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

Indirect adaptive control of flexible structures is considered under harmonic and broadband disturbances. Limited prior modeling information is assumed, and system identification with an input-output model structure is performed online in the presence of the exogenous disturbance. By realizing the input-output model structure in block observable canonical form, the full state is available, which facilitates output-feedback control without the need for an observer. The control input is determined by model predictive control (MPC) using quadratic programming for receding horizon optimization. The resulting sampled-data controller is implemented at a fixed sample rate, where the frequencies of some of the modes may lie above the Nyquist rate, thus emulating spillover. The approach is applied to a truss structure with 16 lightly damped modes.

Predictive Cost Adaptive Control of Flexible Structures with Harmonic and Broadband Disturbances

Although the Kalman filter is often presented within a continuous-time context [1]-[4], the original derivation was provided in discrete time [5]. In practice, controllers and observers are invariably implemented digitally; thus discrete-time algorithms deserve special attention. This article focuses on the real-time implementation of the discrete-time Kalman filter and its relation to the discrete-time Kalman predictor. These algorithms differ in subtle ways, as highlighted in this article. Equations for the discrete-time Kalman filter and predictor are given in [6]. Unfortunately, the covariance matrix becomes indefinite after a few steps, which shows that those equations are erroneous. The present article corrects and extends the results from [6].

Real-Time Implementation of the Optimal Predictor and Optimal Filter: Accuracy Versus Latency [Lecture Notes]

Recursive least squares (RLS) is widely used in identification and estimation. An unfortunate weakness of RLS is the divergence of its covariance matrix in cases where the data are not sufficiently persistent. To solve this problem, [1] introduced the exponential forgetting and resetting algorithm (EFRA), whose covariance update equation is modified so that the covariance matrix remains bounded. Unfortunately, EFRA does not include RLS as a special or limiting case, and cannot easily approximate RLS estimates. In this paper, we derive a modified RLS variant of EFRA that includes RLS without forgetting as a limiting case, and that can closely approximate RLS with forgetting. An additional advantage of MRLS relative to EFRA is greater ease in choosing parameters to set the covariance bounds.

A Modified Recursive Least Squares Algorithm with Forgetting and Bounded Covariance

This paper investigates the emergence of thermodynamic energy flow for collections of coupled oscillators. Unlike previous work based on averaging over stochastic forcing, a stochastic model ensemble, or time, thermodynamic energy flow is viewed as a deterministic physical phenomenon arising solely due to dimension and thus consistent with the physically observed laws of thermodynamics. For two collections of undamped coupled oscillators, the kinetic and potential energy of the respective subsystems is shown to exhibit pointwise-in-time thermodynamic energy flow before energy reversal begins. However, the rate of energy flow is found to be inconsistent with the classical exponential profile corresponding to Newton's law of cooling, whose dynamics are linear. Instead, the rate of energy flow is shown to be captured by a thermodynamic energy-flow model with quadratically nonlinear dynamics.

An Emergent Nonlinear Thermodynamic Energy-Flow Model for Collections of Coupled Oscillators

The authors consider the finite-dimensional approximation of the infinite-dimensional optimal projection theory of Bernstein and Hyland (1984, 1986), the purpose being model and controller order reduction. The approach yields fixed-finite-order controllers which are optimal with respect to high-order, approximating, finite-dimensional plant models. The authors illustrate the technique by computing a sequence of first-order controllers, for a one-dimensional, single-input/single-output, parabolic (heat/diffusion) system using a spline-based, Ritz-Galerkin, finite-element approximation. The numerical studies indicate convergence of the feedback gains with less than 2% performance degradation over full-order LQG controllers. >

Dennis S. Bernstein

Papers

Predictive Cost Adaptive Control of Flexible Structures with Harmonic and Broadband Disturbances

Real-Time Implementation of the Optimal Predictor and Optimal Filter: Accuracy Versus Latency [Lecture Notes]

A Modified Recursive Least Squares Algorithm with Forgetting and Bounded Covariance

An Emergent Nonlinear Thermodynamic Energy-Flow Model for Collections of Coupled Oscillators

An approximation technique for computing optimal fixed-order controllers for infinite-dimensional systems