Root-Locus and Boundary Feedback Design for a Class of Distributed Parameter Systems

Abstract

In this paper, a fairly complete parallel of the finite-dimensional root locus theory is presented for quite general, nonconstant coefficient, even order ordinary differential operators on a finite interval with control and output boundary conditions representative of a choice of colocated point actuators and sensors. Root-locus design methods for linear distributed parameter systems have also been studied for some time and the primary difficulties in rigorously interpreting root locus conclusions for distributed parameter systems are well known. First, the transfer function of a distributed parameter system may not be meromorphic at infinity so that many of the standard Rouche arguments, required even in the lumped case to determine the asymptotic behavior of the root loci, are not generally valid. Another difficulty is that the infinitesimal generator in the state-space model for a closed-loop system may not be selfadjoint, accretive or even satisfy the spectrum determined growth condition. Thus, regardless of whether the root loci---interpreted as closed-loop eigenvalues---lie in the open left-half plane, additional analysis would be required to conclude that the closed-loop system would be asymptotically stable. Formulating the systems in the classical format of a boundary control problem, the asymptotic analysis of the root loci can be based on the pioneering work by Birkhoff on eigenfunction expansions for boundary value problems, work that predated and indeed motivated the development of spectral theory in Hilbert space. Birkhoff's work also contains an asymptotic expansion of eigenfunctions in the spatial variable, generalizing the earlier Sturm--Liouville theory for second-order operators. By further extending this general asymptotic analysis to also include expansions in the gain parameter, a rigorous treatment of the open and closed loop transfer functions and of the corresponding return difference equation can be presented. The asymptotic analysis of the return difference equation forms the basis for both our rigorous formulation of the basic problem and its solution.