Open Journal of Applied Sciences, 2013, 3, 65-70
doi:10.4236/ojapps.2013.32B013 Published Online June 2013 (http://www.scirp.org/journal/ojapps)
UAV Flight Path Control Using Contraction-Based
Backstepping Control
Tan Chun Kiat1, Hungsun Son1, Paw Yew Chai2
1School of Mechanical and Aerospace Engineering, Nanyang Technological University, Singapore
2DSO National Laboratories, Singapore
Received 2013
ABSTRACT
In this paper, a contraction-based backstepping nonlinear control technique was proposed. The proposed controller syn-
thesis technique utilizes both the recursive nature of backstepping control and of contraction analysis. This approach
results in a contracting closed-loop dynamics, with exponential stability. The use of the hierarchical contraction form in
the control problem formulation also results in the exponential convergence of controlled variables and can be easily
applied to non-autonomous systems. A flight path angle controller was synthesized and simulated using the proposed
technique to demonstrate the exponential convergence achieved by the backstepping controller design.
Keywords: Backstepping; Contraction Theory
1. Introduction
In recent years, missions performed by unmanned aerial
vehicle (UAV) has increasingly emphasized on high
agility and maneuverability. An example is the tailsitter
UAV [1,2] which performs maneuver that covers a large
pitch angle. These maneuvers typically require the UAVs
to operate outside their linear operating regime, and results
in nonlinear flight dynamics characteristics. Nonlinear
control techniques, such as backstepping control, were
developed to control and improve the performance of
such UAVs [3-8].
The stability criterion for traditional backstepping
control makes use of the Lyapunov stability theory [3-8].
However, the Lyapunov stability theory often result in
complex Lyapunov control function and requires the
knowledge of an equilibrium point or trajectory. This
often hinders controller synthesis for trajectory tracking
control as the equilibrium trajectory is often unknown
and the system is often non-autonomous.
The contraction theory [9,10] is a nonlinear stability
theory and was applied to control formulation in several
instances [11,12]. Stability in contraction theory is de-
scribed as the diminishing of the effects from any per-
turbation to any trajectory in the state space. The con-
traction theory can be easily applied to non-autonomous
systems without any added mathematical complexity and
does not require the knowledge of any equilibrium point
or trajectory. The contraction theory also provided a
stronger form of stability as compared to Lyapunov sta-
bility theory which can thus be applied to systems that
require fast convergence.
The use of contraction theory as the stability criteria in
backstepping control was introduced by Jouffryo [13]
with the use of standard contraction form in backstepping
control and later by Sharma et al. [14] and Zamani et al.
[15]. Standard contraction forms refer to the arrange-
ment/ connections of contracting subsystems to form a
global contracting system. The contraction form used by
[13-15] belongs to the feedback interconnection form [9].
In this paper, a hierarchical form [9] was used for the
backstepping controller synthesis. This hierarchical form
formulation, which was not commonly found in litera-
tures, provided the basis for a subsystem level control. In
this way, the backstepping control law can be designed
for each subsystem recursively and satisfies global sys-
tem contraction conditions with exponential convergence.
Different from the feedback interconnection form used
by [13-15], the hierarchical form also results in the ex-
ponential convergence of individual controlled variables.
The proposed controller synthesis technique will be ap-
plied to a flight path angle control of a generic UAV.
This paper is organized as follows. Section 2 will
present the formulation of a contraction-based backstep-
ping control. Section 3 will describe the controller syn-
thesis on the UAV flight path angle control. Simulation
results for the closed-loop control with the contrac-
tion-based backstepping controller is presented in section
4 and conclusion of the paper will be given in section 5.

T. C. KIAT ET AL.
66
2. Contraction-Based Backstepping
This section provides the formulation of the contrac-
tion-based backstepping control.
2.1. Contraction Theory
The contraction theory is a stability tool for examining
the stability of trajectories in the state space. According
to contraction theory, a trajectory is stable if it “forgets”
the effects from perturbations to that trajectory.
Consider a nonlinear system,
( )
,t=x fx

, where
n
∈x
is the state of the system and t is the time. By
using the concept of virtual displacement on trajectories,
the virtual dynamics can be written as
( )
,t
δδ
∂
=∂
fx
xx
x

(1)
where δx is the virtual displacement of the trajectory.
We can further describe the time variation of δxTδx as
( )
( )
,.
TT
t
d
dt
δδ δδ
∂
=∂
fx
xx xx
x (2)
Here, δxTδx represents the squared distance between infi-
nitesimally close trajectories. If the Jacobian,
( )
,/ ,t∂∂ fx x
is
uniformly negative definite, then the squared distance
between trajectories reduces exponentially to zero. Hence,
a trajectory is stable as neighboring trajectories con-
verges into each other. The state space region where the
Jacobian is uniformly negative definite is known as the
contraction region.
2.2. Hierarchical Connection Structure
Contraction results can also be extended to different sys-
tem connections. Contracting subsystems can be con-
nected in different ways to obtain a globally contracting
system. In this way, we can prove global contraction by
examining the contraction behavior of smaller subsys-
tems. The system that will be used in this paper is the
hierarchical combination, which resemblances the series
connection.
A system with a virtual dynamics of the form
1 111
221 222
0
d
dt
δδ
δδ
 
=
 
 
xF x
xFF x
(3)
is known as the hierarchical combination. The system
will be converging if the submatrices F11 and F22 are un-
iformly negative definite and F21 is bounded.
This is because subsystems
111 1
δδ =x Fx

(4)
and
221 1222
δ δδ
= +
xFx Fx
 (5)
are converging since x1 is converging and the term F21δx1
in equation (5) is bounded and decreases exponentially to
zero due to the convergence of x1. Therefore the trajecto-
ries in the entire state space in equation (3) are converg-
ing [9].
This implies that we can analyze or design a larger
system by considering the contraction property of smaller
subsystems and extending the results to the original sys-
tem. We will use this principle in the control algorithm
design.
2.3. Control Algorithm
Consider a strict feedback system in the form of
( )
( )
( )
11 12
22 123
12
,
,,
,,, ,
nn n
xf xx
xf xxx
xf xxxu
=
=
=




(6)
where x1,x2,…,xn are the states of the system, for the re-
cursive backstepping control formulation.
A coordinate transformation will be carried out to
transform the system into the hierarchical contraction
form.
Define the transformation
( )
1 11
r
z xx
= − (7)
whe re
( )
1
r
x
is the reference trajectory for x1. The z1-dy-
namics becomes
( )
( )
11 121
,
r
zf xxx=− 
 (8)
whe re
( )
2 11
x gx=
is defined as the virtual control such
that the virtual dynamics is in the form of equation (9)
with
( )
1
r
x
as a particular solution.
( )()
11111121 22
,zh zzhzzz
δδ δ = + 
(9)
whe re
( )
22 11
zxgx= −
with h11 uniformly negative
definite and h12 bounded.
It is clear that if δz2 reduces to zero, then
( )
11
r
xx
→. A
similar transformation procedure is applied to the z2-
dyna mic s.
() ( )
22 1231
,, d zfxx xgx
dt
= −

(10)
whe re
( )
32 12
,xg xx=
is defined as the virtual control
such that the virtual dynamics is in the form of equation
(11) with
( )
2 11
x gx=
as a particular solution.
( )()
22222232 33
,zh zzhzzz δδ δ = + 
(11)
whe re
( )
332 12
,zxg xx= −
with h22 is uniformly nega-
tive definite and h23 bounded.
The recursive transformation procedure is performed
until the control input is obtained. The virtual dynamics
for the global feedback in equation (6) can be expressed
in the following form.

T. C. KIAT ET AL.
67
11 12
11
22 23
22
33
00
00
00 0
000
nn
nn
hh
zz
hh
zz
dh
dt
zz
h
δδ
δδ
δδ

 

 

 

=
 

 

 






  

(12)
This is in the hierarchical contraction form that pro-
vides the global feedback system with exponential stabil-
ity.
In addition, the hierarchical contraction form provides
exponential convergence for each individual state, zi
whe re
1, 2,,in=
. This is because the virtual dynamics
for each state is
,1 1iiiiiii
zhz hz δδ δ
++
= +

(13)
where the second term can be regarded as a bounded
disturbance term that diminishes to zero. In particular,
the controlled variable, z1, converges to the reference
value exponentially which can be used for fast conver-
gence control of a particular variable.
3. UAV Flight Path Control
3.1. Dynamical Model
This section provides a description of the contrac-
tion-based backstepping controller synthesis with a sim-
plified UAV model for flight path angle control.
Figure 1 shows the forces acting on the longitudinal
axis of the UAV, the dynamical model is given by equa-
tion (14) [16] .
( )
1sin cos g
LT
mV V
q
qMu
γ αγ
θ
=+−
=
= =



(14)
where γ is the flight path angle, θ is the pitch angle, q is
the pitch rate, m is the aircraft mass, V is the air speed, T
is the thrust, α is the angle of attack, g is the gravitational
acceleration, L is the aerodynamic lift and M is the aero-
dynamic pitch moment which is defined as the control
input u.
The lift force, L, is expressed in the following form.
( )
2
1
2
L
LV SC
ρα
= (15)
Figure 1. UAV dynamical model.
where ρ is the air density, S is the wing area and CL is the
lift coefficient and is a function of α.
The following assumptions were made to simplify the
model and the control formulation.
• The control surfaces only produce aerodynamic
moments. The aerodynamic forces produced were as-
sumed to be small and neglected.
• The speed of the aircraft was maintained at a con-
stant value independently.
• The control actuator dynamics were sufficiently fast,
thus neglected.
Therefore, equation (14) was reduced to the form
shown in equation (16).
( )
, cos
g
ft
V
q
qu
γα γ
θ
= −
=
=



(16)
whe re
()()( )
( )
1
, sinf tLTt
mV
α αα = +
.
Equation (16) is a simplified dynamical model for the
purpose of key concept demonstration and forms the ba-
sis for the control formulation which will be presented in
the next section.
3.2. Flight Path Angle Tracking
The above control algorithm was applied to a flight path
tracking problem of an UAV.
Figure 2 shows a block diagram of the closed-loop
system, highlighting the subsystem nature of the algo-
rithm.
Step 1:
Define the transformation
*
1
z γγ = −
(17)
where γ* is a reference flight path angle defined in equa-
tion (28). The z1-dynamics becomes
( )
( )
**
11
, cos
g
zf tz
V
α γγ = −+−


(18)
To define a suitable virtual control, we assume that
( )
21
z Kz α = −−
, where
0K>
, and consider the virtual
dynamics of equation (18).
Figure 2. Block diagram of the closed-loop system for the
flight path angle control problem.

T. C. KIAT ET AL.
68
( )
( )
( )
*
1211 1
21 2
, sin
,
g
zKfzKz tzz
V
fzKz tz
δ γδ
δ

′
=− −++


′
+−

(19)
For the subsystem in equation (19) to be in the con-
traction region, the operation region is limited so that
1
min
1g
KK
fV

= +

′
where 10K> , then the subsystem
given by equation (19) will be in the contraction region.
Step 2:
Hence the z2-dynamics becomes
21
1
.
z Kz
q Kz
α
γ
= +
=−+
 

(20)
If we define
( )
3
d
z qq= −
wher e
( )
22 1
d
qK zKz δγ =− +−

(21)
and
20K>
, then the virtual dynamics will become
2223
zKz z δ δδ
=−+
 (22)
which implies a contraction region.
Step 3:
Now, consider z3-dyna mics
( )
3
d
z uq=−

(23)
If we let the control input u be
( )
33
d
u qKz= −

(24)
and
3
0K>, then the virtual dynamics will become
3 33
z Kz δδ = −

(25)
which implies a contraction region.
Hence the virtual dynamics for the entire system will
be
111121
2 22
3 33
0
01
00
z hhz
dz Kz
dt z Kz
δδ
δδ
δδ


= −


−

(26)
whe re
( )
( )
*
112 11
, sin
g
hKfzKz tz
V
γ
′
=− −++
and
( )
122 1
,hfzKz t
′
= −
, is in the hierarchical contrac-
tion form.
Step 4:
For the tracked angle, γ(r), to be a particular solution so
that all trajectories contract onto it, the reference angle, γ*,
is defined as follows. Consider the resultant dynamics,
( ) ( )
**
11
222 3
3 33
, cos
g
zf tz
V
zKz z
z Kz
α γγ = −+−
=−+
= −
 


(27)
It can be seen that
23
0zz== is a particular solution.
Putting the actual tracked flight path angle, γ(r), as a par-
ticular solution into the z1-dy namics ,
( )( )
( )
( )
( )
( )
( )
( )
( )
( )
( )( )
* **
*
*
cos
1
r rr
rr
rr
g
fK V
f fK
K
γγγγγγ
α γγ
γ αγ
−=−− −−
=−−
= +
 
(28)
where we had used the fact that
( )( )
( )
( )
cos
rr r
g
fV γα γ = −

,
2
0z=
and the symbol t is omitted in equation (28) for neatness.
Note that the difference between γ* and γ(r) is due to the
nonzero angle of attack, α, at equilibrium so a nonzero γ*
is required to account for this difference.
Hence the above formulated control law and reference
fli ght path angle, γ*, tracks the flight path angle, γ(r).
4. Simulation Results
The synthesized control law was implemented and tested
in Matlab for flight path angle stabilization and tracking
problem.
To study the convergence of the implemented control-
ler, an initial perturbation (away from equilibrium) was
applied to the system and the system closed-loop re-
sponse was examined.
4.1. Flight Path Angle Regulation
The flight path angle regulation problem is the stabiliza-
tion of the flight path angle to a reference value. It dem-
onstrated the ability for the controller to reject perturba-
tion exponentially fast. Figure 2 to 5 are obtained from
the flight path stabilization simulation.
Figure 3. Flight path angle response for the stabilization
problem.

T. C. KIAT ET AL.
69
Figure 4. Pitch angle response for stabilization the problem.
Figure 5. Control input for the stabilization problem.
4.2. Flight Path Tracking
In the flight path tracking problem, the ability for the
controller to track time-varying reference flight path an-
gle was demonstrated. A sinusoidal reference was chosen
as an example of time-varying signal. Figure 6 to 9
shows the results for the flight path angle tracking simu-
lation.
In both examples, it was shown that exponential sta-
bility was achieved by the control algorithm. This is due
to the presence of a contraction region.
Furthermore, the controlled variable was made to be
exponentially stable subjected to bounded disturbances
due to the errors in virtual controls.
Although simplified, this example demonstrated the
potential of the contraction-based backstepping technique
as an alternative to the Lyapunov backstepping technique
in the formulation of a control algorithm that achieves
exponential stability in a system.
Figure 6. Flight path angle error for the stabilization prob-
l em.
Figure 7. Flight path angle for the tracking problem.
Figure 8. Pitch angle for the tracking problem.

T. C. KIAT ET AL.
70
Figure 9. Control input for the tracking problem.
Figure 10. Flight path angle error for the tracking problem.
5. Conclu sions
In this paper, a contraction-based backstepping technique
using the hierarchical contraction structure was demon-
strated. The unique hierarchical contraction structure
with backstepping control formulation provides recursive
control law that was formulated systematically. Expo-
nential stability of the closed-loop system and individual
controlled variable was achieved. The control algorithm
was demonstrated on a flight path angle stabilization and
tracking problem.
6. Acknow l edgements
The authors would like to thank Nanyang Technological
University and DSO National Laboratories for their sup-
port on this project.
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