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

Within a stochastic noise framework, the validation of a model yields an ellipsoidal parameter uncertainty set, from which a corresponding uncertainty set can be constructed in the space of transfer functions We display the role of the experimental conditions used for validation on the shape of this validated set, and we connect a measure of the size of this set to the stability margin of a controller designed from the nominal model This allows one to check stability robustness for the validated model set and to propose guidelines for validation design

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The role of experimental conditions in model validation for control

Iterative feedback tuning (IFT) is a data-based method for the iterative tuning of restricted complexity controllers. A "special experiment" in which a batch of previously collected output data is fed back at the reference input allows one to compute an unbiased estimate of the gradient of the control performance criterion. We show that, by performing an optimal filtering of the data that are fed back, one can minimize the asymptotic variability of the control performance cost and, hence, minimize the average performance degradation that results from the randomness of the data. The expression of the optimal filter is derived, and a simulation illustrates the benefits that result from using this optimal filter as compared to the use of the classical constant filter.

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Optimal prefiltering in iterative feedback tuning

This paper focuses on the validation of a controller that has been designed from an unbiased model of the true system, identified either in open-loop or in closed-loop using a prediction error framework. A controller is said to be validated if it stabilizes all models in a parametric uncertainty set containing the parameters of the true system with some prescribed probability. This uncertainty set is deduced from the covariance matrix of the parameters of the identified model. Our contribution is to embed this set in the smallest possible overbounding coprime factor uncertainty set. This then allows us to use the results of mainstream robust control theory such as the Vinnicombe gap between plants and its related stability theorems.

https://ieeexplore.ieee.org/iel5/6713/18008/00831360.pdf

Controller validation based on an identified model

Necessary and sufficient conditions for uniqueness of the minimum in Prediction Error Identification

The convergence of the solutions of the differential difference Riccati equations to the strong solution of the corresponding algebraic Riccati equation (ARE) is addressed. Detectability only is required in the analysis, and no assumption is made on the eigenvalues on the real imaginary axis (on the unit circle, in the discrete-time case). In particular, from the results, it follows that under the sole assumption of detectability, a positive definite initial condition guarantees convergence to the strong solution, even in the presence of unreachable eigenvalues on the imaginary axis or on the unit circle. >

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Michel Gevers

Papers

The role of experimental conditions in model validation for control

Optimal prefiltering in iterative feedback tuning

Controller validation based on an identified model

Necessary and sufficient conditions for uniqueness of the minimum in Prediction Error Identification

Difference and differential Riccati equations: a note on the convergence to the strong solution