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

In this paper we develop explicit formulas for induced convolution operator norms and their bounds. These results generalize established induced operator norms for linear dynamical systems with various classes of input-output signal pairs.

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Induced convolution operator norms of linear dynamical systems

In this paper we extend results of Wyatt, Siebert and Tan by deriving an energy flow model in terms of thermodynamic energy rather than stored energy as in the standard Statistical Energy Analysis (SEA) approach to energy flow modeling. This modified energy flow model shows that energy flows according to thermodynamic energy and that this property holds for an arbitrary number of nonidentical subsystems independently of the strength of the coupling. These results are compared with SEA by means of illustrative examples.

Thermodynamic Modeling of Interconnected Systems: Conservative Coupling

Although novel aircraft designs can be quickly conceived and built, the ability to develop and test these concepts is impeded by the inability to rapidly prototype autopilots that can accommodate unconventional airframes with nonstandard sensor/actuator configurations. For vehicles that are complex or expensive and that are intended for military or civil certification and production, autopilots are traditionally designed based on a combination of multi-body modeling, computational fluid dynamics, and wind tunnel testing. However, detailed models are not available for experimental flight vehicles, and thus manual autopilot tuning is needed without guarantees of success. To overcome this problem, the present paper focuses on an adaptive control technique that requires minimal modeling and limited tuning. In particular, we apply retrospective cost adaptive (RCAC) control to three aircraft simulation models. The first aircraft is a quadcopter, and the second is a fixed-wing vehicle. Next, we consider a VTOL vehicle that can fly in both quadcopter and fixed-wing modes. The RCAC tunings for the VTOL aircraft in quadcopter and fixed-wing mode are chosen to be identical to the tunings of the quadcopter and fixed-wing aircraft despite the differences in vehicle geometry. The goal is to assess the ability of RCAC to control different airframes with the same tunings, while also assessing the ability to control the VTOL aircraft during mode transition.

Retrospective Cost Adaptive PID Control of Quadcopter/Fixed-Wing Mode Transition in a VTOL Aircraft

For an LQG-type sampled-data regulator problem which accounts for computational delay and utilizes an averaging A/D device, the equivalent discrete-time problem is shown to be of increased order due to the inclusion of delayed measurement states. The optimal projection equations for reduced-order, discrete-time compensation are applied to the augmented problem to characterize low-order controllers. The design results are illustrated on a 10th-order flexible beam example.

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The Optimal Projection Equations for Fixed-Order, Sampled-Data Dynamic Compensation with Computation Delay

Models for a 3D pendulum, consisting of a rigid body that is supported at a frictionless pivot, were introduced in a recent 2004 CDC paper [1]. Control problems were posed based on these models. A subsequent paper, in the 2005 ACC [2], developed stabilizing controllers for a 3D rigid pendulum assuming three independent control inputs. In the present paper, stabilizing controllers are developed for a 3D rigid pendulum assuming that the pendulum has a single axis of symmetry that is uncontrollable. This assumption allows development of a reduced model that forms the basis for controller design and closed loop analysis; this reduced model is parameterized by the constant angular velocity component of the 3D pendulum about its axis of symmetry. Several different controllers are proposed. The first controller, based on angular velocity feedback only, asymptotically stabilizes the hanging equilibrium. Then controllers are introduced, based on angular velocity and reduced attitude feedback, that asymptotically stabilize either the hanging equilibrium or the inverted equilibrium. These problems can be viewed as stabilization of a Lagrange top. Finally, if the angular velocity about the axis of symmetry is assumed to be zero, controllers are introduced, based on angular velocity and reduced attitude feedback, that asymptotically stabilize either the hanging equilibrium or the inverted equilibrium. This problem can be viewed as stabilization of a spherical pendulum.

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Dennis S. Bernstein

Papers

Induced convolution operator norms of linear dynamical systems

Thermodynamic Modeling of Interconnected Systems: Conservative Coupling

Retrospective Cost Adaptive PID Control of Quadcopter/Fixed-Wing Mode Transition in a VTOL Aircraft

The Optimal Projection Equations for Fixed-Order, Sampled-Data Dynamic Compensation with Computation Delay

Stabilization of a 3D Axially Symmetric Rigid Pendulum