ROOT-LOCUS AND BOUNDARY FEEDBACK DESIGN FOR A CLASS OF DISTRIBUTED-PARAMETER SYSTEMS

In this paper, a fairly complete parallel of the finite-dimensional ro ot locus theory is presented for quite general, nonconstant coefficien t, even order ordinary differential operators on a finite interval wit h control and output boundary conditions representative of a choice of collocated 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-locu s conclusions for distributed parameter systems are well known. First, the transfer function of a distributed parameter system may not be me romorphic at infinity so that many of the standard Rouche arguments, r equired even in the lumped case to determine the asymptotic behavior o f the root loci, are not generally valid. Another difficulty is that t he 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-plan e, additional analysis would be required to conclude that the closed-l oop system would be asymptotically stable. Formulating the systems in the classical format of a boundary control problem, the asymptotic ana lysis of the root loci can be based on the pioneering work by Birkhoff on eigenfunction expansions for boundary value problems, work that pr edated and indeed motivated the development of spectral theory in Hilb ert space. Birkhoff's work also contains an asymptotic expansion of ei genfunctions in the spatial variable, generalizing the earlier Sturm-L iouville theory for second-order operators. By further extending this general asymptotic analysis to also include expansions in the gain par ameter, a rigorous treatment of the open- and closed-loop transfer fun ctions and of the corresponding return difference equation can be pres ented. The asymptotic analysis of the return difference equation forms the basis for both the rigorous formulation of the basic problem and its solution.