Approximate Matrix Diagonalization for Use in Distributed Control Networks

Abstract

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.<p>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. <p>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). <p>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. <p>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.