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             DYNAMICAL SYSTEMS WITH SATURATION NONLINEARITIES:
                            ANALYSIS AND DESIGN

                 Derong Liu, General Motors NAO R&D Center
                 Anthony N. Michel, University of Notre Dame


Lecture Notes in Control and Information Sciences, Vol. 195
Springer-Verlag London Ltd 1994
xiv + 191 pages

ISBN 0-387-19888-1


This monograph which consists of three parts addresses topics in the
areas of control systems, signal processing, and neural networks.

In the first part, we investigate two fundamental issues for a class of
dynamical systems with saturation nonlinearities. For zero-input
continuous-time and discrete-time dynamical systems with saturation
nonlinearities, we establish several results for the global asymptotic
stability of the null solution. For the discrete-time case, we utilize
Lyapunov's Second Method to arrive at our results and we establish
necessary and sufficient conditions under which positive definite
matrices can be used to construct quadratic form Lyapunov functions.
For discrete-time dynamical systems with state saturation and control
constraints, we establish several sufficient conditions for null
controllability.

In the  second part, we establish  results for the stability analysis
of digital filters implemented in finite wordlength format. We consider
both one-dimensional and multidimensional state-space digital filters
endowed with overflow nonlinearities. We utilize the Second Method of
Lyapunov in our analysis. Our results greatly improve existing results
and appear to be the least conservative criteria for the subject
systems. The generalized overflow nonlinearity considered herein
includes the usual types of overflow characteristics used in practice
as special cases. Results of this part build on the developments of the
first part.

In the  third part, we consider analysis and design of a class of
feedback neural networks--neural networks with linear saturation
activation functions. We establish results which enable us to determine
all the equilibrium points and their qualitative properties in a
systematic manner for a given neural network, and results which enable
us to determine allowable upper bounds for parameter perturbations. We
also develop synthesis procedures for networks with various connectivity
constraints. Our focus is on various problems which arise in the
analog VLSI implementations of neural networks which are of great
practical interest. This work constitutes the first successful synthesis
procedure for associative memories by means of artificial neural
networks with arbitrarily prespecified full or partial interconnecting
structure and with or without symmetry constraints for the connection
matrix.


        TABLE OF CONTENTS

Chapter 1:   INTRODUCTION TO DYNAMICAL SYSTEMS WITH SATURATION
             NONLINEARITIES

1.1     Dynamical Systems and Saturation Nonlinearities
1.2     Control Systems with Saturation Nonlinearities
1.3     Digital Filters with Overflow Corrections
1.4     Feedback Neural Networks and Associative Memories

PART I:  QUALITATIVE THEORY OF CONTROL SYSTEMS WITH CONTROL CONSTRAINTS
         AND STATE SATURATION: TWO FUNDAMENTAL ISSUES

Chapter 2:   INTRODUCTION TO PART I

2.1     Concepts of Stability and Lyapunov Functions
2.2     Principal Lyapunov Stability Theorems
2.3     Controllability

Chapter 3:   ASYMPTOTIC STABILITY OF DYNAMICAL SYSTEMS WITH STATE
             SATURATION

3.1     Introduction to Continuous-Time Systems
3.2     Notation
3.3     Main Result for Continuous-Time Systems
3.4     Introduction to Discrete-Time Systems
3.5     A General Result for Discrete-Time Systems
3.6     Results Involving Quadratic Form Lyapunov Functions
3.7     Examples
3.8     Concluding Remarks

Chapter 4:   NULL CONTROLLABILITY OF DISCRETE-TIME DYNAMICAL SYSTEMS
             WITH CONTROL CONSTRAINTS AND STATE SATURATION

4.1     Introduction
4.2     Main Results
4.3     Examples
4.4     Concluding Remarks

PART II:  STABILITY ANALYSIS OF ONE-DIMENSIONAL AND MULTIDIMENSIONAL
          STATE-SPACE DIGITAL FILTERS ENDOWED WITH OVERFLOW
          NONLINEARITIES

Chapter 5:   INTRODUCTION TO PART II

5.1     Fixed-Point Digital Filters and Overflow Nonlinearities
5.2     Limit Cycles in Fixed-Point Digital Filters
5.3     Multidimensional Digital Filters

Chapter 6:   CRITERIA FOR THE ABSENCE OF OVERFLOW OSCILLATIONS IN
             FIXED-POINT DIGITAL FILTERS USING GENERALIZED OVERFLOW
             CHARACTERISTICS

6.1     Introduction
6.2     Some Existing Results
6.3     Digital Filters Using Saturation Arithmetic
6.4     Digital Filters Using Generalized Overflow Characteristics
6.5     Algorithms for Determining the Matrix H
6.6     Examples
6.7     Concluding Remarks

Chapter 7:   STABILITY ANALYSIS OF STATE-SPACE REALIZATIONS FOR
             MULTIDIMENSIONAL FILTERS WITH OVERFLOW NONLINEARITIES

7.1     Introduction
7.2     General Results for Two-Dimensional Digital Filters
7.3     Main Result for Two-Dimensional Digital Filters
7.4     Multidimensional Digital Filters with Overflow Nonlinearities
7.5     Examples
7.6     Concluding Remarks

PART III:  ANALYSIS AND SYNTHESIS OF A CLASS OF NEURAL NETWORKS WITH
           INTERCONNECTION CONSTRAINTS WITH APPLICATIONS TO ASSOCIATIVE
           MEMORIES

Chapter 8:   INTRODUCTION TO PART III

8.1     Models of Feedback Neural Networks
8.2     Design Methods for Associative Memories
8.3     Analog VLSI Implementations and Problems
8.4     Cellular Neural Networks

Chapter 9:   ANALYSIS AND SYNTHESIS OF A CLASS OF NEURAL NETWORKS WITH
             PIECEWISE LINEAR SATURATION ACTIVATION FUNCTIONS

9.1     Introduction
9.2     Notation
9.3     Analysis Results
9.4     Synthesis Procedures
9.5     An Example
9.6     Concluding Remarks

Chapter 10:   SPARSELY INTERCONNECTED NEURAL NETWORKS FOR ASSOCIATIVE
              MEMORIES WITH APPLICATIONS TO CELLULAR NEURAL NETWORKS

10.1    Introduction
10.2    Sparse Design Procedures
10.3    Applications to Cellular Neural Networks
10.4    Examples
10.5    Chinese Character Recognition
10.6    Concluding Remarks

Chapter 11:   ROBUSTNESS ANALYSIS OF A CLASS OF SPARSELY INTERCONNECTED
              NEURAL NETWORKS WITH APPLICATIONS TO THE DESIGN PROBLEM

11.1    Introduction
11.2    Robustness Analysis
11.3    Applications to the Design Problem
11.4    Examples
11.5    Concluding Remarks

CLOSING REMARKS

BIBLIOGRAPHY

ABOUT THE AUTHORS