Sensitivity Shaping under Degree Constraint: Nevanlinna-Pick Interpolation for Multivariable and Time-Delay Systems

It is well known that analytic interpolation theory has found various applications in systems and control, e.g., H∞ control, covariance extension problem for spectral estimation, filtering and gain equalization. For the engineering point of views, it is desirable to synthesize interpolants with bounded degree, where the interpolants correspond to the transfer functions, representing devices of systems. In this context, the theory of the analytic interpolation with complexity constraint has been developed in [9, 6, 5, 10, 4, 2]. The solutions to various analytic interpolation problems are completely and smoothly parameterized by the spectral zeros of the interpolants, and they are uniquely determined by solving a convex optimization problem. The theory is applied to various engineering problems mentioned above, manifesting the potential of this theory. This thesis is a collection of papers about the theory and the applications of the analytic interpolation with complexity constraint. The first paper is about a generalization of the theory of the Nevanlinna-Pick interpolation with degree constraint [2] to the bi-tangential interpolation problems [16]. A smooth and complete parameterization of interpolants is given by solving a convex optimization problem similar to a previous work [2]. It turns out the McMillan degree of interpolants may be lower than the McMillan degree of interpolants given by the theory in [2] as long as free parameters belong to a class of scalar positive real functions. This new feature is discussed through an application to the H∞ sensitivity shaping problem. A subset of this paper is also reported in the CDC paper [12]. In the second paper [15], the theory in [16] is generalized to the two-sided Nudelman interpolation problem. The interpolation problem includes Carathéodoy-Fejer, Nevanlinna-Pick, bi-tangential Nevanlinna-Pick with multiple interpolation points. A benchmark problem of the H∞ control in the multivariable system in [17] is considered in terms of the generalized theory. The performance of the closed loop systems are compared with the conventional weighted H∞ minimization. It is a continuation of [2], where the original bi-tangential interpolation problem was transformed to a matrix-valued interpolation problem at the expense of an increase of the McMillan degree of controllers. It turns out that the McMillan degree of the controllers can be lower than the McMillan degree of the controllers in [2], while the maximum amplitude of the control inputs is unrealizably high for the implementation. In the third paper [13], the unweighted H∞ sensitivity shaping of systems with time delays is considered. It was pointed out in [7] that the unweighted H∞ sensitivity minimization can be cast as a finite-dimensional Nevanlinna-Pick interpolation problem. This formulation is adopted and studied in terms of the theory of Nevanlinna-Pick interpolation with degree constraint. It turns