# Enhanced Second-Order Sliding Mode Control Technique for a Five-Phase Induction Motor

Authors: Mahmoud A. Mossa; Hamdi Echeikh; Najib El Ouanjli; Hassan Haes Alhelou
Journal: International Transactions on Electrical Energy Systems
 (2022)

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## Abstract

Recently, several research papers have addressed multiphase induction motor (IM) drives, owing to their several benefits compared to the three-phase motors, including increasing the torque pulsations frequency and reducing the rotor harmonic current losses. Thus, designing a robust controller to ensure the proper operation of such motors became a challenge. The present study reports the design of an effective second-order sliding mode control (SO-SMC) approach for a five-phase IM drive. The proposed control approach finds its strongest justification for the problem of using a law of nonlinear control robust to the system uncertainties of the model without affecting the system’s simplicity. The formulation of the proposed SO-SMC approach is a prescribed process to ensure the stability and proper dynamics of the five-phase IM. A detailed stability analysis is also presented for this purpose. To validate the effectiveness of the proposed controller, the five-phase IM drive is tested under different dynamic situations, including load changes and system uncertainties. The presented numerical results prove the ability of the designed SO-SMC to handle high system nonlinearities and maintain high robustness against uncertainties.


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## Body

## 1. Introduction

The control of electric machines always raises the interest of the scientific communities. This is due to the diversity of industrial applications requiring speed, torque, or position control. Starting with the continuous current machine (DC), the evolutionary trend took place in the 80s towards the direction of substituting this machine by that with alternating current (AC). Therefore, the three-phase machine associated with a static converter constituted a variator whose use is largely present in the industrial sector. This interest has arisen on the one hand thanks to the development of power electronics and on the other hand thanks to the technical and economic advantages offered by this machine compared to the DC machine [1]. Nevertheless, the three-phase electric actuator remains restricted to the lower limit of the range of high power up to a few megawatts due to the electrical stresses experienced by the semiconductors. Thus, for applications requiring a higher level of power, a segmentation of the power by multiplication of the number of machine phases is necessary. Indeed, the distribution of the power in a number of phases greater than three helped significantly in handling high-power applications, and this solution resulted in what is known as “multiphase machines.”Even though the conception of variable speed multiphase machines drives dates from the 1960s, the multiphase actuators were not seriously utilized in the industry until the mid-1990s. The industrial applications include the propulsion systems of ships, traction locomotives, hybrid and electric vehicles, avionics, and industrial applications of high power. As a result, there has been an escalating interest in these systems around the world, resulting in a massive research work over the past decade [2–16]. In comparison with conventional three-phase machines, multiphase machines have exceptional outstanding features. Firstly, the increase in the number of phases makes it possible to reduce electromagnetic torque ripples [17, 18]. Multiphase machines are as a result an attractive selection for demanding applications in respect of acoustic or vibratory discretion. Then, the multiplication of the number of phases increases the possibilities of functioning in degraded (faulty) mode while preserving an acceptable quality of torque in terms of ripples and amplitude. This problem is fundamental for applications that guarantee excellent continuity of service [19–21]. Indeed, the treatment resulting from the degradation of the power supply of the three-phase machines requires material modifications, which must be reduced to the minimum to ensure the continuation of operation at a minimum cost. Nevertheless, the approach is different when choosing and using polyphase machines.Indeed, the continuation of operation for these machines in degraded mode goes through a control strategy taking into account additional degrees of freedom without the need to add additional equipment. In addition, a structure with a large number of phases means power segmentation, which decreases the rating constraints on the power electronics units supplying each phase. This is why multiphase machines are used in very high-power applications [22, 23]. In addition, the complementary degrees of freedom available for multiphase machines are utilized for torque production. In other words, these extra degrees of freedom can be exploited to provide improvements in torque production through the injection of high-order current harmonics. They also allow the inverter to drive multiple multiphase motors simultaneously. As a result, multiphase drives are becoming more common in certain variable speed utilizations, such as marine, avionics, automotive, and railroad traction [24, 25].In the formulation of any control problem, there are typical anomalies between the real system and the mathematical model developed for the control design. This distinction may be due to the variation of the parameters of the system dynamics or to the approximation of the complex system behavior by a model. This has led to intense interest in developing robust control methods that seek to address this problem. The most known control strategy of the machine is the flux-oriented control [26, 27]. The FOC is a control algorithm that is based on integral proportional action. The FOC control may be sufficient if the requirements on the accuracy and performance of the system are not too strict. Nevertheless, in the opposite case and particularly when the ordered part is subjected to strong nonlinearities and to temporal variations, it is necessary to design control algorithms ensuring the robustness of the behavior of the process vis-à-vis the uncertainties on the parameters and their variations. Since the simple PI regulator presents insufficient performance for AC current regulation, several control techniques using different approaches were proposed in the literature. Several nonlinear state feedback strategies, especially sliding mode control (SMC), eventually are an interesting substitution to the FOC technique for three-phase motor drives to carry out great achievements [28]. This strategy has been extensively adapted to the traditional three-phase motor drives justifying diverse attractive lineaments in comparison with the traditional FOC.The main prosperities of SMC are its robustness in case of parameter uncertainties and disturbance. This strategy of control attempts great stability performance and very fast dynamic response [29]. Nonetheless, the chattering phenomenon of the SMC remains the main limitation of such a technique, which has been exposed in several studies [30, 31]. The study in [30] proposed an adaptive SMC based on recurrent RBFN applied to an induction motor servo drive to decrease the chattering generated by the estimation of the unmodeled uncertainties. In [29], an adaptive control is developed to change the saturation function to avoid the chattering issue. In [31], an original adaptive SMC is presented to obtain a synchronized control algorithm for several PMSMs, which shows promising results in the point of view of decreasing the chattering problem and high-performance control.Different control strategies for multiphase machines and specially five-phase induction motors have been applied as an illustration of the two recognized control methods of the FOC [32] and the direct torque control (DTC) [16, 33–35], owing to their simple implementation. Nonetheless, the considerable inconvenience of these strategies of control is their notable sensitivity due to the existence of torque ripples and the change of motor parameters. Several researches can be found in the literature dealing with those methods. The study in [36] proposed indirect rotor flux-oriented control of an FPIM with novel rotor resistance estimation utilizing a sliding mode estimator to avoid the effects that come from the rotor resistance variation. Also, the same authors presented in [37] an adaptive DTC using sliding mode-Luenberger observer for online stator resistance observation for FPIM drives to overcome the stator resistance variation for DTC. Moreover, it is well-known that the nonlinear control is less sensitive against the system parameters; however, few researchers applied a nonlinear technique to a multiphase machine. In [38], the authors proposed a nonlinear backstepping control of FPIM drive at low-speed conditions, and it was verified through the experimental implementation.In [39], a backstepping strategy articulated on space vector pulse width modulation (SVPWM) is developed. This utilization associates the proposed control strategy with the presented motor method. The presented control strategy gives an admirable performance and a rapid speed tracking response. Additionally, it carries out some key improvements like the enhancement of the robustness to the inevitable motor parameters variation, which aid first in enhancing the dynamic performance of the motor and second in decreasing the torque ripples. However, designing a robust controller for the five-phase IMs and testing their ability to encounter the parameters change without incorporating adaptive observers are quite rare, and for this reason, the present paper introduces an analytical study of a new adaptive SMC for a sinusoidal winding distribution five-phase IM. The essential improvement of this paper is the design of an adaptive second-order SMC of a five-phase IM through which the shortages in classic SMC are avoided, and the system robustness is effectively enhanced.The study contributions can be itemized as follows:(i)
A robust adaptive second-order sliding mode control is designed and analyzed in order to ensure the robustness of a five-phase IM drive system against uncertainties and load disturbance as well.(ii)
The design procedure is accomplished in a systematic manner in order to visualize the base principle of the controller.(iii)
Different from the control approaches, which considered the sliding mode theory to only design one or two control loops (i.e., speed and flux) and used a PI controller for the remaining loops (i.e., current), the proposed SO-SMC considers designing all control loops together.(iv)
The proposed SMC control system is considered an effective tool for handling high nonlinearities.(v)
The validation of the proposed SMC system is achieved using extensive evaluation tests.The structure of the paper is formulated as follows: Section 2 introduces the model of FPIM. Then, Section 3 presents the design of a novel method to determine the current references, respectively, for thed-q and w-u planes. The voltage references for both planes d-q and w-u are also determined in Section 4. The evaluation results are introduced and discussed in Section 5. Meanwhile, Section 6 is dedicated to the research outcomes.


## 2. Modeling of Five-Phase IM

A layout of the drive system utilized in this paper is shown in Figure1. The system is formed by an FPIM of a two-level IGBT-established power converter. In the first step, the modeling equations taking account a distributed winding five-phase IM are defined, and the phase variables model is described [17]. The model is evaluated in the (α-β, x-y) stator frame adopting the Concordia transformation. Then, for control purposes, the model needs to be inspected in the (d-q, w-u) rotating frame, investigating the Park transformation.Figure 1
Configuration of five-phase IM drive system.The FPIM under study consists of a stator with five windings which are uniformly distributed over the circumference of the stator with an electrical displacement ofυ = 2π/5; meanwhile, the rotor has a squirrel-cage structure. Two different methods can be contemplated in the time of modeling an electrical motor, particularly the vector space decomposition algorithm and phase variable modeling. The motor reluctance is independent of the rotor position, the motor is composed of identical windings uniformly distributed over the rotor and the stator, and finally, the rotor has a squirrel-cage structure. Implementing Concordia and Park transformations given in [17], the model of five-phase IM in the (d, q, w, u) rotating frame with sinusoidal electromotive force under balanced circumstances is defined by the following equation:(1)Ω˙=BmJmΩ+pMLrφisq−1JmTL,isd˙=−RtσLsisd+ωsisq+μσLsTrφ+1σLsVsd,isq˙=−RtσLsisq−ωsisd−μσLsωφ+1σLsVsq,isw˙=−1Tlsisw+1LlsVsw,isu˙=−1Tlsisu+1LlsVsu,φ˙=MTrisd−1Trφ,where isd, isq, isw, and isu design, respectively, the stator currents in the (d, q, w, u) rotating frame, Ω is the mechanical speed of the five-phase IM, TL represents the load torque; meanwhile, the developed torque considering the flux orientation state is given by Tem=pM/Lrφisq. p denotes the pole-pairs, and Bm and Jm are the friction and shaft momentum. ω=pΩ designs the electrical motor speed, φ is the magnitude of the rotor flux, M denotes the coupling inductance, Ls and Lr are the cyclic inductances, respectively, of the stator and rotor, Lls presents the stator leakage inductance, and finally, Vsd, Vsq, Vsw, and Vsu design, respectively, the stator voltage in the rotating (d, q, w, u) frame. In this rotating frame, the FPIM model can be projected in two different 2D coordinates, specifically (d-q) and (w-u), and one axis, namely, o, which presents zero relative components. Consequently, the model of FPIM represented by (1) can be rewritten by(2)Ω˙=aωΩ+bωφisq−cωTL,isd˙=adisd+ωsisq+bdφ+ddVsd,isq˙=adisq−ωsisd+cdωφ+ddVsq,isw˙=awisw+dwVsw,isu˙=awisu+dwVsu,φ˙=aφisd−bφφ,where(3)σ=1−M2LsLr,Rt=11−Ts+1−σσTr,Tr=LrRr,Ts=LsRs,μ=LmLr,aω=BmJm,Tls=LlsRs,bω=pMLr,cω=1Jm,ad=−RtσLs,bd=μσLsTr,dd=1σLs,cd=−μσLs,aw=−1Tls,dw=1Lls,aφ=MTr,bφ=1Tr.


## 3. Design Procedure of the Proposed SO-SMC

### 3.1. Definition

The study of the sliding mode began in Yugoslavia in the 1960s, after this work was repeated elsewhere, either to complete the theoretical study or to study some possible applications. However, it is only in the 1980s that the sliding mode control has become interesting and attractive. It has been considered one of the control approaches for nonlinear systems and systems with inaccurate models. The theory of sliding mode systems is a nonlinear control mechanism featured by the interruption of control when passing through a switching surface called the sliding surface. The technique of sliding modes stands on moving the state trajectory of a system towards the sliding surface and making it switch with proper switching around it to the point of equilibrium. In general, the sliding control consists of two essential steps:(i)
Determining a state space area such that once the system is in that area, it has the required dynamic.(ii)
Derivation of a control law that drives the system to the specified state space area.Figure2 presents the sliding systems. The formulation of the SMC considers the problems of stability and appropriate performance in a systematic way in its procedure which is executed principally in three interrelated stages identified by the following:(i)
Selection of sliding surfaces.(ii)
Definition of the conditions of existence and convergence of the sliding regime.(iii)
Derivation of the control law.Figure 2
Sliding systems dynamics.The existence conditions and convergence are the criteria that permit the convergence of different system dynamics towards the sliding surface and to stay there independently of the perturbation. There are two categories of conditions which are as follows:(i)
It is presented and analyzed by Emilyanov and Utkin. It is a question of giving to the surface a dynamic convergent towards zero.(ii)
It is a question of choosing a candidate function of Lyapunov (positive scalar function) for the variables of state of the system and choosing a law of command which will decrease this function.


### 3.2. Design Steps

The implementation of SMC on the three-phase motor is extensively presented and analyzed [25–27]. Nonetheless, concerning multiphase motor in the literature, one can find only a few works attending to it. In this situation, a novel SMC is designed to accomplish the control of FPIM in this section. The idea of the formulated SMC originated from the introduced definition of the sliding mechanism shown in Figure 2. In this effor, in the current study, the SMC is designed to obtain the control laws for managing speed, current and flux of five phase IM dissimilar to the control presented in [25], in which the SMC is applied only for speed control while using a PI controller for the current regulation. In the present study, one will simply apply the technique of adjustment by sliding mode to the five-phase IM, and we will establish the command value expressions based on the model established in the first section.


#### 3.2.1. Design of Sliding Surface

One can introduce the sliding surface in theR2 state as follows:(4)Sk=hk⋅ckzk+∫0tzkτ⋅dτ,where zk denotes the variable error for speed, current, and flux and the parameters ck and hk are two positive constants. In consideration that the control is adapted to five-phase IM, then six sliding surfaces are presented as follows:(5)S1=h1c1z1+∫0tz1τdτ,S2=h2c2z2+∫0tz2τdτ.(6)S3=h3c3z3+∫0tz3τdτ,S4=h4c4z4+∫0tz4τdτ.(7)S5=h5c5z5+∫0tz5τdτ,S6=h6c6z6+∫0tz6τdτ,where(8)z1=Ω−Ω∗,z2=φ−φ∗,z3=isd−isd∗,z4=isq−isq∗,z5=isw−isw∗,z6=isu−isu∗,where the symbol “∗” is utilized to denote the reference value. ck and hk are positive constants and design the control parameters (k = 1, 2, 3, 4, 5, 6). z1, z2, z3, z4, z5, and z6 are presented as the tracking errors of speed, flux, d-current, q-current, w-current, and u-current, in turn.


#### 3.2.2. Design of Control Law

The application of the proposed SO-SMC is performed in three steps: speed control, flux control, and current control. When the sliding regime reaches the dynamics of the system, which is independent of the control law and which is intended to maintain the sliding conditions (the attractiveness of the surface), the surface can be determined independently of the order. Now, it remains to identify the reference required to pull the state trajectory towards the surface and then to its point of equilibrium while guaranteeing the conditions of existence of the sliding mode. In order to obtain the nominal model of the FPIM model, the nominal parameter values on the external disturbance should be considered. Accordingly, the model given by (2) of FPIM turns out to be(9)Ω˙=a¯ωΩ+b¯ωφisq,isd˙=a¯disd+ωsisq+b¯dφ+d¯dVsd,isq˙=a¯disq−ωsisd+c¯dωφ+d¯dVsq,isw˙=a¯wisw+d¯wVsw,isu˙=a¯wisu+d¯wVsu,φ˙=a¯φisd−b¯φφ,where a¯ω, b¯ω, a¯d, b¯d, c¯d, d¯d, a¯w, d¯w, a¯φ, and b¯φ present the nominal values, respectively, of aω, bω, ad, bd, cd, dd, aw, dw, aφ, and bφ. Taking into consideration the parameters variation, external load disturbance, and unexpected uncertainties, the FPIM model can be modeled as follows:(10)Ω˙=a¯ω+ΔaωΩ+b¯ωΔbωφisq+cωTL+δ1,isd˙=a¯d+Δadisd+ωsisq+b¯d+Δbdφ+d¯d+ΔddVsd+δ2,isq˙=a¯d+Δadisq−ωsisd+c¯d+Δcdωφ+d¯d+ΔddVsq+δ3,isw˙=a¯ω+Δawisw+d¯w+ΔdwVsw+δ4,isu˙=a¯w+Δawisu+d¯w+ΔdwVsu+δ5,φ˙=a¯φ+Δaφisd−b¯φ+Δbφφ+δ6.Then, the expressions in (10) can be simplified as follows:(11)Ω˙=a¯ωΩ+b¯ωφisq+L1,isd˙=a¯disd+ωsisq+b¯dφ+d¯dVsd+L2,isq˙=a¯disq−ωsisd+c¯dωφ+d¯dVsq+L3,isw˙=a¯wisw+d¯wVsw+L4,isu˙=a¯wisu+d¯wVsu+L5,where(12)L1=aωΩ+bωφisq+cωTL+δ1,L2=adisd+ωsisq+bdφ+ddVsd+δ2,L3=adisq−ωsisd+cdωφ+ddVsq+δ3,L4=awisw+dwVsw+δ4,L5=awisu+dwVsu+δ5,L6=aφisd−bφφ+δ6.Li (i = 1, 2, 3, 4, 5, 6) is the lumped uncertainty and pretended to be limited by Li<χi (i = 1, 2, 3, 4, 5, 6), where χi is a positive constant.(1) Speed Control Procedure. The speed error z1 and its derivative z1˙ can be expressed by (13)z1=Ω−Ω∗,(14)z1˙=Ω˙−Ω˙∗.Using the first equality in (11), relation (14) can be expressed by(15)z1˙=a¯ωΩ+b¯ωφisq+L1−Ω˙∗.The dynamic surface of the sliding mode ofz1 in the state R2 is defined by(16)S1=h1c1z1+∫0tz1τdτ.The differentiation ofS1 can then be expressed by(17)S˙1=h1c1z˙1+z1.Replacing (15) with (17), the derivative of S1 can be evaluated by(18)z1˙=h1c1a¯ωΩ+bω¯φisq+L1−Ω˙∗+z1.The desired performance can be achieved by solving (18) under a nominal machine model without taking into account the lumped uncertainties L1=0 and S˙1=0.(19)isqn∗=−c1b¯ω−1⋅c1b¯ω+1z1+c1a¯ωΩ∗−c1Ω˙∗.In order to obtain a satisfactory performance of the control strategy against the uncertainties on the system dynamic system (lumped uncertainties), then a required discontinuous function named “reaching controller” needs to be joined to the control part to endure uncertainties beyond the sliding surface. Consequently, reaching control is defined by(20)isqr∗=−bω¯φ−1⋅k1signS1,where k1 designs the switching gain. Thus, a relevant control of q-current considering uncertainties and unmodeled dynamics can be modeled by the following equation:(21)isq∗=isqn∗+isqr∗.By replacing (19) and (20) in (21), the command q-current component can be written as follows:(22)isq∗=−c1b¯ω−1⋅c1a¯ω+1z1+c1a¯ωΩ∗−c1Ω˙∗=−b¯ωφ−1⋅k1signS1.(2) Flux Control Procedure. The flux error z2 and its derivative z2˙ can be defined by (23)z2=φ−φ∗,(24)z2˙=φ˙−φ˙∗=a¯φisd−bφ¯φ+L6−φ˙∗.The dynamic surface of the sliding mode ofz2 in the state R2 is itemized by(25)S2=h2c2z2+∫0tz2τdτ.Setting (24) into (25), the derivative of S2 can be formulated by(26)S˙2=h2c2a¯φisd−bφ¯φ+L6−φ˙∗+z2.The desired performance can be obtained by solving (18) without contemplating the lumped uncertainties L6=0 and S˙2=0. Then,(27)isdn∗=−c2b¯φ−1⋅c1b¯φ+1z2+c2a¯φφ∗−c2φ∗.To get a satisfactory operation of the controller against the uncertainties in the system, then a reaching controller needs to be combined with the control to handle these uncertainties beyond the sliding surface. Thus, the reaching control can be formulated by(28)isdr∗=−b¯φ−1⋅k2signS2,where k2 designs the switching gain. Thus, a relevant control of d-current considering the uncertainties and unmodeled dynamics can be represented as follows:(29)isd∗=isdn∗+isdr∗.Substituting (27) and (28) into (29), the command q-current component can be written as follows:(30)isd∗=−c2b¯φ−1⋅c1b¯φ+1z2+c2a¯φφ∗−c2φ∗−b¯φ−1⋅k2signS2.The referencew and u current components are controlled to be null for the five-phase induction motor in order to minimize the losses.(31)isw∗=isu∗=0.0.(3) Current Control Procedure. The derivatives of the (d-q-w-u) current deviations z3, z4, z5, and z6 are, respectively, presented as (32)z˙3=isd˙−isd∗˙=a¯disd+ωsisq+b¯dφ+d¯dVsd+L2−isd∗˙,z˙4=isq˙−isq∗˙=a¯disq−ωsisd+cd¯ωφ+d¯dVsq+L3−isq∗˙,z˙5=isw˙−isw∗˙=a¯wisw+d¯wVsw+L4−isw∗˙,z˙6=isu˙−isu∗˙=a¯wisu+d¯wVsu+L5−isu∗˙.By setting (32) in (5)–(7), the derivatives S3, S4, S5, and S6 are obtained as follows:(33)S˙3=h3⋅c3z˙3+z3,S˙4=h4⋅c4z˙4+z4,S˙5=h5⋅c5z˙5+z5,S˙6=h6⋅c6z˙6+z6and(34)S˙3=h3⋅c3a¯disd+ωsisq+b¯dφ+d¯dVsd+L2−isd∗˙+z3,S˙4=h4⋅c4a¯disq−ωsisd+c¯dωφ+d¯dVsq+L3−isq∗˙+z4,S˙5=h5⋅c5a¯wisw+d¯wVsw+L4−isw∗˙+z5,S˙6=h6⋅c6a¯wisu+d¯wVsu+L5−isu∗˙+z6.The nominal (d, q, w, u) voltages axis can be evaluated by setting S˙3=S˙4=S˙5=S˙6=0 without the application of model uncertainties L3=L4=L5=L6=0 as follows:(35)Vsdn∗=−c3d¯d−1c3a¯d+1z3+c3a¯disd∗+c3ωsisq+c3b¯dφ−c3isd∗˙,(36)Vsqn∗=−c4d¯d−1c4a¯d+1z4+c4a¯disq∗−c4ωsisd+c4c¯dωφ−c4isq∗˙,(37)Vswn∗=−c5d¯w−1c5a¯w+1z5+c5a¯wisw∗−c5isw∗˙,(38)Vsun∗=−c6d¯u−1c6a¯w+1z6+c6a¯wisu∗−c6isu∗˙.Similar to what is considered when designing the speed and flux controllers, a reaching controller is incorporated within the control part to endure uncertainties behind the sliding surface. Then, the reaching control is defined by(39)Vsdr∗=−d¯d−1⋅k3signS3,Vsqr∗=−d¯d−1⋅k4signS4,Vswr∗=−b¯w−1⋅k5signS5,Vsur∗=−d¯w−1⋅k6signS6,where k3, k4, k5, and k6 are the switching factors. Thus, a relevant control of (d,q,w,u) current components taking into account the uncertainties and unmodeled dynamics can be formulated by(40)Vsd∗=Vsdn∗+Vsdr∗,Vsq∗=Vsqn∗+Vsqr∗,Vsw∗=Vswn∗+Vswr∗,Vsu∗=Vsun∗+Vsur∗.In conclusion, the command (d, q, w, u) voltages are given by(41)Vsd∗=−c3d¯d−1c3a¯d+1z3+c3a¯disd∗+c3ωsisq+c3b¯dφ−c3isd∗˙−d¯d−1⋅k3signS3,Vsq∗=−c4d¯d−1c4a¯d+1z4+c4a¯disq∗−c4ωsisd+c4c¯dωφ−c4isq∗˙−d¯d−1⋅k4signS4,Vsw∗=−c5d¯w−1c5a¯w+1z5+c5a¯wisw∗−c5isw∗˙−b¯w−1⋅k5signS5,Vsu∗=−c6d¯u−1c6a¯w+1z6+c6a¯wisu∗−c6isu∗˙−d¯w−1⋅k6signS6.


### 3.3. Stability Check

To analyze the stability of current and speed regulation, one can contemplate the Lyapunov functionV as given as follows:(42)V=12∑16Sk2=12S12+S22+S32+S42+S52+S62.The stability state is achieved from the Lyapunov stability approach as follows:(43)V˙=∑16SkS˙k=S1S˙1+S2S˙2+S3S˙3+S4S˙4+S5S˙5+S6S˙6≤−∑16hkckαkSk=−h1c1α1S1−h2c2α2S2−h3c3α3S3−h4c4α4S4−h5c5α5S5−h6c6α6S6,where (αi,i=1,2,3,4,5,6) need to be selected as small positive constants. Then by replacing (22) with (18), it results in(44)S˙1=h1c1a¯ωΩ+b¯ωφisqn+isqr+L1−Ω∗˙+z1=h1−c1k1signS1+c1L1.Substituting from (30) in (26), it yields(45)S˙2=h2c2a¯φisdn+isdr+b¯φφ+L6−φ∗˙+z2=h2−c2k2signS2+c2L6.By setting (41) in (34), one obtains(46)S˙3=h3c3a¯disd+ωsisq+b¯dφ+d¯dVsdn+Vsdr+L2−isd∗˙+z3,S˙4=h4c4a¯disq−ωsisd+c¯dωφ+d¯dVsqn+Vsqr+L3−isq∗˙+z4,S˙5=h5c5a¯wisw+d¯wVswn+Vswr+L4−isw∗˙+z5,S˙6=h6c6a¯wisu+d¯wVsun+Vsur+L5−isu∗˙+z6.The relationships in (46) turn out to be as follows:(47)S˙3=h3−c3k3signS3+c3L2,S˙4=h4−c4k4signS4+c4L3,S˙5=h5−c5k5signS5+c5L4,S˙6=h6−c6k6signS6+c6L5.Substituting (44), (45), and (47) into (43), the expression of the Lyapunov function V can be determined as follows:(48)V˙=S1S˙1+S2S˙2+S3S˙3+S4S˙4+S5S˙5+S6S˙6=S1h1−c1k1signS1+c1L1+S2h2−c2k2signS2+c2L6+S3h3−c3k3signS3+c3L2+S4h4−c4k4signS4+c4L3+S5h5−c5k5signS5+c5L4+S6h6−c6k6signS6+c6L5.Then using (48) and (43), the following condition is obtained:(49)V˙=S1S˙1+S2S˙2+S3S˙3+S4S˙4+S5S˙5+S6S˙6=S1h1−c1k1signS1+c1L1+S2h2−c2k2signS2+c2L6+S3h3−c3k3signS3+c3L2+S4h4−c4k4signS4+c4L3+S5h5−c5k5signS5+c5L4+S6h6−c6k6signS6+c6L5≤h1c1S1L1−k1+h2c2S2L6−k2+h3c3S3L2−k3+h4c4S4L3−k4+h5c5S5L4−k5+h6c6S6L5−k6≤h1c1S1n1−k1+h2c2S2n6−k2+h3c3S3n2−k3+h4c4S4n3−k4+h5c5S5n4−k5+h6c6S6n5−k6≤−h1c1α1S1−h2c2α2S2−h3c3α3S3−h4c4α4S4−h5c5α5S5−h6c6α6S6.Finally, the stability proof is obtained if the following conditions are obtained:(50)Li<ni,i=1,2,3,4,5,6,ki≥ni+α,i=1,2,3,4,5,6.


### 3.4. Overall Formulation of the Proposed SO-SMC

As demonstrated in (50), the switching constants ki are calculated on the basis of the magnitudes of the lumped uncertainties. Indeed, limiting the lumped uncertainties in real applications cannot be usually ensured. Thus, to estimate the lumped uncertainties’ upper bounds, an adaptively selected gain is proposed and represented by(51)ki=hiciri∫0tSiτdτ,i=1,2,3,4,5,6,where ri is a positive constant.Based upon this hypothesis, the expressions of (20), (28), and (39) can be adjusted in the following manners:(52)isqr∗=−b¯w−1h1c1r1∫0tS1τdτ,(53)isdr∗=−b¯φ−1h2c2r2∫0tS2τdτ,(54)Vsdr∗=−d¯d−1h3c3r3∫0tS3τdτ,Vsqr∗=−d¯d−1h4c4r4∫0tS4τdτ,Vswr∗=−d¯w−1h5c5r5∫0tS5τdτ,Vsur∗=−d¯w−1h6c6r6∫0tS6τdτ.To check the control stability of the speed, flux, and current, an adequate Lyapunov’s formula is needed. The Lyapunov formulaV is preferred as(55)V=12S12+S22+S32+S42+S52+S62+1212r1k˜12+12r2k˜22+12r3k˜32+12r4k˜42+12r5k˜52+12r6k˜62,where k˜i for (i = 1, 2, 3, 4, 5, 6) is the estimated errors of ki which are presented as follows: k˜i=ki−k^i for (i = 1, 2, 3, 4, 5, 6), where k^i is the estimation of ki. Differentiating formula (55), this results in(56)V˙=S1S˙1+S2S˙2+S3S˙3+S4S˙4+S5S˙5+S6S˙6+12r1k˜1k˜1˙+12r2k˜2k˜2˙+12r3k˜3k˜3˙+12r4k˜4k˜4˙+12r5k˜5k˜5˙+12r6k˜6k˜6˙.Using (22), (30), (31), and (33), the equality in (56) becomes(57)V˙=h1c1S1−k1signS1+L1+h2c2S2−k2signS2+L6+h3c3S3−k3signS3+L2+h4c4S4−k4signS4+L3+h5c5S5−k5signS5+L4+h6c6S6−k6signS6+L5+12r1k1−k^1k1˙+12r2k2−k^2k2˙+12r3k3−k^3k3˙+12r4k4−k^4k4˙+12r5k5−k^5k5˙+12r6k6−k^6k6˙.Setting (57) into (43) and considering (50), the following is obtained:(58)V˙<h1c1S1L1−k1+h2c2S2L6−k2+h3c3S3L2−k3+h4c4S4L3−k4+h5c5S5L4−k5+h6c6S6L5−k6+h1c1k1−k^1S1+h2c2k2−k^2S2+h3c3k3−k^3S3+h4c4k4−k^4S4+h5c5k5−k^5S5+h6c6k6−k^6S6<h1c1S1n1−k1h1c1S1n1−k1+h2c2S2n2−k2+h3c3S3n3−k3+h4c4S4n4−k4+h5c5S5n5−k5+h6c6S6n6−k6+h1c1k1−k^1S1+h2c2k2−k^2S2+h3c3k3−k^3S3+h4c4k4−k^4S4+h5c5k5−k^5S5+h6c6k6−k^6S6<−h1c1α1S1−h2c2α2S2−h3c3α3S3−h4c4α4S4−h5c5α5S5−h6c6α6S6.To maintain the stability of the control, the speed and current controllers should be selected as follows:(59)k^i≥ni+αii=1,2,3,4,5,6.The control technique based on the sliding mode theory described in the previous paragraph provides a desired behavior of the closed-loop system. The actuators, on the other hand, have an infinite commutation in the ideal scenario. As a result, no control member is capable of performing this operation. This results in high-frequency dynamics that are not accounted for in the system’s modeling, resulting in the development of “reluctance” or “chatter,” which is characterized by significant oscillations around the sliding surface.To mitigate the problem impact, a zone around the sliding surface is designated, as illustrated in Figure3, within which a less rigorous slip condition than the “sign” condition is enforced. Indeed, the previously employed sign function is replaced by a smoother function, which is the Sigmoid function expressed as(60)SigmoidSi=21+e−aSi−1.Figure 3
Variable boundary layers per the speed.


## 3.1. Definition

The study of the sliding mode began in Yugoslavia in the 1960s, after this work was repeated elsewhere, either to complete the theoretical study or to study some possible applications. However, it is only in the 1980s that the sliding mode control has become interesting and attractive. It has been considered one of the control approaches for nonlinear systems and systems with inaccurate models. The theory of sliding mode systems is a nonlinear control mechanism featured by the interruption of control when passing through a switching surface called the sliding surface. The technique of sliding modes stands on moving the state trajectory of a system towards the sliding surface and making it switch with proper switching around it to the point of equilibrium. In general, the sliding control consists of two essential steps:(i)
Determining a state space area such that once the system is in that area, it has the required dynamic.(ii)
Derivation of a control law that drives the system to the specified state space area.Figure2 presents the sliding systems. The formulation of the SMC considers the problems of stability and appropriate performance in a systematic way in its procedure which is executed principally in three interrelated stages identified by the following:(i)
Selection of sliding surfaces.(ii)
Definition of the conditions of existence and convergence of the sliding regime.(iii)
Derivation of the control law.Figure 2
Sliding systems dynamics.The existence conditions and convergence are the criteria that permit the convergence of different system dynamics towards the sliding surface and to stay there independently of the perturbation. There are two categories of conditions which are as follows:(i)
It is presented and analyzed by Emilyanov and Utkin. It is a question of giving to the surface a dynamic convergent towards zero.(ii)
It is a question of choosing a candidate function of Lyapunov (positive scalar function) for the variables of state of the system and choosing a law of command which will decrease this function.


## 3.2. Design Steps

The implementation of SMC on the three-phase motor is extensively presented and analyzed [25–27]. Nonetheless, concerning multiphase motor in the literature, one can find only a few works attending to it. In this situation, a novel SMC is designed to accomplish the control of FPIM in this section. The idea of the formulated SMC originated from the introduced definition of the sliding mechanism shown in Figure 2. In this effor, in the current study, the SMC is designed to obtain the control laws for managing speed, current and flux of five phase IM dissimilar to the control presented in [25], in which the SMC is applied only for speed control while using a PI controller for the current regulation. In the present study, one will simply apply the technique of adjustment by sliding mode to the five-phase IM, and we will establish the command value expressions based on the model established in the first section.


### 3.2.1. Design of Sliding Surface

One can introduce the sliding surface in theR2 state as follows:(4)Sk=hk⋅ckzk+∫0tzkτ⋅dτ,where zk denotes the variable error for speed, current, and flux and the parameters ck and hk are two positive constants. In consideration that the control is adapted to five-phase IM, then six sliding surfaces are presented as follows:(5)S1=h1c1z1+∫0tz1τdτ,S2=h2c2z2+∫0tz2τdτ.(6)S3=h3c3z3+∫0tz3τdτ,S4=h4c4z4+∫0tz4τdτ.(7)S5=h5c5z5+∫0tz5τdτ,S6=h6c6z6+∫0tz6τdτ,where(8)z1=Ω−Ω∗,z2=φ−φ∗,z3=isd−isd∗,z4=isq−isq∗,z5=isw−isw∗,z6=isu−isu∗,where the symbol “∗” is utilized to denote the reference value. ck and hk are positive constants and design the control parameters (k = 1, 2, 3, 4, 5, 6). z1, z2, z3, z4, z5, and z6 are presented as the tracking errors of speed, flux, d-current, q-current, w-current, and u-current, in turn.


### 3.2.2. Design of Control Law

The application of the proposed SO-SMC is performed in three steps: speed control, flux control, and current control. When the sliding regime reaches the dynamics of the system, which is independent of the control law and which is intended to maintain the sliding conditions (the attractiveness of the surface), the surface can be determined independently of the order. Now, it remains to identify the reference required to pull the state trajectory towards the surface and then to its point of equilibrium while guaranteeing the conditions of existence of the sliding mode. In order to obtain the nominal model of the FPIM model, the nominal parameter values on the external disturbance should be considered. Accordingly, the model given by (2) of FPIM turns out to be(9)Ω˙=a¯ωΩ+b¯ωφisq,isd˙=a¯disd+ωsisq+b¯dφ+d¯dVsd,isq˙=a¯disq−ωsisd+c¯dωφ+d¯dVsq,isw˙=a¯wisw+d¯wVsw,isu˙=a¯wisu+d¯wVsu,φ˙=a¯φisd−b¯φφ,where a¯ω, b¯ω, a¯d, b¯d, c¯d, d¯d, a¯w, d¯w, a¯φ, and b¯φ present the nominal values, respectively, of aω, bω, ad, bd, cd, dd, aw, dw, aφ, and bφ. Taking into consideration the parameters variation, external load disturbance, and unexpected uncertainties, the FPIM model can be modeled as follows:(10)Ω˙=a¯ω+ΔaωΩ+b¯ωΔbωφisq+cωTL+δ1,isd˙=a¯d+Δadisd+ωsisq+b¯d+Δbdφ+d¯d+ΔddVsd+δ2,isq˙=a¯d+Δadisq−ωsisd+c¯d+Δcdωφ+d¯d+ΔddVsq+δ3,isw˙=a¯ω+Δawisw+d¯w+ΔdwVsw+δ4,isu˙=a¯w+Δawisu+d¯w+ΔdwVsu+δ5,φ˙=a¯φ+Δaφisd−b¯φ+Δbφφ+δ6.Then, the expressions in (10) can be simplified as follows:(11)Ω˙=a¯ωΩ+b¯ωφisq+L1,isd˙=a¯disd+ωsisq+b¯dφ+d¯dVsd+L2,isq˙=a¯disq−ωsisd+c¯dωφ+d¯dVsq+L3,isw˙=a¯wisw+d¯wVsw+L4,isu˙=a¯wisu+d¯wVsu+L5,where(12)L1=aωΩ+bωφisq+cωTL+δ1,L2=adisd+ωsisq+bdφ+ddVsd+δ2,L3=adisq−ωsisd+cdωφ+ddVsq+δ3,L4=awisw+dwVsw+δ4,L5=awisu+dwVsu+δ5,L6=aφisd−bφφ+δ6.Li (i = 1, 2, 3, 4, 5, 6) is the lumped uncertainty and pretended to be limited by Li<χi (i = 1, 2, 3, 4, 5, 6), where χi is a positive constant.(1) Speed Control Procedure. The speed error z1 and its derivative z1˙ can be expressed by (13)z1=Ω−Ω∗,(14)z1˙=Ω˙−Ω˙∗.Using the first equality in (11), relation (14) can be expressed by(15)z1˙=a¯ωΩ+b¯ωφisq+L1−Ω˙∗.The dynamic surface of the sliding mode ofz1 in the state R2 is defined by(16)S1=h1c1z1+∫0tz1τdτ.The differentiation ofS1 can then be expressed by(17)S˙1=h1c1z˙1+z1.Replacing (15) with (17), the derivative of S1 can be evaluated by(18)z1˙=h1c1a¯ωΩ+bω¯φisq+L1−Ω˙∗+z1.The desired performance can be achieved by solving (18) under a nominal machine model without taking into account the lumped uncertainties L1=0 and S˙1=0.(19)isqn∗=−c1b¯ω−1⋅c1b¯ω+1z1+c1a¯ωΩ∗−c1Ω˙∗.In order to obtain a satisfactory performance of the control strategy against the uncertainties on the system dynamic system (lumped uncertainties), then a required discontinuous function named “reaching controller” needs to be joined to the control part to endure uncertainties beyond the sliding surface. Consequently, reaching control is defined by(20)isqr∗=−bω¯φ−1⋅k1signS1,where k1 designs the switching gain. Thus, a relevant control of q-current considering uncertainties and unmodeled dynamics can be modeled by the following equation:(21)isq∗=isqn∗+isqr∗.By replacing (19) and (20) in (21), the command q-current component can be written as follows:(22)isq∗=−c1b¯ω−1⋅c1a¯ω+1z1+c1a¯ωΩ∗−c1Ω˙∗=−b¯ωφ−1⋅k1signS1.(2) Flux Control Procedure. The flux error z2 and its derivative z2˙ can be defined by (23)z2=φ−φ∗,(24)z2˙=φ˙−φ˙∗=a¯φisd−bφ¯φ+L6−φ˙∗.The dynamic surface of the sliding mode ofz2 in the state R2 is itemized by(25)S2=h2c2z2+∫0tz2τdτ.Setting (24) into (25), the derivative of S2 can be formulated by(26)S˙2=h2c2a¯φisd−bφ¯φ+L6−φ˙∗+z2.The desired performance can be obtained by solving (18) without contemplating the lumped uncertainties L6=0 and S˙2=0. Then,(27)isdn∗=−c2b¯φ−1⋅c1b¯φ+1z2+c2a¯φφ∗−c2φ∗.To get a satisfactory operation of the controller against the uncertainties in the system, then a reaching controller needs to be combined with the control to handle these uncertainties beyond the sliding surface. Thus, the reaching control can be formulated by(28)isdr∗=−b¯φ−1⋅k2signS2,where k2 designs the switching gain. Thus, a relevant control of d-current considering the uncertainties and unmodeled dynamics can be represented as follows:(29)isd∗=isdn∗+isdr∗.Substituting (27) and (28) into (29), the command q-current component can be written as follows:(30)isd∗=−c2b¯φ−1⋅c1b¯φ+1z2+c2a¯φφ∗−c2φ∗−b¯φ−1⋅k2signS2.The referencew and u current components are controlled to be null for the five-phase induction motor in order to minimize the losses.(31)isw∗=isu∗=0.0.(3) Current Control Procedure. The derivatives of the (d-q-w-u) current deviations z3, z4, z5, and z6 are, respectively, presented as (32)z˙3=isd˙−isd∗˙=a¯disd+ωsisq+b¯dφ+d¯dVsd+L2−isd∗˙,z˙4=isq˙−isq∗˙=a¯disq−ωsisd+cd¯ωφ+d¯dVsq+L3−isq∗˙,z˙5=isw˙−isw∗˙=a¯wisw+d¯wVsw+L4−isw∗˙,z˙6=isu˙−isu∗˙=a¯wisu+d¯wVsu+L5−isu∗˙.By setting (32) in (5)–(7), the derivatives S3, S4, S5, and S6 are obtained as follows:(33)S˙3=h3⋅c3z˙3+z3,S˙4=h4⋅c4z˙4+z4,S˙5=h5⋅c5z˙5+z5,S˙6=h6⋅c6z˙6+z6and(34)S˙3=h3⋅c3a¯disd+ωsisq+b¯dφ+d¯dVsd+L2−isd∗˙+z3,S˙4=h4⋅c4a¯disq−ωsisd+c¯dωφ+d¯dVsq+L3−isq∗˙+z4,S˙5=h5⋅c5a¯wisw+d¯wVsw+L4−isw∗˙+z5,S˙6=h6⋅c6a¯wisu+d¯wVsu+L5−isu∗˙+z6.The nominal (d, q, w, u) voltages axis can be evaluated by setting S˙3=S˙4=S˙5=S˙6=0 without the application of model uncertainties L3=L4=L5=L6=0 as follows:(35)Vsdn∗=−c3d¯d−1c3a¯d+1z3+c3a¯disd∗+c3ωsisq+c3b¯dφ−c3isd∗˙,(36)Vsqn∗=−c4d¯d−1c4a¯d+1z4+c4a¯disq∗−c4ωsisd+c4c¯dωφ−c4isq∗˙,(37)Vswn∗=−c5d¯w−1c5a¯w+1z5+c5a¯wisw∗−c5isw∗˙,(38)Vsun∗=−c6d¯u−1c6a¯w+1z6+c6a¯wisu∗−c6isu∗˙.Similar to what is considered when designing the speed and flux controllers, a reaching controller is incorporated within the control part to endure uncertainties behind the sliding surface. Then, the reaching control is defined by(39)Vsdr∗=−d¯d−1⋅k3signS3,Vsqr∗=−d¯d−1⋅k4signS4,Vswr∗=−b¯w−1⋅k5signS5,Vsur∗=−d¯w−1⋅k6signS6,where k3, k4, k5, and k6 are the switching factors. Thus, a relevant control of (d,q,w,u) current components taking into account the uncertainties and unmodeled dynamics can be formulated by(40)Vsd∗=Vsdn∗+Vsdr∗,Vsq∗=Vsqn∗+Vsqr∗,Vsw∗=Vswn∗+Vswr∗,Vsu∗=Vsun∗+Vsur∗.In conclusion, the command (d, q, w, u) voltages are given by(41)Vsd∗=−c3d¯d−1c3a¯d+1z3+c3a¯disd∗+c3ωsisq+c3b¯dφ−c3isd∗˙−d¯d−1⋅k3signS3,Vsq∗=−c4d¯d−1c4a¯d+1z4+c4a¯disq∗−c4ωsisd+c4c¯dωφ−c4isq∗˙−d¯d−1⋅k4signS4,Vsw∗=−c5d¯w−1c5a¯w+1z5+c5a¯wisw∗−c5isw∗˙−b¯w−1⋅k5signS5,Vsu∗=−c6d¯u−1c6a¯w+1z6+c6a¯wisu∗−c6isu∗˙−d¯w−1⋅k6signS6.


## 3.2.1. Design of Sliding Surface

One can introduce the sliding surface in theR2 state as follows:(4)Sk=hk⋅ckzk+∫0tzkτ⋅dτ,where zk denotes the variable error for speed, current, and flux and the parameters ck and hk are two positive constants. In consideration that the control is adapted to five-phase IM, then six sliding surfaces are presented as follows:(5)S1=h1c1z1+∫0tz1τdτ,S2=h2c2z2+∫0tz2τdτ.(6)S3=h3c3z3+∫0tz3τdτ,S4=h4c4z4+∫0tz4τdτ.(7)S5=h5c5z5+∫0tz5τdτ,S6=h6c6z6+∫0tz6τdτ,where(8)z1=Ω−Ω∗,z2=φ−φ∗,z3=isd−isd∗,z4=isq−isq∗,z5=isw−isw∗,z6=isu−isu∗,where the symbol “∗” is utilized to denote the reference value. ck and hk are positive constants and design the control parameters (k = 1, 2, 3, 4, 5, 6). z1, z2, z3, z4, z5, and z6 are presented as the tracking errors of speed, flux, d-current, q-current, w-current, and u-current, in turn.


## 3.2.2. Design of Control Law

The application of the proposed SO-SMC is performed in three steps: speed control, flux control, and current control. When the sliding regime reaches the dynamics of the system, which is independent of the control law and which is intended to maintain the sliding conditions (the attractiveness of the surface), the surface can be determined independently of the order. Now, it remains to identify the reference required to pull the state trajectory towards the surface and then to its point of equilibrium while guaranteeing the conditions of existence of the sliding mode. In order to obtain the nominal model of the FPIM model, the nominal parameter values on the external disturbance should be considered. Accordingly, the model given by (2) of FPIM turns out to be(9)Ω˙=a¯ωΩ+b¯ωφisq,isd˙=a¯disd+ωsisq+b¯dφ+d¯dVsd,isq˙=a¯disq−ωsisd+c¯dωφ+d¯dVsq,isw˙=a¯wisw+d¯wVsw,isu˙=a¯wisu+d¯wVsu,φ˙=a¯φisd−b¯φφ,where a¯ω, b¯ω, a¯d, b¯d, c¯d, d¯d, a¯w, d¯w, a¯φ, and b¯φ present the nominal values, respectively, of aω, bω, ad, bd, cd, dd, aw, dw, aφ, and bφ. Taking into consideration the parameters variation, external load disturbance, and unexpected uncertainties, the FPIM model can be modeled as follows:(10)Ω˙=a¯ω+ΔaωΩ+b¯ωΔbωφisq+cωTL+δ1,isd˙=a¯d+Δadisd+ωsisq+b¯d+Δbdφ+d¯d+ΔddVsd+δ2,isq˙=a¯d+Δadisq−ωsisd+c¯d+Δcdωφ+d¯d+ΔddVsq+δ3,isw˙=a¯ω+Δawisw+d¯w+ΔdwVsw+δ4,isu˙=a¯w+Δawisu+d¯w+ΔdwVsu+δ5,φ˙=a¯φ+Δaφisd−b¯φ+Δbφφ+δ6.Then, the expressions in (10) can be simplified as follows:(11)Ω˙=a¯ωΩ+b¯ωφisq+L1,isd˙=a¯disd+ωsisq+b¯dφ+d¯dVsd+L2,isq˙=a¯disq−ωsisd+c¯dωφ+d¯dVsq+L3,isw˙=a¯wisw+d¯wVsw+L4,isu˙=a¯wisu+d¯wVsu+L5,where(12)L1=aωΩ+bωφisq+cωTL+δ1,L2=adisd+ωsisq+bdφ+ddVsd+δ2,L3=adisq−ωsisd+cdωφ+ddVsq+δ3,L4=awisw+dwVsw+δ4,L5=awisu+dwVsu+δ5,L6=aφisd−bφφ+δ6.Li (i = 1, 2, 3, 4, 5, 6) is the lumped uncertainty and pretended to be limited by Li<χi (i = 1, 2, 3, 4, 5, 6), where χi is a positive constant.(1) Speed Control Procedure. The speed error z1 and its derivative z1˙ can be expressed by (13)z1=Ω−Ω∗,(14)z1˙=Ω˙−Ω˙∗.Using the first equality in (11), relation (14) can be expressed by(15)z1˙=a¯ωΩ+b¯ωφisq+L1−Ω˙∗.The dynamic surface of the sliding mode ofz1 in the state R2 is defined by(16)S1=h1c1z1+∫0tz1τdτ.The differentiation ofS1 can then be expressed by(17)S˙1=h1c1z˙1+z1.Replacing (15) with (17), the derivative of S1 can be evaluated by(18)z1˙=h1c1a¯ωΩ+bω¯φisq+L1−Ω˙∗+z1.The desired performance can be achieved by solving (18) under a nominal machine model without taking into account the lumped uncertainties L1=0 and S˙1=0.(19)isqn∗=−c1b¯ω−1⋅c1b¯ω+1z1+c1a¯ωΩ∗−c1Ω˙∗.In order to obtain a satisfactory performance of the control strategy against the uncertainties on the system dynamic system (lumped uncertainties), then a required discontinuous function named “reaching controller” needs to be joined to the control part to endure uncertainties beyond the sliding surface. Consequently, reaching control is defined by(20)isqr∗=−bω¯φ−1⋅k1signS1,where k1 designs the switching gain. Thus, a relevant control of q-current considering uncertainties and unmodeled dynamics can be modeled by the following equation:(21)isq∗=isqn∗+isqr∗.By replacing (19) and (20) in (21), the command q-current component can be written as follows:(22)isq∗=−c1b¯ω−1⋅c1a¯ω+1z1+c1a¯ωΩ∗−c1Ω˙∗=−b¯ωφ−1⋅k1signS1.(2) Flux Control Procedure. The flux error z2 and its derivative z2˙ can be defined by (23)z2=φ−φ∗,(24)z2˙=φ˙−φ˙∗=a¯φisd−bφ¯φ+L6−φ˙∗.The dynamic surface of the sliding mode ofz2 in the state R2 is itemized by(25)S2=h2c2z2+∫0tz2τdτ.Setting (24) into (25), the derivative of S2 can be formulated by(26)S˙2=h2c2a¯φisd−bφ¯φ+L6−φ˙∗+z2.The desired performance can be obtained by solving (18) without contemplating the lumped uncertainties L6=0 and S˙2=0. Then,(27)isdn∗=−c2b¯φ−1⋅c1b¯φ+1z2+c2a¯φφ∗−c2φ∗.To get a satisfactory operation of the controller against the uncertainties in the system, then a reaching controller needs to be combined with the control to handle these uncertainties beyond the sliding surface. Thus, the reaching control can be formulated by(28)isdr∗=−b¯φ−1⋅k2signS2,where k2 designs the switching gain. Thus, a relevant control of d-current considering the uncertainties and unmodeled dynamics can be represented as follows:(29)isd∗=isdn∗+isdr∗.Substituting (27) and (28) into (29), the command q-current component can be written as follows:(30)isd∗=−c2b¯φ−1⋅c1b¯φ+1z2+c2a¯φφ∗−c2φ∗−b¯φ−1⋅k2signS2.The referencew and u current components are controlled to be null for the five-phase induction motor in order to minimize the losses.(31)isw∗=isu∗=0.0.(3) Current Control Procedure. The derivatives of the (d-q-w-u) current deviations z3, z4, z5, and z6 are, respectively, presented as (32)z˙3=isd˙−isd∗˙=a¯disd+ωsisq+b¯dφ+d¯dVsd+L2−isd∗˙,z˙4=isq˙−isq∗˙=a¯disq−ωsisd+cd¯ωφ+d¯dVsq+L3−isq∗˙,z˙5=isw˙−isw∗˙=a¯wisw+d¯wVsw+L4−isw∗˙,z˙6=isu˙−isu∗˙=a¯wisu+d¯wVsu+L5−isu∗˙.By setting (32) in (5)–(7), the derivatives S3, S4, S5, and S6 are obtained as follows:(33)S˙3=h3⋅c3z˙3+z3,S˙4=h4⋅c4z˙4+z4,S˙5=h5⋅c5z˙5+z5,S˙6=h6⋅c6z˙6+z6and(34)S˙3=h3⋅c3a¯disd+ωsisq+b¯dφ+d¯dVsd+L2−isd∗˙+z3,S˙4=h4⋅c4a¯disq−ωsisd+c¯dωφ+d¯dVsq+L3−isq∗˙+z4,S˙5=h5⋅c5a¯wisw+d¯wVsw+L4−isw∗˙+z5,S˙6=h6⋅c6a¯wisu+d¯wVsu+L5−isu∗˙+z6.The nominal (d, q, w, u) voltages axis can be evaluated by setting S˙3=S˙4=S˙5=S˙6=0 without the application of model uncertainties L3=L4=L5=L6=0 as follows:(35)Vsdn∗=−c3d¯d−1c3a¯d+1z3+c3a¯disd∗+c3ωsisq+c3b¯dφ−c3isd∗˙,(36)Vsqn∗=−c4d¯d−1c4a¯d+1z4+c4a¯disq∗−c4ωsisd+c4c¯dωφ−c4isq∗˙,(37)Vswn∗=−c5d¯w−1c5a¯w+1z5+c5a¯wisw∗−c5isw∗˙,(38)Vsun∗=−c6d¯u−1c6a¯w+1z6+c6a¯wisu∗−c6isu∗˙.Similar to what is considered when designing the speed and flux controllers, a reaching controller is incorporated within the control part to endure uncertainties behind the sliding surface. Then, the reaching control is defined by(39)Vsdr∗=−d¯d−1⋅k3signS3,Vsqr∗=−d¯d−1⋅k4signS4,Vswr∗=−b¯w−1⋅k5signS5,Vsur∗=−d¯w−1⋅k6signS6,where k3, k4, k5, and k6 are the switching factors. Thus, a relevant control of (d,q,w,u) current components taking into account the uncertainties and unmodeled dynamics can be formulated by(40)Vsd∗=Vsdn∗+Vsdr∗,Vsq∗=Vsqn∗+Vsqr∗,Vsw∗=Vswn∗+Vswr∗,Vsu∗=Vsun∗+Vsur∗.In conclusion, the command (d, q, w, u) voltages are given by(41)Vsd∗=−c3d¯d−1c3a¯d+1z3+c3a¯disd∗+c3ωsisq+c3b¯dφ−c3isd∗˙−d¯d−1⋅k3signS3,Vsq∗=−c4d¯d−1c4a¯d+1z4+c4a¯disq∗−c4ωsisd+c4c¯dωφ−c4isq∗˙−d¯d−1⋅k4signS4,Vsw∗=−c5d¯w−1c5a¯w+1z5+c5a¯wisw∗−c5isw∗˙−b¯w−1⋅k5signS5,Vsu∗=−c6d¯u−1c6a¯w+1z6+c6a¯wisu∗−c6isu∗˙−d¯w−1⋅k6signS6.


## 3.3. Stability Check

To analyze the stability of current and speed regulation, one can contemplate the Lyapunov functionV as given as follows:(42)V=12∑16Sk2=12S12+S22+S32+S42+S52+S62.The stability state is achieved from the Lyapunov stability approach as follows:(43)V˙=∑16SkS˙k=S1S˙1+S2S˙2+S3S˙3+S4S˙4+S5S˙5+S6S˙6≤−∑16hkckαkSk=−h1c1α1S1−h2c2α2S2−h3c3α3S3−h4c4α4S4−h5c5α5S5−h6c6α6S6,where (αi,i=1,2,3,4,5,6) need to be selected as small positive constants. Then by replacing (22) with (18), it results in(44)S˙1=h1c1a¯ωΩ+b¯ωφisqn+isqr+L1−Ω∗˙+z1=h1−c1k1signS1+c1L1.Substituting from (30) in (26), it yields(45)S˙2=h2c2a¯φisdn+isdr+b¯φφ+L6−φ∗˙+z2=h2−c2k2signS2+c2L6.By setting (41) in (34), one obtains(46)S˙3=h3c3a¯disd+ωsisq+b¯dφ+d¯dVsdn+Vsdr+L2−isd∗˙+z3,S˙4=h4c4a¯disq−ωsisd+c¯dωφ+d¯dVsqn+Vsqr+L3−isq∗˙+z4,S˙5=h5c5a¯wisw+d¯wVswn+Vswr+L4−isw∗˙+z5,S˙6=h6c6a¯wisu+d¯wVsun+Vsur+L5−isu∗˙+z6.The relationships in (46) turn out to be as follows:(47)S˙3=h3−c3k3signS3+c3L2,S˙4=h4−c4k4signS4+c4L3,S˙5=h5−c5k5signS5+c5L4,S˙6=h6−c6k6signS6+c6L5.Substituting (44), (45), and (47) into (43), the expression of the Lyapunov function V can be determined as follows:(48)V˙=S1S˙1+S2S˙2+S3S˙3+S4S˙4+S5S˙5+S6S˙6=S1h1−c1k1signS1+c1L1+S2h2−c2k2signS2+c2L6+S3h3−c3k3signS3+c3L2+S4h4−c4k4signS4+c4L3+S5h5−c5k5signS5+c5L4+S6h6−c6k6signS6+c6L5.Then using (48) and (43), the following condition is obtained:(49)V˙=S1S˙1+S2S˙2+S3S˙3+S4S˙4+S5S˙5+S6S˙6=S1h1−c1k1signS1+c1L1+S2h2−c2k2signS2+c2L6+S3h3−c3k3signS3+c3L2+S4h4−c4k4signS4+c4L3+S5h5−c5k5signS5+c5L4+S6h6−c6k6signS6+c6L5≤h1c1S1L1−k1+h2c2S2L6−k2+h3c3S3L2−k3+h4c4S4L3−k4+h5c5S5L4−k5+h6c6S6L5−k6≤h1c1S1n1−k1+h2c2S2n6−k2+h3c3S3n2−k3+h4c4S4n3−k4+h5c5S5n4−k5+h6c6S6n5−k6≤−h1c1α1S1−h2c2α2S2−h3c3α3S3−h4c4α4S4−h5c5α5S5−h6c6α6S6.Finally, the stability proof is obtained if the following conditions are obtained:(50)Li<ni,i=1,2,3,4,5,6,ki≥ni+α,i=1,2,3,4,5,6.


## 3.4. Overall Formulation of the Proposed SO-SMC

As demonstrated in (50), the switching constants ki are calculated on the basis of the magnitudes of the lumped uncertainties. Indeed, limiting the lumped uncertainties in real applications cannot be usually ensured. Thus, to estimate the lumped uncertainties’ upper bounds, an adaptively selected gain is proposed and represented by(51)ki=hiciri∫0tSiτdτ,i=1,2,3,4,5,6,where ri is a positive constant.Based upon this hypothesis, the expressions of (20), (28), and (39) can be adjusted in the following manners:(52)isqr∗=−b¯w−1h1c1r1∫0tS1τdτ,(53)isdr∗=−b¯φ−1h2c2r2∫0tS2τdτ,(54)Vsdr∗=−d¯d−1h3c3r3∫0tS3τdτ,Vsqr∗=−d¯d−1h4c4r4∫0tS4τdτ,Vswr∗=−d¯w−1h5c5r5∫0tS5τdτ,Vsur∗=−d¯w−1h6c6r6∫0tS6τdτ.To check the control stability of the speed, flux, and current, an adequate Lyapunov’s formula is needed. The Lyapunov formulaV is preferred as(55)V=12S12+S22+S32+S42+S52+S62+1212r1k˜12+12r2k˜22+12r3k˜32+12r4k˜42+12r5k˜52+12r6k˜62,where k˜i for (i = 1, 2, 3, 4, 5, 6) is the estimated errors of ki which are presented as follows: k˜i=ki−k^i for (i = 1, 2, 3, 4, 5, 6), where k^i is the estimation of ki. Differentiating formula (55), this results in(56)V˙=S1S˙1+S2S˙2+S3S˙3+S4S˙4+S5S˙5+S6S˙6+12r1k˜1k˜1˙+12r2k˜2k˜2˙+12r3k˜3k˜3˙+12r4k˜4k˜4˙+12r5k˜5k˜5˙+12r6k˜6k˜6˙.Using (22), (30), (31), and (33), the equality in (56) becomes(57)V˙=h1c1S1−k1signS1+L1+h2c2S2−k2signS2+L6+h3c3S3−k3signS3+L2+h4c4S4−k4signS4+L3+h5c5S5−k5signS5+L4+h6c6S6−k6signS6+L5+12r1k1−k^1k1˙+12r2k2−k^2k2˙+12r3k3−k^3k3˙+12r4k4−k^4k4˙+12r5k5−k^5k5˙+12r6k6−k^6k6˙.Setting (57) into (43) and considering (50), the following is obtained:(58)V˙<h1c1S1L1−k1+h2c2S2L6−k2+h3c3S3L2−k3+h4c4S4L3−k4+h5c5S5L4−k5+h6c6S6L5−k6+h1c1k1−k^1S1+h2c2k2−k^2S2+h3c3k3−k^3S3+h4c4k4−k^4S4+h5c5k5−k^5S5+h6c6k6−k^6S6<h1c1S1n1−k1h1c1S1n1−k1+h2c2S2n2−k2+h3c3S3n3−k3+h4c4S4n4−k4+h5c5S5n5−k5+h6c6S6n6−k6+h1c1k1−k^1S1+h2c2k2−k^2S2+h3c3k3−k^3S3+h4c4k4−k^4S4+h5c5k5−k^5S5+h6c6k6−k^6S6<−h1c1α1S1−h2c2α2S2−h3c3α3S3−h4c4α4S4−h5c5α5S5−h6c6α6S6.To maintain the stability of the control, the speed and current controllers should be selected as follows:(59)k^i≥ni+αii=1,2,3,4,5,6.The control technique based on the sliding mode theory described in the previous paragraph provides a desired behavior of the closed-loop system. The actuators, on the other hand, have an infinite commutation in the ideal scenario. As a result, no control member is capable of performing this operation. This results in high-frequency dynamics that are not accounted for in the system’s modeling, resulting in the development of “reluctance” or “chatter,” which is characterized by significant oscillations around the sliding surface.To mitigate the problem impact, a zone around the sliding surface is designated, as illustrated in Figure3, within which a less rigorous slip condition than the “sign” condition is enforced. Indeed, the previously employed sign function is replaced by a smoother function, which is the Sigmoid function expressed as(60)SigmoidSi=21+e−aSi−1.Figure 3
Variable boundary layers per the speed.


## 4. General System Layout

The general system configuration of the designed SO-SMC is illustrated in Figure4. The control measures the stator currents iabcde and rotor speed ω, and then the angle θs is calculated and used for the coordinates transformation. The designed speed controller is used to develop the reference current isq∗, which is compared with the real current isq to get the current deviation z3.Figure 4
Control system layout for the SO-SMC.On the other hand, the designed flux controller is utilized to obtain the reference currentisd∗, which is then subtracted from the real current isd to finally obtain the current error z4. After that, both current errors z3 and z4 are fed to the designed current regulators to finally obtain the reference d-q voltages Vsd∗ and Vsq∗. The reference currents isw∗ and isu∗ are also compared with the actual currents isw and isu, and the resultant errors are applied to their specified current regulators to obtain the reference voltage components Vsw∗ and Vsu∗. After obtaining all reference voltages, they are transformed using the angle θs to be applied to the PWM scheme, which is used to fire the gating of the inverter switches. All the necessary parameters for the designed adaptive SO-SMC are presented in Table 1. Meanwhile, Table 2 assigns the parameters of the FPIM.Table 1
Parameters of the proposed SO-SMC.
ParametersValueParametersValueParametersValueParametersValueh10.0022c150r1500k11200h20.1738c210r2100k22000h30.0765c310r3100k33000h40.0765c410r4100k44000h50.04c510r5100k52000h60.04c610r6100k61000Table 2
Parameters of five-phase IM.
SymbolValueRs10 ΩRr6.3 ΩLs0.46 HLr0.46 HM0.42 HLls0.04 HLlr0.04 HJm0.01 kg·m2p2Tn8.33 nm


## 5. Test Results

In order to prove and show the performance and effectiveness of the advanced SO-SMC technique, the control strategy is tested and evaluated under different operating conditions: the first test shown in Figure5 is concerned to speed inversion with the application of the rated load toque during a steady state. The operation consists of the following: starting using a zero reference speed, at instant t = 0.15 s, the reference speed variates to a value of 55 rad/s (525 rpm); after that, at instant t = 0.45 s, a full load torque is applied; and lastly, at time t = 0.8 s, the speed is reversed to a value of −525 rpm. One can observe that the speed control presents a precise reference tracking showing firstly a fast dynamic response without any detectable overshoot or undershoot. It can also be noticed that the designed adaptive SO-SMC has an agreeable performance from the point of view of settling time. It is also noticed that the designed SO-SMC displayed good performance from the point of view of disturbance. A slight speed dip is noticed after the load torque application; however, it does not hold any important effects on the control tracking performance. The stator currents isd and isq show appropriate tracking to their reference values. The component isq follows the torque change; meanwhile, the component isd is kept constant. The stator phase current “a” is also highly sinusoidal, even with a variable switching frequency.Figure 5
Dynamic response: motor speed; electromagnetic torque; direct stator current; quadrature stator current; phase “a” stator current.A zoomed view of the load impact is also shown in Figure6, where at the time t = 0.45 s, the nominal load torque is applied. This result shows a fast disturbance rejection without steady-state error, which confirms the effectiveness of the designed SO-SMC.Figure 6
: Load torque effect-dynamic response: motor speed; electromagnetic torque; direct stator current; quadrature stator current; phase “a” stator current.Figure7 highlights the response of the FPIM drive under triangular reference speed. The maneuver consists of the following: starting using a zero reference speed, at instant t = 0.2 s, the reference speed variates to a value of 525 rpm; after that, at instant t = 0.4 s, a reversal speed of −525 rpm is applied. Simulation results demonstrate that the envisaged SO-SMC has significant performance, and this is illustrated through the fast and fast convergence of d-axis and q-axis current components to their reference values.Figure 7
: Speed inversion effects: dynamic response: motor speed; electromagnetic torque; quadrature stator current; direct stator current; phase a stator current.Moreover, the robustness of the designed SO-SMC faces the variation of parameters (in the current study, one will consider two mechanical parameters and one electrical parameter). The stator resistance variation is considered, in addition to a simultaneous variation of the friction coefficient and moment of inertia. A ramp speed variation is applied where the stator resistance is chosen at 200% of its nominal value. The speed,d-current, q-current, developed torque, and stator phase current are presented in Figure 8. One observes that the designed control is insensitive from the point of view of stator resistance variation, inertia moment, and friction coefficient. Good dynamic tracking is shown of the d–q currents components. It is pretended that the five-phase IM starts by applying 200% of the nominal values of both inertia moment and friction coefficient. Obtained results in Figure 8 approve the insensitivity of the proposed SO-SMC to the parameter variations; this is also accompanied by high dynamic performance during steady-state operation.Figure 8
: Parameters variation effects: dynamic response: motor speed; electromagnetic torque; direct stator current; quadrature stator current; phase “a” stator current.The ISO flux profiles for the (αβxy) flux components are also presented for the three cases introduced above: loading case with speed reversal, no-load operation with triangular speed inverse variation, and ramp speed variation with parameters variation, respectively, in Figures 9–11. The shown flux loci exhibit circular waveforms for the three operating conditions, which confirms the effectiveness of the designed SO-SMC in keeping a uniform distribution of magnetic flux, whatever the operating conditions.Figure 9
Stator flux loci for the full-load operation case with speed reversal.Figure 10
Stator flux loci for the no-load load operation case with triangular speed reversal.Figure 11
Stator flux loci for the no-load load operation case with ramp speed profile and parameters variation.At last, a performance comparison test between the designed SO-SMC and classic SMC employed in the literature is performed. The test is carried out for a speed change of0⟶525⟶−525 RPM at times of 0⟶0.15⟶0.8 sec. The rated torque is applied to the motor at time t = 0.45 sec. This test is also carried out considering a variation in the stator resistance of 200% of its nominal value to test the robustness of the adopted system. As shown from the illustrated profiles in Figure 12, the actual speed is tracking appropriately its reference even under load disturbance and resistance variation, and this is ensured for both controllers (classic SMC and designed SO-SMC). The developed torque also tracks precisely the desired value during the speed change for both controllers. The value of the d-axis stator current component is maintained effectively fixed at its reference value; meanwhile, the q-axis term effectively followed the change in the motor torque. On the other hand, it is very obvious from the comparison that the designed SO-SMC maintained a lower chattering rate causing a reduction in the accompanied ripples in comparison with the classic SMC, and this strongly validates the superiority of the designed SO-SMC over the classic SMC in maintaining high dynamic performance and at the same time limiting the chattering if possible. The superiority of the designed SO-SMC is also confirmed via analyzing the stator currents THD, and this is illustrated in Figures 13 and 14, which illustrate, respectively, the current spectrums for the two controllers (classic SMC and proposed SO-SMC). The THD percentage is effectively reduced under the proposed SO-SMC, and the values are numerically reported in Table 3.Figure 12
Dynamic response: motor speed; electromagnetic torque; direct stator current; quadrature stator current; phase “a” stator current and ISO stator flux for both classic SMC and designed SO-SMC for five-phase IM drive.Figure 13
: Stator currents spectrum analysis under classic SMC.Figure 14
: Stator currents spectrum analysis under designed SO-SMC.Table 3
Current THD analysis under the two control schemes.
PhaseClassic SMCDesigned SO-SMC“A”Fundamental (1.6878 A) THD (3.02%)Fundamental (1.8716 A) THD (2.50%)“B”Fundamental (1.8430 A) THD (5.26%)Fundamental (1.8960 A) THD (2.47%)“C”Fundamental (1.7149 A) THD (3.78%)Fundamental (1.8448 A) THD (2.45%)“d”Fundamental (1.761 A) THD (4.09%)Fundamental (1.8699 A) THD (2.87%)“e”Fundamental (1.8273 A) THD (4.94%)Fundamental (1.8727 A) THD (2.36%)


## 6. Conclusion

The paper presented a detailed design for a robust adaptive second-order sliding mode control (SO-SMC) approach for a five-phase IM drive. The control design is performed and described in a systematic matter and accompanied by a system stability check to validate the effectiveness of the controller. Comprehensive evaluation tests for the five-phase IM drive are carried out under different operating conditions and considering system uncertainties as well. The obtained results reveal that the designed SO-SMC presents high system robustness against disturbance and parameter variation while keeping appropriate system dynamics and maintaining a circular distribution for the magnetic flux inside the machine. The calculated machine variables are also exhibiting high tracking precision to the reference signals with negligible steady-state deviation.



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