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

Authors: Eduardo Hernández-Márquez; José Rafael García-Sánchez; Ramón Silva-Ortigoza; Mayra Antonio-Cruz; Victor Manuel Hernández-Guzmán; Hind Taud; Mariana Marcelino-Aranda
Journal: Complexity
 (2018)

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## Abstract

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.


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## Body

## 1. 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 [5–10]. 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.


### 1.1. Unidirectional Rotation

Lyshevski [11] proposed mathematical models for DC/DC power converters connected to DC motors. The topologies there considered were the Buck, Boost, and Cuk. Also, Lyshevski designed a nonlinear PI control for regulating the angular velocity of the DC/DC Buck converter-DC motor system. On the other hand, Boldea and Nasar [12] presented position, velocity, and torque controls to solve the regulation problem of DC/DC power converter-DC motor systems. Later, for the first time, Linares-Flores in [13] proposed a differential flatness-based control, an average GPI control, and a passivity-based control. These three controls solve the trajectory tracking 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 [16–21]. 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 [25–29]. Additional works where other topologies of DC/DC power converters driven DC motors are [30–32] for the Boost converter, [33] for the Buck-Boost converter, and [34] for the Sepic and Cuk converters.


### 1.2. 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] by García-Rodríguez et al. and [38] by Ortigoza et al., respectively. On the other hand, Márquez et al. reported the modeling and experimental validation of the DC/DC Buck-Boost converter-inverter-DC motor system in [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 [11–34] and (ii) for bidirectional rotation of the motor shaft [35–41]. From the point of view of the practical and industrial applications, [11–34] are limited when compared with [35–41]. 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 converter-inverter-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 Section2, the design of the robust controllers is developed. The experimental results are shown in Section 3. Finally, conclusions are presented in Section 4.


## 1.1. Unidirectional Rotation

Lyshevski [11] proposed mathematical models for DC/DC power converters connected to DC motors. The topologies there considered were the Buck, Boost, and Cuk. Also, Lyshevski designed a nonlinear PI control for regulating the angular velocity of the DC/DC Buck converter-DC motor system. On the other hand, Boldea and Nasar [12] presented position, velocity, and torque controls to solve the regulation problem of DC/DC power converter-DC motor systems. Later, for the first time, Linares-Flores in [13] proposed a differential flatness-based control, an average GPI control, and a passivity-based control. These three controls solve the trajectory tracking 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 [16–21]. 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 [25–29]. Additional works where other topologies of DC/DC power converters driven DC motors are [30–32] for the Boost converter, [33] for the Buck-Boost converter, and [34] for the Sepic and Cuk converters.


## 1.2. 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] by García-Rodríguez et al. and [38] by Ortigoza et al., respectively. On the other hand, Márquez et al. reported the modeling and experimental validation of the DC/DC Buck-Boost converter-inverter-DC motor system in [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 [11–34] and (ii) for bidirectional rotation of the motor shaft [35–41]. From the point of view of the practical and industrial applications, [11–34] are limited when compared with [35–41]. 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 converter-inverter-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 Section2, the design of the robust controllers is developed. The experimental results are shown in Section 3. Finally, conclusions are presented in Section 4.


## 2. 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.


### 2.1. System Model

The electronic circuit of the system under study is shown in Figure1. 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 [11–29], such configuration allows the bidirectional driving of the motor shaft.Figure 1
DC/DC Buck converter-inverter-DC motor system.The system average model presented in Figure1, which was deduced and experimentally validated in [35], is given by
(1)Ldidt=Eu1av−υ,(2)Cdυdt=i−υR−iau2av,(3)Ladiadt=υu2av−Raia−keω,(4)Jdωdt=kmia−bω,where i represents the inductor current, υ is the capacitor output voltage, ia 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, La is the armature inductance, Ra is the armature resistance, J is the moment of inertia of the rotor and motor load, ke is the counterelectromotive force constant, km is the motor torque constant, and b is the viscous friction coefficient of the motor. Whereas u1av and u2av represent the system average controls, under the restrictions u1av∈0,1 and u2av∈−1,1, also known as duty cycles.


### 2.2. 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 F1=υ and F2=ω. Thus, the system differential parametrization is determined by
(5)υ=F1,(6)ω=F2,(7)i=CḞ1+1RF1+iau2av,(8)ia=1kmJḞ2+bF2,(9)u1av=LECF¨1+1RḞ1+diau2avdt+1LF1,(10)u2av=1F1JLakmF¨2+1kmbLa+JRaḞ2+bRakm+keF2.From the parametrization ofu1av (9) and u2av (10), the following controls are proposed:
(11)u1av=LECη+1RḞ1+diau2avdt+1LF1,(12)u2av=1F1JLakmμ+1kmbLa+JRaḞ2+bRakm+keF2.After replacing (11) and (12) in (9) and (10), respectively, the trajectory tracking problem, related to F1 and F2, is reduced to control the following:
(13)η=F¨1,(14)μ=F¨2.With the intention of achievingF1→F1∗ and F2→F2∗, with F1∗ being a desired voltage and F2∗ a desired angular velocity. Proposals for the auxiliary controls, η and μ, that achieve such objectives are the following:
(15)η=F¨1∗−β2Ḟ1−Ḟ1∗−β1F1−F1∗−β0∫0tF1−F1∗dτ,(16)μ=F¨2∗−γ2Ḟ2−Ḟ2∗−γ1F2−F2∗−γ0∫0tF2−F2∗dτ.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:
(17)0=e⃛υ+β2e¨υ+β1ėυ+β0eυ,(18)0=e⃛ω+γ2e¨ω+γ1ėω+γ0eω,where the tracking errors are defined as
(19)eυ=F1−F1∗,eω=F2−F2∗.The characteristic polynomials associated with (17) and (18) are determined by
(20)pυs=s3+β2s2+β1s+β0,(21)pωs=s3+γ2s2+γ1s+γ0.A convenient selection of the controls gainsβ2,β1,β0 and γ2,γ1,γ0 are given by
(22)β2=a1+2ξ1ωn1,β1=2ξ1ωn1a1+ωn12,β0=a1ωn12,γ2=a2+2ξ2ωn2,γ1=2ξ2ωn2a2+ωn22,γ0=a2ωn22,which are imposed when matching (20) and (21), term to term, with the following Hurwitz polynomials:
(23)pυds=s+a1s2+2ξ1ωn1s+ωn12,(24)pωds=s+a2s2+2ξ2ωn2s+ωn22,with a1,a2>0, ξ1,ξ2>0, and ωn1,ωn2>0.In consequence, the controls (11) and (12), as long as υ>0, accomplish υ→υ∗ and ω→ω∗, respectively.


### 2.3. 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∗.
(2)
Low-level control: this flatness control executes the DC/DC Buck converter output voltage tracking, that is,υ→υ∗.
(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
(25)Ladiadt=ϑ−Raia−keω,Jdωdt=kmia−bω,where
(26)ϑ=υu2av.According to [18], the system (25) has as flat output to ω. In consequence, the control ϑ allows the following representation:
(27)ϑ=JLakmω¨+1kmbLa+JRaω̇+bRakm+keω.From (27), after proposing the motor control as
(28)ϑ=JLakmμ+1kmbLa+JRaω̇+bRakm+keω,the angular velocity tracking problem is reduced to control the system:
(29)μ=ω¨.Ifω∗ is the desired angular velocity, a selection of μ that accomplishes ω→ω∗ is given by
(30)μ=ω¨∗−γ2ω̇−ω̇∗−γ1ω−ω∗−γ0∫0tω−ω∗dτ,where γ0, γ1, and γ2 are the gains of the auxiliary control μ. Defining the tracking error as eω=ω−ω∗, after replacing (30) in (29), the error dynamics in closed-loop is obtained
(31)e⃛ω+γ2e¨ω+γ1ėω+γ0eω=0,whose characteristic polynomial is
(32)pωs=s3+γ2s2+γ1s+γ0,which is imposed to be stable, via its matching with the Hurwitz polynomial (24), that is,
(33)pωds=s+a2s2+2ξ2ωn2s+ωn22,with a2>0, ξ2>0, and ωn2>0. Therefore, the gains γ2, γ1, and γ0 are given by
(34)γ2=a2+2ξ2ωn2,γ1=2ξ2ωn2a2+ωn22,γ0=a2ωn22.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, ia=0, from (1) and (2) the Buck converter model is
(35)Ldidt=Eu1av−υ,Cdυdt=i−υR.According to [18], the flat output of the system (35) is υ. Thus, u1av can be written as:
(36)u1av=LCEυ¨+LREυ̇+1Eυ.Choosing the control inputu1av, from (36), as
(37)u1av=LCEη+LREυ̇+1Eυ.Then, the voltage tracking in the converter output is reduced to(38)η=υ¨.Thus, a convenient proposal ofη so that υ→υ∗, with υ∗ being the converter desired voltage, is
(39)η=υ¨∗−β2υ̇−υ̇∗−β1υ−υ∗−β0∫0tυ−υ∗dτ,where β2, β1, and β0 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 eυ=υ−υ∗, the closed-loop dynamics is obtained
(40)e⃛υ+β2e¨υ+β1ėυ+β0eυ=0,whose characteristic polynomial is
(41)pυs=s3+β2s2+β1s+β0,which is imposed to be stable through matching it with the Hurwitz polynomial
(42)pυds=s+a1s2+2ξ1ωn1s+ωn12,where a1>0, ξ1>0, and ωn1>0. Hence, the auxiliary control gains are determined by
(43)β2=a1+2ξ1ωn1,β1=2ξ1ωn1a1+ωn12,β0=a1ωn12.(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 controlϑ achieving ω→ω∗ is determined by
(44)ϑ=JLakmμ+1kmbLa+JRaω̇+bRakm+keω,with μ defined as
(45)μ=ω¨∗−γ2ω̇−ω̇∗−γ1ω−ω∗−γ0∫0tω−ω∗dτ.In turn,ϑ defined in (26) as
(46)ϑ=υu2av,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
(47)u2av=ϑυ.On the other hand, the control accomplishingυ→υ∗ was defined as
(48)u1av=LCEη+LREυ̇+1Eυ,with η given by
(49)η=υ¨∗−β2υ̇−υ̇∗−β1υ−υ∗−β0∫0tυ−υ∗dτ.Thus, the hierarchical controller is determined by (48) and (47), as long as υ>0, obtaining υ→υ∗ and ω→ω∗, respectively.


## 2.1. System Model

The electronic circuit of the system under study is shown in Figure1. 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 [11–29], such configuration allows the bidirectional driving of the motor shaft.Figure 1
DC/DC Buck converter-inverter-DC motor system.The system average model presented in Figure1, which was deduced and experimentally validated in [35], is given by
(1)Ldidt=Eu1av−υ,(2)Cdυdt=i−υR−iau2av,(3)Ladiadt=υu2av−Raia−keω,(4)Jdωdt=kmia−bω,where i represents the inductor current, υ is the capacitor output voltage, ia 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, La is the armature inductance, Ra is the armature resistance, J is the moment of inertia of the rotor and motor load, ke is the counterelectromotive force constant, km is the motor torque constant, and b is the viscous friction coefficient of the motor. Whereas u1av and u2av represent the system average controls, under the restrictions u1av∈0,1 and u2av∈−1,1, also known as duty cycles.


## 2.2. 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 F1=υ and F2=ω. Thus, the system differential parametrization is determined by
(5)υ=F1,(6)ω=F2,(7)i=CḞ1+1RF1+iau2av,(8)ia=1kmJḞ2+bF2,(9)u1av=LECF¨1+1RḞ1+diau2avdt+1LF1,(10)u2av=1F1JLakmF¨2+1kmbLa+JRaḞ2+bRakm+keF2.From the parametrization ofu1av (9) and u2av (10), the following controls are proposed:
(11)u1av=LECη+1RḞ1+diau2avdt+1LF1,(12)u2av=1F1JLakmμ+1kmbLa+JRaḞ2+bRakm+keF2.After replacing (11) and (12) in (9) and (10), respectively, the trajectory tracking problem, related to F1 and F2, is reduced to control the following:
(13)η=F¨1,(14)μ=F¨2.With the intention of achievingF1→F1∗ and F2→F2∗, with F1∗ being a desired voltage and F2∗ a desired angular velocity. Proposals for the auxiliary controls, η and μ, that achieve such objectives are the following:
(15)η=F¨1∗−β2Ḟ1−Ḟ1∗−β1F1−F1∗−β0∫0tF1−F1∗dτ,(16)μ=F¨2∗−γ2Ḟ2−Ḟ2∗−γ1F2−F2∗−γ0∫0tF2−F2∗dτ.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:
(17)0=e⃛υ+β2e¨υ+β1ėυ+β0eυ,(18)0=e⃛ω+γ2e¨ω+γ1ėω+γ0eω,where the tracking errors are defined as
(19)eυ=F1−F1∗,eω=F2−F2∗.The characteristic polynomials associated with (17) and (18) are determined by
(20)pυs=s3+β2s2+β1s+β0,(21)pωs=s3+γ2s2+γ1s+γ0.A convenient selection of the controls gainsβ2,β1,β0 and γ2,γ1,γ0 are given by
(22)β2=a1+2ξ1ωn1,β1=2ξ1ωn1a1+ωn12,β0=a1ωn12,γ2=a2+2ξ2ωn2,γ1=2ξ2ωn2a2+ωn22,γ0=a2ωn22,which are imposed when matching (20) and (21), term to term, with the following Hurwitz polynomials:
(23)pυds=s+a1s2+2ξ1ωn1s+ωn12,(24)pωds=s+a2s2+2ξ2ωn2s+ωn22,with a1,a2>0, ξ1,ξ2>0, and ωn1,ωn2>0.In consequence, the controls (11) and (12), as long as υ>0, accomplish υ→υ∗ and ω→ω∗, respectively.


## 2.3. 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∗.
(2)
Low-level control: this flatness control executes the DC/DC Buck converter output voltage tracking, that is,υ→υ∗.
(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
(25)Ladiadt=ϑ−Raia−keω,Jdωdt=kmia−bω,where
(26)ϑ=υu2av.According to [18], the system (25) has as flat output to ω. In consequence, the control ϑ allows the following representation:
(27)ϑ=JLakmω¨+1kmbLa+JRaω̇+bRakm+keω.From (27), after proposing the motor control as
(28)ϑ=JLakmμ+1kmbLa+JRaω̇+bRakm+keω,the angular velocity tracking problem is reduced to control the system:
(29)μ=ω¨.Ifω∗ is the desired angular velocity, a selection of μ that accomplishes ω→ω∗ is given by
(30)μ=ω¨∗−γ2ω̇−ω̇∗−γ1ω−ω∗−γ0∫0tω−ω∗dτ,where γ0, γ1, and γ2 are the gains of the auxiliary control μ. Defining the tracking error as eω=ω−ω∗, after replacing (30) in (29), the error dynamics in closed-loop is obtained
(31)e⃛ω+γ2e¨ω+γ1ėω+γ0eω=0,whose characteristic polynomial is
(32)pωs=s3+γ2s2+γ1s+γ0,which is imposed to be stable, via its matching with the Hurwitz polynomial (24), that is,
(33)pωds=s+a2s2+2ξ2ωn2s+ωn22,with a2>0, ξ2>0, and ωn2>0. Therefore, the gains γ2, γ1, and γ0 are given by
(34)γ2=a2+2ξ2ωn2,γ1=2ξ2ωn2a2+ωn22,γ0=a2ωn22.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, ia=0, from (1) and (2) the Buck converter model is
(35)Ldidt=Eu1av−υ,Cdυdt=i−υR.According to [18], the flat output of the system (35) is υ. Thus, u1av can be written as:
(36)u1av=LCEυ¨+LREυ̇+1Eυ.Choosing the control inputu1av, from (36), as
(37)u1av=LCEη+LREυ̇+1Eυ.Then, the voltage tracking in the converter output is reduced to(38)η=υ¨.Thus, a convenient proposal ofη so that υ→υ∗, with υ∗ being the converter desired voltage, is
(39)η=υ¨∗−β2υ̇−υ̇∗−β1υ−υ∗−β0∫0tυ−υ∗dτ,where β2, β1, and β0 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 eυ=υ−υ∗, the closed-loop dynamics is obtained
(40)e⃛υ+β2e¨υ+β1ėυ+β0eυ=0,whose characteristic polynomial is
(41)pυs=s3+β2s2+β1s+β0,which is imposed to be stable through matching it with the Hurwitz polynomial
(42)pυds=s+a1s2+2ξ1ωn1s+ωn12,where a1>0, ξ1>0, and ωn1>0. Hence, the auxiliary control gains are determined by
(43)β2=a1+2ξ1ωn1,β1=2ξ1ωn1a1+ωn12,β0=a1ωn12.(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 controlϑ achieving ω→ω∗ is determined by
(44)ϑ=JLakmμ+1kmbLa+JRaω̇+bRakm+keω,with μ defined as
(45)μ=ω¨∗−γ2ω̇−ω̇∗−γ1ω−ω∗−γ0∫0tω−ω∗dτ.In turn,ϑ defined in (26) as
(46)ϑ=υu2av,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
(47)u2av=ϑυ.On the other hand, the control accomplishingυ→υ∗ was defined as
(48)u1av=LCEη+LREυ̇+1Eυ,with η given by
(49)η=υ¨∗−β2υ̇−υ̇∗−β1υ−υ∗−β0∫0tυ−υ∗dτ.Thus, the hierarchical controller is determined by (48) and (47), as long as υ>0, obtaining υ→υ∗ and ω→ω∗, respectively.


## 3. Experimental Results in Closed-Loop

Here, the experimental results in closed-loop of the flatness controls designed in Section2 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.


### 3.1. Experimental Setup

The experimental setup was developed using the general connection diagram shown in Figure2. In the diagram presented in Figure 2 the following blocks are distinguished:
(i)
Bidirectional DC/DC Buck converter-DC motor system: this block corresponds to the system under study and is composed of three stages, namely, a Buck power converter, an inverter, and a DC motor. The nominal values of the Buck converter parameters are as follows:
(50)L=4.94mH,R=64Ω,C=114.4μF,E=42V.
The inverter allows obtaining the bidirectional angular velocities in the motor shaft. The driving of the inverter transistors is carried out by two IR2113. Lastly, the used DC motor corresponds to the GNM5440E-G3.1 whose nominal parameters are as follows:(51)La=2.22mH,ke=120.1×10−3N⋅m/A,Ra=0.965Ω,km=120.1×10−3V⋅s/rad,J=118.2×10−3kg·m2,b=129.6×10−3N⋅m⋅s/rad.
(ii)
Flatness controls: in this block, the differential flatness controls are programmed via MATLAB-Simulink. That is, the controls (11) and (12) (based on the complete dynamics) and (47) and (48) (based on the separate dynamics of the system) are experimentally implemented. The gain parameters associated with the differential flatness controls, (11), (12) and (47), (48), are defined as
(52)a1=30,ξ1=1,ωn1=1000,a2=40,ξ2=1.5,ωn2=90.
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:
(53)ω∗t=Asin2πPt,with a period P=20/3s and an amplitude A=13rad/s. Whereas for the Buck converter, the trajectory is defined as
(54)υ∗t=υ¯iti+υ¯ftf−υ¯itiψt,ti,tf.
That is, the desired trajectoryυ∗ smoothly interpolates between the initial voltage υ¯i=24V and the final one υ¯f=30V in the time interval ti,tf=1s,2s through the Bézier polynomial ψ, defined by
(55)ψt,ti,tf=0,t≤ti,t−titf−ti320−45t−titf−ti+36t−titf−ti2−10t−titf−ti3,t∈ti,tf,1,t≥tf.
(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 u1 and u2, respectively. Also, in this block, the measurement of the variables i,υ,ia, and ω is carried out. Two current probes A622 are used to measure i and ia. While the voltage probe P5200A is employed to read the voltage υ. Lastly, ω is obtained via the E6B2-CWZ6C encoder and Simulink blocks.Figure 2
General diagram of the experimental setup connections.


### 3.2. Experimental Results

In order to obtain the experimental results, the connection diagram shown in Figure2 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 parametersE, R, L, and C have been considered:
(56)Em=E,0s≤t<2.5s,70%E,2.5s≤t<5s,E,5s≤t≤20s,Rm=R,0s≤t<7.5s,14%R,7.5s≤t<10s,R,10s≤t≤20s,Lm=L,0s≤t<12.5s,30%L,12.5s≤t<15s,L,15s≤t≤20s,Cm=C,0s≤t<17.5s,300%C,17.5s≤t≤20s.Figure3 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 8s≤t≤15s. In both figures, the results associated with the control based on the complete dynamics, that is, (11) and (12), correspond to ωcd, iacd, u2avcd, υcd, icd, and u1avcd. While the results related to the control based on the separate dynamics, that is, (47) and (48), are ωsd, iasd, u2avsd, υsd, isd, and u1avsd.Figure 3
Experimental results of controls (11), (12) and (47), (48) when abrupt changes in Em, Rm, Lm, and Cm are considered.
(a)
(b)
(c)
(d)
(e)
(f)Figure 4
Experimental results of controls (11), (12) and (47), (48) when an abrupt load change is applied.
(a)
(b)
(c)
(d)
(e)
(f)As can be observed in Figures3 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 u1av and u2av are not saturated when executing the system tracking task. That is, u1av∈0,1 and u2av∈−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.


## 3.1. Experimental Setup

The experimental setup was developed using the general connection diagram shown in Figure2. In the diagram presented in Figure 2 the following blocks are distinguished:
(i)
Bidirectional DC/DC Buck converter-DC motor system: this block corresponds to the system under study and is composed of three stages, namely, a Buck power converter, an inverter, and a DC motor. The nominal values of the Buck converter parameters are as follows:
(50)L=4.94mH,R=64Ω,C=114.4μF,E=42V.
The inverter allows obtaining the bidirectional angular velocities in the motor shaft. The driving of the inverter transistors is carried out by two IR2113. Lastly, the used DC motor corresponds to the GNM5440E-G3.1 whose nominal parameters are as follows:(51)La=2.22mH,ke=120.1×10−3N⋅m/A,Ra=0.965Ω,km=120.1×10−3V⋅s/rad,J=118.2×10−3kg·m2,b=129.6×10−3N⋅m⋅s/rad.
(ii)
Flatness controls: in this block, the differential flatness controls are programmed via MATLAB-Simulink. That is, the controls (11) and (12) (based on the complete dynamics) and (47) and (48) (based on the separate dynamics of the system) are experimentally implemented. The gain parameters associated with the differential flatness controls, (11), (12) and (47), (48), are defined as
(52)a1=30,ξ1=1,ωn1=1000,a2=40,ξ2=1.5,ωn2=90.
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:
(53)ω∗t=Asin2πPt,with a period P=20/3s and an amplitude A=13rad/s. Whereas for the Buck converter, the trajectory is defined as
(54)υ∗t=υ¯iti+υ¯ftf−υ¯itiψt,ti,tf.
That is, the desired trajectoryυ∗ smoothly interpolates between the initial voltage υ¯i=24V and the final one υ¯f=30V in the time interval ti,tf=1s,2s through the Bézier polynomial ψ, defined by
(55)ψt,ti,tf=0,t≤ti,t−titf−ti320−45t−titf−ti+36t−titf−ti2−10t−titf−ti3,t∈ti,tf,1,t≥tf.
(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 u1 and u2, respectively. Also, in this block, the measurement of the variables i,υ,ia, and ω is carried out. Two current probes A622 are used to measure i and ia. While the voltage probe P5200A is employed to read the voltage υ. Lastly, ω is obtained via the E6B2-CWZ6C encoder and Simulink blocks.Figure 2
General diagram of the experimental setup connections.


## 3.2. Experimental Results

In order to obtain the experimental results, the connection diagram shown in Figure2 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 parametersE, R, L, and C have been considered:
(56)Em=E,0s≤t<2.5s,70%E,2.5s≤t<5s,E,5s≤t≤20s,Rm=R,0s≤t<7.5s,14%R,7.5s≤t<10s,R,10s≤t≤20s,Lm=L,0s≤t<12.5s,30%L,12.5s≤t<15s,L,15s≤t≤20s,Cm=C,0s≤t<17.5s,300%C,17.5s≤t≤20s.Figure3 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 8s≤t≤15s. In both figures, the results associated with the control based on the complete dynamics, that is, (11) and (12), correspond to ωcd, iacd, u2avcd, υcd, icd, and u1avcd. While the results related to the control based on the separate dynamics, that is, (47) and (48), are ωsd, iasd, u2avsd, υsd, isd, and u1avsd.Figure 3
Experimental results of controls (11), (12) and (47), (48) when abrupt changes in Em, Rm, Lm, and Cm are considered.
(a)
(b)
(c)
(d)
(e)
(f)Figure 4
Experimental results of controls (11), (12) and (47), (48) when an abrupt load change is applied.
(a)
(b)
(c)
(d)
(e)
(f)As can be observed in Figures3 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 u1av and u2av are not saturated when executing the system tracking task. That is, u1av∈0,1 and u2av∈−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.


## 4. 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.



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*Source: 1260743-2018-08-23.xml*