Primal robustness and semidefinite cones

This paper reformulates and streamlines the core tools of robust stability and performance for LTI systems using now-standard methods in convex optimization. In particular, robustness analysis can be formulated directly as a primal convex (semidefinite program or SDP) optimization problem using sets of gramians whose closure is a semidefinite cone. This allows various constraints such as structured uncertainty to be included directly, and worst-case disturbances and perturbations constructed directly from the primal variables. Well known results such as the KYP lemma and various scaled small gain tests can also be obtained directly through standard SDP duality. To readers familiar with robustness and SDPs, the framework should appear obvious, if only in retrospect. But this is also part of its appeal and should enhance pedagogy, and we hope suggest new research. There is a key lemma proving closure of a grammian that is also obvious but our current proof appears unnecessarily cumbersome, and a final aim of this paper is to enlist the help of experts in robust control and convex optimization in finding simpler alternatives.Comment: A shorter version submitted to CDC 1

Algebraic Distributed Differential Space-Time Codes with Low Decoding Complexity

The differential encoding/decoding setup introduced by Kiran et al, Oggier-Hassibi and Jing-Jafarkhani for wireless relay networks that use codebooks consisting of unitary matrices is extended to allow codebooks consisting of scaled unitary matrices. For such codebooks to be usable in the Jing-Hassibi protocol for cooperative diversity, the conditions involving the relay matrices and the codebook that need to be satisfied are identified. Using the algebraic framework of extended Clifford algebras, a new class of Distributed Differential Space-Time Codes satisfying these conditions for power of two number of relays and also achieving full cooperative diversity with a low complexity sub-optimal receiver is proposed. Simulation results indicate that the proposed codes outperform both the cyclic codes as well as the circulant codes. Furthermore, these codes can also be applied as Differential Space-Time codes for non-coherent communication in classical point to point multiple antenna systems.Comment: To appear in IEEE Transactions on Wireless Communications. 10 pages, 5 figure

Wealth Distributions in Models of Capital Exchange

A dynamical model of capital exchange is introduced in which a specified amount of capital is exchanged between two individuals when they meet. The resulting time dependent wealth distributions are determined for a variety of exchange rules. For ``greedy'' exchange, an interaction between a rich and a poor individual results in the rich taking a specified amount of capital from the poor. When this amount is independent of the capitals of the two traders, a mean-field analysis yields a Fermi-like scaled wealth distribution in the long-time limit. This same distribution also arises in greedier exchange processes, where the interaction rate is an increasing function of the capital difference of the two traders. The wealth distribution in multiplicative processes, where the amount of capital exchanged is a finite fraction of the capital of one of the traders, are also discussed. For random multiplicative exchange, a steady state wealth distribution is reached, while in greedy multiplicative exchange a non-steady power law wealth distribution arises, in which the support of the distribution continuously increases. Finally, extensions of our results to arbitrary spatial dimension and to growth processes, where capital is created in an interaction, are presented.Comment: 10 pages, RevTeX, 4 figures, to be submitted to PR

Controlling extended systems with spatially filtered, time-delayed feedback

We investigate a control technique for spatially extended systems combining spatial filtering with a previously studied form of time-delay feedback. The scheme is naturally suited to real-time control of optical systems. We apply the control scheme to a model of a transversely extended semiconductor laser in which a desirable, coherent traveling wave state exists, but is a member of a nowhere stable family. Our scheme stabilizes this state, and directs the system towards it from realistic, distant and noisy initial conditions. As confirmed by numerical simulation, a linear stability analysis about the controlled state accurately predicts when the scheme is successful, and illustrates some key features of the control including the individual merit of, and interplay between, the spatial and temporal degrees of freedom in the control.Comment: 9 pages REVTeX including 7 PostScript figures. To appear in Physical Review

Hermite matrix in Lagrange basis for scaling static output feedback polynomial matrix inequalities

Using Hermite's formulation of polynomial stability conditions, static output feedback (SOF) controller design can be formulated as a polynomial matrix inequality (PMI), a (generally nonconvex) nonlinear semidefinite programming problem that can be solved (locally) with PENNON, an implementation of a penalty method. Typically, Hermite SOF PMI problems are badly scaled and experiments reveal that this has a negative impact on the overall performance of the solver. In this note we recall the algebraic interpretation of Hermite's quadratic form as a particular Bezoutian and we use results on polynomial interpolation to express the Hermite PMI in a Lagrange polynomial basis, as an alternative to the conventional power basis. Numerical experiments on benchmark problem instances show the substantial improvement brought by the approach, in terms of problem scaling, number of iterations and convergence behavior of PENNON

Analysis of Implicit Uncertain Systems. Part I: Theoretical Framework

This paper introduces a general and powerful framework for the analysis of uncertain systems, encompassing linear fractional transformations, the behavioral approach for system theory and the integral quadratic constraint formulation. In this approach, a system is defined by implicit equations, and the central analysis question is to test for solutions of these equations. In Part I, the general properties of this formulation are developed, and computable necessary and sufficient conditions are derived for a robust performance problem posed in this framework

Investigation of a 2-Colour Undulator FEL Using Puffin

Initial studies of a 2-colour FEL amplifier using one monoenergetic electron beam are presented. The interaction is modelled using the unaveraged, broadband FEL code Puffin. A series of undulator modules are tuned to generate two resonant frequencies along the FEL interaction and a self-consistent 2-colour FEL interaction at widely spaced non-harmonic wavelengths at 1nm and 2.4nm is demonstrated.Comment: Submitted to The 35th International Free-Electron Laser Conference, Manhattan, New York (2013