内容简介：　　　

　　For nonlinear dynamical systems, which represent the majority of real devices, any study of stability requires the investigation of the domain of attraction of an equilibrium point, i.e. the set of initial conditions from which the trajectory of the system converges to equilibrium. Unfortunately, both estimating and attempting to control the domain of attraction are very difficult problems, because of the complex relationship of this set with the model of the system. Domain of Attraction addresses the estimation and control of the domain of attraction of equilibrium points via SOS programming, i.e. optimization techniques based on the sum of squares of polynomials (SOS) that have been recently developed and that amount to solving convex problems with linear matrix inequality constraints. A unified framework for addressing these issues is presented for in various cases depending on the nature of the nonlinear systems considered, including the cases of polynomial, non-polynomial, certain and uncertain systems. The methods proposed are illustrated various example systems such as electric circuits, mechanical devices, and nuclear plants. Domain of Attraction also deals with related problems that can be considered within the proposed framework, such as characterizing the equilibrium points and bounding the trajectories of nonlinear systems, and offers a concise and simple description of the main features of SOS programming, which can be used for general purpose in research and teaching.

图书目录：

Notation
Abbreviations
Part I: SOS Programming
1 SOS Polynomials
　　1.1 Polynomials and Power Vectors
　　　　1.1.1 General Case
　　　　1.1.2 Special Cases
　　1.2 SMR
　　　　1.2.1 General Case
　　　　1.2.2 Special Cases
　　1.3 SOS Polynomials
　　　　1.3.1 General Case
　　　　1.3.2 Special Cases
　　1.4 SOS Parameter-Dependent Polynomials and SOS Matrix Polynomials
　　　　1.4.1 SOS Parameter-Dependent Polynomials: General Case
　　　　1.4.2 SOS Parameter-Dependent Polynomials: Special Cases
　　　　1.4.3 SOS Matrix Polynomials
　　1.5 Extracting Power Vectors from Linear Subspaces
　　　　1.5.1 General Case
　　　　1.5.2 Special Cases
　　1.6 Gap between Positive Polynomials and SOS Polynomials
　　1.7 Notes and References
2 Optimization with SOS Polynomials
　　2.1 Unconstrained Optimization
　　　　2.1.1 Positivity and Non-positivity: General Case
　　　　2.1.2 Positivity and Non-positivity: Special Cases
　　　　2.1.3 Minimum of Rational Functions
　　2.2 Constrained Optimization 
　　　　2.2.1 Positivstellensatz
　　　　2.2.2 Minimum of Rational Functions
　　　　2.2.3 Solving Systems of Polynomial Equations and Inequalities
　　　　2.2.4 Positivity of Matrix Polynomials
　　2.3 Optimization over Special Sets
　　　　2.3.1 Positivity over Ellipoids
　　　　2.3.2 Positivity over the Simplex
　　2.4 Rank Constrained SMR Matrices of SOS Polynomials
　　2.5 Notes and References
Part II: Domain of Attraction
3 Dynamical Systems Background
　　3.1 Equilibrium Points of Nonlinear Systems
　　3.2 Stability
　　3.3 DA
　　3.4 Controlled Systems
　　3.5 Common Equilibrium Points of Uncertain Nonlinear Systems
　　3.6 Robust Stability
　　3.7 RDA
　　3.8 Robustly Controlled Systems
　　3.9 Notes and References
4 DA in Polynomial Systems
　　4.1 Polynomial Systems
　　4.2 Estimates via LFs
　　　　4.2.1 Establishing Estimates
　　　　4.2.2 Choice of the LF
　　　　4.2.3 LEDA
　　　　4.2.4 Estimate Tightness
　　4.3 Estimates via Quadratic LFs
　　　　4.3.1 Establishing Estimates
　　　　4.3.3 LEDA
　　　　4.3.4 Estimate Tightness
　　4.4 Optimal Estimates
　　　　4.4.1 Maximizing the Volume of the Estimate
　　　　4.4.2 Enlarging the Estimate with Fixed Shape Sets
　　　　4.4.3 Establishing Global Asymptotical Stability
　　4.5 Controller Design
　　4.6 Notes and References
5 RDA in Uncertain Polynomial Systems
　　5.1 Uncertain Polynomial Systems
　　5.2 Estimates via Common LFs
　　　　5.2.1 Establishing Estimates
　　　　5.2.2 Choice of the LF
　　　　5.2.3 Parameter-Dependent LEDA and LERDA
　　　　5.2.4 Estimate Tightness
　　5.3 Estimates via Parameter-Dependent LFs
　　　　5.3.1 Establishing Parameter-Dependent Estimates
　　　　5.3.2 Choice of the LF
　　　　5.3.3 Parameter-Dependent LEDA
　　　　5.3.4 LERDA
　　5.4 Optimal Estimates
　　　　5.4.1 Maximizing the Volume of the Estimate
　　　　5.4.2 Enlarging the Estimate with Fixed Shape Sets
　　　　5.4.3 Establishing Robust Global Asymptotical Stability
　　5.5 Controller Design
　　5.6 Notes and References
6 DA and RDA in Non-polynomial Systems
　　6.1 Non-Polynomial Systems
　　6.2 Estimates via LFs
　　　　6.2.1 Establishing Estimates
　　　　6.2.2 Bounding the Remainders
　　　　6.2.3 Choice of the LF
　　　　6.2.4 LEDA
　　　　6.2.5 Estimate Tightness
　　6.3 Optimal Estimates
　　　　6.3.1 Maximizing the Volume of the Estimate
　　　　6.3.2 Enlarging the Estimate with Fixed Shape Sets
　　　　6.3.3 Establishing Global Asymptotical Stability
　　6.4 Controller Design
　　6.5 Uncertain Non-polynomial Systems
　　6.6 Estimate for Uncertain Non-polynomial Systems
　　　　6.6.1 Establishing Estimates
　　　　6.6.2 Choice of the LF
　　　　6.6.3 LERDA
　　　　6.6.4 Extensions
　　6.7 Notes and References
7 Miscellaneous
　　7.1 Union of Estimates
　　7.2 Equilibrium Points in Uncertain Systems
　　　　7.2.1 Estimate Computation
　　　　7.2.2 Estimate Tightness
　　　　7.2.3 Variable Shape Estimates
　　7.3 Trajectory Bounds for Given Sets of Initial Conditions
　　　　7.3.1 Analysis
　　　　7.3.2 Synthesis and Tightness
　　7.4 A Note on Degenerate Polynomial Systems
　　7.5 Notes and References
A  LMI Problems
B  Determinant and Rank Constraints via LMIs
C  MATLAB Codes: SMRSOFT
References
Author Biography

Index