# Finite-Time Convergence ESO-Based Nonsingular Fast Terminal Sliding Mode Control for PMSM with Unknown Parameters and Time-Varying Load

Authors: Chong-yi Gao; Jun Dai; Jing Li; Jian-xiong Li
Journal: Mathematical Problems in Engineering
 (2022)

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## Abstract

In this paper, an extended state observer (ESO)-based nonsingular fast terminal sliding mode (NFTSM) control method is proposed for the speed tracking system of permanent magnet synchronous motor (PMSM) in the present of unknown parameters and time-varying load. Firstly, three finite time convergence ESO (FTCESO) are employed to estimate the lumped disturbances in one speed loop and two current loops, respectively, and the estimates are fed forward to controllers for compensation. Secondly, a NFTSM controller is designed, which can drive the speed of PMSM track in desired speed in finite time. The stability of resulting closed-loop system is proved by using Lyapunov theory. Finally, the simulation and experimental results are compared with the existing methods, and the results show that the proposed method has better performance in tracking different kind of desired speed curves, and also performs better in reducing the chattering phenomenon.


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## Body

## 1. Introduction

Due to the high-power density, high efficiency, and large torque-to-inertia ratio, permanent magnet synchronous motor (PMSM) has been widely used in various fields, such as robotics, aerospace actuators, and electric vehicles [1–3]. The field-orientation vector control is widely employed for high performance of PMSM, and it results a cascade control structure with an inner current control loop and an outer speed or torque control loop. It is well known that the PI control technique is still popular due to its simple implementation [4]. However, the PMSM system is nonlinear and strong coupling system with unavoidable uncertainties and parameter variations, as well as unknown load disturbances [2, 5]. In some cases, some parameters of PMSM are even unknown, for example, the stator resistance, inductance and flux linkage of motors of some old equipment are lost or changed considerably. In these cases, it would be difficult to achieve a high performance in the entire operating rage by only using linear control methods like PI control algorithm [6–8].Aiming to achieve desired control performance, plenty of advanced control algorithms have been put forward for the PMSM drive system, such as adaptive control [1, 9], robust control [2], model predictive control [10, 11], sliding mode control (SMC) [7, 8], backstepping control [12], active disturbance rejection control (ADRC) [13, 14], disturbance observer-based control (DOBC) [2, 8, 15], and intelligent control [16–18]. These advanced control methods have improved the performance of the PMSM system from different aspects. Among these methods, the SMC has been considered as one of the most effective methods to deal with the uncertain nonlinear systems due to its insensitivity to uncertainties and disturbances [8, 19]. Terminal sliding mode control (TSMC) is a nonlinear SMC method and can achieve finite time stability [20]. The TSMC and its subsequent forms of improvement, nonsingular terminal sliding mode control (NTSMC), and nonsingular fast terminal sliding mode control (NFTSMC) have been applied to control PMSMs as well to improve the dynamic performance and disturbance rejection, see [8, 19, 21] and therein.However, the robustness of SMC or TSMC can only be guaranteed by the selection of large control gains, which will lead to the well-known chattering phenomenon. In particular, in some cases the exact upper bound of uncertainties or disturbances is difficult to obtain, so a particularly large control switching gain is naturally adopted to stabilize the system, but more serious chattering phenomenon will occur. For the case that the upper bound of the uncertainties of disturbances is unknown, the adaptive estimation technique has often been employed to estimate the upper bounds, and some adaptive SMC methods have been developed, such as [22–24].A more effective strategy is to estimate and compensate the disturbances. In detail, the lumped disturbances are estimated by disturbance estimator and feed forward compensated in controller design [3, 25–27]. In [26], an aperiodic sampling based ESO was designed to estimate the disturbances for mobile robots with uncertainties, and then a robust antidisturbance kinetic control protocol was developed such that all error variables in the closed-loop system were bounded. In [27], a novel event-triggered unknown system dynamics estimator was designed to learn the uncertainties for quadrotors, and an adaptive event-triggered attitude control method with an appointed-time prescribed performance control was proposed. In [28], an unknown input observer was employed to estimate the unknown system dynamics of quadrotors. These works show that this strategy is available and effective. Moreover, under this strategy, it will be an alternative to use SMC for controller design. Since the lump disturbances are estimated and compensated, and a smaller gain will be selected to offset the estimation error instead of the disturbance in SMC, which will reduce the chattering phenomenon of SMC. In [8], a disturbance observer (DOB) was used to estimate the lumped disturbances, and a composite control method combing NTSMC with DOB, for short TSM + DOB method, was proposed for the speed loop of PMSM. In [29], a novel DOB was constructed based on the inverse model of the speed loop of PMSM to estimate the lumped disturbance, and the estimate value was introduced into the subsequent super-twisting sliding mode controller design. The current loops were governed still by two standard PI controllers in [8, 29]. In [30], the mismatched inductance of the PMSM was estimated by an ESO, and the estimate value was then introduced into a robust predictive controller for current control of PMSM. However, the load torque and other parameters of PMSM were considered to be certain and known. In [31], the speed and q-axis current were regulated in one loop, and a speed-current single-loop PMSM drive system was derived. A three-order ESO was designed to estimate the lumped disturbances for feedforward compensation, and a backstepping sliding mode controller was then proposed. Although TSMC method has been used in some of the literatures mentioned above, all of them cannot guarantee that the closed-loop system achieves finite time stability. Due to unknown load and parameter uncertainties, there is mismatched disturbances in PMSM, in which the mismatched disturbance means that the disturbance and the control input are not in the same channel. In [32], a finite time disturbance observer (FTDO) based NTSMC approach, short for “NTSMC-FTDO,” was proposed for system with mismatched disturbance, and the PMSM was used as an example for experimental verification. Although the NTSMC-FTDO method solves the problem of mismatched disturbance, it could be found from the experimental results that there is still room for improvement in steady-state accuracy of speed tracking control of PMSM.Based on the aforementioned analysis, this paper focuses on the finite-time control for speed tracking of PMSM in the presence of unknown parameters and uncertain parameters and time-varying load torque, and proposes a finite time convergence extend state observer (FTCESO) based NFTSM control method. Firstly, a finite time convergence extend state observer is employed to estimate the lumped disturbances including identification errors of some parameters, unknown load, and external disturbances. Secondly, the field-orientation vector control strategy is adopted, and a novel NFTSM control method is proposed to guarantee the closed loop system in finite time stability. The main contribution of this paper can be summarized as follows:(1)
The finite-time stability of the closed-loop system is guaranteed by using FTCESO-based NFTSM controller despite of mismatched uncertainties and disturbances. With the well estimation performance of the FTCESO for disturbances, the chattering phenomenon is weakened, and driven by the NFTSM controller, the high tracking performance of PMSM is achieved.(2)
The speed tracking control experiment is carried out on the PMSM experimental platform, in which some motor parameters are unknown. By comparing with the existing method, the experimental results show that the proposed method has better tracking accuracy while ensuring the transient performance.The organization of this paper is as follows: in Section2, the mathematical model of PMSM and the control problem formulation are given. The FTCESOs are designed to estimate the total disturbances in Section 3. Section 4 focuses on the NFTSM control design and its stability proof. In Section 5, the simulation and experimental results are presented, followed by conclusion in Section 6.


## 2. Problem Formulations and Preliminaries

Assume that magnetic circuit is unsaturated, hysteresis and eddy current loss are ignored, and the distribution of the magnetic field is sine space. Ind-q coordinates, the model of surface mounted PMSM can be expressed as follows [8]:(1)ω˙=3pnψf2Jiq−BJω+d1,i˙q=−RsLqiq−pnωid−pnψfLqω+1Lquq+d2,i˙d=−RsLdid+pnωiq+1Ldud+d3,where ω is the speed of rotor, id and iq are stator currents in d- and q- axis, respectively, ud and uq denote stator voltages in d- and q- axis, respectively, J is the moment of inertia, B is the viscosity friction coefficient, Ld and Lq are the stator d- and q- axis inductances, respectively, Ld=Lq=L, Rs is the stator resistance, pn is the number of pole-paris, ψf is the magnetic flux linkage, and d1, d2, and d3 denote the perturbations caused by parameter uncertainties and disturbances.In some cases, due to a long time of application, some parameters would change greatly, or even be forgotten. In this paper, the values ofJ and B are considered to be uncertain and the parameters L, Rs, and ψf are regarded to be unknown. In addition, the load torque TL is usually to be unknown. The unknown parameters could be identified offline. Let us put together the terms containing the uncertain and unknown parameters in the three channels of system (1), respectively, denoted by d1, d2, and d3 as follows:(2)d1=3pnJΔψf−ψfΔJ2JJ+ΔJiq−JΔB−BΔJJJ+ΔJω−TLJ+ΔJ,d2=RsΔL−LΔRsLL+ΔLiq−LΔψf−ψfΔLLL+ΔLpnω−ΔLLL+ΔLuq,d3=−LΔRs−RsΔLLL+ΔLid−ΔLLL+ΔLud,where ΔRs, ΔL, and Δψf are identification errors of RS, L, and ψf, respectively, and ΔJ and ΔB denote the errors between the real value and the nominal value of J and B, respectively.In practice, these disturbancesd1, d2, and d3 are bounded due to physical limitations, and there first order time derivatives are also bounded. So, these disturbances satisfy the following assumption.Assumption 1.
The disturbancedi is first order differentiable and assume that di and d˙i are bounded, that is, there exist some positive constants d¯i and d¯ti such that di≤d¯i and d˙i≤d¯ti, i=1,2,3.
The main objective of this paper is to design controllersuq and ud so that the speed ω can track the desired speed ωd in finite time in the presence of unknown parameters and time-varying load.
To end this section, a definition and three lemmas which will be used in finite time control and analysis later are as follows:Lemma 1 (see [33]).
If there exists a Lyapunov functionVxt such that the following inequality holds.(3)V˙xt+aVxt+bVcxt≤0,∀t≥t0,then the system state xt will converge to the equilibrium in finite time, and the settling time t1 can be given by the following equation:(4)t1≤t0+1a1−clnaV1−ct0+bb,where a>0, b>0, 0<c<1, and t0≥0 is the initial time.Lemma 2 (see [34]).
For the following nonsingular fast terminal sliding mode, when the system statex reaches the sliding mode surface s=0, the system state can converge to the origin in a finite time T1.(5)s=x+λ1sigσ1x+λ2sigσ2x˙,where sigσix=signxxσi, sign⋅ is the sign function, λ1>0, λ2>0, 1<σ2<2, and σ1>σ2. The settling time T1 is given by the following equation:(6)T1=∫0x0k21/σ2x+k1xσ11/σ2dx=σ2x01−1/σ2k1σ2−1×F1σ2;σ2−1σ1−1σ2;1+σ2−1σ1−1σ2;−k1x0σ1−1,where x0 represents the initial value of x just after reaching the sliding surface and F⋅ denotes Gauss’ Hypergeometric function [35].


## 3. Finite-Time-Convergence Extended State Observers

In order to weaken the influence of the lumped disturbances on the control performance, three FTCESOs are used to estimatedi and i=1,2,3, respectively.Letx11=ω, x21=iq, x31=id, xi2=di, and i=1,2,3, then from equation (1), we have the following equation:(7)x˙i1=aixi1+giui+ψi+xi2x˙i2=d˙ii=1,2,3,,where a1=−B/J, a2=a3=−Rs/L, u1=0, u2=uq, u3=ud, g1=0, g2=g3=1/L, ψ1=3pnψfiq/2J, ψ2=−pnωid−pnψfω/L, and ψ3=pnωiq.The FTCESO can be designed for equation (7) as follows:(8)x^˙i1=aix^i1+giui+ψi+x^i2−ηi1signx˜i1−κisigαi1x˜i1+sigβi1x˜i1,x^˙i2=−κi2sigαi2x˜i1+sigβi2x˜i1−ηi2signx˜i1i=1,2,3,where x^ij is the estimate of xij, x˜ij=x^ij−xij denotes estimation error, κi>1, 0.5<αi1<1, αi2=2αi1−1, βi1=1/αi1, βi2=1/βi1+βi1−1, ηij>0, i=1,2,3, and j=1,2.The estimate of disturbancedi can be obtained as d^i=x^i2 and the estimation error is d˜i=d^i−di and i=1,2,3.According to [36], if d¯ti is known and ηi2>d¯ti is chosen, then the estimation errors x˜i1 and x˜i2 will converge to zero in finite time, and the settling time is denoted as Toi. Due to space limitations, the convergence analysis of the ESO (8) and the explicit expression of Toi are omitted here. For detailed, please see [36].Otherwise,DTi¯ is unknown, following the analysis of [36], it is not difficult to deduce that the estimation errors x˜i1 and x˜i2 will converge to a neighborhood of the origin. The boundary of the neighborhood depends on DTi¯ and ηi2, and a large ηi2 will lead to a small boundary.


## 4. Nonsingular Fast Terminal Sliding Mode Controller Design

In field-oriented control strategy, the model of PMSM can be divided into three loops, one speed loop and two current loops. It can also be regarded as composed of two subsystems, a second-order subsystem consisting of speed loop andq-axis current loop and a first-order subsystem that is the d-axis current loop. The control structure diagram is shown in Figure 1.Figure 1
Structure of the FTCESO-based NFTSM control system.For the second-order subsystem, the errors are defined as follows:(9)e1=ωd−ωe2=ω˙d−biq−a1ω−d^1,where b=3pnψf/2J and d^1=x^12 obtained from equation (8).From equations (1) and (9), one can get the following equation:(10)e˙1=e2+d˜1e˙2=ω¨d−ba2iq+g2uq+ψ2+d2−a1ω˙d−e2−d˜1−d^˙1..The nonsingular fast terminal sliding mode surface is chosen as follows:(11)s=e1+λ1sigσ1e1+λ2sigσ2e2.And the controlleruq is designed as follows:(12)uq=uqeq+uqn,(13)uqeq=Lbω¨d−a1ω˙d+a1e2−d^˙1−a2biq−bψ2−bd^2+1+λ1σ1e1σ1−1λ2σ2sig2−σ2e2,(14)uqn=Lbk1signs+k2s,where λ1>0, λ2>0, 1<σ2<2, σ1>σ2, k1>0, and k2>0.It should be pointed out thatd1 is the mismatched disturbance which is not in the same channel as the control input uq into the system. To eliminate the influence of mismatched disturbance on the system, the disturbance estimation d^1 is introduced into the sliding mode surface (11), which is inspired by [24]. If the estimation error d˜1=0, according to Lemma 2, when the error states of (10) reach into the surface s=0, e1 and e2 could converge to zero in finite time.Taking the time derivative ofs and substituting (10) into it, one yields the following equation:(15)s˙=1+λ1σ1e1σ1−1e2+d˜1+λ2σ2e2σ2−1ω¨d−b×a2iq+g2uq+ψ2+d2−a1ω˙d−e2−d˜1−d^˙1.Substituting equations (12)–(14) into equation (15), one obtains the following equation:(16)s˙=λ2σ2e2σ2−1−k1signs−k2s+a1d˜1+bd˜2+1+λ1σ1e1σ1−1d˜1.We first consider the case that the upper boundd¯ti is known, the estimation error d˜i=0 is achieved in finite time by using the FTCESOs (8), then the following main result is obtained.Theorem 1.
For the error dynamics (10) with nonsingular fast terminal sliding mode surface (11), the controller (12)–(14) will drive the tracking error e1 to converge to zero in finite time if the estimation errors of disturbances converge to zero in finite time.Proof.
The proof will be divided into two steps, the first step is to prove that the system states do not escape in finite time, and the second step is to prove that the system states will converge to the origin in finite time.(1)
Following [24, 37], we define a finite time bounded function as V1=1/2s2+e12+e22. Taking the time derivative of V1 along with equations (10) and (15), and substituting equations (12)–(16) into it, we can get the following equation:(17)V˙1=−λ2σ2e2σ2−1k1s+k2s2+λ2σ2e2σ2−1sa1d˜1+bd˜2+1+λ1σ1e1σ1−1sd˜1+e1e2+d˜1+e2a1d˜1+bd˜2−k1signs−k2s−1+λ1σ1e1σ1−1λ2σ2e23−σ2.Note that the inequalityxα<1+x holds for x∈R and 0<α<1, then we have the following equation:(18)V˙1≤−k1λ2σ2e2σ2−1s−k2λ2σ2e2σ2−1s2+λ2σ2e2σ2−1×bd˜2+a1d˜1s+sd˜1+λ1σ1sd˜1e1σ1−1+e1e2+e1d˜1+e2d˜2+k1e2+k2e2s−1+λ1σ1e1σ1−1λ2σ2e23−σ2≤123+r1+r2+k2s2+122+r2e12+123+r1+k2e22+12r12+r22+r32+k12+d˜12≤kVV1+kc,wherer1=λ2σ2bd˜2+a1d˜1, r2=λ1σ1d˜1, r3=d˜2+a1/bd˜1, kV=max3+r1+r2+k2,2+r2,3+r1+k2, and kc=sup1/2r12+r22+r32+k12+d˜12.Since the estimation errorsd˜1 and d˜2 are bounded, kV and kc are bounded constants, and it can be concluded from equation (18) that V1, s, e1, and e2 will not escape in finite time [37].(2)
When the disturbance estimation errord˜1 and d˜2 converge to zero in a finite time, then equation (16) can be rewritten as follows:(19)s˙=λ2σ2e2σ2−1−k1signs−k2s.Choose Lyapunov function candidate asV2=1/2s2, the time derivative of V2 along with equation (19) is obtained as follows:(20)V˙2=sλ2σ2e2σ2−1−k1signs−k2s≤−φ1V21/2−φ2V2,whereφ1=2k1λ2σ2e2σ2−1 and φ2=2k2λ2σ2e2σ2−1.
Sinceφ1 and φ2 contain error term e2, phase trajectories are discussed in the following two cases.Case 1.
Whene2≠0, one has φ1>0 and φ2>0. According to Lemma 1, V2 can converge to the sliding surface s=0 within finite time T3, and(21)T3≤2φ1lnφ1V21/2t0+φ2φ2,where V21/2t0 is the initial value of the Lyapunov function V2.Case 2.
Whene2=0, substituting the control law (12)–(14) into (10), yields the following equation:(22)e˙2=−k1signs−k2s,which implies that e˙2<−k1 and e˙2>k1 for the case that s>0 and s<0, respectively. It can be seen that the system states will not always stay on the point (e2=0 and e1≠0), so it is reasonable to assume that there exists a small positive constant δ such that for a vicinity of e2=0, i.e., e2≤δ satisfies e˙2<−k1 and e˙2>k1 for s>0 and s<0, respectively. Thus, it yields that the crossing of trajectories between two boundaries of e2≤δ is performed in finite time, and the dynamics from the region e2>δ converge to the boundaries in finite time as well [20, 34]. Therefore, the sliding mode s=0 can be reached from anywhere in the phase plane in finite time.
When the error states reach the sliding surfaces=0, equation (11) can be written as follows:(23)s=e1+λ1sigσ1e1+λ2sigσ2e˙1=0.
By Lemma2, it can be concluded that e1 can converge to the origin in finite time T4, and T4 is given as follows:(24)T4=∫0e10λ21/σ2e1+λ1e1σ11/σ1dx=σ2e101−1/σ2λ1σ2−1F1σ2;ϑ;1+ϑ;λ1e10σ1−1,where ϑ=σ2−1/σ1−1σ2 and e10 represents the initial value of the error state e1 at the surface of the sliding mode s=0. The proof is completed.
Theorem1 establishes under the condition of d˜1=0 and d˜2=0. Next, we will discuss the case of d˜i≠0 and i=1,2.
As mentioned above, whend¯t1 and d¯t2 (the boundaries of d˙1 and d˙2, respectively), are unknown, the FTCESOs (8) cannot guarantee that the estimation errors d˜1 and d˜2 converge to zero in finite time, but converge to the neighborhood of the origin. Rewriting equation (20) along with equation (16) yields the following equation:(25)V˙2=λ2σ2e2σ2−1s−k1signs−k2s+a1d˜1+bd˜2+1+λ1σ1e1σ1−1d˜1s≤−λ2σ2e2σ2−1k1−a1d˜1+bd˜2s−λ2σ2e2σ2−1k2s2+1+λ1σ1e1σ1−1d˜1s.
Sinced˜1 and d˜2 will be very small by choosing large parameters η12 and η22 in FTCESOs (8), it is not difficult to select a large enough k1 such that k1−a1d˜1+bd˜2=ρ1>0.
Then, letρ2=ρ1λ2σ2e2σ2−1−1+λ1σ1e1σ1−1d˜1; following the above analysis, it is known that e1 and e2 will not escape in finite time, the point e2=0 and e1≠0 is not a stable equilibrium, and e2 will converge the boundaries of e2≤δ. It is reasonable to select a large enough k1 such that ρ2>0, then (25) would turn into V˙2≤−φ2V2−2ρ2V21/2 which implies Theorem 1 still holds.
In the case ofρ2≤0, let ρ2′=−ρ2≥0, one can obtain that,(26)V˙2≤−λ2σ2e2σ2−1k2s2+ρ2′s=−λ2σ2e2σ2−1k2s−ρ2′s,s≤1+λ1σ1e1σ1−1d˜1k2λ2σ2e2σ2−1−k1−a1d˜1+bd˜2k2.
Therefore, bounded stability will be achieved,e1 and e2 will converge to converge to a neighborhood of the origin.Remark 1.
In order to reduce the chattering phenomenon, the signum functionsign⋅ used in (14) could be replaced by a continuous function, such as hyperbolic tangent function tanhkthx=ekthx−e−kthx/ekthx+e−kthx, and for any kth>0 and x∈R, the following inequality holds [38].(27)0≤xsignx−tanhkthx≤δkth,where δ is a constant that satisfies δ=e−δ+1 and δ=0.2785. Therefore, if a large parameter kth is chosen, the hyperbolic tangent function is approximately equivalent to the signum function.
At last, lete3=id∗−id=−id, the d-axis current loop can be rewritten as follows:(28)e˙3=−a3e3−ψ3−g3ud−d3.
The controllerud is designed as follows(29)ud=L−a3e3−ψ3−d^3+k3signe3+k4e3,where k3>0 and k4≥0.
Choosing Lyapunov functionV3=1/2e32, taking its time derivative along with equation (28) and substituting equation (29) into it, it follows.(30)V˙3=e3−k3signe3−k4e3−d˜3≤−k4e32−k3−d˜3e3=−2k4V3−2k3−d˜3V31/2.
Sinced˜3 is bounded, and it will be small by choosing appropriate parameter η32. Therefore, it is not difficult to choose a large enough k3 such that k3−d˜3>0. Then, it can be concluded that e3 will converge to zero in finite time by using Lemma 2.


## 5. Simulation and Experimental Results

In order to illustrate the effectiveness and practicability of the proposed method, the proposed method and NTSMC-FTDO method proposed in [32] are carried out in simulation and experiment.


### 5.1. PMSM Parameters Identification

In the experimental platform, some parameters of the PMSM are unknown, such asRs, L, and ψf. These parameters are identified by offline methods.


#### 5.1.1. Identification ofRs

The stator windings of the motor in the experiment are star connection. The stator resistanceRs is estimated by an offline experiment. Short-circuit the motor phase B and phase C terminals together, and apply a DC voltage ud between A and B terminals, the stator resistance Rs could be estimated by the following relational expression.(31)Rs=2ud3id,where id is the current stable value of the closed loop circuit.


#### 5.1.2. Identification ofL

SinceLd=Lq=L in surface PMSM, we just identify one of them.The motor stator is stationary and the d-axis current changes over time until it reaches saturation. It could be found that the rise of the current from zero to steady value is exponential, so we can calculate the d-axis inductorLd by taking the time for the current to rise. Specifically, recording the time when the current rises from zero to 0.632 times of steady value, that is, t0.632, then we can get the following equation:(32)Ld=t0.632Rs.


#### 5.1.3. Identification ofψf

The magnetic flux linkage of rotor is essentially equal to the back electromotive force (EMF) coefficient, but in different units. The relationship between them is as follows:(33)Ke=pψf10002π60,where Ke is the back EMF coefficient, and it could be obtained by the following empirical formula in engineering.(34)Ke=KPNpfNIN,where PN, fN, and IN are the rated power, rated frequency, and rated current of the motor, respectively. K is the correction factor which is taken as K=9.62 in engineering.The parameters of the motor are shown in Table1.Table 1
The parameters of PMSM.
ParametersKnown/nominal valueUnknownUsed in simulation modelIdentifiedRated power (P/W)180Rated voltage (V)380Pole-pairs (p)2Rated speed (n/(r/min))1500Inertia coefficient (J/kg·m2)0.00080.0009Viscosity coefficient (B)0.00250.002Stator resistance (Rs/Ω)12.412.5Inductance (L/H)0.180.1875Flux linkage (ψf/Wb)0.950.945


### 5.2. Simulation Results

The two comparison methods and their design parameters are as follows:(1)
The proposed method (NFTSMC-FTCESO).FTCESOs:κ1=κ2=κ3=4000, η11=η21=η31=200, η12=η22=η32=10, and α11=α21=α31=0.6.NFTSMC:λ1=0.5, λ2=0.001, σ1=7/5, σ2=9/7, k1=1, k2=20, k3=10, k4=0.1, and kth=20.(2)
NTSMC-FTDO [30].In this method, the controller is designed as follows:(35)uq2=−Lb−βqpe22q−p/q−ω¨d+ba1iq+ψ2+d^2+a1ω˙+d^˙1−k1′sigσ3s−k2′s.Andud controller is the same as in this paper. The parameters are chosen as: n0=20, n1=50, n2=15, H1=2×104, H2=H3=1×104, β=1000, p/q=9/7, k1′=1, k2′=20, and σ3=0.9.In the simulation, the following time-varying load is consideredTL=1+0.2sin2πt,t≤0.3s2+0.2sin2πt,t>0.3sFigures2 and 3 show that the motor tracks a step reference signal and a time-varying signal, respectively, where the step reference speed nd is changed from 300 to 800 r/min at 0.5 s, and the time-varying signal is as follows: nd=60/2πωd=500+50sinπt r/min.Figure 2
Simulation results of step signal tracking. (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)Figure 3
Simulation results of time-varying signal tracking. (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)It could be found from Figures1 and 3 that both control methods can track the reference signals well in the presence of uncertain parameters and unknown time-varying load. In contrast, the proposed control method performs better in terms of the disturbance rejection, tracking accuracy, and dynamic adjustment in case of sudden change in reference signal or load, see Figures 2(a) and 3(a), the tracking error is within 2 r/min under the proposed method, but 4 r/min or more under the NTSMC-FTDO method.In addition, by analyzing the two algorithms, it could be concluded that the influence of the estimation of the mismatched disturbance on the tracking performance is far greater than that of the estimation of the matched disturbances. Specifically, in the proposed method,d^1 is introduced into the sliding mode surface to eliminate the mismatched disturbance d1. While d^2 and d^3 are introduced into the controllers uq and ud for feedforward compensation the matched disturbances d2 and d3, respectively. Obviously, the estimation errors d˜2 and d˜3 could be offset by the two switching terms k1signs and k3signe3, respectively, but eliminating the influence of d1 on tracking error could only depend on the accuracy of the disturbance observer. Therefore, the curves of d^1 are given in the simulation results, as well as in the following experimental results. As can be seen from Figures 2(b) and 3(b), both two methods have little difference in the estimation effect of d1 in the simulation, and the results of the proposed method are relatively smooth.


### 5.3. Experimental Results

The experimental platform is shown in Figure4, where DS1104 control card of dSPACE is used as the controller, and CP1104 is the interface between DS1104 and relevant devices such as power module, encoder, and load drive unit. In the power module, PS21265-AP IPM made by Mitsubishi Electric Corporation is used. The load torque is supplied by a magnetic powder brake, and the torque is proportional to the reference signal voltage generated by the DS1104 and driven by a load drive unit to drive the magnetic powder brake.Figure 4
PMSM experimental platform.The experiment is carried out in the following four cases:Case 1: Tracking the reference signal 1 under load 1Case 2: Tracking the reference signal 1 under load 2Case 3: Tracking the reference signal 2 under load 1Case 4: Tracking the reference signal 2 under load 2Where the reference signals and loads are as follows:Reference signal 1 (step signal):(36)nd=602πωd=100r/mint<5s500r/mint≥5s.Reference signal 2 (time-varying signal):(37)nd=602πωd=500+50sinπtr/min,Load 1 (abrupt changing load):(38)TL=0.3N⋅m,t≤5s1.04N⋅m,t>5s.Load 2 (time-varying load):(39)TL=0.5+0.2sinπtN⋅m.Same as simulation, the two methods, the proposed control method, and the NTSMC-FTDO method [32] are tested on the experimental platform and the experimental results are shown in Figures 5–8, in which the subfigures (a) show the speed tracking and tracking errors and subfigures (b) present the estimations of mismatched disturbance d1 in the four cases, respectively.It can be found that the proposed method has better performance on speed tracking and disturbance rejection than that of the NTSMC-FTDO method in all the four cases, see the speed tracking in subfigure (a) of Figures5–8. In detail, from Figure 5(a), the speed tracking error is within 15 r/min under the proposed method while 40 r/min under the NTSMC-FTDO method of [32] in case 1. And the proposed method also has better transient performance than that of NTSMC-FTDO method. In case 2, when the load is time-varying, the tracking error under the proposed method is within 10 r/min, while 20 r/min under the NTSMC-FTDO method at lower speed, see Figure 6(a). When the speed goes up to 500 r/min, the tracking errors of both methods increase, but the performance of the proposed method is relatively better, see Figure 5(a). When the reference is a time-varying signal, the comparison of the two methods is similar to the case of step signal, see Figure 6(a).Figure 5
Experimental results of step signal tracking under abrupt changing load (case 1). (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)Figure 6
Experimental results of step abrupt signal tracking under time-varying load (case 2). (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)Figure 7
Experimental results of time-varying signal tracking under abrupt changing load (case 3). (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)Figure 8
Experimental results of time-varying signal tracking under time-varying load (case 4). (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)Moreover, it could be found that the tracking performance of the proposed method in tracking time-varying signal is better than that in tracking step signal, but the NTSMC-FTDO method is different, see Figures7(a) and 8(a). A potential reason is that the estimation of the lumped disturbance d1 could affect the tracking error. The estimation accuracy of the FTCESO employed in the proposed method is better than that of the other, see Figures 7(b) and 8(b). In fact, the performance of disturbance estimation and tracking are mutually influenced, since the tracking performance also affects the disturbance estimation through the estimate of the states.


## 5.1. PMSM Parameters Identification

In the experimental platform, some parameters of the PMSM are unknown, such asRs, L, and ψf. These parameters are identified by offline methods.


### 5.1.1. Identification ofRs

The stator windings of the motor in the experiment are star connection. The stator resistanceRs is estimated by an offline experiment. Short-circuit the motor phase B and phase C terminals together, and apply a DC voltage ud between A and B terminals, the stator resistance Rs could be estimated by the following relational expression.(31)Rs=2ud3id,where id is the current stable value of the closed loop circuit.


### 5.1.2. Identification ofL

SinceLd=Lq=L in surface PMSM, we just identify one of them.The motor stator is stationary and the d-axis current changes over time until it reaches saturation. It could be found that the rise of the current from zero to steady value is exponential, so we can calculate the d-axis inductorLd by taking the time for the current to rise. Specifically, recording the time when the current rises from zero to 0.632 times of steady value, that is, t0.632, then we can get the following equation:(32)Ld=t0.632Rs.


### 5.1.3. Identification ofψf

The magnetic flux linkage of rotor is essentially equal to the back electromotive force (EMF) coefficient, but in different units. The relationship between them is as follows:(33)Ke=pψf10002π60,where Ke is the back EMF coefficient, and it could be obtained by the following empirical formula in engineering.(34)Ke=KPNpfNIN,where PN, fN, and IN are the rated power, rated frequency, and rated current of the motor, respectively. K is the correction factor which is taken as K=9.62 in engineering.The parameters of the motor are shown in Table1.Table 1
The parameters of PMSM.
ParametersKnown/nominal valueUnknownUsed in simulation modelIdentifiedRated power (P/W)180Rated voltage (V)380Pole-pairs (p)2Rated speed (n/(r/min))1500Inertia coefficient (J/kg·m2)0.00080.0009Viscosity coefficient (B)0.00250.002Stator resistance (Rs/Ω)12.412.5Inductance (L/H)0.180.1875Flux linkage (ψf/Wb)0.950.945


## 5.1.1. Identification ofRs

The stator windings of the motor in the experiment are star connection. The stator resistanceRs is estimated by an offline experiment. Short-circuit the motor phase B and phase C terminals together, and apply a DC voltage ud between A and B terminals, the stator resistance Rs could be estimated by the following relational expression.(31)Rs=2ud3id,where id is the current stable value of the closed loop circuit.


## 5.1.2. Identification ofL

SinceLd=Lq=L in surface PMSM, we just identify one of them.The motor stator is stationary and the d-axis current changes over time until it reaches saturation. It could be found that the rise of the current from zero to steady value is exponential, so we can calculate the d-axis inductorLd by taking the time for the current to rise. Specifically, recording the time when the current rises from zero to 0.632 times of steady value, that is, t0.632, then we can get the following equation:(32)Ld=t0.632Rs.


## 5.1.3. Identification ofψf

The magnetic flux linkage of rotor is essentially equal to the back electromotive force (EMF) coefficient, but in different units. The relationship between them is as follows:(33)Ke=pψf10002π60,where Ke is the back EMF coefficient, and it could be obtained by the following empirical formula in engineering.(34)Ke=KPNpfNIN,where PN, fN, and IN are the rated power, rated frequency, and rated current of the motor, respectively. K is the correction factor which is taken as K=9.62 in engineering.The parameters of the motor are shown in Table1.Table 1
The parameters of PMSM.
ParametersKnown/nominal valueUnknownUsed in simulation modelIdentifiedRated power (P/W)180Rated voltage (V)380Pole-pairs (p)2Rated speed (n/(r/min))1500Inertia coefficient (J/kg·m2)0.00080.0009Viscosity coefficient (B)0.00250.002Stator resistance (Rs/Ω)12.412.5Inductance (L/H)0.180.1875Flux linkage (ψf/Wb)0.950.945


## 5.2. Simulation Results

The two comparison methods and their design parameters are as follows:(1)
The proposed method (NFTSMC-FTCESO).FTCESOs:κ1=κ2=κ3=4000, η11=η21=η31=200, η12=η22=η32=10, and α11=α21=α31=0.6.NFTSMC:λ1=0.5, λ2=0.001, σ1=7/5, σ2=9/7, k1=1, k2=20, k3=10, k4=0.1, and kth=20.(2)
NTSMC-FTDO [30].In this method, the controller is designed as follows:(35)uq2=−Lb−βqpe22q−p/q−ω¨d+ba1iq+ψ2+d^2+a1ω˙+d^˙1−k1′sigσ3s−k2′s.Andud controller is the same as in this paper. The parameters are chosen as: n0=20, n1=50, n2=15, H1=2×104, H2=H3=1×104, β=1000, p/q=9/7, k1′=1, k2′=20, and σ3=0.9.In the simulation, the following time-varying load is consideredTL=1+0.2sin2πt,t≤0.3s2+0.2sin2πt,t>0.3sFigures2 and 3 show that the motor tracks a step reference signal and a time-varying signal, respectively, where the step reference speed nd is changed from 300 to 800 r/min at 0.5 s, and the time-varying signal is as follows: nd=60/2πωd=500+50sinπt r/min.Figure 2
Simulation results of step signal tracking. (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)Figure 3
Simulation results of time-varying signal tracking. (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)It could be found from Figures1 and 3 that both control methods can track the reference signals well in the presence of uncertain parameters and unknown time-varying load. In contrast, the proposed control method performs better in terms of the disturbance rejection, tracking accuracy, and dynamic adjustment in case of sudden change in reference signal or load, see Figures 2(a) and 3(a), the tracking error is within 2 r/min under the proposed method, but 4 r/min or more under the NTSMC-FTDO method.In addition, by analyzing the two algorithms, it could be concluded that the influence of the estimation of the mismatched disturbance on the tracking performance is far greater than that of the estimation of the matched disturbances. Specifically, in the proposed method,d^1 is introduced into the sliding mode surface to eliminate the mismatched disturbance d1. While d^2 and d^3 are introduced into the controllers uq and ud for feedforward compensation the matched disturbances d2 and d3, respectively. Obviously, the estimation errors d˜2 and d˜3 could be offset by the two switching terms k1signs and k3signe3, respectively, but eliminating the influence of d1 on tracking error could only depend on the accuracy of the disturbance observer. Therefore, the curves of d^1 are given in the simulation results, as well as in the following experimental results. As can be seen from Figures 2(b) and 3(b), both two methods have little difference in the estimation effect of d1 in the simulation, and the results of the proposed method are relatively smooth.


## 5.3. Experimental Results

The experimental platform is shown in Figure4, where DS1104 control card of dSPACE is used as the controller, and CP1104 is the interface between DS1104 and relevant devices such as power module, encoder, and load drive unit. In the power module, PS21265-AP IPM made by Mitsubishi Electric Corporation is used. The load torque is supplied by a magnetic powder brake, and the torque is proportional to the reference signal voltage generated by the DS1104 and driven by a load drive unit to drive the magnetic powder brake.Figure 4
PMSM experimental platform.The experiment is carried out in the following four cases:Case 1: Tracking the reference signal 1 under load 1Case 2: Tracking the reference signal 1 under load 2Case 3: Tracking the reference signal 2 under load 1Case 4: Tracking the reference signal 2 under load 2Where the reference signals and loads are as follows:Reference signal 1 (step signal):(36)nd=602πωd=100r/mint<5s500r/mint≥5s.Reference signal 2 (time-varying signal):(37)nd=602πωd=500+50sinπtr/min,Load 1 (abrupt changing load):(38)TL=0.3N⋅m,t≤5s1.04N⋅m,t>5s.Load 2 (time-varying load):(39)TL=0.5+0.2sinπtN⋅m.Same as simulation, the two methods, the proposed control method, and the NTSMC-FTDO method [32] are tested on the experimental platform and the experimental results are shown in Figures 5–8, in which the subfigures (a) show the speed tracking and tracking errors and subfigures (b) present the estimations of mismatched disturbance d1 in the four cases, respectively.It can be found that the proposed method has better performance on speed tracking and disturbance rejection than that of the NTSMC-FTDO method in all the four cases, see the speed tracking in subfigure (a) of Figures5–8. In detail, from Figure 5(a), the speed tracking error is within 15 r/min under the proposed method while 40 r/min under the NTSMC-FTDO method of [32] in case 1. And the proposed method also has better transient performance than that of NTSMC-FTDO method. In case 2, when the load is time-varying, the tracking error under the proposed method is within 10 r/min, while 20 r/min under the NTSMC-FTDO method at lower speed, see Figure 6(a). When the speed goes up to 500 r/min, the tracking errors of both methods increase, but the performance of the proposed method is relatively better, see Figure 5(a). When the reference is a time-varying signal, the comparison of the two methods is similar to the case of step signal, see Figure 6(a).Figure 5
Experimental results of step signal tracking under abrupt changing load (case 1). (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)Figure 6
Experimental results of step abrupt signal tracking under time-varying load (case 2). (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)Figure 7
Experimental results of time-varying signal tracking under abrupt changing load (case 3). (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)Figure 8
Experimental results of time-varying signal tracking under time-varying load (case 4). (a) Speed and tracking error. (b) Estimate ofd1.
(a)(b)Moreover, it could be found that the tracking performance of the proposed method in tracking time-varying signal is better than that in tracking step signal, but the NTSMC-FTDO method is different, see Figures7(a) and 8(a). A potential reason is that the estimation of the lumped disturbance d1 could affect the tracking error. The estimation accuracy of the FTCESO employed in the proposed method is better than that of the other, see Figures 7(b) and 8(b). In fact, the performance of disturbance estimation and tracking are mutually influenced, since the tracking performance also affects the disturbance estimation through the estimate of the states.


## 6. Conclusion

The speed tracking problem of PMSM with unknown parameters and time-varying load is investigated in this paper, and a FTCESO-based-NFTSMC method is proposed to improve the tracking performance of the PMSM against the matched and mismatched uncertainties and disturbances. The lumped disturbances are well estimated by using FTCESOs and then compensated by feedforward. Finite time stability could be achieved when the upper bounds of the derivatives of the lumped disturbances are known, and the speed tracking error could converge to a neighborhood of the zero in finite time when the bounds are unknown. Finally, simulation and experiment are carried out to illustrate the effectiveness of the proposed method, and the simulation and experimental results show that the proposed method has high tracking accuracy, fast convergence speed, and strong robustness.



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*Source: 9080445-2022-11-24.xml*