Bidirectional Tracking Robust Controls for a DC/DC Buck Converter-DC Motor System

Two differential flatness-based bidirectional tracking robust controls for a DC/DC Buck converter-DC motor system are designed. To achieve such a bidirectional tracking, an inverter is used in the system. First control considers the complete dynamics of the system, that is, it considers the DC/DC Buck converter-inverter-DC motor connection as a whole. Whereas the second separates the dynamics of the Buck converter from the one of the inverter-DC motor, so that a hierarchical controller is generated. The experimental implementation of both controls is performed via MATLAB-Simulink and a DS1104 board in a built prototype of the DC/DC Buck converter-inverter-DC motor connection. Controls show a good performance even when system parameters are subjected to abrupt uncertainties. Thus, robustness of such controls is verified.


Introduction
The electrical energy control with power electronic devices has been an important research topic during the last years [1]. Where the driving of new classes of motors has been possible thanks to the power electronics [2]. Likewise, the use of the control engineering and power electronics [3], among other areas, glimpses the possibility of achieving a driving more efficient of the new forms of electrical energy generation [4]. In that direction, the different forms of electrical conversion result very important in the energy processing. Thus, the appropriate supply of multiple applications of the new energy forms is achieved [5][6][7][8][9][10]. In such applications, some involving motion generations are found whose operation is reduced to the design of drivers for motors (see [11,12]). Among others, the DC motors have been benefited by the design of drivers based on DC/DC power electronic converters. Thus, according to the rotation of the motor shaft, works related to the control of DC/DC power converters-driven DC motors can be divided in two fashions: (a) with unidirectional rotation and (b) with bidirectional rotation. Thus, the state-of-the-art review is as follows.
task in the DC/DC Buck converter-DC motor system. Also, for the same system, Ahmad et al. in [14] presented a performance assessment of the PI, fuzzy PI, and LQR controls for the angular velocity tracking problem. While Bingöl and Paçaci [15] reported a virtual laboratory based on neural networks to control angular velocity of such system. Recent works dealing with the angular velocity trajectory tracking for the DC/DC Buck converter-DC motor system have been reported in [16][17][18][19][20][21]. Sira-Ramírez and Oliver-Salazar [16] proposed a robust control based on the active disturbance rejection and differential flatness for two configurations of the DC/DC Buck converter-DC motor system. Silva-Ortigoza et al. introduced robust hierarchical controls based on differential flatness [17,18] and sliding mode-PI with flatness in [19]. Likewise, Hernández-Guzmán et al. [20] proposed a sliding mode control and PI for controlling the converter voltage, the armature current, and the angular velocity of the motor. On the other hand, via a sensorless load torque estimation, a control based on the exact tracking error dynamics passive output feedback methodology was proposed by Kumar and Thilagar in [21]. More recently, Khubalkar et al. in [22] presented a stand-alone digital fractional order PID tracking control for the DC/DC Buck converter-DC motor system. Rigatos et al. in [23] designed a flatness-based control to solve the trajectory tracking problem. Another solution was proposed by Nizami et al. in [24], where a neuro-adaptive backstepping control for the angular velocity tracking was developed for the aforementioned system. Other interesting works reported in the last months, on tracking control design for the DC/DC Buck-DC motor system, are [25][26][27][28][29]. Additional works where other topologies of DC/DC power converters driven DC motors are [30][31][32] for the Boost converter, [33] for the Buck-Boost converter, and [34] for the Sepic and Cuk converters.

Bidirectional Rotation.
Due to the principle of operation of the DC/DC power converters, the bidirectional velocity tracking control in DC/DC converters-DC motor systems is not possible using only DC/DC converters. Thus, to overcome this restriction, which has been a topic of interest in the last years, an inverter circuit has been introduced in such systems, leading to DC/DC converter-inverter-DC motor systems. In this direction, Ortigoza et al. presented in [35] the modeling and experimental validation of the DC/DC Buck converter-inverter-DC motor system. While in [36], Ortigoza et al. designed and tested a passivity-based tracking control for a built prototype of the same system. Furthermore, the model and a passivity-based tracking control for the DC/DC Boost converter-inverter-DC motor system were reported in [37] [39]. Also, for the same system, Hernández-Márquez et al. in [40] solved the regulation problem via a sensorless passivity-based control. Lastly, for the DC/DC Sepic converter-inverter-DC motor system Linares-Flores et al. in [41] solved the regulation problem through a passive control.
After reviewing the literature of the DC motors driven by DC/DC power converters, it was found that different controls have executed the angular velocity regulation and trajectory tracking tasks in two fashions: (i) for unidirectional rotation of the motor shaft  and (ii) for bidirectional rotation of the motor shaft [35][36][37][38][39][40][41]. From the point of view of the practical and industrial applications,  are limited when compared with [35][36][37][38][39][40][41]. Regarding the latter, to the authors' knowledge, for the DC/DC converter-inverter-DC motor systems, robust solution for the trajectory tracking has not been proposed. Thus, the main contribution of this paper is to propose two robust controls based on flatness for the tracking task in the DC/DC Buck converterinverter-DC motor system. The first control considers the system complete dynamics; whereas the second takes into account the dynamics conforming the system separately (i.e., Buck converter and inverter-DC motor), producing a hierarchical controller. The designed controls are experimentally verified on a prototype, showing robustness under parametric uncertainty.
The remainder of the paper is organized as follows. In Section 2, the design of the robust controllers is developed. The experimental results are shown in Section 3. Finally, conclusions are presented in Section 4.

Tracking Controls Based on Differential Flatness
In this section, two controls are designed with the purpose of carrying out the bidirectional angular velocity trajectory tracking task of a DC/DC Buck converter-DC motor system.

System
Model. The electronic circuit of the system under study is shown in Figure 1. This system, in general, is composed of three stages, namely, a DC/DC Buck converter, a complete bridge inverter, and a DC motor. Unlike the arrangements presented in [11][12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29], such configuration allows the bidirectional driving of the motor shaft. The system average model presented in Figure 1, which was deduced and experimentally validated in [35], is given by where i represents the inductor current, υ is the capacitor output voltage, i a is the armature current, ω is the angular velocity, the voltage source has the constant value E, L is the inductance of the input circuit, C is the capacitance of the output filter, R is the output load, L a is the armature inductance, R a is the armature resistance, J is the moment of inertia of the rotor and motor load, k e is the 2 Complexity counterelectromotive force constant, k m is the motor torque constant, and b is the viscous friction coefficient of the motor. Whereas u 1av and u 2av represent the system average controls, under the restrictions u 1av ∈ 0, 1 and u 2av ∈ −1, 1 , also known as duty cycles.

Flatness Control: Complete Dynamics.
Here, the first flatness control that solves the bidirectional velocity tracking for the DC/DC Buck converter-inverter-DC motor system is designed. This first control considers all the system dynamics, that is, it is based on the fourth-order nonlinear model (1), (2), (3), and (4). Departing from the work experience related to the DC/DC Buck converter-DC motor, particularly [17,18], and [19], and from the control objectives, the flat outputs of the system (1), (2), (3), and (4) are proposed as F 1 = υ and F 2 = ω. Thus, the system differential parametrization is determined by From the parametrization of u 1av (9) and u 2av (10), the following controls are proposed: After replacing (11) and (12) in (9) and (10), respectively, the trajectory tracking problem, related to F 1 and F 2 , is reduced to control the following: With the intention of achieving F 1 → F * 1 and F 2 → F * 2 , with F * 1 being a desired voltage and F * 2 a desired angular velocity. Proposals for the auxiliary controls, η and μ, that achieve such objectives are the following: Thus, it is clear that with the selection (15) and (16), the control objective is achieved. Since, after replacing (15) and (16) in (13) and (14), respectively, the tracking errors' dynamics of the closed-loop system are: where the tracking errors are defined as The characteristic polynomials associated with (17) and (18) are determined by A convenient selection of the controls gains β 2 , β 1 , β 0 and γ 2 , γ 1 , γ 0 are given by which are imposed when matching (20) and (21), term to term, with the following Hurwitz polynomials: p ω d s = s + a 2 s 2 + 2ξ 2 ω n2 s + ω 2 n2 , 24 with a 1 , a 2 > 0, ξ 1 , ξ 2 > 0, and ω n1 , ω n2 > 0.

Flatness-Based Hierarchical Controller: Separate
Dynamics. In this section, the second flatness control strategy that executes the bidirectional angular velocity trajectory tracking for the DC/DC Buck converter-inverter-DC motor system (1), (2), (3) and (4) is designed. This strategy is on the basis of a hierarchical control scheme that takes into account the system dynamics separately, that is, a flatness control for the motor and another for the converter. By initially considering that the DC/DC Buck converter and the DC motor operate separately, the hierarchical control structure is the following: (1) High-level control: this flatness control carries out the bidirectional angular velocity trajectory tracking in the DC motor shaft, that is, ω → ω * .
(3) Integration: this allows the connection between the controls (1) and (2) through the hierarchical control scheme.
(1) DC Motor Control. Since, initially, it is assumed that the Buck converter and DC motor operate separately. Then, from (3) and (4), the motor model is transformed in According to [18], the system (25) has as flat output to ω. In consequence, the control ϑ allows the following representation: From (27), after proposing the motor control as the angular velocity tracking problem is reduced to control the system: If ω * is the desired angular velocity, a selection of μ that accomplishes ω → ω * is given by where γ 0 , γ 1 , and γ 2 are the gains of the auxiliary control μ. Defining the tracking error as e ω = ω − ω * , after replacing (30) in (29), the error dynamics in closedloop is obtained ⃛ e ω + γ 2 e ω + γ 1 e ω + γ 0 e ω = 0, 31 whose characteristic polynomial is which is imposed to be stable, via its matching with the Hurwitz polynomial (24), that is, with a 2 > 0, ξ 2 > 0, and ω n2 > 0. Therefore, the gains γ 2 , γ 1 , and γ 0 are given by With the aforementioned approach, ω → ω * when t → ∞ is achieved.
(2) Buck Converter Control. As it is assumed that there is no connection between the converter and motor. Then, i a = 0, from (1) and (2) the Buck converter model is According to [18], the flat output of the system (35) is υ. Thus, u 1av can be written as: Choosing the control input u 1av , from (36), as Then, the voltage tracking in the converter output is reduced to Thus, a convenient proposal of η so that υ → υ * , with υ * being the converter desired voltage, is where β 2 , β 1 , and β 0 are the gains of the auxiliary control η. Thus, the control (37), with (39), achieves υ → υ * when t → ∞. This is quickly verified when (39) is replaced in (38). Since, after defining the tracking error e υ = υ − υ * , the closed-loop dynamics is obtained ⃛ e υ + β 2 e υ + β 1 e υ + β 0 e υ = 0, 40 whose characteristic polynomial is which is imposed to be stable through matching it with the Hurwitz polynomial where a 1 > 0, ξ 1 > 0, and ω n1 > 0. Hence, the auxiliary control gains are determined by

43
(3) Hierarchical Controller. Having carried out the separate control of the subsystems integrating the DC/DC Buck converter-inverter-DC motor system. Now, the hierarchical controller that executes the bidirectional angular velocity trajectory tracking is proposed.
For the DC motor, it was found that the control ϑ achieving ω → ω * is determined by with μ defined as In turn, ϑ defined in (26) as considers that the voltage υ is the power supply of the DC motor inverter circuit. Therefore, the control associated with the inverter is determined by On the other hand, the control accomplishing υ → υ * was defined as with η given by Thus, the hierarchical controller is determined by (48) and (47), as long as υ > 0, obtaining υ → υ * and ω → ω * , respectively.

Experimental Results in Closed-Loop
Here, the experimental results in closed-loop of the flatness controls designed in Section 2 are presented. The controls robustness is verified under abrupt variations in the system parameters. In the experimental development, MATLAB-Simulink and a dSPACE DS1104 board are used. Notice that the gain definitions related to the controls, given by (22), (34), and (43) are equal.
(iii) Desired trajectories: in this block, the desired trajectories ω * and υ * are programmed. For the DC motor, the following bidirectional trajectory is proposed:   6 Complexity with a period P = 20/3 s and an amplitude A = 13 rad/s. Whereas for the Buck converter, the trajectory is defined as That is, the desired trajectory υ * smoothly interpolates between the initial voltage υ i = 24 V and the final one υ f = 30 V in the time interval t i , t f = 1 s, 2 s through the Bézier polynomial ψ, defined by (iv) Board and conditioning circuit: this block electrically isolates the DS1104 board from the power stage, through the NTE3087 and TLP250 optocouplers. Furthermore, this block properly drives the Buck converter and inverter when generating, by means of the PWM1 and PWM2, the switched inputs u 1 and u 2 , respectively. Also, in this block, the measurement of the variables i, υ, i a , and ω is carried out. Two current probes A622 are used to measure i and i a . While the voltage probe P5200A is employed to read the voltage υ. Lastly, ω is obtained via the E6B2-CWZ6C encoder and Simulink blocks.

Experimental Results.
In order to obtain the experimental results, the connection diagram shown in Figure 2 has been used. This shows the interconnections of the bidirectional DC/DC Buck converter-DC motor system with MATLAB-Simulink and the DS1104 board.
With the intention of evaluating the performance of the designed controls, the following abrupt changes in the system parameters E, R, L, and C have been considered:  Figure 3 shows the experimental results in closed-loop when the abrupt changes (56) are considered. Whereas, Figure 4 depicts the experimental results when an abrupt load variation, via a brake system, is applied in 8 s ≤ t ≤ 15 s. In both figures, the results associated with the control based on the complete dynamics, that is, (11) and (12), correspond to ω cd , i a cd , u 2av cd , υ cd , i cd , and u 1av cd . While the results related to the control based on the separate dynamics, that is, (47) and (48), are ω sd , i a sd , u 2av sd , υ sd , i sd , and u 1av sd .
As can be observed in Figures 3 and 4, the results are satisfactory for both controls, since it is achieved that ω → ω * and υ → υ * even when abrupt changes in the system parameters and an abrupt load variation are taken into account. Thus, the robustness of the controls is shown. On the other hand, the controls u 1av and u 2av are not saturated when executing the system tracking task. That is, u 1av ∈ 0, 1 and u 2av ∈ −1, 1 in all the experimental results.
Lastly, although in general the two controls show good performance, the authors consider that the scheme based on the hierarchical approach, that is, (47) and (48), is more simple to design because the system (1), (2), (3), and (4) is separated in two subsystems.

Conclusions
In this paper, two flatness-based robust controls for the bidirectional angular velocity trajectory tracking problem of the DC/DC Buck converter-inverter-DC motor system were proposed. The first control considers the system complete dynamics. Whereas, the second separates the dynamics of the system, which later are joined through a hierarchical controller. The performance of the proposed controls was verified through experiments with a built prototype, MATLAB-Simulink, and a DS1104 board. The results shown that the proposed controls satisfactorily execute the trajectory tracking task even when abrupt changes on the system parameters are considered. Thus, the robustness of the flatness-based controls was exhibited. It is noteworthy that the tested uncertainties do not simultaneously occur in practice. However, they were presented with the purpose of showing the possible industrial application of the herein designed controls.

Data Availability
The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest
The authors declare that the research was conducted in the absence of any commercial, financial, or personal relationships that could be construed as a potential conflict of interests.   Complexity