Sliding Mode and PI-Based Control for the SISO “Full-Bridge Buck Inverter–DC Motor” System Powered by Renewable Energy

This paper presents the design of a smooth starter for a DC motor that uses a full-bridge Buck inverter powered by renewable energy. For this aim, a renewable energy power supply is considered in the mathematical model of the “full-bridge Buck inverter–DC motor” system. Whereas, for the design of the smooth starter, a control based on sliding-mode and proportional-integral control is proposed. The performance of the developed control scheme is experimentally validated using the system prototype, the TDK-Lambda G100-17 renewable energy emulator source, the DS1104 board, and the Matlab-Simulink software. Experimental tests are conducted with two different power supply scenarios: 1) a time-varying input voltage and 2) a commercial photovoltaic panel emulator. The obtained experimental results show the robustness of the controller proposed here to abrupt disturbances in the system parameters and torque.


I. INTRODUCTION
The discipline of automatic control is essential for the technological development that provides the well-being that we enjoy today.Its applications cover almost all areas in which human beings develop daily: from the operation of robots that are monitoring Mars to the air conditioning at home, among others.It should be noted that the first applications of control date back more than two millennia.
The associate editor coordinating the review of this manuscript and approving it for publication was Jiann-Jong Chen .
However, the milestone that allowed the scientific and technological development of the discipline was the industrial revolution.Since this historical event, the industrial sector has adopted automatic control as a powerful tool that allows improving the quality of products and services of all kinds.In the industrial sector there are different types of plants that are of interest for automatic control; some of these systems are thermal, chemical, electrical, mechanical, electromechanical, etc.
Electromechanical systems are a type of plant that combines electrical and mechanical elements to perform a specific function.Some examples of electromechanical systems are: DC motors, AC motors, robots, machine-tools, refrigeration systems, vehicles, etc.These type of plants is easy to feed and control by means of the pulse width modulation (PWM) technique.Nevertheless, the use of this technique inherently involves a high switching frequency that stresses the electromechanical systems and causes abrupt changes in their dynamic behavior.This fact is reflected in sudden variations in voltages and currents, deteriorating the components of such systems in the long term.One way to avoid the anomalous behavior in the electromechanical systems due to the use of PWM is through power electronic converters.Therefore, in the following, a review of works related to the application of control algorithms in DC motors that are powered by power converters is presented.

A. STATE OF ART
In this section, the literature related to DC motors powered by power converters is reviewed.In this sense, at the beginning of the 21st century, Lyshevski studied the electromechanical system formed by a Buck converter that feeds a DC motor, and proposed a nonlinear PI control for this plant [1].This work was the first of many contributions that have studied the control problems related to a DC motor powered by some topology of power converters.In this regard, the system formed by a Buck power converter connected in series to a DC motor has been mostly exploited by the expert community in automatic control [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23].However, there have also been a significant number of contributions that take advantage of the Boost topology [24], [25], [26], [27], [28], [29], [30].Likewise, systems that consider the use of the Buck-Boost [31], [32] Ćuk [33], and Luo [28], [34] converters as drivers for the DC motors have been designed.Derived from the dynamic nature of the power converters used in the works described so far, the movement of the DC motor shaft is only executed unidirectionally.To overcome this limitation, arrangements that consider the inclusion of an H bridge between the power converter and the DC motor have been proposed to achieve a bidirectional movement of the motor shaft.Works of this nature have taken advantage of the Buck converters [35], [36], [37], [38], [39], Boost [40], [41], [42], Buck-Boost [43], [44], [45], [46], and Sepic [47].Another alternative to achieve the bidirectional rotation of the DC motor is by means of the full-bridge Buck inverter [48], [49], [50], [51].This topology has been used for the handling of AC motors [52], [53] and in the actuators of a wheeled mobile robot [54].

B. MOTIVATION AND CONTRIBUTION
As reviewed in the previous section, DC motor systems powered by power converters are plants widely studied by the scientific community.In addition, the control of DC/DC converters and the control of DC motors are two distinct problems that have been of great interest worldwide.This work is a continuation of two previous studies that examined the full-bridge Buck inverter-DC motor system: [49], which analyzed a passivity-based control and [50], that introduced a differential flatness-based control.Motivated by the previous comments, in this research, a control based on sliding modes and PI, which does not require the use of PWM, is proposed for the system presented in [48] when it is powered by renewable energy.The control method in this paper, which solves the problem mentioned above, has several proportional-integral (PI) loops and a sliding mode loop that controls the current in the converter inductor.Experiments show that this approach produces a robust control that can cope with both variations in plant parameters and external disturbances.Also, in this contribution, asymptotic stability in closed loop is demonstrated by the Lyapunov approach.
Finally, some notation that we use in the present paper is the following.Symbol ∥h∥ 1 = n i=1 |h i | stands for the 1-norm of a vector h ∈ R n , where | • | represents the absolute value.Also, ∥h∥ = n i=1 h 2 i is the Euclidean norm of vector h.All eigenvalues of a n × n symmetric matrix A are real and λ min (A) stands for the smallest eigenvalue of A.

II. MATHEMATICAL MODEL
In Fig. 1, the full-bridge Buck inverter-DC motor system is presented.By applying Kirchhoff's Laws and Newton's Second Law the following mathematical model is found: The control input u is associated with the on/off switching of the transistors Q 1 , Q 2 , Q 1 , and Q 2 , which can take the discrete values 1, 0, and −1.The system parameters R, C, L, R a , L a , B, J , k e , k m , and T L represent a load electric resistance, converter capacitance, converter inductance, armature resistance, armature inductance, viscous friction coefficient, the motor inertia, the motor back electromotive force constant, the motor torque constant, and an unknown but constant load torque, respectively.The states i, υ, i a , and ω correspond to the electric current flowing through L, the voltage at the terminals of C, the electric current passing through the motor armature, and the motor angular velocity, respectively.It is worth mentioning that the element LC act as a low-pass filter that provides a continuous voltage signal at the terminals of the DC motor.Finally, E(t) represents a power supply of renewable energy.

III. CONTROL OF THE FULL-BRIDGE BUCK INVERTER-DC MOTOR SYSTEM
The main result derived from this study is stated in the following proposition.
Then, the origin of the closedloop system state is guaranteed to be asymptotically stable if the positive constant values k p1 , k i1 , k p2 , k i2 , f , r a , and γ are properly chosen and the following condition is met: Remark 3.1: Controller ( 5)-( 10), described in proposition 3.1, is depicted in the diagram of Fig. 2. Note that the proposed controller contains four principal loops.a) The innermost loop uses a control based on sliding mode and drives the electric current i to its desired profile i * through the inductor of the converter.b) The desired profile i * corresponds to the output of a PI control, ensuring that the output voltage v of the converter converges to its desired profile v. c) v is found by using a PI control, driven by the electric current error of the motor's armature, along with two terms involving the error of the velocity and its integral.d) The desired profile īa related to the electric current of the motor's armature is obtained by integrating the velocity error.Therefore, the proposal in this paper incorporates key components, namely proportional and integral ones, found in industrial DC motor control over 1) the armature electric current error and 2) the velocity error.The closed-loop system is expected to exhibit robustness due to the presence of PI schemes on the converter output voltage error, along with sliding modes control applied to the electric current via the converter inductor.The robustness is verified through experiments in section IV.

A. ATTAINING THE SLIDING SURFACE
Let us define the function V (s) = 1 2 s 2 , which is scalar, positive definite, and radially unbounded.Its time derivative along trajectories of ( 1) is given as, where ( 5) was used, if dt > 0 and −υ − L di * dt < 0 are analyzed, then it is easy to verify that ( 12) implies (11).When considering the sliding condition ṡ = 0, the dynamics of the system (1), and the asymptotic stability condition (11) then the equivalent control fulfills the following, meaning that the sliding regime is possible.Note that (12) ensures that s = i − i * = 0 is accomplished, i.e. that i = i * is verified.Therefore, the next step is to analyze the stability of the closed-loop system dynamics, ( 2)-( 4) and ( 7)- (10), when i = i * .

B. CLOSED-LOOP DYNAMICS OVER THE SLIDING SURFACE
After substituting i = i * and ( 7) into (2), introducing additional terms i a , C dυ dt , 1 k m (Bω d + T L ), utilizing the expression for i a from ( 9), and replacing the definitions with, Now, if the terms υ, k e ω d , L a di a dt are added and subtracted in (3), ( 9) is used, and the expressions ω , e a = ρ + σ , and r = R a + r a are defined, the following can be written, Finally, if the terms J ωd , k m i a , Bω d are now added and subtracted in (4), the definition of i a in ( 9) is used, ω d (t) = ω * d (t) + ω d and e a = ρ + σ , then it is also possible to write, where ξ is defined in (15).Thus, the system in closed-loop over the sliding surface s = 0 is defined by ( 13)-( 22) and the state vector of the system dynamics is given as

C. STABILITY ANALYSIS OVER THE SLIDING SURFACE
Take into consideration the scalar function M (y s ), which turns out to be positive definite and radially unbounded if α > 0, After executing some math and considering ±qw ≤ |q| |w|, ∀q, w ∈ R, ∥h∥ 1 ≤ √ n∥h∥, ∀h ∈ R n , it is found that the derivative of M (y s ) with respect to time through the trajectories of the closed-loop system dynamics on the sliding surface s = 0, i.e. ( 13)-( 22 The eight leading principal minors of matrix P can be consistently made positive through the following approach. The positivity of the first principal minor is achieved by selecting a sufficiently small α > 0 given that B > 0. While the positivity of the second principal minor is guarantied by using a suitably large k i > 0. The third principal minor is positive if a sufficiently large r > 0 and a small enough β > 0 are defined.In a similar manner, making the fourth principal minor positive requires a sufficiently large r > 0 and a small enough p > 0. For any f 1 > 0, a sufficiently large γ > 0 is adequate to make both the fifth and the sixth principal minors positive.The seventh principal minor can be made positive by selecting a sufficiently large k p1 > 0, and the eighth principal minor is positive when a sufficiently large k i1 > 0 and a small enough δ > 0 are chosen.Hence, it is possible to always ensure that λ min (P) > 0 so (24) can be written as, for some 0 < < 1.
On the other hand, due to This means that |x(t)| = 0, ∀t ≥ t f , as exposed before (25).This also means that y s → 0 when t → ∞.Thus, the proof of proposition 3.1 is completed.
It is worth noting that the previous results are guaranteed if conditions α > 0, β k m and the appropriate selection of α > 0, k i > 0, r > 0, β > 0, r > 0, p > 0, γ > 0, k p1 > 0, k i1 > 0 and δ > 0 are considered in order to ensure the positivity of the eight principal minors related to matrix P defined in (25).
Remark 3.2: Authors highlight that regardless of equalities k p2 = k p /k m and k i2 = k i /k m , previously introduced ( 13), the knowledge of value k m is not necessary but k p and k i must satisfy the conditions of stability met above.Nevertheless, this does not mean that the exact value for both is required.Consequently, expressions k p2 = k p /k m and k i2 = k i /k m can be validly established if sufficiently large values for k p2 , k i2 , and the other controller gains are used.Also, based on the first equality in ( 9), the value of R a must be known.This latter can be eased by choosing a large controller gain r a ensuring that r a ≫ R a , i.e. that r ≈ r a .Likewise, the strict knowledge of the resistance R, i.e., the output of the Buck converter, required in (7), can be mitigated by also selecting a large enough controller gain k p1 .Even more, as detailed in Remark 3.1, this control strategy is anticipated to be robust, thereby compensating for changes in both R a and R through the action of the PI controllers on the output voltage error and the electric current of the Buck converter and the motor armature, respectively.Experiments supporting these observations are presented in Section IV.
Remark 3.3: The process of achieving that the principal minors of matrix P (refer to (25)) be positive, is a relatively straightforward process.Note that the principal minors of P are influenced by the elements over the diagonal of P. Therefore, the i−th principal minor is consistently positive if the (i − 1)−th principal minor is positive, and P ii is increased sufficiently by selecting a large enough value for the controller gain specified in P ii .This iterative process continues until the 8−th principal minor is made positive.
Remark 3.4: It is worth noting that condition ( 11) is satisfied if a suitable desired velocity profile ω d is designed.In this regard, and since the system under study is differentially flat with ω the flat output, the following relationships can be established [56], Considering that ω and i match ω d and i * , respectively, then a perfect tracking can be assumed and ω = ω d , i = i * are used in the latter expressions.From there, the expression υ + L di * dt is numerically evaluated, for a given ω d , and it will be observed if condition ( 11) is whether or not satisfied.This process must be performed off-line.

IV. EXPERIMENTAL RESULTS
In this section, the experimental tests demonstrating the performance capabilities of the control strategy exposed in Proposition 3.1 are presented.In Fig. 3, the experimental setup for the system under study is shown, which allowed to carry out the experimental tests of the closed-loop system.The experimental setup was designed and built at the facilities of the ''Laboratory of Mechatronics and Renewable Energy'' at CIDETEC-IPN in Mexico City.

A. BLOCK DIAGRAM OF THE FULL-BRIDGE BUCK INVERTER-DC MOTOR SYSTEM IN CLOSED LOOP
In order to describe how the experimental results were obtained, the block diagram of the software and hardware components of the used experimental setup is illustrated in Fig. 4.This block diagram includes three sub-blocks that are described below.
• Full-bridge Buck inverter-DC motor system.This sub-block contains the full-bridge Buck inverter-DC motor system to be controlled and the renewable energy power supply, E(t).The latter is implemented by using the TDK-Lambda G100-17.Regarding the measurement of the variables associated with the system, that is, the  Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.

B. STUDY CASES ASSOCIATED WITH E (t ), DISTURBANCES IN THE LOAD RESISTANCE R, AND DEFINITION OF THE DESIRED TRAJECTORY ω d
To verify the performance of the full-bridge Buck inverter-DC motor system after applying the control presented in ( 5)-( 10), experimental tests are conducted.In these tests, the power supply, E(t), considers the following case studies: i) a time-varying input voltage and ii) the emulation of the behavior of a commercial photovoltaic panel.Additionally, for each of these two case studies for E(t), experimental results will be shown with the following three variants related to the system parameters: a) When the nominal parameters associated with the system components are given by ( 27).b) When there are perturbations in the load resistance R, which are experimentally simulated as: c) When there are disturbances in the torque T L .It should be noted that for both primary power sources, E(t), the following piecewise function is considered as the desired angular velocity trajectory ω d : where ϕ 1 t, t i 1 , t f 1 and ϕ 2 t, t i 2 , t f 2 are Bézier polynomials defined as and s , ω f 1 = 13 rad s , ω i 2 = 13 rad s , and ω f 2 = −13 rad s .After mentioning the case studies to be consider for E(t) and defining ω d , the obtained experimental results in the closed-loop system are presented.

C. EXPERIMENTAL TESTS WHEN E (t ) IS A TIME-VARYING INPUT VOLTAGE
The results obtained in these experimental tests use a time-varying voltage function from renewable energy as the power source for the system.According to [57], a waveform associated with E(t) is determined by the following formula: [11.008 + 0.5504 sin(5t) + 0.5848 sin(10t)].(30) When considering the desired angular velocity trajectory (29), the time varying input voltage (30), and the nominal parameters associated with the system components ( 27), the obtained experimental results are shown in Fig. 5.

FIGURE 5.
Experimental plots of the closed-loop system under nominal conditions (27); when the desired trajectory ω d is considered as (29) and the power source of the system E (t ) is given by the time-varying input voltage (30).
On the other hand, when considering ω d given by ( 29), E(t) given by (30), and the perturbations in the load resistance R given by ( 28), the experimental results illustrated in Fig. 6 are obtained.FIGURE 6. Experimental plots of the closed-loop system when there are variations in the load R described in (28); when the desired trajectory ω d is given by (29), and the power source of the system E (t ) is given by the time-varying input voltage (30).
Whereas, in Fig. 7 the obtained experimental graphs in closed-loop when perturbations are introduced in the torque T L in t = 5 s and t = 15 s are presented.In these experimental results, again, the desired angular velocity, ω d , was considered to be the one given by ( 29) and the primary energy source, E(t), to be the one determined by (30).

D. EXPERIMENTAL TESTS WHEN E (t ) EMULATES THE BEHAVIOR OF A COMMERCIAL PHOTOVOLTAIC PANEL
In this section, the obtained experimental results use the voltage supplied by an LG215P1W photovoltaic panel from FIGURE 7. Experimental graphs of the closed-loop system when there are torque perturbations at t = 5 s and t = 15 s; when the desired trajectory ω d is given by ( 29) and the power source of the system E (t ) is given by the time-varying input voltage (30).
LG Electronics as the power source E(t) for the system.The V-I graph of the photovoltaic panel is illustrated in Fig. 8.The experimental implementation of the LG215P1W photovoltaic panel is achieved by using the TDK-Lambda G100-17 source.When considering the desired angular velocity trajectory (29), the LG215P1W photovoltaic panel as the power source of the system, and the nominal parameters associated with the system components (27), the obtained experimental results are shown in Fig. 9.
On the other hand, when considering ω d given by ( 29), the LG215P1W photovoltaic panel as the power source of the system, and the perturbations in the load resistance R given by ( 28), the experimental results illustrated in Fig. 10 are obtained.
Finally, in Fig. 11 the obtained experimental graphs in closed-loop when perturbations are introduced in the torque T L at t = 5 s and t = 15 s are presented.In these experimental results, again, the desired angular velocity ω d , the power source E(t), the LG215P1W photovoltaic panel, and the nominal parameters associated with the system components (27) were considered.

E. DISCUSSION OF THE EXPERIMENTAL RESULTS
According to the obtained experimental results, shown in Figs.5-7 and 9-11, the good performance of the controller proposed in ( 5)-( 10) was demonstrated, since the control objective was achieved, i.e., ω → ω d ; even when abrupt FIGURE 9. Experimental graphs of the closed-loop system under nominal conditions (27); when the desired trajectory ω d is given by ( 29) and the power source of the system E (t ) is given by the photovoltaic panel LG215P1W emulated via the TDK-Lambda G100-17.

FIGURE 10.
Experimental graphs of the closed-loop system; when the desired trajectory ω d is given by (29), variations in the load R are described in (28), and the primary power source E (t ) is the emulation of the LG215P1W photovoltaic panel via the TDK-Lambda G100-17.

FIGURE 11.
Experimental graphs of the closed-loop system when there are torque perturbations at t = 5 s and t = 15 s and when the desired trajectory ω d is given by (29) and the primary power source E (t ) is the emulation of the LG215P1W photovoltaic panel via the TDK-Lambda G100-17.
perturbations were introduced in some parameters of the full-bridge Buck inverter-DC motor system and time-varying sources E(t) were considered.Based on the obtained results depicted in Figs.5-7 and 9-11, it was shown that, despite the variations in the supply voltage E(t) generated from a renewable energy emulation, there is a good tracking of the angular velocity.On the other hand, in Figs. 6 and 10 the abrupt changes in the load resistance of the converter generate an important increase in the current i when the nominal value of R is decreased.However, the above does not affect ω → ω d to be achieved in a good way.It was also shown in Figs.7 and 11 that a torque disturbance directly affects the dynamic behavior of ω.However, the controller compensates such a disturbance.As a final experimental test of the closedloop system, the case is presented below when the nominal parameters of the system are given by ( 27) and the desired angular velocity ω d is a step-like trajectory defined as: The experimental results associated with the time-varying input voltage (30) and the voltage supplied by the LG215P1W photovoltaic panel are shown in Fig. 12, denoted as ω f and ω PV , respectively.It should be emphasized that the system exhibits a fast response even with abrupt changes in ω d , demonstrating the effectiveness of the proposed control strategy.

FIGURE 12.
Experimental results considering the nominal parameters of the system (27) and ω d (t ) defined in (31).On the one hand, ω f corresponds to a power supply E (t ) given by (30).On the other hand, ω PV corresponds to a power supply E (t ) coming from the emulation of an LG215P1W photovoltaic panel.

V. CONCLUSION
In this work, a full-bridge Buck inverter powered by renewable energy as a smooth starter for a DC motor based on sliding mode and PI control was presented.The closed-loop system exhibited asymptotic stability when supplied with the required voltage by the renewable power source to the full-bridge Buck inverter.That is, when condition (11) is met.In this sense, it was shown that such a condition is met no matter if a time-varying power supply or an emulated commercial photovoltaic panel are used.This can be observed in the experimental graphs plotted in Figs.5-7 and 9-11, since the control objective is achieved, i.e., ω → ω d .Likewise, it is observed that the control task is achieved, even when there are abrupt perturbations in some parameters of the system, in the torque of the DC motor, and abrupt changes in the profile of the desired trajectory.Additionally, due to the switched nature of the sliding mode control, the experimental implementation of the control did not require the use of a modulator for its synthesis.Finally, derived from the presented results in this work, the use of different renewable energy sources to power the system under study is proposed as a future work.

FIGURE 3 .
FIGURE 3. Experimental setup associated with the full-bridge Buck inverter-DC motor system.

•
voltages υ and E(t) are acquired through a pair of Tektronix P5200A voltage probes.While the electric currents i and i a are acquired via two Tektronix A622 current probes.In addition, the Omron E6B2-CWZ6C encoder is used to measure the angular velocity ω.The DC motor used in the experiments is a GNM 5440E-G3.1 from Engel.The parameters of the full-bridge Buck inverter-DC motor system are the following: R = 48 , C = 114.4µF, L = 4.94 mH, L a = 2.22 mH, k m = 120.1 × 10 −3 N•m A , R a = 0.965 , k e = 120.1 × 10 −3 V•s rad , J = 118.2×10−3 kg•m 2 , b = 129.6×10−3 N•m•s rad .Sliding mode and PI tracking control.In this sub-block, the desired trajectory ω d is programmed and the controller exhibited in Proposition 3.1 is implemented, both through Matlab-Simulink.The gains of the PI controller that are used in the execution of the experimental tests are the following: k p1 = 29, k p2 = 0.8326, k i1 = 2, k i2 = 9.159, r a = 0.5, r = 1.465, γ = 50, f = 1, β = 0.15, p = 0.5, α = 0.78, f 1 = 0.4003, δ = 0.0005.• Controller board and signal conditioning.This sub-block shows the connections between the DS1104 board and the system to be controlled, i.e., the full-bridge Buck inverter-DC motor system.On the one hand, the control signals u and u are calculated by the DS1104 board and implemented by means of the digital I/O signals.In the first instance, the signals u and u are introduced each via two optocouplers 6N137 and two drivers IR2101.Subsequently, these signals are applied to the transistors Q 1 , Q 2 , Q 1 , and Q 2 to properly drive the full-bridge Buck inverter.On the other hand, the measurements of E(t), υ, i, i a , and θ are processed by means of the signal conditioning blocks.

FIGURE 4 .
FIGURE 4. Block diagram of the closed-loop system.

FIGURE 8 .
FIGURE 8. V-I graphic of the LG215P1W photovoltaic panel.