Agile Missile Autopilot Design for High Angle of Attack

Agile Missile Autopilot Design for High Angle of Attack Maneuveringwith Aerodynamic Uncertainty*

Yue-Yue MA, Sheng-Jing TANG, Jie GUO, and Yao ZHANG

School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

The paper proposes a new autopilot design for agile missiles flying at a high angle of attack (AoA). A maneuverstrategy applicable to 90° AoA flight for agile turning is described prior to the missile modeling. Accounting for the dis-turbance rejection, the extended state observer (ESO) technique is employed for online estimation of the system uncertain-ties due to the aerodynamic unpredictability at high AoA regimes. Under the circumstances, linearization with dynamiccompensation and non-singular terminal sliding mode control are applied to achieve controllability during 90° AoA flight.Numerical simulation results demonstrate the effectiveness and robustness of the proposed scheme. Additionally, the chat-tering caused by unmodeled dynamics is obviously mitigated with the action of the ESO.

Key Words: Agile Missile, Autopilot, High Angle of Attack, Disturbance Rejection, Extended State Observer

1. Introduction

Current high-performance fighter aircraft are required toachieve omni-directional attack of the launch platform,which implicates the ability to engage targets in the rearhemisphere for an agile missile. The fast 180� turn of a mis-sile is performed under high angle of attack (AoA) maneu-vering. The dynamics of an agile missile at high AoA areinherently nonlinear, fast time varying and extremely uncer-tain. In practice, the uncertainties of missile dynamics are, toa large extent, caused by the unpredictability of aerodynam-ics. When AoA increases, the asymmetric flow separationappearing on the leeside of the missile makes aerodynamicdata unpredictable. The uncertainties in dynamics of the mis-sile increase the difficulty of high AoA flight control. Inaddition, alternative control technology such as thrust vectorcontrol (TVC) or a reaction-jet control system (RCS) mustpossess super-agility for agile missiles because of the ineffec-tiveness of aerodynamic control at high AoA.

Research about agile turn control has progressed in differ-ent perspectives. The dynamics and technical challenges ofan agile missile at high AoA were discussed by Wise andBroy.1) A 90� AoA command was used to turn the missileinto the rear hemisphere, with sideslip angle regulated tozero. The simulation results for the missile using RCS thrust-ers showed the feasibility of such high AoA flight control.Thukral and Innocenti2) proposed an autopilot based on var-iable structure control (VSC) to achieve a fast 180� headingreversal in the vertical plane. Furthermore, robustness of theVSC system for the agile missile was investigated by Inno-centi and Thukral.3) These studies, however, only consideredthe longitudinal dynamics, regardless of the roll and yawchannel control, which is a crucial factor for high AoA

autopilot due to the serious aerodynamic uncertainties.McFarland and Calise4,5) applied neural network-based adap-tive control to agile turning of anti-air missiles, resulting inan improvement in approximate dynamic inversion for thecontrol of uncertain nonlinear systems. A bank-to-turn(BTT) steering technique also used by Wise and Broy1)

was applied in the six-degrees-of-freedom (6-DOF) simula-tion for a head-on merge scenario. Various nonlinear autopi-lot schemes were also considered using the H1 method,6)

pole placement method,7) backstepping method8) and soon. Recently, a novel strategy for agile turn control ofhigh-performance missiles was first proposed by Ratliffet al.9) and further developed by Kim et al.10) In this strategy,fast 180� change in the missile pitch angle was accomplishedby purely aerodynamic control. This aero-surface controlmode, only verified by longitudinal motion simulation with-out the consideration of aerodynamic disturbance, is ques-tionable in practical terms.

As mentioned above, the majority of research has paidclose attention to the nonlinearities of plants to meet chal-lenges of nonlinear characteristics at high AoA, while ignor-ing how to deal with the system uncertainties which actuallyplay a critical role in improving control performance. Theeffects of uncertainties in plant dynamics are counteractedby the robustness of the control system itself. In a distinctway, to address the problem that aerodynamic coefficientscannot be predicted in the high AoA domain, the controllaws designed by Kim and Kim11) did not require the aerody-namic data. In literature, nonetheless, the pitching momentwas assumed to vary according to a sinusoidal functionbounded by a very small constant, which does not conformto real-world conditions.

In a different way than in the past studies, this paperdesigns an autopilot following the thought that the aerody-namic data regarded as the uncertainty of missile dynamicscan be estimated online using an effective mechanism. Theinformation on the true values of aerodynamic coefficients

© 2015 The Japan Society for Aeronautical and Space Sciences+Received 16 June 2014; final revision received 27 February 2015;accepted for publication 13 March 2015.

Trans. Japan Soc. Aero. Space Sci.Vol. 58, No. 5, pp. 270–279, 2015

270

is not used in the control laws, abiding by the viewpoint pre-sented by Kim and Kim.11) What makes the proposed controlscheme different is the disturbance rejection capability result-ing from the real-time acquisition of the unknown system in-formation. As the key part of the active disturbance rejectioncontrol (ADRC) method, extended state observer (ESO) isemployed to estimate the uncertainties of the control systemin real time. As a means of accounting for the unknown in-formation of system dynamics, the ESO proves the possibil-ity of online compensation of the uncertainties for controllerdesign.

This paper begins with a description of the proposed con-trol strategy for 90� AoA maneuvering in Section 2. Further-more, the dynamics of an agile missile equipped with a RCSsystem are studied, followed by a detailed presentation of thethree-channel independent autopilot design using ADRC/ESO and the non-singular terminal sliding mode (NTSM)method in Section 3. In Section 4, simulation results for anagile missile are presented to validate the proposed autopilotdesign technique. Finally, conclusions are summarized inSection 5.

2. Missile Dynamics

2.1. Maneuver descriptionA 90� AoA command1,4,5) is used to achieve agile turning

with a 180� off-boresight trajectory. Traditionally, an aircraftflying in endo-atmosphere performs maneuvers by virtue ofthe normal aerodynamic overload generated by a properAoA. Nevertheless, the normal overload will be enhancedfor the effect of the main engine thrust during a 90� AoAflight (Fig. 1). If the AoA equals exactly 90�, the missile willobtain the maximal normal overload provided by the enginethrust, which is absolutely used in the normal direction.

High AoA flight leads to some control challenges for agilemissiles including strong nonlinearity, model parameter per-turbation and serious aerodynamic coupling. For the purposeof solving the relevant problems in an implementable andreasonable way, a quasi BTT autopilot similar to widely usedBTT is adopted here. Before a detailed description, two atti-tude angles, quasi roll angle �� and quasi yaw angle �, aredefined. �� ð �Þ denotes the angle which the missile body

moves around the x-axis (z-axis) in the body coordinate sys-tem. �� ð �Þ is positive when the rotation vector is in thesame direction as the x-axis (z-axis). At the beginning ofimplementing the quasi BTT maneuver, the roll control ofthe agile missile is performed after separating from the car-rier aircraft to adjust the z-axis of the body into the expectedmaneuvering plane. Then the autopilot controls the missile totrack the AoA command in the pitch channel. In the subse-quent process, only stability control is necessary in the rolland yaw channels, aiming to hold �� and � near zero. Ac-cording to the above approach, the autopilot is synthesizedwith a three-channel independent design to make the missilefly in the preferred orientation as much as possible. In thispaper, the interchannel coupling is treated as an unknowndisturbance whose relevant discussions are described indetail in Section 3.

The quasi BTT proposed in this paper controls the rollingand yawing attitude of the missile without confining the mis-sile sideslip angle. Actually, under the influence of dynamicmodel uncertainties, the missile cannot fly in the expectedmaneuvering plane absolutely, regardless of whether BTTor quasi BTT is chosen for the flight control. However, itis unnecessary to pose the strict requirement that both themissile axis and velocity direction aim at the target afterthe agile turn, since the top priority is to make the missileturn around. Simultaneously, the missile axis and velocitydirection could be adjusted by guidance command in the sub-sequent guidance phase. Compared with BTT control, quasiBTT control is capable of dominating the rolling and yawingattitude of the missile, which is beneficial to enable the mis-sile seeker to capture the target for terminal guidance.2.2. Modeling

The system dynamic models are based on a slender bodywith the RCS system and the rear control dominated by cru-ciform rudders, as shown schematically in Fig. 2. RCS isused for the missile attitude control on the assumption thatthe RCS can provide roll, pitch and yaw moment control re-spectively. The magnitude of the reaction-jet is adjusted bythe RCS nozzle throat area valve and changes continuously.In the current scenario, RCS is the only means to control themissile for an agile maneuver in this paper, as a consequenceof the low effectiveness of rudder control in the high AoAregion.

The motion of a missile can be described using the follow-ing equations:

Fig. 1. 90� AoA flight. Fig. 2. Sketch map of an air-to-air missile.

Trans. Japan Soc. Aero. Space Sci., Vol. 58, No. 5, 2015

271©2015 JSASS

_V ¼ ax cos�þ az sin��

cos �þ ay sin �_� ¼ q� p cos�þ r sin��

tan�

þ �ax sin�þ az cos��

= V cos ��

_� ¼ p sin�� r cos�� ax cos�þ az sin��

sin �=V

þ ay cos �=V_p ¼ L=Ixx

_q ¼ M=Iyy þ 1� Ixx=Iyy� �

pr

_r ¼ N=Iyy � 1� Ixx=Iyy� �

pq

8>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>:

ð1Þ

where V is velocity, � is angle of attack, and � is sideslipangle. ax, ay and az are the body-axis components of acceler-ation. p, q and r are the body-axis angular rates. L, M and Nare aerodynamic moments about the body axis. Ixx and Iyyare rolling and pitching moments of inertia. Additionally, itis assumed that V, �, �, p, q and r are available or effectivelyestimated for control implementation in this paper. Finally,according to the definition proposed above, the differentialequations of the quasi roll angle and the quasi yaw anglecan be expressed as

_�� ¼ p

_ � ¼ r

(ð2Þ

The realization of 90� AoA command flight requires highAoA aerodynamic modeling as the basis for autopilot design.For a missile with a high slenderness ratio, when the AoAincreases from 0� to 90�, four different leeside flow patternsappear orderly in accordance with the effect of the axis flowcomponent. The asymmetric flow separation on the missileleeside induces considerable lateral force and yawingmoment with strong randomness at high AoA, mainly from30� to 60�.12) However, the effect of the axis flow componentobviously declines when the AoA is close to 90�. Comparedto the situation with random lateral loads of 30� to 60� AoA,the uncertainty of 90� AoA aerodynamics is significantlyreduced.

This paper divides the missile aerodynamic mathematicalmodel into two parts: the estimated value and disturbed val-ue. The estimated value is generated by the Missile Datcomcode to estimate a set of general aerodynamic data in the pre-liminary design phase, even for a high AoA region.13,14) Theplots of lift and drag coefficients for a reference velocity ofMach ¼ 0.6 are shown in Fig. 3. It can be concluded thatthe normal load is the leading aerodynamic influence on

the missile for 90� AoA flight. Moreover, other aerodynamicforces are treated as uncertainties including aerodynamicerrors between the missile configuration and the cylinder,aerodynamics induced by the change in missile attitudeand the coupling between channels. Such uncertainties mayinduce lateral force, yaw and roll moments at a high AoA.Because the aerodynamic uncertainties are difficult to meas-ure, the disturbed values need to be set manually and are thenincorporated into the aerodynamic models. The specificforms used to reflect the uncertainties are described inSection 4.

3. Autopilot Design with ESO

3.1. ESO methodologyADRCwas first proposed by Han at the end of the last cen-

tury.15–17) The theory was explained carefully by Han,18) andanalyzed in depth by Gao19) and Huang et al.20) As the resultof years of investigation, the ADRC technique has been ap-plied in real-world applications, especially in the industrialfield. Playing a predominant role in the ADRC method,ESO is able to estimate the total disturbance of plant dynam-ics online. The ESO technique proposes a new viewpoint onhow to deal with system uncertainties. With the action ofESO, ADRC can compensate system uncertainties in real-time to realize linearization for an uncertain nonlinear controlsystem.

Consider an n-dimensional SISO nonlinear system withthe following structure:

_x1ðtÞ ¼ x2ðtÞ_x2ðtÞ ¼ x3ðtÞ...

_xnðtÞ ¼ fðt; x1ðtÞ; � � � ; xnðtÞÞ þ wðtÞ þ uðtÞyðtÞ ¼ x1ðtÞ

8>>>>>>>><>>>>>>>>:

ð3Þ

where u is the input (control), y is the output (measurement),f is a possibly unknown system function, and w is a uncertainexternal disturbance. fþ w models the total disturbance.The ESO is used to provide the approximations of the statexi for i ¼ 1; 2; � � � ; n and the total disturbance fþ w. Akind of nonlinear ESO is written as

_̂x1ðtÞ ¼ x̂2ðtÞ þ "n�1h1yðtÞ � x̂1ðtÞ

"n

!

_̂x2ðtÞ ¼ x̂3ðtÞ þ "n�2h2yðtÞ � x̂1ðtÞ

"n

!

..

.

_̂xnðtÞ ¼ x̂nþ1ðtÞ þ hnyðtÞ � x̂1ðtÞ

"n

!þ uðtÞ

_̂xnþ1ðtÞ ¼1

"hnþ1

yðtÞ � x̂1ðtÞ"n

!

8>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>:

ð4Þ

0 10 20 30 40 50 60 70 80 900

5

10

15

20

25

30

35

40

Angle of attack (˚)

Aer

odyn

amic

coe

ffici

ent

CD

CL

Fig. 3. Aerodynamic forces versus angle of attack.


272©2015 JSASS

where " is a constant gain, and hið�Þ, i ¼ 1; 2; � � � ; nþ 1 arepertinent chosen functions. The states of the observer x̂i,i ¼ 1; 2; � � � ; n and x̂nþ1 can be considered as the approxi-mations of corresponding state xi for i ¼ 1; 2; � � � ; n and to-tal disturbance fþ w, respectively.

The following assumptions21) are presumed:Assumption (H1). The possibly unknown functions f, w

are continuously differentiated with respect to their variables,and

uj j þ f�� þ _wj j þ @f

@t

��þ @f

@xi

�� c0 þ

Xnj¼1

cj xj�� k

for some positive constants cj, j ¼ 0; 1; � � � ; n and positiveinteger k.

Assumption (H2). w and the solution xi of Eq. (3) satisfywj j þ xiðtÞj j � B for some constant B > 0 and alli ¼ 1; 2; � � � ; n, and t � 0.

Assumption (H3). There exist constants R, a > 0 and posi-tive definite, continuous differentiated functions Z,

W :Rnþ1 ! R such thaty ZðyÞ � d��

is bounded for any d > 0,

Xni¼1

@Z

@yiðyiþ1 � giðy1ÞÞ �

@Z

@ynþ1

gnþ1ðy1Þ � �WðyÞ;

@Z

@ynþ1

�� aWðyÞ for y

�� > R:

The weak convergence result stated in the following lem-ma can be obtained.

Lemma 1.21) Under Assumptions (H1), (H2) and (H3), thenonlinear ESO Eq. (4) is convergent in the sense that for any� 2 ð0; 1Þ, there exists "� 2 ð0; 1Þ such that for any" 2 ð0; "�Þ, xiðtÞ � x̂iðtÞj j < �, 8t 2 ðT"; 1Þ, where T" > 0

depends on ", xi; x̂i denote the solutions of Eqs. (3) and (4)respectively, and i ¼ 1; 2; � � � ; nþ 1, xnþ1 ¼ fþ w is theextended state variable for the system Eq. (3).

Specifically, for a first-order nonlinear system

_x1ðtÞ ¼ f0ðt; x1ðtÞÞ þ f1ðt; x1ðtÞÞ þ wðtÞ þ uðtÞyðtÞ ¼ x1ðtÞ

(ð5Þ

where f0 and f1 are known and unknown system func-tions, respectively. By Lemma 1, we can construct an ESOfollowing:

_̂x1ðtÞ ¼ x̂2ðtÞ þ g1yðtÞ � x̂1ðtÞ

"

" #aþf0ðt; yðtÞÞ þ uðtÞ

_̂x2ðtÞ ¼1

"g2

yðtÞ � x̂1ðtÞ"

" #2a�1

8>>>>><>>>>>:

ð6Þ

where �½ �a¼ sgnð�Þ �j ja and a is a constant. g1 and g2 are alsoconstants to be selected. x̂1 and x̂2 are approximations of thestate x1 and f1 þ w.

In this paper, ESO is introduced in the control system toestimate the system uncertainties online (i.e., aerodynamicrelevant terms in system dynamics) to deal with the aerody-

namic unpredictability in the high AoA region. It should beemphasized that aerodynamic data is only used in simulationfor demonstrating the effectiveness of the online estimationscheme, but unknown to autopilot design.3.2. Pitch autopilot design

First, we consider the following missile dynamics of thepitch channel:

_� ¼ f�0 þ f�1 þ q_q ¼ fq0 þ fq1 þ byuRz

(ð7Þ

where

f�0 ¼ � p cos�þ r sin�� tan �þ � sin�T þ cos�uRz

� �= mV cos ��

;

f�1 ¼ ð� sin�X=mþ cos�Z=mþ g sin � sin �þ g cos� cos � cos �Þ=ðV cos �Þ;

fq0 ¼ 1� Ixx=Iyy� �

pr; fq1 ¼ QSDCm=Iyy;

by ¼ �lR=Iyy:

f�0 and fq0 model the known part, while f�1 and fq1 modelthe unknown part of the missile dynamics. g is the gravita-tional acceleration, � and � are Euler angles, uRz is the forceproduced by RCS thrusters in the z-body direction, m is themissile mass, X and Z are the aerodynamic forces, Q is thedynamic pressure, S is the reference area, D is the referencelength, Cm is the pitching moment coefficient, and lR is thedistance between the point of RCS action and the center ofmass.

Owing to estimating the dynamic uncertainties online, theESO translates system Eq. (7) with unmodeled dynamicsinto a certain series-connection integral system based onthe theory of feedback linearization. For the first equationin Eq. (7), as a virtual control, q is available for controlling�. That is, AoA command �c determines the controlled var-iable qc. For the second equation, uRz is controlled to achievethe desired command qc. This approach divides the second-order system into two first-order controllers including theAoA and the pitch rate control loop.

In terms of the first equation (AoA control loop) inEq. (7), ESO Eq. (6) is adopted to estimate the uncertaintyf�1. Here, x1, f0, f1 and u in system Eq. (5) are specifiedas �, f�0, f�1 and q, respectively. Then x̂2 in ESO Eq. (6)becomes the approximation of f�1. A nonlinear state errorfeedback is applicable to form the control input, shown asfollows:

e� ¼ �c � �u� ¼ K�falðe�; c�; ��Þ

(ð8Þ

where falðe; c; �Þ is a continuous power function with alinear segment near the origin:

falðe; c; �Þ ¼e

�1�c; ej j � �

ej jcsgnðeÞ; ej j > �

8><>: ð9Þ

K�, c�, �� are parameters to be designed. The power functionis reorganized into a piecewise function, leading to the more


273©2015 JSASS

excellent characteristics of quick convergence and smallsteady-state error compared to the linear combination offeedback.17) The virtual control input qc is selected as

qc ¼ u� � f�0 � f̂�1 ð10Þwhere f̂�1 is the online approximation of f�1. In the case ofperfect estimation with f̂�1 ¼ f�0, linearization of the AoAcontrol loop can be realized as

_� ¼ u� ð11ÞConsidering the second equation (pitch rate control loop)

in Eq. (7), the same approach is adopted to realize lineariza-tion. Here, x1, f0, f1 and u in system Eq. (5) are specified asq, fq0, fq1 and byuRz, respectively. x̂2 in ESO Eq. (6) is theapproximation of fq1. Then the nonlinear state error feed-back law is obtained as:

eq ¼ qc � quq ¼ Kqfalðeq; cq; �qÞ

(ð12Þ

The control input is

uRz ¼1

byuq � fq0 � f̂q1� � ð13Þ

where f̂q1 is the online approximation of fq1. The pitch ratecontrol loop is also linearized as

_q ¼ uq ð14ÞThe nonlinear system Eq. (7) has been linearized for the

pitch channel control on account of getting the approxima-tions of the high AoA aerodynamic data in real time. Even-tually, the control input is determined by the state error feed-back control technique as a common approach for linearsystem control.3.3. Roll and yaw autopilot design

The missile dynamics of roll and yaw channels involvingquasi roll angle �� and quasi yaw angle � are shown as

_# ¼ !

_! ¼ f0 þ f1 þ u

(ð15Þ

where

# ¼��

�

" #; ! ¼

p

r

" #; f0 ¼

0

� 1� Ixx=Iyy� �

pq

" #;

f1 ¼fp1

fr1

" #¼

QSDCl=Ixx

QSDCn=Iyy

" #; u ¼

uMx=Ixx

lRuRy=Iyy

" #

and f0 and f1 model the system certainty and uncertainty,respectively. Cl is the rolling moment coefficient, and Cn isthe yawing moment coefficient. uMx and uRy, produced byRCS thrusters, are the rolling control moment and the controlforce in the y-body direction.

The roll and yaw autopilot is designed to stabilize �� and � to zero under the disturbance influence of uncertainty f1.The stabilization control is solved by non-singular terminalsliding mode (NTSM) control and the ESO technique in thispaper. According to the proposed agile turn strategy, the lat-eral-directional control determines the maneuvering plane of

the missile, resulting in the roll and yaw channel requiringgreater control precision than the pitch channel. NTSM offerssome superior properties such as fast, finite time conver-gence. This controller is particularly useful for high-preci-sion control as it speeds up the rate of convergence near anequilibrium point.22) Furthermore, the combination of ESOand the SMC method, whose robustness is prominent, hasbetter performance for avoiding the influence of the systemuncertainties. So the proposed approach compares favorablyto both the traditional SMC and ADRC methods for the sta-bilization problem described here.

The sliding mode surface function is selected as

s ¼ s1; s2½ �T¼ #þ B�1sgnð!Þ !j jn ð16Þwhere

B�1 ¼ diag B�11 ; B

�12

� ;

sgnð!Þ ¼ diag sgnðpÞ; sgnðrÞ� ;

!j jn¼ p�� n1 ; rj jn2� T

:

Bi, ni (i=1, 2) are constant, and Bi > 0 and 1 < ni < 2. IfNTSM is applied in the autopilot independently, the NTSMcontrol law based on the exponential approach law is written as

u ¼ � Bn�1sgnð!Þ !j j2�nþf0 þ ksþ �sgnðsÞ� ð17Þwhere k ¼ diag k1; k2½ � and � ¼ diag �1; �2½ �. ki, �i (i ¼ 1; 2)are constant, and ki > 0 and �i > 0. � models the upper boundof absolute values of system uncertainties, i.e.,

fp1�� 1; fr1

�� 2:

The ESO used for online estimating uncertainty f1 isshown as

_X1 ¼ X2 þ G1 "�1 !� X1ð Þ� aþf0 þ u

_X2 ¼ "�1G2 "�1 !� X1ð Þ� 2a�1

(ð18Þ

where

X1 ¼ xp1; xr1� T

; X2 ¼ xp2; xr2� T

;

G1 ¼ diag gp1; gr1�

; G2 ¼ diag gp2; gr2�

;

"�1 ¼ diag "�11 ; "

�12

� ;

Y½ �a¼ diag½sgnðy1Þ; sgnðy2Þ� � y1�� a1 ; y2�� a2� T

;

Y ¼ ½y1; y2�T:X1 and X2 are the approximations of ! and f1. gp1; gp2;gr1; gr2; a1; a2; "1; "2 are constants to be selected. Aug-mented by the approximation of uncertainty f1, the controllaw (17) is restructured as

uESO ¼ � Bn�1sgnð!Þ !j j2�nþf0 þ X2 þ ksþ �ESOsgnðsÞ�

ð19Þwhere �ESO ¼ diag �ESO1; �ESO2½ � and �ESOi > 0 (i ¼ 1; 2).

Comparing control law Eq. (17) with Eq. (19), we draw aconclusion that with the aid of estimating and compensatingthe system uncertainties by ESO, switch gains in the controllaw are clearly decreased. In other words, we can selectsmaller gains as �ESO1 < �1 and �ESO2 < �2. In this case,the control input chattering phenomenon caused by the un-modeled system dynamics will be significantly mitigated.


274©2015 JSASS

Theorem 1. With the non-singular terminal sliding sur-face given by Eq. (16) , ESO obtained by Eq. (18), the tra-jectory of the system Eq. (15) can be driven onto the neigh-borhood around the sliding surface in finite time with thecontrol law Eq. (19). Finally, the system states # and ! con-verge into a residual set of the origin.

Proof. Lemma 1 has shown that the observation errors ofESO converge into the residual set of zero. The observationerrors are defined as e1 ¼ fp1 � xp2 and e2 ¼ fr1 � xr2.

The system Eq. (15) represents the dynamic equationsfor two channels including rolling and yawing motion. Weconsider the roll channel first for the sake of clarity. Nowconsider the Lyapunov function candidate

Vs1 ¼1

2s21 ð20Þ

Its derivative along the system dynamics Eq. (15) can berewritten as follows_Vs1 ¼ s1 _s1

¼ s1 _�� þ n1B�1

1 p�� n1�1

_p �

¼ s1 pþ n1B�11 p�� n1�1

fp1 þ uMx=Ixx� �h i

¼ s1 pþ n1B�11 p�� n1�1

e1 � n�11 B1sgnðpÞ p

�� 2�n1�k1s1h� �ESO1sgnðs1Þ

�i¼ n1B

�11 p�� n1�1 �k1s21 � �ESO1 s1j j þ e1s1

� �� n1B

�11 p�� n1�1 �k1s21 � �ESO1 s1j j þ e1j j s1j j� �

For the case p 6¼ 0, it can be concluded that: 1) If the de-sign parameter �ESO1 meets the condition �ESO1 � e1j j, weget _Vs1 � 0, which means the sliding surface s1 ¼ 0 will bereached. 2) For the condition �ESO1 < e1j j, we have _Vs1 < 0

if s1j j > ð e1j j � �ESO1Þ=k1. Decreasing Vs1 eventually drivesthe system trajectory into a neighborhood around the slidingsurface. Therefore, the trajectory of the system is ultimatelybounded in the region

s1j j � e1j j � �ESO1k1

ð21Þ

For the case p ¼ 0, we get _Vs1 ¼ 0. Then substituting thecontrol law Eq. (19) into the system dynamics Eq. (15) yields

_p ¼ �k1s1 � �ESO1sgnðs1Þ þ e1 ð22ÞUsing Eq. (22), it can be seen that: 1) For s1 > 0 and s1 < 0,_p < 0 and _p > 0 are obtained, respectively, if �ESO1 � e1j j.2) For the condition �ESO1 < e1j j, we analyze _p under the as-sumption s1j j > ð e1j j � �ESO1Þ=k1 in the following. If s1 > 0,we have

s1 >e1j j � �ESO1

k1ð23Þ

Substituting Eq. (22) into the inequation Eq. (23) yields

� 1

k1_pþ �ESO1 � e1� �

>e1j j � �ESO1

k1

_p < e1 � e1j j � 0

If s1 < 0, we have

� s1 >e1j j � �ESO1

k1ð24Þ

Substituting Eq. (22) into the inequation Eq. (24) yields

1

k1_p� �ESO1 � e1� �

>e1j j � �ESO1

k1

_p > e1 þ e1j j � 0

Hence it is clear that for s1 > 0 and s1 < 0, we have _p < 0

and _p > 0, respectively, if s1j j > ð e1j j � �ESO1Þ=k1 for�ESO1 < e1j j, which is the same as that for �ESO1 � e1j j.Under this condition, the system trajectory will cross the linep ¼ 0 and reach the sliding mode s1 ¼ 0.22)

In conclusion, the system trajectory is ultimately boundedin the region Eq. (21). For the yaw channel, the proof proce-dure omitted here is logically the same as that of the rollchannel shown above. It is also concluded that the trajectoryof the yaw channel control system is bounded ultimately as

s2j j � e2j j � �ESO2k2

ð25Þ

According to TSM theory,22) the states in the sliding modewill reach the origin in finite time. For the system Eq. (15),by the above analysis, the states # and ! finally convergeinto a residual set of the origin. The proof is complete. �

Remark 1. Since the ESO cannot completely track thesignal in any practical system, asymptotic stability is lostand it can only guarantee the bounded motion about the slid-ing surface. In Eqs. (21) and (25), it can be seen that theboundary layer about the sliding surface is determined bythe estimation error of ESO. Thus the parameter selectingof ESO is very important, because it not only determinesthe observation performance, but also impacts the behaviorof the sliding surface.

Remark 2. The parameter selecting of ESO is associatedwith the observer error dynamics. For the ESO employed inthe roll channel, after subtracting equation set (18) fromEq. (15), we get an expression for the observer error dynamics:

_e0 ¼ e1 � gp1e0

"1

" #a1

_e1 ¼ _fp1 �gp2

"1

e0

"1

" #2a1�1

8>>>>><>>>>>:

ð26Þ

where _fp1 is the derivative of fp1 and e0 ¼ p� xp1. When theobserver is stable, we have _e0 ¼ 0, _e1 ¼ 0. Then the observa-tion errors can be written as

e0j j ¼ "1"1 _fp1

gp2

��

12a1�1

e1j j ¼ gp1"1 _fp1

gp2

��

a12a1�1

8>>>>>><>>>>>>:

ð27Þ


275©2015 JSASS

It can be seen that the errors of estimation are determined bygp1, gp2, "1 and a1. Via tuning these parameters appropriately,the estimation errors of the observer can be forced smallenough such that uncertainty fp1 can be effectively observedby ESO. Especially, an appropriate "1 can be selected smallenough such that e0j j and e1j j are small enough despite _fp1being unknown. The fundamental selection of the parameterscan be chosen as gp1 > 0, gp2 > 0, 0 < "1 < 1 and 0:5 <

a1 < 1.

4. Simulation Results

In this section, the autopilot synthesis is tested with a 6-DOF simulation of an agile missile. A typical engagementscenario simulated here is the fast 180� turn for attacking atarget in the rear hemisphere. The simulation program is per-formed to test 90� AoA flight performance. For this reason,whole target interception maneuvering is unnecessary to dis-cuss and show here. With RCS control for each channel, thepitch channel follows a 90� AoA command, while the rolland yaw channels perform stabilization control. Here weselect the local vertical plane as the missile expected maneu-vering plane, so the missile need not perform a roll motionbefore turning around. For a simulation with large attitudeangle variation, it is unsuitable for the rotation motion tobe modeled by the Euler angle coordinate system due tothe singular problem. Hence, the quaternion algorithm isemployed to establish the simulation model, which preventsthe degeneration of the dynamic equations in any case.

As previously mentioned, the missile aerodynamic data isobtained by two means. One is Missile Datcom calculationfor the estimated values, and the other is manual setting ofthe disturbed values (i.e., uncertainty f1 in systemEq. (15), reflecting the aerodynamic uncertainty of roll andyaw channels). Time histories ofCl and Cn set as three differ-ent variational forms including constant function, step func-tion and sinusoidal function are shown in Fig. 4. The aerody-namic coefficients change more drastically during 0 to 1 s inorder to show the aerodynamic disturbance with strong var-iation and randomness at rapid changing AoA. Note againthat the rolling and yawing moment coefficient values arescarcely possible to get in advance with the current state-of-the-art of aerodynamics. So it is assumed that the aerody-namic moment coefficients vary according to the artificialfunctions, just as the pitching moment presented by Kim

and Kim.11) Correspondingly, the proposed control lawEq. (19) does not use any true value of aerodynamic infor-mation.

In the simulation, the initial parameters are selected asthe flight altitude 5 km, the initial velocity 0.8Ma, the mainengine thrust 15,000N and the maximum steady thrust ofRCS 2,000N. The simulation time is set as a fixed value,3 s. The main geometric and physical characteristics of theconfiguration are a length of 3m, diameter of 0.15m andmass of 105 kg.

The missile responses in pitch angle �, angle of attack �,quasi roll angle �� and quasi yaw angle � are given inFig. 5. It should be noted that the pitch angle curves andAoA curves for three scenarios are overlapped respectively.AoAs rapidly increase to 90� within about 1 s from the be-ginning of the maneuver and then becomes steady. �� and �

fluctuate near 0� and their absolute values are always keptwithin 0.8�. Certainly, �� and � cannot be kept at 0� invar-iably because of the real-time change in aerodynamic distur-bances. Huge drag and unavailability of the main enginethrust in the direction of velocity lead to a distinct drop invelocity magnitude, as shown in Fig. 6. The velocity curvesare also overlapped.

Estimation errors for uncertainty f�1, fp1, fq1 and fr1, rep-resenting aerodynamic information, are shown in Fig. 7,which illustrates the ESO’s ability to obtain approximationsof uncertainties. All ESOs used in the proposed controlmethod take the same design parameters of g1 ¼ g2 ¼ 1,a ¼ 0:8 and " ¼ 0:01. Furthermore, it is observed that esti-mation errors change vigorously in the early stages of simu-lation. At the beginning, the estimation error is big while " issmall, which leads to large quantities on the right-hand sides

0 0.5 1 1.5 2 2.5 3-0.4-0.2

00.20.4

Coe

ffici

ent C

l

0 0.5 1 1.5 2 2.5 3-20

0

20

Time (s)

Coe

ffici

ent C

n

Scenario1 Scenario2 Scenario3

Fig. 4. Time histories of rolling and yawing moment coefficients.

0 1 2 30

50100150200250

Ang

le q

(˚)

0 1 2 30

30

60

90

Ang

le a

(˚)

0 1 2 3-1

-0.5

0

0.5

1

Time (s)

Ang

le F

(˚)

*

0 1 2 3-0.6-0.4-0.2

00.20.40.6

Time (s)

Ang

le y

(˚)

*

Fig. 5. Time histories of responses.

0 0.5 1 1.5 2 2.5 3

100

150

200

250

300

Time (s)

Vel

ocity

(m/s

)

Fig. 6. Velocity curves.


276©2015 JSASS

of the equalities in ESO dynamics Eq. (6). Consequently, bigderivatives of state variables result in dramatic changes. Fur-thermore, estimation errors for fp1 and fr1 appear suddenlyand change the subsequent process due to the aerodynamicdiscontinuous step disturbance for Scenario 2.

Figure 8 shows the RCS control inputs of the yaw channelwith the sinusoidal changing moment coefficients (Scenario3). If the control law Eq. (17) is selected as the control input,the NTSM approach is used alone to stabilize the missileattitude. At this rate, the control input appears to have a seri-ous chattering phenomenon. If the compound control lawEq. (19) based on NTSM and ESO is applied, the chatteringis lessened significantly and the need for thrust is also re-duced. The advantage associated with the introduction ofthe ESO system is the chattering amplitude decay dependingon the values of switch gains in the control input. However,eliminating the chattering completely, depending on perfectperformance of disturbance rejection, is unpractical and un-necessary. Figure 9 illustrates the time histories of responsesin three kinds of scenarios when the contribution of ESO tothe control system is artificially reduced. Here f̂�1 and f̂q1are replaced with f̂�1=2 and f̂q1=2 in pitch control lawEqs. (10) and (13), respectively, while the standard NTSMcontrol law Eq. (17) without ESO is adopted for roll andyaw control. Compared with the simulation results shownin Fig. 5, the pitch angles and AoAs increase more slowly;meanwhile, the quasi roll angles and quasi yaw angles fluc-tuate far more out of equilibrium.

In order to observe the robustness of ESO to time delay, a

simulation of the longitudinal control is performed by replac-ing yðtÞ ¼ x1ðtÞ with yðtÞ ¼ x1ðt þ �Þ for x1 ¼ � (AoA con-trol loop), x1 ¼ q (pitch rate control loop) and � ¼ 0:03 inScenario 3. The result plotted in Fig. 10 shows that theESO can tolerate a small output time delay. In addition, sim-ulations of the ESO system used in the pitch autopilot withadditive white Gaussian noises are performed for the purposeof evaluating the robustness against high-frequency distur-bances. One of the two kinds of disturbances that we are con-cerned with is measurement noises in the state variable q formean value ¼ 0 and variance ¼ 0.001. The signal withnoises and the estimated result of uncertainty fq1 are shownin Fig. 11. The other disturbance is the high-frequency aero-dynamic disturbance included in pitch rate dynamics Eq. (7)for mean value ¼ 0 and variance ¼ 0.1. The result is plottedin Fig. 12. It is seen that ESO is qualified to be used in theagile missile control system with measurement noises andhigh-frequency disturbances.

Finally, AoAs and pitch angles of a 180� turn under a giv-en 90� AoA command are plotted in Fig. 13 for three kinds

0 1 2 3-0.06

-0.04

-0.02

0

0.02E

stim

atio

n er

ror

of f a

1 (rad

/s)

0 1 2 3-60-30

03060

Est

imat

ion

erro

rof

f p1 (r

ad/s2 )

0 1 2 3

-2

-1

0

Time (s)

Est

imat

ion

erro

rof

f q1 (r

ad/s2 )

0 1 2 3-60

-30

0

30

60

Time (s)

Est

imat

ion

erro

rof

f r1 (r

ad/s2 )

Fig. 7. Estimation of uncertainties via ESO.

0 0.5 1 1.5 2 2.5 3-2,000

-1000

0

1000

2,000

Time (s)

Thr

ust w

ith E

SO (N

)

0 0.5 1 1.5 2 2.5 3-2,000

-1000

0

1000

2,000

Thr

ust w

ithou

t ESO

(N)

Fig. 8. Control input.

0 1 2 30

50100150200250

Ang

le q

(˚)

0 1 2 30

30

60

90

Ang

le a

(˚)

0 1 2 3-6-4-202

Time (s)

Ang

le F

(˚)

*

0 1 2 30

0.5

1

Time (s)

Ang

le y

(˚)

*

Fig. 9. Time histories of responses with low-performance ESO.

0 0.5 1 1.5 2 2.5 3-30

-25

-20

-15

-10

-5

0

5

Time (s)E

stim

atio

n of

f q1 (r

ad/s2 )

Real valueEstimated value

Fig. 10. Estimation results of ESO with time delay.

0 0.5 1 1.5 2 2.5 30

1

2

3

q w

ith m

easu

rem

ent

nois

es (r

ad/s

)

0 0.5 1 1.5 2 2.5 3-30

-20

-10

0

Time (s)

Est

imat

ion

of

f q1

(rad

/s2 )


Fig. 11. Simulation results of ESO with measurement noises.


277©2015 JSASS

of scenarios. Then Fig. 14 illustrates missile trajectoriesaccordingly. The missile inevitably flies out of expectedmaneuvering plane due to the effect of rolling and yawingmoments. However, the missile is able to roughly keep flyingin the vertical plane when under control. In addition, the hor-izontal flight path angles given by numerical calculation atthe end of the simulation are 179.79�, 178.62� and 179.73�

for the three scenarios, respectively, closing to the idealvalue, 180�.

5. Conclusion

A nonlinear missile autopilot based on the ESO methodhas been synthesized to fulfill a fast 180� turn maneuver.High AoA aerodynamic data is so hard to predict preciselythat we treat the aerodynamic forces as uncertainties in thecontrol system. To achieve decoupling control in three chan-nels, an ESO technique is used to estimate the total systemuncertainty including aerodynamic data online. After theapproximations of uncertainties are determined, the pitchingdynamics can be linearized, and consequently high AoA con-

trol is achieved. According to the proposed quasi BTT steer-ing technique, the attitude stabilization for the roll and yawchannels has been addressed successfully using a compoundmethod based on NTSM control and the ESO technique. Theperformance of the proposed autopilot was proven by numer-ical simulation. In addition, the control input chatteringcaused by unmodeled dynamics can be mitigated effectivelythrough the approach of combining SMC and ESO. Signifi-cantly, the proposed method achieves high-precision effec-tive control for low-precision modeling of missile dynamicswithout using an aerodynamics database. This is an outstand-ing feature when plant dynamics are difficult or impossible tomodel accurately for the purpose of control design and im-plementation.

References

1) Wise, K. A. and Broy, D. J.: Agile Missile Dynamics and Control,J. Guid. Control Dynam., 21 (1998), pp. 441–449.

2) Thukral, A. and Innocenti, M.: A SlidingModeMissile Pitch AutopilotSynthesis for High Angle of Attack Maneuvering, IEEE Trans. Con-trol Syst. Technol., 6 (1998), pp. 359–371.

3) Innocenti, M. and Thukral, A.: Robustness of a Variable StructureControl System for Maneuverable Flight Vehicles, J. Guid. ControlDynam., 20 (1997), pp. 377–383.

4) McFarland, M. B. and Calise, A. J.: Neural Networks and AdaptiveNonlinear Control of Agile Antiair Missiles, J. Guid. Control Dynam.,23 (2000), pp. 547–553.

5) McFarland, M. B. and Calise, A. J.: Adaptive Nonlinear Control ofAgile Antiair Missiles Using Neural Networks, IEEE Trans. ControlSyst. Technol., 8 (2000), pp. 749–756.

6) Kang, S., Kim, H. J., Lee, J. I., Jun, B. E., and Tahk, M. J.: Roll-Pitch-Yaw Integrated Robust Autopilot Design for a High Angle-of-AttackMissile, J. Guid. Control Dynam., 32 (2009), pp. 1622–1628.

7) Ryu, S. M., Won, D. Y., Lee, C. H., and Tahk, M. J.: High Angle ofAttack Missile Autopilot Design by Pole Placement Approach, 20103rd International Symposium on Systems and Control in Aeronauticsand Astronautics, Harbin, 2010, pp. 535–539.

8) Kim, K. U., Kang, S., Kim, H. J., Lee, C. H., and Tahk, M. J.: RealtimeAgile-Turn Guidance and Control for an Air-to-Air Missile, AIAAGuidance, Navigation, and Control Conference, Toronto, 2010.

9) Ratliff, R. T., Ramsey, J. A., Wise, K. A., and Lavretsky, E.: Advancesin Agile Maneuvering for High Performance Munitions, AIAA Guid-ance, Navigation, and Control Conference, Chicago, 2009.

10) Kim, Y., Kim, B. S., and Park, J.: Aerodynamic Pitch Control Designfor Reversal of Missile’s Flight Direction, Proc. Inst. Mech. Eng. PartG: J. Aerospace Eng., 228 (2014), pp. 1519–1527.

11) Kim, Y. and Kim, B. S.: Pitch Autopilot Design for Agile Missiles withUncertain Aerodynamic Coefficients, IEEE Trans. Aerospace Elec-tronic Syst., 49 (2013), pp. 907–914.

12) Hemsch, M. J. andMendenhall, M. R.: Tactical Missile Aerodynamics,AIAA, Reston, 1992.

13) Simon, J. M. and Blake, W. B.: Missile Datcom: High Angle of AttackCapability, 24th Atmospheric Flight Mechanics Conference, Portland,1999, pp. 699–708.

14) Abney, E. J. and McDaniel, M. A.: Datcom: High Angle of AttackAerodynamic Predictions Using Missile Datcom, 23rd AIAA AppliedAerodynamics Conference, Toronto, 2005.

15) Han, J. Q.: Control Theory: Is It a Theory of Model or Control, J. Syst.Sci. Math. Sci., 1 (1989), pp. 328–335 (in Chinese).

16) Han, J. Q. and Wang, W.: Nonlinear Tracking Differentiator, J. Syst.Sci. Math. Sci., 14 (1994), pp. 177–183 (in Chinese).

17) Han, J. Q.: From PID to Active Disturbance Rejection Control, IEEETrans. Ind. Electronics, 31 (2009), pp. 900–906.

18) Han, J. Q.: Active Disturbance Rejection Control Technique—TheTechnique for Estimating and Compensating the Uncertainties,National Defense Industry Press, Beijing, 2008 (in Chinese).

0 0.5 1 1.5 2 2.5 3-30

-25

-20

-15

-10

-5

0

5

Time (s)

Est

imat

ion

of f q1

(rad

/s2 )


Fig. 12. Estimation results of ESO with aerodynamic turbulences.

0 1 2 3 4 50

50

100

150

200

Time (s)

Ang

les

(˚)

AoA command

AoAs

Pitch angles

Fig. 13. Time histories of controlled angles.

-5 05 10

15

01002003004005000

5050

5100

5150

5200

5250

5300

5350

Crossrange (m)Downrange (m)

Hei

ght (

m)

Scenario 2

Scenario 1

Scenario 3

Fig. 14. Trajectories.


278©2015 JSASS

19) Gao, Z. Q.: Active Disturbance Rejection Control: a Paradigm Shift inFeedback Control System Design, Proceeding of TJE American Con-trol Conference, Piscataway, 2006, pp. 2399–2405.

20) Huang, Y., Xue, W., and Gao, C.: Active Disturbance Rejection Con-trol: Methodology and Theoretical Analysis, J. Syst. Sci. Math. Sci., 31(2011), pp. 1111–1129 (in Chinese).

21) Guo, B. Z. and Zhao, Z. L.: On the Convergence of an Extended StateObserver for Nonlinear Systems with Uncertainty, Syst. Control Lett.,

60 (2011), pp. 420–430.22) Feng, Y., Yu, X. H., and Man, Z. H.: Non-Singular Terminal Sliding

Mode Control of Rigid Manipulators, Automatica, 38 (2002),pp. 2159–2167.

Y. OchiAssociate Editor


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Agile Missile Autopilot Design for High Angle of Attack