Sensorless Tracking Control for a “Full-Bridge Buck Inverter–DC Motor” System: Passivity and Flatness-Based Design

A sensorless control based on the exact tracking error dynamics passive output feedback (ETEDPOF) methodology is proposed for executing the angular velocity trajectory tracking task on the “full-bridge Buck inverter–DC motor” system. When such a methodology is applied to the system, the tracking task is achieved by considering only the current sensing and by using some reference trajectories for the system. The reference trajectories are obtained by exploiting the flatness property associated with the mathematical model of the “full-bridge Buck inverter–DC motor” system. Experimental tests are developed for different desired angular velocity trajectories. With the aim of obtaining the experimental results in closed-loop, a “full-bridge Buck inverter–DC motor” prototype, Matlab-Simulink, and a DS1104 board from dSPACE are employed. The experimental results show the effectiveness of the proposed control.


I. INTRODUCTION
Permanent magnet DC motors are used in a wide range of applications [1]. Such applications, for mentioning a few, are: domestic life (kitchen equipment, washing machines, toys, cameras, etc.), public life (autobank machines, automatic vending machines, ticketing machines, etc.), and industry (industrial drives, machine tools, robots, etc.). From those applications, and with the aim of achieving a better performance in DC motors, in the last years these systems have been intensively studied by the automatic control community.
The associate editor coordinating the review of this manuscript and approving it for publication was Francisco Perez-Pinal .
In that direction, in the last decade, different topologies of DC/DC power electronics converters have been used as drivers for DC motors. One of those topologies is the DC/DC Buck power electronic converter. This topology has been extensively used as a driver for DC motors; the most relevant contributions regarding the latter has been reported in [2]- [34]. Such contributions can be classified in ''SISO DC/DC Buck converter-DC motor'' [2]- [30] and ''MIMO DC/DC Buck converter-inverter-DC motor'' [31]- [34] for unidirectional and bidirectional movement, respectively. The unidirectional movement is related to the clockwise rotation of the motor shaft, while the bidirectional one implies both, the clockwise and the counter-clockwise rotation VOLUME 9, 2021 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ of the motor shaft. Meanwhile, important contributions related to other DC/DC converters used as drivers for DC motors have been reported in [35] and [36] (for multilevel and parallel Buck topologies, respectively), [37]- [44] (for Boost topology), [45]- [50] (for Buck-Boost topology), [51] (for Sepic topology), [52] (for Luo topology), and [53] (for Cuk topology). In the following, the ''SISO DC/DC Buck converter-DC motor'' and the ''MIMO DC/DC Buck converter-inverter-DC motor'' contributions reported in literature are presented.

A. ''SISO DC/DC BUCK CONVERTER-DC MOTOR'' SYSTEM
For the first time, in 2000 Lyshevski developed the mathematical model and a nonlinear PI control for the angular velocity regulation task for the DC/DC Buck converter-DC motor system [2]. Later, in 2010 Ahmad et al. in [3] analyzed a comparative assessment of the PI, fuzzy PI, and LQR control strategies for the velocity of the motor. In 2012, Bingöl and Paçaci in [4] used a virtual laboratory based on neural networks for controlling the angular velocity of the system. A year later, in 2013 a robust control based on active disturbance rejection and differential flatness for two configurations of the Buck converter-DC motor system was addressed by Sira-Ramírez and Oliver-Salazar in [5]. Whereas Hoyos et al. in [6]- [9] studied the zero average dynamics and fixed-point induced control for the motor speed of a Buck converter and DC motor coupled system. By exploiting the flatness of the system, for the angular velocity tracking task, Silva-Ortigoza et al. explained two hierarchical robust controls; in 2013 a sensorless control approach based on integral reconstructors [10] and in 2014 the experimental verification of a hierarchical control based on flatness [11]. An adaptive robust control based on dynamic surface control and sliding mode control (SMC) was exhibited by Wei et al. [12]. In 2015, Silva-Ortigoza et al. exposed a hierarchical controller composed of two independent controls; one via differential flatness (for the DC motor) and the other one via a cascade scheme through SMC and PI control (for the Buck converter), which were interconnected in order to work as a whole [13]. A control based on the exact tracking error dynamics passive output feedback (ETEDPOF) methodology was presented by Srinivasan and Srinivasan in [14]. Additionally, Hernández-Guzmán et al. in [15] introduced a SMC with PI controls to regulate the converter capacitor voltage, the motor armature current, and the motor velocity. Later, in 2016 Khubalkar et al. in [16] utilized a fractional order PID control whose tuning was executed with the dynamic particle swarm optimization (dPSO) technique. Rigatos et al. in [17] reported a control based on the differential flatness theory and used a derivative-free nonlinear Kalman filter for compensating disturbances. Another solution was implemented in 2017 by Nizami et al. in [18], where a neuro-adaptive backstepping control method, using a single-layer Chebyshev polynomial based neural network, was exposed. In 2018, Rigatos et al. in [19] provided a flatness-based control in successive loops for the tracking task. Meanwhile, Khubalkar et al. in [20] applied a fractional order PID control with an improved dSPO technique. Recently in 2019, Yang et al. in [21] designed a discrete-time robust predictive speed regulation algorithm by using a discrete-time reduced-order generalized proportional-integral (GPI) observer. Likewise, Hanif et al. in [22] proposed a piecewise affine PI control based on the safe experimentation dynamics algorithm. While Kazemi and Montazeri in [23] showed a fault detection system using a combination of bond graph method and a switching observer. More recently, in 2020 an H-infinity adaptive neurofuzzy control method was exploited by Rigatos et al. in [24].  [29] and [30].
In recent years, Silva-Ortigoza et al. described the development and experimental validation of a mathematical model associated with the DC/DC Buck converter-inverter-DC motor system [31]. Likewise, the design of a control based on the ETEDPOF methodology for the bidirectional angular velocity tracking of the system was considered in [32]. Later, Hernández-Márquez et al. in [33] employed two robust controls based on flatness for solving the angular velocity tracking task of the DC/DC Buck-inverter-DC motor system. Lastly, a control based on the adaptive sliding mode technique with neural network estimation for the bidirectional DC motor system consisting of a DC/DC Buck converter, a full bridge inverter, and a DC motor was investigated by Chi et al. in [34].

C. DISCUSSION OF RELATED WORK, MOTIVATION, AND CONTRIBUTION
During the last years, different control strategies for driving a DC motor fed via DC/DC Buck power converters have been proposed. Related to such contributions the following was found: • For the ''SISO DC/DC Buck converter-DC motor'' system [2]- [30], the angular velocity control is unidirectional. This latter, because the output voltage of the Buck converter is unipolar. Some applications that have been profited, although some kind limited, from using this system can be found in mechatronic systems [54], robotic arms [55], and wheeled mobile robots [56]- [58].
• By using the ''MIMO DC/DC Buck converter-inverter-DC motor'' system [31]- [34] the unidirectional rotation restriction associated with the DC motor shaft is overcome, since an inverter circuit is introduced between the Buck converter and the DC motor. An application recently developed that uses such a system can be found in [34], where the problem of renewable energy power source based on fuel cells was presented by Chi et al. in 2020. Motivated by the benefits of the new ''full-bridge Buck inverter-DC motor'' system [59]: • On the one hand, this kind of system is a linear SISO system, and at the same time, that drives bipolar voltage to the DC motor. In consequence, and unlike the contributions presented so far for the ''SISO DC/DC Buck converter-DC motor'' system in [2]- [30], this is the manner to overcome the unidirectional rotation limitation associated with the DC motor shaft.
From the aforementioned discussion of the state-of-the-art and motivation of the work, the main contributions of this study can be summarized as follows: 1) To design a sensorless passivity-based control by using the ETEDPOF methodology for solving the angular velocity trajectory tracking task for the new ''full-bridge Buck inverter-DC motor'' system.
2) To obtain the reference trajectories for the ''full-bridge Buck inverter-DC motor'' system. The implementation of the proposed control, via the ETEDPOF methodology, requires some reference trajectories. Such reference trajectories are generated through the differential flatness concept, since the mathematical model of the system enjoys the flatness property. 3) To experimentally verify the performance of the control, designed via the ETEDPOF methodology and the flatness concept, for different types of desired angular velocity trajectories. For this aim, a built prototype of the ''full-bridge Buck inverter-DC motor'' system, Matlab-Simulink, and a DS1104 board from dSPACE are used. 4) To accomplish a revision of the experimental closedloop tracking errors, that are acquired after implementing the proposed ETEDPOF control on the system, and the experimental open-loop tracking errors (obtained in [59]). Such a revision is carried out with the intention of showing how the system in closed-loop is benefited from the ETEDPOF strategy, rather than for comparative purposes. The remaining part of this paper is organized as follows: Section II provides a brief introduction to the ''full-bridge Buck inverter-DC motor'' system. Section III presents the design of the passive control via ETEDPOF for solving the trajectory tracking task on the system. The reference trajectories of the system are obtained by exploiting the flatness property and are presented in Section IV. Then, Section V discusses the experimental implementation of the proposed control and also presents the experimental validation of its performance. Finally, the paper is concluded in Section VI.

II. A NEW ''FULL-BRIDGE BUCK INVERTER-DC MOTOR'' SYSTEM
The electric circuit of the full-bridge Buck inverter-DC motor system is shown in Fig. 1. Such a circuit can be divided in two systems: The full bridge Buck inverter modulates and supplies bipolar voltage υ to the DC motor via the input u. It comprises a power supply E and an array of four transistors, denoted by Q 1 , Q 1 , Q 2 , and Q 2 that operates according to the clock cycles depicted in Fig. 1. In addition, the first subsystem is composed of an LC filter; where the current i flows through the inductor L and the voltage υ appears over the terminals of capacitor C that is connected in parallel to the load R. The DC motor corresponds to the system actuator, and is made up of the following elements: R a , L a , and i a , corresponding to the armature resistance, armature inductance, and armature current, respectively. Whereas ω is the angular velocity of the motor shaft. Other important parameters of the motor are J , b, k e , and k m , corresponding to the moment of inertia of the rotor and motor load, the viscous friction coefficient of the motor, the counterelectromotive force constant, and the motor torque constant, respectively.
According to [59], the average mathematical model of the full-bridge Buck inverter-DC motor system shown in Fig. 1  FIGURE 1. Full-bridge Buck inverter-DC motor system and clock cycles u associated with Q 1 , Q 1 , Q 2 , and Q 2 , for the positive cycle (υ > 0) and the negative cycle (υ < 0). is given by, with u av ∈ [−1, 1], being u av the duty cycle or average input signal of the system. Whereas the rest of the variables and constants related to the mathematical model (1) were previously defined.

III. DESIGN OF THE PASSIVE CONTROL VIA ETEDPOF
This section presents the design of a passivity-based control for solving the trajectory tracking task on the full-bridge Buck inverter-DC motor system. Such a control is based on the ETEDPOF strategy, whose underlying theoretical considerations were presented in [60]. Whereas some applications of this strategy in connection with traditional power electronics were carried out in [61].

A. ETEDPOF CONTROLLER DESIGN METHODOLOGY
By invoking the seminal work of Sira-Ramírez [60], the derivation of the ETEDPOF methodology will be given here in a general setting for a better understanding. Thus, with the aim of applying such a strategy, the following convenient representation for a system is used, being J (u av ), for all u av , of the form, where A n×n is a symmetric, positive definite, constant, matrix, J (u av ) n×n is a skew symmetric matrix (the matrices J 0 and J i , i = 1, . . . , m, are constant and skew symmetric matrices), R n×n is a symmetric, positive semi-definite constant matrix, x n×1 is the state vector of the system, B n×m is a constant matrix, u m×1 av is the average control input vector (each component u i av takes values on the closed interval [−1, 1] of the real line), and η(t) n×1 is a smooth vector function of time t or, sometimes, a vector of constant entries.
From (2), the desired dynamics of the system is represented as follows: with where x * and u * av are the desired state vector and the desired average control input vector, respectively.
Defining the state error and average control error as: By subtracting (4) from (2), and after using (6), it is obtained: Now, by replacing (3) and (5) into (7) the following is obtained: Then, the exact open-loop tracking error dynamics is given by, According to the ETEDPOF methodology [61], the control e u av that achieves e → 0 is determined by, being = diag[γ 1 , γ 2 , . . . , γ m ] > 0. Thus, the closed-loop tracking error dynamics, that is obtained after replacing (10) into (9), is: In order to assess the stability of the closed-loop dynamics (11), the following positive definite Lyapunov function is proposed, The time derivative of such a positive definite function, along the controlled trajectories of the tracking error dynamics, yieldsV which guaranties that e → 0 as long as,

B. VELOCITY-SENSORLESS TRACKING CONTROL FOR THE FULL-BRIDGE BUCK INVERTER-DC MOTOR SYSTEM
After applying the previous results to the full-bridge Buck inverter-DC motor system, the following is obtained: • First, it is verified that the average model (1) can be represented by means of the energy managing structure (2). For this, by simple inspection it is found that, where J (u av ) complies with being a skew symmetric matrix, due to k m = k e in a DC motor [80]. Whereas x * , as previously mentioned, corresponds to the desired state vector of the system. It is worth mentioning that, for the system under study, x * is composed of i * , υ * , i * a , and ω * representing the desired reference trajectories associated with i, υ, i a , and ω, respectively.
• Second, it must be verified that condition (13) is satisfied. For this, after finding R+B * B * T and by invoking the Sylvester criterion [81] it is shown that, with γ 1 > 0. In this manner, it is concluded that the closed-loop system is asymptotically stable.
• Lastly, after verifying the previous two items and by considering (6) and (10), it is concluded that the proposed ETEDPOF control that solves the trajectory tracking task for the full-bridge Buck inverter-DC motor system is determined by, The reference trajectories i * and u * av will be obtained from exploiting the differential flatness property enjoyed by the system (1).

C. ADVANTAGES AND LIMITATIONS OF THE ETEDPOF CONTROLLER
Having designed the passivity-based control by using the ETEDPOF strategy for solving the trajectory tracking task in the full-bridge Buck inverter-DC motor system, now it is convenient to highlight the advantages and limitations of such a strategy. In this regard, the following are considered to be advantages: • The control (14) requires only the measurement of the converter current, i, and the knowledge of the reference trajectories i * and u * av . Therefore no mechanical sensors, such as tachometers or encoders, are needed for the experimental implementation of the ETEDPOF control.
In fact, when it is applied to electromechanical systems, in general, such a passive control demands only the use of electric variables (see the applications previously mentioned in section I-C).
• The usage of the ETEDPOF methodology invariably results in linear, time-varying, state feedback controllers for trajectory tracking tasks, that achieve asymptotic stability [60], [61].
• The potential applications of this strategy is wide, since most of power electronics devices (such as DC/DC converters, DC/AC converters, and AC/DC converters) and interesting combinations of these devices with DC and AC motors are accurately described by the special ''energy managing'' structure (2). Moreover, nowadays, the ETEDPOF control law has applications in airships [64], mobile robotics [65], renewable energy systems [66], [68], and magnetorheological automotive suspensions [71]. Whereas the following is considered to be a limitation of the ETEDPOF control: • When electrical abrupt changes, or mechanical load perturbations, are introduced in some parameters of the system the results show that the performance of the system in closed-loop is not robust. However, this could be overcome when the ETEDPOF control is modified as in [66] or when it is combined with algebraic estimation methods as in [14], [37], [67]- [69].

IV. GENERATION OF REFERENCE TRAJECTORIES VIA DIFFERENTIAL FLATNESS
The differential flatness property associated with (1) is used for generating the reference trajectories i * and u * av that are required in (14). For this aim, firstly, it is shown that the full-bridge Buck inverter-DC motor system is controllable and, hence, differentially flat. Secondly, after differentially parameterizing all variables of the system in terms of the flat output, the required reference trajectories will be obtained. This, will allow the experimental implementation of the designed ETEDPOF control (14) for solving the trajectory tracking task on the full-bridge Buck inverter-DC motor system. The average mathematical model of the full-bridge Buck inverter-DC motor system, given by (1), can be represented as:ẋ where Based on [82], the controllability matrix C associated with (15) turns out to be, In this way, it is found that Thus, system (15) is controllable and, therefore, flat. From the latter and since C −1 exists, then it is possible to find the flat output of the system, y, from the following mathematical relation: Consequently, and without loss of generality, the flat output is considered to be the angular velocity of the full-bridge Buck inverter-DC motor system, this is, y = ω. In this way, variables i, υ, i a , and the average control input u av , all associated with (15), are determined by the following differential parameterization: with α = bLL a +JR a L+JRL+bRR a LC+k e k m RLC+JRL a Ek m R .

V. HARDWARE AND EXPERIMENTAL RESULTS IN CLOSED-LOOP
In order to validate the performance of the proposed ETEDPOF control, experimental tests are carried out on a built platform of the full-bridge Buck inverter-DC motor system. For this aim, a block diagram of the connections related to the system and the ETEDPOF control is firstly presented. Then, after executing the experimental tests associated with four different desired angular velocity profiles ω * and by considering the system nominal parameters, given by (23) and (24), the obtained results are reported and commented. Later, an experimental assessment of the ETEDPOF control is realized when abrupt changes in the converter load R are introduced into the system. Lastly, the ETEDPOF control is experimentally compared with a PID control with the aim of verifying the dynamic response of the system in closed-loop.  The blocks composing the diagram of Fig. 2 are described below:

A. DIAGRAM OF THE SYSTEM IN CLOSED-LOOP
• Desired trajectory and reference trajectories. In this block, by using Matlab-Simulink, the desired angular velocity ω * , and the reference trajectories i * , υ * , i * a , and u * av are programmed. These latter, are found through ω * and the system differential parameterization given by (19)- (22).
• ETEDPOF control. The sensorless passivity-based control for solving the trajectory tracking task in the system, given by (14), is implemented in this block via Matlab-Simulink. The control gain parameter γ 1 was chosen to be: While the measurement of the converter current, i, demanded by the proposed control is carried out via a Tektronix A622 current probe. Lastly, the reference trajectories i * and u * av required in (14) are obtained from the block previously described.
The measurements of i a , υ, and θ were carried out with a Tektronix A622 current probe, a Tektronix P5200A voltage probe, and an Omron E6B2-CWZ6C incremental rotary encoder, respectively.
Lastly, a picture of the used hardware for starting up the full-bridge Buck inverter-DC motor system in closed-loop is shown in Fig. 3.

B. EXPERIMENTAL RESULTS OF THE SYSTEM IN CLOSED-LOOP
With the aim of verifying the performance of the ETEDPOF control on the built system, in the following, the experimental results for four different desired angular velocity profiles are presented. Each plot shows first the results of the DC motor variables, i.e., ω and i a and then the results associated with the full-bridge Buck inverter variables, i.e., i, υ, and u av . In obtaining all the results of this subsection the nominal parameters of the full-bridge Buck inverter-DC motor system, previously defined in (23) and (24), have been used.

EXPERIMENTAL TEST 1
In this experiment, the proposed desired trajectory is of the Bézier type and is defined as, where ϕ t, t i , t f is given by With this proposal it is achieved that ω * smoothly interpolates between ω i = −10 rad s and ω f = 10 rad s over the times VOLUME 9, 2021 t i = 4 s and t f = 6 s, respectively. The corresponding experimental results are depicted in Fig. 4.

EXPERIMENTAL TEST 2
Now, with the intention of checking the performance of the control when periodic signals are considered as desired trajectories, in this second test the profile for ω * is defined as a sinusoidal function given by, The experimental results in closed-loop of this test are shown in Fig. 5.
Thus, the corresponding experimental results are presented in Fig. 6.

EXPERIMENTAL TEST 4
Lastly, the fourth experiment uses the desired trajectory profile defined in (28). While the obtained experimental results are depicted in Fig. 7.

C. COMMENTS ON THE EXPERIMENTAL RESULTS
Based on the experimental results (see Figs. 4-7) it is concluded that the sensorless passivity-based control, for the angular velocity trajectory tracking task, in the full-bridge Buck inverter-DC motor system achieves the objective, i.e., ω → ω * . Until now, the only experimental work related to the full-bridge Buck inverter-DC motor system, is the one where its mathematical model was developed and experimentally validated [59]. In the experimental tests reported in [59] the four desired trajectories ω * used there were the same ones used in this paper, i.e., (25)- (28). Thus, in order to highlight the performance of the proposed ETEDPOF control (14), in the following, the similarity between ω versus ω * and υ versus υ * is quantified from the experimental results obtained in closed-loop (developed in this paper) and those in open-loop (reported in [59]). This latter is realized with the intention of showing how the system is benefited from the designed ETEDPOF control, rather than for comparative purposes. Since, as it is well known, the closed-loop and the open-loop results cannot be compared; this is because in open-loop there is no feedback and, hence, no regulation is performed.
On the one hand, the closed-loop tracking errors associated with the experimental results of the angular velocity (e ω E cl j ) and the voltage (e υ E cl j ) are defined as: where subscript E cl j, with j ∈ {1, 2, 3, 4}, indicates the experimental result in closed-loop from which the tracking error is obtained. On the other hand, the open-loop tracking errors associated with the experimental results for the angular velocity (e ω E ol k ) and for the voltage (e υ E ol k ) are defined as: where subscript E ol k, with k ∈ {1, 2, 3, 4}, represents the experimental result in open-loop from which the tracking error is obtained [59].
Having defined the experimental closed-loop tracking errors in (29) and the experimental open-loop tracking errors in (30); the plots of the tracking errors for ω and υ, associated with the desired angular velocity profiles ω * (25)- (28), are presented in Fig. 8. As can be observed in Figs . Such a behavior is because the desired trajectories (25) and (28) have less sign changes compared to the desired trajectories (26) and (27), respectively.
From the aforementioned, it is concluded that the designed ETEDPOF control achieves the objective of trajectory tracking, i.e., ω → ω * , and also achieves that υ → υ * . Thus, validation of the performance associated with the designed sensorless passivity-based control, given by (14), for the angular velocity trajectory tracking task in the system, is satisfactory. However, in accordance with the limitation of the ETEDPOF strategy (previously mentioned in section III-C), such a control is not robust when some perturbations are considered into the system; this latter is verified in the following.

EXPERIMENTAL TEST 5
This experimental test is performed when changes in R are introduced into the built experimental platform of the system. Thus, with the aim of evaluating the performance of the system when abrupt variations are executed, the following change in R is proposed: The trajectory ω * considered in Experimental test 5 is defined in (25). The associated experimental results are presented in Fig. 9. Since the control based on ETEDPOF is not robust, a tracking error for υ and ω will remain from t ≥ 7.5 s to the end of the experiment. Thus, motivated by the experimental results when abrupt variations are executed into the system, the design of robust controls are considered as future work. Lastly, a visual comparison between the ETEDPOF control (14) and a PID control is carried out with the aim of verifying the dynamic response of the full-bridge Buck inverter-DC motor system in closed-loop. The experimental results are presented below.

EXPERIMENTAL TEST 6
In this experiment both, the ETEDPOF control and a PID control are experimentally implemented on the built platform of the system when the nominal parameters, defined in (23) and (24), are taken into account. The desired angular velocity trajectory to be followed by the motor shaft is proposed to be, 5 3 s ≤ t < 6 s, 10 6 s ≤ t < 9 s, 6 9 s ≤ t < 12 s, 0 12 s ≤ t < 15 s, The corresponding experimental results are depicted in Fig. 10. As can be observed in such a figure, the dynamic response of the system is apparently better when the PID control is used. However, it is also verified that the currents i a and i and the voltage υ, all of them associated with the ETEDPOF control, are smaller in magnitude than their corresponding ones related to the PID control, i.e., i a PID , i PID , and υ PID , respectively. This latter implies that the PID control could compromise the integrity of the built platform, since the currents are bigger enough to damage the circuitry. Finally, note that the controls u av and u av PID never get saturated and the peaks of both signals are practically instantaneous.  (14) versus a PID control, both experimentally implemented on the full-bridge Buck inverter-DC motor system when ω * is given by (32) and considering nominal parameters. Here, signals ω, i a , i , υ, and u av are related to the ETEDPOF control. Whereas, signals ω PID , i a PID , i PID , υ PID , and u av PID correspond to the PID control.

VI. CONCLUSION
A velocity-sensorless tracking control approach based on the exact tracking error dynamics passive output feedback (ETEDPOF) methodology for a new full-bridge Buck inverter-DC motor system was presented in this paper. With such a methodology, it is only required the measurement of current i and the knowledge of the reference trajectories i * and u * av for solving the tracking related to ω. Therefore no mechanical sensors, such as tachometers and encoders, are needed for the implementation of the control. The nominal reference trajectories demanded by the ETEDPOF control are obtained by exploiting the differential flatness property of the mathematical model associated with the full-bridge Buck inverter-DC motor system. Regarding the implementation of the ETEDPOF control, its realization was executed by using Matlab-Simulink and the DS1104 board on a prototype of the system. According to the experimental results it is concluded that, in general, the trajectory tracking task is solved satisfactorily when system nominal parameters are considered. However, when abrupt variations in the system nominal parameters are introduced, ω and υ no longer track to ω * and υ * , respectively.
Finally, motivated by the obtained experimental results, in particular those in Fig. 9, the design of a robust control that takes into account abrupt perturbations in parameters of the system is considered as future work. Also, applications in AC motors, mechatronic systems, robotic arms, wheeled mobile robots, underactuated mechanical systems, and renewable energy are considered as future work. important prize granted by the Instituto Politécnico Nacional of Mexico to its students. Also, three of these students in 2015, 2017, and 2020, have also been honored with The Best Master Thesis Award by the Instituto Politécnico Nacional of Mexico. He has been the leader in more than ten research projects and collaborated in nine additional research projects. His research interests include mechatronic control systems, mobile robotics, control in power electronics, development of educational technology, and mathematical physics.
Dr. Silva-Ortigoza's research work has been cited over 1000 times. He has been a reviewer in several JCR indexed journals. He has been a referee in several awards of research and engineering in Mexico, the National Program of Quality Postgraduate, and research projects of CONACYT. He was an Editor of the book Mechatronics (in Spanish; Mexico City: CIDETEC-IPN, 2010). His research interests include the theory and application of automatic control in mobile robotics. He is also interested in automation, embedded systems, and power electronics. She has been a Professor and a Researcher with CIDETEC-IPN, since 2007. She has been an Advisor of several postgraduate students and the Leader of research projects with IPN. She belongs to SNI-CONACYT, Mexico, and the Mexican Societies of Physics and Mathematics. She has published several articles in JCR indexed journals. Her research interests include dynamic systems, control theory, and mathematical physics.