Fractional order adaptive kalman filter for sensorless speed control of dc motor

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International Journal of Electronics ISSN: (Print) (Online) Journal homepage: www.tandfonline.com/journals/tetn20 Fractional order adaptive Kalman filter for sensorless speed control of DC motor Ravi Pratap Tripathi, Ashutosh Kumar Singh & Pavan Gangwar To cite this article: Ravi Pratap Tripathi, Ashutosh Kumar Singh & Pavan Gangwar (2023) Fractional order adaptive Kalman filter for sensorless speed control of DC motor, International Journal of Electronics, 110:2, 373-390, DOI: 10.1080/00207217.2021.2025452 To link to this article: https://doi.org/10.1080/00207217.2021.2025452 Published online: 05 Feb 2022 Submit your article to this journal Article views: 278 View related articles View Crossmark data Citing articles: View citing articles Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=tetn20 INTERNATIONAL JOURNAL OF ELECTRONICS 2023, VOL 110, NO 2, 373–390 https://doi.org/10.1080/00207217.2021.2025452 Fractional order adaptive Kalman filter for sensorless speed control of DC motor Ravi Pratap Tripathi , Ashutosh Kumar Singh and Pavan Gangwar Electronics and Communication Engineering Department, Indian Institute of Information Technology Allahabad, Prayagraj, India ABSTRACT ARTICLE HISTORY Received 31 July 2021 State estimation is a challenging and most crucial issue in the Accepted December 2021 industry for proper monitoring and controlling of the plants These kinds of control systems have the requirement of costly KEYWORDS measurement sensors/equipment for measurable and unmeasur­ DC motor; Kalman filter; able state variables of the dynamical plants These drawbacks can fractional calculus; sensorless be overcome by designing a sensorless system to estimate the state control variables In the proposed work, sensorless speed control of DC motor is implemented by using a fractional-order adaptive Kalman filter (FOAKF) The FOAKF algorithm uses a fractional feedback loop of the previous Kalman gain along with the current Kalman gain The motor shaft speed is estimated by the FOAKF state estimator Furthermore, a performance comparison of extended Kalman filter (EKF) and FOAKF estimator under a similar condition is realised using MATLAB/Simulink environment To validate the performance of the FOAKF estimator, a hardware prototype model has been presented with the help of the arduino board In the proposed work, the root-mean-square error (RMSE) and Euclidean distance error between reference speed and estimated speed have been used for the performance metric The performance comparison result shows that FOAKF is a more robust and accurate state esti­ mator in comparison with EKF Introduction In all industrial work, mechanical movement is carried out with the use of electric motors, hydraulics and pneumatic systems To drive these systems, motors are required For motion transmission systems generally, AC motors are used while DC motors are used in robotic manipulators and in industrial applications, where high load torque is needed To imple­ ment the industrial task, various control approaches are used, in general emphasis is given to controlling the speed of the machines To achieve effective speed control, a closed-loop control system is used, in which motor state variables like current, position and rotor speed are feedbacked A suitable controller can be implemented when all the state variables of the system are known To measure the entire state variable, it uses various sensors; thereby cost increases and makes the system more complex To reduce the number of sensors, hardware complexity and system cost, state observers/estimators are used in the control loops of CONTACT Ravi Pratap Tripathi rse2018510@iiita.ac.in; Indian Institute of Information Technology Allahabad, Prayagraj, Uttar Pradesh, India © 2022 Informa UK Limited, trading as Taylor & Francis Group 374 R P TRIPATHI ET AL electrical motors (Jang et al., 2003; Mohan et al., 2020) The advantages of state observers over sensors are that they are not affected by any environmental causes and any machine distortions (Wang et al., 2019) Some sensorless speed control methods were performed (Lascu et al., 2006; Raca et al., 2008) An observer is a mathematical tool that estimates states with the help of system dynamical models and some measured state variables Some works have been presented in previously published literature in which sensors are not mounted on the motor shaft and stochastic observer/estimator are used to estimate the speed of the DC motor In the previous works, the state estimation was performed by using the sliding mode observer (Lascu et al., 2009; Qiao et al., 2013) Kalman Filter (KF) (Rigatos, 2009), Extended Kalman Filter (EKF; Boizot & Busvelle, 2007; Terzic & Jadric, 2001; E Zerdali, 2019) and the particle filter (Ristic et al., 2004) KF is an elementary concept and an extensively used mathematical tool for state estimation KF is applicable only for linear systems and gaussian noise In a nonlinear dynamic system generally, EKF is used (O Aydogmus & Talu, 2012; Z Aydogmus & Aydogmus, 2015; Habibullah & Lu, 2015; Tiwari et al., 2017; Yin et al., 2019; Zhou et al., 2016) EKF is applicable only when the system is differentiable, well defined, noises are gaussian and their covariance is known; otherwise, it starts diverging In such cases, the other versions of KF like UKF (Jafarzadeh et al., 2013), CDKF (Zhao et al., 2020), Adaptive KF/EKF (Jiancheng & Sheng, 2011; Emrah Zerdali, 2020; ZhiWen et al., 2013) were pro­ posed In the open literature, EKF and its derivative-based sensorless speed control algorithms are present but they could not able to provide sufficient contribution for sensorless speed control when the system is diverging and not well defined To incorpo­ rate the above difficulties in state estimation FOAKF algorithm has been proposed FOAKF is a robust adaptive state estimation technique, which is more adaptive in the field of signal and image processing, thermodynamics and tracking applications (Kaur & Sahambi, 2016) In the presented FOAKF algorithm, modified steady-state Kalman gain is proposed that is obtained by inserting a fractional feedback loop of previous Kalman gain with current steady-state Kalman gain The main advantage of FOAKF is that its gain will never be diverse because of the fractional derivative of the previous Kalman gain The main contribution of this work is to develop a MATLAB/Simulink model for analysing the performance of the EKF and FOAKF estimator at nonlinear load in the closed-loop sensorless speed control Furthermore, a simulink model and its experimental validation have been presented of a FOAKF estimator-based sensorless speed control mechanism for DC motor at nonlinear and sudden load conditions The paper is organised in the following sections: In section 2, a DC motor mathematical model has been presented The state estimation algorithm is presented in section In section 4, a stability analysis of the FOAKF estimator has been discussed The simulation and experimental results are described in section and at last, the concluding remarks are presented in section DC motor mathematical model DC motor mathematical model in the time domain form is required for state estimation algorithm The DC motor model has two main equations, one is electrical and the other one is mechanical (Kothari & Nagrath, 2010) On applying Kirchhoff’s law in the armature circuit shown in Figure 1(a), the electrical equation is given as: INTERNATIONAL JOURNAL OF ELECTRONICS 375 Figure DC motor circuit diagram and load model (a) Circuit diagram of DC motor (b) Nonlinear load model d vtị ẳ eatị ỵ Raiatị ỵ La iatị (1) dt where Ra is armature resistance, ia is armature current, La is the self-inductance produced by the armature flux, vt is armature supply voltage, and eatị ẳ Keif mị is back emf A separately excited DC motor is shown in Figure 1(a), which requires an extra DC voltage vf to produce a magnetic field Rf and Lf represent the resistance and inductance of the field winding respectively It is well known, when vf constant and steady-state exists on the field circuit, then if (field current) is constant, therefore Equation (1) can be written as: vtị ẳ K0emtị ỵ Raiatị ỵ La d iaðtÞ (2) dt For motoring operation, the dynamic equation for the mechanical system is d Ttị ẳ Ktif tịiatị ẳ J mtị ỵ TLtị ỵ Tf tị ỵ Dmtị (3) dt where J is moment of inertia of motor and load, ωm is the motor speed, D viscous damping coefficient, Tf ðtÞ and TL is coulomb friction torque and load torque, respectively The load torque is shown in Figure 1(b) and can be mathematically written as: TL ẳ mgL cos ỵ mL2 d mtị (4) dt where L is arm length, m is the mass of load, g is gravitational constant and θ is angular displacement (integral of ωm) As if constant, then Equation (3) can be written as: Ttị ẳ K0tiatị ẳ J d mtị ỵ TLtị ỵ Tf tị ỵ Dmtị (5) dt 376 R P TRIPATHI ET AL With the help of Equations (2) and (5), the state-space model can be implemented The model comprises two state variables armature current (ia) and motor speed (ωm) The state-space model of DC motor in the time domain is given as: � � "D K0 # t � � " mgL cos θÀ Tf #� � _ m mL2ỵJ mL2ỵJ m 01 ¼ À K0t ỵ mL2 ỵJ (6) _ À Ra À Ra ia La La ia La vt From the state-space model, θ produces nonlinearity in the system For digital implemen­ tation, the above Equation (6) must be discretised The discrete-time model of the above system is represented by equations given below: xkỵ1 ẳ Adxk ỵ Bduk ỵ vk (7) yk ẳ Cdxk ỵ wk (8) In the above equation, state variable x ẳ ẵm ia t, input vector u ¼ ½1 vt�t, and output vector y ¼ ½0 ia�t whereas v (mean 0, covariance Q) and w (mean 0, covariance R) are the process noise, and measurement noise, respectively After putting the motor parameters value given in Appendix in Equation (6), the discretised matrix coefficients Ad; Bd and Cd at sampling interval ðTsÞ can be written as: � � Ad ẳ I ỵ ATs ẳ 0:005 0:007 0:9828 (9) � � Bd ¼ BTs ¼ À 0:016 0 0:0067 (10) Cd ẳ C ẳ ẵ (11) With the help of controllability c ẳ ẵBd AdBd ị and observability O ẳ ẵCd AdCdt ị test, we find that c and O are full rank (n = 2) matrix, which implies that we can design a controller and observer The details of EKF and FOAKF observer and their performance comparison are present in the next sections State estimation algorithm The state-space model obtained in section is used by the EKF and FOAKF algorithm to estimate the motor speed ðωmÞ and armature current ðiaÞ from the measurement ia 3.1 EKF algorithm KF is a state estimator that uses an iterative mathematical equation to compute the best possible state from the noisy state The KF is not applicable for the nonlinear system, as in the calculation of θ using ωm leads the DC motor model to nonlinear, so EKF is used to overcome this difficulty EKF is the extended version of KF that uses first-order Taylor series approximation of nonlinear dynamical function to estimate the state by noisy measured data The EKF has two main steps one is state prediction and the other one is state correction In the prediction step, a dynamical model of the system with process INTERNATIONAL JOURNAL OF ELECTRONICS 377 noise covariance Q is used In the correction, the predicted state is corrected with the help of continuously measured data and Kalman gain ðKÞ However in the application of EKF, the system dynamical model and initial values of all the covariance matrices, namely, Q; R and P must be known accurately If these are not defined correctly then the system may become unstable and can diverge The standard EKF estimation has a two-step process, one is prediction and the other one is corrections, which are presented below The prediction step is ^xkỵ1=k ẳ f ð^xk; ukÞ (12) Pkỵ1=k ẳ AdPkAtd ỵ Q (13) ^xkỵ1=k is predicated state of xkỵ1 The correction step is Kkỵ1 ẳ Pkỵ1=kCdt CdPkỵ1=kCdt þ RÞÀ (14) ^xkþ1 ẳ ^xkỵ1=k ỵ Kkỵ1ykỵ1 Cd^xkỵ1=kị (15) Pkỵ1 ẳ Pkỵ1=kI Kkỵ1Cdị (16) ^x0kỵ1 is the updated states, I is the identity matrix and Kkỵ1 is Kalman gain Equations (12)- (16) are one complete cycle of the EKF algorithm 3.2 FOAKF algorithm The mathematical model of the system is nonlinear, and generally, the system model and their covariance Q; R and P are not normally well defined Therefore, the other adaptive state estimation algorithm known as FOAKF has been presented in this section The main concept behind FOAKF is to introduce a fractional derivative of previous Kalman gain as feedback in the Kalman gain loop Grunawald-Letnikov’s definition (Sierociuk & Dzieliński, 2006) has been used for the computation of fractional feedback Kalman gain in the proposed technique Based on Grunawald-Letnikov definition, the fractional difference of signal xk can be given as: α Xk j Δ xk ẳ 1ị jxk j (17) h j¼0 α is fractional order ð0 < α < 1ÞÞ, k number of samples in signal xk, h (Sampling interval = 1) In the proposed work, for all the simulation we assume α ¼ 0:5 because if we take α close to zero then the time taken to stabilising xk will be more and for α close to one it takes less time to stabilise The γj (gamma function) can be given as: � � � αðαÀ 1ị :: jỵ1ị α for j>0 γj ¼ ¼ j j for j ¼ The FOAKF gain is calculated by using fractional calculus For calculating FOAKF gain, following step has been used as given below: 378 R P TRIPATHI ET AL Figure Control strategy of sensorless speed control of DC motor (1) Calculate standard Kalman gain Kkỵ1ị (2) Calculate fractional difference gain ðfkÞ of prior standard Kalman gain (3) FOAKF gain is the algebraic sum of fractional difference gain and standard Kalman gain It is well known that the working of EKF is a two-step process The first step of EKF is prediction while the second step is correction Similar to EKF, FOAKF is also a two-step process which is given below: Predicted state of Equations (7) and (8) is: ^xkỵ1=k ẳ Ad^xk ỵ Bduk (18) Pkỵ1=k ẳ AdPkAtd þ Q (19) In the correction step, first calculate standard Kalman gain Kkỵ1ị using Equation (14) Applying the theory of fractional calculus presented in Equation (17) on prior Kalman gain ðKkÞ the fractional difference gain ðfkÞ can be calculated as: (Xk ) fk ¼ E ðÀ 1ịjỵ1jKk j (20) j¼0 fk presented in Equation (20) is the mean of the fractional derivative of the prior Kalman gain The proposed FOAKF gain ðKmodÞ is given by adding Equations (14) and (20) Kmod ẳ Kkỵ1 ỵ fk (21) Now the corrected, states and error covariance of discrete time system can be calculated similarly as Equations (15) and (16) using Kmod, which are given below: ^xkỵ1 ẳ ^xkỵ1=k ỵ Kmodykỵ1 Cd^xkỵ1=kị (22) Pkỵ1 ẳ Pkỵ1=kI KmodCdị (23) Equations (18)-(23) are the one complete cycle of the FOAKF algorithm The FOAKF gain ðKmodÞ can be calculated by minimising the error covariance Pkỵ1: INTERNATIONAL JOURNAL OF ELECTRONICS 379 Figure Detailed simulink model of the system (a) EKF and FOAKF block (b) DC motor model n o Pkỵ1 ẳ E xkỵ1 ^xkỵ1ị2 (24) nÀ 2o Pkỵ1 ẳ E xkỵ1 ^x0kỵ1=k Kkỵ1 ỵ fkị ykỵ1 Cd^x0kỵ1=k (25) For minimising the Pkỵ1 put @Pkỵ1 ẳ 0, and after solving we get FOAKF gain ðKmodÞ: @K ( Xk ) Kmod ẳ Kk ỵ E 1ịjỵ1jKk j (26) j¼0 380 R P TRIPATHI ET AL Stability analysis For FOAKF estimator stability analysis, the error between the actual state ðxkÞ and the estimated state ð^xkÞ should be considered Because if error reaches zero, then ^xk will reach to xk The dynamics of the xk and ^xk are given in Equations (7) and (23), respectively Now the error ðeÞ and its derivatives ðe_Þ can be given as: e ¼ xk À ^xk (27) e_ ẳ Axk ỵ Buị A^xk ỵ Bu ỵ KmodCxk C^x0kị (28) ẳ A KmodCịe Equation (28) will be stable only if ðA À KmodCÞ is hurwitz matrix If it is hurwitz then ^xk will keep becoming a better estimation of xk as time goes on and the system become stable Therefore to prove stability, a Lyapunov function can be considered as: VðeÞ ¼ e0Pe (29) V_ eị ẳ e0Pe_ ỵ e_0Pe (30) V_ eị ẳ e0PA KmodCịe ỵ e0ðA À KmodCÞ0Pe (31) V_ eị ẳ e0PA0 ỵ AP 2KmodCPịe (32) According to Lyapunov stability concept for asymptotic stability, V_ ðeÞ < and using the riccati equation given in (Aminifar, 2016): PA0 ỵ AP KmodCP ỵ B0B ẳ (33) Therefore, Equation (32) can be written as PA0 ỵ AP 2KmodCP ¼ À B0B À KmodHP (34) Now substitute the Equation (34) value into Equation (32), V_ ðeÞ will be: V_ eị ẳ e0B0B ỵ KmodHPịe (35) From Equation (35), it is clear that VðeÞ will be a valid quadratic Lyapunov of Equation (29), which implies that the corresponding matrix must be hurwitz Simulation and experimental results The block diagram of the sensorless speed control strategy is shown in Figure In the block diagram, a PI controller, motor driver, DC motor and a FOAKF state estimator are present The FOAKF state estimator estimates the speed, which is feedbacked to perform the sensorless speed control of the DC motor INTERNATIONAL JOURNAL OF ELECTRONICS 381 Figure Performance comparison of EKF and FOAKF estimator at variable load TL ẳ mgL cos ỵ mL2 dt dωmÞ (a) Speed estimations result and actual motor shaft speed (b) Current estimations result and measured current speed (c) Euclidean distance speed estimation error of EKF and FOAKF (d) Root mean square error of EKF and FOAKF (e) Load torque 5.1 Simulation results A sensorless PI-based speed control system is built using Simulink/MATLAB code generator to compare the EKF and FOAKF performance The detailed MATLAB simu­ link structure of the system is shown in Figure The Simulink structure has two 382 R P TRIPATHI ET AL modules, ‘module-1’ and ‘module-2’ Module-1 consists of a PI controller for speed and current, a PWM generator, EKF, and FOAKF algorithms The sampling interval of module-1 is 200 µs Module-2 has a separately excited DC motor and an H-bridge built by four IGBT using MATLAB/Simscape toolbox A PWM-controlled H-bridge is used to drive the DC motor as shown in module-2 The sampling interval of module- is µs In this work, armature current ðiaÞ of the DC motor is a known or measured state variable, which is used for estimating the other states using EKF and FOAKF algorithms In a real-time application, armature current ðiaÞ is calculated by the use of a current transducer with machine noise, A/D quantisation noise and measurement inaccuracy Therefore in simulation, armature current is obtained by adding white gaussian noise In the controller block shown in Figure 3, the current and speed PI controller is used The parameter of speed PI controller are kps ¼ 10 and kis ¼ 0:5, current controller parameters are kpc ¼ 100 and kic ¼ 10 5.1.1 Comparison of EKF and FOAKF estimator To compare the EKF and FOAKF estimator performance, the manual switch presents in Figure is connected to work-1 In a real-time application, the exact parameters of the motor cannot be perfectly known, therefore to perform simulation gaussian noise INTERNATIONAL JOURNAL OF ELECTRONICS 383 is added to the motor shaft speed ðωacÞ The obtained ωac is used for closed-loop speed control and to carry out the performance comparison between EKF and FOAKF estimator for constant reference speed ðωref ¼ 50 rad= secÞ tracking The speed and current estimation results of the EKF and FOAKF estimator at variable load are shown in Figure 4(a,b), respectively From Figure 4(a), it can be stated that the estimated speed by FOAKF is closer to ωref in comparison with EKF, i.e the tracking capability of ωref is better by FOAKF estimator In Figure 4(c,d), the Euclidean distance speed estimation error and root-mean-square error (RMSE) between the estimated motor speed and reference speed is shown The average Euclidean estimation error is obtained by subtracting the corresponding value of estimated speed and reference speed RMSE is the root-mean-square error between a reference speed and estimated speed, over a range of observations The RMSE is obtained with the help of the RMS block present in the simulink library From Figure 4(c,d), it can be observed that FOAKF has the better error (nearer to zero is better) response in comparison with the EKF estimator 5.1.2 Speed control with FOAKF estimator In the second work, the DC motor speed control is performed by FOAKF-based speed estimation technique The FOAKF estimated speed is used for the closed- loop speed control mechanism The simulink model for the present work is similar to the same model used in section 5.1; the only difference is the manual switch has been connected to work-2 in the place of work-1 In work-2, the FOAKF estimated shaft speed ðωfoakf Þ is used in closed-loop control to perform the sensorless con­ stant speed control The FOAKF-based sensorless closed-loop speed control perfor­ mances are shown in Figure at the same load condition as used in section 5.1.1 The speed and current estimation results are presented in Figure 5(a,b), respec­ tively In Figure 5(a), the FOAKF estimated speed and actual motor shaft speed (controller shaft speed) are shown From Figure 5(a), it can be observed that the actual shaft speed and FOAKF estimated shaft speed both track the reference speed very quickly and produce minimum transient On comparing the results obtained in Figures (a) with 5(a), it can be stated that the FOAKF based speed controller has better overshoot, settling time and it has very good ωref tracking ability In Figure 5(b), the FOAKF estimated armature current and measured arma­ ture current are shown On comparing the results obtained in Figures 4(b) to 5(b), we are finding that FOAKF based estimation method takes less current for the same load torque In Figure 5(c,d), the average Euclidean error and RMSE error between reference speed and FOAKF estimated speed is present In the perusal of this, it can be observed that the obtained error by the FOAKF estimator is closer to zero; therefore, it can be said that FOAKF has a better error response To check the robustness of the FOAKF estimator, the variable load is replaced by a sudden load of 20 N À m and reference speed is varied from 50 rad= sec to 130 rad= sec To create a sudden load in the simulink environment, a step function is used with step value 20 at t = 0.5 sec The performance of the FOAKF estimator at sudden load is shown in Figure In Figure 6(a), the speed estimation perfor­ mance is shown From this Figure, it can be observed that when there is no load, then after some momentary oscillation FOAKF estimator estimates the reference 384 R P TRIPATHI ET AL Figure Simulation results of DC motor using FOAKF estimator for nonlinear load TL ẳ mgL cos ỵ mL2 dt dmị a) Closed-loop speed control of DC motor using FOAKF (b) Armature current (c) Euclidean distance speed estimation error (d) Root mean square error INTERNATIONAL JOURNAL OF ELECTRONICS 385 Figure Simulation results of DC motor using FOAKF estimator for sudden constant load TL ẳ 20 N mị (a) Closed-loop speed control of DC motor using FOAKF (b) Armature current (c) Step load torque 386 R P TRIPATHI ET AL Figure Hardware in loop system design (a) Schematic of hardware in loop system design (b) Arduino model for experimental setup interfacing speed perfectly before the EKF estimator At t = 0.5 sec, load torque is changed from to 20 N- m From Figure 6(a), it can be observed that as the load changes (0 to 20 N-m), the estimated speed by FOAKF and EKF estimator deviates slightly But after some transient, the FOAKF estimator again starts tracking to reference speed, while EKF does not From the discussion of the above results, we can state that FOAKF-based sensorless speed controller has better overshoot and settling time and it has very good reference tracking ability along with any load condition 5.2 Experimental validation To validate FOAKF-based sensorless speed control performance experimentally, a hardware prototype model has been designed The schematic of the hardware in loop (HIL) configuration, Arduino communication setup for HIL simulation, and INTERNATIONAL JOURNAL OF ELECTRONICS 387 Figure Experimental setup of sensorless speed control of DC motor its experimental setup is shown in Figures and 8, respectively The experimental setup consists of four major components, which are a motor with a voltage and current sensor, motor driver module, PWM inbuilt Arduino board, and Personal Computer (PC) To apply the load torque on the shaft, manual friction has been provided In PC, PI controller and FOAKF estimator are present The armature current and voltage across the motor winding are sensed by the voltage and current sensor and read by the Arduino analog pin A1 and A2 This sensed value is used to estimate the speed of the motor in the next iteration by the FOAKF estimator The difference between reference speed and estimated speed generates the error signal for the controller block The controller block generates the corre­ sponding duty cycle, which is responsible for generating the PWM signal The generated PWM signal is applied to Arduino digital pin 9, to generate an appro­ priate voltage with the help of the L298N motor driver to control the speed of the DC motor The experimental results of the prototype with nonlinear load at constant and variable speed have been given in Figures 9(a,b), respectively From the figures, it can be observed that the FOAKF estimator showing remarkable reference speed tracking capability in both cases 388 R P TRIPATHI ET AL Figure Experimental results on constant and variable speed (a) Experimental validation of FOAKF estimator on constant speed (b) Experimental validation of FOAKF estimator on variable speed Conclusion In this paper, EKF- and FOAKF-based sensorless speed control of DC motor have been performed The entire simulation test has been carried out at the same condition to compare the state estimation performance The result shows that EKF’s average Euclidean estimation error and average RMSE are 0.9641 and 1.52, respectively, while FOAKF’s average Euclidean estimation error and average RMSE are 0.5641 and 1.15, respectively The computational time of FOAKF is 23 µs and in EKF 19 µs The computational complexity of FOAKF is slightly greater than EKF because in FOAKF the effect of previous Kalman gain is added by inserting the fractional feedback loop in the standard Kalman gain However, this amount of computational complexity does not create any problem because of INTERNATIONAL JOURNAL OF ELECTRONICS 389 modern fast computational processors and GPU (graphics processing unit) The one more advantage of FOAKF, it is derivative-free that makes it more suitable and adaptive for any load variation In the second work, FOAKF-based closed-loop speed control has been performed The results of this work indicate that the sensorless speed control of DC motor using FOAKF has better accuracy in comparison to EKF in all the cases The result shows that FOAKF is more robust and more stable for any low-speed application and load variation Disclosure statement No potential conflict of interest was reported by the author(s) ORCID Ravi Pratap Tripathi 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