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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)
"
#
$$$$$$$$
%
&
''''''''