There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

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 paper will very briefly review the history of the relationship between modern optimal control and robust control. The latter is commonly viewed as having arisen in reaction to certain perceived inadequacies of the former. More recently, the distinction has effectively disappeared. Once-controversial notions of robust control have become thoroughly mainstream, and optimal control methods permeate robust control theory. This has been especially true in H-infinity theory, the primary focus of this paper.

http://www.gbv.de/dms/goettingen/18606599X.pdf

Robust and Optimal Control

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Convex Analysisの二,三の進展について

In this book, Andrew Harvey sets out to provide a unified and comprehensive theory of structural time series models. Unlike the traditional ARIMA models, structural time series models consist explicitly of unobserved components, such as trends and seasonals, which have a direct interpretation. As a result the model selection methodology associated with structural models is much closer to econometric methodology. The link with econometrics is made even closer by the natural way in which the models can be extended to include explanatory variables and to cope with multivariate time series. From the technical point of view, state space models and the Kalman filter play a key role in the statistical treatment of structural time series models. The book includes a detailed treatment of the Kalman filter. This technique was originally developed in control engineering, but is becoming increasingly important in fields such as economics and operations research. This book is concerned primarily with modelling economic and social time series, and with addressing the special problems which the treatment of such series poses. The properties of the models and the methodological techniques used to select them are illustrated with various applications. These range from the modellling of trends and cycles in US macroeconomic time series to to an evaluation of the effects of seat belt legislation in the UK.

Forecasting, Structural Time Series Models and the Kalman Filter

Nonlinear estimation methods, such as the extended Kalman filter, unscented Kalman filter, and ensemble Kalman filter, can be used for parameter estimation by viewing the unknown parameters as cons...

Gradient-, Ensemble-, and Adjoint-Free Data-Driven Parameter Estimation

An adaptive algorithm is developed for the MIMO tracking problem. The MIMO system and controller are represented as ARMARKOV/Toeplitz models, and the parameter matrix of the compensator is updated online by means of a gradient algorithm. The algorithm does not require any knowledge of the plant. Simulation results on a fourth order system are presented.

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Adaptive tracking using ARMARKOV/Toeplitz models

We use retrospective cost adaptive control (RCAC) with a constant forgetting factor (CFF), variable forgetting factor (VFF), and Kalman Filter (KF) to control a planar missile with nonlinear dynamics and aerodynamics. RCAC/CFF, RCAC/VFF, and RCAC/KF are used within an inner-loop/outer-loop control architecture, where the normal acceleration command is used to update the pitch-rate command for use by the pitch-rate servo loop. This control architecture is necessitated by the fact that, except for circular arcs with constant velocity, the appropriate pitch-rate command cannot be inferred from the normal acceleration command. Aggressive commands are used to compare RCAC/VFF and RCAC/KF with RCAC/CFF.

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An inner-loop/outer-loop architecture for an adaptive missile autopilot

This paper considers fixed-structure stable ℋ2-optimal controller synthesis using a multiobjective optimization technique which provides a trade-off between closed-loop performance and the degree of controller stability. The problem is presented in a decentralized static output feedback framework developed for fixed-structure dynamic controller synthesis. A quasi-Newton/continuation algorithm is used to compute solutions to the necessary conditions. To demonstrate the approach, two numerical examples are considered. The first example is a second-order spring–mass–damper system and the second example is a fourth-order two-mass system, both of which are considered in the stable stabilization literature. The results are then compared with other methods of stable compensator synthesis. Copyright © 2000 John Wiley & Sons, Ltd.

Stable ?2-optimal controller synthesis

In this paper, we present a sliding-window variable-regularization recursive least squares algorithm. In contrast to standard recursive least squares, the algorithm presented in this paper operates on a finite window of data, where old data are discarded as new data become available. This property can be beneficial for estimating time-varying parameters. Furthermore, standard recursive least squares uses time-invariant regularization. More specifically, the inverse of the initial covariance matrix in standard recursive least squares can be viewed as a regularization term, which weights the difference between the next estimate and the initial estimate. This regularization is fixed for all steps of the recursion. The algorithm derived in this paper allows for time-varying regularization. In particular, the present paper allows for time-varying regularization in the weighting as well as what is being weighted. Specifically, the regularization term can weight the difference between the next estimate and a time-varying vector of parameters rather than the initial estimate.

Dennis S. Bernstein

Papers

Gradient-, Ensemble-, and Adjoint-Free Data-Driven Parameter Estimation

Adaptive tracking using ARMARKOV/Toeplitz models

An inner-loop/outer-loop architecture for an adaptive missile autopilot

Stable ?2-optimal controller synthesis

Sliding window recursive quadratic optimization with variable regularization