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As a result:
V
=
S
T
(
2
sgn(
S
)
2
S
+
k
1
˙
−
λ
−
γ
δ
)
(17)
Owing to:
λ
21
>
k
1
δ
and
λ
22
>
0,
γ
2
>
0 in the Theorem 1, so the inequation can be
obtained as follows:
−
λ
22
S
−
γ
2
S
<
0
2
V
<
S
T
(
−
λ
22
S
−
γ
2
sgn(
S
)) =
As the above formulas, the system states can reach the integral manifold
S
= 0 in finite
time. Moreover, the system is moved around the sliding manifold, namely
S
=
S
= 0, ,
the equation (5) can be obtained as:
k
1
e
ω
(
t
)+
k
2
e
ω
dt
= 0
(18)
According to the principles of ordinary differential equations, the root of the equation
(18) can be obtained as follows:
e
ω
= exp(
−
k
i
t
)+
ζ
(19)
where
>
0,
k
i
=
k
1
/
k
2
is the positive constant.
From the equation (19), it is known that the tracking error
e
ω
can converge to zero
exponentially if the constant coefficient
k
1
,
k
2
is selected properly and equation (18)
is strictly Hurwitz, which is a polynomial whose roots lie strictly in open left half of
the complex plane, namely lim
x
ζ
e
ω
= 0. Therefore, the tracking error system (4) can
coverage to zero and is globally stable.
In order to avoid the windup phenomenon, the proposed algorithm is designed as
follow (20) according to the anti-reset windup controller theory.
→
∞
(
1
k
1
α
2
sign
(
S
)+(
2
S
)
k
c
ω
(
i
∗
q
−
i
q
))
dt
−
i
sw
=
λ
1
sign
(
S
)+
λ
γ
1
S
+
γ
(20)
where
k
c
ω
is compensate constant and
i
q
is the input of the limited amplitude.
4
Simulation and Experiment
In order to validate the feasibility and effectiveness of the proposed method, computer
simulations are conducted. The computer simulation is mainly used to verify the per-
formance of the novel second order sliding mode controller for PMSM. The simulation
environment is the Matlab/Simulink. The Simulink model of the PMSM vector control
system, which includes the proposed second order sliding mode controller, has been
constructed. The parameters of PMSM are shown as the following table.
The results of the proposed sliding mode observer simulations are shown in Fig.2,
Fig.3 and Fig.4. The parameters of the novel second order sliding mode controller are
designed as follows:
γ
2
= 30000 and
k
1
= 1,
k
2
= 14.
Fig.2 shows the simulation experimental results of PMSM start response by two
control algorithms. From the results, the system response is faster by the SOSMC than
by PI control.
λ
1
= 50,
λ
2
= 1200,
γ
1
= 15000,
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