Preface 1. Introduction Overview A Brief History of LMIs in Control Theory Notes on the Style of the Book Origin of the Book 2. Some Standard Problems Involving LMIs. Linear Matrix Inequalities Some Standard Problems Ellipsoid Algorithm Interior-Point Methods Strict and Nonstrict LMIs Miscellaneous Results on Matrix Inequalities Some LMI Problems with Analytic Solutions 3. Some Matrix Problems. Minimizing Condition Number by Scaling Minimizing Condition Number of a Positive-Definite Matrix Minimizing Norm by Scaling Rescaling a Matrix Positive-Definite Matrix Completion Problems Quadratic Approximation of a Polytopic Norm Ellipsoidal Approximation 4. Linear Differential Inclusions. Differential Inclusions Some Specific LDIs Nonlinear System Analysis via LDIs 5. Analysis of LDIs: State Properties. Quadratic Stability Invariant Ellipsoids 6. Analysis of LDIs: Input/Output Properties. Input-to-State Properties State-to-Output Properties Input-to-Output Properties 7. State-Feedback Synthesis for LDIs. Static State-Feedback Controllers State Properties Input-to-State Properties State-to-Output Properties Input-to-Output Properties Observer-Based Controllers for Nonlinear Systems 8. Lure and Multiplier Methods. Analysis of Lure Systems Integral Quadratic Constraints Multipliers for Systems with Unknown Parameters 9. Systems with Multiplicative Noise. Analysis of Systems with Multiplicative Noise State-Feedback Synthesis 10. Miscellaneous Problems. Optimization over an Affine Family of Linear Systems Analysis of Systems with LTI Perturbations Positive Orthant Stabilizability Linear Systems with Delays Interpolation Problems The Inverse Problem of Optimal Control System Realization Problems Multi-Criterion LQG Nonconvex Multi-Criterion Quadratic Problems Notation List of Acronyms Bibliography Index.

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Linear Matrix Inequalities in System and Control Theory

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

Algorithms that must deal with complicated global functions of many variables often exploit the manner in which the given functions factor as a product of "local" functions, each of which depends on a subset of the variables. Such a factorization can be visualized with a bipartite graph that we call a factor graph, In this tutorial paper, we present a generic message-passing algorithm, the sum-product algorithm, that operates in a factor graph. Following a single, simple computational rule, the sum-product algorithm computes-either exactly or approximately-various marginal functions derived from the global function. A wide variety of algorithms developed in artificial intelligence, signal processing, and digital communications can be derived as specific instances of the sum-product algorithm, including the forward/backward algorithm, the Viterbi algorithm, the iterative "turbo" decoding algorithm, Pearl's (1988) belief propagation algorithm for Bayesian networks, the Kalman filter, and certain fast Fourier transform (FFT) algorithms.

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Factor graphs and the sum-product algorithm

Monte Carlo methods are revolutionizing the on-line analysis of data in fields as diverse as financial modeling, target tracking and computer vision. These methods, appearing under the names of bootstrap filters, condensation, optimal Monte Carlo filters, particle filters and survival of the fittest, have made it possible to solve numerically many complex, non-standard problems that were previously intractable. This book presents the first comprehensive treatment of these techniques, including convergence results and applications to tracking, guidance, automated target recognition, aircraft navigation, robot navigation, econometrics, financial modeling, neural networks, optimal control, optimal filtering, communications, reinforcement learning, signal enhancement, model averaging and selection, computer vision, semiconductor design, population biology, dynamic Bayesian networks, and time series analysis. This will be of great value to students, researchers and practitioners, who have some basic knowledge of probability. Arnaud Doucet received the Ph. D. degree from the University of Paris-XI Orsay in 1997. From 1998 to 2000, he conducted research at the Signal Processing Group of Cambridge University, UK. He is currently an assistant professor at the Department of Electrical Engineering of Melbourne University, Australia. His research interests include Bayesian statistics, dynamic models and Monte Carlo methods. Nando de Freitas obtained a Ph.D. degree in information engineering from Cambridge University in 1999. He is presently a research associate with the artificial intelligence group of the University of California at Berkeley. His main research interests are in Bayesian statistics and the application of on-line and batch Monte Carlo methods to machine learning. Neil Gordon obtained a Ph.D. in Statistics from Imperial College, University of London in 1993. He is with the Pattern and Information Processing group at the Defence Evaluation and Research Agency in the United Kingdom. His research interests are in time series, statistical data analysis, and pattern recognition with a particular emphasis on target tracking and missile guidance.

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Sequential Monte Carlo methods in practice

Planning algorithms are impacting technical disciplines and industries around the world, including robotics, computer-aided design, manufacturing, computer graphics, aerospace applications, drug design, and protein folding. This coherent and comprehensive book unifies material from several sources, including robotics, control theory, artificial intelligence, and algorithms. The treatment is centered on robot motion planning but integrates material on planning in discrete spaces. A major part of the book is devoted to planning under uncertainty, including decision theory, Markov decision processes, and information spaces, which are the “configuration spaces” of all sensor-based planning problems. The last part of the book delves into planning under differential constraints that arise when automating the motions of virtually any mechanical system. Developed from courses taught by the author, the book is intended for students, engineers, and researchers in robotics, artificial intelligence, and control theory as well as computer graphics, algorithms, and computational biology.

Planning Algorithms: Introductory Material

This paper studies the necessary and sufficient conditions for a pth-order observer to observe linear functions of the states of a linear dynamical system. The conditions are a set of multivariable polynomial equations which must be satisfied for some variable set in order for a pth-ncder observer to exist. It is possible to test for the existence of such a variable set in a finite number of steps via decision methods and thereby to construct an observer with the aid of polynomial factorization. To minimize the computational effort, the necessary and sufficient conditions are expressed in terms of the minimum number of variables.

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Minimal order observers for estimating linear functions of a state vector

This paper develops output reachability characterizations of linear finite dimensional systems, so as to translate excitation properties of system inputs to excitation properties of system outputs, states, or associated regression vectors. Such properties are of fundamental concern for convergence of algorithms involving on-line identification, adaptive state estimation, prediction and control. The case of adaptive control is of particular interest since it is desirable that persistence of excitation of associated regression vectors be maintained in the presence of time-varying feedback controllers. Persistence of excitation guarantees convergence without a priori stability assumptions and ensures robustness properties. The theory of the paper suggests modifications to standard adaptive schemes to ensure the required persistence of excitation.

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Persistence of Excitation in Linear Systems

The performance of adaptive state estimators for linear dynamic systems is investigated. The adaptive state estimates are formed under the assumption that the unknown system parameter belongs to a finite set and is thus readily implementable. It is shown that, for the true parameter value in a prescribed region in the parameter space, the corresponding a posteriori probablity (or weighting coefficient in the adaptive estimator) converges exponentially in the vth mean (v > 1) and almost surely to unity. The analysis is based on the asymptotic per sample formula for the Kullback information function, which is derived in this paper. The significance of the analysis for applications is also examined.

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Performance bounds for adaptive estimation

When stabilizing linear discrete-time finite dimensional systems in control and estimation either optimally or suboptimally, technical difficulties arise in the conventional stability theories for coping with state transition matrices which are permitted to be singular or with eigenvalues arbitrarily small. In overcoming these difficulties, earlier results for feedback stabilization of linear systems and for Kalman filters and regulators are generalized in this paper, with proofs being in fact more direct than those explored earlier.

Coping with singular transition matrices in estimation and control stability theory

The standard discrete-time autoregressive model is poorly suited for modeling series obtained by sampling continuous-time processes at fairly rapid rates. Large computational errors can occur when the Levinson algorithm is used to estimate the parameters of this model, because the Toeplitz covariance matrix is ill-suited for inversion. An alternative model is developed based on an incremental difference operator, rather than the shift operator. It is shown that, as the sampling period goes to zero, unlike the standard autoregressive parameters, the coefficients of this model converge to certain parameters that depend directly on the statistics of the continuous-time process. A Levinson-type algorithm for efficiently estimating the parameters of this model is derived. Numerical examples are given to show that when the sampling interval is small this algorithm is considerably less sensitivity to arithmetic roundoff errors than the Levinson algorithm. >

John B. Moore

Papers

Minimal order observers for estimating linear functions of a state vector

Persistence of Excitation in Linear Systems

Performance bounds for adaptive estimation

Coping with singular transition matrices in estimation and control stability theory

A Levinson-type algorithm for modeling fast-sampled data