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Missile Guidance Law Considering Constraints on Impact Angleand Terminal Angle of Attack

by

HYEONGGEUN Kim

THESIS

Presented to the Faculty of the Graduate School of

Seoul National University

in Partial Fulfillment

of the Requirements

for the Degree of

MASTER OF SCIENCE

Department of Mechanical and Aerospace Engineering

Seoul National University

Supervisor : Associate Professor H. Jin Kim

JANUARY 2014

to my

MOTHER, FATHER, and SISTER

with love

ii

Abstract

This paper proposes a guidance law considering constraints on impact angle and terminal

angle of attack for a homing missile. In the proposed structure, the guidance law generates

angle of attack command and the controller tracks the generated command. For deriving the

angle of attack command, the differential game problem with terminal boundary conditions

is proposed and solved. Then, the sliding mode control is applied in order to derive the

actual command input from the guidance command. Because the guidance command is

the angle of attack, the terminal angle of attack constraint can be easily handled and

the controller needs not deal with non-minimum phase characteristics. This capability to

control the terminal angle of attack is the main contribution of the paper. The performance

of the proposed law is evaluated using a two-dimensional nonlinear simulation. The result

demonstrates that the proposed law allows the missile to intercept the maneuvering target

with the constraints on impact angle and terminal angle of attack.

iii

Table of Contents

Page

Table of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vi

Chapter

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Thesis Contribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.1 Guidance Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2.2 Missile Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3 Guidance Law and Autopilot Design . . . . . . . . . . . . . . . . . . . . . . . . 10

3.1 Derivation of the Terminal Angle of Attack Constrained Guidance Law . . 10

3.2 Angle of Attack Controller Design . . . . . . . . . . . . . . . . . . . . . . . 14

3.3 Angle of Attack Observer Design . . . . . . . . . . . . . . . . . . . . . . . 15

3.4 Synthesis of Guidance Law, Controller and Observer . . . . . . . . . . . . 16

4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

4.1 Performance Analysis of TAACG . . . . . . . . . . . . . . . . . . . . . . . 18

4.2 Comparison with Other Guidance Laws . . . . . . . . . . . . . . . . . . . . 19

5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

iv

List of Tables

4.1 Homing Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4.2 Approximated Missile Parameters . . . . . . . . . . . . . . . . . . . . . . . 22

4.3 Average Result of Terminal Values . . . . . . . . . . . . . . . . . . . . . . 22

v

List of Figures

2.1 Two dimensional engagement geometry. . . . . . . . . . . . . . . . . . . . . 6

3.1 Angle of attack response in controller-observer synthesis . . . . . . . . . . . 17

4.1 Missile and target trajectories for various terminal impact angle constraints. 23

4.2 Flight path angle difference for various terminal impact angle constraints. . 23

4.3 Angle of attack for various terminal impact angle constraints. . . . . . . . 24

4.4 Missile angle for various terminal impact angle constraints. . . . . . . . . . 24

4.5 Missile and target trajectories by BPNG, TACG and TAACG. . . . . . . . 25

4.6 Flight path angle differences by BPNG, TACG and TAACG. . . . . . . . . 25

4.7 Guidance command and corresponding response by BPNG. . . . . . . . . . 26

4.8 Guidance command and corresponding response by TACG. . . . . . . . . . 26

4.9 Guidance command and corresponding response by TAACG. . . . . . . . . 27

4.10 Angle of attack by BPNG, TACG and TAACG. . . . . . . . . . . . . . . . 27

4.11 Guidance command and corresponding response by TACG at terminal stage

of the homing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.12 Guidance command and corresponding response by TAACG at terminal

stage of the homing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.13 Tail-fin deflection by TACG and TAACG at terminal stage of the homing. 29

vi

1Introduction

Proportional navigation guidance (PNG) has been widely used because it is known to be an

optimal solution for minimizing the flight energy and terminal miss distance to a stationary

target [1]. However, simple PNG is not enough to be applied if the target is armored

or has a missile defense system like CIWS (close-in weapon system). To incapacitate

this defense system and intercept the target effectively, guidance laws that perform the

additional capability, e.g. controlling impact angle and impact time, have been investigated

[2-11]. Particularly, control of the impact angle is an important factor in order to destroy

the weak point on the target effectively, and various approaches have been proposed for

fulfilling it. Also, concerned studies for the impact angle with additional task have been also

proposed so as to satisfy more performance capability. But in many impact angle control

guidance laws, the actual terminal attitude angle was not considered directly because the

missile is assumed to be a particle model. Therefore, an extended approach that considers

the terminal angle of attack is needed to be studied.

1

1.1 Literature Review

Studies about controlling impact angle have been proposed over the past several decades.

In [2], the impact attitude angle control law was proposed for reentry vehicles using linear

quadratic control problem. In [3], biased PNG (BPNG) form guidance law was proposed for

intercepting a moving and maneuverable target. The capturability of the guidance law with

the desired impact angle is evaluated by Lyapunov candidate function. In [4], a generalized

form of the optimal guidance law considering the terminal impact angle constraint was

studied. A corresponding time-to-go estimation method was also provided. A composite

guidance law that adjusts the navigation gain of PNG for satisfying the impact angle

constraint was proposed in [5]. The works in [6] proposed an impact angle constrained

guidance law considering nonlinear dynamics. For deriving the guidance law, the biased

proportional navigation form was applied.

Techniques for impact angle control have been also applied to the missile guidance law

with additional tasks. The study in [7] suggested the guidance law that can control both

impact angle and impact time simultaneously. The works in [8] devised this law for the

engagement situations considering moving target. A guidance law for the same purpose

was proposed in [9]. In this work, the line-of-sight rate shaping technique and second-order

sliding mode approach are applied for satisfying the impact angle and time constraints.

In many impact angle control guidance laws such as those mentioned above, the terminal

attitude angle of the missile is assumed to be equal to the terminal flight path angle by

neglecting the angle of attack. However in many engagement situations with a maneuvering

target, the guidance law usually generates a large lateral acceleration command at the

terminal stage of the homing, and it means that the large terminal angle of attack is

induced. Therefore, the terminal attitude angle is not equal to the terminal flight path

angle, which may make the missile glance or bounce off the armored target such as tank

or warship, instead of effectively penetrating

For handling this problem, impact angle control guidance laws with terminal accel-

2

eration constraint have been proposed because the angle of attack is approximately pro-

portional to the lateral acceleration for an axis symmetric missile. The study in [10] dealt

with an optimal guidance law which has impact angle and terminal acceleration constraints

against a fixed target. In [11], the guidance law based on time-to-go polynomial was sug-

gested for the same purpose. But these laws are limited in application because they are

designed for intercepting the fixed target. In addition, the approximation on the relation-

ship between the angle of attack and the lateral acceleration may not be accurate due to

the effect of the tail-fin at the terminal stage of the homing.

1.2 Thesis Contribution

In this paper, we propose a guidance and control law that can satisfy both impact angle and

the terminal angle of attack constraints against a maneuvering target. For deriving this law

which will be called TAACG (Terminal angle of attack constrained guidance), the angle

of attack is included in the engagement kinematics equations. With these equations, the

linear quadratic cost function with terminal boundary conditions is defined. Miss distance,

the flight path angle, and the angle of attack at the impact moment are used as boundary

conditions. The use of the angle of attack rather than the lateral acceleration allows more

accurate control of the terminal angle of attack, compared with the traditional approaches.

Finally, the angle of attack command is derived as a guidance command like lateral

acceleration in conventional guidance laws, and an autopilot for tracking the angle of attack

command is connected to the guidance loop. In this regard the performance of the autopilot

is expected to be enhanced in comparison with conventional acceleration-tracking autopilot

because the angle of attack has a minimum phase characteristics in a tail-fin controlled

missile while the acceleration has a non-minimum phase characteristics. Therefore, more

accurate satisfaction of the terminal boundary conditions is expected with the proposed

law.

3

1.3 Thesis Outline

The rest of this paper consists of four chapters as follows. In Chapter 2, the homing

problem and boundary conditions are formulated. Next, the guidance law and the autopilot

are designed in Chapter 3, and numerical simulations are performed to demonstrate the

proposed in Chapter 4. Finally, conclusions of this work are made in Chapter 5.

4

2Problem Formulation

In this chapter, engagement kinematics and the missile configuration are considered. En-

gagement is assumed to be in two-dimensional plane and the missile configuration is treated

as longitudinal dynamics. In Section 2.1, the equations of engagement kinematics are for-

mulated and modified to derive the guidance law. The angle of attack is involved in the

equations of motion to deal with its terminal constraint. In Section 2.2, the short period

approximation of the longitudinal dynamics is considered for the tail-fin controlled missile.

2.1 Guidance Formulation

XZ plane is considered for engagement kinematics between the missile and the target as

shown in Fig. 2.1. The position of the missile and the target is denoted as (xM , zM)

and (xT , zT ) respectively in inertial coordinate frame OIXIZI . The missile moves with

constant speed VM and maneuvers with acceleration aM which is perpendicular to the

velocity vector. Thus the flight path angle of the missile γM changes in proportion to the

maneuvering acceleration aM .

In the fin-controlled missile dynamics, maneuvering acceleration is induced by aerody-

5

Mg

Tg

IZ

( , )M Mx z

MV

TV

Ma

IO

dg

DO

DX

DZ

IX

( , )T Tx z

Ta

Figure 2.1: Two dimensional engagement geometry.

namic forces. Particularly, the acceleration normal to the velocity is directly induced by

the aerodynamic lift forces expressed by

L = QS (CLαα + CLδδ) (2.1)

where L, α and δ represent the life force, the angle of attack and fin deflection angle, and S

and Q represent the missile reference area and the dynamic pressure respectively. CLα and

CLδdenote the aerodynamic coefficient of the lift force. The missile form is assumed to be

symmetric. During the flight, the lift force by the fin deflection is negligible in comparison

to the total lift force. Also, variation of each term is negligible except for α. Then the

maneuvering acceleration in Fig. 1 can be written as

aM = Lαα (2.2)

where

Lα , QSCLα/m. (2.3)

The variation of the missile mass m is negligible too. Therefore defined term Lα is consid-

ered as constant.

The target is assumed to move with constant speed VT and maneuvering acceleration

aT . Thus, the flight path angle of the target γT changes in proportion to the acceleration

6

aT . Then, the equations of motion in this plane are expressed as

xM = VM cos γM , zM = −VM sin γM (2.4)

xT = VT cos γT , zT = −VT sin γT (2.5)

γM = Lαα/VM , γT = aT/VT . (2.6)

In this homing problem, the objective of the missile is intercepting the target with the

desired impact angle and the zero terminal angle of attack. Thus, the terminal value

constraints are as follows.

xM(tf ) = xT (tf ), zM(tf ) = zT (tf )

γM(tf ) = γT + γd, α(tf ) = 0 (2.7)

where γd represents the desired impact angle between the missile and the target. In here,

for considering the angle of attack constraint, the angle of attack rate η can be introduced

as

α = η (2.8)

From the above equations and boundary conditions, we can reformulate the equations of

motion in the desired impact frame ODXDZD expressed in Fig. 2.1. The frame ODXDZD

is rotated γT + γd from the inertial frame and has the origin on the position of the target,

which is expressed as xM

zM

=

cos (γT + γd) − sin (γT + γd)

sin(γT + γd) cos(γT + γd)

xM

zM

−

xT

zT

(2.9)

γM = γM − (γT + γd) . (2.10)

where the bar¯denotes the variables in the desired impact frame. Using the above desired

impact frame, terminal boundary conditions of the missile in (2.7) become zero value con-

straints, which allows to use the small angle assumption on the flight path angle. Then,

7

the reformulated equations with respect to the position of the missile in the desired impact

frame are linearized as follows.

ξ′ = Aξ +Bη + C (2.11)

where the state vector ξ is

ξ = [ zM γM α ]T (2.12)

and the corresponding terminal boundary conditions are

zM(xMf) = γM(xMf

) = α(xMf) = 0. (2.13)

The matrices A, B and C are given by

A =

0 VM/Vr,x 0

0 0 −Lα/(VMVr,x)

0 0 0

, B =

0

0

−1/Vr,x

, C =

VT,z

aT/(VTVr,x)

0

(2.14)

where VT,x and VT,z are the target velocity components of x-axis and z-axis direction re-

spectively and Vr,x is the relative velocity component of x-axis direction. This velocity

components can be expressed as

VT,x = VT cos γd, VT,z = VT sin γd (2.15)

Vr,x = VT cos γd − VM cos γM ≈ VT,x − VM . (2.16)

Also, Lα represents the estimated value of the lift coefficient Lα and ′ denotes the derivative

which respect to xM . Applying the downrange derivative to the equations, we do not have

to estimate the time-to-go which is more difficult to estimate than the range-to-go.

2.2 Missile Dynamics

Prior to designing the guidance law and the autopilot, let us set the missile dynamics

equations. The short period approximation of the longitudinal dynamics is considered for

the tail-fin controlled missile. Thus, the missile has one input channel and the total speed

8

of the missile is assumed to be constant. Then, the missile equations of motion in the XZ

pitch plane are expressed as

α = q − L/(mVM) (2.17)

θ = q (2.18)

q = M/Iyy (2.19)

where the lift force L and the pitch moment M can be approximated as

L/m = Lαα + Lδδ (2.20)

M/Iyy = Mαα + Mqq + Mδδ. (2.21)

θ, q and Iyy represent the pitch angle, pitch rate and the moment of inertia of pitch axis

respectively. L(·) and M(·) are weighted values of each state about lift and pitch moment

respectively, and the hatˆdenotes the approximated values which can be estimated.

In the longitudinal missile dynamics, it is difficult to measure the maneuvering accelera-

tion aM directly so that it is converted to measurable z-direction acceleration approximated

as

az = −aM cosα = −L

mcosα. (2.22)

In our work, the acceleration az is used to track other guidance laws for performance

comparison with proposed law which generates the angle of attack command.

The actuator systems for the tail-fin deflection of the missile incurs time lag. Thus the

actuator dynamics can be approximated as a first-order time delay system expressed as

δ ≈ δc − δ

τδ. (2.23)

Here, δc denotes the fin deflection command from a controller and τδ denotes the time delay

constant.

9

3Guidance Law and Autopilot Design

In this chapter, TAACG and corresponding autopilot is designed. The angle of attack

command is derived from the guidance law and the controller is designed to track the

command. Because the angle of attack is difficult to be measured, the observer is designed

and synthesized to the guidance law and the controller. Section 3.1 deals with the derivation

of the angle of attack command by solving the differential game problem. Then controller

and observer are designed in order to track the command and estimate the angle of attack

in section 3.2 and section 3.3 respectively, and these are combined with the guidance law

in section 3.4.

3.1 Derivation of the Terminal Angle of Attack Constrained

Guidance Law

We can formulate the differential game problem for satisfying the terminal value constraints

in (2.13) with the following cost function:

J =1

2bz2Mf

+1

2cγ2Mf

+1

2dα2f +

1

2

∫ xMf

xM0

a2M − µa2T ds. (3.1)

10

where the subscript f denotes the final state variables, and the nonnegative constants b,

c and d represent the weights on the terminal miss, flight path angle and angle of attack

respectively. By modulating the weights, the intensity of the corresponding constraints can

be adjusted. The design parameter µ represents the weight on the target maneuverability.

Like the preceding weights, the characteristics of the guidance law can be adjusted according

to the target maneuverability. Note that optimizing the cost function in (3.1) minimizes or

maximizes miss, flight path angle error and angle of attack at the end of the homing and

the control input of each agent. That is, the control input of the missile can be derived by

minimizing the cost function, and the unknown acceleration of the target can be assumed

by letting the cost function maximize.

In order to obtain the optimal solution of the problem, the Hamiltonian can be given

by

H =1

2a2M − 1

2µa2T + λz

(VM

Vr,x

γM +VT,z

Vr,x

)+ λγ

(− Lα

VMVr,x

α +1

VTVr,x

aT

)+ λα

(− 1

Vr,x

α

)(3.2)

where the acceleration rate can be approximated as aM = Lαα. From the necessary condi-

tions for optimality, the adjoint equations are written as

λ′z = 0, λ′

γ = − VM

Vr,x

, λ′α =

Lα

VMVr,x

(3.3)

where corresponding terminal boundary conditions are

λz(xMf) =

1

bzMf

, λγ(xMf) =

1

cγMf

, λα(xMf) =

1

dαf . (3.4)

Integrating (3.3) from the terminal conditions in (3.4) yields

λz =1

bzMf

, λγ =VM

bVr,x

xgozMf+

1

cγMf

, λα = − Lα

2V 2r,xb

− Lα

cVMVr,x

+1

dαf (3.5)

where xgo ≡ xT − xM = −xM . From the optimality condition and solutions of the adjoint

equations, the optimal solution of the differential game is obtained as

α = − Lα

2bV 3r,x

x2gozMf

− Lα

cVMV 2r,x

xgoγMf+

1

dVr,x

αf (3.6)

aT =VM

µbVTV 2r,x

xgozMf+

1

µcVTVr,x

γMf(3.7)

11

For the implementation of the above optimal solution, it needs to be expressed in terms of

the current state variables instead of the final state variables. Substituting α into (2.11)

and integrating backward from the terminal conditions give the current state variables

expressed as

zM =

(1

120bV 6r,x

x5go +

V 2M

6µbV 2T V

4r,x

x3go + 1

)zMf

+

(1

24cVMV 5r,x

x4go +

VM

2µcV 2T V

3r,x

x2go −

VM

Vr,x

xgo

)γMf

+

(− 1

6dLαV 4r,x

x3go −

Lα

2V 2r,x

x2go

)αf −

VT,z

Vr,x

xgo (3.8)

γM =

(− 1

24bVMV 5r,x

x4go −

VM

2µbV 2T V

3r,x

x2go

)zMf

+

(− 1

6cV 2MV 4

r,x

x3go −

1

µcV 2T V

2r,x

xgo + 1

)γMf

+

(1

2dLαVMV 3r,x

x2go +

Lα

VMVr,x

xgo

)αf (3.9)

α =

(− 1

6bLαV 4r,x

x3go

)zMf

+

(− 1

2cVM LαV 3r,x

x2go

)γMf

+

(1

dL2αV

2r,x

xgo + 1

)αf . (3.10)

Solving the above linear simultaneous equation, the final state variables can be expressed

in terms of the current state variables. Therefore, the optimal solution in (3.6) can be

expressed in terms of the current state values as

α = KT ξ +DVT,z. (3.11)

12

where

K = k−1

−60µ2V 4T V

3r,xx

6go − 360µ2dL2

αV4T V

5r,xx

5go + 720µV 4

MV 2T V

5r,xx

4go

−2880µ2cV 2MV 4

T V7r,xx

3go − 4320µcdL2

αV2MV 4

T V9r,xx

2go

−36µ2VMV 4T V

2r,xx

7go − 252µ2dL2

αVMV 4T V

4r,xx

6go + 720µV 3

MV 2T V

4r,xx

5go

+(720µdL2

αV3MV 2

T V6r,x − 2880µ2cV 3

MV 4T V

6r,x

)x4go − 4320µ2cdL2

αV3MV 4

T V8r,xx

3go

−4320µ2bVMV 4T V

8r,xx

2go − 8640µ2bdL2

αVMV 4T V

10r,xxgo

9µ2LαV4T Vr,xx

8go + 72µ2dL3

αV4T V

3r,xx

7go − 312µLαV

2MV 2

T V3r,xx

6go

+(1152µ2cLαV

2MV 4

T V5r,x − 720µdL3

αV2MV 2

T V5r,x

)x5go +

(2160µ2cdL3

αV2MV 4

T V7r,x

+720LαV4MV 5

r,x

)x4go +

(2880µ2bLαV

4T V

7r,x − 2880µcLαV

4MV 2

T V7r,x

)x3go

+8640µ2bdL3αV

4T V

9r,xx

2go − 8640µbLαV

2MV 2

T V9r,xxgo + 8640µ2bcLαV

2MV 4

T V11r,x

(3.12)

D = k−1[− 60µ2V 4

T V2r,xx

7go − 360µ2dL2

αV4T V

4r,xx

6go + 720µV 2

MV 2T V

4r,xx

5go

−2880µ2cV 2MV 4

T V6r,xx

4go − 4320µ2cdL2

αV2MV 4

T V8r,xx

3go

](3.13)

k = µ2LαV4T x

9go + 9µ2dL3

αV4T V

2r,xx

8go − 72µLαV

2MV 2

T V2r,xx

7go +

(192µ2cLαV

2MV 4

T V4r,x

−312µdL3αV

2MV 2

T V4r,x

)x6go +

(432µ2cdL3

αV2MV 4

T V6r,x + 720LαV

4MV 4

r,x

)x5go

+(720dL3

αV4MV 6

r,x − 2880µcLαV4MV 2

T V6r,x + 720µ2bLαV

4T V

6r,x

)x4go(

2880µ2bdL3αV

4T V

8r,x − 2880µcdL3

αV4MV 2

T V8r,x

)x3go − 8640µbLαV

2MV 2

T V8r,xx

2go(

8640µ2bcLαV2MV 4

T V10r,x − 8640µbdL3

αV2MV 2

T V10r,x

)xgo + 8640µ2bcdL3

αV2MV 4

T V12r,x. (3.14)

Then, the angle of attack command is derived by integrating the optimal solution in (3.11)

as follows.

αc =

∫ t

0

α dτ. (3.15)

We obtain the guidance law by solving the differential game problem. Note that the

guidance law does not generate the acceleration command but generates the angle of attack

command. Commonly, the missile dynamics where the angle of attack is controlled has

minimum phase characteristics contrary to the acceleration-controlled dynamics which is

13

known to be the non-minimum phase. Thus, we can apply the straightforward feedback

linearization approach for tracking the angle of attack command.

3.2 Angle of Attack Controller Design

Now, we can design the sliding mode controller because the straightforward feedback lin-

earization can be applied. Let us define the sliding surface for tracking the angle of attack

command as follows.

s = e+ c1e+ c2

∫edτ (3.16)

where the error e is defined as

e = α− αc (3.17)

Taking the first derivative of the sliding surface leads to

s = α− αc + c1 (α− αc) + c2 (α− αc)

= f + gδc (3.18)

where

f = q − Lα

VM

α +Lδ

τδVM

δ − αc + c1 (α− αc) + c2 (α− αc)

g = − Lδ

τδVM

. (3.19)

Then, the equivalent control command can be given by

δceq = −g−1f. (3.20)

As a result, the fin deflection command is derived as

δc = −g−1 (f + k1s+ k2sgn(s)) (3.21)

where constants k1 and k2 are chosen to be positive. In order to analyze the stability of

the controller, Lyapunov candidate function can be given by

V =1

2s2. (3.22)

14

The time derivative of the candidate function is

V = ss = −k1s2 − k2|s| ≤ 0 (3.23)

Therefore, the control input guarantees the tracking of the angle of attack to the command

in (3.15).

3.3 Angle of Attack Observer Design

The guidance law and the autopilot developed in preceding sections are connected by

the angle of attack command. In this structure, information about the angle of attack is

necessary; however, it is difficult to measure the angle of attack in an actual missile system.

This section gives the observer design that can provide the approximated estimation of the

unknown angle of attack based on measurement of other variables such as the pitch rate

and the output acceleration. Simple technique in [12] is used to design the observer. This

structure is given by

˙α = q − Lαα + Lδδ

VM

−Ka

(az +

L

mcos α

)(3.24)

y = α (3.25)

where az represents the output acceleration having z-direction in the missile coordinate

frame, and Ka represents the gain constant selected as positive. Through the small angle

assumption that allows cos α = cosα, the error dynamics of the observer is expressed as

˙α + Lα

(1

VM

+Ka cosα

)α =

(1

VM

+Ka cosα

)∆α (3.26)

where the angle of attack error is

α = α− α, (3.27)

and the disturbance term is given by

∆α = −(Lα − Lα)α− (Lδ − Lδ)δ. (3.28)

15

If the lift coefficients Lα and Lδ are estimated with satisfactory accuracy, the disturbance

term becomes negligible. Then, the solution of the error dynamics converge to zero because

the coefficient Lα has positive value.

3.4 Synthesis of Guidance Law, Controller and Observer

In an actual missile system, the observer loop is connected to the control loop in order to

estimate the unmeasurable state variables. Likewise, the guidance law and the controller

developed in the preceding sections are combined with the observer for using the angle of

attack estimates. Then, the guidance input in (3.15) and control input in (3.21) can be

expressed as

αc =

∫ t

0

(KT ξ +DVT,z

)dτ (3.29)

δc = −g−1(f + k1s+ k2sgn(s)) (3.30)

˙α = q − Lαα + Lδδ

VM

−Ka

(az +

L

mcos α

)(3.31)

where ξ, f and s are same as ξ, f and s in which argument α substitutes for α. Fig. 3.1

shows the angle of attack response in controller-observer synthesis. It is observed that the

output is biased from the command when an uncertainty term exists. Combined simulation

about synthesis of guidance law, controller and observer is in Chapter 4.

16

0 2 4 6 8 10−4

−2

0

2

4

6

8

Time [sec]

Ang

le o

f atta

ck [d

eg]

CommandOutput (w/o uncertainty)Output (w/ uncertainty)

Figure 3.1: Angle of attack response in controller-observer synthesis .

17

4Simulation Results

In this section, the proposed TAACG is investigated through two subsections. In the

section 4.1, the characteristics of TAACG are analyzed by simulating with various terminal

conditions. The section 4.2 compares TAACG with other impact angle control guidance

laws. A non-stationary and maneuvering target is considered for simulations.

4.1 Performance Analysis of TAACG

Homing conditions and approximated missile parameters are given in TABLE 4.1 and

TABLE 4.2 respectively. Coefficients in the missile dynamics are random variables within

±20% from the approximated values. Nonlinear simulations in four different situations

depending on the desired impact angle are carried out.

Fig. 4.1 shows the missile and the target trajectories for different terminal impact

angles. It can be seen that the missile intercepts the target with impact angle constraints

by using TAACG. More detailed plots for the missile angle histories are given in Figs. 4.2

∼ 4.4. The desired impact angle equals the flight path angle difference between the missile

and the target. Fig. 4.2 shows the flight path angle difference, the terminal value of which

18

is the impact angle. We can see that the terminal values of the difference are approximately

equal to desired impact angle.

The corresponding angle of attack histories are given in Fig. 4.3, which shows that

the terminal angle of attack converges to zero. It means that the missile attitude angle

converges to the flight path angle at the terminal stage of the homing because the angle of

attack is the difference between the flight path angle and the missile attitude angle. Fig.

4.4 shows the time histories of the attitude angle which converges to the flight path angle

at the end of the homing.

4.2 Comparison with Other Guidance Laws

In this section, the performance of TAACG is compared with two guidance laws that

considers the terminal impact angle against the maneuvering target. One is the guidance

law based on the differential game with the following cost function:

J =1

2bz2Mf

+1

2cγ2Mf

+1

2da2Mf

+1

2

∫ xMf

xM0

a2M − µa2T ds. (4.1)

The above cost function is same as the cost function in (3.1) except that the angle of attack

constraint is replaced as the acceleration constraint. Therefore, the corresponding guidance

law has constraints on impact angle and terminal acceleration against the maneuvering

target, which will be called TACG (Terminal acceleration constrained guidance). The other

is BPNG in [3] that considers the impact angle but does not have constraint on terminal

acceleration against the maneuvering target. These two laws generate the acceleration

command so that an autopilot tracking the acceleration command is needed in order to

investigate the terminal angle of attack.

In an tail-fin controlled missile, it is difficult to use straightforward feedback linearization

approach due to its non-minimum characteristics. Thus control techniques that add a

classical PI feedback on the nonlinear angle of attack control were proposed using minimum

phase characteristics of the angle of attack[13-16]. Similar approach applies to the design

19

of autopilot connected to TACG and BPNG by connecting the PI control to the angle of

attack control in Chapter 3.2 as follows.

αc = kP (acz − az) + kI

∫ t

0

(acz − az)dτ (4.2)

where acz is the acceleration command from TACG and BPNG, and αc becomes the reference

command of the angle of attack controller.

To compare TAACG with TACG and BPNG, the same nonlinear simulations whose

conditions are given in TABLE 4.1 and 4.2 are performed with 100 sample runs except

that the desired impact angle is fixed to −90◦ as shown in Fig. 4.5. It is observed that all

three laws intercept the maneuvering target with the desired impact angle approximately,

but the average angle values of each law in TABLE 4.3 give the result that TACG and

TAACG satisfy the constraint on impact angle better then BPNG. Fig. 4.6 also shows the

similar result by plotting flight path angle difference between the missile and the target.

It is because BPNG generates larger acceleration command which cannot be tracked by

the autopilot at the terminal stage of the homing as shown in Fig. 4.7. In contrast,

the acceleration command from TACG and the angle of attack command from TAACG

converge to near zero at the terminal stage as shown in Figs. 4.8 and 4.9 respectively,

which makes the autopilot track the command with relative ease.

Fig. 4.10 provides the angle of attack history of each law. At the early stage of the

homing, TACG and TAACG have the larger angle of attack in comparison with BPNG.

However as time goes on, the necessary angle of attack with BPNG increases and finally

diverges at the terminal stage like the acceleration command while TACG and TAACG

make the angle of attack converge to near zero at the terminal stage. That is, TACG and

TAACG have more uniform distribution of the angle of attack and show the much smaller

terminal angle of attack in comparison with BPNG.

Figs. 4.11 and 4.12 show the guidance command and corresponding response of TACG

and TAACG respectively in one second just before the end of the homing. It can be seen

that the controller tracking the TAACG command has faster the response speed in compari-

20

son with the controller tracking the TACG command. It is because the controller connected

to TACG consists of two control loops, PI controller and angle of attack controller, while

the controller connected to TACG consists of only the angle of attack controller. This

difference on response speed between TACG and TAACG results in the difference on sat-

isfaction of the terminal constraints given in TABLE 4.3 because a deviation between the

actual output and the guidance command causes a discrepancy between the actual state

and the desired state.

Fig. 4.13 gives the tail-fin deflection histories of the missile guided by TACG and

TAACG in 1 second just before the end of the homing. Both guidance laws show a rapid

change of fin deflection due to target maneuver, which may increase the effect of the tail-fin

with respect to the total life force. Then, the angle of attack does not converge to zero

although the acceleration converges to zero due to the effect of the tail-fin. As a result, in

rapid engagement situations, TAACG has the smaller terminal angle of attack because it

generates the angle of attack command near zero while TACG generates the acceleration

command near zero which does not mean the zero angle of attack at terminal stage of the

homing.

21

Table 4.1: Homing Conditions

Missile Target

Initial position (xM , zM) (0,−3000) m (4000, 0) m

Moving speed 300 m/s 100 m/s

Initial flight path angle 0◦ 0◦

Maneuvering acceleration 0.5g

Desired Impact angle 0◦, −30◦, −60◦, −90◦

Table 4.2: Approximated Missile Parameters

Lift coefficients Lα = 970m/s2, Lδ = 120m/s2

Moment coefficients Mα = −250s−2, Mq = −5s−1, Mδ = −180s−2

Fin time constant τδ = 0.05s

Actuator limitation |δ| ≤ 30◦, |δ| ≤ 450◦/s

Table 4.3: Average Result of Terminal Values

BPN TACG TAACG

Impact angle γMf− γTf

−105.3 deg −95.97 deg −88.97 deg

Terminal angle of attack |αf | 21.65 deg 11.97 deg 4.539 deg

22

0 1000 2000 3000 4000 5000 6000−500

0

500

1000

1500

2000

2500

3000

3500

4000

4500

Downrange [m]

Alti

tude

[m]

MissileTarget

γd=−90°

γd=−60°

γd=−30°

γd=0°

Figure 4.1: Missile and target trajectories for various terminal impact angle constraints.

0 5 10 15 20 25 30−140

−120

−100

−80

−60

−40

−20

0

20

40

Time [sec]

γ M −

γT [d

eg]

γd=−90°

γd=−60°

γd=−30°

γd=0°

Figure 4.2: Flight path angle difference for various terminal impact angle constraints.

23

0 5 10 15 20 25 30−6

−4

−2

0

2

4

6

8

Time [sec]

Ang

le o

f atta

ck [d

eg]

γd=−90°

γd=−60°

γd=−30° γ

d=0°

Figure 4.3: Angle of attack for various terminal impact angle constraints.

0 5 10 15 20 25 30−80

−60

−40

−20

0

20

40

60

80

100

Time [sec]

Mis

sile

ang

le [d

eg]

γM

θ γd=−90°

γd=−60°

γd=−30°γ

d=0°

Figure 4.4: Missile angle for various terminal impact angle constraints.

24

0 1000 2000 3000 4000 5000 60000

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

5500

Downrange [m]

Alti

tude

[m]

BPNTACGTAACG

Figure 4.5: Missile and target trajectories by BPNG, TACG and TAACG.

0 5 10 15 20 25−140

−120

−100

−80

−60

−40

−20

0

20

40

Time [sec]

γ M −

γT [d

eg]

BPNTACGTAACG

Figure 4.6: Flight path angle differences by BPNG, TACG and TAACG.

25

0 5 10 15 20 250

5

10

15

20

25

30

35

40

45

50

Time [sec]

Acc

eler

atio

n [g

]

azc

az

Figure 4.7: Guidance command and corresponding response by BPNG.

0 5 10 15 20 25−20

−15

−10

−5

0

5

10

15

20

Time [sec]

Acc

eler

atio

n [g

]

azc

az

Figure 4.8: Guidance command and corresponding response by TACG.

26

0 5 10 15 20 25−4

−2

0

2

4

6

8

10

Time [sec]

Ang

le o

f atta

ck [d

eg]

αc

α

Figure 4.9: Guidance command and corresponding response by TAACG.

0 5 10 15 20 25−25

−20

−15

−10

−5

0

5

10

15

20

25

Time [sec]

Ang

le o

f atta

ck [d

eg]

BPNTACGTAACG

Figure 4.10: Angle of attack by BPNG, TACG and TAACG.

27

24 24.2 24.4 24.6 24.8 25−10

−5

0

5

10

15

20

Time [sec]

Acc

eler

atio

n [g

]

azc

az

Figure 4.11: Guidance command and corresponding response by TACG at terminal stage

of the homing.

24 24.2 24.4 24.6 24.8 25−1

0

1

2

3

4

5

6

7

8

9

Time [sec]

Ang

le o

f atta

ck [d

eg]

αc

α

Figure 4.12: Guidance command and corresponding response by TAACG at terminal stage

of the homing.

28

24 24.2 24.4 24.6 24.8 25−15

−10

−5

0

5

10

Time [sec]

Fin

def

lect

ion

[deg

]

TACGTAACG

Figure 4.13: Tail-fin deflection by TACG and TAACG at terminal stage of the homing.

29

5Conclusion

This paper proposes a guidance law that generates the angle of attack command, unlike

conventional laws that generate the acceleration command. The derived angle of attack

command is tracked by using simple sliding mode control which involves straightforward

feedback linearization. In this approach, the angle of attack in the missile dynamics has

minimum phase characteristics, and the tail-fin deflection command that makes the angle

of attack track the command is derived. As a result, the better tracking performance can

be obtained.

With this law called TAACG, impact angle and the angle of attack can be controlled

simultaneously, and eventually the impact attitude angle can become equal to the desired

impact angle. By satisfying these constraints, the intercept capability against the armored

targets can be increased because the missile warhead collides the target directly.

Numerical simulations demonstrate the performance of TAACG, and it is compared

with other impact angle control guidance laws called BPNG and TACG with a maneuvering

target. Results show that TAACG satisfies the desired terminal conditions and performs

better than BPNG and TACG in terms of flight path angle accuracy and terminal angle

30

of attack magnitude. Particularly, it is shown that the technique using the angle of attack

as a guidance command has advantages by comparing TAACG with TACG, although the

estimated angle of attack is not correct due to uncertain missile parameters.

31

References

[1] A. E. Bryson, Jr. and Y. C. Ho, Applied optimal Control, Washington, DC: Hemi-

sphere, 1975.

[2] M. Kim and K. V. Grider, Terminal guidance for impact attitude angle constrained

flight trajectories, IEEE Transactions on Aerospace and Electronic Systems, Vol. 9,

No. 6, Nov 1973, pp. 852-859.

[3] B. S. Kim, J. G. Lee, and H. S. Han, Biased PNG law for impact with angular con-

straint, IEEE Transactions on Aerospace and Electronic Systems, Vol. 34, No. 1, Jan

1998, pp. 277-288.

[4] C. K. Ryoo, H. Cho, and M. J. Tahk, Optimal guidance laws with terminal impact

angle constraint, Journal of Guidance, Control, and Dynamics, Vol. 28, No. 4, Jul-Aug

2005, pp. 724-732.

[5] A. Ratnoo and D. Ghose, Impact angle constrained interception of stationary targets,

Journal of Guidance, Control, and Dynamics, Vol. 31, No. 6, Nov-Dec 2008, pp. 1816-

1821.

[6] G. Akhil and D. Ghose, Biased PN based impact angle constrained guidance using a

nonlinear engagement model, 2012 American Control Conference, Montreal, Canada,

Jun 27-29, 2012, pp. 950-955.

[7] J. I. Lee, I. S. Jeon, and M. J. Tahk, Guidance law to control impact time and angle,

IEEE Transactions on Aerospace and Electronic Systems, Vol. 43, No. 1, Jan 2007,

pp. 301-310.

32

[8] S. Kang and H. J. Kim, Differential game missile guidance with impact angle and

time constraints, Proceedings of International Federation of Automatic Control World

Congress, Milano, Italy, Aug 28-Sep 2, 2011, pp. 3920-3925.

[9] N. Harl and S. N. Balakrishnan, Impact time and angle guidance with sliding mode

control, IEEE Transaction on Control Systems Technology, Vol. 20, No. 6, Nov 2012,

pp. 1436-1449.

[10] Y. I. Lee, C. K. Ryoo, and E. Kim, Optimal guidance with constraints on impact angle

and terminal acceleration, AIAA Guidance, Navigation, and Control Conference and

Exhibit, Austin, Texas, 11-14, Aug 2003.

[11] C. H. Lee, T. H. Kim, M. J. Tahk, and I. H. Hwang, Polynomial guidance laws

considering terminal impact angle and acceleration constraints, IEEE Transaction on

Aerospace and Electronic Systems, Vol. 49, No. 1, Jan 2013, pp. 74-92.

[12] E. Devaud, H. Siguerdidjane, and Font. S, Nonlinear dynamic autopilot for the non

minimum phase missile, Proceedings of the 37th IEEE Conference on Decision and

Control, Tampa, Florida USA, Dec 1998, pp. 4691-4696.

[13] M. J. Tahk, M. M. Briggs, and P. K. Menon, Application of plant inversion via state

feedback to missile autopilot design, Proceedings of the 27th IEEE Conference on

Decision and Control, Austin, Texas, Dec 1988, pp. 730-735.

[14] M. J. Tahk and M. M. Briggs, An autopilot design technique based on feedback lin-

earization and wind angle estimation of bank-to-turn missile systems, Proceedings of

AIAA Missile Sciences Conference, Monterey, California, Nov 29-Dec 1, 1988.

[15] M. J. Tahk, Robustness characteristics of missile autopilot system based on feedback

linearization, JSASS Aircraft Symposium, Tokyo, Japan, Nov 1990, pp. 366-369.

[16] S. H. Kim and M. J. Tahk, Missile acceleration controller design using proportional-

integral and non-linear dynamic control design method, Proceedings of Institution of

33

Mechanical Engineers, Part G: Journal of Aerospace Engineering, Vol. 225, No. 11,

2011.

34

국 문 초 록

본 논문에서는 충돌각과 종말 받음각 제한조건을 가지는 미사일의 유도법칙을 제안한다. 설계된

유도법칙은 기존의 유도법칙들과 달리 받음각을 미사일의 유도명령으로 생성하며 이를 추종하는 자

동조종장치를 설계한다. 받음각 명령을 생성하기 위해 미분게임 기반의 최적제어문제를 정의하고

이를 통해 받음각 명령을 구한다. 유도된 받음각 명령을 추종하는 자동조종장치를 설계하기 위해서

슬라이딩 모드 제어 기법을 적용하고 이를 통해 미사일 자동조종장치의 실질적인 입력명령을 계

산한다. 유도명령이 받음각형태이기 때문에 종말 받음각 조건이 더 직접적으로 고려될 수 있으며,

자동조종장치의 제어기는 비최소위상 특성을 다룰 필요가 없게 되어 더 용이한 제어가 가능하다. 이

렇게 받음각을 직접적으로 다룰 수 있는 점이 본 유도법칙의 장점으로, 이차원 비선형 시뮬레이션을

통하여그성능을검증하였다.시뮬레이션결과는설계된유도법칙을통해미사일이기동하는표적에

대하여 충돌각과 종말 받음각 제한조건을 만족시킴을 보여준다.

주요어 : 미사일유도, 충돌각, 받음각, 종말 받음각, 받음각 제어, 비최소위상

학번 : 2012-20661