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U  Vd

Vq

The derivative of this surface is given by the expression:

T

𝑆 𝛺 = 𝛺𝑟𝑒𝑓 − 𝑐1 Ω +

𝐿𝑞
𝑅𝑠
𝐼𝑑 + 𝑝Ω𝐼𝑞
𝐿𝑑
𝐿𝑑
𝛷𝑓
𝑅𝑠
𝐿𝑑
− 𝐼𝑞 − 𝑝Ω𝐼𝑑 −
𝑝Ω
𝐹=
𝐿𝑞
𝐿𝑞
𝐿𝑞
3𝑝
𝑓
Tl
(𝐿𝑑 − 𝐿𝑞 𝐼𝑑 𝐼𝑞 + 𝜙𝑓 𝐼𝑞 ] − Ω −
2𝐽
𝐽
J

𝐺=

𝑐1 = −

III.

, 𝑐2 =

𝑝 (𝐿𝑑 −𝐿𝑞 )
𝐽

, 𝑐3 =

𝑝𝜙 𝑓
𝐽

𝑇
−𝑐1 Ω+ 𝑙 +𝛺 𝑟𝑒𝑓 +𝑘 Ω 𝑠𝑖𝑔𝑛𝑠 (Ω)
𝐽

(6)

𝑐2 𝐼𝑑 +𝑐3

The components 𝐼𝑑 and 𝐼𝑞 are independently controlled.
(7)

𝑆 𝐼𝑑 = 𝐼𝑑𝑟𝑒𝑓 − 𝐼𝑑 𝑆 𝐼𝑞 = 𝐼𝑞𝑟𝑒𝑓 − 𝐼𝑞

Where:

J
f
𝑉𝑑 𝑉𝑞
𝐼𝑠
Tl
P
𝜙𝑓
Ω

𝐽

With the speed gain 𝑘Ω &gt; 0

1
𝐿𝑑
0

0
(d, q)
𝑅𝑠
𝐿𝑑 , 𝐿𝑞

𝑓𝑟

0

0

(5)

− (𝑐2 𝐼𝑑 + 𝑐3 )𝐼𝑞

The associated control input is given by (6):
𝐼𝑞𝑟𝑒𝑓 =

1
𝐿𝑑

𝐽

With:

And

𝑇𝑙

Axes for direct and quadrate park subscripts.
Stator resistance.
subscripts
Inertia moment of the moving element
Viscous friction and iron-loss coefficient.
Stator voltage in direct and quadrate park subscripts
Stator Currents
Is number of pole pairs
flux.
Rotor speed.

With 𝐼𝑑𝑟𝑒𝑓 = 0
Frequently 𝐼𝑑𝑟𝑒𝑓 is made equal to zero, because its
contribution to the motor torque is almost insignificant.
Flux and torque control are independently made through the
surfaces 𝑆 𝐼𝑑 and 𝑆 𝐼𝑞 respectively.
The derivative of the surface 𝑆 𝐼𝑑 and 𝑆 𝐼𝑞 is given by the
expression:
𝑆 𝐼𝑑 = 𝐼𝑑𝑟𝑒𝑓 − 𝑎1 𝐼𝑑 − 𝑎2 𝐼𝑞 𝛺 +

SLIDING MODE CONTROL DESIGN

1
𝐿𝑑

(8)

𝑉𝑑

𝑆 𝐼𝑞 = 𝐼𝑞𝑟𝑒𝑓 − 𝑏1 𝐼𝑞 − 𝑏2 𝐼𝑑 𝛺 − 𝑏3 𝛺 +

1

𝑉
𝐿𝑞 𝑞

With:
The sliding mode control can be justified and designed
using the notion of Lyapunov stability. By solving the
equation 𝑆 = 0 , the equivalent control 𝑈𝑒𝑞 can be obtained.
The 𝑈𝑛 component satisfies 𝑆𝑆 &lt; 0 and is given by:
𝑈𝑛 = −𝑘𝑠𝑖𝑔𝑛𝑆

(2)

With: 𝑘 &gt; 0

𝑅𝑟
𝐿𝑑

(3)

Surfaces are chosen in order to determine the behavior of
the motor in the transient period. For the speed control, we
propose switching law which depends on the difference
between reference speed and real speed, presented in (4):
𝑆 𝛺 = 𝛺𝑟𝑒𝑓 − 𝛺
(4)

, 𝑎2 =

𝑝𝐿𝑞
𝐿𝑑

, 𝑏1 = −

𝑅𝑟
𝐿𝑞

, 𝑏2 = −

𝑝𝐿𝑑
𝐿𝑞

𝑝𝜙𝑓
𝑏3 = −
𝐿𝑞
The associated control inputs is given by (9):
𝑈𝑑𝑟𝑒𝑓 =

A. Selection of Switching Surfaces and Determination of the
Control Inputs
We use attractivity condition of switched surface 𝑆𝑆 &lt;
0.The vector of control laws can be expressed as:
𝑈 = 𝑈𝑒𝑞 + 𝑈𝑛

𝑎1 = −

𝑈𝑞𝑟𝑒𝑓 =

[𝐼𝑑𝑟𝑒𝑓 −𝑎 1 𝐼𝑑 −𝑎 2 𝐼𝑞 𝛺]+𝑘 d 𝑠𝑖𝑔𝑛𝑠 (𝐼𝑑 )
𝐿𝑑

(9)

[𝐼𝑞𝑟𝑒𝑓 −𝑏1 𝐼𝑞 −𝑏2 𝐼𝑑 𝛺+𝑏3 𝛺]+𝑘 q 𝑠𝑖𝑔𝑛𝑠 (𝐼𝑞 )
𝐿𝑞

Hense𝑘𝑑 , 𝑘𝑞 and 𝑘Ω are positives gains, given as
followed:
𝑘𝑑 = 3000,𝑘𝑞 = 4000, 𝑘Ω = 1
The necessity for high performance in PMSM systems
increases as the demand for precision controlsit is necessary
to estimate the rotor speed and the position. For this,
theLuenberger observer design is presented in the next
section.

With Ω𝑟𝑒𝑓 is the rotor speed reference.