# Robust Flatness Tracking Control for the “DC/DC Buck Converter-DC Motor” System: Renewable Energy-Based Power Supply

Authors: Ramón Silva-Ortigoza; Alfredo Roldán-Caballero; Eduardo Hernández-Márquez; José Rafael García-Sánchez; Magdalena Marciano-Melchor; Victor Manuel Hernández-Guzmán; Gilberto Silva-Ortigoza
Journal: Complexity
 (2021)

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## Abstract

The design of a robust flatness-based tracking control for the DC/DC Buck converter-DC motor system is developed in this paper. The design of the control considers the dynamics of a renewable energy power source that plays the role of the primary power supply associated with the system. The performance and robustness of the control is verified through simulations via MATLAB-Simulink when abrupt changes in some parameters of the system are taken into account. Also, experiments are performed by using a built prototype of the DC/DC Buck converter-DC motor system, a TDK-Lambda G100-17 programmable DC power supply, MATLAB-Simulink, and the DS1104 board from dSPACE. In this regard, the TDK-Lambda G100-17 is implemented with the aim of emulating photovoltaic panels through the solar array mode for generating the power supply of the system. Thus, both simulations and experiments show the effectiveness of the proposed control scheme.


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## Body

## 1. Introduction

The rise of human population all around the globe has caused, as a direct consequence, an overdemand of energy consumption. According to [1], the world population will increase by almost 2.5 billion people by 2050. On the one hand, the demand of energy generated through fossil fuels leads to several environmental issues, such as global warming, acid rain, ozone layer depletion, among others [2]. In this context, the need of new research topics related to renewable energy, as an alternative to conventional energy, arises and is just where the research community should focus its efforts. On the other hand, it is well known that industry and transportation are the largest energy demand-consumption sectors around the world [3] and both share one of the most highly used elements, the DC motor [4]. Because of this, the design of controls for DC motors fed by renewable energy power supplies turns out to be extremely relevant and important when the tracking or regulation tasks, associated with the angular velocity of the motor shaft, need to be solved. In this direction, it is usually to drive the velocity of the shaft by using pulse-width-modulation techniques. However, when using those techniques, some undesirable dynamic behavior emerges; that is, abrupt changes in voltage and current appear. It is stressed that this kind of variations can be reduced by implementing DC/DC power electronic converters. Such converters will attenuate the abrupt behavior due to the inductors and capacitors included in their design.


### 1.1. Related Works

According to the specialized literature, different topologies of DC/DC power electronic converters as drivers for DC motors have been proposed. The most important ones are Buck [5–40], Boost [41–48], Buck-Boost [49–54], Sepic [55, 56], Cuk [57], and Luo [58]. Nevertheless, as can be observed in all these contributions, the topology of the Buck converter-DC motor connection has attracted the greatest attention of the research community. The reason of this latter is as a result of its multiple applications and linear dynamic behavior. In this research, the Buck converter is used as driver for a DC motor. Thus, the corresponding review linked to the related works is presented below.On the one hand, solutions associated with DC motors unidirectionally driven by DC/DC Buck converters to control their angular velocity are the following. Lyshevski, one of the pioneers in developing this type of systems, proposed a nonlinear PI control in [5]. Later, in [6] Ahmad et al. designed the PI, fuzzy PI, and LQR controls for the aforementioned system. In contrast to previous works, Linares-Flores et al. in [7] presented a velocity smooth starter based on flatness. Further, Bingöl and Paçaci in [8] applied a control based on neural networks. In another direction, Sira-Ramírez and Oliver-Salazar [9] exploited the active disturbance rejection and differential flatness concepts. The works by Hoyos et al. [10–13] reported an application of the zero average dynamics strategy and a fixed point induction control. Additionally, controls based on differential flatness were implemented by using hierarchical approaches [14–16], passivity and load torque estimation methods [17], derivative-free nonlinear Kalman filter [18], and successive loops [19]. Subsequently, a robust adaptive control based on sliding mode with dynamic surface was utilized by Wei et al. in [20]. Also, Hernández-Guzmán et al. in [21] introduced a control algorithm by using sliding mode plus PI controls. Likewise, Khubalkar et al. presented fractional-order PID controls in [22, 23] associated with dynamic particle swarm optimization (dPSO) and improved dPSO (IdPSO) techniques, respectively. Another research was exposed by Nizami et al. in [24], where a neuro-adaptive backstepping control using single-layer Chebyshev polynomials based neural network was exhibited. Recently, interesting contributions have been accomplished in [25–27]. Hanif et al. in [25] explained a safe experimentation dynamics algorithm for a data-driven piecewise affine PI control. Yang et al. in [26] studied a robust predictive control via a discrete-time reduced-order generalized proportional-integral (GPI) observer. Meanwhile, a fault detection scheme based on a combination of switching observer and bond graph method was addressed by Kasemi and Montazeri in [27]. More recently, Rigatos et al. in [28] used an adaptive neuro-fuzzy H-infinity control approach. And a continuous dynamic sliding mode control with high-order mismatched disturbance compensation was analyzed by Rauf et al. in [29]. Additionally, Guerrero et al. in [30] developed an active disturbance rejection control based on GPI observer for a DC motor driven by a parallel DC/DC Buck converter. Lastly, other important contributions, recently published, related to the connection of the DC/DC Buck power converter and DC motor have been reported in [31–35].On the other hand, solutions associated with DC motors bidirectionally driven by DC/DC Buck converters are described next. A mathematical model and a sensorless passivity-based control with flatness were investigated by Silva-Ortigoza et al. in [36, 37], respectively. Likewise, Hernández-Márquez et al. considered two robust controls based on flatness [38]; the first control considers the system complete dynamics, whereas the second separates the dynamics of the system for using a hierarchical approach. Concerning this same system, an adaptive backstepping sliding-mode control integrated with Chebyshev neural network estimation was employed by Chi et al. in [39]. Another interesting study has been recently provided by Ismail and Elnady in [40], who described a new drive system for DC motor using multilevel DC/DC Buck converter circuit.


### 1.2. Discussion of Related Works, Motivation, and Contribution

In papers [5–40], different kinds of control schemes have been developed for both the regulation and the tracking tasks of DC motors fed by DC/DC Buck converters. Those control schemes considered the dynamics of the converter and the DC motor when the primary power supply of the converter, E, is obtained either by using a regulated power supply or by using static values. However, when the “DC/DC converter-DC motor” systems are fed through renewable energy power supplies, as presented in [39, 55, 56, 59], the dynamics of the power supply should be considered in the design of the controller with the aim of taking into account voltage and electric current variations related with such power supplies. A relevant approximation regarding the latter was reported by Gil-Antonio et al. in [60], where a flatness control was presented for the DC/DC Boost converter and also considered the dynamics associated with renewable energy sources. Thus, the contribution of this paper consists in developing a robust tracking control against power supply variations that may arise from the use of renewable energy sources in a DC/DC Buck converter-DC motor system. The feasibility and effectiveness of the proposed control are demonstrated via simulation studies and experimental tests through a built prototype. Particularly, this paper considers as the primary power supply the emulation of solar photovoltaic energy [61].The rest of the paper is as follows. In Section2, the generalities of the DC/DC Buck converter-DC motor system, where the dynamics associated with the renewable energy source is taken into consideration in the mathematical model, are presented, whereas the design of the differential flatness-based robust tracking control is described in Section 3. The simulations, which verify the effectiveness of the robust control, are presented in Section 4. Then, in Section 5, the system performance in closed loop is experimentally validated through tests by using a prototype of the system. Additionally, in Section 6, an assessment of the simulation and experimental results in closed loop is presented. Finally, the conclusions are given in Section 7.


## 1.1. Related Works

According to the specialized literature, different topologies of DC/DC power electronic converters as drivers for DC motors have been proposed. The most important ones are Buck [5–40], Boost [41–48], Buck-Boost [49–54], Sepic [55, 56], Cuk [57], and Luo [58]. Nevertheless, as can be observed in all these contributions, the topology of the Buck converter-DC motor connection has attracted the greatest attention of the research community. The reason of this latter is as a result of its multiple applications and linear dynamic behavior. In this research, the Buck converter is used as driver for a DC motor. Thus, the corresponding review linked to the related works is presented below.On the one hand, solutions associated with DC motors unidirectionally driven by DC/DC Buck converters to control their angular velocity are the following. Lyshevski, one of the pioneers in developing this type of systems, proposed a nonlinear PI control in [5]. Later, in [6] Ahmad et al. designed the PI, fuzzy PI, and LQR controls for the aforementioned system. In contrast to previous works, Linares-Flores et al. in [7] presented a velocity smooth starter based on flatness. Further, Bingöl and Paçaci in [8] applied a control based on neural networks. In another direction, Sira-Ramírez and Oliver-Salazar [9] exploited the active disturbance rejection and differential flatness concepts. The works by Hoyos et al. [10–13] reported an application of the zero average dynamics strategy and a fixed point induction control. Additionally, controls based on differential flatness were implemented by using hierarchical approaches [14–16], passivity and load torque estimation methods [17], derivative-free nonlinear Kalman filter [18], and successive loops [19]. Subsequently, a robust adaptive control based on sliding mode with dynamic surface was utilized by Wei et al. in [20]. Also, Hernández-Guzmán et al. in [21] introduced a control algorithm by using sliding mode plus PI controls. Likewise, Khubalkar et al. presented fractional-order PID controls in [22, 23] associated with dynamic particle swarm optimization (dPSO) and improved dPSO (IdPSO) techniques, respectively. Another research was exposed by Nizami et al. in [24], where a neuro-adaptive backstepping control using single-layer Chebyshev polynomials based neural network was exhibited. Recently, interesting contributions have been accomplished in [25–27]. Hanif et al. in [25] explained a safe experimentation dynamics algorithm for a data-driven piecewise affine PI control. Yang et al. in [26] studied a robust predictive control via a discrete-time reduced-order generalized proportional-integral (GPI) observer. Meanwhile, a fault detection scheme based on a combination of switching observer and bond graph method was addressed by Kasemi and Montazeri in [27]. More recently, Rigatos et al. in [28] used an adaptive neuro-fuzzy H-infinity control approach. And a continuous dynamic sliding mode control with high-order mismatched disturbance compensation was analyzed by Rauf et al. in [29]. Additionally, Guerrero et al. in [30] developed an active disturbance rejection control based on GPI observer for a DC motor driven by a parallel DC/DC Buck converter. Lastly, other important contributions, recently published, related to the connection of the DC/DC Buck power converter and DC motor have been reported in [31–35].On the other hand, solutions associated with DC motors bidirectionally driven by DC/DC Buck converters are described next. A mathematical model and a sensorless passivity-based control with flatness were investigated by Silva-Ortigoza et al. in [36, 37], respectively. Likewise, Hernández-Márquez et al. considered two robust controls based on flatness [38]; the first control considers the system complete dynamics, whereas the second separates the dynamics of the system for using a hierarchical approach. Concerning this same system, an adaptive backstepping sliding-mode control integrated with Chebyshev neural network estimation was employed by Chi et al. in [39]. Another interesting study has been recently provided by Ismail and Elnady in [40], who described a new drive system for DC motor using multilevel DC/DC Buck converter circuit.


## 1.2. Discussion of Related Works, Motivation, and Contribution

In papers [5–40], different kinds of control schemes have been developed for both the regulation and the tracking tasks of DC motors fed by DC/DC Buck converters. Those control schemes considered the dynamics of the converter and the DC motor when the primary power supply of the converter, E, is obtained either by using a regulated power supply or by using static values. However, when the “DC/DC converter-DC motor” systems are fed through renewable energy power supplies, as presented in [39, 55, 56, 59], the dynamics of the power supply should be considered in the design of the controller with the aim of taking into account voltage and electric current variations related with such power supplies. A relevant approximation regarding the latter was reported by Gil-Antonio et al. in [60], where a flatness control was presented for the DC/DC Boost converter and also considered the dynamics associated with renewable energy sources. Thus, the contribution of this paper consists in developing a robust tracking control against power supply variations that may arise from the use of renewable energy sources in a DC/DC Buck converter-DC motor system. The feasibility and effectiveness of the proposed control are demonstrated via simulation studies and experimental tests through a built prototype. Particularly, this paper considers as the primary power supply the emulation of solar photovoltaic energy [61].The rest of the paper is as follows. In Section2, the generalities of the DC/DC Buck converter-DC motor system, where the dynamics associated with the renewable energy source is taken into consideration in the mathematical model, are presented, whereas the design of the differential flatness-based robust tracking control is described in Section 3. The simulations, which verify the effectiveness of the robust control, are presented in Section 4. Then, in Section 5, the system performance in closed loop is experimentally validated through tests by using a prototype of the system. Additionally, in Section 6, an assessment of the simulation and experimental results in closed loop is presented. Finally, the conclusions are given in Section 7.


## 2. DC/DC Buck Converter-DC Motor System

The circuit of the system under study is shown in Figure1. This system is divided, for the purpose of this research, into three stages: a renewable energy power supply, a DC/DC Buck converter, and a DC motor. The power supply, due to its nature, is considered to be a function that depends on time (i.e., Et). The converter is built by using a transistor Q, a diode D, an inductor L, a capacitor C, and a load R. The input signal u allows the current i to flow through the inductor so that the voltage υ can be generated in the load terminals. The motor is composed of a resistance Ra, an inductor La, a moment of inertia J, a friction coefficient b, and two electromechanical constants ke and km. The flowing of current ia through La allows the rotation of the motor shaft and, consequently, an angular velocity ω is originated.Figure 1
DC/DC Buck converter-DC motor system.By using the results reported in [21], the average mathematical model of the system depicted in Figure 1 is given as(1)x˙=Ax+Buav,where(2)A=0−1L001C−1RC−1C001La−RaLa−keLa00kmJ−bJ,B=EtL000,x=iυiaω,with uav∈0,1.


## 3. Flatness-Based Tracking Control

In differential algebra, a system is considered to be differentially flat if all the state variables and the inputs associated with vector statex and vector u, respectively, can be expressed in terms of a specific set of variables, called flat outputs, without using any kind of mathematical integration [62]. Based on [9], the flat output of the DC/DC Buck converter-DC motor system is determined by the angular velocity of the motor, that is, y=ω. Thus, by performing some algebraic operations, the vector state and input can be strictly defined as functions of the flat output; this is(3)x=iωυωiaωω,uav=uavω,where(4)iω=CJLakmω3+bCLaR+CJRRa+JLakmRω¨+bLa+JR+JRa+bCRRa+kekmCRkmRω˙+bR+bRa+kekmkmRω,(5)υω=JLakmω¨+bLa+JRakmω˙+bRakm+keω,(6)iaω=Jkmω˙+bkmω,and(7)uavω=CJLLakmEtω4+bCLLaR+CJLRRa+JLLakmREtω3+αω¨+bLR+bLRa+bLaR+kekmL+JRRakmREtω˙+bRa+kekmkmEtω,with(8)α=bLLa+JLR+JLRa+bCLRRa+kekmCLR+JLaRkmREt.On one hand, in order to avoid a division by zero inuav (equation (7)) and the DC/DC Buck converter to operate adequately, then the voltage Et must be considered strictly positive. On the other hand, the differential parameterization of the unidirectional DC/DC Buck converter-DC motor system, determined by equations (4)–(7), turns out to be similar to the case when the power supply is considered to be constant [7]. It is worth mentioning that in some other systems when the power supply is considered to be dynamic (i.e., as Et), in the control strategy appears explicitly Et and its derivatives with respect to time. An example of this is the DC/DC Boost converter [60], whose average control uav depends on Et and E˙t when such a control is based on differential flatness.After obtaining the differential parameterization of the DC/DC Buck converter-DC motor system (equation (1)), given by equations (4)–(7), the next step is to design the robust flatness-based tracking control. Hence, from equation (7), the following auxiliary control is proposed:(9)ω4=μ.The calculation of the auxiliary controlμ is performed through equation (10) and achieves exponential asymptotic tracking of the trajectory ω, that is, ω⟶ω∗, with ω∗ being the desired angular velocity to be defined later:(10)0=μ−ω∗4+k4ω3−ω∗3+k3ω¨−ω¨∗+k2ω˙−ω˙∗+k1ω−ω∗+k0∫0tω−ω∗dτ.The integral component ensures zero static error in steady state and compensates the abrupt variations that can be generated in some parameters of the system. As can be observed, the error signale has been defined as the difference between ω and ω∗. Thus, after introducing equation (10) in equation (9) in the frequency domain, the following is obtained:(11)Ps=s5+k4s4+k3s3+k2s2+k1s+k0,where k0,k1,k2,k3,k4 is the set of control parameters and has to be chosen so that the roots of the closed loop characteristic polynomial in the complex variable s,(12)PHs=s+as2+2ζωns+ωn22,be a Hurwitz polynomial (i.e., a, ζ, and ωn be positive constants). In this way, the set of control parameters are defined as(13)k0=aωn4,k1=4aζωn3+ωn4,k2=2aωn2+4aζ2ωn2+4ζωn3,k3=4aζωn+2ωn2+4ζ2ωn2,k4=a+4ζωn.Once the stability of subsystem equation (9) is ensured, the control signal uav must be proposed. In this context, and with the aim of achieving that ω⟶ω∗ in the DC/DC Buck converter-DC motor system (equation (1)), the control is now proposed by introducing equation (9) in equation (7); that is,(14)uav=CJLLakmEtμ+bCLLaR+CJLRRa+JLLakmREtω3+αω¨+bLR+bLRa+bLaR+kekmL+JRRakmREtω˙+bRa+kekmkmEtω,where μ is obtained from equation (10) and is given by(15)μ=ω∗4−k4ω3−ω∗3−k3ω¨−ω¨∗−k2ω˙−ω˙∗−k1ω−ω∗−k0∫0tω−ω∗dτ.


## 4. Simulations of the Flatness-Based Tracking Control

This section presents the implementation of the proposed control (equation (14)) in the DC/DC Buck converter-DC motor system described by equation (1). The performance and robustness of the differential flatness-based control is verified when the system is subjected to some abrupt changes in its parameters.Figure2 shows the block diagram that has been implemented in MATLAB-Simulink for acquiring the simulation results of the system in closed loop.Figure 2
Block diagram associated with the implementation, in MATLAB-Simulink, of the flatness-based control in the DC/DC Buck converter-DC motor system.The reference inputuav∗ and variables i∗,υ∗, and ia∗ are obtained after substituting the desired angular velocity ω∗ in the differential parameterization of the system, that is, in equations (4)–(7). The parameters of the DC/DC Buck converter and DC motor for obtaining the simulation results are(16)L=286.5mH,C=114.4μF,R=250Ω,La=2.22mH,km=120.1×10−3N⋅mA,Ra=0.965Ω,ke=120.1×10−3V⋅srad,J=118.2×10−3kg⋅m2,b=129.6×10−3N⋅m⋅srad.While the gains of the control (k0, k1, k2, k3, k4) are obtained after choosing and replacing the parameters a=2, ζ=0.707, and ωn=900 in equation (13).


### 4.1. Simulation Results

With the intention of verifying the performance of the proposed flatness-based control (equation (14)), the simulation results are now presented and were developed in 4 different scenarios. For each one of these scenarios, two functions have been selected for the emulation of the renewable energy power supply associated with the system, Et:(i)
The first two simulations consider the waveform reported by Gil-Antonio et al. in [60] (see Figure 3(a)) and is defined as(17)Et=511.008+0.5504sin5t+0.5848sin10t.(ii)
The last two simulations consider, as a function of time, the emulation of a photovoltaic module that is subjected to constant radiation [63, 64]. The behavior of such a module can be modeled (see Figure 3(b)) as(18)Et=611−e−30t+0.5sin100t+0.001.Figure 3
Renewable energy emulation of the power supplyEt. (a) Waveform described by equation (17) and contains variations that could emerge from the usage of a renewable energy source. (b) Time-varying function proposed in equation (18) with variations that may appear in a photovoltaic module with constant radiation.
(a)(b)Simulation 1.
The first simulation contemplates the power supply described by equation (17). Also, the following desired angular velocity of the Bézier type is proposed:(19)ω∗t=ω¯iti+ω¯ftf−ω¯itiφt,ti,tf,where φt,ti,tf is defined by(20)φt,ti,tf=0,t≤ti,20t−titf−ti3−45t−titf−ti4+36t−titf−ti5−10t−titf−ti6t∈ti,tf,1,t≥tf.
Thus, from equation (19), ω∗ is smoothly interpolated from ω¯iti to ω¯ftf. Additionally, it has been selected that ω¯i=0rad/s, ti=2s, ω¯f=13rad/s, and tf=6s. In this simulation, whose results are depicted in Figure 4, the abrupt variations Rp are introduced in the load resistance R as follows:(21)Rp=R0s≤t<3s,200%R3s≤t<5s,R5s≤t<7s,20%R7s≤t<10s.Figure 4
DC/DC Buck converter-DC motor system response when the time-variant waveform and abrupt variations are defined by equations (17) and (21), respectively.Simulation 2.
In this simulation, the abrupt variationsCp are applied to the capacitance C and are defined as (22)Cp=C0s≤t<3s,200%C3s≤t<5s,C5s≤t<7s,50%C7s≤t<10s.
Here, for the power supplyEt, the function proposed by Gil-Antonio et al. (equation (17)) and the desired angular velocity ω∗ (equation (19)) are used again. Thus, the corresponding simulation results are presented in Figure 5.Figure 5
Response of the DC/DC Buck converter-DC motor system when the time-variant waveform and the abrupt variations are defined by equations (17) and (22), respectively.Simulation 3.
In this simulation, the power supplyEt of the DC/DC Buck converter-DC motor system is proposed to be the emulation of a photovoltaic module that is subjected to constant radiation. Likewise, as in the previous simulation results, the desired angular velocity corresponds to the one defined in equation (19). After taking into account the previous considerations and introducing the abrupt variations in load R given by Rp (equation (21)), the simulation results presented in Figure 6 are obtained.Figure 6
Time-response of the DC/DC Buck converter-DC motor system when using the waveform (equation (18)) and abrupt variations (equation (21)) in R.Simulation 4.
Lastly, in this simulation, the power supplyEt and the desired angular velocity ω∗ are defined again as in equations (18) and (19), respectively. Hence, when the abrupt perturbations Cp (equation (22)) are applied in C, the behavior depicted in Figure 7 is achieved.Figure 7
Dynamic behavior of the DC/DC Buck converter-DC motor system whenEt is given by equation (18) and when the abrupt changes (equation (22)) are introduced in C.


### 4.2. General Comments on the Simulation Results

The simulation results presented in Figures4–7, associated with the numeric implementation of the flatness-based tracking control on the DC/DC Buck converter-DC motor system, show that the control objective is accomplished, that is, ω⟶ω∗. The trajectory tracking task is achieved even when abrupt variations in R and C are taken into account. Moreover, the tracking task is also carried out even if abrupt variations appear in Et as a consequence of using a renewable energy source. Note that the signal uav compensates all variations in the power supply.With the aim of demonstrating the performance of tracking control (equation (14)), the similarity of signals ω and ω∗ is quantified through the following tracking error definition:(23)eSi=ω−ω∗,where subscript Si, with i∈1,2,3,4, denotes the simulation test from which the error has been calculated. These errors are depicted within Figure 8, where it is easily observed that the dynamic behavior of the system in closed loop when Et is defined as in equation (17), Simulations 1 and 2, is very similar when Et is defined as in equation (18), Simulations 3 and 4. Such a similarity comes from the fact that the control uav is robust enough to compensate all kinds of variations in Et and, consequently, the angular velocity ω will be less affected by those variations. On the other hand, when perturbations in R are introduced into the system, the tracking errors eS1 and eS3 are larger than those associated with perturbations introduced through C, that is, the errors eS2 and eS4. Those differences are due to the fact that current i is directly affected by any kind of change in load R. Because of this, the power delivered by the DC/DC Buck power converter-DC motor system must be greater to keep the same value of angular velocity ω.Figure 8
Angular velocity tracking errors obtained from the simulation results depicted in Figures4–7 when abrupt variations are applied in parameters R and C of the DC/DC Buck converter-DC motor system and when Et emulates a renewable energy power supply given by equations (17) and (18).


## 4.1. Simulation Results

With the intention of verifying the performance of the proposed flatness-based control (equation (14)), the simulation results are now presented and were developed in 4 different scenarios. For each one of these scenarios, two functions have been selected for the emulation of the renewable energy power supply associated with the system, Et:(i)
The first two simulations consider the waveform reported by Gil-Antonio et al. in [60] (see Figure 3(a)) and is defined as(17)Et=511.008+0.5504sin5t+0.5848sin10t.(ii)
The last two simulations consider, as a function of time, the emulation of a photovoltaic module that is subjected to constant radiation [63, 64]. The behavior of such a module can be modeled (see Figure 3(b)) as(18)Et=611−e−30t+0.5sin100t+0.001.Figure 3
Renewable energy emulation of the power supplyEt. (a) Waveform described by equation (17) and contains variations that could emerge from the usage of a renewable energy source. (b) Time-varying function proposed in equation (18) with variations that may appear in a photovoltaic module with constant radiation.
(a)(b)Simulation 1.
The first simulation contemplates the power supply described by equation (17). Also, the following desired angular velocity of the Bézier type is proposed:(19)ω∗t=ω¯iti+ω¯ftf−ω¯itiφt,ti,tf,where φt,ti,tf is defined by(20)φt,ti,tf=0,t≤ti,20t−titf−ti3−45t−titf−ti4+36t−titf−ti5−10t−titf−ti6t∈ti,tf,1,t≥tf.
Thus, from equation (19), ω∗ is smoothly interpolated from ω¯iti to ω¯ftf. Additionally, it has been selected that ω¯i=0rad/s, ti=2s, ω¯f=13rad/s, and tf=6s. In this simulation, whose results are depicted in Figure 4, the abrupt variations Rp are introduced in the load resistance R as follows:(21)Rp=R0s≤t<3s,200%R3s≤t<5s,R5s≤t<7s,20%R7s≤t<10s.Figure 4
DC/DC Buck converter-DC motor system response when the time-variant waveform and abrupt variations are defined by equations (17) and (21), respectively.Simulation 2.
In this simulation, the abrupt variationsCp are applied to the capacitance C and are defined as (22)Cp=C0s≤t<3s,200%C3s≤t<5s,C5s≤t<7s,50%C7s≤t<10s.
Here, for the power supplyEt, the function proposed by Gil-Antonio et al. (equation (17)) and the desired angular velocity ω∗ (equation (19)) are used again. Thus, the corresponding simulation results are presented in Figure 5.Figure 5
Response of the DC/DC Buck converter-DC motor system when the time-variant waveform and the abrupt variations are defined by equations (17) and (22), respectively.Simulation 3.
In this simulation, the power supplyEt of the DC/DC Buck converter-DC motor system is proposed to be the emulation of a photovoltaic module that is subjected to constant radiation. Likewise, as in the previous simulation results, the desired angular velocity corresponds to the one defined in equation (19). After taking into account the previous considerations and introducing the abrupt variations in load R given by Rp (equation (21)), the simulation results presented in Figure 6 are obtained.Figure 6
Time-response of the DC/DC Buck converter-DC motor system when using the waveform (equation (18)) and abrupt variations (equation (21)) in R.Simulation 4.
Lastly, in this simulation, the power supplyEt and the desired angular velocity ω∗ are defined again as in equations (18) and (19), respectively. Hence, when the abrupt perturbations Cp (equation (22)) are applied in C, the behavior depicted in Figure 7 is achieved.Figure 7
Dynamic behavior of the DC/DC Buck converter-DC motor system whenEt is given by equation (18) and when the abrupt changes (equation (22)) are introduced in C.


## 4.2. General Comments on the Simulation Results

The simulation results presented in Figures4–7, associated with the numeric implementation of the flatness-based tracking control on the DC/DC Buck converter-DC motor system, show that the control objective is accomplished, that is, ω⟶ω∗. The trajectory tracking task is achieved even when abrupt variations in R and C are taken into account. Moreover, the tracking task is also carried out even if abrupt variations appear in Et as a consequence of using a renewable energy source. Note that the signal uav compensates all variations in the power supply.With the aim of demonstrating the performance of tracking control (equation (14)), the similarity of signals ω and ω∗ is quantified through the following tracking error definition:(23)eSi=ω−ω∗,where subscript Si, with i∈1,2,3,4, denotes the simulation test from which the error has been calculated. These errors are depicted within Figure 8, where it is easily observed that the dynamic behavior of the system in closed loop when Et is defined as in equation (17), Simulations 1 and 2, is very similar when Et is defined as in equation (18), Simulations 3 and 4. Such a similarity comes from the fact that the control uav is robust enough to compensate all kinds of variations in Et and, consequently, the angular velocity ω will be less affected by those variations. On the other hand, when perturbations in R are introduced into the system, the tracking errors eS1 and eS3 are larger than those associated with perturbations introduced through C, that is, the errors eS2 and eS4. Those differences are due to the fact that current i is directly affected by any kind of change in load R. Because of this, the power delivered by the DC/DC Buck power converter-DC motor system must be greater to keep the same value of angular velocity ω.Figure 8
Angular velocity tracking errors obtained from the simulation results depicted in Figures4–7 when abrupt variations are applied in parameters R and C of the DC/DC Buck converter-DC motor system and when Et emulates a renewable energy power supply given by equations (17) and (18).


## 5. Testbed and Closed Loop Experimental Results

In this section, the experimental results in closed loop are presented. With the aim of showing the effectiveness of the proposed control, given by equation (14), perturbations are considered in some parameters of the DC/DC Buck converter-DC motor system.


### 5.1. Experimental Testbed

The general diagram of the experimental testbed, for implementing the design control on the DC/DC Buck converter-DC motor system that considers the emulation of renewable energy, is shown in Figure9.Figure 9
Diagram of the experimental testbed associated with the DC/DC Buck converter-DC motor system.The blocks composing the experimental testbed, depicted in Figure9, are as follows:(i)
Desired Trajectory and Reference Variables. In this block, the desired trajectory ω∗ and the reference variables i∗, υ∗, ia∗, and uav, obtained when ω∗ is replaced in equations (4)–(7), are programmed through MATLAB-Simulink.(ii)
Flatness-Based Control. Here, the flatness-based control (equation (14)) is programmed also via MATLAB-Simulink. It is worth mentioning that such a control requires only the information associated with the flat output of the system, ω, and the power supply Et. The gains of the control scheme k0,k1,k2,k3,k4, previously declared, are obtained after the parameters a=1, ζ=10, and ωn=1200 are introduced in equation (13).(iii)
Board and Signal Conditioning Circuit. The connection of the DC/DC Buck converter-DC motor system along with the DS1104 board from dSPACE is realized in this block. For such an aim, the signals Et and υ are measured through two Tektronix P5200 A voltage probes, whereas i and ia are measured by using two Tektronix A622 current probes, and ω is measured via an E6B2-CWZ6C encoder. The signal conditioning is executed with blocks SC. Also, the control signal u, allowing the appropriate commutation of transistor Q, is obtained after the signal uav is modulated through a PWM port of the DS1104 board. The electrical isolation between the DS1104 board and the power system is achieved by implementing a TLP250 photocoupler.(iv)
DC/DC Buck Converter-DC Motor. This block corresponds to the built DC/DC Buck converter-DC motor system prototype. Here, a TDK-Lambda G100-17 programmable DC power supply is used to provide the signal Et. With such a power supply, it is possible to generate several DC arbitrary waveforms. As a matter of fact, it is possible also to emulate photovoltaic panels through the solar array mode option. Regarding the prototype, the parameters of the DC/DC Buck converter-DC motor system built are exactly the same as those used in simulations and are given by equation (16).Additionally, a photograph of the experimental testbed associated with the DC/DC Buck converter-DC motor system, depicted in Figure9, is shown in Figure 10.Figure 10
Photograph of the testbed associated with the DC/DC Buck converter-DC motor system. (1) PC, (2) DS1104 board from dSPACE, (3) DC/DC Buck converter, (4) Tektronix A622 current probes, (5) Tektronix P5200A voltage probes, (6) TDK-Lambda G100-17 programmable DC power supply, (7) power supply of the instrumentation stage, (8) electronic board to apply perturbations, and (9) DC motor.


### 5.2. Experimental Results in Closed Loop

In correspondence to the four different scenarios for which the simulations of the system in closed loop were obtained (see Section1.1), here, the experimental results linked to those simulations are presented.As it was previously stated, the emulation of the renewable energy power supplyEt was achieved by proposing the following two functions:(i)
For the first two experiments, the wave formEt was proposed as the one reported by Gil-Antonio et al. in [60], whose behavior is depicted in Figure 3(a); this is,(24)Et=511.008+0.5504sin5t+0.5848sin10t.The experimental implementation of equation (24) was carried out by programming the TDK-Lambda G100-17 power supply. Thus, the waveform of signal Et had the form presented in Figure 11(a).(ii)
Regarding the third and fourth experiments, the waveform ofEt was generated by emulating a photovoltaic module subjected to constant radiation and whose dynamic behavior was shown in Figure 3(b). The mathematical model for such an Et was given by equation (18); that is,(25)Et=611−e−30t+0.5sin100t+0.001.In this case, the experimental implementation of equation (25) was achieved through the solar array simulator of the TDK-Lambda G100-17 power supply by selecting the TS-S425 solar panel, whose V-I graphic and dynamic behavior are depicted in Figures 11(b) and 11(c), respectively.Figure 11
Renewable energy emulation ofEt generated via the TDK-Lambda G100-17 programmable DC power supply. (a) Experimental realization of the waveform (equation (24)) that contains some variations that could arise from the usage of a renewable energy source. Experimental implementation when the programmable DC power supply emulates the behavior of solar panels with constant radiation (equation (25)). (b) V-I curve of the TS-S425 solar panel and (c) dynamic behavior of the TS-S425 solar panel.
(a)(b)(c)Experiment 1.
In correspondence to the results ofSimulation 1 (see Figure 4), in this section, the experimental results linked to those simulations are presented. Thus, for Et, the experimental realization of equation (24), depicted in Figure 11(a), is introduced, whereas the angular velocity of the Bézier type (equation (19)), previously defined, is used. That is, (26)ω∗t=ω¯iti+ω¯ftf−ω¯itiφt,ti,tf,with φt,ti,tf defined by equation (20). Then, from equation (26), the signal ω∗ smoothly interpolates from ω¯i to ω¯f in ti and tf, respectively. In this way, the experimental results associated with perturbations Rp equation (27) are shown in Figure 12.(27)Rp=R0s≤t<3s,200%R3s≤t<5s,R5s≤t<7s,20%R7s≤t<10s.Figure 12
Dynamic response of the DC/DC Buck converter-DC motor system whenEt, given by equation (24), is experimentally implemented through the TDK-Lambda G100-17 power supply and when the abrupt variations (equation (27)) are considered in R.Experiment 2.
In regard with the results ofSimulation 2 (see Figure 5), in this section, the experimental results associated with those simulations are presented.
In this experiment, the waveform forEt shown in Figure 11(a), representing a renewable energy source, is again implemented via the TDK-Lambda G100-17 programmable DC power supply. Then, after using the desired angular velocity ω∗ (equation (26)) and introducing the variations Cp (equation (28)) in capacitance C of the DC/DC Buck converter, the obtained results are depicted in Figure 13.(28)Cp=C0s≤t<3s,200%C3s≤t<5s,C5s≤t<7s,50%C7s≤t<10s.Figure 13
Dynamic response of the DC/DC Buck converter-DC motor system whenEt is defined as in equation (24), experimentally implemented through the TDK-Lambda G100-17 programmable power supply, and considering variations equation (28) in C.Experiment 3.
In this section, the experimental results linked to the results ofSimulation 3 (see Figure 6) are presented.
In order to obtain the experimental results presented in Figure14, the power supply Et feeds the system with the waveform shown in Figure 11(c), corresponding to the emulation of a signal delivered by a solar panel with constant radiation. Such an Et is generated through the TDK-Lambda G100-17 programmable DC power supply by using its solar array mode, as a result of having used the solar panel TS-S425 for the experimental implementation of equation (25). Likewise, the results presented in Figure 14 consider the desired angular velocity of the Bézier type (equation (26)) and perturbations Rp (equation (27)) for the load R.Figure 14
Experimental results of the DC/DC Buck converter-DC motor system in closed loop whenEt is given by equation (25) and carried out through the TDK-Lambda G100-17 power supply via the solar panel TS-S425, and the variations equation (27) are taken into account in R.Experiment 4.
Lastly, this section shows the experimental results associated withSimulation 4 (see Figure 7). In this experiment, the power supply Et feeds the system with a similar waveform as the one used in Experiment 3. Such a waveform is depicted in Figure 11(c) and generated through the solar panel TS-S425 of the programmable DC power supply. Then, after using ω∗ (equation (26)) and introducing perturbations Cp (equation (28)) in capacitance C of the Buck converter, the obtained experimental results are depicted in Figure 15.Figure 15
Experimental results of the DC/DC Buck converter-DC motor system in closed loop whenEt (equation (25)) is generated through the TDK-Lambda G100-17 programmable power supply, via the solar panel TS-S425, and the perturbations (equation (28)) are imposed in C.


### 5.3. General Comments on the Experimental Results

In correspondence to the results of the numeric implementation presented in Section1.1, the experimental implementation associated with the control scheme based on differential flatness (equation (14)) was carried out and the obtained results are shown in Figures 12–15. The experimental realization was executed by using a built DC/DC Buck converter-DC motor system, where variations on the power supply Et, which could emerge from the usage of renewable energy sources, were considered. Similar to simulation results, the experimental ones demonstrated that ω⟶ω∗ when Et is described by equations (24) and (25) and even when abrupt variations (equations (27) and (28)) are introduced in parameters R and C, respectively.The experimental results depicted in Figures12 and 14 show a bigger variation in current i, when perturbations in load R (equation (27)) are taken into account, with respect to those presented in Figures 13 and 15, associated with perturbations C (equation (28)). Also, it is worth noting that voltage Et in the experimental results presented in Figures 14 and 15 drops when the load of the system rises. Such a voltage drop is due to an increase in current, which represents a natural behavior in solar panels. In this direction, the experimental results presented in Figures 14 and 15 were obtained by generating Et (equation (25)) through the solar panel TS-S425 (see Figures 11(b) and 11(c)) of the TDK-Lambda G100-17 programmable power supply. On the other hand, the tracking error related to the control based on differential flatness (equation (14)) needs to be quantified, that is, the difference between ω and ω∗. Thus, in correspondence to the definition given by equation (23), the tracking error of the experimental results is defined as(29)eEj=ω−ω∗,where subscript Ej, with j∈1,2,3,4, denotes the experimental result from which the error has been calculated. These errors are depicted within Figure 16, and as can be observed the tracking errors of the experimental results are larger in magnitude in comparison with the simulation ones shown in Figure 8. The difference between the tracking errors of the simulation results and the experimental results is because the mathematical model of the system does not take into account some dynamics, such as the internal resistance of the solid-state devices, heat losses, nonlinearities, among others. Nevertheless, the tracking errors associated with the experimental results are small enough regarding the angular velocity magnitude. Thus, it can be concluded that the control objective is satisfactorily achieved, that is, ω⟶ω∗.Figure 16
Angular velocity tracking errors obtained from the experimental results shown in Figures12–15 when abrupt variations are considered in parameters R and C of the DC/DC Buck converter and Et emulates a renewable energy power supply. The plotted results associated with eE1 and eE2 are achieved when the experimental implementation of Et (equation (24)) is programmed through the TDK-Lambda G100-17 power supply, while the plotted results related to eE3 and eE4 are obtained when the experimental implementation of Et (equation (25)) is realized via the solar panel TS-S425 of the TDK-Lambda G100-17 power supply.


## 5.1. Experimental Testbed

The general diagram of the experimental testbed, for implementing the design control on the DC/DC Buck converter-DC motor system that considers the emulation of renewable energy, is shown in Figure9.Figure 9
Diagram of the experimental testbed associated with the DC/DC Buck converter-DC motor system.The blocks composing the experimental testbed, depicted in Figure9, are as follows:(i)
Desired Trajectory and Reference Variables. In this block, the desired trajectory ω∗ and the reference variables i∗, υ∗, ia∗, and uav, obtained when ω∗ is replaced in equations (4)–(7), are programmed through MATLAB-Simulink.(ii)
Flatness-Based Control. Here, the flatness-based control (equation (14)) is programmed also via MATLAB-Simulink. It is worth mentioning that such a control requires only the information associated with the flat output of the system, ω, and the power supply Et. The gains of the control scheme k0,k1,k2,k3,k4, previously declared, are obtained after the parameters a=1, ζ=10, and ωn=1200 are introduced in equation (13).(iii)
Board and Signal Conditioning Circuit. The connection of the DC/DC Buck converter-DC motor system along with the DS1104 board from dSPACE is realized in this block. For such an aim, the signals Et and υ are measured through two Tektronix P5200 A voltage probes, whereas i and ia are measured by using two Tektronix A622 current probes, and ω is measured via an E6B2-CWZ6C encoder. The signal conditioning is executed with blocks SC. Also, the control signal u, allowing the appropriate commutation of transistor Q, is obtained after the signal uav is modulated through a PWM port of the DS1104 board. The electrical isolation between the DS1104 board and the power system is achieved by implementing a TLP250 photocoupler.(iv)
DC/DC Buck Converter-DC Motor. This block corresponds to the built DC/DC Buck converter-DC motor system prototype. Here, a TDK-Lambda G100-17 programmable DC power supply is used to provide the signal Et. With such a power supply, it is possible to generate several DC arbitrary waveforms. As a matter of fact, it is possible also to emulate photovoltaic panels through the solar array mode option. Regarding the prototype, the parameters of the DC/DC Buck converter-DC motor system built are exactly the same as those used in simulations and are given by equation (16).Additionally, a photograph of the experimental testbed associated with the DC/DC Buck converter-DC motor system, depicted in Figure9, is shown in Figure 10.Figure 10
Photograph of the testbed associated with the DC/DC Buck converter-DC motor system. (1) PC, (2) DS1104 board from dSPACE, (3) DC/DC Buck converter, (4) Tektronix A622 current probes, (5) Tektronix P5200A voltage probes, (6) TDK-Lambda G100-17 programmable DC power supply, (7) power supply of the instrumentation stage, (8) electronic board to apply perturbations, and (9) DC motor.


## 5.2. Experimental Results in Closed Loop

In correspondence to the four different scenarios for which the simulations of the system in closed loop were obtained (see Section1.1), here, the experimental results linked to those simulations are presented.As it was previously stated, the emulation of the renewable energy power supplyEt was achieved by proposing the following two functions:(i)
For the first two experiments, the wave formEt was proposed as the one reported by Gil-Antonio et al. in [60], whose behavior is depicted in Figure 3(a); this is,(24)Et=511.008+0.5504sin5t+0.5848sin10t.The experimental implementation of equation (24) was carried out by programming the TDK-Lambda G100-17 power supply. Thus, the waveform of signal Et had the form presented in Figure 11(a).(ii)
Regarding the third and fourth experiments, the waveform ofEt was generated by emulating a photovoltaic module subjected to constant radiation and whose dynamic behavior was shown in Figure 3(b). The mathematical model for such an Et was given by equation (18); that is,(25)Et=611−e−30t+0.5sin100t+0.001.In this case, the experimental implementation of equation (25) was achieved through the solar array simulator of the TDK-Lambda G100-17 power supply by selecting the TS-S425 solar panel, whose V-I graphic and dynamic behavior are depicted in Figures 11(b) and 11(c), respectively.Figure 11
Renewable energy emulation ofEt generated via the TDK-Lambda G100-17 programmable DC power supply. (a) Experimental realization of the waveform (equation (24)) that contains some variations that could arise from the usage of a renewable energy source. Experimental implementation when the programmable DC power supply emulates the behavior of solar panels with constant radiation (equation (25)). (b) V-I curve of the TS-S425 solar panel and (c) dynamic behavior of the TS-S425 solar panel.
(a)(b)(c)Experiment 1.
In correspondence to the results ofSimulation 1 (see Figure 4), in this section, the experimental results linked to those simulations are presented. Thus, for Et, the experimental realization of equation (24), depicted in Figure 11(a), is introduced, whereas the angular velocity of the Bézier type (equation (19)), previously defined, is used. That is, (26)ω∗t=ω¯iti+ω¯ftf−ω¯itiφt,ti,tf,with φt,ti,tf defined by equation (20). Then, from equation (26), the signal ω∗ smoothly interpolates from ω¯i to ω¯f in ti and tf, respectively. In this way, the experimental results associated with perturbations Rp equation (27) are shown in Figure 12.(27)Rp=R0s≤t<3s,200%R3s≤t<5s,R5s≤t<7s,20%R7s≤t<10s.Figure 12
Dynamic response of the DC/DC Buck converter-DC motor system whenEt, given by equation (24), is experimentally implemented through the TDK-Lambda G100-17 power supply and when the abrupt variations (equation (27)) are considered in R.Experiment 2.
In regard with the results ofSimulation 2 (see Figure 5), in this section, the experimental results associated with those simulations are presented.
In this experiment, the waveform forEt shown in Figure 11(a), representing a renewable energy source, is again implemented via the TDK-Lambda G100-17 programmable DC power supply. Then, after using the desired angular velocity ω∗ (equation (26)) and introducing the variations Cp (equation (28)) in capacitance C of the DC/DC Buck converter, the obtained results are depicted in Figure 13.(28)Cp=C0s≤t<3s,200%C3s≤t<5s,C5s≤t<7s,50%C7s≤t<10s.Figure 13
Dynamic response of the DC/DC Buck converter-DC motor system whenEt is defined as in equation (24), experimentally implemented through the TDK-Lambda G100-17 programmable power supply, and considering variations equation (28) in C.Experiment 3.
In this section, the experimental results linked to the results ofSimulation 3 (see Figure 6) are presented.
In order to obtain the experimental results presented in Figure14, the power supply Et feeds the system with the waveform shown in Figure 11(c), corresponding to the emulation of a signal delivered by a solar panel with constant radiation. Such an Et is generated through the TDK-Lambda G100-17 programmable DC power supply by using its solar array mode, as a result of having used the solar panel TS-S425 for the experimental implementation of equation (25). Likewise, the results presented in Figure 14 consider the desired angular velocity of the Bézier type (equation (26)) and perturbations Rp (equation (27)) for the load R.Figure 14
Experimental results of the DC/DC Buck converter-DC motor system in closed loop whenEt is given by equation (25) and carried out through the TDK-Lambda G100-17 power supply via the solar panel TS-S425, and the variations equation (27) are taken into account in R.Experiment 4.
Lastly, this section shows the experimental results associated withSimulation 4 (see Figure 7). In this experiment, the power supply Et feeds the system with a similar waveform as the one used in Experiment 3. Such a waveform is depicted in Figure 11(c) and generated through the solar panel TS-S425 of the programmable DC power supply. Then, after using ω∗ (equation (26)) and introducing perturbations Cp (equation (28)) in capacitance C of the Buck converter, the obtained experimental results are depicted in Figure 15.Figure 15
Experimental results of the DC/DC Buck converter-DC motor system in closed loop whenEt (equation (25)) is generated through the TDK-Lambda G100-17 programmable power supply, via the solar panel TS-S425, and the perturbations (equation (28)) are imposed in C.


## 5.3. General Comments on the Experimental Results

In correspondence to the results of the numeric implementation presented in Section1.1, the experimental implementation associated with the control scheme based on differential flatness (equation (14)) was carried out and the obtained results are shown in Figures 12–15. The experimental realization was executed by using a built DC/DC Buck converter-DC motor system, where variations on the power supply Et, which could emerge from the usage of renewable energy sources, were considered. Similar to simulation results, the experimental ones demonstrated that ω⟶ω∗ when Et is described by equations (24) and (25) and even when abrupt variations (equations (27) and (28)) are introduced in parameters R and C, respectively.The experimental results depicted in Figures12 and 14 show a bigger variation in current i, when perturbations in load R (equation (27)) are taken into account, with respect to those presented in Figures 13 and 15, associated with perturbations C (equation (28)). Also, it is worth noting that voltage Et in the experimental results presented in Figures 14 and 15 drops when the load of the system rises. Such a voltage drop is due to an increase in current, which represents a natural behavior in solar panels. In this direction, the experimental results presented in Figures 14 and 15 were obtained by generating Et (equation (25)) through the solar panel TS-S425 (see Figures 11(b) and 11(c)) of the TDK-Lambda G100-17 programmable power supply. On the other hand, the tracking error related to the control based on differential flatness (equation (14)) needs to be quantified, that is, the difference between ω and ω∗. Thus, in correspondence to the definition given by equation (23), the tracking error of the experimental results is defined as(29)eEj=ω−ω∗,where subscript Ej, with j∈1,2,3,4, denotes the experimental result from which the error has been calculated. These errors are depicted within Figure 16, and as can be observed the tracking errors of the experimental results are larger in magnitude in comparison with the simulation ones shown in Figure 8. The difference between the tracking errors of the simulation results and the experimental results is because the mathematical model of the system does not take into account some dynamics, such as the internal resistance of the solid-state devices, heat losses, nonlinearities, among others. Nevertheless, the tracking errors associated with the experimental results are small enough regarding the angular velocity magnitude. Thus, it can be concluded that the control objective is satisfactorily achieved, that is, ω⟶ω∗.Figure 16
Angular velocity tracking errors obtained from the experimental results shown in Figures12–15 when abrupt variations are considered in parameters R and C of the DC/DC Buck converter and Et emulates a renewable energy power supply. The plotted results associated with eE1 and eE2 are achieved when the experimental implementation of Et (equation (24)) is programmed through the TDK-Lambda G100-17 power supply, while the plotted results related to eE3 and eE4 are obtained when the experimental implementation of Et (equation (25)) is realized via the solar panel TS-S425 of the TDK-Lambda G100-17 power supply.


## 6. Assessment of the Simulation and Experimental Results

Lastly, this section presents a direct visual comparison between the simulation and experimental results associated withω. Such a comparison is realized through a visual assessment that is achieved by including, simultaneously, the simulation results and the corresponding experimental ones in the same graphic. This is executed with the aim of highlighting the good performance of the tracking control based on differential flatness.To carry out the comparison, the variablesωS1, ωS2, ωS3, and ωS4 are defined as the angular velocities previously obtained at simulation level and shown in the first graphic of Figures 4–7, respectively. Also, the variables ωE1, ωE2, ωE3, and ωE4 are now defined as the angular velocities previously obtained at experimental level and depicted in the first graphic of Figures 12–15, respectively. Thus, after having redefined the simulation and experimental results for the angular velocity, in Figure 17, the direct visual comparison between the simulation and experimental results for ω is presented.Figure 17
Assessment of the trajectory tracking task for the angular velocityω associated with the DC/DC Buck converter-DC motor system, via a direct visual comparison between the simulation and experimental results.In Figure17, it is observed that the experimental results are very similar to their corresponding simulation ones. This is confirmed through the graphics of the tracking errors for ω depicted in Figure 18. Such a figure emerges as a result of overlapping the plots associated with the simulation closed loop tracking errors esi (shown in Figure 8) and the plots related to the experimental closed loop tracking errors eEj (shown in Figure 16). Thus, again, it can be concluded that the control objective is satisfactorily achieved, i.e., ω⟶ω∗.Figure 18
Tracking errors of the system in closed loop forω. In these graphics, the tracking errors associated with the simulation results are denoted as eS1, eS2, eS3, and eS4, while the tracking errors related to the experimental results are represented by eE1, eE2, eE3, and eE4.


## 7. Conclusion

This paper considered, for the first time, the dynamics of the primary power supply in the design of a robust tracking control for the DC/DC Buck converter-DC motor system. Particularly, the emulation of solar photovoltaic energy was introduced in this work. Also, the flatness property associated with the mathematical model was exploited with the aim of describing the control signal in terms of the flat output, conveniently determined by the angular velocity of the motor shaft, allowing the design of a robust tracking control. The robustness of the control was verified, via simulations in MATLAB-Simulink, when abrupt changes are considered in some parameters of the system.The performance and robustness of the control was verified via simulations in MATLAB-Simulink when abrupt changes were considered in some parameters of the system. While the experimental implementation was carried out on a built prototype of the DC/DC Buck converter-DC motor system, a TDK-Lambda G100-17 programmable DC power supply, MATLAB-Simulink, and the DS1104 board from dSPACE. The TDK-Lambda G100-17 power supply allowed to generate DC arbitrary waveforms and to emulate solar panels forEt. It is worth mentioning that the approach presented in this paper can be applied in many types of systems whose power supply is based on renewable energy sources. Thus, it would not be necessary to redesign the control strategy when changing the primary power supply.Finally, motivated by the obtained results of this research, the design of controls that considers the important problem of active disturbance rejection [65, 66] and applications in mobile robotics and subactuated systems are considered as future work.



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*Source: 2158782-2021-10-14.xml*