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

We apply retrospective cost adaptive control (RCAC) with auxiliary nonlinearities to a command-following problem for uncertain Hammerstein-Wiener systems with memoryless and hysteretic nonlinearities. The only required modeling information of the linear plant is a single Markov parameter. To account for the nonlinearities, RCAC uses auxiliary nonlinearities that reflect the monotonicity properties of the nonlinearity. Various memoryless nonlinearities such as deadband, cubic, and saturation are considered. The hysteresis nonlinearity is modeled using the Prandtl-Ishlinskii model.

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Retrospective-Cost Adaptive Control of Uncertain Hammerstein-Wiener Systems with Memoryless and Hysteretic Nonlinearities

Broadband adaptive disturbance rejection is demonstrated on the artificial white-light source of the deployable optical telescope (DOT) developed by the Air Force Research Laboratory (AFRL). Large deployable optical systems are envisioned as the future of space telescopes. These systems are require active control for disturbance rejection. In this paper, we consider a multi-input, multi-output discrete-time adaptive disturbance rejection algorithm. The algorithm requires knowledge of some Markov parameters of the system from the control signal inputs to the performance variables. No information about the disturbance spectrum is required. We demonstrate broadband disturbance rejection numerically on an identified model of DOT's white-light source and experimentally on the system itself.

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Broadband adaptive disturbance rejection for a deployable optical telescope testbed

Physical dimensions and units, such as mass (kg), length (m), time (s), and charge (C), provide the link between mathematics and the physical world. It is well known that careful attention to physical dimensions can provide valuable insight into relationships among physical quantities. In this regard, the Buckingham Pi theorem, which is essentially an application of the fundamental theorem of linear algebra on the sum of the rank and defect of a matrix, has been extensively applied.

Dimensional Analysis of Matrices State-Space Models and Dimensionless Units [Lecture Notes]

Real parameter uncertainty and phase information play a key role in the analysis and synthesis of robust controllers for lightly damped flexible structures. The purpose of this study is to examine the impact of these issues on structural control, their interrelationship, and their manifestation within the analysis and synthesis of feedback systems. The discussion is illustrated by examining robust controllers designed for the ACES structure at Marshall Space Flight Center. These controllers were designed by means of the maximum entropy generalized linear quadratic Gaussian methodology. >

Real parameter uncertainty and phase information in the robust control of flexible structures

In this paper we develop frequency-domain methods for approximating IIR plants with FIR transfer functions. The underlying goal is to increase the performance and robustness of Retrospective-Cost Adaptive Control (RCAC), which is applicable to MIMO possibly nonminimum-phase (NMP) plants without the need to know the locations of the NMP zeros. The only required modeling information is an FIR approximation of the plant, which may be based on a limited number of Markov parameters, or possibly noisy frequency response data. In this paper we investigate the resulting phase mismatch between the true plant and the FIR approximation obtained through linear and nonlinear approximation methods. We consider degradation in the phase mismatch due to uncertainty in the frequency response data.

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Dennis S. Bernstein

Papers

Retrospective-Cost Adaptive Control of Uncertain Hammerstein-Wiener Systems with Memoryless and Hysteretic Nonlinearities

Broadband adaptive disturbance rejection for a deployable optical telescope testbed

Dimensional Analysis of Matrices State-Space Models and Dimensionless Units [Lecture Notes]

Real parameter uncertainty and phase information in the robust control of flexible structures

FIR-based phase matching for robust Retrospective-Cost Adaptive Control