鲁棒滑模预测控制：PMSM驱动器的电流控制与扰动补偿
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本文针对电动驱动系统中永久磁同步电机(PMSM)的电流控制问题，着重探讨了在模型不确定性以及外界干扰存在的情况下，如何实现快速响应和强健性的控制策略。研究者Xudong Liu、Chenghui Zhang、Ke Li 和 Qi Zhang 来自山东大学控制科学与工程学院，他们在2016年5月13日首次提交了这篇论文，并在2017年7月2日进行了修订，最终于8月27日接受，于同年9月7日在线发表。
论文的核心贡献是提出了一种结合广义预测控制（Generalized Predictive Control, GPC）和滑模扰动补偿方法的鲁棒电流控制策略。GPC理论通常基于连续时间模型，该方法在设计时并未考虑扰动因素，这使得其在数字控制器中实现起来较为简便。然而，实际应用中，电机驱动系统经常面临如负载变化、电机参数漂移等不确定性，这些都可能导致控制系统性能下降。
为了克服这些问题，论文提出了一种改进的GPC，它能够在预测控制过程中考虑到这些扰动。通过引入滑模控制技术，能够有效地抑制扰动对系统的影响，提高控制系统的鲁棒性。滑模控制利用其固有的“无穷维”结构，能够实现对快速变化的动态环境有很好的适应性，确保在存在外部扰动时仍能保持系统稳定和高精度的电流控制。
在实施这一方法时，首先对电机模型进行建模，然后利用GPC的优化算法预测未来状态，制定出最优电流指令。在执行过程中，通过实时监测并补偿由扰动引起的误差，保持电流跟踪目标值。这种方法不仅提升了控制系统的响应速度，也显著增强了其抵抗外部干扰的能力，从而满足了电动驱动系统对于高性能和稳定性的要求。
这篇研究论文提供了一个实用且具有理论基础的解决方案，对于提升PMSM在电动驱动中的控制性能，尤其是在复杂工况下的应用具有重要意义。未来的研究可能继续探索如何进一步优化滑模补偿算法，以减少计算负担，或者将此方法推广到更广泛的电机类型中。
disturbance and adapting parameter variations and model un-
certainties. Thus a robust and adaptive current controller based on
generalized predictive control with sliding mode disturbance
compensation is proposed in this paper.
3.1. The design of predictive current controller
Inspired by the generalized predictive control theory in [27],
the design of current controller for ideal system is presented as
follows.
According to (4), neglecting the disturbance part, the nominal
model can be expressed as
=()+()
=()=()
()
xfxgxu
yhxgxx
,
,
5
1
1
2
where
=[ ]uuu
dq
T
111
.
Let
*
i
d
,
*
i
q
be the reference current, and
=[
**
]yii
r
dq
T
. First of all,
the cost function is defined as
∫
τττττ
=(
^
(+)−
^
(+))(
^
(+)−
^
(+))
()
J ytytytytd
1
2
,
6
o
T
r
T
r
where T is the predictive horizon,
τ
^
(+
)
yt
and
τ
^
(+
)
yt
r
represent
the predictive current and the predictive reference current,
respectively.
According to the nominal PMSM system (5), each of the output
current has the same relative degree
ρ
= 1
. The total relative de-
gree of the system is equal to the systems order. Consequently,
there is no zero dynamics. The output current is measurable. In
addition, the current is taken as the smooth variable, so it satisfies
continuously differentiable with respect to the time. Therefore,
four assumptions given in [27] can be satisfied. In order to alleviate
the computation burden, the control order
=r
0
is chosen, thus
the control input
τ(+
)
ut
in the moving time frame is constant.
Repeated differentiation up to
ρ(+
)
r
th of the output
^
(
)
yt
with
respect to time
^
()= ()
^
()= ()+ ()
()
yt Lhx
yt Lhx Lhxu
,
,
7
f
fg
0
11
1
1
where
()= ()=[ ]Lhx hx i i
fdq
T0
,
()= ()= (
)
∂( ( ))
∂
Lhx f x f x
f
hx
x
1
,
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
()=
∂( ( ))
∂
()=Lhx
hx
x
gx
0
0
.
g
L
L
1
1
1
1
d
q
1
Because
ρ
= 1
and
=r
0
, at least
ρ
+=r 1
terms should be
maintained in the truncated Taylor's expansion of
τ
^
(+
)
yt
, i.e.
ττ
^
(+)=
^
()+
^
() =
()
yt yt y tm,1,2.
8
mmm
where m is defined as an element index of multidimensional
output
^
(
)
yt
, with
^
()=[
^
()
^
()] =[ ]yt y t y t i i
T
dq
T
12
.
Then
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
τ
τ
τ
^
(+)=
^
()
^
()
()
yt
yt
yt
10 0
010
.
9
Let
⎡
⎣
⎢
⎤
⎦
⎥
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
Γτ
τ
τ
()=
¯
()=
^
()
^
()
()
Yt
yt
yt
10 0
010
,.
10
It follows that
τΓτ
^
(+)= ()
¯
()
()
yt Yt.
11
Similarly, the Taylor's expansion of the reference output current
is
τΓτ
^
(+)= ()
¯
()
()
yt Yt,
12
r
r
where
¯
()=[ ()
()]
Y
tytyt
r
r
T
r
TT
and
(
)
yt
r
is the derivative of the re-
ference current.
Let
∫
ΓΓτΓττ
¯
()= ()()Td
T
T
0
, and according to the Appendix A in
[27], the
(
)
ij,
th element in the matrix
Γ
¯
()T
can be represented as
⎡
⎣
⎢
⎤
⎦
⎥
Γ
¯
() =
( − )!( − )!( + − )
=
()
+−
T
ijij
T
T
ij
1
11 1
0
0
, , 1, 2.
13
ij
ij
,
1
Then, allow for the truncation of
τ
^
(+
)
yt
and
τ
^
(+
)
yt
r
, the cost
function is approximatively equal to
Γ
=[
¯
()−
¯
()]
¯
()[
¯
()−
¯
()]
()
JYtYtTYtYt
1
2
.
14
r
T
r
To minimize the current tracking error in finite horizon, the
generalized predictive control law [18] for PMSM is derived as
=− ()( + ()−
)
()
ρ
−
uGxKMLhxy,
15
f
r
1
11
where
⎡
⎣
⎤
⎦
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎡
⎣
⎤
⎦
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
()=
() ()
==
()− ()
=
−
*
−
*
ρ
Gx
LhxLhx
M
hx y t
ii
ii
0
0
,,
gg
L
L
r
dd
qq
11
1
12
1
1
1
d
q
g
g,
11 12
represent the column vector of
(
)g
x
1
.
∈
×
KR
2
2
is the first
2 rows of the matrix
Γ
¯
−
rr
1
Γ
¯
ρ
r
T
, and
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
Γ
ΓΓ
ΓΓ
¯
()=
¯¯
¯¯
ρρ ρ
ρ
T ,
r
r
T
rr
where
Γ
¯
∈
ρρ
×
R
22
,
Γ
¯
∈
ρ
×
R
r
2
2
,
Γ
¯
∈
×
R
rr
22
. Then, K can be get as
⎡
⎣
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
=K
0
0
.
T
T
3
2
3
2
The control laws
u
d1
and
u
q1
calculated in (15) are the reference
voltage in the PMSM control system based on GPC. It's worth
noting that the controller in (15) can solve the GPC control pro-
blem in general without any additional supporting means, and the
influence triggered by model uncertainties in the PMSM drive
system is disregarded. The current regulator performance may be
affected significantly by the strong disturbance. For this reason, it's
necessary to consider the disturbance to strength the robustness
of the current controller.
3.2. The design of sliding mode disturbance compensation controller
To improve the robustness of the current controller and guar-
antee the fast current response, a sliding mode disturbance com-
pensation [32,33] combining the generalized predictive controller
is studied. The objective is to find a control voltage u such that the
trajectories of the real motor system (4) coincide with the nominal
system (5). A kind of method is to add another control
=[ ]uuu
dq
T
222
to the control
u
1
of the nominal system, i.e.
=+uu u
12
. The structure of the proposed current controller is
shown in Fig. 1. The sliding mode control
u
2
will be used to
eliminate the disturbance d, and the predictive control
u
1
provides
the preliminaries for the design of sliding mode controller.
u
2
is a
discontinuous control. According to (4) and (5) yields
() =−
()
gxu d,
16
eq
1
2
where
u
eq2
is the equivalent control of
u
2
.
X. Liu et al. / ISA Transactions 71 (2017) 542–552544
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