Uncertain Models and Robust Control

Functions.- 37 System Analysis Via Pulse Functions and Piecewise Linear Functions.- 38 Orthogonal Decomposition Applications.- A Matrix Algebra and Control.- A.1 Matrix Multiplication.- A.2 Properties of Matrix Operations.- A.3 Diagonal Matrices.- A.4Triangular Matrices.- A.5 Column Matrices (Vectors) and Row Matrices.- A.6 Reduced Matrix, Minor, Cofactor, Adjoint.- A.7 Similar Matrices.- A.8 Some Properties of Determinants.- A.9 Singularity.- A.10 System of Linear Equations.- A.11 Stable Matrices.- A.12 Range Space. Rank. Null Space.- A.13 Trace.- A.14 Matrix Functions.- A.15 Metzler Matrices.- A.16 Projectors.- A.17 Projectors and Rank.- A.18 Projectors. Left-Inverse and Right-Inverse.- A.19 Trigonal Operator.- A.20 Transfer Function Zeros and Initial Step Transients.- A.21 Convolution Sum and TrigonalOperator.- B Eigenvalues and Eigenvectors.- B.1 Right-Eigenvectors.- B.2 Left-Eigenvectors.- B.3 Complex-Conjugate Eigenvalues.- B.4 Modal Matrix of Eigenvectors.- B.5 Complex Matrices.- B.6 Modal Decomposition.- B.7 Linear Differential Equations and Modal Transformations.- B.8 Eigenvalue Assignment.- B.9 Eigensystem Assignment.- B.10 Complete Modal Synthesis.- B.11 Vandermonde Matrix.- B.12 Decompostion into Eigenvectors.- B.13 Properties of Eigenvalues.- B.13.1 Smallest and Largest Eigenvalue of Symmetrie Matrices.- B.13.2 Eigenvalues and Trace.- B.13.3 Maximum Real Part of an Eigenvalue.- B.13.5 Adding the Identity Matrix.- B.13.6 Eigenvalues of Matrix Products.- B.13.7 Eigenvalue of a Matrix Polynomial.- B.13.8 Weyl Inequality.- B.14 Rayleigh's Theorem.- B.15 Eigenvalues and Eigenvectors of the Inverse.- B.16 Dyadic Decomposition (Spectral Representation).- B.17 Spectral Representation of the Exponential Matrix.- B.18 Perron-Frobenius Theorem.- B.19 Multiple Eigenvalues. Generalized Eigenvectors.- B.20 Jordan Canonical Form and Jordan Blocks.- B.21 Special Cases.- B.22 Fundamental Matrix.- B.23 Eigenvector Assignment.- B.23.1 Assignable Subspaces. Parametrization of Controllers.- B.23.2 Single Real or Complex-Conjugate Eigenvalues.- B.23.3 Multiple Eigenvalues and Linearly Independent Eigenvectors.- B.23.4 Multiple Eigenvalues and Generalized Eigenvectors.- B.23.5 Assignable Subspace. Concluding Remarks.- C Matrix Inversion.- C.1 Matrix Inversion Using Cayley-Hamilton Theorem.- C.2 Matrix Inversion Lemma.- C.3 Simplified Version of the Matrix Inversion Lemma.- C.4 Matrices in Partitioned Form.- C.4.1 Algebraic Properties.- C.4.2 Inversion of a Partitioned Matrix.- C.4.3 Inversion of a Partitioned Matrix. Nonsingular Submatrices.- C.4.4 Inversion of a Block-Diagonal Matrix.- C.4.5 Determinants of Matrices in Partitioned Form.- C.4.6 Reducible Matrix.- C.5 Right-Inverse.- C.6 Left-Inverse.- C.7 Pseudo-Inverse.- C.7.1 General Pseudo-Inverse.- C.7.2 General Pseudo-Inverse and a General Matrix Equation.- C.7.3 Right-Pseudo-Inverse.- C.7.4 Left-Pseudo-Inverse.- C.8 General System Inverse.- C.9 Pseudo-Inverse and Singular-Value Decomposition.- C.1O Pseudo-Inverse of a Matrix Partitioned into Submatrices.- C.11 Pseudo-Inverse of a Matrix Partitioned into Columns.- C.12 Successive Application of Right and Left-Pseudo-Inverse Operator.- C.13 Conditioning and Scaling.- C.13.1 Condition Number of a Matrix.- C.13.2 General Spectral Decomposition.- C.13.3 Eigenvalue Decomposition.- C.13.4 Orthogonal Transformation.- C.13.5 Scaled Decomposition.- C.13.6 Square Root Decomposition.- C.13.7 Cholesky Decomposition.- C.14 Orthogonalizing.- D Linear Regression and Estimation.- D.1 Parameter Demarcation.- D.2 Interpolation.- D.3 Weighted Least Squares Approximation.- D.4 Ordinary Least Squares Approximation.- D.5 Left Inverse and Right Inverse. Mnemonic Aid.- D.7 Sum of Errors and Residual Sum in Parameter Space.- D.8 Successive Estimation in Large-Scale Systems.- D.9 Recursive Least-Squares Estimation.- D.10 Recursive Instrumental Variable Method.- D.11 Linear Estimation.- D.11.1 Parametrie Models. Markov Processes.- D.11.2 Observation as a Random Process.- D.11.3 Minimum Variance Estimator. Gauss-Markov Theorem.- D.11.4 Estimation Sensitivity.- E Notations.- E.1 General Conventions.- E.2 Abbreviations and General Symbols.- E.3 Superscripts.- E.4 Subscripts.- E.5 Glossary of Symbols in Alphabetic Order.- F Author Index.- G Index.