Uncertain Models and Robust Control

Control systems particularly designed to manage uncertainties are called robust control system. Choosing appropriate design methods, the influence of uncertainties on the closed-loop behaviour can be reduced to a large extent. Most of the important areas of robust control are covered. The aim of the book is to provide an introduction to the theory and methods of robust control system design, to present a coherent body of knowledge, to clarify and unify presentation of significant derivations and proofs. The book contains a thorough treatment of important material of uncertainties and robust control which is scattered throughout the literature.

I Introduction.- 1 Introductory Survey.- 2 Vector Norm. Matrix Norm. Matrix Measure.- 3 FUnctional Analysis, Function Norms and Control Signals.- II Differential Sensitivity. Small-Scale Perturbation.- 4 Kronecker Calculus in Control Theory.- 5 Analysis Using Matrices and Control Theory 79.- 6 Eigenvalue and Eigenvector Differential Sensitivity.- 7 Transition Matrix Differential Sensitivity.- 8 Characteristic Polynomial Differential Sensitivity.- 9 Optimal Control and Performance Sensitivity.- 10 Desensitizing Control.- III Robustness in the Time Domain.- 11 General Stability Bounds in Perturbed Systems.- 12 Robust Dynamic Interval Systems.- 13 Lyapunov-Based Methods for Perturbed Continuous-Time Systems.- 14 Lyapunov-Based Methods for Perturbed Discrete-Time Systems.- 15 Robust Pole Assignment.- 16 Models for Optimal and Interconnected Systems.- 17 Robust State Feedback Using Ellipsoid Sets.- 18 Robustness of Observers and Kalman-Bucy Filters.- 19 Initial Condition Perturbation, Overshoot and Robustness.- 20 Lpn-Stability and Robust Nonlinear Control.- IV Robustness in the Frequency Domain.- 21 Uncertain Polynomials. Interval Polynomials.- 22 Eigenvalues and Singular Values of Complex Matrices.- 23 Resolvent Matrix and Stability Radius.- 24 Robustness Via Singular-Value Analysis.- 25 Generalized Nyquist Stability of Perturbed Systems.- 26 Block-Structured Uncertainty and Structured Singular Value.- 27 Performance Robustness.- 28 Robust Controllers Via Spectral Radius Technique.- V Coprime Factorization and Minimax Frequency Optimization.- 29 Robustness Based on the Internal Model Principle.- 30 Parametrization and Factorization of Systems.- 31 Hardy Space Robust Design.- VI Robustness Via Approximative Models.- 32 Robust Hyperplane Design in Variable Structure Control.- 33 Singular Perturbations. Unmodelled High-Frequency Dynamics.- 34 Control Using Aggregation Models.- 35 Optimum Control of Approximate and Nonlinear Systems.- 36 System Analysis via Orthogonal 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.