0004

Trajectory Tracking Error Using PID Control Law for

a 2 DOF Helicopter Model via Adaptive Time-Delay

Neural Networks

Joel Perez�, Jose P. Perezy, Jose J. Hdzz, Santiago ArroyoxAngel Flores{

Abstract

This paper deals with the problem of trajectory tracking for

adaptive time-delay neural networks. The tracking error is

globally asymptotically stabilized by a control law derived on

the basis of a Lyapunov-Krasovsky functional.

The main methodologies, on which the approach is based,

are time-delay neural networks and the Lyapunov-Krasovskii

functions metodology and the PID control for nonlinear sys-

tems. The proposed new control scheme is applied via simu-

lations to control a 2 DOF Helicopter model [1], [2]. Experi-

mental results show the usefulness of the proposed approach.

To verify the analytical results, an example of a dynamical

network is simulated and a theorem is proposed that ensures

the tracking of the nonlinear system.

Keywords. Adaptive Time-Delay Neural networks, Trajec-

tory tracking, Adaptive control, Lyapunov-Krasovskii func-

tion Stability and PID control.

1. Introduction

In order to solve problems in the �elds of optimization,

pattern recognition, signal processing and control systems,

among others, recurrent neural networks have to be designed

with the property that there is only one equilibrium point,

and this point has be globally asymptotically stable. There-

fore analysis of this kind of stability has been intensively in-

vestigated (see [3] and references therein). In biological and

arti�cial neural networks, time delays arise in the processing

of information storage and transmission. It is known that

�Facultad de Ciencias Fisico Matematicas, Universidad Au-tonoma de Nuevo Leon (UANL) Monterrey, Mexico. e-mail:[email protected]

yFacultad de Ciencias Fisico Matematicas, Universidad Au-tonoma de Nuevo Leon (UANL) Monterrey, Mexico. e-mail:[email protected]

zFacultad de Ciencias Fisico Matematicas, Universidad Au-tonoma de Nuevo Leon (UANL) Monterrey, Mexico. e-mail:[email protected]

xFacultad de Ciencias Fisico Matematicas, Universidad Au-tonoma de Nuevo Leon (UANL) Monterrey, Mexico. e-mail: [email protected]

{Facultad de Ciencias Fisico Matematicas, Universidad Au-tonoma de Nuevo Leon (UANL) Monterrey, Mexico. e-mail: a�[email protected]

these delays can create oscillatory or even unstable trajecto-

ries [4]. Results regarding stability of delayed neural networks

can be found in [5], [6].

In [7], analysis of Recurrent Neural Networks for identi�-

cation, estimation and control are developed, with applica-

tions to chaos control, robotics and chemical processes. Con-

trol methods that are applicable to general nonlinear systems

have been intensely developed since the early 1980´s. Main

approaches include, for example, the use of di¤erential geom-

etry theory [8]. The passivity approach has, at the same time,

generated interest for synthesizing control laws [9]. In [10]

and [11], this methodology was modi�ed for stabilization and

trajectory tracking of an unknown chaotic dynamical system.

The proposed adaptive control scheme is composed of a

recurrent neural identi�er and a controller, see �g. (1),

Fig. 1 Adaptive Time-Delay Recurrent Neural Network

scheme

where the former is used to build an on-line model for the

unknown plant and the latter, to ensure that the unknown

plant tracks the reference trajectory.

Robot manipulators [12] present a practical challenge for

control purposes due to the nonlinear and variable nature

of their dynamical behavior. Motion control in joint space

is the most fundamental task in robot control; it has moti-

vated extensive research work in synthesizing di¤erent control

1

Congreso Nacional de Control Automático 2013Ensenada, Baja California, Mexico, Octubre 16-18, 2013

methods such as fuzzy computed torque control [13], PI+PD

fuzzy control [14] and static neural network control [15]. An

important problem for developing control algorithms is that

most robots models neglect practical aspects such as actuator

dynamics, sensor noise, and friction, which if are not consid-

ered in the design may cause performance deterioration.

Even though conditions of global asymptotic stability for

delayed neural networks have been established, as far as we

know, no publication presents results on trajectory tracking

of such systems. It is relevant to procure this tracking in

order to allow the implementation of important tasks for au-

tomatic control such as: high speed target recognition and

tracking, real-time visual inspection, and recognition of con-

text sensitive and moving scenes, among others, by means of

recurrent neural networks.

In this paper we present the design of a control law to

ensure tracking of a general nonlinear system by delayed re-

current neural networks.

The tracking error stability is analyzed by means of a

Lyapunov-Kraosvkii functional ([5], [16], [17]). The applica-

bility of the approach is illustrated by one example.

2. Modelling of the PlantThe unknown nonlinear plant is given by:

�xp = Fp(xp; u) , fp(xp) + gp(xp)u (1)

where xp, fp 2 Rn; u 2 Rm and gp 2 R

nxm. xp is the plant,

u is the control input and both fp and gp are unknown. We

propose to model (1) by the time-delay neural network state

space representation�xp = Ax +W ��z[x(t � �)] + u, plus

one more term modelling error [5], [17]. A is a n�m matrix;

without loss of generality we can assume that A = ��I,

� > 0. x are the neural states, giving an approximation to

the real plant by a neural network, W � is the weights matrix,

�z(x) is the hyperbolic tangent and is a n�m matrix that

modi�es the input u.

We de�ne the modelling error between the time-delay

neural network and the plant by:

wper = x� xp (2)

We assume the following:

Hypotheses 1. (Objective of Modelling): The modelling

error is exponentially stable, that is:

�wper = �kwper (3)

In this work we consider k = 1, and now, from (2) we

have�wper=

�x�

�xp and using the hypothesis we get:

�xp=

�x+

wper

The unknown plant can be modelled as:�xp =

�x+ wper = Ax+W

��z[x(t� �)] + wper +u (4)

W � are the �xed weights but unknown from the time-delay

neural network. They minimize the modelling error.

3. Trajectory Tracking

We proceed now to analyze the modelling error between the

unknown plant, modelled by (4) and the reference signal de-

�ned by:

�xr = fr(xr; ur); with ur and xr 2 R

n (5)

where xr are the reference states, ur is the input and fr is

a nonlinear function

For this purpose, we de�ne the control error between the

plant and the reference signal by:

e = xp � xr (6)

whose derivative with respect to time is

�e =

�xp�

�xr= Ax+W ��z[x(t��)]+wper+u�fr(xr; ur) (7)

Adding and subtracting, to the right hand side of (7)

the terms_

W�z[xr(t � �)]; �r(t;_

W ); Ae; where_

W is the

estimate of W � and considering that e = xp � xr, we have

�e = Ax+W

��z[x(t� �)] + x� xp +u� fr(xr; ur) +_

W�z[xr(t� �)]�_

W�z[xr(t� �)] + �r(t;_

W )�

�r(t;_

W ) +Ae�Ae

�e = Ae+W

��z[x(t� �)] + u� fr(xr; ur) +_

W�z[xr(t� �)] + �r(t;_

W )� (8)

�r(t;_

W )� e� xr �Ae+ x+Ax_

� W �z[xr(t� �)]

In this part, we consider the next supposition:

The time-delay neural network will follow the reference sig-

nal, even with the presence of disturbances, if:

Axr +_

W�z[xr(t� �)] + xr � xp +�r(t;_

W ) = fr(xr; ur):

Then

�r(t;_

W ) = fr(xr; ur)�Axr�_

W�z[xr(t��)]�xr+xp (9)

and we get

�e = Ae+W

��z[x(t� �)]�_

W�z[xr(t� �)]�Ae+

(A+ I)(x� xr) + (u� �r(t;_

W )) (10)

Now, adding and subtracting in (10) the term_

W

�z[x(t��)] and de�ning �z[x(t��)] = �(z[x(t��)]�z[xr(t�

�)]) we have

2

CNCA 2013, Ensenada B.C. Octubre 16-18 142

�e = Ae+ (W �

�_

W )�z[x(t� �)] +_

W�(z[x(t� �)]� z[xr(t� �)]) + (11)

(A+ I)(x� xr)�Ae+(u� �r(t;_

W ))

We de�ne,

�

W =W��

_

W and�u = u� �r(t;

_

W ) (12)

and replacing (12) in (11), we obtain

�e = Ae+

�

W�z[x(t� �)] +_

W�(z[x(t� �)]� z[xr(t� �)]) +

(A+ I)(x� xr)�Ae+�u

�e = Ae+

�

W�z[x(t� �)] +

_

W�

(

z[x(t� �)]� z[xp(t� �)]+

z[xp(t� �)] + z[xr(t� �)]

)

+ (13)

(A+ I)(x� xp + xp � xr)�Ae+�u

Now we can set:

�u = u1 + u2 (14)

and de�ne:

u1 = �_

W�(z[x(t��)]�z[xp(t��)])�(A+I)(x�xp) (15)

so (13) is reduced to:

�e = Ae+

�

W�z[x(t� �)] +_

W�(z[xp(t� �)]� z[xr(t� �)]) +

(A+ I)(xp � xr)�Ae+u2

Considering that e = xp � xr, shortening notation a

little bit by setting � = �z and de�ning:

��(t� �) = �(xp(t� �))� �(xr(t� �))

we get

�e = (A+ I)e+

�

W�[x(t� �)] +_

W��(t� �) + u2 (16)

Now, the problem is to �nd the control law u2 that

stabilizes the system (16). We will obtain the control law by

using the Lyapunov-Krasovskii methodology.

4. Stability of the Tracking Er-

rorOnce (16) is obtained, we consider its stabilization in feedfor-

ward. We note (e;_

W )= 0 is an asymptotically stable equilib-

rium point of the undisturbed autonomous system (A = ��I

and � > 0). For its stability, we propose the next PID control

law:

u2 = Kpe+Kv�e+Ki

Z t

0

e(�)d��(1

2+1

2

_

W

2

L2�)e (17)

The parameters Kp; Kv and Ki will be determined later,

L2� is the Lipschitz constant of � [18], � > 0. This control

law (17) is similar to the control law in [19]:

We will show the feedback system is asymptotically stable.

Replacing (17) in (16) and setting a = (1�Kv), we get:

�e =

1

a(A+ I)e+

1

a

�

W�[x(t� �)] +1

a

_

W��(t� �) +1

aKpe+

1

aKi

Z t

0

e(�)d� ��

a(1

2+1

2

_

W

2

L2�)e (18)

and assuming A = ��I:

�e =

�1

a(�� 1�Kp)e+

1

a

�

W�[x(t� �)] +1

a

_

W��(t� �) +

1

aKi

Z t

0

e(�)d� ��

a(1

2+1

2

_

W

2

L2�)e (19)

If we set w = 1aKi

R t

0e(�)d� , with derivative

�w = 1

aKie(t),

then (19) can be written as

�e =

�1

a(�� 1�Kp)e+

1

a

�

W�[x(t� �)] +1

a

_

W��(t� �) +

w ��

a(1

2+1

2

_

W

2

L2�)e (20)

We will show the new state (e; w)T is asymptotically stable

and that the equilibrium point is (e; w)T = (0; 0)T ; when�

W�[xr(t��)] = 0, which is taken as an external disturbance.

Let V be the candidate Lyapunov-Krasovskii function

([16], [17]) given by

V =1

2(eT ; wT )(e; w)T +

1

2atr

�

�

WT~W

�

+

1

a

Z t

t��

[�T� (s)_

WT _

W��(s)]ds (21)

The time derivative of (21) along the trajectories of (20)

is:

�

V = eT �e+ w

T �w +

1

atr

8

<

:

�

�

W

T�

W

9

=

;

+

1

a[�T� (t)

_

WT _

W��(t)� �T� (t� �)

_

WT _

W��(t� �)] (22)

and using (20) we get:

3


�

V = eT [�1

a(�� 1�Kp)e+

1

a

�

W�[x(t� �)] +

1

a

_

W��(t� �) + w �

�

a(1

2+1

2

_

W

2

L2�)e] +

1

awTKie (23)

+1

atr

8

<

:

�

�

W

T�

W

9

=

;

+1

a[�T� (t)

_

WT _

W��(t)�

�T� (t� �)

_

WT _

W��(t� �)]

In this part, we select the next learning law for the neural

network weights as in [20] and [21]:

tr

8

<

:

�

�

W

T�

W

9

=

;

= �eT�

W�[x(t� �)] (24)

Then (23) is reduced to

�

V =�1

a(�� 1�Kp)e

Te+

eT

a

_

W��(t� �) +

(1 +Ki

a)eTw �

�

a(1

2+1

2

_

W

2

L2�)e

Te+ (25)

1

a[�T� (t)

_

WT _

W��(t)� �T� (t� �)

_

WT _

W��(t� �)]

If we apply the inequality

xTy �

1

2xTx+

1

2yTy (26)

to the second term in the right hand side of (25) then:�

V ��1

a(�� 1�Kp)e

Te+

1

a[eT e

2+1

2�T� (t� �)

_

WT _

W��(t� �)] +

(1 +Ki

a)eTw �

�

a(1

2+1

2

_

W

2

L2�)e

Te+

1

a[�T� (t)

_

WT _

W��(t)� �T� (t� �)

_

WT _

W��(t� �)]

�

V ��1

a(�� 1�Kp)e

Te+

1

a(1

2+1

2

_

W

2

L2�)e

Te+

(1 +Ki

a)eTw �

�

a(1

2+1

2

_

W

2

L2�)e

Te (27)

Here, we select (1+ Ki

a) = 0, so KV = Ki+1; and KV � 0

when Ki � �1. With this selection of parameters (27) is

reduced to:

�

V ��1

a(��1�Kp)e

Te�

1

a(��1)(

1

2+1

2

_

W

2

L2�)e

Te (28)

In this part, if ��1�Kp > 0; a > 0 and ��1 > 0; then�

V

< 0, 8 e; w;_

W 6= 0; the error tracking is asymptotically stable

and it converges to zero for every e 6= 0, i.e., the plant will

follow the reference asymptotically . Finally, the control law,

which a¤ects the plant and the time-delay neural network, is

given by:

u = y[�_

W�(z[x(t� �)]� z[xp(t� �)])� (A+ I)(x� xp) +

Kpe+Kv�e+Ki

Z t

0

e(�)d� ��(1

2+1

2

_

W

2

L2�)e+

fr(xr; ur)�Axr �_

W�z[xr(t� �)]� xr + xp] (29)

This control law gives asymptotic stability of error dynam-

ics and thus ensures the tracking of the reference signal. The

results obtained can be summarized as follows.

Theorem 1 For the unknown nonlinear system modeled by

(4), the learning law (24) and the control law (29) ensure the

tracking of the nonlinear reference model (5).

Remark 2 From (28) we have

�

V ��1

a(�� 1�Kp)e

Te�

1

a(�� 1)(

1

2+1

2

_

W

2

L2�)e

Te < 0; 8e 6= 0; 8

_

W

and therefore V is decreasing and bounded from below

by V (0). Since V = 12(eT ; wT )(e; w)T + 1

2atr

�

�

WT~W

�

+

1a

R t

t��[�T� (s)

_

WT _

W��(s)]ds;

we conclude that e;�

W 2 L1; this means that the weights

remain bounded.

5. Simulations

The 2 DOF Helicopter is pivoted around the Pitch axis by

the angle �; and around the Yaw axis by the angle [2], see

�g. 2.

The plant is stated in [1],[2], and it is given by:

�

X1 = X3

�

X2 = X4

�

X3 = �Mheli[

sin 2�(l2mc

�h2)

2� lmch cos 2�]

�

2

(Jeqp +Mheli(l2mc + h2))�

Mhelig(lmc cos � + h sin �) +Bp�

� � (KppVmp + Fcpp)

(Jeqp +Mheli(l2mc + h2))�

(KpyVmy + Fcpy)

(Jeqp +Mheli(l2mc + h2))(30)

�

X4 = �Mheli[sin 2�(h

2 � l2mc) + 2lmch cos 2�]�

��

�By�

[Jeqy +Mheli[cos2 �(l2mc � h2) + lmch sin � + h2]]�

(KypVmp + Fcyp) cos � � (KyyVmy + Fcyy) cos �

[Jeqy +Mheli[cos2 �(l2mc � h2) + lmch sin � + h2]]

4


Fig. 2 Two DOF Helicopter Model

With the goal of supporting the e¤ectiveness of the pro-

posed controller we have used a Du¢ng equation:

The time-delay neural network is modelled by the di¤eren-

tial equation:

_x = Ax +W�[x(t � �)] + u;with � = 10 sec, A = ��I;

I 2 R4x4; and � = 20; W is estimated using the learning law

given in (24),

�[x(t��)] = (tanh[x1(t��)]; tanh[x2(t��)]; :::; tanh[xn(t��)])T;

=

0 0 1 0

0 0 0 1

!

T ; and the u is calculated using

(29).

The time evolution for the position angles and applied

torque are shown in Figs. 4-7. As can be seen in Figs. 4

and 5, the trajectory tracking is successfully obtained where

the plant and reference signals are the same.

We try to force this manipulator to track a reference signal

given by undamped Du¢ng equation:

�x� x+ x3 = 0:114 cos(1:1t); x(0) = 1; _x(0) = 0:114 (31)

and it�s phase space trajectory, it is given in Fig. 3

1.5 1 0.5 0 0.5 1 1.51

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

X1PlantX1Reference

X2P

lanr

X2R

efe

rence

Fig. 3 A phase space trajectory of Du¢ng equation.

0 20 40 60 80 100 120 140 160 180 2001.5

1

0.5

0

0.5

1

1.5

Time (Sec)

Pitch

Time evolution for pitch position

Plant

Reference

Fig. 4 Time evolution for the angular position Pitch

(rad)

0 20 40 60 80 100 120 140 160 180 2001

0.8

0.6

0.4

0.2

0

0.2

0.4

0.6

0.8

1

Time (Sec)

Yaw

Time evolution for taw position

Plant

Reference

Fig. 5 Time evolution for the angular position yaw (rad)

5


0 20 40 60 80 100 120 140 160 180 200100

50

0

50

100

150

200

250

300

350

Time (Sec.)

T1

Applied torque

Fig. 6 Torque (Nm) applied to the pitch

0 20 40 60 80 100 120 140 160 180 20012000

10000

8000

6000

4000

2000

0

2000

Time (Sec.)

T2

Applied torque

Fig. 7 Torque (Nm) applied to the yaw

We can see that the Recurrent Neural Controller ensures

rapid convergence of the system outputs to the reference tra-

jectory. The controller is robust in presence of disturbances

applied to the system. Another important issue of this ap-

proach related to other neural controllers, is that most neural

controllers are based on indirect control, �rst the adaptive

time-delay neural network identi�es the unknown system and

when the identi�cation error is small enough, the control is

applied. In our approach, direct control is considered, the

learning laws for the adaptive time-delay neural networks

depend explicitly of the tracking error instead of the iden-

ti�cation error. This approach results in faster response of

the system.

Acknowledgements. The �rst and second authors would

like to thank the support from CONACYT and the Matem-

aticas Aplicadas group of the Facultad de Ciencias Fisico-

Matematicas, Universidad Autonoma de Nuevo Leon, Mex-

ico.

6. Conclusions

We have extended the adaptive recurrent neural control pre-

viously developed in [10], [11] and [22] for trajectory tracking

control problem in order to consider less inputs than states.

Stability of the tracking error is analized via Lyapunov-

Krasovskii control functions and the control law is obtained

based on the PID approach. A 2 DOF Helicopter model with

friccion terms and unknown external disturbances is used to

verify the design for trajectory tracking, with satisfactory

performance. Research along this line will continue to imple-

ment the control algorithm in real time and to further test

it in a laboratory enviroment as well as to consider the 2

DOF Helicopter with an unknown load, see example 4.9-pp

151-154 [23].

7. References

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