# A root locus methodology for parabolic boundary control systems

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## Abstract

Recently, C. I. Byrnes and D. S. Gilliam initiated an interest in the development of a root locus methodology for distributed parameter systems. Along with the author, a fairly complete root locus theory was established for an important class of parabolic distributed parameter control problems with spatial part given by general even order ordinary differential operators with real C°° coefficients on a bounded interval. Due to the technically complicated nature of such problems, this preliminary work was restricted to systems with "separated" boundary conditions with an equal number at highest order at each end of the interval. Moreover, it was also assumed that the actuator and sensor, i.e., the boundary input and output, were collocated. It is important to note that, even in this case, the differential operators as well as the associated boundary conditions almost never correspond to self-adjoint problems. However, with a relative degree assumption that the order of the input exceeds the order of the output, it was shown that the open-loop transfer function exists, is strictly proper and, in fact, is in the Callier-Desoer class. A complete root locus analysis was given for a closed-loop system obtained via a proportional error feedback control law.

In this thesis, we extend these results in several directions including the more general class of systems governed by so-called "Birkhoff regular" boundary conditions. We note that separated boundary conditions with an equal number at each end are always Birkhoff regular. More precisely, we introduce the class of Birkhoff regular distributed parameter feedback control systems and investigate the root loci for these systems. This class of problems is considerably more complicated than the separated case and includes the case of non-collocated actuator and sensor pairs. General root locus results are established for these problems. In particular, the quantities that specify and characterize the asymptotic behavior of the closed-loop poles are analyzed in detail and the resulting root loci are described and illustrated in a classified way.

Another class of distributed parameter feedback control systems is also considered, in which the boundary operators are "completely separated" but do not necessarily have an equal number of conditions at each endpoint. We note that a system of this type can never be either Birkhoff regular or self-adjoint in the case of unequal numbers of conditions at the endpoints. We are able to show that, nevertheless, the return difference equation for these systems can be written in the same asymptotic form as for Birkhoff regular systems. Thus, many of the results obtained for Birkhoff regular systems are also true in this case. Indeed, the conclusions for these systems are parallel to what was established for systems with separated boundary conditions with an equal number of conditions at each endpoint.