Interval Analysis

About the Author viii 1 Preface and Overview 1 Chapter 1: Preface and Overview 1 Chapter 2: Introduction 2 Chapter 3: Literature Review 2 Chapter 4: Introduction to Interval Analysis and Solving the Problems with Interval Uncertainties 2 Chapter 5: Stability and Controllability Based on Interval Analysis 3 Chapter 6: Optimal Control of the Systems with Interval Uncertainties 3 Chapter 7: Conclusions 3 2 Introduction 5 2.1 Background 5 2.2 Relationship Between Error and Uncertainty 7 2.3 Expert Perspectives on Interval Analysis 8 2.4 Precision Analysis of Interval-Based Models (Self-Validated Numeric) 10 2.5 Concepts of the Ordinary and Interval-based Optimal Control 11 2.6 Orthogonal Spectral Methods 13 2.7 Desired Confidence Interval 13 2.8 Conclusions 14 3 Literature Review 17 4 Introduction to Interval Analysis and Solving the Problems with Interval Uncertainties 29 4.1 Introduction 29 4.2 Introduction to IA 30 4.3 The Algebra of Interval Sets 31 4.3.1 Classic Interval Algebra (Minkowski Method) 31 4.3.2 Mathematical Norm and Distance in the IA 33 4.3.3 Kaucher Extended Interval Analysis 33 4.3.4 Modal Interval Analysis 34 4.3.5 Hukuhara Difference Method 35 4.4 Interval Representations 36 4.5 Interval Complex Integers 39 4.6 Interval Vector and Matrix 41 4.6.1 Vector and Matrix IA 41 4.6.2 Interval Vector Norm 42 4.6.3 Interval Matrix Analysis 42 4.6.3.1 Conceptional Example: Wrapping Effect 43 4.6.4 Interval Matrices Norm 44 4.6.5 Interval Determination of a Matrix 45 4.6.6 The Inverse of a Regular Interval Matrix 45 4.6.7 Eigenvalues and Eigenvectors of an Interval Matrix 45 4.7 Solving Linear Systems with Interval Parameters 46 4.8 Interval Functions 46 4.8.1 Overestimated Interval Function 46 4.8.2 Minimal Interval Function 48 4.9 Determining the Minimal Interval 49 4.9.1 Uniform Interval Functions 49 4.9.2 Nonuniform Interval Functions 49 4.9.3 Interval Power Series 50 4.10 Interval Derivative and Integral Functions 52 4.10.1 Interval Derivative 52 4.10.2 Interval Integration 53 4.11 Centered Inclusion Method 54 4.11.1 Linearized Interval Functions Around the Center 54 4.11.2 Taylor Inclusion Functions 54 4.12 Interval Nonlinear Systems 56 4.13 Analysis of the Interval Dynamic Systems in the Presence of Interval Uncertainties 57 4.13.1 Solving the Interval Initial Value Problems 58 4.14 The Interval Runge–Kutta Method (IRKM) for Interval Differential Equations 60 4.14.1 Introduction 60 4.14.2 Generalized IRKM (GIRKM) Based on Switching Points 63 4.14.3 Numerical Examples 66 4.15 Interval Uncertainty Analyses based on Orthogonal Functions 78 4.15.1 Interval Ε-orthogonal 79 4.15.2 Interval Weierstrass's Theorem 79 4.16 Interval Orthogonal Polynomials 79 4.16.1 Legendre Polynomials 80 4.16.2 Chebyshev Polynomials 80 4.16.3 Interval Orthogonal Functions 82 4.17 Piecewise Extension of the Interval Orthogonal Functions 83 4.18 Conclusion 85 5 Stability and Controllability Based on Interval Analysis 91 5.1 Introduction 91 5.1.1 Classical Control Theory 91 5.1.2 Advanced Modern Control Systems Theory 92 5.1.3 Optimal Control 93 5.1.4 Robust Control 94 5.1.5 Adaptive Control Theory 95 5.2 Interval Stability and Controllability 95 5.3 Interval Stability 96 5.4 Characteristic Polynomial 97 5.5 Routh–Hurwitz Stability Test 98 5.6 Kharitonov's Theorem (Interval Routh–Hurwitz Stability Test) 99 5.6.1 Kharitonov Polynomial Theory 99 5.6.2 A Centered Representation of the Interval Routh–Hurwitz Stability Criterion 102 5.7 Interval Stability Based on Linear Matrix Inequalities 102 5.7.1 The Positive Matrix of the Interval Matrix 102 5.7.2 Stability Analysis of the Interval Systems 103 5.7.3 Linear Matrix Inequalities 104 5.8 Controllability and Observability 107 5.9 Controllability and Observability Based on Interval Criteria 107 5.9.1 Singular Values for Analyzing the Controllability 111 5.10 Conclusions 117 6 Optimal Control of the Systems with Interval Uncertainties 121 6.1 Introduction 121 6.2 Indirect Methods 123 6.3 Direct Methods 123 6.4 Optimal Control Problem in the Presence of Interval Uncertainties 127 6.5 Interval Optimal Control Based on the Indirect Method 128 6.5.1 Analysis of the Standard Interval Calculus of Variations 129 6.6 Analysis of the Problem of the Interval Optimal Control Based on Euler–Lagrange Equations 131 6.7 Solving Optimal Control Problems with Interval Uncertainties: Interval Runge–Kutta Method 132 6.8 Optimal Control of Problems with Interval Uncertainties Using the Chebyshev Inclusion Method 146 6.9 Piecewise Interval Chebyshev Method for OCPs 152 6.10 Solving Quadratic Optimal Control Problems with Interval Uncertainties Based on Indirect Method: Interval Quadratic Regulator 159 6.11 Problem Statement (Interval Quadratic Regulator) 173 6.12 Interval Optimal Control Based on Direct Method 179 6.13 Applied Simulations 183 6.14 Conclusion 192 References 193 7 Conclusions 197 Index 199