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.

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