The traditional first-order ultra-local model can be obtained as follows (Fliess and Join, 2013): 2y˙=H+αu, where, u is the control variable of the system and y is the output variable of the system, α is a nonphysical scale factor of the model, and H is the interference part of the system.

By rewriting Eq. (), u can be expressed as shown in Eq. (): 3u=y˙∗-H^+ζα, where y∗ is the expected output of the system; H^ is the estimated value of H in the system; and ζ is given by the proportional controller. In Eq. (), 4e+ζ=0, where e=y∗-y represents the tracking error of the system.

When using a PI controller, Eq. () can be rewritten as Eq. (): 5u=y˙∗-H^+Kpeα, where Kp  is the proportional gain.

The Laplace transform of Eq. () is obtained: 6sY=H^s+αU+y0.

Differentiate both sides of Eq. () with respect to s, and obtain the following: 7y=αdUds-H^s2-sdyds.

After eliminating the influence of noise in the time domain and multiplying Eq. () by s2, the inverse Laplace transform is carried out, and the estimated value of the unknown part of the control system in the time domain is 8H^=-6Z∫t-Zt((Z-2ξ)y(ξ)+αξ(Z-ξ)u(ξ))dξ, where Z is a value within the sampling period, depending on the sampling period and noise, where t-Z≤ξ≤Z.

The ultra-local model in the PMLSM is constructed based on Eq. () as follows: 9v˙=Hq+αqiq, where Hq represents the unknown part of the internal and external interference factors of the PMLSM, αq represents the stator current coefficient of the q axis in the PMLSM. The following equation can be obtained by rewriting Eq. (): 10H^q=-6Ts3∫0Ts((Ts-2ξ)ν(ξ)+αqξ(Ts-ξ)iq(ξ))dξ, where H^q is the estimated value of H^ in the PMLSM system, and Ts represents the sampling time in the PMLSM system.

The value H^q is estimated and calculated online by Eq. (). Meanwhile, the d-axis reference value in the PMLSM system is set to 0 and the complex trapezoid equation is used for calculation: 11H^q=-3c3Ts∑n=1c{(c-2(n-1))v[n-1]+αq(n-1)×Ts(c-(n-1))ĩq[n-1]+(c-2n)v[n]+αqnTs(c-n)ĩq[n]}, where c represents the sampling step size of the PMLSM system, ĩq[n] represents the current sampling value in the PMLSM and ν[n] represents the speed sampling value in the PMLSM.

The current estimation of the q axis for the MFSC in the PMLSM system can be calculated by Eq. (): 12i^q=v˙-H^q+Kp(ṽ-v)aq, where i^q represents the current estimation of the q axis; ṽ and v are the reference value and actual value of the operating speed of the PMLSM system, respectively. Figure 2 shows the block diagram of the MFSC system based on the PMLSM ultra-local model.

Since the motor used in this paper is a surface-mounted PMLSM, there is Ld=Lq=L. Equation () can be rewritten to obtain the following equation: 13uα=(Rs+qL)iα+eαuβ=(Rs+qL)iβ+eβ, where q denotes the differential operator, eα and eβ are the extended BEMFs in the PMLSM. Meanwhile, the following equation is satisfied: 14eα=-ωcψfsin⁡θeeβ=ωcψfcos⁡θe.

To observe the BEMF of the PMLSM with SMO, the voltage equation of Eq. () is rewritten into the state equation of current: 15ddtiα=uαL-RsLiα-eαLddtiβ=uβL-RsLiβ-eβL.

To obtain the estimated value of the extended BEMF, Eq. () is rewritten: 16ddti^α=uαL-RsLi^α-sαLddti^β=uβL-RsLi^β-sβL, where i^α and i^β are observations of the stator current. The error of the current in the PMLSM can be obtained by subtraction between Eqs. () and (): 17ddtĩα=-RsLĩα+eα-sαLddtĩβ=-RsLĩβ+eβ-sβL, where ĩα=i^α-iα and ĩβ=i^β-iβ represent the observation errors of the current. The SMC law is designed as follows: 18sα=ksgn(ĩα)sβ=ksgn(ĩβ), where k>max-Rs|ĩα|+eαsgn(ĩα),-Rs|ĩβ|+eβsgn(ĩβ).

When the state variables of the observer reach the sliding surfaces ĩα=0 and ĩβ=0, the observer state will always remain on the sliding surface. According to the equivalent control principle of the SMC, the control quantity at this time can be regarded as the equivalent control quantity. The observed values of the BEMF in the two-phase stationary coordinate system are as follows: 19eα-eq=ksgn(ĩα)eβ-eq=ksgn(ĩβ).

By taking the derivative of Eq. (), the following equation can be obtained: 20ddteα=-dωcdtψfsin⁡θe-ωc2ψfcos(θe)ddteβ=-dωcdtψfcos⁡θe-ωc2ψfsin(θe).

Since the change speed of the motor is very slow, it can be determined that dωc/dt=0. In combination with Eq. (), the above equation can be simplified as follows: 21ddteα=-ωceβddteβ=ωceα.

Taking Eq. () as the reference model of the adaptive law of the BEMF, the adjustable model is established as follows: 22ddte^α=-ω^ce^β-l(e^α-eα)ddte^β=ω^ce^α-l(e^β-eβ), where ω^c is the estimate value of ωc, e^α and e^β are expressed by the following equation: 23e^α=ωcs+ωceα-eqe^β=ωcs+ωceβ-eq, where l>0. The following equation can be obtained by making a difference between Eqs. () and (): 24ddte‾αe‾β=-l-ωcωcle‾αe‾β-e^β-e^α(ω^c-ωc), where, e‾α=e^α-eα,e‾β=e^β-eβ.

According to Popov's hyperstability theory, to prove whether the system is stable, the following two conditions must be met: (1) The transfer function matrix H(s)=(sI-A)-1 of the system is strictly a positive definite; (2) Popov's integral inequality η(0,t1)=∫0t1VTWdt≥-γ02, (t1≥0,0≤γ0≤∞) holds, where V=e‾αe‾β, W=e^β-e^αT(ω^c-ωc), γ02 is any finite real number.

For condition (1), the transfer function matrix of the system is 25H(s)=1(s+l)2+ωc2s+l-ωcωcs+l.

Equation () shows that the transfer function is strictly a positive definite and condition (1) holds.

For condition (2), it is assumed that the mechanism of the adaptive law of the MRAS is as follows: 26ω^c=∫0tF1(ve,t,σ)dσ+F2(ve,t)+ω^c(0).

When substituting V and W into η(0,t1) of condition (2), the following is obtained: 27η(0,t1)=∫0t1e‾αe^β-e‾βe^αω^c-ωcdt.

The combination of Eqs. () and () obtains the following: 28η(0,t1)=η1(0,t1)+η2(0,t1)=∫0t1(e‾αe^β-e‾βe^α)×∫0tF1(ve,t,σ)dσ+ω^c(0)-ωcdt+∫0t1e‾αe^β-e‾βe^αF2(ve,t)dt.

For η1(0,t1) in Eq. (), 29ddtf(t)=e‾αe^β-e‾βe^αkef(t)=∫0tF1(ve,t,σ)dσ+ω^c(0)-ωc, where ke is an arbitrary value and satisfies 0<ke<1. According to inequality, 30∫0t1kef(t)df(t)dtdt=12kef2(t1)-f2(0)≥-12kef2(0), where it can be proved that η1(0,t1)≥-γ12.

For η2(0,t1) in Eq. (), let (1-ke)f(t)=F2(ve,t). According to Eq. () and the arbitrary value of F2(ve,t), it can also be proved that η2(0,t1)≥-γ22. From the above proof process, it can be concluded that condition (2) of Popov's hyperstability theory holds, proving that the system is stable. According to Eq. (), (1-ke)f(t)=F2(ve,t) and dωc/dt=0, the adaptive law of the system can be obtained as follows: 31dω^cdt=e‾αe^β-e‾βe^α.

By selecting an appropriate value of l, the adjustable model shown in Eq. (), which is established by taking the observed value of the BEMF output by the traditional SMO as the reference model, can obtain a relatively smooth BEMF after being adjusted by the adaptive law of Eq. (), thereby further reducing the chattering phenomenon of the SMO. The SMO system block diagram based on the MRAS optimization is shown in Fig. 3.

In this paper, the speed information in the BEMF is extracted by a phase-locked loop (PLL). Set 32g=(Ld-Lq)(ωcid-qid)+ω^cψf=ω^cψf.

The error signal in the PLL is 33Δe=-eαcosθ^e-eβsinθ^e=gsinθe-θ^e.

The value of |θe-θ^e| is minimal when the system approaches the steady state. Assuming sin(θe-θ^e)=θe-θ^e, after normalization, the error signal of the observer is 34Δe‾=Δeeα2+eβ2=θe-θ^e.

The closed-loop transfer equation of the PLL from θ^e to θe is as follows: 35G(s)=θ^eθe=2ϱωns+ωn2s2+2ϱωns+ωn2, where ϱ=gki, ωn=(kp/2)g/ki. The PI controller's bandwidth is represented by ωn and kpki represent normal numbers. Figure 4 shows the system block diagram of the PLL.

The design is verified by the MATLAB/Simulink simulation platform. The model of this control strategy in the simulation platform is constructed according to the motor parameters in Table 1. By comparing the speed waveform and load thrust waveform of the MFSC controller based on the ultra-local model and the traditional PI controller designed in this design under variable speed motion and variable load motion,the designed MFSC controller is shown to have better dynamic response ability. On this basis, the SMO based on the MRAS optimization is added and compared with the traditional SMO to observe the speed tracking performance of the two observers under variable speed motion. Figure 5 shows the establishment of this design. The system sampling time is 1 µs.

This design is verified by using a TMS320F28335 digital processing chip to achieve control, as shown in Fig. 11. During the operation of the PMLSM, the built-in mechanical sensor transmits the current signal to the control board for closed-loop control operation. Compare the waveform used by the observer and the performance of the motor in the process of variable speed tracking. The PMLSM's parameters in the experimental process are the same as the linear motor parameters set in the simulation.

Given the fact that the system speed is 1 ms-1, and the speed will be increased to 1.5 ms-1 at 0.4 s, Figs. 12 and 13 are velocity waveforms of the PI controller and the MFSC controller under variable velocity motion, respectively. Figures 14 and 15, respectively show the speed overshoot and settling time of MFSC controller and PI controller under variable speed movement through bar charts. It can be seen in Figs. 14 and 15 that the speed overshoot and settling time of the MFSC controller in the two stages of the variable velocity motion are less than those of the PI controller. Relative to the PI controller, the MFSC controller has better dynamic response ability.

Given the system speed of 1 ms-1, apply 60 N load at 0.4 s. Figures 16 and 17 show the speed waveforms of the two controllers under the load motion. Figures 18 and 19, respectively show the speed overshoot, dynamic landing and settling time of the two controllers in the load movement through bar charts. It can be seen from Figs. 18 and 19 that the speed overshoot, dynamic landing and settling time of the MFSC controller in the two stages of load motion are smaller than those of the PI controller, and the MFSC controller has better anti-disturbance ability and stability than the PI controller.

Based on the MFSC controller, the MRAS–SMO and the SMO are added, respectively, and the velocity tracking ability of the two observers is compared when the given velocity of the PMLSM is 1 ms-1. Figures 20 and 21 show the velocity tracking errors of the SMO and MRAS–SMO, respectively. Compared with Figs. 20 and 21 and relative to the SMO, the MRAS–SMO has less of a velocity tracking error. This indicates that the MRAS–SMO has a smaller tracking error than the SMO.