Abstract
AbstractThis paper discusses the pole assignment problem for the three‐dimensional system which is described by a state‐space model. The problem is to provide a feedback and input transformation using two‐dimensional dynamic compensators to the given three‐dimensional system. In this way, the denominator polynomial of the transfer function of the closed‐loop system coincides with the product of three one‐dimensional polynomials with roots in the unit circle.First, the three‐dimensional system is considered as a one‐dimensional one over a two‐dimensional rational functional field, and the desired pole assignment is realized by providing the two‐dimensional dynamic feedback and the input transformation. Concretely, the dynamic compensators for the feedbak system and the input transformation are regarded as two‐dimensional transfer functions. By providing the minimal realizations of those in two steps, the feedback system and the input transformation system are realized. Then, a detailed discussion is given to the realizability of those two‐dimensional dynamic compensators, i.e., the possibility that the dynamic compensators are proper rational functions with respect to both variables.A counter‐measure for the nonproper case is also proposed. A condition for the obtained three‐dimensional system to be stable is derived. The dimension needed in the realization of those two‐dimensional dynamic compensators, i.e., the dynamical dimension needed in the pole assignment, is calculated. A method for realizing the pole assignment with the lowest dynamical dimension is discussed.
Published Version
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