Controllability of Singularly Perturbed Linear Time Delay Systems (Systems & Control: Foundations & Applications)

✍ Scribed by Valery Y. Glizer

• Publisher: Birkhäuser
• Year: 2021
• Tongue: English
• Leaves: 429
• Category: Library

✦ Synopsis

This monograph provides a comprehensive analysis of the control of singularly perturbed time delay systems. Expanding on the author’s previous work on controllability of linear systems with delays in the state and control variables, this volume’s comprehensive coverage makes it a valuable addition to the field. Each chapter is self-contained, allowing readers to study them independently or in succession.
After a brief introduction, the book systematically examines properties of different classes of singularly perturbed time delay systems, including linear time-dependent systems with multiple point-wise and distributed state delays. The author then considers more general singularly perturbed systems with state and control delays. Euclidean space controllability for all of these systems is also discussed, using numerous examples from real-life models throughout the text to illustrate the results presented. More technically complicated proofs are presented in separate subsections. The final chapter includes a section dedicated to non-linear time delay systems.
This book is ideal for researchers, engineers, and graduate students in systems science and control theory. Other applied mathematicians and researchers working in biology and medicine will also find this volume to be a valuable resource.

✦ Table of Contents

Contents
1 Introduction
1.1 Real-Life Models
1.1.1 Neurosystem Model
1.1.2 Sunflower Equation
1.1.3 Model of Nuclear Reactor Dynamics
1.1.4 Model of Controlled Coupled-Core Nuclear Reactor
1.1.5 Car-Following Model: Lane as a Simple Open Curve
1.1.6 Car-Following Model: Lane as a Simple Closed Curve
References
2 Singularly Perturbed Linear Time Delay Systems
2.1 Introduction
2.2 Singularly Perturbed Systems with Small Delays
2.2.1 Original System
2.2.2 Slow–Fast Decomposition of the Original System
2.2.3 Fundamental Matrix Solution
2.2.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Small Delays
2.2.5 Example 1
2.2.6 Example 2: Tracking Model with Delay
2.2.7 Example 3: Analysis of Neurosystem Model
2.2.8 Example 4: Analysis of Sunflower Equation
2.2.9 Proof of Lemma 2.2
2.2.10 Proof of Theorem 2.1
2.2.10.1 Technical Proposition
2.2.10.2 Main Part of the Proof
2.3 Singularly Perturbed Systems with Delays of Two Scales
2.3.1 Original System
2.3.2 Slow–Fast Decomposition of the Original System
2.3.3 Fundamental Matrix Solution
2.3.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Delays of Two Scales
2.3.5 Example 5
2.3.6 Example 6: Dynamics of Nuclear Reactor
2.3.7 Example 7: Analysis of Car-Following Model in a Simple Closed Lane
2.3.8 Proof of Theorem 2.2
2.4 One Class of Singularly Perturbed Systems with NonsmallDelays
2.4.1 Original System
2.4.2 Slow–Fast Decomposition of the Original System
2.4.3 Fundamental Matrix Solution
2.4.4 Estimates of Solutions to Singularly Perturbed Matrix Differential Systems with Nonsmall Delays
2.4.5 Example 8
2.4.6 Proof of Lemma 2.4
2.4.7 Proof of Theorem 2.4
2.5 Concluding Remarks and Literature Review
References
3 Euclidean Space Output Controllability of Linear Systems with State Delays
3.1 Introduction
3.2 Systems with Small Delays: Main Notions and Definitions
3.2.1 Original System
3.2.2 Asymptotic Decomposition of the Original System
3.3 Auxiliary Results
3.3.1 Output Controllability of a System with State Delays: Necessary and Sufficient Conditions
3.3.2 Linear Control Transformation in Systems with Small Delays
3.3.2.1 Control Transformation in the Original System
3.3.2.2 Asymptotic Decomposition of the Transformed System (3.30)–(3.31), (3.3)
3.3.3 Hybrid Set of Riccati-Type Matrix Equations
3.3.4 Proof of Lemma 3.1
3.3.4.1 Sufficiency
3.3.4.2 Necessity
3.3.5 Proof of Lemma 3.5
3.3.6 Proof of Lemma 3.7
3.3.7 Proof of Lemma 3.8
3.3.8 Proof of Lemma 3.9
3.4 Parameter-Free Controllability Conditions for Systems with Small Delays
3.4.1 Case of the Standard System (3.1)–(3.2)
3.4.2 Case of the Nonstandard System (3.1)–(3.2)
3.4.3 Proofs of Theorems 3.1, 3.2, and 3.3
3.4.3.1 Proof of Theorem 3.1
3.4.3.2 Proof of Theorem 3.2
3.4.3.3 Proof of Theorem 3.3
3.5 Special Cases of Controllability for Systems with Small Delays
3.5.1 Complete Euclidean Space Controllability
3.5.2 Controllability with Respect to x(t)
3.5.3 Controllability with Respect to y(t)
3.6 Examples: Systems with Small Delays
3.6.1 Example 1
3.6.2 Example 2
3.6.3 Example 3
3.6.4 Example 4
3.6.5 Example 5
3.6.6 Example 6: Pursuit-Evasion Engagement with Constant Speeds of Participants
3.6.7 Example 7: Pursuit-Evasion Engagement with Variable Speeds of Participants
3.6.8 Example 8: Analysis of Controlled Coupled-Core Nuclear Reactor Model
3.7 Systems with Delays of Two Scales: Main Notionsand Definitions
3.7.1 Original System
3.7.2 Asymptotic Decomposition of the Original System
3.8 Linear Control Transformation in Systems with Delays of Two Scales
3.8.1 Control Transformation in the Original System
3.8.2 Asymptotic Decomposition of the Transformed System (3.196)–(3.197), (3.187)
3.9 Parameter-Free Controllability Conditions for Systems with Delays of Two Scales
3.9.1 Case of the Validity of the Assumption (AIII)
3.9.2 Case of the Validity of the Assumption (AIV)
3.9.3 Special Cases of Controllability
3.9.3.1 Complete Euclidean Space Controllability
3.9.3.2 Controllability with Respect to x(t)
3.9.3.3 Controllability with Respect to y(t)
3.9.4 Example 9
3.9.5 Example 10
3.9.6 Example 11: Controlled Car-Following Model in a Simple Open Lane
3.10 Concluding Remarks and Literature Review
References
4 Complete Euclidean Space Controllability of Linear Systems with State and Control Delays
4.1 Introduction
4.2 System with Small State Delays: Main Notions and Definitions
4.2.1 Original System
4.2.2 Asymptotic Decomposition of the Original System
4.3 Preliminary Results
4.3.1 Auxiliary System with Small State Delays and Delay-Free Control
4.3.2 Output Controllability of the Auxiliary System and Its Slow and Fast Subsystems: Necessary and Sufficient Conditions
4.3.2.1 Equivalent Forms of the Auxiliary System
4.3.2.2 Output Controllability of the Auxiliary System
4.3.2.3 Output Controllability of the Slow and Fast Subsystems Associated with the Auxiliary System
4.3.3 Linear Control Transformation in the Original System with Small State Delays
4.3.4 Stabilizability of a Parameter-Dependent System with State and Control Delays by a Memory-Less Feedback Control
4.3.5 Proof of Lemma 4.8
4.4 Parameter-Free Controllability Conditions for Systems with Small State Delays
4.4.1 Case of the Standard System (4.1)–(4.2)
4.4.2 Case of the Nonstandard System (4.1)–(4.2)
4.4.3 Proof of Main Lemma (Lemma 4.9)
4.4.3.1 Auxiliary Propositions
4.4.3.2 Main Part of the Proof
4.4.4 Alternative Approach to Controllability Analysis of the Nonstandard System (4.1)–(4.2)
4.4.4.1 Linear Control Transformation in the Auxiliary System (4.40)–(4.42)
4.4.4.2 Proof of Lemma 4.10
4.4.4.3 Hybrid Set of Riccati-Type Matrix Equations
4.4.4.4 Parameter-Free Controllability Conditions of the Nonstandard System (4.1)–(4.2)
4.5 Examples: Systems with Small State and Control Delays
4.5.1 Example 1
4.5.2 Example 2
4.5.3 Example 3
4.6 Systems with State Delays of Two Scales: Main Notions and Definitions
4.6.1 Original System
4.6.2 Asymptotic Decomposition of the Original System
4.7 Auxiliary System with State Delays of Two Scales and Delay-Free Control
4.7.1 Description of the Auxiliary System and Some of Its Properties
4.7.2 Asymptotic Decomposition of the Auxiliary System (4.180)–(4.181)
4.7.3 Linear Control Transformation in the Auxiliary System (4.180)–(4.181)
4.8 Parameter-Free Controllability Conditions for Systems with Delays of Two Scales
4.8.1 Case of the Validity of the Assumption (AV)
4.8.2 Case of the Validity of the Assumption (AVI)
4.8.3 Example 4
4.8.4 Example 5
4.8.5 Example 6: Analysis of Car-Following Model with State and Control Delays
4.9 Concluding Remarks and Literature Review
References
5 First-Order Euclidean Space Controllability Conditions for Linear Systems with Small State Delays
5.1 Introduction
5.2 Singularly Perturbed System: Main Notions and Definitions
5.2.1 Original System
5.2.2 Asymptotic Decomposition of the Original System
5.3 Auxiliary Results
5.3.1 Estimates of Solutions to Some Singularly Perturbed Linear Time Delay Matrix Differential Equations
5.3.2 Proof of Lemma 5.1
5.3.2.1 Technical Proposition
5.3.2.2 Main Part of the Proof
5.3.3 Complete Controllability of the Original System and Its Slow Subsystem: Necessary and SufficientConditions
5.4 Parameter-Free Controllability Conditions
5.4.1 Formulation of Main Assertions
5.4.2 Proof of Theorem 5.1
5.4.3 Proof of Lemma 5.2
5.4.4 Proof of Theorem 5.2
5.4.4.1 Euclidean Space Controllability of a Pure Fast System
5.4.4.2 Main Part of the Proof
5.5 Examples
5.5.1 Example 1
5.5.2 Example 2
5.5.3 Example 3
5.5.4 Example 4
5.5.5 Example 5
5.5.6 Example 6
5.5.7 Example 7: Analysis of Controlled Car-Following Model in a Simple Open Lane
5.6 Concluding Remarks and Literature Review
References
6 Miscellanies
6.1 Introduction
6.2 Euclidean Space Controllability of Linear Time Delay Systems with High Gain Control
6.2.1 High Gain Control System: Main Notionsand Definitions
6.2.1.1 Initial System
6.2.1.2 Transformation of the System (6.1)
6.2.2 High Dimension Controllability Condition for the System (6.5)
6.2.3 Asymptotic Decomposition of the System (6.5)
6.2.4 Auxiliary Results
6.2.4.1 Linear Control Transformation in the System (6.13)–(6.14) and Some of its Properties
6.2.4.2 Asymptotic Decomposition of the Transformed System (6.13), (6.21)
6.2.4.3 Block-Wise Estimate of the Solution to the Terminal-Value Problem (6.23)
6.2.5 Lower Dimension Parameter-Free Controllability Condition for the System (6.5)
6.2.6 Example
6.3 Euclidean Space Controllability of Linear Systems with Nonsmall Input Delay
6.3.1 Original System
6.3.2 Discussion on the Slow–Fast Decomposition of the Original System
6.3.3 Auxiliary Results
6.3.3.1 Necessary and Sufficient Controllability Conditions of the Original System
6.3.3.2 Block-Wise Estimate of the Solution to the Terminal-Value Problem (6.58)
6.3.3.3 Asymptotic Analysis of the Controllability Matrix W(tc,)
6.3.4 Parameter-Free Controllability Conditions
6.3.5 Example 1
6.3.6 Example 2
6.4 Functional Null Controllability of Some Nonlinear Systems with Small State Delays
6.4.1 Problem Formulation
6.4.2 Preliminary Results
6.4.2.1 Euclidean Space Controllability of Singularly Perturbed Linear Time Delay System
6.4.2.2 Euclidean Space Controllability of the Original System (6.113)–(6.114)
6.4.3 System of the First Type: Formulation and Some Auxiliary Results
6.4.3.1 Asymptotic Decomposition of the System (6.124)–(6.125)
6.4.3.2 Controllability Conditions for the Slow and Fast Subsystems
6.4.3.3 Parameter-Free Conditions for the Complete Euclidean Space Controllability of the Linear System (6.124)–(6.125)
6.4.3.4 Euclidean Space Null Controllability of the Nonlinear System (6.122)–(6.123): Parameter-Free Conditions
6.4.4 System of the First Type: Parameter-Free Conditions for the Functional Null Controllability
6.4.5 System of the Second Type: Formulation and Some Auxiliary Results
6.4.5.1 Asymptotic Decomposition of the System (6.155)–(6.156)
6.4.5.2 Parameter-Free Conditions for the Complete Euclidean Space Controllability of the Linear System (6.155)–(6.156)
6.4.5.3 Euclidean Space Null Controllability of the Nonlinear System (6.153)–(6.154): Parameter-Free Conditions
6.4.6 System of the Second Type: Parameter-Free Conditions for the Functional Null Controllability
6.4.7 Example 1
6.4.8 Example 2: Analysis of Controlled Sunflower Equation
6.4.9 Example 3
6.5 Some Open Problems
6.5.1 Complete Euclidean Space Controllability of Linear Systems with State Delays and Nonsmall ControlDelays
6.5.2 Euclidean Space Controllability of Linear Systems with Nonsmall Delays
6.5.3 Complete Euclidean Space Controllability of One Class of Nonlinear Systems with Small State Delays
6.6 Concluding Remarks and Literature Review
References
Index