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

We consider coupled Riccati equations that arise in jump linear systems. We show how to reliably solve these equations using convex optimization over linear matrix inequalities (LMIs). The result extends to other nonstandard Riccati equations, such as those arising in the optimal control of linear systems subject to state-dependent multiplicative noise.

Solving non-standard Riccati equations using LMI optimization

We consider a financial decision problem involving dynamic investment decisions on a single risky instrument over multiple and discrete time periods. Investment returns are assumed stochastic and possibly dependent over time, and proportional transaction costs are considered in the model. In this setting, the investor?s goal is to determine investment policies that maximize the net profit while maintaining the associated risk under control. We propose approximations of the ensuing stochastic multistage optimization problem that are based on affine recourse strategies and that lead to efficiently solvable second order cone or semidefinite programs.

https://www.seas.upenn.edu/~utopcu/pubs/TCE-cdc08.pdf

Multistage investments with recourse: A single-asset case with transaction costs

We study a class of uncertain linear estimation problems in which the data are affected by random uncertainty. In this setting, we consider two estimation criteria, one based on minimization of the expected l1 or l2 norm residual and one based on minimization of the level within which the l1 or l2 norm residual is guaranteed to lie with an a-priori fixed probability (residual at risk). The random uncertainty affecting the data is characterized by means of its first two statistical moments, and the above criteria are intended in a worst-case probabilistic sense, that is worst-case expectations and probabilities over all possible distribution having the specified moments are considered. The ensuing estimation problems can be solved efficiently via convex programming, yielding exact solutions in the l2 norm case and upper-bounds on the optimal solutions in the l1 case.

https://staff.polito.it/giuseppe.calafiore/Documenti/Papers/Parameter%20estimation%20with%20expected%20and%20residual%20at%20risk%20criteria_SYSCONLET.pdf

Parameter estimation with expected and residual-at-risk criteria

Letter to the Editor on the Paper ‘Output Feedback Control with Input Saturations: LMI Design Approach’ by G. Scorletti, J.-P. Folcher, and L. El Ghaoui

L. El Ghaoui

Papers

Solving non-standard Riccati equations using LMI optimization

Multistage investments with recourse: A single-asset case with transaction costs

Parameter estimation with expected and residual-at-risk criteria

Letter to the Editor on the Paper ‘Output Feedback Control with Input Saturations: LMI Design Approach’ by G. Scorletti, J.-P. Folcher, and L. El Ghaoui

Discussion on:‘Output Feedback Control with Input Saturations: LMI Design Approaches’ by G. Scorletti, J. P. Folcher and L. El Ghaoui