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

This paper describes a linear matrix inequality (LMI)-based algorithm for the static and reduced-order output-feedback synthesis problems of nth-order linear time-invariant (LTI) systems with n/sub u/ (respectively, n/sub y/) independent inputs (respectively, outputs). The algorithm is based on a "cone complementarity" formulation of the problem and is guaranteed to produce a stabilizing controller of order m/spl les/n-max(n/sub u/,n/sub y/), matching a generic stabilizability result of Davison and Chatterjee (1971). Extensive numerical experiments indicate that the algorithm finds a controller with order less than or equal to that predicted by Kimura's generic stabilizability result (m/spl les/n-n/sub u/-n/sub y/+1). A similar algorithm can be applied to a variety of control problems, including robust control synthesis.

A cone complementarity linearization algorithm for static output-feedback and related problems

A method for estimating unknown node positions in a sensor network based exclusively on connectivity-induced constraints is described. Known peer-to-peer communication in the network is modeled as a set of geometric constraints on the node positions. The global solution of a feasibility problem for these constraints yields estimates for the unknown positions of the nodes in the network. Providing that the constraints are tight enough, simulation illustrates that this estimate becomes close to the actual node positions. Additionally, a method for placing rectangular bounds around the possible positions for all unknown nodes in the network is given. The area of the bounding rectangles decreases as additional or tighter constraints are included in the problem. Specific models are suggested and simulated for isotropic and directional communication, representative of broadcast-based and optical transmission respectively, though the methods presented are not limited to these simple cases.

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Convex position estimation in wireless sensor networks

This paper describes a linear matrix inequality (LMI)-based algorithm for the static and reduced-order output-feedback synthesis problems of n-th order linear time invariant (LTI) systems with n/sub u/ (resp. n/sub y/) independent inputs (resp. outputs). The algorithm is based on a "cone complementarity" formulation of the problem, and is guaranteed to produce a stabilizing controller of order m/spl les/n-max(n/sub u/,n/sub y/), matching a generic stabilizability result of Davison and Chatterjee (1971). Extensive numerical experiments indicate that the algorithm finds a controller with order less or equal to that predicted by Kimura's (1994) generic stabilizability result (m/spl les/n-n/sub u/-n/sub y/+1). A similar algorithm can be applied to a variety of control problems, including robust control synthesis.

/pdf/a-cone-complementarity-linearization-algorithm-for-static-2v7541dmi7.pdf

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 (namely, radial distributions) on the data, the probability constraints can be converted explicitly into convex second-order cone constraints; hence, the probability-constrained linear program can be solved exactly with great efficiency. Next, we analyze the situation where the probability distribution of the data is not completely specified, but 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.

On Distributionally Robust Chance-Constrained Linear Programs

This note presents a new approach to finite-horizon guaranteed state prediction for discrete-time systems affected by bounded noise and unknown-but-bounded parameter uncertainty. Our framework handles possibly nonlinear dependence of the state-space matrices on the uncertain parameters. The main result is that a minimal confidence ellipsoid for the state, consistent with the measured output and the uncertainty description, may be recursively computed in polynomial time, using interior-point methods for convex optimization. With n states, l uncertain parameters appearing linearly in the state-space matrices, with rank-one matrix coefficients, the worst-case complexity grows as O(l(n + l)/sup 3.5/) With unstructured uncertainty in all system matrices, the worst-case complexity reduces to O(n/sup 3.5/).

https://staff.polito.it/giuseppe.calafiore/Documenti/Papers/Robust%20Filtering_TAC-01.pdf

L. El Ghaoui

Papers

A cone complementarity linearization algorithm for static output-feedback and related problems

Convex position estimation in wireless sensor networks

A cone complementarity linearization algorithm for static output-feedback and related problems

On Distributionally Robust Chance-Constrained Linear Programs

Robust filtering for discrete-time systems with bounded noise and parametric uncertainty