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 develop a method for identifying SISO Hammerstein systems with an unknown static nonlinearity, linear dynamics, white input noise and colored output noise We use least squares with a μ-Markov model to estimate the Markov parameters of the linear time-invariant dynamical system Since the input to the linear system is not available, we use a substitute (ersatz) nonlinearity to transform the input for use in the regressor matrix We prove that the Markov parameters of the system can be estimated consistently up to a constant scalar as the amount of data increases This method is demonstrated with several numerical examples

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Consistent identification of Hammerstein systems using an ersatz nonlinearity

Education must be both conceptual and experiential. Abstract concepts are elegant and powerful, but learning is always enhanced by direct experience. The author believes that control education can benefit by becoming more specific. Control theory and much of control education is highly conceptual. In fact, control engineering tends to be the least tangible of all subjects in the engineering curriculum. In the hope of tipping the balance from the conceptual to the experiential, the author offers several suggestions.

http://ieeexplore.ieee.org/iel5/6343/16963/00783610.pdf

Concretizing control education

We apply the forward-integrating Riccati-based feedback controller, which has been developed in our previous work for stabilization of time-varying systems, to a maneuvering spacecraft in an elliptic orbit around the Earth. We simulate rendezvous maneuvers on Molniya and Tundra orbits. We demonstrate that the controller performs well under thrust constraints, in the case where the spacecraft can thrust in only the orbital tangential direction, in the case where the thrusters may operate only intermittently due to faults or power availability, with thrust direction errors, and, finally, in an output feedback configuration where only relative position measurements are available.

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Forward-integration Riccati-based feedback control for spacecraft rendezvous maneuvers on elliptic orbits

The objective of input reconstruction is to use knowledge of the output of a system to estimate the input. An estimate of the input can be used in an observer to provide more accurate state estimates. This paper presents a technique for combined state and input estimation for discrete-time, linear time-varying systems. The algorithm is based on the analysis of the rank of the time-dependent matrix that relates vectors of states and input values to a vector of outputs. The key contribution is a characterization of the time-varying delay under which the state and input can be estimated. The input and state estimate are given in terms of the generalized inverse of a partitioned matrix. The approach is demonstrated on discrete-time examples with linear periodically time-varying dynamics.

Deadbeat Input Reconstruction and State Estimation for Discrete-Time Linear Time-Varying Systems

This paper presents an extension of the adaptive virtual autobalancing controller for a rigid, statically unbalanced rotor, previously proposed by the authors (1995), to the case of a rigid, dynamically unbalanced rotor. The state equations of the controller are based on the equations of motion of a multiple-plane autobalancer, and the interaction between rotor and autobalancer is emulated using a pair of magnetic actuators. It is shown in simulation that the adaptive virtual autobalancing controller can achieve stabilization of rotor motion as well as adaptation to changes in imbalance.

Dennis S. Bernstein

Papers

Consistent identification of Hammerstein systems using an ersatz nonlinearity

Concretizing control education

Forward-integration Riccati-based feedback control for spacecraft rendezvous maneuvers on elliptic orbits

Deadbeat Input Reconstruction and State Estimation for Discrete-Time Linear Time-Varying Systems

Adaptive virtual autobalancing for a magnetic rotor with unknown dynamic imbalance