Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 11, No. 2, 2025 

 

89 

Higher	Order	Predefined‐time	Sliding	Mode	Control	for	
PMSM	With	Uncertain	Disturbance	
Shitian Chen, Chengyue Su 

School of Physics and Optoelectronic Engineering, Guangdong University of Technology, Guangzhou Guangdong, 510006, China 

 

Abstract: Permanent magnet synchronous motors (PMSM) offer several inherent advantages, such as higher power density, 
greater efficiency and reliability, more precise and rapid torque control, higher power factor, and longer bearing and insulation 
lifespans. As a result, they have been widely used in adjustable-speed traction motor drives. However, during operation, these 
motors are often subject to external load disturbances and parameter variations, which severely affect the robustness of the 
control system. To address this, an adaptive predefined-time disturbance observer (APTDO) is designed to track the disturbances 
in real-time and feed them back into the controller for disturbance compensation. Furthermore, to improve the control 
performance, a higher order predefined-time sliding mode controller (HO-PTSMC) is designed for the PMSM control system. 
Compared to traditional super-twisting sliding mode control (STSMC) methods, the field-oriented sliding mode control approach 
proposed in this paper significantly improves dynamic torque and speed responses and enhances the system's robustness under 
uncertain disturbances. The new sliding mode algorithm, being a fully continuous smooth function, effectively suppresses the 
chattering phenomenon inherent in traditional sliding mode algorithms, thereby significantly improving the steady-state 
performance of the system. Finally, simulations are conducted on the MATLAB Simulink platform, which validate the superiority 
of the proposed algorithm. 

Keywords: Permanent Magnet Synchronous Motors (PMSM); Predefined-Time Sliding Mode Control; Field-oriented Control. 
 

1. Introduction 
Permanent Magnet Synchronous Motors (PMSM) are 

widely used in applications such as electric vehicles, rail 
transportation, and marine propulsion, owing to their high 
efficiency, high power density, excellent overload capability, 
and outstanding performance[1]. However, the dynamic 
model of a PMSM is a highly coupled, multivariable, and 
nonlinear complex system. Under practical operating 
conditions, the performance of the control system is highly 
susceptible to significant disturbances such as parameter 
variations, external disturbances, and unmodeled 
dynamics[2]. To achieve robust speed control performance, 
various nonlinear algorithms have been proposed, including 
Model Predictive Control [3], Fuzzy Control [4], Finite-Time 
Control [5], Robust Control [6], and Sliding Mode Control 
(SMC) [7]. 

Among various control methods, Sliding Mode Control 
(SMC) is considered one of the most effective approaches for 
handling system uncertainties. Due to its fast response, high 
accuracy, and strong robustness, it has been widely applied in 
motor control fields. In [8], a novel Higher-Order Super 
Twisting Sliding Mode Control (HO-STSMC) was proposed 
to achieve speed control of the Permanent Magnet 
Synchronous Motor (PMSM) with finite-time convergence. 
However, the upper bound of the settling time depends on the 
system's initial conditions, which limits its applicability for 
improving productivity in practical applications. To achieve 
better convergence performance, [9] designed an improved 
Second-Order Fixed-Time Sliding Mode Control (FTSMC) to 
ensure global fixed-time stability of the closed-loop system, 
where the settling time is bounded and independent of the 
initial conditions. In [10], a composite control strategy was 
proposed, combining an enhanced Non-Singular Fast 
Terminal Sliding Mode Control (NFTSMC) with a 
disturbance observer. This method aims to ensure fixed-time 

convergence while enhancing the system’s disturbance 
rejection capabilities. 

However, the Fixed-Time Sliding Mode Control (FTSMC) 
algorithm has two main issues: first, the estimation of the 
convergence time is conservative, and second, there is no 
clear relationship between the convergence time and the 
adjustable parameters of the controller. To address these 
issues, the application and development of predefined-time 
control methods are necessary [11-12]. Compared to fixed-
time control, predefined-time control effectively resolves 
these problems. Firstly, the adjustable parameters of the 
controller determine the upper bound of the predefined 
settling time, avoiding the overestimation of the convergence 
time. Secondly, the predefined stable convergence time is 
bounded by the controller's adjustable parameters, facilitating 
the design of controllers that meet specific requirements. In 
[13], a novel predefined-time sliding mode control scheme 
was proposed for chaotic systems. Simulation results show 
that this method has a faster convergence rate compared to 
traditional finite-time SMC and fixed-time SMC, which is 
particularly suitable for systems like Permanent Magnet 
Synchronous Motors (PMSM) that require high dynamic 
performance. However, the introduction of the sign function 
on the sliding surface may lead to undesirable chattering 
phenomena. In [14], an adaptive practical predefined-time 
sliding mode control strategy was proposed for second-order 
MAS systems under uncertain disturbances, effectively 
addressing the issues of chattering and singularity. By 
introducing a tanh function, the strategy suppressed the 
chattering effect and improved the system's steady-state 
performance. These controllers, due to their excellent 
performance, are particularly suitable for complex nonlinear 
control systems such as PMSM. Currently, the application of 
predefined-time sliding mode control algorithms in PMSM 
control systems holds great promise. 

On the other hand, in practical drive systems with high 



 

90 

utilization, Permanent Magnet Synchronous Motors (PMSM) 
experience nonlinear effects such as rapidly changing 
external loads, inaccurate measurement of nominal 
parameters, and parameter perturbations. In such cases, the 
presence of uncertain disturbances significantly reduces the 
performance and stability of the system, becoming a major 
weakness in PMSM drives. To address these issues, 
disturbance observers have attracted significant attention 
from researchers, leading to many interesting results, such as 
nonlinear disturbance observers [15], fuzzy disturbance 
observers [16], and extended state observers [17]. However, 
the convergence of the disturbance observation errors in these 
approaches is typically achieved over an infinite time span. 
To obtain precise disturbance estimation, finite-time and 
fixed-time control techniques have been employed to design 
disturbance observers. In [18], a finite-time sliding mode 
disturbance observer was designed, where the observation 
error converges to zero within a finite time. However, these 
observers use discontinuous sign functions, which can lead to 
chattering phenomena. In [19], a novel nonlinear fixed-time 
disturbance observer (FTDO) was proposed, which improves 
convergence speed while avoiding the use of discontinuous 
functions like the sign function, thereby reducing chattering 
and improving observation accuracy. Subsequently, research 
on predefined-time sliding mode disturbance observers has 
gained attention. In [20], a predefined-time attitude 
stabilization strategy based on a disturbance observer was 
proposed for rigid spacecraft with external disturbances and 
inertial uncertainties. The method achieves disturbance 
suppression by constructing a predefined-time nonlinear 
disturbance observer (PTNDO) and ensures the rapid and 
accurate convergence of the disturbance estimation error. It is 
important to note that the aforementioned observers assume 
an upper bound for disturbances, which is not suitable for 
PMSM drive systems, as the motor load disturbances can vary 
greatly in practical applications. Using a large upper bound as 
a parameter for the observer can cause chattering, which 
negatively impacts control accuracy. Adaptive strategies can 
effectively resolve this issue. 

Inspired by the above studies, this paper proposes a high-
order predefined-time sliding mode control (HO-PTSMC) 
method based on an adaptive predefined-time disturbance 
observer (APTDO) for PMSM drive systems. Simulation 
results demonstrate that the new control method effectively 
enhances the anti-disturbance performance of PMSMs under 
uncertain disturbances. The highlights of this work are as 
follows: 

1. The APTDO is designed to track the uncertain 
disturbances present in the motor, ensuring that disturbances 
converge within a predefined time. Moreover, this method 
adopts an adaptive control strategy that does not require prior 
knowledge of the disturbance upper bound. The adaptive gain 
varies with the system state, preventing overestimation. 

2. The HO-PTSMC is designed to track the speed and 
current loop errors of the PMSM. This controller 
demonstrates a better convergence speed and anti-disturbance 
performance. Compared to traditional sliding mode 
algorithms, it uses a fully continuous smooth function, 
effectively suppressing the chattering effect of sliding mode 
algorithms and improving the steady-state performance of the 
system. 

3. Both HO-PTSMC and APTDO employ a novel practical 
predefined-time sliding mode algorithm. This algorithm treats 
time as an explicit design parameter, allowing more precise 
determination of the sliding surface and switching control 
timing during system design. As a result, the design process 
is simplified, and the design becomes more controllable and 
adjustable. 

2. SPMSM Mathematical Model 

2.1. Conventional Model Description 
The dynamical model of a PMSM in synchronous reference 

frame is given by the following: 

 
r e r L

e n f d q q

J T B T

T 1.5P (L L ) i

  

  

 

 





       (1) 

q
d d n r q d

d d d

LR 1
i i P i u

L L L
             (2) 

d f
q q r d r q

q q q q

LR 1
i i p i p u

L L L L


          (3) 

   Surface-mounted PMSM(SPMSM) meets 

d q e n f q

3
L L L ,T P i

2
    , it can be rewritten as: 

r n f q r LJ 1.5P i B T                (4) 

d d n r q d

R 1
i i P i u

L L
                 (5) 

f
q q r d r q

R 1
i i p i p u

L L L


              (6) 

   Here, r   is the mechanical angular velocity, f   is 

the magnetic chain, nP  is the number of pole pairs, dL , qL  

is the shaft stator inductance, B   is the damping friction 

coefficient, qi   is the q-axis stator current, LT   is the load 

torque, and J  is the moment of inertia.   

2.2. Improved Model Description with 
Uncertain Disturbance  

In practical systems, measurement inaccuracies always 
lead to discrepancies between nominal and actual parameters. 
Moreover, during motor operation, the parameters of the 
motor may change. These factors result in models that may 
not accurately reflect the actual behavior of the system, and 
under certain operating conditions, the model may differ from 
the system's actual response. Therefore, the control system 
designed must exhibit robustness in the presence of parameter 
uncertainties. To address these uncertainties, the model takes 
into full account the uncertainties in parameters such as flux 
linkage, stator inductance, moment of inertia, damping 
friction coefficient, and stator resistance. 

The improved model, considering parameter uncertainties, 
is expressed as follows:  

 

     r n f f q r LJ J 1.5P i B B T                                      (7) 

 



 

91 

 
The improved model, considering parameter uncertainties, 

is expressed as follows: 
 

  

*
r q

*
q q r L n f q r

i G

1
G a(i i ) B B T 1.5P i J

J

   



           





                  (8) 

 

  Here, n f J1.5P /    , G is the total uncertain 

disturbance, which is influenced by motor speed, external 
load, current errors, and unmatched parameters that are 
limited by the rated values of SPMSM. 

3. Controller Design 
This section first introduces an adaptive predefined-time 

disturbance observer (APTDO) to capture the uncertain 

disturbances during the motor operation, employing an 
adaptive control strategy that does not require knowledge of 
the upper bound of the disturbances. Subsequently, a higher 
order predefined-time sliding mode controller (HO-PTSMC) 
is proposed for controlling the motor's speed and current 
loops, with the disturbance estimate from the observer fed 
back to the controller. This approach enables precise speed 
control of the motor in the presence of uncertain disturbances. 
Fig. (1) presents the block diagram of the novel PMSM 
system. 

 
Fig 1. the block diagram of the novel PMSM system 

 

3.1. Design of Adaptive Predefined-time 
Disturbance Observer 

We define a speed estimation error as the sliding mode 

surface: r ro ˆs     , through equation (4), we design 

APTDO as follows: 

   

*r
q

1 1
1 2 3

o
o o

1

ˆd ˆi G
dt

sˆ ˆG a sig s a sig s a tanh 

   
          

 (9) 

Here,  sig ( ) sign
      , 

1 1
2 2

o
2

o
1

1 1 1 1
0 1,a ,a

T 2 T 2

 
                

, 1  is a 

small positive constant. The adaptive parameter 3â   is 

designed as follows: 

3 o 3( s )ˆ ˆa a                 (10) 

Here,  1 /2 1 /2
1 2max 2a / ,2a /      ,    is a 

positive constant. 
According to [19], we can obtain that there exists a positive 

constant such that 3 3â a   and 3 3a a  . Considering the 

non-linear dynamical system in equation (4), if the 
disturbance observer is designed as equation (5,6), the 

estimated speed error s  will converge to a narrow region 

d   around zero within the predefined-time oT  , d   is 

expressed as follows: 
 

2
2 2 2

2
d

2
d d

d
d2 T 2 T

2min ,


 

                    

                                 (11) 



 

92 

 

Here,  2 2
d 3 3 d 3

1
aa a

2
         , 

     1 /2 / /2 1/ /2
d (1 / 2) (1 / 2)          ,

0.2785  . 

When ot T  , ˆG G G    will exist an upper bound 

G  which can be expressed as follows: 

 1 1 d
3 1 d 2 d 3

1

G â ˆa sig ( ) a sig a tanh( )  
     


                        (12) 

 
So, we can conclude that the novel disturbance observer 

can quickly track the uncertain disturbance of the PMSM in a 

predefined-time period oT . 

3.2. Design of Higher Order Predefined-time 
Sliding Mode Controller 

The sliding manifold    T

d qs x, t s ,s ,s   of the 

field-oriented sliding mode control is governed by: 
*

d d ds i i                        (13) 
*

q q qs i i                        (14) 

*
r rs                         (15) 

The effect of uncertain disturbance is considered in First-
Order Velocity Control and not in Higher-Order Current 
Control. 

3.2.1. First-Order Velocity Control 
We define the q-axis velocity control law as 

* * *
q q,eq q,Ni i i                    (16) 

Here, *
qi  is the quadrature-axis stator current, *

q,eqi  is the 

equivalent control and *
q,Ni  is the switching control.  

From the sliding surface s , s  is found to be 
* *
q rs i G                    (17) 

The equivalent control becomes: 

 r
*
q,eq

*1
Gi 


                (18) 

Substituting the estimated term Ĝ  by the above designed 
disturbance observer, equation (18) becomes: 

 r
*
q,eq

*1
i Ĝ


               (19) 

By applying a predefined time sliding mode algorithm, 
switch control becomes: 

 

   1 1

2

*
q,N 1 2 3

1
i s

s
sig s b sig b tanhb  

 
 

 
 

   
  




                        (20) 

 
Here, 

1 1
2 2

1
c

2
1 c1

1 1 1 1
0 1,b ,b

T 2 T 2


 




           
   

 , 2  

is a small positive constant.  

The output of q-axis velocity control *
qi  serves as the input 

to q-axis current sliding mode control in next Section.  

3.2.2. Higher-Order Current Control 
We define the qd-axis current control law as 

q q,eq q,Nu u u                 (21) 

d d,eq d,Nu u u                 (22) 

Here, q du ,u  is the qd-axis stator voltage, q,eq d,equ ,u  is 

the equivalent control and q,N d,Nu ,u   is the switching 

control. 

From the sliding surface q, ds s , q, ds s   is found to be 

*f
q n r d n r q q

*
d d n r q d d

q

R 1
i P i P u i

L L L
R 1

i P i u i
L

s

L
s

 


     

     








 (23) 

The equivalent control becomes: 

fq,eq n

d,e

* f
q q r d n r r

*
d n r qq d

u

L

i Ri i P

i

L p
L

R

LP

iu PLi


    

 

  

 




 (24) 

By applying a predefined time sliding mode algorithm, 
switch control becomes: 

 

   1 1
1 2 3

d,q
qd,N d,q d

3
,qu

s
sig s c sig s cL t nhc a 


 




 
  


 

 


                     (25) 

 
Here, 

1 1
2 2

1
c

2
2 c2

1 1 1 1
0 1,c ,c

T 2 T 2

 
                

, 3  is a 

small positive constant.  

The output of qd-axis voltage control q,eq d,equ , u  serves 

as the input to SVPWM module to complete the convergence 
of current. Finally, we can calculate the total convergence 

time of the speed tracking error as c c1 c2T T T  . 

4. Simulation Experiment 
In this section, the proposed control method for the PMSM 



 

93 

will be validated through real-time experiments to 
demonstrate its effectiveness and advantages. The PMSM 
parameters of the simulation model are shown by Table 1. The 
sampling frequency of simulation is set to 100 kHz, and the 
initial states of the PMSM are specified as follows:  

 T T

q d r ei , i , , 0,0,0,0             (26) 

The indicator RMSE and MAXE used in the experiment 
are defined by the following: 

   

 
2

1



 
N

i

MAXE e max e ( i )

e ( i )
RMSE e

N

 




       (27) 

Here, N is the number of the sampled tracking error, 

e ( i )  is the speed tracking error of the i-th sampling time. 

Meanwhile, the settling time and maximum speed drop are 
considered to evaluate the dynamic performance of the 
proposed method. 

 
Table 1. Parameters of PMSM 

 

 2.875 

  8.5 

 0.003 

  0.175 

 
0.008 

 
 4 

4.1. Comparison of the APTDO 
In this experiment to verify the effectiveness of the 

proposed APTDO, three types of loads are considered: step 
disturbance, time-varying disturbance, and high-frequency 

noise disturbance. From the Fig. (2), it can be seen that the 
APTDO designed in this paper can track these three types of 
disturbance quickly with good special effects. 

  

 
Fig 2. The disturbance estimation performance of APTDO 

 
 



 

94 

 

4.2. Comparison of the HO-PTSMC 
The effect of the proposed HO-PTSMC is verified in this 

experiment and compared with the HO-STSMC proposed in 
paper [8]. As can be seen from Fig. (3-4), the novel proposed 
algorithm has better dynamic performance at both step and 
time-varying reference speed, and is able to complete speed 

tracking quickly. The data in Table 2 further validate the 
above conclusion, as the new algorithm adopts the fully 
continuous smooth function, it can effectively suppress the 
jitter oscillations and thus achieve a smaller RMSE effect, 
which indicates that the new algorithm also has a significant 
improvement in the steady state performance. 

 
Fig 3. the dynamic performance comparison of step speed 

 

 
Fig 4. the dynamic performance comparison of time-varying speed 

 
Table 2. Parameters of PMSM 

Controller method 
Step speed Time-varying speed 

RMSE MAXE RMSE MAXE 

HO-STSMC 0.2498 2.3671 0.3594 4.4399 

HO-PTSMC 0.0781 0.6352 0.0698 0.5468 

 



 

95 

 

4.3. Comparison of the HO-PTSMC+APTDO 
Feedback 

The effect of the proposed HO-PTSMC with or without 
perturbation feedback is compared in this experiment. In 

considering the three-disturbance proposed above, it can be 
found from Fig. (5) that the control system with the addition 
of APTDO feedback has a huge improvement in the anti-
perturbation capability, which verifies the effectiveness of the 
perturbation observer. 

 

 
Fig 5. the dynamic performance comparison of disturbance feedback 

 

5. Summary 
This paper develops an improved sliding mode control 

scheme using APTDO and HO-PTSMC techniques for 
permanent magnet synchronous motor (PMSM) control, 
aiming to enhance disturbance rejection and achieve speed 
tracking convergence within a predefined time. An adaptive 
strategy is employed in APTDO, enabling dynamic 
adjustment to adapt to varying uncertain disturbances. 
Subsequently, an HO-PTSMC is proposed to ensure 
predefined time convergence and robustness of the speed 
control. Real-time experimental results effectively validate 
the excellent speed estimation performance of HO-PTSMC 
and the superior speed control performance of the proposed 
method. Future work will focus on the DSP hardware 
validation of the PMSM system. 

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