Adaptive Uncerntainty Identification with Neural Network
ZHANG Yumin, TENG Fei, LIANG Guiqin and WANG Zhiqiang
School of Instrumentation and Opto-Electronics Engineering, Beihang University, 100191 Beijing,P. R. China
E-mail: zhyminus@yahoo.com.cn, tengfei@aspe.buaa.edu.cn, 274794440@qq.com
Abstract:
This paper provides an identification method for uncertainties in system via dynamic neural networks, where the uncertainties
include parameter uncertainty, disturbances, faults or system load. The incertainties here are translated into the weight matrices
to be identified. To idenfication purpose, a dynamic neural network observer is designed, where weight matrices are adaptive
tuned. The numerical simulation shows that the given idenificatuion algorithm is more suitable for disturbances, faults or system
load. For given system load, the present algorithm can model system into multimodel mode.
Key Words: Adaptive Learning, Observer, Neural Network
1 Introduction
It is well known that the modelling problem concerns
uncertainties such as constructure unceratinty [1-3, 9], ex-
ogenous disturbances [16-19],the nonlinearities [4-6, 9-11],
unmolled dynamics and system noises [4, 6], which may
result in model uncertainty. The usual solution is ˙x(t)=
f (x(t),u(t),Δ(t)), where Δ(t) represents the system uncer-
tainty. The interval system is a type of linear system with
parameters drift, which resulted robust stability problem and
the H
∞
optimization is often concerned [2]. The multi-
model system is employed for system with multi-work-
mode, which results in mode switching [3]. Inpractice, ex-
ogenous disturbances [2, 11], faults or unknown system load
[16-19] have been more reguardede, where observers (filters)
such as Lunberger-type observer [1, 18, 19], disturbance ob-
server [2], neural network (NN) observer [7-9, 13-15], are
empolyed for online or real-time identification purpose.
As a class of well known identification tools, dynamic
neural network is effective for nonlinear system dynam-
ics identification and uncertainties [15].It is noted that a
neural network observer well designed can deal with rapid
time-varying disturbances, which is more efficient to the
Lunberger-type observer and the disturbance observer.
In this contribution, the uncertainty modeling problem
will be further investigated via the dynamic NN. Under given
tracking control rule, the parameter uncertainty and exoge-
nous disturbance will be estimated through illustrative sim-
ulation example. Certainly, the Lipsihitz condition is em-
ployed for NN basis function. As a result, an adaptive filter
with dynamic NN will be established for identification pur-
pose, where adaptive learning rules for weight matrices are
given.
In this contribution, the real symetrix matrix P > 0(P ≥ 0)
represents P is a definite positive (semi-definite positive). I
and 0 represent identity matrix and zero matrix, respectively.
For any matrix M
nxn
, sym(M)=M +M
T
.
This work was supported by the National Natural Science Foundation
of China under Grants 61374131 and 61333005.
2 Problem Formulation and Preliminaries
Assume that a dynamic system ˙x(t)= f (x(t),u(t)) is
firstly modelled in the following
˙x(t)=Ax(t)+Bu(t)+Δ
y(t)=Cx(t)
(1)
where x(t) ∈ R
m
is the state, y(t) is the output and u(t) is
the input with bound u
T
(t)u(t) ≤ ¯u. A, B and C are coef-
ficient matrices with compatible dimensions. A is a stable
matrix,(A,B) is controllable, (C,A) is observable. Δ is mod-
elling error including some nonlinearities, uncertainties or
disturbances. For more precision control purpose, system
(1) can be further modelled with neural networks
⎧
⎨
⎩
˙x(t)=Ax(t)+Bu(t)+B
1
W
∗
1
σ(x(t))
+BW
∗
2
φ(x(t))u(t)
y(t)=Cx(t)
(2)
where B
1
is a coefficient matix, W
∗
1
and W
∗
2
are unknown
weight matrices to be designed of the given neural network,
σ(·) and φ (·) are assumed to be basis functions with the el-
ements increasing monotonically. σ(·) is a vector function
and φ (·) is a diagnal matrix function. Typical presentation
of the elements σ
i
(x
i
) and φ(·) are sigmoid vector functions,
such as, σ
i
(x
i
)=φ
i
(x
i
)=aarctan(bx
i
) − c, where a, b and c
are real numbers.
It is known that the neural network basis function σ(·)
and φ(·) are Lipschitz functions, that is
Assumption 1: The elements of vector functions σ(·)
and φ(·) are ascent and there exist known positive-definite
matrices E
σ
and E
φ
such that
(σ(x
1
) − σ (x
2
))
T
(σ(x
1
) − σ (x
2
)) ≤ (x
1
− x
2
)
T
E
σ
(x
1
− x
2
)
(φ(x
1
) − φ (x
2
))
T
(φ(x
1
) − φ (x
2
)) ≤ (x
1
− x
2
)
T
E
φ
(x
1
− x
2
)
(3)
for x
1
= x
2
.
For modelling purpose, the following observer is de-
signed for system (2)
⎧
⎪
⎪
⎨
⎪
⎪
⎩
˙
ˆx(t)= A ˆx(t)+Bu + B
1
ˆ
W
1
σ( ˆx(t))
+B
ˆ
W
2
φ( ˆx(t))u(t)+u
f
(t)+Lr(t)
r(t)= y(t) − ˆy(t)
ˆy(t)= C ˆx(t)
(4)
Proceedings of the 34th Chinese Control Conference
Jul
28-30, 2015, Han
zhou, China
2055