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

Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Simple state-space formulas are derived for all controllers solving the following standard H/sub infinity / problem: For a given number gamma >0, find all controllers such that the H/sub infinity / norm of the closed-loop transfer function is (strictly) less than gamma . It is known that a controller exists if and only if the unique stabilizing solutions to two algebraic Riccati equations are positive definite and the spectral radius of their product is less than gamma /sup 2/. Under these conditions, a parameterization of all controllers solving the problem is given as a linear fractional transformation (LFT) on a contractive, stable, free parameter. The state dimension of the coefficient matrix for the LFT, constructed using the two Riccati solutions, equals that of the plant and has a separation structure reminiscent of classical LQG (i.e. H/sub 2/) theory. This paper is intended to be of tutorial value, so a standard H/sub 2/ solution is developed in parallel. >

State-space solutions to standard H/sub 2/ and H/sub infinity / control problems

In this paper, we consider the application of linear quadratic Gaussian (LQG) control to the problem of optimizing the level of squeezing in one of the quadratures of an optical field. Squeezed states of light can be generated when two optical fields (at fundamental and second-harmonic frequencies) interact in an optical cavity in the presence of a second-order nonlinear crystal. Our system is an optical squeezer which is modelled as a nonlinear quantum system. Suitable models for the quantum and classical noises present in the system are used and laser phase noise which arises due to mechanical vibration of the mirror(s) in the beam path is modeled as (approximately) integrated white noise. An LQG controller is synthesized and the closed loop system is simulated to validate our design.

LQG control of an optical squeezer

The Robust Fixed Interval Smoothing Problem and Robust Observability of Uncertain Time-Varying Systems

We introduce a novel adaptive damping technique for an inertial gradient system which finds application as a gradient descent algorithm for unconstrained optimisation. In an example using the non-convex Rosenbrock's function, we show an improvement on existing momentum-based gradient optimisation methods. Also using Lyapunov stability analysis, we demonstrate the performance of the continuous-time version of the algorithm. Using numerical simulations, we consider the performance of its discrete-time counterpart obtained by using the symplectic Euler method of discretisation.

A Closed Loop Gradient Descent Algorithm applied to Rosenbrock's function.

This paper presents the design and implementation of a multi-input multi-output (MIMO) controller using a resonant controller, an integral controller, and a velocity feedback controller for damping, tracking and cross coupling reduction of a nanopositioner used in most of the scanning probe microscopes (SPMs). The MIMO dynamics identification of the system are done by using measured data and the design of the controller is based on to achieve large bandwidth and small cross coupling effects between the axes of the nanopositioner. The controller design presented in this paper is able to achieve a bandwidth close to the first resonance frequency of a nanopositioner which is five times greater than the bandwidth that can achieved by using a standard integral controller. Experimental images at scanning rates of 62.5 Hz and 125 Hz obtained by using the proposed controller and the built-in proportional-integral (PI) controller in a SPM are given to illustrate the effectiveness of the proposed controller.

Ian R. Petersen

Papers

LQG control of an optical squeezer

The Robust Fixed Interval Smoothing Problem and Robust Observability of Uncertain Time-Varying Systems

A Closed Loop Gradient Descent Algorithm applied to Rosenbrock's function.

A MIMO controller design for damping, tracking, and cross coupling reduction of nanopositioners

A systems theory approach to the synthesis of non-reciprocal phase-insensitive quantum amplifiers