This paper considers observer design problem of singularly perturbed systems (SPSs) with multirate sampled and delayed measurements. The outputs are classified into two sets which are measured at different sampling rates and subject to transmission delays. The error system is modeled as a continuous-time SPS with a slow-varying delay and a fast-varying delay. A new Lyapunov functional taking the delay properties into account is constructed. Based on the Lyapunov–Krasovskii functional, sufficient conditions for stability of the error system are proposed by which an observer design method is proposed. A realistic example is used to illustrate the obtained results.

References

References
1.
Kokotovic
,
P. V.
,
Khalil
,
H. K.
, and
O'Reilly
,
J.
,
1986
,
Singular Perturbation Methods in Control: Analysis and Design
,
Academic
,
New York
.
2.
Aditya
,
K.
,
Panagiotis
,
D.
, and
Prodromos
,
D.
,
1998
, “
Singular Perturbation Modeling of Nonlinear Processes With Nonexplicit Time-Scale Multiplicity
,”
Chem. Eng. Sci.
,
53
(
8
), pp.
1491
1504
.
3.
Yang
,
C.
,
Zhang
,
Q.
, and
Zhou
,
L.
,
2013
,
Stability Analysis and Design for Nonlinear Singular Systems
,
Springer-Verlag
,
Berlin
.
4.
Asemani
,
M.
,
Yazdanpanah
,
M.
,
Majd
,
V.
, and
Golabi
,
A.
,
2013
, “
H∞ Control of T-S fuzzy Singularly Perturbed Systems Using Multiple Lyapunov Functions
,”
Circuits, Syst., Signal Process.
,
32
(
5
), pp.
2243
2266
.
5.
Mei
,
P.
,
Cai
,
C.
, and
Zou
,
Y.
,
2009
, “
A Generalized KYP Lemma-Based Approach for H∞ Control of Singularly Perturbed Systems
,”
Circuits, Syst. Signal Process.
,
28
(
6
), pp.
945
957
.
6.
Kim
,
B.
,
Kim
,
Y.
, and
Lim
,
M.
,
2005
, “
LQG Control for Nonstandard Singularly Perturbed Discrete-Time Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
126
(
4
), pp.
860
864
.
7.
Khosravi
,
M.
, and
Taghirad
,
H. D.
,
2014
, “
Dynamic Modeling and Control of Parallel Robots With Elastic Cables: Singular Perturbation Approach
,”
IEEE Trans. Rob.
,
30
(
3
), pp.
694
704
.
8.
Amjadifard
,
R.
,
Beheshti
,
M.
, and
Yazdanpanah
,
M.
,
2011
, “
Robust Stabilization for a Class of Nonlinear Singularly Perturbed Systems
,”
ASME J. Dyn. Syst., Meas., Control
,
133
(
5
), p.
051004
.
9.
Porter
,
B.
,
1977
, “
Singular Perturbation Methods in the Design of Full-Order Observers for Multivariable Linear Systems
,”
Int. J. Control
,
26
(
4
), pp.
589
594
.
10.
O'Reilly
,
J.
,
1979
, “
Full-Order Observers for a Class of Singularly Perturbed Linear Time-Varying Systems
,”
Int. J. Control
,
30
(
5
), pp.
745
756
.
11.
Yoo
,
H.
,
2014
, “
Design of Observers for Systems With Slow and Fast Modes
,” M.S. thesis, Rutgers University—Graduate School, New Brunswick, NJ.
12.
Lin
,
K. J.
,
2010
, “
Composite Observer-Based Feedback Design for Singularly Perturbed Systems Via LMI Approach
,”
SICE Annual Conference 2010
, Aug. 18–21, pp.
3056
3061
.
13.
Kando
,
H.
, and
Iwazumi
,
T.
,
1985
, “
Design of Observers and Stabilising Feedback Controllers for Singularly Perturbed Discrete Systems
,”
IEE Proc. D
,
132
(
1
), pp.
1
10
.
14.
Oloomi
,
H.
, and
Sawan
,
M. E.
,
1987
, “
The Observer-Based Controller Design of Discrete-Time Singularly Perturbed Systems
,”
IEEE Trans. Autom. Control
,
32
(
3
), pp.
246
248
.
15.
Shouse
,
K.
, and
Taylor
,
D.
,
1995
, “
Discrete-Time Observers for Singularly Perturbed Continuous-Time Systems
,”
IEEE Trans. Autom. Control
,
40
(
2
), pp.
224
235
.
16.
Kando
,
H.
,
Aoyama
,
T.
, and
Iwazumi
,
T.
,
1989
, “
Multirate Observer Design Via Singular Perturbation Theory
,”
Int. J. Control
,
50
(
5
), pp.
2005
2023
.
17.
Bidani
,
M.
, and
Djemai
,
M.
,
2002
, “
A Multirate Digital Control Via a Discrete-Time Observer for Non-Linear Singularly Perturbed Continuous-Time Systems
,”
Int. J. Control
,
75
(
8
), pp.
591
613
.
18.
Litkouhi
,
B.
, and
Khalil
,
H.
,
1985
, “
Multirate and Composite Control of Two-Time-Scale Discrete-Time Systems
,”
IEEE Trans. Autom. Control
,
30
(
7
), pp.
645
651
.
19.
Naidu
,
D.
,
2002
, “
Singular Perturbations and Time Scales in Control Theory and Applications: An Overview
,”
Dyn. Contin. Discrete Impulsive Syst. Ser. B
,
9
(
2
), pp.
233
278
.
20.
Dong
,
J.
, and
Yang
,
G.
,
2007
, “
Robust H∞ Control for Standard Discrete-Time Singularly Perturbed Systems
,”
IET Control Theory Appl.
,
1
(
4
), pp.
1141
1148
.
21.
Wang
,
G.
,
Zhang
,
Q.
, and
Sreeram
,
V.
,
2010
, “
H∞ Control for Discrete-Time Singularly Perturbed Systems With Two Markov Processes
,”
J. Franklin Inst.
,
347
(
5
), pp.
836
847
.
22.
Dong
,
J.
, and
Yang
,
G.
,
2008
, “
H∞ Mathcontainer Loading Mathjax Control for Fast Sampling Discrete-Time Singularly Perturbed Systems
,”
Automatica
,
44
(
5
), pp.
1385
1393
.
23.
Xu
,
S.
, and
Feng
,
G.
,
2009
, “
New Results on H∞ Control of Discrete Singularly Perturbed Systems
,”
Automatica
,
45
(
10
), pp.
2339
2343
.
24.
Naghshtabrizi
,
P.
,
Hespanha
,
J.
, and
Teel
,
A.
,
2008
, “
Exponential Stability of Impulsive Systems With Application to Uncertain Sampled-Data Systems
,”
Syst. Control Lett.
,
57
(
5
), pp.
378
385
.
25.
Fridman
,
E.
,
Seuret
,
A.
, and
Richard
,
J.
,
2004
, “
Robust Sampled-Data Stabilization of Linear Systems: An Input Delay Approach
,”
Automatica
,
40
(
8
), pp.
1441
1446
.
26.
Fridman
,
E.
,
2010
, “
A Refined Input Delay Approach to Sampled-Data Control
,”
Automatica
,
46
(
2
), pp.
421
427
.
27.
Moarref
,
M.
, and
Rodrigues
,
L.
,
2014
, “
Observer Design for Linear Multi-Rate Sampled-Data Systems
,”
Am. Control Conf.
, pp.
5319
5324
.
28.
Liu
,
K.
, and
Fridman
,
E.
,
2012
, “
Wirtinger's Inequality and Lyapunov-Based Sampled-Data Stabilization
,”
Automatica
,
48
(
1
), pp.
102
108
.
29.
Fridman
,
E.
,
2014
, “
Tutorial on Lyapunov-Based Methods for Time-Delay Systems
,”
Eur. J. Control
,
20
(
6
), pp.
271
283
.
30.
Yang
,
C.
, and
Zhang
,
Q.
,
2009
, “
Multi-Objective Control for T-S Fuzzy Singularly Perturbed Systems
,”
IEEE Trans. Fuzzy Syst.
,
17
(
1
), pp.
104
115
.
31.
Liu
,
K.
,
Suplin
,
V.
, and
Fridman
,
E.
,
2010
, “
Stability of Linear Systems With General Sawtooth Delay
,”
IMA J. Math. Control Inf.
,
27
(
4
), pp.
419
436
.
32.
Gu
,
K.
,
Kharitonov
,
V.
, and
Chen
,
J.
,
2003
,
Stability of Time-Delay Systems
,
Birkhäuser
,
Boston, MA
.
33.
Boyd
,
S.
,
Ghaoui
,
L.
,
Feron
,
E.
, and
Balakrishnan
,
V.
,
1994
,
Linear Matrix Inequalities in Systems and Control Theory
,
SIAM
,
Philadelphia, PA
.
34.
Chaichanavong
,
P.
, and
Banjerdpongchai
,
D.
,
1999
, “
A Case Study of Robust Control Experiment on One-Link Flexible Robot Arm
,”
IEEE Conf. Decis. Control
, pp.
4319
4324
.
35.
Pota
,
H. R.
,
1992
, “
A Prototype Flexible Robot Arm—An Interdisciplinary Undergraduate Project
,”
IEEE Trans. Educ.
,
35
(
1
), pp.
83
89
.
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