非线性扰动系统自适应扩展状态观测器设计
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"一类非线性扰动系统的自适应扩展状态观测器"
这篇研究论文深入探讨了非线性扰动系统中的自适应扩展状态观测器（AESOs）设计。该研究提出了一个新颖的线性时变形式的AESO，旨在将传统线性ESO（LESO）的理论完备性和非线性ESO（NESO）的良好实际性能相结合，以扩大ESO在非线性扰动系统中的应用范围。
在论文中，作者首先将AESO的误差动力学转换为规范的（相位变量）形式。这种转换是通过差分代数谱理论来实现的，它允许对规范系统的时间-varying PD特征值进行分配。通过这种方式，可以调整观察器的增益以适应系统的动态变化。
在存在未知扰动的情况下，论文提供了关于AESO稳定性及其估计误差界限的定理。这些定理不仅确保了观测器的稳定运行，还为PD特征值的分配提供了重要的指导原则。通过这种方法，可以有效地估计系统状态并抑制由非线性和扰动引起的不确定性。
为了证明AESO的有效性，论文中可能包含了数值模拟或实际系统的案例分析。这些示例可能展示了在不同条件和扰动下，AESO如何准确地跟踪系统状态并提供可靠的估计，从而增强系统的控制性能和鲁棒性。
此外，论文可能还讨论了与其他现有观测器方法的比较，突出了AESO在处理非线性扰动问题上的优势。这可能会包括对不同观测器性能的评估，如收敛速度、估计精度和对系统模型不完全知识的容忍度。
这篇研究为非线性系统的状态估计提供了一个强大的工具，其自适应特性使得AESO能够在不断变化的环境中适应并有效地估计系统状态，即使面临未知扰动。这为工业电子和其他领域中非线性系统的控制策略设计提供了新的思路。
5860 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 62, NO. 9, SEPTEMBER 2015
function forms of g
i
, while the observer gain l
i
is set as
constant. The novelty of AESO lies in that g
i
is still chosen as
g
i
(e)=e, kept in a simple linear form as that in LESO, while
the observer gain is set as time varying, that is, l
i
= l
i
(t),to
obtain more design flexibilities. In this seminal idea, AESO is
designed in the form of
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙
ˆx
1
=ˆx
2
− l
1
(t)(ˆx
1
− x
1
)
.
.
.
˙
ˆx
n−1
=ˆx
n
− l
n−1
(t)(ˆx
1
− x
1
)
˙
ˆx
n
=ˆx
n+1
+ b
0
u − l
n
(t)(ˆx
1
− x
1
)
˙
ˆx
n+1
= −l
n+1
(t)(ˆx
1
− x
1
).
(5)
Here, we first transform the estimate error dynamics of
AESO (5) into a canonical form that is convenient for theoret-
ical analysis. Then, we give a brief introduction of DAST and,
based on that, develop an adaptive method to tune the time-
varying observer gain l
i
(t). Finally, we conduct a comprehen-
sive analysis of the stability and estimate error bounds of the
proposed AESO (5).
A. Transformation of AESO Error Dynamics
In view of (2) and (5), AESO estimate error dynamics can be
obtained as
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
˙e
1
= e
2
− l
1
(t)e
1
.
.
.
˙e
n−1
= e
n
− l
n−1
(t)e
1
˙e
n
= e
n+1
− l
n
(t)e
1
˙e
n+1
= −l
n+1
(t)e
1
− h(x, w)
(6)
or written in a vector differential equation form as
˙e = A(t)e + b (−h(x, w)) (7)
where
A(t)=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
−l
1
(t)10··· 0
−l
2
(t)01··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
−l
n
(t)0··· 01
−l
n+1
(t)0··· 00
⎤
⎥
⎥
⎥
⎥
⎥
⎦
,b=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0
0
.
.
.
0
1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
and e =[e
1
,...,e
n+1
]
T
, e
i
=ˆx
i
− x
i
denoting the estimate
erroroftheith state, i =1, 2,...,n+1. Obviously, (7) is an
LTV system with an unknown but bounded input −h(x, w).In
fact, (7) is a canonical form of LTV systems that are uniformly
controllable [29]. Next, we need to design proper gain l
i
(t) to
guarantee the stability of (6) and (7) and, additionally, to obtain
good estimate performance. The starting point is to transform
(7) into an equivalent canonical (phase-variable) form [30]
given below, and the AESO properties will be discussed in the
new canonical form. Thus
˙z = A
c
(t)z + b
c
(−h(x, w)) (8)
where
A
c
(t)=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
010··· 0
001··· 0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
00··· 01
−a
1
(t) −a
2
(t) ··· −a
n
(t) −a
n+1
(t)
⎤
⎥
⎥
⎥
⎥
⎥
⎦
b
c
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
0
0
.
.
.
0
1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
.
Here, the element a
i
(t) is assumed to be smooth (it owns con-
tinuous derivatives at any order, if needed in the later derivation)
and bounded. Clearly, the canonical form (8) is a realization of
the scalar linear differential system, i.e.,
ξ
(n+1)
+ a
n+1
(t)ξ
(n)
+ ···+a
2
(t)
˙
ξ+a
1
(t)ξ = b
c
(−h) (9)
where the elements a
i
(t) in A
c
(t) are directly related to the
coefficients of (9). These representations (8) and (9) are central
to investigating LTV systems, which will be addressed later.
Here, we first discuss how to transform (7) into (8).
The controllability matrix of LTV system (7) is defined as
M =[p
1
p
2
··· p
n+1
] (10)
where M and p
i
, i =1,...,n+1, are, respectively, a square
matrix and a column vector of dimension n +1, and
p
k+1
= −A(t)p
k
+
d
dt
p
k
,p
1
= b, k =1,...,n. (11)
The controllability matrix M
c
of LTV system (8) is similarly
defined. With the definitions (10) and (11), it can be verified by
direct construction that
M =
⎡
⎢
⎢
⎢
⎢
⎢
⎣
00··· 0(−1)
n
00··· (−1)
n−1
0
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0(−1)
1
··· 00
(−1)
0
0 ··· 00
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(12)
M
c
=
⎡
⎢
⎢
⎢
⎢
⎢
⎣
00··· 0(−1)
n
00··· (−1)
n−1
q
n,n
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
0(−1)
1
··· q
n−1,2
q
n,2
(−1)
0
q
1,1
··· q
n−1,1
q
n,1
⎤
⎥
⎥
⎥
⎥
⎥
⎦
(13)
where q
i,k
is defined as in the equation
q
i,k
=
⎧
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎩
−q
i−1,k−1
+˙q
i−1,k
, 1 <k<i≤ n
(−1)
i+1
a
n−i+2
+
i−2
j=0
a
n−j+1
q
i−1,j+1
+˙q
i−1,1
,k=1<i≤ n
(−1)
i+1
a
n+1
, 1 ≤ k = i ≤ n.
It is seen that the controllability matrix M of the LTV system
(7) is constant, whereas the controllability matrix M
c
of the
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