Control and Intelligent Systems, Vol. 36, No. 4, 2008

THE DESIGN OF SLIDING MODE CONTROL

SYSTEM BASED ON BACKSTEPPING

THEORY FOR BTT UAV

J. Yao,∗,∗∗ X. Zhu,∗∗∗ and Z. Zhou∗∗∗

Abstract

A novel control system design approach is proposed based on

backstepping theory and sliding mode control (SMC) for Bank-

To-Turn (BTT) Unmanned Aerial Vehicle (UAV). It ensures BTT

UAV stable and accurate flight under large parametric perturbation.

In addition, the aerodynamic coefficients are not necessary to be

identified online. This approach is based on backstepping theory

and the whole system is not divided into slow and fast subsystems.

However, backstepping cannot ensure the robustness of the closed-

loop system. To solve this problem, SMC is employed, which

is designed in terms of the bounds of aerodynamic coefficients.

To evaluate the performance of the flight control system using

the proposed approach, the three channels united simulation is

conducted considering the actuators’ rate and magnitude saturation.

The results show that the proposed approach is capable of handling

serious nonlinear and large model uncertainty.

Key Words

Unmanned Aerial Vehicle, Bank-To-Turn, backstepping, sliding

mode

1. Introduction

In the past few years, a variety of control theories have beenapplied to the flight control system design of UnmannedAerial Vehicle (UAV). In the flight control, there are twocontrol modes: Skid-To-Turn (STT) and Bank-To-Turn(BTT). In the STT mode, the UAV is controlled in pitchand yaw separately, and the roll channel is stabilized. Inthe BTTmode, the command of acceleration is produced inthe pitch channel and the direction is controlled by the rollchannel. Meanwhile, the sideslip angle is required as smallas possible. BTT mode can generate bigger acceleration

∗ School of Astronautics, Northwestern Polytechnical Univer-sity, Xi’an, Shaanxi 710072, P.R.China; e-mail: jianguo@cs.mcgill.ca

∗∗ School of Computer Science, McGill University, Montreal, QCH3A2A7, Canada

∗∗∗ UAV Research and Development Centre, Northwestern Poly-technical University, Xi’an, Shaanxi 710072, P.R.China; e-mail:{zhuxiaoping, zhouzhou}@nwpu.edu.cn

Recommended by Prof. Jonathan Wu(paper no. 201-2010)

and faster response than STT mode. However, the modelof BTT UAV is seriously nonlinear and coupled, and theaerodynamic coefficients vary greatly in the different flightaltitudes and speeds. Although the linear control theoryhas been mature, it can’t handle BTT UAV because itsmodel can’t be linearized employing traditional method.

In the flight control system design for BTT UAV,nonlinear control theories have more advantages than thelinear control theories. Among the nonlinear control theo-ries, the dynamic inversion (DI) has been applied to designflight control system [1]. In the approach employing DI,the whole system is divided into two subsystems on thebasis of time-scale theory [2, 3]. The fast subsystem shouldrespond much faster than the slow subsystem. It is anapproximated method and the gains of the subsystems arehard to choose because the analysis of stability is verycomplicated [4].

Backstepping is another nonlinear control theory com-bined with DI, which is an approach guaranteeing the sta-bility based on Lyapunov function [5–8]. It does not con-sider zero-dynamics and has more advantages than DI. Inthe original backstepping algorithm, it assumes the modelis accurate without uncertainty. In addition, backsteppingis easy to combine PID to design control system [9, 10].However, it has small robustness to attenuate the modeluncertainty. In the real flight of BTT UAV, a complete andaccurate dynamic model is difficult to obtain. Althoughthe aerodynamic coefficients can be identified online, it isnot as accurate as we expect because the system is nonlin-ear. To get the robust control system, backstepping the-ory is usually combined with the robust control theories.Many of these approaches have been studied. Backstep-ping theory combined with Neural Network (NN) methodsare presented in [8, 11, 12]; Backstepping theory combinedwith sliding mode control (SMC) methods are presentedin [13, 14]; Backstepping theory combined with adaptivecontrol methods are presented in [8, 15–17].

In this paper, we propose a novel nonlinear controlapproach of flight control system design based on back-stepping and SMC for BTT UAV. In this approach, it doesnot require the separation of system variables and identi-fication of aerodynamic coefficients online. The effects ofparametric perturbation are compensated by SMC. Being

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different from the existing methods combining backstep-ping and SMC, the design of SMC in our approach is basedon the bounds of aerodynamic coefficients. Finally, the sta-bility of the proposed approach is proven using Lyapunovfunction.

The remainder of this paper is organized as follows. InSection 2, the problem of design BTT UAV flight controlsystem using backstepping is stated. In Section 3, anew nonlinear control system design approach combiningbackstepping and SMC is presented. In Section 4, theproposed approach is applied to BTT UAV and the controllaw is derived. A simulation study is conducted and theresults are summarized in Section 5. Finally, conclusionsare discussed in Section 6.

2. Problem Statement

This paper focuses on the flight control system designfor BTT UAV. In general, BTT guidance law generatesthe commands including rolling angle φc, and accelera-tions Ayc, Azc. However, the system with the outputs[φc, Ayc, Azc] is a non-minimum phase phenomenon, whichresults in instability when we exert DI. One way to solvethis problem is to redefine the outputs as [φc, αc, βc] [18].However, when BTT UAV flights with big angle of attack,the control of φc may lead to big sideslip angle. To ensurethe sideslip angle in the range of a small value, φc is re-placed by the bank angle command μc. So the outputs ofBTT UAV is [μc, αc, βc]. Furthermore, this modificationensures that the BTT UAV is stabler under the large angleof attack. The detailed dynamic model can be referred inthe Appendix.

To present the problem conveniently, the model ofBTT UAV is described as:

x1 = f1(x1) + g1(x1)x2 + d1(x1, x2)

x2 = f2(x2) + g2u+ d2(x1, x2) (1)

where x1 = [α β μ]T , x2 = [p q r]T , u= [l m n]T andδa, δr, δe are calculated from the input u= [l m n]T in (57).d1(x1, x2) and d2(x1, x2) are the parametric uncertaintyaffected by the aerodynamic coefficients. The variation ofaerodynamic coefficients may be great in the flight of BTTUAV, and we assume that they have bounds as follows:

|d1(x1, x2)| < dmax1 , |d2(x1, x2)| < dmax

2 (2)

where dmax1 and dmax

2 are constant vectors.We define the error states variables z1, z2 ∈ R3 as:

z1 = x1 − xd1

z2 = x2 − xd2 (3)

where xd1 and xd

2 are the desired commands of x1 and x2.Using (1) and (3), we can get that:

z1 = f1(x1) + g1(x1)x2 − xd1 + d1(x1, x2)

z2 = f2(x2) + g2u− xd2 + d2(x1, x2) (4)

The control law employing the original backsteppingand PID is described as:

xd2 = g−1

1 (x1)[−k1z1 − f1(x1) + xd1]

u = g−12

[−k2z1 − f2(x) + xd2

](5)

where k1, k2 ∈ R are positive parameters. We can prove thesystem with the above control law is stable when the modeluncertainty d1(x1, x2) and d2(x1, x2) is zero or very small[19]. Note that the PID controller has small robustnessto the model uncertainty. When the model uncertainty isvery large, the system performance may deteriorate. Tomake the control system more robust, we employ SMC tomake the controller compensate the model uncertainty. Inthe next sections, we will discuss our detailed approach.

3. A Novel Control System Design CombiningBackstepping with SMC

In this section, we propose a novel control system designapproach based on backstepping and SMC for a generalnonlinear system under model uncertainty. Backsteppingis used to handle the model nonlinearity and SMC makesthe system robust in presence of the model uncertainty. Inthe preliminary design in Section 3.1, there are derivativesof the pseudo control variables. However, these derivativesdo not exist because they are not continuous. To dealwith this problem, we modify the pseudo-control signal inSection 3.2. Finally, the stability of the control system isproven in Section 3.3.

3.1 The Preliminary Design

Considering the following nonlinear system:

xi = xi+1 +Δxi, i = 1, 2, . . . , n− 1

xn = f(x) + g(x)u+ d(x)

y = x1 (6)

where x= [x1, x2, . . . , xn]T is the state variable vector,

u, y are the input and output, respectively. Δxi is themodel uncertainty. f(x) and g(x) are continuous anddifferentiable with respect to x. d(x) is the disturbance.

Defining sliding planes:

si = xi − xic, i = 1, 2, . . . , n (7)

where x1c = y, x2c, x3c, . . . , xic are pseudo control variables.To make the control system design approach easy to

understand, we introduce it in n steps.Step 1. Defining the pseudo-control variable x2c as:

x2c = −k1s1 + x1c − ud1 (8)

where ud1 is the item corresponding to SMC, k1 > 0.Calculating the time derivative of s1:

s1 = s2 + x2c +Δx1 − x1c (9)

348

Substituting (8) into (9), we can get:

s1 = s2 − k1s1 +Δx1 − ud1 (10)

Defining Lyapunov function as:

V1 =1

2s21 (11)

Calculating the time derivative of V1 and substituting(10), we can get:

V1 = −k1s21 + s1s2 + s1(Δx1 − ud1) (12)

We select ud1 in terms of the following inequation:

s1(Δx1 − ud1) < 0 (13)

Apparently, if s2 =0, the time derivative of Lyapunovfunction V1 < 0 so that the variable s1 is stable. However,the assumption s2 =0 is not ensured actually. So we shoulddesign pseudo-control variable x3c to make s2 converge to0 asymptotically in the next step.

Step i (i=2,. . . ,n−1): Defining the pseudo-controlvariable x(i+1)c as:

x(i+1)c = −si−1 − kisi + xic − udi (14)

where udi is the item corresponding to SMC, ki > 0.Calculating the time derivative of si:

si = si+1 + x(i+1)c +Δxi − xic (15)

Substituting (14) into (15), we can get:

si = si+1 − si−1 − kisi +Δxi − udi (16)

Defining Lyapunov function as:

Vi =1

2s2i +

i−1∑j=1

Vj (17)

Calculating the time derivative of Vi and substituting(16), we can get:

Vi = −i∑

j=1

kjs2j + sisi+1 +

i∑j=1

sj(Δxi − udj) (18)

We select ud1 in terms of the following inequation:

sj(Δxj − udj) < 0 (19)

Apparently, if si+1 =0, the time derivative of Lya-punov function Vi < 0 so that the variables s1, s2, . . . , siare stable. Similar with Step 1, we should design pseudocontrol variable xi+1,c to make si+1 converge to 0 asymp-totically.

Step n: Defining the control input u as:

u = g−1(x)[−f(x) + xnc − knsn − sn−1 − udn] (20)

where udn is the item corresponding to SMC, kn > 0.Calculating the time derivative of sn:

sn = f(x) + g(x)u+ d(x)− xnc (21)

Substituting (20) into (21), we can get:

sn = −knsn − sn−1 + d(x)− udn (22)

Lyapunov function is defined as:

Vn =1

2s2n +

n−1∑j=1

Vj (23)

Calculating the time derivative of Vn and substituting(22), we can get:

Vn = −n∑

j=1

kjs2j+

n−1∑j=1

sj(Δxj − udj)+sn(d(x)−udn) (24)

Selecting udn in terms of the following inequation:

sn(d(x)− udn) < 0 (25)

Vn is negative definitive on the condition of (13),(19) and (25). So the control system is stable and theperformance of system completely depends on the selectionof parameters ki and ui, i=1, 2, . . . , n.

3.2 The Modification of Pseudo Control

In the control law design of Section 3.1, there are deriva-tives of the pseudo control variables. However, thesepseudo control variables are not differentiable because theyare not continuous actually. To deal with this problem, thepseudo control variables should be redefined.

In this subsection, we redefine pseudo control vari-ables as:

xi+1,c = −si−1−kisi+xic, i = 1, 2, . . . , n−1, s0 = 0 (26)

and define:

u = g−1(x)[−f(x) + xnc − knsn − sn−1 − ud] (27)

where ud is the item corresponding to SMC. The modeluncertainty of the whole system can be compensated inthe last step of the control system design. The pseudocontrol variables will have derivatives with the above mod-ifications.

To get the SMC item ud, we define it in the followingformulation:

349

ud =n∑

j=1

u′dj = u′

dn +n−1∑j=1

u′dj = u′

dn+n−1∑j=1

snwj(x) (28)

Selecting control law as follows:

wj(x) =

⎧⎨⎩>

sjΔxj

s2n, sn > δ

= 0, sn < δ, j = 1, 2, . . . , n− 1 (29)

u′dn =

⎧⎨⎩> |d(x)|, sn > 0

< |d(x)|, sn < 0(30)

To prevent saturation of the actuators, wj(x) is set aszero in (29) when sn <δ. Note that sj , j=1, 2, . . . , n− 1,is also small when sn is small, so there is no need to addSMC item.

Although the modification of pseudo control cannotensure Vi < 0, i=1, 2, . . . , n− 1, the stability of the wholesystem is ensured employing the proposed control law. Inthe next part, the stability analysis will be given.

3.3 The Stability Analysis

Defining Lyapunov function of the whole system as:

V =1

2STS =

1

2

n∑j=1

s2j (31)

where

S = [s1 s2 · · · sn] (32)

Calculating the time derivation of Lyapunov function,we can get:

V = −2n∑

j=1

kjs2j + 2

n−1∑j=1

sjΔxj + 2snd(x)− 2snud (33)

Substituting (28) and in terms of (29) and (30), we canget, if sn >δ so that V < 0. It implies that the system isstable; if sn <δ so sj , j=1, 2, . . . , n− 1, is also small.

V < −2n∑

j=1

kjs2j + 2

n−1∑j=1

sjΔxj (34)

If sj >Δxj/kj , j=1, 2, . . . , n− 1, so that V < 0. Itimplies the system is Uniformly Ultimately Bounded(UUB) [20].

4. The Flight Control System Design for BTT UAV

In this section, we design the flight control system exertingthe proposed approach in Section 3. In this control law,there are two gain matrix K1 and K2 to choose. The itemof SMC is designed in terms of the bounds of aerodynamiccoefficients.

We define sliding planes as:

S = [eT1 eT2 ]T = [xT

1 − xT1c xT

2 − xT2c]

T (35)

where x1c = [αc βc μc]T is input command vector.

Defining pseudo control as:

x2c = g−11 (x1)[−K1e1 − f1(x1) + x1c] (36)

where K1 is diagonal and positive definite gain matrix.The control law is defined as:

u = g−12 [−K2e2 − f2(x2) + x2c − gT1 (x1)e1 − ud] (37)

where K2 is diagonal and positive definite gain matrix,ud is the item corresponding to SMC to compensate thevariation of aerodynamic coefficients.

Substituting the control law (36) and (37) into thesystem model (1), we can get:

e1 = −K1e1 + d1(x1, x2)

e2 = −K2e2 − gT1 (x1)e1 + d2(x1, x2)− ud (38)

Lyapunov function is defined as:

V =1

2STS (39)

Calculating the time derivative of V and substituting(38), we can get:

V = −eT1 K1e1 − eT2 K2e1 + eT1 d1 + eT2 d2 − eT2 ud (40)

To make sure V < 0, we should choose the SMC itemin terms of the following inequation:

eT1 d1 + eT2 d2 − eT2 ud < 0 (41)

To this end, we define the formulation of ud as:

ud = ud1 + ud2 = e2ω(x) + ud2 (42)

Substituting (42) into (41):

eT1 d1 − eT2 e2ω(x) + eT2 (d2 − ud2) < 0 (43)

We choose the sufficient condition of (43) as follows:

eT1 d1 − eT2 e2ω(x) < 0 (44)

eT2 (d2 − ud2) < 0 (45)

350

Calculating the parameters ω(x) of the SMC item from(44). It is defined as:

ω(x) >eT1 d1‖e2‖ (46)

Considering the bounds of aerodynamic coefficients, so(46) is written as:

ω(x) =

⎧⎨⎩>

|eT1 d1|max

‖e2‖ , ‖e2‖ > δ

= 0 ,‖e2‖ < δ

|eT1 d1|max =1

MV|e11||ΔL|max +

1

MV|e12|(|ΔD|max|β|

+ |ΔY |max)+1

MV|e12|(|ΔD|max|β cosμ tan γ|

+ |ΔY |max|cosμ tan γ|+ |ΔL|max

× |sinμ tan γ + β|) (47)

where e1i e2i are ith element of the error vectors e1 and e2.Calculating the parameters ud2 of the SMC item from

(45), it is presented as follows:

ud2(1) =

⎧⎨⎩< −c3|Δl|max − c4|Δn|max, e21 < 0

> c3|Δl|max + c4|Δn|max, e21 > 0

ud2(2) =

⎧⎨⎩< −c7|Δm|max, e22 < 0

> c7|Δm|max, e22 > 0

ud2(3) =

⎧⎨⎩< −c4|Δl|max − c9|Δn|max, e23 < 0

> c4|Δl|max + c9|Δn|max, e23 > 0(48)

The maximum perturbation of aerodynamic forces andmoments are defined in Appendix. From the above deriva-tion, we can get the control law consisting of (36), (37),(47) and (48).

5. Numerical Simulation

In this section, we make a simulation to evaluate the per-formance of the proposed control law based on backstep-ping and SMC applied to BTT UAV. The results showthat the backstepping combined with SMC controller canhandle serious nonlinearity and large model uncertainty.

5.1 The Simulation Setup

The simulation is conducted in Matlab [21]. In the simula-tion, the parameters are set as follows: the UAV flights atthe altitude of 12,000m and the speed of 0.9Ma; the mag-nitude limitation of actuators is ±20◦, and the maximumrate limitation is 300◦/s; the perturbation of aerodynamiccoefficients is in the bound of ±30%; the angle of attackcommand is a step signal of 10◦, the bank angle commandis a step signal of 90◦ and the sideslip angle is expected atzero.

Figure 1. Tracking responses of angle of attack.

5.2 The Results Evaluation

Fig. 1 shows the tracking responses using the controllerbased on backstepping and SMC. As we can see, the controlsystem ensures that the angle of attack tracks the referencecommand without static error in both the conditions.

Figure 2. The responses of sideslip angle.

The responses of sideslip angle using the controllerbased on backstepping and SMC are illustrated in Fig.2. We can get that, when there is perturbation, thesideslip angle is a bit bigger than the situation there isno perturbation, but both of them are controlled to beless than ±2◦. Note that one of the demanded controlperformances in BTT UAV is to ensure sideslip in therange of ±3◦, so that the performance of the controller iseligible.

From Fig. 3, we can get the tracking responses of bankangle using the controller based on backstepping and SMC.

Figure 3. Tracking responses of bank angle.

351

When there is no perturbation, the response is a littlebetter than the situation perturbation exists. However,the overshoot is less than 11% in both the conditions, sothe control performance is acceptable. Fig. 4 shows thedeflection responses of actuators using the controller basedon backstepping and SMC.

Figure 4. Tracking responses of actuator deflections.

6. Conclusions

An SMC system combined with backstepping for BTTUAV has been proposed. It can deal with nonlinearity andmodel uncertainty. In this approach, it doesn’t require thatthe system be separated into slow and fast subsystems andthe aerodynamic coefficients be identified online. In theSMC design, the bounds of the aerodynamic coefficients areneeded to make the system robust. To prove the stability ofthe proposed approach, Lyapunov function is designed andthe proof is given. Finally, the numerical simulation showsthat the BTT UAV employing the proposed approachmaneuvers with good performance in the presence of largeparametric perturbation.

Acknowledgement

This work was supported in part by an NCET GrantNo.05-0867 and a National Study-Abroad Scholarship ofP.R.China under Grant No. [2007] 3020.

Appendix

When the UAV is in the BTT mode, the sideslip angleis usually in the range of ±3◦, on this condition cosβ ≈1, sinβ≈ tanβ=β. The attack angle is usually 0–20◦, sosinα sinβ≈ tanα sinβ≈ 0. The function (1) is defined asfollows:

d1(x1, x2) =1

MV

⎡⎢⎢⎢⎢⎢⎢⎣

−ΔL

ΔDβ +ΔY

ΔDβ cosμ tan γ +ΔY cosμ tan γ

+ΔL(sinμ tan γ + β)

⎤⎥⎥⎥⎥⎥⎥⎦(49)

d2(x1, x2) =

⎡⎢⎢⎢⎣c3Δl + c4Δn

c7Δm

c4Δl + c9Δn

⎤⎥⎥⎥⎦ (50)

g1(x1) =

⎡⎢⎢⎢⎣−β cosα 1 0

sinα 0 cosα

cosα 0 sinα

⎤⎥⎥⎥⎦ (51)

g2 =

⎡⎢⎢⎢⎣c3 0 c4

0 c7 0

c4 0 c9

⎤⎥⎥⎥⎦ (52)

f1(x1) =

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

− 1MV (L+ Px sinα− Pz cosα)

+ g cosμ cos γV

−−−−−−−−−−−−−−−−−−−1

MV (Dβ + Y − Pxβ cosα+ Pyβ)

+ g sinμ cos γV

−−−−−−−−−−−−−−−−−−−

1MV

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

Dβ cosμ tan γ + Y cosμ tan γ

+L(sinμ tan γ + β) + PX(sinα sinμ

· tan γ − β cosα cosμ tan γ)

+PY β cosμ sinα− PZ(β sinα cosμ

· tan γ + cosα sinμ tan γ + β cosα)

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

−βg cosμ cos γV

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(53)

f2(x2) =

⎡⎢⎢⎢⎣

(c1r + c2p)q

c5pr − c6(p2 − r2)

(c8p− c2r) q

⎤⎥⎥⎥⎦ (54)

The nominal values of aerodynamic forces are de-fined as:

L =

(CLαΔα+ CLqq

CA

2V+ CL �αα

CA

2V+ CLδeδe

)qS

Y =

(CY ββ + CY pp

b

2V+ CY rr

b

2V+ CY δaδa + CY δrδr

)qS

D = (CD0 + CDδe)qS (55)

The perturbation value of lift force is defined as:

ΔL =

(ΔCLαΔα+ΔCLqq

CA

2V+ΔCL �αα

CA

2V

+ΔCLδeδe

)qS (56)

ΔY,ΔD are calculated similar with ΔL.The nominal values of aerodynamic moments are de-

fined as:

352

l=

(Clββ + Clpp

b

2V+ Clrr

b

2V+ Clδaδa + Clδrδr

)qSb

m=

(Cm0+Cmαα+Cmqq

CA

2V+ Cm �αα

CA

2V+Cmδeδe

)qSCA

n=

(Cnββ + Cnpp

b

2V+ Cnrr

b

2V+ Cnδaδa + Cnδrδr

)qSb

(57)

The perturbation value of roll moment is defined as:

Δl =

(ΔClββ +ΔClpp

b

2V+ΔClrr

b

2V+ΔClδaδa

+ΔClδrδr

)qSb (58)

Δm,Δn are calculated similar with Δl.

The maximum perturbation of lift force is defined as:

|ΔL|max =

(|ΔCLα|max|Δα|+ |ΔCLq|max|q|

CA

2V

+|ΔCL �α|max|α|CA

2V+ |ΔCLδe |max|δe|

)qS

(59)

|ΔY |max, |ΔD|max are calculated similar with |ΔL|max.

The maximum perturbation of roll moment is de-fined as:

|Δl|max =

(|ΔClβ |max|β|+ |ΔClp|max|p|

b

2V+ |ΔClr|max|r|

× b

2V+ |ΔClδa |max|δa|+ |ΔClδr |max|δr|

)qSb

(60)

|Δm|max, |Δn|max are calculated similar with |Δl|max.

Nomenclature

Constants:

C#: aerodynamic coefficient of #m: massg: vertical component of gravityIxx: roll moment of inertiaIyy: pitch moment of inertiaIzz: yaw moment of inertiaIxz: product moment of inertiaci: element of inverse-inertial matrixCA: mean geometric chordb: wing span

Functions:L,D, Y : lift, drag, and side-forceL, D, Y : nominal values of lift,

drag, and side-forceΔL,ΔD,ΔY : perturbations of lift,

drag, and side-force|ΔL|max, |ΔD|max, |ΔY |max: maximum perturbations

of lift, drag, and side-forcel,m, n: roll, pitch, and yaw

momentl, m, n: nominal values of roll,

pitch, and yaw momentΔl,Δm,Δn: perturbations of roll,

pitch, and yaw moment|Δl|max, |Δm|max, |Δn|max: maximum perturbations

of roll, pitch, and yawmoment

Px, Py, Pz: thrust on three body axis

Variables:γ: flight-path angleV : air speedq: dynamic pressureμ: roll angle about velocity

vector or bank angleα: angle of attackβ: sideslip anglep, q, r: roll, pitch, yaw angular

rates (body axis)φc: command of roll angleAyc, Azc: command of acceleration

along the body axisμc: command of bank angleαc: command of angle of attackβc: command of sideslip angleδe, δa, δr: elevator, aileron, rudder

angle

References

[1] D.J. Bugajski, D.F. Enns, & M.R. Elgersma, A dynamicinversion based control law with application to high angleof attack research vehicle, Proc. AIAA Conf. on Guidance,Navigation and Control, Portland, OR, 1990, 826–839.

[2] P.K.A. Menon, M. Badgett, & R.A. Walker, Nonlinear flighttest trajectory controllers for aircraft, Journal of Guidance,Control, and Dynamics, 10 (1), 1987, 67–72.

[3] R. Srivastava, A. Sarkar, & S. Gollakota, Nonlinear inversionflight control for a super maneuverable aircraft, Journal ofGuidance, Control, and Dynamics, 15 (4), 1992, 976–984.

[4] C. Schumacher & P.P. Khargonekar, Stability analysis of amissile control system with a dynamic inversion Controller,Journal of Guidance, Control, and Dynamics, 21 (3), 1998,508–515.

[5] O. Harkegard & S. Torkel Glad, A backstepping design forflight path angle control, Proc. 39th IEEE Conf. on Decisionand Control, Sydney, Australia, 2000, 3570–3575.

[6] M. Xin & S.N. Balakrishan, Nonlinear missile autopilot designtheta-D with technique, Journal of Guidance Control andDynamics, 27 (3), 2004, 406–417.

[7] M.A. Khan & P. Lu, New technique for nonlinear control ofaircraft, Journal of Guidance, Control, and Dynamics, 17 (5),1994, 1055–1060.

[8] M.M. Polycarpou & P.A. Ioannou, A robust adaptive nonlinearcontrol design, Automatica, 32 (3), 1996, 423–427.

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[9] B. Hemici, L. Nezli, M. Tadjine, & M.S. Boucherit, RobustPID/backstepping control design for permanent magnet syn-chronous motor drive, Control and Intelligent Systems, 34 (3),2006, 194–204.

[10] R. Skjetne & T.I. Fossen, On integral control in backstep-ping: analysis of different techniques, Proc. American ControlConference, Boston, MA, 2004, 1899–1904.

[11] J.Y. Choi & J.A. Farrell, Adaptive observer backsteppingcontrol using neural networks, IEEE Transactions on NeuralNetworks, 12 (5), 2001, 1103–1112.

[12] O. Kuljaca, N. Swamy, F.L. Lewis, & C.M.A.K.C.M. Kwan,Design and implementation of industrial neural network con-troller using backstepping, IEEE Transactions on IndustrialElectronics, 50 (1), 2003, 193–201.

[13] J. Li, D. Xu, B. Song, & W. Yan, An easily implementablebackstepping sliding mode controller for nonlinear system withunmatched uncertainties, Journal of Northwestern Polytech-nical University, 22 (2), 2004, 145–148.

[14] P. Carbonell, Z.P. Jiang, & D.W. Repperger. Nonlinear controlof a pneumatic muscle actuator: Backstepping vs. sliding-mode, Proc. IEEE Conf. on Control Applications, Mexico City,Mexico, 2001, 167–172.

[15] Y. Zhang, B. Fidan, & P.A. Ioannou, Backstepping controlof linear time-varying systems with known and unknown pa-rameters, IEEE Transactions on Automatic Control, 48 (11),2003, 1908–1925.

[16] J. Zhou, C. Wen, & Y. Zhang, Adaptive backstepping control ofa class of uncertain nonlinear systems with unknown backlash-like hysteresis, IEEE Transactions on Automatic Control,49 (10), 2004, 1751–1759.

[17] Y. Zhang, C. Wen, & Y.C. Soh, Adaptive backstepping controldesign for systems with unknown high-frequency gain, IEEETransactions on Automatic Control, 45 (12), 2000, 2350–2354.

[18] C.-K. Lin & S.-D. Wang, An adaptive H/sup /spl infin//controller design for bank-to-turn missiles using ridge Gaussianneural networks, IEEE Transactions on Neural Networks,15 (6), 2004, 1507–1516.

[19] A.R. Benaskeur & A. Desbiens, Backstepping-based adap-tive PID control, IEE Transactions on Control Theory andApplications, 149 (1), 2002, 54–59.

[20] F.L. Lewis, C.T. Abdallah, & D.M. Dawson, Control of robotmanipulators (Englewood Cliffs, NY: Macmillan, 1993).

[21] B. Stevens & F. Lewis, Aircraft control and simulation (En-glewood Cliffs, NY: Wiley, 1992).

Biographies

Jianguo Yao was born in Hebei,P.R. China. He received hisB.E. and M.E. degrees from theNorthwestern Polytechnical Uni-versity (NPU), Xi’an, Shaanxi,P.R. China, respectively in 2004and 2007. He received the Chi-nese State Scholarship to StudyAbroad in 2007. Currently, he isa Ph.D. Student at the NPU, anda Joint Education Ph.D. Studentat the McGill University, Mon-

treal, QC, Canada. His research interests are robust andnonlinear control, stochastic control and guidance systemfor UAV.

Xiaoping Zhu was born in Hu-nan, P.R. China. He receivedhis Ph.D. degree from the North-western Polytechnical Univer-sity (NPU), Xi’an, Shaanxi, P.R.China, in 1992. He did postdoc-toral research on flight control atthe Nanjing University of Aero-nautics and Astronautics, Nan-jing, Jiangsu, P.R.China, from1993 to 1995. And then he didpostdoctoral research on rocket

design at the NPU from 1995 to 1997. Currently, he is aProfessor at the NPU. His research interests are conceptualdesign, flight control and simulation for UAV.

Zhou Zhou was born in Hunan,P.R. China. She received herPh.D. degree from the North-western Polytechnical Univer-sity (NPU), Xi’an, Shaanxi, P.R.China, in 1992. She did postdoc-toral research on UAV conceptualdesign at the Nanjing Universityof Aeronautics and Astronautics,Nanjing, Jiangsu, P.R. China,from 1993 to 1995. And thenshe did postdoctoral research on

integrated optimization design of aircraft aerodynamicsand stealth performances at the NPU. Currently, she is aProfessor at the NPU. Her research interests are conceptualdesign, flight dynamics and flight control for UAV.

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