10
15
20
25
30
35
40
45
50
k
0.3
0.25
0.2
| e 0.15
k|
0.1
0.05
00
5
10
15
20
25
30
35
40
45
50
k
Fig. 9. Time behavior of yk and | ek| in blue solid lines and model response (top) and estimated
H∞ guaranteed cost (bottom) in red dashed lines, for vertex 1 and delays from 2 to 13.
0.2
0
−0.2
uk
−0.4
−0.6
−0.80
5
10
15
20
25
30
35
40
45
50
k
Fig. 10. Control signals used in time simulations presented in Figure 9.
as decentralized control, switched systems, actuator failure, output feedback and following
model conditions.
It has been shown that the proposed convex conditions can be systematically obtained by
i) defining a suitable positive definite parameter dependent Lyapunov-Krasovskii function;
ii) calculating an over bound for Δ V( k) < 0 and iii) applying Finsler’s Lemma to get a set
of LMIs, formulated in a enlarged space, where cross products between the matrices of the
system and the matrices of the Lyapunov-Krasovskii function are avoided. In case of robust
design conditions, they are obtained from the respective analysis conditions by congruence
transformation and, in the H∞ guaranteed cost design, by replacing some matrix blocs by
their over bounds. Numerical examples are given to demonstrated some relevant aspects of
the proposed conditions.
324
Discrete Time Systems
0.7
0.6
0.5
loop
0.4
en
op0.3
) )
( z ( z 0.2
E
W
0.1
00
0.5
1
1.5
2
2.5
3
3.5
ω[rad/s]
0.7
0.6
0.5
-loopd0.4
0.3
close
) )( z 0.2
( zE W 0.1
00
0.5
1
1.5
2
2.5
3
3.5
ω[rad/s]
Fig. 11. Gain frequency response between signals ek and wk for the open loop (top) and
closed-loop (bottom) cases for delays from 2 to 13 and a sweep on α ∈ [0, 1].
The approach used in this proposal can be used to deal with more complete
Lyapunov-Krasovskii functions, yielding less conservative conditions for both robust stability
analysis and design, including closed-loop performance specifications as presented in this
chapter.
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18
Stability Analysis of Grey Discrete Time
Time-Delay Systems: A Sufficient Condition
Wen-Jye Shyr1 and Chao-Hsing Hsu2
1Department of Industrial Education and Technology,
National Changhua University of Education
2Department of Computer and Communication Engineering
Chienkuo Technology University
Changhua 500, Taiwan,
R.O.C.
1. Introduction
Uncertainties in a control system may be the results modeling errors, measurement errors,
parameter variations and a linearization approximation. Most physical dynamical systems
and industrial process can be described as discrete time uncertain subsystems. Similarly, the
unavoidable computation delay may cause a delay time, which can be considered as time-
delay in the input part of the original systems. The stability of systems with parameter
perturbations must be investigated. The problem of robust stability analysis of a nominally
stable system subject to perturbations has attracted wide attention (Mori and Kokame, 1989).
Stability analysis attempts to decide whether a system that is pushed slightly from a steady-
state will return to that steady state. The robust stability of linear continuous time-delay
system has been examined (Su and Hwang, 1992; Liu, 2001). The stability analysis of an
interval system is very valuable for the robustness analysis of nominally stable system
subject to model perturbations. Therefore, there has been considerable interest in the
stability analysis of interval systems (Jiang, 1987; Chou and Chen, 1990; Chen, 1992).
Time-delay is often encountered in various engineering systems, such as the turboject
engine, microwave oscillator, nuclear reactor, rolling mill, chemical process, manual control,
and long transmission lines in pneumatic and hydraulic systems. It is frequently a source of
the generation of oscillation and a source of instability in many control systems. Hence,
stability testing for time-delay has received considerable attention (Mori, et al., 1982; Su, et
al., 1988; Hmamed, 1991). The time-delay system has been investigated (Mahmoud, et al.,
2007; Hassan and Boukas, 2007).
Grey system theory was initiated in the beginning of 1980s (Deng, 1982). Since then the
research on theory development and applications is progressing. The state-of-the-art
development of grey system theory and its application is addressed (Wevers, 2007). It aims
to highlight and analysis the perspective both of grey system theory and of the grey system
methods. Grey control problems for the discrete time are also discussed (Zhou and Deng,
1986; Liu and Shyr, 2005). A sufficient condition for the stability of grey discrete time
systems with time-delay is proposed in this article. The proposed stability criteria are simple
328
Discrete Time Systems
to be checked numerically and generalize the systems with uncertainties for the stability of
grey discrete time systems with time-delay. Examples are given to compare the proposed
method with reported (Zhou and Deng, 1989; Liu, 2001) in Section 4.
The structure of this paper is as follows. In the next section, a problem formulation of grey
discrete time system is briefly reviewed. In Section 3, the robust stability for grey discrete
time systems with time-delay is derived based on the results given in Section 2. Three
examples are given to illustrate the application of result in Section 4. Finally, Section 5 offers
some conclusions.
2. Problem formulation
Considering the stability problem of a grey discrete time system is described using the
following equation
x( k + 1) = (
A ⊗) x( k) (1)
where ( )
n
x k ∈ R represents the state, and (
A ⊗) represents the state matrix of system (1).
The stability of the system when the elements of (
A ⊗) are not known exactly is of major
interest. The uncertainty can arise from perturbations in the system parameters because of
changes in operating conditions, aging or maintenance-induced errors.
Let ⊗
=
⊗
ij ( i, j
1,2,..., n) of (
A ⊗) cannot be exactly known, but ij are confined within the
intervals
≤ ⊗ ≤
⊗ ∈ ⎡⊗ ⊗⎤
i
e j
ij
fij . These e and
ij
fij are known exactly, and
,
ij
⎣
⎦ . They are called
white numbers, while ⊗ ij are called grey numbers. (
A ⊗) has a grey matrix, and system (1)
is a grey discrete time system.
For convenience of descriptions, the following Definition and Lemmas are introduced.
Definition 2.1
From system (1), the system has
⎡⊗
⊗
" ⊗
11
12
1 n ⎤
⎢
⎥
⊗
⊗
" ⊗
⎢ 21
22
2
(
A ⊗) = [
]
n ⎥
⊗
=
ij n× n
⎢
(2)
#
#
# ⎥
⎢
⎥
⊗
⊗
" ⊗
⎣ n 1
n 2
nn ⎦
⎡ 11
e
1
e 2 " 1
e n ⎤
⎢ e
e
" e ⎥
⎢ 21
22
2
E = [ e ]
n ⎥
=
ij n× n
⎢
(3)
#
#
# ⎥
⎢
⎥
⎣ n
e 1
n
e 2 " n
e n ⎦
⎡ f 11 f 12 " f 1 n ⎤
⎢ f
f
" f ⎥
⎢ 21
22
2
F = [ f ]
n ⎥
=
ij n× n
⎢
(4)
#
#
# ⎥
⎢
⎥
⎣ fn 1 fn 2 " fnn ⎦
where E and F represent the minimal and maximal punctual matrices of (
A ⊗) , respectively.
Suppose that A represents the average white matrix of (
A ⊗) as
Stability Analysis of Grey Discrete Time Time-Delay Systems: A Sufficient Condition
329
⎡ e + f ⎤
E + F
A = [ a ]
ij
ij
= ⎢
⎥
=
ij n× n
(5)
⎣
2
⎦
2
n× n
and
A = [ a ]
× = [⊗ − a ] × =
(
A ⊗) −
G
gij n n
ij
ij n n
A (6)
M = [ m ] × = [ f − a ]
= −
ij n n
ij
ij n× n
F A (7)
where G
A represents a bias matrix between (
A ⊗) and A; M represents the maximal bias
matrix between F and A. Then we have
≤
G
A m M m (8)
where M m represents the modulus matrix of M; r[ M] represents the spectral radius of
matrix M; I represents the identity matrix, and λ( M) is the eigenv