Inter national J our nal of P o wer Electr onics and Dri v e System (IJPEDS) V ol. 16, No. 2, June 2025, pp. 840 ∼ 850 ISSN: 2088-8694, DOI: 10.11591/ijpeds.v16.i2.pp840-850 ❒ 840 Single-neur on adapti v e double-po wer super -twisting sliding mode contr ol f or induction motor Siham Mencou, Majid Ben Y akhlef, El Bachir T azi Engineering Sciences Laboratory , Polydisciplinary F aculty of T aza, Sidi Mohamed Ben Abdellah Uni v ersity , Fez, Morocco Article Inf o Article history: Recei v ed Oct 19, 2024 Re vised Apr 9, 2025 Accepted May 6, 2025 K eyw ords: Direct torque control Double-po wer -super -twisting algorithm Induction motor Single-neuron adapti v e control Sliding mode control ABSTRA CT Direct torque control is a widely used control method for induction motors be- cause it of fers rapid dynamic response and relati v ely simple implementation. Ho we v er , it presents high torque and ux ripples and v ariable switching frequen- cies. T o o v ercome these constraints, the doubl e-po wer super -twisting sliding mode (DPSTSM) control approach has been propose d, inte grating the adv an- tages of the super -twisting algorithm designed to reduce chattering with those of the double po wer con v er gence la w aimed to impro v e system speed a nd dy- namic quality . Ho we v er , the optimal tuning of the sliding mode g ains of the double-po wer super -twisting sliding mode controller represents a considerable challenge. T o address this issue, we proposed an impro v ement to the DPSTSM algorithm through the inte gration of a single-neuron adapti v e algorithm. The single-neuron adapti v e double-po wer super -twisting sliding mode control ap- proach aims to dynamically adjust the cont roller g ains, while deli v ering supe- rior performance in terms of chattering reduction, impro v ed dynamic response, and enhanced rob ustness to load disturbances. A detailed in v estig ation w as car - ried out via MA TLAB/Simulink simulations to determine the ef fecti v eness of the proposed control strate gy . This is an open access article under the CC BY -SA license . Corresponding A uthor: Siham Mencou Engineering Sciences Laboratory , Polydisciplinary F aculty of T aza, Sidi Mohamed Ben Abdellah Uni v ersity Fez, Morocco Email: siham.mencou@usmba.ac.ma 1. INTR ODUCTION Induction motors (IM) ha v e attained considerable acclaim in industrial and transportation sectors due to their cost-ef fecti v eness, durability , and minimal maintenance requirements [1]–[4]. It accounts for more than 60% of the total electrical ener gy consumption within the industrial sectors of de v eloped countries [5]. Analyses predict that the induction motor industry is e xpected to achie v e a compound annual gro wth rate of 3.72% from 2019 through 2028, culminating in a mark et w orth $20,316 million by the year 2028 [6]. Ho we v er , these motors pose distinct challenges re g arding control o wing to their non-linear dynam- ics and susceptibility to parameter v ariations [7]–[9]. Their control requires adv anced strate gies to ensure optimal operational performance. Direct torque control (DTC) is frequently emplo yed in induction machine applications, as it pro vides a swift dynamic response and f acilitates straightforw ard implementation [10], [11]. Ho we v er , traditional DTC e xhibits notable dra wbacks, including high torque and ux ripples alongside v ari- able switching frequencies [12]–[14]. This v ariability stems from its h ysteresis-based control system, which triggers switching at irre gular interv als [15]. These issues detrimentally inuence system performance, induce mechanical vibrations, and increase component wear . J ournal homepage: http://ijpeds.iaescor e .com Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 841 In order to address these limitations, the double-po wer super -twisting slidi n g mode (DPSTSM) algo- rithm has been introduced [16], inte grating the benecial characteristics of the super -twisting algorithm (ST A), which mitig ates the chattering phenomenon [17]–[20], alongside those of the double po wer sliding mode reach- ing la w (DPSMRL), which enhances both the reaching speed and the dynamic quality of the sys tem [21], [22]. As delineated in the study [16], the ef cac y of the proposed controller is assessed within the frame w ork of DTC for induction motor -dri v en elec tric v ehicles, and is juxtaposed with the performance of proportional-inte gral (PI), fuzzy logic, and super -twisting sliding mode controllers. The results indicate that the DPSTSM control enhances the rob ustness of the control system and substantially mitig ates chattering, whilst pre serving ele v ated dynamic performance. Ho we v er , since the operating conditions of IMs v ary considerably due to v arious f actors such as load changes, temperature v ariations and component ageing, determining the optimum sliding-mode g ains of the DPSTSM controller is a major challenge in maintaining optimum performance. Incorrect setting of these parameters can lead to de graded dynamic performance and e v en system instability , which is unacceptable in critical industrial en vironments. In response to the aforementioned challenges, this paper introduces an enhancement of the DP STSM algorithm that incorporates an adapti v e single-neuron approach. The single-neuron adapti v e method pro vides an ef fecti v e solution for dynamically adjusting controller parameters. This methodology le v erages the learning and adapti v e prociencies inherent in neural netw orks, e v en when represented in a simplied architecture comprising a single neuron, to ne-tune controller parameters in alignment with uctuations in operational circumstances without unnecessarily complicating the controller [23]–[25]. In contrast to con v entional multi- layer neural netw orks, the utilization of a single neuron diminishes computational comple xity and promotes practical applicability , while simultaneously preserving a rob ust capacity for adaptation [26], [27]. Recent researches suggest that applying articial intelligence strate gies, lik e neural netw orks and adapti v e algorithms, can bring about note w orth y impro v ements in dynamic response and stability indicators. In [28]–[31], researchers emplo yed neural adapti v e control mechanisms to calibrate the three es sential PID control parameters, specically the proportional, inte gral, and deri v ati v e coef cients, thereby addressing the challenge posed by the comple x g ain paramet rization inherent in traditional PID controllers. The ndings demonstrated that neural adapti v e control signicantly ele v at es the performance of con v entional PID control systems, pro viding augmented rob ustness, stability , and dynamic operational ef cac y . By incorporating a single adapti v e neuron within the frame w ork of the DPSTSM algorithm, our ob- jecti v e is to enhance the rob ustness and dynamic responsi v eness of the system while concurrently mitig ating torque and ux ripples. This paper is or g anized as follo ws: The second section elucidates the principles un- derlying the single-neuron adapti v e methodology and its inte grati on into the DPSTSM controller tailored for the direct torque control approach. The third section presents the methodologies used for system modeling and simulation. Subsequently , we present the simulat ion results and a detailed performance analysis of the proposed system. At the end of the study , we conclude by summarizing the main contrib utions. 2. SINGLE-NEUR ON AD APTIVE DOUBLE-PO WER SUPER-TWISTING CONTR OLLER DE- SIGN 2.1. Double-po wer super -twisting contr oller The DPSTSM algorithm has been designed to enhance the performance of the ST A algorithm by e xploiting the properties of the DPSMRL [16]. The main idea is to replace the switching mechanism of the ST A with that of DPSMRL. The fundamental structure of this algorithm is delineated as (1). dx 1 dt = − k 1 ϕ 1 ( x 1 ) + x 2 + φ 1 ( x 1 , t ) dx 2 dt = − k 2 ϕ 2 ( x 1 ) + φ 2 ( x 2 , t ) (1) W ith: ϕ 1 ( x ) = | x | 1 / 2 sign( x ) + λ | x | 3 / 2 sign( x ) ϕ 2 ( x ) = ϕ 2 1 ( x ) ′ = 2 ϕ 1 ( x ) · ϕ 1 ( x ) ′ = sign( x ) + 4 λ | x | sign( x ) + 3 2 λ 2 | x | 2 sign( x ) ; λ ≥ 0 Single-neur on adaptive double-power super -twisting sliding mode contr ol for ... (Siham Mencou) Evaluation Warning : The document was created with Spire.PDF for Python.
842 ❒ ISSN: 2088-8694 where k 1 and k 2 are sliding mode g ains coef cients. If for constants δ 1 ≥ 0 and ξ 1 = 1 ϕ ′ 1 ( x 1 ) as (2). | φ 1 | ≤ δ 1 | ζ 1 | and φ 2 = 0 ; ∀ t ≥ 0 (2) The system will approach the equilibrium point x (0 , 0) within a nite temporal duration, pro vided that the sliding mode g ains k 1 and k 2 satisfy the conditions as (3) [16]. ( k 1 > 2 δ 1 k 2 > k 1 5 δ 1 k 1 +4 δ 2 1 2( k 1 − 2 δ 1 ) (3) As detailed in [16], the DPSTSM control command is e xpressed as (4). u ( t ) = − k 1  | s ( t ) | 1 2 + λ | s ( t ) | 3 2 sign( s ( t )) − Z k 2  1 + 4 λ | s ( t ) | + 3 2 λ 2 | s ( t ) | 2 sign( s ( t )) (4) W ith s ( t ) the sliding v ariable. Although the controller ensures system s tability as long as t h e g ains sati sfy the st ability conditions as (3), these conditions allo w a wide range of v alues, which mak es it dif cult to calculate the specic g ain v alues k 1 and k 2 . This causes problems for the design of optimal controller parameters. T o solv e this problem, we designed a single-neuron adapti v e double-po wer super -twisting sliding mode (SN A-DPSTSM) controller for adjusting the g ain of the DPSTSM controller in real-time. This approach mer ges the benets of the single-neuron adapti v e control method, kno wn for its simplicity and adaptability , with the principles of DPSTSM control theory . The specic steps in v olv ed in designing this control algorithm are described in 2.2. 2.2. Single-neur on adapti v e double-po wer super -twisting sliding mode contr oller design Figure 1 sho ws the o v erall block diagram of the SN A-DPSTSM controller approach proposed in this study , where the control u ( k ) is dynamically adjusted as a function of the sliding v ariable s ( k ) . Since the single- neuron controller operates using a numerical control, it is essential to discretize the DPSTSM control command described in (4). W e used the direct nite dif ference met ho d for this discretization. This approach allo wed us to transform the continuous controller into an incremental controller , adapted for a numerical en vironment. The discrimination of u ( t ) is gi v en by (5): u ( k ) = k ′ 1 | s ( k ) | 1 2 sign( s ( k )) + k ′ 2 | s ( k ) | 3 2 sign( s ( k )) + k ′ 3 k X i =0 sign( s ( i )) + k ′ 4 k X i =0 | s ( i ) | sign( s ( i )) + k ′ 5 k X i =0 | s ( i ) | 2 sign( s ( i )) (5) W ith (6): k ′ 1 = k 1 k ′ 2 = k 1 λ k ′ 3 = k 2 k ′ 4 = 4 k 2 λ k ′ 5 = 3 2 k 2 λ 2 (6) The incremental controller is gi v en by (7). ∆ u ( k ) = u ( k ) − u ( k − 1) = k ′ 1 | s ( k ) | 1 2 sign( s ( k )) − | s ( k − 1) | 1 2 sign( s ( k − 1)) + k ′ 2 | s ( k ) | 3 2 sign( s ( k )) − | s ( k − 1) | 3 2 sign( s ( k − 1)) + k ′ 3 sign( s ( k )) + k ′ 4 | s ( k ) | sgn( s ( k )) + k ′ 5 | s ( k ) | 2 sign( s ( k )) (7) Int J Po w Elec & Dri Syst, V ol. 16, No. 2, June 2025: 840–850 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 843 W e dene the set v ariables as (8). v 1 ( k ) = | s ( k ) | 1 2 sign( s ( k )) − | s ( k − 1) | 1 2 sign( s ( k − 1)) v 2 ( k ) = | s ( k ) | 3 2 sign( s ( k )) − | s ( k − 1) | 3 2 sign( s ( k − 1)) v 3 ( k ) = sign( s ( k )) v 4 ( k ) = | s ( k ) | sign( s ( k )) v 5 ( k ) = | s ( k ) | 2 sign( s ( k )) (8) These state v ariables are the i nputs to our single neuron, as sho wn in Figure 1. T o each input v ariable v i ( k ) , we assign a weighting f actor ω i ( k ) . Based on the input v ariables and the weighting f actors, the neuron calculates the incremental controller as in (9). ∆ u ( k ) = K 5 X i =1 ω i ( k ) v i ( k ) (9) Where K is the neuron g ain coef cient K > 0 , and k ′ i = K ω i ( k ) for i = 1 , . . . , 5 . Adapti v e learning w as chosen for the adjustment of the weighting f actors because it allo ws the wei ghts to be dynamically modied in response to v ariati ons in the error . This approach guarantees a f aster and more stable con v er gence of the model. W e dene the object i v e function J as a measure of the squared error J = s ( k ) 2 2 , gi v en as (10). ∆ ω i ( k ) = − η i ∂ J ∂ ω i (10) Using impro v ed Hebb-supervised learning, we obtain (11). ∆ ω i ( k ) = η i s ( k ) u ( k − 1)(2 s ( k ) − s ( k − 1)) (11) T o a v oid uncontrolled weight g ain, the output is obtained after normalizing the weighting f actors as in (12). ∆ u ( k ) = K 5 X i =1 ω ′ i ( k ) v i ( k ) with: ω ′ i ( k ) = ω i ( k ) P 5 i =1 ω i ( k ) (12) Whereas in (13). ω i ( k ) = ω i ( k − 1) − η i s ( k ) u ( k − 1)(2 s ( k ) − s ( k − 1)) (13) The SN A DPSTSM control command will be e xpressed as (14). u ( k ) = u ( k − 1) + K 5 X i =1 ω ′ i ( k ) v i ( k ) (14) Figure 1. Block diagram of SN A-DPSTSM controller Single-neur on adaptive double-power super -twisting sliding mode contr ol for ... (Siham Mencou) Evaluation Warning : The document was created with Spire.PDF for Python.
844 ❒ ISSN: 2088-8694 3. SYSTEM MODELING AND SIMULA TION METHODOLOGY 3.1. Integration the SN A DPSTSM contr oller in DTC contr ol In order to v erify the accurac y and ef cienc y of the proposed SN A DPSTSM controller , we t ested it in the conte xt of DTC control of an induction motor [16]. Figure 2 sho ws the block diagram of the DTC control of an IM based on a SN A DPSTSM controller . This control algorithm is inte grated into the closed-loop speed control. The SN A DPSTSM controller calculates the control command u(k) based on the speed error . The reference torque T ∗ em is calculated according to (15) [16]. T ∗ em = f v ω ∗ m + J ω ∗ m + T L − J u (15) 3.2. System modeling and simulation methodology T o test in detail the performances of the SN A DPSTSM proposed controller , simulations were per - formed usi n g MA TLAB/Simulink. F igure 3 sho ws the sim u l ation model structure of the induction motor (IM) controlled by the DTC strate gy based on the SN A-DPSTSM speed controller . The detailed modeling of the DTC control of IM is presented in [16]. Figure 2. Block diagram DTC control of IM based on a SN A DPSTSM controller Figure 3. General system modeling structure of DTC based on SN A DPSTSM controller Int J Po w Elec & Dri Syst, V ol. 16, No. 2, June 2025: 840–850 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 845 4. SIMULA TIONS AND RESUL TS In this section, we delineate the simulation outcomes acquired to assess and scrutinize the ef cac y of the proposed SN A DPSTSM controller . These performance metrics are e v aluated alongside those of the DPSTSM and PI controllers [16]. F or the simulation, we chose a 4 KW induction motor with tw o pairs of poles, whose parameters are summarized in T able 1. The PI and DPSTSM controller parameters used for the simulation were K p = 8 , K i = 32 , K 1 = 35 , K 2 = 15 , and λ = 3 / 2 , which were optimized using the trial- and-error approach. The neuron’ s g ain coef cient w as set to K = 25 , and the initial v alues of the weighting f actors were 0 . 1 . T able 1. Induction machine parameters P arameter V alue P ara meter V alue Nominal po wer 4 kW Stator resistance ( R s ) 1.405 Ω Nominal v oltage 400 V Rotor resistance ( R r ) 1.395 Ω Frequenc y 50 Hz Stator inductance ( L s ) 0.005839 H Inertia ( J ) 0.0131 k g .m 2 Rotor inductance ( L r ) 0.005839 H Friction f actor ( f v ) 0.002985 N.m.s Mutual inductance ( L m ) 0.1722 H Firstly , we e xamine the controller’ s responses to a reference speed setpoint of 75 rad/s applied at 0.05 s and a load torque of 28 Nm introduced at 0.3 s. The reference v alue of the stator ux module is set to 1.1 Wb . Simulation results are presented in Figure 4. The speed response of the three controllers, illustrated in Figure 4(a), sho ws that, under the a ction of the PI control, the motor speed follo ws the reference v alue with an o v ershoot of 5.84% and a steady-state error of 0.12 rad/s in the absence of load and 0.43 rad/s when load is applied. On the other hand, under the command of the DPSTSM controlle r , the motor operates without o v ershoot, b ut has a relati v ely longer re gulation time: 0.15 s for speed v ariation and 0.03 s for torque v ariation. In contrast, the SN A DPSTSM proposed cont roller demonstrated superior performance in terms of both response time and steady-stat e error . Under the control of the SN A DPSTSM controller , the motor accurately follo ws the setpoint signal without displacement , with a v ery f ast response time of 12 ms for speed v ariation and just 2 ms for torque v ariation adjustment. As sho wn in Figures 4(b) and 4(c), the inte gration of the single-neuron adapti v e algorithm into the DPSTSM has signicantly reduced stator ux and torque ripples. This approach decreases steady-state torque ripples to 14.64%, compared with 15.88% for the DPSTSM and 17.83% for the PI. In addition, the proposed SN A-DPSTSM controller minimizes ux uctuations by 60% compare d with the DPSTSM. T o further assess the ef fecti v eness of the proposed controller , we test its rob ustness and stability under more demanding operating conditions. W e apply a reference speed setpoint v arying between 0 rad/s and 157 rad/s o v er a period of 22 s, with a rectangular load torque signal of 14 Nm amplitude applied between 5 s and 17 s. V ariations of the controller parameters k 1 , k 2 , k 3 , k 4 and k 5 are sho wn in Figures 5(a)-5(e). The results clearly demonstrate the ability of the proposed single-neuron adapti v e algorithm to dynamically and optimally adjust these parameters according to the operating conditions of the system. Figure 5(f), which illustrates the v ariation of the objecti v e function J , sho ws that the latter con v er ges to 0 in steady state. This sho ws that the proposed algorithm succeeds in ef ciently minimizing the error . The simulation re sults, presented in Figure 6, illustrate that under the action of the S N A-DPSTSM controller , the m otor speed follo ws the reference signal quickly and ef ciently , without o v ershoot, while main- taining o v erall system stability . Unlik e the PI and DPSTSM controllers, the adapti v e neural netw ork controller sho wed rob ustness to load disturbances and the ability to adapt dynamically to changing operating conditions. In addition, a comparati v e study w as conducted using minimization criteria, specically the inte gral square er - ror (ISE) and the inte gral absolute error (IAE). These tw o statistical parameters are commonly used in control systems to e v aluate and compare the performance of closed-loop systems. According to the v alues of the e v aluation parameters sho wn in T able 2, the PI controller sho wed the least satisf actory beha vior compared to the other controllers, with high v alues for IAE ( 0 . 410 ) and ISE ( 2 . 340 ). In contrast, the DPSTSM controller sho wed superior performance, displaying lo wer v alues for the statistical parameters (IAE: 0 . 367 and ISE: 0 . 798 ). Ho we v er , the proposed SN A DPSTSM controller sho wed the most promising results compared with the other controllers, with an IAE of 0 . 021 and an ISE of 0 . 177 . The use of an SN A controller sho wed e xceptional ability to minimize error o v er time. Single-neur on adaptive double-power super -twisting sliding mode contr ol for ... (Siham Mencou) Evaluation Warning : The document was created with Spire.PDF for Python.
846 ❒ ISSN: 2088-8694 (a) (b) (c) Figure 4. Simulation results of the PI, DPSTSM, and SN A DPSTSM controllers: (a) rotor speed, (b) electromagnetic torque, and (c) stator ux Int J Po w Elec & Dri Syst, V ol. 16, No. 2, June 2025: 840–850 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 847 (a) (b) (c) (d) (e) (f) Figure 5. V ariations of the controller parameters: (a) k 1 , (b) k 2 , (c) k 3 , (d) k 4 , (e) k 5 , and (f) the objecti v e function J Single-neur on adaptive double-power super -twisting sliding mode contr ol for ... (Siham Mencou) Evaluation Warning : The document was created with Spire.PDF for Python.
848 ❒ ISSN: 2088-8694 Figure 6. Rotor speed response for a speed v ariation from 0 rad/s to 157 rad/s T able 2. Ev aluation of ISE end IAE parameters P arameter P I DPSTSM SN A-DPSTSM ISE 0.410 0.367 0.021 IAE 2.340 0.798 0.177 These results highlight the impro v ements brought by the inte gration of the single-neuron adapti v e algorithm into the DPSTSM controller in terms of self-adjustment of the controller , reduced chattering, im- pro v ed dynamic response, and rob ustness to v ariations in system parameters. This guarantees more ef cient and reliable control, e v en under v ariable operating conditions, making it an ideal solution for applications re- quiring high precision and increased rob ustness, such as motor control in electric v ehicles or other demanding industrial systems. The ndings elucidate the enhancements f acilitated by the incorporation of the single-neuron adapti v e algorithm into the DPSTSM controller , particularly re g ardi ng the self-adjustment of controller parameters, the mitig ation of chattering, the enhancement of dynamic response, and the resilience to uctuations in system parameters. This ensures a more ef cacious and dependable control mechanism, e v en in the f ace of v ariable operational conditions, thereby rendering it an optimal solution for applications necessitating high precision and augmented rob ustness, such as motor control within electric v ehicles or other high-demand industrial systems. 5. CONCLUSION This paper introduces an enhancement of the DPSTSM algorithm through the incorporation of a single-neuron adapti v e algorithm, specically designed to address the issue of optimal controller g ain ad- justment. Simulation results sho wed that the single-neuron adapti v e controller f acilitates dynamic and optimal adjustment of control parameters under dif ferent operating conditions. The proposed SN A DPSTSM controller has e xhibited superior performance compared to both the DPSTSM and PI control lers by deli v ering prompt and precise responses to setpoint and load v ariations, signicantly mitig ating the chattering phenomenon, reducing torque and ux ripples and enhancing resilience ag ai n s t disturbances and uctuations in operational conditions. Furthermore, the inherent simplicity of the single-neuron algorithm promotes the practical deplo yment of the controller , while concurrently diminishing the computational comple xity in relation to con v entional multi-layer neural netw orks. These adv ancements render the SN A DPSTSM controller e xceptionally well-suited for ap- plications necessitating high le v els of precision and rob ustness, such as motor control in electric v ehicles and in rigorous industrial en vironments. FUNDING INFORMA TION This research did not recei v e an y specic grant from funding agencies in the public, commercial, or not-for -prot sectors. Int J Po w Elec & Dri Syst, V ol. 16, No. 2, June 2025: 840–850 Evaluation Warning : The document was created with Spire.PDF for Python.
Int J Po w Elec & Dri Syst ISSN: 2088-8694 ❒ 849 A UTHOR CONTRIB UTIONS ST A TEMENT This journal uses the Contrib utor Roles T axonomy (CRediT) to recognize indi vidual author contrib u- tions, reduce authorship disputes, and f acilitate collaboration. Name of A uthor C M So V a F o I R D O E V i Su P Fu Siham Mencou ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ Majid Ben Y akhlef ✓ ✓ ✓ ✓ El Bachir T azi ✓ ✓ ✓ ✓ C : C onceptualization I : I n v estig ation V i : V i sualization M : M ethodology R : R esources Su : Su pervision So : So ftw are D : D ata Curation P : P roject Administration V a : V a lidation O : Writing - O riginal Draft Fu : Fu nding Acquisition F o : F o rmal Analysis E : Writing - Re vie w & E diting CONFLICT OF INTEREST ST A TEMENT Authors state no conict of interest. D A T A A V AILABILITY The authors conrm that the data supporting the ndings of this study are a v ailable within the art icle and/or its supplementary materials. REFERENCES [1] P . Dinolo v a, V . Ruse v a, and O. 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