UAV Flight Path Control Using Contraction-Based Back-stepping Control

doi:10.4236/ojapps.2013.32B013

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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

Email: ckta[email protected], [email protected], [email protected]

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.

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

∈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

( )

δδ

∂

=∂



(1)

where δx is the virtual displacement of the trajectory.

We can further describe the time variation of δxTδx as

( )

δδ δδ

∂

=∂

xx xx

x (2)

Here, δxTδx represents the squared distance between infi-

nitesimally close trajectories. If the Jacobian,

( )

,/ ,t∂∂

fx x

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

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

,,, ,

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

z xx

= − (7)

whe re

( )

is the reference trajectory for x1. The z1-dy-

namics becomes

( )

11 121

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

( )

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

( )

→. A

similar transformation procedure is applied to the z2-

dyna mic s.

() ( )

22 1231

,, d

zfxx xgx

= −



(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.

11 12

22 23

00 0

000

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

mV V

qMu

γ αγ

=+−

= =



(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.

( )

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

γα γ

= −



(16)

whe re

()()( )

( )

, sinf tLTt

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

γγ

= −

(17)

where γ* is a reference flight path angle defined in equa-

tion (28). The z1-dynamics becomes

( )

, cos

zf tz

α γγ

= −+−



(18)

To define a suitable virtual control, we assume that

( )

z Kz

= −−

, 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.

( )

1211 1

21 2

, sin

zKfzKz tzz

fzKz tz

δ γδ



′

=− −++





′

+−



(19)

For the subsystem in equation (19) to be in the con-

traction region, the operation region is limited so that

min



= +



′

where 10K> , then the subsystem

given by equation (19) will be in the contraction region.

Step 2:

Hence the z2-dynamics becomes

z Kz

q Kz

= +

=−+





(20)

If we define

( )

z qq= −

wher e

( )

22 1

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

( )

z uq=−



(23)

If we let the control input u be

( )

u qKz= −



(24)

and

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

111121

2 22

3 33

z hhz

dz Kz

dt z Kz

δδ





= −



−



(26)

whe re

( )

112 11

, sin

hKfzKz tz

′

=− −++

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,

( )

222 3

3 33

, cos

zf tz

zKz z

z Kz

α γγ

= −+−

=−+

= −



(27)

It can be seen that

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

r rr

fK V

f fK

γγγγγγ

α γγ

γ αγ

−=−− −−

=−−

= +

 

(28)

where we had used the fact that

( )( )

( )

cos

rr r

γα γ

= −



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.

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.

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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