Mathematical Modeling of Guided Weapons
for Autopilot Designing
Puneet Joshi1, S.Arora 2,
1
Department of Electrical Engineering, G.B. Pant University, Pantnagar-263145, India
2
Professor, Department of Electrical Engineering, G.B. Pant University, Pantnagar-263145, India
Abstract— This paper considers the derivation of mathematical
model for a missile autopilot in State Space form. The basic
model developed is highly nonlinear and is difficult to analyze
analytically, therefore Jacobian linearization technique is used
to linearize the model. A coupled Multi-Input-Multi-Output
(MIMO) model is derived suitable for both the application of
the modern control techniques as well as the classical timedomain and frequency domain techniques. The models
developed are useful for further research on precision
optimum guidance and control. The model will provide more
accurate presentations of missile autopilot dynamics and can
be used for adaptive and integrated guidance and control of
agile missiles.
Keywords-State Space Analysis, Euler’s Equations of Motion,
Closed Loop Simulation, Open loop Simulation
I. INTRODUCTION
T
HE Proliferation of Guided Weapons (missiles) is
increasing, and supporters of the concept believe
that these systems represent the future of air
warfare. Integrating the guidance and control
function is a synthesis approach that is being
pursued as it allows the optimization of the overall system
performance. This approach requires a more complete
representation of the airframe dynamics and the guidance
system. Therefore, Autopilot designing is one of the
essential functions for flight controls of these weapons. The
performance of the autopilot system directly affects the
performance and mission success of the weapons. The
motion of a missile in free flight is extremely complicated.
It contains three translation motions (vertical, horizontal,
and transverse), three rotational motions (pitch, yaw, and
roll) and numerous structural coupled elastic motions. To
reduce the complexity of analysis, we usually assume that
the missile is rigid-body and the missile’s motion consists of
small deviation from its equilibrium (trimmed) flight
condition. In addition, we assume that the missile equations
of motion can be separated into two groups, namely,
longitudinal equations and lateral equations.
Typically, for a missile autopilot, the input is the
demanded control surface deflection and outputs are the
achieved airframe (lateral) accelerations and body rates
measured about the body axes. Other input/output variables
(such as: the flight path angle and angle rate or the body
angles) can be derived directly from lateral accelerations
and body rates. This paper considers the derivation of the
mathematical model for a missile autopilot in state space
form. The basic equations defining the airframe dynamics
are non-linear, however, since the non-linearities are
“structured” (in the sense that the states are of quadratic
form) a novel approach of expressing this non-linear
dynamics in state space form is given. This should provide a
useful way to implement the equations in a computer
simulation program and possibly for future application of
non-linear analysis and synthesis techniques.
This paper also considers a locally linearised state space
model that lends itself to better known linear techniques of
the modern control theory. A coupled multi-input multioutput (MIMO) model is derived suitable for both the
application of the modern control techniques as well as the
classical time-domain and frequency domain techniques.
II. NOTATION AND CONVENTION
Conventions and notations for vehicle body axes systems
as well as the forces, moments and other quantities are
shown in Figure 1 and defined in Table I.
Figure 1. Motion variable notations
Mass of the missile is considered as constant and denoted by
m. The variables shown in figure 2.1 are defined as:
α- Incidence in the pitch plane
β- Incidence in the yaw plane
- Incidence plane angle.
- Total incidence, such that: tan α = tancos,
Aileron deflection: = 1 +2+3+4, if all four control
surfaces are active; or
1
1
 = (1+3) or  = (2+4) if only two surfaces are active.
and tanβ= tan sin, T - Thrust.
2
2
Positive control defection () causes negative roll.
1
Elevator deflection: = (1-3). Positive control deflection
2
(η) causes negative pitch.
1
Rudder deflection: = (2-4). Positive control deflection
2
(ζ) causes negative yaw.
III. EULER’S EQUATIONS OF MOTION
The dynamics of a missile motion are defined using
Euler’s equations of motion. The six equations of motion for
a body with six degree of freedom may be written as:
𝑚(𝑢̇ + 𝑤𝑞 − 𝑣𝑟) = 𝑋 + 𝑇 + 𝑔𝑥 𝑚
𝑚(𝑣̇ + 𝑢𝑟 − 𝑤𝑝) = 𝑌 + 𝑔𝑦 𝑚
𝑚(𝑤̇ + 𝑢𝑞 − 𝑣𝑝) = 𝑍 + 𝑔𝑥
(1)
Figure 2. Control surfaces seen from the rear of a missile
TABLE I.
MOTION VARIABLES
Vehicle Body Axes
System
Roll
axis X
Pitch
axis Y
Yaw
axis Z
The above set of equations (1) represent the force
equations of a generalized rigid body and describe the
translational motion of its centre of gravity(c.g.) since the
origin of the vehicle body axes is assumed to be co-located
with the body c.g.
p
q
r
Component of
vehicle velocity along
each axis
𝐼𝑥𝑥 𝑝̇ − (𝐼𝑦𝑦 − 𝐼𝑧𝑧 )𝑞𝑟 + 𝐼𝑦𝑧 (𝑟 2 − 𝑞2 ) − 𝐼𝑧𝑥 (𝑝𝑞 + 𝑟̇ ) + 𝐼𝑥𝑦 (𝑟𝑝 + 𝑞̇ ) = 𝐿
𝐼𝑦𝑦 𝑞̇ − (𝐼𝑧𝑧 − 𝐼𝑥𝑥 )𝑟𝑝 + 𝐼𝑧𝑧 (𝑝2 − 𝑟 2 ) − 𝐼𝑥𝑦 (𝑞𝑟 + 𝑝̇ ) + 𝐼𝑦𝑧 (𝑝𝑞 + 𝑟̇ ) = 𝑀
𝐼𝑧𝑧 𝑟̇ − (𝐼𝑥𝑥 − 𝐼𝑦𝑦 )𝑝𝑞 + 𝐼𝑥𝑦 (𝑞2 − 𝑝2 ) − 𝐼𝑦𝑧 (𝑟𝑝 + 𝑞̇ ) + 𝐼𝑧𝑥 (𝑞𝑟 + 𝑝̇ ) = 𝑁
u
v
w
Component of
aerodynamic forces
acting on vehicle along
each axis
X
Y
Z
(2)
The above equations set (2) represent the
momentum equations of a generalized rigid body and
describe the rotational motion about the body axes through
its c.g.
The set of equations (1) can be rewritten as:
Moments acting on
vehicle about each axis
L
M
N
Angular rates
Moments of inertia
about each axis
Ixx
Iyy
Izz
Products of each
inertia
Iyz
Izx
Ixy
Longitudinal and
lateral acceleration
ax
ay
az
Euler angles



Gravity along each
axis
gx
gy
gz
Vehicle thrust along
the body axis
T
Figure 2 defines the control surface convention.
Here the control surfaces are numbered as shown and the
deflections (1, 2, 3, 4) are taken to be positive if
clockwise, looking outwards along the individual hinge axis.
Thus:
uq
ur
gx
u
0 0
0 1 0 −1 vp
𝑋̌ + 𝑇̌
d
gy]
v ] = [0 −1
̌
+
+
[
]
[
]
[
0
0
1
0
Y
vr
dt
gz
w
1 0 −1 0 0 0 wp
𝑍̌
[wq]
(3)
Where,
̌=
X
X
Y
Z
T
̌ = ; Ž = ; T
̌= ;
; Y
m
m
m
m
And, similarly equation set (2), as:
𝑝2
𝑝𝑞
𝑝
𝐿
𝑝𝑟
𝑑
[𝐴] [𝑞] = [𝐵] 2 + [𝑀]
𝑑𝑡
𝑞
𝑟
𝑁
𝑞𝑟
[ 𝑞2 ]
(4)
Where,
Ixx
[A] = [−Ixy
−Izx
−Ixy
Iyy
−Iyz
−Izx
−Iyz ]
−Izz
&
0
[B] = [−Izx
Ixy
𝐼𝑧𝑥
−Iyz
−Ixy
Iyz
(Izz − Ixx ) 0
−Ixy
Iyz
(Ixx − Iyy )
[2]
[0]
[𝐼]
𝑥1
⋯ ⋯ ] [… . . .] + [… …
[[𝐴]−1[𝐵]] 𝑥2[2]
[0]
⋮
……
⋮
[1]
𝑔
[0]
𝑢1
⋯ ⋯ ] [… . . .] + [⋯]
0
[[𝐴]−1] 𝑢2[1]
⋮
……
⋮
(5)
Where,
0
[C] = [0
1
0
−1
0
0
0
−1
1 0
0 1
0 0
[1]
𝑢1
= [𝑋̌ + 𝑇̌
𝑌̌
𝑍̌]
𝑇
[1]
𝑢1
= [𝑋̌ + 𝑇̌
𝑌̌
𝑔𝑦
𝑔𝑧 ]𝑇
𝑇
[1]
𝑢2
= [𝐿
𝑀
𝑁
(6)
Where,
L̃ =
Ixx
L
Ixx
̃ =
, M
̃Iyy
M
Iyy
Izz
Izz
Izz
,
Equations (5) and (6) completely represent the non-linear
description of the full 6 DOF autopilot model. The state
variables of the matrix are the components of missile
velocities along each axis and angular body rates. The
equations are classified as the quadratic dynamic model.
IV. LINEARISING MODEL FOR A TWO-AXIS
SYMMETRICAL AIRFRAME
For linearization process it is assumed that X, Y, Z, L, M
and N are the functions of u, v, w, p, q, r, , and . Then
first order linearization about the nominal values of u 0, v0,
w0, p0, q0, r0, 0, 0 and 0 can be done and all the
aerodynamic derivatives are defined as:
̃u =
X
̃
∂X
∂u
̃v =
,X
̃
∂X
̃
∂X
∂v
∂w
̃w =
,X
̃
̃
̃
∂ξ
∂η
∂ζ
,
̃p =
X
̃
∂X
∂p
̃q =
,X
̃
∂X
̃
∂X
∂q
∂r
̃r =
, X
̃ ξ = ∂X, X
̃ η = ∂X, X
̃ ζ = ∂X,
X
̃
̃
̃
∂Y
∂u
∂v
∂w
∂Y
∂Y
̃
Yu = , ̃
Yv = , ̃
Yw =
̃ξ =
Y
,
̃
∂Y
̃
∂Y
̃
∂Y
∂ξ
∂η
∂ζ
̃η =
, Y
̃ζ =
,Y
̃
̃
̃
∂p
∂q
∂r
∂Y
∂Y
∂Y
̃
Yp = , ̃
Yq = , ̃
Yr = ,
,
̃
∂r
,
∂L
∂L
∂L
L̃p = , L̃q = , L̃r = ,
̃
̃
̃
∂L
∂v
̃
∂L
∂w
̃
∂L
L̃ξ =
∂ξ
∂η
, L̃η =
, L̃ζ =
̃
∂ζ
̃
̃
∂u
̃
∂M
∂v
̃
∂M
∂w
̃
∂M
∂ξ
∂η
∂ζ
∂u
̃
𝜕N
𝜕𝜉
̃𝜂 =
, N
̃ζ =
,M
̃
̃
∂N
∂v
̃
𝜕N
∂w
̃
𝜕N
𝜕𝜂
, N𝜁 =
̃
̃
∂q
∂r
,
̃ u = ∂M, M
̃ v = ∂M, M
̃ w = ∂M,
M
̃η =
, M
̃
∂p
𝜕𝜁
,
̃
̃
̃
∂p
∂q
∂r
̃ p = ∂M, M
̃ q = ∂L, M
̃ r = ∂M,
M
,
̃
̃
̃
∂p
∂q
∂r
̃ p = ∂N, N
̃ q = ∂N, N
̃ r = ∂N,
N
,
∆𝑢̇ = 𝑟0 ∆𝑣 + 𝑣0 ∆𝑟 − 𝑞0 ∆𝑤 − 𝑤0 ∆𝑞 + (𝑋̃𝑢 ∆𝑢 + 𝑋̃𝑣 ∆v + 𝑋̃𝑤 ∆w +
𝑋̃𝑝 ∆p + 𝑋̃𝑞 ∆q + 𝑋̃𝑟 ∆r + 𝑋̃𝜉 ∆ξ + 𝑋̃𝜂 ∆η + 𝑋̃𝜍 ∆ς)∆𝑇̃ + ∆𝑔𝑥
And the lateral equation of motion to:
∆𝑣 ̇ = 𝐼 ̃_𝑥𝑥 (𝑞_0 ∆𝑟 + 𝑟_0 ∆𝑞) + (𝐼 ̃_𝑢 ∆𝑢 + 𝐼 ̃_𝑣 ∆v + 𝐼 ̃_𝑤 ∆w +
𝐼 ̃_𝑝 ∆p + 𝐼 ̃_𝑞 ∆q + 𝐼 ̃_𝑟 ∆r + 𝐼 ̃_𝜉 ∆ξ + 𝐼 ̃_𝜂 ∆η + 𝐼 ̃_𝜍 ∆ς)
The rest of the equations can be changed likewise and can
be written in matrix notations as:
∆𝑢̇
∆𝑣̇
∆𝑤̇
∆𝑝̇
∆𝑞̇
[ ∆𝑟̇ ]
Ĩ
xx
−Iyz ]
0
0
pr qr]𝑇
[1]
̃ 𝑁
̃]𝑇
u2 = [𝐿̃ 𝑀
I −I
Izz −Ixx
̃
=
,
Ixx = xx yy,
N
̃=
, N
̃
∂q
∂Z
∂Z
∂Z
Z̃p = , Z̃q = , Z̃r = ,
The longitudinal equation of motion was changed to:
]𝑇
[2]
[2]
𝑔
𝑑
[𝑐] [0] 𝑥1
[𝐼] [0] 𝑢1
[… . . .] = [
][
][
]+[
]+[ ]
0
[0] [𝐻] 𝑥 [2]
[0] [𝐼] 𝑢[1]
𝑑𝑡 [1]
2
2
𝑥1
̃Ixx = Iyy −Izz,
̃
∂ζ
̃
[1]
𝑥1
0
Ĩ
yy
̃
∂η
̃ u = ∂N, N
̃ v = ∂N, N
̃w =
N
For a two-axis symmetrical airframe, Iyz = Izx = Ixy
= 0. These values when substituted in (3.3) and (3.4) and
the new matrix is obtained.
0
[H] = [ 0
Ĩ
zz
[2]
x2 = [pq
̃
∂ξ
̃
∂p
,
∂u
̃
∂L
̃ξ =
M
−1
0]
0
𝑍̌] ,
̃
∂Z
∂w
∂L
∂L
L̃u = , L̃v = , Z̃w =
̃𝜉 =
N
[1]
[1]
𝑥1 = [𝑢 𝑣 𝑤]𝑇
𝑥2 = [𝑝 𝑞 𝑟]𝑇
[21]
𝑤𝑝 𝑤𝑞]𝑇
𝑣𝑝
𝑣𝑟
𝑥1 = [𝑢𝑞 𝑢𝑟
[2]
2
2
𝑥2 = [𝑝 𝑝𝑞 𝑝𝑟 𝑞 𝑞𝑟 𝑟 2 ]𝑇 𝑔 = [𝑔𝑥
̃
∂v
∂Z
∂Z
∂Z
Z̃ξ = , Z̃η = , Z̃ζ = ,
The MIMO system formed by combining (3) and (4):
[1]
[𝑐]
𝑑 𝑥1
[ … . . .] = [ … …
𝑑𝑡 [1]
[0]
𝑥1
̃
∂u
∂Z
∂Z
Z̃u = , Z̃v = , Z̃w =
(Iyy − Izz ) −Iyz
𝐼𝑥𝑦
Izx ]
−Izx
0
,
𝑋̃𝑝
(−𝑤0 + 𝑋̃𝑞 )
(𝑣0 + 𝑋̃𝑟 )
𝑋̃𝑢
(𝑟0 + 𝑋̃𝑣 ) (−𝑞0 + 𝑋̃𝑤 )
̃
̃
̃
+
𝑌
(𝑤
)
𝑌
̃
(−𝑟0 + 𝑌𝑢 )
(−𝑢0 + 𝑌̃𝑟 )
0
𝑝
𝑞
𝑌𝑣
(𝑝0 + 𝑌̃𝑤 )
̃
̃
̃
̃
+
𝑍
(−𝑣
)
+
𝑍
(𝑢
)
𝑍̃𝑟
(𝑞0 + 𝑍𝑢 ) (−𝑝0 + 𝑍𝑣 )
0
𝑝
0
𝑞
𝑍̃𝑤
=
̃
̃
̃
̃
̃𝐿𝑢
̃𝐿𝑣
̃𝐿𝑤
𝐿̃𝑝
𝑞
(𝐼
(𝐼𝑥𝑥 𝑟0 + 𝐿𝑞 )
𝑥𝑥 0 + 𝐿𝑟
̃𝑢
̃𝑣
̃𝑤
̃
̃
̃
̃𝑟
̃
𝑀
𝑀
𝑀
𝑟
+
𝑀
(𝐼𝛼 0
𝑀𝑞
(𝐼𝑦𝑦 𝑞0 + 𝑀
𝑝)
̃
̃
̃
̃
̃
̃
̃
̃
𝑁
𝑁
𝑁
𝑁𝑟
(𝐼𝑧𝑧 𝑞0 + 𝑁𝑝 ) 𝐼𝑧𝑧 𝑝0 + 𝑁𝑞
𝑢
𝑣
𝑤
[
𝑋̃𝜉 𝑋̃𝜂 𝑋̃𝜍
∆𝑇̃ + ∆𝑔𝑥
𝑌̃𝜉 𝑌̃𝜂 𝑌̃𝜍
∆𝑔𝑦
∆𝜉
𝑍̃𝜉 𝑍̃𝜂 𝑍̃𝜍
∆𝑔𝑧
+
[∆𝜂] +
𝐿̃𝜉 𝐿̃𝜂 𝐿̃𝜍 ∆𝜍
0
0
̃𝜉 𝑀
̃𝜂 𝑀
̃𝜍
𝑀
[
]
0
̃
̃
̃
[ 𝑁𝜉 𝑁𝜂 𝑁𝜍 ]
The above equation can be represented in State Space form
as:
d
∆x = [F1 ]∆x + [G1 ]∆u1 + ∆w1
dt
Where,
∆u
∆𝑇̃ + ∆𝑔𝑥
∆v
∆𝑔𝑦
∆ε
∆w
∆𝑔𝑧
∆𝑥 =
, ∆u1 = [∆η] , ∆w1 =
∆p
0
∆𝜁
∆q
0
[ ∆r ]
[
]
0
[𝐹1 ]
=
𝑋̃𝑢
(−𝑟0 + 𝑌̃𝑢 )
(𝑞0 + 𝑍̃𝑢 )
(𝑟0 + 𝑋̃𝑣 )
𝑌̃𝑣
(−𝑝0 + 𝑍̃𝑣 )
𝐿̃𝑢
̃𝑢
𝑀
̃𝑢
𝑁
𝐿̃𝑣
̃𝑣
𝑀
̃𝑣
𝑁
[
(−𝑞0 + 𝑋̃𝑤 )
(𝑝0 + 𝑌̃𝑤 )
𝑍̃𝑤
𝐿̃𝑤
̃𝑤
𝑀
̃𝑤
𝑁
𝑋̃𝑝
(𝑤0 + 𝑌̃𝑝 )
(−𝑣0 + 𝑍̃𝑝 )
(−𝑤0 + 𝑋̃𝑞 )
𝑌̃𝑞
[𝐻1 ]
(𝑣0 + 𝑋̃𝑟 )
(−𝑢0 + 𝑌̃𝑟 )
(𝑢0 + 𝑍̃𝑞 )
𝑍̃𝑟
(𝐼̃𝑥𝑥 𝑟0 + 𝐿̃𝑞 ) (𝐼̃𝑥𝑥 𝑞0 + 𝐿̃𝑟 )
̃𝑝 )
̃𝑟 )
̃𝑞
(𝐼̃𝛼 𝑟0 + 𝑀
(𝐼̃𝑦𝑦 𝑞0 + 𝑀
𝑀
̃𝑟
̃𝑞
̃𝑝 ) 𝐼̃𝑧𝑧 𝑝0 + 𝑁
𝑁
(𝐼̃𝑧𝑧 𝑞0 + 𝑁
]
𝐿̃𝑝
𝑋̃𝜉
𝑌̃𝜉
𝑋̃𝜂
𝑌̃𝜂
𝑋̃𝜍
𝑌̃𝜍
𝑍̃𝜉
[𝐺1 ] =
𝐿̃𝜉
𝑍̃𝜂
𝐿̃𝜂
𝑍̃𝜍
𝐿̃𝜍
̃𝜉
𝑀
̃𝜉
[𝑁
̃𝜂
𝑀
̃𝜂
𝑁
̃𝜍
𝑀
̃𝜍 ]
𝑁
And,
𝐓 = 𝐓𝟎 + ∆𝐓, 𝐠 𝐱 = 𝐠 𝐱𝟎 + ∆𝐠 𝐱 , 𝐠 𝐲 = 𝐠 𝐲𝟎 + ∆𝐠 𝐲 ,
𝐠 𝐳𝟎 = 𝐠 𝐳 + ∆𝐠 𝐳 ,
V. INCORPORATION OF ACCELEROMETER AND
GYRO MEASUREMENT MODEL
Generally, not all state variables in the state equations are
accessible or measurable, thus it is needed to convert the
state equations. The commonly measurable variables, in
most missiles, are the angular rate components (roll rate-p,
pitch rate-q, and yaw rate-r) and the acceleration
components (ax, ay, az). It is assumed that gyros provided
ideal readings of angular rates:
pm=p qm=q rm=r, where pm , qm and rm are the measured
body rates. The errors due to drifts and noises may be
included in the above readings as additional additive terms.
In contrast to the readings of the angular rate components,
the readings of the acceleration components are dependent
on the location of the accelerometers, with respect to the c.g.
of the body, and therefore:
ax = u̇ + qW − rV − d𝑥 (q2 + r 2 ) + d𝑦 (pq − ṙ ) + d𝑧 (pr + q̇ )
ay = v̇ + ru − pw + d𝑥 (pq + ṙ ) − d𝑦 (p2 + r 2 ) + d𝑧 (qr − ṗ )
az = ẇ + pv − qu + d𝑥 (pr − q̇ ) + d𝑦 (qr + ṗ ) − d𝑧 (p2 + q2 )
The acceleration components are assumed to be measured
at a point O, where O is at a distance of d x, dy and dz from
the central point of gravity, along the x-, y- and z-axis,
respectively. Usually the accelerometers are mounted along
the x-axis and therefore, dy=0 and dz=0. And therefore the
above equations become:
𝑎_𝑥 = 𝑢 ̇ + 𝑞𝑤 − 𝑟𝑣 − 𝑑_𝑥 (𝑞^2 + 𝑟^2 )
= 𝑋 ̃ − 𝑇 ̃ + 𝑔 ̃ − 𝑑_𝑥 (𝑞^2 + 𝑟^2 )
𝑎𝑦 = 𝑣̇ + 𝑟𝑢 − 𝑝𝑤 + 𝑑𝑥 (𝑝𝑞 + 𝑟̇ ) = 𝑌̃ − 𝑇̃ + 𝑔𝑥 + 𝑑𝑥 (𝑝𝑞 + 𝑟̇ )
𝑎𝑧 = 𝑤̇ + 𝑞𝑤 − 𝑝𝑣 − 𝑞𝑢 + 𝑑𝑥 (𝑝𝑟 − 𝑞̇ ) = 𝑍̃ + 𝑔𝑧 + 𝑑𝑥 (𝑝𝑟 + 𝑞̇ )
Note that the right hand side of the above equations comes
directly from equation set (1). Linearising these equations
and using the relationship (7) gives us:
0
0
0
=
1
0
0
[
𝑋̃𝑢
𝑋̃𝑣
𝑋̃𝑤
𝑋̃𝑝
0
0
0
0
0
0
0
0
1
0
𝑋̃𝑞 − 2𝑞0 𝑑𝑥
𝑋̃𝑟 − 2𝑟0𝑑𝑥
0
1
T
∆y(t) = [∆p ∆q ∆r ∆ax ∆ay ∆az ]
, is the output vector
̃𝑢 𝑑 𝑥
𝑍̃𝑢 + 𝑀
̃𝑢 𝑑 𝑥
𝑍̃𝑣 + 𝑀
̃𝑤 𝑑 𝑥
𝑍̃𝑤 + 𝑀
̃𝑝 ) 𝑑 𝑥 ]
[𝑍̃𝑝 + (𝑟0 + 𝐼̃𝑦𝑦 𝑟0 − 𝑀
𝑍̃𝑞
𝑇
̃𝑟 ) 𝑑 𝑥 ] ]
[𝑍̃𝑟 + (𝑝0 + 𝐼̃𝑦𝑦 𝑝0 − 𝑀
[H1] is the state output matrix
0
0
0
̃ξ
X
[J1 ] =
0
0
0
̃η
X
0
0
0
̃𝜁
X
̃ξ + N
̃ ξ 𝑑𝑥
̃η + N
̃ η 𝑑𝑥 ̃
̃ 𝜁 𝑑𝑥
Y
Y
Y𝜁 + N
̃
̃
̃
̃
̃
̃ 𝜁 𝑑𝑥 ]
Z
−
M
𝑑
Z
−
M
𝑑
Z
−
M
[ ξ
ξ 𝑥
η
η 𝑥
𝜁
is the matrix related to inputs in the measurement matrix
∆v1 (t) = [0 0 0 ∆𝑇/𝑚 + ∆𝑔_(𝑥 ) ∆𝑔_𝑦 ∆𝑔_𝑧 ]𝑇 , is the disturbance
vector.
VI. ADDING FIN SERVOS TRANSFER FUNCTION
Assuming that the servo dynamics for the aileron, elevator
and rudder can be described adequately by a second order
lag as:
𝑘𝑠
∆
= 2
2𝜇𝑠 𝑠
∆𝑑
𝑠
+1
2 + 𝜔
𝜔𝑠
𝑠
𝑘𝑠η
∆η
= 2
2𝜇𝑠η 𝑠
∆η𝑑
𝑠
+1
2 + 𝜔
𝜔𝑠η
𝑠η
𝑘𝑠ζ
∆ζ
= 2
2𝜇𝑠ζ 𝑠
∆ζ𝑑
𝑠
+1
2 + 𝜔
𝜔𝑠ζ
𝑠ζ
Where,
∆d , ∆ηd and ∆ζd are the demand aileron, elevator
and rudder deflections respectively.
k s , k sη and k sζ are the servo gain for the aileron,
elevator and rudder respectively.
μs , 2μsη and 2μsζ are the damping factors for the
aileron, elevator and rudder respectively.
ωs , ωsη and ωsζ are the natural frequency for the
aileron, elevator and rudder respectively.
Above transfer functions can be converted into differential
equations as follows:
2
2 2
∆̈ = −𝜔𝑠
∆ − 2𝜇𝑠 𝜔𝑠 ∆̇ + 𝑘𝑠
𝜔𝑠 ∆ ,
𝑑
2
2
2
∆η̈ = −𝜔𝑠η
∆η − 2𝜇𝑠η 𝜔𝑠η ∆η̇ + 𝑘𝑠η
𝜔𝑠η
∆η𝑑 ,
2
2 2
∆ζ̈ = −𝜔𝑠ζ
∆ζ − 2𝜇𝑠ζ 𝜔𝑠ζ ∆ζ̇ + 𝑘𝑠ζ
𝜔𝑠ζ ∆ζ𝑑 ,
Hence, the state-space model for the autopilot of a missile
including the servos and airframe is:
∆ẋ 2 (t) = [A2 ]∆x2 (t) + [B2 ]∆u2 (t) + ∆w2 (t)
Where,
[𝑨𝟐 ]
[𝑭𝟏 ]
∆y = [H1 ]∆x(t) + [J1 ]∆u(t) + ∆v(t)
Where,
̃𝑢 𝑑𝑥
𝑌̃𝑢 + 𝑁
̃𝑣 𝑑𝑥
𝑌̃𝑣 + 𝑁
̃𝑤 𝑑𝑥
𝑌̃𝑤 + 𝑁
̃𝑝 ) 𝑑𝑥 ]
[𝑌̃𝑝 + (𝑞0 + 𝐼̃𝑧𝑧 𝑞0 + 𝑁
̃𝑞 ) 𝑑𝑥 ]
[𝑌̃𝑝 + (𝑝0 + 𝐼̃𝑧𝑧 𝑝0 + 𝑁
̃𝑢 𝑑𝑥
𝑌̃𝑟 + 𝑁
= ⋯
𝟎
𝟎
𝟎
𝟎
𝟎
[𝟎
⋯
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
⋯
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
⋯
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
⋯
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
⋯
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
⋮
⋮
⋮
⋮
⋮
⋮
⋯
⋯
𝟎
⋮
⋮
𝟎
⋮
𝟎
⋮ −𝝎𝟐𝝃
⋮
𝟎
⋮
𝟎
[𝑮𝟏 ]
⋯
⋯
𝟎
𝟎
𝟎
𝟎
−𝝎𝜼𝟐
𝟎
⋯
⋯
𝟎
𝟎
𝟎
−𝝎𝟐𝜻
⋯
𝟎
⋮
𝟎
⋮
𝟎
⋮
𝟎
⋮
𝟎
⋮
𝟎
⋮
⋯
⋯
𝟎
⋮
𝟎
⋮
𝟎
⋮
⋮ −𝟐𝝁𝒔𝝃 𝝎𝟐𝒔𝝃
⋮
𝟎
⋮
𝟎
⋯
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
⋯
𝟎
𝟎
𝟎
𝟎
−𝟐𝝁𝒔𝜼 𝝎𝟐𝒔𝜼
𝟎
⋯
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
⋯
𝟎
𝟎
𝟎
𝟎
𝟎
𝟐
−𝟐𝝁𝒔𝜻 𝝎𝒔𝜼
⋯
]
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
[B2 ] =
0
0
0
0
0
0
0
0
0
0
0
−k s ωs 2
0
−k sη ωsη 2
0
−k
ωsζ 2 ]
[
sζ
0
0
As the aileron, rudder and elevator deflection now became
the state variables the dimension of the state vector increase
to [12*1].
T
∆x2 (t) = [∆u ∆v ∆w ∆p ∆q ∆r ∆ ∆η ∆ζ ∆̇ ∆η̇ ∆ζ̇]
And the inputs now become the demanded aileron, rudder
and elevator deflection, as shown below
𝑇
∆𝑢 (𝑡) = [∆𝑑 ∆η𝑑 ζ𝑑 ]
T
∆T
∆w2 (t) = [ + ∆g x ∆g y ∆g z 0 0 0 0 0 0 0 0 0 ]
m
The output vector is represented by:
∆y2 (t) = [∆𝑝𝑚 ∆𝑞𝑚 ∆𝑟𝑚 ∆𝑎𝑥𝑚 ∆𝑎𝑦𝑚 ∆𝑎𝑧𝑚 ]
⋮
⋮
⋮
[H2 ] = [H1 ] [J1 ]
⋮
⋮
⋮
[
⋮0
⋮0
⋮0
⋮0
⋮0
⋮0
∆v2 (t) = [0 0 0 0 0 0 0 0 0
0
0
0
0
0
0
∆T
m
∆r = [∆pd ∆azd ∆ayd ]
T
For a case of lateral directional control, the control input
signal for the fin servos can be derived as follows:
∆ηd = ∆azd − K az ∆az − K gq ∆q
∆ζd = ∆ayd − K ay ∆ay − K gr ∆r
∆y2 (t) = H2 ∆x2 (t) + ∆v2 (t) ,
Where,
outputs) can be expressed as inputs to the gyros multiplied
by the gyro gains, Kgr, Kgp and Kgy, respectively. Similarly,
the measured longitudinal acceleration, ax, and lateral
accelerations, ay and az, are inputs to the accelerometers
multiplied by the accelerometer gains, Kax, Kay and Kaz,
respectively. The accelerometer gains affect the steady state
response and may be set to 1 for transient tests. Rescaling
accelerometer gains, after selecting gyro gains, allows a
unity gain autopilot to be designed.
The reference signals, generally used for testing the
transient time response of the autopilot, are the desired
accelerations in yaw direction, ayd, the pitch direction, azd,
and roll rate, pd. The reference roll rate was set as zero to
assess the missile dynamics in roll against spurious
disturbances. Hence, the reference vector, Δr, may be
written as:
0
0
0
0
0
0
0
0
0
0
0
0
T
0
0
0
0
0
0]
For sake of simplicity, pd was set to zero since this case only
considers the lateral directional control. As a result, the
control input vector becomes:
∆u (t) = ∆r(t) − K∆y2 (t),
Where, K is the feedback matrix:
0 0
K = [0 K gq
0 0
0 0 0
0 0 0
K gr 0 K ay
0
K az ]
0
T
+ ∆g x ∆g y ∆g z ]
Figure 3. block diagram of an open-loop autopilot system.
VII. DESIGNING LATERAL AUTOPILOT
For the case of small perturbation, it may be assumed that
(u0, v0, w0, p0, q0, r0) are identically zero. In this case, the
airframe model decouples into two lateral dynamics (pitch
and yaw) and one roll dynamics. Lateral autopilot dynamics
is considered to validate the model derived.
Figure 4. block diagram of a closed-loop auto-pilot system
Ignoring the instrument (gyro, accelerometer) dynamics, the
measured roll, pitch and yaw angular rates (the gyro
VIII. SIMULATION OF THE DEVELOPED MODEL
In order to verify the developed model, the statespace model was converted into transfer function form using
Matlab symbolic toolbox. Also the numeric values were
substituted into the state space of all the variables. Then
time analysis was done for the transfer functions obtained.
For this step input was given to rudder and elevator and both
open loop and closed loop simulations were performed. The
results obtained were found to be identical to those already
published [4].
IX. SIMULATION RESULTS
Figure 5 and 6 show the lateral accelerations of
the missile due to a step input to the rudder and elevator,
respectively, for an open loop simulation. As can be seen
from these figures, there are large steady state errors.
However, the steady state errors can be reduced with a
feedback loop as can been seen in Figures 7 and 8. These
simulation results are similar to the results presented in [4].
Figure 5 Open loop simulation: Lateral autopilot response to a step demand
acceleration of rudder
Figure 6 Open loop simulation: Lateral autopilot response to a step demand
acceleration of elevator.
Figure 7. Closed loop simulation: Lateral autopilot response to a step
demand acceleration.
Figure 8. Closed loop simulation: Lateral autopilot response to a step
demand acceleration.
X. CONCLUSIONS
Both the non-linear and linearised autopilot models
have been derived. The state-space model of a missile
autopilot was validated by comparing the model with the
other published results, and through both open and closedloop systems simulation. The non-linear dynamics model
presented as structural quadratic algebraic system is novel
and can be used for developed non-linear control techniques
suitable for missile systems high g-maneuvers and operating
of a range of aerodynamics conditions. The models
developed are useful for further research on precision
optimum guidance and control. It is hoped that the higher
order model with motion and inertial coupling will provide
more accurate representation of missile autopilot dynamics
and should be used for adaptive and integrated guidance and
control of agile missiles.
REFERENCES
[1]. Pallett, E.H.J., Automatic Flight Control, 3rd Edition, BSP
Professional Books, 1987.
[2]. Kuo, B.C., Automatic Control Systems, 6th Edition, Prentice-Hall
International Editions, 1991.
[3]. Blakelock, J.H., Automatic Control of Aircraft and Missiles, John
Wiley & Sons, Inc., 1965.
[4]. Faruqi, F.A., On the Algebraic Structure of a Class of Nonlinear
Dynamical Systems, DSTO report.
[5]. Cook, M.V., Flight Dynamics Principles, Arnold, 1997.
[6]. Babister, A.W., Aircraft Dynamic Stability and Response,
Pergamon, 1980.