Controlling Delayed Dynamics: Advances in Theory, Methods and Applications

Preface
Contents
The Twin Semigroup Approach Towards Periodic Neutral Delay Equations
	1 Introduction
	2 Introduction to NFDE
	3 Norming Dual Pairs and Twin Semigroups
	4 The Norming Dual Pair (B,NBV)
	5 The Twin Semigroup Approach to NFDE
	6 The Variation-of-Constants Formula for NFDE
	7 Bounded Time-Dependent Perturbation of a Twin Semigroup
	8 A Perturbation Approach Towards Periodic NFDE
	A Review of Functions of Bounded Variation
	References
Characteristic Matrix Functions and Periodic Delay Equations
	1 Introduction
	2 Equivalence and Jordan Chains
	3 Introduction to the Theory of Characteristic Matrix Functions
	4 The Period Map of a Neutral Periodic Delay Equation
	5 Scalar Periodic Delay Equations of Period One
	6 Scalar Periodic Delay Equations (Two Periodic)
	References
Pseudospectral Methods for the Stability Analysis of Delay Equations. Part I: The Infinitesimal Generator Approach
	1 Introduction
	2 Semigroups of Solution Operators and Infinitesimal Generator
		2.1 Delay Differential Equations
		2.2 Renewal Equations
		2.3 Coupled Equations
		2.4 Principle of Linearized Stability
	3 Basics of Polynomial Interpolation and Pseudospectral Methods
	4 Pseudospectral Discretization of the Infinitesimal Generator
		4.1 Delay Differential Equations
		4.2 Renewal Equations
		4.3 Coupled Equations
		4.4 Convergence and Other Issues
	5 Extension to Nonlinear Problems
	6 Implementation and Results
		6.1 Example 1: A Delay Differential Equation.
		6.2 Example 2: A Renewal Equation
		6.3 Example 3: A Coupled Equation.
		6.4 Example 4: The Nonlinear Approach for a Delay Differential Equation
	7 Conclusions
	References
Pseudospectral Methods for the Stability Analysis of Delay Equations. Part II: The Solution Operator Approach
	1 Introduction
	2 Evolution Families
		2.1 Autonomous Problems and Stability of Equilibria
		2.2 Periodic Problems and Stability of Periodic Orbits
		2.3 Generic Problems and Detection of Chaos
	3 Pseudospectral Discretization of the Evolution Family
		3.1 Computation of Characteristic Roots
		3.2 Computation of Floquet Multipliers
		3.3 Computation of Lyapunov Exponents
	4 Extension to Renewal and Coupled Equations
	5 Results and Applications
		5.1 Tests on the Mackey-Glass Equation
		5.2 Tests on a Logistic RE
	6 Conclusions
	References
Counting Characteristic Roots of Linear Delay Differential Equations. Part I: Frequency-Sweeping Stability Tests and Applications
	1 Introduction
	2 Preliminaries and Prerequisites
		2.1 Linear Time-Delay Systems
		2.2 Characteristic Roots and Delay Parameter
		2.3 Stability Problem and the Delay Parameter
	3 Frequency-Sweeping Curves
	4 Asymptotic Behavior of a Critical Imaginary Root at a Critical Delay
	5 Invariance Property of Asymptotic Behavior
	6 A Unified Frequency-Sweeping Approach for Complete Stability Problem
		6.1 Computation of NU(+ ε)
		6.2 Explicit NU(τ) Expression
		6.3 Further Classification
		6.4 Procedure for Complete Stability Analysis
	7 Further Extensions of the Frequency-Sweeping Approach
		7.1 Neutral Delay Differential Equations
		7.2 Distributed Delay Differential Equations with Uniform Distribution
		7.3 Delay Differential Equations with Multiple Incommensurate Delays
	8 Applications
		8.1 Neural Network Dynamical Systems
		8.2 Lotka-Volterra Systems
	9 Notes and Comments
	References
Counting Characteristic Roots of Linear Delay Differential Equations. Part II: From Argument Principle to Rightmost Root Assignment Methods
	1 Introduction
	2 Preliminaries and Prerequisites
		2.1 On Integration Contours for Quasipolynomials Corresponding to DDEs of Retarded Type
		2.2 Pólya-Szegö Theorem: Counting Quasipolynomial Roots
	3 Characterizing Multiplicity Using Structured Matrices
		3.1 Singularity Codimension May Exceed the Model Order:Bogdanov-Takens Singularity
		3.2 Codimension Counting: A Vandermonde/Birkhoff-Based Framework
		3.3 Codimension of Zero Singularities of DDEs
	4 Multiple Induced Dominancy and Partial Pole Assignment: Comprehensive Examples
		4.1 Exponential Decay Rate of a Scalar DDE with a Single Delay
		4.2 Multiple Spectral Values for DDEs Systems are Not Necessarily Dominant
		4.3 Stabilizing an Oscillator Via a Delayed Output-Feedback
	5 Parametric MID for Second-Order Systems
		5.1 Open-Loop Systems with One Oscillating Mode
		5.2 Open-Loop Systems with Non Oscillating Modes
	6 The Generic MID Property
		6.1 Degenerate Hypergeometric Functions
		6.2 Spectral Values of Maximal Multiplicity are Dominant
	7 Software: Partial Pole Placement via Delay Action
	8 Active Vibration Control in a Mechanical Flexible Structure
		8.1 Modeling of the Vibrating Beam
		8.2 Vibration Damping
	9 Notes and Comments
	References
Bifurcation Analysis of Systems With Delays: Methods and Their Use in Applications
	1 Introduction
	2 DDEs as Dynamical Systems
		2.1 General Theory for DDEs With Constant Delays
		2.2 General Theory for DDEs With State-Dependent Delays
	3 Capabilities of DDE-BIFTOOL Demonstrated for the Controlled Inverted Pendulum
		3.1 DDE Model of the Controlled Inverted Pendulum
		3.2 Continuation of Branches of Equilibria
		3.3 Linear Stability Analysis of Equilibria
		3.4 Continuation of Codimension-One Bifurcations of Equilibria
		3.5 Codimension-Three Singularity of the Inverted Pendulum
		3.6 Continuation of Periodic Orbits
		3.7 Linear Stability Analysis for Periodic Orbits
		3.8 Continuation of Codimension-One Bifurcations of Periodic Orbits
		3.9 Symmetric and Non-Symmetric Chaos in the Pendulum
		3.10 Some Experimental Features of DDE-BIFTOOL
		3.11 DDE-BIFTOOL Formulation for Other Types of DDEs
	4 An ENSO DDE Model With State Dependence
		4.1 The Delayed Action Oscillator Paradigm
		4.2 The GZT Model
		4.3 State Dependence Due to Upwelling and Ocean Adjustment
		4.4 The GZT Model With Upwelling and Ocean Adjustment
	5 Resonance Phenomena in a Scalar DDE With Two State-Dependent Delays
		5.1 Hopf-Hopf Bifurcation as an Organizing Center
		5.2 Finding and Representing Smooth Invariant Tori
		5.3 Locked Nonsmooth Invariant Tori
	6 Conclusions and Outlook
	References
Design of Structured Controllers for Linear Time-Delay Systems
	1 Introduction
	2 Solving Analysis Problems
		2.1 Computation of Characteristic Roots and the Spectral Abscissa
		2.2 Computation of mathcalHinfty Norms
		2.3 Computation of mathcalH2 Norms
	3 Making the Leap From Analysis to Synthesis
		3.1 Stabilization
		3.2 Optimizing mathcalHinfty and mathcalH2 Norms
	4 Case Studies
	5 Equations of Neutral Type and Delay Differential Algebraic Equations
		5.1 Preliminaries and Assumptions
		5.2 Spectral Properties and Stability
		5.3 Robust Stabilization by Eigenvalue Optimization
		5.4 Examples
		5.5 Note on the Strong mathcalH2 and mathcalHinfty Norm
	6 Concluding Remarks
	References
A Scalable Controller Synthesis Method for the Robust Control of Networked Systems
	1 Introduction
	2 Computing the H-Infinity Norm of Networked Systems
		2.1 System Description and Control Objective
		2.2 Upper Bound for the H-Infinity Norm
	3 Computing the Robust H-Infinity Norm
		3.1 Relation With the Robust Stability Radius
		3.2 Numerical Algorithm
	4 A Scalable H-Infinity Controller Synthesis Method
	5 Example
	6 Conclusion
	References
Regenerative Machine Tool Vibrations
	1 Introduction
	2 Turning Operations
		2.1 Mechanical Model
		2.2 Stability Analysis
		2.3 Phase Shift Along the Stability Lobes
	3 Milling Operations
		3.1 Mechanical Model
		3.2 Stability Analysis by Semidiscretization
		3.3 Chatter Frequencies in Milling Operations
		3.4 Stability Lobe Diagrams for Milling
		3.5 Milling with Helical Tools
	4 Final Comments
	References
Dynamics of Human Balancing
	1 Introduction
	2 Mechanical Models
		2.1 Human Postural Sway: The Pinned Inverted Pendulum Model
		2.2 Stick Balancing: The Pendulum-Cart Model
		2.3 Unifying the Two Models
	3 Time-Delayed Feedback Control
		3.1 Proportional-Derivative (PD) Feedback
		3.2 Proportional-Derivative-Acceleration (PDA) Feedback
		3.3 Predictor Feedback (PF)
		3.4 PD and PDA as Predictor Feedback
	4 Stability Analysis
		4.1 PD Feedback
		4.2 PDA Feedback
		4.3 Predictor Feedback
	5 Critical Parameters
		5.1 PD Feedback
		5.2 PDA Feedback
		5.3 PF Feedback
	6 Conclusions
	References