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

This book is downloadable from http://www.irisa.fr/sisthem/kniga/. Many monitoring problems can be stated as the problem of detecting a change in the parameters of a static or dynamic stochastic system. The main goal of this book is to describe a unified framework for the design and the performance analysis of the algorithms for solving these change detection problems. Also the book contains the key mathematical background necessary for this purpose. Finally links with the analytical redundancy approach to fault detection in linear systems are established. We call abrupt change any change in the parameters of the system that occurs either instantaneously or at least very fast with respect to the sampling period of the measurements. Abrupt changes by no means refer to changes with large magnitude; on the contrary, in most applications the main problem is to detect small changes. Moreover, in some applications, the early warning of small - and not necessarily fast - changes is of crucial interest in order to avoid the economic or even catastrophic consequences that can result from an accumulation of such small changes. For example, small faults arising in the sensors of a navigation system can result, through the underlying integration, in serious errors in the estimated position of the plane. Another example is the early warning of small deviations from the normal operating conditions of an industrial process. The early detection of slight changes in the state of the process allows to plan in a more adequate manner the periods during which the process should be inspected and possibly repaired, and thus to reduce the exploitation costs.

Detection of abrupt changes: theory and application

Networked control systems (NCSs) are spatially distributed systems for which the communication between sensors, actuators, and controllers is supported by a shared communication network. We review several recent results on estimation, analysis, and controller synthesis for NCSs. The results surveyed address channel limitations in terms of packet-rates, sampling, network delay, and packet dropouts. The results are presented in a tutorial fashion, comparing alternative methodologies

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A Survey of Recent Results in Networked Control Systems

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard because it contains vector cardinality minimization as a special case.

In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum-rank solution can be recovered by solving a convex optimization problem, namely, the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability, provided the codimension of the subspace is sufficiently large.

The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this preexisting concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization. We also discuss several algorithmic approaches to minimizing the nuclear norm and illustrate our results with numerical examples.

https://faculty.washington.edu/mfazel/low-rank-v2.pdf

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

This survey and taxonomy of location systems for mobile-computing applications describes a spectrum of current products and explores the latest in the field. To make sense of this domain, we have developed a taxonomy to help developers of location-aware applications better evaluate their options when choosing a location-sensing system. The taxonomy may also aid researchers in identifying opportunities for new location-sensing techniques.

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Location systems for ubiquitous computing

In this paper we show how to compute, via semidefinite programming, ellipsoids of confidence for linear equations with structured (unknown-but-bounded) perturbations. Our results are exact in the "unstructured" perturbations case, enabling us to recover in particular the classical case when the perturbation is additive. The approach can be used for example in the identification of structured, dynamical systems.

Confidence ellipsoids for uncertain linear equations with structure

Robustness in model predictive control (MPC) is the main focus of this work. After a definition of the conceptual framework and of the problem's setting, we analyze how a technique developed for studying robustness in convex optimization can be applied to address the problem of robustness in the MPC case. Therefore, exploiting this relationship between control and optimization, we tackle robustness issues for the first setting through methods developed in the second framework. Proofs for our results are included. As an application of this robust MPC result, we consider a path planning problem and discuss some simulations thereabout.

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Robust model predictive control through adjustable variables: an application to path planning

In this paper, we discuss linear programs in which the data that specify the constraints are subject to random uncertainty. A usual approach in this setting is to enforce the constraints up to a given level of probability. We show that for a wide class of probability distributions (i.e. radial distributions) on the data, the probability constraints can be explicitly converted into convex second order cone (SOC) constraints, hence the probability constrained linear program can be solved exactly with great efficiency. We next analyze the situation when the probability distribution of the data in not completely specified, but it is only known to belong to a given class of distributions. In this case, we provide explicit convex conditions that guarantee the satisfaction of the probability constraints, for any possible distribution belonging to the given class.

Linear Programming with Probability Constraints - Part 1

The LASSO is a widely used shrinkage method for linear regression. We propose an online homotopy algorithm to solve a generalization of the LASSO in which the l1 regularization is applied on a linear transformation of the solution, allowing to input prior information on the structure of the problem and to improve interpretability of the results. The algorithm takes advantage of the sparsity of the solution for computational efficiency and is promising for mining large datasets.

Online Homotopy Algorithm for a Generalization of the LASSO

We describe an interior point algorithm for computing the upper bound for the structured singular value described in [1]. We demonstrate the performance of the algorithm on a simple example.

L. El Ghaoui

Papers

Confidence ellipsoids for uncertain linear equations with structure

Robust model predictive control through adjustable variables: an application to path planning

Linear Programming with Probability Constraints - Part 1

Online Homotopy Algorithm for a Generalization of the LASSO

Computing Bounds for the Structured Singular Value via an Interior Point Algorithm