1

Output Feedback Dynamic Surface Controller forQuadrotor UAV with Actuator Dynamics

Asad Ullah Awan

Department of Mechatronics Engineering

National University of Sciences and Technology, Pakistan

Email: a.awan@ceme.nust.edu.pk

Abstract—In this work, we develop an output feedbackaltitude-attitude controller for quadrotor UAV in the presence ofuncertainties in UAV and actuator dynamics. Controller designfor the quadrotor UAV is a difficult task due to its uncertainnonlinear dynamics. Unlike most previous works, we also con-sider uncertain actuator dynamics into the model construction ofthe UAV. For state estimation, a nonlinear observer using neuralnetworks is designed. For the controller, the dynamic surfacecontrol technique has been used, which has the advantage ofless complexity as compared to the conventional backsteppingtechnique. The closed loop stability is proved using Lyapunovstability analysis. Unlike previously published techniques, we donot assume actuator signals are available for measurement inthe observer/controller design. Simulation results are presentedto demonstrate the effectiveness of the controller in presence ofuncertainties in quadrotor UAV and actuator dynamics.

Index Terms—Quadrotor, UAV, Output feedback, Actuator,Dynamic Surface Control, Neural Networks

I. INTRODUCTION

Quadrotor UAVs are have generated alot of interest among

researchers, especially for experimental validation of algo-

rithms. This is due to the ability to perform complicated

flight maneuvers similar to helicopters while having much

simpler mechanical design. Autonomous control of quadrotor

UAV has been a subject of much research. The quadrotor

UAVs exhibit highly nonlinear dynamics, and the dynamic

charchteristics . Researchers have successfully applied a host

of control techniques to quadrotor UAV control, including

linear control [1], robust nonlinear control [2], sliding mode

control, feedback linearization [3], [4], neural network based

control [5] to name a few.

Backstepping, a widely used techinque for controlling non-

linear system in strict feedback form, has been successfully

applied to quadrotor control [6], [7]. However, conventional

backstepping has the inherent problem of ’explosion of com-

plexity’, which is caused by repeated differentiation of the

so-called virutal control nonlinear functions. To circumvent

this problem, dynamic surface control (DSC) was introduced

in [8], which introduced first-order filters, through which

the virtual control functions is passed at each step, thereby

avoiding repeated differentiations. The DSC technique was

extended to nonlinear adaptive systems using neural networks

in [9].

Most of the works [1], [2], [5], [3], [4] aimed at designing

controllers for the quadrotor UAV do not take into account

the dynamics of the actuators. By actuators, we mean the

individual motor-propeller subsystems generating the down-

ward thrusts (see Figure 1). A number of controllers for robot

manipulator with actuator dynamics have been proposed by

various researchers. In [10], [11], adaptive controllers for

electrically driven robot manipulators were proposed. In [12],

the authors have developed a fuzzy-neural-network controller

for an n-link robot manipulator to achieve position tracking in

presence of model uncertainties. Similar work in done in [13].

The adaptive output feedback DSC controllers was developed

for flexible-joint robot manipulator in [14]. In these works,

the motor dynamics were incorporated in the robot model for

the design of the controller. However, in all these works the

derived control law was a function of actuator signals such

as armature current or torque produced by actuators, which

was assumed to be measured requiring additional sensors. In

the case of the quadrotor UAV, the problem of designing an

adaptive output feedback controller for the UAV model with

actuator dynamics, without using any actuator signals such

as thrust or motor armature current, is not addressed to the

best of the author’s knowledge. In this work, we apply output

feedback DSC technique for designing stabilizing controller

for quadrotor UAV. The actuator is considered to be a motor-

propeller subsystem producing thrust. This is modeled as a first

order uncertain dynamic model, which captures the actuator

delay and conversion between armature currents and thrust.

While first order models are commonly used for DC motors,

quadrotor UAV usually employ BLDC motors. Modeling of

BLDC motors is very complicated, however, when coupled

with internal speed controllers, the system can be modelled

as a first order system [15]. To deal with uncertain terms

in both UAV dynamic and actuator models, NN’s powerful

universal approximation ability is used [16]. The NN weights

are adapted online, using the adaptation laws based on gradient

descent with the so-called ε-modification [17]. By using

Lyapunov stability theory, we arrive at conditions for obtaining

uniformly ultimately bounded (UUB) stability of the closed

loop system. Numerical simulations are used to demonstrate

the validity and performance of the proposed technique.

This paper is organized as follows. In Section 2, we discuss

the mathematical model of the quadrotor UAV with actuator

dynamics. Section 3 descibres the observer and controller

design. Section 4 provides the stability analysis, simulation

results are presented in Section 5, followed by the conclusion

in Section 6.

Proceedings of 2014 11th International Bhurban Conference on Applied Sciences & Technology (IBCAST)Islamabad, Pakistan, 14th – 18th January, 2014

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978-1-4799-2319-9/14/$31.00 © 2014 IEEE

2

X

Y

Z

zb

yb

xb

Fig. 1: The quadrotor UAV consists of four motor-propellors

generating downward thrust.

II. QUADROTOR DYNAMICS

Consider the quadrotor shown in 1. Let us define the

state vector ξ1 = [z, φ, θ, ψ] ∈ R4 to denote the altitude

and Euler angles of the UAV with respect to the inertial

frame I . Also define X = [x, y, ξ1] ∈ R6, and V =

[vx, vy, vz, wx, wy, wz] ∈ R6 denote the translational velocity

and angular velocities in body frame B. We can write the

dynamics as follows

We can write [5]

X = Ao(t)V (1)

V = fo(xo) + G+M−1U + τd (2)

(3)

The details of the terms fo(xo) can be seen in [?]. Grepresents the gravitation accelration term, M represents the

inertia matrix, which is assumed to be known and constant, Uis the force and moment input vector for the quadrotor UAV

and τd represents unknown external disturbances.

Here

Ao(t) =

[R(ξ1) 00 T (ξ1)

]

where Ao ∈ R6×6, R(ξ1) and T (ξ1) are rotation and

transformation matrices from body to fixed frame as defined in

[?]. We consider a first-order linearized actuator model with

lumped uncertainty Δa. If we introduce actuator dynamics,

we can describe the quadrotor UAV model by the following

equations

ξ1 = ξ2 (4)

ξ2 = AA−1ξ2 +Afo(ξ1, ξ2) +AG+AM−1ξ3 +Aτd (5)

ξ3 = αξ3 +Bvt +Δa(ξ3, vt) (6)

Here A ∈ R4×4, it differs from Ao in that it ignores the

terms involve vx and vy i.e. the body frame translational rates

(because they are not available for measurement), vt is the

input voltage vector to the electric motors, B is the non-

singular mapping from the electric voltages to the force and

moment vector. Also α ≺ 0. We can define fo1(ξ1, ξ2) =AA−1ξ2 +Afo(ξ1, ξ2). Here,

A =4∑

i=0

∂A

ξ1iξ2i = Γ(ξ1, ξ2) (7)

So we can finally write

ξ1 = ξ2 (8)

ξ2 = fo1(ξ1, ξ2) +AG+AM−1ξ3 +Aτd (9)

ξ3 = αξ3 +Bvt +Δa(ξ3, vt) (10)

Due to the NN universal approximation property [18], the

unknown terms fo1 and Δa can be approximated as follows

fo1(xo) � Woσo(VTo xo) + εo(xo) (11)

Δa � Waσa(VTa xa) + εa(xa) (12)

The outer weights Vo and Va are fixed, whereas the inner

weights Wo and Wa are adapted. In the following section,

the adaptive observer design with adaptive laws for the NN

weights are presented.

III. NEURAL NETWORK OBSERVER DESIGN

Suppose we define a change of variables for state estimates

ξ1 = z1 (13)

ξ2 = z2 + l1(ξ1 − z1) (14)

ξ3 = z3 + l2(ξ1 − z1) (15)

The NN based observer is defined as

z1 = z2 + l1ξ1 + d1ξ1 (16)

z2 = fo1(xo) +AG (17)

+AM−1(z3 + l2ξ1) + d2ξ1 (18)

z3 = αz3 +Bvt + d3ξ1 + Δa(xa) (19)

where l1, l2, l3, d1, d2, l3 are positive gains,

xa = [z3 + l1(ξ1 − z1), u], xo = [ξ1, z2 + l1ξ1].Due to the universal approximation property of NNs [19], we

can approximate the unknown terms as

fo1 = Woσo(VTo xo)

Δa = Waσa(VTa xa). (20)

The observer error dynamics in the original coordinates are

then as follows

˙ξ1 = ξ2 − d1ξ1 (21)

˙ξ2 = WT

o σ(V To xo) + yo +AM−1ξ3 + l1ξ1 − l1ξ2 (22)

˙ξ3 = αξ3 + WT

o σ(V Ta xa) + ya + l2ξ1 − l2ξ2 (23)

where l2 = αl2−d3+ l2d1, l1 = l1d1−d2, Wj =Wj−Wj ,

yj = WTj [σj(V

Tj xj)− σj(V

Tj xj)] + εj(xj), σj = σj(V

Tj xj)

where j = o, a.

It can be established that ||yj || ≤ kj [14].

Proceedings of 2014 11th International Bhurban Conference on Applied Sciences & Technology (IBCAST)Islamabad, Pakistan, 14th – 18th January, 2014

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A. Observer Stability Analysis

Let us consider the following Lyapunov candidate function

for observer dynamics

Vo =1

2(ξT1 ξ1 + ξT2 ξ2 + ξT3 ξ3) +

1

2tr(WT

o Wo)

+1

2tr(WT

a Wa) (24)

Taking time derivative of (24), we have

Vo = ξT1 (ξ2 − d1ξ1) + ξT2 (WTo σ(V T

o xo) + yo (25)

+AM−1ξ3 − d2ξ2 − l1ξ2) + ξT3 (αξ3 + Waσa + ya (26)

+ l2ξ1 − l2ξ2) + tr(WTo˙Wo) + tr(WT

a˙Wa) (27)

Vo = −d1||ξ1||2 − l1||ξ2|| − α∗||ξ3||+ ξT1 ξ2 + ξT2 WTo σo

+ ξT2 yo + ξ2AM−1ξ3 + l1ξT2 ξ1 + l2ξ

T3 ξ1 − l2ξ

T3 ξ2

+ ξT3 ya + ξT3 WTa σa − tr(WT

o˙Wo)− tr(WT

a˙Wa) (28)

where α∗ = −α. If we select l1 = −1 and use the following

NN adaptation laws,

˙Wo = σoξ1 − koWo (29)

˙Wa = σaξ1 − kaWa (30)

we obtain,

Vo = −d1||ξ1||2 − l1||ξ2|| − α∗||ξ3||+ ξT2 yo + ξ2AM−1ξ3+ l2ξ

T3 ξ1 − l2ξ

T3 ξ2 − tr(WT

o (σ0ξ1 − koWo − σoξT2 ))

− tr(WTa (σaξ1 − kaWa − σaξ

T3 )) (31)

Now we use the following inequalities [20], [5]

tr(WTo (Wo − Wo)) ≤ ||Wo||FWMo − ||Wo||2F (32)

||y|| < ky (33)

||a||||b|| ≤ ||a||22

+||b||22

(34)

||σ|| ≤√

No (35)

||A||F ≤ AM (36)

where (34) is the Young’s inequality. We can write

Vo ≤ −d1||ξ1||2 − l1||ξ2||2 − α∗||ξ3||2 + ||ξ2||||yo||+ ||ξ2||||ξ3||||AM−1||F (37)

+ l2||ξ3||||ξ1||+ l2||ξ3||||ξ2||+ ||ξ3||||ya||+ ko||Wo||WMo − ko||Wo||2 + ||Wo||||ξ2||

√No

+ ||Wo||||ξ1||√

No + ka||Wa||WMa

− ka||Wa||2 + ||Wa||||ξ1||√

Na + ||Wa||||ξ3||√

Na (38)

Now select the following

c1 = d1 − l22

(39)

c2 = l1 − l2 +AMo||M−1||F + 12

(40)

c3 = α∗ − AMo||M−1||F + l2 − l2 − 12

(41)

We can then write

Vo ≤ −c1||ξ1||2 − c2||ξ2||2 − c3||ξ3||2 + ||Wo||||ξ2||√

No

+ ko||Wo||WMo − ko||Wo||2 + ||Wo||||ξ1||√

No

+ ||Wa||||ξ1||√

No + ka||Wa||WMa − ka||Wa||2+ ||Wa||||ξ3||

√No + k (42)

where k =k2o

2 +k2a

2

Now completing the square with respect to ||ξ1||, ||ξ2|| and

||Wo||, we obtain

Vo ≤ −m1||ξ1||2 −m2||ξ2||2 −m3||ξ3||2 − ko4||Wo||2

− ka4||Wa||2 + k (43)

where the parameters m1 = c1 − (No

ko+ Na

ka),m2 = c2 −

No

ko,m3 = c3 − Na

kaare designed to be positive real numbers.

IV. CONTROLLER DESIGN

In this section, we present the design for the dynamic sur-

face controller (DSC) [9], [14]. The conventional backstepping

technique commonly employed to deal with nonlinear systems

in feedback form results in the so-called ’explosion of com-

plexity’ in the terms due to computation of derivative of virtual

control terms. DSC circumvents this drawback by introducing

a first-order filter in the virtual control, thereby eliminating

the need for computing the derivative. The objective of the

controller is to ensure that the trajectory error is made as small

as possible. We make the following assumption

Assumption 1: The desired trajectories ξd are bounded as:

||ξd||+ ||ξd||+ ||ξd|| ≤ M1 (44)

Define the error surface variable S1 as

S1 = ξ1 − ξd

S1 = ξ2 + ξ2 − ξd (45)

where ξd is the desired trajectory ; zd, φd, θd, ψd. The first

virtual control is designed as follows

ξ2 = −k1S1 + ξd (46)

This virtual control is passed through a first order filter as

follows

τ2ξ2f + ξ2f = ξ2 (47)

Proceedings of 2014 11th International Bhurban Conference on Applied Sciences & Technology (IBCAST)Islamabad, Pakistan, 14th – 18th January, 2014 99

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We define the second error surface

S2 = ξ2 − ξ2f

S2 = AG+AM−1ξ3 + WTo σo + l1ξ1 − l1ξ2 − ξ2f (48)

The second virtual control ξ3 is passed through another first-

order filter

ξ3 =MA−1(−AG− WT

o σo − l1ξ1 +

(ξ2 − ξ2f

τ2

)− k2S2

)

(49)

ξ3 = τ3ξ3f + ξ3f (50)

The third error surface is defined as

S3 = ξ3 − ξ3f (51)

S3 = αξ3 − αl2ξ1 +Bvt + d3ξ1 + WTa σa + l2ξ2

− l2d1ξ1 − ξ3f (52)

Finally, the true control input vt is selected as

vt = B−1((−αξ3 + αl2ξ1 − d3ξ1 − WT

a σa

+l2d1ξ1 + (ξ3 − ξ3f

τ3)− k3S3

)(53)

Define the boundary layer errors as

y2 = ξ2f − ξ2 (54)

y3 = ξ3f − ξ3 (55)

We can now write the surface error dynamics as

S1 = S2 + y2 − k1S1 + ξ2 (56)

S2 = AM−1(S3 + y3)− k2S2 − l1ξ2 (57)

S3 = −k3S3 + l2ξ2 (58)

Also the boundary error dynamics can be written as

y2 = −y2τ2+ k1S1 − ξd

= −y2τ2+Ω2(S1, S2, ξ1, ξ2, ξ3) (59)

y3 = −y3τ3+Ω3(S1, S2, S3, ξ1, ξ2, ξ3) (60)

A. Controller Stability AnalysisLet us define the Lyapunov candidate function

Vc =1

2(ST

1 S1 + ST2 S2 + ST

3 S3 + yT2 y2 + yT3 y3) (61)

(62)

Differentiating and utilizing some basic inequalities and the

fact that ||A|| ≤ AM , ||Ω2|| ≤ Ω2M , ||Ω3|| ≤ Ω3M

Vc = ST1 (S2 + y2 − k1S1 + ξ2)

+ ST2 (AM−1(S3 + y3)− l1ξ2 − k2S2)

+ ST3 (−k3S3) + yT2 (−

y2τ2+Ω2)

+ yT3 (−y3τ3+Ω3) (63)

≤ −b1||S1||2 − b2||S2||2 − b3||S3||2 − q1||y1||2

− q2||y2||2 + 32||ξ2||2 + Ω2M

4+Ω3M

4(64)

where b1 = k1 +94 , b2 = k2 − 1

4 − AM ||M−1||2 (1 + l1) −

l12 , b3 = k3− AM ||M−1||

2 , q1 =1τ2− 5

4 , q2 =1τ3−1− AM ||M−1||

2

V. CLOSED LOOP STABILITY

Define the composite Lyapunov function V

V = Vo + Vc (65)

V = Vo + Vc (66)

≤ −m1||ξ1 − (m2 − 33)||ξ2||2 −m3||ξ3||2 − ko

4||Wo||2

− ka4||Wa||2 − b1||S1||2 − b2||S2||2 − b3||S3||2

− q1||y1||2 − q2||y2||2 + Ω2M

4+Ω3M

4+ k (67)

(68)

Lemma 1: The closed loop system is unformly ultimately

bounded.

Proof: This is concluded from the afore stability analysis.

VI. RESULTS

In this section, the effectiveness of the proposed technique

on quadrotor UAV is verified via numerical simulations in the

MATLAB/Simulink environment with the Runge-Kutta ODE

solver. Table I provides the parameters used for the UAV

model. In the simulation, all inner NN weights, i.e. Wo and

Wa, are initialized to zero, while inner weights, i.e. Vo and Vaare randomly initialized from a uniform distribution between

0 and 1. Simulation sample time was set to 0.001s. In order to

demonstrate the effectiveness of the technique in the presence

of actuator uncertainities, a time varying term 0.3 sin t, was

added to the actuator dynamics. The desired trajectory for the

simulation was set as ξd := [−10 + 2 sin t, 0, 0, 0.2 sin t)].Figure 2, 3, 4 and 5 show the performance of the quadrotor

UAV during the simulation in the state variables. As is

evident from the plots, the controller yields good tracking

performance, with low RMSE (see figures). 8 shows the

estimation error of the observer for ξ1. Figures 6 and 7 show

the behaviour of the NN weights that are adapted online. The

input history in terms of voltage signals to individual motors

of the quadrotor UAV, is shown in 9

VII. CONCLUSION

In this paper, we proposed a outout dynamic surface con-

troller for a quadrotor with uncertain model and actuator

dynamics. Previous work incorporating actuator dynamics for

robot control have assumed that actuator signals are available

for measurement. In this paper, we only use the altitude and

attitude measurements for observer-controller design. Bounded

stability of the closed loop observer-controller system has

been derived via the Lyapunov stability theory. Numerical

simulation shows that the propsed technique shows good

tracking performance. In future work we hope to extend the

technique to control the x and y translational positions of the

UAV, which is not trivial due to the underactuated nature of

the UAV dynamics.

Proceedings of 2014 11th International Bhurban Conference on Applied Sciences & Technology (IBCAST)Islamabad, Pakistan, 14th – 18th January, 2014 100

5

0 5 10 15 20 25 30 35 40 45 5012

11.5

11

10.5

10

9.5

9

8.5

8RMSE =0.038538

time (in sec)

z (m

)

Fig. 2: Tracking performance of altitude z

0 5 10 15 20 25 30 35 40 45 500.02

0

0.02

0.04

0.06

0.08

0.1RMSE =0.023311

time (in sec)

φ (m

/s)

Fig. 3: Tracking performance of roll φ

TABLE I: Quadrotor UAV parameters used in simulation

Parameter Value Units

Mass m 0.586 kg

Moment of inertia about x-axis Ixx 7.5× 10−3 kg m−2

Iyy 7.5× 10−3 kg m−2

Izz 1.3× 10−2 kg m−2

Arm length l 0.165 m

Rotor inertia Jr 6× 10−5 kg m−2

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0 5 10 15 20 25 30 35 40 45 500.5

1

1.5

2

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3

3.5

4

|| W

o ||F

Frobenium Norm of Wo

Fig. 6: Plot showing change in NN weights due to adaptation

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0 5 10 15 20 25 30 35 40 45 500

1

2

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|| W

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Frobenium Norm of Wa

Fig. 7: Plot showing change in NN weights due to adaptation

during simulation, ||Wa||F

0 5 10 15 20 25 30 35 40 45 5012

10

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6

4

2

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2

Time (sec)

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ξ1 Estimation Error

Fig. 8: NN adaptive observer performance for state ξ1, plot

showing ξ1

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec)

v 1

Input signal

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec)

v 2

Input signal

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1.5

2

2.5

3

3.5

4

4.5

5

Time (sec)

v 3

Input signal

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0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Time (sec)

v 4

Input signal

Fig. 9: History of inputs (voltage applied to each motor). The

input signal is limited between 0 and 5.

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