International Journal of Control, Automation and Systems 15(X) (2017) 1-9 http://dx.doi.org/10.1007/s12555-015-0177-x
ISSN:1598-6446 eISSN:2005-4092 http://www.springer.com/12555
A Robust Controller for Trajectory Tracking of a DC Motor Pendulum System Carlos Aguilar-Ibañez*, Julio Mendoza-Mendoza, Jorge Davila, Miguel S. Suarez-Castanon, and Ruben Garrido M. Abstract: This work presents a solution to the output feedback trajectory tracking problem for an uncertain DC motor pendulum system under the effect of an unknown bounded disturbance. The proposed algorithm uses a Proportional Derivative (PD) controller plus a novel on-line estimator of the unknown disturbance. The disturbance estimator is obtained by coupling a standard second-order Luenberger observer with a third-order sliding modes differentiator. The Luenberger observer provides estimates of the motor angular position and velocity. Moreover, an ideal disturbance estimator in terms of the Luenberger observer error and its first and second time derivatives is obtained from the observer error formulae; these time derivatives are not available from measurements. Subsequently, the sliding modes third-order differentiator allows obtaining estimates of these time derivatives in finite time. The estimates replace the real values of the first and second time derivatives in the ideal disturbance estimator thus producing a practical disturbance estimator, and also permit obtaining an estimate of the motor angular velocity. A depart from previous approaches is the fact that the disturbance is not directly estimated by the Luenberger observer or the third-order differentiator. Numerical simulations and real-time experiments validate the effectiveness of the proposed approach. Keywords: Finite time observer, PD controller, servomechanism, variable structure control.
1.
INTRODUCTION
The DC-motor pendulum (DCMP) system has been extensively used as a test bed to verify the performance of several control algorithms, for instance sliding modes control, PID control, adaptive control, robust control, neuronal networks control among others. Note that the DCMP model is similar to more complex systems including industrial robots [1, 2]. In [3] a Generalized ProportionalIntegral controller using only position measurements is used to solve the tracking control problem for a linear model of the DCMP system. A sliding-mode supertwisting observer (STBO) in conjunction with a super twisting controller is proposed in [4] and [5] for controlling the DCMP. A solution to the output trajectory tracking problem for a perturbed DCMP system is proposed in [6]. It is based on a Proportional Derivative (PD) controller
plus an on-line nonlinear smooth estimator of the disturbance. The DCMP system angular velocity is estimated by means of a second order supertwisting observer. References [7,8] solves the problem of servo control by using the estimation of an equivalent input disturbance. An exhaustive review on second-order sliding mode control of mechanical systems is presented in [9]. Generally speaking, trajectory output feedback control for an uncertain system under the effect of disturbances is a challenging problem [10–12]. An obstacle in pursuing a solution to this problem is the lack of knowledge about bounds on the output time derivatives. Consider the output y of a system and its time derivatives { } y(1) , ..., y(n) , y(n+1) , where n is the system order. Mathematically, it is not
Manuscript received April 23, 2015; revised December 23, 2015 and July 12, 2016; accepted September 2, 2016. Recommended by Associate Editor Yingmin Jia under the direction of Editor Myo Taeg Lim. This research was supported by the Centro de Investigación en Computación of the Instituto Politécnico Nacional (CIC-IPN), and by the Secretaría de Investigación y Posgrado of the Instituto Politécnico Nacional (SIP-IPN), under Research Grants 20160268 and 20161637. This research was done while Dr. Carlos Aguilar Ibañez was on sabbatical leave from the Departamento de Mecatrónica of the CINVESTAV. Carlos Aguilar-Ibañez is with the Centro de Investigación en Computación, Instituto Politécnico Nacional, Ciudad de México 07738, México (e-mail:
[email protected]). Julio Mendoza-Mendoza is a doctoral student at the the Centro de Investigación en Computación, Instituto Politécnico Nacional, Ciudad de México 07738, México (e-mail:
[email protected]). Jorge Davila is with the Escuela Superior de Ingeniería Mecánica y Eléctrica, Instituto Politécnico Nacional, Ciudad de México 07340, México (e-mail:
[email protected]). Miguel S. Suarez-Castanon is with the Escuela Superior de Cómputo, Instituto Politécnico Nacional, Ciudad de México 07738, México (email:
[email protected]). Ruben Garrido M. is with the Departamento de Control Automático, Centro de Investigación y Estudios Avanzados, Instituto Politécnico Nacional, Ciudad de México 07360, México (e-mail:
[email protected]). * Corresponding author.
c ⃝ICROS, KIEE and Springer 2017
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Carlos Aguilar-Ibañez, Julio Mendoza-Mendoza, Jorge Davila, Miguel S. Suarez-Castanon, and Ruben Garrido M.
always possible to compute the best k > 0, such that (i) y < k < ∞, for all t ≥ 0 and i ∈ In = (1, 2, ..., n + 1) [13]. Fortunately, in many applications including servomechanisms and robot manipulators, velocities and accelerations are physically limited due to actuator and structural constraints. This fact is helpful in solving the time derivative estimation problem by assuming that there is a k∗ ∈ R+ large enough, such that (i) y < k∗ , for all t ≥ 0 and i ∈ In . Under this assumption, it is possible to estimate the i-th time derivative of the output using an observer. This issue has motivated the development of several control techniques based on output feedback, for instance active disturbance rejection [14–16], neural networks control [17–19], sliding mode observers [3, 9, 13, 20, 21], and high-gain observers, [22–25]. This work proposes a smooth controller for solving the output-feedback trajectory tracking problem of a DCMP system. It is composed of two parts, a standard PD controller plus a novel on-line disturbance estimator. A second-order Luenberger observer produces estimates of the motor angular position and velocity but it does not directly estimate the disturbance. However, the observer error formulae provides an expression for estimating the disturbance in terms of the observation error and its first and second order time derivatives. Subsequently, a thirdorder sliding modes differentiator [26] allows obtaining estimates of these time derivatives in finite time. The estimates replace the real values of the first and second time derivatives in the ideal disturbance estimator and also permit obtaining an estimate of the motor angular velocity. It is worth commenting that unlike previous approaches [27] the disturbance is not directly estimated by the observer or the sliding mode differentiator. The remaining of this work is as follows. Section 2 describes the model of the DCMP system. Section 3 is devoted to defining the control problem and describes the proposed control algorithm as well as its stability analysis. Section 4 presents numerical simulations and experiments using a laboratory testbed. Section 5 ends the work with some concluding remarks. 2. DC-MOTOR PENDULUM DYNAMIC MODEL Consider the DCMP system composed by a direct current servomotor driving a pendulum, a servo-amplifier and a position sensor. The corresponding model of this system has the following form x¨ =
1 (− fd x˙ − fc ϕ [x] ˙ + η − gmL sin x + ku τ ) , J
(1)
where x, x˙ and x¨ are respectively the pendulum angular position, velocity and acceleration. The control input is denoted by τ and ku is a constant related to the servoamplifier gain and the servomotor torque constant. The parameters m and L correspond respectively to the pendulum mass and length. The term g is the gravity constant. The constants fd and fc are in that order the pendulum viscous and Coulomb friction coefficients, where ϕ [x] ˙ is a smooth approximation of the Coulomb friction signum function, which is modeled as follows ([28], Section 5): x˙ ϕ [x] ˙ =√ , 2 x˙ + ε
(2)
where ε > 0 is a small constant. The parameter J stands for the system total inertia composed by the arm and the rotor inertias. Finally, η is the unknown bounded disturbance that may account for model uncertainties and external disturbances. It is worth noting that in practice the disturbance η , the Coulomb friction coefficient fc and the gravitational torque gmL sin x may be unknown. Furthermore, the pendulum angular velocity may not be available from measurements. 3. PROPOSED CONTROL ALGORITHM Before establishing the control problem it is useful to express the DCMP system (1) in a suitable state-space representation x˙1 = x2 , x˙2 = − f0 x2 + w + u, y = x1 ,
(3)
where fd f0 = , J ku u = τ, J
(4)
Note that the term 1 w = (− fc ϕ [x2 ] + η − gmL sin x1 ), J
(5)
is considered as a disturbance. Problem statement: Consider the uncertain nonlinear system (3) with its corresponding disturbance (5) bounded as follows 1 |w| ≤ (η + gmL + f c ) ≤ w, J
(6)
where the positive constants η , gmL and f c correspond to upper bounds defined as |η | ≤ η , |gmL sin x| ≤ gmL and | fc ϕ [x2 ]| ≤ f c , respectively. Then, the control objective is
A Robust Controller for Trajectory Tracking of a DC Motor Pendulum System
to design a continuous controller where the angular pendular position tracks a given smooth reference trajectory xr . To this end the following assumptions are needed: Assumption 1: The output y = x1 is available from measurements. Assumption 2: The desired reference trajectory xr is bounded as well as its first and second order time derivatives x˙r , x¨r . Assumption 3: The first three derivatives of y, i.e., y, ˙ y,y ¨ (3) , are bounded in some large enough compact set. Assumption 4: The parameters fd , J are known. Comment 1: Assumption 1 and 2 are needed to solve the output feedback trajectory control problem, while Assumption 3 is a necessary condition for reconstructing the output time derivatives y, ˙ y,y ¨ (3) . Assumption 4 is related to the prior knowledge about the DCMP. Finally, function ϕ [x2 ] is an idealization of the Coulomb friction signum function. This assumption guarantees that the signal y(3) (t) is well defined for all t ≥ 0. Compared to previous works the main contributions here are highlighted in the following paragraphs: • Past works have dealt with the DCMP control problem; however, in order to simplify the stability analysis many of them neglect key issues like the Coulomb friction, unmodeled bounded uncertainties, and external bounded perturbations. Compensating the effect of these terms is one of the motivations of this work. • The proposed control law follows the active disturbance rejection controller (ADRC) philosophy [29, 30] in the sense that it is equipped with an estimator that evaluates the lumped effect of unmodelled terms and external disturbances. The estimator is constructed using a standard Luenberger observer coupled with a high-order sliding mode differentiator. However, unlike previous approaches the observer and the differentiator do not directly estimate the disturbances. Instead, the Luenberger observer error formulae provides an ideal disturbance estimator that is a function of the observer error and its first and second time derivatives. Since the later are not available from measurements, a third-order sliding mode differentiator [13, 31] provides finite time estimates of these variables. • The effectiveness of the obtained controller is evaluated through numerical simulations and using an experimental testbed. The proposed controller is also experimentally compared with a two-well established control methodologies.
3
3.1. Luenberger observer and ideal disturbance estimator Consider the following second order Luenberger observer for (3) [ ] [ ] [ ] 0 1 0 L1 z˙ = z+ u+ (x1 − z1 ), (7) 0 − f0 1 L2 with L1 > 0 and L2 > 0. Define the observation errors e1 = x1 − z1 and e2 = x2 − z2 . Then, it is easy to conclude from (3) and (7), that the observer error dynamics has the following expression [ ] [ ] 0 −L1 1 e˙ = e+ w = Ae + Bw. (8) −L2 − f0 1 From the above it follows that |e(t)| ≤ f , for all t ≥ t0 > 0, where t0 < ∞, and the bound f depends on the fact that the observer gains L = [L1 , L2 ]T produce a Hurwitz stable A matrix and on the disturbance upper bound w. ¯ It is worth noting that the estimates provided by the Luenberger observer do not converge to the true ones because of the effect of the disturbance w. However, it is possible to obtain an expression for the disturbance from the second equation of the observer error formulae (8) w = f0 e2 + L2 e1 + e˙2 .
(9)
The first equation in (8) allows obtaining e2 = e˙1 + L1 e1 ,
(10)
thus implying e˙2 = e¨1 + L1 e˙1 .
(11)
Substituting the above expressions into (8) produces w = ( f0 L1 + L2 )e1 + ( f0 + L1 )e˙1 + e¨1 .
(12)
The above ideal disturbance estimator depends on the time derivatives e˙1 and e¨1 of the observation error that are not available from measurements. This problem is circumvented by estimating these derivatives by means of a sliding mode differentiator. 3.2. Sliding mode differentiator and practical disturbance estimator The next sliding mode differentiator [13, 26, 32] provides estimates of the time derivatives of the observation error e1 v˙1 = −λ1 M 1/3 |v1 − e1 |2/3 sign (v1 − e1 )) + v2 , v˙2 = −λ2 M 1/2 |v2 − v˙1 |1/2 sign (v2 − v˙1 ) + v3 , v˙3 = −λ3 M sign (v3 − v˙2 ) ,
(13)
where the scalar variable vi denotes the estimate of the (i−1) time derivative e1 , for i = {1, 2, 3}. Assumptions 2 and 3 allows establishing the existence of a constant M > 0
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Carlos Aguilar-Ibañez, Julio Mendoza-Mendoza, Jorge Davila, Miguel S. Suarez-Castanon, and Ruben Garrido M.
(i−1) such that e1 (t) ≤ M; for i = {1, 2, 3}. Therefore, the sliding mode differentiator gains λi are chosen as follows: 1
λ3 = 1.1M, λ2 = 1.5M 2 , λ1 = 1.5M.
(14)
Then, according to [13] for a finite time t∗ > 0 the following holds lim |e1 − v1 | → 0,
t→t∗
lim |e˙1 − v2 | → 0,
t→t∗
(15)
i.e. the separation principle is trivially satisfied. The above implies that the stability of the DCMP (3) in closed loop with control law (19) may be studied without worrying about the dynamics of the Luenberger observer and the sliding mode differentiator. Define X=
[
]T
.
(20)
Substituting (19) into (3) and using definition (20) yields
lim |e¨1 − v3 | → 0.
[
t→t∗
Replacing the time derivative estimates provided by the sliding mode differentiator (13) into (12) produces the next practical disturbance estimator
x1 − xr , x2 − x˙r
X˙ =
0 −k p
1 −kd
]
[ X+
0 1
] ∆(t) = Ac X + B∆(t), (21)
where wˆ = ( f0 L1 + L2 )e1 + ( f0 + L1 )v1 + v2 .
(16)
An expression for x2 follows from (10) and the fact that e2 = x2 − z2 , i.e., x2 = z2 + e˙1 + L1 e1 .
(17)
From (17) it is possible to obtain an estimate of x2 by replacing e˙1 by its estimate v2 xˆ2 = z2 + v2 + L1 e1 .
(18)
The following propositions summarize the previous results. Proposition 1: Let {λ1 , λ2 , λ3 } be selected according to (14). Then, the differentiator (13) provides an exact estimate of the time derivatives of the observation error (i−1) e1 after a finite time transient, i.e., vi (t) = e1 (t), for i = {1, 2, 3} and t ≥ t∗ . Proposition 2: Let wˆ and xˆ2 be respectively defined according to (16) and (18) under conditions in Proposition 1. Then, the estimates of (xˆ2 , w) ˆ converge in a finite time transient to their real values (x2 , w). Notice that unlike preceding approaches the disturbance estimate wˆ is not directly produced by an observer or a sliding mode differentiator. 3.3. Robust control design Consider the following control law u = −wˆ + f0 xˆ2 − k p (x1 − xr ) − kd (xˆ2 − x˙r ) + x¨r . (19) The term xr is the desired angular position; wˆ and xˆ2 are obtained respectively from (16) and (18), and k p > 0, kd > 0. According to Proposition 1, after a finite time transient the variables (xˆ2 , w) ˆ converge to their real values (x2 , w). On the other hand, note that the structure of the observer does not depends on the proposed control law,
∆(t) = w − wˆ − kd (x2 − xˆ2 ).
(22)
Assumptions 1 to 4 and the bound (6) guarantee that ∆(t) is uniformly bounded. Therefore. the solutions of (21) are well-defined. Furthermore, according to Proposition 2 the estimates (xˆ2 , w) ˆ converge in finite time to (x2 , w). The above implies that ∆(t) converge to zero in finite time. The above argument and the fact that matrix Ac is Hurwitz allow concluding that X converges exponentially to zero. 4. NUMERICAL SIMULATIONS AND EXPERIMENTAL RESULTS This section shows the results obtained from numerical simulations and experiments performed to asses the effectiveness of the proposed controller. The parameters of the DCMP laboratory prototype are the following f0 = 0.1, gmL = 14.03, J
f1 = 1.3, ku = 5.4, J
η = 0.84, J ε = 10−3 .
The reference is defined as xr = 0.25 sin(t/6). The parameters of the observer (8) and the high-order differentiator (13) are set as follows L1 = 49/10, L2 = 551/100, M = 1000, λ1 = 2, λ2 = 1.5, λ3 = 1.1, where M is selected, such that M >> w. The PD-control parameters are set to k p = 4000 and kd = 130. The initial conditions in the simulation are chosen as x = 0. The initial condition of the observer and the high-order differentiator are fixed at the origin. The simulations are performed using the Euler method with a sampling integration interval of 1 × 10−3 s.
A Robust Controller for Trajectory Tracking of a DC Motor Pendulum System Velocity error
Estimated perturbation 0.1
5
e2
w b 0.05 [rad/s]
Newtons
5
0
0 -0.05
-5
0
5
10
15
-0.1
0
Control signal
5
5
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Position error
-3
x 10
e1
[rad]
Volts
10
u
0
5 0
-5
0
5
10
15
Time[sec]
-5
0
5
10
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Time[sec]
Fig. 1. Simulation results. Disturbance estimate and control signal.
Fig. 3. Simulation results. Velocity and position errors.
Velocity
[rad/s]
0.2 0.1
x2 x˙r
0
x b2
-0.1 -0.2 0
5
10
15
Position
[rad]
0.2 0.1
x1 xr
0
x b1
Fig. 4. Laboratory testbed.
-0.1 -0.2 0
5
10
15
Time[sec]
Fig. 2. Simulation results. Closed-loop response. 4.1. Numerical simulations Fig. 1 depicts the disturbance estimate and the control signal. Fig. 2 shows the closed- loop response, and Fig, 3 portraits the velocity and position errors. These results show that the proposed control law produces small position tracking errors when applied to the DCMP system 4.2. Experimental results The performance of the proposed control law is also evaluated through a laboratory testbed (see Fig. 4). It is composed of a Moog servomotor (C34-L80-W40) driven
by a Copley Controls power servoamplifier (423) configured in current mode. A BEI optical encoder (L15) with 2500 ppr allows measuring the servomotor position. The algorithms are coded using the MatLab/Simulink software platform under the QUARC real-time suite from Quanser Consulting. A Quanser Consulting data acquisition board (Q8) performs data acquisition and is allocated in a PCI slot inside a personal computer running the software. The board electronics increases the optical encoder resolution up to 10,000 ppr. The control signal produced by the data acquisition board passes through a galvanic isolation box. The first experiment is carried out under the same setup employed in the numerical simulations. The estimated disturbance and the control signal are depicted in Fig. 5. Fig. 6 and 7 show respectively the closed loop response and the velocity and position errors. These results are similar to those obtained in the simulation results; the proposed control law is able to track the desired reference
Carlos Aguilar-Ibañez, Julio Mendoza-Mendoza, Jorge Davila, Miguel S. Suarez-Castanon, and Ruben Garrido M.
6
Estimated perturbation
Velocity error 0.1
e2
w b 0.05 [rad/s]
Newtons
5
0
0 -0.05
-5 -0.1
0
5
10
15
0
5
10
x 10
e1
0
[rad]
Volts
5
u
-5
0
5
10
15
Time[secs]
Fig. 5. First experiment. Disturbance estimate and control signal. Velocity 0.2 [rad/s]
5 0
-5
0.1 0
x2 x˙r
-0.1
x b2
-0.2 0
5
10
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Position 0.2 [rad]
15
Position error
-3
Control signal
10
0.1
x1 xr
0
x b1
-0.1 -0.2 0
5
10
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Time[sec]
Fig. 6. First experiment. Closed-loop response. with a small tracking error. A comparative study is also performed between the proposed approach and two well-known methodologies described in the literature, namely the Algebraic Method (AM) [33] based on nested integrations, and the HighGain Observer (HGO) based method [34]. The desired position reference in the three cases is the sinusoidal signal xr = 0.1 sin(2π t/10) For comparison purposes, the gains used in the first experiment for the proposed control law are maintained here. In all the cases the initial conditions are set to zero. Fig. 8, 9, and 10 depict the results obtained with the three control laws. Concerning the estimation of
0
5
10
15
Time[sec]
Fig. 7. First experiment. Velocity and position errors. the perturbation in Fig. 8 the proposed approach and the AM exhibit a smooth profile whereas the estimation obtained through the HGO shows chattering. In order to understand this issue, note that the HGO essentially behaves like a linear differentiator; this fact combined with the discrete nature of the position measurements provided by the optical encoder used in the experiments produce large levels of chattering. Moreover, increasing the HGO gains also increases its bandwidth; therefore, the estimated disturbance has more high-frequency components. On the other hand, note that the control signal in Fig. 8 for the HGO inherits the chatter of the estimated perturbation since it is used to compute the control law. Fig. 9 depicts the velocity and position signals for all the three controllers. In all the cases the position signal is plotted using the position measurement provided by the optical encoder without further processing. In the case of the HGO; the estimated velocity signal also contains some chatter ; the velocity signals provided by the proposed approach and the AM are chatter-free. Interestingly enough, the velocity and position tracking errors in Fig. 10 corresponding to the proposed method and the AM are relatively free of high frequency components. This is not the case of the HGO which portrays high levels of chattering in the velocity signal. Note also that the velocity error for the proposed method is smaller than the corresponding error for the AM. In the case of the position error, the proposed approach slightly outperforms the AM, in particular, for negative values of the tracking error. The results from the two sets of experiments allow concluding the following. The proposed approach is theoretical sound and is able to produce an acceptable tracking performance by compensating the unknown disturbances in the DCMP. Furthermore, it compares favorably
A Robust Controller for Trajectory Tracking of a DC Motor Pendulum System
Velocity error
The estimated perturbations ω b∗ . 0.01
ω bO ω bA ω bH
[rad/s]
Newtons
0.1 0
eO 2 eA2 eH2
0
-0.01
-0.1 0
10 20 The control signals u∗ .
0
30
-3
x 10
uO uA uH
0.4 0.2 0 -0.2 -0.4 0
10 20 Position error
30 eO1 eA1 e H1
2 [rad]
Volts
7
0 -2
10 20 Time [sec]
30
Fig. 8. Comparative experimental study. Disturbance estimate and control signal: O = Proposed approach, A = algebraic approach, H = high-gain approach.
0
10 20 Time [sec]
30
Fig. 10. Comparative experimental study. Velocity and position errors: O = Proposed approach, A = algebraic approach, H = high-gain approach.
Velocity x˙ r xA2 xH2 xO2
[rad/s]
0.05 0 -0.05 0
10
Position
20
[rad]
0.1
30 xr xA1 xH1 xO1
0
-0.1
0
10 20 Time [sec]
30
Fig. 9. Comparative experimental study. Closed-loop response: O = Proposed approach, A = algebraic approach, H = high-gain approach. with existing robust control laws in terms of performance. Moreover, it does not produce control signals with highfrequency content thus avoiding stress and overheating in the DC motor. 5.
CONCLUSIONS
This work proposes a solution for the output-feedback trajectory tracking problem of a DC motor pendulum sys-
tem. The proposed algorithm is composed of two parts, a standard PD controller plus a novel on-line disturbance estimator. The later is produced by coupling a standard Luenberger observer and a third order sliding mode differentiator. The Luenberger observer only produces estimates of the motor angular position and velocity but its formulae provides an expression for constructing an ideal disturbance estimator in terms of the Luenberger observer estimation error and its first and second order time derivatives. Subsequently, a third-order sliding mode differentiator estimates these time derivatives in finite time, which replace the non available values of the first and second time derivatives in the ideal disturbance estimator. It is worth commenting that unlike previous approaches the disturbance is not directly estimated by the observer or the sliding mode differentiator. Numerical simulations and experiments using a laboratory testbed allows assessing the performance of the proposed approach. The experiments also show that the proposed approach is chatter-free and it outperforms well-established control algorithms regarding tracking performance and control signal chattering. REFERENCES [1] J. J. Slotine and W. Li, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, NJ, New Jersey, 1991. [2] M. W. Spong and M. Vidyasagar, Robot Dynamics and Control, John Wiley & Sons, 2008. [3] V. M. Hernandez and H. Sira-Ramirez, “Sliding mode generalized pi tracking control of a dc-motor pendulum sys-
Carlos Aguilar-Ibañez, Julio Mendoza-Mendoza, Jorge Davila, Miguel S. Suarez-Castanon, and Ruben Garrido M.
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Carlos Aguilar-Ibañez was born in Tuxpan, Veracruz, Mexico. He graduated in Physics at the Higher School of Physics and Mathematics of the National Polytechnic Institute (IPN), Mexico City 1990. From the Research Center and Advanced Studies of the IPN (Cinvestav- IPN) he received the M.S. degree in Electrical Engineering in 1994, and a Ph.D. in Automatic Control in 1999. Ever since he has been a researcher at the Center of Computing Research of the IPN (CIC-IPN). As of 2000 he belongs to the National System of Researchers (SNI) of Mexico. His research focuses in nonlinear systems, system identification, observers, automatic control, and chaos theory. Julio Mendoza-Mendoza concluded his computing doctoral degree at CIC IPN in 2016, where he specialised in underactuated robotics, UAS, and intelligent and nonlinear control. Also he achieved his advanced technologies master degree and mechatronics-engineering bachelor at UPIITA IPN, in 2011 and 2008 respectively. Since that time he developed his interest areas and research in robot manipulators, mobile robots, humanoids, underactuated systems, UAVs, and various control techniques. Nowadays he works in 2 patents related with his research field and develops the flying serial-robot manipulator theory.
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Jorge Davila was born in Mexico City, Mexico, in 1977. He received the BSc and MSc degrees from the National Autonomous University of Mexico (UNAM) in 2000 and 2004, respectively. In 2008, he received the Ph.D. degree and was awarded with the Alfonso Caso medal to the best Graduate student from the UNAM. He is Professor in the School of Mechanical and Electrical Engineering - Ticoman of the National Polytechnic Institute of Mexico. His professional activities are concentrated in the fields of observation of systems with unknown inputs, observers for switching systems, robust control design, consensus of multi-agents, high-order sliding-mode control and their application to the aeronautic and aerospace technologies. Miguel S. Suarez-Castanon was born in Mexico City, Mexico, in 1967. He received a B.S. degree in Cybernetics and Computer Science from the School of Engineering of the Lasalle University in 1989. From the Research Institute of Applied Mathematics and Systems he received the M.S. degree in Computer Sciences in 2001. In 2005 he received a Ph.D. in Computer Sciences from the CIC-IPN. Since 2007 he is a member of the SNI of Mexico. Ruben Garrido M. received the B.S. degree in Mechanical and Electrical Engineering from the School of Mechanical and Electrical Engineering in 1983. From the CINVESTAV he recieved the M.S. degree in automatic Control in 1987. In 1993 he received a Ph.D. from the Université de Technologie de Compiègne, Francia, in 1993. Currently, Dr. Garrido is a member of the National Research System of Mexico, level II.