Integrated 3‐D Flight Trajectory Tracking Control with

Asian Journal of Control, Vol. 20, No. 5, pp. 1891–1906, September 2018Published online 15 December 2017 in Wiley Online Library (wileyonlinelibrary.com) DOI: 10.1002/asjc.1696

INTEGRATED 3-D FLIGHT TRAJECTORY TRACKING CONTROL WITHAERODYNAMIC CONSTRAINTS ON ATTITUDE AND

CONTROL SURFACES

Xueyuan Wang, Hao Fang Lihua Dou, Jie Chen, and Bin Xin

ABSTRACT

In this paper, a new 3-D trajectory tracking problem for an uncertain high fidelity six-degree-of-freedom (6-DOF)aerodynamic system is considered. Instead of designing controllers for each subsystem separately, an integrated tra-jectory tracking control algorithm is proposed to exploit beneficial relationships among interacting subsystems. Thehigh-order aerodynamic model is first transformed into a quasi-strict-feedback form. Then, backstepping technique isutilized to resolve the coupling effect problem of three control channels resulting from the bank-to-turn (BTT) con-trol mode. In addition, command filters are introduced to handle state and actuator constraints caused by the physicallimitations and the coordinated turn requirement. Furthermore, the uncertain aerodynamic force and moment coeffi-cients are reconstructed by using the B-spline neural network approximation and adaptive learning approaches. WithLyapunov stability analysis, all the states in the closed-loop system are shown to be semi-globally uniformly ultimatelybounded (SUUB), and the tracking errors will asymptotically converge into a small compact set around zero by prop-erly adjusting the control parameters. Finally, numerical simulations are conducted to demonstrate the effectiveness ofthe proposed algorithm.

Key Words: Trajectory tracking, high-order nonlinear system, backstepping, B-spline neural network.

I. INTRODUCTION

Faced with ever-changing tasks and complex envi-ronments, more complicated problems have to be takeninto consideration for modern missile control systemsbesides the traditional target oriented guidance andcontrol problem. Nowadays, it is becoming more andmore important to accomplish the spatial-temporalflight trajectory manipulation tasks, as it allows mis-siles to, for example, avoid threats and penetrate withhigh probability [1]. Under these circumstances, track-ing a desired time-parameterized trajectory accuratelybecomes essential. In addition, the ability to control mis-siles in the manner of space-time is meaningful to achieve

Manuscript received December 8, 2016; revised July 27, 2017; acceptedSeptember 2, 2017.

Xueyuan Wang, Hao Fang, Lihua Dou, Jie Chen and Bin Xin are with the KeyLaboratory of Intelligent Control and Decision of Complex Systems, School ofAutomation, Beijing Institute of Technology, Beijing, 100081, China.

Hao Fang is the corresponding author (e-mail: [email protected]).This work was supported by Projects of Major International (Regional)

Joint Research Program NSFC (Grant no. 61720106011), NSFC (Grant no.61573062, 61621063, 61673058, and 61603094), Program for Changjiang Schol-ars and Innovative Research Team in University (under Grant IRT1208), BeijingEducation Committee Cooperation Building Foundation Project (Grant No.2017CX02005), Beijing Advanced Innovation Center for Intelligent Robots andSystems (Beijing Institute of Technology), Key Laboratory of Biomimetic Robotsand Systems (Beijing Institute of Technology), Ministry of Education, Beijing,100081, China.

cooperation among a team of missiles in some newlyemerged tasks [2–4].

As for tail controlled missiles (not including the bal-listic missile), to the best of our knowledge, the researchon the trajectory tracking control (TTC) problem hasnot been reported. Most of the existing works focus onthe target oriented guidance and control problem, whichcan be typically solved by designing the guidance lawand the autopilot controller separately. Although suchcascaded control method is efficient in some practicalapplications, it often leads to conservative design of theon-board systems and unfeasible stability analysis for theoverall systems, since the synergistic relationships amongthe interacting subsystems are not fully exploited [5]. Toovercome such problems, the integrated guidance andcontrol (IGC) approach was first proposed by Williamset al. [6] aiming at establishing design trade-offs betweensubsystem specifications. Since then, the IGC approacheshave been studied extensively in 2-D plane with the pla-nar engagement geometry dynamics [7–9], and extendedto the 3-D situation [10–12] to fully exploit the coopera-tive relationship among the pitch, yaw, and roll channels.Moreover, in order to enhance the lethality of missile’swarhead, terminal impact angle constraints are consid-ered in the design of the 2-D [13–15] and 3-D [16,17] IGCcontrol law. However, these controllers cannot be applied

© 2017 Chinese Automatic Control Society and John Wiley & Sons Australia, Ltd

,

http://orcid.org/0000-0002-9627-0325

1892 Asian Journal of Control, Vol. 20, No. 5, pp. 1891–1906, September 2018

to the trajectory tracking problem because the objec-tive of TTC is to follow a desired time-parameterizedtrajectory, while the IGC controller aims at manip-ulating the missile so that there exists an intersec-tion point between the trajectories of the missile andthe target.

For the TTC problem, many efforts have beenmade for several kinds of vertical take-off and landingunmanned aerial vehicles (VTOL-UAVs), such as heli-copter, quadrotor, ducted-fan UAV, etc. Through modellinearization, many linear control methods [18–20] havebeen successfully applied to VTOL-UAVs. Moreover,considering the inherent nonlinear and coupling prop-erties of VTOL-UAVs, a variety of nonlinear controlmethods were also proposed to solve the TTC problem,for example, backstepping approach [21,22], fuzzy adap-tive control [23], sliding mode technique [24,25], modelpredictive control [26], differential flatness method [27],etc. However, since the aerodynamics of VTOL-UAV isquite different from that of missiles, these controllers can-not be applied to missiles directly. The TTC problem hasalso been studied for other types of aircrafts. Amongthem, fixed wing UAVs have the most similar dynamics tomissiles. However, the TTC problem of fixed wing UAVsis often treated as the lateral guidance problem underthe assumption that the dynamics can be approximatedby the integrator dynamic systems [28], or the autopilotsystem has been well designed [29,30]. Therefore, thesecontrollers are not applicable to the TTC problem ofmissiles either.

In this paper, the TTC problem for a missile withbank-to-turn (BTT) control mode is considered. The turnof the BTT missile is achieved by orienting the plane ofmaximum aerodynamic normal force to the desired direc-tion with substantially large roll rate, and by producingthe required maneuvering acceleration with pitch controlsystem in that plane. A high roll rate may lead to severeundesirable couplings among pitch, yaw, and roll controlchannels, so it is difficult to design the attitude controllerfor each channel separately. In addition, the coordinatedturn of BTT missile requires the value of side-slip angleto locate in a small neighborhood of zero (ideally be keptat zero). The deflections of control surfaces (aileron, rud-der, and elevator) and angular rates are also restrictedby some physical limitations. Such characteristics driveus to design an integrated trajectory tracking controllerwhich can not only handle the inherent high-order andcoupling effect problem, but also satisfy the constraintson the attitude and control surfaces.

The main contributions of this paper are as fol-lows. First, a new time-parameterized trajectory trackingproblem of the tail controlled BTT missile with con-straints on the attitude and control surfaces is considered.

Second, an integrated adaptive backstepping controlscheme combined with command filters and B-splineneural networks is proposed, such that the boundednessof the tracking errors are guaranteed. Third, instead ofassuming the side-slip angle to be zero, the derived con-troller is able to stabilize the side-slip angle into a propersmall region so that the coordinated turn can be achieved.

The rest of this paper is organized as follows. Thetrajectory tracking problem is formulated and prelimi-nary knowledge is given in Sect. II. Sect. III presents anovel adaptive backstepping approach to solve the TTCproblem and the closed-loop stability is analyzed. In Sect.IV, simulation results are presented. Finally, we give theconclusions in the last section.

II. PROBLEM FORMULATION ANDPRELIMINARIES

In this section, some reference frames, missilekinematic and dynamic models, and B-spline neuralnetworks, which are employed to approximate the uncer-tain aerodynamic coefficients are introduced. The prob-lem formulation of the flight trajectory tracking controlwith aerodynamic constraints is given at the end of thissection.

2.1 Missile dynamic model

Four reference frames are involved in this paper.The earth-fixed reference frame FE(A − XEYEZE) isan inertial frame with its x-axis (XE) pointing to thenorth, z-axis (ZE) pointing to the east. The x-axes of thevelocity-axis reference frame FV (O − XV YV ZV ) and thetrajectory-axis reference frame FT (O − XT YT ZT ) sharethe same direction with the velocity vector. The x-axisof the body-axis reference frame FB(O − XBYBZB) coin-cides with the body longitudinal centerline and pointstoward the nose of the missile. The reference frames men-tioned above are all depicted in Fig. 1. More details aboutthe reference frames and transformation matrices can befound in the book [31].

Assumption 1. [32] The rotation of the Earth is neglected,that is to say, the curved surface of the earth is treatedas a flat plane. Missile dynamics is built based on theequations of motion for a single rigid body, neglecting theeffects of structural deflections (aeroelasticity) and therelative motion of the control surfaces. The mass of themissile is assumed to be constant. The air is assumed tobe at rest, neglecting any effect caused by wind.


X. Wang et al.: Trajectory Tracking with Aerodynamic Constraints 1893

Fig. 1. Missile reference frames. [Color figure can be viewedat wileyonlinelibrary.com]

Assumption 2. The desired trajectory Xc1 (t) and its

first time derivative X c1 (t) are continuous, bounded and

known. The flight path angle does not equal to± 𝜋

2during

the flight trajectory tracking.

Based on Assumptions 1 and 2, the 6-DOF nonlin-ear model of missiles are given as follows [31]

X1 = G1(X2) (1)

X2 = H1 + G2

(X3,P)

(2)

X3 = A2F1 (X ) + H2 + B1X4 (3)

X4 = A3F2 (X ) + H3 + B2U (4)

where

G1(X2) =⎡⎢⎢⎣

V cos 𝜃 cos𝜓vV sin 𝜃

−V cos 𝜃 sin𝜓v

⎤⎥⎥⎦ H1 =⎡⎢⎢⎣

−g sin 𝜃−g cos 𝜃∕V

0

⎤⎥⎥⎦G2(X3,P) =

⎡⎢⎢⎢⎣P cos 𝛼 cos 𝛽−D

mP(sin 𝛼 cos 𝛾V+cos 𝛼 sin 𝛽 sin 𝛾V )+L cos 𝛾V−Y sin 𝛾V

mV

−P(sin 𝛼 sin 𝛾V−cos 𝛼 sin 𝛽 cos 𝛾V )+L sin 𝛾V+Y cos 𝛾V

mV cos 𝜃

⎤⎥⎥⎥⎦A2 = 1

mV

⎡⎢⎢⎣0 0 −1∕ cos 𝛽0 1 00 tan 𝜃 cos 𝛾V tan 𝛽 + tan 𝜃 sin 𝛾V

⎤⎥⎥⎦

H2 = 1mV

⎡⎢⎢⎢⎣(mg cos 𝜃 cos 𝛾V − P sin 𝛼

)∕ cos 𝛽

mg cos 𝜃 sin 𝛾V − P cos 𝛼 sin 𝛽−mg tan 𝛽 cos 𝜃 cos 𝛾V + P(tan 𝛽 sin 𝛼−

tan 𝜃 cos 𝛾V cos 𝛼 sin 𝛽 + tan 𝜃 sin 𝛼 sin 𝛾V )

⎤⎥⎥⎥⎦B1 =⎡⎢⎢⎣− tan 𝛽 cos 𝛼 sin 𝛼 tan 𝛽 1

sin 𝛼 cos 𝛼 0sec 𝛽 cos 𝛼 − sec 𝛽 sin 𝛼 0

⎤⎥⎥⎦ F1(X ) =⎡⎢⎢⎣

DYL

⎤⎥⎥⎦A3 =⎡⎢⎢⎣

Jx 0 00 Jy 00 0 Jz

⎤⎥⎥⎦−1

H3 =⎡⎢⎢⎣(Jy − Jz)𝜔z𝜔y∕Jx(Jz − Jx)𝜔x𝜔z∕Jy(Jx − Jy)𝜔y𝜔x∕Jz

⎤⎥⎥⎦F2(X ) =

⎡⎢⎢⎣Mx0My0Mz0

⎤⎥⎥⎦ = qS

⎡⎢⎢⎢⎣l[m𝛽

x(𝛽)𝛽 + m𝜔xx (V )𝜔x

]l[m𝛽

y(𝛽)𝛽 + m𝜔yy (V )𝜔y

]bA

[m𝛼

z (𝛼)𝛼 + m𝜔zz (V )𝜔z

]⎤⎥⎥⎥⎦

B2 = qS

⎡⎢⎢⎢⎣m𝛿x

x (V )l∕Jx 0 0

0 m𝛿yy (V )l∕Jy 0

0 0 m𝛿zz (V )bA∕Jz

⎤⎥⎥⎥⎦The state vectors are defined as X1 = [x, y, z]T, X2 =[V , 𝜃, 𝜓V ]T, X3 = [𝛼, 𝛽, 𝛾V ]T, X4 = [𝜔x, 𝜔y, 𝜔z]T, andX = [XT

1 ,XT2 ,X

T3 ,X

T4 ]

T and the control surface deflectionU = [𝛿x, 𝛿y, 𝛿z]T. x, y, and z are the coordinates in theearth-fixed inertial frame with the origin A located at thelaunch point on the ground. V is the velocity. The direc-tion of the missile movement, that is, the direction of thevelocity, is represented by the flight path angle 𝜃 and theheading angle 𝜓V . Note that the transformation from thereference frame FE to FT is achieved by rotating through𝜓V and 𝜃 about YE-axis and ZT -axis successively. 𝛼, 𝛽,and 𝛾V denote the angle of attack, side-slip angle, andbank angle respectively. The reference frame FB can betransformed from FV by two successive rotations of 𝛽 and𝛼 about YV -axis and ZB-axis, respectively (see Fig. 1).The angle between YT and YV is 𝛾V . 𝜔x, 𝜔y, and 𝜔z arethe roll, yaw, and pitch rates with respect to the body-axisreference frame. 𝛿x, 𝛿y, and 𝛿z represent the deflections ofthe aileron, rudder, and elevator.

As for the forces involved in the model, P is theengine thrust. D, Y , and L denote the drag, side force, andlift respectively which can be expressed in the followingform:

⎧⎪⎨⎪⎩D = qSCDY = qSCYL = qSCL

(5)

where q is the dynamic pressure and S is the aerody-namic reference area of the missile. q can be calculatedthrough q = 1

2𝜌V2, where 𝜌 is the average density of air.


http://onlinelibrary.wiley.com/


CD, CY , and CL are the drag, lift, and side force coeffi-cients respectively, which can be expressed in terms of theaerodynamic derivatives:

⎧⎪⎨⎪⎩CD = CD0

(V ) + C𝛼D(𝛼,V ) |𝛼|

CY = C𝛽

Y (𝛼,V )𝛽CL = CL0

(V ) + C𝛼L(𝛼,V )𝛼

(6)

where CD0= CD|𝛼=0 and CL0

= CL|𝛼=0. C𝛼D is defined as

C𝛼D = 𝜕CD∕𝜕𝛼 and other coefficients are defined similarly.

In (4), Jx, Jy, and Jz denote the roll, yaw, and pitchmoments of inertia. l and bA are the wing span and wingmean aerodynamic chord respectively. m∗

x, m∗y, and m∗

zare the rolling, yawing, and pitching moment coefficients,in which m𝛽

x is defined as the partial derivative of therolling moment coefficient mx with respect to 𝛽 and othermoment coefficients have similar definitions. The aerody-namic force and moment coefficients in (4) and (6) canbe modeled as functions with respect to the current flightconditions (e.g., V , 𝛼, and 𝛽).

Assumption 3. The functions, representing the variationof the aerodynamic force and moment coefficients in (4)and (6), are continuous in certain compact sets.

2.2 Approximation of the aerodynamic coefficients

Since the aerodynamic force and moment coeffi-cients involved in the flight dynamics are uncertain, theyneed to be approximated before designing the trajectorytracking controller. B-spline neural network (BSNN) issuitable for providing on-line aerodynamic coefficientapproximation [33] due to the local support property [34]and the flexible choice on the order of the B-splines [35].Therefore, the aerodynamic coefficients will be trans-formed into linear-in-the-parameters form using B-splinefunctions as the basis vectors, and then the parameterswill be on-line learned by the adaptive update laws in thispaper.

Noting that the aerodynamic coefficients whichneed to be approximated have similar characteristics, wetake C𝛼

D(𝛼,V ) in (6) as an example. Let C𝛼D be continuous

in a compact set Ω, then there exists a vector 𝜃∗D𝛼

∈ Rd

such that

𝜃∗D𝛼= arg min

𝜃D𝛼

{supΩ

|||C𝛼D(𝛼,V ) − 𝜙T

D𝛼𝜃D𝛼

|||}

(7)

where 𝜃∗D𝛼

is the optimal vector that minimizes theapproximation error in a norm sense, and𝜙D𝛼

(𝛼,V ) ∶ R×R → Rd is the B-spline basis vector in which the bivari-ate B-spline basis functions are defined as the tensor

products of the univariate B-spline basis functions withrespect to the lattice space. The univariate B-spline basisfunctions are defined as follows

𝜙j(x) = g3

(N

x − ab − a

− j)

j = −2,… ,N − 1 (8)

where 𝜙j(x)s represent the 3rd order B-spline basis func-tions defined in the interval [a, b] with N equal subinter-vals, and g3(⋅) denotes the 3rd order Cardinal B-splinefunction [35] which is given by

g3(x) =

⎧⎪⎪⎨⎪⎪⎩

x2

2for 0 ⩽ x < 1,

−x2 + 3x − 32

for 1 ⩽ x < 2,(3−x)2

2for 2 ⩽ x < 3,

0 otherwise.

(9)

The true function C𝛼D(𝛼,V ) can be written as

C𝛼D(𝛼,V ) = 𝜙T

D𝛼𝜃∗D𝛼

+ 𝜀D𝛼(10)

where 𝜀D𝛼denotes the approximation error which can be

made smaller by increasing the number of nodes in theBSNN. Then C𝛼

D(𝛼,V ) is approximated in the followingform

C𝛼D(𝛼,V ) = 𝜙T

D𝛼(𝛼,V )��D𝛼

(11)

where ��D𝛼is the estimate of the weight vectors which will

be learned by the update laws. Other coefficients can beapproximated in similar forms as (11).

Therefore, F1, F2, and B2 are reconstructed as fol-lows:

F1 = ΦTF1ΘF1

=⎡⎢⎢⎣𝜙T

D 0 00 𝜙T

Y 00 0 𝜙T

L

⎤⎥⎥⎦⎡⎢⎢⎣��D

��Y

��L

⎤⎥⎥⎦ (12)

F2 = ΦTF2ΘF2

=⎡⎢⎢⎢⎣𝜙T

Mx00 0

0 𝜙TMy0

0

0 0 𝜙TMz0

⎤⎥⎥⎥⎦⎡⎢⎢⎣��Mx0

��My0

��Mz0

⎤⎥⎥⎦ (13)

B2 = ΦTB2ΘB2

= diag{𝜙T

M𝛿xx

, 𝜙TM

𝛿yy

, 𝜙TM𝛿z

z

}diag{��M𝛿x

x, ��

M𝛿yy, ��M𝛿z

z

}(14)

where 𝜙TD = qS[𝜙T

D0, 𝜙T

D𝛼

|𝛼|], ��D = [��TD0, ��TD𝛼

]T, and other

𝜙∗s and ��∗s have similar definitions according to (4) and(6). Moreover, the optimal weight vectors Θ∗

F1, Θ∗

F2, and

Θ∗B2j

exist such that the following inequalities hold.



‖‖‖F1 − ΦTF1Θ∗

F1

‖‖‖ = ‖‖‖EF1

‖‖‖ ⩽ e1 (15)

‖‖‖F2 − ΦTF2Θ∗

F2

‖‖‖ = ‖‖‖EF2

‖‖‖ ⩽ e2 (16)

‖‖‖B2j − ΦTB2jΘ∗

B2j

‖‖‖ = ‖‖‖EB2j

‖‖‖ ⩽ e3j j = 1, 2, 3 (17)

where EF1, EF2

, and EB2jare the reconstruction errors,

which are bounded over the compact sets ΩF1, ΩF2

, andΩB2j

respectively. The upper bounds e1, e2, and e3j arepositive constants.

2.3 Problem formulation

Eqs. (1) and (2) are often called the point-massmodel describing the translational motion of the missile,while (3) and (4) represent the rotational motion. For abank-to-turn missile, the allowable side-slip angle is themost significant constraint. Ideally, the side-slip angleshould be zero, which would lead to completely coordi-nated turns. If the side-slip angle becomes too large, itcan, in conjunction with a large angle of attack, producelarge induced rolling moments, which may cause con-trol saturation resulting in possible instability or reducedmaneuverability. In addition, the angular rates and con-trol surface deflections are constrained due to some phys-ical limitations, and the other attitude angles are alsolimited to meet the requirements of BTT control mode,which will be discussed later in Sect. III. To reckon withthe effects of the cross-coupling derived from the largeroll rate𝜔x, the three channels are regarded together as anmulti-input multi-output (MIMO) system in this paper.

The objective of this 3-D flight trajectory track-ing control is to design proper control inputs, includ-ing the control surface deflection U and thrust P, tomake ||X1(t) − Xc

1 (t)|| < 𝜖p and ||X1(t) − X c1 (t)|| < 𝜖v

for any 𝜖p, 𝜖v > 0, where Xc1 (t) denotes the predeter-

mined time-parameterized trajectory and X c1 (t) denotes

the desired velocity, while the state X and the controlsurface deflections U are restricted in the compact setΩX ∈ R12 and ΩU ∈ R3 respectively.

III. INTEGRATED 3-D FLIGHTTRAJECTORY TRACKING CONTROL

In this section, taking the synergistic relationshipsbetween the interacting subsystems into consideration,we will design an integrated control law that drivesthe missile to track a smooth prescribed trajectory

Xc1 = (xc, yc, zc)T. For the high-order aerodynamic mod-

els ((1)-(4)), four control processes including positiontracking control, speed and its orientation angle con-trol, speed-axis angle control, and angular rate controlare effectively integrated under the backstepping frame-work by combining the techniques of command filtersand BSNN approximation.

3.1 The position tracking control with command filter

At first, the position tracking problem in 3-D spacedescribed by (1) is considered with Xc

1 as its referencesignal. The virtual controller Xd

2 is chosen to satisfy:

G1(Xd2 ) = X c

1 − K1Z1 (18)

where K1 is a diagonal positive definite matrix, X c1 =

(xc, yc, zc)T is the desired velocity components projectedon the axes of the earth-fixed reference frame, Z1 = X1 −Xc

1 is the position tracking error. From (1) and (18), thetime derivative of Z1 is

Z1 = X1 − X c1 = −K1Z1 + G1(X2) − G1(Xd

2 ) (19)

Note that the first virtual controller Xd2 = (Vd , 𝜃d , 𝜓d

V )T

is not affine in G1(Xd2 ). So we need to determine the com-

ponents (Vd , 𝜃d , 𝜓dV ) by applying (20)-(22) based on the

definition of G1 = (g11, g12, g13)T in which the range of

the flight path angle 𝜃 and the heading angle 𝜓V are(−𝜋∕2, 𝜋∕2) and (−𝜋, 𝜋], respectively.

Vd =√

g211 + g2

12 + g213 (20)

𝜓dV =⎧⎪⎨⎪⎩− arctan (g13∕g11) if g11 > 0−𝜋 − arctan (g13∕g11) if g11 < 0, g13 > 0𝜋 − arctan (g13∕g11) if g11 < 0, g13 < 0

(21)

𝜃d = arctang12√

g211 + g2

13

. (22)

To get the desired reference state and its derivativefor the next subsystem, the virtual controller Xd

2 wouldbe passed though a command filter [36–38], which couldprovide the derivatives of the virtual control signals with-out computing the partial derivatives, so the explosion ofcomplexity problem for high order systems can be suc-cessfully avoided. In this study, the following commandfilter [39] is used.



xc = 2𝜁𝜔n

(SR

{𝜔2

n

2𝜁𝜔n

[SM(xo) − xc]} − xc

)(23)

ST (x) =⎧⎪⎨⎪⎩

Tmax x ⩾ Tmaxx Tmin < x < Tmax−Tmin x ⩽ −Tmin

T = M,R

(24)

where xo is the input of the command filter, xc and xc arethe output, 𝜁 and 𝜔n represent the damping factor andthe natural frequency respectively, SM(⋅) and SR(⋅) denotethe magnitude and rate limit function, which enforce xc

to stay in the defined limits. The limits can be set to beinfinity if no constraint is imposed on the state.

Note that if xo is bounded, then xc and xc arebounded and continuous [39]. In the linear range of thefunction SM(⋅) and SR(⋅), the transfer function for thecommand filter from xo to xc is a second-order linearfilter with unit dc gain defined as

Xc(s)Xo(s)

=𝜔2

n

s2 + 2𝜁𝜔n + 𝜔2n

. (25)

Thus, when the limit functions are not in effect, the errorxc−xo can be made arbitrarily small by selecting𝜔n to besufficiently large. When the limit functions are in effect,the error xc − xo is bounded because both xc and xo arebounded.

Remark 1. Note that extensive researches have beenmade on the nonlinear system control under the stateand actuator constraints. Some of them [40–42] only con-sidered the actuator constraints, while others focused onhandling the partial state constraints [43] or full stateconstraints [44]. The nonlinear model predictive control(NMPC) [45] may be the most common way to deal withboth state and actuator constraints for the nonlinear sys-tem in the existing literatures. However, it may not satisfythe real-time requirement of the flight control systembecause the optimization process in NMPC will be slowfor the high-order nonlinear system. Therefore, the com-mand filter backstepping method is adopted in this studyso that the state and actuator constraints will not be vio-lated and the real-time requirement of the TTC problemis also satisfied.

For the first subsystem, Xd2 will be passed though

the command filter which derives Xc2 and X c

2 . Then inorder to eliminate the effect of the difference between Xc

2

and Xd2 appeared when the limit functions in the com-

mand filter come into effect, the compensating signal 𝜉1is introduced as

��1 = −K1𝜉1 + G1(X2) − G1(Xd2 ). (26)

Then define the compensated tracking error Z1 asZ1 = Z1 − 𝜉1. Differentiating Z1 and substituting (19)into it, we get

Z1 = Z1 − ��1 = −K1Z1. (27)

3.2 The control of speed and its direction

In this step, based on subsystem (2), the virtual con-troller Xd

3 = (𝛼d , 𝛽d , 𝛾dV )

Tand thrust P will be designed

to eliminate the tracking error of the velocity and its ori-entation angles defined by Z2 = X2 − Xc

2 . Note thatwe don’t assume 𝛽 = 0 in this work, so the couplingbetween the roll and pitch channels has to be consideredwhich obviously complicates the controller design. SelectG2(Xd

3 ,P) = (g21, g22, g23)T such that

G2(Xd3 ,P)=−H1 + X c

2−K2Z2

=

⎡⎢⎢⎢⎢⎣P cos 𝛼d cos 𝛽d−D

mP(sin 𝛼d cos 𝛾d

V+cos 𝛼d sin 𝛽d sin 𝛾dV )+L cos 𝛾d

V−Y sin 𝛾dV

mV

−P(sin 𝛼d sin 𝛾dV−cos 𝛼d sin 𝛽d cos 𝛾d

V )+L sin 𝛾dV+Y cos 𝛾d

V

mV cos 𝜃

⎤⎥⎥⎥⎥⎦(28)

where K2 is positive definite. Since P and Xd3 are not affine

in G2 either, we will derive them according to (28). Notethat, to achieve a coordinated turn of BTT missile, thedesired side-slip angle 𝛽d should be set to zero. Therefore,it is reasonable to set sin 𝛽d = 0 and Y = 0 in (28). Andsince D and L can be decomposed as D = D0 + D𝛼|𝛼d|and L = L0 + L𝛼𝛼

d according to (5) and (6), we can get

⎧⎪⎨⎪⎩g21 = 1

m(P cos 𝛼d − D0 − D𝛼|𝛼d|)

g22 = 1mV

[(L0 + L𝛼𝛼d + P sin 𝛼d) cos 𝛾d

V ]g23 = − 1

mV cos 𝜃[(L0 + L𝛼𝛼

d + P sin 𝛼d) sin 𝛾dV ].

(29)

The missile considered in this study uses BTT 90 air-frame [46], in which the lifting surface can be commandedto roll up to ±90◦. Then 𝛾d

V , 𝛼d , and P will be derivedthrough

𝛾dV =⎧⎪⎨⎪⎩−𝜋∕2 if gx = 0, gy < 0𝜋∕2 if gx = 0, gy > 0arctan(gy∕gx) otherwise.

(30)



tan 𝛼d [mg21 + D(𝛼d)]+ L(𝛼d) =

mg22V

cos 𝛾dV

(31)

P =mg21 + D(𝛼d)

cos 𝛼d(32)

where gx = g22mV , gy = −g23mV cos 𝜃, D(𝛼d) = D0 +D𝛼|𝛼d|, and L(𝛼d) = L0 + L𝛼𝛼

d . Note that the inter-mediate control variable 𝛼d can be obtained throughnumerically solving (31) according to the method in [47].After 𝛼d is derived, P can be calculated by substituting 𝛼d

into (32).Combining (2) and (28), the derivative of Z2 is

Z2 = X2 − X c2 = G2(X3,P) − K2Z2 − G2(Xd

3 ,P)

= −K2Z2 + G2(X3,P) − G2(X3,P)

+ G2(X3,P) − G2(Xd3 ,P).

(33)

Let

A1 = 1mV

⎡⎢⎢⎣−V 0 0

0 − sin 𝛾V cos 𝛾V

0 − cos 𝛾V

cos 𝜃− sin 𝛾V

cos 𝜃

⎤⎥⎥⎦ (34)

F1 = F1 − F1 = ΦTF1Θ∗

F1+ EF1

− ΦTF1ΘF1

= ΦTF1ΘF1

+ EF1

(35)

then A1F1 = G2(X3,P) − G2(X3,P), and (33) can bewritten as

Z2 = −K2Z2 + A1F1 + G2(X3,P) − G2(Xd3 ,P). (36)

According to (3), if 𝜔x is too large, 𝛼 should berestricted to make �� relatively small which enables the realvalue of 𝛽 to stay in a small region around zero so thatthe coordinated turn is able to be achieved. Xc

3 and X c3

can be derived by passing Xd3 though the command filter

in which Xd3 is modified to meet the aforementioned con-

straints on X3. The corresponding compensating signal𝜉2 is given by

��2 = −K2𝜉2 + G2(X3,P) − G2(Xd3 ,P). (37)

Remark 2. Note that different from the models withstrict-feedback form in [36–38], the models in this paperare in quasi-strict-feedback form, i.e., X2 and X3 are notaffine in nonlinear functions G1 and G2 respectively inthe subsystems (1) and (2). In addition, the uncertainaerodynamic forces are included in G2. The non-affine

feature and the uncertainties make it difficult for the vir-tual controller design. To cope with this problem, thecompensating signals 𝜉1 and 𝜉2 are constructed and theadaptive law, which is utilized to estimate the weights ofBSNN approximating the aerodynamic force coefficients,will be proposed in the next subsection.

The compensated tracking error Z2 is defined as

Z2 = Z2 − ��2 = −K2Z2 + A1F1

= −K2Z2 + A1ΦTF1ΘF1

+ A1EF1.

(38)

3.3 The missile attitude adjustment

In the following steps, the missile will be treated asa rigid body instead of a point with mass. In order toachieve the desired attitude, the required angular rateswill be derived using the speed-axis angle control basedon subsystem (3). Since the reference signal Xc

3 and itsderivative X c

3 are available, the third virtual controller

Xd4 = (𝜔d

x, 𝜔dy , 𝜔

dz )

Tis designed as

Xd4 = B−1

1

[−A2F1 (X ) − H2 + X c

3 − K3Z3

](39)

where K3 is positive definite. 𝛽 is guaranteed to stay in asmall region around zero in the last subsection, so that|B1| = − sec 𝛽 ≠ 0 and B1 is nonsingular. Then, thederivative of the tracking error Z3 is given by

Z3 = X3 − X c3 = A2F1 − K3Z3 + B1(X4 − Xd

4 ). (40)

Now we will derive Xc4 and X c

4 by passing Xd4 though

the command filter defined in (23), in which Xd4 is mod-

ified to satisfy the limits of the angular rates. Then thethird compensator of the command filter is defined as

��3 = −K3𝜉3 + B1𝜉4 + B1(Xc4 − Xd

4 ). (41)

Differentiating Z3 = Z3 − 𝜉3 and substituting (40)and (41) into it, we will get

Z3 = −K3Z3 + A2F1 + B1(X4 − Xc4 ) − B1𝜉4

= −K3Z3 + B1Z4 + A2ΦTF1ΘF1

+ A2EF1.

(42)

Considering the modeling errors, the update law ofthe estimated weight ΘF1

can be designed as

ΘF1= ΓF1

[ΦF1

(AT

1 Z2 + AT2 Z3

)− 𝜇F1

ΘF1

](43)

where ΓF1is a positive definite matrix and 𝜇F1

is a smallconstant. Since the existence of reconstruction error EF1

may cause ΘF1to drift further from the optimal value



Θ∗F1

and possibly to infinity. The term −ΓF1𝜇F1

ΘF1is

introduced to prevent ΘF1from drifting.

3.4 The angular rate control with input saturation

In the final step, the desired angular rate Xc4 can be

achieved by directly manipulating the control surfaces ofthe missile (i.e. 𝛿x, 𝛿y, and 𝛿z) according to subsystem(4). Since the derivative of Xc

4 can be derived from thelast step and B2 is nonsingular, the control law can beconstructed as

Ud = B−12

(−K4Z4 − A3F2 − H3 + X c

4 − BT1 Z3

)(44)

where K4 is a positive definite diagonal matrix, the term−BT

1 Z3 is used to cancel out B1Z4 in (42), which wouldfacilitate the following Lyapunov stability analysis. Sub-stituting (4) and (44) into Z4 = X4 − X c

4 , we have

Z4 = −K4Z4 + A3F2 − BT1 Z3 + B2U − B2Ud

= −K4Z4 + A3F2 − BT1 Z3 + B2U + B2(U − Ud).

(45)

The desired control law (44) may not be applica-ble for the actuators due to the physical limitations onthe servo motors. Hence the executable control surfacedeflection U can be obtained by passing Ud through theforth command filter. And the corresponding compensat-ing signal 𝜉4 is defined as

��4 = −K4𝜉4 + B2(U − Ud). (46)

With (45) and (46), the compensated tracking errorZ4 can be calculated by

Z4 = Z4 − ��4 = −K4Z4 + A3F2 − BT1 Z3 + B2U

= −K4Z4 − BT1 Z3 + A3ΦT

F2ΘF2

+ A3EF2

+3∑

j=1

ΦTB2jΘB2j

uj +3∑

j=1

EB2juj

(47)

where U = (u1, u2, u3)T, |uj| ⩽ umax. And the adaptive

laws of ΘF2and ΘB2j

( j = 1, 2, 3) are selected as

ΘF2= ΓF2

(ΦF2

AT3 Z4 − 𝜇F2

ΘF2

)(48)

ΘB2j= ΓB2j

(ΦB2j

Z4uj − 𝜇B2jΘB2j

)j = 1, 2, 3 (49)

where ΓF2and ΓB2j

are positive definite matrices, 𝜇F2and

𝜇B2jare small constants.

Remark 3. As stated in this section, we proposed anintegrated controller to solve the 3-D trajectory track-ing problem and corresponding attitude control prob-lem once for all, instead of solving them separately.The advantage of the integrated design method is thatit enables each interacting subsystem to work syner-gistically, and theoretically guarantees the closed-loopstability which will be discussed in the next subsection.

3.5 Stability analysis

Theorem 1. For the system formulated by (1)-(4), underassumptions 2 and 3, with the control laws (18), (28),(39), and (44), the adaptive laws (43), (48), and (49), andcommand filters (23), it can be guaranteed that

1. The weight parameter errors ΘF1, ΘF2

, ΘB2jand the

compensated tracking errors Zi, i = 1, · · · , 4, areSUUB.

2. The compensated tracking error Z1 will converge toa small region around zero by a proper choice of thecontrol parameters.

3. The compensating signals 𝜉i, i = 1, · · · , 4, arebounded and the tracking errors Zi, i = 1, · · · , 4,are SUUB.

Proof. The Lyapunov function for the overall system ischosen as

V = 12

⎛⎜⎜⎝4∑

i=1

ZTi Zi +

∑△=F1,F2,B2j

ΘT△Γ−1△Θ△

⎞⎟⎟⎠ (50)

where j = 1, 2, 3. Taking derivative of V with respect to tand substituting (27), (38), (42), and (47), we have

V = −4∑

i=1

ZTi KiZi + ΘT

F1ΦF1

(AT

1 Z2 + AT2 Z3

)−

2∑k=1

ΘTFkΓ−1

Fk

ΘFk+ ΘT

F2ΦF2

AT3 Z4 + ZT

3 A2EF1

+3∑

j=1

ΘTB2j

(ΦB2j

Z4uj − Γ−1B2j

ΘB2j

)+ ZT

4 A3EF2

+ ZT4

3∑j=1

EB2juj + ZT

2 A1EF1.

(51)



Then, substituting the adaptive laws (43), (48), and (49)into (51), we obtain

V = −4∑

i=1

ZTi KiZi + 𝜇F1

ΘTF1ΘF1

+ 𝜇F2ΘT

F2ΘF2

+3∑

j=1

𝜇B2jΘT

B2jΘB2j

+ ZT2 A1EF1

+ ZT3 A2EF1

+ ZT4

(A3EF2

+3∑

j=1

EB2juj

).

(52)

Let ki0 = 𝜆min{Ki}, i = 1, · · · , 4, where 𝜆min{⋅} denotesthe smallest eigenvalue of a matrix, then the followinginequalities hold

− ZT1 K1Z1 ⩽ −k10

‖‖Z1‖‖2, (53)

− ZT2 K2Z2 + ZT

2 A1EF1

⩽ −k20

2‖‖Z2‖‖2 − k20

2

‖‖‖‖‖Z2 −A1EF1

k20

‖‖‖‖‖2

+‖‖‖A1EF1

‖‖‖22k20

⩽ −k20

2‖‖Z2‖‖2 + ‖‖‖A1EF1

‖‖‖22k20

,

(54)

− ZT3 K3Z3 + ZT

3 A2EF1⩽ −

k30

2‖‖Z3‖‖2 + ‖‖‖A2EF1

‖‖‖22k30

,

(55)

− ZT4 K4Z4 + ZT

4

(A3EF2

+3∑

j=1

EB2juj

)

⩽ −k40

2‖‖Z4‖‖2 + 1

2k40

‖‖‖‖‖‖A3EF2+

3∑j=1

EB2juj

‖‖‖‖‖‖2

.

(56)

From Cauchy-Schwarz inequality, we have

𝜇F1ΘT

F1ΘF1

⩽ 𝜇F1

(‖ΘF1‖‖Θ∗

F1‖ − ‖ΘF1

‖2)⩽𝜇F1

2

(‖Θ∗F1‖2 − ‖ΘF1

‖2) , (57)

𝜇F2ΘT

F2ΘF2

⩽𝜇F2

2

(‖Θ∗F2‖2 − ‖ΘF2

‖2) , (58)

3∑j=1

𝜇B2jΘT

B2jΘB2j

⩽3∑

j=1

𝜇B2j

2

(‖Θ∗B2j‖2 − ‖ΘB2j

‖2) .(59)

Select a positive definite constant 𝜂 satisfying 𝜂 ⩽min{2k10, k20, k30, k40}, and choose 𝜇F1

, 𝜇F2, 𝜇B2j

, ΓF1,

ΓF2, ΓB2j

, j = 1, 2, 3, such that

𝜇F1⩾ 𝜂𝜆max{Γ−1

F1}, 𝜇F2

⩾ 𝜂𝜆max{Γ−1F2}

𝜇B2j⩾ 𝜂𝜆max{Γ−1

B2j}

(60)

where 𝜆max{⋅} is the largest eigenvalue of a matrix. Thenconsidering (53)-(60), (52) will be written as

V ⩽ −12

4∑i=1

ki0‖Zi‖2 − k10

2‖Z1‖2 − 𝜇F1

2‖ΘF1‖2

−𝜇F2

2‖ΘF2‖2 − 3∑

j=1

𝜇B2j

2‖ΘB2j‖2 + Q

⩽ −𝜂2

4∑i=1

ZTi Zi −

𝜂

2ΘT

F1Γ−1

F1ΘF1

− 𝜂

2ΘT

F2Γ−1

F2ΘF2

− 𝜂

2

3∑j=1

ΘTB2jΓ−1

B2jΘB2j

+ Q

= −𝜂V + Q

(61)

where Q = 12k20

𝜎21 + 1

2k30𝜎2

2 + 12k40

𝜎23 + 1

2𝜇F1

𝜎2F1

+12𝜇F2

𝜎2F2

+ 12

∑3j=1 𝜇B2j

𝜎2B2j

, 𝜎1 = max{‖A1EF1‖}, 𝜎2 =

max{‖A2EF1‖}, 𝜎3 = max{‖A3EF2

+∑3

j=1 EB2jumax‖},

𝜎F1= max{‖Θ∗

F1‖}, 𝜎F2

= max{‖Θ∗F2‖}, and 𝜎B2j

=max{‖Θ∗

B2j‖}.

𝜎1, 𝜎2, and 𝜎3 exist, since the side-slip angle 𝛽 isguaranteed to be small near 0, U is bounded, and thereconstruction errors EF1

, EF2, and EB2j

are bounded. Inaddition, assumption 3 implies that the optimal weightvectors Θ∗

F1, Θ∗

F2, and Θ∗

B2jare bounded, so 𝜎1, 𝜎2,

and 𝜎3 exist.According to (61), Zi and Θ△ will converge into

the domain Ωzi={

Zi

||||‖Zi‖ <√ 2Qki0

}(i = 1, 2, 3, 4)

and Ω𝜃△=

{Θ△

|||||‖Θ△‖ <√ 2Q𝜇△

}(△ = F1,F2,B2j)

respectively. Therefore, the semi-global uniform ultimateboundedness of Zi and Θ△ are proved, which completesthe proof of item 1.

Then from (61), we have

0 ⩽ V (t) ⩽Q𝜂+[

V (0) −Q𝜂

]e−𝜂t <

Q𝜂+ V (0)e−𝜂t.

(62)



Combined with (50), the following inequality is satisfied

12‖Z1‖2 < V (t) <

Q𝜂+ V (0)e−𝜂t. (63)

This implies that for 𝜒 >√(2Q∕𝜂) there exists T such

that for all t ⩾ T , the compensated tracking error Z1satisfies ‖Z1‖ < 𝜒 . The convergence domain factor𝜒 depends on the controller parameters (Ki, Γ△, 𝜇△,i = 1, · · · , 4, △ = F1,F2,B2j) and the BSNN recon-struction errors E△, △ = F1,F2,B2j. It is obvious thatbetter tracking performance can be achieved by increas-ing 𝜆min{Ki} and 𝜆min{Γ△}, and applying BSNN withmore nodes. This completes the proof of item 2.

Considering the command filter compensatorsdescribed by (26), (37), (41), and (46), they are actu-ally BIBO stable linear filters with Ki positive definite.Since each input of these filters are bounded, the outputof these filters are bounded as well (i.e., the signals 𝜉i,i = 1, · · · , 4, are bounded). Therefore, with the theoremproved above, we can also conclude that the actual track-ing errors Zi are SUUB. This completes the proof ofitem 3.

Remark 4. The position tracking error Z1 will convergeinto a small neighborhood near 𝜉1. If the limit functionsof the command filters are not in effect, 𝜉1 will convergeto zero and Z1 will have the same convergence propertiesas Z1.

Remark 5. The stability results derived in this paper aresemi-global, because the BSNN approximation is onlyvalid on some compact sets. In other words, the BSNNbased controllers in this paper are effective within the pre-determined flight envelope, which is acceptable since therange of the missile flight envelope must be considered inproducing the desired trajectory.

IV. NUMERICAL SIMULATION

In this section, numerical simulations are imple-mented to illustrate the performance of the proposedintegrated 3-D trajectory tracking algorithm.

The reference trajectory (dashdot line in Fig. 2)is generated according to Xc

1 (0) = (0, 100, 3000)T,

X c1 (0) = (250, 50, 0)T, X c

1 (0) = (0, 0, 0)T,...X

c

1(t) =(...x c(t),

...y c(t),

...z c(t))T where

...x c(t) = 0 (64)

...y c(t) =

{−3 16s < t ⩽ 20s0 otherwise

(65)

Table I. Nominal parameters.

Parameter Value Parameter Value

m 400 kg g 9.8 kg∕m2

l 0.5m bA 0.4mS 0.42m2 Jx 100 kg ⋅ m2

Jy 1300 kg ⋅ m2 Jz 1200 kg ⋅ m2

...z c(t) =

⎧⎪⎨⎪⎩10 0s < t ⩽ 2s, 14s < t ⩽ 16s−10 6s < t ⩽ 10s−4 22s < t ⩽ 24s0 otherwise.

(66)

The nominal values of the missile parameters arelisted in Table I. The average air density 𝜌 is determinedby expression 𝜌 = 1.225(1 − y

44300)4.2533, where y is the

flight altitude of the missile. The initial position of themissile is set as (0, 100, 3050) in the inertial referenceframe. The initial velocity, flight path angle, and headingangle are respectively set as V (0) = 254 m/s, 𝜃(0) = 11.3◦,and 𝜓V (0) = 0◦. The initial angle of attack, side-slipangle and bank angle are respectively set as 𝛼(0) = 3◦,𝛽(0) = 𝛾V (0) = 0◦. The initial roll, yaw, and pitch rates areset as 𝜔x(0) = 𝜔y(0) = 𝜔z(0) = 0◦. The initial deflectionsof the aileron, rudder and elevator are all set as 0◦.

The 3rd order B-spline functions transformed from(9) are used as the basis functions of BSNN in this sim-ulation. We set that the range of the angle of attack 𝛼 is[−8◦, 14◦] with 12 knots spaced every 2◦; the range of theside-slip angle 𝛽 is [−5◦, 5◦] with 6 knots spaced every 2◦;the speed V , expressed in Mach numbers, is in the rangeof [0.5, 0.9] with 5 knots spaced every 0.1 Ma. Accord-ing to (8), the 3rd order Spline-basis functions with nnodes consist of (n + 1) pieces. The coefficient CD0

(V )is approximated by BSNN with univariate B-spline basisfunctions owning 6 nodes in the hidden layer. The coef-ficient C𝛼

D(𝛼,V ) is approximated by BSNN with bivari-ate B-spline basis functions whose hidden layer is of 78nodes. Other coefficients are approximated similarly. Theaerodynamic data we use in this simulation come from[48]. The initial values of the estimated aerodynamic coef-ficients are displayed in Table II. Each aerodynamic coef-ficient has +10% uncertainty in the form of C∗(t) sin( t

5),

where C∗(t) is a generic marker to represent any of theaerodynamic coefficients.

The 𝜔n in four command filters are selected as(15, 10, 10)T, (25, 20, 20)T, (35, 35, 35)T, and (40, 40, 40)T.The damping factor in each filter is 0.707. The con-straints on the states and control surfaces adopted in thecommand filters are listed in Table III.

The control parameters are chosen as K1 =diag (1.2, 1.5, 1.2), K2 = diag (2.1, 0.8, 0.5), K3 =



Table II. Initial values of the estimated coeffi-cients.

Coefficient Value Coefficient Value

CD0(0) 0.05 C𝛼

D(0) 4.9C𝛽

Y (0) −1.9 CL0(0) 0.1

C𝛼L(0) 2.4 m𝛽

x(0) −0.3

m𝜔xx (0) −0.29 m𝛿x

x (0) 16m𝛽

y(0) −14.5 m𝜔yy (0) −2.9

m𝛿yy (0) −17 m𝛼

z (0) −18.7

m𝜔zz (0) −2.9 m𝛿z

z (0) −24

Table III. Constraints on states and control surfaces.

Variable Magnitude limit Rate limit

𝛼 [−8, 14] deg -𝛽 [−5, 5] deg -𝛾V [−90, 90] deg [−130, 130] deg/s𝜔x [−130, 130] deg/s -𝜔y [−30, 30] deg/s -𝜔z [−30, 30] deg/s -𝛿x [−15, 15] deg [−80, 80] deg/s𝛿y [−15, 15] deg [−60, 60] deg/s𝛿z [−15, 15] deg [−60, 60] deg/s

diag (2.5, 3, 3), and K4 = diag (5, 5, 5). And set 𝜇F1= 0.03,

𝜇F2= 𝜇B2j

= 0.01( j = 1, 2, 3), ΓF1= diag {100 ⋅ 178, 800 ⋅

172, 100 ⋅16, 600 ⋅172}, ΓF2= diag {100 ⋅17, 50 ⋅16, 70 ⋅131},

ΓB21= diag {50 ⋅ 16, 012}, ΓB22

= diag {06, 50 ⋅ 16, 06}, andΓB23

= diag {012, 50 ⋅ 16}, where 1N denotes a 1 × N rowvector of ones, and 0N denotes a 1×N row vector of zeros.The sampling period is 0.02 second.

The time-parameterized reference trajectory andthe actual flight trajectory driven by the proposed inte-grated trajectory tracking control law are depicted inFig. 2, which involve four typical flight actions such asclimbing up, diving, turning left and right. So the effec-tiveness of the controller can be fully verified by the suc-cessful tracking of the reference trajectory. The solid linein Fig. 3 demonstrates that the position tracking errorconverges into a range of 3m, which indicates that theposition tracking error is bounded even when the BSNNreconstruction errors and uncertainties of the unknownaerodynamic coefficients exist. The speed tracking errorconverges to a small region near zero which is shown bythe solid line in Fig. 4.

As depicted in Fig. 5, the virtual control signal 𝛼d

has violated the constraints for several times over theperiod from 0s to 5s, but the actual value of 𝛼 neverviolates the constraints benefiting from the use of thecommand filters. From Fig. 6, we notice that the bank

Fig. 2. Actual and reference trajectories of missile. [Colorfigure can be viewed at wileyonlinelibrary.com]

Fig. 3. The position tracking errors. [Color figure can beviewed at wileyonlinelibrary.com]

0 5 10 15 20 25 30−15

−10

−5

0

5

10

Time (s)

Tra

ckin

g er

ror

of th

e sp

eed

(m/s

)

The CLF methodThe proposed method

Fig. 4. The speed tracking errors. [Color figure can be viewedat wileyonlinelibrary.com]






Fig. 5. Desired and actual values of angle of attack. [Colorfigure can be viewed at wileyonlinelibrary.com]

Fig. 6. Desired and actual values of bank angle. [Color figurecan be viewed at wileyonlinelibrary.com]

0 5 10 15 20 25 30−2

−1.5

−1

−0.5

0

0.5

1

1.5

Time (s)

Sid

e−sl

ip a

ngle

(de

g)

Fig. 7. Actual side-slip angle. [Color figure can be viewed atwileyonlinelibrary.com]

angle goes to about 70◦ when turning left near 5s andturns to the opposite direction when turning right from10s to 14s. Meanwhile, the angle of attack cooperateswith the bank angle by providing required lift force in

0 5 10 15 20 25 30−130

−100

−50−30

0

3050

100

130

Time (s)

Ang

luar

vel

ocity

(de

g/s)

x

y

z

Fig. 8. Actual roll, yaw, and pitch rates. [Color figure can beviewed at wileyonlinelibrary.com]

0 5 10 15 20 25 300

0.5

1

1.5

2

2.5

3x 104

Time (s)

Thu

st (

N)

Fig. 9. Total engine thrust. [Color figure can be viewed atwileyonlinelibrary.com]

0 5 10 15 20 25 30−15

−10

−5

0

5

10

Time (s)

Con

trol

sur

face

def

lect

ion

(deg

)

a

r

e

Fig. 10. Deflections of aileron, rudder, and elevator. [Colorfigure can be viewed at wileyonlinelibrary.com]

the plane of maximum aerodynamic norm force as shownin Fig. 5, which indicates that the desired coordinatedturn is achieved with the proposed integrated controller.Moreover, note that the side-slip angle 𝛽 was successfullystabilized near 0◦ to achieve a coordinated turn as shown









in Fig. 7. When required to make a diving turn, the missilewould change the sign of its angle of attack to generate anegative lift force, and the bank angle would make a sud-den change to accommodate the directional change of thelift force if bank angle was not zero. This phenomenoncan be seen from Fig. 5 and Fig. 6 at about 20s. As shownin Fig. 8, the roll, yaw, and pitch rates are all maintainedin accordance with the constraints (see Table III). Thecontrol inputs such as the thrust P and control surfacedeflections 𝛿x, 𝛿y, 𝛿z (i.e. 𝛿a, 𝛿r, 𝛿e) are respectively plottedin Fig. 9 and Fig. 10. The deflections of aileron, rudderand elevator are not more than 15◦ in both directions.

Further simulations with different types of uncer-tainties in each aerodynamic coefficient are then

0 5 10 15 20 25 300

4

15

25

35

45

55

Time (s)

Pos

ition

trac

king

err

or (

m)

With uncertainty type 1With uncertainty type 2With uncertainty type 3

Fig. 11. The position tracking errors with different types ofuncertainties. [Color figure can be viewed atwileyonlinelibrary.com]

0 5 10 15 20 25 30−10

−5

0

5

10

15

Time (s)

Tra

ckin

g er

ror

of th

e sp

eed

(m)

With uncertainty type 1With uncertainty type 2With uncertainty type 3

Fig. 12. The speed tracking errors with different types ofuncertainties. [Color figure can be viewed atwileyonlinelibrary.com]

conducted and the results are shown in Figs 11 and 12.Three types of uncertainties are considered:

1. +10% uncertainty in the form of C∗(t) sin( t5);

2. +5% uncertainty in the form of C∗(t)(sin( t5) +

cos( t2));

3. +10% uncertainty in the form of C∗(t) sin(t) cos( t2),

in which type 1 is the same condition as in the previoussimulation, and types 2 and 3 are introduced for compar-ison. It can be observed from Figs. 11 and 12 that theaverage position and speed tracking errors, with respectto the uncertainties of types 2 and 3, are larger. How-ever, all the steady-state position tracking errors convergeinto a range of 4m and all the steady-state speed trackingerrors converge to zero. The results imply that the trajec-tory tracking control can still be accomplished even whenuncertainties exist in the aerodynamic coefficients, andgood tracking performance is also maintained.

Another simulation is carried out using the CLFmethod in [28] with the same initial conditions usedbefore. Simulation results, including the flight trajectory,the position tracking error, and the speed tracking error,are presented in Figs. 2, 3 and 4. The flight trajectory con-trolled by our proposed method is smoother than thatcontrolled by the CLF method as shown in Fig. 2. It isseen from Fig. 3 that the position tracking error of ourproposed method is smaller when turning left or right.This is because the two control loops, trajectory trackingcontrol loop and autopilot loop, are designed separatelyin the CLF method, which causes the two loops to be lesscoordinated especially when making a turn. In addition,the pitch, roll, and yaw channels are controlled separatelyneglecting the coupling effects in the CLF method, whichmakes the commands to the autopilots inaccurate dur-ing the high-speed maneuvering bank-to-turn, and what’sworse, degrades the tracking performance. However, ourproposed integrated method is able to make the two con-trol loops work synergistically and achieve a coordinatedturn, which ensures good maneuverability.

V. CONCLUSION

The flight trajectory tracking problem in 3-D spaceis studied in this paper. With the consideration of con-straints imposed on the states and actuators, couplingsamong pitch, yaw, and roll channels and reconstructionerrors, an integrated adaptive backstepping method withcommand filters and BSNN is proposed. The control lawguarantees semi-global uniform ultimate boundedness ofall the states in the closed-loop system, as well as the





stability of the overall system. The tracking errors areproven to converge into a small neighborhood aroundzero. It is promising to extend this method to solve thecooperative control problems for multiple missiles in thefuture work.

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Xueyuan Wang received the B.S. degreefrom the Xi’an University of Posts &Telecommunications, China, in 2012. Heis currently working toward the Ph.D.degree at the School of Automation,Beijing Institute of Technology, Beijing,China.

His current research interests include flight control,multi-agent systems, and nonlinear control applications.

Hao Fang received the B.S. degree fromthe Xi’an University of Technology, in1995, and the M.S. and Ph.D. degree fromthe Xi’an Jiaotong University, in 1998 and2002, respectively.He held two postdoctoral appointmentsat the INRIA/France research group of

COPRIN and at the LASMEA (UNR6602 CNRS/BlaisePascal University, Clermont-Ferrand, France). Since2011 he has been a professor at Beijing Institute of Tech-nology. His research interests include all-terrain mobilerobots, robotic control and parallel manipulators.

Lihua Dou received the B.S., M.S., andPh.D. degrees in control theory and con-trol engineering from Beijing Institute ofTechnology, Beijing, China, in 1979, 1987,and 2001, respectively.She is currently a Professor of control sci-ence and engineering at Key Laboratory

of Complex System Intelligent Control and Decision,School of Automation, Beijing Institute of Technol-ogy. Her research interests include multiobjective opti-mization and decision, pattern recognition, and imageprocessing.

Jie Chen received the B.S., M.S., andPh.D. degrees in control theory and con-trol engineering from the Beijing Instituteof Technology, in 1986, 1996, and 2001respectively. From 1989 to 1990, he wasa visiting scholar in the California StateUniversity, U.S.A. From 1996 to 1997, he

was a research fellow in the school of E&E, the Universityof Birmingham, U.K.He is currently a professor of control science and engi-neering, the Beijing Institute of Technology, P.R. China.He is also the head of the State Key Laboratory ofIntelligent Control and Decision of Complex Systems(Beijing Institute of Technology), the Peoples Republicof China. He serves as a managing editor for the JOUR-NAL OF SYSTEMS SCIENCE AND COMPLEX-ITY (2014-2017) and an associate editor for the IEEETRANSACTIONS ON CYBERNETICS (2016-2018)and many other international journals. His main researchinterests include multi-objective optimization and deci-sion in complex systems, intelligent control, nonlinearcontrol, and optimization methods. He has co-authored3 books and more than 200 research papers.

Bin Xin received the B.S. degree in infor-mation engineering and the Ph.D. degreein control science and engineering, bothfrom the Beijing Institute of Technology,Beijing, China, in 2004 and 2012, respec-tively.He is currently an associate professor with

the School of Automation, Beijing Institute of Technol-ogy. His current research interests include search andoptimization, evolutionary computation, combinatorialoptimization, and multi-agent systems.


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Integrated 3‐D Flight Trajectory Tracking Control with