This work proposes a solution for the output feedback trajectory-tracking problem in the case of an uncertain DC servomechanism system. The system consists of a pendulum actuated by a DC motor and subject to a time-varying bounded disturbance. The control law consists of a Proportional Derivative controller and an uncertain estimator that allows compensating the effects of the unknown bounded perturbation. Because the motor velocity state is not available from measurements, a second-order sliding-mode observer permits the estimation of this variable in finite time. This last feature allows applying the Separation Principle. The convergence analysis is carried out by means of the Lyapunov method. Results obtained from numerical simulations and experiments in a laboratory prototype show the performance of the closed loop system.

1. ISA Transactions 51 (2012) 801–807
Contents lists available at SciVerse ScienceDirect
ISA Transactions
journal homepage: www.elsevier.com/locate/isatrans
Output feedback trajectory stabilization of the uncertainty
DC servomechanism system$
˜
Carlos Aguilar-Ibanez a,n, Ruben Garrido-Moctezuma b, Jorge Davila c
a
´
´
´
CIC-IPN, Av. Juan de Dios Batiz s/n Esq. Manuel Othon de M., Unidad Profesional Adolfo Lopez Mateos, Col. Nueva Industrial Vallejo, Del. Gustavo,
A. Madero, C.P. 07738 D.F., Mexico
b
´
Departamento de Control Automatico, CINVESTAV-IPN, Av. IPN 2508, 07360 D.F., Mexico
c
National Polytechnic Institute (IPN), School of Mechanical and Electrical Engineering (ESIME-UPT), Section of Graduate Studies and Research, Mexico
a r t i c l e i n f o
abstract
Article history:
Received 29 March 2012
Received in revised form
19 June 2012
Accepted 29 June 2012
Available online 9 August 2012
This paper was recommended for
publication by Dr. Jeff Pieper
This work proposes a solution for the output feedback trajectory-tracking problem in the case of an
uncertain DC servomechanism system. The system consists of a pendulum actuated by a DC motor and
subject to a time-varying bounded disturbance. The control law consists of a Proportional Derivative
controller and an uncertain estimator that allows compensating the effects of the unknown bounded
perturbation. Because the motor velocity state is not available from measurements, a second-order
sliding-mode observer permits the estimation of this variable in finite time. This last feature allows
applying the Separation Principle. The convergence analysis is carried out by means of the Lyapunov
method. Results obtained from numerical simulations and experiments in a laboratory prototype show
the performance of the closed loop system.
& 2012 ISA. Published by Elsevier Ltd. All rights reserved.
Keywords:
Servomechanism
PD controller
Finite time observer
Variable structure control
1. Introduction
The direct current motor-pendulum system (DCMP) is widely
used as a test bed for assessing the effectiveness of several control
techniques. This choice is due to the fact that its model captures
some of the features found in more complex systems, as in the case
of industrial robot manipulators [1,2]. Related to this topic we
mention some interesting works; for instance, Ref. [3] applies the
well known Generalized Proportional–Integral controller to the
tracking control problem for a linear DCMP. In [4] the authors solve
the regulation problem using the sliding-mode super-twisting
based observer (STBO) method, in conjunction with a twisting
controller. An interesting work dealing with the control of the
DCMP system using the STBO observer, combined with an identification scheme can be found in [5, Chapter 2]. A close related work
is developed by Davila et al. [6]. A closed-loop input error approach
for on-line estimation of a continuous-time model of the DCMP
$
´
´
This research was supported by the Centro de Investigacion en Computacion
´
of the Instituto Politecnico Nacional (CIC-IPN), and by the Secretarıa de Investiga´
cion y Posgrado of the Instituto Politecnico Nacional (SIP-IPN), under Research
Grant 20121712.
n
´
Corresponding author at: CIC-IPN, Av. Juan de Dios Batiz s/n Esq. Manuel
´
´
Othon de M., Unidad Profesional Adolfo Lopez Mateos, Col. Nueva Industrial
Vallejo, Del. Gustavo, A. Madero, C.P. 07738 D.F., Mexico.
˜
E-mail address: caguilar@cic.ipn.mx (C. Aguilar-Ibanez).
was developed in [7]; while in [8] a parameter identification
methodology based on the discrete-time Least Squares algorithm
and a parameterization using the Operational Calculus is proposed.
In [9] an adaptive neural output feedback controller design is used
to solve the tracking problem of the system studied here, having
the advantage of including the model of the actuator. Ref. [10]
employs H1 techniques to deal with uncertainties; performance of
the proposed approach is evaluated through numerical simulations. Another interesting work [11] reports the application of
second order sliding mode control applied to an uncertain DC
motor; the motor under control receives disturbance torques
produced by another motor directly coupled to its shaft of the first
motor. A smooth hyperbolic switching function eliminates chattering phenomena. The Hoekens mechanical system is the subject of
research in [12]; here the authors apply a sliding mode control
technique with uncertainty estimation combined with a learning
technique. A key feature of this approach is the fact that it applies
the switching to the plant indirectly through the learning process
and the disturbance estimator thus reducing the occurrence of
chattering; experiments validate the findings.
In this context, perhaps one of the most challenging control
problems consists in designing a smooth output-feedback stabilization algorithm for an uncertain and perturbed DCMP [13,14].
Generally speaking, this problem is by no means easy because it
is not possible to fully compensate the effects of the system
uncertainty without having information about the time derivative
0019-0578/$ - see front matter & 2012 ISA. Published by Elsevier Ltd. All rights reserved.
http://dx.doi.org/10.1016/j.isatra.2012.06.015

2. ˜
C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807
802
of this uncertainty [15]. This control problem has been solved using a
combination of neuronal networks and adaptive control theory, as in
[9,16]. On the other hand, the problem is easily solved using a sliding
mode controller (SMC), a methodology ensuring robustness against
disturbances and parameter variations. However, its main drawback
is that it may generate high frequency violent control signals, a
behavior known as chattering. Moreover, the presence of chattering
may excite unmodeled high-frequency dynamics, resulting in unforeseen instability and damage to the actuators. [17–19]. There exists
essentially three main approaches to eliminate the chattering effects.
The first approach uses continuous approximations, as the saturation
function, of the sign function appearing in the sliding mode controllers [1,20,21]. This approach trades the robustness on the sliding
surface and the system convergences to a small domain [22].
Observed-based approaches are another way of overcoming chattering; it consists in bypassing the plant dynamics by a chattering loop,
then reducing the robust control problem to an exact robust estimation problem. However, this action can deteriorate the robustness
with respect to the plant uncertainties and disturbances [19,23,24].
The last approach, based on the high-order sliding-mode method
guaranties convergence to the origin of the sliding variable and its
corresponding derivatives; the high-order sliding-mode algorithms
translate the discontinuity produced by the sign function to the
higher order derivatives, producing continuous control signals; however, these algorithms require a great computing effort [6,25–28].
In this work we introduce a smooth controller for output
feedback trajectory tracking in an uncertain DCMP. The solution
consists of a PD controller and a robust uncertain estimator. A
super-twisting second-order sliding-mode observer estimates
motor velocity. The observer finite time of convergence ensures
that the estimation error will vanish after a finite time transient,
the allowing the use of the Separation Principle. The corresponding convergence analysis is carried out using the Lyapunov
method. This work continues with Section 2, where the model
of the DCMP system and the problem statement are presented. In
Section 3 the control strategy and the corresponding convergence
analysis are developed. Sections 4 and 5 are devoted respectively
to the numerical and experimental results and the conclusions.
perturbation, defined as
wðx,tÞ ¼
1
_
ðÀf c sign½xŠ þ ZÀgmL sin xÞ,
J
ð2Þ
which is a bounded function. In order to simplify the forthcoming
developments, the system (1) is re-written in its state space form
as
_
x 1 ¼ x2 ,
_
x 2 ¼ Àf 0 x2 þ wþ u,
ð3Þ
_
where x1 ¼ x and x2 ¼ x, and
f0 ¼
fd
,
J
u¼
ku
t:
J
ð4Þ
Problem statement. Consider the uncertain nonlinear system (3)
and the corresponding state x1 regarded as the measured system
output, where the perturbation is uniformly bounded by
9w9 r
1
ðZ þgmL þ f c Þ r c:
J
ð5Þ
Then, the control goal is to design a controller such that the
angular pendular position tracks a given continuous reference
trajectory xr(t), with their first and second time derivatives being
also continuous. In other words, we want to control the pendular
angular position x towards a pre-specified desired trajectory xr ðtÞ,
in spite of the perturbation w.
For simplicity, we use the symbol q to denote the upper bound of
the term q 4 0, i.e. 0 o q rq. It is important to notice that the system
(3) can be seen as a general electro-mechanical system, because a
wide range of robots admit this configuration. Consequently, the
solution that we propose in this work can be straightforwardly
applied to more complex configurations, as for example a robot
manipulator. On the other hand, it is important to remark that
reference xr has to be continuous, and the corresponding first and
second time derivative being at least piece-wise continuous ([29].
3. The control strategy
2. DCMP dynamic model
Consider the actuated second order DCMP system composed of
a servomotor driving a pendulum, a servo-amplifier and a position sensor. The corresponding model of this system has the
following form:
€
x¼
1
_
_
ðÀf d xÀf c sign½xŠ þ ZÀgmL sin x þ ku tÞ,
J
ð1Þ
_
Variables x and x are, correspondingly, the pendular angle position and the pendular angle velocity; t is the control input
voltage; parameters m, L and g are respectively the pendulum
mass, the pendulum arm length and the gravity constant; the
terms fd and fc are in that order the pendulum viscous friction and
Coulomb friction coefficients. The parameter J stands for the
pendulum and the rotor inertias; the parameter ku is related to
the amplifier gain and to the constant motor torque. Finally, Z is
the unknown time varying disturbance.
We underscore that in many cases found in practice the
disturbance Z, and the gravitational and Coulomb friction torques
are unknown. On the other hand, most of the amplifiers used in
practice only accept smooth control voltages and has a relatively
low bandwidth; thus, they could not withstand the switching
signals produced by a classic sliding mode controller. In this
context, we solve the trajectory tracking control problem, assuming that we do not know, both the velocity position y and the
In this section we first develop a robust control scheme based
on the approximation of the sign function. Then, we introduce a
robust observer able to estimate, in finite-time, the motor
velocity. Finally, we propose an output-feedback controller that
solves the trajectory tracking control problem for the DCMP.
Finally, the convergence analysis of the closed-loop system is
carried out using the Lyapunov method.
3.1. Robust control scheme
In this section we propose a simple control scheme, which
emulates the robust behavior of a first order sliding mode
controller. To this end, we approximate the sign function by means
of a saturation function in conjunction with a first order filter. The
saturation function is a smooth approximate to the sign function;
the filter attenuates the high frequencies produced by the saturation function.
Consider the following system:
_
y ¼ ÀayÀv þ rðy,tÞ,
ð6Þ
where a 4 0 is a constant, y A R and v A R are the state and the
input. Function rðnÞ A R is unknown and continuous satisfying
9rðy,tÞ9 r r, for a known constant r and for all, t Z 0. It is well
known that if we select v as
v ¼ k sign½yŠ,
k 4r,
ð7Þ

3. ˜
C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807
then the state y globally converges to the origin in finite-time.
Unfortunately, in practice using this scheme would produce
chattering together with its undesirable effects. A way to emulate
the robust property of the sliding-mode method consists of using
a nonlinear smooth saturation together with a first order filter. Let
define the following auxiliary signal:
s ¼ vÀk tanhðLyÞ,
ð8Þ
1
where L 4 1 is a constant large enough. Intuitively, if s % 0, then
v % k sign½yŠ implying that all the solutions of the system (6) will
be confined to moved close enough to the origin. In order to
design a filter that emulates the behavior of the sign function, we
compute the time derivative of s which produces
2
_
s ¼ ur þ kL sech ðLyÞðay þvÞ þ Rðy,tÞ:
ð9Þ
_
In this last dynamics, ur ¼ v plays the role of a control signal and
2
Rðy,tÞ ¼ kL sech ðLyÞrðy,tÞ is the uncertainty with 9Rðy,tÞ9 rkLr.
Hence, to make the variable s to move inside a small neighborhood of the origin, we propose ur as
2
ur ¼ ÀK M sÀkL sech ðLyÞðay þvÞ,
ð10Þ
where K M 4 0. Consequently, substituting (10) into (9), we have
the following stable dynamics:
_
s ¼ ÀK M s þ Rðt,yÞ:
803
where k ¼ br, K M 4 bLr 2 and b,L 41. Then, y(t) is uniformly
ultimately bounded with the estimation bound given by (11). &
Comment 1. If we directly introduce in Eq. (6) v ¼ k tanhðLyÞ with
k4 r and L 40 large enough, evidently the closed-loop solutions
are confined to move inside of a small origin vicinity, but having
the presence of chattering phenomena. We underscore that the
proposed control scheme can be seen as a nonlinear version of a PI
controller, because the auxiliary variable v adds an integral term.
Remark 1. As mentioned in the Introduction, there are several
SMO based control methods with application results especially in
motor control. However, the majority of these works are based on
the usage of function sign, which involves some difficulties such
as the sign of zero is not well defined or high frequencies are
present in the control. These inconveniences make very difficult
to build an actual implementation of these methods. In our
control scheme we approximated the sign function by means of
continuous and derivable functions (see the auxiliary signal ‘‘s’’
defined in (8)). Effectively, this solution brought a small stationary error, however it can be as small as desired. On the other
hand, the high frequency problem can be overcome by introducing auxiliary high order time derivatives of s. For instance, the
performance of our control strategy can be improved if we assure
_
€
that the auxiliary signals s, s and s are almost equal to zero.
The above implies the following inequality when t-1:
9sðtÞ9 r eÀK M T 9sð0Þ9 þ
3.2. State observer
kLr
kLr
ð1ÀeÀK M T Þ r K c 9sð0Þ9 þ
¼
:
KM
KM
If the initial conditions s(0) are set to zero, then the following
inequality holds
9sðtÞ9 r r M ¼
kLr
:
KM
Now, to compute the final value of y, in (6) we use the following
positive function:
V ¼ 1y2 ,
2
whose time derivative evaluated along the solutions of (6)
leads to
_
V ¼ yðÀayÀk tanhðLyÞÀs þ rðy,tÞÞ,
_
where s is defined in (8). Hence, V can be upper bounded by:
_
V r 9y9ðÀa9y9Àk9 tanhðLyÞ9 þr þ 9s9Þ:
Since the values of 9s9 continuously decrease until 9sðtÞ9 rr M for
some t 4 T, then we have
kLr
_
, t 4T,
V ðtÞ r9y9 Àa9y9Àk9 tanhðLyÞ9 þ r þ
KM
_
which implies that V is negative outside of BM ¼ fyA R : 9y9 rMg,
where M is the single root of
kLr
pðyÞ c a9y9 þ ktanhðL9y9ÞÀrÀ
¼
,
KM
ð11Þ
that is pðMÞ ¼ 0. Therefore, y is bounded and converges to the
smallest level set of V that includes BM. In other words, there is a
finite time T after which y(t) is confined to move inside the
compact set BM. The following lemma summarizes the stability
result previously presented:
Lemma 1. Consider the scalar system (6) with 9rðy,tÞ9 r r in closed
loop with the nonlinear controller (10)
2
_
v ¼ ÀK M ðvÀk tanhðLyÞÞÀkL sech ðLyÞðay þvÞ,
1
For simplicity, we use the tanh(s) as a smooth nonlinear saturation function;
however, we can use any kind of smooth saturation function.
In order to estimate the velocity x2 in (3) in a short period, we
use the super-twisting based observer (STBO) [4], defined by the
following second order system:
_
b 1 ¼ Àa1 9e1 91=2 sign½e1 Š þ b2 ,
x
x
_
b 2 ¼ Àa2 sign½e1 Š þf 0 b2 þ u,
x
x
x
e1 ¼ b1 Àx1 :
ð12Þ
x
Variables b1 and b2 are, respectively, the estimates of the state
x
variables x1 and x2; a1 and a2 are strictly positive constants.
Defining the observation errors as e1 ¼ b1 Àx1 and e2 ¼ b2 Àx2 , the
x
x
error equation takes the following form:
1=2
_
sign½e1 Š þ e2 ,
e 1 ¼ Àa1 9e1 9
_
e 2 ¼ Àa2 sign½e1 ŠÀf 0 e2 þw,
ð13Þ
where wðÁÞ can be considered as the system perturbation and is
defined in (2). From the definition of (2), we have that the above
term can be bounded by:
9w9 o d0 c
¼
1
ðZ þgmL þf c Þ:
J
ð14Þ
The following theorem gives a tuning rule for the observer gains.
Lemma 2. Under condition (14) and assuming that a1 and a2 satisfy
the following inequalities:
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
2 ða1 þ d0 Þð1 þ pÞ
,
a1 4 d0 , a2 4
ð1ÀpÞ
a1 Àd0
with 0 o p o 1, then the observer (12) assures finite time convergence of the estimated states ðb1 , b2 Þ to the actual states ðx1 ,x2 Þ.
x x
The corresponding proof can be found in [4,6,30,26].
3.3. Perturbation estimation and robust controller
In this section we focus our attention on a robust control
strategy to compensate the effects of the perturbation w appearing in (3). Lemma 2 assures that, after a finite time, the estimated
state variables ðb1 , b2 Þ converge to the corresponding actual values
x x

4. ˜
C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807
804
is able to make s being stable and ultimately bounded in a finite
time, T 4 0. Besides, variable s remains inside the compact set:
Bm ¼ fs A R : 9s9 r mg, where m is the single root of:
ðx1 ,x2 Þ. Hence, the separation principle holds and the controller
can be designed without taking into account the observers
dynamics. Therefore, the stability analysis of the system (3)
together with the observer (12) is equivalent to examine the
stability of the system (3), for the case where b2 ¼ x2 . The formal
x
and detailed stability analysis of the closed-loop system can be
found in the Appendix.
Let us consider the smooth reference signal xr and its time
_
_
derivatives x r and x r . We propose the following tracking control
law:
We must underscore that b is required to be slightly larger than one,
while L being much more greater than one. We end this section
introducing some assumptions for establishing Proposition 1, which
resumes the main result of this work.
b
_
€
u ¼ ÀwÀkp ðx1 Àxr ÞÀkd ðx2 Àx r Þ þ f 0 x2 þ x r ,
Assumptions.
ð15Þ
pðxÞ ¼ ki 9x9 þ br tanhðL9x9ÞÀrÀ
Lbr 2
:
KM
ð25Þ
b
where kp 4 0 and kd 4 0 are the positive constants, w is an
auxiliary signal that allows us to compensate for the effects of
the disturbance w. Therefore, substituting (15) into (3), we have
the following closed-loop system:
(A1) b2 is obtained by using the STBO (12).
x
(A2) The set of positive control parameters fL,K M ,ki , b, lg
is selected according to the previous discussion.
_
x 2 ¼ x2 ,
_ 2 ¼ ue ðx1 ,x2 , b2 , wÞ þw,
x b
x
Proposition 1. Consider the uncertain DCMP system (3), under the
above assumptions, and in closed-loop with
ð16Þ
where
€
b
ue ðx1 ,x2 , b2 , wÞ ¼ Àkp ðx1 Àxr ÞÀkd ðb2 Àx r Þ þ x r Àf 0 ðx2 Àb2 ÞÀw:
x b
x _
x
We must clarify that the solutions of the system (3) under the
control law (15) are understood in Filippov’s sense [31], because
the discontinuities appearing in the closed loop system.
_
Define the tracking errors x1r ¼ x1 Àxr and x2r ¼ x2 Àx r . Using
the above definitions, and substituting (15) into (3), it is not
difficult to show that the following equations describe the tracking
errors dynamics:
_
x 1r ¼ x2r ,
b
_
x 2r ¼ Àkp x1r Àðkd þ f 0 Þx2r Àw þ w,
ð17Þ
Let us define the following auxiliary variable:
s ¼ x2r þ lx1r ,
ð18Þ
where l 4 0. Evidently, if we assure that s converges to zero, or at
least remains close enough to zero, then the tracking errors stay
very close to zero too. Computing the time derivative of s along the
trajectories of (17), and using (17) we have, after some simple
algebra, the following:
k
b
_
s ¼ Àki x2r þ p x1r Àw þw,
ð19Þ
ki
where:
ki ¼ kd þ f 0 Àl 4 0:
ð20Þ
Hence, fixing l ¼ kp =ki it is not difficult to show that the parameter
l is given by
l¼
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
1
kd þ f 0 À ðkd þ f 0 Þ2 À4kp
2
ð21Þ
ð23Þ
Notice that control parameters kp and kd have to be selected taking
into account ðkd þ f 0 Þ2 4 4kp to assure the feasibility of the parameters ki and l, respectively. Therefore, applying Lemma 1 to (22),
we can assure that the following controller:
2
_
b
b
b
w ¼ ÀK M ðwÀk tanhðLsÞÞ þkL sech ðLsÞðki s þ wÞ,
where
r ¼ d0 ,
Remark 2. Our control strategy is based on the assumption that
the parameter f0 is known. However, if this parameter was
unknown, then it can be easily estimated online or identified by
some identification method, like the one found in [4,6]. In general,
the parameter value is very small in comparison with the actual
kd, and can be neglected.
We want to point out that our control strategy was developed
ad hoc to be applied to an uncertain second order nonlinear
system, based on using the traditional PI controller with a correction or a compensation term, which emulates a smooth sliding
mode controller. But we should not forget that there are more
general strategies which have been successfully designed for
systems with more than two state variables.
4. Numerical and experimental results
This section shows the effectiveness of the proposed controller
applied to the output-feedback trajectory tracking problem of the
DCMP. To this end we carried out a numerical simulation and an
experiment using a laboratory prototype.
4.1. Numerical simulation
f 0 ¼ 0:1,
where
k ¼ br,
where ðkd þf 0 Þ 44kp . Then, the closed-loop system is uniformly
ultimately bounded, with the estimation bound given by (25).
ð22Þ
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ki ¼ 1 ðkd þ f 0 Þ þ 1 ðkd þf 0 Þ2 À4kp :
2
2
2
b,L 4 1, K M 4Lbr , a ¼ ki
ð26Þ
2
The physical parameters of the DCMP are set to the following
values:
with ðkd þf 0 Þ2 44kp . So, system (19) can be written as
b
_
s ¼ Àki sÀw þw,
b b €
b _
u ¼ Àkp ðx1 Àxr ÞÀkd ðx 2 Àx r Þ þ f 0 x 2 Àw þ x r ,
d
2
b
b
b
w ¼ ÀK M ðwÀk tanhðLsÞÞ þ kL sech ðLsÞðki s þ wÞ,
dt
s ¼ ðb2 Àx r Þ þ lðx1 Àxr Þ,
x _
ð24Þ
fc
¼ 1:3,
J
Z
J
¼ 0:84,
gmL
¼ 14:03,
J
ku
¼ 5:4:
J
These values are close to the one proposed in [7]. In order to make
the experiment more interesting, we added an external perturbation to the parameter Z=J, indeed Z=J ¼ 0:84 þ 0:2 sinðt=5Þ.
The objective is to track the reference xr ¼ sinðt=2Þ; for this
system, the bound in the disturbance is fixed to r ¼ 15:32. Hence,
the control parameters are set as
b ¼ 4, kp ¼ 16, kd ¼ 8:4, L ¼ 40, K M ¼ 12,000, l ¼ 2:81:
According to (25), it is quite easy to show numerically that the
manifold s satisfies 9s ¼ x2r þ lx1r 9 o 8 Â 10À3 , which implies that
the steady state error is given by 9x1r 9 r 2:3 Â 10À3 rad. Notice

5. Velocity [rad/s]
Position [rad]
˜
C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807
2
-2
x
0.2
0
2
4
6
8
10
12
14
16
18
6
4
2
0
-2
0.24
20
x2
vr
0.1
xr
0.238
0
0
Input [Nw m]
Reference output
x1
xr
0
805
2
4
6
8
10
12
14
16
18
20
10
-0.1
τ
-0.2
0
-10
0
2
4
6
8
10
12
Time [sec]
14
16
18
20
x 10-3
Output tracking error
Fig. 1. Closed-loop responses of the uncertain DC servo pendulum.
[rad]
2 x 10
2
Tracking position error
-3
0
-2
[rad/s]
2
6
8
-4
5 x 10
10
12
14
Tracking velocity error
16
18
20
0
-2
0
-5
2
[rad/s]
4
4
6
8
10
12
14
Observed velocity error
16
18
20
-4
2 x 10
Control action
1
0
-2
2
4
6
8
10
12
Time [sec]
14
16
18
20
0.5
Fig. 2. State errors of position, velocity and observed velocity, from 2 to 20.5 s.
0
-0.5
-1
0
5
10
15
Time [sec]
Fig. 4. Closed-loop response to the proposed control law (26), when applied to the
laboratory prototype.
of the reference. In Fig. 2, the tracking position, velocity position
and observed velocity errors are illustrated in steady state. For
this case, we only show the simulations from 2 s to 20.5 s.
4.2. Experimental result
Fig. 3. DC servo pendulum prototype used to carried out the actual experiments.
that for this case d0 ¼ 17:4. On the other hand, according to
Lemma 2, the STBO parameters were fixed as
a1 ¼ 3:7, a2 ¼ 41:6, p ¼ 0:5:
The DCMP initial conditions are chosen as x1 ¼ À1:57 rad and
x2 ¼ 0 rad=s, the corresponding state variables of the STBO are set
at the origin. The simulation was carried out by using the Runge–
Kutta algorithm with a sampling integration interval of, 1 Â 10 À 4 s.
Fig. 1 shows the closed-loop response of the whole state. From
this figure, we can see that the proposed controller effectively
makes the pendulum to follow the reference signal xr ¼ sinðt=2Þ
after one second. Moreover, the state x2 tracks the time derivative
In order to carry out this experiment we used a laboratory
prototype, which consists of a DC servomotor from Moog, model
C34-L80-W40 (Fig. 3) driven by a Copley Controls power servoamplifier model 423 configured in current mode. A BEI optical encoder
model L15 with 2500 pulses per revolution allows measuring the
servomotor position. The algorithms are coded using the MatLab/
Simulink software platform under the program Wincon from Quanser
Consulting, and a Quanser Consulting Q8 board performs data
acquisition. The data card electronics increases four times the optical
encoder resolution up to 2500 Â 4¼10,000 pulses per revolution. The
control signal produced by the Q8 board passes through a galvanic
isolation box. The Q8 board is allocated in a PCI slot inside a personal
computer, which runs the control software.

6. ˜
C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807
806
and second time derivatives, as
Reference output
0.24
0.2
€
_
xð3Þ þ 3a x r þ 3a 2 x r þ a 3 xr ¼ a 3 r,
r
xr
x
0.22
0.2
0.1
0.18
2.5
0
3
3.5
4
-0.1
where, a ¼ 20, and, r ¼ 0:25 sinðptÞ.
Fig. 4 shows the reference and the DCMP output, the tracking
position error when the proposed control law (26) was used. Fig. 5
b
shows the same results but in this case the compensator w is set to
zero. From these figures it is clear that the proposed control law
effectively compensates the uncertain term, w, and tracks the
reference signal with an error position ranging in, 10 À 3 rad.
-0.2
5. Conclusions
This work proposes a new output feedback control scheme for
solving the stabilization and trajectory tracking problem in an
uncertain DC servomechanism system. A Proportional Derivative
algorithm plus a robust uncertain compensator composes the control
law. A super-twisting second-order sliding-mode observer recovers
the pendulum velocity. Owing to the finite time convergence of this
observer, the Separation Principle holds thus making possible to
develop the control law without taking into account the observer
dynamics. The effectiveness of the controller is tested by performing
a numerical simulation, and an experiment using a laboratory
prototype. These results show that the proposed algorithm is able
to compensate for the disturbances affecting the DC servomechanism system without generating chattering in the control signal. As a
future extension of the obtained control scheme, it can be generalized to a higher order system.
Output tracking error
0.01
0
Control Action
1
^
Appendix A. Finite time convergence of x 2 to x2
0.5
Proof. First of all notice that the output-feedback controller (15)
with (3) is given by
_
x 1 ¼ x2 ,
_
x 2 ¼ Àf 0 x2 þ wþ ua ,
0
ð27Þ
where ua is in fact the actual controller:
b
€
ua ¼ Àwðx1 , b2 ÞÀkp ðx1 Àxr ÞÀkd ðb2 Àx r Þ þ f 0 b2 þ x r :
x
x _
x
-1
0
5
b
Remember that w, kp and kd are control parameters, previously
defined. Now, by simple algebra, from (27), (28) and (12) we have
the following dynamic error:
10
Time [sec]
Fig. 5. Closed-loop response to the classical PD controller, when applied to the
laboratory prototype.
The physical parameters are fixed as
f 0 ¼ 0:13,
fc
¼ 1:27,
J
Z
J
¼ À0:17,
gmL
¼ 14:21,
J
ku
¼ 18:56:
J
These parameters were previously identified in Ref. [7]. For
this case, the uncertainty in the pendulum model is given by
9wðx,tÞ9 rr ¼ 16. The controller parameters used during the
experiments are
k ¼ 20,
kp ¼ 1600,
kd ¼ 80,
L ¼ 15,
K M ¼ 6000,
ð28Þ
l ¼ 38:04:
From (25), we have that s satisfies 9s ¼ x2r þ lx1r 9 o 0:06, implying that the steady state error is given by 9x1r 9 r 1:5 Â 10À3 rad.
The STBO parameters were fixed as, a1 ¼ 25, and, a2 ¼ 47. The
reference applied to the closed loop system consists of a filtered
sinusoid. The filter allows generating the reference and its first
_
b 1 ¼ Àa1 9e1 91=2 sign½e1 Š þ b2 ,
e
e
_
b 2 ¼ Àa2 sign½e1 Š þ f 0 b2 þ w,
e
e
where 9w9 o d0 . So, selecting a1 and a2 according to Lemma 2, we
can always assure that e1 and e2 asymptotically converge to zero
in a finite time. In other words, b2 ¼ x2 þ fðtÞ, with fðtÞ being a
x
continuous functions, and provided that limt-tf fðtÞ ¼ 0, with tf as
short as needed. Therefore, the actual controller can be read as
b
€
ua ¼ Àwðx1 , b2 ÞÀkp x1r Àðkd þf 0 Þx2r þ x r þ ðkd þ f 0 ÞfðtÞ:
x
Now, we must remember that ua is proposed such that the values
_
of s and s are always almost zero. Therefore, computing the time
derivative of s along the trajectories of (27) and (28), we have
b
_
s ¼ Àki sÀw þ w þðkd þ f 0 ÞfðtÞ,
where ki is defined in (20) and fðtÞ converges to zero in a short
period of time. This discussion justify that the filter variable b2 is
x
not needed to carry out the stability analysis. It means that b2 is
x
equivalent to x2. Formally, the last equation converts to
b
_
s ¼ Àki sÀw þ w,

7. ˜
C. Aguilar-Ibanez et al. / ISA Transactions 51 (2012) 801–807
which coincides with expression in (22). Thus, steps in the
stability analysis introduced in Section 3.3 are justified.
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