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

http://control.disp.uniroma2.it/galeani/CO/materiale/automatica2000.pdf

Survey Constrained model predictive control: Stability and optimality

A new approach is presented to the problem of detecting the number of signals in a multichannel time-series, based on the application of the information theoretic criteria for model selection introduced by Akaike (AIC) and by Schwartz and Rissanen (MDL). Unlike the conventional hypothesis testing based approach, the new approach does not requite any subjective threshold settings; the number of signals is obtained merely by minimizing the AIC or the MDL criteria. Simulation results that illustrate the performance of the new method for the detection of the number of signals received by a sensor array are presented.

Detection of signals by information theoretic criteria

Many problems in the sciences and engineering can be rephrased as optimization problems on matrix search spaces endowed with a so-called manifold structure. This book shows how to exploit the special structure of such problems to develop efficient numerical algorithms. It places careful emphasis on both the numerical formulation of the algorithm and its differential geometric abstraction--illustrating how good algorithms draw equally from the insights of differential geometry, optimization, and numerical analysis. Two more theoretical chapters provide readers with the background in differential geometry necessary to algorithmic development. In the other chapters, several well-known optimization methods such as steepest descent and conjugate gradients are generalized to abstract manifolds. The book provides a generic development of each of these methods, building upon the material of the geometric chapters. It then guides readers through the calculations that turn these geometrically formulated methods into concrete numerical algorithms. The state-of-the-art algorithms given as examples are competitive with the best existing algorithms for a selection of eigenspace problems in numerical linear algebra. Optimization Algorithms on Matrix Manifolds offers techniques with broad applications in linear algebra, signal processing, data mining, computer vision, and statistical analysis. It can serve as a graduate-level textbook and will be of interest to applied mathematicians, engineers, and computer scientists.

http://www.eeci-institute.eu/GSC2011/Photos-EECI/EECI-GSC-2011-M5/book_AMS.pdf

Optimization Algorithms on Matrix Manifolds

https://www.itk.ntnu.no/fag/fordypning/TK16-filer/Samling1_MorariLee.pdf

Model predictive control: past, present and future

Sufficient experimental conditions for the convergence of an adaptive observer for nonlinear biochemical processes

In this work we extend the scope of the classical Cram´er-Rao lower bound, or information inequality, from Euclidean to function spaces. In other words we derive a tight lower bound on the autocovariance function of a function estimator. We do this in the context of system identification. Two key elements of system identification are experiment design and model selection. The novel information inequality on function spaces is important for model selection because it allows the user to compare estimators using different model structures. We provide a consistent treatment of the case where the Fisher information matrix is singular. This makes it possible to take into account that in optimal experiment design one tries to mask those parts of the system non-identifiable, which are irrelevant for the application.

The information inequality for function spaces given a singular information matrix

Properties of the Parametrization of Monic ARMA Systems

When one random variable is estimated from another measured random variable through a nonlinear mapping constituting the estimator, then any independent additive noise present in the measured variable creates a bias error in the estimated variable. This occurs even if the added noise has zero mean and symmetric density. This bias error can be computed approximately using the second derivative of the mapping when this mapping is available analytically, and hence a bias-corrected estimate can be constructed. We show that this idea can be extended to the case where the mapping is implicitly defined as the solution of a minimization problem, such as in Maximum Likelihood estimation. We also analyze the effect of this bias correction when applied to the estimation of a first order transfer function at one frequency on the basis of a noisy measurement of that transfer function at some other frequency.

Bias reduction in transfer function identification

Optimal input design is an important step of the identification process in order to reduce the model variance. In this work a D-optimal input design method for finite-impulse-response-type nonlinear systems is presented. The optimization of the determinant of the Fisher information matrix is expressed as a convex optimization problem. This problem is then solved using a dispersion-based optimization scheme, which is easy to implement and converges monotonically to the optimal solution. Without constraints, the optimal design cannot be realized as a time sequence. By imposing that the design should lie in the subspace described by a symmetric and non-overlapping set, a realizable design is found. A graph-based method is used in order to find a time sequence that realizes this optimal constrained design. These methods are illustrated on a numerical example of which the results are thoroughly discussed. Additionally the computational speed of the algorithm is compared with the general convex optimizer cvx.

Michel Gevers

Papers

Sufficient experimental conditions for the convergence of an adaptive observer for nonlinear biochemical processes

The information inequality for function spaces given a singular information matrix

Properties of the Parametrization of Monic ARMA Systems

Bias reduction in transfer function identification

D-Optimal Input Design for Nonlinear FIR-type Systems:A Dispersion-based Approach