Preface

**Section 3 Applications 161**

**VI** Contents

Gou Nakura

**Stability Issues 183**

**Study Data 213**

Chapter 7 **An Approach to Hybrid Smoothing for Linear Discrete-Time Systems with Non-Gaussian Noises 163**

Saïd Guermah, Saïd Djennoune and Maâmar Bettayeb

**Estimation of Nursing Tasks on the Basis of Time**

Atsue Ishii, Takashi Nakamura, Yuko Ohno and Satoko Kasahara

Chapter 8 **Discrete-Time Fractional-Order Systems: Modeling and**

Chapter 9 **Investigation of a Methodology for the Quantitative**

This volume brings about the contemporary results in the field of discrete-time systems. It covers technical reports written on the topics of robust control, nonlinear systems and recent applications. Although the research views are different, they all geared towards focusing on the up-to-date knowledge gain by the researchers and providing effective developments along the systems and control arena. Each topic has a detailed discussions and suggestions for future perusal by interested investigators.

The book is divided into three sections: Section I is devoted to 'robust control', Section II deals with 'nonlinear control' and Section III provides 'applications'

Section I 'robust control' comprises of three chapters. In what follows we provide brief ac‐ count of each. In the first chapter titled "Stochastic mixed LQR/H control for linear dis‐ crete-time systems" Xiaojie Xu considered the static state feedback stochastic mixed LQR/ Hoo control problem for linear discrete-time systems. In this chapter, the author established sufficient conditions for the existence of all admissible static state feedback controllers solv‐ ing this problem. Then, sufficient conditions for the existence of all static output feedback controllers solving the discrete-time stochastic mixed LQR/ Hoo control problem are presen‐ ted.

In the second chapter titled "Robust control design of uncertain discrete-time descriptor sys‐ tems with delays" by Yoneyama, Uchida, and Takada, the authors looked at the robust H∞ non-fragile control design problem for uncertain discrete-time descriptor systems with timedelay. The controller gain uncertainties under consideration are supposed to be time-vary‐ ing but norm-bounded. The problem addressed was the robust stability and stabilization problem under state feedback subject to norm-bounded uncertainty. The authors derived sufficient conditions for the solvability of the robust non-fragile stabilization control design problem for discrete-time descriptor systems with time-delay obtained with additive con‐ troller uncertainties.

In the third chapter, the authors Jun-min, Jiang-rong and Zhi-le of "Delay-dependent gener‐ alized H2 control for discrete-time fuzzy systems with infinite-distributed delays" examined the generalized H2 control problem for a class of discrete time T-S fuzzy systems with infin‐ ite-distributed delays. They constructed a new delay-dependent piecewise Lyapunov-Kra‐ sovskii functional (DDPLKF) and based on which the stabilization condition and controller design method are derived. They have shown that the control laws can be obtained by solv‐ ing a set of LMIs. A simulation example has been presented to illustrate the effectiveness of the proposed design procedures.

care management. They focused on the critical issues including estimates of ward task times based on time study data, creation of a computer-based virtual ward environment using the estimated values and test experiment on a plan for work management using the virtual

To sum up, the collection of such variety of chapters presents a unique opportunity to re‐ search investigators who are interested to catch up with accelerated progress in the world of

> **Magdi S. Mahmoud** KFUPM, Saudi Arabia

Preface IX

ward environment

discrete-time systems.

Section II 'nonlinear control' is subsumed of three chapters. In the first chapter of this sec‐ tion, Dai, Xia, Fu and Mahmoud, in an overview setting, wrote the chapter "Discrete modelpredictive control" and introduced the principles, mathematical formulation and properties of MPC for constrained dynamic systems, both linear and nonlinear. In particular, they ad‐ dressed the issues of feasibility, closed loop stability and open-loop performance objective versus closed loop performance. Several technical issues pertaining to robust design, sto‐ chastic control and MPC over networks are stressed.

The authors Jordan, Bustamante and Berger presented "Stability Analysis of Nonlinear Dis‐ crete-Time Adaptive Control Systems with Large Dead-Times" as the second chapter in this section. They looked at the guidance, navigation and control systems of unmanned under‐ water vehicles (UUVs) which are digitally linked by means of a control communication with complex protocols and converters. Of particular interest is to carefully examine the effects of time delays in UUVs that are controlled adaptively in six degrees of freedom. They per‐ formed a stability analysis to obtain guidelines for selecting appropriate sampling periods according to the tenor of perturbations and delay.

In the third chapter "Adaptive step-size orthogonal gradient-based per-tone equalization in discrete multitone systems" by Suchada Sitjongsataporn, the author focused on discrete multitone theory and presented orthogonal gradient-based algorithms with reduced com‐ plexity for per-tone equalizer (PTEQ) based on the adaptive step-size approaches related to the mixed-tone criterion. The convergence behavior and stability analysis of the proposed algorithms are investigated based on the mixed-tone weight-estimated errors.

Section III provides 'applications' in terms of three chapters. In one chapter "An approach to hybrid smoothing for linear discrete-time systems with non-Gaussian noises" by Gou Na‐ kura, the author critically examined hybrid estimation for linear discrete-time systems with non- Gaussian noises and assumed that modes of the systems are not directly accessible. In this regard, he proceeded to determine both estimated states of the systems and a candidate of the distributions of the modes over the finite time interval based on the most probable trajectory (MPT) approach.

In the following chapter "Discrete-time fractional-order systems: modeling and stability is‐ sues" by Guermah, Djennoune and Bettayeb, the authors reviewed some basic tools for modeling and analysis of fractional-order systems (FOS) in discrete time and introduced state-space representation for both commensurate and non commensurate fractional orders. They revealed new properties and focused on the analysis of the controllability and the ob‐ servability of linear discrete-time FOS. Further, the authors established testable sufficient conditions for guaranteeing the controllability and the observability.

In the third chapter "Investigation of a methodology for the quantitative estimation of nurs‐ ing tasks on the basis of time study data" by Atsue Ishii, Takashi Nakamura, Yuko Ohno and Satoko Kasahara, the authors concentrated on establishing a methodology for the pur‐ pose of linking the data to the calculation of quantities of nursing care required or to nursing care management. They focused on the critical issues including estimates of ward task times based on time study data, creation of a computer-based virtual ward environment using the estimated values and test experiment on a plan for work management using the virtual ward environment

ing a set of LMIs. A simulation example has been presented to illustrate the effectiveness of

Section II 'nonlinear control' is subsumed of three chapters. In the first chapter of this sec‐ tion, Dai, Xia, Fu and Mahmoud, in an overview setting, wrote the chapter "Discrete modelpredictive control" and introduced the principles, mathematical formulation and properties of MPC for constrained dynamic systems, both linear and nonlinear. In particular, they ad‐ dressed the issues of feasibility, closed loop stability and open-loop performance objective versus closed loop performance. Several technical issues pertaining to robust design, sto‐

The authors Jordan, Bustamante and Berger presented "Stability Analysis of Nonlinear Dis‐ crete-Time Adaptive Control Systems with Large Dead-Times" as the second chapter in this section. They looked at the guidance, navigation and control systems of unmanned under‐ water vehicles (UUVs) which are digitally linked by means of a control communication with complex protocols and converters. Of particular interest is to carefully examine the effects of time delays in UUVs that are controlled adaptively in six degrees of freedom. They per‐ formed a stability analysis to obtain guidelines for selecting appropriate sampling periods

In the third chapter "Adaptive step-size orthogonal gradient-based per-tone equalization in discrete multitone systems" by Suchada Sitjongsataporn, the author focused on discrete multitone theory and presented orthogonal gradient-based algorithms with reduced com‐ plexity for per-tone equalizer (PTEQ) based on the adaptive step-size approaches related to the mixed-tone criterion. The convergence behavior and stability analysis of the proposed

Section III provides 'applications' in terms of three chapters. In one chapter "An approach to hybrid smoothing for linear discrete-time systems with non-Gaussian noises" by Gou Na‐ kura, the author critically examined hybrid estimation for linear discrete-time systems with non- Gaussian noises and assumed that modes of the systems are not directly accessible. In this regard, he proceeded to determine both estimated states of the systems and a candidate of the distributions of the modes over the finite time interval based on the most probable

In the following chapter "Discrete-time fractional-order systems: modeling and stability is‐ sues" by Guermah, Djennoune and Bettayeb, the authors reviewed some basic tools for modeling and analysis of fractional-order systems (FOS) in discrete time and introduced state-space representation for both commensurate and non commensurate fractional orders. They revealed new properties and focused on the analysis of the controllability and the ob‐ servability of linear discrete-time FOS. Further, the authors established testable sufficient

In the third chapter "Investigation of a methodology for the quantitative estimation of nurs‐ ing tasks on the basis of time study data" by Atsue Ishii, Takashi Nakamura, Yuko Ohno and Satoko Kasahara, the authors concentrated on establishing a methodology for the pur‐ pose of linking the data to the calculation of quantities of nursing care required or to nursing

algorithms are investigated based on the mixed-tone weight-estimated errors.

conditions for guaranteeing the controllability and the observability.

the proposed design procedures.

VIII Preface

chastic control and MPC over networks are stressed.

according to the tenor of perturbations and delay.

trajectory (MPT) approach.

To sum up, the collection of such variety of chapters presents a unique opportunity to re‐ search investigators who are interested to catch up with accelerated progress in the world of discrete-time systems.

> **Magdi S. Mahmoud** KFUPM, Saudi Arabia

**Section 1**

**Robust Control**

**Section 1**
