**Contents**



## **About the Special Issue Editor**

**Luis Martinez-Salamero** taught circuit theory, analog electronics, and power processing from 1978 to 1992 at the Escuela Tecnica Superior de Ingenieros de Telecomunicaci ´ on de Barcelona, Polytechnic ´ University of Catalonia, Barcelona, Spain. From 1992 to 1993, he was visiting scholar at the Center for Solid State Power Conditioning and Control, Department of Electrical Engineering, Duke University, Durham, NC. He continued his cooperation with Duke University for three months per year in 1995 and 1996. From 2003 to 2004, 2010 to 2011, and March–August of 2018, he was visiting scholar at the Laboratory of Architecture and Systems Analysis (LAAS), National Agency for Scientific Research (CNRS), Toulouse, France. Since 1995 he has been a full professor with the Department of Electrical Electronic and Automatic Control Engineering ,School of Electrical and Computer Engineering, Rovira i Virgili University (URV), Tarragona, Spain, where he managed the Research Group in Industrial Electronics and Automatic Control (GAEI) in the period 1998–2018 (http://deeea.urv.cat/gaei/index.php). His research interests include the structure and control of power conditioning systems, namely, the electrical architecture of satellites, power distribution in hybrid and electric vehicles, as well as the nonlinear control of converters and drives, and power conditioning for renewable energy. He has published a large number of papers in scientific journals and conference proceedings in the fields of modelling, simulation, and control of power converters, and holds a U.S. patent on dual-voltage electrical distribution in vehicles. He was Guest Editor of the IEEE Transactions on Circuits and Systems Special Issue on Simulation, Theory and Design of Switched-Analog Networks (Aug. 1997). In cooperation with the European Space Agency (ESA), he organized the 5th European Space Power Conference (ESPC-98) in Tarragona and served during two terms (1996–2002) as a Dean of the School of Electrical and Computer Engineering in URV. He has supervised 15 doctoral theses and given lectures in the universities of Toulouse, Bordeaux, and Tel-Aviv. He has conducted numerous seminars on the use of non-linear control in power converters in both Spanish and foreign universities, and has been invited speaker in four plenary sessions in international conferences. He was president of the Spanish Joint Chapter of the IEEE Power Electronics and Industrial Electronics Societies from 2005 to 2008, and distinguished lecturer of the IEEE Circuits and Systems Society in the period 2001–2002. He served as a president of the committee for Communications, Electronics and Computer Science for the Research Activity Evaluation (CNEAI) of the Spanish Ministry of Education in 2012, and as a national coordinator in the area of Electrical, Electronic and Automatic Control Engineering of the National Agency of Prospect and Evaluation (ANEP) of the Spanish Ministry of Science and Innovation in the period 2008–2011. He is currently a distinguished professor of Rovira i Virgili University.

## **Preface to "Sliding Mode Control of Power Converters in Renewable Energy Systems"**

The first paper on dc-to-dc switching converters published in a scientific journal had only one page and did not mention the term power converter in the description of that emergen<sup>t</sup> family of circuits (1). Almost fifty years later, around 7000 patents have been issued on dc–dc conversion in the USA alone, making us consider whether the interdisciplinary field known as Power Electronics reached maturity long time ago, and if there is still space for innovation.

Some data can help us to answer affirmatively to the second question. They are supported by the concern for the global warming, which has renewed the interest in renewable energy technologies. Underpinning a sustainable environment implies the use of more efficient devices and better control strategies in the generation, transport, and conversion of electric energy. Two vectors catalyze the new research: advances in technology and the new paradigm in power distribution. In the first case, the increasing use of wide-gap power devices and digital processors is contributing to reduce the size, weight, and internal losses of power converters in parallel with better dynamic performances. In the second case, electric vehicles and smart grids are the main industrial actors of the new research, in which the notion of a single converter has been substituted by the concept of multi-converters (i.e., an important number of converters interacting with the energy sources or the loads, using intermediate elements called buses for ac or dc power distribution).

In this context, sliding-mode control continues to become established in the control of power converters since it constitutes a reliable and efficient solution to many open problems. It offers robustness in the face of parametric uncertainty, fast response, and systematic approach in the control design, which feeds on a rigorous general theory that has been successfully applied in other engineering fields like process control and electromechanical systems. This Special Issue of Energies reflects the state of the art of the sliding-mode control of power converters, and the selected papers are representative of different applications in the field of renewable energies.

The first paper illustrates the control design of a three-phase voltage source supplying dc loads without a rectifier. In this work, Alsmadi, Chairez, and Utkin develop switching commands for the power devices of a three-phase PWM AC/DC voltage source converter in such a way that the output voltage can track a desired positive time-varying function.

In the second paper, Yang and Tan exhaustingly cover the recent development of sliding-mode control applications for renewable energy systems and examine the current trends to improve their efficiency and load protections in the face of large-signal variations. A comparative study between sliding-mode control and proportional–integral control is presented in three cases (i.e., a low-power wind energy conversion system, a series–series compensated wireless power transfer system, and a multiple energy storage system in a DC microgrid).

The next paper addresses the design of the input filter of a power converter supplying a constant power load (CPL). Anderson, More, and Puleston present a Lyapunov analysis to tackle the ´ nonlinear internal dynamics of the sliding-mode controlled converter with CPL. Using a Lienard-type description, the authors establish the stability conditions and provide a secure operation region, which is eventually translated into filter design guidelines.

Tuning the parameters of a second-order sliding-mode control of the grid-side converter of a doubly-fed induction generator (DFIG) under unbalanced and harmonically distorted grid voltage is the subject of the fourth paper by Susperregui, Herrero, Martinez, Tapia-Otaegui, and Blasco. A multi-objective optimization is applied for tuning, and two versions of the control algorithm are compared. A Pareto front is eventually derived to help the designer to understand the trade-off among objectives and select the final solution.

A novel adaptive-gain second-order sliding mode direct power control strategy for a wind-turbine-driven DFIG is the subject of the fifth paper. Han and Ma present the control scheme in detail, derive the adaptive gains from a Lyapunov stability approach, and show the resulting reduction of the rotor voltage chattering. Active and reactive power regulation is attained under a two-phase stationary reference frame for both balanced and unbalanced grid voltage.

The dual-stator winding generator (DWIG) is a promising electrical machine for wind energy systems in the low/mid power range. In the sixth paper, Talpone, Puleston, Cendoya, and Barrado-Rodrigo propose a super-twisting algorithm sliding-mode control with moderate real-time computation burden to maximize power extraction during low wind regimes. The results show an extended range of operation of the machine, fast finite convergence for maximum power tracking, reduction of both chattering and mechanical stress, and robustness against parameter uncertainties.

Efficient lighting has become a must for either domestic or industrial applications, the discharge lamps being a representative example of the devices involved in the conversion of electricity into light. In the seventh paper, Valderrama-Blavi, Leon-Masich , Olalla and Cid-Pastor present a versatile ballast for discharge lamps of different type. The ballast has two stages. The first stage is a boost converter exhibiting loss-free resistor characteristics imposed by a sliding-mode control. The second stage is a resonant inverter supplying the discharge lamp at high frequencies. Successful ignition, warm-up and both nominal and dimming operations are illustrated in the paper.

An example of an inductive device with potential use in sea-wave energy harvesting is introduced in the next paper by Garriga-Castillo, Valderrama-Blavi, Barrado-Rodrigo, and Cid-Pastor. The energy obtained by a magnetic pick-up is stored in a battery and used to supply a dc load. An interface circuit for impedance matching between the magnetic pick-up and the battery is used. The interface is a dc-to-dc switching converter exhibiting loss-free resistor characteristics imposed by a sliding-mode control. Two candidates for the role of interface, i.e., a SEPIC converter and the cascade connection of a buck and a boost converters show similar performances after being compared on equal experimental basis.

The control of both output and circulating currents in a modular multilevel converter by means of sliding-mode control is described in the ninth paper. Uddin, Zeb, M.A. Khan, Ishfaq, I. Khan, S.ul Islam, Kim, Park , and Lee use a first-order switching strategy to control the output current, and a second-order switching law-based super-twisting algorithm to control the circulating current and suppress its second harmonic. The proposed control shows better performances than the conventional proportional-resonant regulation scheme.

Digital implementation of sliding-mode control is an important research area today because it has recently opened the way to implement sliding-mode controllers resulting in constant switching frequency. One of these ways is analyzed in detail in the tenth paper by Vidal-Idiarte, Restrepo, El Aroudi, Calvente, and Giral. They interpret a digital input-output linearization strategy in a buck converter under the optics of discrete-time sliding-mode control theory, and their theoretical approach is verified by means of simulations and experiments.

Designing boost inverters is a well-known problem for engineers due to the difficulty of the analytic understanding of how these circuits operate in either autonomous or grid-connected mode. Lopez-Caiza, Flores-Bahamonde, Kouro, Santana, Muller, and Chub present in the eleventh ¨ paper in a dual boost inverter a combination of a resonant control and a sliding-mode control to regulate the current injected to the grid and the balance between the inductor currents respectively. The experimental results validate the theoretical predictions and meet the requirements of power quality standards.

Micro-inverters development is an example of recent directions in power conversion for photovoltaic systems. A design based on the cascade connection of two boost converters and a full-bridge circuit is analyzed in the twelfth paper by Valderrama-Blavi, Rodriguez-Ramos, Olalla, and Genaro-Munoz. Two sliding-mode control alternatives are analyzed and their performances ˜ measured in an experimental prototype. The first one regulates the system energy through the control of the input current while the second one enforces a self-oscillating transformer behavior with variable transformer ratio.

Rivera, Ortega-Cisneros, and Chavira address the control of a boost converter for output tracking of a DC biased sinusoidal signal in the thirteenth paper. They apply discontinuous output regulation based on the use of a sliding function made of a linear combination of tracking errors and an integral term. Experimental results show good tracking of the output voltage with THD less than the 5% standard limit, and excellent rejection of input voltage disturbances.

Multiphase converters have become a practical solution in industry to improve efficiency while reducing size of passive components. In the fourteenth paper, Pajer, Chowdhury, and Rodic propose a multiphase buck converter for battery emulation, which is regulated by a cascade control scheme. The inner loop uses a sliding-mode control for phase currents while the outer loop applies a proportional controller with output current feedforward. Disturbance observers are used in both loops for mismatch compensation. The theoretical analysis is corroborated by experimental results in a 4-phase synchronous prototype.

Power converters supplying a constant power load (CPL) are open-loop unstable, so they can only operate in closed-loop with an appropriate control scheme. El Aroudi, Martinez-Trevino, ˜ Vidal-Idiarte, and Cid-Pastor approach the problem in the fifteenth paper by proposing a digital sliding mode -based control with PWM that results in inrush current limitation. The paper covers exhaustingly the design of a cascade control that eventually yields excellent output voltage regulation and the suppression of inrush current in a boost converter experimental prototype feeding a CPL of 1 kW.

Lopez-Santos, Cabeza-Cabeza, Garcia, and Martinez-Salamero illustrate in the last paper the use of sliding-mode control to obtain high power factor in the bridgeless isolated version of the SEPIC converter, which is used as unidirectional isolated interface between an AC source and a low voltage DC distribution bus. Zero-crossing points are considered here as an additional mode, which is analyzed in detail to demonstrate how the switching surface is reached and the sliding motions ensured. The simplicity of the implementation and the low level of resulting THD show that the proposed control is comparable to the best strategies reported in the technical literature.

Finally, I want to express my gratitude to the large number of reviewers who carried out manuscript reviews in record time. The quality of this special issue is also due to their deep and thorough work.

> **Luis Martinez-Salamero** *Special IssueEditor*

## *Article* **Design of a Continuous Signal Generator Based on Sliding Mode Control of Three-Phase AC-DC Power Converters**

#### **Yazan M. Alsmadi 1,\*, Isaac Chairez 2 and Vadim Utkin 3**


Received: 11 October 2019; Accepted: 18 November 2019; Published: 23 November 2019

**Abstract:** In recent years, hundreds of technical papers have been published which describe the use of sliding mode control (SMC) techniques for power electronic equipment and electrical drives. SMC with discontinuous control actions has the potential to circumvent parameter variation e ffects with low implementation complexity. The problem of controlling time-varying DC loads has been studied in literature if three-phase input voltage sources are available. The conventional approach implies the design of a three-phase AC/DC converter with a constant output voltage. Then, an additional DC/DC converter is utilized as an additional stage in the output of the converter to generate the required voltage for the load. A controllable AC/DC converter is always used to have a high quality of the consumed power. The aim of this study is to design a controlled continuous signal generator based on the sliding mode control of a three-phase AC-DC power converter, which yields the production of continuous variations of the output DC voltage. A sliding mode current tracking system is designed with reference phase currents proportional to the source voltage. The proportionality time-varying gain is selected such that the output voltage is equal to the desired time function. The proposed new topology also o ffers the capability to ge<sup>t</sup> rid of the additional DC/DC power converter and produces the desired time-varying control function in the output of AC/DC power converter. The e ffectiveness of the proposed control design is demonstrated through a wide range of MATLAB/Simulink simulations.

**Keywords:** sliding mode control (SMC); power converter; continuous signal generator; equivalent control; AC-DC power converter

## **1. Introduction**

Sliding mode control (SMC) has high order reduction property, good dynamic performance, low sensitivity to disturbances, and plant parameter variations, allowing SMC to handle nonlinear systems with uncertain dynamics and disturbances. Additionally, SMC is decoupled into independent lower dimensional subsystems, simplifying feedback control design. These properties allow SMC to be used in a wide range of applications such as automotive control, robotics, aviation, power systems, power electronics, and electric motors [1–5].

Power electronic converters are controlled by switching electrical components, which can produce two dissimilar values at the gating terminals [6,7]. Their controlled variables may take values from a two valued discrete set. Moreover, linearization is not required [1,2,8,9]. Hence, SMC is a preferred method to realize the control of power converter devices.

Within the wide diversity of available power electronic devices, the well-known three-phase AC/DC is commonly applied in energy conversion plants. Nevertheless, inherent complications appear with regards to reactive power generation, as well as the higher harmonic content in the input current. These characteristics appear as practical disadvantages that have become more relevant as the AC/DC converter capacity turn out to be larger and larger [10–12].

The idealized AC/DC converter shows up as a constant DC voltage as controlled output (or current) and a sinusoidal input set of currents at unity power factor at the AC line. Nevertheless, the current technology of thyristor phase-controlled converters has two intrinsic disadvantages: First, the larger firing angle, the smaller power factor; and second, the line current has moderately large harmonics components [13,14]. As AC/DC converters are more and more controlled using PWM switching patterns, the input as well as the output performances improve. These PWM AC/DC converters offer numerous advantages compared to some traditional rectifiers [15,16]: Unity power factor, low harmonic components in input current, bidirectional power flow, and low ripple in output voltage. These characteristics make simpler the filtering processes on both AC and DC sides of the proposed converter [1,2,10,16].

Conventional control design techniques for this type of power converter device usually solve the maintaining of the DC output voltage at a given reference level firstly, and at second place, try to seek for the minimization of high order harmonics and reactive power at the input. SMC of AC/DC power converters, presented in [1], o ffers the inverse sequence of actions. First, a current tracking system was designed with sinusoidal current references that are proportional to the AC input voltages with a constant gain of proportionality. This automatically sets the reactive power to zero. Second, it was proven that the output voltage will be constant and only depends on the amplitude of the reference input. However, the proposed control method requires an additional DC/DC converter to control the DC load. This introduces the following control design challenge: Is it possible to avoid using an additional DC/DC converter and generate any arbitrary desired time varying function at the output of the AC/DC converter such that the DC load can be directly controlled?

The main contributions of this study are:

(a) Sliding mode control is an appropriate tool for application for wide range of power converters. The first publications on DC/DC converters [1–4,7,8] demonstrated its e fficiency. The methods of minimization heat losses and of chattering amplitude based on harmonic cancellation principle, switching frequency control for DC/DC converters can be found in [2–4]. Multidimensional sliding modes were utilized in power converters to control AC load with DC energy source [5,6]. The design methodology to control output constant voltage and power factor simultaneously for AC-DC converter was developed in [7]. We are not aware of publications with our problem statement—to have an arbitrary time function (not constant) in the output of AC/DC converter. The attempts to find time of the varying gain as an algebraic state function of state failed, because it should satisfy the algebraic equation.

(b) A SMC has been proposed to generate continuous waveforms based on a controlled switched sequence of a three-phase AC-DC converter. This achievement was a consequence of solving a trajectory tracking of estimated reference currents. The realization of such tracking enforces the production of bounded derivative DC output voltage. The tracking controller implemented a time-varying relationship between the currents and voltages on the AC side of the converter. Such given positive relation between voltage and current justified the positive power efficiency of the controlled power converter.

## **2. Problem Statement**

The design problem considered in this study is the generation of switching commands for the power electronics-switching elements of the AC/DC converter (shown in Figure 1) in such a way that the output voltage can track the desired output of the converter *f*(*t*), which should be positive. This condition agrees with the classical realization of AC/DC, buck, and boost power converters. Since the input AC voltage is bounded, the output capacitor *C* can be charged at a limited velocity, which means that the time derivative *f*(*t*) should be bounded. Therefore, the problem can be described as fixing the switching sequence such that:

.

$$\lim\_{t \to \infty} \left| f(t) - \upsilon\_{dc} \right| = 0 \tag{1}$$

**Figure 1.** Scheme of the three-phase PWM AC/DC voltage source converter.

The power efficiency can be maximized if each input phase current of the power converter is proportional to the corresponding phase voltage with a positive gain. Therefore, the objective of this paper is to design a new control algorithm such that the input phase currents track preselected reference inputs and the positive gain of the proportionality is selected as a time varying function. Accordingly, the output voltage is equal to the desired function *f*(*t*).

#### **3. Circuit Model of the Three-Phase PWM AC**/**DC Voltage Source Converter Scheme**

Figure 1 shows the three-phase PWM AC/DC voltage source converter scheme. *ea*, *eb*,*ec* are the balanced three-phase AC input voltages; *idc* is dc-link current; *RL* is a resistive load connected to the DC side; *iL* is the load current; *ia*, *ib*, *ic* are the three-phase AC input currents; *Cdc* is the dc-link capacitance *vdc* is dc-link voltage; *R g* and *Lg* represent the grid-side resistance and inductance, respectively.

The balanced three-phase AC input currents are given by:

$$L\_{\mathcal{S}} \frac{d\dot{i}\_a}{dt} = \varepsilon\_a - R\_{\mathcal{S}} \dot{i}\_a - \upsilon\_{an} \tag{2}$$

$$L\_{\S} \frac{di\_b}{dt} = \mathcal{e}\_b - R\_{\S} i\_a - \mathcal{v}\_{bn} \tag{3}$$

$$L\_{\mathcal{S}} \frac{di\_{\mathcal{C}}}{dt} = \varepsilon\_{\mathcal{C}} - R\_{\mathcal{S}} i\_{\mathcal{C}} - \upsilon\_{\mathcal{C}n} \tag{4}$$

where *van*, *vbn*, *vcn* are the AC side phase voltages of the converter. The balanced three-phase AC voltages are given by:

$$
\varepsilon\_a = E\_0 \sin(\omega t) \tag{5}
$$

$$x\_b = E\_0 \sin\left(\omega t - \frac{2\pi}{3}\right) \tag{6}$$

$$e\_c = E\_0 \sin\left(\omega t + \frac{2\pi}{3}\right) \tag{7}$$

Here, ω is the AC power source angular frequency and *E*0 is the amplitude of the phase voltages. Assume that *iabc* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *iaibic* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, *eabc* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *eaebec* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, *vs* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *van vbn vcn* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦, then Equations (2)–(4) can be re-written in a compact form:

$$L\_{\circ} \frac{d\dot{l}\_{\text{abc}}}{dt} = \varepsilon\_{\text{abc}} - R\_{\circ}\dot{\iota} - \upsilon\_{\text{s}} \tag{8}$$

Define the switching function *S* of each switch as:

$$S\_j = \begin{cases} \begin{array}{ll} 1, & S\_j \quad \text{is close} \\ -1, & S\_j \quad \text{is open} \end{array} & j = a, b, c \end{cases} \tag{9}$$

As a result, the voltage vector *vs* can be given in terms of the switching functions *S* = ⎡⎢⎢⎢⎢⎢⎢⎢⎢⎣ *Sa Sb Sc* ⎤⎥⎥⎥⎥⎥⎥⎥⎥⎦ as:

$$v\_s = \frac{1}{3} v\_{dc} \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix} \mathbf{S} \tag{10}$$

By substituting Equation (10) into (8), the AC input current equations can be given by:

$$\mathcal{L}\_{\mathcal{S}} \frac{d\dot{q}\_{\rm abc}}{dt} = \varepsilon\_{\rm abc} - R\_{\mathcal{S}} i\_{\rm abc} - \frac{1}{3} \upsilon\_{\rm dc} \left| \begin{array}{cc} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{array} \right| \mathcal{S} \tag{11}$$

In conclusion, the output voltage can be given by:

$$\mathbf{C}\frac{d\mathbf{v}\_{\rm dc}}{dt} = -\mathbf{i}\_{L} + \mathbf{i}^{T}\mathbf{S} \tag{12}$$

#### **4. Sliding Mode Current-Tracking Control**

As previously indicated, a sliding mode-based current tracking system is designed such that sinusoidal reference inputs are tracked by phase currents proportional to input AC voltages. Rewriting Equation (11) as:

$$L\_{\%} \frac{di\_{\rm abc}}{dt} = \varepsilon\_{\rm abc} - R\_{\rm g} i\_{\rm abc} - \frac{1}{3} v\_{\rm dc} \Gamma\_{\rm O} S \tag{13}$$

where, matrix Γ0 is given by:

$$
\Gamma\_0 = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \\ 1 & 1 & 2 \end{bmatrix}, \det \Gamma\_0 = 0 \tag{14}
$$

Since the sum of the three-phase currents is zero, only three state variables should be controlled in a system with a three-dimensional control vector S: Two phase currents and output voltage. However, because matrix Г0 is singular, the conventional sliding mode approach cannot be directly applied. Therefore, a tracking system for two phase currents only is first designed. As a result, the sliding mode should be enforced on the intersection of two surfaces σ*a* = *Lg iare f* − *ia* and σ*b* = *Lg ibre f* − *ib* , or in a vector form:

$$
\sigma\_{ab} = \mathcal{L}\_{\mathfrak{J}} (i\_{abref} - i\_{ab}) \tag{15}
$$

Excluding the phase current *ic* = −*ia* − *ib* yields:

$$L\_{\mathcal{K}} \frac{di\_{ab}}{dt} = \varepsilon\_{ab} - R\_{\mathcal{K}} i\_{ab} - \frac{1}{3} \upsilon\_{dc} \Gamma \mathcal{S} \tag{16}$$

*Energies* **2019**, *12*, 4468

.

$$C\frac{dv\_{dc}}{dt} = -\frac{v\_{dc}}{R\_L} + (i\_{a\prime}i\_{b\prime} - i\_a - i\_b)S\tag{17}$$

where Γ is a 2 × 3 matrix given by:

$$
\Gamma = \begin{bmatrix} 2 & 1 & 1 \\ 1 & 2 & 1 \end{bmatrix} \tag{18}
$$

The ideal tracking system is based on the Lyapanov function:

$$V = \frac{1}{2} \,\,\sigma\_{ab}\,^T \,\sigma\_{ab} \tag{19}$$

 *V* has to be negative definite and calculated on the system trajectory when selecting discontinuous control:

$$\dot{V} = \sigma\_{ab}\,^T F(.) - \, (\mathfrak{u}\_{dc}/L) (\mathfrak{a}\_\prime \mathfrak{z}\_\prime \gamma) S \tag{20}$$

where *<sup>F</sup>*(.) is state function, which does not depend on control. α, β, and γ are given by:

$$\begin{cases} \alpha = (2\sigma\_a + \sigma\_b) \\ \beta = (\sigma\_a + 2\sigma\_b) \\ \gamma = (\sigma\_a + \sigma\_b) \end{cases} \tag{21}$$

If *vdc*/*L* is large enough, *<sup>F</sup>*(.) can be suppressed with control *S* given by:

$$S = \begin{cases} S\_{\mathfrak{a}} = \operatorname{sign}\left(\alpha\right) \\ S\_{\mathfrak{b}} = \operatorname{sign}\left(\beta\right) \\ S\_{\mathfrak{c}} = \operatorname{sign}\left(\gamma\right) \end{cases} \tag{22}$$

such that . *V* = <sup>σ</sup>*abTF*(.) − (*udc*/*L*)<sup>|</sup>α<sup>|</sup> + --β--- + --γ--- < 0. As a result,<sup>σ</sup>*ab* tends to zero and σ*ab* becomes zero after a finite time interval [17–19]. Consequently, sliding mode occurs with *iab* = *iabre f* . Therefore, the current tracking system is developed with sinusoidal current references proportional to the input AC voltages as:

$$i\_{ab} = K(t) \times \varepsilon\_{ab} \tag{23}$$

Calculate the equivalent control [2]:

$$(\Gamma S)\_{\alpha\eta} = \left(-L\_{\mathcal{g}}\frac{d\dot{i}\_{abref}}{dt} + \mathfrak{e}\_{ab} - R\_{\mathcal{g}}\dot{i}\_{ab}\right)\frac{\mathfrak{Z}}{\mathfrak{u}\_{\mathfrak{c}}}\tag{24}$$

The three phase input currents can also be expressed as:

$$
\begin{pmatrix} i\_a & i\_{b\_\succ} \ -i\_a \ -i\_b \end{pmatrix} = \frac{1}{3} (\dot{i}\_a - \ i\_{c\succ} \ \dot{i}\_b - \ i\_c) \ \Gamma \tag{25}
$$

The sliding mode equation can be obtained by substituting Equations (24) and (25) into (17):

$$\mathcal{L}\left(\frac{d\upsilon\_{dc}}{dt}\right) = \left(-\frac{\upsilon\_{dc}}{R}\right) + \frac{1}{\upsilon\_{dc}}\left(i\_a - i\_{c\prime}i\_b - i\_c\right)\left(-L\_{\text{g}}\frac{d}{dt}i\_{ab\upsilon\prime f} + c\_{ab} - R\_{\text{g}}i\_{ab}\right) \tag{26}$$

After sliding mode occurs (*iab* = *<sup>K</sup>*(*t*)*eab*):

$$i\_{ab} = i\_{ab,ref} \\\frac{d}{dt}i\_{ab} = \frac{d}{dt}i\_{ab,ref}$$

Therefore,

$$\frac{d}{dt}\dot{\iota}\_{ab} = K(t)\frac{d}{dt}\varepsilon\_{ab} + \varepsilon\_{ab}\frac{d}{dt}k(t) \tag{27}$$

where, *ia* − *ic* = *<sup>K</sup>*(*t*)(*ea* − *ec*,*eb* − *ec*).

> It can be shown that the output voltage *vdc* is given by:

$$\mathbf{C}\frac{d\upsilon\_{dc}}{dt} = \frac{-\upsilon\_{dc}}{R\_L} + \frac{1}{\upsilon\_{dc}} \Big( -\frac{3}{2} L\_\text{\space{R}} K E\_0^2 \frac{d}{dt} \mathbf{K} + \frac{3}{2} K E\_0^2 - \frac{3}{2} R\_\text{\space{R}} K^2 E\_0^2 \Big) \tag{28}$$

If *y* = *<sup>v</sup>*2*dc*, then the gain *K* should satisfy the differential equation

$$\left(\frac{1}{2}\mathbb{C}\frac{dy}{dt} + \frac{1}{R\_L}y\right)\frac{2}{3KE\_0^2} = -L\_\% \frac{dK}{dt} + 1 - R\_\% K\tag{29}$$

Assume that *f* is the voltage reference and *y*ˆ = *f* 2, then the equation

$$\left(\frac{1}{2}\mathbb{C}\frac{d\boldsymbol{\mathcal{Y}}}{dt} + \frac{1}{R\_L}\boldsymbol{\mathcal{Y}}\right)\frac{2}{3\mathcal{K}\mathcal{E}\_0^2} = -L\frac{d\mathcal{K}}{dt} + 1 - R\_{\mathcal{K}}\boldsymbol{\mathcal{K}}\tag{30}$$

can be simulated in the controller,

$$
\Delta y = \hat{y} - y
$$

$$\frac{1}{2}\mathbb{C}\frac{d\Delta y}{dt} + \frac{1}{R\_L}\Delta y = 0\tag{32}$$

Equation (32) shows that Δ*y* → 0 and the output voltage tends to be the reference input *vdc* → *f* . Assume that

$$\begin{aligned} \left| \frac{3}{2} \left( \frac{1}{2} \mathbb{C} \frac{d}{dt} f^2 + \frac{1}{R\_L} f^2 \right) \right| &\le M\_\prime \\\\ L\dot{K} &\ge 1 - R\_{\mathcal{S}} K - \frac{M}{K E\_0^2} \end{aligned} \tag{33}$$

If *RgK* = ε < 1, then

$$\lim\_{K\_0^2 \to \infty} \left( 1 - R\_\% K - \frac{M}{KE\_0^2} \right) < 1\tag{34}$$

It means that

then:

$$L\frac{dK}{dt}\_{\mathcal{R}\_\delta K=\varepsilon} > 0\tag{35}$$

and *K* is always positive if *RgK*(0) > ε. This corresponds to a power factor equal to 1 (*iabc* = *Keabc*) for a high enough amplitude of the input source voltage. It is important that *L* . *K* in (29) will be negative for high enough values of *K*. Hence, the gain *K* is bounded.

Remark: The implementation of the proposed controller in embedded systems requires the online measurement of the dc voltage at the capacitor and the possibility of realizing fast enough oscillations on the switching electronic elements. This part of the problem can be solved using available fast dedicated microcontrollers devices. Notice that the accuracy of the produced signal is a function of the relative relationship between the switching devices operation frequency and the main frequency components of the desired signals. Evidently, the exact reconstruction of the desired signal cannot be acquired. Nevertheless, the studies regarding discrete implementation of sliding mode has shown that the accuracy of the sliding mode realization is proportional to the square power of the sampling period if the explicit discretization is considered. For more details on the implementation issues, please see references [20–23]. Remark: The proposed control strategy offers an alternative to some other sliding mode controllers designs considering adaptive pulse width modulation [24], sub-optimal regulation [25], and multitype restrictive [26] approaches which have been applied on DC-DC power converters to obtain arbitrary signals. However, not one of them has been tested on the AC-DC device.

## **5. Simulation Results**

In order to evaluate the proposed sliding mode control design procedure several computer simulations have been conducted using MATLAB/Simulink software. The control algorithm is represented in the following flow diagram (Figure 2):

**Figure 2.** Flow diagram describing the sliding mode control realization.

Different generated signals have confirmed the abilities of the proposed continuous waveform generators.

The simulation was performed for power converter governed by Equation (10) with control (21) and different desired functions *f(t)* in the converter output. The differential equation for time varying gain *K(t)* in Equation (29).

#### *5.1. Sinusoidal Waveform Generation*

The first signal is a pure sinusoidal waveform, which corresponds to a traditional signal used in diverse signal generators. The selected reference waveform is:

$$f(t) = 250(\sin(\omega t) + 1) \ge 0 \text{ (35)}\tag{36}$$

The results of the simulation are shown in Figure 3.

**Figure 3.** *Cont*.

**Figure 3.** Comparison of the reference and the generated sinusoidal continuous waveform for a fixed period: (**a**) (0,10.0) and (**b**) (0,0.1) s.

The input currents obtained by the application of the suggested first sliding-mode controller show a modulated sinusoidal shape in all three branches (Figure 4).

**Figure 4.** *Cont*.

**Figure 4.** Time variation of the input currents at the three branches of the AC source on the period (2.0–2.5) s (closer view) for the three branches: (**a**) *ia*, (**b**) *ib*, and (**c**) *ic* with the a sinusoidal reference.

The time dependence of the current ˆ *is* also shows the expected modulation with the frequency of the desired output current, which is 15 Hz (Figure 5).

**Figure 5.** Time variation of the controller gain K.

The phase relations between the input currents hold both in the transient and the steady state periods. In the period between 0.0 and 0.1 s, the current decreases exponentially to the steady state, which is detected after 0.1 s. A phase shift of 2/3 is evidenced, which also confirms the efficiency of the suggested controller. Notice that the simultaneous dependence of *K* with respect to the reference waveform as well as its derivative does not relate it to the reference voltage form. The gain variation is continuous but not necessarily differentiable, considering the gain structure estimated in this study.

#### 5.1.1. Variable Frequency Sinusoidal Waveform Generation

The second proposed reference signal is a composite sinusoidal waveform, which corresponds to a class of simplified chirp signal. Such waveforms can be used for testing the spectral response of diverse systems for calibration purposes. The selected composite sinusoidal reference waveform is:

$$f\_2(t) = \begin{cases} 200(\sin(5t) + 1) & if \quad 0 \le t < 2\\ 100(\sin(t) + 1) & if \quad 2 \le t < 8\\ 200(\sin(5t) + 1) & if \quad 8 \le t \le 10 \end{cases} \tag{37}$$

Once more, the selected bias constants mean a positive waveform. The selected transition times between the sinusoidal forms can design a continuous composite waveform with a bounded derivative. The simulation results are shown in Figure 6.

**Figure 6.** Comparison of the reference and the generated sinusoidal continuous waveform for a fixed period: (**a**) (0,10.0) and (**b**) (0,0.1) s.

The input currents agree with the variation of the sinusoidal frequency by the application of the suggested first sliding mode controller with the corresponding modulated sinusoidal shape in all three branches (Figure 7).

When looking at the time variation of the gain *K* for the controller, notice that the simultaneous dependence of *K* with respect to the reference waveform, as well as its derivative does not relate it to the reference voltage form. The second waveform considered in this study produces a smoother variation of the gain *K*. The gain variation is continuous considering the gain structure estimated in this study (Figure 8).

**Figure 7.** Time variation of the input currents at the three branches of the AC source on the period (2.0–2.5) s (closer view) for the three branches: (**a**) *ia*, (**b**) *ib*, and (**c**) *ic* with the reference sinusoidal.

**Figure 8.** Time variation of the controller gain *K*.

#### 5.1.2. Triangular Waveform Generation

The third suggested reference signal is a triangular signal, which is also a common signal used in the calibration of diverse devices. Notice that this signal has a bounded but not continuous derivative. Consequently, the suggested controllers are applicable (Figure 9).

**Figure 9.** Comparison of the reference and the generated triangular waveform for a fixed period: (**a**) (0,10.0) and (**b**) (0,0.1) s.

For the class of triangular signal, the controller succeeded at reconstructing the suggested reference signal as shown in Figure 9a. Consequentially, the tracking error of the reference voltage is reduced to less than 0.05% over a period of 0.08 s (Figure 9b).

The time variation of the gain *K* for the controller with the reference triangular signal appears in Figure 10. The gain variation function is continuous.

**Figure 10.** Time variation of the controller gain *K*.

Even if the exact sliding motion can be acquired if and only if the switches in the power converter oscillate at the infinite frequency, the current available technology allows to oscillate at such high frequencies ensuring the existence of the practical sliding motion. On the other hand, the required high frequency oscillations of the sliding mode may produce heath losses which could damage the switching circuit. In all the presented cases, chattering phenomenon should mentioned always when applying sliding mode control. The set of chattering suppression methods has been developed in the framework of sliding mode control theory. They are surveyed in [27]. The harmonics cancellation principle is the most efficient for power converters and can be applied for our case. The design idea consists in using several parallel converters with controlled phases such that high order harmonics can be cancelled.

## **6. Conclusions**

This paper has presented the control design procedure to directly control DC loads using a three-phase voltage source without a rectifier. It consists of two steps. First, the current tracking problem is solved with reference currents proportional to phase voltages. Then, the time varying proportionality coefficient is selected such that the output voltage is equal to the desired time function. It has been shown that the proportionality gain should satisfy the first order differential equation, which is implemented in the controller. The behavior of the system with a positive coefficient is equivalent to having a unity power factor. Stability of the complete system (the power converter and controller dynamics) was also proved. A wide range of computer simulations were provided to demonstrate efficiency of the proposed control design for different types of sinusoidal and triangular functions as voltage reference inputs.

**Author Contributions:** Formal analysis, Y.M.A., I.C., and V.U.; Investigation, Y.M.A., I.C., and V.U.; Methodology, Y.M.A., I.C., and V.U.; Supervision, V.U.; Validation, V.U.; Writing—Original draft, Y.M.A., I.C., and V.U.

**Funding:** This research work was partially funded by the [Consejo Nacional de Ciencia y Tecnología] gran<sup>t</sup> number [CB-20181457] and the [Instituto Politécnico Nacional] gran<sup>t</sup> number [SIP-20191724].

**Conflicts of Interest:** The authors declare no conflict of interest.
