Chapter 8
Linearization
8.1 General
We have derived twelve nonlinear, coupled, rst order, ordinary dierential
equations that describe the motion of rigid aircraft in a stationary atmo-
sphere over a at Earth: the equations of motion. Analytical solutions to
the equations of motion are obviously not forthcoming, so other means of
solving them must be sought.
One such means is through numerical integration. There are several al-
gorithms available that will allow the dierential equations to be propagated
forward in discrete time steps. At the beginning of each time step the entire
right-hand side of each equation x = . . . is evaluated, yielding the rate of
change of x at that instant. Then, loosely, dx/dt is replaced by x/t and
the change x is approximated over the interval t. With suciently fast
computers this can be done in real-time, and is the basis for ight simula-
tion. Numerical integration will generate time-histories of the aircraft motion
in response to arbitrary initial conditions and control inputs. These time-
histories may then be analyzed to characterize the response in familiar terms
such as its frequency and damping.
Alternatively, one may consider small inputs and variations in initial con-
ditions applied at and about some reference (usually steady) ight condition.
The advantage to this approach is that for suitably small regions about the
steady condition, all the dependencies may be considered linear. This will re-
sult in twelve linear (though still coupled) ordinary dierential equations for
which analytical solutions are available. The process is called linearization
131
132 CHAPTER 8. LINEARIZATION
of the equations.
Numerical integration does not require special inititial conditions or con-
trol inputs; it simply approximates what happens in real life. Linearization
brings with it the assumption that all ensuing motion will be close to the
reference steady-state ight condition. (This assumption is sometimes wan-
tonly disregarded and the linearized equations are treated as if they were
the equations of motion.)
Because linearization requires us to stay close to the steady ight condi-
tion, it is generally reasonable to neglect altitude eects in the equations of
motion. Therefore, none of the three navigation equations need be consid-
ered, nor does the kinematic equation for
(or ). We will therefore focus
on the six motion variables and the remaining two Euler angles for linear
analysis.
8.2 Taylor Series
A familiar example of the linearization of a nonlinear ordinary dierential
equation involves a pendulum of length , for which motion about = 0 is
described by
m
= mg sin
This second order equation may be replaced by two coupled rst order
equations by introducing the angular velocity variable , whence
=
= g sin /
The only nonlinearity is in the term sin , which is normally linearized by
applying the small angle approximation, sin , so
=
= g/
More formally, such results as the small angle approximation may be
obtained from a power series representation of a function by retaining only
8.3. NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS 133
terms through the rst power of the variables. Thus the standard series
expansion of sin about = 0 is
sin =
3
/3! +
5
/5!
sin
In order to deal with functions of several variables it is convenient to use
for the series expansion the Taylor series representations of the functions.
For a function of a single variable, f(x), about some reference value x
Ref
we
have
f(x) = f(x
Ref
) +
df(x)
dx
Ref
x +
d
2
f(x)
dx
2
Ref
x
2
2!
+
d
3
f(x)
dx
3
Ref
x
3
3!
+
In this expression, x = x
Ref
+ x. So for example, f(x) = sin x about
x
Ref
= 0 yields
sin x = sin x
Ref
+ cos x
Ref
x sin x
Ref
x
2
/2! cos x
Ref
x
3
/3! +
= x x
3
/3! + x
5
/5!
Or, through the rst order term, sin x x.
A Taylor series for a function of n independent variables, f(x
1
, x
2
, . . . , x
n
)
about reference conbditions (x
1
, x
2
, . . . , x
n
)
Ref
, through the rst order terms
is:
f(x
1
, x
2
, . . . , x
n
) f(x
1
, x
2
, . . . , x
n
)
Ref
+
f
x
1
Ref
x
1
+
f
x
2
Ref
x
2
+
+
f
x
n
Ref
x
n
+ H.O.T.
0
(8.1)
8.3 Nonlinear Ordinary Dierential Equations
To be as general as possible we will dene the function to be linearized by
moving everything to the left-hand side of the equation and equating it to
134 CHAPTER 8. LINEARIZATION
zero. For example, the body axis kinematic equation for bank angle will be
dened as
f
(
, , , p, q, r)
p (q sin + r cos ) tan 0
There are two advantages to this approach. First, the function evaluates
to zero for any reference condition, and therefore the rst term in the Taylor
series will always vanish. Second, even though we have solved the equations
of motion for the explicit derivatives of the variables, some of the forces and
moments may depend on these derivatives. By dening the functions as
shown all occurences of such terms will be accounted for when derivatives
are taken.
The reference conditions we will use will be those previously discussed
in 7.4 for steady ight. Mathematically this is not necessary, but for our
purposes it is. In any event the meaning of terms such as x should always
be construed to mean (dx/dt) and not d(x)/dt or the reference conditions
of x will be lost, since
d(x)/dt = d(x x
Ref
)/dt = dx/dt
and on the other hand,
(dx/dt) = dx/dt (dx/dt)
Ref
Of course, if (dx/dt)
Ref
= 0 the two are the same.
8.4 Systems of Equations
The formal procedure for dealing with several equations in several variables
is as follows. First, three vectors x, x, and u are dened. The x vector
represents each of the variables that appear as derivatives, the x vector rep-
resents each of those variables as a derivative, and the u vector represents
all of the controls. If n state variables and m controls are being considered,
then x and x are nvectors and u is an mvector. There will be n ordinary
dierential equations which are placed in the form f
i
( x, x, u) = 0, i = 1 . . . n.
A vector-valued function f( x, x, u) is formed by considering the n functions
8.4. SYSTEMS OF EQUATIONS 135
thus dened as a vector. We then write the Taylor series of f( x, x, u) through
the rst order terms,
f( x, x, u) =
f
x
Ref
x +
f
x
Ref
x +
f
u
Ref
u = 0
Here we have used f( x, x, u) = 0 and f( x, x, u)
Ref
= 0 by denition. The
derivative of the nvector of functions f(v) with respect to a pvector v is
a Jacobian and is dened as
f
v
_
_
f
1
/v
1
f
1
/v
2
f
1
/v
p
f
2
/v
1
f
2
/v
2
f
2
/v
p
.
.
.
.
.
.
.
.
.
.
.
.
f
n
/v
1
f
n
/v
2
f
n
/v
p
_
_
Thus, f/x and f/ x are square nn matrices and f/u is an nm
matrix. The matrix f/ x is generally non-singular. We then solve for the
vector x,
x =
_
f
x
_
1
Ref
_
_
f
x
_
Ref
x +
_
f
u
_
Ref
u
_
(8.2a)
This equation is often written
x = Ax + Bu (8.2b)
With the obvious meaning of A and B.
To see how this works, consider our pendulum problem, to which we add
an externally applied torque M to be used as a control,
=
= g sin / + M
The two derivative terms are
and (n = 2) and the single control is M
(m = 1). We therefore dene
136 CHAPTER 8. LINEARIZATION
x =
_
x
1
x
2
_
_
, u = {u
1
} {M}
Two scalar functions are dened to constitute the vector-valued function
f( x, x, u):
= 0 f
1
( x, x, u) x
1
x
2
= 0
+ g sin / M = 0 f
2
( x, x, u) x
2
+ g sin x
1
/ u
1
= 0
f( x, x, u)
_
f
1
( x, x, u)
f
2
( x, x, u)
_
=
_
x
1
x
2
x
2
+ g sin x
1
/ u
1
_
=
_
0
0
_
The derivatives are calculated and then evaluated at reference conditions
x
Ref
= x
Ref
= 0 and M
Ref
= 0 (from which also x = x, x = x, and
u = u):
f
x
_
f
1
/ x
1
f
1
/ x
2
f
2
/ x
1
f
2
/ x
2
_
=
_
1 0
0 1
_
f
x
_
f
1
/x
1
f
1
/x
2
f
2
/x
1
f
2
/x
2
_
=
_
0 1
g cos x
1
/ 0
_
,
f
x
Ref
=
_
0 1
g/ 0
_
f
u
_
f
1
/u
1
f
2
/u
1
_
=
_
0
1
_
The nal result becomes
x =
_
0 1
g/ 0
_
x +
_
0
1
_
u
8.5. EXAMPLES 137
This is identical to the two linearized scalar equations
=
= (g/) + M
The motivation for this approach proceeds from the similarity of the set
of linear, ordinary dierential equations in x = Ax+Bu with its scalar
counterpart with forcing function u, x = ax + bu. Since solutions to the
scalar equations are quite well known, it is reasonable to think that we can
nd similar solutions to the systems of equations.
The problem with this approach from our point of view is that we have
eight equations with four controls to linearize, so f/ x and f/x will be
8 8 matrices and f/u will be an 8 4 matrix. However, in the form
we have derived the equations of motion, in which we have already solved
for the explicit derivative term, the matrix f/ x will be very nearly the
identity matrix (non-unity on the diagonal and non-zero o-diagonal entries
will come from force and moment dependencies on , and possibly
).
It is far simpler to linearize each equation as a scalar function of several
variables and deal with and
dependencies as special cases. On the other
hand the vector-matrix form will be very useful later, so after treating each
equation as a scalar problem, we will assemble them into the form of x =
Ax + Bu.
8.5 Examples
8.5.1 General
In order to proceed we need to specify two things:
1. The state and control variables to be used, and
2. The reference ight condition at which the equations are to be evalu-
ated.
For purposes of discussion we will select the body axis velocities u, v, w,
p, q, and r plus the two body axis Euler angles and for our states, and
the four generic controls
T
,
,
m
, and
n
.
138 CHAPTER 8. LINEARIZATION
Since we have picked a body axis system it is important to state which
one. The analysis is simplied somewhat by using stability axes. In that case,
x
S
is the projection of the velocity vector in the reference ight condition onto
the plane of symmetry. Thus in the reference condition the angle between
the velocity vector and the xaxis is zero, we have
Ref
= w
Ref
= 0 (Stability axes)
This is true for any reference condition if stability axes are chosen. For
our reference ight condition we take steady, straight, symmetric ight. As
a consequence we have for reference conditions
u
Ref
= v
Ref
= w
Ref
= p
Ref
= q
Ref
= r
Ref
=
Ref
=
Ref
= 0
v
Ref
=
Ref
= p
Ref
= q
Ref
= r
Ref
=
Ref
= 0
Because v
Ref
= w
Ref
= 0, and because V
Ref
=
_
u
2
Ref
+ v
2
Ref
+ w
2
Ref
, we
also have
u
Ref
= V
Ref
We assume the speed and altitude at which the analysis is to be performed
have been specied, as has the climb (or descent) angle
Ref
. The use of
stability axes in steady, straight, symmetric ight also means that
Ref
=
Ref
In the force and moment dependencies we need the denitions
q =
1
2
V
2
p =
pb
2V
r =
rb
2V
q =
q c
2V
=
c
2V
8.5. EXAMPLES 139
Because our linear velocity variables are u, v, and w, we also need, for
our force and moment dependencies,
V =
u
2
+ v
2
+ w
2
= tan
1
(w/u)
= sin
1
_
v/
u
2
+ v
2
+ w
2
_
In taking the partial derivatives we will apply the chain rule frequently,
needing to evaluate in the end some partial derivatives repeatedly. Most are
zero when evaluated in steady, straight, symmetric ight (see appendix B.1).
The derivatives are as follows:
Ref
V p q r
q
u 1 0 0 0 0 0 0
Ref
V
Ref
v 0
1
V
Ref
0 0 0 0 0 0
w 0 0
1
V
Ref
0 0 0 0 0
p 0 0 0
b
2V
Ref
0 0 0 0
q 0 0 0 0
c
2V
Ref
0 0 0
r 0 0 0 0 0
b
2V
Ref
0 0
0 0 0 0 0 0
c
2V
Ref
0
We will encounter derivatives relating to thrust and velocity in the force
equations. The basic relationship is
T = qSC
T
_
V ,
T
_
Recall that the functional dependency of C
T
on
V was introduced to allow
for the fact that thrust itself may sometimes be assumed to be invariant
with airspeed (e.g., rockets and jets). An analogous result obtains for certain
engines assumed to have constant thrust-horsepower, modeled as a constant
product TV . The derivations for C
T
V
for both cases are at appendix B.1,
and result in:
140 CHAPTER 8. LINEARIZATION
C
T
V
= 2C
T
Ref
(Constant Thrust)
C
T
V
= 3C
T
Ref
(Constant Thrust-Horsepower)
We will meet a few other nondimensional groupings in the process of
linearizing the equations of motion:
t
t
c/ (2V
Ref
)
(Time)
D()
c
2V
Ref
d()
dt
(Dierentiation)
m
m
S c/2
(Mass)
I
yy
I
yy
S ( c/2)
3
(Moments of Inertia)
I
xx
I
xx
S (b/2)
3
I
zz
I
zz
S (b/2)
3
I
xz
I
xz
S (b/2)
3
A b/ c (Aspect ratio)
We will adopt the following convention to represent partial derivatives of
forces and moments with respect to their independent variables, evaluated
at reference ight conditions. If X is any force or moment that is a function
of y, then:
X
y
X
y
Ref
This notation is by no means standard; see below, 8.6 Customs and
Conventions.
8.5. EXAMPLES 141
8.5.2 A Kinematic Equation
To complete the linearization of the function previously dened,
f
(
, , , p, q, r)
p (q sin + r cos ) tan 0
The linearization proceeds using equation 8.1 as
f
= f
Ref
+
f
Ref
+
f
Ref
+ +
f
Ref
r
Since f
Ref
= 0 by denition, and since all the s except are measured
from zero reference conditions, we have the linearized equation,
f
= (0) + (1)
+ [(q cos + r sin ) tan ]
Ref
+[(q sin r cos ) sec
2
]
Ref
(1)p (sin tan )
Ref
q (cos tan )
Ref
r
Evaluating the reference conditions yields
= p + r tan
Ref
(8.3)
In terms of the nondimensional roll and yaw rates,
=
2V
Ref
b
_
pb
2V
Ref
+
rb
2V
Ref
tan
Ref
_
=
2V
Ref
b
( p + r tan
Ref
)
This may be written as
_
d/dt
2V
Ref
/b
_
= p + r tan
Ref
The term in parentheses represents one way to nondimensionalize the
operator d/dt. In fact, Etkin in the second edition of Dynamics of Flight
Stability and Control did exactly that, dividing time by b/2V
Ref
in the
142 CHAPTER 8. LINEARIZATION
lateral-directional and by c/2V
Ref
in the longitudinal equations. Here we have
adopted the Etkins earlier denition (in Dynamics of Atmospheric Flight) of
nondimensionalizing time by the divisor c/2V
Ref
. This introduces the aspect
ratio A into the equation, since
_
d/dt
2V
Ref
/b
_
=
_
d/dt
2V
Ref
/ c
__
b
c
_
= AD()
The completely nondimensional form of the bank angle equation is there-
fore
D() =
1
A
( p + r tan
Ref
) (8.4)
8.5.3 A Moment Equation
For this example we consider the body-axis rolling equation (multiplied
through by I
D
),
I
D
p = I
zz
[L + I
xz
pq (I
zz
I
yy
) qr] + I
xz
[N I
xz
qr (I
yy
I
xx
) pq]
Part of the linearization is easy: the explicit p, q, and r dependencies pose
no problem. The rolling and yawing moments are problematic. We need to
get all the dependencies down to either states, controls, or constants. Since
the aerodynamic data are normally available in coecient form, we also need
to relate to those. The rolling and yawing moments are:
L = qSbC
(, p, r,
,
n
)
N = qSbC
n
(, p, r,
,
n
)
The only states that do not appear are and , and the two controls on
which we have dependencies are
and
n
. We therefore dene the function
for linearization as
f
p
( p, u, v, w, p, q, r,
,
n
) I
D
p I
zz
L I
xz
N
+ I
xz
(I
xx
+ I
yy
I
zz
) pq +
_
I
zz
(I
yy
+ I
zz
) I
2
xz
qr = 0
8.5. EXAMPLES 143
Before beginning the derivatives, we note that dependencies on u and w
appear only through the moments L and N (either through q, , p, or r). We
have already determined derivatives of q, , p, and r with respect to u and
w, and in particular only q/u is non-zero in the reference ight condition.
However, that term will be multiplied by C
Ref
, which is zero (L
Ref
= 0). We
conclude therefore that when evaluated in steady, straight, symmetric ight,
f
p
/u = f
p
/w = 0
The other partial derivatives go as follows:
f
p
p
Ref
= I
D
= I
xx
I
zz
I
2
xz
f
p
v
Ref
= I
zz
L
v
I
xz
N
v
f
p
p
Ref
= I
zz
L
p
I
xz
N
p
(Using q
Ref
= 0)
f
p
q
Ref
= 0 (Using p
Ref
= r
Ref
= 0)
f
p
r
Ref
= I
zz
L
r
I
xz
N
r
(Using q
Ref
= 0)
f
p
Ref
= I
zz
L
I
xz
N
f
p
Ref
= I
zz
L
n
I
xz
N
n
We may now write out the linearization of the rolling moment equation
in dimensional form as follows:
p =
1
I
D
[(I
zz
L
v
+ I
xz
N
v
) v + (I
zz
L
p
+ I
xz
N
p
) p
+(I
zz
L
r
+ I
xz
N
r
) r + (I
zz
L
+ I
xz
N
+ (I
zz
L
n
+ I
xz
N
n
)
n
]
(8.5)
144 CHAPTER 8. LINEARIZATION
If the stability axes coincide with the principal axes this simplies to
p =
1
I
xp
(L
v
v + L
p
p + L
r
r + L
+ L
n
n
) (Principal Axes)
The partial derivatives of the moments are in dimensional form. We
may at this point take the linearized equation as-is, and evaluate each of the
factors on the right-hand side at some given altitude and speed. Alternatively
we may re-write the equation using the nondimensional derivatives. The
required derivatives are straightforward; for example, we have:
L
v
=
L
v
Ref
=
[ qSbC
(, p, r,
,
n
)]
v
Ref
= Sb
_
C
q
v
+ q
_
C
v
+
C
p
p
v
+
C
r
r
v
__
Ref
=
_
q
Ref
Sb
V
Ref
_
C
In L
v
, C
Ref
= 0, p/v|
Ref
= r/v|
Ref
= 0, and /v|
Ref
= 1/V
Ref
.
The end result of derivatives of L and N with respect to the states and
controls is that the non-zero expressions are:
Ref
L N
v
_
q
Ref
Sb
V
Ref
_
C
_
q
Ref
Sb
V
Ref
_
C
n
p
_
q
Ref
Sb
2
2V
Ref
_
C
p
_
q
Ref
Sb
2
2V
Ref
_
C
np
r
_
q
Ref
Sb
2
2V
Ref
_
C
r
_
q
Ref
Sb
2
2V
Ref
_
C
nr
( q
Ref
Sb) C
( q
Ref
Sb) C
n
n
( q
Ref
Sb) C
n
( q
Ref
Sb) C
n
n
(8.6)
When related to the nondimensional coecients we have (see appendix
B.2):
8.5. EXAMPLES 145
D( p) =
1
A
I
D
__
I
zz
C
+
I
xz
C
n
_
+
_
I
zz
C
p
+
I
xz
C
np
_
p
+
_
I
zz
C
r
+
I
xz
C
nr
_
r +
_
I
zz
C
+
I
xz
C
n
+
_
I
zz
C
n
+
I
xz
C
n
n
_
n
_
(8.7)
In this expression A is the aspect ratio and
I
D
=
I
xx
I
zz
I
2
xz
. If the
stability axes coincide with the principal axes,
D( p) =
1
A
I
xp
_
C
+ C
p
p + C
r
r + C
+ C
n
_
(Principal Axes)
8.5.4 A Force Equation
For this example we take the body-axis Z-force equation,
w =
1
m
(Z + T sin
T
) + g cos cos + qu pv
Dene the function
f
w
( w, u, v, w, p, q, r, , ,
T
,
m
,
n
) = w
1
m
(Z + T sin
T
)
g cos cos qu + pv
Since this is a longitudinal equation, the presence of dependencies on
lateral-directional independent variables v, p, r, , and
n
is not desired.
The dependencies on v and are explicit. The dependencies on v, p, r, and
n
arise through the force dependencies, which are
T = qSC
T
_
V ,
T
_
Z = qSC
Z
The latter is technically complicated by the mixed system of wind- and
body-axis forces (see equation 6.1). It may be shown, however, that lineariz-
ing the correct relationship for C
Z
in a symmetric ight conditions, yields
146 CHAPTER 8. LINEARIZATION
the same result as linearizing the simplied relationship (equation 6.2) in the
same ight condition. That is, the linearization of
C
Z
= C
D
sin sec C
Y
sin tan C
L
cos
leaves no terms involving lateral-directional independent variables (v, p, r,
,
, or
n
) when evaluated at the reference ight condition, and gets the
same result as setting = 0 and linearizing
C
Z
= C
D
sin C
L
cos
In general, however, one should never apply reference conditions prior to
taking the derivatives.
The remaining derivatives evaluate as follows:
f
w
w
Ref
= 1 Z
w
/m
f
w
u
Ref
= (Z
u
+ T
u
sin
T
) /m (Using q
Ref
= 0)
f
w
w
Ref
= Z
w
/m (Using V/w = 0 in C
T
)
f
w
q
Ref
= Z
q
/mV
Ref
(Using u
Ref
= V
Ref
)
f
w
Ref
= g sin
Ref
(Using cos
Ref
= 1)
f
w
Ref
= T
T
sin
T
/m
f
w
Ref
= Z
m
/m
The dimensional form of the linearized equation is then
8.6. CUSTOMS AND CONVENTIONS 147
w =
1
mZ
w
[(Z
u
+ T
u
sin
T
) u + Z
w
w + (Z
q
+ mV
Ref
) q
mg sin
Ref
+ T
T
sin
T
T
+ Z
m
m
]
(8.8)
Nondimensional derivatives are evaluated at appendix B.3, and are:
Ref
Z T
w
_
q
Ref
S c
2V
2
Ref
_
C
L
0
u
_
q
Ref
S
V
Ref
_
_
2C
L
Ref
M
Ref
C
L
M
_
_
q
Ref
S
V
Ref
_
_
2C
T
Ref
+ C
T
V
_
w
_
q
Ref
S
V
Ref
_
_
C
D
Ref
+ C
L
_
0
q
_
q
Ref
S c
2V
Ref
_
C
Lq
0
m
( q
Ref
S) C
L
m
0
T
0 ( q
Ref
S) C
T
T
(8.9)
The nondimensional form of the equation (see B.4) becomes
D() =
1
2 m + C
L
_
(2C
W
cos
Ref
+ C
T
V
sin
T
M
Ref
C
L
M
)
_
C
D
Ref
+ C
L
_
+
_
2 mC
Lq
_
q C
W
sin
Ref
+C
T
T
sin
T
T
C
L
m
m
_
(8.10)
8.6 Customs and Conventions
8.6.1 Omission of .
In the linearized equations it is customary to drop the symbol whether
the reference conditions are zero or not. In general any linear dierential
equation appearing in ight dynamics will have been obtained through the
linearization of more complicated equations, and it may be assumed that all
the variables are small perturbations from some reference. After the equations
are solved it is important to remember to add the reference values to the s
obtained in order to get the actual values.
148 CHAPTER 8. LINEARIZATION
8.6.2 Dimensional Derivatives.
The convention we adopted for dimensional derivatives is consistent with
Etkins usage, in which
X
y
X
y
Ref
Several authors use the same notation, but include in the denition di-
vision by mass or some moment of inertia. Thus one frequently sees, for
example,
Z
w
Z/w
m
Ref
, L
p
L/p
I
xx
Ref
,
In the force equations, and in the pitching moment equation, such nota-
tion marginally simplies the expressions. However, in the rolling and yawing
equations, unless the analysis is performed in principal axes, the notation ac-
tually complicates the resulting expressions.
8.6.3 Added Mass.
Any time a force or moment is dependent upon a state-rate (such as or
) the result is a modication to the mass or moment of inertia factor of
the derivative term in the linearized equation. Thus we had the expression
(mZ
w
) w instead of simply m w in the Zforce equation. Such terms as Z
w
are often referred to as added mass parameters (they are usually negative) for
obvious reasons. In ship dynamics several such terms arise, and the intuitive
description usually oered is that the terms represent the mass of water being
displaced by the ships motion.
8.7 The Linear Equations
It is important to remember that the linearization was performed at a partic-
ular reference ight condition: steady, straight, and symmetric. If any other
ight condition is to be analyzed the linearization will have to be performed
over again.
8.7. THE LINEAR EQUATIONS 149
8.7.1 Linear Equations
Dimensional Longitudinal Equations
u =
1
m
[(X
u
+ T
u
cos
T
) u + X
w
w mg cos
Ref
+T
T
cos
T
T
+ X
m
m
] (8.11a)
w =
1
mZ
w
[(Z
u
+ T
u
sin
T
) u + Z
w
w + (Z
q
+ mV
Ref
) q
mg sin
Ref
+ T
T
sin
T
T
+ Z
m
m
] (8.11b)
q =
1
I
yy
__
M
u
+
M
w
(Z
u
+ T
u
sin
T
)
mZ
w
_
u +
_
M
w
+
M
w
Z
w
mZ
w
_
w
+
_
M
q
+
M
w
(Z
q
+ mV
Ref
)
mZ
w
_
q
_
mgM
w
sin
Ref
mZ
w
_
+
_
M
w
T
T
sin
T
mZ
w
_
T
+
_
M
m
+
M
w
Z
m
mZ
w
_
m
_
(8.11c)
= q (8.11d)
Dimensional Lateral-Directional Equations
v =
1
m
[Y
v
v + Y
p
p + (Y
r
mV
Ref
) r + mg cos
Ref
+ Y
n
n
] (8.12a)
p =
1
I
D
[(I
zz
L
v
+ I
xz
N
v
) v + (I
zz
L
p
+ I
xz
N
p
) p
+ (I
zz
L
r
+ I
xz
N
r
) r + (I
zz
L
+ I
xz
N
+(I
zz
L
n
+ I
xz
N
n
)
n
] (8.12b)
150 CHAPTER 8. LINEARIZATION
r =
1
I
D
[(I
xz
L
v
+ I
xx
N
v
) v + (I
xz
L
p
+ I
xx
N
p
) p
+ (I
xz
L
r
+ I
xx
N
r
) r + (I
xz
L
+ I
xx
N
+(I
xz
L
n
+ I
xx
N
n
)
n
] (8.12c)
= p + r tan
Ref
(8.12d)
Nondimensional Longitudinal Equations
D(
V ) =
1
2 m
_
(2C
W
sin
Ref
M
Ref
C
D
M
+ C
T
V
cos
T
)
V
+
_
C
L
Ref
C
D
_
C
W
cos
Ref
+C
T
T
C
D
m
m
_
(8.13a)
D() =
1
2 m + C
L
_
(2C
W
cos
Ref
+ C
T
V
sin
T
M
Ref
C
L
M
)
V
_
C
D
Ref
+ C
L
_
+
_
2 mC
Lq
_
q C
W
sin
Ref
+sin
T
C
T
T
C
L
m
m
_
(8.13b)
D( q) =
1
I
yy
_
_
M
Ref
C
m
M
+
C
m
(2C
W
cos
Ref
+ C
T
V
sin
T
M
Ref
C
L
M
)
2 m + C
L
_
V
+
_
C
m
C
m
_
C
D
Ref
+ C
L
_
2 m + C
L
_
+
_
C
mq
+
C
m
_
2 mC
Lq
_
2 m + C
L
_
q
C
W
C
m
sin
Ref
2 m + C
L
+
_
C
m
m
C
m
C
L
m
2 m + C
L
_
m
_
(8.13c)
D() = q (8.13d)
8.7. THE LINEAR EQUATIONS 151
Nondimensional Lateral-Directional Equations
D() =
1
2 m
_
C
Y
+ C
Yp
p + (C
Yr
2 m/A) r
+C
W
cos
Ref
+ C
Y
n
(8.14a)
D( p) =
1
A
I
D
__
I
zz
C
+
I
xz
C
n
_
+
_
I
zz
C
p
+
I
xz
C
np
_
p
+
_
I
zz
C
r
+
I
xz
C
nr
_
r +
_
I
zz
C
+
I
xz
C
n
+
_
I
zz
C
n
+
I
xz
C
n
n
_
n
_
(8.14b)
D( r) =
1
A
I
D
__
I
xz
C
+
I
xx
C
n
_
+
_
I
xz
C
p
+
I
xx
C
np
_
p
+
_
I
xz
C
r
+
I
xx
C
nr
_
r +
_
I
xz
C
+
I
xx
C
n
+
_
I
xz
C
n
+
I
xx
C
n
n
_
n
_
(8.14c)
D() =
1
A
( p + r tan
Ref
) (8.14d)
8.7.2 Matrix Forms of the Linear Equations
Before we place these equations in the form x = Ax + Bu, we note
that equations involving derivatives of longitudinal states are functions only
of longitudinal states and controls, and that equations involving derivatives
of lateral/directional states are functions only of lateral/directional states
and controls. That is, for the dimensional equations, we have dierential
equations for u, w, q, and
that are functions of the states u, w, q, and
the controls
T
and
m
. Similarly, the dierential equations for v, p, r, and
are functions of the states v, p, r, , and the controls
and
n
. The same
holds for the nondimensional equations except the states are
V , , q, and
for the longitudinal equations, and , p, r, and for the lateral/directional
equations.
152 CHAPTER 8. LINEARIZATION
Thus the longitudinal equations are decoupled from the lateral/directional
equations, and vice-versa. Instead of dealing with eight equations all at
once, we may break them up into two sets of four equations each. Therefore
dene the longitudinal state and control vectors x
Long
and u
Long
, and the
lateral/directional state and control vectors x
LD
and u
LD
:
x
Long
_
_
u
w
q
_
, u
Long
_
m
_
x
LD
_
_
v
p
r
_
, u
LD
_
n
_
Likewise we have the nondimensional states x
Long
and x
LD
:
x
Long
_
_
x
LD
_
p
r
_
Control deections are normally dened as non-dimensional quantities
(throttle from zero to one, and radians for apping surfaces, for example),
so there is no need for separate denitions of u
Long
or u
LD
.
Further assumptions. For our subsequent purposes it is sucient to fur-
ther simplify the linear equations of motion with four assumptions:
1. The aircraft is in steady, straight, symmetric, level ight (SSSLF), or
Ref
= 0.
2. The engine thrust-line is aligned with x
B
, so
T
= 0.
3. The body-xed coordinate system is aligned with the principal axes,
so I
xz
= 0.
4. There are no aerodynamic dependencies on (or w).
While these assumptions are not necessary, they do permit us to focus on
the more signicant eects typically seen in the study of aircraft dynamics
and control.
8.7. THE LINEAR EQUATIONS 153
Dimensional Longitudinal Equations (SSSLF,
T
= 0)
x
Long
= A
Long
x
Long
+ B
Long
u
Long
(8.15a)
x
Long
_
_
u
w
q
_
, u
Long
_
m
_
(8.15b)
A
Long
=
_
_
Xu+Tu
m
Xw
m
0 g
Zu
m
Zw
m
Zq+mV
Ref
m
0
Mu
Iyy
Mw
Iyy
Mq
Iyy
0
0 0 1 0
_
_
(8.15c)
B
Long
=
_
_
T
T
m
X
m
m
0
Z
m
m
0
M
m
Iyy
0 0
_
_
(8.15d)
Nondimensional Longitudinal Equations (SSSLF,
T
= 0)
x
Long
=
A
Long
x
Long
+
B
Long
u
Long
(8.16a)
x
Long
_
_
(8.16b)
154 CHAPTER 8. LINEARIZATION
A
Long
=
_
_
M
Ref
C
D
M
+C
T
V
2 m
C
L
Ref
C
D
2 m
0
C
W
2 m
2C
W
M
Ref
C
L
M
2 m
C
D
Ref
C
L
2 m
2 mC
Lq
2 m
0
M
Ref
Cm
M
Iyy
Cm
Iyy
Cmq
Iyy
0
0 0 1 0
_
_
(8.16c)
B
Long
=
_
_
C
T
T
2 m
C
D
m
2 m
0
C
L
m
2 m
0
Cm
m
Iyy
0 0
_
_
(8.16d)
Dimensional Lateral-Directional Equations (SSSLF,I
xz
= 0)
x
LD
= A
LD
x
LD
+ B
LD
u
LD
(8.17a)
x
LD
_
_
v
p
r
_
, u
LD
_
n
_
(8.17b)
A
LD
=
_
_
Yv
m
Yp
m
YrmV
Ref
m
g
Lv
Ixx
Lp
Ixx
Lr
Ixx
0
Nv
Izz
Np
Izz
Nr
Izz
0
0 1 0 0
_
_
(8.17c)
B
LD
=
_
_
0
Y
n
m
L
Ixx
L
n
Ixx
N
Izz
N
n
Izz
0 0
_
_
(8.17d)
8.7. THE LINEAR EQUATIONS 155
Nondimensional Lateral-Directional Equations (SSSLF, I
xz
= 0)
x
LD
=
A
LD
x
LD
+
B
LD
u
LD
(8.18a)
x
LD
_
p
r
_
(8.18b)
A
LD
=
_
_
Cy
2 m
Cyp
2 m
Cyr
2 m/A
2 m
C
W
2 m
C
Ixx
C
p
A
Ixx
C
r
A
Ixx
0
Cn
Izz
Cnp
A
Izz
Cnr
A
Izz
0
0
1
A
0 0
_
_
(8.18c)
B
LD
=
_
_
0
C
Y
n
2 m
C
Ixx
C
Ixx
Cn
Izz
Cn
n
Izz
0 0
_
_
(8.18d)
156 CHAPTER 8. LINEARIZATION
