The Kalman--Yakubovich--Popov inequality for passive discrete time-invariant systems

TL;DR

This paper investigates the KYP inequality for passive discrete-time systems, providing new equivalent forms, properties of solutions, and conditions for uniqueness, with a focus on solutions within the operator interval [0, I].

Contribution

It introduces several equivalent formulations of the KYP inequality, characterizes the minimal solution, and establishes conditions for the uniqueness of solutions using advanced operator theory techniques.

Findings

Multiple equivalent forms of the KYP inequality are derived.

The minimal solution satisfies an algebraic Riccati equation.

Conditions for the uniqueness of solutions are established.

Abstract

We consider the Kalman - Yakubovich - Popov (KYP) inequality \[ \begin{pmatrix} X-A^* XA-C^*C & -A^*X B- C^*D\cr -B^*X A-D^* C & I- B^*X B-D^*D \end{pmatrix} \ge 0 \] for contractive operator matrices $ \begin{pmatrix} A&B\cr C &D \end{pmatrix}:\begin{pmatrix}\mathfrak{H}\cr\mathfrak{M} \end{pmatrix}\to\begin{pmatrix}\mathfrak{H}\cr\mathfrak{N} \end{pmatrix}, $ where $\mathfrak{H},$ $\mathfrak{M}$, and $\mathfrak{N}$ are separable Hilbert spaces. We restrict ourselves to the solutions $X$ from the operator interval $[0, I_\mathfrak{H}]$. Several equivalent forms of KYP are obtained. Using the parametrization of the blocks of contractive operator matrices, the Kre\u{\i}n shorted operator, and the M\"obius representation of the Schur class operator-valued function we find several equivalent forms of the KYP inequality. Properties of solutions are established and it is proved that the minimal solution of the KYP inequality satisfies the corresponding algebraic Riccati equation and can be obtained by the iterative procedure with the special choice of the initial point. In terms of the Kre\u{\i}n shorted operators a necessary condition and some sufficient conditions for uniqueness of the solution are established.