1

Agile Missile Autopilot Design Using Nonlinear Backstepping Control with Time-Delay Adaptation

Chang-Hun Lee1, Tae-Hun Kim2 and Min-Jea Tahk3 Korea Advanced Institute of Science and Technology(KAIST), Daejeon, 305-701, Korea

This paper deals with a nonlinear adaptive autopilot design for agile missile systems. In

advance of the autopilot design, an investigation of the agile turn maneuver, based on the

trajectory optimization, is performed to determine state behaviors during the agile turn

phase. This investigation shows that there exist highly nonlinear, rapidly changing dynamics

and aerodynamic uncertainties. To handle of these difficulties, we propose a longitudinal

autopilot for angle-of-attack tracking based on backstepping control methodology in

conjunction with the time-delay adaptation scheme. The performance of the proposed

method is investigated through nonlinear 6-DOF simulations. An intercept scenario is

performed to explore the applicability of the proposed control methodology to agile missile

systems.

Keywords

Agile missile, missile autopilot, backstepping control, adaptive control, time-delay adaptation

Notice

The modified version of this paper has been accepted for publication in Transactions of the Japan Society for

Aeronautical and Space Sciences.

1 Ph. D, Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology(KAIST), Kuseong Yuseong, Daejeon, 305-701, Korea/chlee@fdcl.kaist.ac.kr. 2 Ph. D, Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology(KAIST), Kuseong Yuseong, Daejeon, 305-701, Korea/thkim@fdcl.kaist.ac.kr. 3 Professor, Department of Aerospace Engineering, Korea Advanced Institute of Science and Technology(KAIST), Kuseong Yuseong, Daejeon, 305-701, Korea/mjtahk@fdcl.kaist.ac.kr.

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I. Introduction

The demand for highly maneuverable missile systems has grown recently because of their usefulness in air-to-air

combat scenarios. These missile systems, called agile missiles, are generally operating in the high angle-of-attack

mode to achieve agile maneuverability. The flight envelope of the agile missile can be generally classified into three

phases [1, 2]: launch and separation from the aircraft, the near-180 degree heading reversal maneuver (agile turn),

and terminal homing. For each flight phase, suitable autopilot design strategies and techniques are necessary. The

autopilot design problems for the separation and the terminal homing phases have been extensively studied and are

well-understood via the autopilot design of conventional missile systems. However, the autopilot design for the agile

turn phase is relatively less well-understood, and there exist crucial difficulties: the handling of highly nonlinear,

rapidly changing missile dynamics and aerodynamics parameter uncertainties due to the poor measurement of

aerodynamic data and the un-quantified coupling effect of aerodynamics in the high angle-of-attack. Over the course

of several years, there have been various approaches to the autopilot design of the agile turn in order to solve these

difficulties, ranging from linear control to nonlinear control methodologies.

In reference [3], the authors proposed an autopilot design method for an aggressively maneuvering missile

system, based on linear control in conjunction with velocity-based gain scheduling. Sliding mode control (SMC),

well-recognized robust control, was considered as a longitudinal autopilot design method for agile missiles in [2].

For highly agile missile systems, other robust control methodologies were applied to the design of missile autopilot

based on H∞ control [4], μ -synthesis [5-6], and mixed 2 /H H∞ control synthesis [7]. The authors in [8] proposed

missile autopilot design, using feedback linearization and a two-time scale separation technique. Approximate

feedback linearization and the asymptotic output tracking method were used for the design of high angle-of-attack

missile systems in [9]. In order to cancel feedback linearization errors due to model uncertainties, feedback

linearization with uncertainty adaptation based on neural networks was applied to the autopilot design for agile anti-

air missile systems in [10]. The backstepping control methodology was applied to agile missile systems in [11], and

an adaptive backstepping control based on neural network was studied for the missile autopilot design method in

[12]. The time-delay-control was considered as the agile missile control method in [13].

In addition, for each flight phase, suitable command structures for autopilots are required in order to attain high

overall performance. For the separation and the terminal homing phase, the body rate and the body acceleration

commands are typically recommended, respectively. For the agile turn phase, two options such as the body

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acceleration [3-9] and the angle-of-attack [1, 10-13], are possible autopilot command structures. According to

reference [1], during the agile turn phase, commanding the body acceleration may not be appropriate compared with

commanding the angle-of-attack, because large amounts of the body acceleration command are needed in order to

achieve the required turn rate, due to the significant difference between the missile acceleration perpendicular to

velocity and to the body in the high angle-of-attack. As a result, from simulation studies in [1], commanding body

acceleration causes a large reduction in a missile’s velocity during the agile turn.

In this paper, we deal with the angle-of-attack controller design for the agile turn phase in the pitch plane, using

nonlinear backstepping control [14, 15] and time-delay control [16] methodologies. The backstepping control has

been successfully applied to the flight control system from previous studies [11, 12, 17]. Accordingly, the

backstepping control methodology is adopted for the main structure of the proposed autopilot. In the proposed

method, the time-delay control is used for the uncertainty adaptation scheme, rather than for the role of the control

methodology itself. From previous applications of time-delay control [13, 18, 19], it has been shown the time-delay

approximation, which is the core idea of time-delay control, is a practical and efficient estimation method for

unmodeled dynamics and uncertainties. In this paper, the trajectory optimization of the agile turn is performed in

advance of autopilot design in order to discover the behavior of state variables and the practical considerations for

autopilot design during the agile turn. These results can also be used for the reference trajectory during the agile turn

phase in simulation studies. In order to investigate the performance of the proposed method, the proposed method is

tested in 6-DOF nonlinear simulations with a step command and angle-of-attack profile obtained from the trajectory

optimization. Furthermore, an intercept scenario, including the agile turn and the terminal homing phases, is also

performed. For this purpose, the acceleration controller is also constructed based a cascaded control structure [20,

21]; the angle-of-attack controller is augmented by the outer proportional-integral (PI) controller; and at end of the

agile turn phase, the autopilot controlling the angle-of-attack is reconfigured to control the body acceleration, using

a simple control command blending logic.

The outline of this paper is as follows: Section II presents a nonlinear missile model considered in this work. In

Section III, we analyze the agile turn maneuver in order to determine the behavior of state variables. We discuss the

agile missile autopilot design, using the nonlinear backstepping control with time-delay adaptation methodology, in

Section IV. In Section V, nonlinear simulations are performed to investigate the performance of the proposed

method. Section VI summarizes this work.

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II. Missile Model

A nonlinear missile model with a boost [22] is considered, as shown in Fig. 1. It is assumed that the agile missile

is maneuvering in the horizontal plane with the 90 degrees roll angle after its launch from an aircraft. Accordingly,

the longitudinal gravity component is not considered. The longitudinal equation of motion in the body axis is written

as

( ) /

//

x

z

y yy

u wq F T mw uq F mq M I

= − + +

= +=

(1)

where u , w , and q represent the longitudinal body velocity, the vertical body velocity, and the body pitch angular

velocity, respectively. m , yyI , and T denote mass, pitching moment of inertia, and thrust, respectively. The

aerodynamic forces and moment, denoted by xF , zF , and yM , are given as

, ,x A z N y MF QSC F QSC M QSlC= − = − = (2)

with

( ) ( ) ( )( ) ( )( ) ( )

( ) ( ) ( ) ( )

0

0

0

2

,

/ 2

, ,

, , /2

T

q

A A A A A

N N N N

M M M M N cg ref cg M

C C M C M C M C M

C C M C M C

qlC C M C M C M C x x l CV

α δ

δ φ

δ φ

α δ

α α δ

α α δ

+ + + Δ

+ + Δ

+ + − − + Δ

(3)

where Q , S , l , M , and V are dynamic pressure, reference area, reference length, Mach number, and velocity,

respectively. α and δ represent the angle-of-attack and the control fin deflection angle. ,cg refx and cgx denote the

reference location of CG(center of gravity) at launch and the true location of CG during the booting phase. These

parameters are given in Table. 1. In this missile model, m , yyI , and cgx are linearly varied during the boosting

phase due to the exhaustion of propellant.

The aerodynamic coefficients 0AC , AC

α, AC

δ,

TACΔ , 0NC , NC

δ,

0MC , qMC , and MC

δ are given by nonlinear

functions of the angle-of-attack and Mach number and are obtained from wind tunnel measurements. These

coefficients are mostly assumed to be continuous functions of their arguments and contain 20~30% maximal

admissible uncertainties compared with the true values as

( ) ( ) ( )1pertPertC C= + Δi i (4)

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where ( )i represents their subscripts. The term TACΔ represents an additional axial coefficient related to the rocket

motor on/off as plume effects, and it has a value of zero value during the boosting phase. The term

( ), /N cg ref cgC x x l− in Eq. (3) is the adjustment of the pitching moment for the shifts in . .c gx during the boosting

phase. The notions of NCφ

Δ and MCφ

Δ are the perturbation terms due to the aerodynamic roll coupling effect, which

are expressed as

( )( ) ( ) ( )

2

2 2,

, sin 2

, sin 2 , sin 2 /

N N

M M N cg ref cg

C C M

C C M C M x x lφ φ

φ φ φ

α φ

α φ α φ

Δ =

Δ = − − (5)

where φ is the aerodynamic roll angle. These perturbations reach their maximum values at 45φ = ° and vanish at

0φ = ° or 90φ = ° . From Eq. (5), the aerodynamic coefficients NCφ

and MCφ

are given by nonlinear functions of the

angle-of-attack and Mach number, and those are mostly assumed to be continuous functions of their arguments.

Using the definition of angle-of-attack as ( )1tan /w uα −= , the time-derivative of angle-of-attack is rewritten as

( )2 2 2 2

1 cos sinwu wu w uu w u w

α α α−= = −

+ + (6)

Combining Eqs. (1), (2), and (6) with 2 2V u w= + yields as

[ ] sincos sinN A

Myy

QS Tq C CmV mV

QSlq CI

αα α α= − − −

= (7)

From Eq. (7), it is observed that the equation of motion given in Eq. (1) can be rewritten in terms of α and q .

In many studies of missile autopilot design [3, 5, 6, 9, 10, 12, 13], small angle-of-attack approximations, i.e.,

cos 1α ≈ and sin 0α ≈ , are widely used to simplify the equation of motion as shown in (7). However, as will be

shown in Section III, a small angle approximation is not valid for agile missile systems, because the missile is

operating in the high angle-of-attack mode. Hence, the original form of Eq. (7) will be used for autopilot design in

the remainder of this paper. In the above equation, it is assumed that state variables are available through a built-in

inertial navigation system (INS) and angle-of-attack observer, as studied in [13, 21, 23].

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III. Agile Turn Maneuver Analysis

Agile missile systems have the capability to attack a target in all direction. In order to intercept a target located in

the rear of the missile, the agile turn, a more-than-180-degree heading reversal maneuver, is needed until the missile

locks onto a target and obtains an advantageous homing position. Then, terminal homing follows the agile turn. In

order for the missile to be located at an advantageous homing position for a target, it is desirable to match the flight

path angle at the end of the agile turn to the line-of-sight (LOS) angle at the start of terminal homing. In addition, it

is obvious that high missile velocity is generally essential to enhance the probability of intercepting a target during

the terminal engagement. Therefore, we hold that the objective of the agile turn in this study is to accomplish the

specified flight path angle while maximizing the missile velocity after the agile turn maneuver.

Based on this strategy, the agile turn trajectory, based on the point-mass model, is determined by optimization,

and its characteristics are analyzed in this section. The primary objective of this analysis is observing the behaviors

of state variables, such as the angle-of-attack, velocity, and flight path angle, during the agile turn and determining

the agile turn maneuverability of the missile model used in this study. The results can give useful and fundamental

information about what we should consider in the design of the autopilot for agile missiles. Therefore, based on the

results, the proposed autopilot will be designed in Section IV. In addition, the optimal solution obtained from this

section can be used for the reference trajectory for the agile turn maneuver.

A. Problem Description

To construct the trajectory optimization problem for agile turn maneuver analysis, the equation of motion for the

agile missile and the proper cost function are needed. Consider the missile geometry in the horizontal plane as

shown in Fig. 2. The equation of motion for trajectory optimization is represented as

( )( ) ( )

cos /

sin /

cossin

V T D m

T L mV

X VY V

α

γ α

γ

γ

= −

= +

=

=

(8)

where γ is the flight path angle. X and Y represent the missile position in the inertial reference frame, and the

origin is regarded as the launch position from an aircraft. The lift force and drag force, denoted by L and D , are

given as

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( )( )

cos sin

sin cosN A

N A

L QS C C

D QS C C

α α

α α

= −

= + (9)

In order to alleviate the complexity of the optimization problem, the angle-of-attack is chosen for the control input

of Eq. (8) and is modeled as a first-order lag system:

( )( )1/ cα τ α α= − (10)

where τ is the time constant. To address the problem of achieving a high missile velocity and the desired flight path

angle simultaneously, we define the cost function with the terminal constraint as

( ) ( )min f f fuJ V t subject to tγ γ= − = (11)

where ft represents the terminal time. In order to use the thrust for the agile turn maneuver, it is assumed that the

agile turn is performed until the boost burns out. Hence, we set ft to the thrust off time as offt . Accordingly, this

optimization problem is given by a class of ft fixed problem.

B. Numerical Trajectory Optimization

In order to find the angle-of-attack profile while optimizing Eq. (11), one of the numerical algorithms for solving

optimization problem, which is called CEALM (co-evolutionary augmented Lagrangian method) [24], is employed

with the above equation of motion and the cost function. CEALM is one of the direct methods, based on

evolutionary strategy, for obtaining the optimal solution from the initial guessing of parameterized control input

through several times iterations. It is assumed that the missile is launched from an aircraft at an altitude of 2000 m .

The missile parameters are given in Table 1. The following initial conditions and terminal conditions are used for

this trajectory optimization:

0 0 0 0250 m/ s, 0 , 0m, 0mV X Zγ= = ° = = (12)

120 240 60f for everyγ° ≤ ≤ ° ° (13)

The trajectory optimization results are presented in Fig. 3, which shows the missile velocity profiles, the angle-

of-attack profiles, the flight path angle profiles, the flight trajectories, the turn rate γ , and the ratio of thrust to lift

force for generating of turn rate.

C. Turn Maneuver Analysis

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The terminal velocity decreases and turn rate increases as the terminal specified flight path angles increase, as

shown in Figs. 3(a) and 3(e). Since the velocity at launch is not high, the thrust force is dominantly used at the initial

time until the velocity increases to generate a sufficient lift force for 120 degree and 180 degree turns. For a very

large turn, i.e., a 240 degree turn, the thrust force is dominantly used throughout the flight, as shown in Fig. 3(f).

Accordingly, velocity profiles for 120 degree and 180 degree turns linearly increase with respect to time. However,

240 degree turn holds a constant velocity until 2 sec in Fig. 3(a). The missile velocity is significantly changed during

the agile turn. Since thrust force is used to generate turn rate for all cases, due to the insufficient lift force at the

initial time, we can observe that the high angle-of-attack maneuver of about 35~55 degrees is required for the agile

turn phase, as shown in Fig. 3(b).

In such a high angle-of-attack operation and varying missile velocity, the missile dynamics are highly nonlinear

and rapidly changing, as shown in Eq. (7). In addition, induced roll moment, i.e., the roll coupling effect resulting

from vortices impinging on the tail fin, is generated in the high angle-of-attack. As a result, the perturbation terms in

Eq. (5) may be significant during the agile turn. In the autopilot design, therefore, these perturbations are also taken

into consideration.

IV. Autopilot Design

In this section, we mainly discuss the angle-of-attack controller for the agile turn phase. The proposed method is

formulated by backstepping control methodology with the time-delay estimation technique for the adaptation of

uncertainties. At the end of this section, we also deal with the acceleration controller and control command blending

logic for a planar intercept scenario.

A. Angle-of-Attack Controller Design

From Eq. (7), new variables are defined for convenience as follows:

1 2, ,x x q uα δ= = = (14)

Substituting Eq. (3) into Eq. (7), we rewrite the dynamics in terms of state variables and control input separately:

( )1 1 2 1 1

2 2 2 2

x f x h u gx f h u g= + + +

= + + (15)

where

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( )

( ) ( )

0 0 0 0

,1 2

2 ,1 2

1 2

sincos sin ,2

cos / 2 sin ,

cos ,

T q

cg sN A A A M M N

yy

cg sN A M N

yy

N Myy

xQS T QSl qlf C C C C f C C CmV mV I V l

xQS QSlh u C C h C CmV I l

QS QSlg C g CmV I

α

δ δ δ δ

φ φ

αα α α

δ α δ α

α

⎡ ⎤⎡ ⎤− − + + Δ − + −⎢ ⎥⎣ ⎦ ⎣ ⎦

⎡ ⎤⎡ ⎤− − −⎢ ⎥⎣ ⎦ ⎣ ⎦

− Δ Δ

(16)

and , ,cg s cg ref cgx x x− . In the above equation, the contribution of ( )1h u in the 1x equation can be ignored because

of ( )2 1h u h u , generally, in the missile systems. The terms 1g and 2g are related with the aerodynamic roll

coupling effect and reach their maximum values at 45φ = ° . In this study, the maximum values of these terms, a

worst case scenario, are considered as the additive aerodynamics perturbations due to the induced roll effect in the

high angle-of-attack. Also, taking the multiplicative aerodynamic uncertainties in Eq. (4) into consideration, the

dynamic equation can be rewritten in a strict feedback structure with unknown terms as

1 1 2 1

2 2 2 2

x f xx f h u= + + Δ= + + Δ

(17)

where 1Δ and 2Δ can be regarded as the aerodynamic uncertainties caused by Eqs. (4) and (5), and the unmodeled

dynamic related to ( )1h u . To construct the angle-of-attack controller based on the backstepping control

methodology, let new residual variables be defined as

1 1 1 2 2 2,d dz x x z x x− − (18)

Taking the time-derivative of the residual variables yields

1 1 2 1 1

2 2 2 2 2

d

d

z f x xz f h u x= + + Δ −= + + Δ −

(19)

where 1dx represents the desired values of 1x , which are determined by the reference command of the angle-of-

attack. 2dx is the desired value of 2x , and it forces 1z to converge with zero in a finite time for backstepping control.

Consider the following theorem:

Theorem 1. For the dynamics as shown in (19), the following desired value of 2x forces 1z to approach zero in

a finite time.

2 1 1 1 1 1d dx f K z x= − − + −Δ (20)

with 1 0K > .

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Proof. We consider the following Lyapunov function candidate as

21 1

12

V z= (21)

Suppose that 2x has tracked its command, 2dx . Substituting Eqs. (19) and (20) into the time-derivative of 1V yields

( )

1 1 12

1 1 2 1 1 1 1 0d d

V z z

z f x x K z

=

= + + Δ − = − < (22)

which completes the proof.

Then, the control law for backstepping, which enforces 1x and 2x , tracks their commands, 1dx and 2dx , in a

finite time, as can be determined by the following theorem.

Theorem 2. The control law

( )12 2 2 2 2 1 2du h f K z x z−= − − + − − Δ (23)

ensures that 1z and 2z converge at the origin. In this control law, 2K takes a positive value.

Proof. Let the Lyapunov function candidate be defined as

2 22 1 2

1 12 2

V z z= + (24)

Taking the time-derivative of 2V and substituting Eq. (19) into 2V , we can obtain the following equation:

( ) ( )

2 1 1 2 2

1 1 2 2 1 1 2 2 2 2 2d d d

V z z z zz f z x x z f h u x

= +

= + + − + Δ + + − + Δ (25)

Substituting Eqs. (20) and (23) into Eq. (25) gives

2 22 1 1 2 2 0V K z K z= − − < (26)

Based on the Lyapunov stability theorem [14, 15], the closed-loop system of Eq. (19) is asymptotically stable.

Therefore, 1z and 2z approach zero in a finite time, which completes the proof.

From Theorem 1 and 2, it is noted that the control law and the desired command of 2x contain the unknown

terms 1Δ and 2Δ , which causes the degradation of the tracking performance. In order to improve the tracking

performance, 1Δ and 2Δ should be estimated and compensated for in regard to control commands as

( )2 1 1 1 1 1

12 2 2 2 2 1 2

ˆ

ˆd d

d

x f K z x

u h f K z x z−

= − − + − Δ

= − − + − −Δ (27)

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where 1Δ̂ and 2Δ̂ are the estimates of 1Δ and 2Δ . Accordingly, we propose an efficient and practical adaptation

scheme based on time-delay approximation [16]. The basic idea of time-delay approximation is that if a function

( )f t is continuous with regard to time for 0 t c≤ ≤ , then it allows the following for a sufficiently small time-delay

L :

( ) ( )f t f t L≅ − (28)

In order to ensure the validity of this approximation, the unknown terms under consideration, 1Δ and 2Δ , are also

given by continuous functions of time. This is shown by the following lemma.

Lemma 1. If control input u is continuous over the interval 0 t c≤ ≤ , then 1Δ and 2Δ , as under consideration in

Eq. (17), are continuous functions of time for 0 t c≤ ≤ .

Proof. Since the system equations in Eq. (17) are well-defined over the interval 0 t c≤ ≤ , this implies that 1x and

2x are bounded for 0 t c≤ ≤ . Accordingly, through calculus, it is shown that the state variables 1x and 2x are given

by continuous functions for 0 t c≤ ≤ . Since 1Δ and 2Δ are modeled as the aerodynamic uncertainties resulting

from Eqs. (4) and (5) as well as the unmodeled dynamics ( )1h u , they consist of multiplicative and additive

aerodynamic coefficients, state variables, and control inputs. All parameters are continuous functions of time over

0 t c≤ ≤ , 1Δ and 2Δ result in continuous functions over the interval 0 t c≤ ≤ , which completes the proof.

According to Lemma 1, the unknown terms 1Δ and 2Δ can be estimated using the time-delay approximation as

( ) ( )1 1 2 2ˆ ˆ,t L t LΔ = Δ − Δ = Δ − (29)

Using Eq. (17), the above equation can be written as

( ) ( ) ( )( ) ( ) ( ) ( )

1 1 1 2

2 2 2 2

ˆ

ˆx t L f t L x t L

x t L f t L h t L u t L

Δ = − − − − −

Δ = − − − − − − (30)

As shown in Eq. (30), this adaptation scheme requires the time-delayed information regarding the state variables,

control input, and models. The time-delayed information can be obtained from the single-lag system as

( ) ( )11d

f t L f tsτ

− =+

(31)

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where dτ is the time constant value. The approximation error decreases as the time-delay constant value dτ

decreases. Since the estimation of 1Δ and 2Δ consists of continuous functions from Eq. (30), the control input in

conjunction with 1Δ̂ and 2Δ̂ is also given by a continuous form, as shown in Eq. (27).

To determine the behavior of the system in conjunction with the proposed adaptation scheme, we consider the

same Lyapunov function candidate as given in Eq. (24). Substituting Eq. (27) into Eq. (25), we have

2 22 1 1 2 2 1 1 2 2V K z K z z z= − − + Δ + Δ (32)

where 1 1 1ˆΔ Δ − Δ and 2 2 2

ˆΔ Δ −Δ represent the estimation errors. The above equation is rewritten as

2V A B= − + (33) where

2 2 2 2

1 2 1 21 1 2 2

1 2 1 2

,2 2 4 4

A K z K z BK K K K

⎛ ⎞ ⎛ ⎞Δ Δ Δ Δ− + − +⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ (34)

From the Lyapunov stability theorem, since 2V is not a negative definite for A B≤ from Eq. (33), the asymptotic

stability is not guaranteed, due to the estimation errors of 1Δ and 2Δ . However, it is shown that the system equation

of Eq. (19) with the proposed control law always converges to the bounded ellipsoid, i.e., A B= , because 2 0V ≤

for A B≥ and 2 0V > for A B< . The convergent ellipsoid, as shown in Fig. 4, can be expressed as

2 2

1 21 22 2

1 2

1 1 12 2

z zK Ka b

⎛ ⎞ ⎛ ⎞Δ Δ− + − =⎜ ⎟ ⎜ ⎟

⎝ ⎠ ⎝ ⎠ (35)

where

2 2 2 21 2 1 2

2 21 2 1 21 2

,4 44 4

a bK K K KK K

Δ Δ Δ Δ+ + (36)

As the estimation errors decrease and the gains 1K and 2K increase, the center of ellipsoid approaches the origin

and the region of ellipsoid shrinks.

For the ideal case, i.e., one in which the unknown terms are perfectly estimated, the behavior of the residual

variables is governed by

1 1 1

2 2 2

00

z K zz K z⎡ ⎤ ⎡ ⎤ ⎡ ⎤

=⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(37)

Then, the characteristic equation is written as

13

( )( )1 2 0s K s K+ + = (38)

Since the control gains are identical to the pole locations, the desired control gains can be determined from (38) in

order to create the desired tracking performance.

B. Acceleration Controller Design

For a pursuit scenario, including terminal homing, it is necessary to control the body acceleration in standard

skid-to-turn missile systems. We have designed the angle-of-attack controller to track the angle-of-attack command

in the previous section. In order to track desired acceleration and handle a non-minimum phase system for

acceleration control, a cascaded control structure, as shown in Fig. 5, is used; the PI controller for the outer loop is

augmented to generate the reference angle-of-attack signal, and the inner loop is constructed by the angle-of-attack

controller, as studied in previous section.

Consider the following PI controller structure:

( ) IPI P

KG s K

s= + (39)

If we regard the angle-of-attack controller for the inner loop as the linear subsystem, then its transfer function is

approximated as

( )( )

11c

ss s

αα τ

=+

(40)

where cα and τ are the angle-of-attack command and the time constant value. Since the acceleration is generally

proportional to the angle-of-attack for an axis-symmetric configuration, the relationship between angle-of-attack and

acceleration can be approximated as

( )( )

na sK

sα= (41)

where na is the acceleration. In order to track the reference response, the control gains for the outer loop can be

determined by

22 1

,ref ref refP IK K

K Kζ ω τ τω−

= = (42)

with the following reference closed-loop characteristics:

14

( )2

2 22ref

refref ref ref

G ss s

ωζ ω ω

=+ +

(43)

where refω and refζ denote the reference natural frequency and damping.

C. Control Command Blending Method

For an overall intercept scenario, the angle-of-attack controller is changed to control the body acceleration after

the agile turn phase. In order to avoid an abrupt command change from the flight phase transition between the agile

turn and the terminal homing, we propose a simple control command blending method as follows:

( )1nau Wu W uα= + − (44)

where uα and nau represent the control command from the angle-of-attack and the acceleration controller,

respectively. The weighting function W is varied as described in Fig. 6. 1Tt and 2Tt are the start time and end time

of the transition.

V. Simulation Results

In this section, the proposed autopilots for agile missiles are tested through three different 6-DOF nonlinear

simulations. First, the basic characteristics and performance of the proposed autopilots are investigated by imposing

step command. The second simulation analyzes the tracking performance of the proposed autopilots during the agile

turn maneuvers. In this simulation, the angle-of-attack profile obtained from the trajectory optimization is used for

the reference command of the angle-of-attack. Finally, the proposed autopilots are applied to engage a target in the

rear hemisphere of the missile. In all simulations, it is assumed that the missile is launched at an altitude of 2000 m

from an aircraft. The missile parameters shown in Table 1 are used. The second-order actuator models with natural

frequency 180 rad/ sactω = , damping ratio 0.7actζ = , control fin limit limit 30δ = ± ° , and control fin rate limit

limit 450 / secδ = ± ° are included. The design parameters for controllers are chosen as follows: 1 2 25K K= = ,

0.02dτ = , 0.0098PK = , and 0.34IK = . In addition, the second-order linear command shaping filter is used for

obtaining differential commands.

A. Step Responses

15

In this simulation, it is assumed that the missile starts on the agile turn with the initial velocity V = 250 m/ s and

is then boosted after a safe separation from an aircraft. Fig. 7 presents the step responses for angle-of-attack control,

without considering the aerodynamic uncertainties. In Fig. 7(a), the solid lines and the dotted lines represent the

response values and the command values, respectively. The results show the sound tracking performances of the

proposed controller.

Fig. 8 gives the step responses for angle-of-attack control with 30% multiplicative uncertainties ( 0.3PertΔ = ) in

aerodynamic coefficients and additive aerodynamic uncertainties from the coupling effect in high angle-of-attack. In

Fig. 8(a), the dotted line, the dashed line, and the solid line represent the angle-of-attack command, the angle-of-

attack response without adaptation, and the angle-of-attack response with adaptation, respectively. The results

indicate that even in the presence of aerodynamic uncertainties, the proposed controller can track the desired

command quite well. The time-delay adaptation scheme can accurately estimate the unknown term, as shown in Figs.

8(c) and 8(d).

B. Agile Turn Maneuver

In order to determine the control performance during the agile turn, we use the trajectory optimization results of

a 180 degree turn for the reference command of angle-of-attack. The aerodynamic uncertainties, as discussed in the

above section, are also considered. Fig. 9 presents the response for the agile turn maneuver. If the adaptation is not

performed, the degradation of tracking performance due to aerodynamic uncertainties increases as the operating

angle-of-attack increases. The results show that the proposed autopilot has good tracking performance, even if large

aerodynamic uncertainties exist. From Fig. 9(c), the flight path angle at the end of the agile turn reaches 182 degrees

when the proposed autopilot is used. It shows a similar value compared with the optimization results. However, the

terminal velocity after turn maneuvering decreases compared with the optimal value, due to the aerodynamic

uncertainties, as shown in Fig. 9(d). It is also observed that the uncertainties are estimated quite well from Figs. 9(e)

and 9(f).

C. Engagement Scenario

In order to determine the applicability of the proposed autopilots, a planar engagement scenario is performed in

the horizontal plane for a maneuvering target located at the rear of the missile. The angle-of-attack profile of a 180

degree turn, obtained from the trajectory optimization, is used for the guidance command during the agile turn phase.

16

Proportional navigation guidance (PNG) is employed for the terminal homing phase. In this intercept scenario, we

assume that an aircraft launches the missile when it is pursued by a target. Then, the target senses a danger and takes

a 4g− evasive maneuver 0.5 seconds after the missile is launched. In order to reduce the LOS error, i.e., the

subtraction of the LOS angle from the flight path angle, the missile initially tracks the reference angle-of-attack for

the agile turn, until the magnitude of LOS error is reduced to below 30 degrees. After that, the missile begins

terminal homing with PNG and acceleration control. The initial conditions for the engagement scenario are given in

Table 2.

Fig. 10 provides the results for the engagement scenario. In Fig. 10(a), the solid lines represent the intercept

trajectory and the triangles show the missile attitudes at various points. Fig. 10(b) presents the LOS angle and the

flight path angle. Since the magnitude of LOS error becomes smaller than 30 degrees at 2.14 sect = after launch,

the transition from the agile turn phase to the terminal homing phase is made from 1 2.14 secTt = until

2 3.14 secTt = in order to avoid an abrupt command change. Figs. 10(c) and 10(d) show the acceleration and the

control fin deflection angle responses, respectively. The outer loop for acceleration control is first employed at

1 2.14 secTt = with PNG command. After the end of the transition at 2 3.14 secTt = , the acceleration controller can

track its command quite well. Fig. 10 (e) is the response of angle-of-attack. Although the outer loop command is

changed from the reference angle-of-attack profile to the acceleration produced by PNG during the intercept of a

target, the inner loop angle-of-attack controller shows good tracking performance, as shown in Fig. 10(e). The

velocity response is given by Fig. 10(f). The velocity increases until boost burnout at 2.69 secofft = . After that, it

decreases during the terminal homing phase. As shown in Fig. 10(g) and 10(h), the uncertainties are also estimated

quite well during the intercept of a target.

VI. Conclusion

In this research, after investigating the agile turn maneuver based on the trajectory optimization, which revealed

that the agile missile system can undergo highly nonlinear, rapidly changing dynamics and aerodynamic

uncertainties during agile turns, we designed a nonlinear adaptive autopilot for controlling the angle-of-attack. The

proposed controller was constructed based on the backstepping control methodology in conjunction with a practical

time-delay adaptation scheme and tested in 6-DOF nonlinear simulations. Through the simulation studies, it was

shown that the proposed controller can guarantee good tracking performance, even in the presence of prescribed

17

difficulties, and time-delay adaptation scheme can estimate model uncertainties quite well. Furthermore, a planar

intercept simulation, including the angle-of-attack, acceleration controller, and control command blending logic,

showed that the proposed control methodology can be applied to the challenging issues of agile missile autopilots.

Acknowledgments

This research was supported by Agency for Defense Development under the contract ADD-09-01-03-03.

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Captions for Figures and Tables

Fig. 1 The missile axes, the pitch plane, and the definition of aerodynamic coefficients in the pitch plane

Fig. 2 The missile geometry for trajectory optimization

Fig. 3 The results for trajectory optimization

Fig. 4 The description of the convergent ellipsoid

Fig. 5 The structure of the acceleration controller, augmented by linear control loop

Fig. 6 The weighing function profile for the control command blending method

Fig. 7 The step responses for angle-of-attack control without uncertainties

Fig. 8 The step responses for angle-of-attack control with uncertainties

Fig. 9 The responses for a 180 degree turn with uncertainties

Fig. 10 The responses for an intercept scenario

Table 1 The Missile Model Parameters

20

Table 2 The Initial Conditions for the Engagement Scenario

Figures

Fig. 1 The missile axes, the pitch plane, and the definition of aerodynamic coefficients in the pitch plane

21

Fig. 2 The missile geometry for trajectory optimization

0 0.5 1 1.5 2 2.5 3200

300

400

500

600

700

800

900

1000

Time (sec)

V (m

/s)

Velocity Profiles

γf = 120°

γf = 180°

γf = 240°

0 0.5 1 1.5 2 2.5 30

10

20

30

40

50

60

Time (sec)

α ( °

)

Angle-of-attack Profiles

γf = 120°

γf = 180°

γf = 240°

(a) Velocity Profiles (b) Angle-of-attack Profiles

22

0 0.5 1 1.5 2 2.5 30

50

100

150

200

250

Time (sec)

γ ( °

)

Flight Path Angle Profiles

γf = 120°

γf = 180°

γf = 240°

-600 -400 -200 0 200 400 600 8000

200

400

600

800

1000

1200

X (m)

Y (m

)

Missile Trajectories

γf = 120°

γf = 180°

γf = 240°

(c) Flight Path Angle Profiles (d) Flight Trajectories

0 0.5 1 1.5 2 2.5 30

20

40

60

80

100

120

Time (sec)

Turn

Rat

e ( °/

sec)

Turn Rate Profiles

γf = 120°

γf = 180°

γf = 240°

0 0.5 1 1.5 2 2.5 3

0

0.5

1

1.5

2

2.5

3

3.5

4

Time (sec)

T *

sin(

α )

/ LRatio of Thrust to Lift for Generating Turn Rate

γf = 120°

γf = 180°

γf = 240°

(e) Turn Rate Profiles (f) Ratio of Thrust to Lift for Generation of Turn Rate

Fig. 3 The results for trajectory optimization

23

Fig. 4 The description of the convergent ellipsoid

Fig. 5 The structure of the acceleration controller, augmented by linear control loop

Fig. 6 The weighing function profile for the control command blending method

24

0 1 2 3 4 5 6 7 80

5

10

15

Time (sec)

α ( °

)

Angle-of-attack Response

CommandResponse

0 1 2 3 4 5 6 7 8

-15

-10

-5

0

5

Time (sec)

δ ( °

)

Fin Deflection Angle Response

(a) Angle-of-attack (b) Control Fin Deflection Angle

Fig. 7 The step responses for angle-of-attack control without uncertainties

0 1 2 3 4 5 6 7 80

5

10

15

Time (sec)

α ( °

)

Angle-of-attack Response

CommandAdaptation OffAdaptation On

0 1 2 3 4 5 6 7 8

-15

-10

-5

0

5

10

Time (sec)

δ ( °

)

Fin Deflection Angle Response

Adaptation OffAdaptation On

(a) Angle-of-attack (b) Control Fin Deflection Angle

0 1 2 3 4 5 6 7 8-0.2

-0.15

-0.1

-0.05

0

0.05

Time (sec)

Δ1

Estimation of Δ1

TrueEstimation

0 1 2 3 4 5 6 7 8

-50

-40

-30

-20

-10

0

10

20

Time (sec)

Δ2

Estimation of Δ2

TrueEstimation

(c) Estimation of 1Δ (d) Estimation of 2Δ

Fig. 8 The step responses for angle-of-attack control with uncertainties

25

0 0.5 1 1.5 2 2.5 30

5

10

15

20

25

30

35

40

Time (sec)

α ( °

)

Angle-of-attack Response

CommandAdaptation OffAdaptation On

0 0.5 1 1.5 2 2.5 3

-30

-25

-20

-15

-10

-5

0

5

10

15

Time (sec)

δ ( °

)

Fin Deflection Angle Response

Adaptation OffAdaptation On

(a) Angle-of-attack (b) Control Fin Deflection Angle

0 0.5 1 1.5 2 2.5 3-50

0

50

100

150

200

Time (sec)

γ ( °

)

Flight Path Angle Response

Adaptation OffAdaptation On

0 0.5 1 1.5 2 2.5 3200

300

400

500

600

700

800

900

Time (sec)

V (m

/s)

Velocity Response

Adaptation OffAdaptation On

(c) Flight Path Angle (d) Velocity

0 0.5 1 1.5 2 2.5 3-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Time (sec)

Δ1

Estimation of Δ1

TrueEstimation

0 0.5 1 1.5 2 2.5 3

-40

-30

-20

-10

0

10

20

Time (sec)

Δ2

Estimation of Δ2

TrueEstimation

(e) Estimation of 1Δ (f) Estimation of 2Δ

Fig. 9 The responses for a 180 degree turn with uncertainties

26

-2000 -1500 -1000 -500 0

-400

-200

0

200

400

600

800

1000

1200

1400

X (m)

Y (m

)

Missile

Target

0 1 2 3 4 5 6 7

-50

0

50

100

150

200

250

300

Time (sec)

Flight Path and LOS Angle Response

γ &

LO

S ( °

)

Fligh Path AngleLine-of-sight Angle

Agile Turn Terminal HomingTransition

(a) Trajectory (b) Flight Path Angle and LOS Angle

0 1 2 3 4 5 6 7-10

0

10

20

30

40

50

60

70

Time (sec)

a n (g)

Normal Acceleration Response

CommandResponse

Agile Turn Transition Terminal Homing

0 1 2 3 4 5 6 7-30

-25

-20

-15

-10

-5

0

5

10

Time (sec)

δ ( °

)

Fin Deflection Angle Response

Agile Turn Terminal HomingTransition

(c) Acceleration (d) Control Fin Deflection Angle

0 1 2 3 4 5 6 70

5

10

15

20

25

30

35

40

Time (sec)

α ( °

)

Angle-of-attack Response

CommandResponse

Agile Turn Terminal HomingTransition

0 1 2 3 4 5 6 7200

250

300

350

400

450

500

550

600

650

Time (sec)

V (m

/s)

Velocity Response

Terminal HomingAgile Turn Transition

(e) Angle-of-attack (f) Velocity

27

0 1 2 3 4 5 6 7-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

Time (sec)

Δ1

Estimation of Δ1

TrueEstimation

0 1 2 3 4 5 6 7-40

-30

-20

-10

0

10

20

Time (sec)

Δ2

Estimation of Δ2

TrueEstimation

(g) Estimation of 1Δ (h) Estimation of 2Δ

Fig. 10 The responses for an intercept scenario

28

Tables

Table 1. The Missile Model Parameters

Parameters Values

Mass, m 91.96 ~ 56.30 kg

Pitching Moment of Inertia, yyI 59.80 ~ 44.36 2kg m

True Location of CG, cgx 1.536 ~ 1.221 m

Reference Location of CG, ,cg refx 1.536 m

Reference Length, l 0.1524 m

Reference Area, S 0.01824 2m

Averaging Thrust, T 34,000 N

Thrust Off Time, offt 2.69 sec

Table 2. The Initial Conditions for the Engagement Scenario

Parameters Values

Missile position at launch, ( )0, 0X Y ( )0,0 m

Missile flight path angle at launch, 0γ 0°

Missile velocity at launch, 0V 250 m/ s

Target initial position, ( ),0, ,0T TX Y ( )2000,1000 m−

Target initial flight path angle, ,0Tγ ( ) ( )10 ,0 0 ,0tan /T TY Y X X− − −

Target velocity, TV 200 m/ s