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# Systems with Delays. Analysis, Control, and Computations. Edition No. 1

• ID: 5225027
• Book
• September 2015
• 184 Pages
• John Wiley and Sons Ltd
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The main aim of the book is to present new constructive methods of delay differential equation (DDE) theory and to give readers practical tools for analysis, control design and simulating of linear systems with delays.  Referred to as “systems with delays” in this volume, this class of differential equations is also called delay differential equations (DDE), time-delay systems, hereditary systems, and functional differential equations.  Delay differential equations are widely used for describing and modeling various processes and systems in different applied problems

At present there are effective control and numerical methods and corresponding software for analysis and simulating different classes of ordinary differential equations (ODE) and partial differential equations (PDE). There are many applications for these types of equations, because of this progress, but there are not as many methodologies in systems with delays that are easily applicable for the engineer or applied mathematician.  there are no methods of finding solutions in explicit forms, and there is an absence of generally available general-purpose software packages for simulating such systems.

Systems with Delays fills this void and provides easily applicable methods for engineers, mathematicians, and scientists to work with delay differential equations in their operations and research.

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Preface ix

1 Linear time-delay systems 1

1.1 Introduction 1

1.1.1 Linear systems with delays 1

1.1.2 Wind tunnel model 2

1.1.3 Combustion stability in liquid propellant rocket motors 3

1.2 Conditional representation of differential equations 5

1.2.1 Conditional representation of ODE and PDE 5

1.2.2 Conditional representation of DDE 6

1.3 Initial Value Problem. Notion of solution 9

1.3.1 Initial conditions (initial state) 9

1.3.2 Notion of a solution 10

1.4 Functional spaces 11

1.4.1 Space C[−τ,0] 12

1.4.2 Space Q[−τ,0] 12

1.4.3 Space Q[−τ,0) 13

1.4.4 Space H = Rη × Q[−τ,0) 14

1.5 Phase space H. State of time-delay system 15

1.6 Solution representation 16

1.6.1 Time-varying systems with delays 16

1.6.2 Time-invariant systems with delays 20

1.7 Characteristic equation and solution expansion into a series 24

1.7.1 Characteristic equation and eigenvalues 24

1.7.2 Expansion of solution into a series on elementary solutions 26

2 Stability theory 39

2.1 Introduction 29

2.1.1 Statement of the stability problem 30

2.1.2 Eigenvalues criteria of asymptotic stability 31

2.1.3 Stability via the fundamental matrix 32

2.1.4 Stability with respect to a class of functions 33

2.2 Lyapunov-Krasovskii functionals 36

2.2.1 Structure of Lyapunov-Krasovskii quadratic functionals 36

2.2.2 Elementary functionals and their properties 37

2.2.3 Total derivative of functionals with respect to systems with delays 40

2.3 Positiveness of functionals 46

2.3.1 Definitions 46

2.3.2 Sufficient conditions of positiveness 47

2.3.3 Positiveness of functionals 47

2.4 Stability via Lyapunov-Krasovskii functionals 49

2.4.1 Stability conditions in the norm - - ・ - - H 50

2.4.2 Stability conditions in the norm - - ・ - - 51

2.4.3 Converse theorem 52

2.4.4 Examples 53

2.5 Coefficient conditions of stability 54

2.5.1 Linear system with discrete delay 54

2.5.2 Linear system with distributed delays 56

3.1 Introduction 59

3.2 Statement of the problem 60

3.3 Explicit solutions of generalized Riccati equations 67

3.3.1 Variant 1 67

3.3.2 Variant 2 68

3.3.3 Variant 3 69

3.4 Solution of Exponential Matrix Equation 73

3.4.1 Stationary solution method 73

3.5 Design procedure 75

3.5.1 Variants 1 and 2 75

3.5.2 Variant 3 76

3.6 Design case studies 76

3.6.1 Example 1 76

3.6.2 Example 2 78

3.6.3 Example 3 78

3.6.4 Example 4 80

3.6.5 Example 5: Wind tunnel model 82

3.6.6 Example 6: Combustion stability in liquid propellant rocketmotors 84

4 Numerical methods 89

4.1 Introduction 89

4.2 Elementary one-step methods 91

4.2.1 Euler’smethod 92

4.2.2 Implicit methods (extrapolation) 95

4.2.3 Improved Euler’smethod 96

4.2.4 Runge-Kutta-like methods 97

4.3 Interpolation and extrapolation of the model pre-history 98

4.3.1 Interpolational operators 98

4.3.2 Extrapolational operators 100

4.3.3 Interpolation-Extrapolation operator 101

4.4 Explicit Runge-Kutta-like methods 102

4.5 Approximation orders of ERK-like methods 104

4.6 Automatic step size control 106

4.6.1 Richardson extrapolation 106

4.6.2 Automatic step size control 107

4.6.3 Embedded formulas 108

5 Appendix 111

5.1 i-Smooth calculus of functionals 111

5.1.1 Invariant derivative of functionals 111

5.1.2 Examples 116

5.2 Derivation of generalized Riccati equations 124

5.3 Explicit solutions of GREs (proofs of theorems) 134

5.3.1 Proof of Theorem 3.2 134

5.3.2 Proof of Theorem 3.3 137

5.3.3 Proof of Theorem 3.4 139

5.4 Proof of Theorem 1.1. (Solution representation) 139

Bibliography 143

Index 164

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A. V. Kim
A. V. Ivanov
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