Derivation of Linearization Small Deviation Motion Equations of Blended Control System

Zhai Hua1, Liu Juan2, Gu Zhi-jun1,Zhou Bo-zhao1 1College of Aerospace and Material Engineering

National University of Defense Technology, Changsha, Hunan, 410073, China 2Beijing Institute of Space Systems Engineering, Beijing 100076, China

Abstract

The linearization small deviation motion model is the basis of stability analysis to attitude control system. Endoatmospheric tactics missile weapons widely adopt aerodynamic rudders as actuators, but only depending on aerodynamic rudders can’t satisfy with control requirements at the initial flight phase where the missile’s dynamic pressure is relatively minor. On the other hand, gas rudders are usually used at the starting control phase to augment the control force. In practice, the linkage mode with gas rudders and aerodynamic rudders is the better way to control the missile’s attitude. In this paper, the integrated dynamic model of the above blended control missile is established. And complete linearization small deviation motion equations on pitching, yawing and roll loops are derived in details. For this kind of tactics missiles, the research can provide technology supply to dynamic characteristic analysis and designing attitude control system, and establish the basis of the missile’s motion stability analysis. 1. Introduction

At present, frequency domain analysis means is the most universal method to analyze the stability of attitude control system, such as root locus means and logarithmic frequency domain means et al [1]. These methods are based on transfer function to synthesize the system. Whereas, a missile’s movement consists of a series of motion, including rigid body motion and elasticity vibration et al. The equations describing the missile’s movement are multidimensional nonlinear variable coefficient equations. In principle, these equations can’t change to algebra formulas by the Laplace transform, further make up the transfer function and settled by frequency domain analysis method. Therefore, the missile’s motion differential equations need some appropriate simplification when analyzing the stability of attitude control system.

The basic simplification treatments [2] are: (1) Under the condition of small deviation, the missile’s dimensional

motion can separated into patching, yawing and roll movement unattached each other. (2)Based on the supposing of small disturbance, the nonlinear equations can be dealt with linearization. (3) Using “freezing coefficient” method, the variable coefficients of the equations can regarded as the constant coefficients. The first two steps of the above method just are the linearization small deviation handling of the motion equations [3]. And then the linearization small deviation motion equations can obtained as the stability analysis basis of attitude control system.

Endoatmospheric tactics missile weapons widely adopt aerodynamic rudders as executive mechanism [4], but just depending on aerodynamic rudders can’t satisfy with control demands at the initial flight phase where the missile’s dynamic pressure is relatively minor. On the other hand, gas rudders are usually added at the starting control phase to augment the control force. Namely, the linkage mode with gas rudders and aerodynamic rudders is the better way to control the missile’s attitude. In this article, the complete linearization small deviation motion equations on pitching, yawing and roll loops are derived in details. These equations provide a basis for the missile’s motion stability analysis and the further design of attitude control system. 2. Dynamic model of blended control missile 2.1 Rigid body motion model

The missile is a mass changing system, but according to the “solidification” principle, the motion equations of this system can be equal to the rigid body motion equations by solidifying at the corresponding time [5]. Therefore, the missile is considered as a rigid body with fixed mass m . By the momentum and momentum moment principle, the motion equations of the rigid body are as follows:

dQ Fdt

= dG Mdt

= (1)

Here, Q and G is respectively the centroidal momentum vector and the momentum moment vector circling around

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the centroid; F and M separately is the outside force vector and the outside moment vector.

For convenience to the research, the missile’s motion analysis is in the active coordinate system (ACS). Therefore the foregoing formulas are rewritten in the ACS:

( )m v t w v F∂ ∂ + × = (2) H t w H M∂ ∂ + × = (3)

Where, v is the velocity vector of the centroid, F is the sum of the outside forces, H is the momentum moment, M is the sum of the outside moment.

In order to get the formally simplified quantity equations, the equation(2) is projected on the velocity coordinate system(VCS) and equation(3) is projected on the body coordinate system(BCS). Because the tactics missiles commonly are axisymmetric, the products of inertia are all zeros. If taking V to express the fight speed of the missile, the following formula can be set up in the VCS:

[ ,0,0]v V= (4) Then the motion equations of the rigid body are given:

.

.

cos( )

x

y

z

mV F

mV F

mV F

θ σ

σ

⋅

⎧ =⎪⎪

=⎨⎪⎪− =⎩

(5)

.

1 1 1 1

.

1 1 1 1

.

1 1 1 1

( )

( )

( )

xx z y y z x

yy x z z x y

zz y x x y z

J J J M

J J J M

J J J M

ω ω ω

ω ω ω

ω ω ω

⎧ + − =⎪⎪⎪ + − =⎨⎪⎪ + − =⎪⎩

(6)

Hereinto, xF , yF , zF are the projection of the outside force on the VCS; 1xω , 1yω , 1zω are the projection on the BCS of the angle velocity that in the BCS relative to the launch coordinate system(LCS), xJ , yJ , zJ are the moments of inertia relative to the BCS, 1xM , 1yM , 1zM are projection of the outside moment on the BCS.

As a rule, after linearization, dimensional movement can be decompounded to the plane motion unattached each other. But for the axisymmetric ballistic missile, its dimensional movement can be decompounded firstly to simplify the calculation and then separately make a linearization on the three equation groups describing a certain direction. Moreover, the missile is always flying close to the trajectory plane and all of the lateral parameters are very small. Then the simplified three loops equations are as follows:

.

.

1 1

x

y

zz z

mV F

mV F

J M

θ

ωϕ θ α

⋅

⎧ =⎪⎪⎪ =⎨⎪ =⎪⎪ = +⎩

,

.

.

1 1

z

yy y

mV F

J M

σ

ωφ σ β

⎧− =⎪⎪

=⎨⎪ = +⎪⎩

.

1 1xx xJ Mω = (7) 2.2 Decomposition of force and moment

The forces on the missile mainly include the earth’s gravitation, engine thrust, aerodynamic force, control power by gas rudders and aerodynamic rudders and disturbance et al [6]. The moments on the missile mostly include aerodynamic moment, control moment by gas rudders and aerodynamic rudders and disturbance moment et al.

(1) Gravitation The gravitation is transformed from the LCS to the

VCS:

3

3

3

0 sin coscos

0 sin sin

x

y G

z

G mgG V mg mg

mgG

θ σθ

θ σ

⎡ ⎤ −⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= − = −⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦ (8)

0 2( )e

e

Rg gR H

=+ (9)

Where, H is the flight height of the missile; GV express the transfer matrix from the LCS to the VCS.

(2) Thrust Assume the engine thrust is along x-axis of the BCS,

well then it is transformed from the BCS to the VCS:

3

3

3

cos cos0 sin0 cos sin

x

y B

z

P P PP V P

PP

α βα

α β

⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= =⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥−⎣ ⎦ ⎣ ⎦⎢ ⎥⎣ ⎦ (10)

Here, BV express the transfer matrix from the BCS to the VCS.

(3) Aerodynamic force and aerodynamic moment The aerodynamic force is transformed from the BCS to

the VCS: ( cos sin )cos sin

sin cos ( cos sin )sin cos

B M M

X ca ca cn czY V cn qS ca cn qSZ cz ca cn cz

α α β βα α

α α β β

− − + +⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥ ⎢ ⎥= = − +⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥ ⎢ ⎥+ +⎣ ⎦ ⎣ ⎦ ⎣ ⎦ (11) The expressions of the aerodynamic moment are

presented: M k

M k

M k

CMX cmx qS lCMY cmy qS lCMZ cmz qS l

⋅⎡ ⎤ ⎡ ⎤⎢ ⎥ ⎢ ⎥= ⋅⎢ ⎥ ⎢ ⎥⎢ ⎥ ⎢ ⎥⋅⎣ ⎦ ⎣ ⎦ (12)

In which, ca , cn , cz respectively is the axial, normal, lateral coefficient; cmz , cmy , cmx is separately the

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pitching, yawing, roll moment coefficient; q expresses the dynamic pressure; MS is the characteristic acreage;

kl expresses the characteristic length; α express the angle of attack; β is the sideslip angle; ϕδ , ψδ , γδ respectively express the equivalent pitching, yawing and roll rudder deflexion angle. All of the aerodynamic parameters are functions of α , β , ϕδ , ψδ , γδ .

Furthermore, aerodynamic force engenders inverted draft moment in proportion with the angle velocity of the missile [7].

2

1 /dz M dz k zM qS m l Vω= − (13) Where, dzm is the damp moment coefficient by the laboratory. Because the draft moment counteracts the disturbance, the actual system control will has more stability margins if the draft moment is not considered.

(4)Control power and control moment This missile’s actuators comprise gas rudders and

aerodynamic rudders in term of “ × ”pattern. Because the control power by aerodynamic rudders is discussed with aerodynamic force, here the control power is only by gas rudders. If 'R represents the lift grads of one rudder, then the control force in the BCS on pitching and yawing loops is given:

''

''

24 2 222 224

2

y

z

R RLL R

R

ϕϕ

ψψ

δ δδ

δ

⎡ ⎤⋅ ⋅⎢ ⎥ ⎡ ⎤⋅⎡ ⎤ ⎢ ⎥= = ⎢ ⎥⎢ ⎥ ⎢ ⎥ − ⋅⎢ ⎥⎣ ⎦ ⎣ ⎦− ⋅ ⋅⎢ ⎥

⎣ ⎦ (14) The expression of the control force in the VCS is

following: 3

3

sin cos( cos sin )sin cos

y x yyB

x y zzz

L L LLV

L L LLLα α

α α β β⎡ ⎤ +⎡ ⎤⎡ ⎤

= =⎢ ⎥ ⎢ ⎥⎢ ⎥ − + +⎢ ⎥ ⎣ ⎦ ⎣ ⎦⎣ ⎦ (15) Supposing that '

DR expresses the draft coefficient, δ expresses the deflexion angle of gas rudder, the draft of gas rudders in the BCS can be obtained:

'4 0 0T

DD R δ⎡ ⎤= −⎣ ⎦ (16) Counting separately the draft on pitching, yawing and

roll loop, δ in equation(16) should be ϕδ , ψδ and γδ in turn. The draft of gas rudders is transformed form the BCS

to the VCS: ' '

3'

3'

3

4 4 cos cos0 4 sin0 4 cos sin

x D D

y B D

z D

D R RD V RD R

δ δ α βδ α

δ α β

⎡ ⎤ ⎡ ⎤− −⎡ ⎤⎢ ⎥ ⎢ ⎥⎢ ⎥ = = −⎢ ⎥ ⎢ ⎥⎢ ⎥⎢ ⎥ ⎢ ⎥⎢ ⎥⎣ ⎦ ⎣ ⎦ ⎣ ⎦

(17)

Not difficult to gain the control moment of gas rudders:

'

'

'

4

24 ( )224 ( )

2

r dx

y R c

z

R c

R xMM R x xM

R x x

ψ

ϕ

δ

δ

δ

⎡ ⎤⎢ ⎥− ⋅ ⋅⎢ ⎥⎡ ⎤⎢ ⎥⎢ ⎥ = − ⋅ ⋅ −⎢ ⎥⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦ ⎢ ⎥− ⋅ ⋅ −⎢ ⎥⎣ ⎦

(18)

Hereinto, dx represents the distance from the pressure center on rudder plane to x-axis of the BCS, Rx expresses the distance from the pressure center on rudder plane to theoretic top end, cx expresses the distance from the centroid to theoretic top end. 3. Derivation of the linearization small deviation motion equations

Based on the supposing of little disturbance, namely the trajectory parameters warp between the motion with disturbance and the motion without disturbance at the same time, the dynamic equations can be linearized[1].

Illustrating with the pitching loop, considering the slow change of the centroidal speed describing by the first formula of equation(7)and the fast change of circling the centroidal velocity, the disturbance of the former can be ignored. And the centroidal speed V , the mass m ,the thrust P are all functions of time. In this way, it only needs the last three formulas of the equation(7) to linearize.

Puts the expression of every force and every moment into equation(7):

' '

'1

sin cos ( sin cos )

4 sin 2 2 cos

2 2 ( )

M

D yj

z M k R c

mv P mg ca cn qS

R R F

J cmz qS l R x xϕ ϕ

ϕ

θ α θ α αδ α δ α

ϕ δϕ θ α

⎧ = − + − +⎪

− + + ∑Δ⎪⎨

= ⋅ − −⎪⎪ = +⎩

(19)

Gives the Taylor expansion of the first formula in Equation(19) and ignores the infinitesimal of the second order or higher and compares with the ideal trajectory equations:

' '

''

sin [ cos ( )sin

( )cos 4 cos 2 2 sin ]

( sin cos ) [ ( sin cos )

2 2 cos 4 sin ]

M

M D

M M

D

camv mg P qS cn

cnqS ca R R

qS ca cn qS ca cnmv mv

RRmv

ϕ ϕ

ϕ ϕ

θ θ θ α αα

α δ α δ α αα

α α β α αβ β δ δ

αα

∂Δ = ⋅ Δ + − + +∂

∂ − − − Δ +∂

∂ ∂ ∂ ∂− + Δ + − + −∂ ∂ ∂ ∂

+ Δ ( sin cos )

( sin cos )

M

Myj

qS ca cnmv

qS ca cn Fmv

ϕ ψψ ψ

γγ γ

δ α α δδ δ

α α δδ δ

∂ ∂+ − + Δ +∂ ∂

∂ ∂− + Δ +∑Δ∂ ∂

(20) The above formulas are rewritten to the form of

differential equation and the parameters are unified, at the same time added the attack angle by the wind. Then, the linearization small deviation equation in the linkage mode with gas rudders and aerodynamic rudders is given:

1 2 3 4 5

6 1 5 f f f f f

f f w f w yj

c c c c c

c c c Fϕ γ

ψ

θ α θ δ δ β

δ α β

= Δ + Δ + Δ + Δ + Δ +

Δ + + +∑Δ (21)

Here, wα and wβ express the disturbance by the wind,

yjF∑Δ represents the sum of the longitudinal disturbance

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except the wind. The coefficient expressions in equation(21) are just as follows:

1

' '

2

''

3

4

5

cos ( )sin ( )cos

4 cos 2 2 sin

sin

2 2 cos( sin cos ) 4 sin

( sin cos )

( sin

Mf

D

f

Mf D

Mf

Mf

P qS ca cnc cn camv mv

R Rmv

gcv

qS ca cn Rc Rmv mvqS ca cncmv

qS cacmv

ϕ ϕ

ϕ ϕ

γ γ

α α αα α

δ α δ α

θ

αα α αδ δ

α αδ δ

αβ

∂ ∂⎡ ⎤= + − + + − −⎢ ⎥∂ ∂⎣ ⎦

+

=

∂ ∂= − + − +∂ ∂∂ ∂= − +∂ ∂∂= −∂

6

cos )

( sin cos )Mf

cn

qS ca cncmv ψ ψ

αβ

α αδ δ

∂+∂

∂ ∂= − +∂ ∂

Similarly, gives the Taylor expansion of the second formula in Equation(19) and unifies the parameters:

1 2 3 4 5 6

2 5 1

f f f f f f

f w f w z j

b b b b b b

b b Mϕ γ ψϕ ϕ α δ δ β δ

α β

+ Δ + Δ + Δ + Δ + Δ + Δ

= − − + ∑Δ (22)

Where, yjF∑Δ represents the sum of the longitudinal disturbance moment except the wind.

The coefficient expressions in equation(22) are just as follows:

1

21

'3

1 1

41

51

61

0

1

1 12 2 ( )

1

1

1

f

f M kz

f M k R cz z

f M kz

f M kz

f M kz

b

cmzb qS lJ

cmzb qS l R x xJ J

cmzb qS lJ

cmzb qS lJ

cmzb qS lJ

ϕ

γ

ψ

α

δ

δ

β

δ

=

∂= −∂

∂= − + −∂

∂= −∂

∂= −∂

∂= −∂

Likewise, the expression of the third formula is given: ϕ θ αΔ = Δ + Δ (23)

To sum up, the linearization small deviation equation on pitching loop is presented:

1 2 3 4

5 6 1 5

1 2 3 4 5 6

2 5 1

f f f f

f f f w f w yj

f f f f f f

f w f w z j

c c c cc c c c F

b b b b b b

b b M

ϕ γ

ψ

ϕ γ ψ

θ α θ δ δβ δ α β

ϕ ϕ α δ δ β δ

α βϕ θ α

⎧Δ = Δ + Δ + Δ + Δ +⎪ Δ + Δ + + + ∑Δ⎪⎪Δ + Δ + Δ + Δ + Δ + Δ + Δ⎨⎪ = − − + ∑Δ⎪⎪Δ = Δ + Δ⎩

In the same way, the linearization small deviation equation on pitching and yawing loop can be gained. Because of the length, it is not listed in this paper.

4. Conclusion

For the tactics missile blended controlled by gas rudders and aerodynamic rudders, the integrated dynamic model is set up. According to the hypothesis of little disturbance, the linearization small deviation equations are detailedly derived. On this basis, multiple characteristic points on typical trajectories can be selected and calculated the coefficients on the points. By the method of solidify, the variational coefficient equations can be regarded as constant coefficient equations [8]. Well then, the control system can be designed after the dynamic characteristic analysis of these characteristic points. References [1] Qing Xin-fang, Lin Zhui-xiong, Zhao Ya-nan. Missile

Flight Dynamics. Beijing. Science and engineering university Press, 2006.

[2] Zhang Zui-laing. Guidance and control of ballistic missiles. National University of Defense Technology press, 1981.

[3] Rui Hirokawa, Koichi Sato. “Autopilot design for a missile with reaction-jet using coefficient diagram method”. AIAA-2001-4162.

[4] Feng Guan-yi. Design of control and guide system of Air Defense missile weapons. Astronautics press, 1996.

[5] Li Xin-guo, Fang Qun. Winged Missile Dynamic. National University of Defense Technology Press, 1993.

[6] Jia Pei-ran, Chen Ke-jun, He Li. Long-Distance Rocket Ballistics. National University of Defense Technology Press, 1993.

[7] Zhao Shan-you. Collectivity design of Air Defense seeking-missile. Astronautics press, 1992.

[8] Long Le-hao. Collectivity design (Middle). Astronautics press, 1993.

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