p

r

)

*

!

"

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

$

%

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

First six elements ofthe state arelongitudinal variables

Second six elementsof the state are lateral-directional variables

Longitudinal

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

angular rate in the vertical plane

!u = X / m ! gsin" + rv ! qw = !x1= f

1

!w = Z / m + gcos# cos" + qu ! pv = !x2= f

2

!xI = cos" cos$( )u + ! cos# sin$ + sin# sin" cos$( )v + sin# sin$ + cos# sin" cos$( )w = !x3 = f3!zI = ! sin"( )u + sin# cos"( )v + cos# cos"( )w = !x4 = f4

!q = M ! Ixx ! Izz( ) pr ! Ixz p2 ! r2( )%& '( Iyy = !x5 = f5

!" = qcos# ! r sin# = !x6= f

6

!xLon(t) = f[x

Lon(t),u

Lon(t),w

Lon(t)]

!v = Y / m + gsin! cos" # ru + pw = !x7= f

7

!yI = cos" sin$( )u + cos! cos$ + sin! sin" sin$( )v + # sin! cos$ + cos! sin" sin$( )w = !x8 = f8

!p = IzzL + IxzN # Ixz Iyy # Ixx # Izz( ) p + Ixz2 + Izz Izz # Iyy( )%& '(r{ }q( ) Ixx Izz # Ixz2( ) = !x9 = f9!r = IxzL + IxxN # Ixz Iyy # Ixx # Izz( )r + Ixz2 + Ixx Ixx # Iyy( )%& '( p{ }q( ) Ixx Izz # Ixz2( ) = !x10 = f10!! = p + qsin! + r cos!( ) tan" = !x

11= f

11

!$ = qsin! + r cos!( )sec" = !x12= f

12

Lateral-Directional

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

angular rate out of the vertical plane

!xLD(t) = f[x

LD(t),u

LD(t),w

LD(t)]

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

F(t) =

! f1

!x1

! f1

!x2

...! f

1

!x12

! f2

!x1

! f2

!x2

...! f

2

!x12

... ... ... ...

! fn!x

1

! fn!x

2

...! fn

!x12

"

#

$$$$$$$$

%

&

''''''''

=

! f1

!u! f

1

!w ...! f

1

!(

! f2

!u! f

2

!w ...! f

2

!(

... ... ... ...

! f12

!u! f

12

!w ...! f

12

!(

"

#

$$$$$$$$

%

&

''''''''

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

F =FLon

FLat!DirLon

FLon

Lat!DirFLat!Dir

"

#

$$

%

&

''

Effects of longitudinal perturbationson longitudinal motion

Effects of longitudinal perturbations

on lateral-directional motion

Effects of lateral-directional

perturbations on longitudinal motion

Effects of lateral-directional perturbations

on lateral-directional motion

Sensitivity to

Control Inputs Control input vector and perturbation

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

G =G

LonG

Lat!DirLon

GLon

Lat!DirG

Lat!Dir

"

#

$$

%

&

''

Effects of longitudinal controls onlongitudinal motion

Effects of longitudinal controls on

lateral-directional motion

Effects of lateral-directional controls

on longitudinal motion

Effects of lateral-directional controls on

lateral-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

Side Force Panels, deg or rad

!u(t) =

!"E(t)

!"T (t)

!"F(t)

!"A(t)

!"R(t)

!"SF(t)

#

$

%%%%%%%%

&

'

((((((((

DecouplingApproximation for

Small Perturbations

Restrict the Nominal Flight Path

to the Vertical Plane

Nominal longitudinal equations reduce to

Nominal lateral-directional motions are zero

x1

x2

x3

x4

x5

x6

x7

x8

x9

x10

x11

x12

!

"

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

$

%

&&&&&&&&&&&&&&&&&&N

=xLon

xLat'Dir

!

"

##

$

%

&&N

=

uN

wN

xN

zN

qN

(N

0

0

0

0

0

0

!

"

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

$

%

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

!uN= X / m ! gsin"

N! q

NwN

!wN= Z / m + gcos"

N+ q

NuN

!xIN= cos"

N( )uN + sin"N( )wN!zIN= ! sin"

N( )uN + cos"N( )wN

!qN=M

Iyy

!"N= q

N

!xLat!Dir

N

= 0

xLat!Dir

N

= 0

Nominal State Vector

BUT, Lateral-directional perturbations need not bezero

!!xLat"Dir

N

# 0

!xLat"Dir

N

# 0

Restrict the Nominal Flight Path to

Steady, Level Flight

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

solution method

0 = X / m ! gsin"N! q

NwN

0 = Z / m + gcos"N+ q

NuN

VN= cos"

N( )uN + sin"N( )wN0 = ! sin"

N( )uN + cos"N( )wN

0 =M

Iyy

0 = qN

Trimmed State Vector isconstant

Specify nominal airspeed (VN) and altitude (hN = zN)

u

w

x

z

q

!

"

#

$$$$$$$

%

&

'''''''Trim

=

uTrim

wTrim

VN(t

zN

0

!Trim

"

#

$$$$$$$$

%

&

''''''''

Small Perturbation Effects are Uncoupled inSteady, Symmetric, Level Flight

Assume the airplane is symmetric and its nominal pathis steady, level flight

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 =FLon

0

0 FLat!Dir

"

#

$$

%

&

''

G =G

Lon0

0 GLat!Dir

"

#

$$

%

&

''

L =L

Lon0

0 LLat!Dir

"

#

$$

%

&

''

!!x

Lon(t) = F

Lon!x

Lon(t) +G

Lon!u

Lon(t) + L

Lon!w

Lon(t)

!xLon

=

!x1

!x2

!x3

!x4

!x5

!x6

"

#

$$$$$$$$

%

&

''''''''Lon

=

!u

!w

!x

!z

!q

!(

"

#

$$$$$$$

%

&

'''''''

(6 x 6) LTI LongitudinalPerturbation Model

!uLon

=

!"T

!"E

!"F

#

$

%%%

&

'

(((

!wLon =

!uwind

!wwind

!qwind

"

#

$$$

%

&

'''

(6 x 6) LTI Lateral-DirectionalPerturbation 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)

!xLat"Dir =

!x1

!x2

!x3

!x4

!x5

!x6

#

$

%%%%%%%%

&

'

((((((((Lat"Dir

=

!v!y

!p

!r!)

!*

#

$

%%%%%%%%

&

'

((((((((

!uLat"Dir =

!#A

!#R

!#SF

$

%

&&&

'

(

)))

!wLon =

!vwind

!pwind

!rwind

"

#

$$$

%

&

'''

Frequency DomainDescription of LTISystem Dynamics

Fourier Transform ofa Scalar Variable

Transformation from time domain to frequency domain

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

#$

$

% dt, " = frequency, rad / s

!x(t)

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

!x(t) : real variable

!x( j" ) : complex variable

= a(" ) + jb(" )

= A(" )e j# (" )

A : amplitude

! : phase angle

j! : Imaginary operator, rad/s

Laplace Transform of

a Scalar Variable Laplace transformation from time domain to frequency domain

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

0

#

$ dts = ! + j"

= Laplace (complex) operator, rad/s

Laplace transformation is a linear operation

L !x1(t) + !x

2(t)[ ] = L !x1(t)[ ] + L !x2 (t)[ ] = !x1(s) + !x2 (s)

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

!x(t) : real variable

!x(s) : complex variable

= a(s) + jb(s)

= A(s)e j" (s )

Laplace Transforms of

Vectors and Matrices Laplace transform of a vector variable

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

!x1(s)

!x2(s)

...

"

#

$$$

%

&

'''

Laplace transform of a matrix variable

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

a11(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 of

a 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 of

a Dynamic System

Rearrange Laplace transform of dynamic equation

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

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

!x(s) = sI " F[ ]"1!x(0) +G!u(s) + L!w(s)[ ]

Modes of Motion

Characteristic Polynomial

of a LTI Dynamic System

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

sI ! F(n x n)

Characteristic polynomial of the system

is a scalar

defines the system!s modes of motion

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

= sn+ a

n!1sn!1

+ ...+ a1s + a

0

!x(s) = sI " F[ ]"1!x(0) +G!u(s) + L!w(s)[ ]

Eigenvalues (or Roots) of

a Dynamic System

!(s) = sn+ a

n"1sn"1

+ ...+ a1s + a

0= 0

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

Characteristic equation of the system

... where !i are the eigenvalues of F or the rootsof the characteristic polynomial

Eigenvalues are complex numbers thatcan be plotted in the s plane

s Plane

!i= "

i+ j#

i

Complex conjugate

!*

i= "

i# j$

iPositive real part

represents instability

LongitudinalModes of Motion

!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$P%nPs +%nP2( ) s2 + 2$SP%nSP s +%nSP

2( ) = 0

LongitudinalModes of Motion

Eigenvalues determine the damping and natural frequenciesof 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 characteristic equationhas 6 eigenvalues

4 eigenvalues normally appear as 2complex pairs

Range and height modes usuallyinconsequential

Lateral-DirectionalModes of Motion

!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( ) s2+ 2$

DR%

nDRs +%

nDR

2( ) = 0

Lateral-DirectionalModes of Motion

Lateral-directional characteristicequation has 6 eigenvalues

2 eigenvalues normally appear as acomplex pair

Crossrange and heading modesusually inconsequential

!cr : crossrange mode " 0

!head :heading mode " 0

!S : spiral mode

!R : roll mode

#DR ,$nDR( ) :Dutch roll mode

Next Time:Longitudinal Dynamics

SupplementalMaterial

Sensitivity to Small Control and

Disturbance Perturbations

Control-effect matrix

G(t) =!f

!u x=xN (t )u=uN (t )w=wN (t )

=

! f1

!u1

! f1

!u2

...! f

1

!um

! f2

!u1

! f2

!u2

...! f

2

!um

... ... ... ...

! fn!u

1

! fn!u

2

...! fn

!um

"

#

$$$$$$$$

%

&

''''''''x=xN (t )u=uN (t )w=wN (t )

F(t) =!f

!x x=xN (t )u=u

N(t )

w=wN(t )

; G(t) =!f

!u x=xN (t )u=u

N(t )

w=wN(t )

; L(t) =!f

!w x=xN (t )u=u

N(t )

w=wN(t )

Disturbance-effect matrix

L(t) =!f

!w x=xN (t )u=uN (t )w=wN (t )

=

! f1

!w1

! f1

!w2

...! f

1

!ws

! f2

!w1

! f2

!w2

...! f

2

!ws

... ... ... ...

! fn!w

1

! fn!w

2

...! fn

!ws

"

#

$$$$$$$$

%

&

''''''''x=xN (t )u=uN (t )w=wN (t )

How Do We Calculate the

Partial Derivatives?

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

Analytically Symbolic evaluation of analytical

models of F, G, and L

F(t) =!f

!x x=xN (t )u=u

N(t )

w=wN(t )

G(t) =!f

!u x=xN (t )u=u

N(t )

w=wN(t )

L(t) =!f

!w x=xN (t )u=u

N(t )

w=wN(t )

Numerical Estimation of theJacobian Matrices

F(t) !

f1

x1+ "x

1( )

x2

!

xn

#

$

%%%%%

&

'

(((((

) f1

x1) "x

1( )

x2

!

xn

#

$

%%%%%

&

'

(((((

2"x1

f1

x1

x2+ "x

2( )

!

xn

#

$

%%%%%

&

'

(((((

) f1

x1

x2+ "x

2( )

!

xn

#

$

%%%%%

&

'

(((((

2"x2

!

f1

x1

x2

!

xn + "xn( )

#

$

%%%%%

&

'

(((((

) f1

x1

x2

!

xn + "xn( )

#

$

%%%%%

&

'

(((((

2"xn

f2

x1+ "x

1( )

x2

!

xn

#

$

%%%%%

&

'

(((((

) f2

x1) "x

1( )

x2

!

xn

#

$

%%%%%

&

'

(((((

2"x1

f2

x1

x2+ "x

2( )

!

xn

#

$

%%%%%

&

'

(((((

) f2

x1

x2+ "x

2( )

!

xn

#

$

%%%%%

&

'

(((((

2"x2

! !

! ! ! !

fn

x1+ "x

1( )

x2

!

xn

#

$

%%%%%

&

'

(((((

) fn

x1) "x

1( )

x2

!

xn

#

$

%%%%%

&

'

(((((

2"x1

! !

fn

x1

x2

!

xn + "xn( )

#

$

%%%%%

&

'

(((((

) fn

x1

x2

!

xn + "xn( )

#

$

%%%%%

&

'

(((((

2"xn

#

$

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

&

'

((((((((((((((((((((((((((x=xN (t)u=u

N(t)

w=wN(t)

Matrix Inverse

A[ ]!1=Adj A( )

A=Adj A( )

detA

(n " n)

(1 " 1)

=C

T

detA; C = matrix of cofactors

Cofactors are signed

minors of A

ijth minor of A is the

determinant of A with

the ith row and jth

column removed

y = Ax; x = A!1ydim(x) = dim(y) = (n !1)

dim(A) = (n ! n)

Matrix Inverse

A =a11

a12

a21

a22

!

"

##

$

%

&&; A

'1=

a22

'a21

'a12

a11

!

"

##

$

%

&&

T

a11a22

' a12a21

=

a22

'a12

'a21

a11

!

"

##

$

%

&&

a11a22

' a12a21

A =

a11

a12

a13

a21

a22

a23

a31

a32

a33

!

"

###

$

%

&&&; A

'1=

a22a33

' a23a32( ) ' a21a33 ' a23a31( ) a21a32 ' a22a31( )

' a12a33

' a13a32( ) a11a33 ' a13a31( ) ' a11a32 ' a12a31( )

a12a23

' a13a22( ) ' a11a23 ' a13a21( ) a11a22 ' a12a21( )

!

"

####

$

%

&&&&

T

a11a22a33+ a

12a23a31+ a

13a21a32

' a13a22a31

' a12a21a33

' a11a23a32

=

a22a33

' a23a32( ) ' a12a33 ' a13a32( ) a12a23 ' a13a22( )

' a21a33

' a23a31( ) a11a33 ' a13a31( ) ' a11a23 ' a13a21( )

a21a32

' a22a31( ) ' a11a32 ' a12a31( ) a11a22 ' a12a21( )

!

"

####

$

%

&&&&

a11a22a33+ a

12a23a31+ a

13a21a32

' a13a22a31

' a12a21a33

' a11a23a32

dim(A) = (2 ! 2)

dim(A) = (3! 3)

A = a; A!1=1

a

dim(A) = (1!1)

Sensitivity toDisturbance Inputs

Disturbance input vector and perturbation

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

L =L

LonL

Lat!DirLon

LLon

Lat!DirL

Lat!Dir

"

#

$$

%

&

''

Effects of longitudinal disturbanceson longitudinal motion

Effects of longitudinal disturbances

on lateral-directional motion

Effects of lateral-directional

disturbances on longitudinal motion

Effects of lateral-directional disturbances

on 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) =

!uw(t)

!ww(t)

!qw(t)

!vw(t)

!pw(t)

!rw(t)

"

#

$$$$$$$$

%

&

''''''''