Approximate Matrix Diagonalization for Use in Distributed Control Networks

Distributed control networks are rapidly emerging as aviable and important alternative to centralized control. In a typical distributed control network, a number of spatially distributed nodescomposed of "smart" sensors and actuators are used to take measurements and apply control inputs to some physical plant. The nodes have local processing power and the ability to communicate with the other nodes via a network. The challenge is to compute and implement a feedback law for the resulting MIMO system in a distributed manner on the network.Our approach to this problem is based on plant diagonalization.To do this, we search for basis transformations for the vector of outputs coming from the sensors and the vector of inputs applied to the actuators so that, in the new bases, the MIMOsystem becomes a collection of decoupled SISO systems.This formulation provides a number of advantages for the synthesis and implementation of a feedback control law,particularly for systems where the number of inputs and outputs is large. Of course, in order for this idea to be feasible,the required basis transformations must have properties which allow them to be implemented on a distributed control network. Namely, they must be computed in a distributed manner which respects the spatial distribution of the data(to reduce communication overhead) and takes advantage of the massive parallel processing capability of the network (to reduce computation time). In this thesis, we present some tools which can be used to find suitable transforms which achieve "approximate"plant diagonalization. We begin by showing how to search the large collection of orthogonal transforms which are contained in the wavelet packet to find the one which most nearly, or approximately, diagonalizes a given real valued matrix.Wavelet packet transforms admit a natural distributed implementation,making them suitable for use on a control network.We then introduce a class of linear operators called recursive orthogonal transforms (ROTs) which we have developed specifically for the purpose of signal processing on distributed control networks. We show how to use ROTs to approximately diagonalize fixed real and complex matricesas well as transfer function matrices which exhibit a spatial invariance property. Numerical examples of allproposed diagonalization methods are presented and discussed

Degrees of Freedom of Two-Hop Wireless Networks: "Everyone Gets the Entire Cake"

We show that fully connected two-hop wireless networks with K sources, K relays and K destinations have K degrees of freedom both in the case of time-varying channel coefficients and in the case of constant channel coefficients (in which case the result holds for almost all values of constant channel coefficients). Our main contribution is a new achievability scheme which we call Aligned Network Diagonalization. This scheme allows the data streams transmitted by the sources to undergo a diagonal linear transformation from the sources to the destinations, thus being received free of interference by their intended destination. In addition, we extend our scheme to multi-hop networks with fully connected hops, and multi-hop networks with MIMO nodes, for which the degrees of freedom are also fully characterized.Comment: Presented at the 2012 Allerton Conference. Submitted to IEEE Transactions on Information Theor

A joint-channel diagonalization for multiuser MIMO antenna systems

In this paper, we address the problem of improving the performance of multiuser space-division multiplexing (SDM) systems where multiple independent signal streams can be transmitted in the same frequency and time slot. The problem is important in multiuser multiple-input multiple-output systems where communication from one base station to many mobile stations can occur simultaneously. Our objective is to devise a multiuser linear space-time precoder for simultaneous channel diagonalization of the multiuser channels enabling SDM. Our new approach is based on diagonalizing the multiuser channel matrices and we use a variation of successive Jacobi rotations. In addition to the diagonalization, our approach attempts to optimize the resultant channel gains for performance enhancement. Our method is valid for both frequency-flat and frequency-selective fading channels but we assume that the base station knows all the channels and that they are quasi-stationary