LMI Conditions of Strictly Bounded Realness On A State-space Realization To Bi-tangential Rational Interpolation

In this paper, we present the LMI conditions to characterize the strictly bounded realness of the state-space realization of the solution to the bi-tangential rational interpolation problem, i.e., they give the solution to the bitangential Nevanlinna-Pick interpolation problem [13], [3]. The bi-tangential Nevanlinna-Pick interpolation problem is the generalization of the classical interpolation problems by Carathéodory, Fejér, Nevanlinna, Pick, see, e.g., [3], [4]. The conventional approach to the interpolation problem is by means of the J-lossless function, which enables one to describe the set of the solutions to the interpolation problem by the linear fractional transformation [3]. Clearly, the rational interpolation problem is independently interesting problem, where the constraint of the bounded realness on the interpolant is removed [1]. A state-space realization is shown in [8], which gives a parameterization of the solutions to the tangential rational interpolation problem. The approach to the bi-tangential Nevanlinna-Pick interpolation problem in this paper is based on the linear matrix inequality (LMI) [5], [11], [7], [2], [10], [12]. For a rational function, which satisfies the interpolation condition, a convex characterization of the free parameter of the statespace realization of the rational function is given by an LMI. We present the theory for the continuous-time and discretetime systems. We use the state-space realization, which gives the solutions to the tangential rational interpolation problem without the bounded realness [9], [8]. However, we show that the bi-tangential Nevanlinna-Pick interpolation problem can be equivalently transformed to a tangential NevanlinnaPick interpolation interpolation problem under the condition of the strictly bounded realness. Thus, we can use the statespace realization to obtain the solution to the bi-tangential Nevanlinna-Pick interpolation problem. We discuss the set of strictly bounded real interpolants, which satisfies additional interpolation conditions. The solvability condition of the interpolation problem is given by a matrix inequality, which is not linear. Thus, we give a convex relaxation of the matrix inequality by an LMI, following the scalar case [6].