时间迭代变化参数下MIMO非线性系统中的自适应迭代学习控制
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本文探讨了多输入多输出（MIMO）离散时间非线性系统在存在迭代时间变化参数情况下的自适应迭代学习控制（Adaptive Iterative Learning Control, AILC）问题。研究关注的焦点在于处理那些在迭代域内初始状态和参考轨迹可能随机变化的系统。系统受到未知的时间迭代变化参数的影响，这些参数由一个已知的高阶内部模型（High-Order Internal Model, HOIM）生成，HOIM被设计成连接两个连续迭代周期之间的多项式表达。
HOIM的引入对于控制器的设计至关重要，它能够确保学习过程的收敛性。作者通过严格的Lyapunov理论分析，证明了在这样的条件下，AILC方法能够在保证系统稳定性的前提下，有效地引导系统逼近期望的行为。Lyapunov稳定性理论是一种常用的数学工具，用于分析系统的稳定性，尤其是在控制理论中，它能够帮助评估系统的长期行为和稳定性特性。
该研究论文发表于《非线性动力学》(Nonlinear Dynamics)杂志，2017年第89卷，第2561-2571页，DOI:10.1007/s11071-017-3604-0，是一篇原创研究。研究者们提出了一个创新的框架，即通过结合高阶内部模型来设计适应性强的控制器，这在实际应用中具有显著的优势，因为许多工业系统中可能存在的参数不确定性或者环境变化都可能导致传统控制方法失效，而AILC在这种情况下展现出其独特的适应性和学习能力。
为了验证这一理论，文中给出了一个实例，展示了自适应迭代学习控制方法在处理具有迭代时间变化参数的MIMO非线性系统时的有效性和实用性。这个例子不仅提供了理论分析的直观证据，也为实际工程中的此类系统提供了实用的控制策略参考。这篇论文对迭代学习控制在复杂动态环境中的应用进行了深入探索，并为未来的研究和工程实践开辟了新的可能性。
On iterative learning control for MIMO nonlinear systems 2563
AILC approach to an illustrative example in two cases.
Finally, conclusions are given in Sect. 6.
2 Problem formulation
In this paper, we consider a class of MIMO discrete-
time nonlinear systems as follows
x
k
(
t + 1
)
=
p
i=1
θ
(i)
k
(
t
)
ξ
(i)
(
x
k
(
t
)
, t
)
+ B
(
t
)
u
k
(
t
)
+ D
(
t
)
,
(1)
where t ∈
{
0, 1,...,T
}
is the discrete-time index and
k = 1, 2,... denotes the kth iteration. x
k
(
t
)
∈ R
n
is
the measurable state vector with random initial value
x
k
(
0
)
. u
k
(
t
)
∈ R
n
is the system input. θ
(i)
k
(
t
)
∈ R, i =
1, 2,..., p, are the unknown time-iteration-varying
parameters. p is an appropriate integer specifying the
dimension. ξ
(i)
(
x
k
(
t
)
, t
)
∈ R
n
, i = 1, 2,..., p,are
the known vector-valued functions. B
(
t
)
∈ R
n×n
is
the known gain matrix of the system input. D
(
t
)
∈ R
n
is an unknown external disturbance.
The reference trajectories can vary from iteration to
iteration randomly. The reference trajectory in the kth
iteration is denoted as x
d,k
(
t
)
∈ R
n
.
Remark 1 The prior information of control direction is
that B
(
t
)
is known for all t ∈
{
0, 1,...,T
}
. Different
from unknown control gains [37,38], this condition can
be satisfied in real application such as one-link robotic
manipulator [30]. Furthermore, according to the discus-
sions in [39], the permanent magnet synchronous motor
in continuous-time form can be expressed in the model
(1) with the definition of B
(
t
)
∈ R
n×n
. That means
that the MIMO system model (1) can be achieved in
the practical example.
For the system (1), we have the following assump-
tions.
Assumption 1 The input gain matrix B
(
t
)
is left
invertible.
Assumption 2 The nonlinear function ξ
(i)
(
x
k
(
t
)
, t
)
satisfies the linear growth condition, that is, ∀t ∈
{
0, 1,...,T
}
, ∀k ∈ Z
+
and ∀i ∈
{
1, 2,..., p
}
,we
have
ξ
(i)
(
x
k
(
t
)
, t
)
≤ a
1
+ a
2
x
k
(
t
)
, (2)
where 0 < a
1
< ∞ and 0 < a
2
< ∞ are some positive
constants.
Assumption 3 B
(
t
)
, x
k
(
0
)
, D
(
t
)
and x
d,k
(
t
)
are uni-
formly bounded for t ∈
{
0, 1,...,T
}
and k ∈ Z
+
.
Assumption 4 θ
(i)
k
(
t
)
is time-iteration-varying un-
known parameter generated by a m
i
-order internal
model as
θ
(i)
k
(
t
)
= h
(i)
1
θ
(i)
k−1
(
t
)
+ h
(i)
2
θ
(i)
k−2
(
t
)
+···+h
(i)
m
i
θ
(i)
k−m
i
(
t
)
(3)
where h
(i)
l
, l = 1, 2,...,m
i
, are known constant coef-
ficients of a neutrally stable HOIM, which could be
described by a polynomial
H
(i)
ω
−1
= h
(i)
1
+ h
(i)
2
ω
−1
+···+h
(i)
m
i
ω
−m
i
+1
, (4)
where ω is a shift operator in the iteration domain that is
defined as ω
−1
g
k
(t) = g
k−1
(t). Thus the difference of
θ
(i)
k
(
t
)
, i = 1, 2,..., p, between two consecutive iter-
ations is expressed as θ
(i)
k
(
t
)
= H
(i)
ω
−1
θ
(i)
k−1
(
t
)
.
Define δ
(i)
k
(
t
)
=
θ
(i)
k+1−m
i
(
t
)
··· θ
(i)
k−1
(
t
)
θ
(i)
k
(
t
)
T
,
from [25], we have
δ
(i)
k
(
t
)
= F
(i)
δ
(i)
k−1
(
t
)
=··· =
F
(i)
k
δ
(i)
0
(
t
)
(5)
where F
(i)
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
01 0... 00
00 1... 00
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
00 0... 01
h
(i)
m
i
h
(i)
m
i
−1
h
(i)
m
i
−2
... h
(i)
2
h
(i)
1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,
δ
(i)
0
(
t
)
=
θ
(i)
−m
i
+1
(
t
)
··· θ
(i)
0
(
t
)
T
is the time-
varying vector consisting of the unknown basis func-
tions of HOIM. The unknown basis functions are lin-
early independent. Denote the last row of matrix
F
(i)
k
as f
(i)
k
=
f
(i)
1,k
f
(i)
2,k
··· f
(i)
m
i
,k
. According to (5), the
following relationship can be obtained
θ
(i)
k
(
t
)
= f
(i)
1,k
θ
(i)
1−m
i
(
t
)
+···+ f
(i)
m
i
−1,k
θ
(i)
−1
(
t
)
+ f
(i)
m
i
,k
θ
(i)
0
(
t
)
.
(6)
Rewrite the above equation as
θ
(i)
k
(
t
)
= f
(i)
k
δ
(i)
0
(
t
)
(7)
123
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