Recursive Filtering for 2-D Shift-Varying Systems with Communication Constraints

Jinling Liang, Zidong Wang, Fan Wang

• Publisher: CRC Press
• Year: 2021
• Print ISBN: 9781032038179
• eBook ISBN: 9781000429305
• Edition: 1
• Topic: Computer Science
• Subtopic: Computer Networking
• Index: Computer Science

This book presents up-to-date research developments and novel methodologies regarding recursive filtering for 2-D shift-varying systems with various communication constraints. It investigates recursive filter/estimator design and performance analysis by a combination of intensive stochastic analysis, recursive Riccati-like equations, variance-constrained approach, and mathematical induction. Each chapter considers dynamics of the system, subtle design of filter gains, and effects of the communication constraints on filtering performance. Effectiveness of the derived theories and applicability of the developed filtering strategies are illustrated via simulation examples and practical insight.

Features: -

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• Covers recent advances of recursive filtering for 2-D shift-varying systems subjected to communication constraints from the engineering perspective.

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• Includes the recursive filter design, resilience operation and performance analysis for the considered 2-D shift-varying systems.

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• Captures the essence of the design for 2-D recursive filters.

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• Develops a series of latest results about the robust Kalman filtering and protocol-based filtering.

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• Analyzes recursive filter design and filtering performance for the considered systems.

This book aims at graduate students and researchers in mechanical engineering, industrial engineering, communications networks, applied mathematics, robotics and control systems.

1

Introduction

1.1 2-D Systems

Two-dimensional (2-D) systems arise primarily from the practical requirements for depicting the information broadcast of the target plants over two directions. In contrast with the traditional one-dimensional (1-D) systems whose states evolve along a single direction, the states of 2-D systems propagate along two independent directions leading to more complicated dynamics [58]. Thanks to the inherent feature of two-directional propagations, 2-D systems have shown their promising applications in many engineering fields, such as manufacturing, industrial automation, grid sensor networks, and environment monitoring.

It is worth noting that 2-D systems provide a powerful tool to describe the system dynamics with multiple independent variables. Typical examples of practical 2-D systems include, but are not limited to, iterative circuits, batch processes, thermal processes, digital image processing, seismographic data analysis, and multi-variable network visualization [14, 51, 62, 83, 123]. Particularly, unlike a unilateral or sequential circuit with one-directional evolution, signals of the bilateral circuit flow in two different directions. States of the batch process transmit along both the time and batch directions in industry. The reactor temperature of the heating process varies with different spatial and temporal positions. A general 2-D system, from a mathematical point of view, has a formulation of partial differential/difference equations with evolutions of two-variable functions and diffusions, which can represent the dynamical evolution with respect to the fluid, heat, and electrodynamics. To date, both theoretical developments and practical applications of the 2-D systems have received considerable research interest. Moreover, a number of basic concepts and theories have been developed for various 2-D systems, which mainly embrace the controllability and observability issues [84, 91, 132, 181, 234], the model reduction or approximation [35, 63, 90, 231, 236], the stability analysis [48, 100, 133, 148, 152, 171], and the control and filtering issues [47, 103, 108, 168, 176, 177, 229].

1.1.1 Some Classical 2-D Models

Introduction of the 2-D state-space models dates back to the 1970s. One of the earliest 2-D state-space models has been developed in [68] for a multidimensional linear iterative circuit, where the general response formula has been obtained on the strength of two-tuple powers of certain matrix. Based on a similar approach, a linear discrete state-space model has been generalized in [162] from the model defined in a single-dimensional time to that in a 2-D space, thereby being able to describe the linear image processing. The corresponding generalization contains the novel state transition matrix, observability, and controllability in the 2-D framework. Ever since then, some classical 2-D models have been proposed in the existing literature, and the relationships between them can be clearly stated [10, 57, 58, 83, 162].

Consider the Roesser model described as follows [162]:

$$\begin{array}{ll}\left[\begin{array}{c}{x}^h(i+1,j)\\ x^v(i,j+1)\end{array}\right]=\hfill & \left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]\left[\begin{array}{c}{x}^h(i,j)\\ x^v(i,j)\end{array}\right]+\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right]u(i,j),\hfill \end{array}$$(1.1)

where $x^h(i,j)$ and $x^v(i,j)$ denote the horizontal and vertical states, respectively, $u(i,j)$ is the input vector, and A₁₁, A₁₂, A₂₁, A₂₂, B₁, and B₂ are real-valued matrices with appropriate dimensions.

The Attasi model is given as [10]

$$\begin{array}{ll}x(i,j)=\hfill & A_{1}x(i,j-1)+A_{2}x(i-1,j)\hfill \\ \hfill & -A_1{A}_{2}x(i-1,j-1)+Bu(i-1,j-1)\hfill \end{array}$$(1.2)

with $A_1{A}_2=A_2{A}_1$, where $x(i,j)$ is the system state, A₁, A₂, and B are real-valued matrices.

Further, consider the following first Fornasini-Marchesini (FM-I) model [58]:

$$\begin{array}{ll}\overrightarrow{x}(i,j)=\hfill & {\overrightarrow{A}}_1\overrightarrow{x}(i-1,j-1)+{\overrightarrow{A}}_2\overrightarrow{x}(i,j-1)+{\overrightarrow{A}}_3\overrightarrow{x}(i-1,j)+\overrightarrow{B}u(i-1,j-1),\hfill \end{array}$$(1.3)

where $\overrightarrow{x}(i,j)$ is the system state and ${\overrightarrow{A}}_1$, ${\overrightarrow{A}}_2$, ${\overrightarrow{A}}_3$, and $\overrightarrow{B}$ are known parameter matrices. In addition, the second Fornasini-Marchesini (FM-II) model is expressed by [57]:

$$\begin{array}{ll}\overline{x}(i,j)=\hfill & {\overline{A}}_1\overline{x}(i,j-1)+{\overline{A}}_2\overline{x}(i-1,j)\hfill \\ \hfill & +{\overline{B}}_{1}u(i,j-1)+{\overline{B}}_{2}u(i-1,j),\hfill \end{array}$$(1.4)

where $\overline{x}(i,j)$ is the state vector and ${\overline{A}}_1$, ${\overline{A}}_2$, ${\overline{B}}_1$, and ${\overline{B}}_2$ are parameter matrices with appropriate dimensions.

1.1.2 Relationships between the Models

Apparently, the Attasi model (1.2) can be derived from the FM-I model (1.3) when ${\overrightarrow{A}}_1=-A_1{A}_2$, ${\overrightarrow{A}}_2=A_1$, ${\overrightarrow{A}}_3=A_2$, and $\overrightarrow{B}=B$. By defining $x^h(i,j)=\overrightarrow{x}(i,j+1)-{\overrightarrow{A}}_2\overrightarrow{x}(i,j)$ and $x^v(i,j)=\overrightarrow{x}(i,j)$, model (1.3) is converted into the Roesser model (1.1) with

$$\begin{array}{l}\left[\begin{array}{cc}{A}_{11}& A_{12}\\ A_{21}& A_{22}\end{array}\right]=\left[\begin{array}{cc}{\overrightarrow{A}}_3& {\overrightarrow{A}}_1+{\overrightarrow{A}}_3{\overrightarrow{A}}_2\\ I& {\overrightarrow{A}}_2\end{array}\right],\left[\begin{array}{c}{B}_1\\ B_2\end{array}\right]=\left[\begin{array}{c}\overrightarrow{B}\\ 0\end{array}\right].\hfill \end{array}$$

With the aid of some routine manipulations, model (1.1) can be recast into the FM-II model (1.4) with

$$\begin{array}{l}{\overline{A}}_1=\left[\begin{array}{cc}0& 0\\ A_{21}& A_{22}\end{array}\right],{\overline{A}}_2=\left[\begin{array}{cc}{A}_{11}& A_{12}\\ 0& 0\end{array}\right]\hfill \\ \overline{x}(i,j)=\left[\begin{array}{c}{x}^h(i,j)\\ x^v(i,j)\end{array}\right],{\overline{B}}_1=\left[\begin{array}{c}0\\ B_2\end{array}\right],{\overline{B}}_2=\left[\begin{array}{c}{B}_1\\ 0\end{array}\right].\hfill \end{array}$$

It is observed that the FM-I model is a special case of the Roesser model, and the FM-II model can be recognized as a more general one which covers the Roesser model. Owing to their broad applications, both the FM-II model and the Roesser model have drawn considerable research interest when dealing with the analysis and synthesis issues of 2-D systems.

1.1.3 Linear Repetitive Processes

As a particular class of 2-D systems, the linear repetitive processes (LRPs) have been gaining momentum owing mainly to their practical insights in industry areas such as machining learning, metal rolling operations, coal mining, and digital allpass filters [13, 93, 94, 144, 163, 164, 205]. A typical repetitive process consists of a series of sweeps (known as passes) which are described by certain differential or difference dynamics over a finite duration (known as the pass length). On each pass, the process output, termed as the pass profile, is developed which performs as a forcing function and contributes to the dynamics of the next pass profile. The distinct feature of an LRP lies in that the state dynamics exhibits along each pass over a finite duration and the pass profile evolves along the pass-to-pass direction. Owing to such a bidirectional transmission, repetitive processes have been greatly investigated with the aid of 2-D theory and some elegant results have appeared [18, 161, 184, 225]. For instance, sufficient criterion has been presented in [161] to ensure that the stability along the pass of the underlying repetitive processes is equivalent to the bounded-input/bounded-output stability of the Roesser model. The observer-based sliding mode control problem has been addressed in [225] for a class of differential LRPs with unknown input disturbance by using the 2-D Lyapunov function.

The research of repetitive processes is also bound up with iterative control algorithms that pursue to achieve a favorable tracking performance in terms of repetitive operation from circle to circle. There are some classical iterative learning control (ILC) strategies proposed and further applied in many practical...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Dedication
• Contents
• Preface
• Acknowledgments
• Author Biographies
• List of Figures
• List of Tables
• Symbols
• 1 Introduction
• 2 Minimum-Variance Recursive Filtering for 2-D Systems with Degraded Measurements: Boundedness and Monotonicity
• 3 Robust Kalman Filtering for 2-D Systems with Multiplicative Noises and Measurement Degradations'
• 4 Robust Finite-Horizon Filtering for 2-D Systems with Randomly Varying Sensor Delays
• 5 Recursive Filtering for 2-D Systems with Missing Measurements Subject to Uncertain Probabilities
• 6 Resilient State Estimation for 2-D Shift-Varying Systems with Redundant Channels
• 7 Recursive Distributed Filtering for 2-D Shift-Varying Systems Over Sensor Networks Under Random Access Protocols
• 8 Resilient Filtering for Linear Shift-Varying Repetitive Processes under Uniform Quantizations and Round-Robin Protocols
• 9 Event-Triggered Recursive Filtering for Shift-Varying Linear Repetitive Processes
• 10 Conclusions and Future Topics
• Bibliography
• Index