Continuous Fast Terminal Sliding Surface-Based Sensorless Speed Control of PMBLDCM Drive

Variable frequency drives (VFDs) have been used for position sensors for many years. Alternatively, reliable VFDs for smart autonomous systems require the simultaneous configuration of sensors and sensorless operation. This article details the design and implementation of a continuous fast terminal (CFT) sliding mode controller (SMC)-based speed controller and speed estimation through the CFT sliding mode observer that accounts for iron loss while calculating total disturbances. The rotor speed dynamics of the permanent magnet brushless dc motor drive system is first investigated considering lumped disruption (interference, parametric complexity, and nonlinear dynamics). The built-in SMC can control the rotor speed in real time, as well as analyze and compensate for aggregated interruptions in rotor speed dynamics. In addition, the field-oriented control (FOC) method is developed to maximize torque production while drastically minimizing torque ripples. With this approach, the zero-torque pulsation constraint is gradually achieved at a wide speed range. The validation of the proposed CFT-SMC with FOC is carried out using simulation and experiments. According to the comparative study, the proposed sensorless control outperforms conventional methods due to the inclusion of iron loss.

increase the torque-weight ratio, improve control sensitivity, and reduce vibration. As a result of magnetic alignment and winding configuration, PMBLDCM is characterized by a trapezoidal back electromagnetic force (EMF), which demands a specific control strategy to maintain the electromagnetic torque. Normally, PMBLDCM works in a six-step conduction mode, commuting the voltage source inverter (VSI) every 60 electrical degrees. Meanwhile, PMBLDCM exhibits a torque ripple during operation, specifically a periodic commutation torque ripple, which is crucial to ensure smooth operation [2]. In high-precision applications, torque ripple causes vibrations and acoustic noise. In addition, the torque ripple causes more current ripples and higher copper losses in the PMBLDCM drive, resulting in a reduction in the overall drive efficiency. Consequently, reducing copper loss is a significant focus of current research to improve motor torque performance with reduced torque ripple [3].
An accurate PMBLDCM modeling, in which material and design parameters are taken into account, is essential to reduce torque ripple [4]. Permanent magnets are particularly susceptible to performance degradation due to temperature and magnetic saturation [5]. A decrease in PMBLDCM performance is attributed to iron loss when hysteresis and eddy current losses are considered [6]. In the literature, a variety of PMBLDCM models based on iron loss are examined to predict and assess the effects of iron loss [3]. Iron loss directly or indirectly affects the electromagnetic torque performance of the PMBLDCM drive [7]. Therefore, a desirable electromagnetic torque reference must be evaluated to compensate for the effect of iron loss. Electromagnetic torque reference is traditionally generated from a proportional-integral (PI)-based speed regulator with a motor-dependent gain factor [8]. This stage of outer loop control does not include any iron loss compensation. It is only possible that the speed feedback should be included with the disturbance corresponding to iron loss. This type of disturbance inclusion may not be possible with traditional rotor position sensor implementation. Again, the reliability and sensitivity of the rotor position sensor have a significant impact on the performance of PMBLDCM [2], [9]. This challenge of sensitivity can be overcome by high-quality encoders and resolvers in electrical drives for the electric vehicle propulsion system. However, sensor devices are sensitive to mechanical and electrical malfunctions, susceptible to electromagnetic interference [6], which could also cause the vehicle to go out of operation or sometimes lead to severe situations. Therefore, this motivates the research demand for sensorless motor drive operation that includes iron loss compensation [10].
Rotor speed dynamics is a time-varying nonlinear mechanism that is attributed to lumped disturbances [11]. These external disturbances constitute the load torque, the friction torque, and the uncertainties of the parameters. Adopting a traditional controller to address the problem of rotor speed tracking in the PMBLDCM drive system results in poor dynamic response [12]. Therefore, it is effective to utilize the nonlinear control technique that incorporates external disturbances and large nonlinear effects to achieve robust rotor speed control [13]. Sliding mode observer (SMO)-based rotor position estimation to achieve sensorless operation can positively contribute to enhance the reliability and drives performance. Due to the superior ability to maintain good dynamic performance and improve the ability to reject disturbances, the SMO is deployed in [14]. In the conventional SMO method, a higher switching gain is required to approach the maximum limit of total disturbance, causing steady error and chattering. After analyzing the state-of-the-art SMO for PMBLDCM speed estimation, it can be concluded that the PMBLDCM control problem remains a relatively open topic and requires further research. Hence, the objectives of this article are as follows.
1) To develop an SMO for PMBLDCM speed estimation with improved disturbance rejection capability for better stability. For this, precise dynamic modeling of the PMBLDCM containing parameters and load torque is required for accurate performance. 2) To generate continuous control signals and eliminate the chattering phenomenon of the SMO. 3) To achieve fast time convergence of the sliding surface to the equilibrium state in a specific time limit. SMO is often used as a nonlinear control theory as a result of its desirable advantages, such as stability toward mismatched perturbations and reduced-order compensated dynamics [15]. As a result, the SMO-based speed controller [i.e., sliding mode controller (SMC)] has gained considerable interest [16]. SMC takes the change in load torque as a disturbance for PM-BLDCM [17]. However, this article used hall sensors where the use of sensors has several limitations. Fuzzy SMC for PM-BLDCM with linear sliding surface and switching gain function is proposed in [18]. Due to the linear surface of traditional SMC and asymptotic stability problems with higher switching gain, it cannot be considered a robust feature for PMBLDCM applications [15]. Furthermore, fuzzy SMC is reported using an integral sliding surface for electric vehicle application [11], for water pump application [19] using PMBLDCM. Conventional SMCs demand a level of control gain to improve effective accuracy and stability, even though they have considerable chattering issues [12]. To overcome the stability problem, the controller is built by reconfiguring the sliding surface of the traditional SMC approach. Terminal SMC (TSMC) and nonsingular fast TSMC (FTSMC) for permanent magnet synchronous motor (PMSM) are explored in [20] and [21], respectively. The TSMC approach, among all other sliding procedures devised to deal with this problem, ensures limited time convergence while monitoring errors to the origin [22]. However, TSMC has the limitation of discontinuous control action, leading to the chattering phenomenon. To reduce such issues, a nonsingular FTSMC for the automotive throttle system is reported in [21], [23], and [24]. Therefore, the objectives of this article are extended further to the following: 1) design high-precision speed tracking through SMC, considering the effect of iron loss and unknown external disturbance and 2) improve the torque ripple content during steady-state dynamics and transient conditions, considering a better speed profile at low stator phase currents total harmonic distortion (THD). Without a suitable control action, PMBLDCM with iron loss consideration prevails in instability and disruption [7]. This often results in ineffective closed-loop control performance, such as increased torque ripple and reduced speed. Several sensorless high-performance control strategies, such as fieldoriented control (FOC) [25], direct torque control (DTC) [26], and SMC [11], are proposed to control the PMBLDCM speed. A three-level neutral point clamped (NPC) VSI-based torque ripple suppression circuit is proposed in [27]. This approach reduces the commutation torque ripple at both low and high speeds, but requires a high switching frequency. In [28], a finite control set model predictive control approach was presented, which provides a fast torque response and good steady-state performance; however, it is completely dependent on motor parameters. FOC method presented in [25] provides satisfactory performance with a reduced torque ripple. This method is completely derived as a parameter-dependent strategy that can degrade the performance of the drive with variation in the operating motor parameters. Therefore, a sensorless FOC scheme is proposed to address the issues resulting from variations in internal parameters and unknown external disturbances. Sensorless FOC offers significant benefits in terms of efficiency and torque quality if the precise angular rotor orientation signal of PMBLDCM is available. To make the technique acceptable for implementation to achieve the exact rotor position, the SMO is integrated into an FOC architecture that guarantees a stable system with reduced computational burden. During PMBLDCM drive control, the objective is to achieve maximum torque per ampere (MTPA) while regulating the currents in the synchronously rotating dq-frame. The complete cycles of the electrical rotor angle are divided into six sectors to execute this procedure. In each sector, pulsewidth modulated (PWM) signals are used to regulate one leg of the VSI at a specified duty cycle, whereas a fixed voltage (usually low) is used for the other. In the proposed sensorless FOC, a prerequisite parameter information or monitoring system is required specifically for SMO, which is again considered as a disturbance if any parameter variation occurs during motor operation. The significant contributions of this article can be summarized as follows.
1) Dynamic modeling with the FOC controller is suggested for the speed control of the PMBLDCM drives, which yields better system stability considering the effect of iron loss. The MTPA control is achieved with the suggested FOC controller that is independent of parameter variation. 2) A continuous fast terminal (CFT) SMC-based speed control method is proposed that compensates for iron loss considerations and other parameter variations with unknown external disturbances. The proposed solutions can easily alleviate the chattering problem and steadystate error. Thus, it improves the robustness of the system with respect to parameter perturbations and external disturbances. 3) A continuous rotor position with rotor speed is estimated by CFT-SMO for precise FOC control. 4) Despite the time-varying loads, the proposed solution can obtain faster response times and smaller ripples while reducing the computational burden. It also improves the start-and-load transient time, as well as the steady-state dynamics. The rest of this article is organized as follows. The PM-BLDCM drives are discussed in Section II with a dynamic model using the FOC controller. Section III discusses the detailed design of CFT-SMC and CFT-SMO, whereas Section IV explores the performance of SMO-FOC using an experimental prototype at a wide range of operating speeds. Finally, Section V concludes this article.

II. MODELING AND CONTROL OF PMBLDCM DRIVE
The PMBLDCM drive topology is shown in Fig. 1 with the complete control configuration. The battery pack is connected to a two-level (2 L) voltage source inverter (VSI) through a dc-link capacitor bank, where the capacitance C 1 and C 2 are connected in series with the midpoint O. The 2L-VSI further supplies power to a PMBLDCM. The proposed control method shown in Fig. 1 generates controlled PWM signals, which are used to operate the 2L-VSI-based PMBLDCM drive. This proposed control strategy is implemented in a DSP-based controller. This section details the PMBLDCM modeling through the equivalent circuit shown in Fig. 2.

A. PMBLDCM Modeling
The modeling of PMBLDCM in the abc-frame can be expressed as in (1), which is coupled through the induced EMF. This relation can be directly obtained using Fig. 2. However, this modeling may not be useful to implement a linearized control law for an accurate estimation of the internal state of the PMBLDCM. Therefore, modeling in the dq-frame can be effectively decoupled as given in (2), considering the effect of iron loss [7], [26], [29] v abc where the superscripts {} abc and {} dq represent the variables in abc and dq-frames, respectively. The variables {v s , i s , i m , e s , λ s } denote the stator terminal voltage, stator current, magnetizing current, back-EMF, and the flux linkage of PMBLDCM, respectively. R s and L s denote the resistance and effective inductance of PMBLDCM of the stator phases, respectively, and contribute to the stator time constant τ s (i.e., L s /R s ). The back EMF (e s ) is the product of the flux linkage (λ s ) and the rotor electrical angular speed (ω r ). It can be derived from the mechanical angular speed (ω m ) and the number of pole pairs (p) of the PMBLDCM, i.e., ω r = pω m . From [7], the variable C = k c e −jθ c can be obtained with magnitude k c = 1/ 1 + (ω r L s /R i ) 2 and angle advancement θ c = tan −1 (ω r L s /R i ) due to the involvement of the iron loss component R i . Here, f dq e = −e dq s /L s has been considered as the unknown disturbance in the rotating reference frame, f dq i = −(C −1 /τ s + jω r )i dq s are the known disturbances obtained from the parameter specification of the motor drives. ω r represents the estimated rotor speed ω r . Furthermore, (2) is remodeled taking disturbances into account and is described as where b v = C −1 /L s denotes the control gain of the control input v dq s .

B. FOC and Overall Control Methodology
Fig . 1 shows the FOC control architecture for the PMBLDCM drive. To achieve better drive stability, FOC has proven to be a better control approach. FOC is considered mostly motor parameter dependent; however, the motor parameters are not used in the proposed study. In this article, a simple FOC structure is incorporated with the following methodology.
Step 1: Initially, the dc-Link voltage v dc and a-phase and bphase stator currents i ab s are measured using one voltage sensor and two current sensors.
Step 2: The voltages (v abc s ) in the stator phases are formulated using the VSI model, that is, v abc s = ±0.5S abc v dc . Here, S abc and S a b c are the switching pulses and their complements to control the IGBT switches, as shown in Fig. 1. Internally, S abc is used as feedback through a unit delay. The c-phase current can be generated by the relation i c s = −(i a s + i b s ). After generating phase voltages and phase currents, v dq s and i dq s can be generated using the following equation, considering the generic variable X ∈ {v s , i s }: Step 3: Here,θ r is the estimated value of the position of the PMBLDCM rotor, which is generated from the CFT-SMO described in Section III-C. CFT-SMO also evaluates the estimated value of the rotor angular speed (ω m ). The inputs to the CFT-SMO are v dq s , i dq s , and the known disturbance, as shown in Fig. 1.
Step 4: The estimated value ofω m is compared with the reference value of the angular speed of the rotor (ω * m ), and the corresponding speed tracking error (ε ω = ω * m −ω m ) is controlled by a CFT-SMC as designed in Section III-B. CFT-SMC also generates the q-axis reference current i q * s . The electromagnetic torque (T em ) vary linearly with i q * s , the desired torque for PMBLDCM can be generated through i q * s control by maintaining d-axis reference current (i d * s ) to zero, i.e., i d * s = 0.
Step 5: Utilize two separate PI regulators to track current error , which further produces v q * s and v d * s , respectively. k d p and k d i (k q p and k q i ) are the proportional and integral constants of the d-axis (q-axis) current controller, respectively.
Step 6: Convert the reference voltage space vector ( v dq * to v αβ * s as the following using the estimated rotor angleθ r : where v αβ * s is utilized further to calculate the sector (k v ) and the dwell angle (δ v ).
Step 7: k v and δ v are used to select the appropriate voltage vectors as in the SVPWM scheme [30] and the switching pulses {S abc , S a b c } are generated accordingly to ensure the operation of the constant switching frequency.

III. PMBLDCM SPEED ESTIMATION AND CONTROL WITH IRON LOSS CONSIDERATION
To achieve higher steady-state tracking accuracy with a faster response, a CFT SMC is presented in this section. Initially, the robustness of CFT-SMC is analyzed to estimate external disturbances through dynamic modeling of PMBLDCM. In addition, a simplified implementation methodology is incorporated to understand the design flow of the proposed CFT-SMC. In this section, a stability analysis is provided to see the feasibility of the proposed method.

A. PMBLDCM Torque and Speed Dynamics
The effective electromagnetic torque (T em ) developed can be represented as (6). It can be seen that T em is completely determined by the flux linkage and the magnetizing component of the current where{} and {} represent the complex conjugate and the real part of a complex variable, respectively. Without considering the effect of iron loss, the abovementioned electromagnetic torque can be modeled as follows: It means that the term T i in (6) represents an additional torque component, which is introduced by considering the effect of iron loss. Now, the iron loss torque component T i can be formulated as follows: where k v ∈ {S 1 . . . S 6 } represents the sectors of the PM-BLDCM back-EMF space vector. Taking into account the characteristics of the system and the load torque, the design of a high-precision speed tracking controller will be a challenging task. FOC essentially involves the regulation of rotational variation between the stator and rotor flux vectors to maximize torque. The FOC technique described in Section II-B has two inner current control loops and one outer speed control loop.
By generating a reference current from the outer loop, the inner loop can be controlled. Considering that T em is a linear function of the q-axis current (i q s ), it is possible to generate a constant torque by regulating the current on the q-axis while maintaining the zero value of d-axis current (i d s ), i.e., i d s = 0. Now, the electromagnetic torque T em can be expressed in terms of the current (i q s ) on the q-axis as follows: where k t is denoted as the torque constant considering the iron loss effect, and its value can be represented as follows: Ignoring mechanical losses, the mechanical dynamic of PM-BLDCM in terms of T em can be described as follows: where J, B, and T L represent the moment of inertia, the damping coefficient, and the applied load torque, respectively. Letω m be the estimated angular speed of PMBLDCM. Finally, by solving (9) and (11), the rate of change in angular speed can be interpreted as follows: where b i (=k t /J) is considered as control gain and f m ω (=(k t (i q * s − i q s ) + T i + T L + Bω m )/J) is the total disturbance due to unknown disturbances and unmodeled dynamics.

B. CFT-SMC Proposed Design
Consider that the speed tracking error (ε ω ) as the difference between the reference speed ω * m and the estimated speedω m of PMBLDCM, i.e., ε ω = ω * m −ω m . Now, the dynamics of the speed tracking error can be expressed as (13). The objective of calculating (13) is to minimize the speed error close to zero.
Solving (12) and (13),ε ω can be further simplified as follows: The sliding surface (s) of conventional SMC, TSMC, and NTSMC can be mentioned as follows: . (17) The TSMC has a slower convergence rate than the SMC when the value of δ is taken into account. The NTSMC has a slower convergence rate but can help to avoid singularity issues when the states are distant from the equilibrium point. To improve the convergence rate against disturbances, the CFT-SMC is developed as indicated by where conditions μ 1 , μ 2 > 0, 0 < δ 1 , δ 2 < 2, δ 2 > δ 1 are satisfied. The operator sgn( * ) represents the signum function and can be defined as follows: Let the dynamics of the sliding surface of the CFT be defined as follows:ṡ where conditions k 1 , k 2 > 0 and 0 < δ 3 < 1 are satisfied. Now, the integral form of (20) can be obtained as follows: By solving (14), (18), and (21), the control input (i q * s ) can be obtained as follows: Furthermore, i q * s can be used for the FOC scheme, as mentioned in Section II-B. The detailed block diagram approach to develop i q * s is shown in Fig. 3. Here, tracking differentiator (TD) is represented as the tracking differentiator, which utilizes the Euler forward approximation.

B. Stability Analysis:
The sliding surface and the control law are two distinct phases in the construction of an SMC controller. On the other hand, the control law can cause chattering in the system, which is crucial in the design of SMCs. The SMC demands a low control gain to reduce the chattering phenomenon. In contrast, a large control gain is necessary to ensure that the SMC is adaptable to massive disturbances with minimal undesirable chattering issue. An attractive solution to this challenge is to make the SMC control law adaptable to measured disturbances. To converge the speed tracking error in (13) to zero, it is necessary to satisfy the SMC reachability criteria in the following: To determine the range of coefficients in the SMC, the stability of the designed SMC should be analyzed. Thus, the Lyapunov stability function is described in the following to prove the stability of the improved CFT-SMC: Now, the dynamics of the Lyapunov stability function can be derived in the following using (20): The dynamics of the PMBLDCM speed error converges to zero in finite time if (25) satisfies, i.e.,V < 0. Now, (27) can be rewritten as follows: It is affirmed from (28) that the speed error with CFT-SMC will be zero in a finite time (t f ) and the speed trajectory will remain on the sliding surface.

C. Proposed CFT-SMO Design
This section focuses on the estimation of the rotor position from the disturbance (f dq e ) corresponding to the back EMF in (3). The extended state observer for (3) can be designed as follows [29]: Authorized licensed use limited to the terms of the applicable license agreement with IEEE. Restrictions apply.
are satisfied. From the sliding surface error dynamics, s dq i can also be defined as follows: where conditions k dq 1 , k dq 2 > 0, 0 < δ dq 3 < 1 satisfy. To improve the convergence rate against disturbances, the CFT-SMO can be developed by solving (31) and (32) and can be indicated as follows: The detailed block diagram approach to estimatef dq e is shown in Fig. 4. It can be seen from [31] that the back-EMF in α-axis is dependent on the rotor electrical angle position. During the transformation of the axes from αβ-frame to dq-frame using (4), both axes of the EMF in dq-frame contain information of rotor position. Fig. 5 shows such a relationship between the dq-frame EMF and the rotor positionθ r . As dq-frame back EMF cannot be estimated directly, the proposed SMC evaluates the disturbancê f dq e corresponding to the back EMF. In other words,f dq e has a direct relationship withθ r as mentioned in the following: Again, the rotor speed of PMBLDCM can be estimated in the following, which provides the minimum tracking error for utilization in the FOC system: (37) where p is the number of pole pairs of PMBLDCM and f dq e denotes the norm off dq e . It is obvious that if the injection of the sliding surface output s dq i is continuous, the estimated speed can be smooth. Otherwise, if there is chattering in s dq i , the output injections contain high-frequency noise and cannot even be used in the PMBLDCM FOC system. The block diagram algorithm of the speed/flux observer proposed in this article based on CFT-SMO is shown in Fig. 4.

IV. RESULTS AND DISCUSSIONS
In this section, the feasibility study of the proposed CFT-SMC estimation techniques with the FOC scheme is carried out by experimental verification. The experimental prototype consists of a 2.5-kW PMBLDCM, a dc generator, and a three-phase 2L-VSI with a battery pack, as shown in Fig. 6. The shaft of PMBLDCM is coupled to a dc generator that is considered a load. The specifications of the PMBLDCM drive are given in Table I. Initially, the PMBLDCM drive control is simulated in MATLAB/Simulink. With satisfactory simulation performance, PWM signals are generated by implementing the proposed control strategy on a dSPACE (DS1104) controller. During the execution process in DS1104, a sample period of 50 μs is maintained for the proposed control action. DS1104 has integrated analog-to-digital (ADC) and digital-to-analog (DAC) units. ADCs are particularly utilized to process measured current and voltage signals, whereas DACs are used to record the performance results of the overall drive through a digital storage oscilloscope. To evaluate the proposed control strategies, several test scenarios are examined as mentioned in the following.  [32]. (d) Waveforms of i q s , i d s , P e , and Q e for comparison study [32].

A. Scenario 1: Rated Speed Performance
In this section, the steady-state operation of PMBLDCM is discussed and compared with that of the approach given in [32]. Torque and current performance is evaluated with the goal of reducing related ripples. The impact of disturbances that include iron loss acting on the drive system has been investigated and compared. It is worth mentioning that existing approaches have not considered iron losses in the SMO to improve the performance of PMBLDCM. The developed motor torque (T es ) with load torque (T L ), estimated motor speed (ω m ) with reference speed command (ω * m ), stator currents (i dq s ) in dq-frame and active power (P e ) with reactive power (Q e ) for the proposed CFT-SMC technique are shown in Fig. 7(a) and (b) during Scenario 1, respectively. The comparative study with Scenario 1 is shown in Fig. 7(c) and (d). Here, a step change inω m from  0 to 1500 r/min with T L = 10 N · m is given in t 1 s. The drive system takes 0.175 and 0.275 s with the proposed CFT-SMC and [32], respectively, to reach steady-state condition. In addition to this, another benefit of the proposed method can be evaluated in terms of reduced ripple content in the T es ,ω m , i q s , profile of PMBLDCM from Fig. 7. The steady-state ripple improvement in T es andω m is found to be 52.02% and 42%, respectively, for Scenario 1. It can be concluded from Fig. 7(b) that P e demand matches the dynamics of T es , and the nearly zero Q e shows reduced losses with perfect synchronization. However, such achievements have not been noted in [32], as shown in Fig. 7(d). i d s is always kept at zero with CFT-SMC, whereas i d s reaches 9.3 A during the transition region and 5.6 A in the steady-state region with [32]. The battery current (i bat ) and the three-phase stator currents i abc s for the proposed method and the method presented in [32] are shown in Fig. 8(a) and (c), respectively. These plots show less ripple for the proposed CFT-SMC method. Synthesized phase currents i αβ s and phase back-EMF e αβ s are shown in Fig. 8(b). The trajectory plots of i αβ s and e αβ s are shown in Fig. 8(b) to guarantee the efficacy of the proposed method. For comparison study, i αβ s and e αβ s with their trajectory plot can be shown in Fig. 8(d). Figs. 7 and 8 are given in Table II for Scenario 1. It reports a numerical comparison of the improvement % in the ripple content of T es , ω m , P e , I dq s , and for comparison study [32]. (d) Waveforms and trajectory plot of i αβ s and e αβ s for comparison study [32].
I bat between the proposed CFT-SMC and the approach described in [32]. To achieve torque ripple suppression during Scenario 1, the proposed CFT-SMC provides better performance when the motor is operating in a steady state. In addition, the proposed CFT-SMC provides better current ripple and speed tracking.

B. Scenario 2: Speed Reversal and Low Speed Performance
The performance of PMBLDCM is also tested and compared during low-speed and speed reversal conditions. In Scenario 2, a variable load with T L = 10 → −10 N · m is considered for the operation of the PMBLDCM drive. Initially, ω * changes from 0 to 300 r/min in t 1 s and gradually settles within 30 ms, as shown in Fig. 9(a). Again, at t 2 s, a speed command of 600 r/min is set, which settles at a rate of 10000 rpm/s. Furthermore, ω * changes to −300 r/min and then reaches 0 r/min at t 3 s and t 4 s, respectively. The overall comparative study of the speed reversal performance of PMBLDCM drives can be shown in Fig. 9(c) and (d). The SMO with [32] has a high torque and current ripple throughout the speed range, whereas the proposed CFT-SMC tracks the rotor speed very precisely with minimal tracking error. It can also be observed that the proposed CFT-SMC can estimate the rotor speed during low-speed operation, as well as the speed reversal period. PMBLDCM has significant torque ripple and spikes at high speeds, causing the speed and current waveforms to be distorted, leading to weak drive performance with [32]. However, the CFT-SMC guarantees better improvement in terms of ripples and improves the drive profile at a higher speed. Once the speed is reduced to -300 r/min, the ripple of the electromagnetic torque steadily increases, resulting in a large current ripple, as can be compared with Fig. 9(b) and (d). The proposed method proves the improved torque and current ripple under wide-speed operation. The experimental findings of i q s , i d s , P e , and Q e are shown in Fig. 11(b). Regenerative action can be seen from P e during the speed reversal operation at t 3 s. The same performance can also be observed from the battery current in Fig. 11(a). Here, the phase currents clearly indicate the petal wave patterns. Using Scenario 2 in Table II, it can be seen that the magnitude of the ripple contents of T es , ω m , P e , I dq s , and I bat has been improved with the proposed method. With [32], the torque ripple is higher considering the iron loss. It is worth noting that the proposed CFT-SMC yields an appreciable reduction of the torque ripple at wide range of speed operation.  [32]. (d) Waveforms of i q s , i d s , P e , and Q e for comparison study [32]. for comparison study [32]. (d) Waveforms and trajectory plot of i αβ s and e αβ s for comparison study [32].  [32]. (d) Waveforms of i q s , i d s , P e , and Q e for comparison study [32].

C. Scenario 3: Load Change Performance
The steady-state performance of the PMBLDCM drives at ω * m = 1500 r/min can be seen in Figs. 11 and 12. It can be observed from Fig. 11(a) that ω * m is closely trackingω m . At t 1 s, a load of T L = 5 N · m is applied to the PMBLDCM drive system. It can be seen that T es can track T L very accurately. However, torque performance with the FOC method in [32] is observed with a significant deviation from its reference value, as seen from the comparative results shown in Fig. 11(c). The ripple improvement in T es andω m with the proposed CFT-SMC is concluded to be 53.1% and 41.1%, respectively. The experimental findings of i q s , i d s , P e , and Q e are shown in Fig. 11(b), and their comparative performance is shown in Fig. 11(d). The steady-state performance of i bat and i abc s at T L = 5, 10, and 5 N · m with step command of ω * m = 1500 r/min is shown in Fig. 12(a). Their performance can be assessed with the comparative results provided in Fig. 12(c). The plots of i αβ s and e αβ s with T L = 5→ 10 →5 N · m with constant speed command of ω * m = 1500 r/min with the proposed method and the comparative method from [32] are shown in Fig. 12(b) and (d), respectively. It has been noted that the trajectory appears like a petal structure and is in line with the theoretical assumptions on reduced ripple contents. In Scenario 3, the torque ripple is more prominent (nearly twice) for [32] compared with the proposed CFT-SMC, as given in Table II and Figs. 11 and 12. Moreover, the proposed CFT-SMC is effective in suppressing the ripple content in the PMBLDCM drives performance.

D. Transient Performance
With the proposed controller, the robustness and fast dynamic characteristics of the PMBLDCM drive are evaluated during transient conditions. During the speed transition, the change in load torque and disturbances caused by variation of the parameters, the overall drive may be subjected to transient conditions. To determine whether the proposed controller is effective, current and torque ripples are used as measures of performance evaluation. The plot of the waveforms of each estimated quantity against its reference value distinguishes the speed at which the FOC settles the desired performance. This performance evaluation can be observed from Figs. 7, 9, and 11. The fast Fourier transform (FFT) analysis of the phase-a stator current is characterized, which has a THD of 5.83% due to its pattern of petal wave current. This THD level of the stator phase current is significantly lower than that of the conventional quasi-square wave pattern. The analysis of the motor torque harmonics reveals a THD of 2.39%, in which the sixth and twelfth harmonics dominate with magnitudes of 2.3% and 0.1%, respectively.

E. Parameter Variation and Comparative Performance
Since the CFT-SMO is dependent on electrical parameters, variations in the parameters can significantly affect the estimated motor speed. It should be noted that the estimation of motor speed is more specifically based on the disturbance associated with the back-EMF (f dq e ), which has all the disturbance associated with the variations of the parameters. Speed tracking performance under steady-state and transient conditions is tested for Scenario 3 under parameter variation, as shown in Fig. 13. Taking into account an uncertain disturbance changing from 0.233 to 0.405 per unit at t = t 1 s, the speed estimation with the tracking error ( ω ) is shown in Fig. 13(a) with the proposed CFT-SMC and in Fig. 13(b) with SMO- [32]. The minimal speed tracking error of 5 r/min is observed with the proposed CFT-SMC. However, ω m decreases by 30 r/min after T L increases to the rated value with SMO- [32]. It can be well observed that the proposed CFT-SMO performs much better than SMO- [32] in terms of disturbance estimation and speed tracking precision. Furthermore, the CFT-SMC and the existing SMC require 80 and 110 ms, respectively, to restore the nominal speed. When the speed for comparison study [32]. (d) Waveforms and trajectory plot of i αβ s and e αβ s for comparison study [32]. drop and recovery period are compared, the proposed method outperforms SMO- [32] in terms of disturbance rejection. For a better comparative evaluation, an experimental validation is performed between the proposed controller, SMO- [32], and MTPA-DTC- [26], as shown in Fig. 14. Fig. 14(a) shows the %THD of the stator phase current (i a s ) with respect to the speed ratio k ω (=ω m /ω n , ω n is the nominal motor angular speed), whereas Fig. 14(b) represents the torque ripple (ΔT es ) and the efficiency (η) profiles of PMBLDCM with respect to k ω . This comparative study validates the superiority of the proposed controller over the existing control strategies SMO- [32] and MTPA-DTC- [26]. At k ω = 0, the %ΔT es for the proposed SMO, SMO- [32], and MTPA-DTC- [26] are recorded as 3.35%, 4.95% and 5.65%, respectively. The efficiencies (η) of 86, 76, and 71 units are observed for the proposed observer, SMO- [32], and MTPA-DTC- [26], respectively. With the increase in speed for k ω = 1, i.e., at rated speed, %ΔT es is observed to be much reduced and η much improved compared with SMO- [32] and MTPA-DTC- [26]. Furthermore, with an increase in speed for k ω > 1, again the PMBLDCM achieves a better torque ripple and higher efficiency with the proposed SMO.

V. CONCLUSION
An approach to estimate the speed of PMBLDCM drives with iron loss based on CFT-SMO was proposed. It was possible to observe the rotor position signals more accurately and efficiently without using any sensors. In the proposed approach, an estimate of the total disturbance was incorporated, reducing the complexity of designing and executing higher order models. Taking into account the unknown disturbances, a CFT-SMC-based speed controller was provided that offered improved speed tracking. In addition, the FOC gave an accurate current control that matches the back-EMF when it used optimal modulation of the VSI drive. In addition to minimizing copper loss and optimizing drive efficiency, iron loss was also used to influence FOC. Laboratory testing showed that drive performance was improved when the suggested technique was used. A petalwave Lissajous pattern was observed in the motor currents of the stationary reference frame, ensuring less ripple, lower copper losses, and smoother motor torque. Based on the comparisons, it seemed that the proposed algorithm performed better than the existing ones. In addition to high-performance applications, such as electric vehicles, the suggested sensorless PMBLDCM algorithm could also be used in the case of low power consumption. Further application of the aforementioned method to enhance the performance of electric cars will involve extending the aforementioned method to field weakening and regenerative braking applications, with an emphasis on regenerative breaking.