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 consider an L2 control problem with an L∞ norm on the closed-loop impulse response. To do this we first construct an upper bound for the L∞ norm of the closed-loop system. To perform controller synthesis, we then modify the standard LQG cost functional by including an additional penalty term that weights the L∞ norm of the impulse response of the closed-loop system. A numerical example is given to illustrate the improved L∞ response of the closed-loop system.

L 2 controller synthesis with L ∞ -bounded closed-loop impulse response

The Total Energy Control System (TECS) is a method to control airplane longitudinal flight dynamics by regulating energy and energy balance. This multiple input, multiple output method accounts for the highly coupled nature of aircraft dynamics and allows for control of airspeed and altitude with a proportional-integral controller. This paper reviews the heuristic TECS control law and then validates it in a MATLAB simulation. Next, an adaptive TECS is designed by augmenting the fixed-gain controllers in a nominal TECS with retrospective cost optimization-based adaptive controllers. It is shown through simulations that adaptive augmentation improves the closed-loop performance and accelerates the tuning process. 

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Adaptive Energy Control of Longitudinal Aircraft Dynamics

In this paper we apply extensions of retrospective cost adaptive control (RCAC) to a 2D missile model considered in prior papers as a benchmark test of adaptive control methods. The dynamics of the missile are highly nonlinear, and instantaneous linearizations are nonminimum phase due to nose sensing and tail actuation. The results that we present in this paper show that the RCAC controller provides results that are comparable to a highly tuned autopilot based on aerodynamic modeling, whereas the RCAC controller uses no knowledge of the missile’s aerodynamics. These results significantly improve the results obtained on the same problem using an earlier version of RCAC, presented at the 2010 GNC.

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Adaptive Control of a Seeker-Guided 2D Missile with Unmodeled Aerodynamics

We apply an extension of retrospective cost adaptive control (RCAC) to a command-following problem for the uncertain electromagnetically controlled oscillator (ECO). We assume that an estimate of the first Markov parameter of the discretized and linearized plant is known, but RCAC does not require knowledge of the inertia, damping, or stiffness of the plant. RCAC uses a setpoint feedback path and an auxiliary nonlinearity to stabilize the unstable ECO at the commanded equilibria.© 2012 ASME

Dennis S. Bernstein

Papers

L 2 controller synthesis with L ∞ -bounded closed-loop impulse response

Adaptive Energy Control of Longitudinal Aircraft Dynamics

Adaptive Control of a Seeker-Guided 2D Missile with Unmodeled Aerodynamics

Setpoint control of the uncertain electromagnetically controlled oscillator

Steady-state Kalman filtering with an H_infinity error bound