时变延迟与死区输入的不确定MIMO系统自适应控制策略
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更新于2024-08-28
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"这篇研究论文探讨了一种针对具有时变延迟和死区输入的不确定多输入多输出（MIMO）动态系统的直接自适应控制方法。该方法旨在解决系统中的非线性参数化、时间延迟以及未知非线性问题，以实现系统的跟踪控制。"
在自动控制领域，MIMO系统因其在复杂工程应用中的广泛使用而备受关注。这些系统通常包含多个输入和多个输出，其行为可能受到多种因素的影响，如时变延迟和输入死区。时变延迟可能导致系统性能下降，甚至不稳定，而输入死区则会导致控制信号在一定范围内不敏感，对控制系统的设计提出挑战。
该论文提出的直接自适应控制器设计是一种克服这些挑战的有效策略。它利用自适应技术来估计系统的未知参数和非线性特性，同时考虑到时变延迟的存在。这种控制器能够实时调整控制输入，以补偿由于延迟和非线性效应引起的性能损失。
论文的关键点包括：
1. **自适应控制算法**：自适应控制是控制系统理论的一个分支，它允许控制器根据系统动态的实时变化进行调整。在本研究中，自适应算法被用来估计和补偿系统的不确定性，包括时变延迟和死区输入的影响。
2. **时变延迟处理**：时变延迟的处理是控制系统设计的关键部分。论文中，研究人员开发了一种策略，可以在线跟踪和补偿由于延迟引起的系统动态变化。
3. **死区输入处理**：对于存在输入死区的系统，控制器需要能够在死区边界附近有效地操作。论文中，作者提出了一种方法，即使在输入死区条件下，也能保证控制器的稳定性和性能。
4. **非线性参数化**：系统模型可能存在非线性参数，这增加了控制设计的复杂性。通过直接自适应控制，论文尝试估计并补偿这些非线性，以确保系统能够跟踪期望的参考轨迹。
5. **理论分析与实验验证**：论文不仅提出了控制策略，还进行了理论分析，证明了控制器的稳定性，并可能通过仿真或实际系统实验验证了其有效性。
这篇论文为不确定MIMO动态系统提供了一种创新的控制解决方案，特别是在面临时变延迟和输入死区的挑战时。这种方法对于提高系统的鲁棒性和性能具有重要意义，为实际工程应用提供了理论基础。
Z. Li et al. / Automatica 63 (2016) 287–291 289
3. Control design and analysis
Define the filtered tracking error s
i
(Slotine & Li, 1991)
s
i
= e
(n−1)
i
+ λ
i1
e
(n−2)
i
+ · · · + λ
i,n−1
e
i
, (15)
where e
i
= y
i
− y
di
, e
(n−1)
i
(i = 1, 2, . . . , m) is the (n −
1)th derivative of e
i
, λ
i1
, . . . , λ
i,n−1
are positive constants and are
appropriately chosen coefficient vectors such that e
i
→ 0 as s
i
→
0 (i.e. r
m−1
+ λ
i1
r
m−2
+ · · · + λ
i,n−1
is Hurwitz).
From (15), we have
˙
S = B
−1
d
(x)
F (x(t − τ (t))) + K
T
ΦV + ∆
P
+ ν, (16)
where S = [s
1
, . . . , s
i
, . . . , s
m
]
T
and ν = [ν
1
, . . . , ν
i
, . . . , ν
m
]
T
with
ν
i
= −y
(n)
di
+ λ
i1
e
(n−1)
i
+ · · · + λ
i,n−1
. (17)
We now construct a new high dimensional Lyapunov–
Krasovskii functional (see Eq. (31)). The first part of the Lyapunov–
Krasovskii functional is chosen as
V
1
= S
T
B
ϑ
S, (18)
where
B
ϑ
=
1
0
ϑB
α
dϑ = diag
1
0
ϑB
αii
(
¯
x
i
)dϑ
(19)
with B
α
= B
d
α = diag[b
dii
α
ii
]
m×m
and matrix α ∈ R
m×m
.
For easy analysis, we choose α
11
= · · · = α
mm
. By exchanging
x
ni
in x with ϑ s
i
+ ζ
i
(i = 1, 2, . . . , m), we define
¯
x
i
= [x
T
1
,
x
T
2
, . . . , x
T
n−1
, x
n1
, x
n2
, . . . , ϑs
i
+ ζ
i
, . . . , x
nm
]
T
∈ R
nm
where ζ
i
=
y
(n−1)
di
− ξ
i
with ξ
i
= λ
i1
e
(n−2)
i
+ · · · + λ
i,n−1
e
i
. ϑ is a scalar and
independent of
¯
x
i
. We can choose suitable B
d
(x) and α, such that
b
dii
α
ii
> 0.
Because B
ϑ
in (18) depends on time t, the time derivative of
V
1
includes the differentiation of matrix B
ϑ
with regard to time
t. To facilitate computation of its derivative, according to Gentle
(2007), we introduce a matrix operator for derivative operation
of matrix-value function with respect to time t, i.e., for a time-
dependent matrix A ∈ R
m×n
and a vector b(t) ∈ R
l
, a matrix
operator M
∂
(A, b) ∈ R
m×n
is defined with the entry of its ith row
and jth column being M
∂ij
(A, b) =
∂A
ij
∂b
T
˙
b with i = 1, 2, . . . , m and
j = 1, 2, . . . , n.
Differentiating (18) with respect to t gives
˙
V
1
= 2S
T
B
ϑ
˙
S + S
T
M
∂
(B
ϑ
, S)S
+ S
T
M
∂
(B
ϑ
, x)S + S
T
M
∂
(B
ϑ
, ζ )S, (20)
where ζ = [ζ
1
, ζ
2
, . . . , ζ
m
]
T
, M
∂
(B
ϑ
, S) ∈ R
m×m
, M
∂
(B
ϑ
, x) ∈
R
m×m
and M
∂
(B
ϑ
, ζ ) ∈ R
m×m
are given below
M
∂
(B
ϑ
, S) = diag
1
0
ϑ
∂B
αii
∂s
i
˙
s
i
dϑ
, (21)
M
∂
(B
ϑ
, x) = diag
1
0
ϑ
nm
j=1,j=i
∂B
αii
∂x
j
˙
x
j
dϑ
, (22)
M
∂
(B
ϑ
, ζ ) = diag
1
0
ϑ
∂B
αii
∂ζ
i
˙
ζ
i
dϑ
, i = 1, . . . , m. (23)
Let σ = ϑs
i
(i = 1, 2, . . . , m), we can obtain
∂B
αii
∂s
i
=
∂B
αii
∂σ
∂σ
∂s
i
= ϑ
∂B
αii
∂σ
, (24)
∂B
αii
∂ϑ
=
∂B
αii
∂σ
∂σ
∂ϑ
=
∂B
αii
∂σ
s
i
. (25)
Noting that ϑ is a scalar and independent of ζ
i
, and the fact
˙
ζ
i
= −ν
i
(i = 1, 2, . . . , m), we have
1
0
ϑ
∂B
αii
∂ζ
i
˙
ζ
i
s
i
dϑ = −ν
i
1
0
ϑ
∂B
αii
∂s
i
dϑ. (26)
It is easy to check that Ψ (z) is well-defined even if S approaches
zero. We design an adaptive control
V = Λ
sgn
u
1
+ u
2
, (27)
u
1
= −β
−1
0
(K
1
+
m
2
α)S
|·|
+
ˆ
W
T
|·|
Ψ
|·|
(z) + Υ
|·|
,
(28)
u
2
= −β
−1
0
S
∥S∥
ρ, (29)
ρ =
γ
1
+ γ
2
∥u
1
∥
1 − γ
2
∥β
−1
0
∥
, if S = 0,
0, if S = 0,
(30)
where Λ
sgn
= diag[sgn(s
i
)]; (∗)
|·|
denotes matrix or vector that
its every element is the absolute value of (∗)’s corresponding
element; ρ is positive when γ
2
< ∥β
−1
0
∥
−1
;
ˆ
W is the estimate
of W ; K
1
is a positive diagonal matrix; Υ will be defined later;
and β
0
= diag[β
10
, . . . , β
m0
] where β
i0
has been defined in
Assumption 2.1.
Consider the following Lyapunov function candidate as
V
2
= V
1
+ V
a
+
m
j=1
V
U
j
(t), (31)
V
a
=
m
i=1
1
2
˜
W
T
i
Ω
−1
i
˜
W
i
, (32)
where
˜
W
i
= W
i
−
ˆ
W
i
.
The adaption law is designed as
˙
ˆ
W
i
= Ω
i
Ψ
i
(z)α
ii
s
i
, (33)
where Ω
i
> 0(i = 1, 2, . . . , m) is a diagonal constant matrix to be
designed.
V
U
j
(t) is introduced to overcome unknown time-delays τ
1
(t),
τ
2
(t), . . . , τ
m
(t) and defined as
V
U
j
(t) =
1
2(1 − ¯τ
max
)
m
k=1
t
t−τ
k
(t)
ϱ
2
jk
(x
k
(τ ))dτ . (34)
From the definition of V
a
, we have
˙
V
a
=
m
i=1
˜
W
T
i
Ω
−1
i
˙
˜
W
i
. (35)
The time derivative of V
U
j
(t) is
˙
V
U
j
(t) =
1
2(1 − ¯τ
max
)
m
k=1
ϱ
2
jk
(x
k
(t))
− ϱ
2
jk
(x
k
(t − τ
k
(t)))(1 − ˙τ
k
(t))
. (36)
Thus, the time derivative of V
2
is
˙
V
2
=
˙
V
1
+
˙
V
a
+
m
j=1
˙
V
U
j
(t)
≤ S
T
α(−K
1
S − Υ ) +
1
2(1 − ¯τ
max
)
m
j=1
m
k=1
ϱ
2
jk
(x
k
(t)), (37)
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