DISCRETE TIME SYSTEMS
Edited by Mario A. Jordán
and Jorge L. Bustamante
Discrete Time Systems
Edited by Mario A. Jordán and Jorge L. Bustamante
Published by InTech
Janeza Trdine 9, 51000 Rijeka, Croatia
Copyright © 2011 InTech
All chapters are Open Access articles distributed under the Creative Commons
Non Commercial Share Alike Attribution 3.0 license, which permits to copy,
distribute, transmit, and adapt the work in any medium, so long as the original
work is properly cited. After this work has been published by InTech, authors
have the right to republish it, in whole or part, in any publication of which they
are the author, and to make other personal use of the work. Any republication,
referencing or personal use of the work must explicitly identify the original source.
Statements and opinions expressed in the chapters are these of the individual contributors
and not necessarily those of the editors or publisher. No responsibility is accepted
for the accuracy of information contained in the published articles. The publisher
assumes no responsibility for any damage or injury to persons or property arising out
of the use of any materials, instructions, methods or ideas contained in the book.
Publishing Process Manager Ivana Lorkovic
Technical Editor Teodora Smiljanic
Cover Designer Martina Sirotic
Image Copyright Emelyano, 2010. Used under license from Shutterstock.com
First published March, 2011
Printed in India
A free online edition of this book is available at www.intechopen.com
Additional hard copies can be obtained from orders@intechweb.org
Discrete Time Systems, Edited by Mario A. Jordán and Jorge L. Bustamante
p. cm.
ISBN 978-953-307-200-5
free online editions of InTech
Books and Journals can be found at
www.intechopen.com
Contents
Preface IX
Part 1
Discrete-Time Filtering 1
Chapter 1
Real-time Recursive State Estimation
for Nonlinear Discrete Dynamic Systems
with Gaussian or non-Gaussian Noise 3
Kerim Demirbaş
Chapter 2
Observers Design for a Class of Lipschitz
Discrete-Time Systems with Time-Delay 19
Ali Zemouche and Mohamed Boutayeb
Chapter 3
Distributed Fusion Prediction for Mixed
Continuous-Discrete Linear Systems 39
Ha-ryong Song, Moon-gu Jeon and Vladimir Shin
Chapter 4
New Smoothers for Discrete-time Linear
Stochastic Systems with Unknown Disturbances 53
Akio Tanikawa
Chapter 5
On the Error Covariance Distribution
for Kalman Filters with Packet Dropouts 71
Eduardo Rohr Damián Marelli, and Minyue Fu
Chapter 6
Kalman Filtering for Discrete Time Uncertain Systems 93
Rodrigo Souto, João Ishihara and Geovany Borges
Part 2
Discrete-Time Fixed Control 109
Chapter 7
Stochastic Optimal Tracking with Preview
for Linear Discrete Time Markovian Jump Systems 111
Gou Nakura
Chapter 8
The Design of a Discrete Time Model Following
Control System for Nonlinear Descriptor System 131
Shigenori Okubo and Shujing Wu
VI
Contents
Chapter 9
Output Feedback Control of Discrete-time
LTI Systems: Scaling LMI Approaches 141
Jun Xu
Chapter 10
Discrete Time Mixed LQR/H∞ Control Problems 159
Xiaojie Xu
Chapter 11
Robust Control Design of Uncertain
Discrete-Time Systems with Delays 179
Jun Yoneyama, Yuzu Uchida and Shusaku Nishikawa
Chapter 12
Quadratic D Stabilizable Satisfactory Fault-tolerant
Control with Constraints of Consistent Indices
for Satellite Attitude Control Systems 195
Han Xiaodong and Zhang Dengfeng
Part 3
Discrete-Time Adaptive Control 205
Chapter 13
Discrete-Time Adaptive Predictive Control
with Asymptotic Output Tracking 207
Chenguang Yang and Hongbin Ma
Chapter 14
Decentralized Adaptive Control
of Discrete-Time Multi-Agent Systems 229
Hongbin Ma, Chenguang Yang and Mengyin Fu
Chapter 15
A General Approach to Discrete-Time
Adaptive Control Systems with Perturbed
Measures for Complex Dynamics - Case Study:
Unmanned Underwater Vehicles 255
Mario Alberto Jordán and Jorge Luis Bustamante
Part 4
Stability Problems 281
Chapter 16
Stability Criterion and Stabilization
of Linear Discrete-time System
with Multiple Time Varying Delay 283
Xie Wei
Chapter 17
Uncertain Discrete-Time Systems with Delayed
State: Robust Stabilization with Performance
Specification via LMI Formulations 295
Valter J. S. Leite, Michelle F. F. Castro, André F. Caldeira,
Márcio F. Miranda and Eduardo N. Gonçalves
Chapter 18
Stability Analysis of Grey Discrete Time
Time-Delay Systems: A Sufficient Condition 327
Wen-Jye Shyr and Chao-Hsing Hsu
Contents
VII
Chapter 19
Stability and L2 Gain Analysis of Switched Linear
Discrete-Time Descriptor Systems 337
Guisheng Zhai
Chapter 20
Robust Stabilization for a Class of Uncertain
Discrete-time Switched Linear Systems 351
Songlin Chen, Yu Yao and Xiaoguan Di
Part 5
Miscellaneous Applications 361
Chapter 21
Half-overlap Subchannel Filtered MultiTone
Modulation and Its Implementation 363
Pavel Silhavy and Ondrej Krajsa
Chapter 22
Adaptive Step-size Order Statistic LMS-based
Time-domain Equalisation in Discrete
Multitone Systems 383
Suchada Sitjongsataporn and Peerapol Yuvapoositanon
Chapter 23
Discrete-Time Dynamic Image-Segmentation System 405
Ken’ichi Fujimoto, Mio Kobayashi and Tetsuya Yoshinaga
Chapter 24
Fuzzy Logic Based Interactive Multiple Model
Fault Diagnosis for PEM Fuel Cell Systems 425
Yan Zhou, Dongli Wang, Jianxun Li, Lingzhi Yi and Huixian Huang
Chapter 25
Discrete Time Systems with Event-Based Dynamics:
Recent Developments in Analysis
and Synthesis Methods 447
Edgar Delgado-Eckert, Johann Reger and Klaus Schmidt
Chapter 26
Discrete Deterministic and Stochastic Dynamical
Systems with Delay - Applications 477
Mihaela Neamţu and Dumitru Opriş
Chapter 27
Multidimensional Dynamics:
From Simple to Complicated 505
Kang-Ling Liao, Chih-Wen Shih and Jui-Pin Tseng
Preface
Discrete-Time Systems comprehend an important and broad research fi eld. The con-
solidation of digital-based computational means in the present, pushes a technological
tool into the fi eld with a tremendous impact in areas like Control, Signal Processing,
Communications, System Modelling and related Applications. This fact has enabled
numerous contributions and developments which are either genuinely original as
discrete-time systems or are mirrors from their counterparts of previously existing
continuous-time systems.
This book att empts to give a scope of the present state-of-the-art in the area of Discrete-
Time Systems from selected international research groups which were specially con-
voked to give expressions to their expertise in the fi eld.
The works are presented in a uniform framework and with a formal mathematical
context.
In order to facilitate the scope and global comprehension of the book, the chapters were
grouped conveniently in sections according to their affi
nity in 5 signifi cant areas.
The fi rst group focuses the problem of Filtering that encloses above all designs of State
Observers, Estimators, Predictors and Smoothers. It comprises Chapters 1 to 6.
The second group is dedicated to the design of Fixed Control Systems (Chapters 7 to
12). Herein it appears designs for Tracking Control, Fault-Tolerant Control, Robust Con-
trol, and designs using LMI- and mixed LQR/Hoo techniques.
The third group includes Adaptive Control Systems (Chapter 13 to 15) oriented to the
specialities of Predictive, Decentralized and Perturbed Control Systems.
The fourth group collects works that address Stability Problems (Chapter 16 to 20).
They involve for instance Uncertain Systems with Multiple and Time-Varying Delays
and Switched Linear Systems.
Finally, the fi ft h group concerns miscellaneous applications (Chapter 21 to 27). They
cover topics in Multitone Modulation and Equalisation, Image Processing, Fault Diag-
nosis, Event-Based Dynamics and Analysis of Deterministic/Stochastic and Multidi-
mensional Dynamics.
X
Preface
We think that the contribution in the book, which does not have the intention to be
all-embracing, enlarges the fi eld of the Discrete-Time Systems with signifi cation in the
present state-of-the-art. Despite the vertiginous advance in the fi eld, we think also that
the topics described here allow us also to look through some main tendencies in the
next years in the research area.
Mario A. Jordán and Jorge L. Bustamante
IADO-CCT-CONICET
Dep. of Electrical Eng. and Computers
National University of the South
Argentina
Part 1
Discrete-Time Filtering
1
Real-time Recursive State Estimation for
Nonlinear Discrete Dynamic Systems with
Gaussian or non-Gaussian Noise
Kerim Demirba¸s
Department of Electrical and Electronics Engineering
Middle East Technical University
Inonu Bulvari, 06531 Ankara
Turkey
1. Introduction
Many systems in the real world are more accurately described by nonlinear models. Since
the original work of Kalman (Kalman, 1960; Kalman & Busy, 1961), which introduces the
Kalman filter for linear models, extensive research has been going on state estimation
of nonlinear models; but there do not yet exist any optimum estimation approaches for
all nonlinear models, except for certain classes of nonlinear models; on the other hand,
different suboptimum nonlinear estimation approaches have been proposed in the literature
(Daum, 2005). These suboptimum approaches produce estimates by using some sorts of
approximations for nonlinear models. The performances and implementation complexities
of these suboptimum approaches surely depend upon the types of approximations which
are used for nonlinear models. Model approximation errors are an important parameter
which affects the performances of suboptimum estimation approaches. The performance of a
nonlinear suboptimum estimation approach is better than the other estimation approaches for
specific models considered, that is, the performance of a suboptimum estimation approach is
model-dependent.
The most commonly used recursive nonlinear estimation approaches are the extended
Kalman filter (EKF) and particle filters. The EKF linearizes nonlinear models by Taylor
series expansion (Sage & Melsa, 1971) and the unscented Kalman filter (UKF) approximates
a posteriori densities by a set of weighted and deterministically chosen points (Julier, 2004).
Particle filters approximates a posterior densities by a large set of weighted and randomly
selected points (called particles) in the state space (Arulampalam et al., 2002; Doucet et al.,
2001; Ristic et al., 2004). In the nonlinear estimation approaches proposed in (Demirba¸s,
1982; 1984; Demirba¸s & Leondes, 1985; 1986; Demirba¸s, 1988; 1989; 1990; 2007; 2010): the
disturbance noise and initial state are first approximated by a discrete noise and a discrete
initial state whose distribution functions the best approximate the distribution functions of the
disturbance noise and initial state, states are quantized, and then multiple hypothesis testing
is used for state estimation; whereas Grid-based approaches approximate a posteriori densities
by discrete densities, which are determined by predefined gates (cells) in the predefined state
space; if the state space is not finite in extent, then the state space necessitates some truncation
of the state space; and grid-based estimation approaches assume the availability of the state
4
Discrete Time Systems
transition density p( x( k) |x( k − 1)), which may not easily be calculated for state models with
nonlinear disturbance noise (Arulampalam et al., 2002; Ristic et al., 2004). The Demirba¸s
estimation approaches are more general than grid-based approaches since 1) the state space
need not to be truncated, 2) the state transition density is not needed, 3) state models can be
any nonlinear functions of the disturbance noise.
This chapter presents an online recursive nonlinear state filtering and prediction scheme for
nonlinear dynamic systems. This scheme is recently proposed in (Demirba¸s, 2010) and is
referred to as the DF throughout this chapter. The DF is very suitable for state estimation of
nonlinear dynamic systems under either missing observations or constraints imposed on state
estimates. There exist many nonlinear dynamic systems for which the DF outperforms the
extended Kalman filter (EKF), sampling importance resampling (SIR) particle filter (which is
sometimes called the bootstrap filter), and auxiliary sampling importance resampling (ASIR)
particle filter. Section 2 states the estimation problem. Section 3 first discusses discrete noises
which approximate the disturbance noise and initial state, and then presents approximate
state and observation models. Section 4 discusses optimum state estimation of approximate
dynamic models. Section 5 presents the DF. Section 6 yields simulation results of two
examples for which the DF outperforms the EKF, SIR, and ASIR particle filters. Section 7
concludes the chapter.
2. Problem statement
This section defines state estimation problem for nonlinear discrete dynamic systems. These
dynamic systems are described by
State Model
x( k + 1) = f ( k, x( k), w( k))
(1)
Observation Model
z( k) = g( k, x( k), v( k)),
(2)
where k stands for the discrete time index; f : R xR mxR n → R m is the state transition function; R m is the m-dimensional Euclidean space; w( k) ∈ R n is the disturbance noise vector at time k; x( k) ∈ R m is the state vector at time k; g : R xR mxR p → R r is the observation function; v( k) ∈ R p is the observation noise vector at time k; z( k) ∈ R r is the observation vector at time k; x(0), w( k), and v( k) are all assumed to be independent with known distribution functions.
Moreover, it is assumed that there exist some constraints imposed on state estimates. The DF
recursively yields a predicted value ˆ x( k|k − 1) of the state x( k) given the observation sequence
from time one to time k − 1, that is, Zk− 1 Δ
= {z(1), z(2), . . . , z( k − 1) }; and a filtered value
ˆ x( k|k) of the state x( k) given the observation sequence from time one to time k, that is, Zk.
Estimation is accomplished by first approximating the disturbance noise and initial state with
discrete random noises, quantizing the state, that is, representing the state model with a time
varying state machine, and an online suboptimum implementation of multiple hypothesis
testing.
3. Approximation
This section first discusses an approximate discrete random vector which approximates a
random vector, and then presents approximate models of nonlinear dynamic systems.
Real-time Recursive State Estimation for Nonlinear
Discrete Dynamic Systems with Gaussian or non-Gaussian Noise
5
3.1 Approximate discrete random noise
In this subsection: an approximate discrete random vector with n possible values of a
random vector is defined; approximate discrete random vectors are used to approximate
the disturbance noise and initial state throughout the chapter; moreover, a set of equations
which must be satisfied by an approximate discrete random variable with n possible values
of an absolutely continuous random variable is given (Demirba¸s, 1982; 1984; 2010); finally, the
approximate discrete random variables of a Gaussian random variable are tabulated.
Let w be an m-dimensional random vector. An approximate discrete random vector with n
possible values of w, denoted by wd, is defined as an m-dimensional discrete random vector
with n possible values whose distribution function the best approximates the distribution
function of w over the distribution functions of all m-dimensional discrete random vectors
with n possible values, that is
w
− 1
d = min
{
[ Fy( a) − Fw( a)]2 da}
(3)
y D
R n
where D is the set of all m-dimensional discrete random vectors with n possible values, Fy( a)
is the distribution function of the discrete random vector y, Fw( a) is the distribution function
of the random vector w, and R m is the m-dimensional Euclidean space. An approximate
discrete random vector wd is, in general, numerically, offline-calculated, stored and then used
for estimation. The possible values of wd are denoted by wd 1, wd 2, ...., and wdn ; and the
occurrence probability of the possible value wdi is denoted by Pw , that is
di
Δ
Pw = Prob{w
di
d = wdi }.
(4)
where Prob{wd(0) = wdi} is the occurrence probability of wdi.
Let us now consider the case that w is an absolutely continuous random variable. Then, wd is
an approximate discrete random variable with n possible values whose distribution function
the best approximates the distribution function Fw( a) of w over the distribution functions of
all discrete random variables with n possible values, that is
w
− 1
d = min
{J( Fy( a)) }
y D
in which the distribution error function (the objective function) J( Fy( a)) is defined by
J( Fy( a)) Δ
=
[ Fy( a) − Fw( a)]2 da
R
where D is the set of all discrete random variables with n possible values, Fy( a) is the
distribution function of the discrete random variable y, Fw( a) is the distribution function of the
absolutely continuous random variable w, and R is the real line. Let the distribution function
Fy( a) of a discrete random variable y be given by
⎧
⎨0 if a < y 1
Fy( a) Δ
= ⎩ Fy if y
i
i ≤ a < yi+1, i = 1, 2, . . . , n