Multivariable Nyquist criteria, root loci, and pole placement: A geometric viewpoint

Classical control theory is concerned with the topics in our title in the context of single-input/single-output systems. There is now a large and growing literature on the extension of these ideas to the multiinput/multioutput case. This development has posed certain difficulties, some due to the intrinsic nature of the problem and some, we would argue, due to an inadequate reflection on what the multivariable problem calls for. In this paper we describe what seems to us to be the natural multivariable analogs of these concepts from classical control theory. A rather satisfactory generalization of the Nyquist criterion will be described, and a clear analog of the asymptotic properties of the root locus will be obtained in the "multiparameter" case. However, an example is given which illustrates the quite surprising fact that the root locus map is not always continuous at infinite gains. This calls for a new ingredient, a compactification of the space of gains, and perhaps the most interesting new feature in this circle of ideas comes in the area of pole placement. This problem is difficult in the multivariable case, but by establishing a correspondence with a classical set of problems in geometry, we are able to understand its main aspects and to derive results on pole placement by output feedback over the real field.