Multi-dimensional Taylor Network Optimal Control of the

Axisymmetric Cruise Missile Flight

Yongyan Ge

, Hongsen Yan

1,2

and Lingchen Xia

School of Automation, Southeast University, Nanjing, China

Key Laboratory of Measurement and Control of Complex Systems of Engineering, Ministry of Education

Nanjing, China

2451418571@qq.com, hsyan@seu.edu.cn, 461563918@qq.com

Keywords: Missile control, Multi-dimensional Taylor network (MTN) controller, Attitude control, Optimal control.

Abstract: In this paper, we present the design of the multi-dimensional Taylor network (MTN) optimal controller in the

flight control of cruise missile. The MTN optimal control, which combines the classical architecture of

feedback control system and the new controller structure, it is not only suitable for the analysis of the stability

of the closed-loop system, but also for the control of nonlinear systems with mechanism known or unknown

models. Firstly, this paper will briefly introduce the theoretical basis of the MTN optimal control. Secondly,

the characteristics of the missile mathematical model and the theory of missile control will be explained,

accompanied with the design of controller. Finally, the feasibility of the method is validated through

numerical simulation of the PID controller, PIDNN controller and the MTN optimal controller. The results

show that the MTN optimal controller has the best control effect of them.

1 INTRODUCTION

As a kind of high lethality tool the missile has

played a very important role in the modern battle

field. In a local war, high precision guided missiles

may affect the war. Because of its powerful

long-range destructive force and precise fixed point

strike capability, the missile plays an irreplaceable

role in the national defense security. The missile

control system is an important part of the missile

overall design, and its performance affects the

capability of the missile directly.

The control system of missile is designed by PID

in the literature (Liu, 2009), and the nonlinear model

of missile is linearized, then the three channels

controller is used in the missile control system. A

new method for establishing the six degree of

freedom simulation model of missile is proposed,

and the general principles and methods of

establishing the simulation model of the six degree

of freedom of the large aircraft are presented in the

literature (Yan, 1998). The paper (Wang, 2008) used

the PID neural network algorithm to control the

missile mathematical model linearized. This kind of

linear model is difficult to accurately describe the

actual nonlinear model. The PID neural network

algorithm is used to control the attitude of the

ballistic missile in the literature (Tang, 2012).

According to the above analysis, most design of

missile control system in the past adopted the small

disturbance linearization method, and then the model

of the missile is linearized, which use the classical

control method of transfer function for missile

control. The defects caused by this method may lead

to inaccurate simulation results. In addition, most

research of missile control system in the past

analyzed the channel in the missile control system.

There are a few researches on the trajectory control

of missile trajectory. In view of the current research

status of the missile controller design, it is necessary

to carry out the simulation research of the missile

trajectory control based on the dynamic model of the

missile.

In this paper, the multi-dimensional Taylor

network (MTN) controller (Yan, 2010&2017) will

be used in the design of missile control. The MTN

controller is a nonlinear control technique. It uses

the nonlinear model of the missile to design the

controller to avoid linear behavior in various state

points. In the meantime, it can be combined with

intelligent control methods to directly identify the

controller parameters through a set of desired output.

This makes the design of the controller simpler and

156

Ge, Y., Yan, H. and Xia, L.

Multi-dimensional Taylor Network Optimal Control of the Axisymmetric Cruise Missile Flight.

In 3rd International Conference on Electromechanical Control Technology and Transportation (ICECTT 2018), pages 156-162

ISBN: 978-989-758-312-4

more efficient. Additionally, with the nonlinear

terms of object considered, the control system can

perform well in a wide range.

2 THE MTN CONTROL

THEORY

A controlled system can be described as follows:

(1) ((),(),)

() ((),(), )

kfxkukk

yk g xk uk k

⎧

⎨

⎩

(1)

where

()

kR∈

is the state variable,

()

uk R∈

is the input variable,

()

yk R∈

is the output variable.

((),())

xk uk and (())

xk are respectively smooth

nonlinear function of relevant variables. We can also

rewrite (1) as (2):

1112

2212

( 1) ( (), (), , (),())

( 1) ( ( ), ( ), , ( ), ( ))

nn n

k fxkxk xkuk

kfxkxkxkuk

⎧

⎪

⎨

⎪

⎩

(2)

For all of the controllable system, under certain

conditions, there must be an optimal control signal

u*, which makes the system achieve the optimal

control effect under the action of u*. For general

control systems, feedback control is usually used to

design the controller, which is based on the

deviation between the expected value and the output

feedback. At the same time, reference Wells Truss

approximation theorem and the Taylor formula, can

be used for quantitative and feedback the high-order

error to infinite approximation of the optimal control

signal. The whole control structure diagram can be

expressed in Figure 1:

()ek

()ek ()



(1)uk+

1,1

1,2

1, ( , )Nnm



Figure 1: The structure of MTN.

The Taylor formula uses a series of infinite terms

to approximate a function. These additional terms

are obtained by the derivative of the function at a

certain point.

The Taylor formula is defined as follows. For a

positive integer n, if the function f (x) in the closed

zone [a, b] can be continuously guided, either take [a,

b] on a certain point x, then the function at the point

where the approximate expression is as follows:

()

() () ()

0! 1! 2!

()

... ( ) ( )

fa f a f a

xxaxa

xa Rx

′′′

=+ −+ −

++ − +

(3)

In the formula,

()

is n derivative of f(x) at

a point, polynomial

()

−

is nth Taylor

expansion of f(x) at a point,

R is the remainder

term. The more Taylor’s series, the more

approximation function point value.

According to the Taylor formula and the Wells

Dreads approximation theorem, we have the

following expressions (Sun, 2014).

(, )

112

(1) (,,,)

Nnm

ji m n

uk p e R ee e

+= +

∑

∏

(4)

In the formula,

is the weight coefficient,

,ij

is the number of errors,

R is the Taylor remainder

which contains the number of the error terms are

greater than m,

(, )Nnmis the highest number of

items is obtained through the combination with the

highest number of dimensions. If the allowable error

conditions and m value is larger, it can be simplified

to formula (5).

(, )

(1)

Nnm

uk p e

∑

∏

(5)

In the formula, x

can be expressed as follows:

211

322

() ()

() () ( 1) () ( 1)

() () (1)[() (1)][(1) (1)]

() () ( 1)

nn n

et et

et et et et et

e t e t e t et et et et

et etet

−−

=−−=−−

= − −= −−− −−−

=−−



(6)

As we can see, the structure of MTN controller is

a forward network of three layers which consists of

input layer, middle layer and output layer. All the

Multi-dimensional Taylor Network Optimal Control of the Axisymmetric Cruise Missile Flight

157

MTN elements of different power and order are

multiplied by a given weight coefficient, adding all

of them as to be the controller output. Combining

with appropriate algorithm, it is easy to determine

the weight of parameters. Detail analysis of MTN is

described in (Zhou, 2013).

3 MISSILE MODEL ANALYSIS

In order to research the change regulation of the

missile trajectory and the change of attitude, it is

necessary to consider the missile motion as a rigid

body motion in space. Missile kinematics equation is

built on this assumption. It mainly describes the

kinematics and motion rules of projectile body

rotating around the center of mass. The position and

attitude angle of the missile at a certain time point

can be solved by using the missile kinematics

equations. The missile motion equations are

composed of missile dynamics, kinematics and

quality variation equation. The relationship between

these equations can describe the missile by process

conveniently and accurately in force, torque and

motion parameters of the missile. Then we can

obtain the equations describing motion of the missile

as shown in the following expressions (Qian, 2000):

cos cos sin

(sin cos cos sin sin ) cos sin cos

cos (sin sin cos sin cos ) sin cos

()

VVVV

xb xb zb yb yb zb

yb yb xb zb xb zb

mP Xmg

mV P Y Z mg

mV P Y Z

JMJJ

αβ θ

αγ αβγ γ γ θ

θαγαβγγγ

ωω

=−−

=+ +−−

−=− ++

=−−

() ()

()

cos cos

sin

cos sin

sin cos

( cos sin ) / cos

tan ( cos sin )

sin cos cos sin sin sin cos sin cos sin

sin cos co

zb yb xb yb xb

yb zb

xb yb zb

ωω

θψ

ωγωγ

ωγωγ ϑ

ωϑωγωγ

βθγψψ ϑγψψ θϑγ

αβ

−−

=−

=− −

=−+−−

⎡⎤

⎣⎦

s [sin cos cos( ) sin sin( )] sin cos cos

sin cos cos sin sin sin sin cos cos cos sin cos

θϑ

γψψ γψψ

θϑ

γθ αβϑαβγϑ βγϑ

⎧

⎪

⎨

⎪

−− − −

⎪

=− +

⎪

=−

⎪

⎩

(7)

where among all of the sixteen state variables,

the missile mass,

V is the center of mass velocity;

is the roll angle,

is the pitch angle,

is the yaw

angle;

is the ballistic inclination,

is the

trajectory angle,

is the angle of attack ,

is the

sideslip angle;

and

are the moment of

inertia of the three coordinate axes of the missile

body;

and

are the component of the

rotational angular velocity

of the body coordinate

system on three axes;

and

are the

components of the total moment of rotation in three

coordinate systems of the missile body coordinate

system;

m is the amount of mass change per unit

time, depending on the engine performance;

(, ,)

yz represents the coordinate value of the

missile in the ground coordinate system.

4 DESIGN OF THE

CONTROLLER

In this paper, we adopt three channels control

method to design of the controller. The

corresponding control system designed in the pitch

channel, the yaw channel and the roll channel. The

control method of the three channels as an example

to introduce the missile three channels. The typical

control principle diagram of three channels is given

in Figure 2:

Figure 2: Missile Three Channels Control System.

The working principle of the three channels

control mode: firstly, the measuring mechanism

measures motion parameters of the missile and the

target, then according to the relative motion between

the missile and the target, the guidance and control

system generates a series of control instructions, and

control the attitude variable calculation, coordinate

transformation, navigation calculation, error

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158

compensation calculation and control instruction the

formation, the state variable feedback signal and the

corresponding channel control instruction, and pass

it to the executive body.

In this paper, the MTN control algorithm is used

to design the missile longitudinal channel control

system, and the pitch angle is controlled by using the

MTN optimal control algorithm. Here the PID

control as a MTN optimal control case, the

proportion, integral and differential as three items of

the MTN optimal control, then arranges and

combines the three items, multiplied the

corresponding control weight in the front of Taylor

items, finally get the controller output. At the same

time, combined with the experimental data, the

parameters are optimized manually, and then

complete the MTN control. Specific control

framework as shown in Figure 3:

upsilon

1,1

1,2

1, ( , )Nnm





(1)

uk+

Figure 3: Control chart of pitch angle of MTN.

Due to limited time and laboratory hardware

conditions, the controller only get the first term, the

second term and the third term as the part number of

the MTN control, the form can be written as (8):

222

123456

78 910

(1) +

up p i d p i d

ipdid pid

kuuuwuwuwuwuww

wu u wu u wuu w u uu

=++ +

++ +

(8)

In the formula,

and

are the control

parameters of the first item,

are the second terms of the control parameters,

is the third terms of the control parameter.

When the system is close to the state of

equilibrium, the influence of high-order (8) type is

far less than the linear term, such as

=0.01,

=0.0001, and the high terms does not affect the

stability of equilibrium. In other words, the influence

of (8) on the stability of equilibrium is equivalent to

PID. When the super unit ball system is out of

balance, especially when it is far from the super unit

ball, the impact of the high order term on the system

is much larger than linear, so through parameter

optimization can make the high terms object has

better dynamic characteristics and anti-interference.

In Figure 4, take the proportional link as an example,

when the error is small, especially when it is close to

the origin, the control effect of the higher order term

is much smaller than that of the first term(Xia,

2016).

Figure 4: Comparison of high-order term and linear term.

In (8), MTN is composed of the linear and high

terms of the error. The comparison of the linear and

high terms is shown in Fig. 4.

According to (8) and Fig. 4, we have the

following discussions:

1) MTN controller is a nonlinear controller,

which includes proportional-integral-derivative (PID)

control and linear controller as the special case, so

we can take the PID parameters as the initial ones of

MTN that makes the closed-loop system stable.

2) For asymptotically stable systems, the

high-order terms of the MTN controller are

high-order infinitesimal in the vicinity of the error

equilibrium point, and do not affect the stability of

the equilibrium point. On the other hand, the farther

away from the equilibrium point the error state, the

greater role the high-order terms are playing. The

effect of the higher order term is much greater than

that of the linear term when the error state is far away

from the equilibrium point-centric hyper-unit sphere.

As the sudden input or large disturbances can cause

large errors and large fluctuations, the main effect of

higher order terms is to affect the dynamic

performance and anti-disturbance performance of the

closed loop system, especially performance of

resisting strong disturbance.

3) MTN optimal controller is equivalent to the

linear controller because it is dominated by linear

parts when the error state is in the vicinity of the

error balance point. When the error state is far from

Multi-dimensional Taylor Network Optimal Control of the Axisymmetric Cruise Missile Flight

159

the equilibrium point-centric hyper-unit sphere, the

nonlinear parts will play a leading role. When the

error is in between, the linear parts and the nonlinear

ones will take effect in the same time. Therefore, the

MTN can be regarded as a kind of sliding mode

control (SMC) without chattering.

In the design of longitudinal channel control

system, the attitude angle of the missile's climbing

section and the last dive section is controlled by PID,

and the double closed loop compound control is

adopted in the flying section. In this paper, the pitch

angle control is used as the inner loop, and the MTN

is used to optimize the control. The outer loop uses a

high degree of feedback, and the PID control

algorithm is adopted. After optimizing the control

parameters of the pitch angle controller, the outer

loop is added to the longitudinal control system.

After adjusting the inner loop parameters, the

controller parameters of the outer loop are adjusted.

The height control structure is shown in figure 5:

•

Figure 5: MTN height control block diagram.

5 RESULTS AND ANALYSIS

Four feedback signals are used in the design of the

height control system, such as the height, the height

change rate, the pitch angle and the pitch angular

velocity. If the flight height is 50m, height control

chart of missile can be obtained as Figure 6 showed.

Figure 6: MTN height control.

As can be seen from figure 6, the missile flights

smoothly from the glide segment to the transition

period, and quickly converge to about 50m. The

result shows that the double closed-loop control

method designed by PID control as the outer loop

and MTN optimal control as the inner loop can

control the missile better.

Figure 7: Pitch angle control of MTN control.

As shown in Figure 7, the pitch angle is

maintained at 60 degrees at first, then decreased

rapidly, which corresponds to the missile in flight

down. In 22s, after a smaller overshoot, the pitch

angle is stable at 0 degree, and the missile gets into

the cruise flight. At the last time in the 50s, the pitch

angle first increases rapidly at less time, and then

decreased. This corresponds to the missile's climb to

a height, and then quickly dive down hit the target

flight process.

Figure 8: Height adjustment comparison curve.

Fig.8 illustrates that when the constant wind is

9m/s, the optimal control effect of multi-dimensional

Taylor network is better than that of PID control and

PID neural network.

In addition, a good many simulations of the

cruise missile counteracting wind disturbance by

three kinds of control methods are conducted, and

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160

their results are summarized as follows: 1)

anti-disturbance capacities of PID control for

constant wind, gust and random wind respectively

are 8m/s, 9m/s and 19m/s; those of PID neural

network control (PIDNN) for the three winds

respectively are 9m/s, 10m/s and 10m/s; those of

MTN optimal control for the three winds

respectively are 9m/s, 11m/s and 22m/s. Under the

same conditions, the MTN optimal control has the

best wind anti-disturbance effect. 2) Under the

condition of the shear wind ratio k1=

−

, the

maximum shear wind stresses b1 borne by MTN,

PIDNN and PID respectively are

16 10

−

16 10

−

and

13 10

−

; under the condition of

shear wind stress b1=

16 10

−

, the maximum shear

wind ratio k1 borne by MTN and PIDNN

respectively are

16 10

−

and

410

−

; Under the

condition of b1=

13 10

−

, the maximum k1 of MTN,

PIDNN and PID can be

108 10

−

97 10

−

and

23 10

−

, respectively. 3) Target striking accuracy:

under the same conditions, the average target hitting

impact degrees of PID, PIDNN and MTN

respectively are 2.020313, 2.364979 and 0.032239.

If results for random wind without waves (in which

the target hitting impact degree of PID is particularly

large and reaches 18.5181) are neglected, the

average target hitting impact degrees of PID,

PIDNN and MTN respectively are 0.520514,

2.390054 and 0.033152. Their corresponding target

deviations respectively are 52.0514m, 239.0054m

and 3.3152m, i.e., the average target hitting accuracy

of MTN is 14.70 and 71.09 times more than those of

PID and PIDNN respectively. Neglecting the results

of flight path divergence by PID and PIDNN (i.e.

out of control), the maximum target hitting impact

degrees of PID, PIDNN and MTN respectively are

18.5181, 6.85152 and 0.186743. Their

corresponding target deviations respectively are

1851.81m, 685.152m and 18.6743m, i.e., the worst

target hitting accuracy of MTN is 98.16 and 35.69

times more than those of PID and PIDNN

respectively (Yan, 2017).

6 CONCLUSIONS

In this paper, we illustrate the optimal control theory

of MTN, establish the missile flight dynamics model

and analyze its characteristics. Among them, the

scheme guidance is adapted in the missile flight path

in the take-off, horizontal and dive directions, in

which gradient method combined with hand

adjustment is used to optimize MTN controller

parameters. The above results show that the MTN

optimal control has better dynamic performance and

external stability, stronger anti-disturbance

performance and 1~2 magnitudes higher target

striking accuracy than PID and PIDNN.

ACKNOWLEDGMENTS

This work was supported by National Natural

Science Foundation of China under Grants

61673112 and 60934008, the Fundamental Research

Funds for the Central Universities under Grants

2242017K10003 and 2242014K10031, and a Project

Funded by the Priority Academic Program

Development of Jiangsu Higher Education

Institutions.

REFERENCES

B. Liu, S. Gao, S. He, 2009. “Missile sub-channel

simulation based on PID control”, Electronic Design

Engineering, 17(11): 97-99

B. Yan, F.L. Deng, 1998. “The Modeling Research of

Six-Degree-of-Freedom Simulation of Large Scale

Space Vehicle”, Journal of system simulation, 10(3):

22-29

Q. Wang, C. Chen, X.E. Wang, 2008. Application of

Neural Network PID In Control of BTT Missile

System , Aeronautical Computing Technique, 38(4):

1-4

Y.F. Tang, B.J. Liu, 2012. Application of PID neural

network in ballistic missile attitude control system.

China science and Technology Review, (17): 424-426

H.S. Yan, 2010. MTN (multi-dimensional Taylor network)

optimal control. Technical Report. Research Institute

of Control and Optimization of Manufacturing

Systems, School of Automation, Southeast University,

Nanjing, China.

H.S. Yan. (2017, Jul.). Multi-dimensional Taylor network

optimal control. School of Automation, Southeast

University, Nanjing, China. [Online]. Available:

http://automation.seu.edu.cn/Articles. Aspx?id=3487

Department of mathematics, Tongji University. Advanced

mathematics and its application. Beijing: Higher

Education Press, 2008.

Sun Q M, Yan H S. Optimal adjustment control of SISO

nonlinear systems based on multi-dimensional Taylor

network only by output feedback. In: 2014

International Conference on Automation, Control and

Information Science (ACIS 2014), Xi'an, 2014.

Advanced Materials Research, 2014, V1049-1050.

Zurich-Durnten, Switzerland: Trans Tech Publications

Ltd, 2014: 1389-1391

Multi-dimensional Taylor Network Optimal Control of the Axisymmetric Cruise Missile Flight

161

B. Zhou, and H.S. Yan, 2013. “Financial time series

forecasting based on wavelet and multi-dimensional

Taylor network dynamics model,” Systems

Engineering-Theory & Practice, vol. 33, issue 10, pp.

2654-2662.

X.F. Qian, R.X. Lin, Y.N. Zhao, 2000. Missile dynamics

of flight, Beijing: Beijing Institute of Technology

Press.

L.C. Xia, 2016. Research into Multi-dimensional Taylor

Network Optimal Control of the Axisymmetric Cruise

Missile Flight for Attacking Static Target. Nanjing:

Southeast University.

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