Missile Longitudinal Autopilot Design Using Backstepping Approach

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Nov 16, 2013 (4 years and 1 month ago)

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1
Missile Longitudinal Autopilot Design
Using Backstepping Approach
FAN Jun-fang
1
, SU Zhong
1, 2

1. Beijing Information Science & Technology University
Beijing, China, 100101
wyhffjf@gmail.com
2. Beijing SinsTek Co., Ltd.
No.2 building, JIA 1#, Sanqindi, Wujiacun Rd., Fengtai District, Beijing, China, 100040
xingjian@bjxj.net

Abstract—The tactical missile autopilot design process is
detailed from a backstepping control perspective. Wherein,
two autopilot topologies are proposed, i.e. the angle of
attack (AOA) autopilot and acceleration autopilot. The
nonlinear missile longitudinal dynamics is dealt with firstly
to meet the strict feedback form. Control parameters of
AOA autopilot are introduced in turn and required to be
positive real numbers during the recursive process, however,
act with some combination form in the final law. Thus a set
of new parameters is presented to simplify the expression
and disclose the conservatism of the aforementioned
autopilot design. The results show that the positive real
requirement on AOA autopilot parameters during step by
step design has an unfavorable effect on closed loop system
performance. An acceleration autopilot as a tracking
problem is then set up and developed. On the one hand, the
derivative of measured acceleration containing much noise
is included in the law, which is thus not benefit to practical
implementation. On the other hand, it’s hard to transform
the design parameters in the control formula into a compact
form similar to the case of AOA autopilot. Two control
gains, i.e. k
1
and k
2
, are determined on the basis of step and
sine command tracking. The results show that k
1
affects
mainly system steady state error, and k
2
affects mainly
response speed. Moreover, k
1
is bounded and its upper
bound has less relevance with k
2
. Compared with the
traditional linear three-loop acceleration topology, the
nonlinear acceleration autopilot based on a backstepping
approach exhibits excellent tracking performance and
robustness. In spite of good performance, the application of
nonlinear autopilot is limited owing to a lack of physical
meaning and complex engineering implementation.
Actually, the exact mathematical model including
aerodynamics and unconventional control strategy of an
advanced missile could hardly be obtained from wind
tunnel testing data or software simulation. Both linear and
nonlinear autopilots could stabilize a static unstable missile.
Through the control usage analysis, it can be concluded
that actuator resource is the crucial factor in controlling a
static unstable missile
12
.
TABLE OF CONTENTS

1
978-1-4244-3888-4/10/$25.00 ©2010 IEEE
2
IEEEAC paper#1416, Version 1, Updated 2009:10:31
1. INTRODUCTION ............................................................. 1
2. MISSILE LONGITUDINAL DYNAMICS ............................. 1
3. AUTOPILOT DESIGN AND SIMULATION .......................... 2
4. CONCLUSION ................................................................ 7
REFERENCES .................................................................... 7
BIOGRAPHY ...................................................................... 7

1. INTRODUCTION
The requirements for the short range air-to-air dogfight
missile and the missile interceptor, particularly with respect
to the capability to engage highly agile fourth-generation
fighter and tactical ballistic missile (TBM), and achieve
precision end-game trajectory in seconds or less, have
prompted a revision and research of the way in which the
guidance and autopilot design is undertaken. Autopilot
design for future missile systems will be dominated by the
requirement of ultimate agility in the entire flight envelope
of the missile. In addition, the new missile configurations
being considered trade smaller aerodynamic surfaces for
increased aerodynamic instability to provide larger
available angle of attack and acceleration capability, which
poses a significant challenge for autopilot design.
2. MISSILE LONGITUDINAL DYNAMICS
The longitudinal missile dynamics can be described using
the short period approximation of the longitudinal
equations of motion. Written in differential equation
notation the basic nonlinear plant is

1 1
2 2
cos
( ( ) )
( ( ) )
y
qS
M d
mV
qSd M d I

   
  
  
 



(1)
where
3 2
( ) 1,2
i i i i
M a b c i    
 
The notation definition and numerical values are shown in
references [1, 2] for brevity. The model described by (1)
2
does not meet the strict-feedback form required by the
backstepping method and thus should be dealt with firstly.
Let
1 2
( )
y
C qS mV C qSd I
 
The first equation can be rewritten as
1 1 1 1
( )cos ( cos )
C f C d
    
  



Note that the factor (C
1
d
1
cos) r
epresents the force effect
of the actuator on angle of attack, and could be omitted
according to its physical meaning and engineering practice.
Such that the strict-feedback system is obtained

1 1 1 1 1 2
2 2 1 2 2 1 2
( ) ( )
(,) (,)
x f x g x x
x f x x g x x u
 
 


(2)
where
1 2
[ ] [ ]
T T
x x
 



u



1 1
( ) 1
g x


2 1 2 2 2
(,)
g x x C d


3 2
1 1 1 1 1 1 1 1 1 1
( ) cos ( )
f x C x a x b x c x
  
3 2
2 1 2 2 2 1 2 1 2 1
(,) ( )
f x x C a x b x c x
  
For the sake of linear autopilot design, the nonlinear state
equations are linearized at the trim operating point (M
y
= 0)
to form linear state space equations of the form

11 1
21 2
11 1
1
0
0
0 1 0
A B
x Ax Bu x u
A B
C D
y Cx Du x u
   
   
   
   
   
   
   
   

(3)
where
1 2
[ ]
T
x x x

[ ]
T
y
y a




2
11 1 0 21 2 2 0 2 0 2
1 1 1 0 2 2 2 11 1 0
1 1 1 0
( ) (3 2 )
cos ( )
cos
A C A C a b c
B C d B C d C CV
D CVd
  
 

 
 
   
 

0 1 0 0 2 0 0
2
1 0 1 0 1 0 1
3 2
2 0 1 2 1 2 0 1 2 1 2 0
1 2 1 2 0
( ) ( ) cos ( ) sin
( ) 3 2
( ) ( ) ( )
( )
N N
N a b c
N a a d d b b d d
c c d d
    
  
  

 
  

  
 

Here a
y
represents the measured acceleration that assumes
the accelerometer is located just at the center of mass. The
following average model is used in the linear design
process. These do not actually correspond to a given
linearization about an operating angle of attack, rather they
represent an average value for each element of the matrices
when examined over the entire angle of attack range, i.e. 
 (0
20deg).
0.99 1 0.12
159.66 0 130.11
936.09 0 111.63
0 1 0
A B
C D
 
 
 
 
   
   
   
   
   
   

Note that from a linear model perspective, when   (0
1
.3deg), the airframe is unstable; when   (1
.3deg 20deg),
the airframe is stable; and the airframe is critically stable
when  =
1.3deg.
3. AUTOPILOT DESIGN AND SIMULATION
3.1 AOA Autopilot
The state command equation for (2) can be assumed as

1
0
des des des
x x x
  
 
(4)
where 
1
, 
2
>
0.
Let
1 1
des
x x
   a
nd select the Lyapunov function

2 2
1 1 1
1 1
( )
2 2
des
V x x  
(5)
where
des
x
is the desired value. Then

1 1 1 1 1 1 2
des
dx
V f g x
dt
  
  
 
 
 


(6)
Selecting
2
x
as the virtual control usage

2 1 1 1
1
1
des
des
dx
x f k
g dt

  
 
 
 
(7)
where
1
0
k

. It can obtain

2
1 1 1 1 1
2 0
V k k V
    

(8)
such that
1
des
x x
 when
t

. The first sub-system of (2)
is asymptotically stable.
Let
2 2 2
des
x x
   a
nd the extended Lyapunov function

2 2 2
2 1 2 1 2
1 1 1
2 2 2
V V
  
    (9)
3
Then
 
2
2
1 1 2 22 1 1 2
2
1
2
1
1
1 1 1 1
1 1
2
des
des
g f g uV k
g
g dx
f d x
g f k
k
t dt
t dt

 


  
 


  
  



  
 
 
 
 
 
 
 


(10)
Note that
1 2 1 1
d dt k
  
  and
1
1
g

, so
 
2
2 1 1
2
2
1
1 1 1 2 2 2 1 22
2
1
(1 )
des
V k
f d x
k k f g u f x
x dt

 
  


     

 
 
 

(11)
If the control law is given as





 
2
1 2
2 1 2 2
1
2
1
1 2
2
1
1
des
k k
g u f
k
f d x
f x
x dt
 

  


  


(12)
then

2 2
2 1 1 2 2
0
V k k 
   

(13)
such that
2 2
des
x x
 when
t

. Therefore the final control
law can be written as

 
   
1
1 2
1 22 2
1 2 1
1
1 2 1 1 2 2
des des
f
k k
f xg u f k k x
x
k k k kx x 

 

     

  
 
 
 

(14)
where the item
2
g
is moved to the left for brevity.
The system described by (1) is globally asymptotically
stable according to La Salle-Yoshizawa theorem.
Though the control parameters in (14), i.e.
1
k
and
2
k
, are
introduced in the design procedure step by step, they appear
in the control law as some combination form.
Since
1 2
,0
k k

, let
1 2
2
1 2
2
0
k k a
a b
k k b
 
 





Given that the AOA command filter is a second-order
system with damping ratio 0.707 and natural frequency
4Hz, then the control law (14) can be rewritten as

   
3 2 2
2
2 2 2
2 2
3 2
1 1 1 1
3 2
1 1 1 1
2
1 1 1 1
2
1
2
1
{ ( )
cos ( )
2 sin ( )
cos (3 2 )
2 }
c c
u C a b c b
C d
C a b c
a C a b c
C a b c
a
b
   
    
   
  
  

 
   

  
   
 
 

 
 
 
 
 


(15)
where
1
35.5
 ,
2
631.7
 ,
and
c

presents the AOA
tracking command, as shown in Fig. 1.
1
t
o
1
s
1

1
s
2




c

c



Figure 1 – AOA command
The backstepping method can provide a systematic
construction process for controller design, but it fails in
determining the optimal values of control parameters.
Given the actuator (fin) as a second-order system with
damping ratio 0.707 and natural frequency 23.9Hz, the
AOA response and fin deflection curves are shown in Fig. 2
– Fig. 7.
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

(deg)
t(
s)
b = 10
a = 10
a = 15
a = 20
a = 50

Figure 2 – AOA response (given b)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
1.3
a = 30
b = 10
b = 30
b = 50
b = 100

(deg)
t(
s)

Figure 3 – AOA response (given a)
4
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
b = 10
a = 10
a = 15
a = 20
a = 50
 (deg)
t(
s)

Figure 4 – Fin deflection (given b)
0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0
-5
-4
-3
-2
-1
0
1
2
3
4
a = 30
b = 10
b = 30
b = 50
b = 100
 (deg)
t(
s)

Figure 5 – Fin deflection (given a)
Let
1
a kb k
 

then

 
2
1 2
2
1 2
max(,)
1
min(,)
k k
k k
k k

 
(16)
If a is much larger than b, such with
5
k

, then the closed
loop response deteriorates, while the system performance
will be very good if a is close to b. In the backstepping
procedure, the requirements that
0
i
k

( 1,2,,)
i n


is
equivalent to
a b

, which perhaps introduces
conservatism. In the final control law,
1
k
and
2
k
appear in
pairs and
a b

is not necessary. If selecting 0
a b
 
,
1
k

and
2
k
may be not real numbers. The results shown in Fig.
6 and Fig. 7 validate the analysis. However, further study is
needed since only the case of n = 2 is detailed here.
0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.2
0.4
0.6
0.8
1.0
1.2
(deg)
t(
s)
a = 20, b = 30
a = 18, b = 20

Figure 6 – AOA response (a<b)
0.0 0.1 0.2 0.3 0.4 0.5
-2
-1
0
1
2
a = 20, b = 30
a = 18, b = 20
(deg)
t(
s)

Figure 7 – Fin deflection (a<b)
3.2 Acceleration Autopilot
3.2.1 Nonlinear Autopilot Based on Backstepping
The nonlinear tracking problem of strict feedback form for
acceleration autopilot design can be written as

1 1 1 1 1 2
2 2 1 2 2 1 2
1
( ) ( )
(,) (,)
( ) ( )
x f x g x x
x f x x g x x u
y h x l u
 
 
 


(17)
where u is the control input and y is output [3, 5, 6].
The design procedure is similar to that for the
aforementioned AOA autopilot. Defining the tracking error
e as
r
e y y
 

where y
r
refers to the input command. Taking the
derivative of y,
 
1 1 2
1
h l
y f g x u
x u
 
  
 
 

Introducing error 
1
e
qual to e and Lyapunov function V
1

2
1 1
2
V 
5
then
 
1 1 2
1 1
1
r
h l
f g x u y
V
x u

 
  
 
 
 
 
 

 

Let
2
x
as the virtual control for the sub-system described by
the first equation in (17)





 
1
1 1 1
2
1
1
des r
h x l u
y f u k
x
h x g

     

 
 
(18)
where
1
0
k

. Thus
2
1 1 1
0
V k 
  

,
r
y y
 when
t

.
Then introducing the error 
2
a
nd the extended Lyapunov
function V
2



2 2
2 2 2 2 1 2
2
des
x x V   
 
Setting the control input u as


2 2 2 2 2
des
u x f k g
  
t
hen
2 2
1 1 1 2 2
0
V k k 
   

,
2 2
des
x x
, when
t

.
The implicit-form control law for the nonlinear tracking
problem (17) can then be expressed as
 
 
 
 
2
2
2
1 1 1 1
2
2
1 1 1 1
1 1
1
1 2 2 1 2
2
1
1 1
2
1
1 1 1 22
1 1 1
1 2
1 1
( )
( ) ( )
( )
( )
r
r
r
r r
l l
y u u
f g f f
u u
xg u
g x x g
h x g
l
y u
f
k k f k k y y
u
x
h x
g g
h h g
g
f g x
x x xl
y k y y u
h xu g
 
 
 
 
  
 
 


     

 
 
  
 
  
  
 
  
  
  
 
 
 
 
 
 
 
 
 
 
  
 
 
(19)
where the item g
2
is moved to the left for brevity.
It is worth noting that the backstepping method provides a
systematic constructive process to formulate the control law.
However, at least four disadvantages exist
a. The control law is so complex that its engineering
implementation will be a significant challenge.
b. The control law depends on a plant mathematic model.
c. It’s difficult to obtain the differential item of missile
normal acceleration or control input as there is much
noise in the measured data.
d. There seems no a simple way to determine the control
parameters, i.e. k
1
and k
2
. Thus it could hardly insure an
optimal result.
The control parameters for nonlinear acceleration autopilot
(19) are determined for both step and sine input commands.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
a
y
(m/s
2
)
t(
s)
k
1
= 10
com.
k
2
= 1
k
2
= 2
k
2
= 3
k
2
= 5

Figure 8 – Response for step input (given k
2
)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
1.1
1.2
k
2
= 2
com.
k
1
= 8
k
1
= 10
k
1
= 12
k
1
= 100
ay
(m/s
2)
t(
s)

Figure 9 - Response for step input (given k
1
)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-2.5
-2.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
a
y
(m/s
2
)
t(
s)
k
1
= 10
com.
k
2
= 1
k
2
= 2
k
2
= 3
k
2
= 5

Figure 10 - Response for sine input (given small k
1
)
6
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
k
1
= 50
com.
k
2
= 1
k
2
= 5
k
2
= 10
k
2
= 20
ay (m/s
2)
t(
s)

Figure 11 - Response for sine input (given larger k
1
)
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
ay (m/s
2)
t(
s)
k
2
= 5
com.
k
1
= 50
k
1
= 100
k
1
= 200
k
1
= 250

Figure 12 - Response for sine input (given k
2
)
In the step input case, as shown in Fig. 8 – Fig. 9, the
relative increment of k
2
is benefit to accelerate system
response. The relative increment of k
1
could decrease the
transient process, but a much larger k
1
will result in
performance deterioration.
In the sine input case, as shown in Fig. 10 – Fig.12, system
tracking performance will be bad for any k
2
if k
1
is small.
There almost always exists a k
2
to improve system
performance when k
1
is large enough. The upper boundary
M(k
1
) of k
1
has a weak relation with k
2
. The system will
diverge when k
1
exceeds M(k
1
).
Considering both system performance and control input,
the decision on the control parameters is
1 2
100 9
k k
 

3.2.2 Linear Autopilot Based on Optimal Control and
Frequency-domain constraint
An optimal control combined with frequency domain
constraint approach is adopted for the common three-loop
acceleration autopilot for comparison [3, 4].
An optimization objective could be to use a weighted sum
of the measured acceleration and the control rate. The cost
function would then be

 
2 2
11
11
0
min ( )
y dc yc
J Q a k a R dt



  




(20)
Compared with the linear system described by (3), this
needs the fin deflection rate as the input to the system.
Therefore, the states are augmented with the control (fin),
the control is replaced with the derivative of the control,
and the outputs are augmented with the angular
acceleration. Note that no new information is introduced
into the plant. The plant can be rewritten as

1 1 1 1 1
1 1 1 1 1 1
dc
x A x Bu
y C x Du k r
 
  


(21)
where
 
1 1
1 1 1
0
[0]
ym dc
dc
dc
a k r
k
x u y k D

  
 

 
  
   
 
   
 
   
 
   
   
   


1 1 1
[0]
[0] 0 1 (2,:) (2,:)
A B C D
A B C
A B
  
     
     
     

The plant now is strictly proper, such that a coordinate
transform can be used to obtain the optimal solution.
Defining
1 1
1 1 1 1 2
x C y C x
 
 
The transformed plant now is

2 2 2 2
2 2
x A x B u
y x
 


(22)
where


1
2 2 2 1 1 1 2 1 1
T
ym
x a u A C AC B C B
  

 
 
  

The solution K can now be determined by calculating the
following Algebraic Riccati Equation (ARE)
1
2 2 2 2 2 2
0
T T
A P PA PB R B P Q

   

The forward loop gain k
dc
can be determined by inspecting
the closed loop system
2 2
2
c c yc
y c
x A x B a
a C x
 



Then the final control law is obtained as
7



ym dc yc
a k a dt
K dt 



 
 
 
 
 




(23)
Selection of weights is the tuning knob in optimal control
design. The rules of thumb are step by step sweeping. An
optimal control method usually tends to take no account of
frequency domain constraints, while the open-loop
crossover frequency ωcr plays a key role in frequency
analysis. A smart choice is to incorporate the crossover
frequency ωcr into autopilot design procedure [3].
3.3 Comparison of Nonlinear & Linear Acceleration
Autopilot
A comparison between the mentioned linear and nonlinear
acceleration autopilots is shown in Fig. 13 – Fig. 14.
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
0.4
0.6
0.8
1.0
1.2
ay (m/s
2)
t(
s)
command
three-loop, 
cr
=
50rad/s
three-loop, 
cr
=
80rad/s
nonlinear autopilot

Figure 13 – Acceleration tracking
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
0.2
0.3
0.4
0.5
three-loop, 
cr
=
50rad/s
three-loop, 
cr
=
80rad/s
nonlinear autopilot
 (deg)
t(
s)

Figure 14 – Fin deflection
A nonlinear autopilot based on the backstepping approach
tracks sine command much better than the common three-
loop topology. However, it also requires more actuator
resource, particularly during the transient process. Thus the
actuator plays an essential limiting role in missile control.
In addition, the longitudinal three-loop autopilots for
tactical missiles have been successfully employed for over
several decades, and have been continually improved.
While the complexity of nonlinear autopilot restricts its
application.
4. CONCLUSION
This paper first analyzed the conservatism introduced by a
backstepping design procedure for an AOA autopilot, and
presented a translation method to improve the system
performance. Then it posed and solved the nonlinear
acceleration autopilot design as a tracking problem using a
backstepping approach. The determination of control
parameters should consider both system performance and
control usage. Compared with the classic three loop
autopilot designed by an optimal control method, the
nonlinear autopilot based on backstepping shows better
tracking capability but requires more actuator resource.
However, complexity and lack of physical meaning restricts
its application in engineering.
REFERENCES
[1] M. B. McFarland, Adaptive nonlinear control of
missiles using neural networks [D]. The School of
Aerospace Engineering, Georgia Institute of
Technology, 1997.
[2] R.T. Reichert, "Robust autopilot design using -
sy
nthesis," Proc. of the American Control Conference,
pp. 2368-2373, 1990.
[3] FAN Jun-fang, Research on autopilot design for static
unstable missiles [D]. School of Aerospace, Beijing
Institute of Technology (BIT), 2008.
[4] C. P. Mracek and D. B. Ridgely, "Missile longitudinal
autopilots: connections between optimal control and
classical topologies," AIAA Guidance, Navigation and
Control Conference, 2005, pp. 1-29.
[5] Ola Härkegård. Flight control design using
backstepping [D]. Department of Electrical Engineering,
Linköpings University, Sweden, 2001.
[6] E.R. van Oort, L. Sonneveldt, Q.P. Chu, and J.A.
Mulder, "Modular Adaptive Input-to-State Stable
Backstepping of a Nonlinear Missile Model," AIAA
Guidance, Navigation and Control Conference, 2007,
pp. 1-14.
BIOGRAPHY
FAN Jun-fang received the Ph.D.
degree in flying vehicle design from
Beijing Institute of Technology (BIT) in
2009. He is currently a lecturer in
School of Automation, Beijing
Information Science & Technology
University (BISTU), China. His research interests lie in
8
flying vehicle control and guidance theory, and advanced
navigation technology.
SU Zhong, senior member of Chinese
Institute of Electronics (CIE), and
director of Chinese Association for
System Simulation (CASS). He
received his bachelor and master’s
degree from Beijing Institute of
Technology (BIT), and Ph.D. degree in
physical electronics from Beijing
Vacuum Electronics Research Institute (BVERI),
respectively. He is currently a professor in School of
Automation, Beijing Information Science & Technology
University (BISTU), China. His research interests include
advanced navigation & guidance technology, and detection
technology & automatic equipment. He has co-authored
more than 60 papers in research journals and conference
proceedings in these areas.