The sliding mode control (SMC) strategy is applied to a permanent magnet synchronous machine vector control system in this study to improve system robustness amid parameter changes and disturbances. In view of the intrinsic chattering of SMC, a current sliding mode control method with a load sliding mode observer is proposed. In this method, a current sliding mode control law based on variable exponent reaching law is deduced to overcome the disadvantage of the regular exponent reaching law being incapable of approaching the origin. A load torquesliding mode observer with an adaptive switching gain is introduced to observe load disturbance and increase the minimum switching gain with the increase in the range of load disturbance, which intensifies system chattering. The load disturbance observed value is then applied to the output side of the current sliding mode controller as feedforward compensation. Simulation and experimental results show that the designed method enhances system robustness amid load disturbance and effectively alleviates system chattering.
I. INTRODUCTION
The permanent magnet synchronous motor (PMSM) is a multivariable, nonlinear, and strongcoupled system that is extremely sensitive to parameters and disturbances; thus, its static and dynamic processes cannot be accurately described by traditional linear control methods, and its high operation performance is difficult to guarantee in a wide speed range
[1]
,
[2]
.
Sliding mode control (SMC) is a special nonlinear control method with advantages such as quick response, strong robustness, and simple implementation. However, this control method inevitably increases frequency chattering, thus making current research on the application of SMC to actual motor drive systems focus on how to alleviate chattering
[3]

[6]
. Current methods that effectively weaken system chattering applied to PMSM include boundary layer
[7]
,
[8]
, integral sliding surface
[9]
,
[10]
, reaching law
[11]

[14]
, sliding mode observer
[15]

[19]
, and intelligent control methods
[20]
[22].
Exponent reaching law can obtain a quick response and weaken chattering, but its sliding motion area does not decay with time; that is, system states cannot reach the origin, whereas chattering does. The highfrequency components in the system model are excited, and the burden on the controller is consequently increased
[11]
. In
[12]
, an SMC method based on the exponent reaching law was applied to the speed regulator of a PMSM vector control system; the method enhanced system robustness and the dynamic characteristics effectively. A variablerate reaching law was proposed in
[13]
,
[14]
; switching gain was designed as the product of a constant item and the state variable. The design solves the problem of the regular exponent reaching law being incapable of approaching the origin. However, when the system state arrives at the sliding band, system chattering remains high. To solve this problem, an improved exponent reaching law with timevarying switching gain is proposed in this study; that is, a modified factor with a small value and convergence to zero is introduced to switching gain to obtain minimal chattering on the arrival of the sliding band and to approach the origin finally.
In the SMC system, only the minimum switching gain that increases with the increase in the range of load disturbances can meet the sliding mode existence and accessibility condition and thus resist the load disturbance effectively. However, an increase in switching gain intensifies system chattering. If the load observer method is introduced to compensate for the load disturbance, the value of switching gain will decrease significantly and system chattering will be greatly reduced. The regular openloop load observer has a simple structure but does not use the output observation error for feedback correction of the load observed value; thus, the observer cannot guarantee its convergence and speed
[15]
. The regular closedloop load observer can ensure the asymptotic stability of observation but depends on the cumbersome online identification of uncertain or timevarying parameters
[16]
,
[17]
. A load observer based on sliding mode method can effectively inhibit the influence of parameter perturbations and load disturbances on system characteristics; however, reports on how to alleviate the chattering of the observed value are rare
[18]
,
[19]
. To solve this problem, an improved extended load sliding mode observer is proposed in this study with load torque and speed as observed objects. An integral sliding surface with integral separation with respect to the speed observation error is adopted to avoid saturation. Switching gain is adjusted by the load disturbance observation error and is thus made convergent to zero. The designed observer can effectively weaken the chattering of the observed objects on the premise of the assurance of system robustness.
II. MATHEMATICAL MODEL OF PMSM
The stator voltage of PMSM in the d–q synchronous rotating reference frame can be expressed as
where
u
_{d}
and
u
_{q}
are the d–q axis stator voltages,
i
_{d}
and
i
_{q}
are the d–q axis stator currents,
L
_{d}
and
L
_{q}
are the d–q axis inductances,
R
_{s}
is the stator phase resistance,
Ψ
_{f}
is the rotor permanent magnet flux linkage, and
ω
_{e}
is the rotor electrical angular speed.
The corresponding electromagnetic torque is
The associated electromechanical equation is as follows:
where
T
_{L}
is the load torque,
p
is the number of pole pairs,
J
is the moment of inertia,
B
_{m}
is the friction coefficient, and
ω
_{m}
is the rotor mechanical angular speed.
III. DESIGN OF CURRENT SMC
 A. State Space Equation
In a current control system, the state variables are d–q axis current errors
e
_{d}
and
e
_{q}
, and the control inputs are d–q axis voltages
u
_{d}
and
u
_{q}
. Thus,
where
i
_{dr}
and
i
_{qr}
are the reference commands of the d–q axis currents.
The state space equation of the d–q axis current control system can be described from Eqs. (1) and (4) as
where
 B. Sliding Surface
The integral sliding surface can eliminate static system errors and enhance control precision. Thus, the integral sliding surfaces with respect to the d–q axis current errors are utilized as follows:
where
c
_{1}
and
c
_{2}
are the integral coefficients of the d–q axis sliding surfaces.
 C. Reaching Law
Given the disadvantage of the regular exponent reaching law, an improved exponent reaching law is proposed in this study, where a timevarying modified factor
ρ
(x) is introduced to switching gain. The modified switching gain is then designed as the product of a constant item and
ρ
(x) .
Modified factor
ρ
(x) is designed as follows:
where
σ
_{d,q}
s is the corresponding coefficient of either the daxis or the qaxis modified factor. The reaching law with respect to the d–q axis current errors can then be expressed as where
ε
_{1}
and
ε
_{2}
are the switching gains of the d–q axis sliding surfaces and
η
_{1}
and
η
_{2}
are the exponent coefficients of the d–q axis sliding surfaces.
Therefore, modified switching gain
ε
_{1}
ρ
(
e
_{d}
) or
ε
_{2}
ρ
(
e
_{q}
) is always smaller than the original
ε
_{1}
or
ε
_{2}
and thus makes system chattering smaller upon arrival of the sliding band than the variable rate reaching law; it decays to zero with the system state to overcome the disadvantage of the regular exponent reaching law.
 D. Control Law (Switching Function)
With
E
_{1}
and
E
_{2}
as disturbance terms, the current SMC law based on the improved exponent reaching law can be derived from Eqs. (4) to (8) as follows:
 E. Continuous Switching Function
The sign functions in the control law in Eq. (9) can be replaced by a continuous smoothing function to alleviate the highfrequency chattering caused by sliding mode switching.
where
δ
_{d,q}
is the smoothing coefficient of either the daxis or the qaxis smoothing function. Chattering cannot be alleviated effectively if
δ
_{d,q}
is too small; otherwise, the dynamic responses of the sliding motion will worsen. Thus, these two conditions should be weighed to select an appropriate value of
δ
_{d,q}
.
 F. Stability Analysis
According to Lyapunov’s stability theory, the sliding mode existence and accessibility condition is expressed as
Substituting Eqs. (4) to (8) into Eq. (11) yields
The minimum switching gain that increases only with the increase in the range of load disturbance can meet Eq. (12). However, the increase in switching gain intensifies system chattering. If the load observer method is introduced to compensate for the load disturbance, a small switching gain can meet the sliding mode existence and accessibility condition and thus weaken system chattering.
IV. DESIGN OF THE LOAD SLIDING MODE OBSERVER
 A. Extended State Space Equation
With electrical angular speed
ω
_{e}
and load torque
T
_{L}
as the state variables, electromechanical torque as the system input, and
ω
_{e}
as the system output, the extended state space equation can be described from Eq. (3) as
 B. Observer Model
With speed
ω
_{e}
and load torque
T
_{L}
as the observed objects, the model of the load sliding mode observer is established from Eq. (13) as follows:
where
l
is the feedback gain of the observer and
represents the sliding mode control function with respect to observation error
which depends on its sliding surface and reaching law.
Subtracting Eq. (14) from Eq. (13) yields the error equation of the observer.
where
 C. Sliding Surface
The integral sliding surface can reduce static system errors but may result in saturation and even system oscillation. An integral separationbased integral sliding surface is designed in this study to solve this problem. The integral function is canceled to avoid saturation when system error
e
_{ω}
is greater than preset threshold
e
_{ω(th)}
.
The integral sliding surface with integral separation with respect to the speed observation error can then be expressed as
where
c
_{ω}
and
e
_{ω(th)}
are the integral coefficient and the separation threshold of the observer integral sliding surfaces, respectively.
 D. Reaching Law
The exponent reaching law with respect to the speed observation error can be expressed as
where
ε
_{ω}
and
η
_{ω}
are the switching gain and the exponent coefficient of the observer reaching law, respectively.
 E. Control Law
By substituting Eq. (15) into Eq. (17) and with
as the disturbance term, the sliding mode control law of the load sliding mode observer can be deduced as follows:
Thus, the structure diagram of the load sliding mode observer can be generated according to Eqs. (13) to (18) as shown in
Fig. 1
.
Structural diagram of the load sliding mode observer.
 F. Stability Analysis
According to Lyapunov’s stability theory in Eq. (11), the sliding mode existence and accessibility condition of the load observer can be derived from Eqs. (14) to (18) as
When load torque observation error
e
_{T}
is large, necessary switching gain
ε
_{ω}
and the resulting chattering of the observed objects are also large.
 G. Switching Gain
To solve the problem above, a method that involves the online adjustment of switching gain is proposed in this study. Switching gain
ε
_{ω}
is adaptively tuned by load torque observation error
e
_{T}
. The designed observer can weaken the observed chattering on the premise of the assurance of system stability.
The adaptive adjustment law with respect to switching gain
ε
_{ω}
is then designed as follows:
where
k_{ε}
is a constant greater than 1. Thus, when load observation error
e
_{T}
is large, switching gain
ε
_{ω}
should also be large to ensure observer robustness; otherwise, when
e
_{T}
converges to zero,
ε
_{ω}
approaches zero and observation chattering decreases.
 H. Load Compensation
Load disturbance observed value
is equivalently compensated to the output side of the current SMC to reduce switching gain in the control law in Eq. (9); thus, we can obtain the current SMC law with load torque feedforward compensation as follows:
where
k
_{cd}
and
k
_{cq}
are the load compensation coefficients of the d–q axis SMC law. In the motor mode (
i
_{q}
> 0 and
i
_{d}
< 0), items
are compensated for items 
L
_{q}
ω
_{e}
i
_{q}
and
L
_{d}
ω
_{e}
i
_{d}
+
ω
_{e}
Ψ
_{f}
of stator voltages
u
_{d}
and
u
_{q}
in Eq. (1) separately. Therefore, 
B
_{1}
^{1}
k
_{cd}
< 0 and 
B
_{2}
^{1}
k
_{cq}
< 0 .
Similarly, according to Lyapunov’s stability theory in Eq. (11), the stability condition of the current SMC can be deduced from Eqs. (4) to (8) as
Adopting the load torque compensation method significantly reduces switching gain and thus effectively alleviates system chattering.
V. SIMULATION RESULTS AND ANALYSIS
A simulation model of the PMSM drive and control system (
Fig. 2
) is built and examined in MATLAB.
The PI control method is applied to the speed outer loop in the system model. The speed controller provides current command
i
_{sr}
, which is distributed in d–q axis current commands
i
_{dr}
and
i
_{qr}
according to the maximum torque per ampere vector control rule. The improved exponent reaching law based on the SMC method is employed in the current inner loop combined with the sliding mode observer based on the load disturbance compensation method to alleviate the system chattering of the SMC.
Structural diagram of the system simulation model.
PARAMETERS OF THE CURRENT SMC AND LOAD SLIDING MODE OBSERVER
PARAMETERS OF THE CURRENT SMC AND LOAD SLIDING MODE OBSERVER
The main parameters of the designed current SMC in the simulation and experiments are shown in
Table I
. The sampling time of the controller is 100 μs in the simulation, similar to the switching time, and 2 ms (20 times the length of the switching time) in the experiments.
Figs. 3
and
4
show the simulation curves of the loading and unloading responses of the observed load torque under a speed command of 3000 r/min. The load torque abruptly changes from 36 N·m to 72 N·m at
t
=10.0 s and from 72 N·m to 36 N·m at
t
=20.0 s. Constant switching gain is employed in the regular method, whereas adaptive switching gain is applied in the improved load observation method as shown in Eq. (20). For the improved method, the observed load torque responds more quickly and has a smaller overshoot; thus, the improved load sliding mode observer has better tracking characteristics.
Simulation curves of the loading responses of observed load torque.
Simulation curves of the unloading responses of observed load torque.
Figs. 5
and
6
provide comparisons of the simulation curves of the current tracking responses of step load in the regular, improved exponent reaching law, and improved load observation and compensation methods under a speed command of 3000 r/min. The load torque changes abruptly from 36 N·m to 72 N·m at
t
=10.0 s and from 72 N·m to 36 N·m at
t
=20.0 s. Constant switching gain is employed in the regular method without the load observer. The modified switching gain with the timevarying factor shown in Eq. (7) is applied to the improved exponent reaching law method without the load observer, whereas the improved exponent reaching law with timevarying switching gain and the improved observer with the adaptive switching gain are both adopted in the improved load observation and compensation method. In the dynamic processes, the d–q axis currents for the improved reaching law method respond more quickly and have a smaller overshoot than the regular one, whereas those for the improved compensation method respond the quickest and have the smallest overshoot; thus, the designed SMC has superior current tracking characteristics with load disturbance. In the steady states, the d–q axis current ripples for the improved reaching law method are smaller than the regular one, whereas those for the improved compensation method are the smallest. This result is consistent with the theoretical analysis on the reaching law and observer in Sections III and IV.
Simulation curves of the loading current tracking responses. (a) Speed responses. (b) Daxis current responses. (c) Qaxis current responses. (d) Phase current responses.
Simulation curves of the unloading current tracking responses. (a) Speed responses. (b) Daxis current responses. (c) Qaxis current responses. (d) Phase current responses.
VI. EXPERIMENTAL RESULTS AND ANALYSIS
Loading and unloading experiments are performed for the proposed reaching law and observer methods to validate the system characteristics of the antidisturbance and chattering inhibition of the current SMC. The hardware configuration of the PMSM drive and control system is shown in
Fig. 7
. The related parameters of PMSM are shown in
Table II
.
Hardware configuration of the PMSM drive and control system.
PARAMETERS OF PMSM
Figs. 8
and
9
provide comparisons of the tracking responses with load change between the regular observer and the improved one under a speed command of 3000 r/min. The load torque increases from 36 N·m to 72 N·m suddenly at
t
=10.0 s and decreases from 72 N·m to 36 N·m abruptly at
t
=10.4 s. The improved load torque observer responds more quickly and has no obvious overshoot; thus, the improved observation method has better tracking characteristics with load disturbance.
Experimental curves of the tracking response with load change for the regular observer.
Experimental curves of tracking response with load change for the improved observer.
Figs. 10
and
11
provide comparisons of the current dynamic index with different step load changes, where Δ
T
_{L}
represents a step change from a different load to the full load with a maximum value of 72 18 = 54(N×m) in actual operating conditions. When the load torque provides a step change within the rated range, the overshoot for the regular method is approximately 30%; that for the improved reaching law method decreases to approximately 15% and that for the improved observer one decreases further to approximately 10%. The adjustment time for the regular method is approximately 490 ms, that for the improved reaching law one decreases to 360 ms, and that for the improved observer one decreases further to 220 ms.
Comparisons of current dynamic index–overshoot with step load.
Comparison of the current dynamic index–adjustment time with step load.
Figs. 12
to
14
provide comparisons of the loading responses of the regular method, improved exponent reaching law method, and improved load observation and compensation method under a speed command of 3000 r/min, where the load torque abruptly increases from 36 N·m to 72 N·m at t=10.0 s.
Figs. 15
to
17
provide comparisons of the unloading responses of the three abovementioned methods under the same speed command, where the load torque abruptly decreases from 72 N·m to 36 N·m at t=20.0 s. The currents and speed for the improved reaching law method respond more quickly and exhibit less chattering in both the dynamic and static states than the regular one, whereas those for the improved compensation method respond the quickest and exhibit the least chattering. Therefore, the designed controller with a load observer enhances the system characteristics under load disturbance and alleviates system chattering significantly.
Experimental curves of the loading responses for the regular method. (a) Speed response. (b) D–q axis current responses. (c) Phase current response.
Experimental curves of the loading responses for the improved reaching law method. (a) Speed response. (b) D–q axis current responses. (c) Phase current response.
Experimental curves of the loading responses for the improved load observer method. (a) Speed response. (b) D–q axis current responses. (c) Phase current response.
Experimental curves of the unloading responses for the regular method. (a) Speed response. (b) D–q axis current responses. (c) Phase current response.
Experimental curves of the unloading responses for the improved reaching law method. (a) Speed response. (b) D–q axis current responses. (c) Phase current response.
Experimental curves of the unloading responses for the improved load observer method. (a) Speed response. (b) D–q axis current responses. (c) Phase current response.
VII. CONCLUSION
SMC has the advantages of quick response, strong robustness, and simple implementation. To improve system robustness amid parameter changes and disturbances, the SMC strategy was applied to the PMSM vector control system. However, this control strategy inevitably results in highfrequency chattering. In view of the problem of intrinsic chattering in the SMC system, a current SMC method with a load sliding mode observer was developed. In this method, an improved exponent reaching law based on the current SMC law was deduced. A load torque sliding mode observer model with adaptive switching gain was built, and the load disturbance observed value was compensated to the output side of the current SMC. Simulation and experimental results show that the designed SMC enhances system robustness amid load disturbance, alleviates system chattering effectively, and thus improves system static and dynamic characteristics under load disturbance.
Acknowledgements
This project is supported by the National Natural Science Foundation of China (No. 51177031) and the National High Technology Research and Development Program (863 Program) of China (No. SQ2010AA1122991001).
BIO
Ningzhi Jin was born in Harbin, China, in 1980. He received his B.S., M.S., and Ph.D. degrees in electrical engineering from Harbin University of Science and Technology, Harbin, China, in 2003, 2006, and 2012, respectively. He is a lecturer of power electronics and power drive in Harbin University of Science and Technology. His research interests include motor drive and power electronics.
Xudong Wang was born in Jixi, China, in 1956. He received his B.S. and M.S. degrees in electrical engineering from the Harbin Institute of Electrical Engineering, Harbin, China, in 1982 and 1987, respectively, and his Ph.D. degree in mechanical electronics from the Harbin Institute of Technology, Harbin, China, in 2000. He is a professor and doctor supervisor of power electronics and power drive in Harbin University of Science and Technology. He has been the director of the Engineering Research Center of the Automotive Electronic Drive Control and System Integration of the Ministry of Education in China since 2006. His research interests include automotive electronics and the traction motors and drives of electric vehicles.
Xiaogang Wu was born in Harbin, China, in 1981. He received his B.S., M.S., and Ph.D. degrees in electrical engineering from Harbin University of Science and Technology, Harbin, China, in 2003, 2006, and 2009, respectively. He is an associate professor of power electronics and power drive in Harbin University of Science and Technology. His research interest is the power train control of electric vehicles.
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