Hierarchical Flatness-Based Control for Velocity Trajectory Tracking of the “DC/DC Boost Converter–DC Motor” System Powered by Renewable Energy

In this investigation, a tracking control is designed for the angular velocity of the DC/DC Boost converter–DC motor system. To this end, the dynamics of the power supply, generated through a renewable energy power source, is considered in both the mathematical model and the designed control. This latter is proposed by using a two-level hierarchical approach, where the dynamics of the DC/DC Boost converter and the one associated with the DC motor not only are treated as two independent subsystems, but also they exploit their differential flatness property. For the DC/DC Boost converter, an alternative first-order mathematical model is obtained for designing the low-level voltage control. Whereas, the well known second order mathematical model of the DC motor is used for developing the high-level angular velocity control. The robustness and performance of the hierarchical tracking control are verified via realistic numerical simulations and experimental results by using Matlab-Simulink, a prototype of the system, the DS1104 board, and the renewable energy emulator TDK-Lambda G100-17. The results demonstrate and validate the effectiveness of the proposed approach.

Related to the unidirectional rotation of the motor shaft, the mathematical model of the DC/DC Buck converter-DC motor system was proposed by Lyshevski in [5]. After this, several control laws have been reported for solving either the regulation or the tracking tasks. In this regard, the control algorithms recently published in specialized literature are commonly designed on the basis of a well known strategy. For example, a nonlinear control [5], a fuzzy logic controller along with a linear quadratic regulator [6], fractional order controls [7], [8], [9], and an affine controller [10], were based on the well known proportional-integral-derivative (PID) control. In a different direction, the zero average dynamics (ZAD) technique [11], [12], [13], [14], sliding modes with dynamic surface [15], a finite-time disturbance observer [16], a continuous nonsingular terminal [17], and three variations of sliding modes [18], are papers where sliding modes were used as the basis control. On the other hand, differential flatness proposals [19] and [20], differential flatness and PI plus sliding modes [21], and linear PI controllers [22], were designed based on a hierarchical approach. Whereas, a generalized proportional integral (GPI) observer control [23], a nonlinear control [24], a control in successive loops [25], and a robust flatnessbased tracking control [26], were developed by considering the differential flatness property. Additionally, a neuronal control [27], neuro-adaptive backstepping controls [28], [29], and an adaptive neurofuzzy H-infinity control [30], were proposed by using the neural networks technique.
Related to the bidirectional rotation of the motor shaft, different topologies of the DC/DC Buck converter have been introduced. One of these corresponds to the DC/DC Buck converter-inverter-DC motor, whose mathematical model was proposed and experimentally validated in [40]. In such a system, the tracking task was solved through a control based on the ETEDPOF methodology in [41], via the proposal of two controls based on differential flatness [42], and by means of an adaptive backstepping using sliding modes control [43]. Another designed topology was the full-bridge Buck inverter-DC motor presented in [44] along with its mathematical model and corresponding experimental validation, while a tracking control based on the ETEDPOF methodology was reported in [45]. Lastly, the bidirectional tracking task was also solved by driving a DC motor through a clamped diode multilevel DC/DC Buck converter in [46].

B. DC/DC BOOST CONVERTER AS A DRIVER FOR A DC MOTOR
Controls for driving the angular velocity of a DC motor by using a DC/DC Boost converter as the driver, can be classified depending on the rotation of the motor shaft. This is, unidirectional rotation [47], [48], [49], [50], [51], [52] and bidirectional one [53], [54], [55], [56]. Related to the unidirectional rotation, the literature reported passivity controls [47], non-linear controls [48], [49], [50], digital controllers [51], and a current control based on fuzzy logic [52]. Regarding the bidirectional rotation, a new topology of the DC/DC Boost converter-DC motor was proposed and its mathematical model was experimentally validated in [53]. Also, for this new topology, a passive control was designed in [54] and a differential flatness-based robust control was developed in [55]. Whereas, a procedure based on sum-of-squares optimization was presented in [56].
From the aforementioned, this work presents a two-level hierarchical robust control that contemplates, for the first time, the dynamics of the renewable energy power source that feeds the system with the aim of solving the angular velocity tracking task in the DC/DC Boost converter-DC motor. The effectiveness and performance of the control are verified via realistic numerical simulations and through its experimental implementation. Both results show the accomplishment of the control task, i.e., ω → ω * , even when abrupt variations in parameters of the system are included.
The remaining of this paper is organized as follows. In Section II, generalities of the DC/DC Boost converter-DC motor system are given. In Section III, the high-level control and the low-level control are developed and then interconnected for generating the two-level hierarchical control. Section IV is devoted to simulation results, whereas Section V presents the corresponding experimental implementation of the proposed approach. Finally, Section VI concludes this paper and describes the future of this research.

II. DC/DC BOOST CONVERTER-DC MOTOR SYSTEM
In this investigation, the DC/DC Boost converter-DC motor system is considered to be the interconnection of two independently controlled subsystems, as can be observed in Fig. 1. On the one hand, the DC/DC Boost converter steps up the input voltage E(t) depending on the input signal u with the aim of generating the output voltage υ. Here, an electric current i flows through the inductance L to be sent right to the parallel connection between the capacitor C and the load R in accordance with the operation of transistor Q and diode D. On the other hand, the angular velocity ω of the DC motor shaft is driven via the voltage ν. In this subsystem, the armature current i a flows through the armature load R a and the inductor L a . The product between the pair constant k m and i a generates the torque of the motor. Since the armature rotates, an induced voltage is generated and corresponds to the product of the counter electromotive force k e and the angular velocity ω. The remaining parameters are the moment of inertia J and the viscous friction coefficient of the motor shaft b.

A. MATHEMATICAL MODELS OF THE DC/DC BOOST CONVERTER AND THE DC MOTOR
The mathematical model of the DC/DC Boost converter when the primary power supply is a renewable energy power source is given by [69] While the mathematical model of the DC motor is given by [21] di a dt = (2)

B. ALTERNATIVE FIRST-ORDER MATHEMATICAL MODEL FOR THE DC/DC BOOST CONVERTER
Since the dynamics associated with the model (1) is a non-minimum phase dynamics [70], for control purposes, this paper will use a reduced model for the DC/DC Boost converter. Such a model is achieved by approximating the second order dynamics (1) to a first order one by using an iterative process [70]. This process exploits the differential flatness property of (1), whose differential parametrization is and the flat output is given by The iterative equations are found from the first derivative with respect to time of (3) and by using (1), so that the following is obtainedḞ and solving for the current i From relation (4) and the flat output (3), the iterative process for obtaining the reduced model of (1) gives as a result where k ∈ N. This process generates a static relation between i and υ. It is worth mentioning that Hernández-Márquez et al.
in [71] demonstrated that only one iteration is required for achieving a dynamic behavior similar to the one described by (1). From the first iteration, where F is considered to be constant, the following relation is obtained Through the time derivative of (5), and by using (1), the following reduced model of first order is obtained for the DC/DC Boost converter that uses a renewable energy power source

III. DESIGN OF THE HIERARCHICAL CONTROL BASED ON DIFFERENTIAL FLATNESS
The design of the two-level hierarchical control considers both the DC/DC Boost converter and the DC motor as independent subsystems, as depicted in Fig. 1. In this proposal, the low level corresponds to the control of the DC/DC Boost converter, whereas the high level is associated with the control of the DC motor.

A. LOW-LEVEL CONTROL
The low-level control is proposed by considering the reduced model of the DC/DC Boost converter (6). Note that in this model the input u av can be represented as Now, (7) is a convenient representation of the DC/DC Boost converter, since it will allow to define a suitable control over the voltage υ. Thus, the robustness of the low-level control will be reflected directly over the output voltage υ. In this manner, the two-level hierarchical control will be capable of compensating variations in this voltage so that the angular velocity tracking task be solved. Instead, if the dynamics (1) be used, with flat output F = 1 2 Li 2 + Cυ 2 , a direct control over the voltage υ cannot be designed. After considering (7), the low-level control for the DC/DC Boost converter is proposed as follows where η is an auxiliary control given by being κ 0 and κ 1 the gains of the control and υ * the desired output voltage of the converter. After equating (7) with (8) and defining the tracking error as e b = υ − υ * , the error dynamics in closed-loop of the DC/DC Boost converter is whose characteristic polynomial is defined as With the aim of achieving that υ → υ * , the polynomial (9) is equated with the following Hurwitz polynomial being ζ b and ω n b the damping factor and the undamped natural frequency of the converter in closed-loop, respectively. Hence, the gains κ 0 and κ 1 of the low-level control are Note that the choice of ζ b and ω n b for tuning the control will ensure that e b → 0 and, consequently, that υ → υ * be achieved.

B. HIGH-LEVEL CONTROL
The high-level control, associated with the DC motor, exploits the differential flatness property of this subsystem. Such a control, in accordance with [21], is where the auxiliary control δ is proposed as where α 0 , α 1 , and α 2 are the control gains of the high-level control and ω * is the desired angular velocity. Based on [21], the error dynamics in closed-loop is given by ... e m + α 2ëm + α 1ėm + α 0 e m = 0, with e m = ω − ω * and whose characteristic polynomial is Similarly to the low-level control, the polynomial P m (s) is also equated with a Hurwitz one with the objective of carrying VOLUME 11, 2023 out the tracking task. To this end, the Hurwitz polynomial for controlling the angular velocity ω is P d m (s) = (s + a) s 2 + 2ζ m ω n m s + ω 2 n m , being 0 < a, ζ m the damping factor and ω n m the undamped natural frequency of the DC motor in closed-loop. Thus, the gains of the high-level control are given as such that, after choosing the parameters a, ζ m , and ω n m , the tracking task in this subsystem is performed, i.e., ω → ω * .

C. HIERARCHICAL CONTROL
With the intention of achieving that ω → ω * through the high-level and low-level controls ν and u av , respectively, the interconnection depicted in Fig. 2 must be realized. The high-level control accomplishes ω → ω * through an appropriate voltage level ν feeding the DC motor. This voltage is generated by the low-level control when υ → υ * . After considering that the output voltage υ of the DC/DC Boost converter feeds the DC motor, it can be concluded the relation between both controls. This is, the desired voltage profile for the low-level control turns out to be the high-level control, i.e., υ * = ν. Thus, the two-level hierarchical control is given by with ν, defined as

IV. SIMULATION RESULTS OF THE HIERARCHICAL CONTROL
With the intention of verifying the performance of the twolevel hierarchical control, four simulations were performed in Matlab-Simulink. The simulations consider the emulation of two renewable energy power sources and also perturbations in some parameters of the DC/DC Boost converter-DC motor.
The following parameters, associated with the system in closed-loop, were used Whereas, the desired angular velocity profile ω * was proposed as a Bézier polynomial type given by where ω i = 12 rad s , t i = 4 s, ω f = 15 rad s , t f = 7 s, and The control gains are obtained after substituting the following values in (10) and (12) ω n m = 500, ζ m = 2.5, a = 0.2, ω n b = 50, ζ b = 2.2.

1) SIMULATION 1: DC/DC BOOST CONVERTER-DC MOTOR IN CLOSED-LOOP AND TIME-VARYING POWER SUPPLY WITH PERTURBATIONS IN LOAD R
The first simulation of the DC/DC Boost converter-DC motor system in closed-loop is shown in Fig. 3. In this result, the waveform for the voltage delivered by the power supply was proposed by Gil-Antonio et al. in [69]. This kind of waveform is very similar to the one rendered by a renewable energy power source. In this case, the waveform is defined by the following function E(t) = 18 + 0.5504 sin(5t) + 0.5848 sin(10t).
Also, for this simulation, the following abrupt variations in load R were considered  The second simulation result, depicted in Fig. 4, takes into account, again, the same function for generating the voltage waveform delivered by the power supply, i.e., the form (16). In this case, abrupt variations were introduced into the capacitor C as follows

3) SIMULATION 3: DC/DC BOOST CONVERTER-DC MOTOR IN CLOSED-LOOP AND EMULATION OF A SOLAR PANEL AS THE POWER SUPPLY WITH PERTURBATIONS IN LOAD R
Unlike the first two simulations, the third simulation shown in Fig. 5 uses a waveform similar to the one generated by a solar panel with constant irradiance, as those described in [72] and [73]. The equation describing such a waveform is given by E(t) = 21(1 − e −30t ) + 0.5 sin(100t) + 0.001. (19) For this simulation the perturbations were introduced over the load R and are described by (17).

4) SIMULATION 4: DC/DC BOOST CONVERTER-DC MOTOR IN CLOSED-LOOP AND EMULATION OF A SOLAR PANEL AS THE POWER SUPPLY WITH PERTURBATIONS IN C
The fourth simulation, presented in Fig. 6, introduces the abrupt changes (18) in capacitor C. This result also uses the power supply generated through (19).  (19) and perturbations in C (18).

A. DISCUSSION ON THE SIMULATION RESULTS
As can be observed in Figs. 3-6, the tracking task is appropriately performed, since ω → ω * ; meaning that also υ → υ * even when abrupt changes in parameters of the system are introduced. It is worth noting, in such results, that in t = 0 s the desired angular velocity profile is different from 0 rad/s. This is due to the output voltage, υ, of the DC/DC

Boost converter lies in the semi open interval [E(t), ∞).
On the other hand, the influence of abrupt variations in load R (see Figs. 3 and 5) is greater than the one associated with the abrupt changes in capacitance C (see Figs. 4 and 6). This, because the former increases (or decreases) the consumption of current i; whereas, the latter affects the voltage ripple.

V. EXPERIMENTAL RESULTS OF THE HIERARCHICAL CONTROL
In this section, the experimental implementation of the twolevel hierarchical control in closed-loop over a platform of the DC/DC Boost converter-DC motor system, is carried out. VOLUME 11, 2023 In this direction, the experimental testbed is firstly described and then the obtained results are presented. The experiments contemplate a time-varying power supply E(t) and the abrupt variations (17) and (18) for parameters R and C, respectively.

A. EXPERIMENTAL TESTBED OF THE DC/DC BOOST CONVERTER-DC MOTOR SYSTEM
In the following, the testbed for executing the two-level hierarchical control implementation is detailed. The experimental setup along with all the required elements are shown in the blocks of Fig. 7. These blocks are described next.
• Renewable energy emulation. This block shows the G100-17 TDK-Lambda power supply that emulates a renewable energy power source. This power supply is useful for obtaining several DC voltage waveforms by programming the required behavior. Also, this device allows to simulate solar panels. In addition, in this block the power supply voltage, E(t), is measured via a Tektronix P5200A voltage probe. • Data acquisition board and conditioning circuit.
The interconnection between the DC/DC Boost converter-DC motor prototype and Matlab-Simulink, via the DS1104 data acquisition board, is performed here. As can be observed, the required signals are treated through a conditioning block (SC) so that the control u be correctly processed and generated. Note that a TLP250 optocoupler is used for electric isolation between the acquisition board and the system.
• Desired trajectory. The desired angular velocity profile ω * is programmed in this block, by using Matlab-Simulink, and is proposed as where ω i = 12 rad s , t i = 4 s, ω f = 15 rad s , t f = 7 s, and λ(t, t i , t f ) defined in (15).
• Hierarchical control. This block contains the programming of the two-level hierarchical control (13) in Matlab-Simulink, i.e., the low-level control (8) and the high-level control (11) with the aim of achieving that ω → ω * .  The desired angular velocity profile ω * is proposed via the Bézier polynomial (20), while the gains (10) and (12)  The first experiment corresponds to simulation 1 and is depicted in Fig. 9. Here, the waveform implemented for the power supply is (16). Also, abrupt changes in load R are introduced and are given by  The second experiment, analogue to simulation 2, uses the time-varying power supply E(t) described by (16). Now, abrupt variations in capacitance C are considered and are described by The experimental results for this experiment are shown in Fig. 10.   Fig. 11. The power supply corresponds to the emulation of the solar panel Ameresco Solar LLC 50 J−50 W (19) through the TDK-Lambda G100-17. In addition, the abrupt variations given by (21) for load R are also taken into account.

4) EXPERIMENT 4: DC/DC BOOST CONVERTER-DC MOTOR IN CLOSED-LOOP AND EMULATION OF A SOLAR PANEL AS THE POWER SUPPLY WITH PERTURBATIONS IN CAPACITANCE C
This last experiment is analogue to simulation 4 and is depicted in Fig. 12. This experiment considers, similar to the previous one, the emulation of solar panel Ameresco Solar LLC 50 J − 50 W (19), as the power supply, through the TDK-Lambda G100-17. In this case, the abrupt variations in capacitance C (22) were introduced.

C. DISCUSSION OF THE EXPERIMENTAL RESULTS
The experimental results of the DC/DC Boost converter-DC motor in closed-loop presented in Figs. 9-12 demonstrated the effectiveness of the proposed two-level hierarchical control (13), since was achieved that υ → υ * and, consequently, ω → ω * . Regarding the experiments of Figs. 9 and 10, these were carried out when the waveform (16), associated with a renewable energy power source, was used as the power supply. Note that the control objective ω → ω * was accomplished even when natural sinusoidsalike variations over the waveform E(t) emerged. Related to the experiments of Figs. 11 and 12, these were obtained by using the waveform (19) as the power supply, via TDK-Lambda G100-17, and corresponds to the emulation of solar panel Ameresco Solar LLC 50 J − 50 W. Here, E(t) slowly drops when the current i rises. Such a behavior is due to the solar panel, whose electric relation V − I is plotted in Fig. 8. However, the average control u av compensates those variations and, thus, the control objective is again achieved, i.e., ω → ω * . Additionally, the robustness of the control (13) was demonstrated after introducing abrupt variations in some parameters of the DC/DC Boost converter-DC motor system. On the one hand, load perturbations R were considered and the flowing current i was directly affected, as can be seen in Figs. 9 and 11. On the other hand, variations in capacitance C were also contemplated and the voltage ripple was slightly affected, as shown in Figs. 10 and 12. Nevertheless, and despite all those changes, the control objective was performed at each instant of time, i.e., ω → ω * .

VI. CONCLUSION
This paper presents, for the first time in literature, a robust hierarchical control that takes into account the dynamics of a renewable energy power source in its design for solving the angular velocity trajectory tracking task in the DC/DC Boost converter-DC motor system. The control considers two levels, a higher and a lower, each one for independently control the subsystems that compose the whole system, i.e., the DC Boost converter and the DC motor. The low-level control, associated with the DC/DC Boost converter, exploits the differential flatness property related to the first order mathematical model of the converter and uses the dynamics of the power supply E(t). This control is capable of achieving that υ → υ * . Whereas, the high-level control also uses the differential flatness property but now the one of the DC motor for performing the angular velocity tracking task, i.e., ω → ω * . Then, both controls were interconnected in order to work as a whole and, thus, to generate the twolevel hierarchical control. Note that, due to the control design, any kind of time-varying power supply can be used without the need of redesigning the control approach. The effectiveness and performance of the proposed control were verified through numerical simulations and by implementing the control on an experimental platform of the system. Abrupt variations were considered in some parameters of the system with the aim of verifying the robustness of the two-level hierarchical control in closed-loop.
Motivated by the obtained results, future work will be focused on considering abrupt changes in other parameters of the system so that the robustness of the proposed approach be verified through simulation and experimental results. Also, the DC/DC Boost converter-DC motor system along with the two-level hierarchical control will be implemented on a mobile robot, where a solar panel will be used as the primary power supply.