Interval Analysis

Navid Razmjooy, PhD, is an independent researcher based in Belgium. He holds a Ph.D. in Electrical Engineering (Control and Automation) from Tafresh University in Iran. His research is focused on renewable energies, interval analysis, image processing, machine vision, data mining, evolutionary algorithms, and system control.

• About the Author viii1 Preface and Overview 1Chapter 1: Preface and Overview 1Chapter 2: Introduction 2Chapter 3: Literature Review 2Chapter 4: Introduction to Interval Analysis and Solving the Problems with Interval Uncertainties 2Chapter 5: Stability and Controllability Based on Interval Analysis 3Chapter 6: Optimal Control of the Systems with Interval Uncertainties 3Chapter 7: Conclusions 32 Introduction 52.1 Background 52.2 Relationship Between Error and Uncertainty 72.3 Expert Perspectives on Interval Analysis 82.4 Precision Analysis of Interval-Based Models (Self-Validated Numeric) 102.5 Concepts of the Ordinary and Interval-based Optimal Control 112.6 Orthogonal Spectral Methods 132.7 Desired Confidence Interval 132.8 Conclusions 143 Literature Review 174 Introduction to Interval Analysis and Solving the Problems with Interval Uncertainties 294.1 Introduction 294.2 Introduction to IA 304.3 The Algebra of Interval Sets 314.3.1 Classic Interval Algebra (Minkowski Method) 314.3.2 Mathematical Norm and Distance in the IA 334.3.3 Kaucher Extended Interval Analysis 334.3.4 Modal Interval Analysis 344.3.5 Hukuhara Difference Method 354.4 Interval Representations 364.5 Interval Complex Integers 394.6 Interval Vector and Matrix 414.6.1 Vector and Matrix IA 414.6.2 Interval Vector Norm 424.6.3 Interval Matrix Analysis 424.6.3.1 Conceptional Example: Wrapping Effect 434.6.4 Interval Matrices Norm 444.6.5 Interval Determination of a Matrix 454.6.6 The Inverse of a Regular Interval Matrix 454.6.7 Eigenvalues and Eigenvectors of an Interval Matrix 454.7 Solving Linear Systems with Interval Parameters 464.8 Interval Functions 464.8.1 Overestimated Interval Function 464.8.2 Minimal Interval Function 484.9 Determining the Minimal Interval 494.9.1 Uniform Interval Functions 494.9.2 Nonuniform Interval Functions 494.9.3 Interval Power Series 504.10 Interval Derivative and Integral Functions 524.10.1 Interval Derivative 524.10.2 Interval Integration 534.11 Centered Inclusion Method 544.11.1 Linearized Interval Functions Around the Center 544.11.2 Taylor Inclusion Functions 544.12 Interval Nonlinear Systems 564.13 Analysis of the Interval Dynamic Systems in the Presence of Interval Uncertainties 574.13.1 Solving the Interval Initial Value Problems 584.14 The Interval Runge–Kutta Method (IRKM) for Interval Differential Equations 604.14.1 Introduction 604.14.2 Generalized IRKM (GIRKM) Based on Switching Points 634.14.3 Numerical Examples 664.15 Interval Uncertainty Analyses based on Orthogonal Functions 784.15.1 Interval Ε-orthogonal 794.15.2 Interval Weierstrass's Theorem 794.16 Interval Orthogonal Polynomials 794.16.1 Legendre Polynomials 804.16.2 Chebyshev Polynomials 804.16.3 Interval Orthogonal Functions 824.17 Piecewise Extension of the Interval Orthogonal Functions 834.18 Conclusion 855 Stability and Controllability Based on Interval Analysis 915.1 Introduction 915.1.1 Classical Control Theory 915.1.2 Advanced Modern Control Systems Theory 925.1.3 Optimal Control 935.1.4 Robust Control 945.1.5 Adaptive Control Theory 955.2 Interval Stability and Controllability 955.3 Interval Stability 965.4 Characteristic Polynomial 975.5 Routh–Hurwitz Stability Test 985.6 Kharitonov's Theorem (Interval Routh–Hurwitz Stability Test) 995.6.1 Kharitonov Polynomial Theory 995.6.2 A Centered Representation of the Interval Routh–Hurwitz Stability Criterion 1025.7 Interval Stability Based on Linear Matrix Inequalities 1025.7.1 The Positive Matrix of the Interval Matrix 1025.7.2 Stability Analysis of the Interval Systems 1035.7.3 Linear Matrix Inequalities 1045.8 Controllability and Observability 1075.9 Controllability and Observability Based on Interval Criteria 1075.9.1 Singular Values for Analyzing the Controllability 1115.10 Conclusions 1176 Optimal Control of the Systems with Interval Uncertainties 1216.1 Introduction 1216.2 Indirect Methods 1236.3 Direct Methods 1236.4 Optimal Control Problem in the Presence of Interval Uncertainties 1276.5 Interval Optimal Control Based on the Indirect Method 1286.5.1 Analysis of the Standard Interval Calculus of Variations 1296.6 Analysis of the Problem of the Interval Optimal Control Based on Euler–Lagrange Equations 1316.7 Solving Optimal Control Problems with Interval Uncertainties: Interval Runge–Kutta Method 1326.8 Optimal Control of Problems with Interval Uncertainties Using the Chebyshev Inclusion Method 1466.9 Piecewise Interval Chebyshev Method for OCPs 1526.10 Solving Quadratic Optimal Control Problems with Interval Uncertainties Based on Indirect Method: Interval Quadratic Regulator 1596.11 Problem Statement (Interval Quadratic Regulator) 1736.12 Interval Optimal Control Based on Direct Method 1796.13 Applied Simulations 1836.14 Conclusion 192References 1937 Conclusions 197Index 199