内容简介：　　　

　　This book presents a number of techniques for robustness analysis of uncertain systems. The theoretical basis for their development is derived from the application of convex optimization tools to problems involving positivity of homogeneous polynomial forms. Usually, stability and performance analysis of dynamic systems affected by structured uncertainties, requires the solution of non-convex optimization problems. Constructing a family of convex relaxations, namely convex optimisations, can provide upper or lower bound to the original problem. In this book convex relaxations for several robustness problems are derived by exploiting and providing new results on the theory of homogeneous polynomial forms. A framework is introduced for dealing with positivity of homogeneous forms via the solution of linear matrix inequalities, with examples including Lyapunov analysis of uncertain systems, computation of the parametric robust stability margin, and robust performance analysis for polytopic systems.

英文目录：
Notation
Abbreviations
1 Positive Forms
　　1.1 Forms and Polynomials
　　1.2 Representation via Power Vectors
　　1.3 Representation via SMR
　　　　1.3.1 Equivalent SMR Matrices
　　　　1.3.2 Complete SMR
　　1.4 SOS Forms
　　　　1.4.1 SOS Tests Based on the SMR
　　　　1.4.2 SOS Index
　　1.5 Matrix Forms
　　　　1.5.1 SMR of Matrix Forms
　　　　1.5.2 SOS Matrix Forms
　　1.6 Positive Forms
　　　　1.6.1 Positivity Index
　　　　1.6.2 Sufficient Condition for Positivity
　　　　1.6.3 Positive Matrix Forms
　　1.7 Positive Polynomials on Ellipsoids
　　　　1.7.1 Solution via Positivity Test on a Form
　　　　1.7.2 Even Polynomials
　　1.8 Positive Matrix Polynomials on the Simplex
　　1.9 Extracting Power Vectors from Linear Spaces
　　　　1.9.1 Basic Procedure
　　　　1.9.2 Extended Procedure
　　1.10 Notes and References
2 Positivity Gap
　　2.1 Hilbert's 17th Problem
　　2.2 Maximal SMR Matrices
　　　　2.2.1 Minimum Eigenvalue Decomposition
　　　　2.2.2 Structure of Maximal SMR Matrices
　　2.3 SMR-tight Forms
　　　　2.3.1 Minimal Point Set
　　　　2.3.2 Rank Conditions
　　2.4 Characterizing PNS Forms via the SMR
　　　　2.4.1 Basic Properties of PNS Forms
　　　　2.4.2 Cones of PNS Forms
　　　　2.4.3 Parametrization of PNS Forms
　　2.5 Notes and References
3 Robustness with Time-varying Uncertainty
　　3.1 Polytopic Systems with Time-varying Uncertainty
　　　　3.1.1 Homogeneous Polynomial Lyapunov Functions
　　　　3.1.2 Extended Matrix
　　3.2 Robust Stability
　　3.3 Robust Performance
　　　　3.3.1 Polytopic Stability Margin
　　　　3.3.2 Best Transient Performance
　　3.4 Rational Parametric Uncertainty
　　　　3.4.1 SMR for LFR Systems
　　　　3.4.2 Conditions for Robust Stability
　　　　3.4.3 Robust Stability Margin for LFR Systems
　　3.5 Examples
　　3.6 Notes and References
4 Robustness with Time-invariant Uncertainty
　　4.1 Polytopic Systems with Time-invariant Uncertainty
　　4.2 Robust Stability
　　　　4.2.1 Parametrization of HPD-QLFs
　　　　4.2.2 Conditions for Robust Stability
　　　　4.2.3 Detecting Instability
　　4.3 Robust Performance
　　　　4.3.1 Robust H-infinity Performance
　　　　4.3.2 Parametric Stability Margin
　　4.4 Rational Parametric Uncertainty
　　4.5 Robustness Analysis via Hurwitz Determinants
　　4.6 Discrete-time Systems
　　4.7 Examples
　　4.8 Notes and References
5 Robustness with Bounded-rate Time-varying Uncertainty
　　5.1 Polytopic Systems with Bounded-rate Time-varying Uncertainty
　　5.2 Robust Stability
　　　　5.2.1 Parametrization of HPD-HLFs
　　　　5.2.2 Robust Stability Condition
　　5.3 Robust Stability Margin
　　5.4 Examples
　　5.5 Notes and References
6 Distance Problems with Applications to Robust Control
　　6.1 QuadraticDistance Problems
　　6.2 Special Cases and Extensions
　　　　6.2.1 The Two-form Case
　　　　6.2.2 Maximum QDPs
　　6.3 Conservatism Analysis
　　　　6.3.1 a posteriori Test for Tightness
　　　　6.3.2 a priori Conditions for Tightness
　　　　6.3.3 An Example of Nontight Lower Bound
　　6.4 Parametric Stability Margin
　　　　6.4.1 Continuous-time Systems
　　　　6.4.2 Discrete-time Systems
　　6.5 Notes and References
A Basic Tools
　　A.1 LMI Optimizations
　　A.2 Hurwitz/SchurMatrices
B SMR Algorithms
References
Authors’ Biography
Index