linearization ch08
TRANSCRIPT
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Chapter 8
Linearization
8.1 General
We have derived twelve nonlinear, coupled, first order, ordinary differentialequations that describe the motion of rigid aircraft in a stationary atmo-sphere over a flat Earth: the equations of motion. Analytical solutions tothe equations of motion are obviously not forthcoming, so other means ofsolving them must be sought.
One such means is through numerical integration. There are several al-gorithms available that will allow the differential equations to be propagated
forward in discrete time steps. At the beginning of each time step the entireright-hand side of each equation x = . . . is evaluated, yielding the rate ofchange of x at that instant. Then, loosely, dx/dt is replaced by x/t andthe change x is approximated over the interval t. With sufficiently fastcomputers this can be done in real-time, and is the basis for flight simula-tion. Numerical integration will generate time-histories of the aircraft motionin response to arbitrary initial conditions and control inputs. These time-histories may then be analyzed to characterize the response in familiar termssuch 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) flight condition.The advantage to this approach is that for suitably small regions about thesteady condition, all the dependencies may be considered linear. This will re-sult in twelve linear (though still coupled) ordinary differential equations forwhich analytical solutions are available. The process is called linearization
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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. Linearizationbrings with it the assumption that all ensuing motion will be close to thereference steady-state flight condition. (This assumption is sometimes wan-tonly disregarded and the linearized equations are treated as if they werethe equations of motion.)
Because linearization requires us to stay close to the steady flight condi-tion, it is generally reasonable to neglect altitude effects in the equations ofmotion. Therefore, none of the three navigation equations need be consid-ered, nor does the kinematic equation for (or ). We will therefore focuson 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 differentialequation involves a pendulum of length , for which motion about = 0 isdescribed by
m = mg sin This second order equation may be replaced by two coupled first 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 beobtained from a power series representation of a function by retaining only
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8.3. NONLINEAR ORDINARY DIFFERENTIAL EQUATIONS 133
terms through the first 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 usefor the series expansion the Taylor series representations of the functions.For a function of a single variable, f(x), about some reference value xRef wehave
f(x) = f(xRef) + df(x)dx
Ref
x + d2
f(x)dx2
Ref
x2
2!+ d
3
f(x)dx3
Ref
x3
3!+
In this expression, x = xRef + x. So for example, f(x) = sin x aboutxRef = 0 yields
sin x = sin xRef + cos xRefx sin xRefx2/2! cos xRefx3/3! + = x x3/3! + x5/5!
Or, through the first order term, sin x x.A Taylor series for a function ofn independent variables, f(x1, x2, . . . , xn)
about reference conbditions (x1, x2, . . . , xn)Ref, through the first order termsis:
f(x1, x2, . . . , xn) f(x1, x2, . . . , xn)Ref+
f
x1
Ref
x1 +f
x2
Ref
x2 +
+f
xn
Ref
xn + H.O.T.
0
(8.1)
8.3 Nonlinear Ordinary Differential Equations
To be as general as possible we will define the function to be linearized bymoving everything to the left-hand side of the equation and equating it to
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134 CHAPTER 8. LINEARIZATION
zero. For example, the body axis kinematic equation for bank angle will be
defined as
f( ,,,p,q,r) p (qsin + r cos )tan 0There are two advantages to this approach. First, the function evaluates
to zero for any reference condition, and therefore the first term in the Taylorseries will always vanish. Second, even though we have solved the equationsof motion for the explicit derivatives of the variables, some of the forces andmoments may depend on these derivatives. By defining the functions asshown all occurences of such terms will be accounted for when derivativesare taken.
The reference conditions we will use will be those previously discussedin 7.4 for steady flight. Mathematically this is not necessary, but for ourpurposes it is. In any event the meaning of terms such as x should alwaysbe construed to mean (dx/dt) and not d(x)/dt or the reference conditionsof x will be lost, since
d(x)/dt = d(x xRef)/dt = dx/dtand on the other hand,
(dx/dt) = dx/dt (dx/dt)RefOf 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 variablesis as follows. First, three vectors x, x, and u are defined. The x vectorrepresents each of the variables that appear as derivatives, the x vector rep-
resents each of those variables as a derivative, and the u vector representsall 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 ordinarydifferential equations which are placed in the form fi(x, x, u) = 0, i = 1 . . . n.A vector-valued function f(x, x, u) is formed by considering the n functions
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8.4. SYSTEMS OF EQUATIONS 135
thus defined as a vector. We then write the Taylor series of f(x, x, u) through
the first 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 definition. Thederivative of the nvector of functions f(v) with respect to a pvector v isa Jacobian and is defined as
fv
f1/v1 f1/v2 f1/vpf2/v1 f2/v2 f2/vp
......
......
fn/v1 fn/v2 fn/vp
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 thevector x,
x =
f
x1
Ref
f
xRef
x + f
uRef
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 define
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136 CHAPTER 8. LINEARIZATION
x =
x1
x2
, u = {u1} {M}
Two scalar functions are defined to constitute the vector-valued functionf(x, x, u):
= 0 f1 (x, x, u) x1 x2 = 0 + g sin / M = 0 f2 (x, x, u) x2 + g sin x1/ u1 = 0
f(x, x, u)
f1 (x, x, u)
f2 (x, x, u)
=
x1 x2
x2 + g sin x1/ u1
=
0
0
The derivatives are calculated and then evaluated at reference conditionsxRef = xRef = 0 and MRef = 0 (from which also x = x, x = x, andu = u):
f
x f1/x1 f1/x2
f2/x1 f2/x2 = 1 0
0 1
f
x
f1/x1 f1/x2
f2/x1 f2/x2
=
0 1
g cos x1/ 0
,
f
x
Ref
=
0 1
g/ 0
f
u
f1/u1
f2/u1
=
0
1
The final result becomes
x =
0 1
g/ 0
x +
0
1
u
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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 setof linear, ordinary differential equations in x = Ax + Bu with its scalarcounterpart with forcing function u, x = ax + bu. Since solutions to thescalar equations are quite well known, it is reasonable to think that we canfind 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 be8 8 matrices and f/u will be an 8 4 matrix. However, in the formwe have derived the equations of motion, in which we have already solvedfor the explicit derivative term, the matrix f/x will be very nearly theidentity matrix (non-unity on the diagonal and non-zero off-diagonal entrieswill come from force and moment dependencies on , and possibly ).
It is far simpler to linearize each equation as a scalar function of severalvariables and deal with and dependencies as special cases. On the otherhand the vector-matrix form will be very useful later, so after treating eachequation 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 flight 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, andthe four generic controls T, , m, and n.
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138 CHAPTER 8. LINEARIZATION
Since we have picked a body axis system it is important to state which
one. The analysis is simplified somewhat by using stability axes. In that case,xS is the projection of the velocity vector in the reference flight condition ontothe plane of symmetry. Thus in the reference condition the angle betweenthe velocity vector and the xaxis is zero, we have
Ref = wRef = 0 (Stability axes)
This is true for any reference condition if stability axes are chosen. Forour reference flight condition we take steady, straight, symmetric flight. Asa consequence we have for reference conditions
uRef = vRef = wRef = pRef = qRef = rRef = Ref = Ref = 0
vRef = Ref = pRef = qRef = rRef = Ref = 0
Because vRef = wRef = 0, and because VRef =
u2Ref + v2Ref + w
2Ref, we
also have
uRef = VRef
We assume the speed and altitude at which the analysis is to be performedhave been specified, as has the climb (or descent) angle Ref. The use ofstability axes in steady, straight, symmetric flight also means that
Ref = Ref
In the force and moment dependencies we need the definitions
q =1
2V2
p =pb
2Vr =
rb
2V
q =qc
2V =
c
2V
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8.5. EXAMPLES 139
Because our linear velocity variables are u, v, and w, we also need, for
our force and moment dependencies,
V =
u2 + v2 + w2
= tan1 (w/u)
= sin1
v/
u2 + v2 + w2
In taking the partial derivatives we will apply the chain rule frequently,needing to evaluate in the end some partial derivatives repeatedly. Most arezero when evaluated in steady, straight, symmetric flight (see appendix B.1).
The derivatives are as follows:
Ref
V p q r q
u 1 0 0 0 0 0 0 RefVRef
v 0 1VRef 0 0 0 0 0 0
w 0 0 1VRef 0 0 0 0 0
p 0 0 0 b2VRef
0 0 0 0
q 0 0 0 0 c2VRef
0 0 0
r 0 0 0 0 0b
2VRef 0 0 0 0 0 0 0 0 c
2VRef0
We will encounter derivatives relating to thrust and velocity in the forceequations. The basic relationship is
T = qSCT
V , T
Recall that the functional dependency ofCT on V was introduced to allow
for the fact that thrust itself may sometimes be assumed to be invariantwith airspeed (e.g., rockets and jets). An analogous result obtains for certainengines assumed to have constant thrust-horsepower, modeled as a constantproduct T V. The derivations for CTV for both cases are at appendix B.1,and result in:
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140 CHAPTER 8. LINEARIZATION
CTV = 2CTRef (Constant Thrust)CTV = 3CTRef (Constant Thrust-Horsepower)
We will meet a few other nondimensional groupings in the process oflinearizing the equations of motion:
t tc/ (2VRef)
(Time)
D() c
2VRef
d()
dt (Differentiation)
m mSc/2
(Mass)
Iyy IyyS(c/2)3
(Moments of Inertia)
Ixx IxxS(b/2)3
Izz IzzS(b/2)3
Ixz IxzS(b/2)3
A b/c (Aspect ratio)
We will adopt the following convention to represent partial derivatives offorces and moments with respect to their independent variables, evaluatedat reference flight conditions. IfX is any force or moment that is a functionof y, then:
Xy XyRef
This notation is by no means standard; see below, 8.6 Customs andConventions.
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8.5. EXAMPLES 141
8.5.2 A Kinematic Equation
To complete the linearization of the function previously defined,
f( ,,,p,q,r) p (qsin + r cos )tan 0
The linearization proceeds using equation 8.1 as
f = fRef +f
Ref
+f
Ref
+ + fr
Ref
r
Since fRef = 0 by definition, and since all the s except are measuredfrom zero reference conditions, we have the linearized equation,
f = (0) + (1) + [(qcos + r sin )tan ]Ref+ [(qsin r cos )sec2 ]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,
=2VRef
b
pb
2VRef+
rb
2VReftan Ref
=
2VRefb
(p + r tan Ref)
This may be written as
d/dt2VRef/b
= p + r tan Ref
The term in parentheses represents one way to nondimensionalize theoperator d/dt. In fact, Etkin in the second edition of Dynamics of Flight Stability and Control did exactly that, dividing time by b/2VRef in the
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142 CHAPTER 8. LINEARIZATION
lateral-directional and by c/2VRef in the longitudinal equations. Here we have
adopted the Etkins earlier definition (inDynamics of Atmospheric Flight
) ofnondimensionalizing time by the divisor c/2VRef. This introduces the aspectratio A into the equation, since
d/dt
2VRef/b
=
d/dt
2VRef/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 (multipliedthrough by ID),
ID p = Izz [L + Ixzpq (Izz Iyy) qr] + Ixz [N Ixzqr (Iyy Ixx)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 toget all the dependencies down to either states, controls, or constants. Sincethe aerodynamic data are normally available in coefficient form, we also needto relate to those. The rolling and yawing moments are:
L = qSbC (, p, r, , n)
N = qSbCn (, p, r, , n)
The only states that do not appear are and , and the two controls onwhich we have dependencies are and n. We therefore define the functionfor linearization as
fp ( p, u, v, w, p, q, r, , n) ID p IzzL IxzN+ Ixz (Ixx + Iyy Izz)pq+
Izz (Iyy + Izz) I2xz
qr = 0
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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). Wehave already determined derivatives of q, , p, and r with respect to u andw, and in particular only q/u is non-zero in the reference flight condition.However, that term will be multiplied by CRef, which is zero (LRef = 0). Weconclude therefore that when evaluated in steady, straight, symmetric flight,
fp/u = fp/w = 0
The other partial derivatives go as follows:
fpp
Ref
= ID = IxxIzz I2xzfpv
Ref
= IzzLv IxzNvfpp
Ref
= IzzLp IxzNp (Using qRef = 0)
fpq
Ref
= 0 (Using pRef = rRef = 0)
fprRef
= IzzLr IxzNr (Using qRef = 0)fp
Ref
= IzzL IxzNfpn
Ref
= IzzLn IxzNn
We may now write out the linearization of the rolling moment equationin dimensional form as follows:
p = 1ID
[(IzzLv + IxzNv) v + (IzzLp + IxzNp)p
+ (IzzLr + IxzNr) r + (IzzL + IxzN)
+ (IzzLn + IxzNn) n]
(8.5)
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144 CHAPTER 8. LINEARIZATION
If the stability axes coincide with the principal axes this simplifies to
p =1
Ixp(Lvv + Lpp + Lrr + L + Lnn) (Principal Axes)
The partial derivatives of the moments are in dimensional form. Wemay at this point take the linearized equation as-is, and evaluate each of thefactors on the right-hand side at some given altitude and speed. Alternativelywe may re-write the equation using the nondimensional derivatives. Therequired derivatives are straightforward; for example, we have:
Lv = Lv
Ref
=[qSbC (, p, r, , n)]
v
Ref
= Sb
C
q
v+ q
C
v+
Cp
p
v+
Cr
r
v
Ref
=
qRefSb
VRef
C
In Lv, CRef = 0, p/v|Ref = r/v
|Ref = 0, and /v
|Ref = 1/VRef.
The end result of derivatives of L and N with respect to the states andcontrols is that the non-zero expressions are:
Ref
L N
vqRefSb
VRef
C
qRefSb
VRef
Cn
pqRefSb
2
2VRef
Cp
qRefSb
2
2VRef
Cnp
r
qRefSb
2
2VRef Cr
qRefSb
2
2VRef Cnr
(qRefSb) C (qRefSb) Cnn (qRefSb) Cn (qRefSb) Cnn
(8.6)
When related to the nondimensional coefficients we have (see appendixB.2):
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8.5. EXAMPLES 145
D(p) = 1AID
IzzC + IxzCn
+
IzzCp + IxzCnp
p
+
IzzCr + IxzCnr
r +
IzzC + IxzCn
+
IzzCn + IxzCnn
n (8.7)
In this expression A is the aspect ratio and ID = IxxIzz I2xz. If thestability axes coincide with the principal axes,
D(p) =1
AIxp C+ Cp p + Cr r + C + Cnn (Principal Axes)8.5.4 A Force Equation
For this example we take the body-axis Z-force equation,
w =1
m(Z+ Tsin T) + g cos cos + qu pv
Define the function
fw ( w,u,v,w,p,q,r,,, T, m, n) = w 1m
(Z+ Tsin T)
g cos cos qu + pvSince 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, andn arise through the force dependencies, which are
T = qSCT V , TZ = qSCZ
The latter is technically complicated by the mixed system of wind- andbody-axis forces (see equation 6.1). It may be shown, however, that lineariz-ing the correct relationship for CZ in a symmetric flight conditions, yields
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146 CHAPTER 8. LINEARIZATION
the same result as linearizing the simplified relationship (equation 6.2) in the
same flight condition. That is, the linearization of
CZ = CD sin sec CY sin tan CL cos
leaves no terms involving lateral-directional independent variables (v, p, r,, , or n) when evaluated at the reference flight condition, and gets thesame result as setting = 0 and linearizing
CZ =
CD sin
CL cos
In general, however, one should never apply reference conditions prior totaking the derivatives.
The remaining derivatives evaluate as follows:
fww
Ref
= 1 Zw/m
fwu
Ref
= (Zu + Tu sin T) /m (Using qRef = 0)
fww
Ref
= Zw/m (Using V/w = 0 in CT)
fwq
Ref
= Zq/m VRef (Using uRef = VRef)
fw
Ref
= g sin Ref (Using cos Ref = 1)
fwT
Ref
= TT sin T/m
fwmRef
= Zm/m
The dimensional form of the linearized equation is then
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8.6. CUSTOMS AND CONVENTIONS 147
w = 1m Zw [(Zu + Tu sin T) u + Zww + (Zq + mVRef) q
mg sin Ref + TT sin TT + Zmm](8.8)
Nondimensional derivatives are evaluated at appendix B.3, and are:
Ref
Z T
w qRefSc
2V2Ref
CL 0
u qRefSVRef 2CLRef MRefCLM qRefS
VRef 2CTRef + CTVw
qRefS
VRef
CDRef + CL
0
q qRefSc
2VRef
CLq 0
m (qRefS) CLm 0T 0 (qRefS) CTT
(8.9)
The nondimensional form of the equation (see B.4) becomes
D() =1
2m + CL (2CW cos Ref + CTV sin T MRefCLM) V CDRef + CL + 2m CLq q CW sin Ref
+CTT sin TT CLmm (8.10)
8.6 Customs and Conventions
8.6.1 Omission of.
In the linearized equations it is customary to drop the symbol whetherthe reference conditions are zero or not. In general any linear differential
equation appearing in flight dynamics will have been obtained through thelinearization of more complicated equations, and it may be assumed that allthe variables are small perturbations from some reference. After the equationsare solved it is important to remember to add the reference values to the sobtained in order to get the actual values.
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148 CHAPTER 8. LINEARIZATION
8.6.2 Dimensional Derivatives.
The convention we adopted for dimensional derivatives is consistent withEtkins usage, in which
Xy Xy
Ref
Several authors use the same notation, but include in the definition di-vision by mass or some moment of inertia. Thus one frequently sees, forexample,
Zw Z/wm
Ref
, Lp L/pIxx
Ref
,
In the force equations, and in the pitching moment equation, such nota-tion marginally simplifies the expressions. However, in the rolling and yawingequations, 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 modification to the mass or moment of inertia factor ofthe derivative term in the linearized equation. Thus we had the expression(m Zw) w instead of simply mw in the Zforce equation. Such terms as Zware often referred to as added mass parameters (they are usually negative) forobvious reasons. In ship dynamics several such terms arise, and the intuitivedescription usually offered is that the terms represent the mass of water beingdisplaced by the ships motion.
8.7 The Linear Equations
It is important to remember that the linearization was performed at a partic-ular reference flight condition: steady, straight, and symmetric. If any otherflight condition is to be analyzed the linearization will have to be performedover again.
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8.7. THE LINEAR EQUATIONS 149
8.7.1 Linear Equations
Dimensional Longitudinal Equations
u =1
m[(Xu + Tu cos T) u + Xww mg cos Ref
+TT cos TT + Xmm] (8.11a)
w =1
m Zw [(Zu + Tu sin T) u + Zww + (Zq + mVRef) q
mg sin Ref + TT sin TT + Zmm] (8.11b)
q =1
Iyy
Mu +
Mw (Zu + Tu sin T)
m Zw
u +
Mw +
MwZwm Zw
w
+
Mq +
Mw (Zq + mVRef)
m Zw
q
mgMw sin Refm Zw
+
MwTT sin T
m Zw
T +
Mm +
MwZmm Zw
m
(8.11c)
= q (8.11d)
Dimensional Lateral-Directional Equations
v =1
m[Yvv + Ypp + (Yr mVRef) r + mg cos Ref + Ynn] (8.12a)
p = 1ID
[(IzzLv + IxzNv) v + (IzzLp + IxzNp)p
+ (IzzLr + IxzNr) r + (IzzL + IxzN)
+ (IzzLn + IxzNn) n] (8.12b)
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150 CHAPTER 8. LINEARIZATION
r = 1ID
[(IxzLv + IxxNv) v + (IxzLp + IxxNp)p
+ (IxzLr + IxxNr) r + (IxzL + IxxN)
+ (IxzLn + IxxNn) n] (8.12c)
= p + r tan Ref (8.12d)
Nondimensional Longitudinal Equations
D(V) = 12m
(2CW sin Ref MRefCDM + CTV cos T) V
+
CLRef CD
CW cos Ref+CTTT CDmm
(8.13a)
D() =1
2m + CL
(2CW cos Ref + CTV sin T MRefCLM) V
CDRef + CL
+
2m CLq
q CW sin Ref
+sin T
CTT
T
CLm
m (8.13b)
D(q) =1
Iyy
MRefCmM
+Cm (2CW cos Ref + CTV sin T MRefCLM)
2m + CL
V
+
Cm
Cm
CDRef + CL
2m + CL
+
Cmq +
Cm
2m CLq
2m + CL
q
CWCm sin Ref2m + CL
+
Cmm CmCLm2m + CL
m
(8.13c)
D() = q (8.13d)
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8.7. THE LINEAR EQUATIONS 151
Nondimensional Lateral-Directional Equations
D() =1
2m
CY+ CYp p + (CYr 2m/A) r
+CW cos Ref + CYnn
(8.14a)
D(p) =1
AID
IzzC + IxzCn
+
IzzCp + IxzCnp
p
+
IzzCr + IxzCnr
r +
IzzC + IxzCn
+
IzzCn + IxzCnn
n
(8.14b)
D(r) =1
AID
IxzC + IxxCn
+
IxzCp + IxxCnp
p
+
IxzCr + IxxCnr
r +
IxzC + IxxCn
+
IxzCn + IxxCnn
n
(8.14c)
D() = 1A (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 notethat equations involving derivatives of longitudinal states are functions onlyof longitudinal states and controls, and that equations involving derivativesof lateral/directional states are functions only of lateral/directional statesand controls. That is, for the dimensional equations, we have differentialequations for u, w, q, and that are functions of the states u, w, q, and
the controls T and m. Similarly, the differential equations for v, p, r, and are functions of the states v, p, r, , and the controls and n. The sameholds for the nondimensional equations except the states are V, , q, and for the longitudinal equations, and , p, r, and for the lateral/directionalequations.
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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 atonce, we may break them up into two sets of four equations each. Thereforedefine the longitudinal state and control vectors xLong and uLong, and thelateral/directional state and control vectors xLD and uLD:
xLong
u
w
q
, uLong
T
m
xLD
v
p
r
, uLD
n
Likewise we have the nondimensional states xLong and xLD:
xLong
V
q
xLD
p
r
Control deflections are normally defined as non-dimensional quantities(throttle from zero to one, and radians for flapping surfaces, for example),so there is no need for separate definitions of uLong or uLD.
Further assumptions. For our subsequent purposes it is sufficient to fur-ther simplify the linear equations of motion with four assumptions:
1. The aircraft is in steady, straight, symmetric, level flight (SSSLF), orRef = 0.
2. The engine thrust-line is aligned with xB, so T = 0.
3. The body-fixed coordinate system is aligned with the principal axes,so Ixz = 0.
4. There are no aerodynamic dependencies on (or w).
While these assumptions are not necessary, they do permit us to focus onthe more significant effects typically seen in the study of aircraft dynamicsand control.
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8.7. THE LINEAR EQUATIONS 153
Dimensional Longitudinal Equations (SSSLF, T = 0)
xLong = ALongxLong + BLonguLong (8.15a)
xLong
u
w
q
, uLong
T
m
(8.15b)
ALong =
Xu+Tum
Xwm 0 g
Zum
Zwm
Zq+mVRefm 0
MuIyy
MwIyy
MqIyy
0
0 0 1 0
(8.15c)
BLong =
TTm
Xmm
0Zmm
0 MmIyy0 0
(8.15d)
Nondimensional Longitudinal Equations (SSSLF, T = 0)
xLong = ALongxLong + BLonguLong (8.16a)
xLong
V
q
(8.16b)
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154 CHAPTER 8. LINEARIZATION
ALong =
MRefCDM+CTV
2mCLRefCD
2m 0 CW
2m2CWMRefCLM
2m
CDRefCL
2m
2mCLq2m
0MRefCmM
Iyy
CmIyy
Cmq
Iyy0
0 0 1 0
(8.16c)
BLong =
CTT2m
CDm2m
0CLm
2m
0CmmIyy
0 0
(8.16d)
Dimensional Lateral-Directional Equations (SSSLF,Ixz = 0)
xLD = ALDxLD + BLDuLD (8.17a)
xLD
v
p
r
, uLD
n (8.17b)
ALD =
Yvm
Ypm
YrmVRefm
gLvIxx
LpIxx
LrIxx
0NvIzz
NpIzz
NrIzz
0
0 1 0 0
(8.17c)
BLD =
0
Ynm
LIxx
LnIxx
NIzz
NnIzz
0 0
(8.17d)
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8.7. THE LINEAR EQUATIONS 155
Nondimensional Lateral-Directional Equations (SSSLF, Ixz = 0)
xLD = ALDxLD + BLDuLD (8.18a)
xLD
p
r
(8.18b)
ALD =
Cy2m
Cyp2m
Cyr2m/A
2mCW2m
C
AIxx
Cp
AIxx
CrAIxx
0Cn
AIzz
Cnp
AIzz
CnrAIzz
0
0 1A
0 0
(8.18c)
BLD =
0
CYn2m
CIxx
CnIxx
CnIzz
CnnIzz
0 0
(8.18d)
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156 CHAPTER 8. LINEARIZATION