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 consider output feedback adaptive stabilization for second-order systems with no zeros. The assumptions we make are standard, namely, that the sign of the high frequency gain is known. However, we complement the existing literature by deriving an explicit expression for the adaptive controller. The controller has the form of a 6th-order dynamic compensator with quadratic, cubic and quartic nonlinearities. The proof of convergence is based on a variation of Lyapunov's method in which the Lyapunov derivative is shown to be asymptotically nonpositive. Application of the controller to the Van der Pol and Duffing oscillators shows that the controller is effective for nonlinear systems as well.

Output feedback adaptive stabilization of second-order systems

Feedback controllers that guarantee asymptotic stabilization of either the hanging equilibrium or the inverted equilibrium of the 3D pendulum are presented and experimentally evaluated using the triaxial attitude control testbed (TACT). Asymptotic stabilization results obtained previously are shown to guarantee almost global response properties, but the local response properties near the equilibrium show non-exponential convergence. These feedback controllers are modified to guarantee almost global response properties and improved local response near the equilibrium. Experimental results illustrate the closed loop time responses for various feedback controllers.

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Experimental results for almost global asymptotic and locally exponential stabilization of the natural equilibria of a 3D pendulum

A state estimation problem for linearized magnetohydrodynamic (MHD) flow is considered. The ideal MHD equations governing the flow of plasma in a two-dimensional channel are linearized about an equilibrium flow. Pseudospectral collocation methods are used to spatially discretize the linear partial differential equations and obtain a state-space model of the linearized dynamics. Three different discrete-time Kalman filtering algorithms are used to estimate the state variables, and their performance is analyzed.

State estimation for linearized MHD flow

This paper develops an adaptive digital autopilot for quadcopters and presents experimental results. The adaptive digital autopilot is constructed by augmenting the PX4 autopilot control system architecture with adaptive digital control laws based on retrospective cost adaptive control (RCAC). In order to investigate the performance of the adaptive digital autopilot, the default gains of the fixed-gain autopilot are scaled by a small factor, which severely degrades its performance. This scenario thus provides a venue for determining the ability of the adaptive digital autopilot to compensate for the detuned fixed-gain autopilot. The adaptive digital autopilot is tested in simulation and physical flight tests, and the resulting performance improvements are examined.

Experimental Implementation of an Adaptive Digital Autopilot

The attitude dynamics of an air spindle and platform testbed, controlled by two proof mass actuators fixed to the platform, are studied assuming that the air spindle is not exactly vertical. A tilt angle of the air spindle induces a gravity moment that has a long term effect on the rotational dynamics. A mathematical model is used to study the free dynamics with the proof masses fixed, as a basis for parameter identification and to explain the longer term dynamics that have been observed in experiments. Feedback compensation for these gravity effects is discussed.

Dennis S. Bernstein

Papers

Output feedback adaptive stabilization of second-order systems

Experimental results for almost global asymptotic and locally exponential stabilization of the natural equilibria of a 3D pendulum

State estimation for linearized MHD flow

Experimental Implementation of an Adaptive Digital Autopilot

Attitude control of a tilted air spindle testbed using proof mass actuators